**Part 2**

**Testing Technology** 

146 Smart Nanoparticles Technology

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Zeng, Q.H., Yu, A.B. & Lu, G.Q. (2008). Multiscale modeling and simulation of polymer

Zhai, Y., Zhai, J., Wang, Y., Guo, S., Ren, W. & Dong, S. (2009). Fabrication of iron oxide

Zhang, S., Niu, H., Hu, Z., Cai, Y. & Shi, Y. (2010). Preparation of carbon coated Fe3O4

Zhang, W.M., Wu, X.L., Hu, J.S., Guo, Y.G. & Wan, L.J. (2008). Carbon coated Fe3O4

Zhang, Z., Zhang, Q., Xu, L. & Xia, Y.-B. (2007). Preparation of nanometer -Fe2O3 by an

Zhong, J. & Adams, J.B. (2008). Adhesive metal transfer at the Al(111)/a-Fe2O3(0001)

Zhong, Z., Ho, J., Teo, J., Shen, S. & Gedanken, A. (2007). Synthesis of porous α-Fe2O3

Zhong, Z., Lin, J., Teh, S.P., Teo, J. & Dautzenberg, F.M. (2007). A rapid and efficient method

*and Reactivity in Inorg. Metal-Org. and Nano-Metal Chem.*, 37: 53–56.

surface-enhanced Raman scattering. *J. Phys. Chem. C* 113(17): 7009-7014. Zhang, J., Liu, X., Guo, X., Wu, S. & Wang, S. (2010). A general approach to fabricate diverse

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*B.*, 115:11693-11699.

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*Mater. Sci. and Eng*., 16: 085001(9 pages).

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at low temperatures. *Adv. Funct. Mater.,* 17(8): 1402-1408.

cetyltrimethylammonium bromide (CTAB)-directed synthesis of goethite nanorods.

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core/shell submicrometer spheres with nanoscale surface roughness for efficient

noble metal (Au, Pt, Ag, Pt/Au)/Fe2O3 hybrid nanomaterials. *Chem.–Eur. J.,* 16(27):

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nanospindles as a superior anode material for lithium-ion batteries. *Adv. Funct.* 

electrochemical method in non-aqueous medium and reaction dynamics. *Synthesis* 

interface: a study with ab initio molecular dynamics, modelling and simulation.

nanorods and deposition of very small gold particles in the pores for catalytic

to deposit gold particles onto catalyst supports and its application for CO oxidation

**7** 

Shengyong Wu

*China* 

**Iron Oxide Nanoparticles Imaging** 

**Dual-Contrast Approaches** 

*Medical Imaging Institute of Tianjin* 

**Tracking by MR Advanced Techniques:** 

Recently a number of imaging modalities have been presented for cellular imaging including magnetic resonance imaging (MRI), optical imaging, and positron emission tomography (PET) based on the background of growing demand for molecular imaging to noninvasively and longitudinally visualize cell migration and track transplanted cells in vivo, also to monitor cell biodistribution. Cellular MRI, with its superb ability of resolving soft tissue anatomies in three-dimensions (3D) with high spatial resolution in comparison to other modalities, is particularly important as a noninvasive tool to provide unique information on the dynamics of cell migration *in vivo* (Modo, 2005; Arbab, 2008a; Zhang,

In vivo MRI of cells is very useful for studying tumors, inflammation, stem cell therapy, and immune response, etc. Cells labeled with commercially available iron oxide nanoparticles (iron particles) can be imaged for weeks with MRI. The labeling procedure does not exhibit any alteration to cell viability or function (Bulte, 2004; Oude Engberink, 2007). Superparamagnetic iron oxides (SPIO) and ultra-small superparamagnetic iron oxide (USPIO) particles are commercial MR contrast agents for cell labeling due to their biocompatibility and strong effects upon T2 and T2\* relaxation. Several labeling methods have been developed to incorporate sufficient quantities of iron into cells. Cellular MRI has now been widely used for tracking transplanted iron-labeled therapeutic cells in vivo (Bulte, 2004; Oude Engberink, 2007). The technique has recently been introduced into the clinic (de Vries, 2005). The effect from iron particles is seen as hypointensity or negative-contrast on T2- and T2\*-weighted images because of the shortening of T2 and T2\* relaxation times. However, concerns have been raised that the negative-contrast could be non-specific and difficult to differentiate from signal hypo-intensities resulting from susceptibility artifacts (i.e. from the presence of air or other field inhomogeneities), flow related signal losses, and calcification. Therefore, several positive-contrast and even dual-contrast imaging techniques have recently been developed for tracking iron-labeled cells. Dual-contrast imaging effectively permits detection of the presence of iron-labeled cells with both negative- and positive-contrast within a single image. This chapter illustrates negative- and positivecontrast MR techniques for tracking iron-labeled cells. Particular attention was paid to

**1. Introduction**

2008).

## **Iron Oxide Nanoparticles Imaging Tracking by MR Advanced Techniques: Dual-Contrast Approaches**

Shengyong Wu *Medical Imaging Institute of Tianjin China* 

## **1. Introduction**

Recently a number of imaging modalities have been presented for cellular imaging including magnetic resonance imaging (MRI), optical imaging, and positron emission tomography (PET) based on the background of growing demand for molecular imaging to noninvasively and longitudinally visualize cell migration and track transplanted cells in vivo, also to monitor cell biodistribution. Cellular MRI, with its superb ability of resolving soft tissue anatomies in three-dimensions (3D) with high spatial resolution in comparison to other modalities, is particularly important as a noninvasive tool to provide unique information on the dynamics of cell migration *in vivo* (Modo, 2005; Arbab, 2008a; Zhang, 2008).

In vivo MRI of cells is very useful for studying tumors, inflammation, stem cell therapy, and immune response, etc. Cells labeled with commercially available iron oxide nanoparticles (iron particles) can be imaged for weeks with MRI. The labeling procedure does not exhibit any alteration to cell viability or function (Bulte, 2004; Oude Engberink, 2007). Superparamagnetic iron oxides (SPIO) and ultra-small superparamagnetic iron oxide (USPIO) particles are commercial MR contrast agents for cell labeling due to their biocompatibility and strong effects upon T2 and T2\* relaxation. Several labeling methods have been developed to incorporate sufficient quantities of iron into cells. Cellular MRI has now been widely used for tracking transplanted iron-labeled therapeutic cells in vivo (Bulte, 2004; Oude Engberink, 2007). The technique has recently been introduced into the clinic (de Vries, 2005). The effect from iron particles is seen as hypointensity or negative-contrast on T2- and T2\*-weighted images because of the shortening of T2 and T2\* relaxation times. However, concerns have been raised that the negative-contrast could be non-specific and difficult to differentiate from signal hypo-intensities resulting from susceptibility artifacts (i.e. from the presence of air or other field inhomogeneities), flow related signal losses, and calcification. Therefore, several positive-contrast and even dual-contrast imaging techniques have recently been developed for tracking iron-labeled cells. Dual-contrast imaging effectively permits detection of the presence of iron-labeled cells with both negative- and positive-contrast within a single image. This chapter illustrates negative- and positivecontrast MR techniques for tracking iron-labeled cells. Particular attention was paid to

Iron Oxide Nanoparticles Imaging Tracking by

applications.

MR Advanced Techniques: Dual-Contrast Approaches 151

stem cells (NSCs) obtained from patients with traumatic brain injury then performed intracerebral injections of either ferumoxide-labeled or unlabeled cells around the injured tissue of them as the first study in the field of noninvasive imaging of stem cell treatment of brain injury, and their serial MRI about 7-10 weeks demonstrated that stem-cell engraftment

Also, in an early study (Kircher, 2003a), a highly derivatized cross-linked iron oxide (CLIO) nanoparticle was used to efficiently label cytotoxic T lymphocytes (CTLs) for in vivo tracking of the injected cells to melanoma cell line at near single-cell resolution, with MRI and optimized the labeling protocol (three-dimensional nature of the calculated T2 maps), showing no cytotoxic and not influencing cell behavior or effector function. Despite the fact that the high spatial resolution given by MRI provides accurate evaluation of morphology of lymphoid organ, the sensitivity and ability to quantify MR data is still limited when compared with nuclear medicine based techniques. For MR cell tracking to be clinically useful, it should be defined for the detection limits of the MR method which will be utilized. The related clinical studies with 3.0 T scanners suggest that negative-contrast techniques possibly detect 150,000 Feridex labeled cells after directly injected into the lymph nodes of patients (de Vries, 2005). Another recent example of study by Laboratory for Gene Transcript Targeting, Imaging and Repair in Massachusetts General Hospital demonstrated that functionalization allows SPIO nanoparticles to be targeted, and it showed that their phosphorothioate-modified DNA probes linked to SPIO could be used to identify differential gene expression due to amphetamine exposure with high reliability using the calculation of rate of signal reduction (R2\*) in T2\*-weighted MR images (Liu, 2009). There are also extensive published works with detailed descriptions of many aspects of labeled cells for detection with negative-contrast MRI (Ferrucci, 1990; Bulte, 2004b; Hsiao, 2007; Gonzalez-Lare, 2009). Those and many of other preclinical studies have provided evidences for the potential translation of iron oxide NPs labeling and cellular MR imaging to the clinic

An important property of USPIO is its ability to shorten T1 and T2 relaxation times (Small, 1993; Li, 2005). USPIO-labeled cells can be tracked in T1 and T2/T2\* weighted images, which should increase the accuracy and the specificity for detection of the labeled cells (Kelloff, 2005), such as in imaging assessment on angiogenesis of tumor (Niu, 2011), atherosclerotic plaques (Metz, 2011), or arthritis (Lefevre, 2011). USPIO nanoparticles recently have shown potential in the imaging of molecular biomarkers, such as integrins that are heterodimeric transmembrane glycoproteins, a family of adhesion molecules playing a major role in

Much of the progress in detecting individual iron-labeled cells has achieved from improvements in contrast agent design that increases targeting and intracellular uptake properties (Cerdan, 1989; Weissleder, 1990; Bulte, 2001; Zhao, 2002). Improvements in MR hardware and pulse sequence design also have played an important role during recent progress in this area of research. Although negative-contrast MRI has shown promise as a means to visualize labeled cells (Hogemann, 2003; Heyn, 2005), some remaining issues may hamper its wide applications: (1) it is difficult to distinguish the signal voids of labeled cells from those of complex background tissue signals; (2) With the resulting signal void as the means for detection, partial-volume effects are often severe and go far beyond the real cell

angiogenesis and tumor metastasis (Chen, 2009; Tan, 2011).

and migration after implantation can be detected noninvasively with the use of MRI.

recently developed positive-contrast cell tracking techniques, the status of dual-contrast approaches of new MRI pulse sequences and image postprocessing techniques and their perspectives. The new advanced technology in imaging contrast of iron oxide NPs on multimodal platform will also be introduced.

## **2. Negative-contrast MRI techniques**

Cellular MRI is a newly emerging field of MR research that allows the "non-invasive, quantitative, and repetitive imaging of targeted macromolecules and biological processes in living organisms" (Herschman, 2003). Cellular MRI requires that cells are labeled with MR contrast agent to make them distinct from the surrounding tissues. Iron oxide nanoparticles are regarded as the most extensively applied contrast agent in cell imaging and cell tracking studies based on the fact of their strong negative contrast effect, biocompatibility, variety in core size and coating surface, as well as ease of detection at microscopic level (Muja, 2009). SPIO and USPIO are currently the predominant MRI contrast agents. The description of the physical and chemical properties of SPIO and USPIO can be found in recent reviews (Herschman, 2003; Thorek, 2006; Muja, 2009). The sizes of monocrystalline iron oxide nanoparticles (MIONs) ≈ 3 nm in diameter, USPIO particles ≈ 15-30 nm, SPIO particles ≈ 60- 180 nm and micron sized iron oxide particles (MPIOs) can be as large as 10 μm (Shapiro, 2005). Some of the SPIO and USPIO agents, such as Endorem (SPIO, Guebert), Ferumoxides (SPIO, Berlex) and Resovist (USPIO, Schering), are already approved by the Food and Drug Administration (FDA) and are extensively used for imaging of the liver, central nervous system (CNS) and lymphatic system (Arbab, 2004b; Helmberger, 2005; Manninger, 2005), etc. Cationic transfection agents such as poly-L-lysine or the FDA-approved protamine sulfate are used to increase labeling efficiency *in vitro*. SPIO particles may decrease T2\* by magnetic susceptibility effect and T2 by dipole-dipole interaction or scalar effect between protons and magnetic centre. A large magnetisation difference occurs as a result of the nonhomogeneous distribution of superparamagnetic particles, which gives rise to local field gradients that accelerate the loss of phase coherence of the spins contributing to the MR signal. Iron-labeled cells cause significant signal dephasing due to the magnetic field inhomogeneity induced in water molecules near the cell such that iron-labeled cells were visualized as signal voids on T2 and T2\* weighted images (negative-contrast MR imaging). Negative-contrast techniques are the most commonly used approach for the detection of the SPIO-labeled cells.

While cell-based therapies have attracted well attention as novel therapeutics for the treatment of so many kinds of diseases, investigations (Zhang, 2005; Heyn, 2005, 2006) have showed that single, living, highly phagocytic large cells, such as macrophages, or human endothelial cells can be tracked over time in MRI using a 3.0 T even 1.5 T scanner. As an example of stem cell-based studies, investigators (Anderson, 2005) demonstrated that MRI of iron-labeled stem cells was directly identified in neovasculature of a glioma model. The cells were labeled using the ferumoxides/poly-L-lysine complex in vitro and the labeled cells were then injected in the model, and their migration toward and incorporation into the tumor neovasculature was visualized in vivo with negative-contrast MRI. Other studies have shown that ferumoxides-TA labeled human MSCs will home to liver (Arbab, 2004a), tumors (Khakoo, 2006), or heart (Kraitchman, 2005), illustrated at negative-contrast imaging with MR scan and confirmed at histologic evaluation. A group (Zhu, 2006) labeled neural

recently developed positive-contrast cell tracking techniques, the status of dual-contrast approaches of new MRI pulse sequences and image postprocessing techniques and their perspectives. The new advanced technology in imaging contrast of iron oxide NPs on

Cellular MRI is a newly emerging field of MR research that allows the "non-invasive, quantitative, and repetitive imaging of targeted macromolecules and biological processes in living organisms" (Herschman, 2003). Cellular MRI requires that cells are labeled with MR contrast agent to make them distinct from the surrounding tissues. Iron oxide nanoparticles are regarded as the most extensively applied contrast agent in cell imaging and cell tracking studies based on the fact of their strong negative contrast effect, biocompatibility, variety in core size and coating surface, as well as ease of detection at microscopic level (Muja, 2009). SPIO and USPIO are currently the predominant MRI contrast agents. The description of the physical and chemical properties of SPIO and USPIO can be found in recent reviews (Herschman, 2003; Thorek, 2006; Muja, 2009). The sizes of monocrystalline iron oxide nanoparticles (MIONs) ≈ 3 nm in diameter, USPIO particles ≈ 15-30 nm, SPIO particles ≈ 60- 180 nm and micron sized iron oxide particles (MPIOs) can be as large as 10 μm (Shapiro, 2005). Some of the SPIO and USPIO agents, such as Endorem (SPIO, Guebert), Ferumoxides (SPIO, Berlex) and Resovist (USPIO, Schering), are already approved by the Food and Drug Administration (FDA) and are extensively used for imaging of the liver, central nervous system (CNS) and lymphatic system (Arbab, 2004b; Helmberger, 2005; Manninger, 2005), etc. Cationic transfection agents such as poly-L-lysine or the FDA-approved protamine sulfate are used to increase labeling efficiency *in vitro*. SPIO particles may decrease T2\* by magnetic susceptibility effect and T2 by dipole-dipole interaction or scalar effect between protons and magnetic centre. A large magnetisation difference occurs as a result of the nonhomogeneous distribution of superparamagnetic particles, which gives rise to local field gradients that accelerate the loss of phase coherence of the spins contributing to the MR signal. Iron-labeled cells cause significant signal dephasing due to the magnetic field inhomogeneity induced in water molecules near the cell such that iron-labeled cells were visualized as signal voids on T2 and T2\* weighted images (negative-contrast MR imaging). Negative-contrast techniques are the most commonly used approach for the detection of the

While cell-based therapies have attracted well attention as novel therapeutics for the treatment of so many kinds of diseases, investigations (Zhang, 2005; Heyn, 2005, 2006) have showed that single, living, highly phagocytic large cells, such as macrophages, or human endothelial cells can be tracked over time in MRI using a 3.0 T even 1.5 T scanner. As an example of stem cell-based studies, investigators (Anderson, 2005) demonstrated that MRI of iron-labeled stem cells was directly identified in neovasculature of a glioma model. The cells were labeled using the ferumoxides/poly-L-lysine complex in vitro and the labeled cells were then injected in the model, and their migration toward and incorporation into the tumor neovasculature was visualized in vivo with negative-contrast MRI. Other studies have shown that ferumoxides-TA labeled human MSCs will home to liver (Arbab, 2004a), tumors (Khakoo, 2006), or heart (Kraitchman, 2005), illustrated at negative-contrast imaging with MR scan and confirmed at histologic evaluation. A group (Zhu, 2006) labeled neural

multimodal platform will also be introduced.

**2. Negative-contrast MRI techniques** 

SPIO-labeled cells.

stem cells (NSCs) obtained from patients with traumatic brain injury then performed intracerebral injections of either ferumoxide-labeled or unlabeled cells around the injured tissue of them as the first study in the field of noninvasive imaging of stem cell treatment of brain injury, and their serial MRI about 7-10 weeks demonstrated that stem-cell engraftment and migration after implantation can be detected noninvasively with the use of MRI.

Also, in an early study (Kircher, 2003a), a highly derivatized cross-linked iron oxide (CLIO) nanoparticle was used to efficiently label cytotoxic T lymphocytes (CTLs) for in vivo tracking of the injected cells to melanoma cell line at near single-cell resolution, with MRI and optimized the labeling protocol (three-dimensional nature of the calculated T2 maps), showing no cytotoxic and not influencing cell behavior or effector function. Despite the fact that the high spatial resolution given by MRI provides accurate evaluation of morphology of lymphoid organ, the sensitivity and ability to quantify MR data is still limited when compared with nuclear medicine based techniques. For MR cell tracking to be clinically useful, it should be defined for the detection limits of the MR method which will be utilized. The related clinical studies with 3.0 T scanners suggest that negative-contrast techniques possibly detect 150,000 Feridex labeled cells after directly injected into the lymph nodes of patients (de Vries, 2005). Another recent example of study by Laboratory for Gene Transcript Targeting, Imaging and Repair in Massachusetts General Hospital demonstrated that functionalization allows SPIO nanoparticles to be targeted, and it showed that their phosphorothioate-modified DNA probes linked to SPIO could be used to identify differential gene expression due to amphetamine exposure with high reliability using the calculation of rate of signal reduction (R2\*) in T2\*-weighted MR images (Liu, 2009). There are also extensive published works with detailed descriptions of many aspects of labeled cells for detection with negative-contrast MRI (Ferrucci, 1990; Bulte, 2004b; Hsiao, 2007; Gonzalez-Lare, 2009). Those and many of other preclinical studies have provided evidences for the potential translation of iron oxide NPs labeling and cellular MR imaging to the clinic applications.

An important property of USPIO is its ability to shorten T1 and T2 relaxation times (Small, 1993; Li, 2005). USPIO-labeled cells can be tracked in T1 and T2/T2\* weighted images, which should increase the accuracy and the specificity for detection of the labeled cells (Kelloff, 2005), such as in imaging assessment on angiogenesis of tumor (Niu, 2011), atherosclerotic plaques (Metz, 2011), or arthritis (Lefevre, 2011). USPIO nanoparticles recently have shown potential in the imaging of molecular biomarkers, such as integrins that are heterodimeric transmembrane glycoproteins, a family of adhesion molecules playing a major role in angiogenesis and tumor metastasis (Chen, 2009; Tan, 2011).

Much of the progress in detecting individual iron-labeled cells has achieved from improvements in contrast agent design that increases targeting and intracellular uptake properties (Cerdan, 1989; Weissleder, 1990; Bulte, 2001; Zhao, 2002). Improvements in MR hardware and pulse sequence design also have played an important role during recent progress in this area of research. Although negative-contrast MRI has shown promise as a means to visualize labeled cells (Hogemann, 2003; Heyn, 2005), some remaining issues may hamper its wide applications: (1) it is difficult to distinguish the signal voids of labeled cells from those of complex background tissue signals; (2) With the resulting signal void as the means for detection, partial-volume effects are often severe and go far beyond the real cell

Iron Oxide Nanoparticles Imaging Tracking by

**agents** 

Ferritin

Ferumoxides

Ferumoxides

mMION/ SPPM

MION-47

MION-47

UTE imaging Ferumoxides *In vitro* and

state free precession; UTE, ultrashort echo-time

Ferumoxides *In vitro* and

**MR sequences Contrast** 

gradientdephasing technique & GRASP

off-resonance (OR) method

Off-resonance saturation

IRON technique

SR-SPSP sequence

FLAPS sequence

SWEET sequence

Techniques

MR Advanced Techniques: Dual-Contrast Approaches 153

Endogeneous ferritin

Embryonic stem cell-derived cardiac precursor

Embryonic stem cell line TL-1

SPIO-luc-mouse embryonic stem

the vβ3-expressing microvasculature

Macrophage

Macrophage

Human bone marrow stromal

GFP-R3230Ac cell

carcinoma cells

cells

line

cell

cell

**Biological target Application and** 

**Results** 

Crush injured rabbit carotid arteries

Myocardial infraction

Injected into the hind limb of mouse

Injection into hindlimbs

molecular imaging of

Atherosclerotic plaque

MR lymphography

Injection into the hind

Injection into the hind

legs of mouse

legs of rat

imaging

tumor cells

Visualization of magnetically labeled

G6 glioma cells Implanted cellular

of mouse

cancer

**Experimental conditions** 

*In vitro* and *in vivo*

*In vitro* and *in vivo*

*In vitro* and *in vivo*

Gel phantom/

*in vivo*

*in vivo*

*In vivo* 

*In vivo* 

*in vivo*

*in vivo*

*in vivo*

Ferumoxides *in vivo* Human epidermal

Table 1. Summary of Previously Published Studies of Positive- and Dual-contrast

Note: GRASP, superparamagnetic particles/susceptibility; IRON, oxide nanoparticles–resonant water suppression; SR-SPSP, self-refocused spatial-spectral; FLAPS, fast low-angle positive contrast steady-

Ferumoxides *In vitro* and

Ferumoxides *In vitro* and

size; (3) it is difficult to discriminate iron-induced susceptibility changes from those caused by other susceptibility artifacts due to i.e. air/tissue interfaces, or peri-vascular effects.

## **3. Positive-contrast and dual-contrast MRI techniques**

The "white-marker imaging" positive-contrast mechanism was introduced by Seppenwoolde et al. in 2003 (Seppenwoolde, 2003). Since then, several groups have developed positive-contrast or dual-contrast pulse sequences for tracking iron-labeled cells *in vitro* and *in vivo* (Table 1).

#### **3.1 Gradient-dephasing technique: "white-marker" imaging**

"White-marker" imaging was initially presented to create positive-contrast around paramagnetic intravascular device markers used in magnetic-resonance-based interventional procedures (Seppenwoolde, 2003). The gradient-dephasing technique uses a slice gradient to dephase the background water signal followed by an incomplete gradient rephasing pulse which was exploited for the depiction and tracking of paramagnetic susceptibility markers. Local magnetic field inhomogeneities were selectively visualized with positive-contrast, such as those created by iron-labeled cells for "white-marker" imaging. Advanced methods were developed to separate magnetic susceptibility effects from partial volume effects in "white marker" imaging in order to avoid compromising the identification of magnetic structures (Seppenwoolde, 2007). However, this method is only sensitive to macroscopic field inhomogeneities caused by paramagnetic material, to a volume surrounding the paramagnetic material that is free of other field variations (Zurkiya, 2006).

A similar gradient depashing technique termed gradient echo acquisition for superparamagnetic particles (GRASP), by dephasing of the background signal, has been used to detect positive-contrast from superparamagnetic particles based on the phenomena that the z-rephasing gradient is reduced so that dipolar fields generated by the cells are rephrased and positive signal can be observed (Mani, 2006a), also to image ferritin deposition in a rabbit model of carotid injury with relatively low concentrations of iron oxides at 1.5 T MR scanner (Mani, 2006b). The GRASP technique was used to successfully image low concentrations of ferumoxides (0.05 mM Fe corresponding to 2.8 μg Fe/mL) and ferritin (5 μg Fe/mL) in gel phantoms (Mani, 2006). GRASP "whitemarker" imaging has several advantages including ease of implementation, high sensitivity, no influence on positive signal due to both B0 and B1 field inhomogeneities, and fast acquisition with various TE values. The feasibility of GRASP was tested to aid in dynamically tracking stem cells in a mouse model of myocardial infraction (Mani, 2008). Using T2\*-GRE and GRASP techniques at 9.4 T scanner, iron-labeled embryonic stem cells were visualized in the border zone of infarcted mice at 24 hours, and 1 week following implantation. The positive signal in areas containing iron-loaded stem cells corresponded precisely with the signal loss detected within images produced with conventional GRE sequences. Regions that contained iron-labeled cells were confirmed by histology (Mani, 2008). The presence of the signal loss because of iron-labeled cells would have been difficult to detect on T2\*-weighted images without using the positive-contrast sequence. The region of the myocardium containing the iron-labeled cells was clearly visible when both GRASP and T2\*-weighted techniques (dual contrast imaging) were applied. Dualcontrast effects act to extend the signal change well beyond the location of the particle or

size; (3) it is difficult to discriminate iron-induced susceptibility changes from those caused by other susceptibility artifacts due to i.e. air/tissue interfaces, or peri-vascular effects.

The "white-marker imaging" positive-contrast mechanism was introduced by Seppenwoolde et al. in 2003 (Seppenwoolde, 2003). Since then, several groups have developed positive-contrast or dual-contrast pulse sequences for tracking iron-labeled cells

"White-marker" imaging was initially presented to create positive-contrast around paramagnetic intravascular device markers used in magnetic-resonance-based interventional procedures (Seppenwoolde, 2003). The gradient-dephasing technique uses a slice gradient to dephase the background water signal followed by an incomplete gradient rephasing pulse which was exploited for the depiction and tracking of paramagnetic susceptibility markers. Local magnetic field inhomogeneities were selectively visualized with positive-contrast, such as those created by iron-labeled cells for "white-marker" imaging. Advanced methods were developed to separate magnetic susceptibility effects from partial volume effects in "white marker" imaging in order to avoid compromising the identification of magnetic structures (Seppenwoolde, 2007). However, this method is only sensitive to macroscopic field inhomogeneities caused by paramagnetic material, to a volume surrounding the paramagnetic

A similar gradient depashing technique termed gradient echo acquisition for superparamagnetic particles (GRASP), by dephasing of the background signal, has been used to detect positive-contrast from superparamagnetic particles based on the phenomena that the z-rephasing gradient is reduced so that dipolar fields generated by the cells are rephrased and positive signal can be observed (Mani, 2006a), also to image ferritin deposition in a rabbit model of carotid injury with relatively low concentrations of iron oxides at 1.5 T MR scanner (Mani, 2006b). The GRASP technique was used to successfully image low concentrations of ferumoxides (0.05 mM Fe corresponding to 2.8 μg Fe/mL) and ferritin (5 μg Fe/mL) in gel phantoms (Mani, 2006). GRASP "whitemarker" imaging has several advantages including ease of implementation, high sensitivity, no influence on positive signal due to both B0 and B1 field inhomogeneities, and fast acquisition with various TE values. The feasibility of GRASP was tested to aid in dynamically tracking stem cells in a mouse model of myocardial infraction (Mani, 2008). Using T2\*-GRE and GRASP techniques at 9.4 T scanner, iron-labeled embryonic stem cells were visualized in the border zone of infarcted mice at 24 hours, and 1 week following implantation. The positive signal in areas containing iron-loaded stem cells corresponded precisely with the signal loss detected within images produced with conventional GRE sequences. Regions that contained iron-labeled cells were confirmed by histology (Mani, 2008). The presence of the signal loss because of iron-labeled cells would have been difficult to detect on T2\*-weighted images without using the positive-contrast sequence. The region of the myocardium containing the iron-labeled cells was clearly visible when both GRASP and T2\*-weighted techniques (dual contrast imaging) were applied. Dualcontrast effects act to extend the signal change well beyond the location of the particle or

**3. Positive-contrast and dual-contrast MRI techniques** 

**3.1 Gradient-dephasing technique: "white-marker" imaging** 

material that is free of other field variations (Zurkiya, 2006).

*in vitro* and *in vivo* (Table 1).


Note: GRASP, superparamagnetic particles/susceptibility; IRON, oxide nanoparticles–resonant water suppression; SR-SPSP, self-refocused spatial-spectral; FLAPS, fast low-angle positive contrast steadystate free precession; UTE, ultrashort echo-time

Table 1. Summary of Previously Published Studies of Positive- and Dual-contrast Techniques

Iron Oxide Nanoparticles Imaging Tracking by

strengths often used in preclinical studies (Sosnovik, 2009).

applied for in vivo tracking of iron-loaded stem cells (Stuber, 2007).

eliminated.

MR Advanced Techniques: Dual-Contrast Approaches 155

iron particles are excited and refocused, the background on-resonance signal is largely

Iron-labeled mouse embryonic stem cells were imaged as positive-contrast through suppression of background tissue with these off-resonance methods (Suzuki, 2008). A spinecho sequence was used with million-fold (120 dB) suppression of on-resonance water by matching the profiles of a 90° excitation and a 180° refocusing pulse. The positive-contrast signal from the volume of cells was affected by how well the excitation profile was defined. The method is therefore inherently limited by the complication associated with unwanted magnetization from the regions that suffer from chemical shift or susceptibility-related artifacts (e.g., from fat/lipid present in the region of interest and/or imperfect B0 shimming, due to air/tissue interfaces, etc.) (Farrar, 2008). Although ORI techniques are being increasingly used to image iron oxide imaging agents such as MION, the diagnostic accuracy, linearity, and field dependence of ORI have not been fully characterized. After the sensitivity, specificity, and linearity of ORI were examined as a function of both MION concentration and magnetic field strength (4.7 and 14 T), and MION phantoms with and without an air interface as well as MION uptake in a mouse model of healing myocardial infarction were imaged, the linear relationship between MION-induced resonance shifts and with MION concentration were illustrated, whereas T2 showed comparable to the TE and then decreasing after increasing initially with MION concentration and the ORI signal/sensitivity being highly non-linear. Improved specificity of ORI in distinguishing MION-induced resonance shifts and linearity can be expected at lower fields (4.7 T, onresonance water linewidths 15 Hz) with on-resonance water linewidths decreased, airinduced resonance shifts reduced, and longer T2 values observed, thus ORI will be likely optimized at low fields with very short TEs choosing and with moderate MION concentrations. Off-resonance approaches generate positive contrast but have a lower sensitivity than T2\*-weighted imaging and are more complex to perform at high field strengths. Superparamagnetic iron-oxide nanoparticles become saturated above 0.5 Tesla and thus have equal sensitivity at clinical field strengths (1.5–3.0 T) and at the higher field

An alternative off-resonance technique termed inversion-recovery with on-resonant water suppression (IRON) sequence was proposed by a serial studies from one lab (Stuber, 2005, 2007). The IRON method used a spectrally-selective saturation pre- pulse to suppress the signal originating from on-resonant protons in the background tissue while preserving the signal from off-resonant spins in proximity to the iron particles. However, since the size of the signal-enhanced region is dependent on the bandwidth of the water suppression pulse, this scheme requires extra steps to adjust the center frequency and bandwidth of the prepulse to locate the exact site proximal to the cells. IRON sequence has been successfully

The utility of IRON method combined with injection of the long-circulating MION-47 has been recently evaluated by investigators in Johns Hopkins University School of Medicine (Korosoglou, 2008a) for developing a novel contrast-enhanced MR angiography technique. One important aspect of the study was fat suppression for the IRON sequence with an initial radiofrequency pulse offset by 440 Hz at 3.0 T, and with spin inversion, to cause zero

cell itself. This form of signal amplification increases sensitivity in detecting the labeled cells within a complex image background. With the use of signal amplification, potential future applications of (U)SPIO include 'doping' of therapeutic cell preparations with a small fraction of labeled cells, to allow cell tracking without altering the majority of the cells. This would allow for better delineation and identification of labeled cells with both techniques. The challenge for both techniques is the difficulty associated with attempting to quantify the concentration of the labeled cells in vivo because of the susceptibility artifact produced via the iron particles.

Generally, to resolve issues associated with volume averaging and other artifacts that may limit the clinical utility of MRI to detect iron labeled cells (especially in tissues other than the brain), GRASP technique has been developed to differentiate between the signal generated by the cells and signal loss cause by various artifacts (Mani, 2006, 2008), and to specifically avoid the signal loss generated by the iron laden cells to be confused with signal caused by other sources (motion, perivascular effects, coil inhomogeneities, etc.). In the recent study (Briley-Saebo, 2010), the GRASP sequence was also used to both detect and confirm the presence of the Feridex labeled dendritic cells (DCs) in the draining lymph nodes of nude mice 24 h after footpad injection. The results showed the possibility to detect and longitudinally track ex vivo human DC vaccines in the spleen of mice for up to 2 weeks, with greater lymphoid targeting observed following i.v. injection, relative to subcutaneous foot-pad injection; also showed good correlation between in vivo R2\* values on a 9.47 Tesla dedicated mouse scanner and Feridex concentration, with detection limits of 3.2% observed for the spleen. But investigators didn't detect the Feridex labeled cells within the liver and spleen using the GRASP sequence while they indicated that, the dipole effects would be limited and signal enhancement would not be observed when the iron particles being homogenously distributed over a large volume (such as the liver or spleen). They further demonstrated the values of nodes the white marker sequence, GRASP, in accurate detection and identification of Feridex labeled DCs in superficial lymph, and indicated that the appropriate utilization of animals models and MR validated imaging strategies might allow for the optimization of human DC vaccine therapies and improved therapeutic success, whereas white marker sequences maybe most effective when the iron laden cells being compartmentalized within a limited volume (such as in lymph nodes, tumors, or myocardium). On the basis of a recent report (Sigovan, 2011) of the feasibility study on a positive contrast technique, GRASP at a relatively high field 4.7 T, for a novel superparamagnetic nanosystem designed for tumor treatment under MRI monitoring, investigators found that the magnetic nanoparticles for drug delivery can be detected using positive contrast, and suggested that the combined negative and susceptibility methods allow good quality images of large magnetic particles and offer their follow-up for theranostic applications.

#### **3.2 Off-resonance Imaging (ORI)**

Off-resonance MRI approaches have also been developed to produce positive-contrast. With this method, a spectrally selective radio frequency (SSRF) pulse was used to excite only the susceptibility-shifted, or 'off-resonance', water signals (Cunningham, 2005; Foltz, 2006), at the frequency shift induced by the iron particles. Since only the off-resonance signal due to eliminated.

154 Smart Nanoparticles Technology

cell itself. This form of signal amplification increases sensitivity in detecting the labeled cells within a complex image background. With the use of signal amplification, potential future applications of (U)SPIO include 'doping' of therapeutic cell preparations with a small fraction of labeled cells, to allow cell tracking without altering the majority of the cells. This would allow for better delineation and identification of labeled cells with both techniques. The challenge for both techniques is the difficulty associated with attempting to quantify the concentration of the labeled cells in vivo because of the susceptibility

Generally, to resolve issues associated with volume averaging and other artifacts that may limit the clinical utility of MRI to detect iron labeled cells (especially in tissues other than the brain), GRASP technique has been developed to differentiate between the signal generated by the cells and signal loss cause by various artifacts (Mani, 2006, 2008), and to specifically avoid the signal loss generated by the iron laden cells to be confused with signal caused by other sources (motion, perivascular effects, coil inhomogeneities, etc.). In the recent study (Briley-Saebo, 2010), the GRASP sequence was also used to both detect and confirm the presence of the Feridex labeled dendritic cells (DCs) in the draining lymph nodes of nude mice 24 h after footpad injection. The results showed the possibility to detect and longitudinally track ex vivo human DC vaccines in the spleen of mice for up to 2 weeks, with greater lymphoid targeting observed following i.v. injection, relative to subcutaneous foot-pad injection; also showed good correlation between in vivo R2\* values on a 9.47 Tesla dedicated mouse scanner and Feridex concentration, with detection limits of 3.2% observed for the spleen. But investigators didn't detect the Feridex labeled cells within the liver and spleen using the GRASP sequence while they indicated that, the dipole effects would be limited and signal enhancement would not be observed when the iron particles being homogenously distributed over a large volume (such as the liver or spleen). They further demonstrated the values of nodes the white marker sequence, GRASP, in accurate detection and identification of Feridex labeled DCs in superficial lymph, and indicated that the appropriate utilization of animals models and MR validated imaging strategies might allow for the optimization of human DC vaccine therapies and improved therapeutic success, whereas white marker sequences maybe most effective when the iron laden cells being compartmentalized within a limited volume (such as in lymph nodes, tumors, or myocardium). On the basis of a recent report (Sigovan, 2011) of the feasibility study on a positive contrast technique, GRASP at a relatively high field 4.7 T, for a novel superparamagnetic nanosystem designed for tumor treatment under MRI monitoring, investigators found that the magnetic nanoparticles for drug delivery can be detected using positive contrast, and suggested that the combined negative and susceptibility methods allow good quality images of large magnetic particles and offer their follow-up for

Off-resonance MRI approaches have also been developed to produce positive-contrast. With this method, a spectrally selective radio frequency (SSRF) pulse was used to excite only the susceptibility-shifted, or 'off-resonance', water signals (Cunningham, 2005; Foltz, 2006), at the frequency shift induced by the iron particles. Since only the off-resonance signal due to

artifact produced via the iron particles.

theranostic applications.

**3.2 Off-resonance Imaging (ORI)** 

iron particles are excited and refocused, the background on-resonance signal is largely

Iron-labeled mouse embryonic stem cells were imaged as positive-contrast through suppression of background tissue with these off-resonance methods (Suzuki, 2008). A spinecho sequence was used with million-fold (120 dB) suppression of on-resonance water by matching the profiles of a 90° excitation and a 180° refocusing pulse. The positive-contrast signal from the volume of cells was affected by how well the excitation profile was defined. The method is therefore inherently limited by the complication associated with unwanted magnetization from the regions that suffer from chemical shift or susceptibility-related artifacts (e.g., from fat/lipid present in the region of interest and/or imperfect B0 shimming, due to air/tissue interfaces, etc.) (Farrar, 2008). Although ORI techniques are being increasingly used to image iron oxide imaging agents such as MION, the diagnostic accuracy, linearity, and field dependence of ORI have not been fully characterized. After the sensitivity, specificity, and linearity of ORI were examined as a function of both MION concentration and magnetic field strength (4.7 and 14 T), and MION phantoms with and without an air interface as well as MION uptake in a mouse model of healing myocardial infarction were imaged, the linear relationship between MION-induced resonance shifts and with MION concentration were illustrated, whereas T2 showed comparable to the TE and then decreasing after increasing initially with MION concentration and the ORI signal/sensitivity being highly non-linear. Improved specificity of ORI in distinguishing MION-induced resonance shifts and linearity can be expected at lower fields (4.7 T, onresonance water linewidths 15 Hz) with on-resonance water linewidths decreased, airinduced resonance shifts reduced, and longer T2 values observed, thus ORI will be likely optimized at low fields with very short TEs choosing and with moderate MION concentrations. Off-resonance approaches generate positive contrast but have a lower sensitivity than T2\*-weighted imaging and are more complex to perform at high field strengths. Superparamagnetic iron-oxide nanoparticles become saturated above 0.5 Tesla and thus have equal sensitivity at clinical field strengths (1.5–3.0 T) and at the higher field strengths often used in preclinical studies (Sosnovik, 2009).

An alternative off-resonance technique termed inversion-recovery with on-resonant water suppression (IRON) sequence was proposed by a serial studies from one lab (Stuber, 2005, 2007). The IRON method used a spectrally-selective saturation pre- pulse to suppress the signal originating from on-resonant protons in the background tissue while preserving the signal from off-resonant spins in proximity to the iron particles. However, since the size of the signal-enhanced region is dependent on the bandwidth of the water suppression pulse, this scheme requires extra steps to adjust the center frequency and bandwidth of the prepulse to locate the exact site proximal to the cells. IRON sequence has been successfully applied for in vivo tracking of iron-loaded stem cells (Stuber, 2007).

The utility of IRON method combined with injection of the long-circulating MION-47 has been recently evaluated by investigators in Johns Hopkins University School of Medicine (Korosoglou, 2008a) for developing a novel contrast-enhanced MR angiography technique. One important aspect of the study was fat suppression for the IRON sequence with an initial radiofrequency pulse offset by 440 Hz at 3.0 T, and with spin inversion, to cause zero

Iron Oxide Nanoparticles Imaging Tracking by

individually shimmed slices (Zurkiya, 2006).

*vivo* tumor target.

2005).

MR Advanced Techniques: Dual-Contrast Approaches 157

and can lead to less erroneous off-resonant signal detection in a multi-slice manner with

The off-resonance saturation method has been developed by Zurkiya and Hu, in which water protons are imaged with and without the presence of an off-resonance saturation pulse (Zurkiya, 2006). This method relies on diffusion-mediated saturation transfer to reduce the on-resonance MRI signal due to the off-resonance saturation (ORS) pulse, similar to chemical exchange saturation transfer techniques (Ward, 2000). This approach has been verified that greatly improved tumor detection accuracy over the conventional T2\*-weighted methods because of its ability to turn "ON" the contrast of superparamagnetic polymeric micelles (SPPM) nanoparticles (Khemtong, 2009). SPPM nanoparticles encoded with cyclic (RGDfK) ligand (arginine-glycine-aspartic acid), cRGD, were able to target the αvβ3-expressing microvasculature in A549 non-small cell lung tumor xenografts in mice. The results suggest that the combination of ORS imaging with cancer-targeted SPPM nanoparticles will show promise in detecting biochemical markers at early stages of non–small cell lung tumor development, and could further enhance the sensitivity of contrast and provide new opportunities in imaging biomarkers setting of *in* 

The study (Zurkiya, 2008) transfected cells with genes from magnetotactic bacteria (i.e., MagA) under doxycycline-regulated gene expression, resulting in the intracellular production of iron oxide nanoparticles similar to synthetic SPION. MagA-expressing cells could be visualized by MRI after transplantation in the mouse brain after 5 d of induction with doxycycline. The generalized implementation of these techniques as treatment strategies in stem cell tracking needs to be explored. Investigators have recently inserted magnetic reporter genes into cells. After the expression of iron storage proteins formed stored iron then MRI can be used to detect it. Another transgene reporter, an adenoviral vector carrying a transgene for light- and heavy-chain ferritin protein to transfect cells has been shown that they could be detected by in vivo magnetic resonance imaging (Genove,

Balchandani et al. recently developed a self-refocused spatial-spectral (SR-SPSP) pulse, which is successful in creating positive-contrast images of SPIO-labeled cells (Balchandani, 2009). This pulse can enable slice-selective, spin-echo imaging of off-resonant spins without an

combined into one pulse. This results in a considerably shorter TE than possible with two separate pulses. The simultaneous spatial and spectral selectivity allows the imaging of offresonant spins while selecting a single slice. The SR-SPSP pulse is also suitable for any application requiring spatial and spectral selectivity, such as tracking metallic devices or replacing standard pulses in MR spectroscopic imaging sequences. More recently a novel combination of off-resonance (ORI) positive-contrast MRI and T(2ρ) relaxation in the rotating frame (ORI-T2ρ method) for positive-contrast MR imaging of USPIO in a mouse model of burn trauma and infection with *Pseudomonas aeruginosa (PA)*, was also reported to have direct implications in the longitudinal noninvasive monitoring of infection, and show promise in testing the new-developed anti-infective compounds (Andronesi, 2010). The same group also reported that ORI-T2ρ method proved to have slightly higher sensitivity than ORI, and MR imaging clearly showed migration and accumulation of labeled MSCs to the burn area which

SPSP pulse

increase in TE, which is essentially a phase-matched 90◦ SPSP pulse and a 180◦

can be confirmed by histology staining for iron labeled cells (Righi, 2010).

longitudinal magnetization of the targeted species for the radiofrequency pulses (105° for fat, 100° for water), which obviously shortened the subsequent recovery time. The usage of MION-47 allowed acquisition of multiple image sets over a 1- or 2-day period with high spatial resolution.

IRON techniques with a commercially available MION-47 were recently employed to detect macrophage-rich atherosclerotic plaques in a rabbit model of atherosclerosis (Korosoglou, 2008b), in which pre-contrast imaging was performed in 7 Watanabe rabbits and 4 control New Zealand rabbits, and post-contrast imaging was repeated on days 1 and 3 after intravenous injection of MION-47. A second injection was performed on day 3 after imaging and post-contrast imaging performed again on day 6. There was a significant increase in signal intensity within aortic atherosclerotic plaques following administration of MION-47 (48% increase on day 3 and 72% increase on day 6) versus hypointensity (negative-contrast) in conventional MR images, but no enhancement was seen in control rabbits that lacked atherosclerosis. The positive-contrast regions corresponded to regions demonstrating deposition of iron particles within macrophage-rich atherosclerotic plaques. These findings not only validated that MION-47 is a successful imaging agent for macrophage-rich atherosclerosis, but also suggested that positive-contrast IRON MRI can be applied to the general class of iron oxide particles. This is significant as USPIO-enhanced MR imaging has been previously studied in human (Trivedi, 2006); enabling IRON MRI sequences to be directly applied to patient care.

Korosoglou et al. also investigated the utility of IRON techniques and MION-47 to create positive-contrast MR-lymphography (Korosoglou, 2008c). After six rabbits received a single bolus injection of 80 mmol Fe/kg MION-47, MRI was performed at baseline, 1 day, and 3 days using conventional T1- and T2\*-weighted sequences and IRON. On T2\*-weighted images, as expected, signal attenuation was observed in areas of para-aortic lymph nodes after MION-47 injection. However, using IRON the para-aortic lymph nodes exhibited very high contrast enhancement, which remained 3 days after injection. IRON in conjunction with iron particles can be therefore used to perform positive-contrast MR-lymphography, particularly 3 days after injection of the contrast agent, when signal is no longer visible within blood vessels. The proposed method may have the potential as an adjunct for nodal staging in cancer screening.

Iron-labeled radioembolization microspheres were visualized for in vivo tracking during trans-catheter delivery to VX2 liver tumors in a rabbit model (Gupta, 2008). The study was performed for real-time observation of microsphere delivery with dual-contrast techniques. The results showed significant changes in post-injection contrast-to-noise ratio (CNR) values from those of pre-injection at positions of microsphere deposition with both negative- and positive-contrast.

The off-resonance MRI method possesses some advantages including no need for dephasing gradients or saturation pulses, high suppression efficiency, and flexible selection of the excited frequency band to encompass spins in the vicinity of the iron particles without fat tissue off-resonance. This technique, however, was not slice-selective such that it can result in interference from insufficiently suppressed background signals or less background signal with regions of greater susceptibility excluded from the selected slice. This technique can also cause less on-resonant signal to be suppressed, has less flexibility in RF pulse design,

longitudinal magnetization of the targeted species for the radiofrequency pulses (105° for fat, 100° for water), which obviously shortened the subsequent recovery time. The usage of MION-47 allowed acquisition of multiple image sets over a 1- or 2-day period with high

IRON techniques with a commercially available MION-47 were recently employed to detect macrophage-rich atherosclerotic plaques in a rabbit model of atherosclerosis (Korosoglou, 2008b), in which pre-contrast imaging was performed in 7 Watanabe rabbits and 4 control New Zealand rabbits, and post-contrast imaging was repeated on days 1 and 3 after intravenous injection of MION-47. A second injection was performed on day 3 after imaging and post-contrast imaging performed again on day 6. There was a significant increase in signal intensity within aortic atherosclerotic plaques following administration of MION-47 (48% increase on day 3 and 72% increase on day 6) versus hypointensity (negative-contrast) in conventional MR images, but no enhancement was seen in control rabbits that lacked atherosclerosis. The positive-contrast regions corresponded to regions demonstrating deposition of iron particles within macrophage-rich atherosclerotic plaques. These findings not only validated that MION-47 is a successful imaging agent for macrophage-rich atherosclerosis, but also suggested that positive-contrast IRON MRI can be applied to the general class of iron oxide particles. This is significant as USPIO-enhanced MR imaging has been previously studied in human (Trivedi, 2006); enabling IRON MRI sequences to be

Korosoglou et al. also investigated the utility of IRON techniques and MION-47 to create positive-contrast MR-lymphography (Korosoglou, 2008c). After six rabbits received a single bolus injection of 80 mmol Fe/kg MION-47, MRI was performed at baseline, 1 day, and 3 days using conventional T1- and T2\*-weighted sequences and IRON. On T2\*-weighted images, as expected, signal attenuation was observed in areas of para-aortic lymph nodes after MION-47 injection. However, using IRON the para-aortic lymph nodes exhibited very high contrast enhancement, which remained 3 days after injection. IRON in conjunction with iron particles can be therefore used to perform positive-contrast MR-lymphography, particularly 3 days after injection of the contrast agent, when signal is no longer visible within blood vessels. The proposed method may have the potential as an adjunct for nodal

Iron-labeled radioembolization microspheres were visualized for in vivo tracking during trans-catheter delivery to VX2 liver tumors in a rabbit model (Gupta, 2008). The study was performed for real-time observation of microsphere delivery with dual-contrast techniques. The results showed significant changes in post-injection contrast-to-noise ratio (CNR) values from those of pre-injection at positions of microsphere deposition with both negative- and

The off-resonance MRI method possesses some advantages including no need for dephasing gradients or saturation pulses, high suppression efficiency, and flexible selection of the excited frequency band to encompass spins in the vicinity of the iron particles without fat tissue off-resonance. This technique, however, was not slice-selective such that it can result in interference from insufficiently suppressed background signals or less background signal with regions of greater susceptibility excluded from the selected slice. This technique can also cause less on-resonant signal to be suppressed, has less flexibility in RF pulse design,

spatial resolution.

directly applied to patient care.

staging in cancer screening.

positive-contrast.

and can lead to less erroneous off-resonant signal detection in a multi-slice manner with individually shimmed slices (Zurkiya, 2006).

The off-resonance saturation method has been developed by Zurkiya and Hu, in which water protons are imaged with and without the presence of an off-resonance saturation pulse (Zurkiya, 2006). This method relies on diffusion-mediated saturation transfer to reduce the on-resonance MRI signal due to the off-resonance saturation (ORS) pulse, similar to chemical exchange saturation transfer techniques (Ward, 2000). This approach has been verified that greatly improved tumor detection accuracy over the conventional T2\*-weighted methods because of its ability to turn "ON" the contrast of superparamagnetic polymeric micelles (SPPM) nanoparticles (Khemtong, 2009). SPPM nanoparticles encoded with cyclic (RGDfK) ligand (arginine-glycine-aspartic acid), cRGD, were able to target the αvβ3-expressing microvasculature in A549 non-small cell lung tumor xenografts in mice. The results suggest that the combination of ORS imaging with cancer-targeted SPPM nanoparticles will show promise in detecting biochemical markers at early stages of non–small cell lung tumor development, and could further enhance the sensitivity of contrast and provide new opportunities in imaging biomarkers setting of *in vivo* tumor target.

The study (Zurkiya, 2008) transfected cells with genes from magnetotactic bacteria (i.e., MagA) under doxycycline-regulated gene expression, resulting in the intracellular production of iron oxide nanoparticles similar to synthetic SPION. MagA-expressing cells could be visualized by MRI after transplantation in the mouse brain after 5 d of induction with doxycycline. The generalized implementation of these techniques as treatment strategies in stem cell tracking needs to be explored. Investigators have recently inserted magnetic reporter genes into cells. After the expression of iron storage proteins formed stored iron then MRI can be used to detect it. Another transgene reporter, an adenoviral vector carrying a transgene for light- and heavy-chain ferritin protein to transfect cells has been shown that they could be detected by in vivo magnetic resonance imaging (Genove, 2005).

Balchandani et al. recently developed a self-refocused spatial-spectral (SR-SPSP) pulse, which is successful in creating positive-contrast images of SPIO-labeled cells (Balchandani, 2009). This pulse can enable slice-selective, spin-echo imaging of off-resonant spins without an increase in TE, which is essentially a phase-matched 90◦ SPSP pulse and a 180◦ SPSP pulse combined into one pulse. This results in a considerably shorter TE than possible with two separate pulses. The simultaneous spatial and spectral selectivity allows the imaging of offresonant spins while selecting a single slice. The SR-SPSP pulse is also suitable for any application requiring spatial and spectral selectivity, such as tracking metallic devices or replacing standard pulses in MR spectroscopic imaging sequences. More recently a novel combination of off-resonance (ORI) positive-contrast MRI and T(2ρ) relaxation in the rotating frame (ORI-T2ρ method) for positive-contrast MR imaging of USPIO in a mouse model of burn trauma and infection with *Pseudomonas aeruginosa (PA)*, was also reported to have direct implications in the longitudinal noninvasive monitoring of infection, and show promise in testing the new-developed anti-infective compounds (Andronesi, 2010). The same group also reported that ORI-T2ρ method proved to have slightly higher sensitivity than ORI, and MR imaging clearly showed migration and accumulation of labeled MSCs to the burn area which can be confirmed by histology staining for iron labeled cells (Righi, 2010).

Iron Oxide Nanoparticles Imaging Tracking by

MR of aorta atherosclerotic rabbit (Crowe, 2005).

report (Liu, 2009).

ideally without background signal.

MR Advanced Techniques: Dual-Contrast Approaches 159

used for UTE imaging, only negligible T2 decay occurs before sampling, and consequently high signal from the short-T2 components can be obtained. Coolen et al. reported that MRI parameters could be optimized for positive-contrast detection of iron-oxide labeled cells using double-echo Ultra-short echo time (d-UTE) sequences (Coolen, 2007). During these studies, there was a linear correlation between signal intensity and concentration USPIO labeled cells. Another group found that the enhancement due to the presence of short T2 USPIO accumulation generally agreed with signal loss within GRE images during ex vivo

Liu et al. recently measured ultrashort T2\* relaxation in tissues containing a focal area of SPIO nanoparticle-labeled cells. MRI experiments in phantoms and rats with iron-labeled tumors demonstrated that these cells can be detected even at ultrashort T2\* down to 1 ms or less (Liu, 2009). The authors suggested that combining ultrashort T2\* relaxometry with the multiple gradient echo T2\* mapping techniques should improve the ability to measure the relaxation of tissues with high densities of implanted iron- labeled cells. In another investigation, T1-weighted positive contrast enhancement from SPIO particles was achieved from the UTE imaging then this sequence, taking advantage of the unique effect of MNPs on relaxation time domain, was also examined to validate its positive contrast imaging capability of "probe" targeting to U87MG human glioblastoma cells through an SPIO conjugated RDG with high affinity to the cells overexpressing integrin αvβ3 (Zhang, 2011). So the study was regarded as providing a dual contrast imaging method from UTE technique plus T2-weighted TSE images in its application of molecular imaging of glioma with potential quantification of SPIO nanoparticles suggested by previously published

The more recent study (Girard, 2011) showed that both contrast mechanisms of optimizing T1 contrast from UTE technique with conventional T2\* contrast of SPIO, even an extra subtraction of a later echo signal from the UTE signal, could be powerful both in improving the specificity by providing long T2\* background suppression and increasing detection sensitivity, in molecular imaging application of tumor-targeted IONPs in vivo. A hybrid sequence, PETRA (Pointwise Encoding Time reduction with Radial Acquisition) (Grodzki, 2011), combined the features of single point imaging with radial projection imaging with no need of hardware changes, to show shorter encoding times over the whole k-space and to enable higher resolution for tissue with very short T2, compared to the UTE sequence, so that it could avoids problems derived from the UTE but with good image quality and might improve e.g. orthopedic MR imaging as well as MR-PET attenuation correction. A 3D imaging technique (Seevinck, 2011) from the group in University Medical Center Utrecht, The Netherlands, applying center-out RAdial Sampling with Off-Resonance reception (co-RASOR) by the using of UTE technique (for the minimization of subvoxel dephasing at locations with high magnetic field gradients in the vicinity of the magnetized objects), and a hard, nonselective RF block pulse and radial sampling of k-space, was also presented to depict and accurately localize small paramagnetic objects with high positive contrast but

**3.5 Others new MRI pulse sequences and image postprocessing techniques** 

Several other new sequences were reported on positive- and dual-contrast methods of MR cell tracking. Kim et al. recently developed simple means of detecting iron-labeled cells by

### **3.3 Fast low angle steady-state free precession (FLAPS) sequence**

FLAPS imaging has been proposed for time-efficient acquisition of off-resonance positivecontrast images (Dharmakumar, 2007). The technique takes advantage of the unique spectral response of the steady-state free precession (SSFP) signal to achieve signal enhancement from off-resonant spins while suppressing signal from on-resonant spins at relatively small flip angles (Dharmakumar, 2006). Besides the positive-contrast generated by the weakly offresonant spins, the spins in and around the core of the local magnetic susceptibility (LMS) shifting media (such as labeled cells) experience large deviations from the central frequency leading to intra-voxel dephasing that was observed as negative-contrast in FLAPS images. So this technique has the capability to identify the presence of labeled cells with both negative- and positive-contrast within a single image.

Zhang et al. recently investigated the feasibility of imaging iron-labeled green fluorescent protein (GFP)-expressing cells with the dual-contrast method and compared its measurements with traditional negative-contrast technique (Zhang, 2009). The GFP-cell was incubated for 24 hours using 20 mg Fe/mL concentration of SPIO and USPIO nanoparticles. The labeled cells were imaged using the FLAPS technique, and FLAPS images with positive-contrast were compared with negative-contrast T2\*-weighted images. The results demonstrated that SPIO and USPIO labeling of GFP cells had no effect on cell function or GFP expression, and the labeled cells were observed as a narrow band of signal enhancement surrounding signal voids in FLAPS images. Positive- and negative-contrast images were both valuable for visualizing labeled GFP-cells. MRI of labeled cells with GFP expression holds great potential for monitoring the temporal and spatial migration of gene markers and cells, and enhances our understanding of cell- and gene-based therapeutic strategies. These findings suggested that the dual-contrast nature of the FLAPS approach offers significant advantages to the field of cellular MRI. A highly sought feature of cellular imaging is the quantification of labeled cells. Past studies have shown that it may be possible to define a relation between number of cells and MR transverse relaxation time constants (apparent T2 or T2\*). However, since the specificity of the labeled cells is often compromised in GRE images, it is often difficult to use the time constant thus derived as a reliable metric to quantify the number of cells. These previous FLAPS investigations showed that local contrast was exponentially related to the number of cells. Furthermore, the dual-contrast filter, using an image metric that is analogous to local contrast, can provide additional quantitative information regarding those regions containing the labeled cells. This technique still could be limited by the magnetic perturbations around MNPs. A careful investigation of how the output of dual-contrast image filters can be used to derive quantitative information regarding the concentration of labeled cells from *in vivo* images has been demonstrated (Dharmakumar, 2009).

#### **3.4 Ultra short echo time methods**

It has been introduced that ultrashort echo-time (UTE) imaging had capability of imaging materials with extremely short T2 and very fast signal decay (Robson, 2006; Rahmer, 2009), and did as a new and promising approach that allowed the detection of short-T2 signal components, such as tendons, ligaments, menisci, periosteum, and cortical bone before signals within these tissues decay to a level where they were not observable with conventional spin echo pulse sequences. Due to the very short TE (on the order of 1/10 ms)

FLAPS imaging has been proposed for time-efficient acquisition of off-resonance positivecontrast images (Dharmakumar, 2007). The technique takes advantage of the unique spectral response of the steady-state free precession (SSFP) signal to achieve signal enhancement from off-resonant spins while suppressing signal from on-resonant spins at relatively small flip angles (Dharmakumar, 2006). Besides the positive-contrast generated by the weakly offresonant spins, the spins in and around the core of the local magnetic susceptibility (LMS) shifting media (such as labeled cells) experience large deviations from the central frequency leading to intra-voxel dephasing that was observed as negative-contrast in FLAPS images. So this technique has the capability to identify the presence of labeled cells with both

Zhang et al. recently investigated the feasibility of imaging iron-labeled green fluorescent protein (GFP)-expressing cells with the dual-contrast method and compared its measurements with traditional negative-contrast technique (Zhang, 2009). The GFP-cell was incubated for 24 hours using 20 mg Fe/mL concentration of SPIO and USPIO nanoparticles. The labeled cells were imaged using the FLAPS technique, and FLAPS images with positive-contrast were compared with negative-contrast T2\*-weighted images. The results demonstrated that SPIO and USPIO labeling of GFP cells had no effect on cell function or GFP expression, and the labeled cells were observed as a narrow band of signal enhancement surrounding signal voids in FLAPS images. Positive- and negative-contrast images were both valuable for visualizing labeled GFP-cells. MRI of labeled cells with GFP expression holds great potential for monitoring the temporal and spatial migration of gene markers and cells, and enhances our understanding of cell- and gene-based therapeutic strategies. These findings suggested that the dual-contrast nature of the FLAPS approach offers significant advantages to the field of cellular MRI. A highly sought feature of cellular imaging is the quantification of labeled cells. Past studies have shown that it may be possible to define a relation between number of cells and MR transverse relaxation time constants (apparent T2 or T2\*). However, since the specificity of the labeled cells is often compromised in GRE images, it is often difficult to use the time constant thus derived as a reliable metric to quantify the number of cells. These previous FLAPS investigations showed that local contrast was exponentially related to the number of cells. Furthermore, the dual-contrast filter, using an image metric that is analogous to local contrast, can provide additional quantitative information regarding those regions containing the labeled cells. This technique still could be limited by the magnetic perturbations around MNPs. A careful investigation of how the output of dual-contrast image filters can be used to derive quantitative information regarding the concentration of labeled cells from *in* 

It has been introduced that ultrashort echo-time (UTE) imaging had capability of imaging materials with extremely short T2 and very fast signal decay (Robson, 2006; Rahmer, 2009), and did as a new and promising approach that allowed the detection of short-T2 signal components, such as tendons, ligaments, menisci, periosteum, and cortical bone before signals within these tissues decay to a level where they were not observable with conventional spin echo pulse sequences. Due to the very short TE (on the order of 1/10 ms)

**3.3 Fast low angle steady-state free precession (FLAPS) sequence** 

negative- and positive-contrast within a single image.

*vivo* images has been demonstrated (Dharmakumar, 2009).

**3.4 Ultra short echo time methods** 

used for UTE imaging, only negligible T2 decay occurs before sampling, and consequently high signal from the short-T2 components can be obtained. Coolen et al. reported that MRI parameters could be optimized for positive-contrast detection of iron-oxide labeled cells using double-echo Ultra-short echo time (d-UTE) sequences (Coolen, 2007). During these studies, there was a linear correlation between signal intensity and concentration USPIO labeled cells. Another group found that the enhancement due to the presence of short T2 USPIO accumulation generally agreed with signal loss within GRE images during ex vivo MR of aorta atherosclerotic rabbit (Crowe, 2005).

Liu et al. recently measured ultrashort T2\* relaxation in tissues containing a focal area of SPIO nanoparticle-labeled cells. MRI experiments in phantoms and rats with iron-labeled tumors demonstrated that these cells can be detected even at ultrashort T2\* down to 1 ms or less (Liu, 2009). The authors suggested that combining ultrashort T2\* relaxometry with the multiple gradient echo T2\* mapping techniques should improve the ability to measure the relaxation of tissues with high densities of implanted iron- labeled cells. In another investigation, T1-weighted positive contrast enhancement from SPIO particles was achieved from the UTE imaging then this sequence, taking advantage of the unique effect of MNPs on relaxation time domain, was also examined to validate its positive contrast imaging capability of "probe" targeting to U87MG human glioblastoma cells through an SPIO conjugated RDG with high affinity to the cells overexpressing integrin αvβ3 (Zhang, 2011). So the study was regarded as providing a dual contrast imaging method from UTE technique plus T2-weighted TSE images in its application of molecular imaging of glioma with potential quantification of SPIO nanoparticles suggested by previously published report (Liu, 2009).

The more recent study (Girard, 2011) showed that both contrast mechanisms of optimizing T1 contrast from UTE technique with conventional T2\* contrast of SPIO, even an extra subtraction of a later echo signal from the UTE signal, could be powerful both in improving the specificity by providing long T2\* background suppression and increasing detection sensitivity, in molecular imaging application of tumor-targeted IONPs in vivo. A hybrid sequence, PETRA (Pointwise Encoding Time reduction with Radial Acquisition) (Grodzki, 2011), combined the features of single point imaging with radial projection imaging with no need of hardware changes, to show shorter encoding times over the whole k-space and to enable higher resolution for tissue with very short T2, compared to the UTE sequence, so that it could avoids problems derived from the UTE but with good image quality and might improve e.g. orthopedic MR imaging as well as MR-PET attenuation correction. A 3D imaging technique (Seevinck, 2011) from the group in University Medical Center Utrecht, The Netherlands, applying center-out RAdial Sampling with Off-Resonance reception (co-RASOR) by the using of UTE technique (for the minimization of subvoxel dephasing at locations with high magnetic field gradients in the vicinity of the magnetized objects), and a hard, nonselective RF block pulse and radial sampling of k-space, was also presented to depict and accurately localize small paramagnetic objects with high positive contrast but ideally without background signal.

## **3.5 Others new MRI pulse sequences and image postprocessing techniques**

Several other new sequences were reported on positive- and dual-contrast methods of MR cell tracking. Kim et al. recently developed simple means of detecting iron-labeled cells by

Iron Oxide Nanoparticles Imaging Tracking by

**3.6 T1 & T2 (T2\*) multi-contrast for cell tracking** 

containing region (Dixon, 2009).

methods is still under evaluation.

within both liver and spleen.

relaxometry mapping.

*vitro* and *in vivo* (Yang, 2011).

MR Advanced Techniques: Dual-Contrast Approaches 161

former derives from the signal decay associated with areas containing contrast SPIO particles (Kuhlpeter, 2007; Rad, 2007; Liu, 2009), assuming that the rate varies linearly with contrast agent concentration; the later derives from the formation of magnetic field by SPIO-

As introduced in as earlier as 1990s, it is possible to achieve positive contrast and dual contrast with superparamagnetic particles by employing T1- and/or T2-weighted sequences (Canet, 1993; Chambon, 1993; Small, 1993). Although most earlier clinical trials with magnetic nanoparticles as contrast agents were evaluated almost exclusively on T2-w fast spin echo (FSE) and T2\*-w gradient echo (GRE) sequences, and the strong T1 contrast enhancement effect of magnetic nanoparticles has rarely been used in clinical and molecular imaging (Reimer, 1995; Yamamoto, 1995; Tang, 1999), the effect of SPIO or USPIO on proton relaxation is not confined to T2 and T2\* effect. They should be considered to influence T1 relaxivity with increased SI on T1-w GRE sequences at low concentrations. For in vivo imaging application of MNPs, optimal combination of negative and positive contrast

Superparamagnetic iron oxide particles (SPIO) were used shortly after gadolinium-chelate magnetic resonance (MR) contrast agent as well known, while USPIO being the strong T2 relaxivity that produces negative contrast also a high T1 relaxivity with an increase in SI on T1-weighted images (Small, 1993), so that a biphasic imaging sequence protocol (only immediate postadministration and 20-24 hr delayed images) in the *in vivo* study allowed visualization of the dynamic enhancement patterns of both normal tissue and potentially tumor based on early T1-shortening effects produced by intravascular USPIO particulate agent (BMS 180549, previously AMI-227) and marked T1-shortening produced following agent uptake by liver and spleen, as well as showed markedly less T2-shortening at 20-24 hr

The more recent investigation (Zhang, 2011) demonstrated that an appropriate SPIO core size and concentration range was paid much attention to obtain positive contrast with UTE imaging, and this technique could be used with the receptor targeted SPIO molecular imaging probe so as to provide an opportunity for monitoring cancer cells with overexpression of integrin αvβ3 in addition to negative contrast by the approach of T2

Investigators recently synthesized a biocompatible water-dispersible Fe3O4–SiO2–Gd– DTPA–RGD nanoparticle with r1 relaxivity of 4.2 mm−1s−1 and r2 relaxivity of 17.4 mm−1s−<sup>1</sup> at the Gd/Fe molar ratio of 0.3:1, indicating the potential to use this multifunctional agent for dual-contrast MR imaging of tumor cells over-expressing high-affinity αvβ3 integrin *in* 

MRI can be commonly used to set up a kind of nanomedicine platform for applications of multimodality probe to obtain information about concomitant anatomic, chemical, and physiological features of body. This kind of approach has been found under the

**4. Imaging contrast of IRON-labeled cell on multimodular platform** 

using susceptibility weighted echo-time encoding technique (SWEET) (Kim, 2006). The subtraction of two sets of image volumes acquired at slightly-shifted echo time generates positive-contrast at the cell position. In a more recent study, the SWEET method was employed to selectively enhance the effect of the magnetic susceptibility caused by SPIOlabeled KB cells (KB cell is a cell line derived from a human carcinoma of the nasopharynx, used as an assay for antineoplastic agent). It was also demonstrated that this method could be used to visualize SPIO-labeled KB cells and their tumor formation in mice for at least a 2 week period (Kim, 2009).

Dual-contrast images can also be achieved by applying T2\*-weighted imaging combined with different post-processing techniques from the magnetic field map (Ward, 2000; Zurkiya, 2006). A susceptibility gradient mapping (SGM) technique has been recently developed, in which a color map of 3D susceptibility-gradient vector for every voxel is generated with calculated echo-shifts, and the map presents a 3D form of a positive-contrast images (Dahnke, 2008; Liu, 2008). Hyperintensities of SGM were seen in areas surrounding the 1×106 ferumoxides/protamine sulfate complex labeled flank C6 glioma cells of experimental rat model. The sensitivity of the method was compared to white-marker and IRON positive-contrast methods for visualizing the proliferation of tumor cells for labeled tumors that were approximately 5mm (small), 10 mm (medium) and 20 mm (large) in diameter along the largest dimension (Liu, 2008). The number of positive voxels detected around small and medium tumors was significantly greater with the SGM technique than those with the other two techniques, while similar as the "white-marker" technique for large tumors that could not be visualized with the IRON technique. The SGM is a post-processing technique and its positive-contrast images can be derived directly from the T2\*-weighted images without requiring dedicated positive-contrast pulse sequences, thereby it can provide the flexibility to display susceptibility gradients or suppress susceptibility artifacts in specific directions; not like the "white marker" or IRON techniques that require specialized pulse sequence designs and extra scans in addition to those obtained for conventional anatomic imaging. With SGM the hyperintense regions on positive-contrast images originating from SPIO labeled cells can be easily differentiated from other signal voids in T2 or T2\*-weighted images.

The phase gradient mapping (PGM) techniques have recently developed independently by two groups, one related derived phase gradient maps from standard phase images also including a phase unwrapping procedure to assist the analysis and characterization of object-induced macroscopic phase perturbations (Bakker, 2008); another one utilized fast Fourier transform (FFT) to form phase gradients and develop positive contrast maps by the use of PGM but without need of phase unwrapping, so as to be appropriate technique for the visualization of magnetic nanoparticulate system (Langley, 2011; Zhao, 2011). By the method introduced recently of dual contrast with therapeutic iron nanoparticles at 4.7 T scanner (Sigovan, 2011), or postprocessing methods, with the measure of the T2\*, an efficient estimation of nanoparticle concentration can be made (Langley, 2011). Applications of two kind of approaches, the traditional relaxometry method and model-based method, have demonstrated that, besides the detection of SPIO nanoparticles by positive contrast methods, quantification of the SPIO concentration also play important role in clinical evaluation of results from different treatments with monitoring cellular therapies, and the

using susceptibility weighted echo-time encoding technique (SWEET) (Kim, 2006). The subtraction of two sets of image volumes acquired at slightly-shifted echo time generates positive-contrast at the cell position. In a more recent study, the SWEET method was employed to selectively enhance the effect of the magnetic susceptibility caused by SPIOlabeled KB cells (KB cell is a cell line derived from a human carcinoma of the nasopharynx, used as an assay for antineoplastic agent). It was also demonstrated that this method could be used to visualize SPIO-labeled KB cells and their tumor formation in mice for at least a 2-

Dual-contrast images can also be achieved by applying T2\*-weighted imaging combined with different post-processing techniques from the magnetic field map (Ward, 2000; Zurkiya, 2006). A susceptibility gradient mapping (SGM) technique has been recently developed, in which a color map of 3D susceptibility-gradient vector for every voxel is generated with calculated echo-shifts, and the map presents a 3D form of a positive-contrast images (Dahnke, 2008; Liu, 2008). Hyperintensities of SGM were seen in areas surrounding the 1×106 ferumoxides/protamine sulfate complex labeled flank C6 glioma cells of experimental rat model. The sensitivity of the method was compared to white-marker and IRON positive-contrast methods for visualizing the proliferation of tumor cells for labeled tumors that were approximately 5mm (small), 10 mm (medium) and 20 mm (large) in diameter along the largest dimension (Liu, 2008). The number of positive voxels detected around small and medium tumors was significantly greater with the SGM technique than those with the other two techniques, while similar as the "white-marker" technique for large tumors that could not be visualized with the IRON technique. The SGM is a post-processing technique and its positive-contrast images can be derived directly from the T2\*-weighted images without requiring dedicated positive-contrast pulse sequences, thereby it can provide the flexibility to display susceptibility gradients or suppress susceptibility artifacts in specific directions; not like the "white marker" or IRON techniques that require specialized pulse sequence designs and extra scans in addition to those obtained for conventional anatomic imaging. With SGM the hyperintense regions on positive-contrast images originating from SPIO labeled cells can be easily differentiated from other signal

The phase gradient mapping (PGM) techniques have recently developed independently by two groups, one related derived phase gradient maps from standard phase images also including a phase unwrapping procedure to assist the analysis and characterization of object-induced macroscopic phase perturbations (Bakker, 2008); another one utilized fast Fourier transform (FFT) to form phase gradients and develop positive contrast maps by the use of PGM but without need of phase unwrapping, so as to be appropriate technique for the visualization of magnetic nanoparticulate system (Langley, 2011; Zhao, 2011). By the method introduced recently of dual contrast with therapeutic iron nanoparticles at 4.7 T scanner (Sigovan, 2011), or postprocessing methods, with the measure of the T2\*, an efficient estimation of nanoparticle concentration can be made (Langley, 2011). Applications of two kind of approaches, the traditional relaxometry method and model-based method, have demonstrated that, besides the detection of SPIO nanoparticles by positive contrast methods, quantification of the SPIO concentration also play important role in clinical evaluation of results from different treatments with monitoring cellular therapies, and the

week period (Kim, 2009).

voids in T2 or T2\*-weighted images.

former derives from the signal decay associated with areas containing contrast SPIO particles (Kuhlpeter, 2007; Rad, 2007; Liu, 2009), assuming that the rate varies linearly with contrast agent concentration; the later derives from the formation of magnetic field by SPIOcontaining region (Dixon, 2009).

## **3.6 T1 & T2 (T2\*) multi-contrast for cell tracking**

As introduced in as earlier as 1990s, it is possible to achieve positive contrast and dual contrast with superparamagnetic particles by employing T1- and/or T2-weighted sequences (Canet, 1993; Chambon, 1993; Small, 1993). Although most earlier clinical trials with magnetic nanoparticles as contrast agents were evaluated almost exclusively on T2-w fast spin echo (FSE) and T2\*-w gradient echo (GRE) sequences, and the strong T1 contrast enhancement effect of magnetic nanoparticles has rarely been used in clinical and molecular imaging (Reimer, 1995; Yamamoto, 1995; Tang, 1999), the effect of SPIO or USPIO on proton relaxation is not confined to T2 and T2\* effect. They should be considered to influence T1 relaxivity with increased SI on T1-w GRE sequences at low concentrations. For in vivo imaging application of MNPs, optimal combination of negative and positive contrast methods is still under evaluation.

Superparamagnetic iron oxide particles (SPIO) were used shortly after gadolinium-chelate magnetic resonance (MR) contrast agent as well known, while USPIO being the strong T2 relaxivity that produces negative contrast also a high T1 relaxivity with an increase in SI on T1-weighted images (Small, 1993), so that a biphasic imaging sequence protocol (only immediate postadministration and 20-24 hr delayed images) in the *in vivo* study allowed visualization of the dynamic enhancement patterns of both normal tissue and potentially tumor based on early T1-shortening effects produced by intravascular USPIO particulate agent (BMS 180549, previously AMI-227) and marked T1-shortening produced following agent uptake by liver and spleen, as well as showed markedly less T2-shortening at 20-24 hr within both liver and spleen.

The more recent investigation (Zhang, 2011) demonstrated that an appropriate SPIO core size and concentration range was paid much attention to obtain positive contrast with UTE imaging, and this technique could be used with the receptor targeted SPIO molecular imaging probe so as to provide an opportunity for monitoring cancer cells with overexpression of integrin αvβ3 in addition to negative contrast by the approach of T2 relaxometry mapping.

Investigators recently synthesized a biocompatible water-dispersible Fe3O4–SiO2–Gd– DTPA–RGD nanoparticle with r1 relaxivity of 4.2 mm−1s−1 and r2 relaxivity of 17.4 mm−1s−<sup>1</sup> at the Gd/Fe molar ratio of 0.3:1, indicating the potential to use this multifunctional agent for dual-contrast MR imaging of tumor cells over-expressing high-affinity αvβ3 integrin *in vitro* and *in vivo* (Yang, 2011).

## **4. Imaging contrast of IRON-labeled cell on multimodular platform**

MRI can be commonly used to set up a kind of nanomedicine platform for applications of multimodality probe to obtain information about concomitant anatomic, chemical, and physiological features of body. This kind of approach has been found under the

Iron Oxide Nanoparticles Imaging Tracking by

**5. Perspectives** 

undoubtedly be essential.

425, ISSN 0006-4971

1172-1183, ISSN 1053-1807

**6. References** 

MR Advanced Techniques: Dual-Contrast Approaches 163

There is an increasing interest in using cellular MRI to monitor behavior and physiologic functions of iron-labeled cells in vivo. Iron particles provide good MR probing capabilities and some of these agents are currently available for clinical applications. Based on the fact that iron particles exhibit unique nanoscale properties of super-paramagnetism and have the potential to be utilized as excellent probes for cellular imaging and molecular imaging, several MR techniques have recently been proposed to increase the detection sensitivity for image contrast generated with iron-labeled cells, including negative-, positive- and dual-

The hyperintense regions on positive-contrast images originating from iron-labeled cells can be easily differentiated from other signal voids on T2 or T2\*-weighted images, therefore providing a greater degree of certainty in the determination of labeled cells. Moreover, the hyperintensities appeared to illustrate a greater sensitivity than the dark spots on regular MR images. Because positive-contrast imaging approaches do not provide sufficient anatomical information, it is necessary to combine positive-contrast techniques with conventional gradient echo or spin echo imaging, to achieve dual-contrast. Also, the combinined gadolinium and SPIO-enhanced imaging in a 'dual contrast' MRI could be the more accurate technique for the detection of rntities, especially of tumors. Additionally, some new applications of agents for MR imaging have been tested so as to obtain dualcontrast agents for noninvasive imaging studies. Dual-contrast MRI techniques for in vivo cell tracking will add to the growing armamentarium for preclinical cellular MR imaging and further demonstrate the value and diagnostic power of molecular MR imaging, and multifunctional iron oxide nanoparticles together with MRI will have unique advantages with diagnostic and therapeutic capabilities. Simutaneously, the "concept" of dual-contrast imaging can be expaned into imaging evaluation on the platform of dual-modality (or even multimodal approach) including the simultaneous MRI-PET of new method for functional

contrast methods for visualization of iron-labeled cells in vitro and in vivo.

and morphological imaging with blooming perspectives for further development.

While much progress has been made to date, many challenges still face cellular MRI approaches aimed at assessing the migration, homing and function of transplanted therapeutic iron-labeled cells in vivo. For cellular MRI techniques to be successful, the combined expertise of basic scientists, clinicians and representatives from industry will

Anderson, S., Glod, J., Arbab, A., Noel, M., Ashari, P., Fine, H., & Frank, J. (2005).

Andronesi, O., Mintzopoulos, D., Righi, V., Psychogios, N., Kesarwani, M., He, J., Yasuhara,

Noninvasive MR Imaging of Magnetically Labeled Stem Cells to Directly Identify Neovasculature in a Glioma Model. *Blood*, Vol. 105, No. 1, (August 2004), pp. 420-

S., Dai, G., Rahme, L., & Tzika, A. (2010). Combined Off-resonance Imaging and T2 Relaxation in The Rotating Frame for Positive Contrast MR Imaging of Infection in a Murine Burn Model. *J Magn Reson Imaging*, Vol. 32, No. 5, (November 2010), pp.

background that, the nanomedicine platform could capitalize on the availability of specific probes, while achieving an theranostic (integrated diagnostic and therapeutic) design to allow for the visualization of therapeutic efficacy by noninvasive imaging methods such as MRI (Guthi, 2010), for example, in the field of tumor imaging researches, the combination of diagnostic capability with therapeutic intervention is critical to address the challenges of cancer heterogeneity and adaptive resistance, also molecular diagnosis by imaging is important to verify the cancer biomarkers in the tumor tissue and to guide target-specific therapy. It has been thought that ideal multimodality imaging probes enhance capabilities from complementary imaging modalities to enable both noninvasive and invasive molecular imaging (e.g, via probes with MRI and NIR fluorescence reporter capabilities) and to facilitate verification of disease detection and deliver additional evidences for the pathology (eg, probes with reporter capabilities for both positron emission tomography and MRI) (Kircher, 2003b; Lee, 2008). As for the establishment and utilizations of multimodular platform, such as optical and multimodality molecular imaging; multifunctional PET/MRI contrast agent; focused ultrasound/magnetic nanoparticle targeting delivery; design magnetic nanoparticles, etc, some topics are beyond of the scope of this chapter, and some good review papers have already published, so readers are recommended to check them (Jaffer, 2009; Chomoucka, 2010; Liu, 2010; Veiseh, 2010).

Guthi et al. recently introduced a multifunctional methoxy-terminated PEG-b-PDLLA micelle system that was encoded with a lung cancer-targeting peptide (LCP) and loaded with SPIO together with doxorubicin for MR imaging and therapeutic delivery in their *in vitro* study of a lung cancer (Guthi, 2010), they presented a significantly increased cell targeting, micelle uptake, superb T2 relaxivity for ultrasensitive MR detection and cell cytotoxicity in αvβ6-expressing lung cancer cells, with confocal laser scanning microscopy of Doxo fluorescence also used to study the targeting specificity of LCP-encoded micelles to αvβ6-expressing H2009 over the αvβ6-negative H460 cells. The same micelles were previously conjugated with a cRGD ligand that can target αvβ3 integrins on tumor endothelial (SLK) cells (Nasongkla, 2006), illustrating growth inhibition of tumor SLK cells with ultrasensitive detection by MRI. The same lab in University of Texas Southwestern Medical Center at Dallas has previously demonstrated a multi-functional micelle design that allows for the vascular targeting of tumor endothelial cells, MRI ultrasensitivity, and controlled release of doxorubicin (Doxo) for therapeutic drug delivery (Nasongkla, 2006; Khemtong, 2009). Investigators (Guthi, 2010) found that SPIO-clustered polymeric micelle design has considerably decreased the MR detection limit to subnanomolar concentrations (< nM) of micelles through the increased T2 relaxivity and high loading of SPIO per micelle particle; suggested that, on that multifunctional platform, the application of positive contrast imaging, such as ORS, could further enhance the contrast sensitivity and allow for the *in vivo* imaging of tumor-specific markers.

The proposed approaches of dual imaging (e.g. with CLIO modified with a NIR fluorophore, therapeutic siRNA sequences, and a cell penetrating peptide for cancer) Medarova, 2007), even multi-modular imaging (e.g. with triple functional iron oxide nanoparticles) (Xie, 2010) demonstrate potential for the creation of targeted multifunctional nanomedicine platforms.

## **5. Perspectives**

162 Smart Nanoparticles Technology

background that, the nanomedicine platform could capitalize on the availability of specific probes, while achieving an theranostic (integrated diagnostic and therapeutic) design to allow for the visualization of therapeutic efficacy by noninvasive imaging methods such as MRI (Guthi, 2010), for example, in the field of tumor imaging researches, the combination of diagnostic capability with therapeutic intervention is critical to address the challenges of cancer heterogeneity and adaptive resistance, also molecular diagnosis by imaging is important to verify the cancer biomarkers in the tumor tissue and to guide target-specific therapy. It has been thought that ideal multimodality imaging probes enhance capabilities from complementary imaging modalities to enable both noninvasive and invasive molecular imaging (e.g, via probes with MRI and NIR fluorescence reporter capabilities) and to facilitate verification of disease detection and deliver additional evidences for the pathology (eg, probes with reporter capabilities for both positron emission tomography and MRI) (Kircher, 2003b; Lee, 2008). As for the establishment and utilizations of multimodular platform, such as optical and multimodality molecular imaging; multifunctional PET/MRI contrast agent; focused ultrasound/magnetic nanoparticle targeting delivery; design magnetic nanoparticles, etc, some topics are beyond of the scope of this chapter, and some good review papers have already published, so readers are recommended to check them (Jaffer, 2009; Chomoucka,

Guthi et al. recently introduced a multifunctional methoxy-terminated PEG-b-PDLLA micelle system that was encoded with a lung cancer-targeting peptide (LCP) and loaded with SPIO together with doxorubicin for MR imaging and therapeutic delivery in their *in vitro* study of a lung cancer (Guthi, 2010), they presented a significantly increased cell targeting, micelle uptake, superb T2 relaxivity for ultrasensitive MR detection and cell cytotoxicity in αvβ6-expressing lung cancer cells, with confocal laser scanning microscopy of Doxo fluorescence also used to study the targeting specificity of LCP-encoded micelles to αvβ6-expressing H2009 over the αvβ6-negative H460 cells. The same micelles were previously conjugated with a cRGD ligand that can target αvβ3 integrins on tumor endothelial (SLK) cells (Nasongkla, 2006), illustrating growth inhibition of tumor SLK cells with ultrasensitive detection by MRI. The same lab in University of Texas Southwestern Medical Center at Dallas has previously demonstrated a multi-functional micelle design that allows for the vascular targeting of tumor endothelial cells, MRI ultrasensitivity, and controlled release of doxorubicin (Doxo) for therapeutic drug delivery (Nasongkla, 2006; Khemtong, 2009). Investigators (Guthi, 2010) found that SPIO-clustered polymeric micelle design has considerably decreased the MR detection limit to subnanomolar concentrations (< nM) of micelles through the increased T2 relaxivity and high loading of SPIO per micelle particle; suggested that, on that multifunctional platform, the application of positive contrast imaging, such as ORS, could further enhance the contrast sensitivity and allow for the *in* 

The proposed approaches of dual imaging (e.g. with CLIO modified with a NIR fluorophore, therapeutic siRNA sequences, and a cell penetrating peptide for cancer) Medarova, 2007), even multi-modular imaging (e.g. with triple functional iron oxide nanoparticles) (Xie, 2010) demonstrate potential for the creation of targeted multifunctional

2010; Liu, 2010; Veiseh, 2010).

*vivo* imaging of tumor-specific markers.

nanomedicine platforms.

There is an increasing interest in using cellular MRI to monitor behavior and physiologic functions of iron-labeled cells in vivo. Iron particles provide good MR probing capabilities and some of these agents are currently available for clinical applications. Based on the fact that iron particles exhibit unique nanoscale properties of super-paramagnetism and have the potential to be utilized as excellent probes for cellular imaging and molecular imaging, several MR techniques have recently been proposed to increase the detection sensitivity for image contrast generated with iron-labeled cells, including negative-, positive- and dualcontrast methods for visualization of iron-labeled cells in vitro and in vivo.

The hyperintense regions on positive-contrast images originating from iron-labeled cells can be easily differentiated from other signal voids on T2 or T2\*-weighted images, therefore providing a greater degree of certainty in the determination of labeled cells. Moreover, the hyperintensities appeared to illustrate a greater sensitivity than the dark spots on regular MR images. Because positive-contrast imaging approaches do not provide sufficient anatomical information, it is necessary to combine positive-contrast techniques with conventional gradient echo or spin echo imaging, to achieve dual-contrast. Also, the combinined gadolinium and SPIO-enhanced imaging in a 'dual contrast' MRI could be the more accurate technique for the detection of rntities, especially of tumors. Additionally, some new applications of agents for MR imaging have been tested so as to obtain dualcontrast agents for noninvasive imaging studies. Dual-contrast MRI techniques for in vivo cell tracking will add to the growing armamentarium for preclinical cellular MR imaging and further demonstrate the value and diagnostic power of molecular MR imaging, and multifunctional iron oxide nanoparticles together with MRI will have unique advantages with diagnostic and therapeutic capabilities. Simutaneously, the "concept" of dual-contrast imaging can be expaned into imaging evaluation on the platform of dual-modality (or even multimodal approach) including the simultaneous MRI-PET of new method for functional and morphological imaging with blooming perspectives for further development.

While much progress has been made to date, many challenges still face cellular MRI approaches aimed at assessing the migration, homing and function of transplanted therapeutic iron-labeled cells in vivo. For cellular MRI techniques to be successful, the combined expertise of basic scientists, clinicians and representatives from industry will undoubtedly be essential.

## **6. References**


Iron Oxide Nanoparticles Imaging Tracking by

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**Nanoscale Electrodeposition of Copper on an AFM Tip and Its Morphological Investigations** 

Electrocrystallisation processes occurring at electrochemical solid/liquid interfaces have attracted the interest of many researchers from both fundamental and applied viewpoints. After the pioneering works of Max Volmer at the beginning of the last century (Volmer, 1934a, 1939b), the processes of electrocrystallisation have been the subject of numerous intensive studies, the results of which have been documented in several books (Bockris & Razumney 1967; Budevski, et al., 1996; Fischer,1954). The electrochemical method offers several advantages over vapour deposition techniques such as molecular beam epitaxy for depositing nanoscale superlattices. Additional technological advantages over the vapour deposition techniques consist in the relatively low processing temperature and the high selectivity. The low processing temperatures minimizes interdiffusion whereas the high selectivity of electrocrystallisation process allows uniform modification of surfaces and structures with complicated profiles. Phase formation and crystal growth phenomena are the most common morphological parameters observed in many technological important cathodic and anodic electrochemical reactions. One of the most frequently studied electrocrystallisation process is the cathodic metal deposition on foreign and native substrates from electrolytes containing complex metal ions (Fleischmann & Thirsk, 1963; Milchev, 2002; Paunovic & Schlesinger, 2006). Some of the typical cited examples are electrocrystallisation of Ag from Ag+ containing electrolytes (Budevski et al., 1980; Fischer, 1969) and the electrodeposition of Cu (Budevski, 1983; Danilov et al., 1994; Hozzle et al., 1995; Michhailova et al. , 1993) which has recently become technologically important for the fabrication of Cu interconnects on integrated circuit chips (Andricacos et al., 1998; Oskam et al. 1998). Since the electrodeposition of metals is a process of great technological importance, a large number of studies have been carried out to understand the mechanism of electrodeposition of metals on conducting surfaces by employing a variety of electrochemical and spectroscopic techniques (Andricacos, 1999; Markovic & Ross, 1993). The conventional electrochemical methods such as cyclic voltammetry, impedance spectroscopy have been used to assess the mechanism and kinetics of metal electrocrystallisation. These techniques however provide information on the whole

**1. Introduction** 

surface.

Udit Surya Mohanty, S. Y. Chen and Kwang-Lung Lin

*Department of Materials Science and Engineering,* 

 *National Cheng Kung University* 


## **Nanoscale Electrodeposition of Copper on an AFM Tip and Its Morphological Investigations**

Udit Surya Mohanty, S. Y. Chen and Kwang-Lung Lin

*Department of Materials Science and Engineering, National Cheng Kung University Tainan, R.O.C, Taiwan* 

## **1. Introduction**

172 Smart Nanoparticles Technology

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Electrocrystallisation processes occurring at electrochemical solid/liquid interfaces have attracted the interest of many researchers from both fundamental and applied viewpoints. After the pioneering works of Max Volmer at the beginning of the last century (Volmer, 1934a, 1939b), the processes of electrocrystallisation have been the subject of numerous intensive studies, the results of which have been documented in several books (Bockris & Razumney 1967; Budevski, et al., 1996; Fischer,1954). The electrochemical method offers several advantages over vapour deposition techniques such as molecular beam epitaxy for depositing nanoscale superlattices. Additional technological advantages over the vapour deposition techniques consist in the relatively low processing temperature and the high selectivity. The low processing temperatures minimizes interdiffusion whereas the high selectivity of electrocrystallisation process allows uniform modification of surfaces and structures with complicated profiles. Phase formation and crystal growth phenomena are the most common morphological parameters observed in many technological important cathodic and anodic electrochemical reactions. One of the most frequently studied electrocrystallisation process is the cathodic metal deposition on foreign and native substrates from electrolytes containing complex metal ions (Fleischmann & Thirsk, 1963; Milchev, 2002; Paunovic & Schlesinger, 2006). Some of the typical cited examples are electrocrystallisation of Ag from Ag+ containing electrolytes (Budevski et al., 1980; Fischer, 1969) and the electrodeposition of Cu (Budevski, 1983; Danilov et al., 1994; Hozzle et al., 1995; Michhailova et al. , 1993) which has recently become technologically important for the fabrication of Cu interconnects on integrated circuit chips (Andricacos et al., 1998; Oskam et al. 1998). Since the electrodeposition of metals is a process of great technological importance, a large number of studies have been carried out to understand the mechanism of electrodeposition of metals on conducting surfaces by employing a variety of electrochemical and spectroscopic techniques (Andricacos, 1999; Markovic & Ross, 1993). The conventional electrochemical methods such as cyclic voltammetry, impedance spectroscopy have been used to assess the mechanism and kinetics of metal electrocrystallisation. These techniques however provide information on the whole surface.

Nanoscale Electrodeposition of Copper on an AFM Tip and Its Morphological Investigations 175

(self assembled monolayer) (SAM). The authors suggested the physical and electrostatic inhibition by the tip, or the diffusion of Cu ions to the area under the tip, even on a bare Au (III) surface. In one of the studies (Koinuma & Uosaki, 1994), AFM and scaling analysis have been employed to investigate the effect of current density, temperature and levelling agent on the morphology of electrolytically produced copper. However, very little information is available on the morphology of the nanostructures formed by the electrodeposition of copper on the AFM tip. In the present chapter an effort has been made to examine various modes of nucleation and growth of copper deposits on an AFM tip of 80 nm. Electron beam lithography techniques have been employed to facilitate selective electrodeposition of copper on the nanosize AFM tip in the presence of photoresist named poly (methyl

In the present study an AFM probe NSC/18 /Ti-Pt was used which consisted of a Si base coated by films of Ti (1st layer, 15 nm) and Pt (2ndlayer) 10 nm. The Cantilevers employed were 230 µm in length and 40 µm in diameter with pyramidal tips of diameter 80 nm. The tip height ranged from 20 to 25 µm. The schematic diagram of the uncoated AFM probe containing the tip is shown in Fig. 1a. The magnified image is demonstrated in Fig. 1b. One of the AFM probes acted as a anode and the other as a cathode. Two connectors were placed on the AFM probes to facilitate the current flow between the power supplier and the probes, as illustrated in Fig. 2. For the preparation of a connector, a Cu plate of diameter 3.4 mm and length 12 mm was first plated with electroless Au (Fig. 3). The following electrolytic composition was used in the study: 2g/L KAu (CN)2; 75 g/L NH4Cl; 50 g/L 6H5Na3O7·2H2O; and 10 g/L NaH2PO2.H2O. The pH of the electrolytic solution was

Fig. 1. SEM micrograph illustrating the schematic representation of an uncoated AFM probe

**a b** 

The layer of electroless gold on the copper plate provided good adhesion. The Cu plate coated with electroless Au was connected to a Cu wire as shown in Fig 3 by means of mechanical force. Finally, the connector was electroplated with a hard layer of Au to provide good abrasion ability and a thicker Au layer. The Si holders and the connectors were fixed together and were placed in a rectangular cell containing the electrolyte of composition 80 g/L CuSO4 and 200 mL/L H2SO4. The power supply was fixed in the range

maintained at 7 and the solution temperature was kept at 92°C.

methacyrlate).

