**3. Plasmonic optical tweezers**

In order to overcome the limit on particle size that can be trapped in conventional optical tweezers, plasmonic optical tweezers have been proposed Novotny et al. [17, 26–33]. Plasmonic optical tweezers employ plasmonic nanoantenna, which can squeeze light to nanoscale volumes comparable to the size of the target particles, thus significantly enhancing the gradient force applied on nanoscale particles. Unlike the light-matter interaction between dielectric materials and electromagnetic wave, plasmonic materials uniquely react to the light field through freeelectron-photon coupling to generate a specific type of surface wave called surface plasmon polariton (SPP). Furthermore, a subwavelength plasmonic particle will efficiently couple to propagating light to generate localized surface plasmon resonance (LSPR) and further enhance the electric field due to this resonance. These phenomena permit light to be strongly localized at deep subwavelength scales. Besides, surface plasmon structures offer additional advantages on lab-on-a-chip manipulation due to its compatibility with integrated photonics devices [34, 35].

There are two main challenges with the use of plasmonic nanoantennas for nearfield nano-optical tweezers. These challenges include the Ohmic loss in the materials, which invariably results in loss-induced heating, and the need to rely on slow Brownian diffusion to transport particles towards the illuminated nanoantenna. The Ohmic loss in plasmonic materials at visible or near-IR is inevitable. It will not only hamper the efficiency of particle trapping due to the need to generate enough trapping potential energy to overcome the Brownian motion and possible thermophoretic force, but the heat generated by Ohmic loss would be problematic in many experiments. Though the light field is tightly confined near an illuminated plasmonic nanoantenna leading to very high field intensity enhancement [16], which are advantageous for particle trapping, the damage from temperature rise due to this field enhancement affects the experimental trapping stability in several ways. Experimentally, bioparticles are vulnerable in an environment with increasing temperature. Furthermore, excessive photothermal heating could deform the plasmonic nanoantenna [36]. In aqueous solutions, this heating effect would even generate bubbles by boiling the water and disrupt the entire experiment. Wang et al. demonstrated that by integrating a heat sink in plasmonic tweezers made up of layers of high thermal conductivity materials, the adverse effects arising from the heating effect can be mitigated [37].

Rather than treating the photothermal heating effect as detrimental in plasmonic nanotweezer experiments, plasmonic heating can be harnessed to enable new

heating. Although, several research works in the literature have considered the loss in plasmonics as detrimental, it is now known that the loss in plasmonics can be beneficial to fast on-chip nano-particle manipulation [17, 20]. The loss-induced heating effect has given rise to the burgeoning field of thermoplasmonic

*American Chemical Society and (e) [15] Copyright © 2019 American Chemical Society.*

*Various applications based on metasurfaces. (a) SEM image of metasurface lens composed of silicon nano-post array. (b) SEM image of a vortex beam generator made of silicon nano-post array. (c) A plasmonic metasurface hologram insensitive to light polarization states. (d) A vortex beam opto-multiplexer and demultiplexer made of gold at terahertz. (e) A titanium dioxide metasurface enhancing third harmonic generation to produce ultraviolet light. Images: (a) and (b) are adapted from [9] Copyright © 2016 American Chemical Society, (c): [13] Copyright © 2018 American Chemical Society, (d) [14] Copyright © 2017*

Optical tweezer, which employs a tightly focused laser to trap microscopic objects have proven to be a versatile tool for many scientific researches such as in biophysics [21] and was recently recognized with a 2018 Physics Nobel prize. The initial experiment to trap particles with a laser beam used two counter-propagating loosely focused beams to localize the particles at a node of the generated standing wave, and this was reported by Arthur Ashkin and his colleagues in 1970 [22]. Subsequently, a single beam optical tweezer, which employs a tightly focused laser beam to achieve three-dimensional manipulation and trapping of microscale particles was demonstrated in 1986 [23]. Dielectric particles are trapped when the refractive index of the particles is higher than the refractive index of the surround-

Generally, the optical force is decomposed into two parts, the gradient force and the scattering force. The scattering force is also called radiation pressure and acts in the direction of light propagation. The gradient force is the significant part in single beam optical trapping procedure, because it is oriented perpendicular to the axis and push particles towards a region with higher optical intensity. Thus, the gradient force ensures that a particle is trapped in an optical tweezer. The total optical force induced on a particle can be determined by Maxwell's Stress Tensor method (MST). When the size of the particle is much smaller than the wavelength of trapping light, the particle can be considered as a dipole, so that the dipole approximation can be

metasurface.

