Preface

Chapter 10 **Liquid-Crystal-Based Phase Gratings and Beam Steerers for**

Chapter 11 **Design and Fabrication of Ultra-Short Throw Ratio Projector**

**Based on Liquid Crystal on Silicon 223**

Chapter 12 **Recent Dispersion Technology Using Liquid Crystal 243**

Ci-Ling Pan, Chia-Jen Lin, Chan-Shan Yang, Wei-Ta Wu and Ru-Pin

**Terahertz Waves 197**

Jiun-Woei Huang

Yuji Yamashita

Pan

**VI** Contents

This book encapsulates varieties of application-oriented phenomena of liquid crystals, contrib‐ uted by authors from different countries. The spotlight shines on the aspects pivoted to the novel properties of this complex-structured medium that can be harnessed for real-life needs. In a sense, some of the chapters describe well-understood phenomena of liquid crystals, whereas the others highlight on how these could be conceived for the development of new devices. In every chapter, the authors review the recent developments in the area reported by the preeminent researchers and, also, touch upon their own contributions. As such, the book essentially provides readers a glimpse of the multitudes of liquid crystal research.

Liquid crystals are optically active in nature and exist in a few different forms of molecular structures, viz., nematic, smectic, and cholesteric. Chapter 1 of this book remains the intro‐ ductory part, wherein the fundamentals behind liquid crystals and the involved research di‐ rectives are briefly described. Within the context of the features of varieties of liquid crystals, the optical textures of cholesterol make the cholesteric kind of liquid crystals greatly interest‐ ing for the study of optical activity. The synthesis of a certain kind of cholesteric liquid crystal phase—the cationic cholesteric liquid crystals—is reported by Méndez in Chapter 2. This chapter shares the important result that such a form of liquid crystal polymer can act as non‐ viral vectors in gene therapy, transfecting DNA to the nucleus cell.

The director of cholesteric liquid crystal molecules exhibits periodic helical structure, the al‐ teration of which due to dopants would modify the liquid crystal phase. In this stream, the blue phase of liquid crystals exhibits outstanding electro-optical properties, which makes it a promising introduction into display-related usage. Chapter 3 by Kemiklioglu discusses the stabilization and electro-optical properties of blue phases of liquid crystals emphasizing their potentials for applications in display technology and other photonic devices.

In the context of phase transition of liquid crystals, Contreras in Chapter 4 focuses on the discus‐ sions of thermotropic liquid crystals of nematic and smectic A types under the external magnet‐ ic field. Exploiting the linear stability theory, the author investigates the effects of thermal phase transition on nematics of finite thickness samples with the conceptual framework of Faraday wave propagation. Here, the author shares his quantitative understanding of the dynamics of surface phenomena in liquid crystals through the theoretical and experimental means.

Chapter 5 by Song et al. reviews the structure and property of lyotropic liquid crystal-based materials. In the ordered phase of liquid crystals, molecules tend to align along a common direction, thereby yielding orderly oriented macroscopic domains, which would provide a way to control the orientation of *additive* materials. Within the context, different kinds of ad‐ ditives, namely, carbon nanotubes, graphene, biomolecules, etc., may be used for investiga‐ tion. This chapter describes the mechanical, electrical, and physicochemical properties of the

hybrids composed of such lyotropics of amphiphilic molecules with different kinds of afore‐ mentioned additives. It is reported that the features of lyotropic liquid crystals are greatly improved upon the usage of additives—the phenomenon that can be prudent in nanotech‐ nology, electrochemical, and biochemical arena.

required design of phase gradient in the grating structure. Good agreement of the experimen‐ tal results with the theoretical predictions puts the investigation on a level where further re‐

The use of liquid crystals in flat panel electronic displays offers several advantages over the traditional ones wherein cathode-ray tubes (CRTs) are implemented. Though varieties of liq‐ uid crystal panels are available in the market, in Chapter 11, J. W. Huang makes an attempt to touch upon the design and fabrication of ultrashort throw projection systems for home cinema and display systems in automobiles. The author discusses the technique of liquid crystal on silicon for generating high-quality images. The emphasis is on the enhancement in