**2. Experimental method** 

To fully understand the process, it is essential to obtain structural information on the substrate and the deposit in pm to the atomic level. Although techniques such as electron microscopy and optical microscopy have been employed to examine the morphology of the substrate and the metal deposit, they can be used only for the ex situ examination. The discovery of scanning tunnelling microscopy and atomic force microscopy (STM and AFM) offered new exciting possibilties for in situ studies of the electrocrystallisation phenomenon down to an atomic level (Binning & Rohrer, 1982; Lustenberger et al., 1988, Sonnenfeld & Hannsma, 1986). The application of these techniques in the last two decades has revolutionized the experimental work in this field and led to significant progress in the understanding of the atomistic aspects of the electrocrystallisation process (Gewirth & Siegenthaler, 1995; Staikov et al., 1994; Stegenthaler, 1992). These processes range from measuring the lateral force using a cantilever tip, measuring magnetic force, electrostatic force, Kelvin potential to the determination of surface conductivity. The invention of scanning probe microscopy (SPM) also provoked a rapid development of the modern nanoscience and nanotechnology dealing with nanoscale structures and objects including single atoms and molecules. Over, the years, many other types of scanning probe microscopic techniques have evolved from the base concept of AFM. Electrochemical fabrication of metal nanostructures has been reported using SPM-based lithography, typically by tip induced electrochemical deposition of metal ions transferred by the STM or AFM tip to the surface. (Allongne, 1995; Benenz et al., 2002). Many studies have been directed towards in situ STM and AFM imaging of metal underdeposition (Hachiya et al.1991; Li et al., 2001) and bulk deposition (Nichols et al., 1992; Yau et al. 1991). Since STM can only be applied to observe conductive surfaces, the existence of anodic oxide films as well as the space charge layer in the depletion condition makes the STM measurements of semiconductor electrodes relatively difficult (Batina & Nichols, 1992). On the other hand, AFM can image even non-conducting surfaces and electrochemical processes on the tip, which causes serious problems in the STM measurements in electrolyte solutions. AFM is also found to be more useful than STM in studying the electrode surface in situ. AFM works the same way as STM and can transfer materials from tip to substrate at a biased voltage. As AFM tips are normally made of silicon or silicon nitide, metallic materials have to be coated onto the AFM tip in order to make the deposition happen. Once it is coated with metals, it becomes no different from a STM tip, and deposition takes place under high electric field. AS AFM can work in liquid solution, it is possible to initiate electrochemical deposition using an AFM tip. Also the electrochemical reaction rate at the interface can be controlled by application of an external potential to the substrate. In particular, the amount of deposit and the kinetics of the metal deposition onto the surface can be controlled. Its because the electrochemical process is sensitive to the surface properties, in situ local deposition of metal can be made selective by tuning the surface characteristics. Copper has been electrochemically deposited onto GaAs surface by immersing the AFM tip into a mixture of CuSO4 + H2SO4 solution (Carlsson et al., 1990).

The electrodeposition of Cu is strongly dependent on the structure of the substrate, applied potential and concentration of Cu2+ ions in the precursor solution. The nanoscale electrodeposition of metal in nanopatterned alkanethiol-modified Au (III) has been reported (Gewirth & Sigenthaler, 1995). Although the interaction between the tip and the sample enhanced Cu deposition on the surface, Cu did not deposit uniformly in the area scanned, but only deposited at the edges of the scanned area as well as in defects in the alkanethiol (self assembled monolayer) (SAM). The authors suggested the physical and electrostatic inhibition by the tip, or the diffusion of Cu ions to the area under the tip, even on a bare Au (III) surface. In one of the studies (Koinuma & Uosaki, 1994), AFM and scaling analysis have been employed to investigate the effect of current density, temperature and levelling agent on the morphology of electrolytically produced copper. However, very little information is available on the morphology of the nanostructures formed by the electrodeposition of copper on the AFM tip. In the present chapter an effort has been made to examine various modes of nucleation and growth of copper deposits on an AFM tip of 80 nm. Electron beam lithography techniques have been employed to facilitate selective electrodeposition of copper on the nanosize AFM tip in the presence of photoresist named poly (methyl methacyrlate).

## **2. Experimental method**

174 Smart Nanoparticles Technology

To fully understand the process, it is essential to obtain structural information on the substrate and the deposit in pm to the atomic level. Although techniques such as electron microscopy and optical microscopy have been employed to examine the morphology of the substrate and the metal deposit, they can be used only for the ex situ examination. The discovery of scanning tunnelling microscopy and atomic force microscopy (STM and AFM) offered new exciting possibilties for in situ studies of the electrocrystallisation phenomenon down to an atomic level (Binning & Rohrer, 1982; Lustenberger et al., 1988, Sonnenfeld & Hannsma, 1986). The application of these techniques in the last two decades has revolutionized the experimental work in this field and led to significant progress in the understanding of the atomistic aspects of the electrocrystallisation process (Gewirth & Siegenthaler, 1995; Staikov et al., 1994; Stegenthaler, 1992). These processes range from measuring the lateral force using a cantilever tip, measuring magnetic force, electrostatic force, Kelvin potential to the determination of surface conductivity. The invention of scanning probe microscopy (SPM) also provoked a rapid development of the modern nanoscience and nanotechnology dealing with nanoscale structures and objects including single atoms and molecules. Over, the years, many other types of scanning probe microscopic techniques have evolved from the base concept of AFM. Electrochemical fabrication of metal nanostructures has been reported using SPM-based lithography, typically by tip induced electrochemical deposition of metal ions transferred by the STM or AFM tip to the surface. (Allongne, 1995; Benenz et al., 2002). Many studies have been directed towards in situ STM and AFM imaging of metal underdeposition (Hachiya et al.1991; Li et al., 2001) and bulk deposition (Nichols et al., 1992; Yau et al. 1991). Since STM can only be applied to observe conductive surfaces, the existence of anodic oxide films as well as the space charge layer in the depletion condition makes the STM measurements of semiconductor electrodes relatively difficult (Batina & Nichols, 1992). On the other hand, AFM can image even non-conducting surfaces and electrochemical processes on the tip, which causes serious problems in the STM measurements in electrolyte solutions. AFM is also found to be more useful than STM in studying the electrode surface in situ. AFM works the same way as STM and can transfer materials from tip to substrate at a biased voltage. As AFM tips are normally made of silicon or silicon nitide, metallic materials have to be coated onto the AFM tip in order to make the deposition happen. Once it is coated with metals, it becomes no different from a STM tip, and deposition takes place under high electric field. AS AFM can work in liquid solution, it is possible to initiate electrochemical deposition using an AFM tip. Also the electrochemical reaction rate at the interface can be controlled by application of an external potential to the substrate. In particular, the amount of deposit and the kinetics of the metal deposition onto the surface can be controlled. Its because the electrochemical process is sensitive to the surface properties, in situ local deposition of metal can be made selective by tuning the surface characteristics. Copper has been electrochemically deposited onto GaAs surface by immersing the AFM tip into a mixture of

The electrodeposition of Cu is strongly dependent on the structure of the substrate, applied potential and concentration of Cu2+ ions in the precursor solution. The nanoscale electrodeposition of metal in nanopatterned alkanethiol-modified Au (III) has been reported (Gewirth & Sigenthaler, 1995). Although the interaction between the tip and the sample enhanced Cu deposition on the surface, Cu did not deposit uniformly in the area scanned, but only deposited at the edges of the scanned area as well as in defects in the alkanethiol

CuSO4 + H2SO4 solution (Carlsson et al., 1990).

In the present study an AFM probe NSC/18 /Ti-Pt was used which consisted of a Si base coated by films of Ti (1st layer, 15 nm) and Pt (2ndlayer) 10 nm. The Cantilevers employed were 230 µm in length and 40 µm in diameter with pyramidal tips of diameter 80 nm. The tip height ranged from 20 to 25 µm. The schematic diagram of the uncoated AFM probe containing the tip is shown in Fig. 1a. The magnified image is demonstrated in Fig. 1b. One of the AFM probes acted as a anode and the other as a cathode. Two connectors were placed on the AFM probes to facilitate the current flow between the power supplier and the probes, as illustrated in Fig. 2. For the preparation of a connector, a Cu plate of diameter 3.4 mm and length 12 mm was first plated with electroless Au (Fig. 3). The following electrolytic composition was used in the study: 2g/L KAu (CN)2; 75 g/L NH4Cl; 50 g/L 6H5Na3O7·2H2O; and 10 g/L NaH2PO2.H2O. The pH of the electrolytic solution was maintained at 7 and the solution temperature was kept at 92°C.

Fig. 1. SEM micrograph illustrating the schematic representation of an uncoated AFM probe

The layer of electroless gold on the copper plate provided good adhesion. The Cu plate coated with electroless Au was connected to a Cu wire as shown in Fig 3 by means of mechanical force. Finally, the connector was electroplated with a hard layer of Au to provide good abrasion ability and a thicker Au layer. The Si holders and the connectors were fixed together and were placed in a rectangular cell containing the electrolyte of composition 80 g/L CuSO4 and 200 mL/L H2SO4. The power supply was fixed in the range

Nanoscale Electrodeposition of Copper on an AFM Tip and Its Morphological Investigations 177

Fig. 4. Schematic diagram demonstrating the electrodeposition of copper on the AFM probe.

**a**

The samples were patterned using a JEOL 6400 thermionic emission scanning electron microscope equipped with the lithography software Elphy Quantum. The polymer used for EBL studies was poly\_methyl methacrylate (PMMA). PMMA was the standard positive ebeam resist dissolved in a casting solvent anisole. The PMMA solution was spin-coated onto the AFM probe at a rotating speed of 1000 rpm for 60 s. Then baking was performed at 220°C to harden the film and to remove the remaining solvent. The EBL system employed a focused electron beam which moved across the sample to selectively expose a pattern in the resist previously designed with the system's built-in computer-assisted design tools.The open area of the AFM tip was selectively exposed to the high energy- beam electrons. The sample was then immersed in the developer solution (3:1 methyl isobutyl ketone: isopropyl alcohol developer) for 30 sec to selectively remove the resist from the exposed areas, whereas the unexposed resist remained insoluble in the developer. The process thus left a patterned resist mask on the sample that could be used for further processing. Finally, Cu

Scanning Electron Microscopy (SEM) was used to investigate the morphologies of the copper deposits nucleated on the AFM probe. SEM micrograph for Cu deposition on the

was electrochemically deposited on the AFM tip.

**2.1 Effect of various electrodeposition parameters** 

**2.1.1 Effect of current density on copper electrodeposition** 

of 10 µA to 100 A. Because the AFM probes cannot bear large amounts of current, a large electrode system consisting of Pt anode and Cu cathode was used as shown in Fig. 4.

Fig. 2. Schematic diagram of the AFM probe placed on the Si holder

The cathode and anode were placed in parallel connection with the AFM probes.The current from the power supply was controlled between 10 and 0.1 A to provide a stable current between the AFM probes. Electron-beam lithography (EBL) techniques were used in our present work.

Fig. 3. Schematic diagram of the connector used in the electrodeposition process

of 10 µA to 100 A. Because the AFM probes cannot bear large amounts of current, a large

AFM probe

The cathode and anode were placed in parallel connection with the AFM probes.The current from the power supply was controlled between 10 and 0.1 A to provide a stable current between the AFM probes. Electron-beam lithography (EBL) techniques were used in our

Fig. 3. Schematic diagram of the connector used in the electrodeposition process

Si holder

Connector

electrode system consisting of Pt anode and Cu cathode was used as shown in Fig. 4.

500μm

**Side view of specimen of specimen**

present work.

Si holder

Fig. 2. Schematic diagram of the AFM probe placed on the Si holder

Fig. 4. Schematic diagram demonstrating the electrodeposition of copper on the AFM probe.

The samples were patterned using a JEOL 6400 thermionic emission scanning electron microscope equipped with the lithography software Elphy Quantum. The polymer used for EBL studies was poly\_methyl methacrylate (PMMA). PMMA was the standard positive ebeam resist dissolved in a casting solvent anisole. The PMMA solution was spin-coated onto the AFM probe at a rotating speed of 1000 rpm for 60 s. Then baking was performed at 220°C to harden the film and to remove the remaining solvent. The EBL system employed a focused electron beam which moved across the sample to selectively expose a pattern in the resist previously designed with the system's built-in computer-assisted design tools.The open area of the AFM tip was selectively exposed to the high energy- beam electrons. The sample was then immersed in the developer solution (3:1 methyl isobutyl ketone: isopropyl alcohol developer) for 30 sec to selectively remove the resist from the exposed areas, whereas the unexposed resist remained insoluble in the developer. The process thus left a patterned resist mask on the sample that could be used for further processing. Finally, Cu was electrochemically deposited on the AFM tip.

#### **2.1 Effect of various electrodeposition parameters**

## **2.1.1 Effect of current density on copper electrodeposition**

Scanning Electron Microscopy (SEM) was used to investigate the morphologies of the copper deposits nucleated on the AFM probe. SEM micrograph for Cu deposition on the

Nanoscale Electrodeposition of Copper on an AFM Tip and Its Morphological Investigations 179

AFM tip might be larger due to the deposition overvoltages. Also, the diameter of the AFM tip, which is around 80 nm, might induce high overpotential for deposition of copper on the

a b

 Fig. 6. SEM image obtained after copper deposition on the AFM tip for plating time of 300 s

The above results can also be explained on the basis of two reaction schemes which govern the Cu electrodeposition process on the AFM probe: one is the electrode surface reaction and the other one is the Cu2+ diffusion from the electrolyte solution to the electrode surface.

Fig. 7. SEM image obtained after copper deposition on the AFM tip for plating time of 300 s

Polarization occurs when the rate of Cu2+ supply from the electrolyte solution is not faster than the rate of reaction at the electrode surface. The film morphology is primarily dependent on the degree of polarization (Seah et al., 1999).Thus higher polarization would make electrodeposition slower resulting in a smoother film. Since the effect of increasing current density is to increase the electrode surface reaction, a faster surface reaction makes Cu2+ undersupplied from the electrolyte solution. Hence, the polarization is higher and smoother film morphology is observed. Nevertheless, when the applied current density is greater than the limiting current density, it is impossible for the electrode to gain any Cu ions from the

electrolyte solution; thereby leading to an increase in the Cu film surface roughness.

and current density of 0.6 A/dm2 (Lin, 2008)

and current density of 0.3 A/dm2 (a) SEI image (b) Magnified image (Lin, 2008)

tip.

investigated AFM probe for a plating time of 300 s and a current density of 0.03 A/dm2 is shown in Fig. 5a . The secondary electron image (SEI) and back scattered electron image (BEI) are displayed in Figs. 5a and 5b. The figure reveals that only a slight amount of copper is electrodeposited on the AFM probe. Further increase in current density to 0.3 A /dm2 enhanced the copper deposition on the AFM probe however the deposits observed are nonuniform and discontinuous (Fig.6a). The magnified image is seen in Fig. 6b. Furthermore, a gradual increase in the current density to 0.6 A /dm2 results in uniform deposition of copper on the probe (Fig.7), nevertheless, no copper deposition is noticed on the AFM tip. Similar observations have been reported (Seah et al., 1998). They visualised this morphology on the basis of the fact that formation of more nucleation sites promoted uniform grain growth. In the present study, the formation of uniform copper deposits on the AFM probe could be attributed to the enhanced mass transfer of copper ions with the increase in current density. Litearture reports (Chang, 2001) describe that increase in plating current density increased the surface roughness and reduced the grain size of copper films due to an increase of plating overpotential. Several other researchers have demonstrated that the polarization overpotential increased with increasing the plating current density leading to high copper nucleation rate (Takahashi & Gross, 1999a, 2000 b; Tean et al., 2003; Teh et al., 2001).

The difficulty in depositing Cu ions on the AFM tip arises due to the local increase of the ion concentration in the electrolyte around the tip, which makes the effective local Nernst potential for deposition at the surface underneath the AFM tip more positive.

Fig. 5. SEM image obtained after copper deposition on the AFM tip for plating time of 300 s and current density of 0.03 A/dm2 (a) SEI image (b) BEI image

Since, the standard electrode potential (ψe) of Cu2+ [ ψe (Cu2+ + /Cu = +0.337 V) is larger than zero (Fu et al., 1990), from the theoretical point of view, the more positive the ψe value, the more easier it is for the reduction of metal ions, and the more negative the ψe value, the more difficult it is to reduce the metal ions. Our results suggest that the copper ions can be reduced to copper atoms more easily on the surface underneath the AFM tip. It might be possible that the effective Nernst potential which is required to initiate nucleation on the

investigated AFM probe for a plating time of 300 s and a current density of 0.03 A/dm2 is shown in Fig. 5a . The secondary electron image (SEI) and back scattered electron image (BEI) are displayed in Figs. 5a and 5b. The figure reveals that only a slight amount of copper is electrodeposited on the AFM probe. Further increase in current density to 0.3 A /dm2 enhanced the copper deposition on the AFM probe however the deposits observed are nonuniform and discontinuous (Fig.6a). The magnified image is seen in Fig. 6b. Furthermore, a gradual increase in the current density to 0.6 A /dm2 results in uniform deposition of copper on the probe (Fig.7), nevertheless, no copper deposition is noticed on the AFM tip. Similar observations have been reported (Seah et al., 1998). They visualised this morphology on the basis of the fact that formation of more nucleation sites promoted uniform grain growth. In the present study, the formation of uniform copper deposits on the AFM probe could be attributed to the enhanced mass transfer of copper ions with the increase in current density. Litearture reports (Chang, 2001) describe that increase in plating current density increased the surface roughness and reduced the grain size of copper films due to an increase of plating overpotential. Several other researchers have demonstrated that the polarization overpotential increased with increasing the plating current density leading to high copper

nucleation rate (Takahashi & Gross, 1999a, 2000 b; Tean et al., 2003; Teh et al., 2001).

potential for deposition at the surface underneath the AFM tip more positive.

The difficulty in depositing Cu ions on the AFM tip arises due to the local increase of the ion concentration in the electrolyte around the tip, which makes the effective local Nernst

**BEI SEI** a b

 Fig. 5. SEM image obtained after copper deposition on the AFM tip for plating time of 300 s

Since, the standard electrode potential (ψe) of Cu2+ [ ψe (Cu2+ + /Cu = +0.337 V) is larger than zero (Fu et al., 1990), from the theoretical point of view, the more positive the ψe value, the more easier it is for the reduction of metal ions, and the more negative the ψe value, the more difficult it is to reduce the metal ions. Our results suggest that the copper ions can be reduced to copper atoms more easily on the surface underneath the AFM tip. It might be possible that the effective Nernst potential which is required to initiate nucleation on the

and current density of 0.03 A/dm2 (a) SEI image (b) BEI image

AFM tip might be larger due to the deposition overvoltages. Also, the diameter of the AFM tip, which is around 80 nm, might induce high overpotential for deposition of copper on the tip.

Fig. 6. SEM image obtained after copper deposition on the AFM tip for plating time of 300 s and current density of 0.3 A/dm2 (a) SEI image (b) Magnified image (Lin, 2008)

The above results can also be explained on the basis of two reaction schemes which govern the Cu electrodeposition process on the AFM probe: one is the electrode surface reaction and the other one is the Cu2+ diffusion from the electrolyte solution to the electrode surface.

Fig. 7. SEM image obtained after copper deposition on the AFM tip for plating time of 300 s and current density of 0.6 A/dm2 (Lin, 2008)

Polarization occurs when the rate of Cu2+ supply from the electrolyte solution is not faster than the rate of reaction at the electrode surface. The film morphology is primarily dependent on the degree of polarization (Seah et al., 1999).Thus higher polarization would make electrodeposition slower resulting in a smoother film. Since the effect of increasing current density is to increase the electrode surface reaction, a faster surface reaction makes Cu2+ undersupplied from the electrolyte solution. Hence, the polarization is higher and smoother film morphology is observed. Nevertheless, when the applied current density is greater than the limiting current density, it is impossible for the electrode to gain any Cu ions from the electrolyte solution; thereby leading to an increase in the Cu film surface roughness.

Nanoscale Electrodeposition of Copper on an AFM Tip and Its Morphological Investigations 181

The mode of instantaneous nucleation is described by the following equation involving the

where N is the number of sites converted into nuclei at time t and A is the nucleation rate constant, N0 is the respective saturation value. Nucleation does not occur simultaneously over the entire cathode surface and a diameter distribution for the crystallites ensues. When A is very high, N ≡ N0, all surface sites are converted immediately into nuclei and the nucleation is said to be instantaneous. The nonhomogeneity and overgrowth of the Cu deposits may be due to the existence of low nucleation overpotential in the area beneath the tip. At low overpotentials, the nucleation is described well by the model of instantaneous

 Fig. 10. SEM micrographs illustrating Cu deposition on the AFM tip at a current density of 0.3 A/dm2 and plating time of 300 s (a) 2000 T (b) 10000 T magnification of the marked area

However, the morphology of copper deposits formed under current density of 0.6 Adm2 and plating time of 900 s were found to be totally different. The copper layer on the AFM probe also shows resemblance to a candle base (Fig. 12), and a thicker layer of copper deposits are grown on the whole of the AFM probe containing the tip. Also, on the basis of instantaneous nucleation model, It has been reported (Thirsk & Harrison, 1972) that under the diffusion controlled three-dimensional growth, the cathodic current density is

The growth of copper layer also takes place slowly and farther away from the tip. Also it can be noticed that the growth rate on the side of tip is faster than on the tip (Fig. 12). From the results it could be established that higher current density and higher plating time increases the mass transfer of Cu2+ ions in the open area beneath the tip, thereby enhancing the rate of Cu deposition between the open area and the tip. The variance of the thickness of copper deposits on the tip and its surrounding area might be attributed to the nanoscale dimension of the AFM tip as compared to the whole of the AFM probe. Literature reports reveal (Seah et al., 1999) that in case of nanocrystalline electrodeposited Cu the pinhole number-density necessary for full coverage on the substrate can be reduced by increasing the current density. However, abnormal crystallite growth-leading to the formation of bimodal grain

N= N0[1 − exp (−At)] (1)

a b

first-order kinetics law (Budevski et al., 1996; Milchev, 1997)

nucleation for reasonably long time scales (Kelber et al., 2006)

in red

proportional to t1/2.

## **2.1.2 Effect of plating time on copper electrodeposition**

The effect of different plating times during copper electrodeposition on the AFM probe is investigated. The plating time was varied from 5 to 900 s for different current densities. The SEM micrograph in Figs. 8-11 illustrates the morphology of copper deposit formed under current density of 0.3 A/dm2 and various plating times namely 5, 60, 300, 540 s respectively. The results reveal a random distribution of copper crystals on the cantilever with no trace of copper deposits on the AFM tip. This morphology clearly suggests the case of instantaneous nucleation.

Fig. 8. SEM micrographs illustrating Cu deposition on the AFM tip at a current density of 0.3 A/dm2 and plating time of 5 s (a) 2000 T (b) 10000 T magnification of the marked area in red

As instantaneous nucleation corresponds to a slow growth of nuclei on a small number of active sites, all activated at the same time. It can be noted from the SEM images displayed in Fig. 8-11, that in most of the samples the nuclei may be nucleated almost simultaneously, as confirmed by their similar size. In other words i.e at high nucleation rates (instantaneous nucleation), all nuclei are formed immediately after imposition of the potential and grow at the same rate. As a result, they are all of the same age and their number remains constant.

Fig. 9. SEM micrographs illustrating Cu deposition on the AFM tip at a current density of 0.3 A/dm2 and plating time of 60 s (a) 2000 T (b) 10000 T magnification of the marked area in red

The effect of different plating times during copper electrodeposition on the AFM probe is investigated. The plating time was varied from 5 to 900 s for different current densities. The SEM micrograph in Figs. 8-11 illustrates the morphology of copper deposit formed under current density of 0.3 A/dm2 and various plating times namely 5, 60, 300, 540 s respectively. The results reveal a random distribution of copper crystals on the cantilever with no trace of copper deposits on the AFM tip. This morphology clearly suggests the case of instantaneous

a b

a b

 Fig. 8. SEM micrographs illustrating Cu deposition on the AFM tip at a current density of 0.3 A/dm2 and plating time of 5 s (a) 2000 T (b) 10000 T magnification of the marked area in red

As instantaneous nucleation corresponds to a slow growth of nuclei on a small number of active sites, all activated at the same time. It can be noted from the SEM images displayed in Fig. 8-11, that in most of the samples the nuclei may be nucleated almost simultaneously, as confirmed by their similar size. In other words i.e at high nucleation rates (instantaneous nucleation), all nuclei are formed immediately after imposition of the potential and grow at the same rate. As a result, they are all of the same age and their number remains constant.

Fig. 9. SEM micrographs illustrating Cu deposition on the AFM tip at a current density of 0.3 A/dm2 and plating time of 60 s (a) 2000 T (b) 10000 T magnification of the marked area in red

**2.1.2 Effect of plating time on copper electrodeposition** 

nucleation.

The mode of instantaneous nucleation is described by the following equation involving the first-order kinetics law (Budevski et al., 1996; Milchev, 1997)

$$\text{N=N\_0[1-\exp\left(-\text{At}\right)]}\tag{1}$$

where N is the number of sites converted into nuclei at time t and A is the nucleation rate constant, N0 is the respective saturation value. Nucleation does not occur simultaneously over the entire cathode surface and a diameter distribution for the crystallites ensues. When A is very high, N ≡ N0, all surface sites are converted immediately into nuclei and the nucleation is said to be instantaneous. The nonhomogeneity and overgrowth of the Cu deposits may be due to the existence of low nucleation overpotential in the area beneath the tip. At low overpotentials, the nucleation is described well by the model of instantaneous nucleation for reasonably long time scales (Kelber et al., 2006)

Fig. 10. SEM micrographs illustrating Cu deposition on the AFM tip at a current density of 0.3 A/dm2 and plating time of 300 s (a) 2000 T (b) 10000 T magnification of the marked area in red

However, the morphology of copper deposits formed under current density of 0.6 Adm2 and plating time of 900 s were found to be totally different. The copper layer on the AFM probe also shows resemblance to a candle base (Fig. 12), and a thicker layer of copper deposits are grown on the whole of the AFM probe containing the tip. Also, on the basis of instantaneous nucleation model, It has been reported (Thirsk & Harrison, 1972) that under the diffusion controlled three-dimensional growth, the cathodic current density is proportional to t1/2.

The growth of copper layer also takes place slowly and farther away from the tip. Also it can be noticed that the growth rate on the side of tip is faster than on the tip (Fig. 12). From the results it could be established that higher current density and higher plating time increases the mass transfer of Cu2+ ions in the open area beneath the tip, thereby enhancing the rate of Cu deposition between the open area and the tip. The variance of the thickness of copper deposits on the tip and its surrounding area might be attributed to the nanoscale dimension of the AFM tip as compared to the whole of the AFM probe. Literature reports reveal (Seah et al., 1999) that in case of nanocrystalline electrodeposited Cu the pinhole number-density necessary for full coverage on the substrate can be reduced by increasing the current density. However, abnormal crystallite growth-leading to the formation of bimodal grain

Nanoscale Electrodeposition of Copper on an AFM Tip and Its Morphological Investigations 183

structures- can be suppressed by increasing the electrodeposition current density. In our present case, the crystal growth variation is seen on the open area below the tip and the tip itself. The morphology observed in Fig. 12 is a case of progressive nucleation followed by

As nucleation progresses, the nuclei begins to overlap. Each nucleus is defined by its own diffusion zone through which copper diffuses, thus representing the mass-supply mechanism for continuation of growth. Progressive nucleation corresponds to fast growth of nuclei on many active sites all activated during the course of electroreduction (Pardave et al., 2000). Fig.12b. shows the SEM micrograph for copper deposition at 0.6 A/dm2 and 900s taken at a tilted angle of 350. Further increase in the plating time to 1200 s for similar current density resulted in an entirely different morphology from the micrograph shown in Fig. 12. The AFM probe containing the nanoscale AFM tip seems to be entirely covered with copper deposits and also a significant increase in growth and thickness of the deposits are observed in Figs. 13a. Fig. 13b represents the SEM image tilted at an angle of 450 for clear depiction of the copper deposition on the AFM probe. The copper deposition process on the AFM probe proceeds through instantaneous and progressive nucleation modes for different values of current density. The mechanisms for instantaneous and progressive nucleation modes are

Once nucleation begins, crystals growth may be determined by the rate of charge- transfer or diffusion process. Simple equations have been described (Harrison & Thirsk, 1971) for two- or three dimensional nucleation and crystal growth processes occurring on a foreign

For two-dimensional (2D) instantaneous nucleation and cylindrical growth, current is

where k2D represents the lateral growth rate constants (mol cm-2 s-1), h is the layer height in cm, N0 represents the total number of active centers (cm-2), A2D the nucleation rate (nuclei cm-2 s-1), M is the atomic weight (g mol-1) and ρ the density ( g cm-3) of the deposit. For these type of mechanisms the current usually increases and then decreases to zero when the surface gets completely covered by two dimensional crystals However, for three dimensional (3D) instantaneous nucleation and growth, the current is depicted by the

Where k and k' signify the lateral and vertical growth rate constants (mol cm2 s-1) and A3D the nucleation rate (nuclei cm2 s-1). Hence nucleation and growth phenomena are affected by

i = 2zFлh N0k2 2D t / ρ exp (-л M2 N0k22D t2 ) / ρ2 (2)

i = z FлhMK2 2DA2Dt2 / ρ exp ( -л M2 k22 DA2D t3 / 3ρ2) (3)

[1 – exp (-л M2k2 N0 t2 /ρ2) ] (4)

[ 1- exp ( -лM2 k2A3D t3 / 3ρ2) ] (5)

growth.

described below.

described by

substrate for charge transfer control reactions.

i = z F K'

i = zFK'

And for 2D progressive nucleation

following equations below.

and for 3D progressive nucleation:

Fig. 11. SEM micrographs illustrating Cu deposition on the AFM tip at a current density of 0.3 A/dm2 and plating time of 540 s (a) 2000 T (b) 10000 T magnification of the marked area in red

Fig. 12. SEM micrographs illustrating Cu deposition on the AFM tip at a current density of 0.6 A/dm2 and plating time of 900 s (a) SEI image (b) image taken at 350 tilt (Lin, 2008)

Fig. 13. SEM micrographs illustrating Cu deposition on the AFM tip at a current density of 0.6 A/dm2 and plating time of 1200 s (a) SEI image (b) image taken at 350 tilt (Lin, 2008)

structures- can be suppressed by increasing the electrodeposition current density. In our present case, the crystal growth variation is seen on the open area below the tip and the tip itself. The morphology observed in Fig. 12 is a case of progressive nucleation followed by growth.

As nucleation progresses, the nuclei begins to overlap. Each nucleus is defined by its own diffusion zone through which copper diffuses, thus representing the mass-supply mechanism for continuation of growth. Progressive nucleation corresponds to fast growth of nuclei on many active sites all activated during the course of electroreduction (Pardave et al., 2000). Fig.12b. shows the SEM micrograph for copper deposition at 0.6 A/dm2 and 900s taken at a tilted angle of 350. Further increase in the plating time to 1200 s for similar current density resulted in an entirely different morphology from the micrograph shown in Fig. 12. The AFM probe containing the nanoscale AFM tip seems to be entirely covered with copper deposits and also a significant increase in growth and thickness of the deposits are observed in Figs. 13a. Fig. 13b represents the SEM image tilted at an angle of 450 for clear depiction of the copper deposition on the AFM probe. The copper deposition process on the AFM probe proceeds through instantaneous and progressive nucleation modes for different values of current density. The mechanisms for instantaneous and progressive nucleation modes are described below.

Once nucleation begins, crystals growth may be determined by the rate of charge- transfer or diffusion process. Simple equations have been described (Harrison & Thirsk, 1971) for two- or three dimensional nucleation and crystal growth processes occurring on a foreign substrate for charge transfer control reactions.

For two-dimensional (2D) instantaneous nucleation and cylindrical growth, current is described by

$$\mathbf{i} = 2\mathbf{z} \mathbf{F} \mathbf{n} \mathbf{h} \, \mathrm{N}\_0 \mathbf{k}^2 \, \_{\mathrm{ZD}}\, ^\dagger \not\propto \exp\left(-\mathbf{n} \, \mathrm{M}^2 \, \mathrm{N}\_0 \mathrm{k}\_2 \mathbf{\hat{r}} \, \mathrm{t}^2\right) / \, \mathrm{p}^2 \tag{2}$$

And for 2D progressive nucleation

182 Smart Nanoparticles Technology

a b

a b

a b

Fig. 11. SEM micrographs illustrating Cu deposition on the AFM tip at a current density of 0.3 A/dm2 and plating time of 540 s (a) 2000 T (b) 10000 T magnification of the marked area

Fig. 12. SEM micrographs illustrating Cu deposition on the AFM tip at a current density of 0.6 A/dm2 and plating time of 900 s (a) SEI image (b) image taken at 350 tilt (Lin, 2008)

Fig. 13. SEM micrographs illustrating Cu deposition on the AFM tip at a current density of 0.6 A/dm2 and plating time of 1200 s (a) SEI image (b) image taken at 350 tilt (Lin, 2008)

in red

$$\mathbf{i} \equiv \mathbf{z} \operatorname{FrhMK}\_2 \mathbf{2}\_\mathrm{D} \mathbf{A}\_2 \mathrm{D} \mathbf{f}^2 \Big/ \mathbf{p} \exp\left( - \mathbf{n} \, \mathbf{M}^2 \, \mathbf{k}\_2 \mathbf{2}\_\mathrm{D} \mathbf{A}\_{\mathrm{2D}} \mathbf{i}^3 / \, \mathbf{3} \mathbf{p}^2 \right) \tag{3}$$

where k2D represents the lateral growth rate constants (mol cm-2 s-1), h is the layer height in cm, N0 represents the total number of active centers (cm-2), A2D the nucleation rate (nuclei cm-2 s-1), M is the atomic weight (g mol-1) and ρ the density ( g cm-3) of the deposit. For these type of mechanisms the current usually increases and then decreases to zero when the surface gets completely covered by two dimensional crystals However, for three dimensional (3D) instantaneous nucleation and growth, the current is depicted by the following equations below.

i = z F K' [1 – exp (-л M2k2 N0 t2 /ρ2) ] (4)

and for 3D progressive nucleation:

$$\mathbf{i} \equiv \mathbf{z} \mathbf{F} \mathbf{K}' \left[ \mathbf{1} \text{-} \exp \left( \text{-} \mathfrak{m} \mathbf{M}^2 \, \mathbf{k}^2 \mathbf{A}\_{\mathfrak{M}} \, \mathbf{t}^3 \, \Big/ \mathfrak{J} \mathbf{p}^2 \right) \right] \tag{5}$$

Where k and k' signify the lateral and vertical growth rate constants (mol cm2 s-1) and A3D the nucleation rate (nuclei cm2 s-1). Hence nucleation and growth phenomena are affected by

Nanoscale Electrodeposition of Copper on an AFM Tip and Its Morphological Investigations 185

convenient alternative to solve this problem (Simon et al., 1997). The fabrication of dense ultra-small magnetic arrays by filling nanoholes with electrodeposited Ni has been

Electron beam lithography technique is used in the present study to enable selective electrodeposition of Cu on the AFM tip and the open area beneath it. The selective electrodeposition of Cu on n-type Si (111) surfaces covered with organic monolayers by using e-beam lithographic techniques has been reported (Balaur et al., 2004). Selective copper deposition on e-beam patterned alkane and biphenylthiols has been reported (Kalten Poth et al., 2002) at suitable deposition potentials. 1-octadecanethiol (ODT) was used as a

4-thiol ( BPT) on the other hand acted as a ''negative template,'' where the irradiated and cross-linked biphenyl layer exhibited a blocking behavior, allowing copper deposition on the non-irradiated parts. In the present study, the open area of the nanosize AFM tip was selectively exposed to the e-beam. It is noticed that copper electrodeposition occurs on the exposed area of the AFM tip. For the copper electrodeposition process, the current density


''positive template'' leading to copper deposition only on the irradiated parts, 1,1'

applied was 0.6 A/dm2, and the electrodeposition time was varied from 300 to 2400s.

Fig. 14. SEM micrograph demonstrating Cu deposition on the AFM tip after EBL treatment under current density of 0.6 A/dm2 and electrodeposition time of 300s (a) BEI image

**Tip site** 

c

a b

demonstrated (Xu et al., 1995).

**2.2.2 EBL induced Copper deposition** 

(b) Image taken at the tip site (c) Exposure area site

**Exposure area site** 

many factors i.e a combination of 2D and 3D growth (Abyaneh & Fleischmann, 1981; Creus et al., 1992), the death and rebirth of nuclei (Abyaneh & Fleischmann, 1981) and the secondary three dimensional (3D) growth on top of the first growth layers (Abyaneh et al., 1982).

## **2.2 Electron Beam lithography studies**

EBL (Electron-Beam lithography) technique followed soon after the development of the scanning electron microscope (SEM) in 1955 (Smith, 1955) and is one of the earliest processes used for IC fabrication (Buck, 1957). To date, EBL is widely exploited to produce structures in the sub-100 nm range (Allee et al., 1991; Matsui et al., 1989; Sun et al., 2005). Also, as compared with photolithography, the lateral resolution achieved by EBL is higher because the beam of electrons can be focused to produce probe size as small as 1 nm. More over, electrons do not suffer from optical thin-film interference. For ICs, where at present low beam energy and thick conventional resists are employed; electron scattering is the most important factor whereas for nanolithography, which utilizes high beam energy and thin resists, secondary electron emission is the most dominant factor. The resolution of EBL is also dependent on the chemical nature of the resist. Recently, new class of resists such as organic self-assembled mono layers (SAMs) has been developed to fabricate structures below 10 nm (Golzhauser et al., 2000; Lercel, 1996) Currently, electron beam lithography is used principally in support of the integrated circuit industry, where it has three niche markets. The first is in maskmaking, typically the chrome-on glass masks used by optical lithography tools. It is the preferred technique for masks because of its flexibility in providing rapid turnaround of a finished part described only by a computer. The ability to meet stringent line width control and pattern placement specifications, on the order of 50 nm each, is a remarkable achievement.

## **2.2.1 Principle of EBL**

The principle of pattern transfer based on EBL consists of several process steps. The process steps are essentially the same as those used for photolithography, except that the pattern on the resist is formed by scanning directly the focused particle beam across the surface. The lithographic sequence usually begins with coating of substrates with a positive or negative resist. Positive resists such as poly (methyl-methacrylate) (PMMA) used in the present chapter become more soluble in a developing solvent after exposure because the radiation causes local bond breakages and thus chain scission. This causes the exposed regions containing material of lower mean molecular weight to dissolve after the development. Nevertheless, negative resists become less soluble in solvent after exposure because crosslinking of polymer chains occurs. If in case, a region of a negative resist-covered film is exposed, only the exposed region will be covered by the resist after development. Subsequently, the resist-free parts of the substrate can be selectively coated with metal or etched before removal of the unexposed resist thus leaving the desired patterns at the surface. Fabrication of metallic nanostructures has been widely explored using conventional EBL and lift off techniques. However, this top-down approach cannot be employed for the fabrication of high aspect ratio vertical structures since gradual accumulation of materials at the top of the resist blocks and closes the opening of the structures during the evaporation of metal. Electrodeposition of metals into the holes formed in presence of PMMA resist is a convenient alternative to solve this problem (Simon et al., 1997). The fabrication of dense ultra-small magnetic arrays by filling nanoholes with electrodeposited Ni has been demonstrated (Xu et al., 1995).

## **2.2.2 EBL induced Copper deposition**

184 Smart Nanoparticles Technology

many factors i.e a combination of 2D and 3D growth (Abyaneh & Fleischmann, 1981; Creus et al., 1992), the death and rebirth of nuclei (Abyaneh & Fleischmann, 1981) and the secondary three dimensional (3D) growth on top of the first growth layers (Abyaneh et al.,

EBL (Electron-Beam lithography) technique followed soon after the development of the scanning electron microscope (SEM) in 1955 (Smith, 1955) and is one of the earliest processes used for IC fabrication (Buck, 1957). To date, EBL is widely exploited to produce structures in the sub-100 nm range (Allee et al., 1991; Matsui et al., 1989; Sun et al., 2005). Also, as compared with photolithography, the lateral resolution achieved by EBL is higher because the beam of electrons can be focused to produce probe size as small as 1 nm. More over, electrons do not suffer from optical thin-film interference. For ICs, where at present low beam energy and thick conventional resists are employed; electron scattering is the most important factor whereas for nanolithography, which utilizes high beam energy and thin resists, secondary electron emission is the most dominant factor. The resolution of EBL is also dependent on the chemical nature of the resist. Recently, new class of resists such as organic self-assembled mono layers (SAMs) has been developed to fabricate structures below 10 nm (Golzhauser et al., 2000; Lercel, 1996) Currently, electron beam lithography is used principally in support of the integrated circuit industry, where it has three niche markets. The first is in maskmaking, typically the chrome-on glass masks used by optical lithography tools. It is the preferred technique for masks because of its flexibility in providing rapid turnaround of a finished part described only by a computer. The ability to meet stringent line width control and pattern placement specifications, on the order of 50

The principle of pattern transfer based on EBL consists of several process steps. The process steps are essentially the same as those used for photolithography, except that the pattern on the resist is formed by scanning directly the focused particle beam across the surface. The lithographic sequence usually begins with coating of substrates with a positive or negative resist. Positive resists such as poly (methyl-methacrylate) (PMMA) used in the present chapter become more soluble in a developing solvent after exposure because the radiation causes local bond breakages and thus chain scission. This causes the exposed regions containing material of lower mean molecular weight to dissolve after the development. Nevertheless, negative resists become less soluble in solvent after exposure because crosslinking of polymer chains occurs. If in case, a region of a negative resist-covered film is exposed, only the exposed region will be covered by the resist after development. Subsequently, the resist-free parts of the substrate can be selectively coated with metal or etched before removal of the unexposed resist thus leaving the desired patterns at the surface. Fabrication of metallic nanostructures has been widely explored using conventional EBL and lift off techniques. However, this top-down approach cannot be employed for the fabrication of high aspect ratio vertical structures since gradual accumulation of materials at the top of the resist blocks and closes the opening of the structures during the evaporation of metal. Electrodeposition of metals into the holes formed in presence of PMMA resist is a

1982).

**2.2 Electron Beam lithography studies** 

nm each, is a remarkable achievement.

**2.2.1 Principle of EBL** 

Electron beam lithography technique is used in the present study to enable selective electrodeposition of Cu on the AFM tip and the open area beneath it. The selective electrodeposition of Cu on n-type Si (111) surfaces covered with organic monolayers by using e-beam lithographic techniques has been reported (Balaur et al., 2004). Selective copper deposition on e-beam patterned alkane and biphenylthiols has been reported (Kalten Poth et al., 2002) at suitable deposition potentials. 1-octadecanethiol (ODT) was used as a ''positive template'' leading to copper deposition only on the irradiated parts, 1,1' -biphenyl-4-thiol ( BPT) on the other hand acted as a ''negative template,'' where the irradiated and cross-linked biphenyl layer exhibited a blocking behavior, allowing copper deposition on the non-irradiated parts. In the present study, the open area of the nanosize AFM tip was selectively exposed to the e-beam. It is noticed that copper electrodeposition occurs on the exposed area of the AFM tip. For the copper electrodeposition process, the current density applied was 0.6 A/dm2, and the electrodeposition time was varied from 300 to 2400s.

Fig. 14. SEM micrograph demonstrating Cu deposition on the AFM tip after EBL treatment under current density of 0.6 A/dm2 and electrodeposition time of 300s (a) BEI image (b) Image taken at the tip site (c) Exposure area site

Nanoscale Electrodeposition of Copper on an AFM Tip and Its Morphological Investigations 187

observed when the electrodeposition time was increased to 1200 s. SEM micrograph in Fig. 16a shows that an uniform layer of copper is deposited on the AFM tip and the open area beneath the tip. These results indicate that the exposure of the tip to the high energy electron beam might have facilitated the electrodeposition of copper on the tip. The micrographs in Fig. 16 b reveal that some copper is being deposited on the edges of the cantilever. This is because the PMMA layers on the edges are found to be thinner than on the platform. Those places are not exposed to the e-beam; therefore the developer could dissolve the PMMA layer on the edges and hence copper deposition took place on the edges. The overpotential required to deposit copper on the edges is lower than on the

**Cantilever Site** 

a b

c d

Fig. 16. SEM micrograph demonstrating Cu deposition on the AFM tip after EBL treatment under current density of 0.6 A/dm2 and electrodeposition time of 1200s (a) SEI image

(b) Magnified to 5000 T (c) Tip site (d) Cantilever site

AFM tip.

**Tip Site** 

SEM micrographs for copper electrodeposition on the AFM tip and the open area beneath it for various deposition times (i.e 300, 600, 1200, 2400 s) and current density of 0.6 A/dm2 are presented in Figs. 14-17.These SEM micrographs were taken after exposure to the electron beam. In Figs. 14 (a) – (c) the micrographs for copper deposition on the AFM tip under current density of 0.6 A dm-2 and electrodeposition time of 300 s are clearly depicted. Copper deposition is found to be minimum and non-uniform in these images. Further increase in the electrodeposition time to 600 s for the similar current density and exposure to the e-beam increases the amount of copper deposits on the nanosize AFM tip and the open area beneath it (Fig. 15 a). SEM micrographs in Fig. 15b and 15 c refers to the magnified images of the AFM tip and the exposed site.

Fig. 15. SEM micrograph demonstrating Cu deposition on the AFM tip after EBL treatmentunder current density of 0.6 A/dm2 and electrodeposition time of 600s (a) BEI image (b) Image taken at the tip site (c) Exposure area site

The micrographs reveal that copper deposition is not uniform in the open area beneath the AFM tip. However a significant change in the morphology of copper deposits is

SEM micrographs for copper electrodeposition on the AFM tip and the open area beneath it for various deposition times (i.e 300, 600, 1200, 2400 s) and current density of 0.6 A/dm2 are presented in Figs. 14-17.These SEM micrographs were taken after exposure to the electron beam. In Figs. 14 (a) – (c) the micrographs for copper deposition on the AFM tip under current density of 0.6 A dm-2 and electrodeposition time of 300 s are clearly depicted. Copper deposition is found to be minimum and non-uniform in these images. Further increase in the electrodeposition time to 600 s for the similar current density and exposure to the e-beam increases the amount of copper deposits on the nanosize AFM tip and the open area beneath it (Fig. 15 a). SEM micrographs in Fig. 15b and 15 c refers to the magnified

**Tip site** 

a b

c

Fig. 15. SEM micrograph demonstrating Cu deposition on the AFM tip after EBL treatmentunder current density of 0.6 A/dm2 and electrodeposition time of 600s

The micrographs reveal that copper deposition is not uniform in the open area beneath the AFM tip. However a significant change in the morphology of copper deposits is

(a) BEI image (b) Image taken at the tip site (c) Exposure area site

**Exposure area site** 

images of the AFM tip and the exposed site.

observed when the electrodeposition time was increased to 1200 s. SEM micrograph in Fig. 16a shows that an uniform layer of copper is deposited on the AFM tip and the open area beneath the tip. These results indicate that the exposure of the tip to the high energy electron beam might have facilitated the electrodeposition of copper on the tip. The micrographs in Fig. 16 b reveal that some copper is being deposited on the edges of the cantilever. This is because the PMMA layers on the edges are found to be thinner than on the platform. Those places are not exposed to the e-beam; therefore the developer could dissolve the PMMA layer on the edges and hence copper deposition took place on the edges. The overpotential required to deposit copper on the edges is lower than on the AFM tip.

Fig. 16. SEM micrograph demonstrating Cu deposition on the AFM tip after EBL treatment under current density of 0.6 A/dm2 and electrodeposition time of 1200s (a) SEI image (b) Magnified to 5000 T (c) Tip site (d) Cantilever site

Nanoscale Electrodeposition of Copper on an AFM Tip and Its Morphological Investigations 189

that the unexposed areas below the AFM tip remain covered with PMMA. However, the resist free parts of the AFM tip are selectively coated with copper. Reports on the selective electrodeposition of Cu (Balaur et al., 2004) on n-type Si (1 1 1) surfaces covered with organic monolayers and e-beam modified using e-beam lithographic techniques have also been established. Copper was electrochemically deposited in the e-beam modified regions and the selectivity of the deposition of copper in these regions was strongly dependent on the applied e-beam dose. The selective deposition of copper on the nanosize AFM tip can be described on the basis of Volmer-Weber approach which states that higher numbers of activation sites are triggered with a higher overvoltage. In the Volmer-Weber model, nucleation and growth are strongly potential dependent. At low cathodic potentials, only a few sites are involved because the energy level is not sufficient whereas at high cathodic voltages more initiation sites contribute to the nucleation process. It implies that at low overpotentials the crystallites have to grow extremely large to reach coalescence and form a homogenous deposit. In the present study, higher overpotential existing on the AFM tip might have increased the number of activation sites, leading to the preferential deposition of

The investigations made in this chapter have highlighted electrodeposition as an attractive approach for the preparation of nanostructured materials. Copper electrodeposition on a nanosize AFM tip of diameter 80 nm was established by varying the magnitude of current densities with electrodeposition time and vice versa. Significant changes in the morphology of copper deposits were observed with changes in the above parameters. Morphological investigations by SEM revealed that a nonuniform layer of copper was formed on the open areas surrounding the tip and the AFM probe; however, deposition of copper on the AFM tip could not be achieved in the absence of photoresist. Electron beam lithography technique facilitated the formation of copper deposits on the nanosize AFM tip of diameter 80 nm in the presence of PMMA. Copper was electrochemically deposited on the e-beam modified regions of the AFM probe at a current density of 0.6 A/dm2 with electrodeposition times ranging from 300 to 2400 s. The most uniform deposition on the AFM tip was noticed after EBL treatment under current density of 0.6 Adm-2 and electrodeposition time of 2400 s.

The authors acknowledge financial support of this study from the National Science Council of China under NSC 94-2811-E-006-021. The Department of Materials Science and Engineering, National Cheng Kung University assisted in meeting the publication costs of

Abyaneh, M.Y.; Hendrikx. J.; Visscher, W. & Barendrecht. E (1982). Studies of Electroplating

129, No.12 (December 1982) pp.2654-2659., ISSN 0013-4651.

using an EQCM. I. Copper and Silver on Gold. *Journal of Electrochemical Society,* Vol.

copper on exposure to the e-beam.

**3. Conclusion** 

**4. Acknowledgment** 

this article.

**5. References** 

Theoretically it has been established (West, 1971) that deposition at low overpotentials is dominated by surface diffusion; hence nucleation and growth occur primarily at step edges and dislocations (Winand, 1975). Fig. 17 illustrates the morphology of copper deposition on the nanosize AFM tip obtained under current density of 0.6 A/ dm2 and electrodeposition time of 2400 s and after exposure to the e-beam. The micrograph in Fig.17a distinctly shows that copper is deposited on the AFM tip and a very thick growth of copper deposits is seen on the open area beneath the tip. From the series of micrographs obtained at different electrodeposition times and current density of 0.6 A /dm2, it is noticed that copper gets deposited both on the AFM tip and the open area beneath it, the most uniform deposition seen at 2400 s of electrodeposition time.

Fig. 17. SEM micrograph demonstrating Cu deposition on the AFM tip after EBL treatment under current density of 0.6 A/dm2 and electrodeposition time of 2400s (a) SEI image (b) Magnified to 5000 T

The PMMA coated on the AFM tip becomes more soluble in a developing solvent after exposure to the e-beam because the radiation causes local bond breakages and thus chain scission (Djenizian et al., 2006) as mentioned above. It could be clearly seen from Fig. 17b that the unexposed areas below the AFM tip remain covered with PMMA. However, the resist free parts of the AFM tip are selectively coated with copper. Reports on the selective electrodeposition of Cu (Balaur et al., 2004) on n-type Si (1 1 1) surfaces covered with organic monolayers and e-beam modified using e-beam lithographic techniques have also been established. Copper was electrochemically deposited in the e-beam modified regions and the selectivity of the deposition of copper in these regions was strongly dependent on the applied e-beam dose. The selective deposition of copper on the nanosize AFM tip can be described on the basis of Volmer-Weber approach which states that higher numbers of activation sites are triggered with a higher overvoltage. In the Volmer-Weber model, nucleation and growth are strongly potential dependent. At low cathodic potentials, only a few sites are involved because the energy level is not sufficient whereas at high cathodic voltages more initiation sites contribute to the nucleation process. It implies that at low overpotentials the crystallites have to grow extremely large to reach coalescence and form a homogenous deposit. In the present study, higher overpotential existing on the AFM tip might have increased the number of activation sites, leading to the preferential deposition of copper on exposure to the e-beam.

## **3. Conclusion**

188 Smart Nanoparticles Technology

Theoretically it has been established (West, 1971) that deposition at low overpotentials is dominated by surface diffusion; hence nucleation and growth occur primarily at step edges and dislocations (Winand, 1975). Fig. 17 illustrates the morphology of copper deposition on the nanosize AFM tip obtained under current density of 0.6 A/ dm2 and electrodeposition time of 2400 s and after exposure to the e-beam. The micrograph in Fig.17a distinctly shows that copper is deposited on the AFM tip and a very thick growth of copper deposits is seen on the open area beneath the tip. From the series of micrographs obtained at different electrodeposition times and current density of 0.6 A /dm2, it is noticed that copper gets deposited both on the AFM tip and the open area beneath it, the most uniform deposition

a b

Fig. 17. SEM micrograph demonstrating Cu deposition on the AFM tip after EBL treatment under current density of 0.6 A/dm2 and electrodeposition time of 2400s (a) SEI image

The PMMA coated on the AFM tip becomes more soluble in a developing solvent after exposure to the e-beam because the radiation causes local bond breakages and thus chain scission (Djenizian et al., 2006) as mentioned above. It could be clearly seen from Fig. 17b

seen at 2400 s of electrodeposition time.

(b) Magnified to 5000 T

The investigations made in this chapter have highlighted electrodeposition as an attractive approach for the preparation of nanostructured materials. Copper electrodeposition on a nanosize AFM tip of diameter 80 nm was established by varying the magnitude of current densities with electrodeposition time and vice versa. Significant changes in the morphology of copper deposits were observed with changes in the above parameters. Morphological investigations by SEM revealed that a nonuniform layer of copper was formed on the open areas surrounding the tip and the AFM probe; however, deposition of copper on the AFM tip could not be achieved in the absence of photoresist. Electron beam lithography technique facilitated the formation of copper deposits on the nanosize AFM tip of diameter 80 nm in the presence of PMMA. Copper was electrochemically deposited on the e-beam modified regions of the AFM probe at a current density of 0.6 A/dm2 with electrodeposition times ranging from 300 to 2400 s. The most uniform deposition on the AFM tip was noticed after EBL treatment under current density of 0.6 Adm-2 and electrodeposition time of 2400 s.

## **4. Acknowledgment**

The authors acknowledge financial support of this study from the National Science Council of China under NSC 94-2811-E-006-021. The Department of Materials Science and Engineering, National Cheng Kung University assisted in meeting the publication costs of this article.

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**9** 

*1Mexico 2Spain* 

**New Trends on the Synthesis of Inorganic** 

Margarita Sanchez-Dominguez1, Carolina Aubery2 and Conxita Solans2 *1Centro de Investigación en Materiales Avanzados, S. C. (CIMAV), Unidad Monterrey;* 

The development of nanotechnology depends strongly on the advances in nanoparticle preparation. Nowadays, there are a number of technologies available for nanoparticle synthesis, from the gas phase techniques such as laser evaporation (Gaertner & Lydtin, 1994), sputtering, laser pyrolisis, flame atomization and flame spray pyrolisis (Kruis et al. 1998), etc, to the liquid phase techniques such as coprecipitation from homogeneous solutions and sol-gel reactions (Qiao et al. 2011), solvothermal processes (Gautam et al. 2002), sonochemical and cavitation processing (Suslick et al. 1996), and surfactant and polymer-templated synthesis (Holmberg, 2004). Amongst the surfactant-based approaches, the microemulsion reaction method is one of the most used techniques for the preparation of very small and nearly monodispersed nanoparticles. This method offers a series of advantages with respect to other methods, namely, the use of simple equipment, the possibility to prepare a great variety of materials with a high degree of particle size and composition control, the formation of nanoparticles with often crystalline structure and high specific surface area, and the use of soft conditions of synthesis, near ambient temperature and pressure. The traditional method is based on water-in-oil microemulsions (W/O), and it has been used for the preparation of metallic and other inorganic nanoparticles since the beginning of the 1980's (Boutonnet et al., 1982). The droplets of W/O microemulsions are conceived as tiny compartments or "nanoreactors". The main strategy for the synthesis of nanoparticles in W/O microemulsions consists in mixing two microemulsions, one containing the metallic precursor and another one the precipitating agent. Upon mixing, both reactants will contact each other due to droplets collisions and coalescence, and they will react to form precipitates of nanometric size (Figure 1). This precipitate will be confined to the interior of microemulsion droplets. Numerous investigations have been published about the use of W/O microemulsions for the preparation of a variety of nanomaterials,

**1. Introduction** 

*2Instituto de Química Avanzada de Cataluña, Consejo Superior de Investigaciones* 

**Nanoparticles Using Microemulsions** 

*GENES-Group of Embedded Nanomaterials for Energy Scavenging* 

**as Confined Reaction Media** 

*Científicas (IQAC-CSIC); CIBER en Biotecnología, Biomateriales y Nanomedicina (CIBER-BBN)* 

Yau, S.L.; Gao, X; Chang, S.C; Schardt, B.C & Weaver , M. J.(1991) Atomic-resolution scanning tunneling microscopy and infrared spectroscopy as combined in situ probes of electrochemical adlayer structure: carbon monoxide on rhodium (111). *Journal of American Chemical Society,* Vol. 113, No.16, pp. 6049-6056, ISSN 0002- 7863.

## **New Trends on the Synthesis of Inorganic Nanoparticles Using Microemulsions as Confined Reaction Media**

Margarita Sanchez-Dominguez1, Carolina Aubery2 and Conxita Solans2 *1Centro de Investigación en Materiales Avanzados, S. C. (CIMAV), Unidad Monterrey; GENES-Group of Embedded Nanomaterials for Energy Scavenging 2Instituto de Química Avanzada de Cataluña, Consejo Superior de Investigaciones Científicas (IQAC-CSIC); CIBER en Biotecnología, Biomateriales y Nanomedicina (CIBER-BBN) 1Mexico 2Spain* 

## **1. Introduction**

194 Smart Nanoparticles Technology

Yau, S.L.; Gao, X; Chang, S.C; Schardt, B.C & Weaver , M. J.(1991) Atomic-resolution

7863.

scanning tunneling microscopy and infrared spectroscopy as combined in situ probes of electrochemical adlayer structure: carbon monoxide on rhodium (111). *Journal of American Chemical Society,* Vol. 113, No.16, pp. 6049-6056, ISSN 0002-

> The development of nanotechnology depends strongly on the advances in nanoparticle preparation. Nowadays, there are a number of technologies available for nanoparticle synthesis, from the gas phase techniques such as laser evaporation (Gaertner & Lydtin, 1994), sputtering, laser pyrolisis, flame atomization and flame spray pyrolisis (Kruis et al. 1998), etc, to the liquid phase techniques such as coprecipitation from homogeneous solutions and sol-gel reactions (Qiao et al. 2011), solvothermal processes (Gautam et al. 2002), sonochemical and cavitation processing (Suslick et al. 1996), and surfactant and polymer-templated synthesis (Holmberg, 2004). Amongst the surfactant-based approaches, the microemulsion reaction method is one of the most used techniques for the preparation of very small and nearly monodispersed nanoparticles. This method offers a series of advantages with respect to other methods, namely, the use of simple equipment, the possibility to prepare a great variety of materials with a high degree of particle size and composition control, the formation of nanoparticles with often crystalline structure and high specific surface area, and the use of soft conditions of synthesis, near ambient temperature and pressure. The traditional method is based on water-in-oil microemulsions (W/O), and it has been used for the preparation of metallic and other inorganic nanoparticles since the beginning of the 1980's (Boutonnet et al., 1982). The droplets of W/O microemulsions are conceived as tiny compartments or "nanoreactors". The main strategy for the synthesis of nanoparticles in W/O microemulsions consists in mixing two microemulsions, one containing the metallic precursor and another one the precipitating agent. Upon mixing, both reactants will contact each other due to droplets collisions and coalescence, and they will react to form precipitates of nanometric size (Figure 1). This precipitate will be confined to the interior of microemulsion droplets. Numerous investigations have been published about the use of W/O microemulsions for the preparation of a variety of nanomaterials,

New Trends on the Synthesis

Friberg, 1981; Cross, 1987).

processes (Fletcher et al., 1987).

**2.2 Microemulsions and phase equilibria** 

of Inorganic Nanoparticles Using Microemulsions as Confined Reaction Media 197

as reaction media for nanoparticle synthesis is first included. In addition, other aspects of nanoparticle synthesis are reviewed, such as study of reaction kinetics; influence of microemulsion dynamics on the characteristics of the obtained materials, as well as phase-

Microemulsions are transparent and thermodynamically stable colloidal dispersions in which two liquids initially immiscible (typically water and oil) coexist in one phase due to the presence of a monolayer of surfactant molecules with balanced hydrophilic-lipophilic properties (Danielsson & Lindman, 1981). They are optically isotropic and transparent. In contrast to emulsions, for which formation requires a considerable energy input, microemulsions form spontaneously upon gentle components mixing, once composition and temperature conditions are right. Depending on the ratio of oil and water and on the hydrophilic-lipophilic balance (HLB) of the surfactant, microemulsions can exist as oilswollen micelles dispersed in water (oil-in-water microemulsions, O/W), or water-swollen inverse micelles dispersed in oil (water-in-oil microemulsions, W/O); at intermediate compositions and HLBs, bicontinuous structures can exist. When a dispersed phase is present, it consists of droplets with a narrow size distribution in the order of 2-50 nm.

The formation of microemulsion depends on the surfactant type and structure, e.g. single hydrocarbon chain ionic surfactants require the incorporation of cosurfactant or electrolytes for microemulsion formation due to their high hydrophilic character; in contrast, double chain ionic surfactants and ethoxylated non-ionic surfactants may form microemulsions without cosurfactant. Lowering the interfacial tension between the oily and aqueous phase (o/w) is the main role of the surfactant (or surfactant/cosurfactant mixture). The extremely low o/w (in the order of 10-2 - 10-3 mN m-1) achieved is one of the main microemulsion characteristics: the decrease on o/w is caused by the surfactant, overcoming the surface energy term caused by the huge increase in interfacial area. In addition, the spontaneous dispersion of numerous water or oil droplets causes an entropy increase, yielding a thermodynamically stable system. The extremely low interfacial tension is decisive for microemulsion formation, and depends on the composition of the system (Kunieda &

Microemulsions are dynamic systems, and it has been shown that droplet content exchange processes can occur in the order of millisecond time scales (Fletcher et al., 1987; Clarke et al., 1990). Collisions are produced due to constant Brownian motion of the droplets. When these collisions are sufficiently violent, the surfactant layer breaks up and the micellar exchange can be produced. It is thought that the micellar exchange process is characterized by an activation energy ( Ea or energy barrier), which is affected by the flexibility or rigidity of the surfactant layer (Fletcher & Horsup, 1992; Lindman & Friberg, 1999}, in addition to diffusion

Phase behavior studies by means of equilibrium phase diagrams of polar solvent/amphiphile/nonpolar solvent systems provide essential information on

transfer and isolation of nanoparticles from the microemulsion reaction media.

**2. General properties and formation of microemulsions** 

**2.1 Microemulsions: Definition and basic properties** 

such as metallic and bimetallic nanoparticles, single metal oxide as well as mixed oxides, quantum dots, and even complex ceramic materials (Boutonnet et al., 1982; Destrée & Nagy, 2006; Eastoe et al. 2006; Holmberg, 2004; López-Quintela et al. 2004; Pileni 1997 and 2003). Materials synthesized in w/o microemulsions exhibit unique surface properties; for example, nano-catalysts prepared by this method show better performance (activity, selectivity) than those prepared by other methods (Boutonnet et al. 2008).

Fig. 1. Scheme of the w/o microemulsion reaction method for the synthesis of inorganic nanoparticles.

In spite of the superior properties and performance of nanoparticles obtained in w/o microemulsions, this method has not found good acceptance at the industrial level, mainly due to the employment of large amounts of oils (solvents) which represent the continuous and hence main component of these systems. In addition, most studies employ relatively low concentration of the metal precursors, leading to small yields of nanoparticles per microemulsion volume. These drawbacks affect negatively from the economic and ecologic point of view. It is the aim of this chapter to review the newest trends in the synthesis of inorganic nanoparticles using microemulsions as confined reaction media, with the objective to identify those alternatives or approaches that make this type of colloidal media more attractive for nanoparticle synthesis from the environmental , economic, technological, and scientific point of view. Some of those approaches are: the synthesis of advanced materials, such as mixed oxides and complex ceramics with nanocrystalline structure, core-shell particles, mixed materials with key nano-heterojunctions, etc, which may be difficult to obtain by other methods; optimization of microemulsion compositions, by making use of advanced phase behaviour knowledge; use of bicontinuous microemulsions in semi-continuous batches, and last but not least, a novel approach based on the use of oil-in-water microemulsions instead of w/o microemulsions as confined reaction media. An introductory section about the generalities and properties of microemulsion systems as well as on the use of microemulsions as reaction media for nanoparticle synthesis is first included. In addition, other aspects of nanoparticle synthesis are reviewed, such as study of reaction kinetics; influence of microemulsion dynamics on the characteristics of the obtained materials, as well as phasetransfer and isolation of nanoparticles from the microemulsion reaction media.

## **2. General properties and formation of microemulsions**

## **2.1 Microemulsions: Definition and basic properties**

196 Smart Nanoparticles Technology

such as metallic and bimetallic nanoparticles, single metal oxide as well as mixed oxides, quantum dots, and even complex ceramic materials (Boutonnet et al., 1982; Destrée & Nagy, 2006; Eastoe et al. 2006; Holmberg, 2004; López-Quintela et al. 2004; Pileni 1997 and 2003). Materials synthesized in w/o microemulsions exhibit unique surface properties; for example, nano-catalysts prepared by this method show better performance (activity,

Fig. 1. Scheme of the w/o microemulsion reaction method for the synthesis of inorganic

In spite of the superior properties and performance of nanoparticles obtained in w/o microemulsions, this method has not found good acceptance at the industrial level, mainly due to the employment of large amounts of oils (solvents) which represent the continuous and hence main component of these systems. In addition, most studies employ relatively low concentration of the metal precursors, leading to small yields of nanoparticles per microemulsion volume. These drawbacks affect negatively from the economic and ecologic point of view. It is the aim of this chapter to review the newest trends in the synthesis of inorganic nanoparticles using microemulsions as confined reaction media, with the objective to identify those alternatives or approaches that make this type of colloidal media more attractive for nanoparticle synthesis from the environmental , economic, technological, and scientific point of view. Some of those approaches are: the synthesis of advanced materials, such as mixed oxides and complex ceramics with nanocrystalline structure, core-shell particles, mixed materials with key nano-heterojunctions, etc, which may be difficult to obtain by other methods; optimization of microemulsion compositions, by making use of advanced phase behaviour knowledge; use of bicontinuous microemulsions in semi-continuous batches, and last but not least, a novel approach based on the use of oil-in-water microemulsions instead of w/o microemulsions as confined reaction media. An introductory section about the generalities and properties of microemulsion systems as well as on the use of microemulsions

nanoparticles.

selectivity) than those prepared by other methods (Boutonnet et al. 2008).

Microemulsions are transparent and thermodynamically stable colloidal dispersions in which two liquids initially immiscible (typically water and oil) coexist in one phase due to the presence of a monolayer of surfactant molecules with balanced hydrophilic-lipophilic properties (Danielsson & Lindman, 1981). They are optically isotropic and transparent. In contrast to emulsions, for which formation requires a considerable energy input, microemulsions form spontaneously upon gentle components mixing, once composition and temperature conditions are right. Depending on the ratio of oil and water and on the hydrophilic-lipophilic balance (HLB) of the surfactant, microemulsions can exist as oilswollen micelles dispersed in water (oil-in-water microemulsions, O/W), or water-swollen inverse micelles dispersed in oil (water-in-oil microemulsions, W/O); at intermediate compositions and HLBs, bicontinuous structures can exist. When a dispersed phase is present, it consists of droplets with a narrow size distribution in the order of 2-50 nm.

The formation of microemulsion depends on the surfactant type and structure, e.g. single hydrocarbon chain ionic surfactants require the incorporation of cosurfactant or electrolytes for microemulsion formation due to their high hydrophilic character; in contrast, double chain ionic surfactants and ethoxylated non-ionic surfactants may form microemulsions without cosurfactant. Lowering the interfacial tension between the oily and aqueous phase (o/w) is the main role of the surfactant (or surfactant/cosurfactant mixture). The extremely low o/w (in the order of 10-2 - 10-3 mN m-1) achieved is one of the main microemulsion characteristics: the decrease on o/w is caused by the surfactant, overcoming the surface energy term caused by the huge increase in interfacial area. In addition, the spontaneous dispersion of numerous water or oil droplets causes an entropy increase, yielding a thermodynamically stable system. The extremely low interfacial tension is decisive for microemulsion formation, and depends on the composition of the system (Kunieda & Friberg, 1981; Cross, 1987).

Microemulsions are dynamic systems, and it has been shown that droplet content exchange processes can occur in the order of millisecond time scales (Fletcher et al., 1987; Clarke et al., 1990). Collisions are produced due to constant Brownian motion of the droplets. When these collisions are sufficiently violent, the surfactant layer breaks up and the micellar exchange can be produced. It is thought that the micellar exchange process is characterized by an activation energy ( Ea or energy barrier), which is affected by the flexibility or rigidity of the surfactant layer (Fletcher & Horsup, 1992; Lindman & Friberg, 1999}, in addition to diffusion processes (Fletcher et al., 1987).

#### **2.2 Microemulsions and phase equilibria**

Phase behavior studies by means of equilibrium phase diagrams of polar solvent/amphiphile/nonpolar solvent systems provide essential information on

New Trends on the Synthesis

Strey et al. (Jakobs et al., 1999).

of Inorganic Nanoparticles Using Microemulsions as Confined Reaction Media 199

Fig. 3. Water / Nonionic surfactant / Oil pseudo bynary phase diagram, as a function of

As for non-ionic surfactant – based systems, Triton® X-100 (octyl-phenol ethoxylate) is one of the most used, however, alkyl-phenol ethoxylate surfactants such as this one have a limited biodegradability. Their metabolites of degradation have low solubility and are toxic, for example, nonylphenol has been proven to be an endocrine disrupter (Jobling & Sumpter, 1993). On the other hand, aliphatic fatty alcohol ethoxylates such as PEGDE (penta(ethylene glycol) dodecyl ether) are more environmentally friendly; for nanoparticle synthesis, the technical-grade options are usually chosen due to their lower cost. A special feature of nonionic surfactant systems is the sensitivity of their hydrophilic-lipophilic properties to temperature, and although sometimes this characteristic is seen as a drawback, the possibility for phase-behavior tuning can be used as an advantage for the formulation of non-ionic microemulsions. In addition, nonionic surfactants have a great capacity of hydration by their ethoxylated (EO) units; hence, an appropriate selection of surfactant, oil and precursor salts/precipitating agent concentration, in combination with the rich structural behavior that such a system may display as a function of temperature, can lead to highly optimized formulations in terms of aqueous phase uptake and hence reactants loading. A good premise to this behavior is the enormous efficiency boost in the formation of middle phase microemulsions by the use of block copolymer surfactants reported by

Although nonionic microemulsion systems are mainly affected by temperature changes, the addition of electrolytes and cosurfactant can also produce shifts in the solubilization and

temperature. Reproduced with permission (Destrée & Nagy, 2006).

**2.3 Effect of precursor salts and additives on the phase behavior** 

microemulsion formation and structure. In 1954, Winsor predicted four types of equilibria which was latter experimentally evidenced: i) Winsor I: oil-in-water (o/w) microemulsions are formed, and the surfactant-rich water phase coexists with the oil phase where surfactant is only present as monomers; ii) Winsor II: water-in-oil (w/o) microemulsions are formed and the surfactant-rich oil phase coexists with the surfactant-poor aqueous phase; iii) Winsor III (middle phase): a three-phase system where a bicontinuous middle-phase microemulsion (rich in surfactant) coexists with both excess water and oil phases; and iv) Winsor IV: a single-phase (isotropic) micellar solution (microemulsion), that forms upon addition of a sufficient quantity of amphiphile. Figure 2 shows how this equilibria can be affected by salinity (for ionic surfactants) or temperature (for non-ionic surfactants), and also illustrates the structural variability of microemulsions (O/W, W/O and bicontinuous (BC)).

Some typical equilibrium phase diagrams are shown in Figure 3 (Destrée & Nagy, 2006). In each of these diagrams L2 denotes a region where one phase W/O microemulsions are formed. AOT (Sodium 2-ethylhexylsulfosuccinate) based systems are amongst the best characterized systems, and it has been found that the size of the inverse microemulsion droplets formed by this type of systems increases linearly with the amount of water added to the system (Pileni, 1998) and can increase from 4 nm to 18 nm with 0.1 M sodium AOT surfactant (water/AOT/isooctane). AOT based systems are probably the most used for the synthesis of inorganic nanoparticles in w/o microemulsions, for two reasons: good control of droplet size as explained above and the large microemulsion regions found in water/AOT/alkane systems, which give rise to a great deal of compositions available for nanoparticle synthesis. Systems based on cetyltrimethylammonium bromide (CTAB), usually combine this surfactant with alcohols such as hexanol as the oil phase. This alcohol can act as co-surfactant, adsorbing at the oil/water interface along with the surfactant. As shown in Figure 3 the microemulsion region of water/CTAB/hexanol system is relatively narrow, however, when shorter alcohols such as butanol are added as cosurfactant, the microemulsion regions are considerably enlarged (Košak et al., 2004).

Fig. 2. Winsor classification of microemulsion equilibria. Microemulsion phase sequence as a function of temperature and salinity for non-ionic and ionic surfactants, respectively.

microemulsion formation and structure. In 1954, Winsor predicted four types of equilibria which was latter experimentally evidenced: i) Winsor I: oil-in-water (o/w) microemulsions are formed, and the surfactant-rich water phase coexists with the oil phase where surfactant is only present as monomers; ii) Winsor II: water-in-oil (w/o) microemulsions are formed and the surfactant-rich oil phase coexists with the surfactant-poor aqueous phase; iii) Winsor III (middle phase): a three-phase system where a bicontinuous middle-phase microemulsion (rich in surfactant) coexists with both excess water and oil phases; and iv) Winsor IV: a single-phase (isotropic) micellar solution (microemulsion), that forms upon addition of a sufficient quantity of amphiphile. Figure 2 shows how this equilibria can be affected by salinity (for ionic surfactants) or temperature (for non-ionic surfactants), and also illustrates the structural variability of microemulsions (O/W, W/O and bicontinuous (BC)). Some typical equilibrium phase diagrams are shown in Figure 3 (Destrée & Nagy, 2006). In each of these diagrams L2 denotes a region where one phase W/O microemulsions are formed. AOT (Sodium 2-ethylhexylsulfosuccinate) based systems are amongst the best characterized systems, and it has been found that the size of the inverse microemulsion droplets formed by this type of systems increases linearly with the amount of water added to the system (Pileni, 1998) and can increase from 4 nm to 18 nm with 0.1 M sodium AOT surfactant (water/AOT/isooctane). AOT based systems are probably the most used for the synthesis of inorganic nanoparticles in w/o microemulsions, for two reasons: good control of droplet size as explained above and the large microemulsion regions found in water/AOT/alkane systems, which give rise to a great deal of compositions available for nanoparticle synthesis. Systems based on cetyltrimethylammonium bromide (CTAB), usually combine this surfactant with alcohols such as hexanol as the oil phase. This alcohol can act as co-surfactant, adsorbing at the oil/water interface along with the surfactant. As shown in Figure 3 the microemulsion region of water/CTAB/hexanol system is relatively narrow, however, when shorter alcohols such as butanol are added as cosurfactant, the

microemulsion regions are considerably enlarged (Košak et al., 2004).

Fig. 2. Winsor classification of microemulsion equilibria. Microemulsion phase sequence as a

function of temperature and salinity for non-ionic and ionic surfactants, respectively.

Fig. 3. Water / Nonionic surfactant / Oil pseudo bynary phase diagram, as a function of temperature. Reproduced with permission (Destrée & Nagy, 2006).

As for non-ionic surfactant – based systems, Triton® X-100 (octyl-phenol ethoxylate) is one of the most used, however, alkyl-phenol ethoxylate surfactants such as this one have a limited biodegradability. Their metabolites of degradation have low solubility and are toxic, for example, nonylphenol has been proven to be an endocrine disrupter (Jobling & Sumpter, 1993). On the other hand, aliphatic fatty alcohol ethoxylates such as PEGDE (penta(ethylene glycol) dodecyl ether) are more environmentally friendly; for nanoparticle synthesis, the technical-grade options are usually chosen due to their lower cost. A special feature of nonionic surfactant systems is the sensitivity of their hydrophilic-lipophilic properties to temperature, and although sometimes this characteristic is seen as a drawback, the possibility for phase-behavior tuning can be used as an advantage for the formulation of non-ionic microemulsions. In addition, nonionic surfactants have a great capacity of hydration by their ethoxylated (EO) units; hence, an appropriate selection of surfactant, oil and precursor salts/precipitating agent concentration, in combination with the rich structural behavior that such a system may display as a function of temperature, can lead to highly optimized formulations in terms of aqueous phase uptake and hence reactants loading. A good premise to this behavior is the enormous efficiency boost in the formation of middle phase microemulsions by the use of block copolymer surfactants reported by Strey et al. (Jakobs et al., 1999).

### **2.3 Effect of precursor salts and additives on the phase behavior**

Although nonionic microemulsion systems are mainly affected by temperature changes, the addition of electrolytes and cosurfactant can also produce shifts in the solubilization and

New Trends on the Synthesis

from Moulik, 1998).

of Inorganic Nanoparticles Using Microemulsions as Confined Reaction Media 201

molecules. The aqueous domain is composed by bounded water (hydrating the surfactant/cosurfactant hydrophilic domains) and free water (forming the droplet core). The exchange process to reach microemulsion equilibria comprises: 1) exchange of water between the bounded and free state; 2) exchange of cosurfactants among the interfacial film, the continuous phase and the dispersed phase (depending on its solubility). If ionic species are solubilized in the aqueous solution, they exchange ions between the bounded and free water. The composition of the aqueous droplets, their concentration and the temperature are

Droplet-droplet interactions depend strongly on droplet concentration, solvent viscosity, temperature, rigidity or flexibility of interfacial layer, and interactions between surfactant tails (Capek, 2004; Lopez-Quintela, 2003). When water or oil droplet dispersions are present, the droplets continuously collide, break apart, aggregate and break apart giving rise to dynamic processes in microemulsions. These dynamic processes allows microemulsion droplets to continuously exchange their content in microsecond scales. The composition of the aqueous droplets, as explained above, has a great influence on droplet interactions. As an example, the interfacial layer plays an important role on the formation, stability and discrete nature of microemulsion droplets. The film rigidity has been observed to increase with the surfactant hydrocarbon chain, whereas it substantially decreases with cosurfactant addition. The surfactant packing capacity can be also affected by the ionic strength of the droplets (Aramaki et al., 2001; Kunieda et al., 1995). The increase of surfactant molecules in

Fig. 4. Schematic representation of W/O microemulsion droplet. (Adapted with permission

Changes on microemulsion dynamics giving rise to structural transitions can be explained in terms of percolation. Figure 5 refers to a percolation process taking place in W/O microemulsions (Borkovec et al., 1988), as the oil to water ratio o is varied. As observed in Figure 5, at high oil concentration, the fraction of water in discrete droplets increases with water composition (decreasing oil concentration) up to a concentration, where it drastically decreases. This concentration is called Percolation Concentration CP. Although clusterization occurs below CP (low water concentrations), these clusters remain finite in size respect to the bulk solution. CP represents the concentration at which the first infinite cluster appears. Further increase on water concentration would lead to the disappearance of discrete water droplets to give rise to an increase of infinite water and oily domains, which

mayor factors defining further interactions between them.

the layer is proportional to the rigidity of the micelles.

THLB (hydrophilic-lipophilic balanced temperature or phase inversion temperature) of the systems (Aramaki et al., 2001; Kunieda et al., 1995; Shinoda, 1968). The use of ionic surfactants may have some drawbacks, as usually the aqueous phase uptake of ionic microemulsions is reduced in the presence of precursor salts due to screening effects, and hence microemulsion regions become smaller (Liu et al., 2000; Gianakas et al., 2006). Additionally, complex species could interfere with particle growth by adsorption to their surface, and contaminations of ceramic nanoparticles with the surfactant counterions aer possible. Often, the effect of addition of precursor salts or precipitating agent on the phase behavior and structure of microemulsion systems is underlooked. Generally two microemulsions with a fixed water/surfactant ratio are prepared without taking into account the influence the added salt has on the size and the structure of the water droplets.

Recently, Stubenrauch et al (Magno et al., 2009) and Sanchez-Dominguez et al. (Aubery et al., 2011) have reported systematic studies on the effects of addition of reactants to nonionic microemulsion systems. It was shown (Magno et al., 2009) that, depending on the aqueous nature of nonionic microemulsion systems, and the salting-in or salting-out effect of the additives, both increase or decrease on the water solubilization could be obtained. The same group studied the effects of different salts on the water solubilization of ionic microemulsions of the system aqueous phase / AOT/butanol /decane (Stubenrauch et al., 2008). They found that depending on the type of precursor (salts of Pt, Bi, or Pb) or the reducing agent (NaBH4), different behaviors can be obtained, and it was necessary to add different amounts of SDS and 1-butanol in order to keep both the w/o nature of the microemulsion droplets as well as their size (which was only assessed theoretically based on microemulsion composition).

In the studies by our group on the effects of addition of precursor salts and precipitating agent to the non-ionic microemulsion system aqueous solution / Synperonic® 13/6.5 / isooctane (Aubery et al., 2011), several factors were taken into account: phase behavior (pseudoternary phase diagrams at constant temperature), dynamicity (presence or absence of percolation in W/O structures, or formation of bicontinuous microemulsions), and droplet size (DLS). It was possible to obtain w/o microemulsions at a wide range of overlapping compositions for both precursor salts and precipitating agent. In fact, the microemulsion regions were considerably enlarged upon addition of precursors and precipitating agent; this behavior is contrary to what is typically obtained with ionic systems which have their microemulsion region reduced with addition of salts. It was difficult to obtain both type of microemulsions in either a non-percolated or percolated state; this was characterized extensively by conductivity, FT PGSE NMR and hydrophilic dye diffusion studies. When pseudobinary phase diagrams as a function of temperature were carried out, there were some compositions and temperatures at which both precursor salts and precipitating agent microemulsions were either percolated, not percolated or bicontinuous.

## **2.4 Dynamic processes**

Among the dynamic processes in microemulsions, interactions of droplets components and droplet- droplet interactions must be taken into account (Fletcher et al, 1987; Fletcher & Horsup, 1992; Moulik & Paul, 1998). Concerning interactions of droplets components in a W/O nonionic microemulsion, a schematic representation is depicted in Figure 4. The example concerns an aqueous droplet stabilized by a mixture of surfactant and cosurfactant

THLB (hydrophilic-lipophilic balanced temperature or phase inversion temperature) of the systems (Aramaki et al., 2001; Kunieda et al., 1995; Shinoda, 1968). The use of ionic surfactants may have some drawbacks, as usually the aqueous phase uptake of ionic microemulsions is reduced in the presence of precursor salts due to screening effects, and hence microemulsion regions become smaller (Liu et al., 2000; Gianakas et al., 2006). Additionally, complex species could interfere with particle growth by adsorption to their surface, and contaminations of ceramic nanoparticles with the surfactant counterions aer possible. Often, the effect of addition of precursor salts or precipitating agent on the phase behavior and structure of microemulsion systems is underlooked. Generally two microemulsions with a fixed water/surfactant ratio are prepared without taking into account the influence the added salt has on the size and the structure of the water droplets. Recently, Stubenrauch et al (Magno et al., 2009) and Sanchez-Dominguez et al. (Aubery et al., 2011) have reported systematic studies on the effects of addition of reactants to nonionic microemulsion systems. It was shown (Magno et al., 2009) that, depending on the aqueous nature of nonionic microemulsion systems, and the salting-in or salting-out effect of the additives, both increase or decrease on the water solubilization could be obtained. The same group studied the effects of different salts on the water solubilization of ionic microemulsions of the system aqueous phase / AOT/butanol /decane (Stubenrauch et al., 2008). They found that depending on the type of precursor (salts of Pt, Bi, or Pb) or the reducing agent (NaBH4), different behaviors can be obtained, and it was necessary to add different amounts of SDS and 1-butanol in order to keep both the w/o nature of the microemulsion droplets as well as their size (which was only assessed theoretically based on

In the studies by our group on the effects of addition of precursor salts and precipitating agent to the non-ionic microemulsion system aqueous solution / Synperonic® 13/6.5 / isooctane (Aubery et al., 2011), several factors were taken into account: phase behavior (pseudoternary phase diagrams at constant temperature), dynamicity (presence or absence of percolation in W/O structures, or formation of bicontinuous microemulsions), and droplet size (DLS). It was possible to obtain w/o microemulsions at a wide range of overlapping compositions for both precursor salts and precipitating agent. In fact, the microemulsion regions were considerably enlarged upon addition of precursors and precipitating agent; this behavior is contrary to what is typically obtained with ionic systems which have their microemulsion region reduced with addition of salts. It was difficult to obtain both type of microemulsions in either a non-percolated or percolated state; this was characterized extensively by conductivity, FT PGSE NMR and hydrophilic dye diffusion studies. When pseudobinary phase diagrams as a function of temperature were carried out, there were some compositions and temperatures at which both precursor salts and precipitating agent microemulsions were either percolated, not percolated or bicontinuous.

Among the dynamic processes in microemulsions, interactions of droplets components and droplet- droplet interactions must be taken into account (Fletcher et al, 1987; Fletcher & Horsup, 1992; Moulik & Paul, 1998). Concerning interactions of droplets components in a W/O nonionic microemulsion, a schematic representation is depicted in Figure 4. The example concerns an aqueous droplet stabilized by a mixture of surfactant and cosurfactant

microemulsion composition).

**2.4 Dynamic processes** 

molecules. The aqueous domain is composed by bounded water (hydrating the surfactant/cosurfactant hydrophilic domains) and free water (forming the droplet core). The exchange process to reach microemulsion equilibria comprises: 1) exchange of water between the bounded and free state; 2) exchange of cosurfactants among the interfacial film, the continuous phase and the dispersed phase (depending on its solubility). If ionic species are solubilized in the aqueous solution, they exchange ions between the bounded and free water. The composition of the aqueous droplets, their concentration and the temperature are mayor factors defining further interactions between them.

Droplet-droplet interactions depend strongly on droplet concentration, solvent viscosity, temperature, rigidity or flexibility of interfacial layer, and interactions between surfactant tails (Capek, 2004; Lopez-Quintela, 2003). When water or oil droplet dispersions are present, the droplets continuously collide, break apart, aggregate and break apart giving rise to dynamic processes in microemulsions. These dynamic processes allows microemulsion droplets to continuously exchange their content in microsecond scales. The composition of the aqueous droplets, as explained above, has a great influence on droplet interactions. As an example, the interfacial layer plays an important role on the formation, stability and discrete nature of microemulsion droplets. The film rigidity has been observed to increase with the surfactant hydrocarbon chain, whereas it substantially decreases with cosurfactant addition. The surfactant packing capacity can be also affected by the ionic strength of the droplets (Aramaki et al., 2001; Kunieda et al., 1995). The increase of surfactant molecules in the layer is proportional to the rigidity of the micelles.

Fig. 4. Schematic representation of W/O microemulsion droplet. (Adapted with permission from Moulik, 1998).

Changes on microemulsion dynamics giving rise to structural transitions can be explained in terms of percolation. Figure 5 refers to a percolation process taking place in W/O microemulsions (Borkovec et al., 1988), as the oil to water ratio o is varied. As observed in Figure 5, at high oil concentration, the fraction of water in discrete droplets increases with water composition (decreasing oil concentration) up to a concentration, where it drastically decreases. This concentration is called Percolation Concentration CP. Although clusterization occurs below CP (low water concentrations), these clusters remain finite in size respect to the bulk solution. CP represents the concentration at which the first infinite cluster appears. Further increase on water concentration would lead to the disappearance of discrete water droplets to give rise to an increase of infinite water and oily domains, which

New Trends on the Synthesis

[C]C

[C]

**3.2 Reaction kinetics** 

out below:

experimentally.

of the alkane.

of Inorganic Nanoparticles Using Microemulsions as Confined Reaction Media 203

critical supersaturation

**microemulsion**

**homogeneous solution**

Fig. 6. Monomer concentration [C] as a function of time in microemulsions, compared to a homogeneous system. (Adapted from La Mer & Dinegar, 1950 and Schmidt, 1999).

time

particle growth

solubility (supersaturation) [C]s

nucleation

Although microemulsions as reaction media for the synthesis of inorganic nanoparticles have been extensively studied, the kinetics of these reactions is still not completely understood. As mentioned above, several types of nanoparticles have been synthesized using a variety of surfactant systems, and relationships between the nanoparticles characteristics and the microemulsion media are not straightforward due to the diversity of variables which can have an influence, and this may be closely related with complex kinetics. An effort to relate the surfactant media with the reaction kinetics was reviewed by Lopez-Quintela et al. (Lopez-Quintela et al., 2004), concerning both inorganic and organic syntheses in microemulsions. Few studies can be cited concerning the follow-up of reactions with time, due to the fast rate of microemulsion reactions. Some of these works are pointed

1. Bandyopadhyaya et al. (Bandyopadhyaya et al., 1997) have modelized CaCO3 formation in microemulsions by carbonation. A time-scale analysis was developed, resulting in a model of reaction kinetic that closely corresponded to results obtained

2. Chew et al. (Chew et al., 1990) have studied the effect of alkanes in the formation of AgBr particles in ionic W/O microemulsions (using AOT as surfactant), where the transmittance of the reactions were followed with time with UV-Vis and Stopped-Flow Spectrophotometry. They have found an increase on reaction rate with the chain lenght

3. Curri et al. (Curri et al., 2000) studied the role of cosurfactant on the synthesis of CdS nanoclusters, using CTAB as surfactant. Stopped-Flow Spectrophotometry was used in order to compare a reaction using CTAB plus cosurfactant and other carried out using AOT. They have summarized two different cosurfactant effects: the influence of the

are characteristic of bicontinuous microemulsions. Water percolation can also be induced by temperature, and is defined as the percolation temperature TP.

Fig. 5. Schematic representation of structural regimes of microemulsions caused by water percolation as a function of the relative amount of oil. Reprinted (adapted) with permission from Borkovec M., Eicke H.-F., Hammerich H., & Das Gupta, B. (1988) *J. Phys. Chem.,* 92, 1, pp. (206-211). Copyrigh (1988) American Chemical Society

## **3. The microemulsion reaction method: introduction and generalities**

A brief description of the synthesis of nanoparticles in W/O microemulsions has already been given in Section 1, along with an explanatory figure (Figure 1). Colloidal nanoparticle formation is a complex process, which includes nucleation and growth steps -giving rise to nanoparticle formation- as well as eventual coagulation and flocculation.

## **3.1 Mechanism of nanoparticle formation**

A model of particle precipitation in a homogeneous aqueous medium has been proposed by La Mer (La Mer & Dinegar, 1950). The model involves particle nucleation at short times. As soon as monomer formation takes place due to chemical reaction, its concentration increases up to the point of spontaneous nucleation, which occurs over a critical supersaturation concentration [C]C. Afterwards, growth takes place (Figure 6). The growing step is mainly controlled by the diffusion of monomers in solution (C) onto the particles surface. Thus, C reaches a maximum and afterwards it begins to decrease. This decrease in monomer concentration is due to the growth of the particles by diffusion. In microemulsions, the number of nucleated sites is expected to be higher, comparing to homogeneous reactions, as illustrated in Figure 6. On the other hand, the diffusion controlled particle growth should occur at lower rate. Another model is based on the thermodynamic stabilization of the particles. In this model the particles are thermodynamically stabilized by the surfactant. The size of the particles remains constant when the precursor concentration and the size of the aqueous droplets vary. Nucleation occurs continuously during the nanoparticle formation.

Fig. 6. Monomer concentration [C] as a function of time in microemulsions, compared to a homogeneous system. (Adapted from La Mer & Dinegar, 1950 and Schmidt, 1999).

## **3.2 Reaction kinetics**

202 Smart Nanoparticles Technology

are characteristic of bicontinuous microemulsions. Water percolation can also be induced by

Fig. 5. Schematic representation of structural regimes of microemulsions caused by water percolation as a function of the relative amount of oil. Reprinted (adapted) with permission from Borkovec M., Eicke H.-F., Hammerich H., & Das Gupta, B. (1988) *J. Phys. Chem.,* 92, 1,

A brief description of the synthesis of nanoparticles in W/O microemulsions has already been given in Section 1, along with an explanatory figure (Figure 1). Colloidal nanoparticle formation is a complex process, which includes nucleation and growth steps -giving rise to

A model of particle precipitation in a homogeneous aqueous medium has been proposed by La Mer (La Mer & Dinegar, 1950). The model involves particle nucleation at short times. As soon as monomer formation takes place due to chemical reaction, its concentration increases up to the point of spontaneous nucleation, which occurs over a critical supersaturation concentration [C]C. Afterwards, growth takes place (Figure 6). The growing step is mainly controlled by the diffusion of monomers in solution (C) onto the particles surface. Thus, C reaches a maximum and afterwards it begins to decrease. This decrease in monomer concentration is due to the growth of the particles by diffusion. In microemulsions, the number of nucleated sites is expected to be higher, comparing to homogeneous reactions, as illustrated in Figure 6. On the other hand, the diffusion controlled particle growth should occur at lower rate. Another model is based on the thermodynamic stabilization of the particles. In this model the particles are thermodynamically stabilized by the surfactant. The size of the particles remains constant when the precursor concentration and the size of the aqueous droplets vary. Nucleation

**3. The microemulsion reaction method: introduction and generalities** 

nanoparticle formation- as well as eventual coagulation and flocculation.

temperature, and is defined as the percolation temperature TP.

pp. (206-211). Copyrigh (1988) American Chemical Society

occurs continuously during the nanoparticle formation.

**3.1 Mechanism of nanoparticle formation** 

Although microemulsions as reaction media for the synthesis of inorganic nanoparticles have been extensively studied, the kinetics of these reactions is still not completely understood. As mentioned above, several types of nanoparticles have been synthesized using a variety of surfactant systems, and relationships between the nanoparticles characteristics and the microemulsion media are not straightforward due to the diversity of variables which can have an influence, and this may be closely related with complex kinetics. An effort to relate the surfactant media with the reaction kinetics was reviewed by Lopez-Quintela et al. (Lopez-Quintela et al., 2004), concerning both inorganic and organic syntheses in microemulsions. Few studies can be cited concerning the follow-up of reactions with time, due to the fast rate of microemulsion reactions. Some of these works are pointed out below:


New Trends on the Synthesis

of Inorganic Nanoparticles Using Microemulsions as Confined Reaction Media 205

and rigidity (Cason et al., 2001). This effect has been observed to produce micellar exchange

*Electrolytes*: Some studies reveal the possible dependence of nanoparticle shape with electrolyte addition (Filankembo et al., 2003). Pileni (Pileni, 2003) has postulated that the selective ion or molecule adsorption over nanocrystal layers can affect their growth in certain directions, which could explain the apparent preference on certain particle shape.