**Figure 1.**

*Nanoplasmonics*

ing medium.

**98**

**2. Optical trapping and optical tweezers**

functionalities in near-field nano-optical trapping. Ndukaife et al. proposed and demonstrated that thermal effect due heating loss can actually promote the trapping process thereby eliminating the need to rely on slow Brownian diffusion to deliver particles towards the plasmonic nanotweezer [18, 20]. This new platform, referred to as electrothermoplasmonic tweezers harnesses the localized photothermal heating from an illuminated plasmonic nanoantenna to establish a thermal gradient in the fluid. The thermal gradient results in a gradient in the permittivity and electrical conductivity of the fluid. An applied AC electrical field acts on these gradients to result in rapid microfluidic flow motion, which transports particles towards the plasmonic hotspot. This typical fluid motion is called electrothermoplasmonic (ETP) flow. Under this condition, heating effect actually speeds up the loading procedure and converts the slow traditional diffusion-based particle loading of conventional plasmonic nano-tweezers into a fast and directional particle loading in electrothermoplasmonic tweezers. The technique works with small temperature rises of a few degrees and only the thermal gradient needs to be optimized, thereby making it suitable for handling biological objects.

*c=c*<sup>0</sup> ¼ exp ½ � �ð Þ *DT=D* ð Þ *T* � *T*<sup>0</sup> (3)

*Ftherm* ¼ �*KbT z*ð Þ*ST*∇*T* (4)

*<sup>T</sup>* represents a high-*T* thermophobic limit and *T*<sup>∗</sup> is the tempera-

*<sup>T</sup>*<sup>∗</sup> � *<sup>T</sup> T*0

(5)

Finally, the thermophoretic force induced by thermal gradients on a particle in

Here, *Kb* stands for Boltzmann's constant and *T z*ð Þ corresponds to the temperature at a given position *z*. Soret coefficient *ST* is defined as the ratio: *ST* ¼ *DT=D*, which depicts how strong the thermodiffusion is at steady state. The Soret coefficient is influenced by many factors including temperature, size of the particles, surface charge of the particles and ions in the solvent. From Eq. (4), we concluded that the thermophoretic force is proportional to Soret coefficient but with opposite sign. It is useful to know that Soret coefficient is also temperature dependent and flips its sign from positive to negative by decreasing temperature, as expressed in

ture where *ST* switches sign, and *T*<sup>0</sup> embodies the strength of temperature effects.

*<sup>T</sup>* 1 � exp

The direction of the thermophoretic force can be tuned from a repulsive to an attractive force by tuning the interfacial permittivity of the electrical double layer (EDL) surrounding the particles in solution. The EDL exists on the surface of an object when it is exposed to a fluid. The object could be a small charged particle floating inside the liquid or a surface coated with fluid. The charged object attracts counterions from the solution to screen the surface charge. This layer is called the Stern layer. This charged object consequently attracts ions from the liquid with opposite charge via Coulombic force to electrically screen the first layer and the particle. The second layer is called diffuse layer. The potential difference across the diffuse layer is defined as Zeta potential, *ζ*. The charges in the diffuse layer are not as tightly anchored to the particle as the first layer and the thickness can be affected

Based on Anderson's model [42], the thermophoretic mobility of the particle is

where *ε* is the solvent permittivity, and *Λ*<sup>1</sup> and *Λ<sup>p</sup>* are the thermal conductivities of the solvent and the particle, respectively. In bulk water, for example, the differ-

1 þ

*∂lnε*

*<sup>∂</sup>lnT <sup>ζ</sup>*<sup>2</sup> (6)