Another newly ventured application of liquid crystals would be in the pharmaceutical indus‐ try. Yuji Yamashita in Chapter 12 describes functional media comprised of unstable colloid dispersion systems using liquid crystals. The author specifically mentions the emulsification technology using liquid crystals constructed by self-assembly of several molecules and, also,

Finally, the book—*Liquid Crystals*: *Recent Advancements in Fundamental and Device Technologies* —overall highlights various research-related aspects that liquid crystal media have come across in recent years. Pioneering scientists from different countries contributed their re‐ search results; these are put in 11 different chapters. The editor hopes the book to be useful for novice as well as expert researchers—the former group of readers would remain abreast of recent research advancements in the relevant area, whereas the latter kind would be fueled

> **Prof. Pankaj Kumar Choudhury** Universiti Kebangsaan, Malaysia

Preface IX

the dispersion of liquid crystals, which would find prominent medicinal applications.

search has to be carried out to eliminate the lacunae.

viewing angle through reducing the throw ratio.

to plan for new research ventures in the liquid crystal arena.

Photonic crystals are known to exhibit band gaps. Liquid crystals also exhibit similar features due to periodicity. However, the physical and chemical features of these are highly dependent on the externally applied fields, and therefore, the band gaps may also be tailored on demand. Apart from this, varying thermal ambience also plays a great role in manipulating the behavior of liquid crystals. Technically speaking, thermal and electrical tuning of liquid crystals would alter the spectral characteristics. Pivoted to this concept, Avendaño et al. present multilayered slabs of nematic liquid crystals in twisted configuration to investigate the tuning property of photonic band gaps; the relevant descriptions are incorporated in Chapter 6. The temperaturedependent defect modes are also exploited in the study. The investigation yields electrical con‐ trol of defect modes in nematic liquid crystal-based multilayer structures through the tuning of transmission and reflection bands. The authors claim the usefulness of study in the develop‐ ment of liquid crystal-based tunable optical filters and waveguides.

Chapter 7 by Lembrikov et al. discusses theoretically some of the interesting nonlinear phenom‐ ena in the case of liquid crystals of the smectic A kind. For this purpose, the authors implement cubic optical nonlinearity, which is determined by the normal displacement of the smectic layer —a particular kind of orientational mechanism of liquid crystal molecules. They claim the in‐ vestigation to be useful in explaining certain nonlinear phenomena in liquid crystals.

The refractive index of liquid crystals depends on the temperature of ambiance—the feature that would greatly affect the birefringent property. This feature is highlighted by Al-Qurainy and Al-Naimee in Chapter 8, wherein they consider five different samples of liquid crystals in pure and mixed (of pure) forms. They share the idea to model the birefringence of liquid crystals and, also, put efforts to demonstrate the validity through experiments. Their study on the bistability of liquid crystals due to temperature reveals the extraordinary refractive index to have larger bistability than the ordinary one.

The use of liquid crystal-based optical fibers in chemical sensors is common. In Chapter 9 Luo et al. discuss micro-/nano-dimensional liquid crystal layer-based tunable optical fiber in‐ terferometers. Two different kinds of interferometers, namely, the optical fiber grating-based and locally bent microfiber taper-based structures, are described on the basis of theories in‐ volved, followed by the experimental investigations of the functional properties of devices. The results indicate potential applications of the suggested forms of interferometers as tuna‐ ble *all-fiber* photonic devices, such as filters and *all-optical* switches.