*Microemulsion structure*: Some studies have claimed about the nanoparticle shape partial dependence on microemulsion structure, where the microemulsion media acts as a template. A particular example is the work carried out by Pileni (Pileni, 2001) on the preparation of Copper nanoparticles from microemulsions by varying the internal structure. Spherical water droplets resulted in spherical particles, water cilinders resulted in cylindrical copper nanocrystals (with spherical particles) and a mixture of W/O microemulsion with lamellar phase resulted in a mixture of particle shape such as spheres, cylinders, etc. It was found that the template was not the only parameter which controls the shape of nanocrystals. There are examples of nonexistent correlation between the microemulsion structure and the nanoparticles obtained, which supports the nanoparticle

Even though there is a diversity of studies carried out in order to relate nanoparticle characteristics with microemulsion properties, there is a gap in the effects of microemulsion dynamic behavior on nanoparticle characteristics, as systematic studies in this direction are scarce. The transport and micellar dynamics influence to some extent the nanoparticle formation, and it is important to take this into account in order to understand the basics of nanoparticles synthesis by this route. This type of studies may give rise to improvements on

**4. Recent advances in the use of microemulsions as confined reaction media** 

There have been a number of advances in different aspects of the synthesis of nanoparticles in microemulsions over the last four years. The main ones are: the use of other types of microemulsions for synthesis (O/W and bicontinuous microemulsions), the preparation of more complex architectures (core/shell and multishell, hybrid nanocrystals), the synthesis of more complex ceramics (spinels, perovskites, etc), modeling of reactions in microemulsions, and novel approaches for the separation of nanoparticles from the reaction

mixtures. The most outstanding examples of each of these aspects are given below.

**4.1 The use of other types of microemulsions for inorganic nanoparticle synthesis**  One of the main drawbacks of the technique reviewed so far (synthesis in W/O microemulsions) and the main reason why it has not been generally accepted for production at the industrial scale, is the fact that these microemulsions employ large quantities of organic solvent, as well as its limited production capacity, since this is restricted to the amount of aqueous phase solubilized and the concentration of precursor which often cannot be that high due to interactions with the surfactant, as discussed in Section 2.3. Some research groups have been working in new approaches to overcome these drawbacks.

decrease and, consequently, smaller particles are obtained.

shape dependence on electrolyte adsorption (Chen & Lin, 2001).

controlled nanoparticle characteristics.

**for the synthesis of inorganic nanoparticles** 

surfactant film flexibility on particle growth and the particles stabilization in solution, determined by the adsorption of cosurfactant onto the particle surface.

4. Lopez-Quintela et al. (De Dios et al., 2009) simulated the kinetics of nanoparticles formation in microemulsions. Simulations were carried out by comparing Ag, Ag-Au and Au formation with experimental data reported by Destrée and Nagy. (Destrée & Nagy, 2006). The detailed comprehension of the kinetics taking place in microemulsion reactions is limited by the experimental data in this direction. Hence, systematic studies focused on reaction rates are greatly encouraged in order to advance in this field.

## **3.3 Parameters influencing on nanoparticle synthesis**

Although complete control of particle characteristics is still far from clear and direct, some results on this field can be pointed out as shown below.

*Aqueous solution concentration*. It have been described in several publications the particle size dependency with water:surfactant molar ratio (w0). In general, it has been observed that, as increasing w0, an increase on particle size is observed (Pileni, 1997; Lopez-Quintela, 2003). However, Cason et al. (Cason et al., 2001) have found that, with different w0, it was possible to obtain constant particle size if the reaction time increases for the synthesis to get completed. They proposed that the growth of the particles is affected by w0. It was considered that for low w0 values, the aqueous solution is not enough to completely hydrate the polar groups of the surfactant and the counterion. As a consequence, the film rigidity is higher compared to higher w0 values. This influences on the micellar exchange and, as a consequence, the growth rate decreases. Increasing w0, the micelle rigidity decreases generating an increase in the growth rate up to a certain concentration, where further increase in w0 simply causes reagent dilution, which causes a decrease in the growth rate. Some studies have indicated a decrease on particle size with w0 (Bagwe & Khilar, 1997).

*Reagent concentration*: Particle size have been determined to be directly dependent on reagent concentration (Lopez-Quintela, 2003). An example is the work carried out by Destrée & Nagy (Destrée & Nagy, 2006). They have synthesized Pt nanoparticles, using different concentrations of K2PtCl4. An increase on particle diameter from 2 to 12 nm was obtained, by increasing the concentration of the precursor. On the other hand, an increase on the precipitating/precursor ratio generally causes a decrease on particle size (Lisiecki & Pileni, 2003). It is thought that increasing precipitating agent concentration, particle nucleation can be favored in a higher extent, which further grow simultaneously, resulting in particles with lower size and polydispersity.

*Surfactant and cosurfactant:* Studies in order to determine the effect of nonionic hydrophilic and lypophilic surfactant groups have been developed. As the lypophilic chain of the surfactant is longer, smaller particles are obtained due to the increased micellar rigidity Generally, the addition of cosurfactant causes an increase micellar exchange, due to the decrease in the interfacial film rigidity. It is thought that the increase in microemulsion droplet size is counteracted with the increase on surfactant film curvature, generating smaller particles than without cosurfactant (Lopez-Quintela et al., 2004).

*Solvent*: Some studies have shown that low weight oil molecules, with low molecular volumes, can penetrate in the sufactant hydrocarbon chains, increasing the film curvature

4. Lopez-Quintela et al. (De Dios et al., 2009) simulated the kinetics of nanoparticles formation in microemulsions. Simulations were carried out by comparing Ag, Ag-Au and Au formation with experimental data reported by Destrée and Nagy. (Destrée & Nagy, 2006). The detailed comprehension of the kinetics taking place in microemulsion reactions is limited by the experimental data in this direction. Hence, systematic studies focused on reaction rates are greatly encouraged in order to advance in this field.

Although complete control of particle characteristics is still far from clear and direct, some

*Aqueous solution concentration*. It have been described in several publications the particle size dependency with water:surfactant molar ratio (w0). In general, it has been observed that, as increasing w0, an increase on particle size is observed (Pileni, 1997; Lopez-Quintela, 2003). However, Cason et al. (Cason et al., 2001) have found that, with different w0, it was possible to obtain constant particle size if the reaction time increases for the synthesis to get completed. They proposed that the growth of the particles is affected by w0. It was considered that for low w0 values, the aqueous solution is not enough to completely hydrate the polar groups of the surfactant and the counterion. As a consequence, the film rigidity is higher compared to higher w0 values. This influences on the micellar exchange and, as a consequence, the growth rate decreases. Increasing w0, the micelle rigidity decreases generating an increase in the growth rate up to a certain concentration, where further increase in w0 simply causes reagent dilution, which causes a decrease in the growth rate. Some studies have indicated a decrease on particle size with w0 (Bagwe & Khilar, 1997).

*Reagent concentration*: Particle size have been determined to be directly dependent on reagent concentration (Lopez-Quintela, 2003). An example is the work carried out by Destrée & Nagy (Destrée & Nagy, 2006). They have synthesized Pt nanoparticles, using different concentrations of K2PtCl4. An increase on particle diameter from 2 to 12 nm was obtained, by increasing the concentration of the precursor. On the other hand, an increase on the precipitating/precursor ratio generally causes a decrease on particle size (Lisiecki & Pileni, 2003). It is thought that increasing precipitating agent concentration, particle nucleation can be favored in a higher extent, which further grow simultaneously, resulting in particles with

*Surfactant and cosurfactant:* Studies in order to determine the effect of nonionic hydrophilic and lypophilic surfactant groups have been developed. As the lypophilic chain of the surfactant is longer, smaller particles are obtained due to the increased micellar rigidity Generally, the addition of cosurfactant causes an increase micellar exchange, due to the decrease in the interfacial film rigidity. It is thought that the increase in microemulsion droplet size is counteracted with the increase on surfactant film curvature, generating

*Solvent*: Some studies have shown that low weight oil molecules, with low molecular volumes, can penetrate in the sufactant hydrocarbon chains, increasing the film curvature

smaller particles than without cosurfactant (Lopez-Quintela et al., 2004).

determined by the adsorption of cosurfactant onto the particle surface.

**3.3 Parameters influencing on nanoparticle synthesis** 

results on this field can be pointed out as shown below.

lower size and polydispersity.

surfactant film flexibility on particle growth and the particles stabilization in solution,

and rigidity (Cason et al., 2001). This effect has been observed to produce micellar exchange decrease and, consequently, smaller particles are obtained.

*Electrolytes*: Some studies reveal the possible dependence of nanoparticle shape with electrolyte addition (Filankembo et al., 2003). Pileni (Pileni, 2003) has postulated that the selective ion or molecule adsorption over nanocrystal layers can affect their growth in certain directions, which could explain the apparent preference on certain particle shape.

*Microemulsion structure*: Some studies have claimed about the nanoparticle shape partial dependence on microemulsion structure, where the microemulsion media acts as a template. A particular example is the work carried out by Pileni (Pileni, 2001) on the preparation of Copper nanoparticles from microemulsions by varying the internal structure. Spherical water droplets resulted in spherical particles, water cilinders resulted in cylindrical copper nanocrystals (with spherical particles) and a mixture of W/O microemulsion with lamellar phase resulted in a mixture of particle shape such as spheres, cylinders, etc. It was found that the template was not the only parameter which controls the shape of nanocrystals. There are examples of nonexistent correlation between the microemulsion structure and the nanoparticles obtained, which supports the nanoparticle shape dependence on electrolyte adsorption (Chen & Lin, 2001).

Even though there is a diversity of studies carried out in order to relate nanoparticle characteristics with microemulsion properties, there is a gap in the effects of microemulsion dynamic behavior on nanoparticle characteristics, as systematic studies in this direction are scarce. The transport and micellar dynamics influence to some extent the nanoparticle formation, and it is important to take this into account in order to understand the basics of nanoparticles synthesis by this route. This type of studies may give rise to improvements on controlled nanoparticle characteristics.

## **4. Recent advances in the use of microemulsions as confined reaction media for the synthesis of inorganic nanoparticles**

There have been a number of advances in different aspects of the synthesis of nanoparticles in microemulsions over the last four years. The main ones are: the use of other types of microemulsions for synthesis (O/W and bicontinuous microemulsions), the preparation of more complex architectures (core/shell and multishell, hybrid nanocrystals), the synthesis of more complex ceramics (spinels, perovskites, etc), modeling of reactions in microemulsions, and novel approaches for the separation of nanoparticles from the reaction mixtures. The most outstanding examples of each of these aspects are given below.

## **4.1 The use of other types of microemulsions for inorganic nanoparticle synthesis**

One of the main drawbacks of the technique reviewed so far (synthesis in W/O microemulsions) and the main reason why it has not been generally accepted for production at the industrial scale, is the fact that these microemulsions employ large quantities of organic solvent, as well as its limited production capacity, since this is restricted to the amount of aqueous phase solubilized and the concentration of precursor which often cannot be that high due to interactions with the surfactant, as discussed in Section 2.3. Some research groups have been working in new approaches to overcome these drawbacks.

New Trends on the Synthesis

of Inorganic Nanoparticles Using Microemulsions as Confined Reaction Media 207

It should be pointed out that in all of these examples, only one microemulsion is used for synthesis, as opposed to what is typically needed with the W/O method (two microemulsions, one bearing the precursors and another one the precipitating agent in the aqueous phase). Since most precipitating agents are water-soluble, it means that it can be added directly to the microemulsion without affecting its O/W structure, and hence only one microemulsion, containing the organometallic precursor is prepared. Hence, the mechanism occurring in this approach is most likely different; possibly, it is an interfacial reaction. Modelization studies in conjunction with kinetic experiments need to be carried out in order to clarify this point. Considering these results, the perspectives of this novel O/W microemulsion reaction approach are very positive, and should complement the W/O microemulsion method, offering a greener alternative. Finally, it must be highlighted that the typical metal loading in the microemulsions reported, and hence the typical production capacity ranges from 2 to 5 grams of nanoparticles per kg of microemulsion, which is comparable and in some cases

An interesting approach to boost the metal loading and hence the nanoparticle production capacity of microemulsions is the use of bicontinuous microemulsions. Lopez et al. (Esquivel et al., 2007; Loo et al., 2008) have reported this approach for the synthesis of magnetic nanoparticles. A microemulsion system based on cationic surfactants was used for the synthesis of a mixture of maghemite/magnetite nanoparticles, using bicontinuous microemulsions at 80°C, with 30-40 wt% of aqueous phase. They obtained small nanoparticles (8 nm) with a narrow size distribution, a nanocrystalline structure and superparamagnetic behavior. Furthermore, the yield of the reactions was as high as 1.16 g of product per 100 grams of microemulsion, which is rather high compared to what can be obtained in most w/o microemulsion systems (0.05 - 0.2 grams per 100 g microemulsion).

Fig. 8. TEM micrographs and related particle size distribution histograms of nanoparticles prepared in O/W microemulsions: (a) Pt, (b) Pd, (c) Rh and (d) CeO2. Scale bar: 50 nm, except d (10 nm) and inset of d (5 nm). Reproduced with permission (Sanchez-Dominguez et al., 2009).

superior to typical metal loadings achieved in W/O microemulsions.

**4.1.2 Synthesis in bicontinuous microemulsions** 

## **4.1.1 Synthesis in oil-in-water microemulsions**

Our research group in collaboration with the group of Boutonnet, has developed a novel and straightforward approach based on O/W microemulsions (Sanchez-Dominguez et al., 2009). From a practical and environmental point of view, the possibility of preparing inorganic nanoparticles using O/W instead of W/O microemulsions may be highly advantageous, since the major (continuous) phase is water. The method consists in the use of organometallic precursors, dissolved in nanometer scale oil droplets of O/W microemulsions (Figure 7), and stabilized by a monolayer of hydrophilic surfactant. The first work reported as a proof of concept the synthesis of metallic (Pt, Pd, and Rh) as well as metal oxide (CeO2) nanoparticles (Sanchez-Dominguez et al., 2009). Small (around 3 nm), nanocrystalline materials with a narrow size distribution were obtained (Figure 8).

It was followed by the synthesis of the following mesoporous nanocrystalline oxides: CeO2, ZrO2, Ce0.5Zr0.5O2, and TiO2 (Sanchez-Dominguez et al., 2010). Small particle size (3 nm), and high specific surface area (200-380 m2 g-1) was obtained for all materials. Nanocrystalline cubic CeO2 and Ce0.5Zr0.5O2 were obtained under soft conditions (35°C). The materials were evaluated as catalyst supports in the CO oxidation reaction by doping them with Au (2 wt%, impregnation technique). The resulting catalysts showed a high Au dispersion (HRTEM/EDX). These materials showed a good activity in CO oxidation at low temperature (T50 of 44°C for TiO2). This study demonstrates the feasibility of this approach for the preparation of highly active catalysts.

In a more recent study by the same group (Tiseanu et al., 2011), Eu-doped luminescent CeO2 nanocrystals were prepared by the same method. Several characterization techniques *(* X-ray diffraction, RAMAN spectroscopy, UV-Vis diffuse-reflectance, FTIR as well as time-resolved photoluminescence spectroscopy) were used to characterize the nanocrystals, and it was shown that there was a surface enrichment of Eu3+, which diffused progressively to the inner Ceria sites upon calcination. Under excitation into the UV and visible spectral range, the calcined europium doped ceria nanocrystals display a variable emission spanning the orange-red wavelengths. A remarkable result was that the surface area of the powders remained as high as 120 m2 g-1 even after calcination at 1000°C.

Fig. 7. TEM micrographs and related particle size distribution histograms of nanoparticles prepared in O/W microemulsions: (a) Pt, (b) Pd, (c) Rh and (d) CeO2. Scale bar: 50 nm, except d (10 nm) and inset of d (5 nm). Reproduced with permission (Sanchez-Dominguez et al., 2009).

Our research group in collaboration with the group of Boutonnet, has developed a novel and straightforward approach based on O/W microemulsions (Sanchez-Dominguez et al., 2009). From a practical and environmental point of view, the possibility of preparing inorganic nanoparticles using O/W instead of W/O microemulsions may be highly advantageous, since the major (continuous) phase is water. The method consists in the use of organometallic precursors, dissolved in nanometer scale oil droplets of O/W microemulsions (Figure 7), and stabilized by a monolayer of hydrophilic surfactant. The first work reported as a proof of concept the synthesis of metallic (Pt, Pd, and Rh) as well as metal oxide (CeO2) nanoparticles (Sanchez-Dominguez et al., 2009). Small (around 3 nm),

nanocrystalline materials with a narrow size distribution were obtained (Figure 8).

It was followed by the synthesis of the following mesoporous nanocrystalline oxides: CeO2, ZrO2, Ce0.5Zr0.5O2, and TiO2 (Sanchez-Dominguez et al., 2010). Small particle size (3 nm), and high specific surface area (200-380 m2 g-1) was obtained for all materials. Nanocrystalline cubic CeO2 and Ce0.5Zr0.5O2 were obtained under soft conditions (35°C). The materials were evaluated as catalyst supports in the CO oxidation reaction by doping them with Au (2 wt%, impregnation technique). The resulting catalysts showed a high Au dispersion (HRTEM/EDX). These materials showed a good activity in CO oxidation at low temperature (T50 of 44°C for TiO2). This study demonstrates the feasibility of this approach

In a more recent study by the same group (Tiseanu et al., 2011), Eu-doped luminescent CeO2 nanocrystals were prepared by the same method. Several characterization techniques *(* X-ray diffraction, RAMAN spectroscopy, UV-Vis diffuse-reflectance, FTIR as well as time-resolved photoluminescence spectroscopy) were used to characterize the nanocrystals, and it was shown that there was a surface enrichment of Eu3+, which diffused progressively to the inner Ceria sites upon calcination. Under excitation into the UV and visible spectral range, the calcined europium doped ceria nanocrystals display a variable emission spanning the orange-red wavelengths. A remarkable result was that the surface area of the powders

Fig. 7. TEM micrographs and related particle size distribution histograms of nanoparticles prepared in O/W microemulsions: (a) Pt, (b) Pd, (c) Rh and (d) CeO2. Scale bar: 50 nm,

**4.1.1 Synthesis in oil-in-water microemulsions** 

for the preparation of highly active catalysts.

remained as high as 120 m2 g-1 even after calcination at 1000°C.

except d (10 nm) and inset of d (5 nm). Reproduced with permission

(Sanchez-Dominguez et al., 2009).

It should be pointed out that in all of these examples, only one microemulsion is used for synthesis, as opposed to what is typically needed with the W/O method (two microemulsions, one bearing the precursors and another one the precipitating agent in the aqueous phase). Since most precipitating agents are water-soluble, it means that it can be added directly to the microemulsion without affecting its O/W structure, and hence only one microemulsion, containing the organometallic precursor is prepared. Hence, the mechanism occurring in this approach is most likely different; possibly, it is an interfacial reaction. Modelization studies in conjunction with kinetic experiments need to be carried out in order to clarify this point. Considering these results, the perspectives of this novel O/W microemulsion reaction approach are very positive, and should complement the W/O microemulsion method, offering a greener alternative. Finally, it must be highlighted that the typical metal loading in the microemulsions reported, and hence the typical production capacity ranges from 2 to 5 grams of nanoparticles per kg of microemulsion, which is comparable and in some cases superior to typical metal loadings achieved in W/O microemulsions.

## **4.1.2 Synthesis in bicontinuous microemulsions**

An interesting approach to boost the metal loading and hence the nanoparticle production capacity of microemulsions is the use of bicontinuous microemulsions. Lopez et al. (Esquivel et al., 2007; Loo et al., 2008) have reported this approach for the synthesis of magnetic nanoparticles. A microemulsion system based on cationic surfactants was used for the synthesis of a mixture of maghemite/magnetite nanoparticles, using bicontinuous microemulsions at 80°C, with 30-40 wt% of aqueous phase. They obtained small nanoparticles (8 nm) with a narrow size distribution, a nanocrystalline structure and superparamagnetic behavior. Furthermore, the yield of the reactions was as high as 1.16 g of product per 100 grams of microemulsion, which is rather high compared to what can be obtained in most w/o microemulsion systems (0.05 - 0.2 grams per 100 g microemulsion).

Fig. 8. TEM micrographs and related particle size distribution histograms of nanoparticles prepared in O/W microemulsions: (a) Pt, (b) Pd, (c) Rh and (d) CeO2. Scale bar: 50 nm, except d (10 nm) and inset of d (5 nm). Reproduced with permission (Sanchez-Dominguez et al., 2009).

New Trends on the Synthesis

shell particles.

of Inorganic Nanoparticles Using Microemulsions as Confined Reaction Media 209

how this silica shell is deposited onto the core; this sophisticated approach is probably the reason for the high control achieved. The core nanoparticles are usually functionalized for one or two purposes: one is in order to be very well dispersed in one of the microemulsion phases (the oily or the aqueous phase), the other is for very controlled deposition of silica via hydrolytic copolymerization with silanized molecules such as (3-aminopropyl)triethoxysilane (APTES), which were covalently linked to the core particles. By this approach, uniform CdTe@silica nanoparticles with a regular core – shell structure, 48±3 nm in diameter were obtained by Dong H. et al. (Dong H. et al., 2009). In their work, the initial core CdTe particles, synthesized by a hydrothermal method, were functionalized with thioglycolic acid, so they could be reacted with APTES and then dispersed in the aqueous phase of the microemulsion. The silica precursor, TEOS, was dissolved in the oily phase of the microemulsion (cyclohexane and octanol), and the silica shell was then formed by addition of ammonia. In the work by Dong B. et al (Dong B. et al., 2009), on the other hand, the core ZnS:Mn particles were functionalized with oleic acid and hence dispersed in the oil phase of the microemulsion, and the silica layer was deposited by reacting TEOS with ammonia in the W/O microemulsion containing the core particles dispersed in the oil. Figure 9 shows TEM results of these core-

Fewer examples deal with the formation of core-shell nanoparticles in which both the core and the shell have been synthesized in a W/O microemulsion (Chung et al., 2011; Takenaka et al., 2007). Takenaka et al. prepared Ni nanoparticles in a W/O microemulsion, and afterwards TEOS and ammonia were added in order to form the silica layer. Core-shell nanoparticles with 20-50 nm diameter and a Ni shell (5 nm) were formed. For comparison, silica nanoparticles were prepared also in W/O microemulsions but the Ni nanoparticles were prepared by impregnation of these silica nanoparticles. Their catalytic activity in the partial oxidation of methane reaction was evaluated, and the core-shell nanoparticles had a better performance than the impregnated ones (Takenaka et al., 2007). On the other hand, Chung et al. prepared silica nanoparticles coated with a thin layer of CeO2, and the material was also prepared in W/O microemulsions in a two-step procedure (Chung et al., 2011). This reaction turned out to be challenging as the formation of CeO2 shell was competing with bulk precipitation. The problem was overcome by coupling two strategies:

Fig. 9. TEM image of ZnS:Mn@silica nanoparticles with a core – shell structure.

(Reproduced with permission, Dong B. et al. 2009).

The same research group reported recently the synthesis of silver nanoparticles by the same approach (Reyes et al., 2010; Sosa et al., 2010), by using a microemulsion system based on AOT/SDS as the surfactant system and toluene as the oil. Depending on the surfactant: oil ratio, the authors found the formation of only globular nanoparticles or a mixture of interconnected, worm-like structures plus globular nanoparticles. The reaction yields for these materials was also remarkably high (up to 1.4 g of silver nanoparticles per 100 g of microemulsion). In all of these works, only one microemulsion was necessary for nanoparticle preparation, as the precipitating agent was added directly, as an aqueous solution, to the microemulsion containing the metallic precursors. This aspect also contributes to the greener quality of this approach as compared to the traditional W/O microemulsion reaction method.

## **4.1.3 Synthesis in microemulsions with an optimized aqueous phase uptake**

In the work carried out by our group concerning a nonionic system (Aubery et al., 2011), large microemulsion regions were obtained when the reactants were incorporated, as mentioned in Section 2.3. Thanks to this high aqueous phase uptake and the overlap of microemulsion regions for both precursor salts and precipitating agent, synthesis of Mn-Zn ferrite nanoparticles could be carried out using a wide range of compositions. Futhermore, different scenarios were available for nanoparticle synthesis: W/O non-percolated, W/O percolated, and bicontinuous microemulsions. Differences were observed in the characteristics of the synthesized nanoparticles depending on the type of microemulsions used, and in all cases spinel nanocrystalline particles with superparamagnetic properties were obtained, directly in the microemulsion, without the need for calcination. The aqueous phase content ranged from 5 wt% to 50 wt%, which represents a boost in the production capacity. This study should encourage further research into optimized non-ionic microemulsion systems, since although the presence of salts affects their phase behavior, it does so in such a way that aqueous solubilization can be significantly increased at a certain temperature, which can be investigated by phase behavior studies.

## **4.2 Preparation of more complex architectures**

In this regard, most of the studies concern core-shell studies, although some other structures include multiple core-shell particles, hollow spheres and nanowires and nanorods.

## **4.2.1 Core/shell nanoparticles**

A large majority of the core-shell nanoparticles synthesized in W/O microemulsions contain silica, usually as the shell material. In the last few years, the W/O microemulsion approach has been gaining popularity over the well-known adaptation of the Stöber method (Nann & Mulvaney, 2004), for coating a diversity of nanoparticles with a silica shell. This is because it has been observed that the microemulsion method results in a better shell thickness control (Dong H, 2009), as compared to the adapted Stöber method, which is based on the sol-gel technique. It must be pointed out that in the majority of the studies, the core material was synthesized in a previous step, by a different method, usually hydrothermal or solvothermal techniques (Dong B. et al., 2009; Dong H et al., 2009; Qian et al., 2009; Vogt et al., 2010; Wang J. et al., 2010). Nevertheless, a very interesting point from these investigations is the strategy on

The same research group reported recently the synthesis of silver nanoparticles by the same approach (Reyes et al., 2010; Sosa et al., 2010), by using a microemulsion system based on AOT/SDS as the surfactant system and toluene as the oil. Depending on the surfactant: oil ratio, the authors found the formation of only globular nanoparticles or a mixture of interconnected, worm-like structures plus globular nanoparticles. The reaction yields for these materials was also remarkably high (up to 1.4 g of silver nanoparticles per 100 g of microemulsion). In all of these works, only one microemulsion was necessary for nanoparticle preparation, as the precipitating agent was added directly, as an aqueous solution, to the microemulsion containing the metallic precursors. This aspect also contributes to the greener quality of this approach as compared to the traditional W/O

**4.1.3 Synthesis in microemulsions with an optimized aqueous phase uptake** 

temperature, which can be investigated by phase behavior studies.

**4.2 Preparation of more complex architectures** 

**4.2.1 Core/shell nanoparticles** 

In the work carried out by our group concerning a nonionic system (Aubery et al., 2011), large microemulsion regions were obtained when the reactants were incorporated, as mentioned in Section 2.3. Thanks to this high aqueous phase uptake and the overlap of microemulsion regions for both precursor salts and precipitating agent, synthesis of Mn-Zn ferrite nanoparticles could be carried out using a wide range of compositions. Futhermore, different scenarios were available for nanoparticle synthesis: W/O non-percolated, W/O percolated, and bicontinuous microemulsions. Differences were observed in the characteristics of the synthesized nanoparticles depending on the type of microemulsions used, and in all cases spinel nanocrystalline particles with superparamagnetic properties were obtained, directly in the microemulsion, without the need for calcination. The aqueous phase content ranged from 5 wt% to 50 wt%, which represents a boost in the production capacity. This study should encourage further research into optimized non-ionic microemulsion systems, since although the presence of salts affects their phase behavior, it does so in such a way that aqueous solubilization can be significantly increased at a certain

In this regard, most of the studies concern core-shell studies, although some other structures

A large majority of the core-shell nanoparticles synthesized in W/O microemulsions contain silica, usually as the shell material. In the last few years, the W/O microemulsion approach has been gaining popularity over the well-known adaptation of the Stöber method (Nann & Mulvaney, 2004), for coating a diversity of nanoparticles with a silica shell. This is because it has been observed that the microemulsion method results in a better shell thickness control (Dong H, 2009), as compared to the adapted Stöber method, which is based on the sol-gel technique. It must be pointed out that in the majority of the studies, the core material was synthesized in a previous step, by a different method, usually hydrothermal or solvothermal techniques (Dong B. et al., 2009; Dong H et al., 2009; Qian et al., 2009; Vogt et al., 2010; Wang J. et al., 2010). Nevertheless, a very interesting point from these investigations is the strategy on

include multiple core-shell particles, hollow spheres and nanowires and nanorods.

microemulsion reaction method.

how this silica shell is deposited onto the core; this sophisticated approach is probably the reason for the high control achieved. The core nanoparticles are usually functionalized for one or two purposes: one is in order to be very well dispersed in one of the microemulsion phases (the oily or the aqueous phase), the other is for very controlled deposition of silica via hydrolytic copolymerization with silanized molecules such as (3-aminopropyl)triethoxysilane (APTES), which were covalently linked to the core particles. By this approach, uniform CdTe@silica nanoparticles with a regular core – shell structure, 48±3 nm in diameter were obtained by Dong H. et al. (Dong H. et al., 2009). In their work, the initial core CdTe particles, synthesized by a hydrothermal method, were functionalized with thioglycolic acid, so they could be reacted with APTES and then dispersed in the aqueous phase of the microemulsion. The silica precursor, TEOS, was dissolved in the oily phase of the microemulsion (cyclohexane and octanol), and the silica shell was then formed by addition of ammonia. In the work by Dong B. et al (Dong B. et al., 2009), on the other hand, the core ZnS:Mn particles were functionalized with oleic acid and hence dispersed in the oil phase of the microemulsion, and the silica layer was deposited by reacting TEOS with ammonia in the W/O microemulsion containing the core particles dispersed in the oil. Figure 9 shows TEM results of these coreshell particles.

Fewer examples deal with the formation of core-shell nanoparticles in which both the core and the shell have been synthesized in a W/O microemulsion (Chung et al., 2011; Takenaka et al., 2007). Takenaka et al. prepared Ni nanoparticles in a W/O microemulsion, and afterwards TEOS and ammonia were added in order to form the silica layer. Core-shell nanoparticles with 20-50 nm diameter and a Ni shell (5 nm) were formed. For comparison, silica nanoparticles were prepared also in W/O microemulsions but the Ni nanoparticles were prepared by impregnation of these silica nanoparticles. Their catalytic activity in the partial oxidation of methane reaction was evaluated, and the core-shell nanoparticles had a better performance than the impregnated ones (Takenaka et al., 2007). On the other hand, Chung et al. prepared silica nanoparticles coated with a thin layer of CeO2, and the material was also prepared in W/O microemulsions in a two-step procedure (Chung et al., 2011). This reaction turned out to be challenging as the formation of CeO2 shell was competing with bulk precipitation. The problem was overcome by coupling two strategies:

Fig. 9. TEM image of ZnS:Mn@silica nanoparticles with a core – shell structure. (Reproduced with permission, Dong B. et al. 2009).

New Trends on the Synthesis

of Inorganic Nanoparticles Using Microemulsions as Confined Reaction Media 211

al. synthesized single-crystalline ZnO nanowire bundles with a length of about 1 m and a diameter of about 20–30 nm (Wang G. et al., 2010). The approach was by reacting zinc acetate with hydrazine in a w/o microemulsion based on water/dodecylbenzene sulfonic acid sodium salt /xylene. The reaction temperature is not mentioned, however for reflux of xylene is achieved around 140°C. The relatively high reaction temperature and the heating time is possibly the driving force for the growth of the nanowires, as different structures were obtained at shorter reaction times. Also, it must be pointed out that the precursor used, zinc acetate, is soluble in both water and the oil phase, which is an unusual approach.

Wu et al. synthesized nanowires of Zn/Co/Fe layered double hydroxides using a w/o microemulsion based on water/CTAB/n-hexanol n-hexane (Wu et al., 2010). In their approach, the sulfate salts were used as precursors, hence these were dissolved in the aqueous phase only, and urea was used as precipitant. The influence of reaction temperature, time, urea concentration and CTAB to water molar ratio on the structure and morphology of Zn/Co/Fe-layered double hydroxides was investigated. The possible reason for nanowire growth is the solvothermal treatment of the reaction mixture which was carried out in an autoclave at 80-180°C during 6-24 hours. The thermal treatment in the autoclave was a key factor for annealing and therefore obtaining both a crystalline structure

The w/o microemulsion reaction method has been used for the synthesis of complex ceramic nanoparticles such as perovskites, spinels, aluminates, and hexaferrites. Often, nanoparticles of precursors such as hydroxides or other amorphous compounds are synthesized in the microemulsions, and these are afterwards calcined at a certain

He et al. synthesized nanoparticles of perovskite-type oxides La0.8Ce0.2Cu0.4Mn0.6O3 and La0.8Ce0.2Ag0.4Mn0.6O3 (He et al., 2007). The microemulsion used was CTAB/butanol /water /heptane, and for comparison purposes, the same materials were synthesized by the sol-gel technique. The precipitation of the precursors was carried out with NaOH for the microemulsion method, whereas citric acid was used for the sol-gel method. The particle size distribution was smaller and more uniform and the specific surface area was higher for the particles synthesized in microemulsions than those synthesized by sol-gel. Furthermore, the catalytic activity in the NO reduction by CO was evaluated. Performance of perovskites synthesized in microemulsion was superior than that of materials synthesized by sol-gel.

Gianakas et al. (Gianakas et al., 2007) reported the synthesis of spinel-type metal aluminates MAl2O4 sutwhere M=Mg, Co, o Zn using w/o and bicontinuous microemulsions. They carried out a very complete phase behavior study, which included pseudoternary phase diagrams for each precursor combination as well as the precipitating agent, ammonia. The microemulsion system was: aqueous solution/ CTAB/butanol/ octane. The spinel structure was achieved after calcination at 800°C. It was found that spinels synthesized in reverse microemulsions showed better surface and textural properties, as well as smaller particle size than spinel synthesized in bicontinuous microemulsions. As for catalytic activity, which was evaluated in the NO reduction by CO, the spinels synthesized in w/o microemulsions was slightly superior. Similar characteristic size was obtained by Wang et al. (Wang Y. et al.,

and formation of high aspect ratio particles (nanowires).

temperature in order to obtain the desired crystalline structure.

**4.3 Synthesis of complex ceramics** 

functionalization of the surface of the core silica nanoparticles with an organoamine group, and step-wise, semi-batch addition of the second microemulsion containing the Ce precursor. In this way, the silica cores were homogenously coated with a CeO2 shell.

As for core-shell nanoparticles made up of materials different from silica, the synthesis of both the core and the shell is usually carried out in W/O microemulsions, either in a two step process by preparing first the core and the later deposition of the shell (by adding more aqueous phase or more microemulsion comprising the second component), or both precursors are incorporated simultaneously, but the different reaction kinetics for each of the products results in a core-shell structure. The following core-shell nanomaterials can be listed: Pt@CeO2 (Yeung & Tsang, 2009 and 2010), Co@Ag (Garcia-Torres et al., 2010), Fe2O3@Au (Iglesias-Silva et al., 2010), Ni@Au (Chiu et al., 2009), Ag@Polystyrene (Li et al., 2009), and CdS@TiO2 (Ghows & Entezari, 2011). So far, the core@shell structures of these materials is not as well defined and controlled as that obtained with core@silica materials.

## **4.2.2 Hollow nanospheres**

Jiang et al. have prepared hollow nanospheres of Ni (Jiang et al., 2010) and CuS (Jiang, 2011), by following an approach which resembles that reported by Sanchez-Dominguez et al. (Sanchez-Dominguez et al., 2009). Jiang et al. used an o/w microemulsion in which the precursor (a naphtenate), was dissolved in the oil phase (dimethylbenzene) of a water/SDS/butanol/dimethylbenzene microemulsion. The precipitating agent was added in the water phase. The authors explain that an interfacial reaction occurs, and hollow nanospheres of about 100-200 nm are formed (Figure 10, for Ni hollow spheres). These are made-up of smaller nanoparticles. One difference between Jiang's method and that reported by Sanchez-Dominguez et al is that in the former, the temperatures used for reaction are higher (85°C for Ni; post-synthesis hydrothermal treatment for CuS), whereas in the latter the temperatures used are near room temperature (25-35°C).

Fig. 10. TEM image of the Ni hollow spheres and (b) a single Ni hollow sphere. Inset: SAED pattern. Reproduced with permission (Jiang et al., 2010).

## **4.2.3 Nanowires, nanorods**

Some works describe the formation of nanowire-like or nanorod structures. Usually, in order to obtain such high aspect ratio structures, it is necessary to carry out the synthesis at a relatively high temperature, or include a certain post-synthesis thermal treatment. Wang et al. synthesized single-crystalline ZnO nanowire bundles with a length of about 1 m and a diameter of about 20–30 nm (Wang G. et al., 2010). The approach was by reacting zinc acetate with hydrazine in a w/o microemulsion based on water/dodecylbenzene sulfonic acid sodium salt /xylene. The reaction temperature is not mentioned, however for reflux of xylene is achieved around 140°C. The relatively high reaction temperature and the heating time is possibly the driving force for the growth of the nanowires, as different structures were obtained at shorter reaction times. Also, it must be pointed out that the precursor used, zinc acetate, is soluble in both water and the oil phase, which is an unusual approach.

Wu et al. synthesized nanowires of Zn/Co/Fe layered double hydroxides using a w/o microemulsion based on water/CTAB/n-hexanol n-hexane (Wu et al., 2010). In their approach, the sulfate salts were used as precursors, hence these were dissolved in the aqueous phase only, and urea was used as precipitant. The influence of reaction temperature, time, urea concentration and CTAB to water molar ratio on the structure and morphology of Zn/Co/Fe-layered double hydroxides was investigated. The possible reason for nanowire growth is the solvothermal treatment of the reaction mixture which was carried out in an autoclave at 80-180°C during 6-24 hours. The thermal treatment in the autoclave was a key factor for annealing and therefore obtaining both a crystalline structure and formation of high aspect ratio particles (nanowires).

## **4.3 Synthesis of complex ceramics**

210 Smart Nanoparticles Technology

functionalization of the surface of the core silica nanoparticles with an organoamine group, and step-wise, semi-batch addition of the second microemulsion containing the Ce

As for core-shell nanoparticles made up of materials different from silica, the synthesis of both the core and the shell is usually carried out in W/O microemulsions, either in a two step process by preparing first the core and the later deposition of the shell (by adding more aqueous phase or more microemulsion comprising the second component), or both precursors are incorporated simultaneously, but the different reaction kinetics for each of the products results in a core-shell structure. The following core-shell nanomaterials can be listed: Pt@CeO2 (Yeung & Tsang, 2009 and 2010), Co@Ag (Garcia-Torres et al., 2010), Fe2O3@Au (Iglesias-Silva et al., 2010), Ni@Au (Chiu et al., 2009), Ag@Polystyrene (Li et al., 2009), and CdS@TiO2 (Ghows & Entezari, 2011). So far, the core@shell structures of these materials is not as well defined and controlled as that obtained with core@silica materials.

Jiang et al. have prepared hollow nanospheres of Ni (Jiang et al., 2010) and CuS (Jiang, 2011), by following an approach which resembles that reported by Sanchez-Dominguez et al. (Sanchez-Dominguez et al., 2009). Jiang et al. used an o/w microemulsion in which the precursor (a naphtenate), was dissolved in the oil phase (dimethylbenzene) of a water/SDS/butanol/dimethylbenzene microemulsion. The precipitating agent was added in the water phase. The authors explain that an interfacial reaction occurs, and hollow nanospheres of about 100-200 nm are formed (Figure 10, for Ni hollow spheres). These are made-up of smaller nanoparticles. One difference between Jiang's method and that reported by Sanchez-Dominguez et al is that in the former, the temperatures used for reaction are higher (85°C for Ni; post-synthesis hydrothermal treatment for CuS), whereas in the latter

Fig. 10. TEM image of the Ni hollow spheres and (b) a single Ni hollow sphere. Inset: SAED

Some works describe the formation of nanowire-like or nanorod structures. Usually, in order to obtain such high aspect ratio structures, it is necessary to carry out the synthesis at a relatively high temperature, or include a certain post-synthesis thermal treatment. Wang et

the temperatures used are near room temperature (25-35°C).

pattern. Reproduced with permission (Jiang et al., 2010).

**4.2.3 Nanowires, nanorods** 

precursor. In this way, the silica cores were homogenously coated with a CeO2 shell.

**4.2.2 Hollow nanospheres** 

The w/o microemulsion reaction method has been used for the synthesis of complex ceramic nanoparticles such as perovskites, spinels, aluminates, and hexaferrites. Often, nanoparticles of precursors such as hydroxides or other amorphous compounds are synthesized in the microemulsions, and these are afterwards calcined at a certain temperature in order to obtain the desired crystalline structure.

He et al. synthesized nanoparticles of perovskite-type oxides La0.8Ce0.2Cu0.4Mn0.6O3 and La0.8Ce0.2Ag0.4Mn0.6O3 (He et al., 2007). The microemulsion used was CTAB/butanol /water /heptane, and for comparison purposes, the same materials were synthesized by the sol-gel technique. The precipitation of the precursors was carried out with NaOH for the microemulsion method, whereas citric acid was used for the sol-gel method. The particle size distribution was smaller and more uniform and the specific surface area was higher for the particles synthesized in microemulsions than those synthesized by sol-gel. Furthermore, the catalytic activity in the NO reduction by CO was evaluated. Performance of perovskites synthesized in microemulsion was superior than that of materials synthesized by sol-gel.

Gianakas et al. (Gianakas et al., 2007) reported the synthesis of spinel-type metal aluminates MAl2O4 sutwhere M=Mg, Co, o Zn using w/o and bicontinuous microemulsions. They carried out a very complete phase behavior study, which included pseudoternary phase diagrams for each precursor combination as well as the precipitating agent, ammonia. The microemulsion system was: aqueous solution/ CTAB/butanol/ octane. The spinel structure was achieved after calcination at 800°C. It was found that spinels synthesized in reverse microemulsions showed better surface and textural properties, as well as smaller particle size than spinel synthesized in bicontinuous microemulsions. As for catalytic activity, which was evaluated in the NO reduction by CO, the spinels synthesized in w/o microemulsions was slightly superior. Similar characteristic size was obtained by Wang et al. (Wang Y. et al.,

New Trends on the Synthesis

(Myakonkaya et al., 2010).

**5. Conclusions and perspectives** 

of Inorganic Nanoparticles Using Microemulsions as Confined Reaction Media 213

two metal ions and the excess of reducing agent. An intermetallic structure is always obtained when both reduction reactions take place at about the same rate. When the metal ions have very different reduction potentials, a core-shell to intermetallic structure transition is found at increasing the excess of the reducing agent. An enhancement of the intermetallic structure at the expense of the core-shell, can be obtained either by decreasing the concentration of both metal salts or by increasing the interdroplet exchange rates. The results obtained by these studies has positive implications in the general formation of

bimetallic nanoparticles with a given structure (core-shell or nano-alloy).

or phase transfer of nanoparticles from microemulsion media.

**4.5 Novel approaches for the separation of nanoparticles from reaction mixtures** 

Often, the nanoparticles formed in a microemulsion are so well dispersed in the reaction media that some solvent has to be added in order to destabilize the microemulsion, which causes desorption of surfactant from the particles, which aggregate and precipitate, making their separation by centrifugation or filtration easier. Sometimes, during this aggressive process the nanoparticles end up so agglomerated that it is difficult to re-disperse them. Some novel and straightforward approaches have been proposed for an improved recovery

Eastoe et al. (Hollamby & Eastoe, 2009; Myakonkaya et al., 2010, 2011; Nazar et al., 2011; Vesperinas et al., 2007) have proposed three approaches for nanoparticle recovery. One of them is based on the use of a photodestructible surfactant for microemulsion formation, and in the final step, irradiation with UV-light induces microemulsion destabilization and hence separation of Au nanoparticles (Vesperinas et al., 2007). In another approach, excess water is added at the end of the reaction, to the microemulsion containing the nanoparticles, inducing a change in phase behavior and hence microemulsion destabilization, followed by phase separation. Interestingly, by this approach, usually the nanoparticles remain in the oil phase, which can be diluted with organic solvents to form stable nanoparticle dispersions (Nazar et al., 2011). This method shows potential benefits for dispersion, storage, application, and recovery of NPs, with the great advantage that it is not necessary to add organic solvents for nanoparticle separation. In other approach by the same group, nanoparticle separation has been achieved by changing the solvent quality, for example, adding squalene to water/AOT/octane microemulsion containing Au nanoparticles

Abecassis et al. have proposed nanoparticle separation by thermally inducing the phase separation of the microemulsion media (Abecassis et al., 2009). This was applied to the synthesis of Au NPs, which upon destabilization remained preferentially in the oil phase.

It has been shown that the microemulsion reaction method is a versatile technique, useful for the controlled synthesis of a large variety of nanomaterials, from metals, metal oxides, ceramics, quantum dots, magnetic nanoparticles, etc. The method has now been extended to the synthesis of other types or architectures, such as core-shell, multishell, hollow spheres and nanowires, in addition to the traditional small globular particles. Although for about 25 years only w/o microemulsions were used for the synthesis of inorganic nanoparticles, in the last five years the use of o/w and bicontinuous microemulsions has also been

2007) for nanoparticles of manganese-doped barium aluminate BaAl12O19: Mn2+; calcination at 1300°C was carried out in order to obtain the crystalline phase expected. The evaluation of photoluminescent properties of this material showed that this phosphor is a good candidate to replace Hg lamps.

Other good examples of ceramic materials obtained in w/o microemulsions include: barium hexaferrite (BaFe12O19) nanoparticles (Xu et al., 2007), tungsten oxide (WO3) nanoparticles (Asim et al., 2007), and rutile TiO2 nanoparticles (Keswani et al., 2010). In the last example, it is remarkable that the rutile phase was obtained at room temperature, without the need for thermal treatment, hence the size of the rutile nanocrystals remained as small as 4 nm.

## **4.4 Modeling of reactions in microemulsions**

There have been a number of studies dealing with the theoretical aspects of nanoparticle formation by the microemulsion reaction method. Most of these studies use the Monte Carlo method. The studies carried out in the last four years are focused on several aspects: kinetics of nanoparticle formation (de Dios et al., 2009), formation of bimetallic nanoparticles (Tojo et al., 2009; Angelescu et al., 2010), droplet exchange (Niemann & Sundmacher, 2010), cluster coalescence (Kuriyedath et al., 2010), and core-shell nanoparticle formation (Viswanadh et al., 2007).

Kinetics of nanoparticle formation in microemulsion were studied for the Ag and Au nanoparticles using Monte Carlo simulations by de Dios et al. (de Dios et al., 2009). It was shown that, although the material interdroplet exchange depends primarily on the flexibility of surfactant film, a slow reaction rate leads to a more effective material interdroplet exchange for a given microemulsion. Two factors contribute to this result. Firstly, a slow reaction implies that autocatalytic growth takes place for a longer period of time, because there are available reactants. If the reaction is faster, the reactants are almost exhausted at early stages of the process. As a consequence, autocatalytic growth is only possible at the beginning. Secondly, a slow reaction rate implies the continuous production of seed nuclei, which can be exchanged between micelles due to their small size, allowing the coagulation of two nanoparticles. This exchange only takes place at early stages of the synthesis. Both factors, autocatalysis and ripening, favor the slow growth of the biggest nanoparticles leading to the production of larger particles when the reaction is slower.

With respect to the formation of bimetallic nanoparticles in microemulsions, the same research group (Tojo et al., 2009), carried out Monte Carlo studies in order to explain the different structures that can be obtained when bimetallic nanoparticles are synthesized in microemulsions. They observed that the difference in reduction rates of both metals is not the only parameter to determine metal segregation; the interdroplet channel size also plays an important role. The reduction rate difference determines nanoparticle structure only in two extreme cases: when both reactions take place at the same rate, a nanoalloy structure is always obtained. In contrast, if both reactions have very different rates, the nanoparticle shows a core-shell structure. However, in the large interval between both extreme cases, the nanoparticle structure is strongly dependent on the intermicellar exchange, which is mainly determined by the flexibility of the surfactant film around the microemulsion droplets. In a related study by Angelescu et al. (Angelescu et al., 2010), it was found that the bimetallic nanoparticle structure is mainly determined by the difference in the reduction rates of the

2007) for nanoparticles of manganese-doped barium aluminate BaAl12O19: Mn2+; calcination at 1300°C was carried out in order to obtain the crystalline phase expected. The evaluation of photoluminescent properties of this material showed that this phosphor is a good

Other good examples of ceramic materials obtained in w/o microemulsions include: barium hexaferrite (BaFe12O19) nanoparticles (Xu et al., 2007), tungsten oxide (WO3) nanoparticles (Asim et al., 2007), and rutile TiO2 nanoparticles (Keswani et al., 2010). In the last example, it is remarkable that the rutile phase was obtained at room temperature, without the need for thermal treatment, hence the size of the rutile nanocrystals remained as small as 4 nm.

There have been a number of studies dealing with the theoretical aspects of nanoparticle formation by the microemulsion reaction method. Most of these studies use the Monte Carlo method. The studies carried out in the last four years are focused on several aspects: kinetics of nanoparticle formation (de Dios et al., 2009), formation of bimetallic nanoparticles (Tojo et al., 2009; Angelescu et al., 2010), droplet exchange (Niemann & Sundmacher, 2010), cluster coalescence (Kuriyedath et al., 2010), and core-shell nanoparticle formation (Viswanadh et

Kinetics of nanoparticle formation in microemulsion were studied for the Ag and Au nanoparticles using Monte Carlo simulations by de Dios et al. (de Dios et al., 2009). It was shown that, although the material interdroplet exchange depends primarily on the flexibility of surfactant film, a slow reaction rate leads to a more effective material interdroplet exchange for a given microemulsion. Two factors contribute to this result. Firstly, a slow reaction implies that autocatalytic growth takes place for a longer period of time, because there are available reactants. If the reaction is faster, the reactants are almost exhausted at early stages of the process. As a consequence, autocatalytic growth is only possible at the beginning. Secondly, a slow reaction rate implies the continuous production of seed nuclei, which can be exchanged between micelles due to their small size, allowing the coagulation of two nanoparticles. This exchange only takes place at early stages of the synthesis. Both factors, autocatalysis and ripening, favor the slow growth of the biggest nanoparticles

With respect to the formation of bimetallic nanoparticles in microemulsions, the same research group (Tojo et al., 2009), carried out Monte Carlo studies in order to explain the different structures that can be obtained when bimetallic nanoparticles are synthesized in microemulsions. They observed that the difference in reduction rates of both metals is not the only parameter to determine metal segregation; the interdroplet channel size also plays an important role. The reduction rate difference determines nanoparticle structure only in two extreme cases: when both reactions take place at the same rate, a nanoalloy structure is always obtained. In contrast, if both reactions have very different rates, the nanoparticle shows a core-shell structure. However, in the large interval between both extreme cases, the nanoparticle structure is strongly dependent on the intermicellar exchange, which is mainly determined by the flexibility of the surfactant film around the microemulsion droplets. In a related study by Angelescu et al. (Angelescu et al., 2010), it was found that the bimetallic nanoparticle structure is mainly determined by the difference in the reduction rates of the

leading to the production of larger particles when the reaction is slower.

candidate to replace Hg lamps.

al., 2007).

**4.4 Modeling of reactions in microemulsions** 

two metal ions and the excess of reducing agent. An intermetallic structure is always obtained when both reduction reactions take place at about the same rate. When the metal ions have very different reduction potentials, a core-shell to intermetallic structure transition is found at increasing the excess of the reducing agent. An enhancement of the intermetallic structure at the expense of the core-shell, can be obtained either by decreasing the concentration of both metal salts or by increasing the interdroplet exchange rates. The results obtained by these studies has positive implications in the general formation of bimetallic nanoparticles with a given structure (core-shell or nano-alloy).

## **4.5 Novel approaches for the separation of nanoparticles from reaction mixtures**

Often, the nanoparticles formed in a microemulsion are so well dispersed in the reaction media that some solvent has to be added in order to destabilize the microemulsion, which causes desorption of surfactant from the particles, which aggregate and precipitate, making their separation by centrifugation or filtration easier. Sometimes, during this aggressive process the nanoparticles end up so agglomerated that it is difficult to re-disperse them. Some novel and straightforward approaches have been proposed for an improved recovery or phase transfer of nanoparticles from microemulsion media.

Eastoe et al. (Hollamby & Eastoe, 2009; Myakonkaya et al., 2010, 2011; Nazar et al., 2011; Vesperinas et al., 2007) have proposed three approaches for nanoparticle recovery. One of them is based on the use of a photodestructible surfactant for microemulsion formation, and in the final step, irradiation with UV-light induces microemulsion destabilization and hence separation of Au nanoparticles (Vesperinas et al., 2007). In another approach, excess water is added at the end of the reaction, to the microemulsion containing the nanoparticles, inducing a change in phase behavior and hence microemulsion destabilization, followed by phase separation. Interestingly, by this approach, usually the nanoparticles remain in the oil phase, which can be diluted with organic solvents to form stable nanoparticle dispersions (Nazar et al., 2011). This method shows potential benefits for dispersion, storage, application, and recovery of NPs, with the great advantage that it is not necessary to add organic solvents for nanoparticle separation. In other approach by the same group, nanoparticle separation has been achieved by changing the solvent quality, for example, adding squalene to water/AOT/octane microemulsion containing Au nanoparticles (Myakonkaya et al., 2010).

Abecassis et al. have proposed nanoparticle separation by thermally inducing the phase separation of the microemulsion media (Abecassis et al., 2009). This was applied to the synthesis of Au NPs, which upon destabilization remained preferentially in the oil phase.

## **5. Conclusions and perspectives**

It has been shown that the microemulsion reaction method is a versatile technique, useful for the controlled synthesis of a large variety of nanomaterials, from metals, metal oxides, ceramics, quantum dots, magnetic nanoparticles, etc. The method has now been extended to the synthesis of other types or architectures, such as core-shell, multishell, hollow spheres and nanowires, in addition to the traditional small globular particles. Although for about 25 years only w/o microemulsions were used for the synthesis of inorganic nanoparticles, in the last five years the use of o/w and bicontinuous microemulsions has also been

New Trends on the Synthesis

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developed, and their usefulness for the synthesis of a variety of nanomaterials has been demonstrated. These developments are greener than the traditional w/o microemulsion method, so it should contribute to an advance towards the industrial use of microemulsions for nanoparticle synthesis. Furthermore, there have been efforts towards boosting the metal loading in microemulsions, in order to increase their production capacity. The investigations on novel approaches for nanoparticle recovery should also be taken into account by more research groups for the improvement of nanoparticle quality and dispersability in different media. The new developments reviewed here should encourage the preparation of novel materials with different architectures, in order to respond quickly to the demands of Nanotechnology and Materials Science. It is hoped that this chapter is useful to students and researchers who start exploring the microemulsion reaction method for nanoparticle synthesis, as well as for those not new to the field but who are looking for the newest trends in this fascinating technique.

## **6. Acknowledgements**

The authors acknowledge financial support by Ministerio Ciencia e Innovación (MICINN Spain, grant number CTQ2008-01979) and Generalitat de Catalunya (Agaur, grant number 2009SGR-961).

## **7. References**


developed, and their usefulness for the synthesis of a variety of nanomaterials has been demonstrated. These developments are greener than the traditional w/o microemulsion method, so it should contribute to an advance towards the industrial use of microemulsions for nanoparticle synthesis. Furthermore, there have been efforts towards boosting the metal loading in microemulsions, in order to increase their production capacity. The investigations on novel approaches for nanoparticle recovery should also be taken into account by more research groups for the improvement of nanoparticle quality and dispersability in different media. The new developments reviewed here should encourage the preparation of novel materials with different architectures, in order to respond quickly to the demands of Nanotechnology and Materials Science. It is hoped that this chapter is useful to students and researchers who start exploring the microemulsion reaction method for nanoparticle synthesis, as well as for those not new to the field but who are looking for the newest trends

The authors acknowledge financial support by Ministerio Ciencia e Innovación (MICINN Spain, grant number CTQ2008-01979) and Generalitat de Catalunya (Agaur, grant number

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**10** 

*Czech Republic* 

**Polymeric Nanoparticles Stabilized** 

*Institute of Macromolecular Chemistry, Academy of Sciences of the Czech Republic,* 

It is accepted nowadays that the self-assembly or self-organization occurs in a system when two types of interactions exist simultaneously between various elements of a system – a short-range attraction and a long-range repulsion. If a combination of such interactions is manifested in a system, equilibrium nanostructures/nanoparticles could occur. This general principle applies for many different systems – *e.g*., liquid crystals, ferrofluids, lyotropic systems, surfactants and polymers. Polymers and copolymers in good solvent are widely used for creation of self-assembled nanoparticles in solution since they offer an extremely wide range of different monomers and compositions, the possibility to vary the polymer chain length and use tailor-made polymers for producing materials with specific properties and functionalities. For such polymers, no additives are required to form equilibrium

This chapter reviews another technique of creating self-assembled and self-organized polymeric nanoparticles **-** *controlled phase separation approach.* Such approach exploits mutual interactions of a polymers and surface active molecules (surfactants or amphiphilic block copolymers) in a common solvent. We shall explore particularly dilute systems where various types of nanoparticles will be investigated. The nanoparticles will be studied keeping in mind their possible applications, especially for biological purposes encapsulation and delivery of active substances in the case of particles and

The common approach applied to all types of physical systems described below is based on controlling the extent of *macrophase* separation that occurs in a mixture of two compounds (solvent and polymer) that became immiscible or incompatible as a result of a change of an external variable. This parameter can be temperature, pH or addition of a another solvent, in principle it could also be a change in pressure but the latter is not very practical since usually large pressure changes are needed to achieve relatively small changes in phase

**1. Introduction**

nanoparticles.

immobilization.

**2. Background**

diagrams.

**by Surfactants: Controlled Phase** 

Sergey K. Filippov, Jiri Panek and Petr Stepanek

**Separation Approach** 


## **Polymeric Nanoparticles Stabilized by Surfactants: Controlled Phase Separation Approach**

Sergey K. Filippov, Jiri Panek and Petr Stepanek *Institute of Macromolecular Chemistry, Academy of Sciences of the Czech Republic, Czech Republic* 

## **1. Introduction**

220 Smart Nanoparticles Technology

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It is accepted nowadays that the self-assembly or self-organization occurs in a system when two types of interactions exist simultaneously between various elements of a system – a short-range attraction and a long-range repulsion. If a combination of such interactions is manifested in a system, equilibrium nanostructures/nanoparticles could occur. This general principle applies for many different systems – *e.g*., liquid crystals, ferrofluids, lyotropic systems, surfactants and polymers. Polymers and copolymers in good solvent are widely used for creation of self-assembled nanoparticles in solution since they offer an extremely wide range of different monomers and compositions, the possibility to vary the polymer chain length and use tailor-made polymers for producing materials with specific properties and functionalities. For such polymers, no additives are required to form equilibrium nanoparticles.

This chapter reviews another technique of creating self-assembled and self-organized polymeric nanoparticles **-** *controlled phase separation approach.* Such approach exploits mutual interactions of a polymers and surface active molecules (surfactants or amphiphilic block copolymers) in a common solvent. We shall explore particularly dilute systems where various types of nanoparticles will be investigated. The nanoparticles will be studied keeping in mind their possible applications, especially for biological purposes encapsulation and delivery of active substances in the case of particles and immobilization.

## **2. Background**

The common approach applied to all types of physical systems described below is based on controlling the extent of *macrophase* separation that occurs in a mixture of two compounds (solvent and polymer) that became immiscible or incompatible as a result of a change of an external variable. This parameter can be temperature, pH or addition of a another solvent, in principle it could also be a change in pressure but the latter is not very practical since usually large pressure changes are needed to achieve relatively small changes in phase diagrams.

Polymeric Nanoparticles Stabilized by Surfactants: Controlled Phase Separation Approach 223

To observe and to prove the formation of well defined nanoparticles we have exploited a variety of methods such as static and dynamic light scattering (SLS/DLS), small-angle

Since kinetic factors are involved into formation of nanoparticles created by controlled phase separation method we have examined the nucleation and growth of polymeric nanoparticles using the stopped-flow technique combined with SAXS on the time scale of

SANS experiments were performed at CEA-Saclay on the spectrometer PAXY of the Laboratoire Leon-Brillouin. Measurements were run on a 128 128 multidetector (pixel size 0.5 0.5 cm) using a non-polarized, monochromatic (wavelength *λ* set by a velocity selector) incident neutron beam collimated with circular apertures for two sample-to-detector distances, namely, 1 m (with *λ* =0.6 nm) and 7 m (with *λ* = 0.8 nm). With such a setup, the investigated range of scattering wave vector modulus was 5.9 10-2 to 4.3 nm-1. In all the cases reported in this paper, the two-dimensional scattering patterns were isotropic so that they were azimuthally averaged to yield the dependence of the scattered intensity *Is*(*q*) on the scattering vector *q*. Data were corrected for background scattering and detector

Static light scattering measurements were carried out on an ALV-6010 instrument equipped with a 22 mW He–Ne laser in the angular range 30–150°. Dynamic light scattering measurements were carried out at 90° angle. The obtained correlation functions were analyzed by REPES (Jakes, 1995) analytical software providing a distribution function, *G*(*R*h) of hydrodynamic radii *R*h. To account for the logarithmic scale on the *R*h axis, all DLS distribution diagrams are shown in the equal area representation, *R*h*G*(*R*h). The static light

2 2 <sup>1</sup>

*G w w*

wave length in the medium, *θ*- scattering angle between the incident and the

(2)

*/2 is the scattering* 

*n/)sin*

() 3

scattered beam, *K* is a contrast factor containing the optical parameters, *c* is a particle concentration, *Mw* is the weight average of the molar mass of the particles, and *RG* is their radius of gyration. The concentration dependence was neglected which was acceptable

*K c R q Rq M M*

 Controlled phase separation induced by a change of solvent Controlled phase separation induced by a change in pH

neutron and X-ray scattering (SANS/SAXS) and Cryo-TEM methods.

efficiency. Intensities of neutron scattering are given in arbitrary units.

where *R*(*q*) is the Rayleigh ratio of the scattering intensity, *q=(4*

**3.2 Dynamic and Static Light Scattering (DLS/SLS)** 

scattering data were analyzed by a Zimm plot:

because of the low concentrations of the solutions.

**3. Experimental methods** 

**3.1 Small-Angle Neutron Scattering (SANS)** 

milliseconds.

*vector,* 

(1)

A phase diagram for a polymer A/polymer B or polymer/solvent system is schematically represented in Fig. 1. In the classical case the energy of the system is given by enthalpic and entropic contributions and the interaction parameter is given by the Flory-Huggins relation *aT b* / where T is absolute temperature and *a* and *b* are specific for the polymer/polymer or polymer/solvent pair. For more complex systems, a third term s has to be included:

*aT b*

Fig. 1. Schematic phase diagram for a polymer A/polymer B or polymer/solvent. is the volume fraction of the first component of the system, N the number of monomers in the polymer chain and the interaction parameter describing the strength of interaction between polymer A and B or between the polymer and solvent.

In this simple representation the third term includes all additional interactions in the system, in particular the effect of different temperature expansions of the system components and that of various specific interactions in the system (hydrogen bonds, ionic interactions, ...) that may be dominant compared to the enthalpic/entropic terms *a*, *b*.

Once a macrophase separation has been initiated, the spatial extent of inhomogeneities produced by nucleation or spinodal decomposition is controlled by addition of amphiphilic molecules chosen in such a way that one part of this amphiphile interacts attractively with the nucleated material while the other part interacts attractively with the surrounding solvent. The surface of the nucleated material then becomes covered with the amphiphilic molecules which effectively terminate the phase separation and stabilizes the system in a dispersed state. The formation of nanoparticles is always driven by thermodynamics (increase of –parameter in Eq. (1)) but controlled by specific factors of interaction with the amphiphilic molecules including kinetic and hydrodynamic effects. Assessment of these effects and their importance for nanoparticles preparation is a primary goal in this manuscript. In the following chapter we will describe several systems where this approach can be realized:

Controlled phase separation induced by a change in temperature.


## **3. Experimental methods**

222 Smart Nanoparticles Technology

A phase diagram for a polymer A/polymer B or polymer/solvent system is schematically represented in Fig. 1. In the classical case the energy of the system is given by enthalpic and entropic contributions and the interaction parameter is given by the Flory-Huggins

polymer/polymer or polymer/solvent pair. For more complex systems, a third term s has

*aT b*

/ *<sup>s</sup>*

N

0

between polymer A and B or between the polymer and solvent.

Controlled phase separation induced by a change in temperature.

5

10

15

20

*aT b* / where T is absolute temperature and *a* and *b* are specific for the

0 0.2 0.4 0.6 0.8 1.0

Fig. 1. Schematic phase diagram for a polymer A/polymer B or polymer/solvent. is the volume fraction of the first component of the system, N the number of monomers in the polymer chain and the interaction parameter describing the strength of interaction

In this simple representation the third term includes all additional interactions in the system, in particular the effect of different temperature expansions of the system components and that of various specific interactions in the system (hydrogen bonds, ionic

Once a macrophase separation has been initiated, the spatial extent of inhomogeneities produced by nucleation or spinodal decomposition is controlled by addition of amphiphilic molecules chosen in such a way that one part of this amphiphile interacts attractively with the nucleated material while the other part interacts attractively with the surrounding solvent. The surface of the nucleated material then becomes covered with the amphiphilic molecules which effectively terminate the phase separation and stabilizes the system in a dispersed state. The formation of nanoparticles is always driven by thermodynamics (increase of –parameter in Eq. (1)) but controlled by specific factors of interaction with the amphiphilic molecules including kinetic and hydrodynamic effects. Assessment of these effects and their importance for nanoparticles preparation is a primary goal in this manuscript. In the following chapter we will describe several systems where this approach

interactions, ...) that may be dominant compared to the enthalpic/entropic terms *a*, *b*.

(1)

relation

to be included:

can be realized:

To observe and to prove the formation of well defined nanoparticles we have exploited a variety of methods such as static and dynamic light scattering (SLS/DLS), small-angle neutron and X-ray scattering (SANS/SAXS) and Cryo-TEM methods.

Since kinetic factors are involved into formation of nanoparticles created by controlled phase separation method we have examined the nucleation and growth of polymeric nanoparticles using the stopped-flow technique combined with SAXS on the time scale of milliseconds.

## **3.1 Small-Angle Neutron Scattering (SANS)**

SANS experiments were performed at CEA-Saclay on the spectrometer PAXY of the Laboratoire Leon-Brillouin. Measurements were run on a 128 128 multidetector (pixel size 0.5 0.5 cm) using a non-polarized, monochromatic (wavelength *λ* set by a velocity selector) incident neutron beam collimated with circular apertures for two sample-to-detector distances, namely, 1 m (with *λ* =0.6 nm) and 7 m (with *λ* = 0.8 nm). With such a setup, the investigated range of scattering wave vector modulus was 5.9 10-2 to 4.3 nm-1. In all the cases reported in this paper, the two-dimensional scattering patterns were isotropic so that they were azimuthally averaged to yield the dependence of the scattered intensity *Is*(*q*) on the scattering vector *q*. Data were corrected for background scattering and detector efficiency. Intensities of neutron scattering are given in arbitrary units.

## **3.2 Dynamic and Static Light Scattering (DLS/SLS)**

Static light scattering measurements were carried out on an ALV-6010 instrument equipped with a 22 mW He–Ne laser in the angular range 30–150°. Dynamic light scattering measurements were carried out at 90° angle. The obtained correlation functions were analyzed by REPES (Jakes, 1995) analytical software providing a distribution function, *G*(*R*h) of hydrodynamic radii *R*h. To account for the logarithmic scale on the *R*h axis, all DLS distribution diagrams are shown in the equal area representation, *R*h*G*(*R*h). The static light scattering data were analyzed by a Zimm plot:

$$\frac{K\,\mathrm{c}}{R(q)} = \frac{1}{M\_w} + \frac{R\_G^2 q^2}{\Im M\_w} \tag{2}$$

where *R*(*q*) is the Rayleigh ratio of the scattering intensity, *q=(4n/)sin/2 is the scattering vector,* wave length in the medium, *θ*- scattering angle between the incident and the scattered beam, *K* is a contrast factor containing the optical parameters, *c* is a particle concentration, *Mw* is the weight average of the molar mass of the particles, and *RG* is their radius of gyration. The concentration dependence was neglected which was acceptable because of the low concentrations of the solutions.

Polymeric Nanoparticles Stabilized by Surfactants: Controlled Phase Separation Approach 225

hydrophilic shell consists of hydrophilic parts of surfactants. An intermediate shell at the core–shell interface contains both the hydrophobic parts of surfactants and PNIPAM chains. The feature of that research is that in contrast to previous studies, where surfactants were

To validate the proposed model, for PNIPAM and SDS system, a contrast variation study was performed by SANS. It is important to note here that the system studied in our research is different from a so-called mesoglobule state that was also observed (Siu et al., 2003; (Aseyev et al., 2005; Kujawa et al., 2006a; Kujawa et al., 2006b) for PNIPAM. It was established in variety of papers that PNIPAM macromolecules of high molar masses on very diluted solutions might undergo through intermediate mesoglobule state with increasing of temperature above LCST. These mesoglobules are aggregates of PNIPAM molecules that consist of one or more macromolecules. They are metastable particles that are stabilized either by electrostatic or steric interactions (Kujawa et al., 2006b). No surfactant is required. Nevertheless when PNIPAM concentration in solution is rather high macroscopic precipitation occurs. In this case, surface active molecules are needed to create stable

Three types of nanoparticles were tested: (*i*) deuterated d7-PNIPAM + protonated SDS in a 72%D2O/28%H2O volume mixture where the coherent scattering originates only from the surfactant. (*ii*) protonated PNIPAM + protonated SDS in pure D2O where the scattering comes from the polymer and the surfactant. The whole nanoparticle should be visible. (*iii*) protonated PNIPAM + deuterated d25-SDS in D2O. The scattering length density of the deuterated surfactant is almost matched by D2O. In this case most of the scattering is

h-SDS/d7-PNIPAM

 1:1 1:10 1:100 pure h-SDS

produced by the polymer. The experiments have been conducted at T=42 oC.

10-2 10-1

PNIPAM/h-SDS sample, *c*d7-PNIPAM=5 g/L; (○) *c*SDS/cPNIPAM =1:1; (•) *c*SDS/cPNIPAM =1:10; (□) *c*SDS/cPNIPAM =1:100; (+) pure SDS. Data are taken from the reference (Lee & Cabane, 1997).

Fig. 2. Scattered intensity *I*s as a function of the scattering vector *q* for systems d7-

*q*, A-1

q\*

used in excess, lower concentrations of surfactants were used.

polymeric nanoparticles.

10-3

10-2

10-1

100

*I*


s, cm

101

102

103

104

## **3.3 Cryo-transmission electron microscopy - Cryo-TEM**

To carry out a Cryo-TEM experiment, a drop of the solution under study was placed on a pretreated copper grid which was coated with a perforated polymer film. Excess solution was removed by blotting with a filter paper. The preparation of the sample film was done under controlled environment conditions, *i.e*., in a chamber at a constant temperature of 25°C and with a relative humidity of 98-99% to avoid evaporation of the liquid. Rapid vitrification of the thin film was achieved by plunging the grid into liquid ethane held just above its freezing point. The sample was then transferred to the electron microscope, a Zeiss 902A instrument (Carl Zeiss NTS, Oberkochen, Germany), operating at an accelerating voltage of 80 kV and in zero-loss bright-field mode. The temperature was kept below -165 °C and the specimen was protected against atmospheric conditions during the entire procedure to prevent sample perturbation and formation of ice crystals. The resolution in this method was 3-5 nm. Digital images were acquired with a BioVision Pro-SM Slow Scan CCD camera (Proscan electronische systeme, GmbH, Germany). iTEM software (Olympus Soft Imaging Solutions, GmbH, Germany) was used for image processing. The polymer concentration used was 510-3 g mL-1.

## **3.4 Small angle X-ray scattering – SAXS**

All time-resolved SAXS (TR-SAXS) experiments were performed on the high brilliance beam line ID02 at the ESRF (Grenoble, France). The SAXS setup is based on a pinhole camera with a beam stop placed in front of a two-dimensional detector (X-ray image intensifier coupled to a CCD camera). The X-ray scattering patterns were recorded on the detector that was located 2 m from the sample, using a monochromatic incident X-ray beam (λ=0.1 nm). The available wave vector range was 0.04 - 2.71 nm-1. Data acquisition and counting of the time *t* was hardware-triggered within 1 ms before the final mixing process was initiated. SAXS data were acquired with an exposure time of 50 ms per frame.

The fast mixing experiments were performed using a stopped-flow device (SFM-3, Bio-Logic) that has been specifically adapted for SAXS experiments. The device was thermostated at 25.0±0.5 °C

## **4. Controlled phase separation induced by a change in temperature**

To fulfill this task we have exploited the phase separation of thermally sensitive PNIPAM polymer on heating above the *lower critical solubility temperature* (LCST). PNIPAM is a typical temperature-sensitive polymer that has LCST around of 32oC (detailed information on pure PNIPAM, can be found in the review (Aseyev et al., 2010) and references therein). Heating of aqueous solution of PNIPAM above LCST will initialize coil-to-globule transformation with following precipitation of the polymer. Such macrophase separation could be terminated if surface active molecules are presented in solution. Earlier we have demonstrated that well defined nanoparticles of PNIPAM could be prepared in presence of ionic and non-ionic surfactants, (SDS, CTAB, Brij98, Brij97) (Konak et al., 2007). The effect of PNIPAM and surfactant concentration, and molecular weight of PNIPAM on nanoparticle parameters and on the phase transition temperature of PNIPAM solutions was investigated. It was proposed that the structure of particles is supposed to be similar to block copolymer micelles. Hydrophobic PNIPAM molecules form the insoluble core of particles and their

To carry out a Cryo-TEM experiment, a drop of the solution under study was placed on a pretreated copper grid which was coated with a perforated polymer film. Excess solution was removed by blotting with a filter paper. The preparation of the sample film was done under controlled environment conditions, *i.e*., in a chamber at a constant temperature of 25°C and with a relative humidity of 98-99% to avoid evaporation of the liquid. Rapid vitrification of the thin film was achieved by plunging the grid into liquid ethane held just above its freezing point. The sample was then transferred to the electron microscope, a Zeiss 902A instrument (Carl Zeiss NTS, Oberkochen, Germany), operating at an accelerating voltage of 80 kV and in zero-loss bright-field mode. The temperature was kept below -165 °C and the specimen was protected against atmospheric conditions during the entire procedure to prevent sample perturbation and formation of ice crystals. The resolution in this method was 3-5 nm. Digital images were acquired with a BioVision Pro-SM Slow Scan CCD camera (Proscan electronische systeme, GmbH, Germany). iTEM software (Olympus Soft Imaging Solutions, GmbH, Germany) was used for image processing. The polymer

All time-resolved SAXS (TR-SAXS) experiments were performed on the high brilliance beam line ID02 at the ESRF (Grenoble, France). The SAXS setup is based on a pinhole camera with a beam stop placed in front of a two-dimensional detector (X-ray image intensifier coupled to a CCD camera). The X-ray scattering patterns were recorded on the detector that was located 2 m from the sample, using a monochromatic incident X-ray beam (λ=0.1 nm). The available wave vector range was 0.04 - 2.71 nm-1. Data acquisition and counting of the time *t* was hardware-triggered within 1 ms before the final mixing process was initiated. SAXS

The fast mixing experiments were performed using a stopped-flow device (SFM-3, Bio-Logic) that has been specifically adapted for SAXS experiments. The device was

To fulfill this task we have exploited the phase separation of thermally sensitive PNIPAM polymer on heating above the *lower critical solubility temperature* (LCST). PNIPAM is a typical temperature-sensitive polymer that has LCST around of 32oC (detailed information on pure PNIPAM, can be found in the review (Aseyev et al., 2010) and references therein). Heating of aqueous solution of PNIPAM above LCST will initialize coil-to-globule transformation with following precipitation of the polymer. Such macrophase separation could be terminated if surface active molecules are presented in solution. Earlier we have demonstrated that well defined nanoparticles of PNIPAM could be prepared in presence of ionic and non-ionic surfactants, (SDS, CTAB, Brij98, Brij97) (Konak et al., 2007). The effect of PNIPAM and surfactant concentration, and molecular weight of PNIPAM on nanoparticle parameters and on the phase transition temperature of PNIPAM solutions was investigated. It was proposed that the structure of particles is supposed to be similar to block copolymer micelles. Hydrophobic PNIPAM molecules form the insoluble core of particles and their

**4. Controlled phase separation induced by a change in temperature** 

**3.3 Cryo-transmission electron microscopy - Cryo-TEM** 

concentration used was 510-3 g mL-1.

thermostated at 25.0±0.5 °C

**3.4 Small angle X-ray scattering – SAXS** 

data were acquired with an exposure time of 50 ms per frame.

hydrophilic shell consists of hydrophilic parts of surfactants. An intermediate shell at the core–shell interface contains both the hydrophobic parts of surfactants and PNIPAM chains. The feature of that research is that in contrast to previous studies, where surfactants were used in excess, lower concentrations of surfactants were used.

To validate the proposed model, for PNIPAM and SDS system, a contrast variation study was performed by SANS. It is important to note here that the system studied in our research is different from a so-called mesoglobule state that was also observed (Siu et al., 2003; (Aseyev et al., 2005; Kujawa et al., 2006a; Kujawa et al., 2006b) for PNIPAM. It was established in variety of papers that PNIPAM macromolecules of high molar masses on very diluted solutions might undergo through intermediate mesoglobule state with increasing of temperature above LCST. These mesoglobules are aggregates of PNIPAM molecules that consist of one or more macromolecules. They are metastable particles that are stabilized either by electrostatic or steric interactions (Kujawa et al., 2006b). No surfactant is required. Nevertheless when PNIPAM concentration in solution is rather high macroscopic precipitation occurs. In this case, surface active molecules are needed to create stable polymeric nanoparticles.

Three types of nanoparticles were tested: (*i*) deuterated d7-PNIPAM + protonated SDS in a 72%D2O/28%H2O volume mixture where the coherent scattering originates only from the surfactant. (*ii*) protonated PNIPAM + protonated SDS in pure D2O where the scattering comes from the polymer and the surfactant. The whole nanoparticle should be visible. (*iii*) protonated PNIPAM + deuterated d25-SDS in D2O. The scattering length density of the deuterated surfactant is almost matched by D2O. In this case most of the scattering is produced by the polymer. The experiments have been conducted at T=42 oC.

Fig. 2. Scattered intensity *I*s as a function of the scattering vector *q* for systems d7- PNIPAM/h-SDS sample, *c*d7-PNIPAM=5 g/L; (○) *c*SDS/cPNIPAM =1:1; (•) *c*SDS/cPNIPAM =1:10; (□) *c*SDS/cPNIPAM =1:100; (+) pure SDS. Data are taken from the reference (Lee & Cabane, 1997).

Polymeric Nanoparticles Stabilized by Surfactants: Controlled Phase Separation Approach 227

Similar features are observed for the system where the PNIPAM is only visible. Again, the formation of nanoparticles could be monitored by the growth of the scattering intensity with decrease of the composition ratio. No peaks at high *q* range observed are visible in this case.

h-PNIPAM/h-SDS

micellar SDS

We conclude that PNIPAM is also uniformly distributed inside a nanoparticle.

0.01 0.1

Fig. 4. Scattered intensity *I*s as a function of the scattering vector *q* for systems h-PNIPAM/h-SDS sample, *c*-PNIPAM=5 g/L; (○)*c*SDS/cPNIPAM=1:1; (•) *c*SDS/cPNIPAM=1:10; (□) *c*SDS/cPNIPAM=1:100; (+) pure SDS. Data are taken from reference (Lee & Cabane, 1997).

q-4

*q*, A-1

The scattering curve at *c*SDS/cPNIPAM =1:100 begins at low *q* at high intensity; then it curves downward and continuous with *q*-4 decay (Fig. 4). This part of the scattering curve corresponds the scattering from colloidal particles. Fitting the scattering curves by formfactor of a hard sphere with Schultz-Zimm distribution provides *R*g values of nanoparticles. Obtained values nanoparticles are 216, and 96 Ǻ for ratios 1:100 and 1:10, respectively, giving corresponding outer radii 279, and 124 Ǻ. Polydispersity value obtained from the fitting routine was 0.37 and 0.44, respectively. Cabanne et. al. reported the similar value of about 0.5. Such high numbers imply strong polydispersity in size for nanoparticles in

At *c*SDS/cPNIPAM =1:1 the scattering is flat at low *q* (Fig. 4, 5a,b) as it could be visible from a comparison with the spectra of samples made at lower ratios (1:10 and 1:100). At high *q*, one can see a plateau and, beyond *q*=0.1 Ǻ-1, a steeper decay. This spectrum is identical to the scattering from a micellar solution of SDS at the same concentration in the absence of polymer (Fig. 4, 5a). In particular, the peak position matches the average intermicellar

In this case both the polymer and the surfactant are visible in SANS.

**4.2 h-PNIPAM/d25-SDS** 

**4.3 h-PNIPAM/h-SDS** 

10-4

distance in pure SDS solutions.

10-3

10-2

 1:1 1:10 1:100 pure h-SDS

10-1

*I*

solution.


s, cm

100

101

102

103

Earlier Cabane and Lee in their pioneer work have investigated similar the PNIPAM-SDS system by SANS (Lee & Cabane, 1997) . The polymer molar mass that have been used in their study was 1106 g/mole and concentration of solution was mainly 30 g/L. To avoid a mesoglobule state we have selected the h-PNIPAM with Mw=1.88105 g/mole and d7- PNIPAM with Mw=3.6105 g/mole that is somewhat smaller than the one used by Cabane *et.al.* For the same reason, concentration of PNIPAM in all solution was kept of 5 g/L. Our work is thus a research on a similar system with different conditions.

Fig. 2-4 represents the data for different surfactant-to-polymer ratios. For all systems, the scattered intensity extrapolated to zero *q* is increasing with decrease of the ratio. In other words, the growth of colloidal nanoparticles is observed with decrease of surfactant-to polymer ratio. One can see continuous evolution of the characteristic features of colloids.

Fig. 3. Scattered intensity *I*s as a function of the scattering vector *q* for systems h-PNIPAM/d25-SDS sample, *c*-PNIPAM=5 g/L; (○)*c*SDS/cPNIPAM =1:1; (•)*c*SDS/cPNIPAM =1:10; (□) *c*SDS/cPNIPAM =1:100

## **4.1 h-SDS/d7-PNIPAM**

When a surfactant is protonated, coherent scattering comes only from the surfactant in a 72%D2O/28%H2O volume mixture. At low *q*, a *q*-4 decay is visible at *c*SDS/cPNIPAM =1:100 (Fig. 2). At high *q* a signal is too low. For comparison reason, the scattering of pure SDS micelles is presented on Fig. 2. No peaks that correspond to the distance between consecutive SDS micelles at high *q* range observed by Cabanne and Lee[8] (located at *q* of about 0.1 Ǻ-1) appear on the graph. We conclude that all surfactant molecules are uniformly incorporated inside of a colloidal particle or on its surface. It is worth to note that such strong q dependence indicates that the surfactant forms big structures. At *c*SDS/cPNIPAM =1:1, one can see that the scattering at low *q* is week and simultaneously a peak at high q range appears (*q*\*=0.11 Ǻ-1). Obviously, colloidal particles are completely dissolved now; a pearl-necklace complex exists with SDS micelles bound to a polymer chain with the distance of 58Ǻ (d=2π/*q*\*). That finding is in good agreement with results of Cabanne where such distance was about 63 Ǻ.

#### **4.2 h-PNIPAM/d25-SDS**

226 Smart Nanoparticles Technology

Earlier Cabane and Lee in their pioneer work have investigated similar the PNIPAM-SDS system by SANS (Lee & Cabane, 1997) . The polymer molar mass that have been used in their study was 1106 g/mole and concentration of solution was mainly 30 g/L. To avoid a mesoglobule state we have selected the h-PNIPAM with Mw=1.88105 g/mole and d7- PNIPAM with Mw=3.6105 g/mole that is somewhat smaller than the one used by Cabane *et.al.* For the same reason, concentration of PNIPAM in all solution was kept of 5 g/L. Our

Fig. 2-4 represents the data for different surfactant-to-polymer ratios. For all systems, the scattered intensity extrapolated to zero *q* is increasing with decrease of the ratio. In other words, the growth of colloidal nanoparticles is observed with decrease of surfactant-to polymer ratio. One can see continuous evolution of the characteristic features of colloids.

> 1:100 1:10 1:1

10-2 10-1

PNIPAM/d25-SDS sample, *c*-PNIPAM=5 g/L; (○)*c*SDS/cPNIPAM =1:1; (•)*c*SDS/cPNIPAM =1:10; (□)

When a surfactant is protonated, coherent scattering comes only from the surfactant in a 72%D2O/28%H2O volume mixture. At low *q*, a *q*-4 decay is visible at *c*SDS/cPNIPAM =1:100 (Fig. 2). At high *q* a signal is too low. For comparison reason, the scattering of pure SDS micelles is presented on Fig. 2. No peaks that correspond to the distance between consecutive SDS micelles at high *q* range observed by Cabanne and Lee[8] (located at *q* of about 0.1 Ǻ-1) appear on the graph. We conclude that all surfactant molecules are uniformly incorporated inside of a colloidal particle or on its surface. It is worth to note that such strong q dependence indicates that the surfactant forms big structures. At *c*SDS/cPNIPAM =1:1, one can see that the scattering at low *q* is week and simultaneously a peak at high q range appears (*q*\*=0.11 Ǻ-1). Obviously, colloidal particles are completely dissolved now; a pearl-necklace complex exists with SDS micelles bound to a polymer chain with the distance of 58Ǻ (d=2π/*q*\*). That finding is in good

Fig. 3. Scattered intensity *I*s as a function of the scattering vector *q* for systems h-

agreement with results of Cabanne where such distance was about 63 Ǻ.

*q*, A-1

10-2

10-1

100

*I*

*c*SDS/cPNIPAM =1:100

**4.1 h-SDS/d7-PNIPAM** 


s, cm

101

102

103

104

work is thus a research on a similar system with different conditions.

h-PNIPAM/d25-SDS

Similar features are observed for the system where the PNIPAM is only visible. Again, the formation of nanoparticles could be monitored by the growth of the scattering intensity with decrease of the composition ratio. No peaks at high *q* range observed are visible in this case. We conclude that PNIPAM is also uniformly distributed inside a nanoparticle.

#### **4.3 h-PNIPAM/h-SDS**

In this case both the polymer and the surfactant are visible in SANS.

Fig. 4. Scattered intensity *I*s as a function of the scattering vector *q* for systems h-PNIPAM/h-SDS sample, *c*-PNIPAM=5 g/L; (○)*c*SDS/cPNIPAM=1:1; (•) *c*SDS/cPNIPAM=1:10; (□) *c*SDS/cPNIPAM=1:100; (+) pure SDS. Data are taken from reference (Lee & Cabane, 1997).

The scattering curve at *c*SDS/cPNIPAM =1:100 begins at low *q* at high intensity; then it curves downward and continuous with *q*-4 decay (Fig. 4). This part of the scattering curve corresponds the scattering from colloidal particles. Fitting the scattering curves by formfactor of a hard sphere with Schultz-Zimm distribution provides *R*g values of nanoparticles. Obtained values nanoparticles are 216, and 96 Ǻ for ratios 1:100 and 1:10, respectively, giving corresponding outer radii 279, and 124 Ǻ. Polydispersity value obtained from the fitting routine was 0.37 and 0.44, respectively. Cabanne et. al. reported the similar value of about 0.5. Such high numbers imply strong polydispersity in size for nanoparticles in solution.

At *c*SDS/cPNIPAM =1:1 the scattering is flat at low *q* (Fig. 4, 5a,b) as it could be visible from a comparison with the spectra of samples made at lower ratios (1:10 and 1:100). At high *q*, one can see a plateau and, beyond *q*=0.1 Ǻ-1, a steeper decay. This spectrum is identical to the scattering from a micellar solution of SDS at the same concentration in the absence of polymer (Fig. 4, 5a). In particular, the peak position matches the average intermicellar distance in pure SDS solutions.

Polymeric Nanoparticles Stabilized by Surfactants: Controlled Phase Separation Approach 229

For a specific polymer/solvent system, phase separation could be induced by a change of the solvent. This particular case of spontaneous macrophase separation leading to formation of nano-sized droplets is frequently referred to as Ouzo effect (Ganachaud & Katz , 2005), although it has a variety of names. Some authors call this process solvent shifting (Brick et al., 2003; Van Keuren, 2004), solvent displacement(Potineni et al., 2003; Trimaille et al., 2003; Lince et al., 2008; Chu et al., 2008; Vega et al., 2008; Nguyen et al., 2008; Beck-Broichsitter et al., 2009) spontaneous emulsification (Gallardo et al., 1993; QuintanarGuerrero et al., 1997; Baimark et al., 2007; Tan et al., 2008; Katas et al., 2009) or micro/nano precipitation (Leroueil-Le Verger et al., 1998; Peracchia et al., 1999; Bilati et al., 2005; Leo et al., 2006; Legrand et al., 2006). In our previous papers (Panek et al., 2011a; Panek et al., 2011b) we have successfully tested this procedure: polymeric nanoparticles were prepared by mixing a polymer - poly(methyl methacrylate) or polystyrene - dispersed in an organic solvent with an aqueous solution of a surfactant (SDS). Since water is a bad solvent for either of the polymers they start to precipitate but the presence of a surfactant terminates the phase separation and nearly monodisperse nanoparticles appear with a typical size in the range of 50 to 300 nm. Finally the organic solvent is evaporated. We demonstrated that a variety of parameters such as polymer molar mass, surfactant hydrophobicity, solution temperature and composition influence the physico-chemical properties of nanoparticles formed in solution (Panek et al., 2011a; Panek et al., 2011b). Nevertheless, detailed information on

Here we describe new experiments with static and dynamic light scattering that were conducted to get further insight on the internal structure of the nanoparticles. Using a combination of both methods, for the first time, we calculated such parameters as a structure factor *R*g/*R*h and density of nanoparticles (*ρ*). *R*g values were measured by static light scattering. In contrast with *R*h (Panek et al., 2011a) there is no detectable difference in the

SDS (-) HDPC (-) CTAB (+) triton (n) B97 (n) B98 (n) F68 (n)

Fig. 7. Histogram of structure factor ρ=*R*g/*R*h obtained from SLS/DLS data for PMMA(1)

factor =*R*g/*R*h of nanoparticles made from ionic and non-ionic surfactants (Fig. 7).

**5. Controlled phase separation induced by a change of solvent** 

**5.1 Density** 

nanoparticles structure is still missing.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

*c*P = 210-4 g mL-1 for various surfactants.

Fig. 5a. Scattered intensity *I*s as a function of the scattering vector *q* for systems PNIPAM/SDS sample cPNIPAM/*c*SDS=1:1, *c*-PNIPAM=5 g/L; (○) d7-PNIPAM/h-SDS; (•) h-PNIPAM/d25-SDS; (□) h-PNIPAM/h-SDS; (+) pure SDS. Data are taken from reference (Lee & Cabane, 1997).

Fig. 5b. Scattered intensity *I*s as a function of the scattering vector *q* for systems PNIPAM/SDS sample cPNIPAM/*c*SDS=1:00, *c*-PNIPAM=5 g/L; (○) h-PNIPAM/h-SDS; (•) d7- PNIPAM/h-SDS; (□) h-PNIPAM/d25-SDS; (+) pure SDS. Data are taken from reference (Lee & Cabane, 1997).

In order to determine the shape and geometric size of the particles we performed Cryo-TEM measurements for the samples at composition 1:100. The Cryo-TEM images are shown in Fig. 6.

Fig. 6. Cryo-TEM micrographs. h-PNIPAM/h-SDS sample *c*PNIPAM=510-3g mL-1; *c*SDS=510-5g mL-1. *c*SDS/cPNIPAM =1:100

Fig. 6 shows objects thate are rather polydisperse in size. The average size of moieties is in agreement with SANS data, giving value roughly 20-30 nm in radius. The TEM images are 2D projection of the particles, observed under different angles. Therefore, we believe that the structures we see in Fig. 6 are of more or less spherical shape. Individual micelles seen in SANS experiments were not imaged in these samples, because of their small size.

## **5. Controlled phase separation induced by a change of solvent**

#### **5.1 Density**

228 Smart Nanoparticles Technology

10-4 10-3 10-2 10-1 100 101 102 103 104

*I*

Fig. 5a. Scattered intensity *I*s as a function of the scattering vector *q* for systems

Fig. 5b. Scattered intensity *I*s as a function of the scattering vector *q* for systems

PNIPAM/SDS sample cPNIPAM/*c*SDS=1:1, *c*-PNIPAM=5 g/L; (○) d7-PNIPAM/h-SDS; (•) h-PNIPAM/d25-SDS; (□) h-PNIPAM/h-SDS; (+) pure SDS. Data are taken from reference (Lee

PNIPAM/SDS sample cPNIPAM/*c*SDS=1:00, *c*-PNIPAM=5 g/L; (○) h-PNIPAM/h-SDS; (•) d7- PNIPAM/h-SDS; (□) h-PNIPAM/d25-SDS; (+) pure SDS. Data are taken from reference (Lee

In order to determine the shape and geometric size of the particles we performed Cryo-TEM measurements for the samples at composition 1:100. The Cryo-TEM images are shown in

Fig. 6. Cryo-TEM micrographs. h-PNIPAM/h-SDS sample *c*PNIPAM=510-3g mL-1; *c*SDS=510-5g

Fig. 6 shows objects thate are rather polydisperse in size. The average size of moieties is in agreement with SANS data, giving value roughly 20-30 nm in radius. The TEM images are 2D projection of the particles, observed under different angles. Therefore, we believe that the structures we see in Fig. 6 are of more or less spherical shape. Individual micelles seen in

SANS experiments were not imaged in these samples, because of their small size.


s, cm

10-2 10-1

cSDS/cPNIPAM=1:100 (b)

*q*, A-1

h-PNIPAM-h-SDS h-PNIPAM-d25-SDS d7-PNIPAM-h-SDS pure h-SDS

10-2 10-1

 h-PNIPAM-h-SDS d7-PNIPAM-h-SDS h-PNIPAM-d25-SDS pure h-SDS

*q*, A-1

cSDS/cPNIPAM =1:1 (a) micellar SDS

10-3

& Cabane, 1997).

& Cabane, 1997).

mL-1. *c*SDS/cPNIPAM =1:100

Fig. 6.

10-2

10-1

*I*


s, cm

100

101

For a specific polymer/solvent system, phase separation could be induced by a change of the solvent. This particular case of spontaneous macrophase separation leading to formation of nano-sized droplets is frequently referred to as Ouzo effect (Ganachaud & Katz , 2005), although it has a variety of names. Some authors call this process solvent shifting (Brick et al., 2003; Van Keuren, 2004), solvent displacement(Potineni et al., 2003; Trimaille et al., 2003; Lince et al., 2008; Chu et al., 2008; Vega et al., 2008; Nguyen et al., 2008; Beck-Broichsitter et al., 2009) spontaneous emulsification (Gallardo et al., 1993; QuintanarGuerrero et al., 1997; Baimark et al., 2007; Tan et al., 2008; Katas et al., 2009) or micro/nano precipitation (Leroueil-Le Verger et al., 1998; Peracchia et al., 1999; Bilati et al., 2005; Leo et al., 2006; Legrand et al., 2006). In our previous papers (Panek et al., 2011a; Panek et al., 2011b) we have successfully tested this procedure: polymeric nanoparticles were prepared by mixing a polymer - poly(methyl methacrylate) or polystyrene - dispersed in an organic solvent with an aqueous solution of a surfactant (SDS). Since water is a bad solvent for either of the polymers they start to precipitate but the presence of a surfactant terminates the phase separation and nearly monodisperse nanoparticles appear with a typical size in the range of 50 to 300 nm. Finally the organic solvent is evaporated. We demonstrated that a variety of parameters such as polymer molar mass, surfactant hydrophobicity, solution temperature and composition influence the physico-chemical properties of nanoparticles formed in solution (Panek et al., 2011a; Panek et al., 2011b). Nevertheless, detailed information on nanoparticles structure is still missing.

Here we describe new experiments with static and dynamic light scattering that were conducted to get further insight on the internal structure of the nanoparticles. Using a combination of both methods, for the first time, we calculated such parameters as a structure factor *R*g/*R*h and density of nanoparticles (*ρ*). *R*g values were measured by static light scattering. In contrast with *R*h (Panek et al., 2011a) there is no detectable difference in the factor =*R*g/*R*h of nanoparticles made from ionic and non-ionic surfactants (Fig. 7).

Fig. 7. Histogram of structure factor ρ=*R*g/*R*h obtained from SLS/DLS data for PMMA(1) *c*P = 210-4 g mL-1 for various surfactants.