*<sup>∂</sup>lnT* ¼ �1*:*4 at room temperature. In

2*Λ*<sup>1</sup> 2*Λ*<sup>1</sup> þ *Λ<sup>p</sup>*

building up the model of ET flow numerical analysis, this term can be taken into account to ensure the accuracy of the results [16]. Inside an EDL, however, the value of *<sup>∂</sup>ln<sup>ε</sup> <sup>∂</sup>lnT* can reach +2.4, which is crucial to reverse the Soret coefficient and induce a negative thermophoresis behavior. Hence, one way to induce negative thermophoresis behavior is to ensure that an EDL has been established, by charging the surface of the particles. Adding surfactants, such as Cetyltrimethylammonium Chloride (CTAC), for example, into particle solution can help to build up an EDL on

At that moment, force applied onto particles switches its orientation from

*ST*ð Þ¼ *<sup>T</sup> <sup>S</sup>*<sup>∞</sup>

*Nanomanipulation with Designer Thermoplasmonic Metasurface*

*DOI: http://dx.doi.org/10.5772/intechopen.91880*

*DT* ¼ � *<sup>ε</sup>*

ential permittivity change with temperature *<sup>∂</sup>ln<sup>ε</sup>*

2*ηT*

the fluid is given by [40]:

Eq. (5) [38, 39]. *S*<sup>∞</sup>

by tangential stress [41].

the particles [27].

**101**

associated with its Zeta potential.

"thermophobic" to "thermophilic" behavior.

By all means, plasmonic optical tweezers are promising and especially useful for manipulation of extremely small particles. By careful thermal engineering to mitigate excessive heating effect, plasmonic optical tweezers can be utilized to enable many applications.

## **4. Electrothermoplasmonic, electro-osmotic and thermophoretic effects**

As mentioned in the previous section, the heating effect from plasmonic nanoantennas can be utilized to enable new capabilities in near-field nano-optical trapping. The physics nature behind this kind of trapping comprises the interplay of multiple physical phenomena. In this section, we describe several forces that can be experienced by objects in electrothermoplasmonic tweezers in the presence of optical illumination and applied AC electric field. To understand how the thermal effect influences the trapping process, we firstly look at the thermophoresis phenomenon, which is the motion of particles or molecules in the presence of thermal gradients. Unlike the normal diffusion process due to Brownian motion, thermophoresis is induced by a temperature gradient ∇*T* to cause the drifting of particles. In most cases, particles prefer to move towards the region of lower temperature, away from the plasmonic nanoantenna. This trend is called "thermophobic" behavior or positive thermophoresis. However, this motion can be reversed in certain instances whereby particles will move towards the region of higher temperature, which is called "thermophilic" behavior, also known as negative thermophoresis.

Quantitatively, in a diluted suspension (particle weight fraction w < < 1), the mass flow *J* can be written as [38, 39]:

$$J \approx -D\nabla c - cD\_T \nabla T,\tag{1}$$

where *D* and *DT* are the Brownian diffusion coefficient and thermodiffusion coefficient, respectively.*c* denotes the concentration. In diluted concentrations, thermodiffusion velocity is generally assumed to be linearly dependent on the temperature gradient ∇*T* with the thermodiffusion coefficient *DT*:

$$
\overrightarrow{v} = -D\_T \nabla T \tag{2}
$$

Under steady-state, the thermodiffusion is balanced by ordinary diffusion and the concentration coefficients are related in an exponential law:

*Nanomanipulation with Designer Thermoplasmonic Metasurface DOI: http://dx.doi.org/10.5772/intechopen.91880*

functionalities in near-field nano-optical trapping. Ndukaife et al. proposed and demonstrated that thermal effect due heating loss can actually promote the trapping process thereby eliminating the need to rely on slow Brownian diffusion to deliver particles towards the plasmonic nanotweezer [18, 20]. This new platform, referred to as electrothermoplasmonic tweezers harnesses the localized photothermal heating from an illuminated plasmonic nanoantenna to establish a thermal gradient in the fluid. The thermal gradient results in a gradient in the permittivity and electrical conductivity of the fluid. An applied AC electrical field acts on these gradients to result in rapid microfluidic flow motion, which transports particles towards the plasmonic hotspot. This typical fluid motion is called electrothermoplasmonic (ETP) flow. Under this condition, heating effect actually speeds up the loading procedure and converts the slow traditional diffusion-based particle loading of conventional plasmonic nano-tweezers into a fast and directional particle loading in electrothermoplasmonic tweezers. The technique works with small temperature rises of a few degrees and only the thermal gradient needs to be optimized, thereby