Polymeric Nanoparticles Stabilized by Surfactants: Controlled Phase Separation Approach 231

Fig. 9. Possible distribution of a surfactant (red color) and polymer (black color) inside of a

an important influence on the size of nanoparticles. Changing the mixing rate from 0.5 mL/min up to 2 mL/min makes two times smaller particles. At higher polymer molecular weights the influence of mixing rate is smaller. We conclude that lower mixing rate reduces the number of surfactant molecules in the neighborhood of polymeric nuclei formed after solvent shifting. Smaller number of surfactant molecules slow down stabilization of

The difference in composition ratio is responsible for molecular weight dependence of nanoparticle dimensions at constant mixing rate and polymer weight concentration (Fig. 10). The bigger molecular weight of a polymer the smaller its molar concentration in mixed solution that leads to a decrease in composition ratio which governs the nanoparticle dimensions. At higher molecular weights of the polymer, the tendency is reverse, showing the growth of sizes (Fig. 10). One of the possible explanations is that macromolecules with

106 107

Fig. 10. Dependence of hydrodynamic radius of nanoparticles on molecular weight of polymer for PS *c*PS = 210-4 g mL-1 and SDS *c*SDS = 510-3 g mL-1 system at different mixing

MW/ g mol-1

polymeric nuclei thus leading to forming bigger nanoparticles.

rate. (■) 2 mL/min; (○) 1 mL/min; (▲) 0.5 mL/min

hydrodynamic radius Rh/ nm

nanoparticle by SANS data.

According to Burchard (Burchard, 1999) this generalized ratio is of special interest for establishing the particle architecture. It is varying in the range 0.8-1.4. The lowest value is for a Triton surfactant, where the particle behaves as a hard sphere ( is close to 0.778). The value of for CTAB, SDS, Brij 97 and 98 is about 1.2-1.4. This is characteristic for several models, in particular for branched polymers, soft spheres and dendrimers.

In contrast, the density of nanoparticles is very sensitive to the nature of a surfactant (Fig. 8). The density of nanoparticles was calculated by dividing their mass obtained from static light scattering by their volume based on the *R*g. Nanoparticles composed of low molecular ionic surfactants have almost two-fold higher density then the ones with polymeric non-ionic surfactants. Since all polymeric surfactants are diblock copolymers, we can assume that polymers can't adopt maximum packing structure due to steric factors.

Fig. 8. Histogram of density obtained from SLS/DLS data for PMMA(1) *c*PMMA = 210-4 g mL-1 for various surfactants.

Obviously some voids should be inside. Such conclusion is in agreement with previous SANS and Cryo-TEM (Panek et al., 2011a). Analysis of the SANS curves supports neither a core-shell structure model of the nanoparticles nor a polymeric sphere with surfactant inclusions. Nevertheless a closer inspection of some micrographs reveals the presence of thin white hallo around a nanoparticle. Possible distribution of surfactant inside of a nanoparticle is presented on Fig. 9. The permanent entrapment of a surfactant inside nanoparticle may occur because the polymer (PMMA or PS) is in the glassy state. Plausibility of such scenario has been proven by J. Kriz *et al*. (Kriz et al., 1996) who demonstrated that the mobility of PS moieties in the core of polystyrene-blockpoly(methacrylic acid) (PS-PMAc) micelles is significantly decreased, which indicates that the polymer including the surfactant inside a micelle is vitrified.

#### **5.2 Influence of mixing rate**

The effect of mixing rate (i.e. the rate at which the water solution of a surfactant is delivered into the organic solution of a polymer) on the self-assembled nanoparticles formed in the PS/SDS mixed solutions was also investigated (Fig. 10). The molecular weight of PS was varied in the range 0.9 - 30 106 g mol-1. At low molecular weights of PS, the mixing rate has

According to Burchard (Burchard, 1999) this generalized ratio is of special interest for establishing the particle architecture. It is varying in the range 0.8-1.4. The lowest value is for a Triton surfactant, where the particle behaves as a hard sphere ( is close to 0.778). The value of for CTAB, SDS, Brij 97 and 98 is about 1.2-1.4. This is characteristic for several

In contrast, the density of nanoparticles is very sensitive to the nature of a surfactant (Fig. 8). The density of nanoparticles was calculated by dividing their mass obtained from static light scattering by their volume based on the *R*g. Nanoparticles composed of low molecular ionic surfactants have almost two-fold higher density then the ones with polymeric non-ionic surfactants. Since all polymeric surfactants are diblock copolymers, we can assume that

> **F68(n) 8400 g/mol**

 **B98(n) 1150 g/mol**

 **B97(n) 709 g/mol**

 **CTAB(+) 384 g/mol**

Fig. 8. Histogram of density obtained from SLS/DLS data for PMMA(1) *c*PMMA = 210-4 g

Obviously some voids should be inside. Such conclusion is in agreement with previous SANS and Cryo-TEM (Panek et al., 2011a). Analysis of the SANS curves supports neither a core-shell structure model of the nanoparticles nor a polymeric sphere with surfactant inclusions. Nevertheless a closer inspection of some micrographs reveals the presence of thin white hallo around a nanoparticle. Possible distribution of surfactant inside of a nanoparticle is presented on Fig. 9. The permanent entrapment of a surfactant inside nanoparticle may occur because the polymer (PMMA or PS) is in the glassy state. Plausibility of such scenario has been proven by J. Kriz *et al*. (Kriz et al., 1996) who demonstrated that the mobility of PS moieties in the core of polystyrene-blockpoly(methacrylic acid) (PS-PMAc) micelles is significantly decreased, which indicates that

The effect of mixing rate (i.e. the rate at which the water solution of a surfactant is delivered into the organic solution of a polymer) on the self-assembled nanoparticles formed in the PS/SDS mixed solutions was also investigated (Fig. 10). The molecular weight of PS was varied in the range 0.9 - 30 106 g mol-1. At low molecular weights of PS, the mixing rate has

 **HDPC(-) 320 g/mol**

 **SDS(-) 288 g/mol**

the polymer including the surfactant inside a micelle is vitrified.

models, in particular for branched polymers, soft spheres and dendrimers.

polymers can't adopt maximum packing structure due to steric factors.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

mL-1 for various surfactants.

**5.2 Influence of mixing rate** 

density, g/cm3

Fig. 9. Possible distribution of a surfactant (red color) and polymer (black color) inside of a nanoparticle by SANS data.

an important influence on the size of nanoparticles. Changing the mixing rate from 0.5 mL/min up to 2 mL/min makes two times smaller particles. At higher polymer molecular weights the influence of mixing rate is smaller. We conclude that lower mixing rate reduces the number of surfactant molecules in the neighborhood of polymeric nuclei formed after solvent shifting. Smaller number of surfactant molecules slow down stabilization of polymeric nuclei thus leading to forming bigger nanoparticles.

The difference in composition ratio is responsible for molecular weight dependence of nanoparticle dimensions at constant mixing rate and polymer weight concentration (Fig. 10). The bigger molecular weight of a polymer the smaller its molar concentration in mixed solution that leads to a decrease in composition ratio which governs the nanoparticle dimensions. At higher molecular weights of the polymer, the tendency is reverse, showing the growth of sizes (Fig. 10). One of the possible explanations is that macromolecules with

Fig. 10. Dependence of hydrodynamic radius of nanoparticles on molecular weight of polymer for PS *c*PS = 210-4 g mL-1 and SDS *c*SDS = 510-3 g mL-1 system at different mixing rate. (■) 2 mL/min; (○) 1 mL/min; (▲) 0.5 mL/min

Polymeric Nanoparticles Stabilized by Surfactants: Controlled Phase Separation Approach 233

We have exploited stopped-flow technique combined with SAXS to monitor early stages of nucleation on the time scale of seconds (Fig. 11). The main difference from solvent-shifting experiments described above is absence of macroscopic fluxes and solution inhomogenuities caused by mixing. In stopped-flow experiments very fast mixing setup provides solution

Fig. 11. The temporal evolution of the SAXS intensity for pNMGL (*c*p/*c*surf=2.0) system.

The aim of the experiment was to measure the kinetics of self-assembly of pH-sensitive polymeric nanoparticles stabilized by surfactants. The four types of pH-sensitive hydrophobic polymers that have been used in our research for the growth of nanoparticles were: (a) poly(N-methacryloyl-L-valine) (pNMV); (b) poly(N-methacryloyl glycyl-Lphenylalanyl-L-leucinyl-glycine) (pNMGPLG), and (c) poly(N-methacryloyl glycyl-Lleucine) (pNMGL). The extent of macrophase separation was controlled by the surfactants Brij 97, and Brij 98. The surfactants were different in the length of hydrophilic PEO chain.

Fig. 11 displays the intensity of scattered X-rays from the mixture of aqueous solution of Brij 98 surfactant and pNMGL (*c*p/*c*surf=2.0) solution as a function of time. The SAXS technique is commonly used to extract information on molecular architecture and size of nano-objects in solution that can be performed by the analysis of Kratky or Guinier plots.(Glatter & Kratky , 1982) The scattering from nanostructures reveals three regions in the dependence of scattering intensity on scattering vector, *I* ~ *q*α with different behaviors characteristic for the various length scales. At low *q-*range, the "Guinier" regime (*qR*g < 1) is usually attained. Middle *q*-range is usually sensitive to the shape of the scattering object; α = -4 stands for hard spheres, -2 stands for planar objects, and α = -1 stands for rod-like structures. It was proved experimentally that in some cases, the α value in middle *q*-range is not integral but rather fractional. This situation corresponds to the so-called fractal structure. In the high *q*range, local stiffness of macromolecules (due to shorter length scales probed) can be revealed with *I*(*q*) ~ *q*-1. Nanoparticles with sharp interface and smooth surface obey a *q*-4 law

with uniform density where nanoparticles are growing in time.

**6.1 The self-assembly of nanoparticles** 

that is usually referred to as "Porod" behavior.

extra large molecular weight have very low diffusion which limits the probability of surfactant molecules to find enough polymer molecules in surroundings. Fast diffusion of water molecules into polymer interface during mixing forms a surfactant-abandoned layer. In these conditions, it's energetically more favorable for a polymer chain to merge with other polymer molecules and form bigger nanoparticles in comparison with solutions of the same surfactant concentration and mixing rate.

The results presented so far show that the self-assembly in the mixed polymer/surfactant systems is rather complex. Mixing of the surfactant solutions with the polymer solutions in organic solvent results in the formation of nanoparticles, whose size can be tuned by changing the relative amounts of surfactant and polymer, as well as mixing rate.

## **6. Controlled phase separation induced by a change in pH**

Changing of pH for a pH sensitive polymer is another way to construct polymeric nanoparticles. The pH of the solution is gradually changed so that the polymer which is in the beginning in an environment where it is molecularly soluble (i.e. in the mostly ionized form), starts to precipitate. Adding a surfactant terminates the phase separation in a controlled way leading to formation of well-defined nanoparticles with low polydispersity. We have demonstrated this procedure using a pH-sensitive hydrophobic polymer – *i.e.* poly(N-methacryloyl-L-valine) (pNMV), the extent of macrophase separation was controlled by the amphiphilic molecule Brij98. We have shown previously (Filippov et al., 2008; Filippov et al., 2010) that in a certain range of concentration and composition of the polymer/amphiphile system very monodisperse particles with size ca. 50 nm could be reproducibly prepared after a change of pH from 7 to 3.5. This change is reversible and the nanoparticles can be repeatedly created and dissolved by variation in pH. This type of particles can be very useful, since they may be able to solubilize hydrophobic drugs in large amounts and release them after a change of pH. For example the pH of stomach is 1 to 3 (nanoparticles associated), while the pH of duodenum is 7 to 8 (particles dissolved, drug released).

The nucleation of these nanoparticles has not yet been investigated. The early stages of nucleation in such systems determine the nanoscopic structure of the particles that is so far unknown, but important for their envisaged applications. Recently, new technical possibilities to study the kinetics of self-assembly were developed. Primary, it concerns socalled stopped-flow experiments combined with small-angle scattering equipments. (Narayanan et al., 2001; Grillo et al., 2003; Panine et al., 2006). A variety of nanostructures were tested by time-resolved light scattering, SAXS, and SANS. The kinetics of micelles-tovesicles (Schmolzer et al., 2002; Weiss et al., 2005; Weiss et al., 2008; Shen et al., 1989) and lamellar-to-microemulsion (Deen et al., 2009; Tabor et al., 2009) phase transition was studied in details. Another challenging areas for time-resolved experiments are the life time of micelles (Lund et al., 2009) monomers-micelles exchange rate (Eastoe et al., 1998; Tucker et al., 2009), and nucleation of gold (Abecassis et al., 2007; Abecassis et al., 2008) and mineral nanoparticles. (Pontoni et al., 2002; Ne et al., 2003; Bolze et al., 2004). For further details on this topic, the reader is referred to reviews on the application of stopped-flow technique in SANS and SAXS.(Grillo et al., 2009; Gradzielski et al., 2003; Gradzielski et al., 2004)

We have exploited stopped-flow technique combined with SAXS to monitor early stages of nucleation on the time scale of seconds (Fig. 11). The main difference from solvent-shifting experiments described above is absence of macroscopic fluxes and solution inhomogenuities caused by mixing. In stopped-flow experiments very fast mixing setup provides solution with uniform density where nanoparticles are growing in time.

Fig. 11. The temporal evolution of the SAXS intensity for pNMGL (*c*p/*c*surf=2.0) system.

The aim of the experiment was to measure the kinetics of self-assembly of pH-sensitive polymeric nanoparticles stabilized by surfactants. The four types of pH-sensitive hydrophobic polymers that have been used in our research for the growth of nanoparticles were: (a) poly(N-methacryloyl-L-valine) (pNMV); (b) poly(N-methacryloyl glycyl-Lphenylalanyl-L-leucinyl-glycine) (pNMGPLG), and (c) poly(N-methacryloyl glycyl-Lleucine) (pNMGL). The extent of macrophase separation was controlled by the surfactants Brij 97, and Brij 98. The surfactants were different in the length of hydrophilic PEO chain.

## **6.1 The self-assembly of nanoparticles**

232 Smart Nanoparticles Technology

extra large molecular weight have very low diffusion which limits the probability of surfactant molecules to find enough polymer molecules in surroundings. Fast diffusion of water molecules into polymer interface during mixing forms a surfactant-abandoned layer. In these conditions, it's energetically more favorable for a polymer chain to merge with other polymer molecules and form bigger nanoparticles in comparison with solutions of the

The results presented so far show that the self-assembly in the mixed polymer/surfactant systems is rather complex. Mixing of the surfactant solutions with the polymer solutions in organic solvent results in the formation of nanoparticles, whose size can be tuned by

Changing of pH for a pH sensitive polymer is another way to construct polymeric nanoparticles. The pH of the solution is gradually changed so that the polymer which is in the beginning in an environment where it is molecularly soluble (i.e. in the mostly ionized form), starts to precipitate. Adding a surfactant terminates the phase separation in a controlled way leading to formation of well-defined nanoparticles with low polydispersity. We have demonstrated this procedure using a pH-sensitive hydrophobic polymer – *i.e.* poly(N-methacryloyl-L-valine) (pNMV), the extent of macrophase separation was controlled by the amphiphilic molecule Brij98. We have shown previously (Filippov et al., 2008; Filippov et al., 2010) that in a certain range of concentration and composition of the polymer/amphiphile system very monodisperse particles with size ca. 50 nm could be reproducibly prepared after a change of pH from 7 to 3.5. This change is reversible and the nanoparticles can be repeatedly created and dissolved by variation in pH. This type of particles can be very useful, since they may be able to solubilize hydrophobic drugs in large amounts and release them after a change of pH. For example the pH of stomach is 1 to 3 (nanoparticles associated), while the pH of duodenum is 7 to 8 (particles dissolved, drug

The nucleation of these nanoparticles has not yet been investigated. The early stages of nucleation in such systems determine the nanoscopic structure of the particles that is so far unknown, but important for their envisaged applications. Recently, new technical possibilities to study the kinetics of self-assembly were developed. Primary, it concerns socalled stopped-flow experiments combined with small-angle scattering equipments. (Narayanan et al., 2001; Grillo et al., 2003; Panine et al., 2006). A variety of nanostructures were tested by time-resolved light scattering, SAXS, and SANS. The kinetics of micelles-tovesicles (Schmolzer et al., 2002; Weiss et al., 2005; Weiss et al., 2008; Shen et al., 1989) and lamellar-to-microemulsion (Deen et al., 2009; Tabor et al., 2009) phase transition was studied in details. Another challenging areas for time-resolved experiments are the life time of micelles (Lund et al., 2009) monomers-micelles exchange rate (Eastoe et al., 1998; Tucker et al., 2009), and nucleation of gold (Abecassis et al., 2007; Abecassis et al., 2008) and mineral nanoparticles. (Pontoni et al., 2002; Ne et al., 2003; Bolze et al., 2004). For further details on this topic, the reader is referred to reviews on the application of stopped-flow technique in

SANS and SAXS.(Grillo et al., 2009; Gradzielski et al., 2003; Gradzielski et al., 2004)

changing the relative amounts of surfactant and polymer, as well as mixing rate.

**6. Controlled phase separation induced by a change in pH** 

same surfactant concentration and mixing rate.

released).

Fig. 11 displays the intensity of scattered X-rays from the mixture of aqueous solution of Brij 98 surfactant and pNMGL (*c*p/*c*surf=2.0) solution as a function of time. The SAXS technique is commonly used to extract information on molecular architecture and size of nano-objects in solution that can be performed by the analysis of Kratky or Guinier plots.(Glatter & Kratky , 1982) The scattering from nanostructures reveals three regions in the dependence of scattering intensity on scattering vector, *I* ~ *q*α with different behaviors characteristic for the various length scales. At low *q-*range, the "Guinier" regime (*qR*g < 1) is usually attained. Middle *q*-range is usually sensitive to the shape of the scattering object; α = -4 stands for hard spheres, -2 stands for planar objects, and α = -1 stands for rod-like structures. It was proved experimentally that in some cases, the α value in middle *q*-range is not integral but rather fractional. This situation corresponds to the so-called fractal structure. In the high *q*range, local stiffness of macromolecules (due to shorter length scales probed) can be revealed with *I*(*q*) ~ *q*-1. Nanoparticles with sharp interface and smooth surface obey a *q*-4 law that is usually referred to as "Porod" behavior.

Polymeric Nanoparticles Stabilized by Surfactants: Controlled Phase Separation Approach 235

To extract further information on the kinetics of nanoparticle formation, the radius of gyration was calculated and compared for different polymers and surfactants. The results are shown in Fig. 13-15. We observe that the radius strongly depends on the composition ratio. Moreover, two distinct regimes separated in time can be observed. During the first seconds, there is a rapid increase in the *R*g value. This behavior could be explained as a nucleation regime when preliminary nuclei are formed. After a short period of time that depends also on composition ratio, the *R*g value of nanoparticles increases by consumption of the remaining surfactant molecules in solution, thus defining the growth regime. The higher the composition ratio, the growth regime is more expressive (Fig. 13, 14). Nevertheless, sometimes a decrease in *R*g is observed at the first seconds. We can assume that such scenario could be realized when several aggregates of pearl-necklace micelles

0 100 200 300

**cBrij98/cpNMGL=2.0 cBrij98/cpNMGL=1.0 cBrij98/cpNMGL=0.5 cBrij98/cpNMGL=0.25**

t, sec

Fig. 14. The temporal evolution of the exponent value α for a pNMV–Brij 98 system at

ratio (1.0) could be arranged into a master-curve (inset of Fig. 15).

The conclusion that the growth regime is governed by a surfactant only is further supported from a comparison of kinetic curves of different polymers but the same composition ratio (Fig. 15). The polymers of different nature but the same surfactant (Brij 98) and composition

disassembling prior to formation of original nuclei.

different composition ratio cBrij98/cpNMGL.

*R*g, nm

Several things should be noted. *I*(*q*) value at the lowest experimental *q* grows with time, which is clearly an indication of particle growth (Fig. 11). For the highest composition ratio cBrij98/cpNMGL =2.0, in the middle *q*-range, the exponent value α is growing from -2.2 at the beginning up to -3.4 for the longer time (Fig. 12). The *q*-2.2 dependence of *I*(*q*) observed on early stages of self-assembly is attributed to the scattering from a loose, fractal structure. In contrast, α value of -3.4 suggests large compact objects. Thus using TRSAXS we can monitor the self-assembly of nanoparticles when particles transform through fractal structure with loose surface into hard spheres with sharp interface.

Fig. 12. The temporal evolution of the exponent value α for a pNMGL–Brij 98 system at different composition ratio cBrij98/cpNMGL.

For the lowest composition ratio cBrij98/cpNMGL =0.25, the behavior changes greatly. *I*(*q*) value at the lowest *q* as well as the α exponent do not evolve with time (Fig. 12). Obviously, nanoparticles have been already formed prior to the first measurement. Those nanoparticles do not have sharp boundaries and have fractal structure that is a characteristic for loose entities. Surfactant molecules are not enough to cover the whole nanoparticles.

Fig. 13. The temporal evolution of the exponent value α for a pNMV –Brij 97 system at different composition ratio cBrij97/cpNMV.

Several things should be noted. *I*(*q*) value at the lowest experimental *q* grows with time, which is clearly an indication of particle growth (Fig. 11). For the highest composition ratio cBrij98/cpNMGL =2.0, in the middle *q*-range, the exponent value α is growing from -2.2 at the beginning up to -3.4 for the longer time (Fig. 12). The *q*-2.2 dependence of *I*(*q*) observed on early stages of self-assembly is attributed to the scattering from a loose, fractal structure. In contrast, α value of -3.4 suggests large compact objects. Thus using TRSAXS we can monitor the self-assembly of nanoparticles when particles transform through fractal structure with

100 101 10<sup>2</sup>

Fig. 12. The temporal evolution of the exponent value α for a pNMGL–Brij 98 system at

entities. Surfactant molecules are not enough to cover the whole nanoparticles.

time, sec

For the lowest composition ratio cBrij98/cpNMGL =0.25, the behavior changes greatly. *I*(*q*) value at the lowest *q* as well as the α exponent do not evolve with time (Fig. 12). Obviously, nanoparticles have been already formed prior to the first measurement. Those nanoparticles do not have sharp boundaries and have fractal structure that is a characteristic for loose

0 50 100 150 200 250

time, sec

Fig. 13. The temporal evolution of the exponent value α for a pNMV –Brij 97 system at

**cBrij97/cPNMV=2.0 cBrij97/cPNMV=1.0 cBrij97/cPNMV=0.5 cBrij97/cPNMV=0.25**

**cBrij98/cpNMGL=2.0 cBrij98/cpNMGL=1.0 cBrij98/cpNMGL=0.5 cBrij98/cpNMGL=0.25**

loose surface into hard spheres with sharp interface.


different composition ratio cBrij97/cpNMV.

*R*g, nm

different composition ratio cBrij98/cpNMGL.

To extract further information on the kinetics of nanoparticle formation, the radius of gyration was calculated and compared for different polymers and surfactants. The results are shown in Fig. 13-15. We observe that the radius strongly depends on the composition ratio. Moreover, two distinct regimes separated in time can be observed. During the first seconds, there is a rapid increase in the *R*g value. This behavior could be explained as a nucleation regime when preliminary nuclei are formed. After a short period of time that depends also on composition ratio, the *R*g value of nanoparticles increases by consumption of the remaining surfactant molecules in solution, thus defining the growth regime. The higher the composition ratio, the growth regime is more expressive (Fig. 13, 14). Nevertheless, sometimes a decrease in *R*g is observed at the first seconds. We can assume that such scenario could be realized when several aggregates of pearl-necklace micelles disassembling prior to formation of original nuclei.

Fig. 14. The temporal evolution of the exponent value α for a pNMV–Brij 98 system at different composition ratio cBrij98/cpNMGL.

The conclusion that the growth regime is governed by a surfactant only is further supported from a comparison of kinetic curves of different polymers but the same composition ratio (Fig. 15). The polymers of different nature but the same surfactant (Brij 98) and composition ratio (1.0) could be arranged into a master-curve (inset of Fig. 15).

Polymeric Nanoparticles Stabilized by Surfactants: Controlled Phase Separation Approach 237

nanoparticles is a two stage process. In the beginning a nucleation stage occurs followed by a growth regime. The hydrophilicity/hydrophobicity of surfactants plays an important role

We gratefully acknowledge the European Synchrotron Radiation Facility (Grenoble, France) for the provision of synchrotron beam time (SC2883 and SC3113). This work was supported by the Grant Agency of the Czech Republic (202/09/2078) and also by Grant No. IAA400500805 of the Grant Agency of the Academy of Sciences of the Czech Republic. Also, we would like to thank Prof. Katarina Edwards and Dr. Goran Karlsson, Uppsala University, Department of Physical and Analytical Chemistry for help with Cryo-TEM

Abecassis, B., Testard, F., Spalla, O. & Barboux, P. (2007). Probing in situ the nucleation and

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study of the effect of a double hydrophilic block-copolymer on the formation of CaCO3 from a supersaturated salt solution*. Journal of Colloid and Interface Science,* 

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in the formation of nanoparticles.

pp. 1723-1727.

**8. Acknowledgements**

experiments.

**9. References** 

178.

7146.

24, pp. 67-75.

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In contrast, the system with Brij 97 (red circles) is undoubtedly couldn't be superimposed into the master curve. We conclude that the number and hydrophobicity of monomeric units of a polymer determine the final size of a nanoparticle whereas the growth nucleation rates are controlled by the nature and amount of a surfactant.

Fig. 15. The temporal evolution of the exponent value α for a pNMV, pNMGL, and pNMGPLG +Brij 98, 97 system at composition ratio 1.0. Inset: master curve.

## **7. General conclusions**

We have systematically investigated nanoparticles prepared by *controlled phase separation approach*. On the basis of our research we have established that the most important parameter for steady-state nanoparticles dimensions is the composition ratio c(surfactant)/c(polymer). Our study demonstrates that full grown nanoparticles have a spherical shape. For the first time we have investigated the architecture of nanoparticles prepared by the solvent shifting method. The density, and factor =*R*g/*R*h measurement together with SANS experiment shows that nanoparticles are entities with uniform density and without internal structure. Polymeric and surfactant molecules are evenly distributed within a nanoparticle.

When macroscopic non-equilibrium hydrodynamic forces are involved into nanoparticle formation, the nature of the surfactant, its hydrophobicity and charge, insignificantly influences the nanoparticles sizes. A mixing rate is of primary importance for that case.

When hydrodynamic fluxes are eliminated by fast mixing again, the surfactant/polymer composition ratio is of primary importance in nanoparticle formation, thus confirming previous results. Excess of a surfactant results in much faster kinetics in comparison with the solution where a polymer is in excess. Our results suggest that the formation of the nanoparticles is a two stage process. In the beginning a nucleation stage occurs followed by a growth regime. The hydrophilicity/hydrophobicity of surfactants plays an important role in the formation of nanoparticles.

## **8. Acknowledgements**

236 Smart Nanoparticles Technology

In contrast, the system with Brij 97 (red circles) is undoubtedly couldn't be superimposed into the master curve. We conclude that the number and hydrophobicity of monomeric units of a polymer determine the final size of a nanoparticle whereas the growth nucleation rates

0 50 100 150 200 250 300

time, sec

We have systematically investigated nanoparticles prepared by *controlled phase separation approach*. On the basis of our research we have established that the most important parameter for steady-state nanoparticles dimensions is the composition ratio c(surfactant)/c(polymer). Our study demonstrates that full grown nanoparticles have a spherical shape. For the first time we have investigated the architecture of nanoparticles

together with SANS experiment shows that nanoparticles are entities with uniform density and without internal structure. Polymeric and surfactant molecules are evenly distributed

When macroscopic non-equilibrium hydrodynamic forces are involved into nanoparticle formation, the nature of the surfactant, its hydrophobicity and charge, insignificantly influences the nanoparticles sizes. A mixing rate is of primary importance for that case.

When hydrodynamic fluxes are eliminated by fast mixing again, the surfactant/polymer composition ratio is of primary importance in nanoparticle formation, thus confirming previous results. Excess of a surfactant results in much faster kinetics in comparison with the solution where a polymer is in excess. Our results suggest that the formation of the

Fig. 15. The temporal evolution of the exponent value α for a pNMV, pNMGL, and pNMGPLG +Brij 98, 97 system at composition ratio 1.0. Inset: master curve.

prepared by the solvent shifting method. The density, and factor

*R*g, nm

100 200

time, sec **cBrij98/cpNMV=1.0 cBrij97/cpNMV=1.0 cBrij98/cpNMGPLG=1.0 cBrij98/cpNMGL=1.0**

**cBrij98/cpNMV=1.0 cBrij97/cpNMV=1.0 cBrij98/cpNMGPLG=1.0 cBrij98/cpNMGL=1.0**

=*R*g/*R*h measurement

are controlled by the nature and amount of a surfactant.

**7. General conclusions** 

within a nanoparticle.

*R*g, nm

We gratefully acknowledge the European Synchrotron Radiation Facility (Grenoble, France) for the provision of synchrotron beam time (SC2883 and SC3113). This work was supported by the Grant Agency of the Czech Republic (202/09/2078) and also by Grant No. IAA400500805 of the Grant Agency of the Academy of Sciences of the Czech Republic. Also, we would like to thank Prof. Katarina Edwards and Dr. Goran Karlsson, Uppsala University, Department of Physical and Analytical Chemistry for help with Cryo-TEM experiments.

## **9. References**


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**0**

**11**

*Japan*

**and a Liquid Crystal**

Akihiko Matsuyama

**Phase Separations in Mixtures of a Nanoparticle**

Liquid crystal suspensions including various micro- and nano-colloidal particles have recently been received great attention for many practical applications such as nanosensors and devices, etc. When large colloidal particles of micronscale are dispersed in a uniform nematic liquid crystal phase, the colloidal particles disturb a long-range orientational order of the nematic phase. For a strong anchoring between the colloidal surface and a liquid crystal, different defect structures such as hedgehogs or Saturn rings can appear around a single colloidal particle, due to strong director deformations.(Fukuda, 2009; Skarabot et.al., 2008; Stark, 2001) Experiments have also shown two-dimensional crystalline structures of colloidal particles.(Loudet et. al., 2004; Musevic et. al., 2006; Nazarenko et. al., 2001; Pouling et.al., 1997; Yada et. al., 2004; Zapotocky et.al., 1999) On the other hand, under a weak surface anchoring between the colloidal surface and a liquid crystal, the coupling to the orientational elasticity of the liquid crystals tends to expel the colloidal particles and the suspension shows a phase separation into an almost pure nematic phase coexisting with a colloidal rich phase.(Anderson et.al., 2001; Pouling et. al., 1994) Such phase separations induced by a nematic ordering have also been discussed in flexible polymers dispersed in a nematic liquid crystal.(Chiu & Kyu, 1999; Das & Ray, 2005; Dubaut et.al., 1980; Matsuyama & Kato, 1996; Shen & Kyu, 1995)

If the colloidal particles are ∼1-10nm in diameter, these "nanoparticles" are too small to distort the nematic director and defects do not form. In this case, the system can show a homogeneous single phase or phase separations,(Anderson et.al., 2001; Anderson & Terentjev, 2001; Caggioni et. al., 2005; Meeker et. al., 2000; Yamamoto & Tanaka, 2001) depending on the interaction between a colloidal particle and a liquid crystal. Although the theoretical progress on the description of a director around colloidal particles with strong anchoring conditions has been noticeable,(Araki & Tanaka, 2004; Fukuda & Yokoyama, 2005; Kuksenok et.al., 1996; Lubensky et. al., 1998; Yamamoto, 2001) little theoretical work exists in phase

In this chapter, we focus on nanoparticles dispersed in liquid crystals and discuss phase separations and phase behaviors in mixtures of a nanoparticle and a liquid crystal. It is mainly based on authors' original theoretical works obtained within recent years. The nanoparticles have a variety in the shape such as spherical and rodlike. In this chapter, we focus on (1) mixtures of a liquid crystal and a spherical nanoparticle and (2) mixtures of a liquid crystal and a rodlike nanoparticle, such as carbon nanotube. The topics are currently interested in the

separations.(Popa-Nita et. al., 2006; Pouling et. al., 1994)

advanced fields of nanoparticles and fundamental sciences.

**1. Introduction**

*Department of Bioscience and Bioinformatics, Kyusyu Institute of Technology*


## **Phase Separations in Mixtures of a Nanoparticle and a Liquid Crystal**

## Akihiko Matsuyama

*Department of Bioscience and Bioinformatics, Kyusyu Institute of Technology Japan*

### **1. Introduction**

240 Smart Nanoparticles Technology

Weiss, T.M., Narayanan, T. & Gradzielski, M. (2008). Dynamics of spontaneous vesicle

Yan, C. H. , Yuan, X. B., Kang, C. S., Zhao, Y. H. , Liu, J., Guo, Y. S. , Lu, J. , Pu P. Y. & Sheng

J. (2008). *Journal of Applied Polymer Science*, Vol. 110, pp. 2446-2452.

pp. 3759-3766.

formation in fluorocarbon and hydrocarbon surfactant mixtures. *Langmuir*, Vol. 24,

Liquid crystal suspensions including various micro- and nano-colloidal particles have recently been received great attention for many practical applications such as nanosensors and devices, etc. When large colloidal particles of micronscale are dispersed in a uniform nematic liquid crystal phase, the colloidal particles disturb a long-range orientational order of the nematic phase. For a strong anchoring between the colloidal surface and a liquid crystal, different defect structures such as hedgehogs or Saturn rings can appear around a single colloidal particle, due to strong director deformations.(Fukuda, 2009; Skarabot et.al., 2008; Stark, 2001) Experiments have also shown two-dimensional crystalline structures of colloidal particles.(Loudet et. al., 2004; Musevic et. al., 2006; Nazarenko et. al., 2001; Pouling et.al., 1997; Yada et. al., 2004; Zapotocky et.al., 1999) On the other hand, under a weak surface anchoring between the colloidal surface and a liquid crystal, the coupling to the orientational elasticity of the liquid crystals tends to expel the colloidal particles and the suspension shows a phase separation into an almost pure nematic phase coexisting with a colloidal rich phase.(Anderson et.al., 2001; Pouling et. al., 1994) Such phase separations induced by a nematic ordering have also been discussed in flexible polymers dispersed in a nematic liquid crystal.(Chiu & Kyu, 1999; Das & Ray, 2005; Dubaut et.al., 1980; Matsuyama & Kato, 1996; Shen & Kyu, 1995)

If the colloidal particles are ∼1-10nm in diameter, these "nanoparticles" are too small to distort the nematic director and defects do not form. In this case, the system can show a homogeneous single phase or phase separations,(Anderson et.al., 2001; Anderson & Terentjev, 2001; Caggioni et. al., 2005; Meeker et. al., 2000; Yamamoto & Tanaka, 2001) depending on the interaction between a colloidal particle and a liquid crystal. Although the theoretical progress on the description of a director around colloidal particles with strong anchoring conditions has been noticeable,(Araki & Tanaka, 2004; Fukuda & Yokoyama, 2005; Kuksenok et.al., 1996; Lubensky et. al., 1998; Yamamoto, 2001) little theoretical work exists in phase separations.(Popa-Nita et. al., 2006; Pouling et. al., 1994)

In this chapter, we focus on nanoparticles dispersed in liquid crystals and discuss phase separations and phase behaviors in mixtures of a nanoparticle and a liquid crystal. It is mainly based on authors' original theoretical works obtained within recent years. The nanoparticles have a variety in the shape such as spherical and rodlike. In this chapter, we focus on (1) mixtures of a liquid crystal and a spherical nanoparticle and (2) mixtures of a liquid crystal and a rodlike nanoparticle, such as carbon nanotube. The topics are currently interested in the advanced fields of nanoparticles and fundamental sciences.

field generated by the nanoparticle interacts with the order parameter of the liquid crystal

Phase Separations in Mixtures of a Nanoparticle and a Liquid Crystal 243

where Δ*�* is the dielectric anisotropy of the aligned liquid crystal, *ρNP* is the number density of nanoparticles, and *p* is the electric dipole moment, *SLC* (*SNP*) is the scalar orientational order parameters of the liquid crystals (nanoparticles). This free energy can predict the enhancement in the isotropic-nematic transition temperature and in the response to an applied electric field. The attractive interaction between the liquid crystal and the nanoparticle through the order parameters is important to understand the phase behaviors. The next section we consider the

We consider mixtures of a spherical nanoparticle and a liquid crystal. By taking into account the ordering of liquid crystals and nanoparticles, we can expect six possible phases in this mixture. Figure 2 shows the schematically illustrated six phases. The isotropic (I) phase means both liquid crystals and nanoparticles have no positional and orientational order. In the nematic (N) phase, liquid crystals have an orientational order, while nanoparticles have no positional order. Similarly, in the smectic *A* (A) phase, liquid crystals have a smectic *A* order, while nanoparticles have no positional order. When the concentration of nanoparticles is high, we may have a crystalline (C) phase of nanoparticles dispersed in an isotopic matrix of liquid crystals. We can also expect a nematic-crystal (NC) phase and a smectic *A*-crystal (AC) phase, where nanoparticles form a crystalline structure dispersed in a nematic and a smectic *A* matrix of liquid crystals. To describe these phases, depending on temperature and concentration, we take into account three scalar order parameters: an orientational order parameter for a nematic phase, one-dimensional translational order parameter for a smectic *A* phase, and a

We consider a binary mixture of *Nc* spherical nano-colloidal particles of the diameter *Rc* and *Nr* low-molecular weight liquid crystal molecules (liquid crystals) of the length *l* and the diameter *d*. The volume of the liquid crystal and that of the nanoparticle are given by

of liquid crystals and colloidal particles with an orientation **u** (or its solid angle Ω) at a position **r**, respectively. The free energy *F* of the dispersion at the level of second virial approximation

*<sup>r</sup>* + ln *ρr*(**r**, **u**) − 1

)*βcr*(**r**;**r** � , **u**� )*d***r***d***r** � *d*Ω�

*<sup>c</sup>* + ln *ρc*(**r**) − 1

� , **u**�

� )*βcc*(**r**;**r** � )*d***r***d***r** � ,

� , **u**�

*<sup>c</sup>* , respectively. Let *ρr*(**u**,**r**) and *ρc*(**r**) be the number density

 *d***r***d*Ω

> � , **u**�

)*d***r***d*Ω*d***r** � *d*Ω� ,

, (2)

 *d***r**

)*βrr*(**r**, **u**;**r**

<sup>180</sup>*π�*0*�*2*R*<sup>3</sup> *SLCSNP*, (1)

*Fint* <sup>=</sup> <sup>−</sup> <sup>Δ</sup>*�ρNP <sup>p</sup>*<sup>2</sup>

**2.2 Phase ordering in mixtures of a spherical nanoparticle and a liquid crystal**

translational order parameter for a crystalline phase of nanoparticles.

is given by(Matsuyama & Hirashima, 2008a; Matsuyama, 2010a)

*ρr*(**r**, **u**) *βμ*◦

*<sup>ρ</sup>r*(**r**, **<sup>u</sup>**)*ρr*(**<sup>r</sup>**

*<sup>ρ</sup>c*(**r**)*ρc*(**<sup>r</sup>**

*<sup>ρ</sup>c*(**r**)*ρr*(**<sup>r</sup>**

**2.3 Free energy of mixtures of a spherical nanoparticle and a liquid crystal**

through the free energy

free energy to describe the phase separations.

*vr* = (*π*/4)*d*2*l* and *vc* = (*π*/6)*R*<sup>3</sup>

*βF*/*V* =

+ *ρc*(**r**) *βμ*◦

+ 1 2

+ 1 2

+

When the nanoparticles are dispersed in isotropic solvents, the system may show phase separations, or sodification, between a liquid and a crystalline phase, depending on temperature and concentration, etc. These phase separations are induced by a balance between steric repulsions and attractive dispersion forces. However, the nature of phase separations of nanoparticles dispersed in liquid crystalline solvents is quite difference. The key point is ordering of nanoparticles induced by liquid crystalline ordering. Depending on the interaction between nanoparticles and liquid crystals, we have a variety of phase separations.

The aim of this chapter is to introduce such a new kind of phase separations. We review recent mean field theories to describe phase separations (or phase diagrams) in mixtures of a nanoparticle and a liquid crystal and summarize the variety of phase separations in such nanoparticle dispersions, where liquid crystalline ordering (nematic and smectic A phases) and nanoparticle ordering compete. In Section 2, We discuss spherical nanoparticles dispersed in liquid crystals. Nanotubes dispersed in liquid crystals are discussed in Section 3. The effects of external forces, such as magnetic and electric fields, on the phase behaviors are also discussed in Section 3.

## **2. Spherical nanoparticles dispersed in liquid crystals**

### **2.1 Ferroelectric nanoparticles dispersed in nematic liquid crystals**

Small nanoparticles do not significantly perturb the nematic director. However, it has been recently discovered that ferroelectric nanoparticles can greatly enhance the physical properties of nematic liquid crystals. Recent experimental(Copic et.al., 2007; Li et. al., 2006) and theoretical(Kralj et.al., 2008; Lopatina & Selinger, 2009) studies have shown that low concentrations of ferroelectric nanoparticles (BaTiO3) increase the orientational order of a liquid crystal and increase the nematic-isotropic transition temperature, due to the coupling between the ferroelectric nanoparticle with electric dipole moment and the orientational order of liquid crystals.(Lopatina & Selinger, 2009)

Fig. 1. Nanoparticles dispersed in liquid crystal. (a) Nanoparticle with no electric dipole moment, in an isotropic phase. (b) Ferroelectric particle with dipole moment, which produces an electric field that interacts with orientational order of the nematic phase. Reproduced with permission from (Lopatina & Selinger, 2009) . Copyright 2009 American Physical Society.

As shown in Fig. 1, the orientational distribution of the nanoparticle dipole moment interacts with the orientational order of liquid crystals and stabilizes the nematic phase. The electric 2 Will-be-set-by-IN-TECH

When the nanoparticles are dispersed in isotropic solvents, the system may show phase separations, or sodification, between a liquid and a crystalline phase, depending on temperature and concentration, etc. These phase separations are induced by a balance between steric repulsions and attractive dispersion forces. However, the nature of phase separations of nanoparticles dispersed in liquid crystalline solvents is quite difference. The key point is ordering of nanoparticles induced by liquid crystalline ordering. Depending on the interaction between nanoparticles and liquid crystals, we have a variety of phase

The aim of this chapter is to introduce such a new kind of phase separations. We review recent mean field theories to describe phase separations (or phase diagrams) in mixtures of a nanoparticle and a liquid crystal and summarize the variety of phase separations in such nanoparticle dispersions, where liquid crystalline ordering (nematic and smectic A phases) and nanoparticle ordering compete. In Section 2, We discuss spherical nanoparticles dispersed in liquid crystals. Nanotubes dispersed in liquid crystals are discussed in Section 3. The effects of external forces, such as magnetic and electric fields, on the phase behaviors are also

Small nanoparticles do not significantly perturb the nematic director. However, it has been recently discovered that ferroelectric nanoparticles can greatly enhance the physical properties of nematic liquid crystals. Recent experimental(Copic et.al., 2007; Li et. al., 2006) and theoretical(Kralj et.al., 2008; Lopatina & Selinger, 2009) studies have shown that low concentrations of ferroelectric nanoparticles (BaTiO3) increase the orientational order of a liquid crystal and increase the nematic-isotropic transition temperature, due to the coupling between the ferroelectric nanoparticle with electric dipole moment and the orientational order

Fig. 1. Nanoparticles dispersed in liquid crystal. (a) Nanoparticle with no electric dipole moment, in an isotropic phase. (b) Ferroelectric particle with dipole moment, which produces an electric field that interacts with orientational order of the nematic phase. Reproduced with permission from (Lopatina & Selinger, 2009) . Copyright 2009 American Physical Society.

As shown in Fig. 1, the orientational distribution of the nanoparticle dipole moment interacts with the orientational order of liquid crystals and stabilizes the nematic phase. The electric

separations.

discussed in Section 3.

**2. Spherical nanoparticles dispersed in liquid crystals**

of liquid crystals.(Lopatina & Selinger, 2009)

**2.1 Ferroelectric nanoparticles dispersed in nematic liquid crystals**

field generated by the nanoparticle interacts with the order parameter of the liquid crystal through the free energy

$$F\_{int} = -\frac{\Delta\varepsilon\rho\_{NP}p^2}{180\pi\epsilon\_0\epsilon^2R^3}S\_{LC}S\_{NP\_\prime} \tag{1}$$

where Δ*�* is the dielectric anisotropy of the aligned liquid crystal, *ρNP* is the number density of nanoparticles, and *p* is the electric dipole moment, *SLC* (*SNP*) is the scalar orientational order parameters of the liquid crystals (nanoparticles). This free energy can predict the enhancement in the isotropic-nematic transition temperature and in the response to an applied electric field. The attractive interaction between the liquid crystal and the nanoparticle through the order parameters is important to understand the phase behaviors. The next section we consider the free energy to describe the phase separations.

#### **2.2 Phase ordering in mixtures of a spherical nanoparticle and a liquid crystal**

We consider mixtures of a spherical nanoparticle and a liquid crystal. By taking into account the ordering of liquid crystals and nanoparticles, we can expect six possible phases in this mixture. Figure 2 shows the schematically illustrated six phases. The isotropic (I) phase means both liquid crystals and nanoparticles have no positional and orientational order. In the nematic (N) phase, liquid crystals have an orientational order, while nanoparticles have no positional order. Similarly, in the smectic *A* (A) phase, liquid crystals have a smectic *A* order, while nanoparticles have no positional order. When the concentration of nanoparticles is high, we may have a crystalline (C) phase of nanoparticles dispersed in an isotopic matrix of liquid crystals. We can also expect a nematic-crystal (NC) phase and a smectic *A*-crystal (AC) phase, where nanoparticles form a crystalline structure dispersed in a nematic and a smectic *A* matrix of liquid crystals. To describe these phases, depending on temperature and concentration, we take into account three scalar order parameters: an orientational order parameter for a nematic phase, one-dimensional translational order parameter for a smectic *A* phase, and a translational order parameter for a crystalline phase of nanoparticles.

#### **2.3 Free energy of mixtures of a spherical nanoparticle and a liquid crystal**

We consider a binary mixture of *Nc* spherical nano-colloidal particles of the diameter *Rc* and *Nr* low-molecular weight liquid crystal molecules (liquid crystals) of the length *l* and the diameter *d*. The volume of the liquid crystal and that of the nanoparticle are given by *vr* = (*π*/4)*d*2*l* and *vc* = (*π*/6)*R*<sup>3</sup> *<sup>c</sup>* , respectively. Let *ρr*(**u**,**r**) and *ρc*(**r**) be the number density of liquid crystals and colloidal particles with an orientation **u** (or its solid angle Ω) at a position **r**, respectively. The free energy *F* of the dispersion at the level of second virial approximation is given by(Matsuyama & Hirashima, 2008a; Matsuyama, 2010a)

$$\begin{split} \beta \mathcal{F}/V &= \int \rho\_{\boldsymbol{r}}(\mathbf{r}, \mathbf{u}) \left[ \beta \mu\_{\boldsymbol{r}}^{\diamond} + \ln \rho\_{\boldsymbol{r}}(\mathbf{r}, \mathbf{u}) - 1 \right] d\mathbf{r} d\Omega \\ &+ \int \rho\_{\boldsymbol{c}}(\mathbf{r}) \left[ \beta \mu\_{\boldsymbol{c}}^{\diamond} + \ln \rho\_{\boldsymbol{c}}(\mathbf{r}) - 1 \right] d\mathbf{r} \\ &+ \frac{1}{2} \iint \rho\_{\boldsymbol{r}}(\mathbf{r}, \mathbf{u}) \rho\_{\boldsymbol{r}}(\mathbf{r}', \mathbf{u}') \beta\_{\boldsymbol{r}\boldsymbol{r}}(\mathbf{r}, \mathbf{u}; \mathbf{r}', \mathbf{u}') d\mathbf{r} d\Omega d\mathbf{r} d\Omega' \\ &+ \frac{1}{2} \iint \rho\_{\boldsymbol{c}}(\mathbf{r}) \rho\_{\boldsymbol{c}}(\mathbf{r}') \beta\_{\boldsymbol{c}\boldsymbol{c}}(\mathbf{r}; \mathbf{r}') d\mathbf{r} d\mathbf{r}' \\ &+ \iint \rho\_{\boldsymbol{c}}(\mathbf{r}) \rho\_{\boldsymbol{r}}(\mathbf{r}', \mathbf{u}') \beta\_{\boldsymbol{c}\boldsymbol{r}}(\mathbf{r}; \mathbf{r}', \mathbf{u}') d\mathbf{r} d\mathbf{r}' d\Omega', \end{split} \tag{2}$$

The orientational order parameter *S* of a nematic phase is given by(Maier & Saupe, 1958)

Phase Separations in Mixtures of a Nanoparticle and a Liquid Crystal 245

where *<sup>P</sup>*2(cos *<sup>θ</sup>*) <sup>≡</sup> <sup>3</sup>(cos2 *<sup>θ</sup>* <sup>−</sup> 1/3)/2. The translational order parameter *<sup>σ</sup><sup>s</sup>* of a smectic A

In the McMillan's model,(McMillan, 1971) the order parameter for the smectic *A* phase is given by �*P*2(cos(*θ*)) cos(2*πz*˜*r*)�. In Eq. (4), we have used the decoupled approximation: �*P*2(cos(*θ*)) cos(2*πz*˜*r*)� = *Sσs*. It has been reported that the decoupled model for the smectic A phase is in quantitative agreement with the original McMillan's theory.(Kventsel et.al., 1985)

For a crystalline phase, we here consider a face-centered cubic (fcc) structure of nanoparticles for example. The translational order parameter for a fcc crystalline phase can be calculated

where *L* is the lattice size of a fcc crystal and we define *x*˜ ≡ *x*/*L*, *y*˜ ≡ *y*/*L*, *z*˜ ≡ *z*/*L*, and *d***˜r** ≡ *dxd*˜ *yd*˜ *z*˜. It is possible to consider the other crystalline structure such as a body-centered

When the interaction between liquid crystals is a short-range attractive interaction, the anisotropic part of the interaction can be given by Fourier components of the

where we have retained the lowest order of the Fourier components. The *ν*(≡ *Ua*/*kBT*) is the orientational dependent (Maier-Saupe) interaction parameter between liquid crystals(Maier & Saupe, 1958) and the *γ* shows the dimensionless interaction of a smectic phase(Matsuyama & Kato, 1998; McMillan, 1971). According to the McMillan theory, the parameter *γ* is given by *<sup>γ</sup>* <sup>=</sup> 2 exp[−(*r*0/*l*)2], which can vary between 0 and 2, and increases with increasing the chain length of alkyl end-chains of a liquid crystal. The smectic condensation is more favored for larger values of *γ*. For the anisotropic interaction between nanoparticles in a fcc crystalline phase, the anisotropic part of the interaction can be given by expanding *βcc* at the lowest order

where the coefficient *βcc* is proportional to the total surface area (*vcRc*) of two particles. The parameter *g*(≡ −*β f*0) is the dimensionless interaction parameter between nanoparticles, where the interaction energy *f*<sup>0</sup> consists of an entropic and enthalpic terms. In this paper, we only consider short-range interactions between particles. The long-range interaction, due to

1 + *γσ<sup>s</sup>* cos(2*πz*/*l*)

*P*2(cos *θ*)*fr*(*θ*)*d*Ω, (8)

cos(2*πz*˜*r*)*fr*(*z*˜*r*)*dz*˜*r*. (9)

cos(2*πx*˜) cos(2*πy*˜) cos(2*πz*˜)*fc*(**˜r**)*d***˜r**, (10)

*βcc* � −(*vcRc*/*d*)*gσ<sup>f</sup>* cos(2*πx*˜) cos(2*πy*˜) cos(2*πz*˜), (12)

*P*2(cos *θ*) (11)

*S* = 

*σ<sup>s</sup>* =

 1 0

In the decoupled model, the smectic *A* phase is defined by *S* �= 0 and *σ<sup>s</sup>* > 0.

phase is given by(McMillan, 1971)

by(Kirkwood & Monroe, 1941)

potential:(McMillan, 1971)

*σ<sup>f</sup>* =

cubic and a simple cubic, etc.(Matsuyama, 2006a;b)

 1 0 1 0 1 0

*βrr* � −(*vrl*/*d*)*νS*

of the Fourier components:(Kirkwood & Monroe, 1941)

Fig. 2. Ordering of nanoparticles dispersed in liquid crystals. We here take into account three scalar order parameters: an orientational order parameter for a nematic phase, one-dimensional translational order parameter for a smectic *A* phase, and a translational order parameter for a crystalline phase of nanoparticles.

where *d*Ω is the solid angle, *μ*◦ *<sup>i</sup>* is the standard chemical potential of a particle *i*, *j*(= *r*, *c*), *β* ≡ 1/*kBT*; *T* is the absolute temperature, *kB* is the Boltzmann constant, *βij* ≡ 1 − exp[−*βuij*] is the Mayer-Mayer function, and *uij* is the interaction energy between two particles *i* and *j*.

Let *fr*(**r**, **u**) be the distribution function of liquid crystals and then the density can be expressed as

$$
\rho\_r(\mathbf{r}, \mathbf{u}) = c\_r f\_r(\mathbf{r}, \mathbf{u}),
\tag{3}
$$

where *cr* ≡ *Nr*/*V* is the average number density of liquid crystals. We here consider a nematic and a smectic *A* phase of liquid crystals and use the decoupled approximation(Kventsel et.al., 1985) for the distribution function:

$$f\_r(\mathbf{r}, \mathbf{u}) = f\_r(\tilde{z}\_r) f\_r(\mathbf{u}),\tag{4}$$

where *z*˜*<sup>r</sup>* ≡ *z*/*l*, *l* is the average distance between smectic layers, *fr*(*z*˜*r*) is the translational distribution function of liquid crystals for a smectic *A* phase, and *fr*(**u**) is the orientational distribution function of liquid crystals for a nematic phase. Similarly, using the translational distribution function *fc*(**r**) of nanoparticles, the density of nanoparticles can be expressed as

$$
\rho\_{\mathcal{C}}(\mathbf{r}) = \mathcal{c}\_{\mathcal{C}} f\_{\mathcal{C}}(\mathbf{r}), \tag{5}
$$

where *cc* ≡ *Nc*/*V* is the average density of nanoparticles. The total number *Nr* of liquid crystals and *Nc* of nanoparticles must be conserved and then we have the normalization conditions:

$$\int \int \rho\_r(\mathbf{r}, \mathbf{u}) d\mathbf{r} d\Omega = \mathcal{N}\_r / V\_\prime \tag{6}$$

and

$$
\int \rho\_{\mathbf{c}}(\mathbf{r})d\mathbf{r} = \mathcal{N}\_{\mathbf{c}}/V.\tag{7}
$$

4 Will-be-set-by-IN-TECH

Fig. 2. Ordering of nanoparticles dispersed in liquid crystals. We here take into account three

*β* ≡ 1/*kBT*; *T* is the absolute temperature, *kB* is the Boltzmann constant, *βij* ≡ 1 − exp[−*βuij*] is the Mayer-Mayer function, and *uij* is the interaction energy between two particles *i* and *j*. Let *fr*(**r**, **u**) be the distribution function of liquid crystals and then the density can be expressed

where *cr* ≡ *Nr*/*V* is the average number density of liquid crystals. We here consider a nematic and a smectic *A* phase of liquid crystals and use the decoupled approximation(Kventsel et.al.,

where *z*˜*<sup>r</sup>* ≡ *z*/*l*, *l* is the average distance between smectic layers, *fr*(*z*˜*r*) is the translational distribution function of liquid crystals for a smectic *A* phase, and *fr*(**u**) is the orientational distribution function of liquid crystals for a nematic phase. Similarly, using the translational distribution function *fc*(**r**) of nanoparticles, the density of nanoparticles can be expressed as

where *cc* ≡ *Nc*/*V* is the average density of nanoparticles. The total number *Nr* of liquid crystals and *Nc* of nanoparticles must be conserved and then we have the normalization

*<sup>i</sup>* is the standard chemical potential of a particle *i*, *j*(= *r*, *c*),

*ρr*(**r**, **u**) = *cr fr*(**r**, **u**), (3)

*fr*(**r**, **u**) = *fr*(*z*˜*r*)*fr*(**u**), (4)

*ρc*(**r**) = *cc fc*(**r**), (5)

*ρr*(**r**, **u**)*d***r***d*Ω = *Nr*/*V*, (6)

*ρc*(**r**)*d***r** = *Nc*/*V*. (7)

one-dimensional translational order parameter for a smectic *A* phase, and a translational

scalar order parameters: an orientational order parameter for a nematic phase,

order parameter for a crystalline phase of nanoparticles.

where *d*Ω is the solid angle, *μ*◦

1985) for the distribution function:

and

as

conditions:

The orientational order parameter *S* of a nematic phase is given by(Maier & Saupe, 1958)

$$S = \int P\_2(\cos \theta) f\_r(\theta) d\Omega,\tag{8}$$

where *<sup>P</sup>*2(cos *<sup>θ</sup>*) <sup>≡</sup> <sup>3</sup>(cos2 *<sup>θ</sup>* <sup>−</sup> 1/3)/2. The translational order parameter *<sup>σ</sup><sup>s</sup>* of a smectic A phase is given by(McMillan, 1971)

$$
\sigma\_s = \int\_0^1 \cos(2\pi \tilde{z}\_r) f\_r(\tilde{z}\_r) d\tilde{z}\_r. \tag{9}
$$

In the McMillan's model,(McMillan, 1971) the order parameter for the smectic *A* phase is given by �*P*2(cos(*θ*)) cos(2*πz*˜*r*)�. In Eq. (4), we have used the decoupled approximation: �*P*2(cos(*θ*)) cos(2*πz*˜*r*)� = *Sσs*. It has been reported that the decoupled model for the smectic A phase is in quantitative agreement with the original McMillan's theory.(Kventsel et.al., 1985) In the decoupled model, the smectic *A* phase is defined by *S* �= 0 and *σ<sup>s</sup>* > 0.

For a crystalline phase, we here consider a face-centered cubic (fcc) structure of nanoparticles for example. The translational order parameter for a fcc crystalline phase can be calculated by(Kirkwood & Monroe, 1941)

$$
\sigma\_f = \int\_0^1 \int\_0^1 \int\_0^1 \cos(2\pi \vec{x}) \cos(2\pi \vec{y}) \cos(2\pi \vec{z}) f\_c(\vec{r}) d\vec{r},\tag{10}
$$

where *L* is the lattice size of a fcc crystal and we define *x*˜ ≡ *x*/*L*, *y*˜ ≡ *y*/*L*, *z*˜ ≡ *z*/*L*, and *d***˜r** ≡ *dxd*˜ *yd*˜ *z*˜. It is possible to consider the other crystalline structure such as a body-centered cubic and a simple cubic, etc.(Matsuyama, 2006a;b)

When the interaction between liquid crystals is a short-range attractive interaction, the anisotropic part of the interaction can be given by Fourier components of the potential:(McMillan, 1971)

$$\beta\_{rr} \simeq - (\upsilon\_r l/d) \upsilon \mathcal{S} \left( 1 + \gamma \sigma\_s \cos(2\pi z/l) \right) P\_2(\cos \theta) \tag{11}$$

where we have retained the lowest order of the Fourier components. The *ν*(≡ *Ua*/*kBT*) is the orientational dependent (Maier-Saupe) interaction parameter between liquid crystals(Maier & Saupe, 1958) and the *γ* shows the dimensionless interaction of a smectic phase(Matsuyama & Kato, 1998; McMillan, 1971). According to the McMillan theory, the parameter *γ* is given by *<sup>γ</sup>* <sup>=</sup> 2 exp[−(*r*0/*l*)2], which can vary between 0 and 2, and increases with increasing the chain length of alkyl end-chains of a liquid crystal. The smectic condensation is more favored for larger values of *γ*. For the anisotropic interaction between nanoparticles in a fcc crystalline phase, the anisotropic part of the interaction can be given by expanding *βcc* at the lowest order of the Fourier components:(Kirkwood & Monroe, 1941)

$$\mathcal{B}\_{\mathfrak{C}} \simeq - (v\_{\mathfrak{C}} \mathbb{R}\_{\mathfrak{C}} / d) \mathfrak{g} \sigma\_f \cos(2\pi \mathfrak{X}) \cos(2\pi \mathfrak{Y}) \cos(2\pi \mathfrak{z}),\tag{12}$$

where the coefficient *βcc* is proportional to the total surface area (*vcRc*) of two particles. The parameter *g*(≡ −*β f*0) is the dimensionless interaction parameter between nanoparticles, where the interaction energy *f*<sup>0</sup> consists of an entropic and enthalpic terms. In this paper, we only consider short-range interactions between particles. The long-range interaction, due to

given by

*n*3

particles:

liquid crystal:

this free energy.

nematic ordering of liquid crystals:

*<sup>η</sup><sup>c</sup>* <sup>≡</sup> *<sup>a</sup>*2*n*<sup>3</sup>

*<sup>a</sup>*3*βFc*/*<sup>V</sup>* <sup>=</sup> *<sup>φ</sup><sup>c</sup>*

*<sup>a</sup>*3*βFnem*/*<sup>V</sup>* <sup>=</sup> *<sup>φ</sup><sup>r</sup>*

*<sup>a</sup>*3*βFsm*/*<sup>V</sup>* <sup>=</sup> *<sup>φ</sup><sup>r</sup>*

*<sup>a</sup>*3*βFanc*/*<sup>V</sup>* <sup>=</sup> *<sup>ω</sup>*

*n*3 *c*

−1 2 *gη*<sup>2</sup> *c σ*2

> *nr*

−1 2 *νφ*<sup>2</sup>

forth term in Eq. (15) shows the free energy for smectic A ordering of liquid crystals:

*nr*

−1 2 *νγφ*<sup>2</sup>

where the first term in Eq. (19) shows the entropy change due to the nematic ordering. The

 1 0

The last term in Eq. (15) shows the anchoring interaction between a colloidal surface and a

<sup>2</sup> *<sup>η</sup>cφ<sup>r</sup>* 

In a thermal equilibrium state, the distribution functions of nanoparticles and liquid crystals are determined by minimizing the free energy (15) with respect to these functions: (*δF*/*<sup>δ</sup> fc*(**˜r**)){ *fr*(*θ*), *fr*(*z*˜*r*)} <sup>=</sup> 0, (*δF*/*<sup>δ</sup> fr*(*θ*)){ *fc* (**˜r**), *fr*(*z*˜*r*)} <sup>=</sup> 0, and (*δF*/*<sup>δ</sup> fz*(*z*˜*r*)){ *fc* (**˜r**), *fr*(*θ*)} <sup>=</sup> 0. The order parameters *S*, *σs*, and *σ<sup>f</sup>* can be determined by Eqs. (8), (9), and (10), respectively. Using these distribution functions and order parameters, we can calculate the free energy of our systems. The chemical potentials of a nanoparticle and a liquid crystal can be obtained from

*<sup>c</sup> Nc a*2(*nrNr* + *n*<sup>3</sup>

Phase Separations in Mixtures of a Nanoparticle and a Liquid Crystal 247

Then the dispersion interaction due to the mixing is given by *χηcφr*. On increasing the diameter *nc* of a colloid, the interaction term decreases with a fixed *φc*. Eq. (16) corresponds to the extended Flory-Huggins free energy for the isotropic mixtures of a liquid crystal, whose the number of segments is *nr*, and a colloidal particle, whose the number of segments is

*<sup>c</sup>* . The second term in Eq. (15) shows the free energy for a crystalline ordering of colloidal

 1 0 1 0 1 0

where the first term in Eq. (18) shows the entropy loss due to the crystalline ordering. When the colloidal particles have no positional order, we have the distribution function *f*(**˜r**) = 1 and the free energy (*Fc*) becomes zero. The third term in Eq. (15) shows the free energy for

*<sup>c</sup> Nc*) <sup>=</sup> *<sup>φ</sup><sup>c</sup> nc*

*f*(**˜r**)ln *f*(**˜r**)*d***˜r**

*fr*(*θ*)ln 4*π fr*(*θ*)*d*Ω

*fr*(*z*˜*r*)ln *fr*(*z*˜*r*)*dz*˜*<sup>r</sup>*

*S*<sup>2</sup> + *ω*1(*Sσs*)<sup>2</sup>

. (17)

*<sup>f</sup>* , (18)

*<sup>r</sup> <sup>S</sup>*2, (19)

*<sup>r</sup>* (*Sσs*)2. (20)

. (21)

the presence of surface charges, is not taken into account. Similarly to Eq. (11), the anisotropic interaction between a nanoparticle and a liquid crystal in a nematic and a smectic*A* phase is given by

$$\beta\_{\mathcal{C}} \simeq \frac{1}{2} (v\_{\mathcal{C}} l / R\_{\mathcal{C}}) \omega S \left( 1 + \omega\_1 \sigma\_{\mathcal{S}} \cos(2\pi z / l) \right) P\_2(\cos \theta), \tag{13}$$

where *βcr* is proportional to the surface area (*vc*/*Rc*) of a nanoparticle(Stark, 1999). The *ω* ≡ *w*0/*kBT* shows the dimensionless interaction parameter between a liquid crystal and a particle surface. When *ω* > 0, or repulsive interaction between a liquid crystal and a nanoparticle, doping nanoparticles disturb the orientational ordering of liquid crystals, or the orientational elasticity of the liquid crystals tends to expel the particles to be lower the elastic free energy of a nematic phase(Pouling et. al., 1994). In the mean field level, the elastic distortion cost of a director is taken into account in the order of *ωS*2. The negative values of *ω*(< 0) indicate the attractive interactions between a liquid crystal and a nanoparticle and the particles tend to disperse into a liquid crystalline matrix as indicated in Fig. 1. The last term *ω*1(> 0) is the coupling between a smectic liquid crystal and a colloidal surface.

We here assume the system is incompressible. Let *φ<sup>r</sup>* = *vrNr*/*V* and *φ<sup>c</sup>* = *vcNc*/*V* be the volume fraction of a liquid crystal and a nano-colloidal particle, respectively. Using the axial ratio *nr*(≡ *l*/*d*) of a liquid crystal and *nc* ≡ *Rc*/*d*, the volume of a particle is given by *vr* = *<sup>a</sup>*3*nr* and *vr* � (*anc*)3, where we define *<sup>a</sup>*<sup>3</sup> <sup>≡</sup> (*π*/4)*d*3. To describe phase behaviors of the incompressible blends, we calculate the free energy of mixing for the binary mixtures of a liquid crystal and a nanoparticle:

$$
\Delta F = F(\mathbf{N\_{c}}, \mathbf{N\_{r}}) - F(\mathbf{N\_{c}}, \mathbf{0}) - F(\mathbf{0}, \mathbf{N\_{r}}), \tag{14}
$$

where the *F*(*Nc*, 0) and *F*(0, *Nr*) are the reference free energy of the pure nanoparticles and the pure liquid crystal in an isotropic phase, respectively. Substituting Eqs. (11)-(13) into (14), the mixing free energy is given by

$$
\Delta F = F\_{\text{mix}} + F\_{\text{c}} + F\_{\text{mem}} + F\_{\text{sm}} + F\_{\text{amc}\prime} \tag{15}
$$

where the each term is given as following.

The first term in Eq. (15) shows the free energy for mixing of colloids and liquid crystals in the isotropic phase:

$$a^3 \beta F\_{\rm mix} / V = \frac{\phi\_r}{n\_r} \ln \phi\_p + \frac{\phi\_c}{n\_c^3} \ln \phi\_c + \chi \eta\_c \phi\_{r\prime} \tag{16}$$

where the first and the second terms in Eq. (16) correspond to the entropy of isotropic mixing for liquid crystals and colloidal particles, respectively. We here have added the third term which shows the isotropic interaction parameter *χ* ≡ *U*0/*kBT* related to the dispersion force between a colloidal particle and a liquid crystal, where *U*<sup>0</sup> is the interaction energy between a colloid and a liquid crystal in an isotropic state. A positive *χ* denotes that the colloid-liquid crystal contacts are less favored compared with the colloid-colloid and liquid crystal-liquid crystal contacts. This interaction parameter is well known as the Flory-Huggins parameter in polymer solutions.(Flory, 1953) For a colloidal particle, its surface only can interact with the surrounding solvents and so the probability for the colloid-liquid crystal contact is proportional to *ηcφr*, where the *η<sup>c</sup>* is the surface fraction of colloidal particles and is given by

6 Will-be-set-by-IN-TECH

the presence of surface charges, is not taken into account. Similarly to Eq. (11), the anisotropic interaction between a nanoparticle and a liquid crystal in a nematic and a smectic*A* phase is

where *βcr* is proportional to the surface area (*vc*/*Rc*) of a nanoparticle(Stark, 1999). The *ω* ≡ *w*0/*kBT* shows the dimensionless interaction parameter between a liquid crystal and a particle surface. When *ω* > 0, or repulsive interaction between a liquid crystal and a nanoparticle, doping nanoparticles disturb the orientational ordering of liquid crystals, or the orientational elasticity of the liquid crystals tends to expel the particles to be lower the elastic free energy of a nematic phase(Pouling et. al., 1994). In the mean field level, the elastic distortion cost of a director is taken into account in the order of *ωS*2. The negative values of *ω*(< 0) indicate the attractive interactions between a liquid crystal and a nanoparticle and the particles tend to disperse into a liquid crystalline matrix as indicated in Fig. 1. The last term *ω*1(> 0) is the

We here assume the system is incompressible. Let *φ<sup>r</sup>* = *vrNr*/*V* and *φ<sup>c</sup>* = *vcNc*/*V* be the volume fraction of a liquid crystal and a nano-colloidal particle, respectively. Using the axial ratio *nr*(≡ *l*/*d*) of a liquid crystal and *nc* ≡ *Rc*/*d*, the volume of a particle is given by *vr* = *<sup>a</sup>*3*nr* and *vr* � (*anc*)3, where we define *<sup>a</sup>*<sup>3</sup> <sup>≡</sup> (*π*/4)*d*3. To describe phase behaviors of the incompressible blends, we calculate the free energy of mixing for the binary mixtures of a

where the *F*(*Nc*, 0) and *F*(0, *Nr*) are the reference free energy of the pure nanoparticles and the pure liquid crystal in an isotropic phase, respectively. Substituting Eqs. (11)-(13) into (14), the

The first term in Eq. (15) shows the free energy for mixing of colloids and liquid crystals in the

where the first and the second terms in Eq. (16) correspond to the entropy of isotropic mixing for liquid crystals and colloidal particles, respectively. We here have added the third term which shows the isotropic interaction parameter *χ* ≡ *U*0/*kBT* related to the dispersion force between a colloidal particle and a liquid crystal, where *U*<sup>0</sup> is the interaction energy between a colloid and a liquid crystal in an isotropic state. A positive *χ* denotes that the colloid-liquid crystal contacts are less favored compared with the colloid-colloid and liquid crystal-liquid crystal contacts. This interaction parameter is well known as the Flory-Huggins parameter in polymer solutions.(Flory, 1953) For a colloidal particle, its surface only can interact with the surrounding solvents and so the probability for the colloid-liquid crystal contact is proportional to *ηcφr*, where the *η<sup>c</sup>* is the surface fraction of colloidal particles and is

ln *<sup>φ</sup><sup>p</sup>* <sup>+</sup> *<sup>φ</sup><sup>c</sup> n*3 *c*

*nr*

1 + *ω*1*σ<sup>s</sup>* cos(2*πz*/*l*)

Δ*F* = *F*(*Nc*, *Nr*) − *F*(*Nc*, 0) − *F*(0, *Nr*), (14)

Δ*F* = *Fmix* + *Fc* + *Fnem* + *Fsm* + *Fanc*, (15)

ln *φ<sup>c</sup>* + *χηcφr*, (16)

*P*2(cos *θ*), (13)

given by

*<sup>β</sup>cr* � <sup>1</sup> 2

liquid crystal and a nanoparticle:

mixing free energy is given by

isotropic phase:

where the each term is given as following.

(*vcl*/*Rc*)*ωS*

coupling between a smectic liquid crystal and a colloidal surface.

*<sup>a</sup>*3*βFmix*/*<sup>V</sup>* <sup>=</sup> *<sup>φ</sup><sup>r</sup>*

$$\eta\_{\mathcal{C}} \equiv \frac{a^2 n\_{\mathcal{C}}^3 N\_{\mathcal{C}}}{a^2 (n\_r N\_r + n\_{\mathcal{C}}^3 N\_{\mathcal{C}})} = \frac{\phi\_{\mathcal{C}}}{n\_{\mathcal{C}}}.\tag{17}$$

Then the dispersion interaction due to the mixing is given by *χηcφr*. On increasing the diameter *nc* of a colloid, the interaction term decreases with a fixed *φc*. Eq. (16) corresponds to the extended Flory-Huggins free energy for the isotropic mixtures of a liquid crystal, whose the number of segments is *nr*, and a colloidal particle, whose the number of segments is *n*3 *<sup>c</sup>* . The second term in Eq. (15) shows the free energy for a crystalline ordering of colloidal particles:

$$a^3 \beta F\_{\varepsilon} / V = \frac{\oint\_{\mathcal{C}}}{n\_{\mathcal{C}}^3} \int\_0^1 \int\_0^1 f(\tilde{\mathbf{r}}) \ln f(\tilde{\mathbf{r}}) d\tilde{\mathbf{r}}$$

$$-\frac{1}{2} g \eta\_{\varepsilon}^2 \sigma\_{f'}^2 \tag{18}$$

where the first term in Eq. (18) shows the entropy loss due to the crystalline ordering. When the colloidal particles have no positional order, we have the distribution function *f*(**˜r**) = 1 and the free energy (*Fc*) becomes zero. The third term in Eq. (15) shows the free energy for nematic ordering of liquid crystals:

$$a^3 \beta F\_{\text{mem}} / V = \frac{\phi\_r}{n\_r} \int f\_r(\theta) \ln 4\pi f\_r(\theta) d\Omega$$

$$-\frac{1}{2} \nu \phi\_r^2 S^2 \, , \tag{19}$$

where the first term in Eq. (19) shows the entropy change due to the nematic ordering. The forth term in Eq. (15) shows the free energy for smectic A ordering of liquid crystals:

$$a^3 \beta F\_{\rm sm} / V = \frac{\phi\_r}{n\_r} \int\_0^1 f\_r(\tilde{z}\_r) \ln f\_r(\tilde{z}\_r) d\tilde{z}\_r$$

$$-\frac{1}{2} \nu \gamma \phi\_r^2 (S \sigma\_s)^2. \tag{20}$$

The last term in Eq. (15) shows the anchoring interaction between a colloidal surface and a liquid crystal:

$$a^3 \beta F\_{\rm nuc} / V = \frac{\omega}{2} \eta\_c \phi\_r \left(\mathcal{S}^2 + \omega\_1 (\mathcal{S} \sigma\_s)^2\right). \tag{21}$$

In a thermal equilibrium state, the distribution functions of nanoparticles and liquid crystals are determined by minimizing the free energy (15) with respect to these functions: (*δF*/*<sup>δ</sup> fc*(**˜r**)){ *fr*(*θ*), *fr*(*z*˜*r*)} <sup>=</sup> 0, (*δF*/*<sup>δ</sup> fr*(*θ*)){ *fc* (**˜r**), *fr*(*z*˜*r*)} <sup>=</sup> 0, and (*δF*/*<sup>δ</sup> fz*(*z*˜*r*)){ *fc* (**˜r**), *fr*(*θ*)} <sup>=</sup> 0. The order parameters *S*, *σs*, and *σ<sup>f</sup>* can be determined by Eqs. (8), (9), and (10), respectively. Using these distribution functions and order parameters, we can calculate the free energy of our systems. The chemical potentials of a nanoparticle and a liquid crystal can be obtained from this free energy.

(a) (b)

Phase Separations in Mixtures of a Nanoparticle and a Liquid Crystal 249

Fig. 3. (a) The first-order phase transition lines (the red dotted-line for NIT (Eq. (22)), the blue dotted-line for ANT (Eq. (24)), and the black dotted-line for ICT (Eq. (25))) on the reduced

appears, where nanoparticles are in an isotropic liquid state but liquid crystals are in a smectic *A* phase. At high temperatures and high concentrations, we have the crystalline (C) phase of colloidal particles. On decreasing temperature, the NC phase appears, where colloidal particles are in a crystalline state and liquid crystals are in a nematic state. Further decreasing temperature the AC phase appears, where colloidal particles are in a crystalline state and liquid crystals are in a smectic *A* phase. The slope of the transition lines depends on the anchoring energy (*αa*) as discussed in Eq. (23). For larger negative values of *αa*, the slopes of the NIT and ANT lines become positive on the temperature-concentration plane and the

Figure 3(b) shows order parameters plotted against the volume fraction *φ<sup>c</sup>* at *T*/*T*◦

first-order phase transition from the NC to C phase appears at *φ<sup>c</sup>* � 0.55.

in Fig. 3(a). On increasing *φc*, we find the first-order phase transition from the smectic A to nematic (N) phase at *φ<sup>c</sup>* � 0.2, where the order parameters *S* and *σ<sup>s</sup>* jump. At *φ<sup>c</sup>* � 0.5, the first-order phase transition from the N to NC phase appears. Further increasing *φ<sup>c</sup>* the

In this subsection we show some phase diagrams calculated from the free energy(15). The coexistence curve (binodal) can be obtained by solving the two-phase coexistence conditions: the chemical potentials of each component are equal in each phase. This binodal curve can also be derived by a double tangent method where the equilibrium volume fractions fall on

Figure 4 shows the phase diagrams for *nr* = 2, *nc* = 3, *ω*<sup>1</sup> = 1, *α<sup>c</sup>* = 0.1, *α<sup>n</sup>* = 5, and *γ* = 0.87 for an example. The value of *α<sup>a</sup>* is changed: (a) *α<sup>a</sup>* = 1; (b) *α<sup>a</sup>* = −2; (c) *α<sup>a</sup>* = −3.5.

*N I*)−concentration (*φc*) plane. (b) Order parameters plotted against the

*N I*=0.92 in Fig. 3(a).

*N I* = 0.92

temperature (*T*/*T*◦

volume fraction of colloidal particles at *T*/*T*◦

nematic and smectic A phase appear at higher temperatures.

**2.5 Phase diagrams of nanoparticle/liquid crystal mixtures**

the same tangent line to the free energy curve.

#### **2.4 Phase transitions in nanoparticle/liquid crystal mixtures**

In a mixture of a nanoparticle and a liquid crystal, we have some phase transitions, depending on temperature and concentration(Matsuyama, 2009).

One is the nematic-isotropic phase transition of this mixture. The nematic-isotropic transition (NIT) temperature is given by

$$
\pi\_{NI} = T/T\_{NI}^{\diamond} = 1 - (1 + \alpha\_a/n\_c)\phi\_{\circ \prime} \tag{22}
$$

where *T*◦ *N I* shows the NIT temperature of a pure liquid crystal. We here define the ratio *α<sup>a</sup>* between the anchoring strength (*w*) and the nematic interaction (*ν*):

$$
\mathfrak{a}\_d \equiv w/\nu. \tag{23}
$$

The value of *α<sup>a</sup>* shows the anchoring strength. The negative sign represents attractive interaction between a nanoparticle and a liquid crystal and thus the nanoparticles tend to disperse in a liquid crystalline matrix. On the other hand, the positive sign represents the repulsive interaction and the liquid crystals tend to expel the nanoparticles. The slope of the NIT line on the *T* − *φ<sup>c</sup>* plane depends on the value of *αa*/*nc*.

The smectic A−nematic phase transition (ANT) is given by

$$\tau\_{AN} = T / T\_{NI}^{\diamond} = 2.27 \gamma \left[ 1 - (1 + \frac{\omega\_1 \mathfrak{a}\_d}{n\_c \gamma}) \phi\_c \right] S^2. \tag{24}$$

Since *γ* > 0 and *ω*<sup>1</sup> > 0, the ANT temperature depends on the sign of *αa*. For larger negative values of *αa*, the ANT temperature increases with increasing *φc*. It can also be obtained the direct phase transition from an isotropic to a smectic A phase(Matsuyama, 2009).

We also have the isotropic fluid-crystal phase transition (ICT). The ICT temperature is given by

$$\tau\_{IC} = T/T\_{NI}^{\diamond} = \frac{0.58 \alpha\_{\text{c}} n\_{\text{c}} \phi\_{\text{c}}}{n\_{\text{r}} (1 - \phi\_{\text{c}}/\phi\_{\text{c}}^{\*})} \,\text{}\tag{25}$$

where

$$\mathfrak{a}\_{\mathfrak{C}} \equiv \mathfrak{P}|e\_{0}| / \nu \,\, \tag{26}$$

shows the strength of the attractive interaction between nanoparticles compared to the nematic interaction parameter *ν*. When *τ* < *τIC* the crystalline phase is stable. The ICT temperature increases with increasing *φ<sup>c</sup>* and diverges at *φ*∗ *<sup>c</sup>* . This corresponds to the entropically driven-liquid-solid transition for hard spherical particles due to the excluded volume interactions.(Alder & Wainwright, 1957; Cates & Evans, 2000)

Figure 3(a) shows the first-order phase transition lines for NIT (red dotted-line, Eq. (22)), ANT (blue dotted-line, Eq. (24)), and ICT (black dotted-line, Eq. (25)) on the reduced temperature (*T*/*T*◦ *N I*)−concentration (*φc*) plane. We set *nc* = 3, *nr* = 2, *αc*=0.1, *α<sup>a</sup>* = −2.5, *αn*(≡ *ν*/*χ*)=5, *γ* = 0.87, *ω*<sup>1</sup> = 1 for a typical example. When *φ<sup>c</sup>* = 0, or pure liquid crystals, the ANT appears at *T*/*T*◦ *N I* ≈ 0.938, which is consistent with the result of the MacMillan theory. At high temperatures and low concentrations, we have the isotropic (I) liquid phase. On decreasing temperature, the N phase appears, where nanoparticles are in an isotropic liquid state but liquid crystals are in a nematic state. Further decreasing temperature the smectic *A* phase 8 Will-be-set-by-IN-TECH

In a mixture of a nanoparticle and a liquid crystal, we have some phase transitions, depending

One is the nematic-isotropic phase transition of this mixture. The nematic-isotropic transition

The value of *α<sup>a</sup>* shows the anchoring strength. The negative sign represents attractive interaction between a nanoparticle and a liquid crystal and thus the nanoparticles tend to disperse in a liquid crystalline matrix. On the other hand, the positive sign represents the repulsive interaction and the liquid crystals tend to expel the nanoparticles. The slope of the

> 1 − (1 +

Since *γ* > 0 and *ω*<sup>1</sup> > 0, the ANT temperature depends on the sign of *αa*. For larger negative values of *αa*, the ANT temperature increases with increasing *φc*. It can also be obtained the

We also have the isotropic fluid-crystal phase transition (ICT). The ICT temperature is given

shows the strength of the attractive interaction between nanoparticles compared to the nematic interaction parameter *ν*. When *τ* < *τIC* the crystalline phase is stable. The

entropically driven-liquid-solid transition for hard spherical particles due to the excluded

Figure 3(a) shows the first-order phase transition lines for NIT (red dotted-line, Eq. (22)), ANT (blue dotted-line, Eq. (24)), and ICT (black dotted-line, Eq. (25)) on the reduced temperature

temperatures and low concentrations, we have the isotropic (I) liquid phase. On decreasing temperature, the N phase appears, where nanoparticles are in an isotropic liquid state but liquid crystals are in a nematic state. Further decreasing temperature the smectic *A* phase

*N I*)−concentration (*φc*) plane. We set *nc* = 3, *nr* = 2, *αc*=0.1, *α<sup>a</sup>* = −2.5, *αn*(≡ *ν*/*χ*)=5, *γ* = 0.87, *ω*<sup>1</sup> = 1 for a typical example. When *φ<sup>c</sup>* = 0, or pure liquid crystals, the ANT

*N I* ≈ 0.938, which is consistent with the result of the MacMillan theory. At high

*N I* <sup>=</sup> 0.58*αcncφ<sup>c</sup> nr*(1 − *φc*/*φ*<sup>∗</sup>

*ω*1*αa nc<sup>γ</sup>* )*φ<sup>c</sup>* 

*c* )

*α<sup>c</sup>* ≡ *β*|*e*0|/*ν*, (26)

*N I* = 2.27*γ*

direct phase transition from an isotropic to a smectic A phase(Matsuyama, 2009).

*τIC* = *T*/*T*◦

ICT temperature increases with increasing *φ<sup>c</sup>* and diverges at *φ*∗

volume interactions.(Alder & Wainwright, 1957; Cates & Evans, 2000)

*N I* shows the NIT temperature of a pure liquid crystal. We here define the ratio *α<sup>a</sup>*

*N I* = 1 − (1 + *αa*/*nc*)*φc*, (22)

*α<sup>a</sup>* ≡ *w*/*ν*. (23)

*S*2. (24)

, (25)

*<sup>c</sup>* . This corresponds to the

**2.4 Phase transitions in nanoparticle/liquid crystal mixtures**

*τN I* = *T*/*T*◦

between the anchoring strength (*w*) and the nematic interaction (*ν*):

NIT line on the *T* − *φ<sup>c</sup>* plane depends on the value of *αa*/*nc*. The smectic A−nematic phase transition (ANT) is given by

*τAN* = *T*/*T*◦

on temperature and concentration(Matsuyama, 2009).

(NIT) temperature is given by

where *T*◦

by

where

(*T*/*T*◦

appears at *T*/*T*◦

Fig. 3. (a) The first-order phase transition lines (the red dotted-line for NIT (Eq. (22)), the blue dotted-line for ANT (Eq. (24)), and the black dotted-line for ICT (Eq. (25))) on the reduced temperature (*T*/*T*◦ *N I*)−concentration (*φc*) plane. (b) Order parameters plotted against the volume fraction of colloidal particles at *T*/*T*◦ *N I*=0.92 in Fig. 3(a).

appears, where nanoparticles are in an isotropic liquid state but liquid crystals are in a smectic *A* phase. At high temperatures and high concentrations, we have the crystalline (C) phase of colloidal particles. On decreasing temperature, the NC phase appears, where colloidal particles are in a crystalline state and liquid crystals are in a nematic state. Further decreasing temperature the AC phase appears, where colloidal particles are in a crystalline state and liquid crystals are in a smectic *A* phase. The slope of the transition lines depends on the anchoring energy (*αa*) as discussed in Eq. (23). For larger negative values of *αa*, the slopes of the NIT and ANT lines become positive on the temperature-concentration plane and the nematic and smectic A phase appear at higher temperatures.

Figure 3(b) shows order parameters plotted against the volume fraction *φ<sup>c</sup>* at *T*/*T*◦ *N I* = 0.92 in Fig. 3(a). On increasing *φc*, we find the first-order phase transition from the smectic A to nematic (N) phase at *φ<sup>c</sup>* � 0.2, where the order parameters *S* and *σ<sup>s</sup>* jump. At *φ<sup>c</sup>* � 0.5, the first-order phase transition from the N to NC phase appears. Further increasing *φ<sup>c</sup>* the first-order phase transition from the NC to C phase appears at *φ<sup>c</sup>* � 0.55.

#### **2.5 Phase diagrams of nanoparticle/liquid crystal mixtures**

In this subsection we show some phase diagrams calculated from the free energy(15). The coexistence curve (binodal) can be obtained by solving the two-phase coexistence conditions: the chemical potentials of each component are equal in each phase. This binodal curve can also be derived by a double tangent method where the equilibrium volume fractions fall on the same tangent line to the free energy curve.

Figure 4 shows the phase diagrams for *nr* = 2, *nc* = 3, *ω*<sup>1</sup> = 1, *α<sup>c</sup>* = 0.1, *α<sup>n</sup>* = 5, and *γ* = 0.87 for an example. The value of *α<sup>a</sup>* is changed: (a) *α<sup>a</sup>* = 1; (b) *α<sup>a</sup>* = −2; (c) *α<sup>a</sup>* = −3.5.

(a) (b)

Phase Separations in Mixtures of a Nanoparticle and a Liquid Crystal 251

(c)

Fig. 4. Phase diagrams for *α<sup>c</sup>* = 0.1, *α<sup>n</sup>* = 5, *γ* = 0.87. The value of *α<sup>a</sup>* is changed: (a) *α<sup>a</sup>* = 1;

a nematic phase at low temperature.(Pouling et. al., 1994) The observed phase diagram are

The binodal lines calculated at high concentrations of nanoparticles may not be experimentally observed because of high viscosity, however, it is important to understand the phase ordering kinetics(Matsuyama et .al., 2000; Matsuyama, 2008b). The cooperative phenomena between liquid crystalline ordering and crystalline ordering induce a variety of

(b) *α<sup>a</sup>* = −2; (c) *α<sup>a</sup>* = −3.5.

phase separations.

qualitatively consistent with Fig. 4(a).

We here discuss the effects of the anchoring strength *α<sup>a</sup>* on the phase behavior. The negative values of *α<sup>a</sup>* mean that the nanoparticles prefer to disperse into liquid crystalline phases. The solid curve shows the binodal curve. The red, blue, and black dotted lines show the NIT, ANT, and ICT line, respectively (see Fig. 3(a)). When *φ<sup>c</sup>* = 0, the smectic *A* phase appears at *T*/*T*◦ *N I* = 0.94. When *α<sup>a</sup>* = 1 [Fig. 4(a)], we have the broad nematic-isotropic (*N* + *I*) phase separation between 1 > *T*/*T*◦ *N I* > 0.94. Below *T*/*T*◦ *N I* < 0.94, we have the smectic *A*-isotropic (*A* + *I*) phase separation. The nematic and smectic *A* phase at the lower concentrations consist of almost pure liquid crystals. The triple point (*A* + *I* + *C*) appears at *T*/*T*◦ *N I* = 0.89, where the smectic *A*, isotropic, and crystalline phases can simultaneously coexist. Below the triple point, we have the two-phase coexistence (*A* + *C*) between a smectic *A* and a crystalline phase. Above the triple point, two-phase coexistence (*I* + *C*) between an isotropic and a crystalline phase appears.

On decreasing the anchoring parameter *α<sup>a</sup>* the phase behavior is drastically changed. When *α<sup>a</sup>* = −2 [Fig. 4(b)], the NIT (Eq. (22)) and ANT (Eq. (24)) lines shift to higher concentrations and the stable single N and A phases appear at low concentrations of nanoparticles. Two tie lines with arrows show the three-phase coexistence: *A* + *N* + *I* and *A* + *I* + *C*. Above the triple point *A* + *N* + *I*, we have two-phase coexistence *A* + *N* and *N* + *I*. Below the triple point *A* + *N* + *I*, we have A+I phase separation. Below the triple point *A* + *I* + *C*, we have the broad A+C phase separation.

Further decreasing *αa*, Fig. 4(c), the nematic and smectic *A* ordering are promoted by adding nanoparticles and the NIT and ANT lines shift to higher temperatures. This increase of the NIT and ANT temperature indicates the attractive interactions between a liquid crystal and a colloidal particle. For example, it has been observed that doping low concentrations of ferroelectric BaTiO3 nanoparticles into liquid crystals increases NIT temperature(Li et. al., 2006a). In this case, ferroelectric nanoparticle with electric dipole moment, which produces an electric field, interacts with orientational order of liquid crystals and stabilizes the nematic phase.(Lopatina & Selinger, 2009) This corresponds to negative anchoring energy in our model. We also have three triple points: *I* + *C* + *NC*, *N* + *NC* + *AC*, and *N* + *A* + *AC*. Above the *I* + *C* + *NC* triple point, we have the *I* + *C* and *C* + *NC* phase separations. Below the *I* + *C* + *NC* triple point, the *I* + *NC* and *NC* + *AC* phase separations appear. Below the triple point *N* + *NC* + *AC*, we have the *I* + *N* and *N* + *AC* phase separations. Below the triple point *N* + *A* + *AC* we have *N* + *A* and *A* + *AC* phase separations. The anchoring energy between liquid crystals and nanoparticles becomes an important parameter to derive a stable N, A, NC, and AC phases in the mixture of nanoparticles and liquid crystals.

Anderson et al. have observed the phase ordering of colloidal (PMMA) particles dispersed in a liquid crystal, 5CB or MBBA.(Anderson et.al., 2001) Particles are covered with chemically grafted short chains, making hairy particles. In a nematic phase, the grafted chains tend to provide a homeotropic (radial) director anchoring. In an isotropic liquid, these particles behave like almost hard spheres and so the *I* + *C* phase separation takes place at high concentrations of the colloidal particles. Such *I* + *C* phase separation, calculated in Fig. 4, has been observed in colloidal dispersions(Pusey & van Megen, 1986) and protein solutions(Tanaka et. al, 2020). At dilute concentrations of the colloidal particles, Anderson et al. observed a decrease in the NIT temperature *TN I* as a function of *φc*, which follows a linear law. This is consistent with Eq. (22). The N+I and N+C phase separations have also been reported in Latex polyballs suspended in an isotropic micellar solution which exhibits 10 Will-be-set-by-IN-TECH

We here discuss the effects of the anchoring strength *α<sup>a</sup>* on the phase behavior. The negative values of *α<sup>a</sup>* mean that the nanoparticles prefer to disperse into liquid crystalline phases. The solid curve shows the binodal curve. The red, blue, and black dotted lines show the NIT, ANT, and ICT line, respectively (see Fig. 3(a)). When *φ<sup>c</sup>* = 0, the smectic *A* phase appears at

*N I* = 0.94. When *α<sup>a</sup>* = 1 [Fig. 4(a)], we have the broad nematic-isotropic (*N* + *I*) phase

(*A* + *I*) phase separation. The nematic and smectic *A* phase at the lower concentrations consist

the smectic *A*, isotropic, and crystalline phases can simultaneously coexist. Below the triple point, we have the two-phase coexistence (*A* + *C*) between a smectic *A* and a crystalline phase. Above the triple point, two-phase coexistence (*I* + *C*) between an isotropic and a crystalline

On decreasing the anchoring parameter *α<sup>a</sup>* the phase behavior is drastically changed. When *α<sup>a</sup>* = −2 [Fig. 4(b)], the NIT (Eq. (22)) and ANT (Eq. (24)) lines shift to higher concentrations and the stable single N and A phases appear at low concentrations of nanoparticles. Two tie lines with arrows show the three-phase coexistence: *A* + *N* + *I* and *A* + *I* + *C*. Above the triple point *A* + *N* + *I*, we have two-phase coexistence *A* + *N* and *N* + *I*. Below the triple point *A* + *N* + *I*, we have A+I phase separation. Below the triple point *A* + *I* + *C*, we have

Further decreasing *αa*, Fig. 4(c), the nematic and smectic *A* ordering are promoted by adding nanoparticles and the NIT and ANT lines shift to higher temperatures. This increase of the NIT and ANT temperature indicates the attractive interactions between a liquid crystal and a colloidal particle. For example, it has been observed that doping low concentrations of ferroelectric BaTiO3 nanoparticles into liquid crystals increases NIT temperature(Li et. al., 2006a). In this case, ferroelectric nanoparticle with electric dipole moment, which produces an electric field, interacts with orientational order of liquid crystals and stabilizes the nematic phase.(Lopatina & Selinger, 2009) This corresponds to negative anchoring energy in our model. We also have three triple points: *I* + *C* + *NC*, *N* + *NC* + *AC*, and *N* + *A* + *AC*. Above the *I* + *C* + *NC* triple point, we have the *I* + *C* and *C* + *NC* phase separations. Below the *I* + *C* + *NC* triple point, the *I* + *NC* and *NC* + *AC* phase separations appear. Below the triple point *N* + *NC* + *AC*, we have the *I* + *N* and *N* + *AC* phase separations. Below the triple point *N* + *A* + *AC* we have *N* + *A* and *A* + *AC* phase separations. The anchoring energy between liquid crystals and nanoparticles becomes an important parameter to derive a stable N, A,

Anderson et al. have observed the phase ordering of colloidal (PMMA) particles dispersed in a liquid crystal, 5CB or MBBA.(Anderson et.al., 2001) Particles are covered with chemically grafted short chains, making hairy particles. In a nematic phase, the grafted chains tend to provide a homeotropic (radial) director anchoring. In an isotropic liquid, these particles behave like almost hard spheres and so the *I* + *C* phase separation takes place at high concentrations of the colloidal particles. Such *I* + *C* phase separation, calculated in Fig. 4, has been observed in colloidal dispersions(Pusey & van Megen, 1986) and protein solutions(Tanaka et. al, 2020). At dilute concentrations of the colloidal particles, Anderson et al. observed a decrease in the NIT temperature *TN I* as a function of *φc*, which follows a linear law. This is consistent with Eq. (22). The N+I and N+C phase separations have also been reported in Latex polyballs suspended in an isotropic micellar solution which exhibits

*N I* < 0.94, we have the smectic *A*-isotropic

*N I* = 0.89, where

*N I* > 0.94. Below *T*/*T*◦

of almost pure liquid crystals. The triple point (*A* + *I* + *C*) appears at *T*/*T*◦

NC, and AC phases in the mixture of nanoparticles and liquid crystals.

*T*/*T*◦

phase appears.

separation between 1 > *T*/*T*◦

the broad A+C phase separation.

Fig. 4. Phase diagrams for *α<sup>c</sup>* = 0.1, *α<sup>n</sup>* = 5, *γ* = 0.87. The value of *α<sup>a</sup>* is changed: (a) *α<sup>a</sup>* = 1; (b) *α<sup>a</sup>* = −2; (c) *α<sup>a</sup>* = −3.5.

a nematic phase at low temperature.(Pouling et. al., 1994) The observed phase diagram are qualitatively consistent with Fig. 4(a).

The binodal lines calculated at high concentrations of nanoparticles may not be experimentally observed because of high viscosity, however, it is important to understand the phase ordering kinetics(Matsuyama et .al., 2000; Matsuyama, 2008b). The cooperative phenomena between liquid crystalline ordering and crystalline ordering induce a variety of phase separations.

(a)

Phase Separations in Mixtures of a Nanoparticle and a Liquid Crystal 253

component is randomly oriented within in the perpendicular plane to the nematic director. The nematic N0 phase shows the nanotube and the liquid crystal are parallel to each other: *S*<sup>1</sup> > 0 and *S*<sup>2</sup> > 0. The nematic N1 phase is defined as that the nanotube and the liquid crystal are perpendicular with each other: *S*<sup>1</sup> > 0 and *S*<sup>2</sup> < 0. In this phase, the nematic director (z axis) can be defined by the orientational direction of the liquid crystals. These perpendicular alignments can be obtained by modifying the surface of a nanotube, or CNT, with polymers or surfactants. The nematic N2 phase is defined as the nanotube and the LC are perpendicular each other with *S*<sup>1</sup> < 0 and *S*<sup>2</sup> > 0. In this phase, the nematic director (z axis) can be defined by the orientational direction of the nanotube. Biaxial nematic phases are discussed in Section 3.3. When an external field (**E**) is applied along to the *z* axis for the particles of the dielectric anisotropy Δ<sup>1</sup> < 0 and Δ<sup>2</sup> < 0, the N3 phase with *S*<sup>1</sup> < 0 and *S*<sup>2</sup> < 0 may appear, where the nanotubes and liquid crystals are randomly oriented on the plane perpendicular to the

We consider a binary mixture of a liquid crystal of the length *L*<sup>1</sup> and the diameter *D*<sup>1</sup> and and a nanotube of the length *L*<sup>2</sup> and the diameter *D*2: *L*<sup>1</sup> < *L*2. The volume of the liquid crystal and

direction (*z* axis) of the external field.

**3.2 Free energy of nanotube/liquid crystal mixtures**

Fig. 5. Schematically illustrated four possible nematic phases. Four nematic phases are defined using the orientational order parameter *S*<sup>1</sup> of the liquid crystal and that *S*<sup>2</sup> of the nanotube: the nematic N0 phase with *S*<sup>1</sup> > 0 and *S*<sup>2</sup> > 0, the nematic N1 phase with *S*<sup>1</sup> > 0 and *S*<sup>2</sup> < 0, and the nematic N2 phase with *S*<sup>1</sup> < 0 and *S*<sup>2</sup> > 0. When an external field (**E**) is applied along to the *z* axis for the particles of the dielectric anisotropy Δ<sup>1</sup> < 0 and Δ<sup>2</sup> < 0, the N3 phase with *S*<sup>1</sup> < 0 and *S*<sup>2</sup> < 0 can appear: the nanotubes and liquid crystals are randomly oriented on the plane perpendicular to the direction of the external field.

## **3. Nanotubes dispersed in liquid crystals**

Since the discovery of carbon nanotubes (CNTs)(Iijima, 1991), extensive studies of physical and chemical properties of CNTs have been received great attention for many practical applications such as nano-sensors and devices. Windle's group first reported the nematic liquid crystalline behavior of an aqueous suspension of CNTs above a certain concentration and the isotropic-nematic phase separations.(Shaffer & Windle, 1999; Song et. al., 2003). Long nanotubes segregate preferentially to the liquid crystalline phase, whereas shorter nanotubes segregate preferentially to the isotropic phase(Zhang et.al., 2006). Recently such nanotubes as liquid crystalline materials become an important to be used in biological applications such as biosensors, biology imaging, artificial muscles, gene delivery, etc(Woltman et al., 2007).

In order to prepare CNT dispersions, strong van der Waals attractions between nanotubes must be screen out. To do this, the surface of nanotubes can be modified by acid oxidation, acid protonation, polymer or surfactant wrapping, etc.(Zhang & Kumar, 2008). For example, a water-soluble polymer, biopolymers such as DNA, and surfactant molecules have been used to wrap CNT and to increase the dispersibility in water(Badaire et. al., 2005). The polymer-wrapped nanotubes can be dispersed in a solvent with a considerable concentration. The excluded volume and electro-static repulsion between polymers can overcome the intermolecular van der Waals attractions and therefore the polymer-wrapped CNT can be dispersed in water. Thus it is possible to change the strength of the intermolecular interaction between nanotubes by using polymer-wrapping and negative or positive charging of nanotube surface, etc.

Alignment of such CNTs, or rigid-rodlike polymers (rods), with the aid of low molecular-weight-liquid crystalline molecules is an alternative approach. Indeed thermotropic(Basu & Iannacchione, 2008; Dierking et.al., 2005; Jayalakshmi & Prasad, 2009; Lynch & Patrick, 2002; Russell et.al., 2006) as well as lyotropic nematic liquid crystals(Courty et.al., 2003; Lagerwall et. al., 2007; Schymura et. al., 2009; Weiss et. al., 2006) have been applied as nematic solvents for the alignment of nanotubes. Anisotropic interactions between the nanotube and liquid crystal drastically change the alignments and physical properties of the mixtures. Duran et.al. have observed the nematic-isotopic phase transition temperature (*TN I*) is enhanced by the incorporation of a multi-wall CNT within a small composition gap(Duran et.al., 2005).

In this section, we discuss phase separations in binary mixtures of a low molecular-weight-liquid crystal and a nanotube, such as CNT. We discuss uniaxial and biaxial nematic phases.

### **3.1 Nematic phases in mixtures of a nanotubes and a liquid crystal**

We here consider the effect of the anisotropic interaction between a nanotube and a liquid crystal and that between rods.(Matsuyama, 2010) Depending on the interaction between a nanotube and a liquid crystal, we can expect various nematic phases. Figure 5 schematically shows the four nematic phases, defined by using the orientational order parameter (*S*1) of a liquid crystal and that (*S*2) of a nanotube. When the orientational order parameter of one component is positive, determining a nematic director, and the orientational order parameter of the second component is negative, we have planer nematic phase, where the second 12 Will-be-set-by-IN-TECH

Since the discovery of carbon nanotubes (CNTs)(Iijima, 1991), extensive studies of physical and chemical properties of CNTs have been received great attention for many practical applications such as nano-sensors and devices. Windle's group first reported the nematic liquid crystalline behavior of an aqueous suspension of CNTs above a certain concentration and the isotropic-nematic phase separations.(Shaffer & Windle, 1999; Song et. al., 2003). Long nanotubes segregate preferentially to the liquid crystalline phase, whereas shorter nanotubes segregate preferentially to the isotropic phase(Zhang et.al., 2006). Recently such nanotubes as liquid crystalline materials become an important to be used in biological applications such as biosensors, biology imaging, artificial muscles, gene delivery, etc(Woltman et al., 2007).

In order to prepare CNT dispersions, strong van der Waals attractions between nanotubes must be screen out. To do this, the surface of nanotubes can be modified by acid oxidation, acid protonation, polymer or surfactant wrapping, etc.(Zhang & Kumar, 2008). For example, a water-soluble polymer, biopolymers such as DNA, and surfactant molecules have been used to wrap CNT and to increase the dispersibility in water(Badaire et. al., 2005). The polymer-wrapped nanotubes can be dispersed in a solvent with a considerable concentration. The excluded volume and electro-static repulsion between polymers can overcome the intermolecular van der Waals attractions and therefore the polymer-wrapped CNT can be dispersed in water. Thus it is possible to change the strength of the intermolecular interaction between nanotubes by using polymer-wrapping and negative or positive charging

Alignment of such CNTs, or rigid-rodlike polymers (rods), with the aid of low molecular-weight-liquid crystalline molecules is an alternative approach. Indeed thermotropic(Basu & Iannacchione, 2008; Dierking et.al., 2005; Jayalakshmi & Prasad, 2009; Lynch & Patrick, 2002; Russell et.al., 2006) as well as lyotropic nematic liquid crystals(Courty et.al., 2003; Lagerwall et. al., 2007; Schymura et. al., 2009; Weiss et. al., 2006) have been applied as nematic solvents for the alignment of nanotubes. Anisotropic interactions between the nanotube and liquid crystal drastically change the alignments and physical properties of the mixtures. Duran et.al. have observed the nematic-isotopic phase transition temperature (*TN I*) is enhanced by the incorporation of a multi-wall CNT within a small composition gap(Duran

In this section, we discuss phase separations in binary mixtures of a low molecular-weight-liquid crystal and a nanotube, such as CNT. We discuss uniaxial and

We here consider the effect of the anisotropic interaction between a nanotube and a liquid crystal and that between rods.(Matsuyama, 2010) Depending on the interaction between a nanotube and a liquid crystal, we can expect various nematic phases. Figure 5 schematically shows the four nematic phases, defined by using the orientational order parameter (*S*1) of a liquid crystal and that (*S*2) of a nanotube. When the orientational order parameter of one component is positive, determining a nematic director, and the orientational order parameter of the second component is negative, we have planer nematic phase, where the second

**3.1 Nematic phases in mixtures of a nanotubes and a liquid crystal**

**3. Nanotubes dispersed in liquid crystals**

of nanotube surface, etc.

et.al., 2005).

biaxial nematic phases.

Fig. 5. Schematically illustrated four possible nematic phases. Four nematic phases are defined using the orientational order parameter *S*<sup>1</sup> of the liquid crystal and that *S*<sup>2</sup> of the nanotube: the nematic N0 phase with *S*<sup>1</sup> > 0 and *S*<sup>2</sup> > 0, the nematic N1 phase with *S*<sup>1</sup> > 0 and *S*<sup>2</sup> < 0, and the nematic N2 phase with *S*<sup>1</sup> < 0 and *S*<sup>2</sup> > 0. When an external field (**E**) is applied along to the *z* axis for the particles of the dielectric anisotropy Δ<sup>1</sup> < 0 and Δ<sup>2</sup> < 0, the N3 phase with *S*<sup>1</sup> < 0 and *S*<sup>2</sup> < 0 can appear: the nanotubes and liquid crystals are randomly oriented on the plane perpendicular to the direction of the external field.

component is randomly oriented within in the perpendicular plane to the nematic director. The nematic N0 phase shows the nanotube and the liquid crystal are parallel to each other: *S*<sup>1</sup> > 0 and *S*<sup>2</sup> > 0. The nematic N1 phase is defined as that the nanotube and the liquid crystal are perpendicular with each other: *S*<sup>1</sup> > 0 and *S*<sup>2</sup> < 0. In this phase, the nematic director (z axis) can be defined by the orientational direction of the liquid crystals. These perpendicular alignments can be obtained by modifying the surface of a nanotube, or CNT, with polymers or surfactants. The nematic N2 phase is defined as the nanotube and the LC are perpendicular each other with *S*<sup>1</sup> < 0 and *S*<sup>2</sup> > 0. In this phase, the nematic director (z axis) can be defined by the orientational direction of the nanotube. Biaxial nematic phases are discussed in Section 3.3. When an external field (**E**) is applied along to the *z* axis for the particles of the dielectric anisotropy Δ<sup>1</sup> < 0 and Δ<sup>2</sup> < 0, the N3 phase with *S*<sup>1</sup> < 0 and *S*<sup>2</sup> < 0 may appear, where the nanotubes and liquid crystals are randomly oriented on the plane perpendicular to the direction (*z* axis) of the external field.

#### **3.2 Free energy of nanotube/liquid crystal mixtures**

We consider a binary mixture of a liquid crystal of the length *L*<sup>1</sup> and the diameter *D*<sup>1</sup> and and a nanotube of the length *L*<sup>2</sup> and the diameter *D*2: *L*<sup>1</sup> < *L*2. The volume of the liquid crystal and

**3.3 Phase diagrams of nanotube/liquid crystal mixtures**

2010).

**3.3.1 Uniaxial nematic** *N*<sup>0</sup> **phase**

respectively. Above *T*/*T*◦

In this subsection, we show some phase diagrams calculated from the free energy(Matsuyama,

Phase Separations in Mixtures of a Nanoparticle and a Liquid Crystal 255

(a) (b)

(c)

We first show the phase diagram for *c*<sup>12</sup> = 0 (see Fig. 6(a)), where the excluded volume interaction between nanotubes only prevails. The solid curve shows the binodal. The red and blue dotted lines show the first-order NIT line of a liquid crystal and that of a nanotube,

concentration of the nanotube due to the excluded volume interaction between nanotubes,

*N I* = 1, the NIT of nanotubes takes place with increasing the

Fig. 6. Phase diagrams for *c*12=0 (a), *c*12=0.3 (b), and *c*12=0.4(c) with *n*<sup>1</sup> = 2 and *n*<sup>2</sup> = 10.

that of the nanotube is given by *v*<sup>1</sup> = (*π*/4)*D*<sup>2</sup> <sup>1</sup>*L*<sup>1</sup> and *<sup>v</sup>*<sup>2</sup> = (*π*/4)*D*<sup>2</sup> <sup>2</sup>*L*2, respectively. We here assume *D* ≡ (*D*<sup>1</sup> = *D*2). Let *ρ*1(**r**, **u**) and *ρ*2(**r**, **u**) be the number density of the liquid crystals and the nanotubes with an orientation **u** (or its solid angle Ω) at a position **r**, respectively. The free energy *F* of the dispersion at the level of second virial approximation is given by Eq. (2). The volume fraction of liquid crystals is given by *φ*<sup>1</sup> = *v*1*ρ*<sup>1</sup> and tat of nanotubes *φ*<sup>2</sup> = *v*2*ρ*2. As discussed in Eq. (14), we here consider the incompressible fluids: *φ*<sup>1</sup> + *φ*<sup>2</sup> = 1.

Consider a uniaxial nematic phase, which is spatially uniform but nonuniform for orientation. Let *fi*(**u**) be the distribution function of the particle *i*(= 1, 2) and then the density can be expressed as

$$
\rho\_i(\mathbf{r}, \mathbf{u}) = c\_i f\_i(\mathbf{u}),
\tag{27}
$$

where *ci* ≡ *Ni*/*V* is the average number density of the particle *i* . The total number *N*<sup>1</sup> of the liquid crystals and *N*<sup>2</sup> of the nanoparticles must be conserved and then we have the normalization conditions:

$$
\int \rho\_i(\mathbf{r}, \mathbf{u}) d\mathbf{r} d\Omega = N\_{\mathbf{i}} / V\_{\mathbf{\cdot}} \tag{28}
$$

where *d*Ω = 2*π* sin *θdθ* for a uniaxial nematic phase.

For the interaction between liquid crystals in Eq. (2), we take the attractive (Maier-Saupe) interaction:

$$\beta\_{11} = -\nu\_1 \upsilon\_1 P\_2(\cos \theta) P\_2(\cos \theta'),\tag{29}$$

where *ν*1(≡ *U*1/*kBT* > 0) and *U*<sup>0</sup> is the anisotropic attractive (Maier-Saupe) interaction between liquid crystals(Brochard et.al., 1984; Maier & Saupe, 1958). (The subscript symbols *c* and *r* in Eq. (2) are changed to 1 and 2, respectively.) For the interaction between nanotubes, we here take into account both the attractive interaction and excluded volume one:(Matsuyama & Kato, 1996)

$$\beta\_{22} = 2L^2 D |\mathbf{u} \times \mathbf{u}'| - \nu\_2 \upsilon\_2 P\_2(\cos \theta) P\_2(\cos \theta'),\tag{30}$$

where the first term is the excluded volume interaction between nanotubes, or rods,(Onsager, 1949) and the *ν*2(≡ *U*2/*kBT* > 0) is the attractive (Maier-Saupe) interaction between nanotubes. The interaction between a liquid crystal and a nanoparticle is given by

$$\beta\_{12} = -\nu\_{12}\nu\_{12}P\_2(\cos\theta)P\_2(\cos\theta'),\tag{31}$$

where the anisotropic interaction *ν*12(≡ *U*12/*kBT*) between a liquid crystal and a rod can be positive or negative value. We here assume that the excluded volume interaction of a liquid crystal can be negligible because the length of liquid crystal is short. The volume *v*<sup>12</sup> = (*π*/4)*L*1*L*2*D* is the average excluded volume between a rod and a liquid crystal in an isotropic phase, Using Eqs. (29), (30), and (31), we can obtain the mixing free energy (15) for nanotube/liquid crystal mixtures. We here define an interaction parameter between a nanotube and a liquid crystal:

$$
\omega\_{12} = \nu\_{12}/\nu\_{1\prime} \tag{32}
$$

which becomes an important parameter in the phase behavior.

#### **3.3 Phase diagrams of nanotube/liquid crystal mixtures**

In this subsection, we show some phase diagrams calculated from the free energy(Matsuyama, 2010).

14 Will-be-set-by-IN-TECH

assume *D* ≡ (*D*<sup>1</sup> = *D*2). Let *ρ*1(**r**, **u**) and *ρ*2(**r**, **u**) be the number density of the liquid crystals and the nanotubes with an orientation **u** (or its solid angle Ω) at a position **r**, respectively. The free energy *F* of the dispersion at the level of second virial approximation is given by Eq. (2). The volume fraction of liquid crystals is given by *φ*<sup>1</sup> = *v*1*ρ*<sup>1</sup> and tat of nanotubes *φ*<sup>2</sup> = *v*2*ρ*2.

Consider a uniaxial nematic phase, which is spatially uniform but nonuniform for orientation. Let *fi*(**u**) be the distribution function of the particle *i*(= 1, 2) and then the density can be

where *ci* ≡ *Ni*/*V* is the average number density of the particle *i* . The total number *N*<sup>1</sup> of the liquid crystals and *N*<sup>2</sup> of the nanoparticles must be conserved and then we have the

For the interaction between liquid crystals in Eq. (2), we take the attractive (Maier-Saupe)

*β*<sup>11</sup> = −*ν*1*v*1*P*2(cos *θ*)*P*2(cos *θ*�

where *ν*1(≡ *U*1/*kBT* > 0) and *U*<sup>0</sup> is the anisotropic attractive (Maier-Saupe) interaction between liquid crystals(Brochard et.al., 1984; Maier & Saupe, 1958). (The subscript symbols *c* and *r* in Eq. (2) are changed to 1 and 2, respectively.) For the interaction between nanotubes, we here take into account both the attractive interaction and excluded volume

where the first term is the excluded volume interaction between nanotubes, or rods,(Onsager, 1949) and the *ν*2(≡ *U*2/*kBT* > 0) is the attractive (Maier-Saupe) interaction between

*β*<sup>12</sup> = −*ν*12*v*12*P*2(cos *θ*)*P*2(cos *θ*�

where the anisotropic interaction *ν*12(≡ *U*12/*kBT*) between a liquid crystal and a rod can be positive or negative value. We here assume that the excluded volume interaction of a liquid crystal can be negligible because the length of liquid crystal is short. The volume *v*<sup>12</sup> = (*π*/4)*L*1*L*2*D* is the average excluded volume between a rod and a liquid crystal in an isotropic phase, Using Eqs. (29), (30), and (31), we can obtain the mixing free energy (15) for nanotube/liquid crystal mixtures. We here define an interaction parameter between a

nanotubes. The interaction between a liquid crystal and a nanoparticle is given by


As discussed in Eq. (14), we here consider the incompressible fluids: *φ*<sup>1</sup> + *φ*<sup>2</sup> = 1.

*<sup>β</sup>*<sup>22</sup> <sup>=</sup> <sup>2</sup>*L*2*D*|**<sup>u</sup>** <sup>×</sup> **<sup>u</sup>**�

which becomes an important parameter in the phase behavior.

where *d*Ω = 2*π* sin *θdθ* for a uniaxial nematic phase.

<sup>1</sup>*L*<sup>1</sup> and *<sup>v</sup>*<sup>2</sup> = (*π*/4)*D*<sup>2</sup>

*ρi*(**r**, **u**) = *ci fi*(**u**), (27)

*ρi*(**r**, **u**)*d***r***d*Ω = *Ni*/*V*, (28)

<sup>2</sup>*L*2, respectively. We here

), (29)

), (30)

), (31)

*c*<sup>12</sup> = *ν*12/*ν*1, (32)

that of the nanotube is given by *v*<sup>1</sup> = (*π*/4)*D*<sup>2</sup>

expressed as

interaction:

normalization conditions:

one:(Matsuyama & Kato, 1996)

nanotube and a liquid crystal:

Fig. 6. Phase diagrams for *c*12=0 (a), *c*12=0.3 (b), and *c*12=0.4(c) with *n*<sup>1</sup> = 2 and *n*<sup>2</sup> = 10.

We first show the phase diagram for *c*<sup>12</sup> = 0 (see Fig. 6(a)), where the excluded volume interaction between nanotubes only prevails. The solid curve shows the binodal. The red and blue dotted lines show the first-order NIT line of a liquid crystal and that of a nanotube, respectively. Above *T*/*T*◦ *N I* = 1, the NIT of nanotubes takes place with increasing the concentration of the nanotube due to the excluded volume interaction between nanotubes,

concentrations.) The two nematic-isotropic phase transitions: 1st-N1IT and 2nd-N1IT, shift to higher temperatures and pass the binodal line of the isotropic phase in Fig. 7(a). The 2nd-N1IT (blue broken line) appears at lower concentrations than the binodal line and we

Phase Separations in Mixtures of a Nanoparticle and a Liquid Crystal 257

Inside the binodal region, we have the 1st-N1IT line (red dotted line). This N1+I phase separation disappears at TCP. At higher concentrations, the N1+N2 phase separation appears. The binodal curve of the coexisting N2 phase exists at *φ*<sup>2</sup> ≈ 0.7, although it is not depicted in this figure. Further decreasing *c*12(< 0), the 1st-N1IT disappears and we have the 2nd-N1IT

(a) (b)

Recent experimental studies of multi-wall carbon nanotube(CNT)/nematic liquid crystal mixtures(Duran et.al., 2005) have observed the NIT temperature of the liquid crystal is enhanced by the incorporation of CNT within a small composition gap and suggested that this enhanced NIT temperature phenomenon is attributed to anisotropic alignment of liquid crystals along the CNT bundles. Our model predicts two kind of phase behavior. When the CNTs and liquid crystals are parallel, the system shows the first-order isotropic-nematic (N0) phase transition. On the other hand, if the CNTs and liquid crystals favor to be perpendicular each other, we have the 1st- and 2nd-N1IT. The appearance of these phase transitions is

To form a nematic *N*<sup>3</sup> phase, external forces such as electric or magnetic fields will be important. When the external magnetic or electric field **E** is applied to the nanotubes and liquid crystals having a dielectric anisotropy <sup>Δ</sup>*�<sup>i</sup>* <sup>≡</sup> *�*||,*<sup>i</sup>* <sup>−</sup> *�*⊥,*<sup>i</sup>* (*<sup>i</sup>* <sup>=</sup> 1, 2), the free energy

(**<sup>n</sup>** · **<sup>E</sup>**)<sup>2</sup> *<sup>f</sup>*1(**u**)*d*<sup>Ω</sup> <sup>−</sup> *<sup>φ</sup>*2*β*Δ*�*<sup>2</sup>

(**<sup>l</sup>** · **<sup>E</sup>**)<sup>2</sup> *<sup>f</sup>*2(**u**)*d*<sup>Ω</sup> (33)

strongly effected by the orientational order of nanotubes and liquid crystals.

changes due to the external field is given by(de Gennes & Prost, 1993)

*<sup>a</sup>*3*βFext*/*<sup>V</sup>* <sup>=</sup> <sup>−</sup>*φ*1*β*Δ*�*<sup>1</sup>

*N I* = 1, the narrow biphasic region N1+I appears.

have the homogeneous N1 phase. Near *T*/*T*◦

and the 2nd-N1IT temperature increases with increasing *φ*2.

Fig. 7. Phase diagrams for *c*12=-0.2 (a) and *c*12=-0.5 (b).

**3.3.3 Effect of external fields**

and we have the isotropic (I)-nematic (N) phase separation (I+N'), which has been obtained by Onsager theory(Onsager, 1949) and Flory's lattice theory(Flory, 1956; 1979). At the low temperatures of the NIT line (red line) of liquid crystals, we have a nematic (N) phase, where liquid crystals are in a nematic state but nanotubes are in an isotropic state. We predict the chimney type phase diagram with a triple point (N+I+N'). Below the triple point, we have the broad nematic-nematic (N+N') phase separation. The nematic N phase at lower concentrations consists of almost pure liquid crystals and the N' phase are formed by the orientational ordering of rods. Near *T*/*T*◦ *N I* < 1, we have the N+I phase separation.

Figure 6(b) shows the phase diagram for *c*<sup>12</sup> = 0.3. On increasing the coupling constant *c*12, the NIT lines shift to higher temperatures and lower concentrations and two NIT curves appeared in Fig. 6(a) merge. Below the NIT line (blue dotted line), we have a nematic N0 phase, where the rods and the liquid crystals are oriented to be parallel to each other (*S*<sup>1</sup> > 0 and *S*<sup>2</sup> > 0). The width of the biphasic region I+N� <sup>0</sup> decreases with decreasing temperature. We find the triple point (N0+I+N� 0), where the nematic N0, isotropic(I), and nematic N� <sup>0</sup> phases simultaneously coexist. The binodal line of the N0 phase shifts to higher concentrations and that of the N� <sup>0</sup> phase shifts to lower concentrations with increasing *c*12. Below the triple point we have the phase separation N0+N� <sup>0</sup>, where the two nematic N0 phases with the different concentrations can coexist.

Figure 6(c) shows the phase diagram for *c*<sup>12</sup> = 0.4. The binodal curve sprits into two parts: one is the phase separation I+N0 with the lower critical solution temperature (LCST) at *T*/*T*◦ *N I* = 1 and the other is the phase separation N0+N� <sup>0</sup> with the upper critical solution temperature (UCST). We have the stable nematic N0 phase between the LCST and UCST. The length of a nanotube is also important to understand the phase diagrams. On increasing the length of the nanotube, the biphasic regions are broadened. Such LCST type phase diagram has been observed in mixtures of a main-chain nematic polyesters (poly[oxy(chloro-1,4-phenylene)oxycarbonyl][(trifluoromethyl)-1,4-phenylene]carbonyl)(PTFC) with a nematic liquid crystal (p-azoxyanisole)(PAAd14)(Ratto et.al., 1991). The theory can qualitatively describe the observed phase diagram.

### **3.3.2 Uniaxial nematic** *N*<sup>1</sup> **and** *N*<sup>2</sup> **phases**

When the coupling parameter *c*<sup>12</sup> is negative, we can expect that the nanotubes and liquid crystals are oriented to be perpendicular with each other.

Figure 7 shows the phase diagram for *c*<sup>12</sup> = −0.2(*a*) and *c*<sup>12</sup> = −0.5(*b*). The binodal line (solid line) is similar to Fig. 6(a), however, the structure of the nematic phases is different. In Fig. 7(a), the red dotted line at low concentrations shows the first-order nematic (N1)-isotropic phase transition (1st-N1IT) and the red dotted line at high concentrations shows the first-order nematic N1-N2 phase transition (N1N2T). The blue broken line corresponds to the second-order N1-I phase transition (2nd-N1IT), where the orientational order parameters continuously change. We also find the tricritical point (TCP) at which the 1st-N1IT meets the 2nd-N1IT. The phase diagram shows the three phase coexistence between N1, I and N2 phases at *T*/*T*◦ *N I* ≈ 0.92. Above the triple point, the I+N2 and N1+I phase separations appear. Below the triple point, we have the N1+N2 phase separation.

On decreasing *c*12(< 0), the system favors more perpendicular alignment. Figure 7(b) shows the phase diagram for *c*<sup>12</sup> = −0.5. (Note that the phase diagram is only shown for low 16 Will-be-set-by-IN-TECH

and we have the isotropic (I)-nematic (N) phase separation (I+N'), which has been obtained by Onsager theory(Onsager, 1949) and Flory's lattice theory(Flory, 1956; 1979). At the low temperatures of the NIT line (red line) of liquid crystals, we have a nematic (N) phase, where liquid crystals are in a nematic state but nanotubes are in an isotropic state. We predict the chimney type phase diagram with a triple point (N+I+N'). Below the triple point, we have the broad nematic-nematic (N+N') phase separation. The nematic N phase at lower concentrations consists of almost pure liquid crystals and the N' phase are formed by the

Figure 6(b) shows the phase diagram for *c*<sup>12</sup> = 0.3. On increasing the coupling constant *c*12, the NIT lines shift to higher temperatures and lower concentrations and two NIT curves appeared in Fig. 6(a) merge. Below the NIT line (blue dotted line), we have a nematic N0 phase, where the rods and the liquid crystals are oriented to be parallel to each other (*S*<sup>1</sup> > 0

simultaneously coexist. The binodal line of the N0 phase shifts to higher concentrations and

Figure 6(c) shows the phase diagram for *c*<sup>12</sup> = 0.4. The binodal curve sprits into two parts: one is the phase separation I+N0 with the lower critical solution temperature

solution temperature (UCST). We have the stable nematic N0 phase between the LCST and UCST. The length of a nanotube is also important to understand the phase diagrams. On increasing the length of the nanotube, the biphasic regions are broadened. Such LCST type phase diagram has been observed in mixtures of a main-chain nematic polyesters (poly[oxy(chloro-1,4-phenylene)oxycarbonyl][(trifluoromethyl)-1,4-phenylene]carbonyl)(PTFC) with a nematic liquid crystal (p-azoxyanisole)(PAAd14)(Ratto et.al., 1991). The theory can

When the coupling parameter *c*<sup>12</sup> is negative, we can expect that the nanotubes and liquid

Figure 7 shows the phase diagram for *c*<sup>12</sup> = −0.2(*a*) and *c*<sup>12</sup> = −0.5(*b*). The binodal line (solid line) is similar to Fig. 6(a), however, the structure of the nematic phases is different. In Fig. 7(a), the red dotted line at low concentrations shows the first-order nematic (N1)-isotropic phase transition (1st-N1IT) and the red dotted line at high concentrations shows the first-order nematic N1-N2 phase transition (N1N2T). The blue broken line corresponds to the second-order N1-I phase transition (2nd-N1IT), where the orientational order parameters continuously change. We also find the tricritical point (TCP) at which the 1st-N1IT meets the 2nd-N1IT. The phase diagram shows the three phase coexistence between N1, I and N2 phases

*N I* ≈ 0.92. Above the triple point, the I+N2 and N1+I phase separations appear. Below

On decreasing *c*12(< 0), the system favors more perpendicular alignment. Figure 7(b) shows the phase diagram for *c*<sup>12</sup> = −0.5. (Note that the phase diagram is only shown for low

*N I* = 1 and the other is the phase separation N0+N�

<sup>0</sup> phase shifts to lower concentrations with increasing *c*12. Below the triple point

*N I* < 1, we have the N+I phase separation.

0), where the nematic N0, isotropic(I), and nematic N�

<sup>0</sup>, where the two nematic N0 phases with the different

<sup>0</sup> decreases with decreasing temperature.

<sup>0</sup> phases

<sup>0</sup> with the upper critical

orientational ordering of rods. Near *T*/*T*◦

We find the triple point (N0+I+N�

concentrations can coexist.

we have the phase separation N0+N�

that of the N�

(LCST) at *T*/*T*◦

at *T*/*T*◦

and *S*<sup>2</sup> > 0). The width of the biphasic region I+N�

qualitatively describe the observed phase diagram.

crystals are oriented to be perpendicular with each other.

the triple point, we have the N1+N2 phase separation.

**3.3.2 Uniaxial nematic** *N*<sup>1</sup> **and** *N*<sup>2</sup> **phases**

concentrations.) The two nematic-isotropic phase transitions: 1st-N1IT and 2nd-N1IT, shift to higher temperatures and pass the binodal line of the isotropic phase in Fig. 7(a). The 2nd-N1IT (blue broken line) appears at lower concentrations than the binodal line and we have the homogeneous N1 phase. Near *T*/*T*◦ *N I* = 1, the narrow biphasic region N1+I appears. Inside the binodal region, we have the 1st-N1IT line (red dotted line). This N1+I phase separation disappears at TCP. At higher concentrations, the N1+N2 phase separation appears. The binodal curve of the coexisting N2 phase exists at *φ*<sup>2</sup> ≈ 0.7, although it is not depicted in this figure. Further decreasing *c*12(< 0), the 1st-N1IT disappears and we have the 2nd-N1IT and the 2nd-N1IT temperature increases with increasing *φ*2.

Fig. 7. Phase diagrams for *c*12=-0.2 (a) and *c*12=-0.5 (b).

Recent experimental studies of multi-wall carbon nanotube(CNT)/nematic liquid crystal mixtures(Duran et.al., 2005) have observed the NIT temperature of the liquid crystal is enhanced by the incorporation of CNT within a small composition gap and suggested that this enhanced NIT temperature phenomenon is attributed to anisotropic alignment of liquid crystals along the CNT bundles. Our model predicts two kind of phase behavior. When the CNTs and liquid crystals are parallel, the system shows the first-order isotropic-nematic (N0) phase transition. On the other hand, if the CNTs and liquid crystals favor to be perpendicular each other, we have the 1st- and 2nd-N1IT. The appearance of these phase transitions is strongly effected by the orientational order of nanotubes and liquid crystals.

#### **3.3.3 Effect of external fields**

To form a nematic *N*<sup>3</sup> phase, external forces such as electric or magnetic fields will be important. When the external magnetic or electric field **E** is applied to the nanotubes and liquid crystals having a dielectric anisotropy <sup>Δ</sup>*�<sup>i</sup>* <sup>≡</sup> *�*||,*<sup>i</sup>* <sup>−</sup> *�*⊥,*<sup>i</sup>* (*<sup>i</sup>* <sup>=</sup> 1, 2), the free energy changes due to the external field is given by(de Gennes & Prost, 1993)

$$a^3 \beta \mathbf{F}\_{\rm ext} / V = -\phi\_1 \beta \Delta \mathbf{c}\_1 \int (\mathbf{n} \cdot \mathbf{E})^2 f\_1(\mathbf{u}) d\Omega - \phi\_2 \beta \Delta \mathbf{c}\_2 \int (\mathbf{1} \cdot \mathbf{E})^2 f\_2(\mathbf{u}) d\Omega \tag{33}$$

to the eternal field and nanotubes favor to be parallel to liquid crystals because of the strong coupling *c*<sup>12</sup> even Δ*�*<sup>2</sup> = 1. The blue dotted line shows the 1st-order N3-N0 phase transition.

Phase Separations in Mixtures of a Nanoparticle and a Liquid Crystal 259

phase separation between N3 and N0 phases. We emphasize that we can control the four

Biaxial nematic phase has been first theoretically predicted by Freiser(Freiser, 1970). Since then, it has been the subject of much experimental(Galerne & Marcerou, 1983; Madsen et. al., 2004; Yu & Saupe, 1980), computational(Biscarini et. al., 1995; Hudson & Larson, 1993), and theoretical(Alben, 1973; Palffy-Muhoray et. al., 1984; Sharma et. al., 1985; Straley, 1974) work (see a recent review(Tschierske & Photinos, 2010)). Biaxiality occurs if anisotropic particles orient along a second axis perpendicular to a main director of the particles(Singh, 2000). Recently it has been experimentally observed a biaxial phase in colloidal dispersions of boardlike particles(van den Pol et. al., 2009). Such biaxiality is expected significant advantages

(a)

Fig. 10. Uniaxial planar nematic phase (N1) and biaxial nematic phase (N1*b*) in mixtures of a long nanotube and a short liquid crystal, which favor perpendicular orientations with each other. Nanotubes on an easy plane induce the additional ordering of nanotubes in the direction **m** perpendicular to the director **n** and yield a biaxial nematic phase N1*b*.

As discussed in Section 3.1, when the order parameter of one component is positive, determining the nematic director, and the order parameter of the second component is negative, we have planar nematic phases (N1 and N2), where the second component is randomly distributed within the perpendicular plane to the director. In these nematic phases

Figure 10 schematically shows a novel biaxial nematic in nanotube/liquid crystal mixtures, where the two components favor a mutually perpendicular orientation.(Matsuyama, 2011) The mutually perpendicular alignments of nanotubes and liquid crystals can be achieved by wrapping polymers or surfactants on nanotube's surface(Badaire et. al., 2005; Zhang &

(N1, N2), we can expect either a uniaxial or a biaxial nematic phase.

*N I* = 0.98. We also find the

Figure 9 shows the order parameters plotted against *φ*<sup>2</sup> at *T*/*T*◦

**3.4 Biaxial nematic ordering in nanotube/liquid crystal mixtures**

in display applications with a fast response.(Luckhurst, 2001)

nematic phases by applying external fields.

where **n** and **l** are the unit orientation vector of the liquid crystal and the nanotube, respectively.

Fig. 8. Phase diagram under an external field for Δ*�*<sup>1</sup> = −1 and Δ*�*<sup>2</sup> = 1, where the liquid crystals tend to align perpendicular to the electric field **E**, while the nanotubes tend to parallel to **E**.

Fig. 9. Order parameters *S*<sup>1</sup> and *S*<sup>2</sup> plotted against *φ*<sup>2</sup> at *T*/*T*◦ *N I* = 0.98 in Fig. 8. We find the phase transition from the N3 phase with *S*<sup>1</sup> < 0 and *S*<sup>2</sup> < 0 to the N0 phase with *S*<sup>1</sup> > 0 and *S*<sup>2</sup> > 0.

We here consider the case of Δ*�*<sup>1</sup> < 0 and Δ*�*<sup>2</sup> > 0: the liquid crystals tend to align perpendicular to the electric field **E**, while the nanotubes tend to parallel to **E**. We apply the external field on Fig. 6(c), where the coupling *c*12(= 0.4) between the liquid crystal and nanotube is strong. Figure 8 shows the phase diagram under an external field for Δ*�*<sup>1</sup> = −1 and Δ*�*<sup>2</sup> = 1. The binodal line is broadened, compared with Fig. 6(c). We find the N3 phase at low concentrations of nanotubes, where most liquid crystals tend to perpendicular 18 Will-be-set-by-IN-TECH

where **n** and **l** are the unit orientation vector of the liquid crystal and the nanotube,

(a)

(a)

phase transition from the N3 phase with *S*<sup>1</sup> < 0 and *S*<sup>2</sup> < 0 to the N0 phase with *S*<sup>1</sup> > 0 and

We here consider the case of Δ*�*<sup>1</sup> < 0 and Δ*�*<sup>2</sup> > 0: the liquid crystals tend to align perpendicular to the electric field **E**, while the nanotubes tend to parallel to **E**. We apply the external field on Fig. 6(c), where the coupling *c*12(= 0.4) between the liquid crystal and nanotube is strong. Figure 8 shows the phase diagram under an external field for Δ*�*<sup>1</sup> = −1 and Δ*�*<sup>2</sup> = 1. The binodal line is broadened, compared with Fig. 6(c). We find the N3 phase at low concentrations of nanotubes, where most liquid crystals tend to perpendicular

*N I* = 0.98 in Fig. 8. We find the

Fig. 9. Order parameters *S*<sup>1</sup> and *S*<sup>2</sup> plotted against *φ*<sup>2</sup> at *T*/*T*◦

Fig. 8. Phase diagram under an external field for Δ*�*<sup>1</sup> = −1 and Δ*�*<sup>2</sup> = 1, where the liquid crystals tend to align perpendicular to the electric field **E**, while the nanotubes tend to

respectively.

parallel to **E**.

*S*<sup>2</sup> > 0.

to the eternal field and nanotubes favor to be parallel to liquid crystals because of the strong coupling *c*<sup>12</sup> even Δ*�*<sup>2</sup> = 1. The blue dotted line shows the 1st-order N3-N0 phase transition. Figure 9 shows the order parameters plotted against *φ*<sup>2</sup> at *T*/*T*◦ *N I* = 0.98. We also find the phase separation between N3 and N0 phases. We emphasize that we can control the four nematic phases by applying external fields.

#### **3.4 Biaxial nematic ordering in nanotube/liquid crystal mixtures**

Biaxial nematic phase has been first theoretically predicted by Freiser(Freiser, 1970). Since then, it has been the subject of much experimental(Galerne & Marcerou, 1983; Madsen et. al., 2004; Yu & Saupe, 1980), computational(Biscarini et. al., 1995; Hudson & Larson, 1993), and theoretical(Alben, 1973; Palffy-Muhoray et. al., 1984; Sharma et. al., 1985; Straley, 1974) work (see a recent review(Tschierske & Photinos, 2010)). Biaxiality occurs if anisotropic particles orient along a second axis perpendicular to a main director of the particles(Singh, 2000). Recently it has been experimentally observed a biaxial phase in colloidal dispersions of boardlike particles(van den Pol et. al., 2009). Such biaxiality is expected significant advantages in display applications with a fast response.(Luckhurst, 2001)

Fig. 10. Uniaxial planar nematic phase (N1) and biaxial nematic phase (N1*b*) in mixtures of a long nanotube and a short liquid crystal, which favor perpendicular orientations with each other. Nanotubes on an easy plane induce the additional ordering of nanotubes in the direction **m** perpendicular to the director **n** and yield a biaxial nematic phase N1*b*.

As discussed in Section 3.1, when the order parameter of one component is positive, determining the nematic director, and the order parameter of the second component is negative, we have planar nematic phases (N1 and N2), where the second component is randomly distributed within the perpendicular plane to the director. In these nematic phases (N1, N2), we can expect either a uniaxial or a biaxial nematic phase.

Figure 10 schematically shows a novel biaxial nematic in nanotube/liquid crystal mixtures, where the two components favor a mutually perpendicular orientation.(Matsuyama, 2011) The mutually perpendicular alignments of nanotubes and liquid crystals can be achieved by wrapping polymers or surfactants on nanotube's surface(Badaire et. al., 2005; Zhang &

(a)

Phase Separations in Mixtures of a Nanoparticle and a Liquid Crystal 261

(b)

*c*<sup>12</sup> = −0.5 (a) and −0.8 (b). The black lines indicate the binodal. The red (blue) lines show a

(continuously) change. The biaxial nematic phase N1*b*, which includes an unstable biaxial phase, a metastable biaxial, and a stable biaxial phase, is indicated by the yellow area. The

first (second)-order phase transition, where the order parameters discontinuously

*N I*)-volume fraction (*φ*2) plane for

Fig. 11. Phase diagrams on the temperature (*τ* ≡ *T*/*T*◦

stable biaxial phase N1*<sup>b</sup>* appears on (b).

Kumar, 2008). To form such mutually perpendicular alignments, the anisotropic interaction (enthalpy) between a nanotube and a liquid crystal is needed. Moreover, in the planar nematic N1 phase, on increasing concentration of nanotubes, we can expect that the excluded volume interaction (entropy) between nanotubes on an easy plane induces the additional ordering of nanotubes in the direction **m** (the second "minor" director) perpendicular to the director **n** (the first "major" director) of liquid crystals and yields a biaxial nematic phase (N1*b*). In the N2 phase, we may have a biaxial nematic phase (N2*b*), where the additional ordering of liquid crystals appears in the direction **m** (minor director) perpendicular to the alignment **n** (major director) of nanotubes. Such a biaxiality in mixtures of two types of rodlike molecules has been first suggested by Alben(Alben, 1973). In this subsection, we introduce phase diagrams including such biaxial nematic phases. The phase diagrams appeared in Fig. 7 are drastically changed.

Using the distribution function *fi*(*θ*, *ϕ*) of the component *i*(= 1, 2), defined by a polar angle *θ* and an azimuthal angle *ϕ*, the biaxial order parameter is given by

$$
\Delta\_{\hat{i}} = \int D(\theta, \varphi) f\_{\hat{i}}(\theta, \varphi) d\Omega,\tag{34}
$$

where *D*(*θ*, *ϕ*) ≡ ( <sup>√</sup>3/2) sin<sup>2</sup> *<sup>θ</sup>* cos(2*ϕ*). Using the tensor order parameter

$$S\_{\dot{\imath},a\beta} = (\Im/2)S\_{\dot{\imath}}(n\_a n\_{\beta} - \delta\_{a\beta}/3),\tag{35}$$

(*α*, *β* = *x*, *y*, *z*), we have Δ*<sup>i</sup>* = *Si*,*yy* − *Si*,*xx* and *Si* = *Si*,*zz*(Singh, 2000). Here *Si*,*zz* describes alignment of molecules along the *z* axis (major director), whereas the nonzero value of Δ*<sup>i</sup>* describes ordering along the *x* or *y* axis. Using the order parameters, we can define an isotropic (I) phase with *Si* = Δ*<sup>i</sup>* = 0, a uniaxial N1 phase: *S*<sup>1</sup> > 0, *S*<sup>2</sup> < 0, Δ*<sup>i</sup>* = 0, a uniaxial N2 phase: *S*<sup>1</sup> < 0, *S*<sup>2</sup> > 0, Δ*<sup>i</sup>* = 0, a biaxial N1*<sup>b</sup>* phase: *S*<sup>1</sup> > 0, *S*<sup>2</sup> < 0, Δ*<sup>i</sup>* �= 0, and a biaxial N2*<sup>b</sup>* phase: *S*<sup>1</sup> < 0, *S*<sup>2</sup> > 0, Δ*<sup>i</sup>* �= 0. Using the additional theorem of a spherical harmonics in Eqs. (29), (30), and (31), we have *P*2(cos *γ*) = *P*2(cos *θ*)*P*2(cos *θ*� ) + *D*(*θ*, *ϕ*)*D*(*θ*� , *ϕ*� ) and can calculate phase separations(Matsuyama, 2011).

Figure 11 shows the phase diagrams numerically calculated for *c*<sup>12</sup> = −0.5 (a) and −0.8 (b). Black lines show the binodal line. The red (blue) lines show a first (second)-order phase transition, where the order parameters discontinuously (continuously) change. The biaxial nematic phase N1*b*, which includes an unstable biaxial phase, a metastable biaxial, and a stable biaxial phase , is indicated by the yellow area. In Fig. 11(a), at high temperatures, we have the phase separation (I+N2) between an isotropic (I) phase at *φ*<sup>2</sup> � 0.14 and a uniaxial N2 phase at *φ*<sup>2</sup> � 0.63. Such a chimney type's phase diagram with a coexistence between I and N phases is induced by the excluded volumes between long rods(Flory, 1956; 1979; Matsuyama & Kato, 1996; Onsager, 1949). Inside the binodal lines, we find the first-order isotropic-biaxial N1*<sup>b</sup>* phase transition at *φ*<sup>2</sup> � 0.22 and the first-order biaxial N1*b*-uniaxial N2 phase transition at *φ*<sup>2</sup> � 0.5. Above *φ*<sup>2</sup> � 0.6, we have a stable uniaxial N2 phase. We also find the three phase coexistence, or triple point (TP), between N1+I+N2 at *τ*(≡ *T*/*T*◦ *N I*) � 0.98. Below the TP, we have the N1+N2 phase separation. At low concentrations, the N1+I phase separation appears. The biaxial nematic phase is hidden inside the binodal lines.

Further increasing *c*<sup>12</sup> (Fig. 11(b)), the coupling between a liquid crystal and a nanotube drastically changes the phase diagram. The biaxial regions shift to lower concentrations and 20 Will-be-set-by-IN-TECH

Kumar, 2008). To form such mutually perpendicular alignments, the anisotropic interaction (enthalpy) between a nanotube and a liquid crystal is needed. Moreover, in the planar nematic N1 phase, on increasing concentration of nanotubes, we can expect that the excluded volume interaction (entropy) between nanotubes on an easy plane induces the additional ordering of nanotubes in the direction **m** (the second "minor" director) perpendicular to the director **n** (the first "major" director) of liquid crystals and yields a biaxial nematic phase (N1*b*). In the N2 phase, we may have a biaxial nematic phase (N2*b*), where the additional ordering of liquid crystals appears in the direction **m** (minor director) perpendicular to the alignment **n** (major director) of nanotubes. Such a biaxiality in mixtures of two types of rodlike molecules has been first suggested by Alben(Alben, 1973). In this subsection, we introduce phase diagrams including such biaxial nematic phases. The phase diagrams appeared in Fig. 7 are drastically

Using the distribution function *fi*(*θ*, *ϕ*) of the component *i*(= 1, 2), defined by a polar angle *θ*

<sup>√</sup>3/2) sin<sup>2</sup> *<sup>θ</sup>* cos(2*ϕ*). Using the tensor order parameter

(*α*, *β* = *x*, *y*, *z*), we have Δ*<sup>i</sup>* = *Si*,*yy* − *Si*,*xx* and *Si* = *Si*,*zz*(Singh, 2000). Here *Si*,*zz* describes alignment of molecules along the *z* axis (major director), whereas the nonzero value of Δ*<sup>i</sup>* describes ordering along the *x* or *y* axis. Using the order parameters, we can define an isotropic (I) phase with *Si* = Δ*<sup>i</sup>* = 0, a uniaxial N1 phase: *S*<sup>1</sup> > 0, *S*<sup>2</sup> < 0, Δ*<sup>i</sup>* = 0, a uniaxial N2 phase: *S*<sup>1</sup> < 0, *S*<sup>2</sup> > 0, Δ*<sup>i</sup>* = 0, a biaxial N1*<sup>b</sup>* phase: *S*<sup>1</sup> > 0, *S*<sup>2</sup> < 0, Δ*<sup>i</sup>* �= 0, and a biaxial N2*<sup>b</sup>* phase: *S*<sup>1</sup> < 0, *S*<sup>2</sup> > 0, Δ*<sup>i</sup>* �= 0. Using the additional theorem of a spherical harmonics in

Figure 11 shows the phase diagrams numerically calculated for *c*<sup>12</sup> = −0.5 (a) and −0.8 (b). Black lines show the binodal line. The red (blue) lines show a first (second)-order phase transition, where the order parameters discontinuously (continuously) change. The biaxial nematic phase N1*b*, which includes an unstable biaxial phase, a metastable biaxial, and a stable biaxial phase , is indicated by the yellow area. In Fig. 11(a), at high temperatures, we have the phase separation (I+N2) between an isotropic (I) phase at *φ*<sup>2</sup> � 0.14 and a uniaxial N2 phase at *φ*<sup>2</sup> � 0.63. Such a chimney type's phase diagram with a coexistence between I and N phases is induced by the excluded volumes between long rods(Flory, 1956; 1979; Matsuyama & Kato, 1996; Onsager, 1949). Inside the binodal lines, we find the first-order isotropic-biaxial N1*<sup>b</sup>* phase transition at *φ*<sup>2</sup> � 0.22 and the first-order biaxial N1*b*-uniaxial N2 phase transition at *φ*<sup>2</sup> � 0.5. Above *φ*<sup>2</sup> � 0.6, we have a stable uniaxial N2 phase. We also find the three phase

have the N1+N2 phase separation. At low concentrations, the N1+I phase separation appears.

Further increasing *c*<sup>12</sup> (Fig. 11(b)), the coupling between a liquid crystal and a nanotube drastically changes the phase diagram. The biaxial regions shift to lower concentrations and

*D*(*θ*, *ϕ*)*fi*(*θ*, *ϕ*)*d*Ω, (34)

) + *D*(*θ*, *ϕ*)*D*(*θ*�

, *ϕ*�

*N I*) � 0.98. Below the TP, we

) and can

*Si*,*αβ* = (3/2)*Si*(*nαn<sup>β</sup>* − *δαβ*/3), (35)

and an azimuthal angle *ϕ*, the biaxial order parameter is given by

Eqs. (29), (30), and (31), we have *P*2(cos *γ*) = *P*2(cos *θ*)*P*2(cos *θ*�

coexistence, or triple point (TP), between N1+I+N2 at *τ*(≡ *T*/*T*◦

The biaxial nematic phase is hidden inside the binodal lines.

calculate phase separations(Matsuyama, 2011).

Δ*<sup>i</sup>* = 

changed.

where *D*(*θ*, *ϕ*) ≡ (

Fig. 11. Phase diagrams on the temperature (*τ* ≡ *T*/*T*◦ *N I*)-volume fraction (*φ*2) plane for *c*<sup>12</sup> = −0.5 (a) and −0.8 (b). The black lines indicate the binodal. The red (blue) lines show a first (second)-order phase transition, where the order parameters discontinuously (continuously) change. The biaxial nematic phase N1*b*, which includes an unstable biaxial phase, a metastable biaxial, and a stable biaxial phase, is indicated by the yellow area. The stable biaxial phase N1*<sup>b</sup>* appears on (b).

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the thermodynamically stable biaxial N1*<sup>b</sup>* phase appears between *φ*<sup>2</sup> ∼ 0.4 and *φ*<sup>2</sup> ∼ 0.6. We find the phase separations: I+N1*b*, N1+N1*b*, I+N1, N1*b*+N2, and the three phase coexistence I+N1+N1*<sup>b</sup>* at *τ* � 1.03. Note that the coexistence region (N1*b*+N2) at *φ*<sup>2</sup> � 0.6 is very narrow. At low concentrations, the I-N1 phase transition temperature increases with increasing *φ*<sup>2</sup> and the TP shifts to higher temperatures. Note that the stable biaxial phase N1*<sup>b</sup>* appears on (b).

Duran et.al have observed in multiwall CNT/liquid crystal mixtures that the NIT temperature of the liquid crystal is enhanced by the incorporation of CNT(Duran et.al., 2005). Our theory demonstrates that this enhanced NIT temperature phenomena is attributed to anisotropic coupling between CNTs and liquid crystals. A mutually perpendicular orientation between rods and LCs can be achieved by wrapping surfactants on nanotube's surface, like a Langmuir-Blodgett film with liquid crystals(Barbero & Durand, 1996), where liquid crystals in contact with the surfactants are oriented by steric interaction with the molecules on rods. These modifications can change the strength of the interaction parameter *ν*<sup>12</sup> in our model and give a possibility of a novel biaxial phase in this mixture. The biaxial N2*<sup>b</sup>* phase does not appear on the phase diagrams because the length of liquid crystal is too short to form the N2*<sup>b</sup>* phase.

## **4. Summary**

In this chapter we have reviewed the possible phase separations in mixtures of a nanoparticle and a liquid crystal, based on the mean field theory. In Section 2, we have introduced mixtures of a spherical nanoparticle and a liquid crystal. Ferroelectric spherical nanoparticles dispersed in liquid crystal have a possibility of various phase separations, discussed in this chapter. In Section 3, we have introduced phase diagrams in mixtures of a nanotube and a liquid crystal. Novel uniaxial and biaxial nematic phases are theoretically predicted. We also discuss the effect of external fields in nanotube/liquid crystal mixtures. Phase diagrams introduced in this chapter have not been experimentally observed yet, however, it will be a challenging subject from both an experimental and theoretical point of view.

## **5. Acknowledgment**

These studies were supported by Grant-in Aid for Scientific Research (C) (Grant No. 23540477) and that on Priority Area "Soft Matter Physics" from the Ministry of Education, Culture, Sports, Science and Technology of Japan (Grant No. 21015025).

#### **6. References**


22 Will-be-set-by-IN-TECH

the thermodynamically stable biaxial N1*<sup>b</sup>* phase appears between *φ*<sup>2</sup> ∼ 0.4 and *φ*<sup>2</sup> ∼ 0.6. We find the phase separations: I+N1*b*, N1+N1*b*, I+N1, N1*b*+N2, and the three phase coexistence I+N1+N1*<sup>b</sup>* at *τ* � 1.03. Note that the coexistence region (N1*b*+N2) at *φ*<sup>2</sup> � 0.6 is very narrow. At low concentrations, the I-N1 phase transition temperature increases with increasing *φ*<sup>2</sup> and the TP shifts to higher temperatures. Note that the stable biaxial phase N1*<sup>b</sup>* appears on (b). Duran et.al have observed in multiwall CNT/liquid crystal mixtures that the NIT temperature of the liquid crystal is enhanced by the incorporation of CNT(Duran et.al., 2005). Our theory demonstrates that this enhanced NIT temperature phenomena is attributed to anisotropic coupling between CNTs and liquid crystals. A mutually perpendicular orientation between rods and LCs can be achieved by wrapping surfactants on nanotube's surface, like a Langmuir-Blodgett film with liquid crystals(Barbero & Durand, 1996), where liquid crystals in contact with the surfactants are oriented by steric interaction with the molecules on rods. These modifications can change the strength of the interaction parameter *ν*<sup>12</sup> in our model and give a possibility of a novel biaxial phase in this mixture. The biaxial N2*<sup>b</sup>* phase does not appear on the phase diagrams because the length of liquid crystal is too short to form the N2*<sup>b</sup>*

In this chapter we have reviewed the possible phase separations in mixtures of a nanoparticle and a liquid crystal, based on the mean field theory. In Section 2, we have introduced mixtures of a spherical nanoparticle and a liquid crystal. Ferroelectric spherical nanoparticles dispersed in liquid crystal have a possibility of various phase separations, discussed in this chapter. In Section 3, we have introduced phase diagrams in mixtures of a nanotube and a liquid crystal. Novel uniaxial and biaxial nematic phases are theoretically predicted. We also discuss the effect of external fields in nanotube/liquid crystal mixtures. Phase diagrams introduced in this chapter have not been experimentally observed yet, however, it will be a challenging

These studies were supported by Grant-in Aid for Scientific Research (C) (Grant No. 23540477) and that on Priority Area "Soft Matter Physics" from the Ministry of Education, Culture,

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**4. Summary**

**5. Acknowledgment**

**6. References**


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**12** 

**Laser-Combined STM and Related Techniques** 

Nanoscale particles and clusters have been attracting considerable attention from researchers and engineers from fundamental and practical viewpoints owing to their high potential for providing an extremely wide range of functional characteristics compared with ordinary solid materials, such as chemical reactivity and electrical, magnetic, optical and mechanical properties [1-5]. In fact, nanoparticles with novel functions have been realized in various fields including catalysis, biology, plasmonics, electronic devices, magnetism and so forth, on the basis of their wide range of properties. The modification of nanoparticle surfaces is producing further advances in the development of functions including those of

For the further development of novel functions based on nanoparticles/clusters and to optimize their use, it is essential to understand the physics and chemistry of such materials in relation to their macroscopic functions. However, because nanoparticles/clusters are generally defined as particles with diameters of 1-100 nm (1-10 nm in the field of nanotechnology), conventional analysis techniques are considered to average the information of nanostructures over the ensemble, limiting the understanding of individual characteristics. Furthermore, using conventional methods, analysis of the effect of local structures in each element such as atomic-scale defects, which are considered to determine the overall characteristics of small materials, is difficult. Therefore, the introduction of new methods for the analysis of these highly functional small materials is eagerly awaited.

Scanning tunneling microscopy (STM) is one of the most promising techniques for such purposes. The characteristics of materials can be obtained at the atomic scale not only for their surface but also for their inner structures including the transient dynamics. Furthermore, external perturbation such as by thermal, mechanical or electromagnetic excitation enables advanced measurements. Among the various STM techniques useful for the study of nanoparticles/clusters, STM in combination with optical technologies, which enables probing of the response of local electronic structures to optical treatment, is an interesting approach for considering the future applications of such materials. On the other

**1. Introduction** 

composite materials.

**for the Analysis of Nanoparticles/Clusters** 

Hidemi Shigekawa1, Shoji Yoshida1,

*1University of Tsukuba* 

*3University of Tokyo* 

*Japan* 

*2Toyota Technological Institute* 

Masamichi Yoshimura2 and Yutaka Mera3


## **Laser-Combined STM and Related Techniques for the Analysis of Nanoparticles/Clusters**

Hidemi Shigekawa1, Shoji Yoshida1,

Masamichi Yoshimura2 and Yutaka Mera3 *1University of Tsukuba 2Toyota Technological Institute 3University of Tokyo Japan* 

## **1. Introduction**

28 Will-be-set-by-IN-TECH

268 Smart Nanoparticles Technology

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For the further development of novel functions based on nanoparticles/clusters and to optimize their use, it is essential to understand the physics and chemistry of such materials in relation to their macroscopic functions. However, because nanoparticles/clusters are generally defined as particles with diameters of 1-100 nm (1-10 nm in the field of nanotechnology), conventional analysis techniques are considered to average the information of nanostructures over the ensemble, limiting the understanding of individual characteristics. Furthermore, using conventional methods, analysis of the effect of local structures in each element such as atomic-scale defects, which are considered to determine the overall characteristics of small materials, is difficult. Therefore, the introduction of new methods for the analysis of these highly functional small materials is eagerly awaited.

Scanning tunneling microscopy (STM) is one of the most promising techniques for such purposes. The characteristics of materials can be obtained at the atomic scale not only for their surface but also for their inner structures including the transient dynamics. Furthermore, external perturbation such as by thermal, mechanical or electromagnetic excitation enables advanced measurements. Among the various STM techniques useful for the study of nanoparticles/clusters, STM in combination with optical technologies, which enables probing of the response of local electronic structures to optical treatment, is an interesting approach for considering the future applications of such materials. On the other

Laser-Combined STM and Related Techniques for the Analysis of Nanoparticles/Clusters 271

Figure 1(a) shows a schematic diagram of the experimental setup of STM-PAS-FT. Figure 1(b) shows a typical photoabsorption spectrum obtained around the band gap of Si by the STM-PAS-FT scheme (a) and the LI scheme (b). Although it took about 100 min to acquire the spectrum for a single sweep of photon energy from 0.68 to 1.55 eV in the LI scheme, it took only 16 min to obtain a. high quality STM-FT-NIR spectrum in the range of 0.25–1.85 eV (1s for each scan and 1000 scans totally) [9]. Compared with the long acquisition time for the LI spectrum, that for a single STM-FT-NIR spectrum is much shorter (1 s), which enables

(a) (b)

Fig. 1. (a) Experimental setup of Fourier transform near-infrared measurement. (b) STM-FT-NIR spectrum and LI spectrum of a Si substrate obtained at 94 K. The acquisition time was 16 min for FT-NIR measurement (1s for each scan and 1000 scans totally) and 100 min for LI measurement (1 scan). The decrease in intensity above 1.3 eV in the LI spectrum is due to

Figure 2 shows an STM topographic image (a) and a two-dimensional map (b) of Si signals integrated from 1.3 to 1.5 eV in the STM-PAS-FT spectra recorded in the framed area in (a) (8 spectra were measured and averaged at each pixel) [9]. The samples were hemispherical

Fig. 2. (a) STM image of GeSn nanodots/Si obtained at 98 K. (b) Map of STM-PAS-FT signal integrated from 1.3 to 1.5 eV for the spectrum obtained from the framed area in (a). [9]\*

us to avoid acquiring spurious spectra.

the cutoff filter used in the grating monochromator. [9]\*

hand, sample preparation is an important factor in the analysis of nanoscale materials. Tunneling current and bias voltage can be used to modify target materials to obtain a deeper understanding of their characteristic properties. In addition, the STM tip plays an essential role in measurement. A Ag tip, for example, is used to enhance the effect of local excitation, and a carbon nanotube (CNT) tip is an excellent probe for observing fine nanoscale structures.

In this chapter, we review and discuss the STM-based techniques developed in combination with optical technologies and their application to the analysis of nanoscale particles and clusters.

## **2. Laser-combined STM and related techniques**

## **2.1 Probing methods**

In this section, we discuss probing methods of STM combined with optical technologies.

## **2.1.1 Photoabsorption spectroscopy**

Photoabsorption spectroscopy is a major branch of optical spectroscopy used to explore the electronic states of materials. Photoabsorption spectroscopy using STM (STM-PAS) provides a high spatial resolution in STM and high spectral accuracy of optical spectroscopy compared with scanning tunneling spectroscopy (STS). The detected signal in STM-PAS is the current flowing through the STM tip in response to the modulation of spectroscopic light. The simplest STM-PAS scheme is based on a lock-in (LI) technique with the intensity modulation of light while the wavelength is swept. Spatially resolved photoabsorption spectroscopy using STM was first demonstrated with a resolution of ~50 nm [6]. It was then shown that STM-PAS enables the nanometer-scale imaging of isolated subsurface defects in semiconductors through the absorption spectra associated with the defects [7].

Here we introduce two advanced STM-PAS schemes.

## **2.1.1.1 Fourier transform STM-PAS**

The simplest STM-PAS scheme has two inherent technical problems. The first is the spurious spectra that are often generated by temporal instabilities or positional drift of the STM tip, which cause the tunneling current to change with time because acquisition for LI detection requires a long time to sweep the wavelength of spectroscopic light. The second problem is an undesirable excess component in the photomodulated tip current due to the photothermal expansion of the tip material [8].

Fourier transform STM-PAS (STM-PAS-FT) based on the Fourier transformation technique was devised to solve such technical problems [9]. This scheme is essentially the same as that used in Fourier transform near-infrared spectroscopy (FT-NIR). Multiple lights modulated with different frequencies corresponding to their wavelengths generate a tip current with an interferogram caused by the superposition of current components modulated at the different frequencies. The photoabsorption spectrum is computed from the interferogram via Fourier transformation. In this case, the photothermal expansion is suppressed by the simultaneous illumination of multiplexed lights.

hand, sample preparation is an important factor in the analysis of nanoscale materials. Tunneling current and bias voltage can be used to modify target materials to obtain a deeper understanding of their characteristic properties. In addition, the STM tip plays an essential role in measurement. A Ag tip, for example, is used to enhance the effect of local excitation, and a carbon nanotube (CNT) tip is an excellent probe for observing fine nanoscale

In this chapter, we review and discuss the STM-based techniques developed in combination with optical technologies and their application to the analysis of nanoscale particles and

In this section, we discuss probing methods of STM combined with optical technologies.

semiconductors through the absorption spectra associated with the defects [7].

Photoabsorption spectroscopy is a major branch of optical spectroscopy used to explore the electronic states of materials. Photoabsorption spectroscopy using STM (STM-PAS) provides a high spatial resolution in STM and high spectral accuracy of optical spectroscopy compared with scanning tunneling spectroscopy (STS). The detected signal in STM-PAS is the current flowing through the STM tip in response to the modulation of spectroscopic light. The simplest STM-PAS scheme is based on a lock-in (LI) technique with the intensity modulation of light while the wavelength is swept. Spatially resolved photoabsorption spectroscopy using STM was first demonstrated with a resolution of ~50 nm [6]. It was then shown that STM-PAS enables the nanometer-scale imaging of isolated subsurface defects in

The simplest STM-PAS scheme has two inherent technical problems. The first is the spurious spectra that are often generated by temporal instabilities or positional drift of the STM tip, which cause the tunneling current to change with time because acquisition for LI detection requires a long time to sweep the wavelength of spectroscopic light. The second problem is an undesirable excess component in the photomodulated tip current due to the

Fourier transform STM-PAS (STM-PAS-FT) based on the Fourier transformation technique was devised to solve such technical problems [9]. This scheme is essentially the same as that used in Fourier transform near-infrared spectroscopy (FT-NIR). Multiple lights modulated with different frequencies corresponding to their wavelengths generate a tip current with an interferogram caused by the superposition of current components modulated at the different frequencies. The photoabsorption spectrum is computed from the interferogram via Fourier transformation. In this case, the photothermal expansion is suppressed by the

**2. Laser-combined STM and related techniques** 

Here we introduce two advanced STM-PAS schemes.

photothermal expansion of the tip material [8].

simultaneous illumination of multiplexed lights.

**2.1.1.1 Fourier transform STM-PAS** 

structures.

clusters.

**2.1 Probing methods** 

**2.1.1 Photoabsorption spectroscopy** 

Figure 1(a) shows a schematic diagram of the experimental setup of STM-PAS-FT. Figure 1(b) shows a typical photoabsorption spectrum obtained around the band gap of Si by the STM-PAS-FT scheme (a) and the LI scheme (b). Although it took about 100 min to acquire the spectrum for a single sweep of photon energy from 0.68 to 1.55 eV in the LI scheme, it took only 16 min to obtain a. high quality STM-FT-NIR spectrum in the range of 0.25–1.85 eV (1s for each scan and 1000 scans totally) [9]. Compared with the long acquisition time for the LI spectrum, that for a single STM-FT-NIR spectrum is much shorter (1 s), which enables us to avoid acquiring spurious spectra.

Fig. 1. (a) Experimental setup of Fourier transform near-infrared measurement. (b) STM-FT-NIR spectrum and LI spectrum of a Si substrate obtained at 94 K. The acquisition time was 16 min for FT-NIR measurement (1s for each scan and 1000 scans totally) and 100 min for LI measurement (1 scan). The decrease in intensity above 1.3 eV in the LI spectrum is due to the cutoff filter used in the grating monochromator. [9]\*

Figure 2 shows an STM topographic image (a) and a two-dimensional map (b) of Si signals integrated from 1.3 to 1.5 eV in the STM-PAS-FT spectra recorded in the framed area in (a) (8 spectra were measured and averaged at each pixel) [9]. The samples were hemispherical

Fig. 2. (a) STM image of GeSn nanodots/Si obtained at 98 K. (b) Map of STM-PAS-FT signal integrated from 1.3 to 1.5 eV for the spectrum obtained from the framed area in (a). [9]\*

Laser-Combined STM and Related Techniques for the Analysis of Nanoparticles/Clusters 273

peak is in good agreement with the optical transition energy between discrete levels theoretically predicted by the size dependence due to the quantum confinement effect.

Conventional electric field modulation spectroscopy (EFMS) techniques, such as electroreflectance and photoreflectance, are established tools used for the accurate measurement of interband transition energies in semiconductors [12]. The principle of EFMS is based on the fact that applying an electric field to a semiconducting material causes an oscillatory change in the optical absorption coefficient depending on the wavelength, i.e., the Franz-Keldysh effect [13]. The spectral line shape of EFMS is closely related to energy derivatives of the unperturbed dielectric function, and represents features corresponding to interband transitions. By combining EMFS with STM (STM-EFMS), we can perform EFMS

Fig. 4. Schematic illustrations of STM-EFMS measurement: (a) OM scheme, (b) BM scheme.[15]\*

Figure 4 illustrates schematics of STM-EFMS measurements using two different schemes for electric field modulation: (a) optical modulation (OM) and (b) bias modulation (BM). In OM, electric field modulation is achieved by an SPV periodically induced by chopped light illumination with energy above the bandgap of the sample from a diode laser. In BM, electric field modulation is achieved by applying a modulated bias voltage to the sample, which directly modulates the tip-induced band bending beneath the tip. In STM-EFMS, the change in is detected as a change in the STM tip current, which is synchronized with the

Figure 5 shows a typical STM-EFMS spectrum (solid curve) and the spectrum obtained by the conventional EFMS method (dashed curve). The STM-EFMS spectrum reproduces the main features of the band structure. Two distinct structures observed at photon energies approximately *hv*=1.41 and 1.78 eV are ascribed to the interband absorption edge of GaAs and the spin split-off band absorption, respectively. It was demonstrated that the spatial resolution of STM-EFMS measurements was of nanometer scale [15, 16]. A typical STM topographic image of a -FeSi2 nanodot sample is shown in Fig. 6(a). The sample was epitaxially grown on an n-Si(111) substrate covered with an ultrathin SiO2 film. After the growth, the surface was terminated with atomic hydrogen. Figure 6(b) shows STM-EFMS spectra at 96 K obtained by the two schemes in an energy range lower than the absorption

**2.1.1.2 Electric field modulation spectroscopy** 

with nanometer spatial resolution [14].

modulation of the electric field.

edge of Si [15].

Ge1-xSnx (x=0.1) nanodots epitaxially grown on Si substrates with an ultrathin SiO2 film. The deposition of Ge and Sn was controlled to 24 monolayers to grow nanodots with diameters of ~40 nm. After the samples were annealed at 770 K, the surface was terminated with atomic hydrogen to suppress surface states. The region of bright contrast in (b) matches the region without nanodots in (a) reasonably well, i.e., the expected part of the Si substrates. The contrast indicates the spatial resolution of STM-PAS-FT to be ~10 nm.

Figure 3(a) shows a set of photoabsorption spectra obtained on different GeSn nanodots with various lateral diameters [10]. The peak indicated in each spectrum by an arrow is observed at an energy lower than the gap energy of Si (~1.2 eV) and exhibits a clear blue shift with decreasing dot size, suggesting that the signal is induced by optical transitions between discrete levels in the quantum dots. The photoabsorption energy of a spherical nanodot with radius *R* is given by

$$E\_{abs} = E\_{bulk} + \frac{\hbar^2 \pi^2}{2\mu R^2} - 1.786 \frac{e^2}{\varepsilon R} - 0.248 \frac{e^2}{8\pi\varepsilon a\_B^\*} \tag{1}$$

Here, \* 22 <sup>B</sup>*a e* 4 / represents the exciton Bohr radius, is Planck's constant, is the dielectric constant of the nanodot material, *e* is the electron charge and is the reduced mass of carriers. The first term *E*bulk is the bandgap energy of the bulk crystal [11]. The solid curve in Fig. 3 shows the theoretical curve calculated from Eq. (1). The energy position of the

Fig. 3. (a) STM-PAS-FT spectra obtained from individual GeSn nanodots of various sizes. (b) Typical topographic STM image of Ge1-xSnx nanodots. The white arrow indicates the position where the spectrum for *d*=30 nm was acquired. (c) Peak energy in (a) as a function of the lateral diameter of the nanodots. The solid curve is calculated from Eq. (1). [10]

peak is in good agreement with the optical transition energy between discrete levels theoretically predicted by the size dependence due to the quantum confinement effect.

## **2.1.1.2 Electric field modulation spectroscopy**

272 Smart Nanoparticles Technology

Ge1-xSnx (x=0.1) nanodots epitaxially grown on Si substrates with an ultrathin SiO2 film. The deposition of Ge and Sn was controlled to 24 monolayers to grow nanodots with diameters of ~40 nm. After the samples were annealed at 770 K, the surface was terminated with atomic hydrogen to suppress surface states. The region of bright contrast in (b) matches the region without nanodots in (a) reasonably well, i.e., the expected part of the Si substrates.

Figure 3(a) shows a set of photoabsorption spectra obtained on different GeSn nanodots with various lateral diameters [10]. The peak indicated in each spectrum by an arrow is observed at an energy lower than the gap energy of Si (~1.2 eV) and exhibits a clear blue shift with decreasing dot size, suggesting that the signal is induced by optical transitions between discrete levels in the quantum dots. The photoabsorption energy of a spherical

2 8 *abs bulk*

of carriers. The first term *E*bulk is the bandgap energy of the bulk crystal [11]. The solid curve in Fig. 3 shows the theoretical curve calculated from Eq. (1). The energy position of the

Fig. 3. (a) STM-PAS-FT spectra obtained from individual GeSn nanodots of various sizes. (b) Typical topographic STM image of Ge1-xSnx nanodots. The white arrow indicates the position where the spectrum for *d*=30 nm was acquired. (c) Peak energy in (a) as a function of the lateral diameter of the nanodots. The solid curve is calculated from Eq. (1). [10]

*e e E E*

dielectric constant of the nanodot material, *e* is the electron charge and

22 2 2 2 \* 1.786 0.248

*R a R*

represents the exciton Bohr radius, is Planck's constant, is the

*B*

is the reduced mass

(1)

The contrast indicates the spatial resolution of STM-PAS-FT to be ~10 nm.

nanodot with radius *R* is given by

Here, \* 22 <sup>B</sup>*a e* 4 / 

Conventional electric field modulation spectroscopy (EFMS) techniques, such as electroreflectance and photoreflectance, are established tools used for the accurate measurement of interband transition energies in semiconductors [12]. The principle of EFMS is based on the fact that applying an electric field to a semiconducting material causes an oscillatory change in the optical absorption coefficient depending on the wavelength, i.e., the Franz-Keldysh effect [13]. The spectral line shape of EFMS is closely related to energy derivatives of the unperturbed dielectric function, and represents features corresponding to interband transitions. By combining EMFS with STM (STM-EFMS), we can perform EFMS with nanometer spatial resolution [14].

Fig. 4. Schematic illustrations of STM-EFMS measurement: (a) OM scheme, (b) BM scheme.[15]\*

Figure 4 illustrates schematics of STM-EFMS measurements using two different schemes for electric field modulation: (a) optical modulation (OM) and (b) bias modulation (BM). In OM, electric field modulation is achieved by an SPV periodically induced by chopped light illumination with energy above the bandgap of the sample from a diode laser. In BM, electric field modulation is achieved by applying a modulated bias voltage to the sample, which directly modulates the tip-induced band bending beneath the tip. In STM-EFMS, the change in is detected as a change in the STM tip current, which is synchronized with the modulation of the electric field.

Figure 5 shows a typical STM-EFMS spectrum (solid curve) and the spectrum obtained by the conventional EFMS method (dashed curve). The STM-EFMS spectrum reproduces the main features of the band structure. Two distinct structures observed at photon energies approximately *hv*=1.41 and 1.78 eV are ascribed to the interband absorption edge of GaAs and the spin split-off band absorption, respectively. It was demonstrated that the spatial resolution of STM-EFMS measurements was of nanometer scale [15, 16]. A typical STM topographic image of a -FeSi2 nanodot sample is shown in Fig. 6(a). The sample was epitaxially grown on an n-Si(111) substrate covered with an ultrathin SiO2 film. After the growth, the surface was terminated with atomic hydrogen. Figure 6(b) shows STM-EFMS spectra at 96 K obtained by the two schemes in an energy range lower than the absorption edge of Si [15].

Laser-Combined STM and Related Techniques for the Analysis of Nanoparticles/Clusters 275

When carriers are injected from an STM tip to a sample, light emission(LE)is induced in some cases. STM luminescence spectroscopy is a measurement scheme in which the emitted

The mechanism of photon emission depends on the process of measurement such as plasmon polariton (SPP) excitation in conductive samples, carrier recombination in semiconductor samples and the HOMO-LUMO transition in molecular samples. Information on molecular vibrations can be obtained by analyzing the spectrum [22], such as by inelastic tunneling spectroscopy [23], which may be used for the analysis of composite

light is collected to explore the local electronic properties of materials (Fig. 7) [19-22].

**2.1.2 Light-emission spectroscopy** 

nanoparticles/clusters combined with organic materials.

Fig. 7. Schematic illustrations of LE-STM setup and basic mechanism.

Fig. 8. (a) is the STM light emission spectrum of a single R6G molecule. (b)

 \*

Photoluminescence (PL) spectrum of R6G on HOPG. The cutoff of the PL spectrum at 2.17

From the distribution of emission intensity on a sample surface (photon map), we can investigate the geometry of the electronic structures of the sample. The photon map also enables us to estimate the transport properties of minority carriers by considering the

eV is due to the short-wavelength-cutoff filter inserted in the collection optics. [24]\*

Reprinted with permission from each reference. Copyright American Institute of Physics.

Fig. 5. STM-EFMS spectrum obtained for a perfect (110) cleaved surface of GaAs. The broken curve indicates the EFMS spectrum measured by the conventional electroreflectance method applied to a macroscopic GaAs sample. E0 and E0+0 denote the interband absorption edge of GaAs and to the spin split-off band absorption, respectively. [14]\*

For epitaxially grown sufficiently strained -FeSi2 nanodots on Si, bandgap crossover, i.e., change from indirect band to direct band, is theoretically predicted [18], which, however, has not been confirmed experimentally despite that the mechanism is of great importance for application. The two spectra exhibit a common feature from 0.72 to 0.76 eV. The energy positions of the signals, 0.72–0.76 eV, closely match the absorption thresholds detected by macroscopic measurements of the photoabsorption coefficient for a bulk -FeSi2 crystal at 100 K [17] and the energy threshold is attributed to optical transitions across the indirect bandgap. Therefore, these findings strongly indicate that the -FeSi2 nanodot sample examined was an indirect-gap semiconductor, instead of the theoretical prediction.

Using the STM-EFMS scheme, the band structure of individual nanodots can be explored with high accuracy.

Fig. 6. (a) Typical STM topographic image of -FeSi2 nanodot sample. Bright contrasts with heights of 5-10 nm are H-terminated -FeSi2 nanodots grown on Si(111). (b) STM-EFMS spectra measured on -FeSi2 nanodots by optical modulation (OM) and bias modulation (BM). The common features near 0.72 – 0.76 eV agree well with the absorption thresholds detected by macroscopic measurements of the photoabsorption coefficient in a bulk-FeSi2 crystal. [15]\*

## **2.1.2 Light-emission spectroscopy**

274 Smart Nanoparticles Technology

Fig. 5. STM-EFMS spectrum obtained for a perfect (110) cleaved surface of GaAs. The broken curve indicates the EFMS spectrum measured by the conventional electroreflectance method applied to a macroscopic GaAs sample. E0 and E0+0 denote the interband absorption edge

For epitaxially grown sufficiently strained -FeSi2 nanodots on Si, bandgap crossover, i.e., change from indirect band to direct band, is theoretically predicted [18], which, however, has not been confirmed experimentally despite that the mechanism is of great importance for application. The two spectra exhibit a common feature from 0.72 to 0.76 eV. The energy positions of the signals, 0.72–0.76 eV, closely match the absorption thresholds detected by macroscopic measurements of the photoabsorption coefficient for a bulk -FeSi2 crystal at 100 K [17] and the energy threshold is attributed to optical transitions across the indirect bandgap. Therefore, these findings strongly indicate that the -FeSi2 nanodot sample

Using the STM-EFMS scheme, the band structure of individual nanodots can be explored

Fig. 6. (a) Typical STM topographic image of -FeSi2 nanodot sample. Bright contrasts with heights of 5-10 nm are H-terminated -FeSi2 nanodots grown on Si(111). (b) STM-EFMS spectra measured on -FeSi2 nanodots by optical modulation (OM) and bias modulation (BM). The common features near 0.72 – 0.76 eV agree well with the absorption thresholds detected by macroscopic measurements of the photoabsorption coefficient in a bulk-FeSi2

examined was an indirect-gap semiconductor, instead of the theoretical prediction.

of GaAs and to the spin split-off band absorption, respectively. [14]\*

with high accuracy.

crystal. [15]\*

When carriers are injected from an STM tip to a sample, light emission(LE)is induced in some cases. STM luminescence spectroscopy is a measurement scheme in which the emitted light is collected to explore the local electronic properties of materials (Fig. 7) [19-22].

The mechanism of photon emission depends on the process of measurement such as plasmon polariton (SPP) excitation in conductive samples, carrier recombination in semiconductor samples and the HOMO-LUMO transition in molecular samples. Information on molecular vibrations can be obtained by analyzing the spectrum [22], such as by inelastic tunneling spectroscopy [23], which may be used for the analysis of composite nanoparticles/clusters combined with organic materials.

Fig. 7. Schematic illustrations of LE-STM setup and basic mechanism.

Fig. 8. (a) is the STM light emission spectrum of a single R6G molecule. (b) Photoluminescence (PL) spectrum of R6G on HOPG. The cutoff of the PL spectrum at 2.17 eV is due to the short-wavelength-cutoff filter inserted in the collection optics. [24]\*

From the distribution of emission intensity on a sample surface (photon map), we can investigate the geometry of the electronic structures of the sample. The photon map also enables us to estimate the transport properties of minority carriers by considering the

 \* Reprinted with permission from each reference. Copyright American Institute of Physics.

Laser-Combined STM and Related Techniques for the Analysis of Nanoparticles/Clusters 277

carriers excited by the first optical pulse remain in the excited state, the absorption of the second optical pulse is suppressed. In such a case, the current *I*\* induced by the second current pulse decreases depending on *td*, reflecting the decay of the excited carriers excitation by the first-pulse. Signal *I* also depends on *td*, because the magnitude difference of the second current pulse changes the temporally averaged value of the tunneling current. Accordingly, the relaxation dynamics of the excited carriers of the target material, namely, the decay of carrier density after excitation by the first optical pulse, can be probed by STM

In SPPX-STM the sophisticated control of delay-time generation and modulation with a pulse-picking procedure is essential. This enables the probing of nanometer-scale structures with a wide range of relaxation lifetimes. Using the pulse-picking method, a large and discrete modification of *td* can be realized by changing the selection of pulses that transmit the pulse pickers, which is suitable for modulating *td* in SPPX-STM. In this method, the

with a high acquisition rate, where *I*(∞) is the tunneling current for a delay time sufficiently long for the excited state to be relaxed. Accordingly, SPPX-STM has made it possible to visualize the carrier dynamics in nanometer-scale structures with a wide range of relaxation

Fig. 10. SPPX-STM signals obtained for various samples. Upper spectra of LT-GaAs, GaNAs and n-GaAs were obtained by optical pump-probe method (R: reflectivity of probe pulse).

, is accurately probed

at the resolution of the pulse width, that is, in the femtosecond range.

delay time dependence of the tunneling current, *It It I d d*

lifetimes. Figure 10 shows the capability of wide timescale measurement.

Fig. 9. Schematic illustration of SPPX-STM

diffusion length. When a local spectrum is analyzed as a photon map, more detailed information such as the distribution of elements can be obtained. However, STM luminescence spectra are affected by various factors other than sample properties, such as the tip shape, tip material and the characteristics of the substrate used for the experiment; thus, careful analysis is necessary to determine the physical properties of the target material from the STM luminescence spectra. In the case of organic materials, damages due to carrier injection must be avoided.

Figure 8(a) shows the STM luminescence spectrum obtained from a single rhodamine 6G (R6G) molecule on an HOPG surface. The features of the spectrum are in good agreement with the photoluminescence spectrum of a layer of rhodamine molecules on HOPG (Fig. 8(b)) [24].

## **2.1.3 Photoexcitation spectroscopy**

Dynamical processes have often been studied by a laser pump-probe method where a pump pulse excites a sample and a subsequently arriving probe pulse with a delay time of *t*d is used to track its temporal evolution [25, 26]. The temporal resolution attainable in such experiments is limited only by the pulse width, which is generally in the femtosecond range. However, the spatial resolution is determined by the optical diffraction limit, which is large compared with the typical size of materials and devices currently being developed, and therefore, the physical observables are obscured by ensemble averaging. Thus, high spatial resolution in pump-probe experiments would provide new insights into nanoscale structures and materials and unveil a rich variety of dynamical features of light-sensitive phenomena in unexplored regimes such as charge transfer, phase transitions, electronic transitions, carrier or spin transport and quantum coherence.

In contrast, STM easily provides atomic-scale spatial resolution despite its low temporal resolution (typically worse than 100 kHz) [27-31]. Therefore, if the tunneling process directly produced by optical excitation can be measured, high temporal and spatial sensitivity can be simultaneously achieved with the atomic-scale resolution of STM [32-41]. A promising setup for achieving this is pulse-pair-excited STM (PPX-STM), in which, in analogy with pump-probe experiments, a sequence of paired laser pulses with a certain delay time *t*d excites the sample surface beneath the STM tip, and the tunneling current *I* is measured as a function of *t*d. To detect a faint time-resolved tunneling current with a high signal-to-noise ratio, the rectangular modulation of delay time with a pulse-picking procedure is used (shaken-pulse-pair excited STM: SPPX-STM), enabling the spatial mapping of time-resolved tunneling current [33].

Figure 9 shows the setup of SPPX-STM. When paired optical pulses arrive at a sample beneath the STM tip, they generate pulses of raw tunneling current *I\**, reflecting the excitation and relaxation of the physical properties of the sample. If these current pulses decay rapidly compared with the time scale of the STM preamplifier bandwidth, they can be temporally averaged in the preamplifier but cannot be detected directly in the signal *I*. Even in this case, the relaxation dynamics can be probed through the *t*d dependence of *I*. When *t*<sup>d</sup> is sufficiently long, paired optical pulses with the same intensity independently induce two current pulses with the same *I\**. In contrast, when *t*d is short and the second pulse illuminates the sample in the excited state induced by the first pulse, the second current pulse may have a different magnitude, depending on *td*. A typical process that can be observed using this mechanism is absorption bleaching in semiconductors; when the carriers excited by the first optical pulse remain in the excited state, the absorption of the second optical pulse is suppressed. In such a case, the current *I*\* induced by the second current pulse decreases depending on *td*, reflecting the decay of the excited carriers excitation by the first-pulse. Signal *I* also depends on *td*, because the magnitude difference of the second current pulse changes the temporally averaged value of the tunneling current. Accordingly, the relaxation dynamics of the excited carriers of the target material, namely, the decay of carrier density after excitation by the first optical pulse, can be probed by STM at the resolution of the pulse width, that is, in the femtosecond range.

Fig. 9. Schematic illustration of SPPX-STM

276 Smart Nanoparticles Technology

diffusion length. When a local spectrum is analyzed as a photon map, more detailed information such as the distribution of elements can be obtained. However, STM luminescence spectra are affected by various factors other than sample properties, such as the tip shape, tip material and the characteristics of the substrate used for the experiment; thus, careful analysis is necessary to determine the physical properties of the target material from the STM luminescence spectra. In the case of organic materials, damages due to carrier

Figure 8(a) shows the STM luminescence spectrum obtained from a single rhodamine 6G (R6G) molecule on an HOPG surface. The features of the spectrum are in good agreement with the photoluminescence spectrum of a layer of rhodamine molecules on HOPG (Fig. 8(b)) [24].

Dynamical processes have often been studied by a laser pump-probe method where a pump pulse excites a sample and a subsequently arriving probe pulse with a delay time of *t*d is used to track its temporal evolution [25, 26]. The temporal resolution attainable in such experiments is limited only by the pulse width, which is generally in the femtosecond range. However, the spatial resolution is determined by the optical diffraction limit, which is large compared with the typical size of materials and devices currently being developed, and therefore, the physical observables are obscured by ensemble averaging. Thus, high spatial resolution in pump-probe experiments would provide new insights into nanoscale structures and materials and unveil a rich variety of dynamical features of light-sensitive phenomena in unexplored regimes such as charge transfer, phase transitions, electronic

In contrast, STM easily provides atomic-scale spatial resolution despite its low temporal resolution (typically worse than 100 kHz) [27-31]. Therefore, if the tunneling process directly produced by optical excitation can be measured, high temporal and spatial sensitivity can be simultaneously achieved with the atomic-scale resolution of STM [32-41]. A promising setup for achieving this is pulse-pair-excited STM (PPX-STM), in which, in analogy with pump-probe experiments, a sequence of paired laser pulses with a certain delay time *t*d excites the sample surface beneath the STM tip, and the tunneling current *I* is measured as a function of *t*d. To detect a faint time-resolved tunneling current with a high signal-to-noise ratio, the rectangular modulation of delay time with a pulse-picking procedure is used (shaken-pulse-pair excited

STM: SPPX-STM), enabling the spatial mapping of time-resolved tunneling current [33].

Figure 9 shows the setup of SPPX-STM. When paired optical pulses arrive at a sample beneath the STM tip, they generate pulses of raw tunneling current *I\**, reflecting the excitation and relaxation of the physical properties of the sample. If these current pulses decay rapidly compared with the time scale of the STM preamplifier bandwidth, they can be temporally averaged in the preamplifier but cannot be detected directly in the signal *I*. Even in this case, the relaxation dynamics can be probed through the *t*d dependence of *I*. When *t*<sup>d</sup> is sufficiently long, paired optical pulses with the same intensity independently induce two current pulses with the same *I\**. In contrast, when *t*d is short and the second pulse illuminates the sample in the excited state induced by the first pulse, the second current pulse may have a different magnitude, depending on *td*. A typical process that can be observed using this mechanism is absorption bleaching in semiconductors; when the

injection must be avoided.

**2.1.3 Photoexcitation spectroscopy** 

transitions, carrier or spin transport and quantum coherence.

In SPPX-STM the sophisticated control of delay-time generation and modulation with a pulse-picking procedure is essential. This enables the probing of nanometer-scale structures with a wide range of relaxation lifetimes. Using the pulse-picking method, a large and discrete modification of *td* can be realized by changing the selection of pulses that transmit the pulse pickers, which is suitable for modulating *td* in SPPX-STM. In this method, the delay time dependence of the tunneling current, *It It I d d* , is accurately probed with a high acquisition rate, where *I*(∞) is the tunneling current for a delay time sufficiently long for the excited state to be relaxed. Accordingly, SPPX-STM has made it possible to visualize the carrier dynamics in nanometer-scale structures with a wide range of relaxation lifetimes. Figure 10 shows the capability of wide timescale measurement.

Fig. 10. SPPX-STM signals obtained for various samples. Upper spectra of LT-GaAs, GaNAs and n-GaAs were obtained by optical pump-probe method (R: reflectivity of probe pulse).

Laser-Combined STM and Related Techniques for the Analysis of Nanoparticles/Clusters 279

In SPPX-STM, the nonlinear interference between the excitations is essential, which depends on the material we measure. In SPPX-STM applied to a semiconductor, tip-induced band bending and surface photovoltage play important roles in the measurement. However, in general, such as dipole formation, charge transfer, changes in conductance, and vibration that causes the change in the tip-sample distance are possible mechanisms for producing SPPX-STM signals. Therefore, SPPX-STM enables the nanoscale probing of transient dynamics over a wide range of time scales, simultaneously with the observation of local

Another promising technique is STM combined with synchrotron radiation (SR-STM), which probes core-level photoemission, enabling the identification of atomic species of the target materials [42]. The spatial resolution has been improved to ~10 nm, and therefore in the near future, in addition to the analysis of isolated nanostructures, probing of the inner

The tunneling current and bias voltage in STM, which are the basic parameters of STM measurement, can be used for the modification of target materials. Probing, for example, the effect of atomic-scale defects on local electronic structures enables the clarification of the fundamental mechanism in each element and its relation to macroscopic functions. For nanoparticles/clusters, such effects are essential for determining the characteristic

Fig. 12. (a) STM image of current-injection-induced polymerized C60 molecules (dark contrasts), (b) schematic of C60 polymerization, (c) nanoscale patterning of polymerized C60

Figure 12(a) shows an STM image of a C60 crystalline film within a thickness of several monolayers grown on an HOPG surfaces in ultrahigh vacuum (UHV). Before acquiring the image, electrons were injected at the point indicated by a cross in Fig. 12(a) at a sample bias voltage of Vs=+4.2 V. The dark contrasts around the point represent intracluster structures with a stripe pattern, suggesting the frozen rotation of C60 molecules despite the room temperature. Namely, the dark sites are C60 molecules polymerized with molecules in the underlayer. The polymerization was induced by the injection of low-energy electrons from the STM probe tip. Figure 12(c) shows a line structure consisting of polymerized C60 clusters confined in a width as small as ~2 nm, which is a good example of nanoscale electron-beam

molecules formed by scanning the tip along the longitudinal direction. [43]†

structures by STM.

structures of targets may become possible.

**2.2 Manipulation for fine measurement** 

† Copyright The Japan Society of Applied Physics.

properties of their total systems.

As an example, SPPX-STM has been applied to the analysis of carrier dynamics in a Co nanoparticle/GaAs(110) system. When Co is deposited on a GaAs, nanoparticles are formed (Fig. 11(a)). In this system, photoexcited minority carriers (holes) captured at the surface are recombined with electrons tunneling from the STM tip via the gap states formed by Co as shown in Fig. 11(c). This is considered to be enhanced by the existence of gap states at the Co nanoparticle sites. Understanding such a charge transport mechanism through nanoparticles is of great importance not only for the development of nanoscale electronic devices but also for their application to the finer control of chemical reactions in catalysis.

Fig. 11. (a) STM image of Co/GaAs, (b) 2D map of decay constant, (c) schematic model of recombination at gap states, (d) cross section along the line in (b), (e) decay constant as a function of tunneling current, (f) decay constant as a function of Co particle size.