By all means, plasmonic optical tweezers are promising and especially useful for manipulation of extremely small particles. By careful thermal engineering to mitigate excessive heating effect, plasmonic optical tweezers can be utilized to

**4. Electrothermoplasmonic, electro-osmotic and thermophoretic effects**

As mentioned in the previous section, the heating effect from plasmonic nanoantennas can be utilized to enable new capabilities in near-field nano-optical trapping. The physics nature behind this kind of trapping comprises the interplay of multiple physical phenomena. In this section, we describe several forces that can be experienced by objects in electrothermoplasmonic tweezers in the presence of optical illumination and applied AC electric field. To understand how the thermal effect influences the trapping process, we firstly look at the thermophoresis phenomenon, which is the motion of particles or molecules in the presence of thermal gradients. Unlike the normal diffusion process due to Brownian motion, thermophoresis is induced by a temperature gradient ∇*T* to cause the drifting of particles. In most cases, particles prefer to move towards the region of lower temperature, away from the plasmonic nanoantenna. This trend is called "thermophobic" behavior or positive thermophoresis. However, this motion can be reversed in certain instances whereby particles will move towards the region of higher temperature, which is

called "thermophilic" behavior, also known as negative thermophoresis.

temperature gradient ∇*T* with the thermodiffusion coefficient *DT*:

the concentration coefficients are related in an exponential law:

*v*

Quantitatively, in a diluted suspension (particle weight fraction w < < 1), the

where *D* and *DT* are the Brownian diffusion coefficient and thermodiffusion coefficient, respectively.*c* denotes the concentration. In diluted concentrations, thermodiffusion velocity is generally assumed to be linearly dependent on the

Under steady-state, the thermodiffusion is balanced by ordinary diffusion and

*J* ≈ � *D*∇*c* � *cDT*∇*T*, (1)

! ¼ �*DT*∇*<sup>T</sup>* (2)

making it suitable for handling biological objects.

enable many applications.

*Nanoplasmonics*

mass flow *J* can be written as [38, 39]:

**100**

$$c/c\_0 = \exp\left[-(D\_T/D)(T - T\_0)\right] \tag{3}$$

Finally, the thermophoretic force induced by thermal gradients on a particle in the fluid is given by [40]:

$$F\_{therm} = -K\_b T(\mathbf{z}) \mathbf{S}\_T \nabla T \tag{4}$$

Here, *Kb* stands for Boltzmann's constant and *T z*ð Þ corresponds to the temperature at a given position *z*. Soret coefficient *ST* is defined as the ratio: *ST* ¼ *DT=D*, which depicts how strong the thermodiffusion is at steady state. The Soret coefficient is influenced by many factors including temperature, size of the particles, surface charge of the particles and ions in the solvent. From Eq. (4), we concluded that the thermophoretic force is proportional to Soret coefficient but with opposite sign. It is useful to know that Soret coefficient is also temperature dependent and flips its sign from positive to negative by decreasing temperature, as expressed in Eq. (5) [38, 39]. *S*<sup>∞</sup> *<sup>T</sup>* represents a high-*T* thermophobic limit and *T*<sup>∗</sup> is the temperature where *ST* switches sign, and *T*<sup>0</sup> embodies the strength of temperature effects. At that moment, force applied onto particles switches its orientation from "thermophobic" to "thermophilic" behavior.

$$\mathcal{S}\_T(T) = \mathcal{S}\_T^\approx \left[ \mathbf{1} - \exp\left(\frac{T^\*-T}{T^0}\right) \right] \tag{5}$$

The direction of the thermophoretic force can be tuned from a repulsive to an attractive force by tuning the interfacial permittivity of the electrical double layer (EDL) surrounding the particles in solution. The EDL exists on the surface of an object when it is exposed to a fluid. The object could be a small charged particle floating inside the liquid or a surface coated with fluid. The charged object attracts counterions from the solution to screen the surface charge. This layer is called the Stern layer. This charged object consequently attracts ions from the liquid with opposite charge via Coulombic force to electrically screen the first layer and the particle. The second layer is called diffuse layer. The potential difference across the diffuse layer is defined as Zeta potential, *ζ*. The charges in the diffuse layer are not as tightly anchored to the particle as the first layer and the thickness can be affected by tangential stress [41].