Figure 11(b) shows the overlap of the STM image in Fig. 11(a) with the map of the decay constant obtained over the surface. The two-dimensional (2D) map of the decay constant shown in color scale indicates the decay constant of the photoinduced carrier density at each point. The positional agreement is good. As shown in the cross section in Fig. 11(d) obtained along the line in Fig. 3(b), the decay is rapid in the Co regions. In such regions, photoinduced holes trapped at the surface are recombined with electrons tunneling from the STM tip at the gap states; thus, there are two limitations in this process: the tunneling current and hole-capture rate. When the tunneling current is sufficient, the hole-capture rate becomes the limiting factor of the recombination process. Figure 11(e) shows the relation between the decay constant and tunneling current. As expected, the decay constant decreases with increasing tunneling current and has a saturated value of 6.9 ns, which corresponds to the hole-capture rate of this system. The decay process should depend on the gap-state density. Figure 11(f) shows the decay constant as a function of the Co nanoparticle size. The time constant increases with decreasing nanoparticle size as expected.

In SPPX-STM, the nonlinear interference between the excitations is essential, which depends on the material we measure. In SPPX-STM applied to a semiconductor, tip-induced band bending and surface photovoltage play important roles in the measurement. However, in general, such as dipole formation, charge transfer, changes in conductance, and vibration that causes the change in the tip-sample distance are possible mechanisms for producing SPPX-STM signals. Therefore, SPPX-STM enables the nanoscale probing of transient dynamics over a wide range of time scales, simultaneously with the observation of local structures by STM.

Another promising technique is STM combined with synchrotron radiation (SR-STM), which probes core-level photoemission, enabling the identification of atomic species of the target materials [42]. The spatial resolution has been improved to ~10 nm, and therefore in the near future, in addition to the analysis of isolated nanostructures, probing of the inner structures of targets may become possible.

## **2.2 Manipulation for fine measurement**

278 Smart Nanoparticles Technology

As an example, SPPX-STM has been applied to the analysis of carrier dynamics in a Co nanoparticle/GaAs(110) system. When Co is deposited on a GaAs, nanoparticles are formed (Fig. 11(a)). In this system, photoexcited minority carriers (holes) captured at the surface are recombined with electrons tunneling from the STM tip via the gap states formed by Co as shown in Fig. 11(c). This is considered to be enhanced by the existence of gap states at the Co nanoparticle sites. Understanding such a charge transport mechanism through nanoparticles is of great importance not only for the development of nanoscale electronic devices but also for their application to the finer control of chemical reactions in

Fig. 11. (a) STM image of Co/GaAs, (b) 2D map of decay constant, (c) schematic model of recombination at gap states, (d) cross section along the line in (b), (e) decay constant as a

Figure 11(b) shows the overlap of the STM image in Fig. 11(a) with the map of the decay constant obtained over the surface. The two-dimensional (2D) map of the decay constant shown in color scale indicates the decay constant of the photoinduced carrier density at each point. The positional agreement is good. As shown in the cross section in Fig. 11(d) obtained along the line in Fig. 3(b), the decay is rapid in the Co regions. In such regions, photoinduced holes trapped at the surface are recombined with electrons tunneling from the STM tip at the gap states; thus, there are two limitations in this process: the tunneling current and hole-capture rate. When the tunneling current is sufficient, the hole-capture rate becomes the limiting factor of the recombination process. Figure 11(e) shows the relation between the decay constant and tunneling current. As expected, the decay constant decreases with increasing tunneling current and has a saturated value of 6.9 ns, which corresponds to the hole-capture rate of this system. The decay process should depend on the gap-state density. Figure 11(f) shows the decay constant as a function of the Co nanoparticle

function of tunneling current, (f) decay constant as a function of Co particle size.

size. The time constant increases with decreasing nanoparticle size as expected.

catalysis.

The tunneling current and bias voltage in STM, which are the basic parameters of STM measurement, can be used for the modification of target materials. Probing, for example, the effect of atomic-scale defects on local electronic structures enables the clarification of the fundamental mechanism in each element and its relation to macroscopic functions. For nanoparticles/clusters, such effects are essential for determining the characteristic properties of their total systems.

Fig. 12. (a) STM image of current-injection-induced polymerized C60 molecules (dark contrasts), (b) schematic of C60 polymerization, (c) nanoscale patterning of polymerized C60 molecules formed by scanning the tip along the longitudinal direction. [43]†

Figure 12(a) shows an STM image of a C60 crystalline film within a thickness of several monolayers grown on an HOPG surfaces in ultrahigh vacuum (UHV). Before acquiring the image, electrons were injected at the point indicated by a cross in Fig. 12(a) at a sample bias voltage of Vs=+4.2 V. The dark contrasts around the point represent intracluster structures with a stripe pattern, suggesting the frozen rotation of C60 molecules despite the room temperature. Namely, the dark sites are C60 molecules polymerized with molecules in the underlayer. The polymerization was induced by the injection of low-energy electrons from the STM probe tip. Figure 12(c) shows a line structure consisting of polymerized C60 clusters confined in a width as small as ~2 nm, which is a good example of nanoscale electron-beam

 † Copyright The Japan Society of Applied Physics.

Laser-Combined STM and Related Techniques for the Analysis of Nanoparticles/Clusters 281

the tube was initially metallic. Figure 14(e) shows the LDOS measured at the defect, which is characterized by a HOMO-LUMO gap that opened across the Fermi level. The HOMO-LUMO gap was observed to be over 2 nm along the long axis of the CNT and is considered to act as a barrier to carrier transport along the metallic SWNT. This result indicates that we can modify the local electronic properties of a single cluster in a controlled manner using the

Another method of manipulation is mechanical deformation of clusters by an STM tip. For example, the change in the HOMO-LUMO gap of C60 molecule due to deformation was observed thorough the measurement of tunneling current under the compression of the

Combination of STM manipulations with optical techniques enables further analysis of

For the STM measurement of nanoscale materials, choosing the most suitable STM tip depending on the specific experiment is important. In this section, silver tips for optical measurement, glass-coated tips for photoemission measurements, molecular tip for chemical

STM combined with a synchrotron radiation light source (SR-STM) has attracted considerable attention owing to the possibility of elemental analysis at nanometer resolution by detecting the core-level electrons of surface atoms. The fabrication of a tip coated with an insulating thin film is the key to achieving high spatial resolution by reducing the photoinduced current impinging to the side wall of the tip [48, 49]. For example, a W tip was coated with glass except for the region less than 5 m from the tip apex using a focused ion beam (FIB) technique [48]. Using this state-of-the-art STM tip, the photoinduced current was dramatically reduced by a factor of ~40 compared with that of an untreated tip. Recently, using this tip in combination with the Lock-in (LI) detection method, a spatial resolution of as high as ~10 nm was demonstrated on checkerboard-patterned Ni and Fe samples [42].

Tip-enhanced Raman spectroscopy (TERS) [50] is a promising method of chemical analysis at the nanometer level. Under external illumination, a sharp tip is used to create a localized light source and excite a specimen surface. According to classical electromagnetic theory, a sharp metal tip is suitable for enhancing the Raman scattering of nearby molecules. It is known that Ag produces greater enhancement than Au in the visible range because the imaginary part of its permittivity is much smaller. The silver tip is also used in STMinduced luminescence (STML), where STML intensities are enhanced by about one order of magnitude compared with those obtained using tungsten tips [51]. There have been many reports on the fabrication of Ag tips by electrochemical etching with various electrolytes such as a mixture of perchloric acid and ethanol [52]. Using such a tip, single-molecule tipenhanced Raman spectra from brilliant cresyl blue (BCB) sub-monolayers deposited on a flat

analysis and CNT probes for high resolution imaging will be described.

**2.3.1 Insulator-coated metal tips for SR-STM** 

**2.3.2 Silver tips for TERS and STML** 

STM modification technique [45, 46].

molecule by STM tip [47].

nanoscale materials.

**2.3 Probe technology** 

patterning [43]. When a template such as nanoscale cavity is used, individual C60 molecules are stabilized in each cavity even at room temperature. Manipulation of a single C60 molecule using STM tunneling current was successfully carried out (Fig.13) [44.].

Fig. 13. (a) STM image of glycine-nanocavity array (template). (b) Schematic illustration of C60 molecule stabilized in a nanocavity. STM images of C60 molecules stabilized by a glycine template before (c) and after (d) the injection of tunnel current on the molecule indicated by arrow. (e) Change in tunnel current upon manipulation. (f) Cross sections along the line in (a).

Figures 14(b) and (c) show topographic STM images of a single-walled carbon nanotube (SWNT). After acquiring the image in Fig. 14(b), the STM tip was fixed at the position marked in Fig. 14(b), and a tunneling current of 0.1 nA with 7.0 V bias voltage was injected. Figure 13(c) shows the defect generated at the probed site. The finite flat LDOS around the Fermi level shown in Fig. 14(c), which was measured before defect generation, indicates that

Fig. 14. (a) Schematic illustration of tunnel current injection. (b) STM image of an SWNT acquired with Vs=1.0 V and It=0.1 nA at 95 K. (c) dI/dV vs bias-voltage curve, obtained at the position marked in (b), exhibiting the features of a metallic SWNT characterized by a finite flat LDOS in the first van Hove gap. (d) STM image acquired after current injection at the marked position. (e) dI/dV vs bias-voltage curve, obtained at the position marked in (d), exhibiting a HOMO-LUMO gap of ~0.7 eV. [45]

the tube was initially metallic. Figure 14(e) shows the LDOS measured at the defect, which is characterized by a HOMO-LUMO gap that opened across the Fermi level. The HOMO-LUMO gap was observed to be over 2 nm along the long axis of the CNT and is considered to act as a barrier to carrier transport along the metallic SWNT. This result indicates that we can modify the local electronic properties of a single cluster in a controlled manner using the STM modification technique [45, 46].

Another method of manipulation is mechanical deformation of clusters by an STM tip. For example, the change in the HOMO-LUMO gap of C60 molecule due to deformation was observed thorough the measurement of tunneling current under the compression of the molecule by STM tip [47].

Combination of STM manipulations with optical techniques enables further analysis of nanoscale materials.

## **2.3 Probe technology**

280 Smart Nanoparticles Technology

patterning [43]. When a template such as nanoscale cavity is used, individual C60 molecules are stabilized in each cavity even at room temperature. Manipulation of a single C60

Fig. 13. (a) STM image of glycine-nanocavity array (template). (b) Schematic illustration of C60 molecule stabilized in a nanocavity. STM images of C60 molecules stabilized by a glycine template before (c) and after (d) the injection of tunnel current on the molecule indicated by arrow. (e) Change in tunnel current upon manipulation. (f) Cross sections along the line in (a).

Figures 14(b) and (c) show topographic STM images of a single-walled carbon nanotube (SWNT). After acquiring the image in Fig. 14(b), the STM tip was fixed at the position marked in Fig. 14(b), and a tunneling current of 0.1 nA with 7.0 V bias voltage was injected. Figure 13(c) shows the defect generated at the probed site. The finite flat LDOS around the Fermi level shown in Fig. 14(c), which was measured before defect generation, indicates that

Fig. 14. (a) Schematic illustration of tunnel current injection. (b) STM image of an SWNT acquired with Vs=1.0 V and It=0.1 nA at 95 K. (c) dI/dV vs bias-voltage curve, obtained at the position marked in (b), exhibiting the features of a metallic SWNT characterized by a finite flat LDOS in the first van Hove gap. (d) STM image acquired after current injection at the marked position. (e) dI/dV vs bias-voltage curve, obtained at the position marked in

(d), exhibiting a HOMO-LUMO gap of ~0.7 eV. [45]

molecule using STM tunneling current was successfully carried out (Fig.13) [44.].

For the STM measurement of nanoscale materials, choosing the most suitable STM tip depending on the specific experiment is important. In this section, silver tips for optical measurement, glass-coated tips for photoemission measurements, molecular tip for chemical analysis and CNT probes for high resolution imaging will be described.

## **2.3.1 Insulator-coated metal tips for SR-STM**

STM combined with a synchrotron radiation light source (SR-STM) has attracted considerable attention owing to the possibility of elemental analysis at nanometer resolution by detecting the core-level electrons of surface atoms. The fabrication of a tip coated with an insulating thin film is the key to achieving high spatial resolution by reducing the photoinduced current impinging to the side wall of the tip [48, 49]. For example, a W tip was coated with glass except for the region less than 5 m from the tip apex using a focused ion beam (FIB) technique [48]. Using this state-of-the-art STM tip, the photoinduced current was dramatically reduced by a factor of ~40 compared with that of an untreated tip. Recently, using this tip in combination with the Lock-in (LI) detection method, a spatial resolution of as high as ~10 nm was demonstrated on checkerboard-patterned Ni and Fe samples [42].

## **2.3.2 Silver tips for TERS and STML**

Tip-enhanced Raman spectroscopy (TERS) [50] is a promising method of chemical analysis at the nanometer level. Under external illumination, a sharp tip is used to create a localized light source and excite a specimen surface. According to classical electromagnetic theory, a sharp metal tip is suitable for enhancing the Raman scattering of nearby molecules. It is known that Ag produces greater enhancement than Au in the visible range because the imaginary part of its permittivity is much smaller. The silver tip is also used in STMinduced luminescence (STML), where STML intensities are enhanced by about one order of magnitude compared with those obtained using tungsten tips [51]. There have been many reports on the fabrication of Ag tips by electrochemical etching with various electrolytes such as a mixture of perchloric acid and ethanol [52]. Using such a tip, single-molecule tipenhanced Raman spectra from brilliant cresyl blue (BCB) sub-monolayers deposited on a flat

Laser-Combined STM and Related Techniques for the Analysis of Nanoparticles/Clusters 283

structures of the top layers. Individual Pt atoms are clearly identified, especially for (c) Pt(8) and (d) Pt(9), indicating that the geometry of the clusters is atomically resolved and the details

Fig. 15. STM images of TiO2(110) surface after deposition of size-selected Ptn+ (n = 4,7–10,15) cluster ions. Images with uppercase letters are 20×20 nm2 views and those with lowercase letters are 3.5×3.5 nm2 views of a cluster on the same surface. TiO2 surface after the

+, [(C)(c)(c′)] Pt8

This method is suitable for the mass production of CNT probes. Chemical vapor deposition (CVD) is commonly used for the synthesis of CNTs [66]. Tips fabricated by the direct growth method sometimes consist of numerous CNTs, and selective growth at the apex is required for stable operation of the tip. For this purpose, several methods of pinpointing catalysis have been reported [67, 68]. The growth direction of CNTs is also important in measurements. Plasma CVD is suitable for controlling the alignment of CNTs [69]. By optimizing the reaction at the sharp apex, CNT probes can be directly grown on the apex of a tungsten probe without reducing its sharpness, as shown in Fig.16 [70]. Thin films of Fe or Co (20–30 nm) are used as a catalyst, and the growth of CNTs with a diameter of ~40 nm has been observed. Because the magnetic nanoparticles are located at the tip of CNTs, this type of probe can also be utilized to study the magnetic properties of nanoclusters with higher spatial resolution [71]. For spin-polarized STM measurement, a magnetic coating of Fe (10- 20 monolayers) on a cleaned tungsten tip is conventionally used [72], which may be

Fig. 16. SEM and TEM images of a grown CNT-STM probe. Two CNTs are grown on the apex of the STM tip. Black contrast corresponds to metal particles used as catalyst.

+, [(D)(d)(d′)] Pt9+, [(E)(e)] Pt10+, [(F)(f)]

can be analyzed with a CNT tip.

deposition of [(A), (a)] Pt4+, [(B)(b)] Pt7

**(b) Direct growth method**

improved using a CNT tip.

Pt15+ [65].

Au surface were obtained [53]. A highly enhanced electric field was created in the gap of 1 nm between the tip and sample. For STM imaging, the tip apex should be free of oxidation or contaminants. Atomically resolved STM imaging and STML spectra with a high signal-tonoise ratio are obtained using an electrochemically etched Ag tip followed by tip cleaning by Ar ion sputtering in UHV [54].

## **2.3.3 Molecular tips**

Carboxyl-terminated SWNTs from solution phases can be attached onto Au tips through self-assembed monolayers for using in STM [55. 56]. In addition to the high-resolution imaging of molecules (such as diether) on a surface, these CNT tips enable chemically selective observation due to electron tunneling through hydrogen-bond interactions between the atached molecule and carboxyl groups at SWNTs. The differentiation of DNA bases and chiral recognition on a single-molecule basis have also been demonstrated using molecular tips [56]. In a similar way, voltage-induced chemical contrast in an STM image was reported using chemically modified tips with hydrogen-bond donors [57]. Moreover, molecular orbitals of metal phthalocyanines on metal surfaces have been clearly imaged with an O2-functionalized STM tip, where the observations were supported by theoretical calculations [58].

## **2.3.4 CNT probes**

CNTs are one of the most intriguing materials in nanotechnology [59]. A CNT has a onedimensional cylindrical structure with sistinct physical characteristics such as a small diameter, high aspect ratio, high stiffness, high conductivity and so forth. In view of the shape and electric conductivity required for a high-resolution STM tip, these properties of CNTs make them ideal as a tip material for probing extremely small objects such as nanoclusters. [60]. Mechanical attachment, direct growth and dielectrophoresis are methods employed to fabricate CNT-STM tips. A single CNT tip can be prepared by mechanical attachment, which enables the high-resolution ghost-free imaging of nanoclusters. However, this method is time-consuming, and the other methods (direct growth and dielectrophoresis) are more suitable for the mass production of CNT tips.

## **(a) Mechanical attachment method**

Following the first approach to fabricating CNT probes under an optical microscope [61]. a more sophisticated method was proposed, where the attachment of a CNT onto a probe is performed in two independent precise stages under SEM observation, where beam-deposited amorphous carbon is used as a glue [62]. Although the thus mechanically prepared CNT-STM tips exhibited atomic resolution on Au(111) reconstruction, the cleaning process of CNTs by heating in UHV was necessary for stable observation. [63]. A metal coating method has been proposed for improving the electric conductivity between a CNT and the supporting metal tip [64]. Automotive exhaust catalysts consist of metal nanoclusters supported on metal oxide surfaces. Since catalytic activity can be altered by controlling the size of the nanoclusters because of its strong size dependence, the precise characterization of metal nanoclusters is essential. High-resolution UHV-STM images of size-selected Pt(n) (n=4,7-10,15) clusters deposited on TiO2(110)-(1x1) surfaces were obtained using a CNT tip (Fig. 15) [65]. Clusters of Pt(7) (Fig.15(b)) and smaller were oriented flat on the surface with a planar structure, and a planar-to-three-dimensional transition was observed at n=8 (Fig.15(c)). Color scale shows the structures of the top layers. Individual Pt atoms are clearly identified, especially for (c) Pt(8) and (d) Pt(9), indicating that the geometry of the clusters is atomically resolved and the details can be analyzed with a CNT tip.

Fig. 15. STM images of TiO2(110) surface after deposition of size-selected Ptn+ (n = 4,7–10,15) cluster ions. Images with uppercase letters are 20×20 nm2 views and those with lowercase letters are 3.5×3.5 nm2 views of a cluster on the same surface. TiO2 surface after the deposition of [(A), (a)] Pt4 +, [(B)(b)] Pt7 +, [(C)(c)(c′)] Pt8 +, [(D)(d)(d′)] Pt9 +, [(E)(e)] Pt10+, [(F)(f)] Pt15+ [65].

## **(b) Direct growth method**

282 Smart Nanoparticles Technology

Au surface were obtained [53]. A highly enhanced electric field was created in the gap of 1 nm between the tip and sample. For STM imaging, the tip apex should be free of oxidation or contaminants. Atomically resolved STM imaging and STML spectra with a high signal-tonoise ratio are obtained using an electrochemically etched Ag tip followed by tip cleaning

Carboxyl-terminated SWNTs from solution phases can be attached onto Au tips through self-assembed monolayers for using in STM [55. 56]. In addition to the high-resolution imaging of molecules (such as diether) on a surface, these CNT tips enable chemically selective observation due to electron tunneling through hydrogen-bond interactions between the atached molecule and carboxyl groups at SWNTs. The differentiation of DNA bases and chiral recognition on a single-molecule basis have also been demonstrated using molecular tips [56]. In a similar way, voltage-induced chemical contrast in an STM image was reported using chemically modified tips with hydrogen-bond donors [57]. Moreover, molecular orbitals of metal phthalocyanines on metal surfaces have been clearly imaged with an O2-functionalized STM tip, where the observations were supported by theoretical

CNTs are one of the most intriguing materials in nanotechnology [59]. A CNT has a onedimensional cylindrical structure with sistinct physical characteristics such as a small diameter, high aspect ratio, high stiffness, high conductivity and so forth. In view of the shape and electric conductivity required for a high-resolution STM tip, these properties of CNTs make them ideal as a tip material for probing extremely small objects such as nanoclusters. [60]. Mechanical attachment, direct growth and dielectrophoresis are methods employed to fabricate CNT-STM tips. A single CNT tip can be prepared by mechanical attachment, which enables the high-resolution ghost-free imaging of nanoclusters. However, this method is time-consuming, and the other methods (direct growth and

Following the first approach to fabricating CNT probes under an optical microscope [61]. a more sophisticated method was proposed, where the attachment of a CNT onto a probe is performed in two independent precise stages under SEM observation, where beam-deposited amorphous carbon is used as a glue [62]. Although the thus mechanically prepared CNT-STM tips exhibited atomic resolution on Au(111) reconstruction, the cleaning process of CNTs by heating in UHV was necessary for stable observation. [63]. A metal coating method has been proposed for improving the electric conductivity between a CNT and the supporting metal tip [64]. Automotive exhaust catalysts consist of metal nanoclusters supported on metal oxide surfaces. Since catalytic activity can be altered by controlling the size of the nanoclusters because of its strong size dependence, the precise characterization of metal nanoclusters is essential. High-resolution UHV-STM images of size-selected Pt(n) (n=4,7-10,15) clusters deposited on TiO2(110)-(1x1) surfaces were obtained using a CNT tip (Fig. 15) [65]. Clusters of Pt(7) (Fig.15(b)) and smaller were oriented flat on the surface with a planar structure, and a planar-to-three-dimensional transition was observed at n=8 (Fig.15(c)). Color scale shows the

dielectrophoresis) are more suitable for the mass production of CNT tips.

by Ar ion sputtering in UHV [54].

**2.3.3 Molecular tips** 

calculations [58].

**2.3.4 CNT probes** 

**(a) Mechanical attachment method** 

This method is suitable for the mass production of CNT probes. Chemical vapor deposition (CVD) is commonly used for the synthesis of CNTs [66]. Tips fabricated by the direct growth method sometimes consist of numerous CNTs, and selective growth at the apex is required for stable operation of the tip. For this purpose, several methods of pinpointing catalysis have been reported [67, 68]. The growth direction of CNTs is also important in measurements. Plasma CVD is suitable for controlling the alignment of CNTs [69]. By optimizing the reaction at the sharp apex, CNT probes can be directly grown on the apex of a tungsten probe without reducing its sharpness, as shown in Fig.16 [70]. Thin films of Fe or Co (20–30 nm) are used as a catalyst, and the growth of CNTs with a diameter of ~40 nm has been observed. Because the magnetic nanoparticles are located at the tip of CNTs, this type of probe can also be utilized to study the magnetic properties of nanoclusters with higher spatial resolution [71]. For spin-polarized STM measurement, a magnetic coating of Fe (10- 20 monolayers) on a cleaned tungsten tip is conventionally used [72], which may be improved using a CNT tip.

Fig. 16. SEM and TEM images of a grown CNT-STM probe. Two CNTs are grown on the apex of the STM tip. Black contrast corresponds to metal particles used as catalyst.

Laser-Combined STM and Related Techniques for the Analysis of Nanoparticles/Clusters 285

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(2012), DOI:10.1039/C2NR11551D.

[42] T. Okuda et al., Phys. Rev. Lett. 102 (2009) 105503.

Shigekawa, Chem. Commun. 47 (2011) 11312.

[48] K. Akiyama et al., Rev. Sci. Instrum. 76 (2005) 083711.

## **(c) Dielectrophoresis**

When an alternating electric field (~MHz) is applied between asymmetric electrodes (for example, a metal probe and counter plane electrode) immersed in CNT solution (where solvent is water, alcohol, dichloroethane, etc.), the CNTs are polarized and become attached to the probe (dielectrophoresis). [73]. The high-yield synthesis of conductive CNT tips for the multiprobe microscope [74-76] was reported using the dielectrophoresis method [77]. After Pt-Ir coating, such a tip were successfully applied for electronic transport measurement by multiprobe STM using the four-terminal method.

## **3. Summary**

Laser-combined STM and related techniques have been reviewed and discussed focusing on the analysis of nanoscale particles and clusters. The addition of optical technologies to STM provides new approaches to the study of nanoscale-material physics and chemistry. Nearfield optical microscopy (NSOM) and other techniques [78-86], which have not been discussed in this chapter, are expected to play complementary roles in understanding and developing the physics and chemistry of new nanoparticles/clusters for realizing novel functional devices.

## **4. References**


[20] C. Chen, C. A. Bobisch, and W. Ho, Science 325 (2009) 981.

284 Smart Nanoparticles Technology

When an alternating electric field (~MHz) is applied between asymmetric electrodes (for example, a metal probe and counter plane electrode) immersed in CNT solution (where solvent is water, alcohol, dichloroethane, etc.), the CNTs are polarized and become attached to the probe (dielectrophoresis). [73]. The high-yield synthesis of conductive CNT tips for the multiprobe microscope [74-76] was reported using the dielectrophoresis method [77]. After Pt-Ir coating, such a tip were successfully applied for electronic transport

Laser-combined STM and related techniques have been reviewed and discussed focusing on the analysis of nanoscale particles and clusters. The addition of optical technologies to STM provides new approaches to the study of nanoscale-material physics and chemistry. Nearfield optical microscopy (NSOM) and other techniques [78-86], which have not been discussed in this chapter, are expected to play complementary roles in understanding and developing the physics and chemistry of new nanoparticles/clusters for realizing novel

[2] P. Jena and A. W. Castleman (Eds), Nanoclusters, A Bridge across Disciplines, Elsevier

[3] J. A. Alonso, Structure and Properties of Atomic Nanoclusters, Imperial College Press

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[5] F. J. Owens and C. P. Poole (Eds), The Physics and Chemistry of Nanosolids, Wiley (2008). [6] J. M. R. Weaver, L. M. Wapita, and H. K. Wickramasinghe, Nature 342 (1989) 783.

[8] S. Grafstrom, P. Schuller, J. Kowalski, and R. Neumann, J. Appl. Phys. 83 (1998) 3453. [9] N. Naruse, Y. Mera, Y. Fukuzawa, Y. Nakamura, M. Ichikawa, and K. Maeda, J. Appl.

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measurement by multiprobe STM using the four-terminal method.

**(c) Dielectrophoresis** 

**3. Summary** 

functional devices.

**4. References** 

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[18] L. Miglio, V. Meregalli, and O. Jepsen, Appl. Phys. Lett. 75 (1999) 385.


**13** 

*Spain* 

**On the Optical Response of Nanoparticles:** 

Braulio García-Cámara1, Francisco González1, Fernando Moreno1,

*1Grupo de Óptica, Departamento de Física Aplicada, Universidad de Cantabria* 

*3Instituto de Ciencias de los Materiales de Madrid, CSIC* 

Raquel Gómez-Medina2, Juan José Sáenz2 and Manuel Nieto-Vesperinas3

*2Departamento de Física de la Materia Condensada, Universidad Autónoma de Madrid* 

Nowadays, miniaturization is a general challenge for technology. Researchers in science and technology claim to study ever smaller systems and develop ever smaller devices. The nanometric range is, at present, an important focus of attention of scientists and engineers following the famous prediction by Prof. Feynman: "There's plenty of room at the bottom". Reduction of dimensions, at this level, involves that more specific and more complex tools

Light has appeared as a convenient solution for these tasks because of its wavelength (hundreds of nanometers) and the large amount of information it contains about systems with which it interacts (Prasad, 2004). The interaction of light with small systems, either particles or structures, gives rise to several scattering phenomena which are strongly dependent on both the characteristics of the incident radiation (frequency, polarization) and those of the object (size, shape, optical properties). These interactions can be used either to obtain information about the interacting object (e.g. particle sizing) (Zhu et al., 2010) or to

At the nanoscopic level, the interaction between an incident beam and a metallic system produces an interesting physical phenomenon which is the base of many technological applications in diverse fields like medicine, biology, communications, information storing, energy transformation, photonics, etc (Anker et al., 2008; Maier et al., 2003). This is the excitation of *localized surface plasmon resonances* (*LSPR*) (Prasad, 2004). For these, the electromagnetic field experiences a high localization in the scatterer and a strong

These advances have stimulated new research devoted to obtain a greater control over how light is scattered by these systems. Researchers have analyzed emerging structures (nanoholes (Gao et al., 2010), nanocups (Mirin & Halas, 2009), etc). But, what it is more interesting, new engineered materials, called *metamaterials* and whose optical properties can be manipulated, have been developed (Boltasseva & Atwater, 2011). The possibility to obtain structures with

produce light scattering phenomena "à la carte" by means of suitable nanoobjects.

**1. Introduction** 

are needed.

enhancement out of the scatterer.

**Directionality Effects and Optical Forces** 


## **On the Optical Response of Nanoparticles: Directionality Effects and Optical Forces**

Braulio García-Cámara1, Francisco González1, Fernando Moreno1, Raquel Gómez-Medina2, Juan José Sáenz2 and Manuel Nieto-Vesperinas3 *1Grupo de Óptica, Departamento de Física Aplicada, Universidad de Cantabria 2Departamento de Física de la Materia Condensada, Universidad Autónoma de Madrid 3Instituto de Ciencias de los Materiales de Madrid, CSIC Spain* 

## **1. Introduction**

286 Smart Nanoparticles Technology

[50] R. M. Stockle, Y. D. Suh, V. Deckert and R. Zenobi, Chem. Phys. Lett. 318 (2000) 131. [51] R. Berndt, J. K. Gimzewski, and P. Johansson, Phys. Rev. Lett. 71 (1993) 3493. [52] M. Iwami, Y. Uehara and S. Ushioda, Rev. Sci. Instrum. 69 (1998) 4010.

[61] H. Dai, J. H. Hafner, A. G. Rinzler, D. T. Colbert, and R. E. Smalley, Nature 384 (1996) 147.

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[74] I. Shiraki, F. Tanabe, R. Hobara, T. Nagao, and S. Hasegawa, Surf. Sci. 493 (2001) 633. [75] J. Onoe , T. Nakayama, M. Aono, and T. Hara, Appl. Phys. Lett. 82 (2003) 595. [76] M. Ishikawa, M. Yoshimura and K. Ueda, Jpn. J. Appl. Phys. 44 (2005) 1502.

[77] H. Konishi, Y. Murata, W. Wongwiriyapan, M. Kishida, K. Tomita, K. Motoyoshi, S.

[78] Y. Terada, S. Yoshida, A. Okubo, K. Kanazawa, M. Xu, O. Takeuchi and H.

[79] S. Yoshida, Y. Kanitani, O. Takeuchi and H. Shigekawa, Appl. Phys. Lett. 92 (2008) 102105. [80] S. Yoshida, Y. Kanitani, R. Oshima, Y. Okada, O. Takeuchi and H. Shigekawa, Phys.

[82] H. Watanabe, Y. Ishida, N. Hayazawa, Y. Inouye and S. Kawata, Phys. Rev. B 69 (1004) 1. [83] S. Yasuda, T. Nakamura, M. Matsumoto and H. Shigekawa, J. Am. Chem. Soc. 125

[84] D. Futaba, R. Morita, M. Yamashita, S. Tomiyama and H. Shigekawa, Appl. Phys. Lett.

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Honda, M. Katayama, S. Yoshimoto, K. Kubo, R. Hobara, I. Matsuda, S. Hasegawa

[53] W. H. Zhang, B. S. Yeo, T. Schmid, R. Zenobi, J. Phys. Chem. C111 (2007) 1733.

[49] A. Saito et al., Surf. Sci. 601 (2007) 5294.

[58] Z. Cheng et al., Nano Res. 4 (2011) 523. [59] S. Iijima, Nature, 354, 56 (1991).

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[56] T. Nishino and Y. Umezawa, Anal. Sci. 26 (2010) 1023. [57] D. Gingery and P. Bühlmann, Surf. Sci. 605 (2011) 1099.

[62] S. Akita et al., J. Phys D: Appl. Phys. 32 (1999) 1044.

[66] W. Wongwiriyapan et al., Jpn. J. Appl. Phys. 45 (2006) 1880. [67] C. L. Cheung, J. H. Hafner, and C. M. Lieber, PNAS 97 (2000) 3809.

[64] T.Ikuno et al., Jpn. J. Appl. Phys. 43 (2004) L644.

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[60] J. M. Marulanda, 2011, *Electronic Properties of Carbon Nanotubes*, InTech.

[65] N. Isomura, X. Wu, and Y. Watanabe, J. Chem Phys. 131(16) (2009) 164707.

[69] M. Yoshimura, S. Jo and K. Ueda, Jpn. J. Appl. Phys. 42 (7B) (2003) 4841. [70] K. Tanaka, M. Yoshimura, and K. Ueda, e-J. Surf. Sci. Nanotech. 4 (2006) 276. [71] K. Tanaka, M. Yoshimura and K. Ueda, J. Nanomaterials 2009 (2009) 147204.

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and M. Yoshimura, Rev. Sci. Instr. 78 (2007) 013703.

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[86] S. Grafstorm (o: umlaut), J. Appl. Phys. 91 (2002) 1717.

Shigekawa, Nano Lett. 8 (11), (2008) 3577-3581.

Nowadays, miniaturization is a general challenge for technology. Researchers in science and technology claim to study ever smaller systems and develop ever smaller devices. The nanometric range is, at present, an important focus of attention of scientists and engineers following the famous prediction by Prof. Feynman: "There's plenty of room at the bottom". Reduction of dimensions, at this level, involves that more specific and more complex tools are needed.

Light has appeared as a convenient solution for these tasks because of its wavelength (hundreds of nanometers) and the large amount of information it contains about systems with which it interacts (Prasad, 2004). The interaction of light with small systems, either particles or structures, gives rise to several scattering phenomena which are strongly dependent on both the characteristics of the incident radiation (frequency, polarization) and those of the object (size, shape, optical properties). These interactions can be used either to obtain information about the interacting object (e.g. particle sizing) (Zhu et al., 2010) or to produce light scattering phenomena "à la carte" by means of suitable nanoobjects.

At the nanoscopic level, the interaction between an incident beam and a metallic system produces an interesting physical phenomenon which is the base of many technological applications in diverse fields like medicine, biology, communications, information storing, energy transformation, photonics, etc (Anker et al., 2008; Maier et al., 2003). This is the excitation of *localized surface plasmon resonances* (*LSPR*) (Prasad, 2004). For these, the electromagnetic field experiences a high localization in the scatterer and a strong enhancement out of the scatterer.

These advances have stimulated new research devoted to obtain a greater control over how light is scattered by these systems. Researchers have analyzed emerging structures (nanoholes (Gao et al., 2010), nanocups (Mirin & Halas, 2009), etc). But, what it is more interesting, new engineered materials, called *metamaterials* and whose optical properties can be manipulated, have been developed (Boltasseva & Atwater, 2011). The possibility to obtain structures with

On the Optical Response of Nanoparticles: Directionality Effects and Optical Forces 289

*p n n sn n*

*m j mx xj x j x mxj mx <sup>b</sup>*

*p n n sn n*

<sup>2</sup> , *<sup>a</sup> x ka* 

In addition, *jn* are the *spherical Bessel functions* and *hn(1)* the *spherical Bessel functions of third kind* or *Hankel functions*. As the electric and magnetic dipolar contributions are weighted by coefficients *a1* and *b1*, respectively, the quadrupolar ones by *a2* and *b2* and so on, Mie coefficients *an* are associated to the electric part of the scattered electromagnetic radiation,

These coefficients contain the relevant information about essential scattering parameters as

<sup>2</sup> (2 1)( ) *sca n n*

<sup>2</sup> (2 1)Re( ) *ext n n*

contributions (*n=1* in Eqs. (1)- (2)) clearly dominate and Mie coefficients of order higher than 1 can be neglected. Thus, the Mie expansion can be simplified and the previous parameters

*C n ab*

*C n ab*

2 2

(6)

(7)

<<1, dipolar

the extinction, *Cext*, and scattering, *Csca*, cross sections. These can be written as

2 1

> 2 1

*k* 

*n*

If particle size is very small compared with the incident wavelength, that is *a/*

*k* 

*n*

*m j mx xh x h x mxj mx*

*n*

*x* being the size parameter, that is defined as

while *bn* are associated to the magnetic one.

Fig. 1. Scheme of the geometry of the problem.

**2.2 Details of Mie theory at the nanoscopic level** 

have simple expressions

2 ' ' ' 2 (1) (1) ' ( ) () () ( )

 

( ) () () ( )

 

(5)

(4)

optical properties "à la carte" allows getting scattering phenomena never observed before in natural media, for instance negative refraction (Shalaev, 2008). The main consequences of negative refraction are the two interesting potential applications: cloacking (Pendry et al., 2006) and perfect lens (Pendry, 2000; Nieto-Vesperinas & Garcia, 2003).

The control over the values of both the electric permittivity and the magnetic permeability of an object gives us a control over the way it scatters light, and in particular, the angular distribution of the scattered radiation. This control could involve a dramatic evolution on the field of nanodevices. For this reason, the objective of this chapter is to analyze directional effects on both light scattering and optical forces of a nanoparticle with convenient optical constants. The structure of the chapter is as follows: while sections 2, 3 and 4 are devoted to the directional features on light scattering by nanoparticles, section 5 summarizes the main results on optical forces. Finally, the most important conclusions about these results are recapitulated in section 6.

#### **2. Light scattering by nanoparticles**

#### **2.1 Mie theory**

The problem of the electromagnetic scattering from an isolated and spherical particle was firstly solved in 1908 by Gustav Mie (Mie, 1908). However, this simple system still involves interesting physical behaviors that are worthy of further study.

Mie theory considers a spherical particle of radius *a* and optical constants given by an electric permittivity, *p*and a magnetic permeability, *<sup>p</sup>*immersed in a homogeneous and isotropic medium. This is illuminated by a linear polarized plane wave, as in Figure 1. Without loss of generality, we assume that the surrounding medium is vacuum (s=s=1). The scattered electromagnetic field (**E***s***, H***s*) can be expressed as a multipole expansion of *Vector Spherical Harmonics (VSH),* called Mie expansion, as follows

$$\mathbf{E}\_s = \sum\_{n=1}^{\omega} E\_n \{ ia\_n \mathbb{N}\_{e \ln}^{\{3\}} - b\_n \mathbb{M}\_{\text{oln}}^{\{3\}} \} \tag{1}$$

$$\mathbf{H}\_S = \frac{k}{\alpha \mu\_p} \sum\_{n=1}^{\infty} E\_n \left( i b\_n \mathbf{N}\_{\text{oln}}^{\{3\}} + a\_n \mathbf{M}\_{\text{eln}}^{\{3\}} \right) \tag{2}$$

where *km cm* / 2/ , being the incident wavelength in vacuum, *m <sup>p</sup> <sup>p</sup>* the refractive index of the particle, *c* the speed of light in vacuum and the angular frequency of the incident wave. *En* is defined as <sup>0</sup> 2 1 ( 1) *<sup>n</sup> <sup>n</sup> <sup>n</sup> E Ei n n* , *E0* being the amplitude of the incident plane wave. The series are characterized by the *an* and *bn* Mie coefficients which are defined as (Bohren & Huffman, 1983)

$$a\_n = \frac{\mu\_s m^2 j\_n(m\mathbf{x}) \left[ \mathbf{x} j\_n(\mathbf{x}) \right]^\cdot - \mu\_p j\_n(\mathbf{x}) \left[ m\mathbf{x} j\_n(m\mathbf{x}) \right]^\cdot}{\mu\_s m^2 j\_n(m\mathbf{x}) \left[ \mathbf{x} h\_n^{(1)}(\mathbf{x}) \right]^\cdot} - \mu\_p h\_n^{(1)}(\mathbf{x}) \left[ m\mathbf{x} j\_n(m\mathbf{x}) \right]^\cdot} \tag{3}$$

$$b\_{n} = \frac{\mu\_{p}m^{2}j\_{n}\text{(m\times)}\left[\text{x}j\_{n}\text{(x)}\right]^{\circ} - \mu\_{s}j\_{n}\text{(x)}\left[\text{m\times}j\_{n}\text{(m\times)}\right]^{\circ}}{\mu\_{p}m^{2}j\_{n}\text{(m\times)}\left[\text{x}h\_{n}^{\text{(1)}}\text{(x)}\right]^{\circ} - \mu\_{s}h\_{n}^{\text{(1)}}\text{(x)}\left[\text{m\times}j\_{n}\text{(m\times)}\right]^{\circ}}\tag{4}$$

*x* being the size parameter, that is defined as

288 Smart Nanoparticles Technology

optical properties "à la carte" allows getting scattering phenomena never observed before in natural media, for instance negative refraction (Shalaev, 2008). The main consequences of negative refraction are the two interesting potential applications: cloacking (Pendry et al.,

The control over the values of both the electric permittivity and the magnetic permeability of an object gives us a control over the way it scatters light, and in particular, the angular distribution of the scattered radiation. This control could involve a dramatic evolution on the field of nanodevices. For this reason, the objective of this chapter is to analyze directional effects on both light scattering and optical forces of a nanoparticle with convenient optical constants. The structure of the chapter is as follows: while sections 2, 3 and 4 are devoted to the directional features on light scattering by nanoparticles, section 5 summarizes the main results on optical forces. Finally, the most important conclusions

The problem of the electromagnetic scattering from an isolated and spherical particle was firstly solved in 1908 by Gustav Mie (Mie, 1908). However, this simple system still involves

Mie theory considers a spherical particle of radius *a* and optical constants given by an

isotropic medium. This is illuminated by a linear polarized plane wave, as in Figure 1. Without loss of generality, we assume that the surrounding medium is vacuum (s=s=1). The scattered electromagnetic field (**E***s***, H***s*) can be expressed as a multipole expansion of

( ) *s nn n e o*

*<sup>k</sup> E ib a*

refractive index of the particle, *c* the speed of light in vacuum and the angular frequency

*<sup>n</sup> <sup>n</sup> <sup>n</sup> E Ei*

incident plane wave. The series are characterized by the *an* and *bn* Mie coefficients which are

*s n n pn n*

*s n n pn n*

*E ia b*

ln ln

 (3) (3) ln ln

> 2 1 ( 1)

*n n*

2 ' ' ' 2 (1) (1) ' ( ) () () ( )

 

*m j mx xj x j x mxj mx*

*m j mx xh x h x mxj mx*

( ) () () ( )

 

**E NM** (1)

**H NM** (2)

, *E0* being the amplitude of the

being the incident wavelength in vacuum, *m <sup>p</sup> <sup>p</sup>*

*<sup>p</sup>*immersed in a homogeneous and

  the

(3)

2006) and perfect lens (Pendry, 2000; Nieto-Vesperinas & Garcia, 2003).

about these results are recapitulated in section 6.

interesting physical behaviors that are worthy of further study.

*Vector Spherical Harmonics (VSH),* called Mie expansion, as follows

(3) (3)

*p*and a magnetic permeability,

1

1 *s nn n o e*

*n*

*p n*

**2. Light scattering by nanoparticles** 

**2.1 Mie theory** 

electric permittivity,

where *km cm* 

 / 2/ , 

defined as (Bohren & Huffman, 1983)

of the incident wave. *En* is defined as <sup>0</sup>

*n*

*a*

$$
\lambda x = ka = \frac{2\pi a}{\lambda},
\tag{5}
$$

In addition, *jn* are the *spherical Bessel functions* and *hn(1)* the *spherical Bessel functions of third kind* or *Hankel functions*. As the electric and magnetic dipolar contributions are weighted by coefficients *a1* and *b1*, respectively, the quadrupolar ones by *a2* and *b2* and so on, Mie coefficients *an* are associated to the electric part of the scattered electromagnetic radiation, while *bn* are associated to the magnetic one.

Fig. 1. Scheme of the geometry of the problem.

These coefficients contain the relevant information about essential scattering parameters as the extinction, *Cext*, and scattering, *Csca*, cross sections. These can be written as

$$\mathcal{C}\_{sca} = \frac{2\pi}{k^2} \sum\_{n=1}^{\infty} (2n+1) \{ \left| a\_n \right|^2 + \left| b\_n \right|^2 \} \tag{6}$$

$$\mathcal{L}\_{\text{ext}} = \frac{2\pi}{k^2} \sum\_{n=1}^{\infty} \{2n + 1\} \text{Re}\{a\_n + b\_n\} \tag{7}$$

#### **2.2 Details of Mie theory at the nanoscopic level**

If particle size is very small compared with the incident wavelength, that is *a/*<<1, dipolar contributions (*n=1* in Eqs. (1)- (2)) clearly dominate and Mie coefficients of order higher than 1 can be neglected. Thus, the Mie expansion can be simplified and the previous parameters have simple expressions

On the Optical Response of Nanoparticles: Directionality Effects and Optical Forces 291

In this section, the main theoretical aspects described by M. Kerker et al. are briefly

When we consider a system, like that of Figure 1, the scattered intensity in the scattering plane can be described by means of two polarized components: *ITE* and *ITM*. While *ITE* corresponds to an incident electric field parallel to the scattering plane, *ITM* corresponds to a

<sup>2</sup> <sup>2</sup>

<sup>2</sup> <sup>2</sup>

functions defined in (Bohren & Huffman, 1983). As we are considering a very small or

expressions. In addition some approximations can be applied to these coefficients in such a

 

4 4 2 2 *TM*

 

being the scattering angle, defined as the angle between the incident and the scattered

 

6 \_ 2 1 1 (180 ) ( )( ) 4 2 2

, or equivalently when

backward direction is zero for both incident polarizations. This is the *zero-backward scattering condition* and we shall call in the following the *first Kerker's condition*. In Figure 2 the

<sup>2</sup> 2 6 26 \_ \_ <sup>2</sup>

<sup>2</sup> 2 6 26 \_ \_ <sup>2</sup>

cos ( )cos ( )

 

 

*e=*

<sup>2</sup> 2 \_\_

<sup>2</sup> 2 \_\_ 6 \_

cos ( ) ( ) cos

2 1 ( )

2 1 ( )

 

 

> *r*

*0*), only the two first Mie coefficients (*a1* and *b1*) are introduced in the

1 1

 

 

1 1

 

 

 

*=180º*) the previous expressions adopt the following

*r* (18)

*<sup>m</sup>*, the scattered intensity in the

(14)

(15)

*>>1*) and

*n* and 

*<sup>n</sup>* are angular

(16)

(17)

(19)

perpendicular one. These components can be written as (Bohren & Huffman, 1983)

<sup>4</sup> ( 1) *TE n n nn n <sup>n</sup> <sup>I</sup> a b r n n*

<sup>4</sup> ( 1) *TM nn n n n <sup>n</sup> <sup>I</sup> a b r n n*

2

2

way that the scattered intensity components can be approximated by

*x x I ab*

 

*I x TM*

*x x I ab*

2 2 1 1

2 2 1 1

2 1 1 (180 ) ( ) ( ) <sup>4</sup> 2 2 *TE I x*

*r*

*r r*

*r r*

4 4 2 2 *TE*

where *r* is the distance from the particle to the observer (*2*

**3.1.1 Zero-backward scattering: First Kerker's condition** 

reviewed.

dipole-like particle (*a*

directions (see Figure 1).

It easy to observe that when

For the backward scattering direction (

forms

$$\mathcal{C}\_{sca} = \frac{2\pi}{k^2} \{ \Im \left( \left| a\_1 \right|^2 + \left| b\_1 \right|^2 \right) \}\tag{8}$$

$$\mathcal{C}\_{\text{ext}} = \frac{2\pi}{k^2} [\Im \text{Re}\{a\_1 + b\_1\}] \tag{9}$$

This is the case of a nanoparticle (*a* < *50 nm*) when it is illuminated by an incident wave in the visible or near infrared (NIR) part of the spectrum (*>500 nm*).

The predominant dipolar conduct, either electric or magnetic, of nanoparticles is usually described by the electric and/or magnetic complex polarizabilities, *e* and *<sup>m</sup>*¸ respectively. Both can also be expressed as a function of the two first Mie coefficients

$$\alpha\_e = \frac{\alpha\_e^{\{0\}}}{1^\cdot i \frac{2}{3} k^3 \alpha\_e^{\{0\}}} = \frac{3i}{2k^3} a\_1 \tag{10}$$

$$\alpha\_m = \frac{\alpha\_m{}^{\{0\}}}{1^{-}i\frac{2}{3}k^3\alpha\_m{}^{\{0\}}} = \frac{3i}{2k^3}b\_1\tag{11}$$

where

$$
\omega\_e^{\{0\}} = 4\pi a^3 \frac{\varepsilon\_p \cdot \mathbf{1}}{\varepsilon\_p + \mathbf{2}} \tag{12}
$$

$$
\alpha\_m^{\{0\}} = 4\pi a^3 \frac{\mu\_p \cdot 1}{\mu\_p + 2} \tag{13}
$$

are the static polarizabilities, defined in the limit *ka0*.

#### **3. Directional effects on light scattering by nanoparticles with arbitrary values of and .**

#### **3.1 Kerker's theory**

In the early eighties, M. Kerker and co-authors (Kerker et al., 1983) presented an interesting study about the scattering properties, in the far field, of a spherical particle much smaller than the incident wavelength, illuminated by a plane wave and without any restriction for the values of its relative optical constants ( and ). Some interesting electromagnetic scattering effects were described in this work such as the zero-backward and the zeroforward scattering. Although the idea of a magnetic permeability different from 1 in the visible range was hypothetical and the described effects were thought to be impossible to be observed when the work was presented, the engineered metamaterials have currently revitalized these electromagnetic studies (Zhedulev, 2010).

In this section, the main theoretical aspects described by M. Kerker et al. are briefly reviewed.

#### **3.1.1 Zero-backward scattering: First Kerker's condition**

290 Smart Nanoparticles Technology

2 1 1 <sup>2</sup> [3( )] *sca C ab*

2 1 1 <sup>2</sup> [3Re( )] *ext C ab*

This is the case of a nanoparticle (*a* < *50 nm*) when it is illuminated by an incident wave in

The predominant dipolar conduct, either electric or magnetic, of nanoparticles is usually

(0)

\_ 3 (0)

<sup>2</sup> <sup>2</sup> <sup>1</sup> 3

(0)

\_ 3 (0)

<sup>2</sup> <sup>2</sup> <sup>1</sup> 3

*m*

(0) 3 1 4

(0) 3 1 4

In the early eighties, M. Kerker and co-authors (Kerker et al., 1983) presented an interesting study about the scattering properties, in the far field, of a spherical particle much smaller than the incident wavelength, illuminated by a plane wave and without any restriction for

> and

scattering effects were described in this work such as the zero-backward and the zeroforward scattering. Although the idea of a magnetic permeability different from 1 in the visible range was hypothetical and the described effects were thought to be impossible to be observed when the work was presented, the engineered metamaterials have currently

 

 

*k i k*

*e*

*k i k*

*k* 

> *k*

*<sup>e</sup> <sup>e</sup>*

*<sup>m</sup> <sup>m</sup>*

*e*

*m*

**3. Directional effects on light scattering by nanoparticles with** 

are the static polarizabilities, defined in the limit *ka*

 **and .** 

the values of its relative optical constants (

revitalized these electromagnetic studies (Zhedulev, 2010).

the visible or near infrared (NIR) part of the spectrum (

where

**arbitrary values of** 

**3.1 Kerker's theory** 

described by the electric and/or magnetic complex polarizabilities,

Both can also be expressed as a function of the two first Mie coefficients

2 2

3 1

3 1

*i b*

3

*i a*

3

\_

\_

*p a* 

*p a* 

> *0*.

2 *p*

2 *p*

*>500 nm*).

(8)

(9)

(10)

(11)

(12)

(13)

). Some interesting electromagnetic

*e* and 

*<sup>m</sup>*¸ respectively.

When we consider a system, like that of Figure 1, the scattered intensity in the scattering plane can be described by means of two polarized components: *ITE* and *ITM*. While *ITE* corresponds to an incident electric field parallel to the scattering plane, *ITM* corresponds to a perpendicular one. These components can be written as (Bohren & Huffman, 1983)

$$I\_{TE} = \frac{\lambda^2}{4\pi r^2} \left| \sum\_{n} \frac{2n+1}{n\{n+1\}} (a\_n \pi\_n + b\_n \tau\_n) \right|^2 \tag{14}$$

$$I\_{TM} = \frac{\lambda^2}{4\pi r^2} \left| \sum\_{n} \frac{2n+1}{n\{n+1\}} (a\_n \tau\_n + b\_n \pi\_n) \right|^2 \tag{15}$$

where *r* is the distance from the particle to the observer (*2r>>1*) and *n* and *<sup>n</sup>* are angular functions defined in (Bohren & Huffman, 1983). As we are considering a very small or dipole-like particle (*a0*), only the two first Mie coefficients (*a1* and *b1*) are introduced in the expressions. In addition some approximations can be applied to these coefficients in such a way that the scattered intensity components can be approximated by

$$I\_{TE} = \frac{\lambda^2 \chi^6}{4\pi r^2} \left| (a\_1 + b\_1 \cos \theta) \right|^2 = \frac{\lambda^2 \chi^6}{4\pi r^2} \left| \{\frac{\varepsilon \cdot \mathbf{1}}{\varepsilon + 2}\} + \{\frac{\mu \cdot \mathbf{1}}{\mu + 2}\} \cos \theta \right|^2 \tag{16}$$

$$I\_{TM} = \frac{\lambda^2 \mathbf{x}^6}{4\pi r^2} \left| (a\_1 \cos \theta + b\_1) \right|^2 = \frac{\lambda^2 \mathbf{x}^6}{4\pi r^2} \left| \left( \frac{\varepsilon \mathbf{-1}}{\varepsilon + \mathbf{2}} \right) \cos \theta + \left( \frac{\mu \cdot \mathbf{1}}{\mu + \mathbf{2}} \right) \right|^2 \tag{17}$$

 being the scattering angle, defined as the angle between the incident and the scattered directions (see Figure 1).

For the backward scattering direction (*=180º*) the previous expressions adopt the following forms

$$I\_{TE} \text{(180^\circ)} = \frac{\lambda^2}{4\pi r^2} \times^6 \left| \left( \frac{\varepsilon^{-1}}{\varepsilon + 2} \right) \because \left( \frac{\mu \cdot 1}{\mu + 2} \right) \right|^2 \tag{18}$$

$$I\_{TM}(180^\circ) = \frac{\lambda^2}{4\pi r^2} \ge 6 \left| \left( -\left(\frac{\varepsilon \cdot 1}{\varepsilon + 2}\right) + \left(\frac{\mu \cdot 1}{\mu + 2}\right) \right)^2 \right| \tag{19}$$

It easy to observe that when , or equivalently when *e=<sup>m</sup>*, the scattered intensity in the backward direction is zero for both incident polarizations. This is the *zero-backward scattering condition* and we shall call in the following the *first Kerker's condition*. In Figure 2 the

On the Optical Response of Nanoparticles: Directionality Effects and Optical Forces 293

symmetry of Mie coefficients (Eqs. (3)-(4)) ensures that all the

) which minimizes the scattered intensity in the forward

*,* which are included in the figure caption,

) = (0.1429; 3), and TM incident polarization.

) with relative optical properties

*<<1,* orders in the Mie expansion

Fig. 3. Scattering diagram of a dipole-like particle (*a = 10-6*

**3.2.1 Size effects on the directionality conditions** 

particle, *a*. When this deviates from the condition *a/*

satisfying the zero-backward scattering condition *(*

on the two Kerker's conditions (García-Cámara et al., 2010a).

Kerker's theory was developed under the far-field approximation and for the very particular case of dipole-like particles for which only the two first Mie coefficients (*a1* and *b1*) are non negligible. However, as particle size increases or the observer approaches, other multipolar terms become important and the directional features, described previously, can be modified.

One of the responsibles for the appearance of multipolar contributions is the size of the

greater than *1* start to be non negligible. The purpose of this section is to analyze size effects

Zero-backward scattering condition can be extended even for large particle sizes. This is

electric and magnetic multipolar contributions are equal and with opposite sign at backward direction. This produces a destructive interferential effect between both contributions for every multipolar order and for a given particle size, *a.* Figure 4 shows the scattering diagrams for several particles with different size (*a*) and optical properties

On the contrary, the zero-forward scattering condition is much more sensitive to size effects. In fact, as *a* increases and multipolar terms, other than the dipolar ones, become important, the electric and magnetic contributions in the forward direction do not interfere destructively anymore, and the zero-forward-scattering tends to disappear. In spite of this, it

direction. In Figure 5, the distribution of the scattered intensity for spherical particles of

were chosen such that a minimum of the scattered intensity in the forward direction

 and  *).* 

fulfilling the zero-forward condition, (

possible because the

is possible to find pairs (

different sizes is plotted. The values of

**3.2 An analysis of Kerker's conditions** 

scattering pattern of a dipole-like particle with relative optical properties,  *= 3* is shown. Only a TM polarization is considered because, from Eqs. (18) and (19), the scattered intensity is equal for both polarizations under this condition.

Fig. 2. Scattering diagram of a dipole-like particle (*a = 10-6*) with relative optical properties fulfilling the zero-backward condition and for a TM incident polarization

#### **3.1.2 Zero-forward scattering: Second Kerker's condition**

For *=0º (*forward scattering direction), Eqs. (16)-(17) become

$$I\_{TE} \{ 0^\circ \} = \frac{\lambda^2}{4\pi r^2} \times^6 \left| \left( \frac{\varepsilon^{-1}}{\varepsilon + 2} \right) + \left( \frac{\mu^{-1}}{\mu + 2} \right) \right|^2 \tag{20}$$

$$I\_{TM}\{0^\circ\} = \frac{\lambda^2}{4\pi r^2} \times \left| \left( \frac{\varepsilon^\circ \mathbf{1}}{\varepsilon + \mathbf{2}} \right) + \left( \frac{\mu^\circ \mathbf{1}}{\mu + \mathbf{2}} \right) \right|^2 \tag{21}$$

In this case, the relation which cancel *ITE(0º)* and *ITM(0º)* is not as evident as before. However, Kerker et al. (Kerker et al., 1983) demonstrated that this happens when

$$
\varepsilon = \frac{4\text{--}\,\mu}{2\,\mu + 1} \tag{22}
$$

which is equivalent to *Re(e)= -Re(e)* and *Im(e)= Im(e)*. This is the *zero-forward scattering condition*, that we shall call the *second Kerker's condition*.

It is interesting to highlight that this condition is symmetric. This means that it remains invariant by interchanging and An example of the angular distribution of the scattered intensity of a very-small particle satisfying this condition is shown in Figure 3 for a TM polarized incident beam (TE polarization produces a similar result).

Fig. 3. Scattering diagram of a dipole-like particle (*a = 10-6*) with relative optical properties fulfilling the zero-forward condition, () = (0.1429; 3), and TM incident polarization.

## **3.2 An analysis of Kerker's conditions**

292 Smart Nanoparticles Technology

Only a TM polarization is considered because, from Eqs. (18) and (19), the scattered

) with relative optical properties

(20)

 *= 3* is shown.

scattering pattern of a dipole-like particle with relative optical properties,

intensity is equal for both polarizations under this condition.

Fig. 2. Scattering diagram of a dipole-like particle (*a = 10-6*

For 

In this case, the

which is equivalent to *Re(*

invariant by interchanging

*e)= -Re(*

*condition*, that we shall call the *second Kerker's condition*.

 and 

**3.1.2 Zero-forward scattering: Second Kerker's condition** 

*=0º (*forward scattering direction), Eqs. (16)-(17) become

*I x TM*

polarized incident beam (TE polarization produces a similar result).

fulfilling the zero-backward condition and for a TM incident polarization

However, Kerker et al. (Kerker et al., 1983) demonstrated that this happens when

*e)* and *Im(*

1 1 (0 ) ( ) ( ) <sup>4</sup> 2 2 *TE I x r*

> 6 2

6 2

1 1 (0 ) ( ) ( ) 4 2 2

It is interesting to highlight that this condition is symmetric. This means that it remains

intensity of a very-small particle satisfying this condition is shown in Figure 3 for a TM

*e)= Im(*

<sup>2</sup> 2 \_\_

 

<sup>2</sup> 2 \_\_

 

relation which cancel *ITE(0º)* and *ITM(0º)* is not as evident as before.

*r* (21)

(22)

An example of the angular distribution of the scattered

*e)*. This is the *zero-forward scattering* 

Kerker's theory was developed under the far-field approximation and for the very particular case of dipole-like particles for which only the two first Mie coefficients (*a1* and *b1*) are non negligible. However, as particle size increases or the observer approaches, other multipolar terms become important and the directional features, described previously, can be modified.

## **3.2.1 Size effects on the directionality conditions**

One of the responsibles for the appearance of multipolar contributions is the size of the particle, *a*. When this deviates from the condition *a/<<1,* orders in the Mie expansion greater than *1* start to be non negligible. The purpose of this section is to analyze size effects on the two Kerker's conditions (García-Cámara et al., 2010a).

Zero-backward scattering condition can be extended even for large particle sizes. This is possible because the symmetry of Mie coefficients (Eqs. (3)-(4)) ensures that all the electric and magnetic multipolar contributions are equal and with opposite sign at backward direction. This produces a destructive interferential effect between both contributions for every multipolar order and for a given particle size, *a.* Figure 4 shows the scattering diagrams for several particles with different size (*a*) and optical properties satisfying the zero-backward scattering condition *().* 

On the contrary, the zero-forward scattering condition is much more sensitive to size effects. In fact, as *a* increases and multipolar terms, other than the dipolar ones, become important, the electric and magnetic contributions in the forward direction do not interfere destructively anymore, and the zero-forward-scattering tends to disappear. In spite of this, it is possible to find pairs () which minimizes the scattered intensity in the forward direction. In Figure 5, the distribution of the scattered intensity for spherical particles of different sizes is plotted. The values of and *,* which are included in the figure caption, were chosen such that a minimum of the scattered intensity in the forward direction

On the Optical Response of Nanoparticles: Directionality Effects and Optical Forces 295

Kerker's conditions, as have been remarked above, were deduced under the far-field

on light scattering are affected. In a recent work (García-Cámara et al. 2010b), it has been shown that directional effects on light scattering of nanoparticles with optical properties under Kerker's conditions tends to disappear as *r* decreases. Figure 6 shows the scattered intensity

direction (Z-axis). Figure 6(a) is devoted to a particle satisfying the first Kerker's condition, while Figure 6(b) shows the same result when its relative optical constants fulfill the second Kerker's condition (eq. (22)). In both cases incident light is P-polarized (an orthogonal polarization produces similar results) and the case of a particle with the same value of

*=1 (*non-directional case*)* is also plotted, for comparison purposes. For observation distances,

intensity of a nanoparticle with directional features tends to that of a nanoparticle which

, the directionality effects appears through a remarkable drop of the scattered intensity in either the backward (Figure 6a) or the forward direction (Figure 6b). However, as the

*>>1)*. If the observer tends to approach (*r*

), the evolution with the observation distance of the scattered

*=3*) or b) the second Kerker's condition

and relative optical constants

) from the backward to the forward

), directional effects

and

**3.2.2 Distance effects on the directionality conditions: From far to near-field** 

approximation, that is (*2*

observer approaches (*r*

(*=3;* 

(*= 3;* 

scattering plane.

*r>0.16* *r/*

measured on a line crossing a nanoparticle (*a~0.01*

*0.16*

optical constants do not satisfied any Kerker's condition.

Fig. 6. Scattered intensity by a nanoparticle of radius *a=0.01*

*=*

the incident direction. For comparison, we have also included the case of a particle with

*=0.1429*) as a function of the distance from the particle surface in a direction parallel to

*= 1*). In both cases the incident beam is polarized with the electric field parallel to the

satisfying a) the first Kerker's condition (

appears. For the smallest value of *a*, the scattered intensity in the forward direction is considerably lower compared to other angles. However, as *a* increases, this minimum becomes less pronounced due to the influence of quadrupolar terms.

Fig. 4. Scattering diagrams, in logarithmic scale, for a spherical particle with relative optical properties () = (-3;-3) and illuminated by a TE-polarized incident light. Several particlesizes have been considered.

In a recent research (García-Cámara et al., 2010a), it was found that these optical constants which minimize forward scattering don't follow Kerker's conditions but can be fitted to a formally similar expression where fitting coefficients are dependent on particle size.

Fig. 5. Scattering diagrams, in logarithmic scale, for a spherical particle illuminated with a TE linearly polarized incident beam. For each particle size, optical properties, in the negative-negative range, are such that the scattered intensity is minimum in the forward direction. In particular, *=-4.55* for every particle size and *=-1.06 (a=0.01=-1.07 (a=0.02)*, *=-1.09 (a=0.03), =-1.11 (a=0.04)* and *=-1.13 (a=0.05).*

appears. For the smallest value of *a*, the scattered intensity in the forward direction is considerably lower compared to other angles. However, as *a* increases, this minimum

Fig. 4. Scattering diagrams, in logarithmic scale, for a spherical particle with relative optical

In a recent research (García-Cámara et al., 2010a), it was found that these optical constants which minimize forward scattering don't follow Kerker's conditions but can be fitted to a

Fig. 5. Scattering diagrams, in logarithmic scale, for a spherical particle illuminated with a TE linearly polarized incident beam. For each particle size, optical properties, in the negative-negative range, are such that the scattered intensity is minimum in the forward

*=-1.13 (a=0.05*

*=-1.06 (a=0.01*

*).* *=-1.07* 

*=-4.55* for every particle size and

*)* and 

*=-1.11 (a=0.04*

) = (-3;-3) and illuminated by a TE-polarized incident light.

formally similar expression where fitting coefficients are dependent on particle size.

becomes less pronounced due to the influence of quadrupolar terms.

properties (

direction. In particular,

*=-1.09 (a=0.03*

*(a=0.02)*,  *),* 

Several particlesizes have been considered.

#### **3.2.2 Distance effects on the directionality conditions: From far to near-field**

Kerker's conditions, as have been remarked above, were deduced under the far-field approximation, that is (*2r/>>1)*. If the observer tends to approach (*r*), directional effects on light scattering are affected. In a recent work (García-Cámara et al. 2010b), it has been shown that directional effects on light scattering of nanoparticles with optical properties under Kerker's conditions tends to disappear as *r* decreases. Figure 6 shows the scattered intensity measured on a line crossing a nanoparticle (*a~0.01*) from the backward to the forward direction (Z-axis). Figure 6(a) is devoted to a particle satisfying the first Kerker's condition, while Figure 6(b) shows the same result when its relative optical constants fulfill the second Kerker's condition (eq. (22)). In both cases incident light is P-polarized (an orthogonal polarization produces similar results) and the case of a particle with the same value of and *=1 (*non-directional case*)* is also plotted, for comparison purposes. For observation distances, *r>0.16*, the directionality effects appears through a remarkable drop of the scattered intensity in either the backward (Figure 6a) or the forward direction (Figure 6b). However, as the observer approaches (*r0.16*), the evolution with the observation distance of the scattered intensity of a nanoparticle with directional features tends to that of a nanoparticle which optical constants do not satisfied any Kerker's condition.

Fig. 6. Scattered intensity by a nanoparticle of radius *a=0.01* and relative optical constants satisfying a) the first Kerker's condition (*==3*) or b) the second Kerker's condition (*=3; =0.1429*) as a function of the distance from the particle surface in a direction parallel to the incident direction. For comparison, we have also included the case of a particle with (*= 3; = 1*). In both cases the incident beam is polarized with the electric field parallel to the scattering plane.

On the Optical Response of Nanoparticles: Directionality Effects and Optical Forces 297

lobe structure with the position of the minimum depending on the particular values of the relative electric permittivity () and the relative magnetic permeability (). Therefore, a suitable tuning of the material optical constants serves to control the angular position of the

in the negative-negative range (labeled in the figure) which produce a minimum scattering at certain scattering angles. The particle is illuminated with a linearly polarized incident plane

Previous analysis presented in this chapter about directional effects of light scattering have been done for nanoparticles with arbitrary values of both the relative electric permittivity (

general, conventional materials do not show any stimulus to the magnetic field of electromagnetic radiation in the visible or the near-infrared region of the electromagnetic spectrum. For this reason, previous analyses have been considered an entelechy, as V. Veselago did in his work to generate and enrich scientific knowledge (Veselago, 1968). Very recently and looking for real situations, it has been shown that submicrometer particles made of Silicon (Evlyukhin et al, 2010 ; García-Extarri et al., 2011) or Germanium (Gómez-Medina et al, 2011b) present both effective electric and magnetic responses, corresponding to the dipolar contributions characterized by their first-order Mie coefficients, in the nearinfrared range. Either of them can be selected by changing the illumination wavelength.

For this kind of nanoparticles, the spectral proximity of both dipolar electric and magnetic responses allows the appearance of coherent effects between dipolar modes. Consequently, under certain conditions, these scatterers are able to satisfy Kerker's conditions. Following the work made by Gómez-Medina et al. (Gómez-Medina et al, 2011b), in Figure 8, the

wave with the electric field perpendicular to the scattering plane (TE polarization).

**4. Directional effects on light scattering by dielectric particles** 

) which do not correspond to any real material. In

and relative optical constants

)

minimum of the scattered intensity.

Fig. 7. Scattering diagrams of a spherical particle with *a = 0.01*

and the relative magnetic permeability (

#### **3.3 A generalization of the Kerker's conditions**

#### **3.3.1 The zero-forward scattering condition and the optical theorem**

In a recent research, Alù et al. (Alù & Engheta, 2011) stated that the *zero-forward scattering condition* (Eq. 22) is incongruent with the Optical Theorem. This relates the extinction efficiency (*Qext*) and the scattering amplitude in the forward direction [*S(0º)*] as follows (Bohren & Huffman, 1983)

$$Q\_{\text{ext}} = \frac{4}{\chi^2} \text{Re}\{\mathcal{S}\{0^\circ\}\} \tag{23}$$

When the *zero-forward scattering condition* holds, *S(0º)=0* and then *Qext=0.* This would imply that the particle would not scatter neither absorb electromagnetic radiation. However, in the examples shown in Figures 3 and 5, while the absorption is null because the optical constants are real, light scattering, and then the extinction efficiency, is non-zero at scattering angles other than *=0º.* 

A first attempt to solve this apparent paradox is found in (Chylek & Pinnick, 1979) where they conclude that the dipolar approximation used by Kerker and co-workers is a nonunitary approximation because *Re(an)|an|2*, *Re(bn)|bn|2* are not satisfied, and therefore the Optical Theorem cannot be applied. However, other more specific solutions to this paradox have been proposed recently. Alù et al (Alù & Engheta, 2011) established that, for a correct estimation of *Qext* it is crucial to include the radiative correction (Draine & Flatau, 1994) into the two first Mie coefficients (*a1* and *b1*). From these considerations, energy conservation is warranted and, although the forward scattering is not zero, it is minimum with respect to other scattering angles. In addition, if the radiative correction is also included in the deduction of the *zero-forward scattering condition* (García-Cámara et al., 2011), a new condition can be found where both the Optical Theorem and the zero scattering at *=0º* hold. This condition follows the equation

 \_ \_ 3\_ \_ 3\_ (4 ) ( 1) (2 1) ( 1) *iVk iVk* (24)

where *V* is the volume of the particle.

#### **3.3.2 Directional effects at scattering angles other than forward and backward directions**

Previous analysis on the distribution of the scattered intensity by a nanoparticle at both the forward and the backward direction can also be extended to other scattering angles. In a previous work (García-Cámara, 2010a), it is shown that by choosing a certain scattering angle different from *0º* and *180º,* there are pairs (), which produce minimum scattered intensity within the scattering plane.

In Figure 7, we plot the scattering diagrams of a nanoparticle (*a = 0.01*) illuminated by a TE polarized incident beam. The optical constants are such that the scattered intensity is minimum at representative angles like *30º*, *60º*, *120º* and *150º*. Each diagram shows a double-

In a recent research, Alù et al. (Alù & Engheta, 2011) stated that the *zero-forward scattering condition* (Eq. 22) is incongruent with the Optical Theorem. This relates the extinction efficiency (*Qext*) and the scattering amplitude in the forward direction [*S(0º)*] as follows

> 2

When the *zero-forward scattering condition* holds, *S(0º)=0* and then *Qext=0.* This would imply that the particle would not scatter neither absorb electromagnetic radiation. However, in the examples shown in Figures 3 and 5, while the absorption is null because the optical constants are real, light scattering, and then the extinction efficiency, is non-zero at

A first attempt to solve this apparent paradox is found in (Chylek & Pinnick, 1979) where they conclude that the dipolar approximation used by Kerker and co-workers is a non-

Optical Theorem cannot be applied. However, other more specific solutions to this paradox have been proposed recently. Alù et al (Alù & Engheta, 2011) established that, for a correct estimation of *Qext* it is crucial to include the radiative correction (Draine & Flatau, 1994) into the two first Mie coefficients (*a1* and *b1*). From these considerations, energy conservation is warranted and, although the forward scattering is not zero, it is minimum with respect to other scattering angles. In addition, if the radiative correction is also included in the deduction of the *zero-forward scattering condition* (García-Cámara et al., 2011), a new condition can be found where both the Optical Theorem and the zero scattering at

 

\_ \_ 3\_ \_ 3\_ (4 ) ( 1) (2 1) ( 1) *iVk*

 

*|an|2*, *Re(bn)*

**3.3.2 Directional effects at scattering angles other than forward and backward** 

Previous analysis on the distribution of the scattered intensity by a nanoparticle at both the forward and the backward direction can also be extended to other scattering angles. In a previous work (García-Cámara, 2010a), it is shown that by choosing a certain scattering

polarized incident beam. The optical constants are such that the scattered intensity is minimum at representative angles like *30º*, *60º*, *120º* and *150º*. Each diagram shows a double-

In Figure 7, we plot the scattering diagrams of a nanoparticle (*a = 0.01*

<sup>4</sup> Re{ (0 )} *Q S ext <sup>x</sup>* (23)

*|bn|2* are not satisfied, and therefore the

*iVk* (24)

), which produce minimum scattered

) illuminated by a TE

*=0º*

**3.3 A generalization of the Kerker's conditions** 

*=0º.* 

(Bohren & Huffman, 1983)

scattering angles other than

unitary approximation because *Re(an)*

hold. This condition follows the equation

where *V* is the volume of the particle.

intensity within the scattering plane.

angle different from *0º* and *180º,* there are pairs (

**directions** 

**3.3.1 The zero-forward scattering condition and the optical theorem** 

lobe structure with the position of the minimum depending on the particular values of the relative electric permittivity () and the relative magnetic permeability (). Therefore, a suitable tuning of the material optical constants serves to control the angular position of the minimum of the scattered intensity.

Fig. 7. Scattering diagrams of a spherical particle with *a = 0.01* and relative optical constants in the negative-negative range (labeled in the figure) which produce a minimum scattering at certain scattering angles. The particle is illuminated with a linearly polarized incident plane wave with the electric field perpendicular to the scattering plane (TE polarization).

## **4. Directional effects on light scattering by dielectric particles**

Previous analysis presented in this chapter about directional effects of light scattering have been done for nanoparticles with arbitrary values of both the relative electric permittivity () and the relative magnetic permeability () which do not correspond to any real material. In general, conventional materials do not show any stimulus to the magnetic field of electromagnetic radiation in the visible or the near-infrared region of the electromagnetic spectrum. For this reason, previous analyses have been considered an entelechy, as V. Veselago did in his work to generate and enrich scientific knowledge (Veselago, 1968). Very recently and looking for real situations, it has been shown that submicrometer particles made of Silicon (Evlyukhin et al, 2010 ; García-Extarri et al., 2011) or Germanium (Gómez-Medina et al, 2011b) present both effective electric and magnetic responses, corresponding to the dipolar contributions characterized by their first-order Mie coefficients, in the nearinfrared range. Either of them can be selected by changing the illumination wavelength.

For this kind of nanoparticles, the spectral proximity of both dipolar electric and magnetic responses allows the appearance of coherent effects between dipolar modes. Consequently, under certain conditions, these scatterers are able to satisfy Kerker's conditions. Following the work made by Gómez-Medina et al. (Gómez-Medina et al, 2011b), in Figure 8, the

On the Optical Response of Nanoparticles: Directionality Effects and Optical Forces 299

nanosphere (*a=240nm*) when the incident wavelengths are those marked by vertical lines in

scattered intensity in this direction (Figure 9a). However, the *zero-forward scattering condition* is strongly affected by size effects (Figure 9b). As was described in Section 3.2.1, the size of the particle prevents scattered intensity to be completely suppressed in the forward direction. However, its value is very small compared with those at other scattering angles

and most part of the scattered intensity is located in the backward hemisphere (

Fig. 9. Scattering diagrams for a Ge nanoparticles (*a=240nm*) illuminated by a linear polarized plane wave. Both polarizations, with the incident electric field parallel (TM or P polarization) or normal (TE or S polarization) to the scattering plane are considered. The

incident wavelength is labeled in the figure. From (Gómez-Medina et al., 2011b).

development of new silicon applications as, for instance, optical nanocircuits.

**5. Optical forces** 

Previous results for Germanium can also be extended to Silicon nanoparticles. These behaviors in Silicon could be even more interesting due to the wide range of applications of this material. Silicon is the base of microelectronics due to its semiconductor character and also to its abundance in Earth. For this reason, the industry of Silicon is very well developed. These new scattering features in the nanometric range could be the base for the

Light carries energy and both linear and angular momenta that can be transferred to atoms, molecules and particles. Demonstration of levitation and trapping of micron-sized particles

*=1823 nm,* and there is no

).

Figure 8. The *zero-backward scattering condition* is satisfied for

electric (*e*) and the magnetic (*<sup>m</sup>*) polarizabilities of a Ge nanoparticle of radius *a=240nm* are plotted as a function of the wavelength () of the incident radiation. In the considered spectral range, Germanium has a refractive index which can be well approximated by a real constant *m=4 (*Palik, 1985*)*. Also the spectral evolution of the extinction efficiency (Q*ext*) has been included in order to show the resonant behaviors that appear in a Ge nanoparticle. A dipolar electric (DE) mode arises at *=1823 nm,* while a dipolar magnetic (DM) resonance is located at *=2193 nm*. The vertical lines point the wavelengths at which either the first (*e =m*) or the second (*Re(e)=-Re(e)* and *Im(e)= Im(e)*) Kerker's condition are fulfilled.

Fig. 8. Real and imaginary parts of the electric (*<sup>e</sup>*) and the magnetic *<sup>m</sup>*) polarizabilities for a Ge nanoparticle (*a=240nm*). The refractive index of Germanium, in the considered range, can be considered as real and constant, *m 4+0i.* The wavelengths at which the first and second Kerker's conditions (*=2193nm* and *=1823 nm*, respectively) are satisfied, are identified with vertical lines. Also, for comparison purposes, the extinction efficiency is plotted identifying the dipolar electric (DE) and the dipolar magnetic (DM) resonances.

The fact that a dielectric and non-magnetic particle (>0 and *=1*) presents both dipolar electric and also dipolar magnetic modes is quite interesting and could be useful for potential applications. For instance, this kind of resonances has been currently used for several tasks in a wide range of fields, ranging from the design of nanodevices (Maier et al, 2003; Anker et al., 2008) to biomedical treatments (Zemp, 2009). Unfortunately, they were observed only in metallic materials which present strong absorption losses. One of the advantages of dielectric materials, like Germanium or Silicon, is that they show negligible absorption in the considered range (Palik, 1985) and then losses are almost absent.

The position and shape of the dipolar resonances shown in Fig. 8 for Ge particles (similarly for Si particles) produces interesting coherent effects between them and consequently a natural way of reproducing Kerker's conditions by means of real materials. In order to verify that these directional features show up, Figure 9 plots the scattering diagrams of a Ge nanosphere (*a=240nm*) when the incident wavelengths are those marked by vertical lines in Figure 8. The *zero-backward scattering condition* is satisfied for *=1823 nm,* and there is no scattered intensity in this direction (Figure 9a). However, the *zero-forward scattering condition* is strongly affected by size effects (Figure 9b). As was described in Section 3.2.1, the size of the particle prevents scattered intensity to be completely suppressed in the forward direction. However, its value is very small compared with those at other scattering angles and most part of the scattered intensity is located in the backward hemisphere ().

Fig. 9. Scattering diagrams for a Ge nanoparticles (*a=240nm*) illuminated by a linear polarized plane wave. Both polarizations, with the incident electric field parallel (TM or P polarization) or normal (TE or S polarization) to the scattering plane are considered. The incident wavelength is labeled in the figure. From (Gómez-Medina et al., 2011b).

Previous results for Germanium can also be extended to Silicon nanoparticles. These behaviors in Silicon could be even more interesting due to the wide range of applications of this material. Silicon is the base of microelectronics due to its semiconductor character and also to its abundance in Earth. For this reason, the industry of Silicon is very well developed. These new scattering features in the nanometric range could be the base for the development of new silicon applications as, for instance, optical nanocircuits.

## **5. Optical forces**

298 Smart Nanoparticles Technology

spectral range, Germanium has a refractive index which can be well approximated by a real constant *m=4 (*Palik, 1985*)*. Also the spectral evolution of the extinction efficiency (Q*ext*) has been included in order to show the resonant behaviors that appear in a Ge nanoparticle. A

> *e)= Im(*

a Ge nanoparticle (*a=240nm*). The refractive index of Germanium, in the considered range,

identified with vertical lines. Also, for comparison purposes, the extinction efficiency is plotted identifying the dipolar electric (DE) and the dipolar magnetic (DM) resonances.

electric and also dipolar magnetic modes is quite interesting and could be useful for potential applications. For instance, this kind of resonances has been currently used for several tasks in a wide range of fields, ranging from the design of nanodevices (Maier et al, 2003; Anker et al., 2008) to biomedical treatments (Zemp, 2009). Unfortunately, they were observed only in metallic materials which present strong absorption losses. One of the advantages of dielectric materials, like Germanium or Silicon, is that they show negligible

The position and shape of the dipolar resonances shown in Fig. 8 for Ge particles (similarly for Si particles) produces interesting coherent effects between them and consequently a natural way of reproducing Kerker's conditions by means of real materials. In order to verify that these directional features show up, Figure 9 plots the scattering diagrams of a Ge

absorption in the considered range (Palik, 1985) and then losses are almost absent.

*=2193nm* and

*<sup>e</sup>*) and the magnetic

>0 and  *4+0i.* The wavelengths at which the first and

*=1823 nm*, respectively) are satisfied, are

*<sup>m</sup>*) polarizabilities for

*=1*) presents both dipolar

*=2193 nm*. The vertical lines point the wavelengths at which either the first

*<sup>m</sup>*) polarizabilities of a Ge nanoparticle of radius *a=240nm* are

*=1823 nm,* while a dipolar magnetic (DM) resonance is

) of the incident radiation. In the considered

*e)*) Kerker's condition are fulfilled.

electric (

located at

(*e =* *e*) and the magnetic (

dipolar electric (DE) mode arises at

*m*) or the second (*Re(*

plotted as a function of the wavelength (

*e)=-Re(*

Fig. 8. Real and imaginary parts of the electric (

The fact that a dielectric and non-magnetic particle (

can be considered as real and constant, *m*

second Kerker's conditions (

*e)* and *Im(*

Light carries energy and both linear and angular momenta that can be transferred to atoms, molecules and particles. Demonstration of levitation and trapping of micron-sized particles

On the Optical Response of Nanoparticles: Directionality Effects and Optical Forces 301

The fields in Eq. (25) are total fields, namely the sum of the incident and scattered (reradiated) fields: *E(i)+E(r), B(i)+B(r)*. **s** is its local outward unit normal. A time dependence *e( iwt)* is assumed throughout. For a small particle, within the range of validity of the dipolar approximation, the scattered field corresponds to that radiated by the induced electric and magnetic dipole moments, *p* and *m*, respectively. In this case, Eq. (25) leads to the

*i i k*

*/c*, 

Equation (26) represents the generalization of the result of (Chaumet & Rahmani, 2009) for the time-averaged force on a particle immersed in an arbitrary medium with refractive

 represents the dyadic product so that the matrix operation: **W V** ( ) has elements *W V jjj* for *i, j* = 1, 2 3. All variables in Eq. (26) are evaluated at a point *r = r0* in the particle. The first term of Eq. (26) is the force *<* **F***e >* exerted by the incident field on the induced electric dipole, the second and third terms *<* **F***m >* and *<* **F***em >* are the force on the induced magnetic dipole and the force due to the interaction between both dipoles (Chaumet &

The question of energy conservation has been recurrently addressed and debated as regards small particles (Chýlek & Pinnick, 1979; Lock et al., 1995), especially in connection with magnetic particles that produce zero-forward scattering intensity (Alù & Engheta, 2011; Nieto-Vesperinas et al., 2011; García-Cámara et al., 2011; Gómez-Medina et al., 2011b). It is thus relevant to explore the formal analogy between the force as momentum "absorption" rate and the optical theorem expressing the conservation of electromagnetic energy. From the Poynting's theorem (Bohren & Huffman, 1983; Jackson, 1998), the rate *–W(a)* at which

> - *<sup>a</sup>* ( )*<sup>i</sup> S*

*<sup>c</sup> dS*

By introducing the incident field as a decomposition of plane wave components and taking the sphere *S* in Eq. (27) so large that <sup>0</sup> *kr r* , and using Jones' lemma based on the principle of the stationary phase, (see Appendix XII of Bohren & Huffman, 1983), and the source-free condition, we get the optical theorem for an arbitrary field (Nieto-Vesperinas et

 

**EsE BsB E B** (28)

4

*m*

1 2 2

 

**p m** . (29)

\*1 \* <sup>1</sup> 2 1 <sup>1</sup> () () 8 2 *<sup>S</sup>*

 

\* - -- -

 

*ai i c k W rr*

0 0 22 3

**p E m B**

**F pE mB p m** (26)

*<sup>W</sup> dS* **SS s** (27)

being the frequency. The symbol

\*\* \* -

. The wavenumber is *k* = *m*

Rahmani, 2009; Nieto-Vesperinas et al., 2010).

energy is being absorbed by the particle is given by

*m*

<sup>4</sup> 1 2 2 3

**5.2 Optical theorem and forces on an electric and magnetic dipolar particle** 

expression

index: *m*

al., 2010):

by radiation pressure dates back to 1970 and the experiments reported by Ashkin and coworkers (Ashkin, 1970). Light forces on small particles are usually described as the sum of two terms: the dipole or gradient force and the radiation pressure or scattering force (Askhin et al., 1986; Neuman & Block, 2004; Novotny & Hecht, 2006; Chaumet & Nieto-Vesperinas 2000b; Gómez-Medina et al., 2001; Chaumet & Nieto-Vesperinas, 2002; Nieto-Vesperinas et al., 2004; Gómez-Medina & Saénz, 2004). There is an additional nonconservative curl force arising in a light field of non-uniform ellipticity that is proportional to the curl of the spin angular momentum of the light field (Albaladejo et al., 2009a; Nieto-Vesperinas et al., 2010). In analogy with electrostatics, small particles develop an electric (magnetic) dipole moment in response to the light electric (magnetic) field. The induced dipole is then drawn by field intensity gradients which compete with radiation pressure due to momentum transferred from the photons in the beam. By fashioning proper optical field gradients it is possible to trap and manipulate small dielectric particles with optical tweezers (Askhin et al, 1986; Neuman & Block, 2004) or create atomic arrays in optical lattices (Verkerk et al., 1992; Hemmerich & H'ansch, 1993). Intense optical fields can also induce significant forces between particles (Burns et al., 1989; Burns et al., 1990; Tartakova et al., 2002; Chaumet & Nieto-Vesperinas, 2001; Gómez-Medina & Saénz, 2004). Some previous work focused on optical forces on macroscopic media, either with electric (Mansuripur, 2004) or magnetic response (Kemp et al., 2005; Mansuripur, 2007), or particles with electric response (Kemp et al., 2006a). Radiation pressure forces on dielectric and magnetic particles under plane wave incidence have been computed for both small cylinders (Kemp et al., 2006b) and spheres (Lakhtakia & Mulholland, 1993; Lakhtakia, 2008). The total force on an electric and magnetic dipolar particle has been shown (Chaumet & Rahmani, 2009; Nieto-Vesperinas et al., 2010; Nieto-Vesperinas et al., 2011; Gómez-Medina et al., 2011a; Gómez-Medina et al., 2011b) to have a similarity with that previously obtained for electric dipoles. Moreover, in the presence of both electric and magnetic responses, the force presents an additional term proportional to the cross product of the electric and magnetic dipoles (Chaumet & Rahmani, 2009; Nieto-Vesperinas et al., 2010; Nieto-Vesperinas et al., 2011; Gómez-Medina et al., 2011a; Gómez-Medina et al., 2011b). The relevance and physical origin of this electric-magnetic dipolar interaction term for a single particle has been recently discussed (Nieto-Vesperinas et al., 2010; Nieto-Vesperinas et al., 2011; Gómez-Medina et al., 2011a; Gómez-Medina et al., 2011b)

#### **5.1 Force on a small particle with electric and magnetic response to an electromagnetic wave**

We consider a dipolar particle embedded in a non-dissipative medium with relative dielectric permittivity and magnetic permeability , subjected to an incident electromagnetic field whose electric and magnetic vectors are *E(i)* and *B(i)*, respectively. The total time-averaged electromagnetic force acting on the particle is (Chaumet & Nieto-Vesperinas, 2000; Jackson, 1998; Nieto-Vesperinas et al., 2010):

$$
\begin{Bmatrix} \mathbf{F} \end{Bmatrix} = \frac{1}{8\pi} \Re \left\{ \int\_{S} \left[ \varepsilon \mathbf{f} \mathbf{E} \cdot \mathbf{s} \right] \mathbf{E}^{\ast} + \mu^{-1} \mathbf{(B} \cdot \mathbf{s}) \mathbf{B}^{\ast} - \frac{1}{2} \left( \varepsilon \left| \mathbf{E} \right|^{2} + \mu^{-1} \left| \mathbf{B} \right|^{-1} \right) \mathbf{s} \right\} \tag{25}
$$

where stands for real part, *dS* denotes the element of any surface *S* that encloses the particle.

by radiation pressure dates back to 1970 and the experiments reported by Ashkin and coworkers (Ashkin, 1970). Light forces on small particles are usually described as the sum of two terms: the dipole or gradient force and the radiation pressure or scattering force (Askhin et al., 1986; Neuman & Block, 2004; Novotny & Hecht, 2006; Chaumet & Nieto-Vesperinas 2000b; Gómez-Medina et al., 2001; Chaumet & Nieto-Vesperinas, 2002; Nieto-Vesperinas et al., 2004; Gómez-Medina & Saénz, 2004). There is an additional nonconservative curl force arising in a light field of non-uniform ellipticity that is proportional to the curl of the spin angular momentum of the light field (Albaladejo et al., 2009a; Nieto-Vesperinas et al., 2010). In analogy with electrostatics, small particles develop an electric (magnetic) dipole moment in response to the light electric (magnetic) field. The induced dipole is then drawn by field intensity gradients which compete with radiation pressure due to momentum transferred from the photons in the beam. By fashioning proper optical field gradients it is possible to trap and manipulate small dielectric particles with optical tweezers (Askhin et al, 1986; Neuman & Block, 2004) or create atomic arrays in optical lattices (Verkerk et al., 1992; Hemmerich & H'ansch, 1993). Intense optical fields can also induce significant forces between particles (Burns et al., 1989; Burns et al., 1990; Tartakova et al., 2002; Chaumet & Nieto-Vesperinas, 2001; Gómez-Medina & Saénz, 2004). Some previous work focused on optical forces on macroscopic media, either with electric (Mansuripur, 2004) or magnetic response (Kemp et al., 2005; Mansuripur, 2007), or particles with electric response (Kemp et al., 2006a). Radiation pressure forces on dielectric and magnetic particles under plane wave incidence have been computed for both small cylinders (Kemp et al., 2006b) and spheres (Lakhtakia & Mulholland, 1993; Lakhtakia, 2008). The total force on an electric and magnetic dipolar particle has been shown (Chaumet & Rahmani, 2009; Nieto-Vesperinas et al., 2010; Nieto-Vesperinas et al., 2011; Gómez-Medina et al., 2011a; Gómez-Medina et al., 2011b) to have a similarity with that previously obtained for electric dipoles. Moreover, in the presence of both electric and magnetic responses, the force presents an additional term proportional to the cross product of the electric and magnetic dipoles (Chaumet & Rahmani, 2009; Nieto-Vesperinas et al., 2010; Nieto-Vesperinas et al., 2011; Gómez-Medina et al., 2011a; Gómez-Medina et al., 2011b). The relevance and physical origin of this electric-magnetic dipolar interaction term for a single particle has been recently discussed (Nieto-Vesperinas et al., 2010; Nieto-Vesperinas et al., 2011; Gómez-Medina et al.,

2011a; Gómez-Medina et al., 2011b)

Vesperinas, 2000; Jackson, 1998; Nieto-Vesperinas et al., 2010):

**electromagnetic wave** 

dielectric permittivity

particle.

**5.1 Force on a small particle with electric and magnetic response to an** 

We consider a dipolar particle embedded in a non-dissipative medium with relative

electromagnetic field whose electric and magnetic vectors are *E(i)* and *B(i)*, respectively. The total time-averaged electromagnetic force acting on the particle is (Chaumet & Nieto-

1 1 \*1 \* 2 1 <sup>1</sup> () () , 8 2 *<sup>S</sup>*

where stands for real part, *dS* denotes the element of any surface *S* that encloses the

 

**<sup>F</sup> EsE BsB E B s** (25)

*dS*

, subjected to an incident

and magnetic permeability

  The fields in Eq. (25) are total fields, namely the sum of the incident and scattered (reradiated) fields: *E(i)+E(r), B(i)+B(r)*. **s** is its local outward unit normal. A time dependence *e( iwt)* is assumed throughout. For a small particle, within the range of validity of the dipolar approximation, the scattered field corresponds to that radiated by the induced electric and magnetic dipole moments, *p* and *m*, respectively. In this case, Eq. (25) leads to the expression

$$\left\langle \mathbf{F} \right\rangle = \frac{1}{2} \Re \left\langle \mathbf{p} \left( \nabla \otimes \mathbf{E}^{(l) \mathbf{r}} \right) + \mathbf{m} \left( \nabla \otimes \mathbf{B}^{(l) \mathbf{r}} \right) \cdot \frac{2k^4}{3} \sqrt{\frac{\mu}{\varepsilon}} \left( \mathbf{p} \times \mathbf{m}^\* \right) \right\rangle \tag{26}$$

Equation (26) represents the generalization of the result of (Chaumet & Rahmani, 2009) for the time-averaged force on a particle immersed in an arbitrary medium with refractive index: *m* . The wavenumber is *k* = *m/c*, being the frequency. The symbol represents the dyadic product so that the matrix operation: **W V** ( ) has elements *W V jjj* for *i, j* = 1, 2 3. All variables in Eq. (26) are evaluated at a point *r = r0* in the particle. The first term of Eq. (26) is the force *<* **F***e >* exerted by the incident field on the induced electric dipole, the second and third terms *<* **F***m >* and *<* **F***em >* are the force on the induced magnetic dipole and the force due to the interaction between both dipoles (Chaumet & Rahmani, 2009; Nieto-Vesperinas et al., 2010).

#### **5.2 Optical theorem and forces on an electric and magnetic dipolar particle**

The question of energy conservation has been recurrently addressed and debated as regards small particles (Chýlek & Pinnick, 1979; Lock et al., 1995), especially in connection with magnetic particles that produce zero-forward scattering intensity (Alù & Engheta, 2011; Nieto-Vesperinas et al., 2011; García-Cámara et al., 2011; Gómez-Medina et al., 2011b). It is thus relevant to explore the formal analogy between the force as momentum "absorption" rate and the optical theorem expressing the conservation of electromagnetic energy. From the Poynting's theorem (Bohren & Huffman, 1983; Jackson, 1998), the rate *–W(a)* at which energy is being absorbed by the particle is given by

$$-\mathcal{W}^{(a)} = \bigcap\_{\mathcal{S}} \left\langle \left\langle \mathbf{S} \right\rangle - \left\langle \mathbf{S}^{(l)} \right\rangle \right\rangle \mathbf{s} dS \tag{27}$$

$$=\frac{c}{8\pi m}\Re\left\{\int\_{S}\varepsilon(\mathbf{E}\cdot\mathbf{s})\mathbf{E}^\* + \mu^{-1}(\mathbf{B}\cdot\mathbf{s})\mathbf{B}^\* - \frac{1}{2}\Big(\varepsilon\left|\mathbf{E}\right|^2 + \mu^{-1}\left|\mathbf{B}\right|^{-1}\right)d\mathbf{S}\right\}\tag{28}$$

By introducing the incident field as a decomposition of plane wave components and taking the sphere *S* in Eq. (27) so large that <sup>0</sup> *kr r* , and using Jones' lemma based on the principle of the stationary phase, (see Appendix XII of Bohren & Huffman, 1983), and the source-free condition, we get the optical theorem for an arbitrary field (Nieto-Vesperinas et al., 2010):

$$\mathbf{\dot{W}}^{(a)} = \frac{\alpha \nu}{2} \Im \left( \mathbf{p} \cdot \mathbf{E}^{(l) \ast} \left( r\_0 \right) \right) \cdot \frac{\alpha \nu}{2} \Im \left( \mathbf{m} \cdot \mathbf{B}^{(l)} \left( r\_0 \right) \right) + \frac{c}{m} \frac{k^4}{3} \left( \varepsilon^{-1} \left| \mathbf{p} \right|^2 + \mu \left| \mathbf{m} \right|^2 \right). \tag{29}$$

On the Optical Response of Nanoparticles: Directionality Effects and Optical Forces 303

 

 

the sum of radiation pressures for a pure electric and a pure magnetic dipole, respectively. The third term, *<* **F***em >,* is the time-averaged scattered momentum rate, and we shall see below that it also contributes to radiation pressure (Nieto-Vesperinas et al., 2010; Gómez-Medina et al., 2011a) and it is related to the asymmetry in the scattered intensity distribution

Fig. 10. Different contributions to the total radiation pressure versus the wavelength, for the

From Eqs. (6) and (34), one derives for the radiation pressure force (Nieto-Vesperinas et al., 2011):

 

*dC dC F C*

6 2

Equation (35) emphasizes the dominant role of the backward scattering on radiation

1 3

*sca sca abs*

first and second generalized Kerker's conditions. Notice that when the first generalized

*e m* 

s 0 3 180

*kd d*

**FF F F** *e m em* . From (Gómez-Medina et al. 2011b).

**5.4 The generalized Kerker's conditions on optical forces** 


0 0

<sup>2</sup> ( ) <sup>0</sup> / 2 *<sup>i</sup> <sup>F</sup>* 

 and 

**F** . (35)

**e** . The vertical lines mark the

 *e m* ,

3 1 \*

2

<sup>3</sup> *e m e m*

**e** . The first two terms, *<* **F***e >* and *<* **F***m >*, correspond to the forces on to

 

*<sup>k</sup> <sup>F</sup>* , (34)


0 0

(Nieto-Vesperinas et al., 2011; Gómez-Medina et al., 2011b)*.* 

Ge particle of Figs. 8-9. Normalization is done by

Kerker's condition is fulfilled, i.e.,

pressure forces.

s

<sup>2</sup> ( ) <sup>0</sup> / 2 *<sup>i</sup> <sup>F</sup>* 

where

 

The first two terms of Eq. (29), coming from the interference between the incident and radiated fields, are the energy analogue of the electric and magnetic dipolar forces given by first two terms in Eq. (26).

The third and fourth terms of Eq. (29) that come from the integral of the third and fourth terms of Eq. (28), now yield the rate *W*(*<sup>s</sup>*) at which the energy is being scattered, which together with the left hand side of this equation contributes to the rate of energy extinction by the particle *W*(*<sup>a</sup>*)+*W*(*<sup>s</sup>*):

$$\mathcal{W}^{(a)} + \mathcal{W}^{(s)} = \frac{\alpha}{2} \Im \left\{ \mathbf{p} \cdot \mathbf{E}^{(l)^\star} \left( r\_0 \right) \right\} + \frac{\alpha}{2} \Im \left\{ \mathbf{m} \cdot \mathbf{B}^{(l)^\star} \left( r\_0 \right) \right\} \,. \tag{30}$$

Analogously as with the rate of scattered energy, the electric-magnetic dipolar interaction term of the force (third term of Eq. (26)) corresponds to the rate at which momentum is being scattered by the particle. We shall explore in some detail this analogy in order to illustrate the physical origin of *<* **F***em >*. We notice that the power density of the scattered field can be written as the sum of two terms (Nieto-Vesperinas et al., 2010)

$$
\left\langle \mathbf{S}^{(r)} \right\rangle dS = \frac{c}{8\pi m} k^4 \left( \varepsilon^{-1} \left| \mathbf{p} \times \mathbf{s} \right|^2 + \mu \left| \mathbf{m} \times \mathbf{s} \right|^2 \right) \mathbf{s} d\Omega$$

$$
+ \frac{c}{4\pi m} k^4 \sqrt{\frac{\mu}{\varepsilon}} \Re \left\langle \left( \mathbf{s} \times \mathbf{p} \right) \cdot \mathbf{m}^\* \right\rangle \mathbf{s} d\Omega \,\tag{31}$$

where the second term of Eq. (31) corresponds to the interference between the electric and magnetic dipolar fields. After integration over the closed surface *S*, that second term does not contribute to the radiated power, while it is the only contribution to the electricmagnetic dipolar interaction term of the force in Eq. (26). Namely, *<* **F***em >* comes from the interference between the fields radiated by **p** and **m**.

#### **5.3 Forces on an electric and magnetic dipolar particle for plane wave incidence**

In order to illustrate the relevance of the different terms in the optical forces, we shall next consider the force from a plane wave 0 0 () () () () , *ii ii iks r iks r E ee B be* with () () <sup>0</sup> / *i i* **e bs** *<sup>m</sup>* on a small dielectric and magnetic spherical particle characterized by its electric and magnetic polarizabilities *<sup>e</sup>* and*<sup>m</sup>*. When the induced dipole moments are expressed in terms of the incident field, i.e.

$$\mathbf{p} = \alpha\_e \mathbf{e}^{(l)}; \quad \mathbf{m} = \alpha\_m \mathbf{b}^{(l)}.\tag{32}$$

For plane wave incidence, the total force is given by (Nieto-Vesperinas et al., 2010):

$$
\begin{aligned}
\begin{Bmatrix} \mathbf{F} \end{Bmatrix} &=& \begin{Bmatrix} \mathbf{F}\_e \end{Bmatrix} + \begin{Bmatrix} \mathbf{F}\_m \end{Bmatrix} + \begin{Bmatrix} \mathbf{F}\_{em} \end{Bmatrix} \\\\ &=& \mathbf{s}\_0 \frac{k}{2} \Im \left\{ \mathbf{p} \cdot \mathbf{e}^{(l) \star} + \mathbf{m} \cdot \mathbf{b}^{(l) \star} \right\} - \frac{m}{c} \Im \left\{ \mathbf{S}^{(r)} \right\} \mathbf{s} \mathbf{S} \end{aligned} \tag{33}$$

The first two terms of Eq. (29), coming from the interference between the incident and radiated fields, are the energy analogue of the electric and magnetic dipolar forces given by

The third and fourth terms of Eq. (29) that come from the integral of the third and fourth terms of Eq. (28), now yield the rate *W*(*<sup>s</sup>*) at which the energy is being scattered, which together with the left hand side of this equation contributes to the rate of energy extinction

*as i <sup>i</sup> W W r r*

Analogously as with the rate of scattered energy, the electric-magnetic dipolar interaction term of the force (third term of Eq. (26)) corresponds to the rate at which momentum is being scattered by the particle. We shall explore in some detail this analogy in order to illustrate the physical origin of *<* **F***em >*. We notice that the power density of the scattered

4 1- 2 2

**S ps ms**

 

**5.3 Forces on an electric and magnetic dipolar particle for plane wave incidence** 

In order to illustrate the relevance of the different terms in the optical forces, we shall next consider the force from a plane wave 0 0 () () () () , *ii ii iks r iks r E ee B be* with () ()

on a small dielectric and magnetic spherical particle characterized by its electric and

*i i* **pem b**

**F FF F** *e m em*

 

 c \* \* ()

*S <sup>k</sup> dS* **pe mb S** (33)

*i i r*

For plane wave incidence, the total force is given by (Nieto-Vesperinas et al., 2010):

*<sup>r</sup> <sup>c</sup> dS k <sup>d</sup>*

*<sup>c</sup> k d*

where the second term of Eq. (31) corresponds to the interference between the electric and magnetic dipolar fields. After integration over the closed surface *S*, that second term does not contribute to the radiated power, while it is the only contribution to the electricmagnetic dipolar interaction term of the force in Eq. (26). Namely, *<* **F***em >* comes from the

 

**p E m B** . (30)

s

**spm s** . (31)

*<sup>m</sup>*. When the induced dipole moments are expressed in

*e m* (32)

 

<sup>0</sup> / *i i* **e bs** *<sup>m</sup>*

\* \* 0 0 2 2

field can be written as the sum of two terms (Nieto-Vesperinas et al., 2010)

8

*m*

 4 \* 4

interference between the fields radiated by **p** and **m**.

*<sup>e</sup>* and

; .

<sup>m</sup>

0s 2 *m*

first two terms in Eq. (26).

by the particle *W*(*<sup>a</sup>*)+*W*(*<sup>s</sup>*):

magnetic polarizabilities

terms of the incident field, i.e.

$$=\mathbf{s}\_0 F\_0 \left[ \mathfrak{T} \left( \varepsilon^{-1} \alpha\_e \right) + \mathfrak{T} \left( \mu \alpha\_m \right) \cdot \frac{2k^3}{3} \frac{\mu}{\varepsilon} \Re \left( a\_e \alpha\_m \right) \right],\tag{34}$$

where <sup>2</sup> ( ) <sup>0</sup> / 2 *<sup>i</sup> <sup>F</sup>* **e** . The first two terms, *<* **F***e >* and *<* **F***m >*, correspond to the forces on to the sum of radiation pressures for a pure electric and a pure magnetic dipole, respectively. The third term, *<* **F***em >,* is the time-averaged scattered momentum rate, and we shall see below that it also contributes to radiation pressure (Nieto-Vesperinas et al., 2010; Gómez-Medina et al., 2011a) and it is related to the asymmetry in the scattered intensity distribution (Nieto-Vesperinas et al., 2011; Gómez-Medina et al., 2011b)*.* 

Fig. 10. Different contributions to the total radiation pressure versus the wavelength, for the Ge particle of Figs. 8-9. Normalization is done by <sup>2</sup> ( ) <sup>0</sup> / 2 *<sup>i</sup> <sup>F</sup>* **e** . The vertical lines mark the first and second generalized Kerker's conditions. Notice that when the first generalized Kerker's condition is fulfilled, i.e., *e m* and *e m* , **FF F F** *e m em* . From (Gómez-Medina et al. 2011b).

#### **5.4 The generalized Kerker's conditions on optical forces**

From Eqs. (6) and (34), one derives for the radiation pressure force (Nieto-Vesperinas et al., 2011):

$$\left\langle \mathbf{F} \right\rangle = \mathbf{s}\_0 \, F\_0 \frac{1}{6k} \left[ \frac{d\mathcal{C}\_{\text{sca}}}{d\Omega} \left( 0^\circ \right) + 3 \frac{d\mathcal{C}\_{\text{sca}}}{d\Omega} \left( 180^\circ \right) \cdot \frac{3}{2\pi} \mathcal{C}\_{\text{abs}} \right]. \tag{35}$$

Equation (35) emphasizes the dominant role of the backward scattering on radiation pressure forces.

On the Optical Response of Nanoparticles: Directionality Effects and Optical Forces 305

This work has been supported by the EU NMP3-SL-2008-214107-Nanomagma, the Spanish MICINN Consolider NanoLight (CSD2007-00046), FIS2010-21984, FIS2009-13430-C01-C02, and FIS2007-60158, as well as by the Comunidad de Madrid Microseres-CM (S2009/TIC-1476). B.G.-C. wants to express his gratitude to the University of Cantabria for his postdoctoral fellowship. Work by R.G.-M. was supported by the MICINN "Juan de la

Albaladejo, S.; Marqués, M.I.; Laroche, M. & Sáenz, J.J. (2009). Scattering forces from the curl

Albaladejo, S.; Marqués, M.I. & Sáenz, J.J. (2011). Light control of silver nanoparticle's

Albaladejo, S.; Marqués, M.I. & Sáenz, J.J. (2011). Light control of silver nanoparticle's

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**7. Acknowledgment** 

Cierva" Fellowship.

6984.

4087.

ISSN 1476-1122.

**8. References** 

At the first generalized Kerker's condition, the interference term of Eqs. (34-35) cancels out the magnetic contribution and we obtain *<* **F** *>* = *<* **F***e >*. At the second generalized Kerker's condition, where the backscattering is enhanced, *<* **F** *>* = 3*<* **F***e >*. Notice that at both generalized Kerker's conditions the scattering cross section is exactly the same; however, the radiation pressures differ by a factor of 3. These properties are illustrated in Figure 10, where we show the different contributions to the total time averaged force on a submicrometer Ge particle.

The strong peak in the radiation pressure force is mainly dominated by the first "magnetic" Mie resonance. This is striking and in contrast with all previous beliefs about optical forces on dipolar dielectric particles, that assumed that these forces would solely be described by the electric polarizability. It is also common to assume that for dielectric particles the real part of the polarizability is much larger than its imaginary part. As a matter of fact, this is behind the development of optical tweezers, in which gradient forces (that are proportional to ( ) *<sup>e</sup>* ), dominate over the radiation pressure or scattering force contribution (which is proportional to ( ) *<sup>e</sup>* ) (Volpe et al., 2006). However, as the size of the particle increases, and for any dielectric particle, there is a crossover from electric to magnetic response as we approach the first Mie resonance, the point at which the response is absolutely dominated by the magnetic dipole. Moreover, just at the resonance, and in absence of absorption, ( )0 *<sup>m</sup>* and <sup>3</sup> ( ) 3/( 2 ) *<sup>m</sup> k* . Then, the radiation pressure contribution of the magnetic term dominates the total force <sup>2</sup> ( ) <sup>3</sup> <sup>0</sup> /2 3/2 *<sup>i</sup> <sup>m</sup> <sup>k</sup>* **FF S e** . Namely, in resonance the radiation pressure force presents a strong peak, the maximum force being independent of both material parameters and particle radius.

## **6. Conclusion**

In this chapter we have analyzed the main aspects of one of the most interesting phenomena of light scattering by nanoparticles: the possibility to control its angular distribution (directionality). As it has been shown, a general magneto-dielectric particle, with suitable values of its relative optical constants (), could present directional effects resulting from a coherent effect between real and imaginary parts of both electric and magnetic polarizabilities. The control of this effect could improve the characteristics of many current applications which employ nanoparticles. Also, it can be the base of new potential applications related with light guidance in low dimensions, as for instance, intra- or interchip optical communications (García-Cámara; 2011b). In addition, we showed that these scattering effects also affect the radiation pressure on these small particles. Thus, the "nonusual" scattering properties discussed before will strongly affect the dynamics of particle confinement in optical traps and vortex lattices (Albaladejo et al., 2009b; Gómez-Medina et al., 2011a; Albaladejo et al., 2011) governed by both gradient and curl forces.

Finally, we have showed that small dielectric particles made of non magnetic materials present scattering properties similar to those previously reported for hypothetical magnetodielectric particles. In particular, it has been shown that submicrometer Germanium particles present these directional phenomena in light scattering in the near-infrared range. These studies could serve as a stimulus for new experiments which implement these nonconventional phenomena.

## **7. Acknowledgment**

304 Smart Nanoparticles Technology

At the first generalized Kerker's condition, the interference term of Eqs. (34-35) cancels out the magnetic contribution and we obtain *<* **F** *>* = *<* **F***e >*. At the second generalized Kerker's condition, where the backscattering is enhanced, *<* **F** *>* = 3*<* **F***e >*. Notice that at both generalized Kerker's conditions the scattering cross section is exactly the same; however, the radiation pressures differ by a factor of 3. These properties are illustrated in Figure 10, where we show the different contributions to the total time averaged force on a

The strong peak in the radiation pressure force is mainly dominated by the first "magnetic" Mie resonance. This is striking and in contrast with all previous beliefs about optical forces on dipolar dielectric particles, that assumed that these forces would solely be described by the electric polarizability. It is also common to assume that for dielectric particles the real part of the polarizability is much larger than its imaginary part. As a matter of fact, this is behind the development of optical tweezers, in which gradient forces (that are proportional

*<sup>e</sup>* ), dominate over the radiation pressure or scattering force contribution (which is proportional to ( ) *<sup>e</sup>* ) (Volpe et al., 2006). However, as the size of the particle increases, and for any dielectric particle, there is a crossover from electric to magnetic response as we approach the first Mie resonance, the point at which the response is absolutely dominated by the magnetic dipole. Moreover, just at the resonance, and in absence of absorption,

> <sup>0</sup> /2 3/2 *<sup>i</sup> <sup>m</sup>*

radiation pressure force presents a strong peak, the maximum force being independent of

In this chapter we have analyzed the main aspects of one of the most interesting phenomena of light scattering by nanoparticles: the possibility to control its angular distribution (directionality). As it has been shown, a general magneto-dielectric particle, with suitable

coherent effect between real and imaginary parts of both electric and magnetic polarizabilities. The control of this effect could improve the characteristics of many current applications which employ nanoparticles. Also, it can be the base of new potential applications related with light guidance in low dimensions, as for instance, intra- or interchip optical communications (García-Cámara; 2011b). In addition, we showed that these scattering effects also affect the radiation pressure on these small particles. Thus, the "nonusual" scattering properties discussed before will strongly affect the dynamics of particle confinement in optical traps and vortex lattices (Albaladejo et al., 2009b; Gómez-Medina et

Finally, we have showed that small dielectric particles made of non magnetic materials present scattering properties similar to those previously reported for hypothetical magnetodielectric particles. In particular, it has been shown that submicrometer Germanium particles present these directional phenomena in light scattering in the near-infrared range. These studies could serve as a stimulus for new experiments which implement these non-

al., 2011a; Albaladejo et al., 2011) governed by both gradient and curl forces.

*<sup>m</sup> k* . Then, the radiation pressure contribution of the magnetic

<sup>2</sup> ( ) <sup>3</sup>

*<sup>k</sup>* **FF S e** . Namely, in resonance the

), could present directional effects resulting from a

submicrometer Ge particle.

 *<sup>m</sup>* and <sup>3</sup> ( ) 3/( 2 ) 

values of its relative optical constants (

conventional phenomena.

term dominates the total force

 

both material parameters and particle radius.

to ( ) 

 ( )0 

**6. Conclusion** 

This work has been supported by the EU NMP3-SL-2008-214107-Nanomagma, the Spanish MICINN Consolider NanoLight (CSD2007-00046), FIS2010-21984, FIS2009-13430-C01-C02, and FIS2007-60158, as well as by the Comunidad de Madrid Microseres-CM (S2009/TIC-1476). B.G.-C. wants to express his gratitude to the University of Cantabria for his postdoctoral fellowship. Work by R.G.-M. was supported by the MICINN "Juan de la Cierva" Fellowship.

## **8. References**


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**1. Introduction**

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It has long been known that atomic radiation processes near a macroscopic body differ from those in free space substantially (Purcell, 1946). In particular, the lifetime of an excited state of an atom or a molecule near surface (Arnoldus & George, 1988a;b; Barnes, 1998; Chance et al., 1978; Drexhage et al., 1968; Ford et al., 1984; Fort & Grésillon, 2008; Garrett et al., 2004; Hellen & Axelrod, 1987; Kreiter et al., 2002; Lukosz & Kunz, 1977; Macklin et al., 1996; Milonni & Knight, 1973; Snoeks et al., 1995; Steiner et al., 2005; Yeung & Gustafson, 1996) or in the vicinity of (or inside) a nanoparticle (Chew, 1987; 1988; Das & Metiu, 1985; Dung et al., 2000; Gersten & Nitzan, 1981; Klimov, Ducloy & Letokhov, 1996; Klimov et al., 2001; Klimov, Ducloy, Letokhov & Lebedev, 1996; Ruppin, 1982) may be increased or decreased depending on specific conditions. This lifetime change is theoretically calculated in many papers. These calculations made in a variety of ways. Nevertheless all of these papers can be divided into two classes. The first class includes the papers that represent an excited atom as a three-dimensional damped oscillator (Chance et al., 1978; Chew, 1987; 1988; Das & Metiu, 1985; Hellen & Axelrod, 1987; Klimov, Ducloy & Letokhov, 1996; Klimov, Ducloy, Letokhov & Lebedev, 1996; Ruppin, 1982). The second class includes the papers that consider an excited atom by means of quantum mechanics (Agarwal, 1975a;b; Arnoldus & George, 1987; 1988a;b;

**Deexcitation Dynamics of a Degenerate** 

**Two-Level Atom near (Inside) a Body** 

*1Institute of Automation and Electrometry, Siberian Branch,* 

*Russian Academy of Sciences, Novosibirsk, 2Novosibirsk State University, Novosibirsk,* 

**14**

Gennady Nikolaev

*Russia* 

Barnes, 1998; Dung et al., 2000; Wylie & Sipe, 1984; 1985; Yeung & Gustafson, 1996).

It is shown in the papers that are in the first class that the atomic oscillator rate of damping take a different value in the case of radial and tangential orientation of the oscillating atomic electric dipole. The magnitude of the rate of damping lies between these values in the case of another atomic dipole orientation. However the atomic or molecule decay rate is measured by the fluorescence detection after light pulse excitation of the atom or molecule. So, fluorescence is two-step process, and hence, orientation of the oscillating atomic dipole in general is not the

In the second class papers the problem of the atomic dipole orientation is either no discussed explicitly or reduced to partitioning of the dipole matrix element on radial and tangential parts as in the case of the classic atomic oscillator. The ratio between these two parts is either no evaluated or assumed to be in the ratio 1:2 as in the case of free space. This approach


## **Deexcitation Dynamics of a Degenerate Two-Level Atom near (Inside) a Body**

## Gennady Nikolaev

*1Institute of Automation and Electrometry, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 2Novosibirsk State University, Novosibirsk, Russia* 

## **1. Introduction**

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on small magnetodielectric particles. *Optics Express,* Vol. 18, No. 11 (May 2010), pp.

and optical forces on submicrometer dielectric particles. *Journal of the Optical Society* 

It has long been known that atomic radiation processes near a macroscopic body differ from those in free space substantially (Purcell, 1946). In particular, the lifetime of an excited state of an atom or a molecule near surface (Arnoldus & George, 1988a;b; Barnes, 1998; Chance et al., 1978; Drexhage et al., 1968; Ford et al., 1984; Fort & Grésillon, 2008; Garrett et al., 2004; Hellen & Axelrod, 1987; Kreiter et al., 2002; Lukosz & Kunz, 1977; Macklin et al., 1996; Milonni & Knight, 1973; Snoeks et al., 1995; Steiner et al., 2005; Yeung & Gustafson, 1996) or in the vicinity of (or inside) a nanoparticle (Chew, 1987; 1988; Das & Metiu, 1985; Dung et al., 2000; Gersten & Nitzan, 1981; Klimov, Ducloy & Letokhov, 1996; Klimov et al., 2001; Klimov, Ducloy, Letokhov & Lebedev, 1996; Ruppin, 1982) may be increased or decreased depending on specific conditions. This lifetime change is theoretically calculated in many papers. These calculations made in a variety of ways. Nevertheless all of these papers can be divided into two classes. The first class includes the papers that represent an excited atom as a three-dimensional damped oscillator (Chance et al., 1978; Chew, 1987; 1988; Das & Metiu, 1985; Hellen & Axelrod, 1987; Klimov, Ducloy & Letokhov, 1996; Klimov, Ducloy, Letokhov & Lebedev, 1996; Ruppin, 1982). The second class includes the papers that consider an excited atom by means of quantum mechanics (Agarwal, 1975a;b; Arnoldus & George, 1987; 1988a;b; Barnes, 1998; Dung et al., 2000; Wylie & Sipe, 1984; 1985; Yeung & Gustafson, 1996).

It is shown in the papers that are in the first class that the atomic oscillator rate of damping take a different value in the case of radial and tangential orientation of the oscillating atomic electric dipole. The magnitude of the rate of damping lies between these values in the case of another atomic dipole orientation. However the atomic or molecule decay rate is measured by the fluorescence detection after light pulse excitation of the atom or molecule. So, fluorescence is two-step process, and hence, orientation of the oscillating atomic dipole in general is not the same as exciting light polarization.

In the second class papers the problem of the atomic dipole orientation is either no discussed explicitly or reduced to partitioning of the dipole matrix element on radial and tangential parts as in the case of the classic atomic oscillator. The ratio between these two parts is either no evaluated or assumed to be in the ratio 1:2 as in the case of free space. This approach

**2.1 Transition rate in dipole approximation vs atomic and fluctuating electric field**

Let the atom and the field be independent at the initial time moment *t*0. Therefore at that instant the quantum state of the system |*mki*�is equal to |*m*�|*ki*�, where |*m*�, |*ki*� are the initial states of the atom and field, respectively. In the first order of the perturbation theory, the amplitude *cnk* (*t*) of the transition into some state |*nk*� is proportional to the matrix element of the interaction operator *<sup>V</sup>*<sup>ˆ</sup> , �*nk*|*V*<sup>ˆ</sup> <sup>|</sup>*mki*�, where <sup>|</sup>*n*�, <sup>|</sup>*k*� are states of the atom and field at the

Deexcitation Dynamics of a Degenerate Two-Level Atom near (Inside) a Body 311

*<sup>d</sup>*(*t*) <sup>ˆ</sup> *E*(*t*), so,

*<sup>d</sup>*(*t*) and the electric field strength <sup>ˆ</sup>

*E*(−) have no contribution into the integral

��)|*m*�

��)|*ki*�, (3)

(+) *<sup>β</sup>* (*t*

<sup>|</sup>*nk* �� *nk*<sup>|</sup> <sup>=</sup> <sup>ˆ</sup>*I*. (4)

)|*mki*�, (1)

(*t*), (2)

 *E*(*t*) are

*<sup>d</sup>*(−) and <sup>ˆ</sup> *E*(−)

We will assume that both the atom and the electromagnetic field are quantized.

final time moment *<sup>t</sup>*, respectively. In the dipole approximation, *<sup>V</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>−</sup> <sup>ˆ</sup>

*t*0 *dt*� �*nk*<sup>|</sup> <sup>ˆ</sup> *d*(*t* � ) ˆ *E*(*t* �

considered in the interaction picture, i.e., without the account for the perturbation.

(*t*) + <sup>ˆ</sup> *d*(+) (*t*) <sup>ˆ</sup> *E*(−)

*<sup>E</sup>*(+) are the positive-frequency parts of the operators, whereas <sup>ˆ</sup>

*<sup>d</sup>*(−) <sup>ˆ</sup> 

are negative-frequency ones. At *t* − *t*<sup>0</sup> ≡ *T* � 1/*ω*¯ , fast oscillating (with approximately twice

The initial atomic state |*m*� has more high energy than the final atomic state |*n*� for the deexcitation process under consideration. That is why only the second term in (2) gives a nonzero contribution for this process. Hence, the probability of the atomic deexcitation is

)|*k*��*k*|*E*ˆ(−)

where we have used the equality �*r*|*A*ˆ(+)|*s*�<sup>=</sup> �*s*|*A*ˆ(−)|*r*�<sup>∗</sup> for the matrix element of an operator *<sup>A</sup>*<sup>ˆ</sup> between states <sup>|</sup>*r*� and <sup>|</sup>*s*�. We also have used the Greek letters in subscripts for the

One should sum the expression (3) over all possible states |*nk*�if we are not interested in what specific state the system under consideration has came. These states constitute the complete

*<sup>β</sup>* (*t*

*<sup>E</sup>*(+) and <sup>ˆ</sup>

*cnk* <sup>=</sup> <sup>−</sup> <sup>1</sup> *ih*¯ *t*

In the rotating-wave approximation (Allen & Eberly, 1975), we have

*d*(−) (*t*) <sup>ˆ</sup> *E*(+)

*<sup>d</sup>*(+) <sup>ˆ</sup> 

<sup>−</sup> *<sup>V</sup>*ˆ(*t*)<sup>≈</sup> <sup>ˆ</sup>

where the operators of the atomic dipole moment <sup>ˆ</sup>

*<sup>P</sup>*(*nk*|*mki*) = <sup>1</sup>

set and satisfy the completeness condition

*h*¯ 2 *t*

notation of the Descartes's components of the vector operators.

*t*0

× �*ki*|*E*ˆ(+) *<sup>α</sup>* (*<sup>t</sup>*

 *t*

*t*0 *dt*� *dt*�� ∑ *αβ* �*m*<sup>|</sup> <sup>ˆ</sup>*<sup>d</sup>* (−) *<sup>α</sup>* (*<sup>t</sup>* � )|*n*��*n*<sup>|</sup> <sup>ˆ</sup>*<sup>d</sup>*

�

∑ *nk*

**correlation functions**

where <sup>ˆ</sup>

given by

(1).

*<sup>d</sup>*(+) and <sup>ˆ</sup> 

the mean frequency *<sup>ω</sup>*¯ ) products <sup>ˆ</sup>

one cannot consider as correct because of anisotropy of the atomic surroundings. The remark about fluorescence as two-step process mentioned above refers equally to the papers.

To rigorous description of the vector nature of the atomic dipole moment it is necessary to take into account the atomic angular degrees of freedom, that is degeneracy of atomic levels. As far as we know, it was done only in the papers (Arnoldus & George, 1987; 1988a;b). In the papers the steady-state fluorescence of the atom near an axial symmetrical surface was theoretically investigated and influence of the surface was expressed in terms of electric field correlation function.

The purpose of the chapter is to present the correct description of deexcitation dynamics of a degenerate two-level atom in the vicinity of arbitrary body.

We start with a quantum mechanical expression for the atomic deexcitation probability expressed in terms of the normal correlation function of the atomic dipole moment operator and the antinormal correlation function of the electric field strength operator. Then the antinormal correlation function is expressed in terms of the field susceptibility by use of the fluctuation-dissipation theorem. The atomic dipole moment operator as well as the atomic density matrix operator is expressed in terms of irreducible tensor operators. Finally, it is shown that the atomic deexcitation rate at the instant immediately after pulse excitation is proportional to a linear combination of the products of the so-called atomic polarization moments, population and alignment, and anisotropic relaxation matrix.

To find out deexcitation dynamics, a master equation for atomic density matrix is derived from an evolution equation for the total density matrix describing both atom and field. A consistent system of linear first-order ordinary differential equations for the atomic polarization moments is obtained from the master equation. Components of the anisotropic relaxation matrix describing the consistent system are expressed in terms of the field susceptibility tensor. Symmetries of the anisotropic relaxation matrix are found. It is shown that atomic deexcitation in general is multi-exponential. The simple exponential decay of the excited energy level takes place only if its total angular momentum is less then one. Deexcitation dynamics is considered in more detail for the case when the total angular momenta of the upper and lower levels are equal to 1 and 0 respectively. It is shown that in this case deexitation dynamics also may be exponential at certain polarizations of the exciting light.

In conclusion, an intriguing issue that is why the simple model of classical oscillating dipole for description of fluorescence is in good agreement with observational evidence(Amos & Barnes, 1997; Chance et al., 1978; Drexhage et al., 1968; Fort & Grésillon, 2008; Kreiter et al., 2002; Snoeks et al., 1995; Vallée et al., 2001), is clarified.

### **2. Atomic transition rate of a degenerate two-level atom in the vicinity of a material body**

To investigate deexcitation of a degenerate two-level atom in the vicinity of a nanoparticle we consider more general problem of deexcitation of the atom in the vicinity of a material body at first.

Our approach to the problem is based on using correlation functions that appear in linear-response theory. It is about the same as used in number of works (Agarwal, 1975a; Wylie & Sipe, 1984) concerning the quantum electrodynamics and life time of a non-generate atom near an interface. It is most of all close to approach developed in (Klyshko, 2011).

2 Will-be-set-by-IN-TECH

one cannot consider as correct because of anisotropy of the atomic surroundings. The remark

To rigorous description of the vector nature of the atomic dipole moment it is necessary to take into account the atomic angular degrees of freedom, that is degeneracy of atomic levels. As far as we know, it was done only in the papers (Arnoldus & George, 1987; 1988a;b). In the papers the steady-state fluorescence of the atom near an axial symmetrical surface was theoretically investigated and influence of the surface was expressed in terms of electric field

The purpose of the chapter is to present the correct description of deexcitation dynamics of a

We start with a quantum mechanical expression for the atomic deexcitation probability expressed in terms of the normal correlation function of the atomic dipole moment operator and the antinormal correlation function of the electric field strength operator. Then the antinormal correlation function is expressed in terms of the field susceptibility by use of the fluctuation-dissipation theorem. The atomic dipole moment operator as well as the atomic density matrix operator is expressed in terms of irreducible tensor operators. Finally, it is shown that the atomic deexcitation rate at the instant immediately after pulse excitation is proportional to a linear combination of the products of the so-called atomic polarization

To find out deexcitation dynamics, a master equation for atomic density matrix is derived from an evolution equation for the total density matrix describing both atom and field. A consistent system of linear first-order ordinary differential equations for the atomic polarization moments is obtained from the master equation. Components of the anisotropic relaxation matrix describing the consistent system are expressed in terms of the field susceptibility tensor. Symmetries of the anisotropic relaxation matrix are found. It is shown that atomic deexcitation in general is multi-exponential. The simple exponential decay of the excited energy level takes place only if its total angular momentum is less then one. Deexcitation dynamics is considered in more detail for the case when the total angular momenta of the upper and lower levels are equal to 1 and 0 respectively. It is shown that in this case deexitation dynamics also may be exponential at certain polarizations of the exciting light. In conclusion, an intriguing issue that is why the simple model of classical oscillating dipole for description of fluorescence is in good agreement with observational evidence(Amos & Barnes, 1997; Chance et al., 1978; Drexhage et al., 1968; Fort & Grésillon, 2008; Kreiter et al.,

**2. Atomic transition rate of a degenerate two-level atom in the vicinity of a material**

To investigate deexcitation of a degenerate two-level atom in the vicinity of a nanoparticle we consider more general problem of deexcitation of the atom in the vicinity of a material body

Our approach to the problem is based on using correlation functions that appear in linear-response theory. It is about the same as used in number of works (Agarwal, 1975a; Wylie & Sipe, 1984) concerning the quantum electrodynamics and life time of a non-generate atom near an interface. It is most of all close to approach developed in (Klyshko, 2011).

degenerate two-level atom in the vicinity of arbitrary body.

2002; Snoeks et al., 1995; Vallée et al., 2001), is clarified.

moments, population and alignment, and anisotropic relaxation matrix.

about fluorescence as two-step process mentioned above refers equally to the papers.

correlation function.

**body**

at first.

#### **2.1 Transition rate in dipole approximation vs atomic and fluctuating electric field correlation functions**

We will assume that both the atom and the electromagnetic field are quantized.

Let the atom and the field be independent at the initial time moment *t*0. Therefore at that instant the quantum state of the system |*mki*�is equal to |*m*�|*ki*�, where |*m*�, |*ki*� are the initial states of the atom and field, respectively. In the first order of the perturbation theory, the amplitude *cnk* (*t*) of the transition into some state |*nk*� is proportional to the matrix element of the interaction operator *<sup>V</sup>*<sup>ˆ</sup> , �*nk*|*V*<sup>ˆ</sup> <sup>|</sup>*mki*�, where <sup>|</sup>*n*�, <sup>|</sup>*k*� are states of the atom and field at the final time moment *<sup>t</sup>*, respectively. In the dipole approximation, *<sup>V</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>−</sup> <sup>ˆ</sup> *<sup>d</sup>*(*t*) <sup>ˆ</sup> *E*(*t*), so,

$$\mathcal{L}\_{nk} = -\frac{1}{i\hbar} \int\_{t\_0}^{t} dt' \langle nk|\hat{\vec{d}}(t')\hat{\vec{E}}(t')|mk\_i\rangle\_{\prime} \tag{1}$$

where the operators of the atomic dipole moment <sup>ˆ</sup> *<sup>d</sup>*(*t*) and the electric field strength <sup>ˆ</sup> *E*(*t*) are considered in the interaction picture, i.e., without the account for the perturbation.

In the rotating-wave approximation (Allen & Eberly, 1975), we have

$$-\hat{V}(t) \approx \hat{\vec{d}}^{(-)}(t)\hat{\vec{E}}^{(+)}(t) + \hat{\vec{d}}^{(+)}(t)\hat{\vec{E}}^{(-)}(t),\tag{2}$$

where <sup>ˆ</sup> *<sup>d</sup>*(+) and <sup>ˆ</sup> *<sup>E</sup>*(+) are the positive-frequency parts of the operators, whereas <sup>ˆ</sup> *<sup>d</sup>*(−) and <sup>ˆ</sup> *E*(−) are negative-frequency ones. At *t* − *t*<sup>0</sup> ≡ *T* � 1/*ω*¯ , fast oscillating (with approximately twice the mean frequency *<sup>ω</sup>*¯ ) products <sup>ˆ</sup> *<sup>d</sup>*(+) <sup>ˆ</sup> *<sup>E</sup>*(+) and <sup>ˆ</sup> *<sup>d</sup>*(−) <sup>ˆ</sup> *E*(−) have no contribution into the integral (1).

The initial atomic state |*m*� has more high energy than the final atomic state |*n*� for the deexcitation process under consideration. That is why only the second term in (2) gives a nonzero contribution for this process. Hence, the probability of the atomic deexcitation is given by

$$\begin{split} P(nk|mk\_{i}) &= \frac{1}{\hbar^{2}} \int \limits\_{t\_{0}}^{t} dt' dt'' \sum\_{\alpha\beta} \langle m|\not\!\!\! \not\!\!/ (t')|n\rangle \langle n|\not\!\!\! \not\!\!/ (t'')|m\rangle \\ &\times \langle k\_{i}|\not\!\!\!\!\!/ \_{\mathfrak{u}} (t')|k\rangle \langle k|\not\!\!\!\!/ \_{\mathfrak{F}} (t'')|k\_{i}\rangle. \end{split} \tag{3}$$

where we have used the equality �*r*|*A*ˆ(+)|*s*�<sup>=</sup> �*s*|*A*ˆ(−)|*r*�<sup>∗</sup> for the matrix element of an operator *<sup>A</sup>*<sup>ˆ</sup> between states <sup>|</sup>*r*� and <sup>|</sup>*s*�. We also have used the Greek letters in subscripts for the notation of the Descartes's components of the vector operators.

One should sum the expression (3) over all possible states |*nk*�if we are not interested in what specific state the system under consideration has came. These states constitute the complete set and satisfy the completeness condition

$$\sum\_{nk} |nk\rangle\langle nk| = \hat{I}.\tag{4}$$

**2.2 Transition rate in terms of electric field susceptibility**

= ∑ *s*� ,*s*=±1

−2 ∞

−∞

*gαβ* (*t*, *t* + *τ*) ≡

) *<sup>α</sup>* (*t*) and *<sup>E</sup>*ˆ(*s*)

by definition, where *θ* (*ω*) is step function.

correlation function {*g*}*αβ* (*τ*) defined by

2 *E*ˆ*α*(0)*E*ˆ

{*g*}*αβ* (*τ*) <sup>≡</sup> <sup>1</sup>

 *E*ˆ(*s*� ) *<sup>α</sup>* (*ω*�

(−)

*s*� = −*s*. Hence, in (14) only two terms are nonzero, and we have

*gαβ* (*t*, *t* + *τ*) ≡ (2*π*)

proportional to Dirac function:

anti-normally ordered one *g*

Expressing *E*ˆ(*s*�

Note, that

correlation function of the electric field strength may be written as

∞

−∞

*<sup>E</sup>*ˆ(+) *<sup>α</sup>* (*t*) + *<sup>E</sup>*ˆ(−) *<sup>α</sup>* (*t*)

*dω*� *dωe*

*E*ˆ(*s*)

)*E*ˆ(*s*) *<sup>β</sup>* (*ω*) 

where spectral density of the normally ordered correlation function *g*

*gαβ* (*τ*) = *g*

Note that from (14), (16), and (17) it is follows that relationship between *g*

*g* (±)

There is a simple Kubo-Martin-Schwinger's boundary condition

*<sup>β</sup>*(*τ*) + *E*ˆ

 *E*ˆ(*s*� ) *<sup>α</sup>* (*t*)*E*ˆ(*s*)

It is known that total correlation function is represented as a sum of normally and anti-normally ordered correlation function in the case of stationary process. Indeed, the total

Deexcitation Dynamics of a Degenerate Two-Level Atom near (Inside) a Body 313

*<sup>E</sup>*ˆ(+)

<sup>+</sup>*ω*)*<sup>t</sup>* ∑ *s*� ,*s*=±1

> *ω*� + *ω*

*αβ* (*ω*) are introduced respectively. In turn, (16) and (15) imply

*αβ* (*ω*) = *θ* (±*ω*) *gαβ* (*ω*) (18)

*gβα* (−*τ*) = *gαβ* (*τ* + *ih*¯ *ξ*), (20)

*gαβ* (*τ*) + *gβα* (−*τ*)

(−)

*<sup>β</sup>* (*t* + *τ*)

*<sup>β</sup>* (*t* + *τ*) in terms of Fourier transforms, we obtain

−*i*(*ω*�

<sup>−</sup>*iωτe*

It is clear that (14) is independent on *t* only when expression in the angle brackets is

≡ 2*πg* (*s*) *αβ* (*ω*) *δ*

(+) *αβ* (*τ*) + *g*

is given by the ordinary formula (12). It is clear also that ordered correlation functions *g*

are expressed in terms of the ordinary correlation function *gαβ* (*ω*) similar to relation (15):

At thermal equilibrium the correlation function *gαβ* (*τ*) is simply related with symmetrized

*<sup>β</sup>*(*τ*)*E*ˆ*α*(0)

 <sup>=</sup> <sup>1</sup> 2 

*<sup>β</sup>* (*<sup>t</sup>* <sup>+</sup> *<sup>τ</sup>*) + *<sup>E</sup>*ˆ(−)

 *E*ˆ(*s*� ) *<sup>α</sup>* (*ω*�

*<sup>α</sup>* (*ω*) <sup>≡</sup> *<sup>θ</sup>* (*sω*) *<sup>E</sup>*ˆ*α*(*ω*) (15)

*<sup>β</sup>* (*t* + *τ*)

)*E*ˆ(*s*) *<sup>β</sup>* (*ω*) 

, (16)

(+)

*αβ* (*τ*). (17)

(±)

*αβ* (*ω*) and *g*

*αβ* (*ω*) and

(±) *αβ* (*τ*)

(±) *αβ* (*ω*)

. (19)

(13)

. (14)

Thus we can represent the total probability of the atomic deexcitation in the following way

$$P = \hbar^{-2} \int\_{t\_0}^{t} \int\_{t\_0}^{t} dt' dt'' \sum\_{a\notin\mathcal{S}} f\_{a\notin\mathcal{S}}^{(+)}\left(t', t''\right) g\_{a\notin}^{(-)}\left(t', t''\right),\tag{5}$$

where

$$\begin{aligned} \left| f\_{a\hat{\boldsymbol{\beta}}}^{(+)} \left( t', t'' \right) \equiv \left< \hat{d}\_{a}^{(-)} (t') \hat{d}\_{\hat{\boldsymbol{\beta}}}^{(+)} (t'') \right>, \\\\ \left| g\_{a\hat{\boldsymbol{\beta}}}^{(-)} \left( t', t'' \right) \equiv \left< \hat{E}\_{a}^{(+)} (t') \hat{E}\_{\hat{\boldsymbol{\beta}}}^{(-)} (t'') \right> \end{aligned} \tag{6}$$

are normally and anti-normally ordered correlation function (CF) of the atomic dipole moment and the electric field strength in an initial state, respectively. The initial state may be pure as well as mixed, of course.

We suppose that initial unperturbed states of both interacting systems are stationary. In this case correlation functions (6) depend only on the difference of their arguments:

$$f\_{a\circledast}^{(\pm)}(\tau) \equiv f\_{a\circledast}^{(\pm)}(t, t + \tau) = \left\langle \vec{d}\_a^{(\mp)}(0)\vec{d}\_\beta^{(\pm)}(\tau) \right\rangle = \left(f\_{\beta a}^{(\pm)}(-\tau)\right)^\*,\tag{7}$$

$$\left(\mathcal{g}\_{a\mathcal{\mathcal{B}}}^{(\pm)}(\tau) \equiv \mathcal{g}\_{a\mathcal{\mathcal{B}}}^{(\pm)}(t, t + \tau) = \left\langle \hat{\mathcal{E}}\_{a}^{(\mp)}(0)\hat{\mathcal{E}}\_{\mathcal{\mathcal{B}}}^{(\pm)}(\tau) \right\rangle = \left(\mathcal{g}\_{\hat{\mathcal{B}}a}^{(\pm)}(-\tau)\right)^{\*}.\tag{8}$$

Hence, the total probability of the atomic deexcitation (5) becomes

$$P = \hbar^{-2} \int\_0^T d\tau \left(T - \tau\right) \sum\_{a\beta} \left[ f\_{a\beta}^{(+)}\left(\tau\right) g\_{a\beta}^{(-)}\left(\tau\right) + \left(\tau \to -\tau\right) \right],\tag{9}$$

where *T* ≡ *t* − *t*<sup>0</sup> is observation time. When it is much more then the atomic and field correlation time, the total probability of the atomic deexcitation (9) becomes proportional to *T*. So, atomic transition rate *W* ≡ *P*/*T* independent on time one may introduce

$$\mathcal{W} = \hbar^{-2} \int\_{-\infty}^{\infty} d\tau \sum\_{a\beta} f\_{a\beta}^{(+)}\left(\tau\right) g\_{a\beta}^{(-)}\left(\tau\right) \,. \tag{10}$$

where limits of integration ±*T* are extended to ±∞. It is convenient rewrite (10) in terms of the Fourier components of the correlation functions in the following way

$$\mathcal{W} = \left(1/2\pi\hbar^2\right) \int\_{-\infty}^{\infty} d\omega \sum\_{a\notin\mathcal{S}} f\_{a\notin}^{(+)}\left(\omega\right) g\_{a\notin}^{(-)}\left(-\omega\right),\tag{11}$$

where the Fourier transform *A* (*ω*) of a function *A* (*τ*) is defined by

$$A\left(\omega\right) = \int\_{-\infty}^{\infty} d\tau \, e^{i\omega\tau} A\left(\tau\right). \tag{12}$$

#### **2.2 Transition rate in terms of electric field susceptibility**

It is known that total correlation function is represented as a sum of normally and anti-normally ordered correlation function in the case of stationary process. Indeed, the total correlation function of the electric field strength may be written as

$$\begin{split} g\_{\mathfrak{a}\mathcal{\mathcal{A}}}(t, t + \tau) &\equiv \left\langle \left[ \hat{\mathcal{E}}\_{\mathfrak{a}}^{(+)}(t) + \hat{\mathcal{E}}\_{\mathfrak{a}}^{(-)}(t) \right] \left[ \hat{\mathcal{E}}\_{\mathfrak{\mathcal{P}}}^{(+)}(t + \tau) + \hat{\mathcal{E}}\_{\mathfrak{\mathcal{P}}}^{(-)}(t + \tau) \right] \right\rangle \\ &= \sum\_{s', s = \pm 1} \left\langle \hat{\mathcal{E}}\_{\mathfrak{a}}^{(s')}(t) \hat{\mathcal{E}}\_{\mathfrak{\mathcal{P}}}^{(s)}(t + \tau) \right\rangle \end{split} \tag{13}$$

Expressing *E*ˆ(*s*� ) *<sup>α</sup>* (*t*) and *<sup>E</sup>*ˆ(*s*) *<sup>β</sup>* (*t* + *τ*) in terms of Fourier transforms, we obtain

$$\mathcal{g}\_{a\not\!\!\!/}(t, t + \tau) \equiv (2\pi)^{-2} \int\_{-\infty}^{\infty} \int d\omega' \, d\omega e^{-i\omega\tau} e^{-i(\omega' + \omega)t} \sum\_{s', s = \pm 1} \left< \triangle\_a^{(s')}(\omega') \triangle\_{\beta}^{(s)}(\omega) \right> . \tag{14}$$

Note, that

4 Will-be-set-by-IN-TECH

are normally and anti-normally ordered correlation function (CF) of the atomic dipole moment and the electric field strength in an initial state, respectively. The initial state may

We suppose that initial unperturbed states of both interacting systems are stationary. In this

(∓) *<sup>α</sup>* (0) <sup>ˆ</sup>*<sup>d</sup>*

*<sup>E</sup>*ˆ(∓) *<sup>α</sup>* (0)*E*ˆ(±)

(±) *<sup>β</sup>* (*τ*) = *f* (±) *βα* (−*τ*)

*<sup>β</sup>* (*τ*) = *g* (±) *βα* (−*τ*)

(−)

(−)

(−)

*αβ* (*τ*) + (*τ* → −*τ*)

ˆ*d*

*αβ f* (+) *αβ* (*τ*) *g*

*T*. So, atomic transition rate *W* ≡ *P*/*T* independent on time one may introduce

∞

*dτ* ∑ *αβ f* (+) *αβ* (*τ*) *g*

where limits of integration ±*T* are extended to ±∞. It is convenient rewrite (10) in terms of

*dω*∑ *αβ f* (+) *αβ* (*ω*) *g*

∞

*dτ e*

−∞

−∞

<sup>∞</sup>

−∞

*A* (*ω*) =

the Fourier components of the correlation functions in the following way

where the Fourier transform *A* (*ω*) of a function *A* (*τ*) is defined by

where *T* ≡ *t* − *t*<sup>0</sup> is observation time. When it is much more then the atomic and field correlation time, the total probability of the atomic deexcitation (9) becomes proportional to

case correlation functions (6) depend only on the difference of their arguments:

, (5)

∗

∗

*αβ* (*τ*), (10)

*αβ* (−*ω*), (11)

*<sup>i</sup>ωτ A* (*τ*). (12)

, (7)

. (8)

, (9)

(6)

Thus we can represent the total probability of the atomic deexcitation in the following way

*P* = *h*¯ <sup>−</sup><sup>2</sup>

where

 *t*  *t*

*t*0 *dt*� *dt*�� ∑ *αβ f* (+) *αβ t* � , *t* �� *g* (−) *αβ t* � , *t* ��

*t*0

*f* (+) *αβ t* � , *t* �� ≡ ˆ*d* (−) *<sup>α</sup>* (*<sup>t</sup>* � ) ˆ*d* (+) *<sup>β</sup>* (*t* ��) ,

*g* (−) *αβ t* � , *t* �� ≡ *<sup>E</sup>*ˆ(+) *<sup>α</sup>* (*<sup>t</sup>* � )*E*ˆ(−) *<sup>β</sup>* (*t* ��) 

(±)

(±)

 *T*

0

*W* = 1/2*πh*¯ <sup>2</sup>

*P* = *h*¯ <sup>−</sup><sup>2</sup>

*αβ* (*t*, *t* + *τ*) =

*αβ* (*t*, *t* + *τ*) =

Hence, the total probability of the atomic deexcitation (5) becomes

*W* = *h*¯ <sup>−</sup><sup>2</sup>

*<sup>d</sup><sup>τ</sup>* (*<sup>T</sup>* − *<sup>τ</sup>*)∑

be pure as well as mixed, of course.

*f* (±) *αβ* (*τ*) ≡*f*

*g* (±) *αβ* (*τ*) ≡*g*

$$
\hat{E}\_{\mathfrak{a}}^{(s)}(\omega) \equiv \theta \left( s\omega \right) \hat{E}\_{\mathfrak{a}}(\omega) \tag{15}
$$

by definition, where *θ* (*ω*) is step function.

It is clear that (14) is independent on *t* only when expression in the angle brackets is proportional to Dirac function:

$$\left\langle \hat{E}\_{\mathfrak{a}}^{(s')}(\omega') \hat{E}\_{\mathfrak{f}}^{(s)}(\omega) \right\rangle \equiv 2\pi \mathfrak{g}\_{a\mathfrak{f}}^{(s)}(\omega) \,\delta\left(\omega' + \omega\right),\tag{16}$$

where spectral density of the normally ordered correlation function *g* (+) *αβ* (*ω*) and anti-normally ordered one *g* (−) *αβ* (*ω*) are introduced respectively. In turn, (16) and (15) imply *s*� = −*s*. Hence, in (14) only two terms are nonzero, and we have

$$\mathbf{g}\_{a\mathcal{B}}\left(\boldsymbol{\tau}\right) = \mathbf{g}\_{a\mathcal{B}}^{(+)}\left(\boldsymbol{\tau}\right) + \mathbf{g}\_{a\mathcal{B}}^{(-)}\left(\boldsymbol{\tau}\right). \tag{17}$$

Note that from (14), (16), and (17) it is follows that relationship between *g* (±) *αβ* (*ω*) and *g* (±) *αβ* (*τ*) is given by the ordinary formula (12). It is clear also that ordered correlation functions *g* (±) *αβ* (*ω*) are expressed in terms of the ordinary correlation function *gαβ* (*ω*) similar to relation (15):

$$\mathcal{g}^{(\pm)}\_{\mathfrak{a}\mathfrak{F}}\left(\omega\right) = \theta\left(\pm\omega\right)\mathcal{g}\_{\mathfrak{a}\mathfrak{F}}\left(\omega\right) \tag{18}$$

At thermal equilibrium the correlation function *gαβ* (*τ*) is simply related with symmetrized correlation function {*g*}*αβ* (*τ*) defined by

$$\{\mathbf{g}\}\_{\mathfrak{a}\mathfrak{F}}(\tau) \equiv \frac{1}{2} \left\langle \mathbf{\hat{E}}\_{\mathfrak{a}}(0)\mathbf{\hat{E}}\_{\mathfrak{F}}(\tau) + \mathbf{\hat{E}}\_{\mathfrak{F}}(\tau)\mathbf{\hat{E}}\_{\mathfrak{a}}(0) \right\rangle = \frac{1}{2} \left\{ \mathbf{g}\_{\mathfrak{a}\mathfrak{F}}(\tau) + \mathbf{g}\_{\mathfrak{f}\mathfrak{a}}(-\tau) \right\}.\tag{19}$$

There is a simple Kubo-Martin-Schwinger's boundary condition

$$\mathbf{g}\_{\mathfrak{R}^d}(-\tau) = \mathbf{g}\_{a\mathfrak{R}}\left(\tau + i\hbar\mathfrak{F}\right),\tag{20}$$

When there is no external magnetic field, tensor *Gαβ* (*r*,*r*�

*αβ* (−*ω*) = *h*¯ *θ* (*ω*)

∞

*dω*∑ *αβ f* (+) *αβ* (*ω*)

0

**2.3 Transition rate of a degenerate two-level atom**

*g* (−)

*W* = (1/2*πh*¯)

where *r*<sup>0</sup> is radius vector of the atom.

The explicit form of the atomic CF *f*

et al., 1988):

*T*ˆ*K*

argument *τ* and, hence in sign of *ω*.

*<sup>Q</sup>*(*Jm Jn*) = ∑

Substituting (28) in (11) we find

imaginary part is odd in *ω*. In this case (27) goes over into (Agarwal, 1975a)<sup>1</sup>

(+)

*v*<sup>0</sup> = *vz*, *<sup>v</sup>*±<sup>1</sup> <sup>=</sup> <sup>∓</sup>

Wigner-Eckart theorem in terms of the so-called unit irreducible tensor operators *T*ˆ *<sup>K</sup>*

√3 *T*ˆ 1

> (−1) *σ* ˆ *d* (+) <sup>−</sup>*<sup>σ</sup>* (*t*)

momentum of the level *j* and its projection on the *z*-axis , respectively.

with the Descartes's one *vi* as follows (Varshalovich et al., 1988):

ˆ*d*

ˆ *d* (−) *<sup>σ</sup>* (*t*) =

*Mm*,*Mn*

(+) *<sup>σ</sup>* (*t*) = *dnm*

(−1)

*Jn*, and *K* of the coefficient obey triangle unequality, so |*Jm* − *Jn*| *K Jm* + *Jn*.

<sup>1</sup> <sup>+</sup> coth *<sup>h</sup>*¯ *ωξ*

Deexcitation Dynamics of a Degenerate Two-Level Atom near (Inside) a Body 315

We are interesting in only local field response because of point atom approximation used.

consider a degenerate two-level atom. Its energy levels are degenerate on the total angular momentum projection on any axis. Suppose the excited upper energy level *m* and lower one *n* have quantum numbers *JmMm* and *JnMn* respectively, where *Jj* and *Mj* label the total angular

It is convenient describe vector or tensor values in terms of the circular components instead of the Descartes's one. The circular components *v<sup>σ</sup>* of a vector *v*, where *σ* = 0,±1, are related

> *vx* ± *vy* / √

The circular components of the atomic dipole operator can be expressed according to the

the following way (Biedenharn & Louck, 1981; Blum, 1996; Fano & Racah, 1959; Varshalovich

where *dmn* and *ωmn* are reduced matrix element of the atomic dipole moment and resonant frequency of the atomic transition, respectively. The irreducible tensor operator *T*ˆ *<sup>K</sup>*

where *K* and *Q* are its rank and component (−*K Q K*) correspondingly, is defined as (Biedenharn & Louck, 1981; Blum, 1996; Fano & Racah, 1959; Varshalovich et al., 1988)

where �*JmMm Jn* − *Mn*|*KQ*� is the vector coupling (Clebsch-Gordan) coefficient. Quantities *Jm*,

<sup>1</sup> Definition of the ordered correlation functions in this paper differs from ours one by sign of the

*<sup>σ</sup>*(*Jn Jm*) exp(−*iωmnt*),

†

2

<sup>1</sup> <sup>+</sup> coth *<sup>h</sup>*¯ *ωξ*

 � *<sup>G</sup>αβ r*� ,*r*; *ω* 

2

 � 

*αβ* (*ω*) depends on the atomic model used. Here we

; *ω*) is symmetrical one, and its

*Gαβ* (*r*0,*r*0; *ω*)

2. (30)

, (31)

*Jn*−*Mn* �*JmMm Jn* <sup>−</sup> *Mn*|*KQ*�|*JmMm*��*JnMn*|, (32)

, (28)

, (29)

*<sup>Q</sup>*(*Jm Jn*) in

*<sup>Q</sup>*(*Jm Jn*),

where *ξ* ≡ 1/ (*kT*), *k* and *T* are Boltzmann's constant and temperature respectively. It is easily proofed by using the invariance of the trace under a cyclic permutation of the operators:

$$g\_{\beta\alpha}\left(-\tau\right) = \left\langle \hat{\mathbf{E}}\_{\beta}(\tau)\hat{\mathbf{E}}\_{a}(0) \right\rangle \equiv \text{tr}\left\{ \beta\_{0}e^{i\hat{H}\tau/\hbar}\hat{\mathbf{E}}\_{\beta}e^{-i\hat{H}\tau/\hbar}\hat{\mathbf{E}}\_{a} \right\}$$

$$= Z^{-1}\text{tr}\left\{ e^{-\xi\hat{H}}e^{i\hat{H}\tau/\hbar}\hat{\mathbf{E}}\_{\beta}e^{-i\hat{H}\tau/\hbar}\hat{\mathbf{E}}\_{a} \right\} \tag{21}$$

$$= Z^{-1}\text{tr}\left\{ \hat{\mathbf{E}}\_{a}e^{i(i\hat{\mathbf{s}}+\tau/\hbar)\hat{H}}\hat{\mathbf{E}}\_{\beta}e^{-i(i\hat{\mathbf{s}}+\tau/\hbar)\hat{H}}e^{-\xi\hat{H}} \right\} = g\_{a\beta}\left(\tau+i\hbar\xi\right),$$

where *<sup>ρ</sup>*ˆ0 <sup>=</sup> *<sup>Z</sup>*−1*e*−*ξH*<sup>ˆ</sup> is the thermal equilibrium density operator, *<sup>Z</sup>* <sup>=</sup> tr *e*−*ξH*<sup>ˆ</sup> , and *H*ˆ is unperturbed Hamiltonian of the system.

Using (20), we rewrite relation (19) as follows

$$\{g\}\_{a\not\!\!\!\!\/}(\tau) = \frac{1}{2} \left\{ g\_{a\not\!\!\/\!\!\/}(\tau) + g\_{a\not\!\!\/\!\/)}(\tau + i\hbar \tilde{\xi}) \right\}.\tag{22}$$

In turn, taking the Fourier transform, we obtain

$$\left\{ \left\{ g \right\} \_{a\notin} \left( \omega \right) = \frac{1}{2} \left\{ 1 + e^{\hbar \omega \xi} \right\} g\_{a\not\!\not\!\!/} \left( \omega \right) . \tag{23}$$

The Fourier transform of symmetrized correlation function {*g*}*αβ* (*r*,*r*� ; *ω*) is related with dynamical value *Gαβ* (*r*,*r*� ; *ω*), the Fourier transform of the electric field susceptibility *Gαβ* (*r*,*r*� ; *τ*), by the fluctuation-dissipation theorem as follows (Bernard & Callen, 1959; Callen et al., 1952; Callen & Welton, 1951; Landau & Lifshitz, 1980)

$$\{\mathbf{g}\}\_{a\not\rhd}\left(\vec{r},\vec{r}';\omega\right) = \frac{1}{2}i\hbar\left[\mathbf{G}\_{\not\not\aslearrow}^{\*}\left(\vec{r}',\vec{r};\omega\right) - \mathbf{G}\_{a\not\rhd}\left(\vec{r},\vec{r}';\omega\right)\right]\coth\left(\frac{\hbar\omega\zeta}{2}\right),\tag{24}$$

where tensor *Gαβ* (*r*,*r*� ; *<sup>ω</sup>*) relates Fourier transforms of the electric dipole <sup>ˆ</sup>*dβ*(*<sup>r</sup>*� ; *ω*) and induced electric field *E*ˆ*α*(*r*; *ω*) as follows

$$
\hat{E}\_{\mathfrak{A}}(\vec{r};\omega) = \sum\_{\beta} \mathcal{G}\_{\mathfrak{A}\beta} \left( \vec{r}, \vec{r}'; \omega \right) \hat{d}\_{\beta}(\vec{r}'; \omega), \tag{25}
$$

and the electric field susceptibility tensor *Gαβ* (*r*,*r*� ; *τ*) is defined by

$$G\_{\mathfrak{a}\mathfrak{f}}\left(\vec{r},\vec{r}';\tau\right)\equiv\frac{i}{\hbar}\theta\left(\tau\right)\left\langle \left[\hat{E}\_{\mathfrak{a}}(\tau),\hat{E}\_{\mathfrak{f}}(0)\right] \right\rangle. \tag{26}$$

Note that the same tensor *Gαβ* (*r*,*r*� ; *ω*) relates classical, not quantum, values *Eα*(*r*; *ω*) and *dβ*(*r*� ; *ω*) by the same way (25). So it can be found from the solution of the classical electrodynamic problem in the same condition.

Using (18), (23), and (24), we obtain

$$g\_{a\not\!\!\!\!\/}^{(-)}\left(-\omega\right) = i\hbar\theta\left(\omega\right)\frac{1}{2}\left[1 + \coth\left(\frac{\hbar\omega\zeta}{2}\right)\right]\left[\mathcal{G}\_{\not\!\!\!\!a}^{\*}\left(\vec{r}',\vec{r};\omega\right) - \mathcal{G}\_{a\not\!\!\!\!\/)}\left(\vec{r},\vec{r}';\omega\right)\right],\tag{27}$$

When there is no external magnetic field, tensor *Gαβ* (*r*,*r*� ; *ω*) is symmetrical one, and its imaginary part is odd in *ω*. In this case (27) goes over into (Agarwal, 1975a)<sup>1</sup>

$$\mathcal{G}\_{a\beta}^{(-)}\left(-\omega\right) = \hbar\theta\left(\omega\right)\left[1 + \coth\left(\frac{\hbar\omega\xi}{2}\right)\right] \otimes \left[\mathcal{G}\_{a\beta}\left(\vec{r}', \vec{r}; \omega\right)\right],\tag{28}$$

We are interesting in only local field response because of point atom approximation used. Substituting (28) in (11) we find

$$\mathcal{W} = \left(1/2\pi\hbar\right) \int\_0^\infty d\omega \sum\_{a\notin\mathcal{S}} f\_{a\notin}^{(+)}\left(\omega\right) \left[1 + \coth\left(\frac{\hbar\omega\xi}{2}\right)\right] \otimes \left[G\_{a\notin}\left(\vec{r\_0}, \vec{r\_0}; \omega\right)\right],\tag{29}$$

where *r*<sup>0</sup> is radius vector of the atom.

6 Will-be-set-by-IN-TECH

where *ξ* ≡ 1/ (*kT*), *k* and *T* are Boltzmann's constant and temperature respectively. It is easily proofed by using the invariance of the trace under a cyclic permutation of the operators:

> *iH*<sup>ˆ</sup> *<sup>τ</sup>*/¯*hE*<sup>ˆ</sup> *βe*

<sup>−</sup>*iH*<sup>ˆ</sup> *<sup>τ</sup>*/¯*hE*ˆ*<sup>α</sup>*

<sup>−</sup>*i*(*iξ*+*τ*/¯*h*)*H*<sup>ˆ</sup>

*gαβ* (*τ*) + *gαβ* (*τ* + *ih*¯ *ξ*)

<sup>1</sup> <sup>+</sup> *<sup>e</sup>h*¯ *ωξ*

; *τ*), by the fluctuation-dissipation theorem as follows (Bernard & Callen, 1959; Callen

; *<sup>ω</sup>*) relates Fourier transforms of the electric dipole <sup>ˆ</sup>*dβ*(*<sup>r</sup>*�

; *τ*) is defined by

*<sup>β</sup>*(0) 

; *ω*) relates classical, not quantum, values *Eα*(*r*; *ω*) and

*E*ˆ*α*(*τ*), *E*ˆ

; *ω*) by the same way (25). So it can be found from the solution of the classical

 *G*∗ *βα r*� ,*r*; *ω* − *Gαβ r*,*r*� ; *ω* 

 *h*¯ *ωξ* 2

<sup>−</sup>*iH*<sup>ˆ</sup> *<sup>τ</sup>*/¯*hE*ˆ*<sup>α</sup>*

*e* <sup>−</sup>*ξH*<sup>ˆ</sup> 

; *ω*), the Fourier transform of the electric field susceptibility

= *gαβ* (*τ* + *ih*¯ *ξ*),

 *e*−*ξH*<sup>ˆ</sup> 

*gαβ* (*ω*). (23)

 *h*¯ *ωξ* 2

; *ω*), (25)

. (26)

. (22)

; *ω*) is related with

, (24)

; *ω*) and

, (27)

(21)

, and *H*ˆ is

*gβα* (−*τ*) =

 *E*ˆ

= *Z*<sup>−</sup>1tr

= *Z*<sup>−</sup>1tr

unperturbed Hamiltonian of the system.

dynamical value *Gαβ* (*r*,*r*�

where tensor *Gαβ* (*r*,*r*�

{*g*}*αβ r*,*r*� ; *ω* <sup>=</sup> <sup>1</sup> 2 *ih*¯ *G*∗ *βα r*� ,*r*; *ω* − *Gαβ r*,*r*� ; *ω* coth

induced electric field *E*ˆ*α*(*r*; *ω*) as follows

Note that the same tensor *Gαβ* (*r*,*r*�

Using (18), (23), and (24), we obtain

*g* (−)

and the electric field susceptibility tensor *Gαβ* (*r*,*r*�

electrodynamic problem in the same condition.

*αβ* (−*ω*) <sup>=</sup> *ih*¯ *<sup>θ</sup>* (*ω*) <sup>1</sup>

*Gαβ r*,*r*� ; *τ* ≡ *i h*¯ *θ* (*τ*) 

> 2

1 + coth

*Gαβ* (*r*,*r*�

*dβ*(*r*�

Using (20), we rewrite relation (19) as follows

In turn, taking the Fourier transform, we obtain

*<sup>β</sup>*(*τ*)*E*ˆ*α*(0)

 *e* <sup>−</sup>*ξH*<sup>ˆ</sup> *e iH*<sup>ˆ</sup> *<sup>τ</sup>*/¯*hE*<sup>ˆ</sup> *βe*

 *E*ˆ*αe*

{*g*}*αβ* (*τ*) <sup>=</sup> <sup>1</sup>

 ≡ tr *ρ*ˆ0*e*

*i*(*iξ*+*τ*/¯*h*)*H*ˆ

2 

> 2

{*g*}*αβ* (*ω*) <sup>=</sup> <sup>1</sup>

The Fourier transform of symmetrized correlation function {*g*}*αβ* (*r*,*r*�

*<sup>E</sup>*ˆ*α*(*<sup>r</sup>*; *<sup>ω</sup>*) = ∑

*β Gαβ r*,*r*� ; *ω* <sup>ˆ</sup>*dβ*(*<sup>r</sup>*�

et al., 1952; Callen & Welton, 1951; Landau & Lifshitz, 1980)

where *<sup>ρ</sup>*ˆ0 <sup>=</sup> *<sup>Z</sup>*−1*e*−*ξH*<sup>ˆ</sup> is the thermal equilibrium density operator, *<sup>Z</sup>* <sup>=</sup> tr

*E*ˆ *βe*

#### **2.3 Transition rate of a degenerate two-level atom**

The explicit form of the atomic CF *f* (+) *αβ* (*ω*) depends on the atomic model used. Here we consider a degenerate two-level atom. Its energy levels are degenerate on the total angular momentum projection on any axis. Suppose the excited upper energy level *m* and lower one *n* have quantum numbers *JmMm* and *JnMn* respectively, where *Jj* and *Mj* label the total angular momentum of the level *j* and its projection on the *z*-axis , respectively.

It is convenient describe vector or tensor values in terms of the circular components instead of the Descartes's one. The circular components *v<sup>σ</sup>* of a vector *v*, where *σ* = 0,±1, are related with the Descartes's one *vi* as follows (Varshalovich et al., 1988):

$$v\_0 = v\_{z, \
u}$$

$$v\_{\pm 1} = \mp \left(v\_x \pm v\_y\right) / \sqrt{2}.\tag{30}$$

The circular components of the atomic dipole operator can be expressed according to the Wigner-Eckart theorem in terms of the so-called unit irreducible tensor operators *T*ˆ *<sup>K</sup> <sup>Q</sup>*(*Jm Jn*) in the following way (Biedenharn & Louck, 1981; Blum, 1996; Fano & Racah, 1959; Varshalovich et al., 1988):

$$
\vec{d}\_{\sigma}^{(+)}(t) = \frac{d\_{nm}}{\sqrt{3}} \hat{T}\_{\sigma}^{1}(f\_{n}I\_{m}) \exp(-i\omega\_{mn}t) \,, $$

$$
\vec{d}\_{\sigma}^{(-)}(t) = \left\{(-1)^{\sigma} \hat{\vec{d}}\_{-\sigma}^{(+)}(t)\right\}^{\dagger} \, , \tag{31}
$$

where *dmn* and *ωmn* are reduced matrix element of the atomic dipole moment and resonant frequency of the atomic transition, respectively. The irreducible tensor operator *T*ˆ *<sup>K</sup> <sup>Q</sup>*(*Jm Jn*), where *K* and *Q* are its rank and component (−*K Q K*) correspondingly, is defined as (Biedenharn & Louck, 1981; Blum, 1996; Fano & Racah, 1959; Varshalovich et al., 1988)

$$\hat{T}\_{\mathcal{Q}}^{\mathcal{K}}(f\_{\mathfrak{m}}f\_{\mathfrak{n}}) = \sum\_{M\_{\mathfrak{m}},M\_{\mathfrak{n}}} (-1)^{J\_{\mathfrak{n}}-M\_{\mathfrak{n}}} \left< f\_{\mathfrak{m}}M\_{\mathfrak{m}}f\_{\mathfrak{n}} - M\_{\mathfrak{n}} |K\mathcal{Q}\rangle \left| f\_{\mathfrak{m}}M\_{\mathfrak{m}}\right> \left< f\_{\mathfrak{n}}M\_{\mathfrak{m}}\right|.\tag{32}$$

where �*JmMm Jn* − *Mn*|*KQ*� is the vector coupling (Clebsch-Gordan) coefficient. Quantities *Jm*, *Jn*, and *K* of the coefficient obey triangle unequality, so |*Jm* − *Jn*| *K Jm* + *Jn*.

<sup>1</sup> Definition of the ordered correlation functions in this paper differs from ours one by sign of the argument *τ* and, hence in sign of *ω*.

where

*G<sup>K</sup>*

*Jn*.

*G*0

tensor **G** are not zero:

and [**G**��(*r*�0,�*r*0; *ωmn*)]

*γK <sup>Q</sup>* ≡ 2

*K*

*G<sup>K</sup>*


with circular components *Gσσ*�(*r*�0,�*r*0; *ωmn*) as follows

is irreducible relaxation tensor of the multipole *ρKQ* (*Jm Jm*),

susceptibility in the *<sup>ω</sup>*−representation. Irreducible spherical tensor *<sup>G</sup><sup>K</sup>*

It is follows from properties of the Clebsch-Gordan coefficient �1*σ*1*σ*�

*<sup>Q</sup>*(*r*�0,�*r*0; *<sup>ω</sup>mn*) ≡ ∑

*<sup>Q</sup>*(*r*�0,�*r*0; *ωmn*). Let us consider these factors in more detail.

of the level. So, deexcitation is governed only by *γ*<sup>0</sup>

*G*0 <sup>0</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup> √3

*G*2 <sup>0</sup> =

*G*2 <sup>±</sup><sup>2</sup> <sup>=</sup> <sup>1</sup>

As is seen from (44), components *G*<sup>2</sup>

boundary or near a spherical particle.

*Wis* <sup>=</sup> <sup>2</sup> 3 2 3  *Jm*+*Jn*

*σσ*�

�1*σ*1*σ*�

Furthermore, symmetry of the tensor *Gσσ*�(*r*�0,�*r*0; *ωmn*) under the interchange *σ σ*� requires that *K* have to be even, so *K* = 0, 2. In other words, deexcitation rate depends on the total population of excited level (*K* = 0) and its alignment (*K* = 2). Their relative contribution depends according to (39) and (40) on quantum numbers of combining levels *m* and *n*, on the excitation type determining the value of *ρKQ* (*Jm Jm*), and on the atom surroundings by

As was noted after (38), *K* is in the range of values defining by 0 *K* 2*Jm*. Consequently, if the total momentum *Jm* of the the excited level is equal to 0, or 1/2, there is no alignment

In the case of *Jm* > 1/2, the ratio of two deexcitation rates corresponding to some two fixed excitation types, differing in initial values of *ρKQ* (*Jm Jm*), is not universal but depends on *Jm*,

One can diagonalize symmetrical tensor *Gαβ*(*r*�0,�*r*0; *ωmn*). Let us label its principal axes of coordinate by *X*, *Y*, *Z*. In this proper basis only the following irreducible components of the

√3

(symmetry axis along *Z*). In particular, this case is realized when atom is near a half-space

When surroundings of the atom is isotropic, the only nonzero component of the tensor **G** is

<sup>0</sup> one. It is just the case of an isotropic infinite medium (in particular, vacuum) or when

*Gii Jm* ∑ *Mm*=−*Jm*

<sup>2</sup> (*GXX* <sup>+</sup> *GYY*)

<sup>2</sup> (*GXX* <sup>−</sup> *GYY*). (44)

<sup>±</sup><sup>2</sup> <sup>=</sup> 0 if surroundings of the atom is axial symmetric

tr(**G**) <sup>=</sup> <sup>−</sup> <sup>1</sup>

*GZZ* <sup>−</sup> <sup>1</sup>

atom is in the center of spherical particle or cavity. In this case (−1)

<sup>−</sup>1/<sup>3</sup> (2*Jm* <sup>+</sup> <sup>1</sup>) in (40). So, using relations (42), (40) we obtain from (39)

� 

∑ *i*=*X*,*Y*,*Z*


¯ (2*Jm* + 1)

 1 1 *K Jm Jm Jn*

Deexcitation Dynamics of a Degenerate Two-Level Atom near (Inside) a Body 317

*<sup>Q</sup>* is irreducible spherical tensor of the imaginary part of the electric field

**G**��(*r*�0,�*r*0; *ωmn*)

 1 1 *K Jm Jm Jn* *K*


<sup>0</sup> and does not depend on excitation type.

(*GXX* + *GYY* + *GZZ*), (42)

, (43)

*Jm*+*Jn*

�*JmMm*|*ρ*ˆ|*JmMm*�. (45)

 110 *Jm Jm Jn*  =

*<sup>Q</sup>* (40)

is 6 − *j* coefficient,

*<sup>Q</sup>*(*r*�0,�*r*0; *ωmn*) is related


#### **2.3.1 Properties of irreducible tensor operators and density matrix multipole components**

The operators *T*ˆ *<sup>K</sup> Q*(*J J*� ) are orthonormal in the following sense

$$\text{tr}\left[\Upsilon\_{Q'}^{K'}(f')|\Upsilon\_Q^{K\dagger}(f')\right] \equiv \sum\_{M'M} \langle f'M'|\Upsilon\_{Q'}^{K'}(f')|fM\rangle \langle fM|\Upsilon\_Q^{K\dagger}(f')|f'M'\rangle = \delta\_{K'k}\delta\_{Q'Q'} \tag{33}$$

where the Hermitian conjugate operator *T*ˆ *<sup>K</sup>*† *<sup>Q</sup>* (*J*� *J*) is expressed in terms of *T*ˆ *<sup>K</sup> Q*(*J J*� ) as follows

$$
\hat{T}\_{\mathbb{Q}}^{K\dagger}(f') \equiv (-1)^{l'-l-\mathcal{Q}} \hat{T}\_{-\mathbb{Q}}^{K}(f\!\!/\!).\tag{34}
$$

Set of the operators *T*ˆ*<sup>K</sup> Q*(*J*� *J*) is complete. So, density operator can be decomposed into irreducible parts as follows

$$\boldsymbol{\rho} = \sum\_{\boldsymbol{J}' \boldsymbol{J} \boldsymbol{K} \boldsymbol{Q}} \rho^{\boldsymbol{K} \boldsymbol{Q}} \left( \boldsymbol{J}' \boldsymbol{J} \right) \, \hat{\boldsymbol{T}}\_{\boldsymbol{Q}}^{\boldsymbol{K}}(\boldsymbol{J}' \boldsymbol{J}). \tag{35}$$

In turn, coefficients *ρKQ* (*J*� *J*) known as multipole components are expressed in terms of *T*ˆ *K Q*(*J*� *J*) and density operator by using (33) and (32) in the following way

$$\rho^{KQ}\left(\left.l'\right|l\right) = \text{tr}\left[\not{\rho}\hat{I}\_Q^{K\dagger}\left(\left.l'\right|l\right)\right] = \sum\_{M'M} \left(-1\right)^{I-M} \left<\not{J}^{\prime}M^{\prime}\right> - M|K\rangle\langle\!\left(\left.l'^{\prime}M^{\prime}\right|\not{\rho}\right)M\rangle.\tag{36}$$

It is seen that multipole components *ρKQ* (*J*� *J*) satisfy the following relations similar to relations (34):

$$\left[\rho^{K\mathcal{Q}}\left(l'l\right)\right]^\* = (-1)^{l-f'-\mathcal{Q}}\rho^{K-\mathcal{Q}}\left(ll'\right),\tag{37}$$

so multipole components *ρK*<sup>0</sup> (*J J*) is real. Note also that *ρKQ* (*J*� *J*) transform under rotations like *T*ˆ *<sup>K</sup>*† *<sup>Q</sup>* (*J*� *J*) , and hence, are contravariant to *T*ˆ *<sup>K</sup> Q*(*J*� *J*) because of property (34).

We are interesting only in states of the excited level *m*, so the relevant density operator *ρ*ˆ (*Jm*) is

$$\oint \left( f\_m \right) = \sum\_{KQ} \rho^{KQ} \left( f\_m f\_m \right) \hat{T}\_Q^K \left( f\_m f\_m \right). \tag{38}$$

In this decomposition the rank *K* is in the range 0 *K* 2*Jm* as was noted after definition (32). All multipole components *ρKQ* (*Jm Jm*) have clear physical sense (see, for example, (Biedenharn & Louck, 1981; Blum, 1996; Omont, 1977; Varshalovich et al., 1988)). In particular, <sup>√</sup>2*Jm* <sup>+</sup> <sup>1</sup>*ρ*<sup>00</sup> (*Jm Jm*) is equal to the total population of the level *<sup>m</sup>*, the *<sup>ρ</sup>*1*<sup>Q</sup>* (*Jm Jm*)'s are the three standard components of what is generally called "orientation" proportional to the mean magnetic dipole of the state, and the *ρ*2*<sup>Q</sup>* (*Jm Jm*)'s are the five standard components of the "alignment" proportional to the mean electric quadrupole moment of the state.

#### **2.3.2 Transition rate and material body symmetry**

Finally, after some manipulation using the relations (7), (31), and (38), and also properties of irreducible tensor operators, one can represent relation (29) in the form

$$W = \frac{1}{2} \left[ 1 + \coth\left(\frac{\hbar \omega\_{mn} \xi}{2}\right) \right] \sum\_{KQ} \gamma\_Q^K \rho^{KQ} \left( f\_m f\_m \right), \tag{39}$$

with circular components *Gσσ*�(*r*�0,�*r*0; *ωmn*) as follows

where

8 Will-be-set-by-IN-TECH

**2.3.1 Properties of irreducible tensor operators and density matrix multipole components**

*<sup>Q</sup>* (*J*�

*ρKQ J* � *J T*ˆ *K <sup>Q</sup>*(*J* �

(−1)

= (−1)

*J*�

*J*) ≡ (−1)

*ρ*ˆ = ∑ *J*�*JKQ*

*J*) and density operator by using (33) and (32) in the following way

*<sup>J</sup>*)|*JM*��*JM*|*T*<sup>ˆ</sup> *<sup>K</sup>*†

<sup>−</sup>*J*−*<sup>Q</sup> T*ˆ *<sup>K</sup>*

*<sup>J</sup>*−*<sup>M</sup>* �*<sup>J</sup>* � *M*�

*J*−*J*�

*Q*(*J*�

*ρKQ* (*Jm Jm*) *T*ˆ *<sup>K</sup>*

We are interesting only in states of the excited level *m*, so the relevant density operator *ρ*ˆ (*Jm*)

In this decomposition the rank *K* is in the range 0 *K* 2*Jm* as was noted after definition (32). All multipole components *ρKQ* (*Jm Jm*) have clear physical sense (see, for example, (Biedenharn & Louck, 1981; Blum, 1996; Omont, 1977; Varshalovich et al., 1988)). In particular, <sup>√</sup>2*Jm* <sup>+</sup> <sup>1</sup>*ρ*<sup>00</sup> (*Jm Jm*) is equal to the total population of the level *<sup>m</sup>*, the *<sup>ρ</sup>*1*<sup>Q</sup>* (*Jm Jm*)'s are the three standard components of what is generally called "orientation" proportional to the mean magnetic dipole of the state, and the *ρ*2*<sup>Q</sup>* (*Jm Jm*)'s are the five standard components of the

Finally, after some manipulation using the relations (7), (31), and (38), and also properties of

 ∑ *KQ γK*

 *h*¯ *ωmnξ* 2

<sup>−</sup>*<sup>Q</sup> ρK*−*<sup>Q</sup>*

*<sup>Q</sup>* (*J* � *J*)|*J* � *M*�

*J*) is expressed in terms of *T*ˆ *<sup>K</sup>*

−*Q*(*J J*�

*J*) is complete. So, density operator can be decomposed into

*J*) known as multipole components are expressed in terms of

*J* − *M*|*KQ*��*J*

*J J*� 

*J*) because of property (34).

� = *δK*�*KδQ*�*Q*, (33)

) as follows


, (37)

*J*) transform under rotations

*<sup>Q</sup>*(*Jm Jm*). (38)

*<sup>Q</sup>ρKQ* (*Jm Jm*), (39)

*Q*(*J J*�

). (34)

*J*). (35)

� *M*�

*J*) satisfy the following relations similar to

) are orthonormal in the following sense

The operators *T*ˆ *<sup>K</sup>*

Set of the operators *T*ˆ*<sup>K</sup>*

irreducible parts as follows

In turn, coefficients *ρKQ* (*J*�

*ρKQ J* � *J* = tr *ρ*ˆ*T*ˆ *<sup>K</sup>*† *<sup>Q</sup>* (*J* � *J*) = ∑ *M*�*M*

relations (34):

like *T*ˆ *<sup>K</sup>*† *<sup>Q</sup>* (*J*�

is

tr *T*ˆ *<sup>K</sup>*� *<sup>Q</sup>*�(*J* � *J*)*T*ˆ *<sup>K</sup>*† *<sup>Q</sup>* (*J* � *J*) ≡ ∑ *M*�*M* �*J* � *M*� <sup>|</sup>*T*<sup>ˆ</sup> *<sup>K</sup>*� *<sup>Q</sup>*�(*J* �

*T*ˆ *K Q*(*J*� *Q*(*J J*�

where the Hermitian conjugate operator *T*ˆ *<sup>K</sup>*†

*Q*(*J*�

It is seen that multipole components *ρKQ* (*J*�

**2.3.2 Transition rate and material body symmetry**

*<sup>W</sup>* <sup>=</sup> <sup>1</sup> 2 

 *ρKQ J* � *J* <sup>∗</sup>

so multipole components *ρK*<sup>0</sup> (*J J*) is real. Note also that *ρKQ* (*J*�

*ρ*ˆ (*Jm*) = ∑

*KQ*

"alignment" proportional to the mean electric quadrupole moment of the state.

irreducible tensor operators, one can represent relation (29) in the form

1 + coth

*J*) , and hence, are contravariant to *T*ˆ *<sup>K</sup>*

*T*ˆ *<sup>K</sup>*† *<sup>Q</sup>* (*J* �

$$\gamma\_Q^K \equiv 2 \frac{|d\_{mn}|^2}{\hbar} (-1)^{l\_n + l\_n} \begin{Bmatrix} 1 & 1 & K \\ J\_m \ I\_m \ I\_n \end{Bmatrix} \left[ \mathbf{G}''(\vec{r\_0}; \vec{r\_0}; \omega\_{mn}) \right]\_Q^K \tag{40}$$

is irreducible relaxation tensor of the multipole *ρKQ* (*Jm Jm*), 1 1 *K Jm Jm Jn* is 6 − *j* coefficient, and [**G**��(*r*�0,�*r*0; *ωmn*)] *K <sup>Q</sup>* is irreducible spherical tensor of the imaginary part of the electric field susceptibility in the *<sup>ω</sup>*−representation. Irreducible spherical tensor *<sup>G</sup><sup>K</sup> <sup>Q</sup>*(*r*�0,�*r*0; *ωmn*) is related

$$\mathbb{G}\_{\mathbb{Q}}^{K}(\vec{r\_{0}}\,\vec{r\_{0}}\,\omega\_{mn}) \equiv \sum\_{\sigma\sigma'} \langle 1\sigma\mathbf{1}\sigma'|K\mathbf{Q}\rangle \mathbb{G}\_{\sigma\sigma'}(\vec{r\_{0}}\,\vec{r\_{0}}\,\omega\_{mn}).\tag{41}$$

It is follows from properties of the Clebsch-Gordan coefficient �1*σ*1*σ*� |*KQ*� that 0 *K* 2. Furthermore, symmetry of the tensor *Gσσ*�(*r*�0,�*r*0; *ωmn*) under the interchange *σ σ*� requires that *K* have to be even, so *K* = 0, 2. In other words, deexcitation rate depends on the total population of excited level (*K* = 0) and its alignment (*K* = 2). Their relative contribution depends according to (39) and (40) on quantum numbers of combining levels *m* and *n*, on the excitation type determining the value of *ρKQ* (*Jm Jm*), and on the atom surroundings by *G<sup>K</sup> <sup>Q</sup>*(*r*�0,�*r*0; *ωmn*). Let us consider these factors in more detail.

As was noted after (38), *K* is in the range of values defining by 0 *K* 2*Jm*. Consequently, if the total momentum *Jm* of the the excited level is equal to 0, or 1/2, there is no alignment of the level. So, deexcitation is governed only by *γ*<sup>0</sup> <sup>0</sup> and does not depend on excitation type. In the case of *Jm* > 1/2, the ratio of two deexcitation rates corresponding to some two fixed excitation types, differing in initial values of *ρKQ* (*Jm Jm*), is not universal but depends on *Jm*, *Jn*.

One can diagonalize symmetrical tensor *Gαβ*(*r*�0,�*r*0; *ωmn*). Let us label its principal axes of coordinate by *X*, *Y*, *Z*. In this proper basis only the following irreducible components of the tensor **G** are not zero:

$$\mathbf{G}\_0^0 = -\frac{1}{\sqrt{3}}\text{tr}\left(\mathbf{G}\right) = -\frac{1}{\sqrt{3}}\left(\mathbf{G}\_{XX} + \mathbf{G}\_{YY} + \mathbf{G}\_{ZZ}\right),\tag{42}$$

$$\mathcal{G}\_0^2 = \sqrt{\frac{2}{3}} \left[ \mathcal{G}\_{ZZ} - \frac{1}{2} \left( \mathcal{G}\_{XX} + \mathcal{G}\_{YY} \right) \right] \tag{43}$$

$$\mathcal{G}\_{\pm 2}^{2} = \frac{1}{2} \left( \mathcal{G}\_{\text{XX}} - \mathcal{G}\_{\text{YY}} \right). \tag{44}$$

As is seen from (44), components *G*<sup>2</sup> <sup>±</sup><sup>2</sup> <sup>=</sup> 0 if surroundings of the atom is axial symmetric (symmetry axis along *Z*). In particular, this case is realized when atom is near a half-space boundary or near a spherical particle.

When surroundings of the atom is isotropic, the only nonzero component of the tensor **G** is *G*0 <sup>0</sup> one. It is just the case of an isotropic infinite medium (in particular, vacuum) or when atom is in the center of spherical particle or cavity. In this case (−1) *Jm*+*Jn* 110 *Jm Jm Jn* = <sup>−</sup>1/<sup>3</sup> (2*Jm* <sup>+</sup> <sup>1</sup>) in (40). So, using relations (42), (40) we obtain from (39)

$$\mathcal{W}\_{\rm is} = \frac{2}{3} \frac{|d\_{mn}|^2}{\hbar \left(2f\_m + 1\right)} \odot \left(\sum\_{i=X,Y,Z} G\_{ii}\right) \sum\_{M\_m = -J\_m}^{J\_m} \langle f\_m M\_m | \beta | f\_m M\_m \rangle. \tag{45}$$

Substituting this expression into Eq. (48), we get the equation for the total density matrix

Deexcitation Dynamics of a Degenerate Two-Level Atom near (Inside) a Body 319

+ (−*i*/¯*h*)<sup>2</sup>

In Eqs. (49) and (50) the lower limit we took 0 since it is assumed that the thermostat and the atom did not interact before this time moment because the atom was unexcited. Consequently, until this moment the thermostat and the atom were uncorrelated, so the total density matrix

where *ρ*ˆ and *ρ*ˆ*th* are the density matrix operator of the atom and thermostat, respectively.

*R* ˆ

*T*ˆ *<sup>K</sup>*† *<sup>Q</sup>* (*Jm Jm*)

This relation is referred to as the main condition of the irreversibility.

the reduced atomic density matrix operator, *<sup>ρ</sup>*ˆ(*t*) <sup>≡</sup> trth*R*<sup>ˆ</sup> (*t*),

*V*ˆ (*t*), *ρ*ˆ(0)*ρ*ˆ*th*(0)

Precisely, let us multiply both sides of (53) by *T*ˆ *<sup>K</sup>*†

<sup>d</sup>*<sup>t</sup>* <sup>=</sup> <sup>−</sup>(*i*/¯*h*)tr*all*

<sup>−</sup>(1/¯*h*)<sup>2</sup>

 *t*

0 *dt*� tr*all T*ˆ *<sup>K</sup>*† *<sup>Q</sup>* (*Jm Jm*)

**3.3 Integro-differential equation for atomic multipole components**

Following the paper (Fano & Racah, 1959) (see also (Blum, 1996)), we will suppose that thermostat is always in the state of the thermal equilibrium because it has a large number of degrees of freedom and, hence, atom almost do not changes its state. The supposition implies that the total density matrix is always equal to the direct product of the density matrices of

Substituting (52) in (50) and taking trace over thermostat variables, we get the equation for

 *t*

0 *dt*� trth

*V*ˆ (*t*), *ρ*ˆ(0)*ρ*ˆ*th*(0)

 *V*ˆ (*t*), *V*ˆ *t* � , *ρ*ˆ(*t* � )*ρ*ˆ*th*(0)

<sup>−</sup> (1/¯*h*)<sup>2</sup>

To obtain dynamics equation for atomic multipole components, we make use of relation (36).

where tr*all* stands for the trace over all isolated system variables including atomic and

We will now transform this equation in such a way that terms include the trace of the product

 *t*

*dt*�

*V*ˆ (*t*), *V*ˆ *t* � , *R*ˆ *t* �

*R*ˆ(0) = *ρ*ˆ(0)*ρ*ˆ*th*(0), (51)

(*t*) = *ρ*ˆ(*t*)*ρ*ˆ*th*(0) (52)

*V*ˆ (*t*), *V*ˆ *t* � , *ρ*ˆ(*t* � )*ρ*ˆ*th*(0)

*<sup>Q</sup>* (*Jm Jm*) and take trace over atomic variable.

. (50)

. (53)

, (54)

0

operator in the following form:

<sup>d</sup>*<sup>t</sup>* = (−*i*/¯*h*)

**3.2 Large thermostat approximation**

the system:

d*ρ*ˆ(*t*)

So, we get

thermostat one.

of *ρ*ˆ(*t* �

<sup>d</sup>*<sup>t</sup>* <sup>=</sup> <sup>−</sup>(*i*/¯*h*)trth

d*ρKQ* (*Jm Jm*)(*t*)

)*ρ*ˆ*th*(0) by an operator.

*V*ˆ (*t*), *R*ˆ (0)

*R*ˆ was equal to the direct product of the density matrices of the system:

d*R*ˆ(*t*)

Since we are here interested primarily in atomic transition energies on the order of a Rydberg that implies *<sup>h</sup>*¯ *<sup>ω</sup>mn<sup>ξ</sup>* <sup>2</sup> � 1 at room temperature, we have replaced the expression in square brackets in Eq. (39) by 2. The total population of the upper level

$$\sum\_{M\_{m}=-J\_{m}}^{J\_{m}} \langle J\_{m}M\_{m}|\hat{\rho}|J\_{m}M\_{m}\rangle = 1$$

because we suppose that atom is excited on level *m* at the initial time. For free space (Barash, 1988; Lifshitz & Pitaevskii, 1980; Nikolaev, 2006), we have

$$\text{G3}\left(\sum\_{i=X,Y,Z} \text{G}\_{ii}\right) = \text{2}\left(\frac{\omega\_{mn}}{\mathcal{c}}\right)^3. \tag{46}$$

Substituting these two expressions in Eq. (45) we immediately obtain the well-known expression for the radiative decay rate of the excited state of an isolated atom (see, i.e., (Berestetskii et al., 2008; Sobelman, 1972)) :

$$\mathcal{W}\_0 = \frac{4}{3} \frac{|d\_{mn}|^2}{\hbar \left(2J\_m + 1\right)} \left(\frac{\omega\_{mn}}{c}\right)^3. \tag{47}$$

It should be noted that Eq. (39) describes deexcitation rate at the initial time moment just following the excitation. Density matrix multipole components *ρKQ* (*Jm Jm*) will be changed with the passage of time. It is reasonable to suggest that the expression opposite in sign to the right-hand side of Eq. (39) describes the decrease of the upper level population per unit of time. To prove the suggestion let us consider more general problem of the dynamics of the density matrix multipole components caused by interaction of the atom with quantized field.