Based on Anderson's model [42], the thermophoretic mobility of the particle is associated with its Zeta potential.

$$D\_T = -\frac{\varepsilon}{2\eta T} \frac{2\Lambda\_1}{2\Lambda\_1 + \Lambda\_p} \left( 1 + \frac{\partial l n \epsilon}{\partial l n T} \right) \zeta^2 \tag{6}$$

where *ε* is the solvent permittivity, and *Λ*<sup>1</sup> and *Λ<sup>p</sup>* are the thermal conductivities of the solvent and the particle, respectively. In bulk water, for example, the differential permittivity change with temperature *<sup>∂</sup>ln<sup>ε</sup> <sup>∂</sup>lnT* ¼ �1*:*4 at room temperature. In building up the model of ET flow numerical analysis, this term can be taken into account to ensure the accuracy of the results [16]. Inside an EDL, however, the value of *<sup>∂</sup>ln<sup>ε</sup> <sup>∂</sup>lnT* can reach +2.4, which is crucial to reverse the Soret coefficient and induce a negative thermophoresis behavior. Hence, one way to induce negative thermophoresis behavior is to ensure that an EDL has been established, by charging the surface of the particles. Adding surfactants, such as Cetyltrimethylammonium Chloride (CTAC), for example, into particle solution can help to build up an EDL on the particles [27].

The photothermal heating of the fluid by the plasmonic nanoantennas can also be utilized to induce strong electro-convection fluidic motion, which is called electrothermal (ET) effect. When the temperature inside fluid is no longer uniform due to a thermal gradient, a gradient in the permittivity and the conductivity of the fluid will induced as well. In such a system, a local free charge distribution must be present if Gauss's Law and charge conservation are to be satisfied simultaneously. When an AC electric field is applied, both free and bounded local charge density responds to the applied electric field so that a non-zero body force is generated on the fluid:

$$f\_{ET} = \rho\_e E - \frac{1}{2}|E|^2 \nabla \varepsilon\_m \tag{7}$$

Finally, the Navier-Stokes equation is solved to find the velocity of the fluid:

where **F** here is the electrothermal force described in the prior section [43]. We have also established the theoretical framework between the ETP flow velocity with laser power and external AC electric field. It is noted that the ETP flow velocity scales linearly with laser power and quadratically with AC electric field

uð Þ¼ **r F**, (12)

E<sup>∥</sup> (13)

<sup>0</sup> (15)

*<sup>j</sup><sup>ω</sup>* (16)

(14)

<sup>ρ</sup>0½ � <sup>u</sup>ð Þ� **<sup>r</sup>** <sup>∇</sup> <sup>u</sup>ð Þþ **<sup>r</sup>** <sup>∇</sup>*p*ð Þ� **<sup>r</sup>** *<sup>η</sup>*∇<sup>2</sup>

*Nanomanipulation with Designer Thermoplasmonic Metasurface*

*DOI: http://dx.doi.org/10.5772/intechopen.91880*

Electro-osmotic flow is another category of body flow happening in a microfluidic channel [45]. Electro-osmosis occurs when tangential electric field acts on the loosely bound charges in the EDL along the channel walls. Because of the existence of electrical double layer, near the charged channel walls the tangential electric field will act on the EDL charges to induce an electro-osmotic flow with

> us ¼ � <sup>ε</sup>w<sup>ζ</sup> η

where E<sup>∥</sup> is the tangential component of the bulk electric field, η is the fluid viscosity and *ζ* is the zeta potential. When there is a plasmonic structure existing inside a microfluid channel on the channel wall, this plasmonic structure perturbs the AC electric field, resulting in a non-zero tangential component of the AC electric field. For instance, Jamshidi or Hwang and their colleagues integrated photoconductive material as the electrode of applied bias inside a microfluidic channel to design a so-called "NanoPen" device for dynamical manipulation on

This perturbed external electric field not only contributes to electro-osmosis, but could also induce a dielectrophoretic force on the suspended particles [48]. Particles experience dielectrophoresis only when the external field is non-uniform and the force that particles feel, the DEP force, does not depend on the polarity of the AC field. DEP can be observed both in AC or DC electric field. There is a crucial

> <sup>K</sup> <sup>¼</sup> <sup>ε</sup><sup>2</sup> � <sup>ε</sup><sup>1</sup> ε<sup>2</sup> þ 2ε<sup>1</sup>

From which, we understand DEP force is proportional to particle volume and magnitude of K. The direction of DEP force is along the gradient of the electric field

In AC field and lossy medium, the Clausius-Mossoti factor is frequency depen-

<sup>1</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>1</sup> <sup>þ</sup> *<sup>σ</sup>*<sup>1</sup>

The Clausius-Mossoti factor determines whether the DEP force is attractive or repulsive ant it depends on the frequency of AC field and electric properties of

*K*∇*E*<sup>2</sup>

*<sup>j</sup><sup>ω</sup>* and *<sup>ε</sup>* <sup>∗</sup>

<sup>2</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>2</sup> <sup>þ</sup> *<sup>σ</sup>*<sup>2</sup>

In terms of Clausius-Mossoti factor, the DEP force is expressed as:

intensity and its sign depends on the sign of Clausius-Mossoti factor.

FDEP <sup>¼</sup> <sup>2</sup>πε1*R*<sup>3</sup>

particles using low-power laser illumination [46, 47].

evaluator for DEP force direction called Clausius-Mossoti factor.

amplitude [16].

slip velocity given by:

dent and it is given by:

particles and medium.

**103**

<sup>K</sup> <sup>¼</sup> <sup>ε</sup> <sup>∗</sup>

ε ∗ <sup>2</sup> þ 2ε <sup>∗</sup> 1 , *ε* <sup>∗</sup>

<sup>2</sup> � <sup>ε</sup> <sup>∗</sup> 1

After perturbative expansion in the limit of small temperature gradient, the force density is expressed as [43]:

$$f\_{ET} = \frac{1}{2} \operatorname{Re} \left\{ \frac{\epsilon (a - \beta) (\nabla T.E) E^\*}{\sigma + i \alpha e} + \frac{1}{2} \epsilon \alpha |E|^2 \nabla T \right\} \tag{8}$$

where *α* ¼ <sup>1</sup>*=ε* d*ε* <sup>d</sup>*<sup>T</sup>* and *β* ¼ <sup>1</sup>*=<sup>σ</sup>* <sup>d</sup>*<sup>σ</sup>* <sup>d</sup>*<sup>T</sup>*. *ε* stands for permittivity and *σ* stands for conductivity. *ω* is the AC frequency. So far, the body force is decomposed into two parts: the first term on the right-hand side of Eq. (7) is the Coulombic force, and it increases as AC frequency goes down. It is worth mentioning that this electrothermal flow is able to generate a fast flow to rapidly transport particles from a long distance of several hundreds of microns to the hot spot. One no longer need to wait for particles to diffuse through Brownian motion into the target region to be successfully trapped.

Based on this concept, a novel plasmonic nano-tweezer called electrothermoplasmonic tweezers using a single plasmonic nanoantenna was invented [18, 44]. In this platform, a plasmonic nanoantenna is illuminated causing the surrounding fluid near the nanoantenna to be slightly heated up due to the Ohmic loss nature of plasmonic materials. Localized heating of fluid medium induces local gradients in the fluid's electrical conductivity and permittivity. An applied AC electric field acts on these gradients to induce an electrothermal microfluidic flow, which acts to transport particles towards the illuminated plasmonic nanoantenna.