#### **3. Master equations for the excited density matrix multipole components**

#### **3.1 Integro-differential equation for total density matrix operator**

Let us consider a large isolated system consisting of an atom, material body and interacting with them quantum electromagnetic field. Atomic surrounding, electromagnetic field and material body that interact among themselves, we will treat as a large subsystem referred to as the thermostat. In the interaction picture representation, the density matrix *R*ˆ of the total isolated system obeys the Liouville equation:

$$i\hbar \frac{d\mathcal{R}\,\hat{\boldsymbol{\ell}}(t)}{\mathbf{d}t} = \left[\hat{\boldsymbol{\mathcal{V}}}\left(t\right), \hat{\boldsymbol{\mathcal{R}}}\left(t\right)\right],\tag{48}$$

where *V*ˆ is the atom-field interaction operator that in the rotating-wave approximation is given by Eq. (2). It is known that this equation can be rewritten in the integro-differential form that is suitable for perturbation technique. Indeed, formal integrating this equation in time, we obtain the integral equation:

$$
\hat{\mathcal{R}}\left(t\right) = \hat{\mathcal{R}}\left(0\right) - \left(i\slash\!\!\right) \int dt' \left[\hat{\mathcal{V}}\left(t'\right), \hat{\mathcal{R}}\left(t'\right)\right].\tag{49}
$$

Substituting this expression into Eq. (48), we get the equation for the total density matrix operator in the following form:

$$\frac{d\hat{\mathcal{R}}(t)}{dt} = \left(-i/\hbar\right) \left[\hat{\mathcal{V}}\left(t\right), \hat{\mathcal{R}}\left(0\right)\right] + \left(-i/\hbar\right)^{2} \int dt' \left[\hat{\mathcal{V}}\left(t\right), \left[\hat{\mathcal{V}}\left(t'\right), \hat{\mathcal{R}}\left(t'\right)\right]\right].\tag{50}$$

In Eqs. (49) and (50) the lower limit we took 0 since it is assumed that the thermostat and the atom did not interact before this time moment because the atom was unexcited. Consequently, until this moment the thermostat and the atom were uncorrelated, so the total density matrix *R*ˆ was equal to the direct product of the density matrices of the system:

$$
\vec{R}(0) = \not p(0)\not p\_{th}(0),
\tag{51}
$$

where *ρ*ˆ and *ρ*ˆ*th* are the density matrix operator of the atom and thermostat, respectively.

#### **3.2 Large thermostat approximation**

10 Will-be-set-by-IN-TECH

Since we are here interested primarily in atomic transition energies on the order of a Rydberg

because we suppose that atom is excited on level *m* at the initial time. For free space (Barash,

Substituting these two expressions in Eq. (45) we immediately obtain the well-known expression for the radiative decay rate of the excited state of an isolated atom (see, i.e.,

> |*dmn*| 2

*h*¯ (2*Jm* + 1)

**3. Master equations for the excited density matrix multipole components**

**3.1 Integro-differential equation for total density matrix operator**

*ih*¯ d*R* ˆ (*t*) <sup>d</sup>*<sup>t</sup>* <sup>=</sup>

*<sup>R</sup>*<sup>ˆ</sup> (*t*) <sup>=</sup> *<sup>R</sup>*<sup>ˆ</sup> (0) <sup>−</sup> (*i*/¯*h*)

It should be noted that Eq. (39) describes deexcitation rate at the initial time moment just following the excitation. Density matrix multipole components *ρKQ* (*Jm Jm*) will be changed with the passage of time. It is reasonable to suggest that the expression opposite in sign to the right-hand side of Eq. (39) describes the decrease of the upper level population per unit of time. To prove the suggestion let us consider more general problem of the dynamics of the density matrix multipole components caused by interaction of the atom with quantized field.

Let us consider a large isolated system consisting of an atom, material body and interacting with them quantum electromagnetic field. Atomic surrounding, electromagnetic field and material body that interact among themselves, we will treat as a large subsystem referred to as the thermostat. In the interaction picture representation, the density matrix *R*ˆ of the total

where *V*ˆ is the atom-field interaction operator that in the rotating-wave approximation is given by Eq. (2). It is known that this equation can be rewritten in the integro-differential form that is suitable for perturbation technique. Indeed, formal integrating this equation in

> *t*

*dt*� *V*ˆ *t* � , *R*ˆ *t* �

0

*V*ˆ (*t*), *R*ˆ (*t*)

brackets in Eq. (39) by 2. The total population of the upper level

1988; Lifshitz & Pitaevskii, 1980; Nikolaev, 2006), we have

(Berestetskii et al., 2008; Sobelman, 1972)) :

isolated system obeys the Liouville equation:

time, we obtain the integral equation:

� 

*<sup>W</sup>*<sup>0</sup> <sup>=</sup> <sup>4</sup> 3

∑ *i*=*X*,*Y*,*Z*

*Gii* = 2

*Jm* ∑ *Mm*=−*Jm*

<sup>2</sup> � 1 at room temperature, we have replaced the expression in square

 *ωmn c* 3

 *ωmn c* 3 . (46)

. (47)

, (48)

. (49)

�*JmMm*|*ρ*ˆ|*JmMm*� = 1

that implies *<sup>h</sup>*¯ *<sup>ω</sup>mn<sup>ξ</sup>*

Following the paper (Fano & Racah, 1959) (see also (Blum, 1996)), we will suppose that thermostat is always in the state of the thermal equilibrium because it has a large number of degrees of freedom and, hence, atom almost do not changes its state. The supposition implies that the total density matrix is always equal to the direct product of the density matrices of the system:

$$R\stackrel{\uparrow}{t} \tag{5} = \not{\mathfrak{p}}(t)\not{\mathfrak{p}}\_{th}(0) \tag{52}$$

This relation is referred to as the main condition of the irreversibility.

Substituting (52) in (50) and taking trace over thermostat variables, we get the equation for the reduced atomic density matrix operator, *<sup>ρ</sup>*ˆ(*t*) <sup>≡</sup> trth*R*<sup>ˆ</sup> (*t*),

$$\frac{d\boldsymbol{\uprho}(t)}{dt} = -\left(\boldsymbol{i}/\hbar\right)\text{tr}\_{\text{th}}\left[\boldsymbol{\uphat{V}}\left(t\right), \boldsymbol{\uphat{\rho}}(0)\boldsymbol{\uphat{\rho}}\_{\text{th}}\left(0\right)\right] - \left(\boldsymbol{1}/\hbar\right)^{2} \int\_{0}^{t} dt' \text{tr}\_{\text{th}}\left[\boldsymbol{\uphat{V}}\left(t\right), \left[\boldsymbol{\uphat{V}}\left(t'\right), \boldsymbol{\uphat{\rho}}\left(t'\right)\boldsymbol{\uphat{\rho}}\_{\text{th}}\left(0\right)\right]\right]. \tag{53}$$

#### **3.3 Integro-differential equation for atomic multipole components**

To obtain dynamics equation for atomic multipole components, we make use of relation (36). Precisely, let us multiply both sides of (53) by *T*ˆ *<sup>K</sup>*† *<sup>Q</sup>* (*Jm Jm*) and take trace over atomic variable. So, we get

$$\begin{split} \frac{d\rho^{K\dot{Q}}\left(\left.\left(\!\left.\left\langle\!\left.\left.\left.\right\langle\!\left.\left.\right|\!\left.\right|\right.\right|\right.\right.\right.\right.\right.}\right) \left(\!\left.\left(\!\left.\left.\left.\left.\left.\right|\!\left.\left.\left.\left.\right|\!\left.\left.\right|\!\left.\left.\right|\right.\right.\right.\right.\right.\right)\right) \right. \left. \left(\!\left.\left.\left.\left.\left.\left.\right|\!\left.\left.\left.\left.\right|\!\left.\left.\right|\!\left.\left.\right|\!\left.\left.\right|\!\left.\left.\right|\!\left.\left.\right|\!\left.\left.\right|\!\left.\left.\right|\!\left.\right|\!\left.\left.\right|\!\left.\left.\right|\!\left.\right|\!\left.\right|\!\right.\right.\right.\right.\right.\right) \right. \right. \right. \right) \right) \right) \right) \right) \right) \right) \right)} \right) \right) \right) \right) \right) \right)} \right) \right) \right) \right) \right) \right) \right) \right) \frac{1}{4}$$

where tr*all* stands for the trace over all isolated system variables including atomic and thermostat one.

We will now transform this equation in such a way that terms include the trace of the product of *ρ*ˆ(*t* � )*ρ*ˆ*th*(0) by an operator.

assume following (Loisell, 1973) that this correlation time is much less then typical variation times of the atomic multipole components. Thus, in the case of free space the lifetime of

Deexcitation Dynamics of a Degenerate Two-Level Atom near (Inside) a Body 321

*Jm* because of its definition (32) and invariance of the trace under a cyclic permutation of the

Taking into account assumptions mentioned above, property (8), and by making the change

∞

*dτg* (−) *αβ* (*τ*)*e*

Now we will show that integral (62) is expressed in terms of retarded Green function

∞

*dτg* (−) *αβ* (*τ*)*e*

<sup>−</sup>*iωmn<sup>τ</sup>* +

∞

*dτg* (−) *αβ* (*τ*)*e*

0

∞

*dτg* (−) *αβ* (*τ*)*e*

0

∗

−∞

Making the change of variable in integration *τ* → −*τ* in the first integral and utilizing relation

∗ *e <sup>i</sup>ωmn<sup>τ</sup>* +

The second integral in (65) is just equal to *Iαβ* (*ωmn*), and the first one to its complex

*αβ* (−*ωmn*) = *Iαβ* (*ωmn*) + *I*

0

*J*; *t*) and to take it out of the integral in (57). It is so-called Markov-type approximation.

*βα* (*ωmn*) *<sup>A</sup>αβ* <sup>+</sup> *<sup>I</sup>αβ* (*ωmn*) *<sup>C</sup>αβ*

(±)

) incoming in (58) and (60) are nonzero only if *J* = *J*� =

*ρK*� *Q*�

<sup>−</sup>*iωmnτ*. (62)

*αβ* (*τ*) is in fact zero at *τ* � *τc*. The

<sup>−</sup>*iωmn<sup>τ</sup>* (63)

*αβ* (−*ωmn*) of the function

<sup>−</sup>*iωmnτ*. (64)

<sup>−</sup>*iωmnτ*. (65)

*βα* (*ωmn*). (66)

(−)

*Q*� (*J*� *J*; *t* � ) by

(*Jm Jm*; *t*), (61)

the atomic excited state much more than *<sup>τ</sup><sup>c</sup>* <sup>≈</sup> 1/*ωmn*. So, we can replace *<sup>ρ</sup>K*�

� in integration, we can represent (57) as

*<sup>Q</sup>*�(*J J*�

*Iαβ* (*ωmn*) ≡

error of this replacement is negligible in Markov-type approximation.

; *ωmn*). To prove that, let as consider Fourier transform *g*

*αβ* (−*ωmn*) =

*dτg* (−) *αβ* (*τ*)*e*

 0

−∞

∞

0 *dτ g* (−) *βα* (*τ*)

3¯*h*<sup>2</sup> ∑ *K*�*Q*� ∑ *αβ I* ∗

In (62) we extended upper limit from *t* to ∞ because of *g*

*g* (−)

*ρK*� *Q*� (*J*�

operators.

where

*Gαβ* (*r*,*r*�

*αβ* (*τ*) defined by (12):

Let us split this integral into two parts

*αβ* (−*ωmn*) =

*αβ* (−*ωmn*) =

conjugation. So, (65) can be rewritten as follows

*g* (−)

*g* (−)

(8), we can rewrite (64) as

*g* (−)

*g* (−)

of variable *τ* ≡ *t* − *t*

d*ρKQ* (*Jm Jm*; *t*)

It is also important to note that *T*ˆ *<sup>K</sup>*�

<sup>d</sup>*<sup>t</sup>* <sup>=</sup> <sup>−</sup>*dmndnm*

To do this, we make use of the identity (Il'inskii & Keldysh, 1994)

$$\text{tr}\left\{\hat{A}\left[\hat{A}\_{1\prime}\left[\hat{A}\_{2\prime}\cdots\left[\hat{A}\_{k\prime}\hat{B}\right]\cdots\cdots\right]\right]\right\} = \text{tr}\left\{\left[\cdots\left[\left[\hat{A}\_{\prime}\hat{A}\_{1}\right],\hat{A}\_{2}\right]\cdots\hat{A}\_{k}\right]\hat{B}\right\}\tag{55}$$

which holds for arbitrary operators *A*ˆ, *A*ˆ 1, *<sup>A</sup>*ˆ2, ··· , *<sup>B</sup>*ˆ.

Using identity (55) and the atomic density matrix decomposition (35), we can rewrite (54) as

$$\frac{d\boldsymbol{\rho}^{K\boldsymbol{\xi}\boldsymbol{\zeta}}\left(\boldsymbol{f}\_{m}\boldsymbol{m};\boldsymbol{t}\right)}{\mathrm{d}\boldsymbol{t}}=-\frac{\mathrm{i}}{\hbar}\sum\_{\begin{subarray}{c}\mathcal{I}'\boldsymbol{\mathcal{K}}'\boldsymbol{\mathcal{Q}}'\\\mathcal{I}'\boldsymbol{\mathcal{K}}'\boldsymbol{\mathcal{Q}}\end{subarray}}\boldsymbol{\rho}^{K'\boldsymbol{\mathcal{Q}}'}\left(\boldsymbol{f}'\boldsymbol{\mathcal{I}};\boldsymbol{0}\right)\operatorname{tr}\_{\mathrm{all}}\left\{\left[\hat{\boldsymbol{\mathcal{I}}}\_{\mathcal{Q}}^{K\dagger}\left(\boldsymbol{f}\_{m}\boldsymbol{I}\_{m}\right),\hat{\boldsymbol{\mathcal{V}}}\left(\boldsymbol{t}\right)\right]\hat{\boldsymbol{\mathcal{I}}}\_{\mathcal{Q}'}^{K'}\left(\boldsymbol{f}'\boldsymbol{I}\right)\hat{\boldsymbol{\rho}}\_{\mathrm{th}}\left(\boldsymbol{0}\right)\right\}$$

$$-\frac{1}{\hbar^{2}}\sum\_{\begin{subarray}{c}\mathcal{I}'\boldsymbol{\mathcal{K}}'\boldsymbol{\mathcal{Q}}'\end{subarray}}\int\_{0}^{t}\mathrm{d}\boldsymbol{t}'\boldsymbol{\rho}^{K'\boldsymbol{\mathcal{Q}}'}\left(\boldsymbol{f}'\boldsymbol{\mathcal{I}};\boldsymbol{t}'\right)\operatorname{tr}\_{\mathrm{all}}\left\{\left[\left[\hat{\boldsymbol{\mathcal{I}}}\_{\mathcal{Q}}^{K\dagger}\left(\boldsymbol{f}\_{m}\boldsymbol{I}\_{m}\right),\hat{\boldsymbol{\mathcal{V}}}\left(\boldsymbol{t}\right)\right],\hat{\boldsymbol{\mathcal{V}}}\left(\boldsymbol{t}'\right)\right]\hat{\boldsymbol{\mathcal{I}}}\_{\mathcal{Q}'}^{K'}\left(\boldsymbol{f}'\right)\hat{\boldsymbol{\rho}}\_{\mathrm{th}}(\boldsymbol{0})\right\}.\tag{56}$$

Substituting in (56) the interaction Hamiltonian (2), using (31), and also taking into account that scalar product *<sup>d</sup><sup>E</sup>*<sup>ˆ</sup> in the circular basis (30) has the form <sup>∑</sup>*<sup>σ</sup>* (−1) *<sup>σ</sup> <sup>d</sup>σE*−*σ*, we obtain

$$\frac{\mathrm{d}\rho^{K\mathcal{Q}}\left(\mathcal{J}\_{\mathrm{m}}\mathrm{f}\_{\mathrm{m}};t\right)}{\mathrm{d}t} = -\frac{d\_{\mathrm{mm}}d\_{\mathrm{mm}}}{\mathfrak{N}^{2}}\sum\_{l'lK'\mathcal{Q}'}\int\_{0}^{t}dt'\rho^{K'\mathcal{Q}'}\left(l'l\,t\,t'\right)\sum\_{a\beta}\left\{e^{i\omega\_{\mathrm{m}}\left(t-t'\right)}\left[\mathcal{g}\_{a\beta}^{(-)}\left(t'-t\right)\,A\_{a\beta}\right]\right\}}{\mathrm{-g}\_{a\beta}^{(+)}\left(t-t'\right)\,\mathrm{B}\_{a\beta}}\left\{\mathcal{g}\_{a\beta}^{(-)}\left(t-t'\right)\,\mathrm{C}\_{a\beta}-\mathcal{g}\_{a\beta}^{(+)}\left(t'-t\right)\,\mathrm{B}\_{a\beta}\right\}},\tag{57}$$

where *g* (±) *αβ* (*τ*) are the ordered correlation functions of the fluctuating electromagnetic field (8),

$$A\_{a\mathcal{B}} \equiv \sum\_{\sigma\sigma'} (-1)^{\sigma+\sigma'} \left\langle a \middle| 1-\sigma \right\rangle \langle \mathcal{S} \middle| 1-\sigma' \right\rangle \text{tr} \left\{ \hat{T}\_Q^{\mathbf{K}\dagger} (f\_{\text{m}} I\_{\text{m}}) \hat{T}\_{\sigma}^{1} (f\_{\text{m}} I\_{\text{n}}) \hat{T}\_{\sigma'}^{1} (f\_{\text{n}} I\_{\text{m}}) \hat{T}\_{Q'}^{K'} (f I') \right\}, \tag{58}$$

$$B\_{a\notin} \equiv \sum\_{\sigma\sigma'} (-1)^{\sigma+\sigma'} \langle a|1-\sigma\rangle \langle \beta|1-\sigma'\rangle \text{tr}\left\{ \hat{T}\_{\sigma}^{1} (f\_{n}I\_{m}) \hat{T}\_{Q}^{K\dagger} (f\_{m}I\_{m}) \hat{T}\_{\sigma'}^{1} (f\_{m}I\_{n}) \hat{T}\_{Q'}^{K'} (fI') \right\}, \tag{59}$$

$$\mathcal{C}\_{\mathsf{a}\mathcal{B}} \equiv \sum\_{\sigma\sigma'} (-1)^{\sigma+\sigma'} \langle a|1-\sigma\rangle \langle \beta|1-\sigma'\rangle \text{tr}\left\{ \hat{T}\_{\sigma}^{1} (f\_{m}I\_{\mathrm{n}}) \hat{T}\_{\sigma'}^{1} (f\_{n}I\_{\mathrm{m}}) \hat{T}\_{\mathbf{Q}}^{K\dagger} (f\_{\mathrm{m}}I\_{\mathrm{m}}) \hat{T}\_{\mathbf{Q}'}^{K'} (fI') \right\}. \tag{60}$$

In the definitions (58) – (60) symbols �*α*|1 − *σ*� and �*β*|1 − *σ*� � are transformation matrices from the circular components to the Descartes's one, that are inverse of that given by (30), and symbol tr {··· } from now on stands for trace over atomic variables. Note that the linear on *V*ˆ (*t*) term in (56) vanishes in our case because of the average fluctuated field is zero at the thermal equilibrium: trth *E*ˆ*α* ≡ �*E*ˆ*α*� <sup>=</sup> 0.

It should be noted that ratio of |*g* (+) *αβ* (*t* − *t* � )| to |*g* (−) *αβ* (*t* − *t* � )| is proportional to the mean number of photons in the thermal equilibrium, �*nph*� ∼ *kT*/¯*hωmn* � 1. Therefore terms that proportional to *g* (+) *αβ* (*t* − *t* � ) can be ignored in (57).

#### **3.4 Master equation for multipole components in Markov-type approximation**

Fluctuating field correlation functions *g* (±) *αβ* (*t* − *t* � ) are nonzero only for the sufficiently small time difference |*τ*|≡|*t* − *t* � | comparable with the typical field correlation time *τc*. We will assume following (Loisell, 1973) that this correlation time is much less then typical variation times of the atomic multipole components. Thus, in the case of free space the lifetime of the atomic excited state much more than *<sup>τ</sup><sup>c</sup>* <sup>≈</sup> 1/*ωmn*. So, we can replace *<sup>ρ</sup>K*� *Q*� (*J*� *J*; *t* � ) by *ρK*� *Q*� (*J*� *J*; *t*) and to take it out of the integral in (57). It is so-called Markov-type approximation.

It is also important to note that *T*ˆ *<sup>K</sup>*� *<sup>Q</sup>*�(*J J*� ) incoming in (58) and (60) are nonzero only if *J* = *J*� = *Jm* because of its definition (32) and invariance of the trace under a cyclic permutation of the operators.

Taking into account assumptions mentioned above, property (8), and by making the change of variable *τ* ≡ *t* − *t* � in integration, we can represent (57) as

$$\frac{\mathbf{d}\rho^{K\mathcal{Q}}\left(\operatorname{J}\_{\mathrm{m}}\mathbf{l}\_{\mathrm{m}};t\right)}{\mathbf{d}t} = -\frac{d\_{\mathrm{mm}}d\_{\mathrm{mm}}}{\mathfrak{H}^{2}}\sum\_{\mathbf{K}'\mathbf{Q}'}\sum\_{\mathbf{a}\mathfrak{F}}\left[I\_{\mathrm{fl}}^{\star}\left(\omega\_{\mathrm{mm}}\right)A\_{\mathrm{a}\mathfrak{F}} + I\_{\mathrm{a}\mathfrak{F}}\left(\omega\_{\mathrm{mm}}\right)\mathbf{C}\_{\mathrm{a}\mathfrak{F}}\right]\rho^{\mathbf{K}'\mathbf{Q}'}\left(\operatorname{J}\_{\mathrm{m}}\mathbf{l}\_{\mathrm{m}};t\right), \tag{61}$$

where

12 Will-be-set-by-IN-TECH

1, *<sup>A</sup>*ˆ2, ··· , *<sup>B</sup>*ˆ. Using identity (55) and the atomic density matrix decomposition (35), we can rewrite (54) as

Substituting in (56) the interaction Hamiltonian (2), using (31), and also taking into account

··· <sup>=</sup> tr ··· *<sup>A</sup>*ˆ, *<sup>A</sup>*<sup>ˆ</sup>

*<sup>Q</sup>* (*Jm Jm*), *<sup>V</sup>*<sup>ˆ</sup> (*t*)

*αβ* (*τ*) are the ordered correlation functions of the fluctuating electromagnetic field

*<sup>Q</sup>* (*Jm Jm*)*T*<sup>ˆ</sup> <sup>1</sup>

*<sup>σ</sup>*(*Jn Jm*)*T*<sup>ˆ</sup> *<sup>K</sup>*†

*<sup>σ</sup>*(*Jm Jn*)*T*<sup>ˆ</sup> <sup>1</sup>

(−) *αβ* (*t* − *t* � 1 , *A*ˆ2 ··· *<sup>A</sup>*<sup>ˆ</sup> *k B*ˆ

> *T*ˆ *<sup>K</sup>*� *<sup>Q</sup>*�(*J* � *J*)*ρ*ˆ*th*(0)

*<sup>Q</sup>* (*Jm Jm*), *<sup>V</sup>*<sup>ˆ</sup> (*t*)

 , *V*ˆ *t* � *T*ˆ *<sup>K</sup>*� *<sup>Q</sup>*�(*J* � *J*)*ρ*ˆ*th*(0)

*iωmn*(*t*−*t* � ) *g* (−) *αβ t* � − *t Aαβ*

*Cαβ* − *g*

*<sup>σ</sup>*(*Jm Jn*)*T*<sup>ˆ</sup> <sup>1</sup>

*<sup>Q</sup>* (*Jm Jm*)*T*<sup>ˆ</sup> <sup>1</sup>

*<sup>σ</sup>*�(*Jn Jm*)*T*<sup>ˆ</sup> *<sup>K</sup>*†

(+) *αβ t* � − *t Bαβ* 

*<sup>σ</sup>*�(*Jn Jm*)*T*<sup>ˆ</sup> *<sup>K</sup>*�

*<sup>σ</sup>*�(*Jm Jn*)*T*<sup>ˆ</sup> *<sup>K</sup>*�

*<sup>Q</sup>* (*Jm Jm*)*T*<sup>ˆ</sup> *<sup>K</sup>*�

*<sup>Q</sup>*�(*J J*� ) , (58)

*<sup>Q</sup>*�(*J J*� ) , (59)

*<sup>Q</sup>*�(*J J*� ) . (60)

� are transformation matrices

)| is proportional to the mean

) are nonzero only for the sufficiently small


(55)

 . (56)

}, (57)

*<sup>σ</sup> <sup>d</sup>σE*−*σ*, we obtain

To do this, we make use of the identity (Il'inskii & Keldysh, 1994)

*A*ˆ *<sup>k</sup>*, *B*ˆ 

*ρK*� *Q*� *J* � *J*; 0 tr*all T*ˆ *<sup>K</sup>*†

that scalar product *<sup>d</sup><sup>E</sup>*<sup>ˆ</sup> in the circular basis (30) has the form <sup>∑</sup>*<sup>σ</sup>* (−1)

�*α*|1 − *σ*��*β*|1 − *σ*�

�*α*|1 − *σ*��*β*|1 − *σ*�

�*α*|1 − *σ*��*β*|1 − *σ*�

≡ �*E*ˆ*α*� <sup>=</sup> 0.

(+) *αβ* (*t* − *t* � )| to |*g*

) can be ignored in (57).

**3.4 Master equation for multipole components in Markov-type approximation**

(±) *αβ* (*t* − *t* �

In the definitions (58) – (60) symbols �*α*|1 − *σ*� and �*β*|1 − *σ*�

 *E*ˆ*α*

�

 *t*

0 *dt*� *ρK*� *Q*� *J* � *J*; *t* � ∑ *αβ* {*e*

−*iωmn*(*t*−*t* � ) *g* (−) *αβ t* − *t* � 

> �tr *T*ˆ *<sup>K</sup>*†

�tr *T*ˆ 1

�tr *T*ˆ 1

from the circular components to the Descartes's one, that are inverse of that given by (30), and symbol tr {··· } from now on stands for trace over atomic variables. Note that the linear on *V*ˆ (*t*) term in (56) vanishes in our case because of the average fluctuated field is zero at the

number of photons in the thermal equilibrium, �*nph*� ∼ *kT*/¯*hωmn* � 1. Therefore terms that

3¯*h*<sup>2</sup> ∑ *J*�*JK*�*Q*�

tr *A*ˆ *A*ˆ 1, *<sup>A</sup>*<sup>ˆ</sup> 2, ···

<sup>d</sup>*<sup>t</sup>* <sup>=</sup> <sup>−</sup> *<sup>i</sup>*

 *t*

0 *dt*� *ρK*� *Q*� *J* � *J*; *t* � tr*all T*ˆ *<sup>K</sup>*†

<sup>d</sup>*<sup>t</sup>* <sup>=</sup> <sup>−</sup>*dmndnm*

−*g* (+) *αβ t* − *t* � *Bαβ* + *e*

(−1) *σ*+*σ*�

(−1) *σ*+*σ*�

(−1) *σ*+*σ*�

It should be noted that ratio of |*g*

(+) *αβ* (*t* − *t* �

Fluctuating field correlation functions *g*

time difference |*τ*|≡|*t* − *t*

d*ρKQ* (*Jm Jm*; *t*)

− 1 *<sup>h</sup>*¯ <sup>2</sup> ∑ *J*�*JK*�*Q*�

d*ρKQ* (*Jm Jm*; *t*)

(±)

*<sup>A</sup>αβ* ≡ ∑ *σσ*�

*<sup>B</sup>αβ* ≡ ∑ *σσ*�

*<sup>C</sup>αβ* ≡ ∑ *σσ*�

thermal equilibrium: trth

proportional to *g*

where *g*

(8),

which holds for arbitrary operators *A*ˆ, *A*ˆ

*<sup>h</sup>*¯ ∑ *J*�*JK*�*Q*�

$$I\_{a\mathcal{G}}\left(\omega\_{mn}\right) \equiv \int\_0^\infty d\tau g\_{a\mathcal{G}}^{(-)}(\tau)e^{-i\omega\_{mn}\tau}.\tag{62}$$

In (62) we extended upper limit from *t* to ∞ because of *g* (±) *αβ* (*τ*) is in fact zero at *τ* � *τc*. The error of this replacement is negligible in Markov-type approximation.

Now we will show that integral (62) is expressed in terms of retarded Green function *Gαβ* (*r*,*r*� ; *ωmn*). To prove that, let as consider Fourier transform *g* (−) *αβ* (−*ωmn*) of the function *g* (−) *αβ* (*τ*) defined by (12):

$$\mathcal{g}\_{a\mathcal{\beta}}^{(-)}\left(-\omega\_{mn}\right) = \int\_{-\infty}^{\infty} d\tau \mathcal{g}\_{a\mathcal{\beta}}^{(-)}\left(\tau\right) e^{-i\omega\_{mn}\tau} \tag{63}$$

Let us split this integral into two parts

$$g\_{a\not\!\!\!\!}^{(-)}\left(-\omega\_{mn}\right) = \int\_{-\infty}^{0} d\tau g\_{a\not\!\!\!\!\!\/}^{(-)}\left(\tau\right)e^{-i\omega\_{mn}\tau} + \int\_{0}^{\infty} d\tau g\_{a\not\!\!\!\/}^{(-)}\left(\tau\right)e^{-i\omega\_{mn}\tau}.\tag{64}$$

Making the change of variable in integration *τ* → −*τ* in the first integral and utilizing relation (8), we can rewrite (64) as

$$\mathcal{g}\_{a\beta}^{(-)}\left(-\omega\_{mn}\right) = \int\_0^\infty d\tau \left(\mathcal{g}\_{\beta a}^{(-)}\left(\tau\right)\right)^\* e^{i\omega\_{mn}\tau} + \int\_0^\infty d\tau \mathcal{g}\_{a\beta}^{(-)}\left(\tau\right) e^{-i\omega\_{mn}\tau}.\tag{65}$$

The second integral in (65) is just equal to *Iαβ* (*ωmn*), and the first one to its complex conjugation. So, (65) can be rewritten as follows

$$\left(\operatorname{g}\_{a\mathcal{\beta}}^{(-)}\left(-\omega\_{mn}\right) = I\_{a\mathcal{\beta}}\left(\omega\_{mn}\right) + I\_{\mathcal{\beta}a}^{\*}\left(\omega\_{mn}\right)\right. \tag{66}$$

and

*<sup>G</sup>αβ KK*� *L* <sup>=</sup> <sup>1</sup> 2 *G*��

*<sup>G</sup>αβ KK*� *L* <sup>=</sup> <sup>1</sup> 2 *G*�

the circu lar one, that given by (30).

**3.4.1 Relaxation matrix symmetry**

symmetrical with respect to *K* and *K*�

*γ*(*KK*�

*L*) and *Gαβ* (*KK*�

*<sup>G</sup>αβ KK*� *L*

*<sup>G</sup>αβ KK*� *L*

*<sup>M</sup>* (*KK*�

 *GL M KK*� *L* <sup>∗</sup>

 *GL M KK*� *L* <sup>∗</sup>

> *γKK*� *QQ*� ∗

 Δ*KK*� *QQ*� ∗

that can be rewrite in terms of *γKK*�

 Γ*KK*� *QQ*� ∗

 *γKK*� *QQ*� ∗

 Δ*KK*� *QQ*� ∗

permutation of *K* and *K*� as follows

*L*) and *Gαβ* (*KK*�

*L*, *Jm Jn*) = (−1)

*<sup>L</sup>*) and *<sup>G</sup><sup>L</sup>*

because of invariance of 6 − *j* symbol as regard to permutation of its columns.

<sup>=</sup> (−1)

<sup>=</sup> <sup>−</sup> (−1)

*<sup>M</sup>* (*KK*�

�1*σ*|*α*�<sup>∗</sup> = (−1)*σ*�<sup>1</sup> <sup>−</sup> *<sup>σ</sup>*|*α*� and Clebsch-Gordan coefficients symmetry, one can show that

= (−1)

= (−1)

This relations allow to find the following symmetry of the relaxation matrix components

*K*�

*K*�

On the other hand, from hermiticity of density matrix and equation (69) it is easy to obtain

= (−1)

= (−1)

= − (−1)

= (−1)

= (−1)

*QQ*� and <sup>Δ</sup>*KK*�

symbols �1*σ*|*α*� and �1*σ*�

Note that *Gαβ* (*KK*�

tensors *Gαβ* (*KK*�

Irreducible tensors *<sup>G</sup><sup>L</sup>*

*βα*(*ωmn*) + (−1)

*βα*(*ωmn*) − (−1)

*K*+*K*�

*K*+*K*�

*<sup>L</sup>*), and consequently, *<sup>G</sup><sup>L</sup>*

. As for the scalar *γ*(*KK*�

2*K*� + 1 <sup>2</sup>*<sup>K</sup>* <sup>+</sup> <sup>1</sup> *<sup>γ</sup>*(*K*�

*L*) have one, as one can see from (77) and (78),

<sup>−</sup>*<sup>L</sup> <sup>G</sup>βα*

*KK*� *L* 

> *KK*� *L*

*L*) in general are complex. Using relation

<sup>−</sup>*<sup>L</sup> <sup>G</sup>βα*

*<sup>L</sup>*+*<sup>M</sup> <sup>G</sup><sup>L</sup>* −*M KK*� *L* 

*<sup>L</sup>*+*<sup>M</sup> <sup>G</sup><sup>L</sup>* −*M KK*� *L* 

−*K*+*Q*−*Q*�

−*K*+*Q*−*Q*�

*Q*−*Q*�

*QQ*� as follows

*Q*−*Q*�

*Q*−*Q*�

*γKK*�

Δ*KK*�

Γ*KK*�

*γKK*�

Δ*KK*�

Symbol *G*� and *G*�� in (77) and (78) denotes real and imaginary part of *G*, respectively, and

Deexcitation Dynamics of a Degenerate Two-Level Atom near (Inside) a Body 323

*K*−*K*� 

Although tensor *Gαβ* in general has no symmetry with respect to permutation of subscripts,

*K*+*K*�

*K*+*K*�

<sup>−</sup>*<sup>L</sup> G*��

<sup>−</sup>*<sup>L</sup> G*�


*αβ*(*ωmn*)

*αβ*(*ωmn*)

 / *ωmn c* 3

 / *ωmn c* 3

*<sup>M</sup>* (*KK*�

*<sup>L</sup>*) and *<sup>G</sup><sup>L</sup>*

*<sup>M</sup>* (*KK*�

*L*, *Jm Jn*), it changes upon

*KL*, *Jm Jn*) (79)

, (80)

. (81)

, (82)

. (83)

<sup>−</sup>*Q*−*Q*� , (84)

<sup>−</sup>*Q*−*Q*� . (85)

<sup>−</sup>*Q*−*Q*� , (86)

<sup>−</sup>*Q*−*Q*� , (87)

<sup>−</sup>*Q*−*Q*� . (88)

*L*), are

, (77)

. (78)

Now comparing right-hand sides of (66) and (27), we obtain desired relation

$$I\_{\text{a\#}}\left(\omega\_{mn}\right) = -i\hbar \frac{1}{2} \left[1 + \coth\left(\frac{\hbar \omega\_{mn}}{2kT}\right)\right] \mathcal{G}\_{\text{a\#}}\left(\vec{r}, \vec{r}'; \omega\_{mn}\right) \tag{67}$$

It is yet mentioned after (37) that multipole components *ρKQ* (*J*� *J*) transform under rotations contravariant to *T*ˆ *<sup>K</sup> Q*(*J*� *J*). It is convenient to introduce co-variant multipole components *ρK <sup>Q</sup>* (*J*� *J*) by convention

$$\rho^K\_{\mathbb{Q}}\left(l'l\right) \equiv (-1)^{I-l'-Q} \rho^{K-Q}\left(l l'\right) = \left[\rho^{KQ}\left(l'l\right)\right]^\*.\tag{68}$$

In these notations, making use of (67) and explicitly calculating traces in (58) and (60), one can finally represent (61) as follows <sup>2</sup>

$$\frac{\mathrm{d}\rho\_Q^K(t)}{\mathrm{d}t} = -\gamma\_0 \sum\_{K'Q'} \Gamma\_{QQ'}^{KK'} \rho\_{Q'}^{K'}(t) \, . \tag{69}$$

where

$$\gamma\_0 = W\_0 = \frac{4}{3} \frac{\left|d\_{mn}\right|^2}{\hbar \left(2J\_m + 1\right)} \left(\frac{\omega\_{mn}}{c}\right)^3 \tag{70}$$

is radiation decay rate of the excited degenerate state of the atom in vacuum, dimensionless relaxation tensor Γ*KK*� *QQ*� can be represented as follows:

$$
\Gamma\_{QQ'}^{KK'} = \gamma\_{QQ'}^{KK'} + i\Delta\_{QQ'}^{KK'} \tag{71}
$$

where *γKK*� *QQ*� and <sup>Δ</sup>*KK*� *QQ*� are in general complex.

> *GL M*

Geometrical part of *γKK*� *QQ*� and <sup>Δ</sup>*KK*� *QQ*� is represented by Clebsch-Gordan coefficient and dynamical one is proportional to retarded Green function:

$$
\gamma\_{QQ'}^{KK'} = \sum\_{LM} \langle \mathbf{K'}Q'LM|\mathbf{KQ}\rangle \overline{\mathbf{G}}\_M^L \left(\mathbf{KK'}L\right) \gamma(\mathbf{K'}L, \mathbf{J}\_{\text{m}}\mathbf{J}\_{\text{n}}),\tag{72}
$$

$$
\Delta\_{\mathbf{Q}\mathbf{Q}'}^{\mathbf{K}\mathbf{K}'} = \sum\_{LM} \langle \mathbf{K}' \mathbf{Q}' LM | \mathbf{K} \mathbf{Q} \rangle \tilde{\mathbf{G}}\_M^L \left( \mathbf{K} \mathbf{K}' \mathbf{L} \right) \gamma \left( \mathbf{K} \mathbf{K}' \mathbf{L}, \mathbf{J}\_m \mathbf{J}\_n \right), \tag{73}
$$

where scalar coefficient *γ*(*KK*� *<sup>L</sup>*, *Jm Jn*) and irreducible tensors *<sup>G</sup><sup>L</sup> <sup>M</sup>* (*KK*� *<sup>L</sup>*) and *<sup>G</sup><sup>L</sup> <sup>M</sup>* (*KK*� *L*) are

$$\gamma(\text{KK}'L, l\_{\text{m}}l\_{\text{m}}) = (-1)^{K + l\_{\text{n}} - l\_{\text{m}}} \frac{3}{2} \left(2l\_{\text{m}} + 1\right) \sqrt{\left(2K' + 1\right) \left(2L + 1\right)}\tag{74}$$

$$\bigcup\_{\text{n}} \left\{ K, K' \perp L \; \right\} \left\{ 1 \; \begin{array}{cccc} 1 & 1 & L \end{array} \right\}$$

$$
\times \left\{ \begin{array}{c} \text{K} \cdot \text{K} \\ \text{J}\_{m} \text{ } f\_{m} \text{ } f\_{m} \end{array} \right\} \left\{ \begin{array}{c} 1 \quad 1 \quad 1 \\ \text{J}\_{m} \text{ } f\_{m} \text{ } f\_{n} \end{array} \right\} \text{ } \prime \text{ }
$$

$$
\left\{ \text{K} \mathbf{K}^{\prime} \mathbf{L} \right\} = \sum \left\langle 1 \sigma \mathbf{1} \sigma^{\prime} \right| \text{L} \mathbf{M} \rangle \left\langle 1 \sigma \left| a \right\rangle \langle 1 \sigma^{\prime} \left| \beta \right\rangle \overline{\mathbf{G}}\_{\mathbf{a} \beta} \left( \mathbf{K} \mathbf{K}^{\prime} \mathbf{L} \right) \,, \tag{75}
$$

$$\left(\tilde{\mathbf{G}}\_{M}^{\mathrm{L}}\left(\mathrm{K}\mathbf{K}^{\prime}\mathrm{L}\right) = \sum\_{\boldsymbol{\alpha}\boldsymbol{\beta}\boldsymbol{\sigma}\boldsymbol{\sigma}^{\prime}}^{\boldsymbol{\alpha}\boldsymbol{\beta}\boldsymbol{\sigma}\boldsymbol{\sigma}^{\prime}} \langle 1\boldsymbol{\sigma}1\boldsymbol{\sigma}^{\prime}|\mathrm{L}M\rangle\langle 1\boldsymbol{\sigma}|\boldsymbol{\alpha}\rangle\langle 1\boldsymbol{\sigma}^{\prime}|\boldsymbol{\beta}\rangle\tilde{\mathbf{G}}\_{\boldsymbol{\alpha}\boldsymbol{\beta}}\left(\mathrm{K}\mathbf{K}^{\prime}\mathrm{L}\right), \tag{76}$$

<sup>2</sup> hereinafter for simplicity we omit the dependence of *ρ<sup>K</sup> <sup>Q</sup>* on *Jm*: *<sup>ρ</sup><sup>K</sup> <sup>Q</sup>* (*t*) <sup>≡</sup> *<sup>ρ</sup><sup>K</sup> <sup>Q</sup>* (*Jm Jm*; *t*) and

14 Will-be-set-by-IN-TECH

<sup>−</sup>*<sup>Q</sup> ρK*−*<sup>Q</sup>*

In these notations, making use of (67) and explicitly calculating traces in (58) and (60), one can

*K*�*Q*�


*h*¯ (2*Jm* + 1)

*QQ*� <sup>+</sup> *<sup>i</sup>*Δ*KK*�

is radiation decay rate of the excited degenerate state of the atom in vacuum, dimensionless

 *h*¯ *ωmn* 2*kT*

> *J J*� = *ρKQ J* � *J* <sup>∗</sup>

Γ*KK*� *QQ*� *<sup>ρ</sup>K*�

 *Gαβ r*,*r*� ; *ωmn*

*J*). It is convenient to introduce co-variant multipole components

 *ωmn c* 3 (67)

. (68)

(70)

*J*) transform under rotations

*<sup>Q</sup>*� (*t*), (69)

*QQ*� , (71)

*L*, *Jm Jn*), (72)

*L*, *Jm Jn*), (73)

*<sup>M</sup>* (*KK*�

, (75)

, (76)

*L*) are

*<sup>L</sup>*) and *<sup>G</sup><sup>L</sup>*

(2*K*� + 1) (2*L* + 1) (74)

*QQ*� is represented by Clebsch-Gordan coefficient and

*<sup>M</sup>* (*KK*�

1 + coth

Now comparing right-hand sides of (66) and (27), we obtain desired relation

1 2 

*Iαβ* (*ωmn*) = −*ih*¯

*Q*(*J*�

*ρK Q J* � *J* 

contravariant to *T*ˆ *<sup>K</sup>*

relaxation tensor Γ*KK*�

*QQ*� and <sup>Δ</sup>*KK*�

where scalar coefficient *γ*(*KK*�

*γ*(*KK*�

*GL M KK*� *L* <sup>=</sup> ∑ *αβσσ*�

*GL M KK*� *L* <sup>=</sup> ∑ *αβσσ*�

<sup>2</sup> hereinafter for simplicity we omit the dependence of *ρ<sup>K</sup>*

Geometrical part of *γKK*�

*J*) by convention

finally represent (61) as follows <sup>2</sup>

*ρK <sup>Q</sup>* (*J*�

where

where *γKK*�

It is yet mentioned after (37) that multipole components *ρKQ* (*J*�

≡ (−1)

d*ρ<sup>K</sup> <sup>Q</sup>* (*t*)

*<sup>γ</sup>*<sup>0</sup> <sup>=</sup> *<sup>W</sup>*<sup>0</sup> <sup>=</sup> <sup>4</sup>

*QQ*� can be represented as follows:

*QQ*� are in general complex.

*QQ*� and <sup>Δ</sup>*KK*�

dynamical one is proportional to retarded Green function:

*L*, *Jm Jn*) = (−1)

×

*γKK*� *QQ*� = ∑ *LM* �*K*� *Q*�

Δ*KK*� *QQ*� = ∑ *LM* �*K*� *Q*�

Γ*KK*� *QQ*� <sup>=</sup> *<sup>γ</sup>KK*�

*J*−*J*�

<sup>d</sup>*<sup>t</sup>* <sup>=</sup> <sup>−</sup>*γ*<sup>0</sup> ∑

3

*LM*|*KQ*�*G<sup>L</sup>*

*LM*|*KQ*�*G<sup>L</sup>*

*<sup>K</sup>*+*Jn*−*Jm* 3

 *K K*� *L Jm Jm Jm*

�1*σ*1*σ*�

�1*σ*1*σ*�

*M KK*� *L γ*(*KK*�

*M KK*� *L γ*(*KK*�

*<sup>L</sup>*, *Jm Jn*) and irreducible tensors *<sup>G</sup><sup>L</sup>*

<sup>2</sup> (2*Jm* <sup>+</sup> <sup>1</sup>)



*<sup>Q</sup>* on *Jm*: *<sup>ρ</sup><sup>K</sup>*

 1 1 *L Jm Jm Jn*  ,

> |*β*�*Gαβ KK*� *L*

> |*β*�*Gαβ KK*� *L*

*<sup>Q</sup>* (*t*) <sup>≡</sup> *<sup>ρ</sup><sup>K</sup>*

*<sup>Q</sup>* (*Jm Jm*; *t*)

$$\overline{\mathcal{G}}\_{a\beta} \left( \mathcal{K}^{\prime} L \right) = \frac{1}{2} \left[ \mathcal{G}\_{\beta a}^{\prime\prime} (\omega\_{mn}) + (-1)^{K+K^{\prime}-L} \mathcal{G}\_{a\beta}^{\prime\prime} (\omega\_{mn}) \right] / \left( \frac{\omega\_{mn}}{c} \right)^{3},\tag{77}$$

$$\tilde{\mathcal{G}}\_{a\beta} \left( \mathcal{K}^{\prime} L \right) = \frac{1}{2} \left[ \mathcal{G}\_{\beta a}^{\prime} (\omega\_{mn}) - (-1)^{K + K^{\prime} - L} \mathcal{G}\_{a\beta}^{\prime} (\omega\_{mn}) \right] / \left( \frac{\omega\_{mn}}{c} \right)^{\beta}. \tag{78}$$

Symbol *G*� and *G*�� in (77) and (78) denotes real and imaginary part of *G*, respectively, and symbols �1*σ*|*α*� and �1*σ*� |*β*� are transformation matrices from the Descartes's components to the circu lar one, that given by (30).

#### **3.4.1 Relaxation matrix symmetry**

Note that *Gαβ* (*KK*� *L*) and *Gαβ* (*KK*� *<sup>L</sup>*), and consequently, *<sup>G</sup><sup>L</sup> <sup>M</sup>* (*KK*� *<sup>L</sup>*) and *<sup>G</sup><sup>L</sup> <sup>M</sup>* (*KK*� *L*), are symmetrical with respect to *K* and *K*� . As for the scalar *γ*(*KK*� *L*, *Jm Jn*), it changes upon permutation of *K* and *K*� as follows

$$\gamma(\text{KK}' \text{L}, \text{J}\_m \text{J}\_n) = (-1)^{K - K'} \sqrt{\frac{2\text{K}' + 1}{2\text{K} + 1}} \gamma(\text{K}' \text{KL}, \text{J}\_m \text{J}\_n) \tag{79}$$

because of invariance of 6 − *j* symbol as regard to permutation of its columns.

Although tensor *Gαβ* in general has no symmetry with respect to permutation of subscripts, tensors *Gαβ* (*KK*� *L*) and *Gαβ* (*KK*� *L*) have one, as one can see from (77) and (78),

$$
\overline{\mathcal{G}}\_{a\beta} \left( \mathcal{K} \mathcal{k}' L \right) = (-1)^{\mathcal{K} + \mathcal{K}' - L} \overline{\mathcal{G}}\_{\beta a} \left( \mathcal{K} \mathcal{k}' L \right) \, , \tag{80}
$$

$$
\tilde{G}\_{a\beta} \left( \mathcal{K} \mathcal{k}' L \right) = - \left( -1 \right)^{\mathcal{K} + \mathcal{K}' - L} \tilde{G}\_{\beta a} \left( \mathcal{K} \mathcal{k}' L \right) \,. \tag{81}
$$

Irreducible tensors *<sup>G</sup><sup>L</sup> <sup>M</sup>* (*KK*� *<sup>L</sup>*) and *<sup>G</sup><sup>L</sup> <sup>M</sup>* (*KK*� *L*) in general are complex. Using relation �1*σ*|*α*�<sup>∗</sup> = (−1)*σ*�<sup>1</sup> <sup>−</sup> *<sup>σ</sup>*|*α*� and Clebsch-Gordan coefficients symmetry, one can show that

$$\left[\overline{\mathbf{G}}\_{M}^{L}\left(\mathbf{K}\mathbf{K}^{\prime}L\right)\right]^{\*} = \left(-1\right)^{L+M}\overline{\mathbf{G}}\_{-M}^{L}\left(\mathbf{K}\mathbf{K}^{\prime}L\right),\tag{82}$$

$$\left[\widetilde{\boldsymbol{G}}\_{M}^{L}\left(\boldsymbol{K}\boldsymbol{K}^{\prime}\boldsymbol{L}\right)\right]^{\*} = \left(-1\right)^{L+M}\widetilde{\boldsymbol{G}}\_{-M}^{L}\left(\boldsymbol{K}\boldsymbol{K}^{\prime}\boldsymbol{L}\right).\tag{83}$$

This relations allow to find the following symmetry of the relaxation matrix components

$$\left[\gamma\_{QQ'}^{KK'}\right]^\* = (-1)^{K'-K+Q-Q'} \gamma\_{-Q-Q'}^{KK'} \tag{84}$$

$$\left[\Delta\_{QQ'}^{KK'}\right]^\* = (-1)^{K'-K+Q-Q'} \Delta\_{-Q-Q'}^{KK'}.\tag{85}$$

On the other hand, from hermiticity of density matrix and equation (69) it is easy to obtain

$$\left[\Gamma\_{QQ'}^{KK'}\right]^\* = (-1)^{Q-Q'} \Gamma\_{-Q-Q'}^{KK'} \tag{86}$$

that can be rewrite in terms of *γKK*� *QQ*� and <sup>Δ</sup>*KK*� *QQ*� as follows

$$\left[\gamma\_{QQ'}^{KK'}\right]^\* = (-1)^{Q-Q'} \gamma\_{-Q-Q'}^{KK'} \tag{87}$$

$$\left[\Delta\_{QQ'}^{KK'}\right]^\* = -(-1)^{Q-Q'} \Delta\_{-Q-Q'}^{KK'}.\tag{88}$$

In case of the atomic surroundings is axial symmetrical in addition, there are only two nonzero

Deexcitation Dynamics of a Degenerate Two-Level Atom near (Inside) a Body 325

*QQ*� <sup>=</sup> *<sup>δ</sup>Q*,*Q*�Γ*KK*�

*K*�

*K*�*Q*�

Hereinafter we suppose that there is no external magnetic field. In this case *γ*0Γ0*K*�

−*Q*�

Let us restrict themselves to the case of axial symmetrical atomic surroundings.

as in (39). To obtain temporal variation of the deexcitation, it is necessary to solve consistent differential equations, involving along with Eq. (100) also differential equations for *ρK*�

As it mentioned above, this case include half-space boundary and spherical particle. From (97), (98), (100), and also (99), it is follows that consistent differential equations, describing deexcitation dynamics in the proper coordinate system, include only multipole components with even *K* and *Q* = 0. The number of such components is [*Jm*] + 1 because of 0 *K* 2*Jm* as noted above (symbol [*Jm*] here and further denotes the integer part of *Jm*). As the relevant

<sup>00</sup> are real in our case, from (96) we obtain that they are symmetrical relative to *K* and *K*�

As is known , the general solution of [*Jm*] + 1 consistent linear homogeneous differential equations is given by a linear combination of [*Jm*] + 1 their eigen vectors, each of them varies in time exponentially with its own rate. The rates are eigen values of the consistent equations. The number of the eigen values is also in general equal to [*Jm*] + 1. So, the atomic deexcitation

*K*

Γ*KK*� <sup>00</sup> = <sup>Γ</sup>*K*�

<sup>+</sup>*L*−*<sup>K</sup>* �*K*�

Γ0*K*� <sup>0</sup>*Q*� *<sup>ρ</sup>K*�

<sup>−</sup>*Q*� is defined by (40), multiplier (−1)

Γ*KK*�

<sup>0</sup> and *<sup>G</sup>*<sup>2</sup>

*QQ* and <sup>Δ</sup>*KK*�

, imaginary for odd *K* + *K*� and

*L*) are nonzero and real in the system (see relations (82) and

<sup>0</sup>. Therefor, only irreducible

*QQ* are also nonzero and real (see

0*L*0|*K*0�. (99)

*<sup>Q</sup>*� (*t*). (100)

*Q*�

<sup>0</sup>, not of population that is <sup>√</sup>2*Jm* <sup>+</sup> <sup>1</sup>*ρ*<sup>0</sup>

<sup>00</sup> . (101)

<sup>00</sup> is ([*Jm*] + 1) × ([*Jm*] + 2) /2.

and denominator <sup>√</sup>2*Jm* <sup>+</sup> 1 reflect the fact that

<sup>0</sup>*Q*� =

0

*<sup>Q</sup>*� (*t*),

transforms covariant

*QQ* . (97)

<sup>00</sup> = 0 (98)

*<sup>M</sup>* in the proper coordinate system, *<sup>G</sup>*<sup>0</sup>

Γ*KK*�

for odd *K* + *K*� because of the following Clebsch-Gordan coefficient symmetry

0*L*0|*K*0� = (−1)

<sup>d</sup>*<sup>t</sup>* <sup>=</sup> <sup>−</sup>*γ*<sup>0</sup> ∑

components of *G<sup>L</sup>*

So, in this case Γ*KK*�

<sup>0</sup> (*KK*�

relations (72 and (73)), hence,

**4. Deexcitation dynamics**

incoming in its right-hand side.

*<sup>L</sup>*) and *<sup>G</sup><sup>L</sup>*

<sup>0</sup> (*KK*�

*QQ* is real for even *K* + *K*�

�*K*�

Deexcitation of upper level is given by (69) with *K* = *Q* = 0

<sup>√</sup>2*Jm* <sup>+</sup> 1, where *<sup>γ</sup>K*�

*<sup>Q</sup>*� into contravariant one *<sup>ρ</sup>K*�

the right-hand side of (100) is variation in time of *ρ*<sup>0</sup>

Hence, the number of different relevant components Γ*KK*�

is also usually expressed as a linear sum of [*Jm*] + 1 exponentials.

d*ρ*<sup>0</sup> <sup>0</sup> (*t*)

(83), and (93) and (94)). Consequently, only *γKK*�

tensors *<sup>G</sup><sup>L</sup>*

(−1) *Q*� *γK*� <sup>−</sup>*Q*�/

Γ*KK*�

component *ρK*�

Comparing (84) and (87) shows that *γKK*� *QQ*� is different from zero only for even *K* + *K*� . Similarly, comparing (85) and (88) shows that Δ*KK*� *QQ*� is different from zero only for odd *K* + *K*� .

These properties can be find more straightforward from symmetries (80) and (81) and definitions (75) and (76) that yield

$$
\overline{\mathbf{G}}\_M^L \left( \mathbf{K} \mathbf{K}' \mathbf{L} \right) = \left( -1 \right)^{\mathbf{K} + \mathbf{K}'} \overline{\mathbf{G}}\_M^L \left( \mathbf{K} \mathbf{K}' \mathbf{L} \right), \tag{89}
$$

$$
\tilde{G}\_M^L \left( K \mathcal{K}' L \right) = \left( -1 \right)^{K + K' + 1} \tilde{G}\_M^L \left( K \mathcal{K}' L \right) \,. \tag{90}
$$

Taking into account these properties that we can reformulate as *K* + *K*� is even for *Gαβ* (*KK*� *L*) and odd for *Gαβ* (*KK*� *L*), one can see from (80) and (81) that part of *Gαβ* which is symmetrical with respect to permutation of subscripts makes a contribution to *Gαβ* (*KK*� *L*) and to *Gαβ* (*KK*� *L*), and hence to Γ*KK*� *QQ*� , only when *L* is even. As for antisymmetrical part of *Gαβ*, it contributes to Γ*KK*� *QQ*� only when *L* is odd.

When tensor *Gαβ* is symmetrical (i.e., no external magnetic field), the form of tensor *Gαβ* (*KK*� *L*) as well of tensor *Gαβ* (*KK*� *L*) is simplified

$$\overline{\mathbf{G}}\_{\mathbf{a}\boldsymbol{\beta}}\left(\mathbf{K}\mathbf{K}^{\prime}\mathbf{L}\right) = \delta\_{\mathbf{L},2\mathbf{l}}\delta\_{\mathbf{K}+\mathbf{K}^{\prime},2\mathbf{n}}\left(\frac{c}{\omega\_{mn}}\right)^{\mathbf{3}}\mathbf{G}\_{\mathbf{a}\boldsymbol{\beta}\prime}^{\prime\prime} \tag{91}$$

$$
\tilde{\mathbf{G}}\_{\mathbf{a}\boldsymbol{\beta}}\left(\mathbf{K}\mathbf{K}^{\prime}\mathbf{L}\right) = \delta\_{\mathbf{L},2\mathbf{l}}\delta\_{\mathbf{K}+\mathbf{K}^{\prime},2\mathbf{n}+1} \left(\frac{c}{\omega\_{mn}}\right)^{3} \mathbf{G}\_{\mathbf{a}\boldsymbol{\beta}\prime}^{\prime} \tag{92}
$$

where *<sup>n</sup>* and *<sup>l</sup>* are integer. As a consequence, *<sup>G</sup><sup>L</sup> <sup>M</sup>* (*KK*� *<sup>L</sup>*) and *<sup>G</sup><sup>L</sup> <sup>M</sup>* (*KK*� *L*) are also simplified

$$\overline{\mathbf{G}}\_{M}^{\rm L} \left( \mathbf{K} \mathbf{K}' \mathbf{L} \right) = \delta\_{\rm L,21} \delta\_{\rm K+K',2\rm n} \left( \frac{c}{\omega\_{\rm mm}} \right)^{\rm 3} \left[ \mathbf{G}'' \right]\_{M'}^{\rm L} \tag{93}$$

$$
\tilde{G}\_M^L \left( \mathbf{K} \mathbf{K}' \mathbf{L} \right) = \delta\_{\mathbf{L}, \mathbf{2} \mathbf{l}} \delta\_{\mathbf{K} + \mathbf{K}', \mathbf{2} \mathbf{n} + 1} \left( \frac{c}{\omega\_{mn}} \right)^3 \left[ \mathbf{G}' \right]\_M^L. \tag{94}
$$

As stated above (see Eqs. (42) -(44) ), in this case there are only four nonzero components of *G<sup>L</sup> <sup>M</sup>* in the proper coordinate system.

There is additional symmetry of the relaxation tensor Γ*KK*� *QQ*� in the case. Using the fact that *GL <sup>M</sup>* (*KK*� *<sup>L</sup>*) and *<sup>G</sup><sup>L</sup> <sup>M</sup>* (*KK*� *L*) are symmetrical with respect to *K* and *K*� , evenness of *L*, relation (79) and also Clebsch-Gordan coefficient symmetry �*K*� *Q*� *LM*|*KQ*� = (−1) *<sup>L</sup>*+*<sup>M</sup>* <sup>2</sup>*K*+<sup>1</sup> <sup>2</sup>*K*�+<sup>1</sup> �*K* − *QLM*|*K*� − *Q*� �, one can obtain

$$
\Gamma\_{QQ'}^{KK'} = (-1)^{K-K'+Q-Q'} \Gamma\_{-Q'-Q}^{K'K} \tag{95}
$$

that we can rewrite using (86) as follows

$$
\Gamma\_{QQ'}^{KK'} = (-1)^{K-K'} \left[ \Gamma\_{Q'Q}^{K'K} \right]^\*. \tag{96}
$$

This is just the symmetry of Γ*KK*� *QQ*� relative to time reversal (Omont, 1977) that is natural in the absence of magnetic field.

In case of the atomic surroundings is axial symmetrical in addition, there are only two nonzero components of *G<sup>L</sup> <sup>M</sup>* in the proper coordinate system, *<sup>G</sup>*<sup>0</sup> <sup>0</sup> and *<sup>G</sup>*<sup>2</sup> <sup>0</sup>. Therefor, only irreducible tensors *<sup>G</sup><sup>L</sup>* <sup>0</sup> (*KK*� *<sup>L</sup>*) and *<sup>G</sup><sup>L</sup>* <sup>0</sup> (*KK*� *L*) are nonzero and real in the system (see relations (82) and (83), and (93) and (94)). Consequently, only *γKK*� *QQ* and <sup>Δ</sup>*KK*� *QQ* are also nonzero and real (see relations (72 and (73)), hence,

$$
\Gamma^{KK'}\_{QQ'} = \delta\_{QQ'} \Gamma^{KK'}\_{QQ'}.\tag{97}
$$

So, in this case Γ*KK*� *QQ* is real for even *K* + *K*� , imaginary for odd *K* + *K*� and

$$
\Gamma\_{00}^{KK'} = 0 \tag{98}
$$

for odd *K* + *K*� because of the following Clebsch-Gordan coefficient symmetry

$$
\langle \mathbf{K}' \mathbf{0} \mathbf{L} \mathbf{0} | \mathbf{K} \mathbf{0} \rangle = (-1)^{K' + L - K} \langle \mathbf{K}' \mathbf{0} \mathbf{L} \mathbf{0} | \mathbf{K} \mathbf{0} \rangle. \tag{99}
$$

#### **4. Deexcitation dynamics**

16 Will-be-set-by-IN-TECH

These properties can be find more straightforward from symmetries (80) and (81) and

*K*+*K*� *GL M KK*� *L* 

*K*+*K*�

<sup>+</sup><sup>1</sup> *<sup>G</sup><sup>L</sup> M KK*� *L* 

*L*), one can see from (80) and (81) that part of *Gαβ* which is symmetrical

*QQ*� , only when *L* is even. As for antisymmetrical part of *Gαβ*,

<sup>3</sup> *G*��

> <sup>3</sup> *G*�

> > *<sup>M</sup>* (*KK*�

*LM*|*KQ*� = (−1)

 *c ωmn*

*<sup>L</sup>*) and *<sup>G</sup><sup>L</sup>*

<sup>3</sup> *G*��*<sup>L</sup>*

> <sup>3</sup> *G*� *L*

 *c ωmn*

*Q*�

Γ*K*� *K*

*QQ*� relative to time reversal (Omont, 1977) that is natural in the

<sup>=</sup> (−1)

<sup>=</sup> (−1)

with respect to permutation of subscripts makes a contribution to *Gαβ* (*KK*�

Taking into account these properties that we can reformulate as *K* + *K*� is even for *Gαβ* (*KK*�

When tensor *Gαβ* is symmetrical (i.e., no external magnetic field), the form of tensor

,2*n c ωmn*

,2*n*+1

*<sup>M</sup>* (*KK*�

,2*n c ωmn*

,2*n*+1

As stated above (see Eqs. (42) -(44) ), in this case there are only four nonzero components of

*L*) are symmetrical with respect to *K* and *K*�

*K*−*K*�

+*Q*−*Q*�

*<sup>K</sup>*−*K*� Γ*K*� *K Q*�*Q* ∗

*L*) is simplified

= *<sup>δ</sup>L*,2*lδK*+*K*�

= *<sup>δ</sup>L*,2*lδK*+*K*�

= *<sup>δ</sup>L*,2*lδK*+*K*�

= *<sup>δ</sup>L*,2*lδK*+*K*�

*QQ*� is different from zero only for even *K* + *K*�

*QQ*� is different from zero only for odd *K* + *K*�

. Similarly,

*L*)

*L*) and to

.

, (89)

*αβ*, (91)

*αβ*, (92)

*L*) are also simplified

*<sup>M</sup>* , (93)

*<sup>M</sup>* . (94)

, evenness of *L*, relation

*<sup>L</sup>*+*<sup>M</sup>* <sup>2</sup>*K*+<sup>1</sup>

<sup>2</sup>*K*�+<sup>1</sup> �*K* −

*QQ*� in the case. Using the fact that

<sup>−</sup>*Q*�−*<sup>Q</sup>* (95)

. (96)

. (90)

Comparing (84) and (87) shows that *γKK*�

comparing (85) and (88) shows that Δ*KK*�

*L*), and hence to Γ*KK*�

*L*) as well of tensor *Gαβ* (*KK*�

*Gαβ KK*� *L*

*Gαβ KK*� *L*

where *<sup>n</sup>* and *<sup>l</sup>* are integer. As a consequence, *<sup>G</sup><sup>L</sup>*

*GL M KK*� *L*

*GL M KK*� *L*

There is additional symmetry of the relaxation tensor Γ*KK*�

Γ*KK*�

*QQ*� = (−1)

*QQ*� = (−1)

Γ*KK*�

(79) and also Clebsch-Gordan coefficient symmetry �*K*�

*<sup>M</sup>* in the proper coordinate system.

*<sup>M</sup>* (*KK*�

�, one can obtain

that we can rewrite using (86) as follows

This is just the symmetry of Γ*KK*�

absence of magnetic field.

*<sup>L</sup>*) and *<sup>G</sup><sup>L</sup>*

*GL M KK*� *L*

*GL M KK*� *L*

*QQ*� only when *L* is odd.

definitions (75) and (76) that yield

and odd for *Gαβ* (*KK*�

it contributes to Γ*KK*�

*Gαβ* (*KK*�

*Gαβ* (*KK*�

*G<sup>L</sup>*

*GL <sup>M</sup>* (*KK*�

*QLM*|*K*� − *Q*�

Deexcitation of upper level is given by (69) with *K* = *Q* = 0

$$\frac{\mathbf{d}\rho\_0^0(t)}{\mathbf{d}t} = -\gamma\_0 \sum\_{\mathbf{K}'\mathbf{Q}'} \Gamma\_{0\mathbf{Q}'}^{0\mathbf{K}'} \rho\_{\mathbf{Q}'}^{\mathbf{K}'}(t) \,. \tag{100}$$

Hereinafter we suppose that there is no external magnetic field. In this case *γ*0Γ0*K*� <sup>0</sup>*Q*� = (−1) *Q*� *γK*� <sup>−</sup>*Q*�/ <sup>√</sup>2*Jm* <sup>+</sup> 1, where *<sup>γ</sup>K*� <sup>−</sup>*Q*� is defined by (40), multiplier (−1) *Q*� transforms covariant component *ρK*� *<sup>Q</sup>*� into contravariant one *<sup>ρ</sup>K*� −*Q*� and denominator <sup>√</sup>2*Jm* <sup>+</sup> 1 reflect the fact that the right-hand side of (100) is variation in time of *ρ*<sup>0</sup> <sup>0</sup>, not of population that is <sup>√</sup>2*Jm* <sup>+</sup> <sup>1</sup>*ρ*<sup>0</sup> 0 as in (39). To obtain temporal variation of the deexcitation, it is necessary to solve consistent differential equations, involving along with Eq. (100) also differential equations for *ρK*� *<sup>Q</sup>*� (*t*), incoming in its right-hand side.

Let us restrict themselves to the case of axial symmetrical atomic surroundings.

As it mentioned above, this case include half-space boundary and spherical particle. From (97), (98), (100), and also (99), it is follows that consistent differential equations, describing deexcitation dynamics in the proper coordinate system, include only multipole components with even *K* and *Q* = 0. The number of such components is [*Jm*] + 1 because of 0 *K* 2*Jm* as noted above (symbol [*Jm*] here and further denotes the integer part of *Jm*). As the relevant Γ*KK*� <sup>00</sup> are real in our case, from (96) we obtain that they are symmetrical relative to *K* and *K*�

$$
\Gamma\_{00}^{KK'} = \Gamma\_{00}^{K'K}.\tag{101}
$$

Hence, the number of different relevant components Γ*KK*� <sup>00</sup> is ([*Jm*] + 1) × ([*Jm*] + 2) /2.

As is known , the general solution of [*Jm*] + 1 consistent linear homogeneous differential equations is given by a linear combination of [*Jm*] + 1 their eigen vectors, each of them varies in time exponentially with its own rate. The rates are eigen values of the consistent equations. The number of the eigen values is also in general equal to [*Jm*] + 1. So, the atomic deexcitation is also usually expressed as a linear sum of [*Jm*] + 1 exponentials.

in the case is given by

� *ρ*<sup>0</sup> <sup>0</sup> (*t*), *ρ*2 <sup>0</sup> (*t*),

elements are following: Γ<sup>00</sup>

<sup>00</sup> <sup>−</sup> <sup>Γ</sup><sup>02</sup> <sup>00</sup>/

Γ<sup>22</sup> <sup>00</sup> <sup>=</sup> <sup>Γ</sup><sup>00</sup> � <sup>=</sup> <sup>1</sup> 2

Substituting (109)-(111) into (105), we obtain

Eigen values *γ*<sup>±</sup> in the case are

*<sup>S</sup>*(*t*) = <sup>1</sup> 3 ⎛

� <sup>1</sup> <sup>−</sup> <sup>Γ</sup><sup>−</sup> Γ �

*e*−*γ*−*<sup>t</sup>* +

Deexcitation Dynamics of a Degenerate Two-Level Atom near (Inside) a Body 327

In the case under consideration that is *Jm* = 1, *Jn* = 0, dimensionless relaxation matrix

Γ<sup>02</sup> 00 Γ �

<sup>00</sup> = (1/2)(*c*/*ωmn*)3tr(**G**��), <sup>Γ</sup><sup>02</sup>

<sup>√</sup>2. So, relevant dimensionless <sup>Γ</sup><sup>±</sup> and <sup>Γ</sup> are

� *c <sup>ω</sup>mn* �<sup>3</sup> � *G*�� *ZZ* + *G*��

� *c <sup>ω</sup>mn* �<sup>3</sup> � *G*�� *ZZ* − *G*��

2*e*−*γ*−*<sup>t</sup>* + *e*−*γ*+*<sup>t</sup>*

*<sup>γ</sup>*<sup>+</sup> <sup>=</sup> <sup>3</sup> 2

*<sup>γ</sup>*<sup>−</sup> <sup>=</sup> <sup>3</sup> 2

that upper level deexcitation is pure exponential. Such cases only three.

� *ρ*<sup>0</sup> <sup>0</sup> (0) *ρ*2 <sup>0</sup> (0)

*c***<sup>0</sup>** ≡

population and alignment of the upper level:

*<sup>e</sup>*−*γ*−*<sup>t</sup>* <sup>−</sup> *<sup>e</sup>*−*γ*+*<sup>t</sup>*

� *c <sup>ω</sup>mn* �<sup>3</sup>

� *c <sup>ω</sup>mn* �<sup>3</sup>

In the case under consideration (i.e., *Jm* = 1, *Jn* = 0) it is possible such excitation conditions

In the first case the atom is excited by light with linear polarization that is collinear to the symmetry axis. Such light excites only one upper sublevel with angular momentum projection

where *ρ*00(0) is population of the sublevel mentioned above. If we multiply fundamental solution matrix (112) on the right by column *c*0, we get the variation in the time of the

�

In the second case the atom is excited by circular polarized light that propagates along symmetry axis. Now the only upper sublevel with angular momentum projection on the

= *c***0***e*

−*γ*+*t*

on the symmetry axis *JmZ* = 0. In this case the initial conditions column is given by

� <sup>=</sup> <sup>1</sup> √3

� *ρ*<sup>0</sup> <sup>0</sup> (*t*) *ρ*2 <sup>0</sup> (*t*)

� *c <sup>ω</sup>mn* �<sup>3</sup> � *G*�� *ZZ* − *G*��

� 1 + Γ− Γ � *e*−*γ*+*<sup>t</sup>*

<sup>−</sup>*e*−*γ*−*<sup>t</sup>* <sup>+</sup> *<sup>e</sup>*−*γ*+*<sup>t</sup>*

�

<sup>00</sup> = −(

*XX*�

*XX*�

� �*e*−*γ*−*<sup>t</sup>* + 2*e*−*γ*+*<sup>t</sup>*

� <sup>√</sup><sup>2</sup> �

*G*��

*G*��

� 1 − √2 �

*XX*�

*<sup>e</sup>*−*γ*−*<sup>t</sup>* <sup>−</sup> *<sup>e</sup>*−*γ*+*<sup>t</sup>*

*ρ*00(0),

�

*ZZ*, (113)

*XX*. (114)

� �

⎞

⎟⎟⎠ *ρ*0

<sup>√</sup>2/2)(*c*/*ωmn*)3(*G*��

, (109)

, (110)

. (111)

. (112)

<sup>0</sup> (0). (108)

*ZZ* − *G*��

*XX*),

⎜⎜⎝

<sup>Γ</sup><sup>+</sup> <sup>=</sup> <sup>3</sup> 4

<sup>Γ</sup><sup>−</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> 4

> <sup>Γ</sup> <sup>=</sup> <sup>3</sup> 4

� �

√2 �

In fact, the eigenvalues are relaxation rates of populations of magnetic sublevels |*Jm* ± *M*� in the case under consideration. Indeed, relevant multipole components *ρ<sup>K</sup>* <sup>0</sup> incoming in the consistent differential equations, describing deexcitation dynamics, are linear combination of the populations of the sublevels |*JmM*� (see (36)). In addition, the sublevels |*JmM*� and <sup>|</sup>*Jm* <sup>−</sup> *<sup>M</sup>*� are transformed one into another (with the sign (−1)*P*, where *<sup>P</sup>* is parity of the level *m*) under reflection in any plane through the symmetry axis (Landau & Lifshitz, 1977). Consequently, the relaxation rates of these sublevels are equal. So, the number of different relaxation rates is [*Jm*] + 1 as stated above with respect to the eigenvalues.

#### **4.1 Deexcitation dynamics in the case of** *Jm* = 1**,** *Jn* = 0

Let us consider in more detail the case when the angular momentums are *Jm* = 1 and *Jn* = 0. In the case under study, deexcitation dynamics is described by only two equations

$$\frac{d\rho\_0^0(t)}{dt} = -\gamma\_0 \left[\Gamma\_{00}^{00} \rho\_0^0(t) + \Gamma\_{00}^{02} \rho\_0^2(t)\right],\tag{102}$$

$$\frac{\mathbf{d}\rho\_0^2(t)}{\mathbf{d}t} = -\gamma\_0 \left[\Gamma\_{00}^{02}\rho\_0^0(t) + \Gamma\_{00}^{22}\rho\_0^2(t)\right].\tag{103}$$

The eigen values *γ*<sup>±</sup> of the consistent equations are

$$
\gamma\_{\pm} = \gamma\_0 \left[ \Gamma\_+ \pm \Gamma \right], \tag{104}
$$

and fundamental solution matrix are

$$S(t) = \frac{1}{2} \begin{pmatrix} \left(1 - \frac{\Gamma\_{-}}{\Gamma}\right) e^{-\gamma\_{-}t} + \left(1 + \frac{\Gamma\_{-}}{\Gamma}\right) e^{-\gamma\_{+}t} & \frac{\Gamma\_{00}^{02}}{\Gamma} \left(-e^{-\gamma\_{-}t} + e^{-\gamma\_{+}t}\right) \\ \frac{\Gamma\_{00}^{02}}{\Gamma} \left(-e^{-\gamma\_{-}t} + e^{-\gamma\_{+}t}\right) & \left(1 + \frac{\Gamma\_{-}}{\Gamma}\right) e^{-\gamma\_{-}t} + \left(1 - \frac{\Gamma\_{-}}{\Gamma}\right) e^{-\gamma\_{+}t} \end{pmatrix} \tag{105}$$

where dimensionless Γ<sup>±</sup> and Γ are defined as

$$
\Gamma\_{\pm} = \frac{1}{2} \left( \Gamma\_{00}^{00} \pm \Gamma\_{00}^{22} \right) , \tag{106}
$$

$$
\Gamma = \sqrt{(\Gamma\_{-})^2 + \left(\Gamma\_{00}^{02}\right)^2}.\tag{107}
$$

Specific solution column of the consistent equations (102)-(103), corresponding to initial conditions given by column *c* = col (*ρ*<sup>0</sup> <sup>0</sup> (0), *<sup>ρ</sup>*<sup>2</sup> <sup>0</sup> (0)) , is obtained by multiplication of fundamental solution matrix on the right by column *c*.

It is known that the excited atomic states that are produced by the absorption of anisotropic resonance light are strongly polarized (Alexandrov et al., 1993; Happer, 1972; Omont, 1977). This atomic polarization results from the directionality or polarization of the light beam. So, immediately after excitation there are nonzero both *ρ*<sup>0</sup> <sup>0</sup> (0) and *<sup>ρ</sup>*<sup>2</sup> <sup>0</sup> (0).

However, let us consider the simplest case of isotropic excitation, when there is only population *ρ*<sup>0</sup> <sup>0</sup> (0) on the upper level at the instant after excitation. So, the solution column in the case is given by

18 Will-be-set-by-IN-TECH

In fact, the eigenvalues are relaxation rates of populations of magnetic sublevels |*Jm* ± *M*�

consistent differential equations, describing deexcitation dynamics, are linear combination of the populations of the sublevels |*JmM*� (see (36)). In addition, the sublevels |*JmM*� and <sup>|</sup>*Jm* <sup>−</sup> *<sup>M</sup>*� are transformed one into another (with the sign (−1)*P*, where *<sup>P</sup>* is parity of the level *m*) under reflection in any plane through the symmetry axis (Landau & Lifshitz, 1977). Consequently, the relaxation rates of these sublevels are equal. So, the number of different

Let us consider in more detail the case when the angular momentums are *Jm* = 1 and *Jn* = 0.

<sup>0</sup> (*t*) <sup>+</sup> <sup>Γ</sup><sup>02</sup>

<sup>0</sup> (*t*) <sup>+</sup> <sup>Γ</sup><sup>22</sup>

*<sup>e</sup>*−*γ*+*<sup>t</sup>* <sup>Γ</sup><sup>02</sup>

1 + Γ− Γ �

� �

<sup>2</sup> + � Γ<sup>02</sup> 00 �2

Specific solution column of the consistent equations (102)-(103), corresponding to initial

It is known that the excited atomic states that are produced by the absorption of anisotropic resonance light are strongly polarized (Alexandrov et al., 1993; Happer, 1972; Omont, 1977). This atomic polarization results from the directionality or polarization of the light beam. So,

However, let us consider the simplest case of isotropic excitation, when there is only

<sup>0</sup> (0) on the upper level at the instant after excitation. So, the solution column

<sup>0</sup> (0), *<sup>ρ</sup>*<sup>2</sup>

00*ρ*<sup>2</sup> <sup>0</sup> (*t*) �

00*ρ*<sup>2</sup> <sup>0</sup> (*t*) �

*γ*<sup>±</sup> = *γ*<sup>0</sup> [Γ<sup>+</sup> ± Γ] , (104)

*e*−*γ*−*<sup>t</sup>* +

<sup>−</sup>*e*−*γ*−*<sup>t</sup>* <sup>+</sup> *<sup>e</sup>*−*γ*+*<sup>t</sup>*

� <sup>1</sup> <sup>−</sup> <sup>Γ</sup><sup>−</sup> Γ � *e*−*γ*+*<sup>t</sup>*

, (106)

. (107)

<sup>0</sup> (0)) , is obtained by multiplication of

00 Γ �

<sup>0</sup> (0) and *<sup>ρ</sup>*<sup>2</sup>

<sup>0</sup> (0).

In the case under study, deexcitation dynamics is described by only two equations

� Γ<sup>00</sup> 00*ρ*<sup>0</sup>

� Γ<sup>02</sup> 00*ρ*<sup>0</sup> <sup>0</sup> incoming in the

, (102)

. (103)

�

⎞

⎟⎟⎠ ,

(105)

in the case under consideration. Indeed, relevant multipole components *ρ<sup>K</sup>*

relaxation rates is [*Jm*] + 1 as stated above with respect to the eigenvalues.

**4.1 Deexcitation dynamics in the case of** *Jm* = 1**,** *Jn* = 0

d*ρ*<sup>0</sup> <sup>0</sup> (*t*) <sup>d</sup>*<sup>t</sup>* <sup>=</sup> <sup>−</sup>*γ*<sup>0</sup>

d*ρ*<sup>2</sup> <sup>0</sup> (*t*) <sup>d</sup>*<sup>t</sup>* <sup>=</sup> <sup>−</sup>*γ*<sup>0</sup>

*e*−*γ*−*<sup>t</sup>* +

� 1 + Γ− Γ �

<sup>−</sup>*e*−*γ*−*<sup>t</sup>* <sup>+</sup> *<sup>e</sup>*−*γ*+*<sup>t</sup>*

<sup>Γ</sup><sup>±</sup> <sup>=</sup> <sup>1</sup> 2 � Γ<sup>00</sup> <sup>00</sup> <sup>±</sup> <sup>Γ</sup><sup>22</sup> 00 �

Γ = � (Γ−)

The eigen values *γ*<sup>±</sup> of the consistent equations are

Γ<sup>02</sup> 00 Γ �

where dimensionless Γ<sup>±</sup> and Γ are defined as

conditions given by column *c* = col (*ρ*<sup>0</sup>

fundamental solution matrix on the right by column *c*.

immediately after excitation there are nonzero both *ρ*<sup>0</sup>

and fundamental solution matrix are

⎛

� <sup>1</sup> <sup>−</sup> <sup>Γ</sup><sup>−</sup> Γ �

⎜⎜⎝

*<sup>S</sup>*(*t*) = <sup>1</sup> 2

population *ρ*<sup>0</sup>

$$
\begin{pmatrix}
\rho\_0^0(t)\,'\_{\,} \\
\rho\_0^0(t)\,'\_{\,}
\end{pmatrix} = \frac{1}{2} \begin{pmatrix}
\left(1 - \frac{\Gamma\_-}{\Gamma}\right)e^{-\gamma\_-t} + \left(1 + \frac{\Gamma\_-}{\Gamma}\right)e^{-\gamma\_+t} \\
\frac{\Gamma\_{00}^{02}}{\Gamma}\left(-e^{-\gamma\_-t} + e^{-\gamma\_+t}\right)
\end{pmatrix} \rho\_0^0(0)\,. \tag{108}
$$

In the case under consideration that is *Jm* = 1, *Jn* = 0, dimensionless relaxation matrix elements are following: Γ<sup>00</sup> <sup>00</sup> = (1/2)(*c*/*ωmn*)3tr(**G**��), <sup>Γ</sup><sup>02</sup> <sup>00</sup> = −( <sup>√</sup>2/2)(*c*/*ωmn*)3(*G*�� *ZZ* − *G*�� *XX*), Γ<sup>22</sup> <sup>00</sup> <sup>=</sup> <sup>Γ</sup><sup>00</sup> <sup>00</sup> <sup>−</sup> <sup>Γ</sup><sup>02</sup> <sup>00</sup>/ <sup>√</sup>2. So, relevant dimensionless <sup>Γ</sup><sup>±</sup> and <sup>Γ</sup> are

$$
\Gamma\_{+} = \frac{3}{4} \left( \frac{c}{\omega\_{mm}} \right)^{3} \left( \mathbf{G}\_{ZZ}^{\prime\prime} + \mathbf{G}\_{XX}^{\prime\prime} \right),
\tag{109}
$$

$$
\Gamma\_- = -\frac{1}{4} \left( \frac{c}{\omega\_{mn}} \right)^3 \left( \mathcal{G}\_{ZZ}^{\prime\prime} - \mathcal{G}\_{XX}^{\prime\prime} \right),
\tag{110}
$$

$$
\Gamma = \frac{3}{4} \left( \frac{c}{\omega\_{\rm mm}} \right)^3 \left( \mathbf{G}\_{\rm ZZ}^{\prime\prime} - \mathbf{G}\_{\rm XX}^{\prime\prime} \right). \tag{111}
$$

Substituting (109)-(111) into (105), we obtain

$$S(t) = \frac{1}{3} \begin{pmatrix} \left[ 2e^{-\gamma\_- t} + e^{-\gamma\_+ t} \right] & \sqrt{2} \left[ e^{-\gamma\_- t} - e^{-\gamma\_+ t} \right] \\ \sqrt{2} \left[ e^{-\gamma\_- t} - e^{-\gamma\_+ t} \right] & \left[ e^{-\gamma\_- t} + 2e^{-\gamma\_+ t} \right] \end{pmatrix}. \tag{112}$$

Eigen values *γ*<sup>±</sup> in the case are

$$\gamma\_{+} = \frac{3}{2} \left(\frac{c}{\omega\_{mn}}\right)^{3} \mathcal{G}''\_{ZZ'} \tag{113}$$

$$
\gamma\_- = \frac{3}{2} \left(\frac{c}{\omega\_{mn}}\right)^3 G\_{XX}''.\tag{114}
$$

In the case under consideration (i.e., *Jm* = 1, *Jn* = 0) it is possible such excitation conditions that upper level deexcitation is pure exponential. Such cases only three.

In the first case the atom is excited by light with linear polarization that is collinear to the symmetry axis. Such light excites only one upper sublevel with angular momentum projection on the symmetry axis *JmZ* = 0. In this case the initial conditions column is given by

$$\mathbf{c\_{0}} \equiv \begin{pmatrix} \rho\_{0}^{0}(0) \\ \rho\_{0}^{2}(0) \end{pmatrix} = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 \\ -\sqrt{2} \end{pmatrix} \rho\_{00}(0).$$

where *ρ*00(0) is population of the sublevel mentioned above. If we multiply fundamental solution matrix (112) on the right by column *c*0, we get the variation in the time of the population and alignment of the upper level:

$$
\begin{pmatrix} \rho\_0^0(t) \\ \rho\_0^2(t) \end{pmatrix} = \mathbf{c\_0} e^{-\gamma\_+ t}
$$

In the second case the atom is excited by circular polarized light that propagates along symmetry axis. Now the only upper sublevel with angular momentum projection on the

with the simple model of the classic scattering dipole, in spite of the fact that fluorescence is

Deexcitation Dynamics of a Degenerate Two-Level Atom near (Inside) a Body 329

It should be noted that consistent equations (102)-(103) describe deexcitation dynamics also in the case *Jm* = 1, *Jn* = 1, or *Jm* = 1, *Jn* = 2, and also in the case *Jm* = 3/2, *Jn* = 1/2, and either *Jm* = 3/2, *Jn* = 3/2, or *Jm* = 3/2, *Jn* = 5/2. Of course, specific values of the dimensionless

It should be pointed out too that in the case *Jm* = 3/2 and *Jn* = 1/2 there is the only exciting light polarization, namely linear polarization along symmetry axis, that leds to the pure exponential decay of the excited state because of the relaxation rate equality of the excited

In the chapter we have proposed a general approach to the problem of deexcitation of a degenerate two-level atom near (inside) a body. On the basis of the approach the master equation for density matrix in the polarization moments representation was obtained.

We have shown that relaxation dynamics of a polarization moment is described in general by a consistent linear equations for all 2*Jm* + 1 polarization moments of the excited level, where *Jm* is the total momentum of the level. We have expressed relaxation matrix elements of the consistent linear equations in terms of the field response tensor that can be found as the electric

An additional relaxation matrix symmetry is recognized in the case when there is no external quasistatic magnetic field, and as a result, the field response tensor is symmetrical one. Therefore, the tensor may be diagonalized. We have shown that relaxation matrix depends only on the trace of the field response tensor, on the difference between the most principal value of the diagonal response tensor and the half-sum of two others, and also on the

Axial symmetric atomic surroundings gives rise to one more additional symmetry of the relaxation matrix. In this case it depends only on the trace of the field response tensor and

We have shown that deexcitation dynamics of the degenerate two-level atom in the conditions under consideration represents multiexponential decay. In the case of the axial symmetric atomic surroundings, the number of the exponential is equal to [*Jm*] + 1, where [*Jm*] is the integer part of *Jm*. So, the simple exponential decay of the atomic excitation is possible only in two cases, namely, when *Jm* = 0 or *Jm* = 1/2. We have shown that simple exponential decay of the atomic excitation is also possible in the case of *Jm* = 1, *Jn* = 0 and on special polarizations of exciting light, namely on the linear polarization that is collinear or orthogonal to the axial symmetry axis, and on the circular polarizations rotating in the plane that is orthogonal to the symmetry axis. In this exceptional cases both the excitation and decay of the corresponding upper states follow the one and the same respective channel. Simple exponential decay of the atomic excitation is possible too in the case *Jm* = 3/2 and *Jn* = 1/2 when exciting light

Our analysis have carried out in the absence of hyperfine structure on the combine energy levels. However, it can be easily expanded straightforward on general case by expanding

<sup>00</sup> in these cases differ from considered above.

the two-step process, rather than scattering.

sublevels (*JmZ* = ±1/2) due to the axial symmetry.

field of the classic oscillating unit dipole situated near the body.

We have found symmetry of the relaxation matrix.

on the difference between its two principal values.

polarization is linear oriented along symmetry axis.

difference between these two.

Γ<sup>00</sup> <sup>00</sup>, <sup>Γ</sup><sup>02</sup>

<sup>00</sup>, and <sup>Γ</sup><sup>22</sup>

**5. Conclusions**

symmetry axis *JmZ* = +1 (or *JmZ* = −1 for the opposite circular polarization) is excited. Initial conditions column in the case is given by

$$\mathbf{c\_1} \equiv \begin{pmatrix} \rho\_0^0(0) \\ \rho\_0^2(0) \end{pmatrix} = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 \\ 1/\sqrt{2} \end{pmatrix} \rho\_{11}(0),\tag{115}$$

where *ρ*11(0) is population of the exited sublevel. The solution corresponding to this column is

$$
\begin{pmatrix} \rho\_0^0(t) \\ \rho\_0^2(t) \end{pmatrix} = \mathbf{c\_1} e^{-\gamma\_- t}.\tag{116}
$$

Lastly, in the third case the atom is excited by light with linear polarization that is orthogonal to the symmetry axis. It has been known that such polarization can be represented by the sum of the opposite circular polarization with the same amplitude, rotating in the plane that is orthogonal to the symmetry axis. This case is reduced to the previous one because of only two upper sublevels with angular momentum projection on the symmetry axis *JmZ* = ±1 are excited independently with equal probability, and hence *<sup>ρ</sup>*11(0) = *<sup>ρ</sup>*−1−1(0). The rates of decay of the both excited sublevels into the only low state are equal due to axial symmetry. Deexcitation dynamics in the case also given by (116).

These three exceptional cases of simple exponetial deexcitation can be physically interpreted as follows. In every case the excited state transforms to the only low state by means of one channel. The decay itself is induced by the optical transition oscillating dipole that arises due to interaction of the excited atom with the electric field quantum oscillations. Both the direction of the dipole oscillation and the direction of the exciting light polarization are the same due to the one and the same channel of excitation and deexcitation (see Fig. 1).

(a) Exciting light is linear polarized along (or transversely to) the symmetry axis passing through the atom and body; Z – axis is along (or transversely to) this axis

(b) Exciting light is circular polarized and propagates along the symmetry axis that is Z – axis

Fig. 1. Exceptional polarizations of the exciting light that led to the pure exponential decay of the excited atomic state (*ω* and *ω<sup>f</sup>* are frequencies of the exciting light and fluorescence respectively)

Precisely owing to this fact, experimental results of the measurement of the decay of the fluorescence signal (Amos & Barnes, 1997; Chance et al., 1978; Drexhage et al., 1968; Fort & Grésillon, 2008; Kreiter et al., 2002; Snoeks et al., 1995; Vallée et al., 2001) are in good agreement with the simple model of the classic scattering dipole, in spite of the fact that fluorescence is the two-step process, rather than scattering.

It should be noted that consistent equations (102)-(103) describe deexcitation dynamics also in the case *Jm* = 1, *Jn* = 1, or *Jm* = 1, *Jn* = 2, and also in the case *Jm* = 3/2, *Jn* = 1/2, and either *Jm* = 3/2, *Jn* = 3/2, or *Jm* = 3/2, *Jn* = 5/2. Of course, specific values of the dimensionless Γ<sup>00</sup> <sup>00</sup>, <sup>Γ</sup><sup>02</sup> <sup>00</sup>, and <sup>Γ</sup><sup>22</sup> <sup>00</sup> in these cases differ from considered above.

It should be pointed out too that in the case *Jm* = 3/2 and *Jn* = 1/2 there is the only exciting light polarization, namely linear polarization along symmetry axis, that leds to the pure exponential decay of the excited state because of the relaxation rate equality of the excited sublevels (*JmZ* = ±1/2) due to the axial symmetry.

## **5. Conclusions**

20 Will-be-set-by-IN-TECH

symmetry axis *JmZ* = +1 (or *JmZ* = −1 for the opposite circular polarization) is excited.

where *ρ*11(0) is population of the exited sublevel. The solution corresponding to this column

Lastly, in the third case the atom is excited by light with linear polarization that is orthogonal to the symmetry axis. It has been known that such polarization can be represented by the sum of the opposite circular polarization with the same amplitude, rotating in the plane that is orthogonal to the symmetry axis. This case is reduced to the previous one because of only two upper sublevels with angular momentum projection on the symmetry axis *JmZ* = ±1 are excited independently with equal probability, and hence *<sup>ρ</sup>*11(0) = *<sup>ρ</sup>*−1−1(0). The rates of decay of the both excited sublevels into the only low state are equal due to axial symmetry.

These three exceptional cases of simple exponetial deexcitation can be physically interpreted as follows. In every case the excited state transforms to the only low state by means of one channel. The decay itself is induced by the optical transition oscillating dipole that arises due to interaction of the excited atom with the electric field quantum oscillations. Both the direction of the dipole oscillation and the direction of the exciting light polarization are the

same due to the one and the same channel of excitation and deexcitation (see Fig. 1).

= *c***1***e*

 1 1/√<sup>2</sup>

−*γ*−*t*

*ρ*11(0), (115)

. (116)

(b) Exciting light is circular polarized and propagates along the symmetry axis that is

Z – axis

Fig. 1. Exceptional polarizations of the exciting light that led to the pure exponential decay of the excited atomic state (*ω* and *ω<sup>f</sup>* are frequencies of the exciting light and fluorescence

Precisely owing to this fact, experimental results of the measurement of the decay of the fluorescence signal (Amos & Barnes, 1997; Chance et al., 1978; Drexhage et al., 1968; Fort & Grésillon, 2008; Kreiter et al., 2002; Snoeks et al., 1995; Vallée et al., 2001) are in good agreement

 <sup>=</sup> <sup>1</sup> √3

 *ρ*<sup>0</sup> <sup>0</sup> (*t*) *ρ*2 <sup>0</sup> (*t*)

Initial conditions column in the case is given by

is

*c***<sup>1</sup>** ≡

Deexcitation dynamics in the case also given by (116).

(a) Exciting light is linear polarized along (or transversely to) the symmetry axis passing through the atom and body; Z – axis is along

(or transversely to) this axis

respectively)

 *ρ*<sup>0</sup> <sup>0</sup> (0) *ρ*2 <sup>0</sup> (0)

> In the chapter we have proposed a general approach to the problem of deexcitation of a degenerate two-level atom near (inside) a body. On the basis of the approach the master equation for density matrix in the polarization moments representation was obtained.

> We have shown that relaxation dynamics of a polarization moment is described in general by a consistent linear equations for all 2*Jm* + 1 polarization moments of the excited level, where *Jm* is the total momentum of the level. We have expressed relaxation matrix elements of the consistent linear equations in terms of the field response tensor that can be found as the electric field of the classic oscillating unit dipole situated near the body.

We have found symmetry of the relaxation matrix.

An additional relaxation matrix symmetry is recognized in the case when there is no external quasistatic magnetic field, and as a result, the field response tensor is symmetrical one. Therefore, the tensor may be diagonalized. We have shown that relaxation matrix depends only on the trace of the field response tensor, on the difference between the most principal value of the diagonal response tensor and the half-sum of two others, and also on the difference between these two.

Axial symmetric atomic surroundings gives rise to one more additional symmetry of the relaxation matrix. In this case it depends only on the trace of the field response tensor and on the difference between its two principal values.

We have shown that deexcitation dynamics of the degenerate two-level atom in the conditions under consideration represents multiexponential decay. In the case of the axial symmetric atomic surroundings, the number of the exponential is equal to [*Jm*] + 1, where [*Jm*] is the integer part of *Jm*. So, the simple exponential decay of the atomic excitation is possible only in two cases, namely, when *Jm* = 0 or *Jm* = 1/2. We have shown that simple exponential decay of the atomic excitation is also possible in the case of *Jm* = 1, *Jn* = 0 and on special polarizations of exciting light, namely on the linear polarization that is collinear or orthogonal to the axial symmetry axis, and on the circular polarizations rotating in the plane that is orthogonal to the symmetry axis. In this exceptional cases both the excitation and decay of the corresponding upper states follow the one and the same respective channel. Simple exponential decay of the atomic excitation is possible too in the case *Jm* = 3/2 and *Jn* = 1/2 when exciting light polarization is linear oriented along symmetry axis.

Our analysis have carried out in the absence of hyperfine structure on the combine energy levels. However, it can be easily expanded straightforward on general case by expanding

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**Part 3** 

**Properties and Applications** 


URL: *http://www.sciencedirect.com/science/article/pii/0030401873902393*


URL: *http://www.sciencedirect.com/science/article/pii/S0009261401011198*