The electrothermal flow from plasmon-induced heating can be modeled through the solution of the electromagnetic wave-equation, heat equation and the Navier-Stokes equation. Mathematically, we can solve the wave equation to find the electric field in the vicinity of the plasmonic nanoantenna:

$$\nabla \times (\nabla \times \mathbf{E}) - k\_0^2 c(\mathbf{r}) \mathbf{E} = \mathbf{0} \tag{9}$$

The heat source density generated by plasmonic heating is calculated using the electric field distribution and the induced current density representing the energy loss:

$$\mathbf{q}(\mathbf{r}) = \frac{1}{2} \operatorname{Re} \left( \mathbf{J} \cdot \mathbf{E}^\* \right) \tag{10}$$

The temperature field distribution is determined by solving the heat equation below:

$$\nabla \cdot \left[ -\kappa \nabla \mathbf{T}(\mathbf{r}) + \rho \mathbf{c}\_P \mathbf{T}(\mathbf{r}) \mathbf{u}(\mathbf{r}) \right] = q(\mathbf{r}) \tag{11}$$

*Nanomanipulation with Designer Thermoplasmonic Metasurface DOI: http://dx.doi.org/10.5772/intechopen.91880*

The photothermal heating of the fluid by the plasmonic nanoantennas can also

be utilized to induce strong electro-convection fluidic motion, which is called electrothermal (ET) effect. When the temperature inside fluid is no longer uniform due to a thermal gradient, a gradient in the permittivity and the conductivity of the fluid will induced as well. In such a system, a local free charge distribution must be present if Gauss's Law and charge conservation are to be satisfied simultaneously. When an AC electric field is applied, both free and bounded local charge density responds to the applied electric field so that a non-zero body force is

*<sup>f</sup> ET* <sup>¼</sup> *<sup>ρ</sup>eE* � <sup>1</sup>

<sup>2</sup> *Re ε α*ð Þ � *<sup>β</sup>* ð Þ <sup>∇</sup>*T:<sup>E</sup> <sup>E</sup>*<sup>∗</sup>

2 j j *E* 2

*<sup>σ</sup>* <sup>þ</sup> *<sup>i</sup>ωε* <sup>þ</sup>

1 2 *εα*j j *<sup>E</sup>* <sup>2</sup> ∇*T*

<sup>d</sup>*<sup>T</sup>*. *ε* stands for permittivity and *σ* stands for conduc-

<sup>0</sup>*ϵ*ð Þ**r E** ¼ 0 (9)

<sup>2</sup> *Re* <sup>J</sup> � **<sup>E</sup><sup>∗</sup>** ð Þ (10)

∇ � �½ κ∇Tð Þþ **r** ρcPTð Þ**r** uð Þ**r** � ¼ *q*ð Þ**r** (11)

After perturbative expansion in the limit of small temperature gradient, the

tivity. *ω* is the AC frequency. So far, the body force is decomposed into two parts: the first term on the right-hand side of Eq. (7) is the Coulombic force, and it increases as AC frequency goes down. It is worth mentioning that this electrothermal flow is able to generate a fast flow to rapidly transport particles from a long distance of several hundreds of microns to the hot spot. One no longer need to wait for particles to diffuse through Brownian motion into the target region to be

Based on this concept, a novel plasmonic nano-tweezer called electrothermoplasmonic tweezers using a single plasmonic nanoantenna was invented [18, 44]. In this platform, a plasmonic nanoantenna is illuminated causing the surrounding fluid near the nanoantenna to be slightly heated up due to the Ohmic loss nature of plasmonic materials. Localized heating of fluid medium induces local gradients in the fluid's electrical conductivity and permittivity. An applied AC electric field acts on these gradients to induce an electrothermal microfluidic flow, which acts to

The electrothermal flow from plasmon-induced heating can be modeled through the solution of the electromagnetic wave-equation, heat equation and the Navier-Stokes equation. Mathematically, we can solve the wave equation to find the electric

transport particles towards the illuminated plasmonic nanoantenna.

<sup>∇</sup> � ð Þ� <sup>∇</sup> � *<sup>E</sup> <sup>k</sup>*<sup>2</sup>

q rð Þ¼ <sup>1</sup>

The heat source density generated by plasmonic heating is calculated using the electric field distribution and the induced current density representing the

The temperature field distribution is determined by solving the heat equation

field in the vicinity of the plasmonic nanoantenna:

∇*ε<sup>m</sup>* (7)

(8)

generated on the fluid:

*Nanoplasmonics*

where *α* ¼ <sup>1</sup>

successfully trapped.

energy loss:

below:

**102**

force density is expressed as [43]:

*=ε* d*ε* *<sup>f</sup> ET* <sup>¼</sup> <sup>1</sup>

<sup>d</sup>*<sup>T</sup>* and *β* ¼ <sup>1</sup>

*=<sup>σ</sup>* <sup>d</sup>*<sup>σ</sup>* Finally, the Navier-Stokes equation is solved to find the velocity of the fluid:

$$
\rho\_0[\mathbf{u}(\mathbf{r}) \cdot \nabla] \mathbf{u}(\mathbf{r}) + \nabla p(\mathbf{r}) - \eta \nabla^2 \mathbf{u}(\mathbf{r}) = \mathbf{F}, \tag{12}
$$

where **F** here is the electrothermal force described in the prior section [43]. We have also established the theoretical framework between the ETP flow velocity with laser power and external AC electric field. It is noted that the ETP flow velocity scales linearly with laser power and quadratically with AC electric field amplitude [16].

Electro-osmotic flow is another category of body flow happening in a microfluidic channel [45]. Electro-osmosis occurs when tangential electric field acts on the loosely bound charges in the EDL along the channel walls. Because of the existence of electrical double layer, near the charged channel walls the tangential electric field will act on the EDL charges to induce an electro-osmotic flow with slip velocity given by:

$$\mathbf{u}\_s = -\frac{e\_\mathbf{w}\zeta}{\eta}\mathbf{E}\_{\parallel} \tag{13}$$

where E<sup>∥</sup> is the tangential component of the bulk electric field, η is the fluid viscosity and *ζ* is the zeta potential. When there is a plasmonic structure existing inside a microfluid channel on the channel wall, this plasmonic structure perturbs the AC electric field, resulting in a non-zero tangential component of the AC electric field. For instance, Jamshidi or Hwang and their colleagues integrated photoconductive material as the electrode of applied bias inside a microfluidic channel to design a so-called "NanoPen" device for dynamical manipulation on particles using low-power laser illumination [46, 47].

This perturbed external electric field not only contributes to electro-osmosis, but could also induce a dielectrophoretic force on the suspended particles [48]. Particles experience dielectrophoresis only when the external field is non-uniform and the force that particles feel, the DEP force, does not depend on the polarity of the AC field. DEP can be observed both in AC or DC electric field. There is a crucial evaluator for DEP force direction called Clausius-Mossoti factor.

$$\mathbf{K} = \frac{\mathbf{e}\_2 - \mathbf{e}\_1}{\mathbf{e}\_2 + 2\mathbf{e}\_1} \tag{14}$$

In terms of Clausius-Mossoti factor, the DEP force is expressed as:

$$\mathbf{F\_{DEP}} = 2\pi \mathbf{e\_1} \mathbf{R^3} K \nabla E\_0^2 \tag{15}$$

From which, we understand DEP force is proportional to particle volume and magnitude of K. The direction of DEP force is along the gradient of the electric field intensity and its sign depends on the sign of Clausius-Mossoti factor.

In AC field and lossy medium, the Clausius-Mossoti factor is frequency dependent and it is given by:

$$\mathbf{K} = \frac{\mathbf{e}\_2^\* - \mathbf{e}\_1^\*}{\mathbf{e}\_2^\* + 2\mathbf{e}\_1^\*}, \mathbf{e}\_1^\* = \mathbf{e}\_1 + \frac{\sigma\_1}{j\rho} \text{ and } \mathbf{e}\_2^\* = \mathbf{e}\_2 + \frac{\sigma\_2}{j\rho} \tag{16}$$

The Clausius-Mossoti factor determines whether the DEP force is attractive or repulsive ant it depends on the frequency of AC field and electric properties of particles and medium.

#### *Nanoplasmonics*

So far, we have briefly introduced the main mechanisms which could occur in plasmonic nanotweezers. The physical phenomena could be harnessed to introduce new capabilities in plasmonic nanotweezers. A recent article [49] has proposed the use of DEP force to promote particle transport towards plasmonic resonators.