Chapter 10 by Pan et al. is pivoted on the investigation of liquid crystal-based terahertz (THz) phase gratings and beam steerers theoretically and experimentally. The authors introduce such phase gratings as capable to function as THz polarizers and tunable THz beam splitters. However, the thickness of the used liquid crystal layers affects the response time of grating, which is reported to be relatively low, thereby causing unsuitability for *fast modulation*-relat‐ ed applications. Instead, the suitability of the device remains in the need of precise control requiring a fixed beam splitting ratio. The ways to overcome the issues are also proposed in the chapter. The developed liquid crystal phase grating-based electrically tunable THz beam steerer can steer broadband THz radiation by 8.5° (with respect to the incident beam). This can be achieved through electrical control, that is, by varying the driving voltages to yield the required design of phase gradient in the grating structure. Good agreement of the experimen‐ tal results with the theoretical predictions puts the investigation on a level where further re‐ search has to be carried out to eliminate the lacunae.

hybrids composed of such lyotropics of amphiphilic molecules with different kinds of afore‐ mentioned additives. It is reported that the features of lyotropic liquid crystals are greatly improved upon the usage of additives—the phenomenon that can be prudent in nanotech‐

Photonic crystals are known to exhibit band gaps. Liquid crystals also exhibit similar features due to periodicity. However, the physical and chemical features of these are highly dependent on the externally applied fields, and therefore, the band gaps may also be tailored on demand. Apart from this, varying thermal ambience also plays a great role in manipulating the behavior of liquid crystals. Technically speaking, thermal and electrical tuning of liquid crystals would alter the spectral characteristics. Pivoted to this concept, Avendaño et al. present multilayered slabs of nematic liquid crystals in twisted configuration to investigate the tuning property of photonic band gaps; the relevant descriptions are incorporated in Chapter 6. The temperaturedependent defect modes are also exploited in the study. The investigation yields electrical con‐ trol of defect modes in nematic liquid crystal-based multilayer structures through the tuning of transmission and reflection bands. The authors claim the usefulness of study in the develop‐

Chapter 7 by Lembrikov et al. discusses theoretically some of the interesting nonlinear phenom‐ ena in the case of liquid crystals of the smectic A kind. For this purpose, the authors implement cubic optical nonlinearity, which is determined by the normal displacement of the smectic layer —a particular kind of orientational mechanism of liquid crystal molecules. They claim the in‐

The refractive index of liquid crystals depends on the temperature of ambiance—the feature that would greatly affect the birefringent property. This feature is highlighted by Al-Qurainy and Al-Naimee in Chapter 8, wherein they consider five different samples of liquid crystals in pure and mixed (of pure) forms. They share the idea to model the birefringence of liquid crystals and, also, put efforts to demonstrate the validity through experiments. Their study on the bistability of liquid crystals due to temperature reveals the extraordinary refractive

The use of liquid crystal-based optical fibers in chemical sensors is common. In Chapter 9 Luo et al. discuss micro-/nano-dimensional liquid crystal layer-based tunable optical fiber in‐ terferometers. Two different kinds of interferometers, namely, the optical fiber grating-based and locally bent microfiber taper-based structures, are described on the basis of theories in‐ volved, followed by the experimental investigations of the functional properties of devices. The results indicate potential applications of the suggested forms of interferometers as tuna‐

Chapter 10 by Pan et al. is pivoted on the investigation of liquid crystal-based terahertz (THz) phase gratings and beam steerers theoretically and experimentally. The authors introduce such phase gratings as capable to function as THz polarizers and tunable THz beam splitters. However, the thickness of the used liquid crystal layers affects the response time of grating, which is reported to be relatively low, thereby causing unsuitability for *fast modulation*-relat‐ ed applications. Instead, the suitability of the device remains in the need of precise control requiring a fixed beam splitting ratio. The ways to overcome the issues are also proposed in the chapter. The developed liquid crystal phase grating-based electrically tunable THz beam steerer can steer broadband THz radiation by 8.5° (with respect to the incident beam). This can be achieved through electrical control, that is, by varying the driving voltages to yield the

vestigation to be useful in explaining certain nonlinear phenomena in liquid crystals.

nology, electrochemical, and biochemical arena.

VIII Preface

ment of liquid crystal-based tunable optical filters and waveguides.

index to have larger bistability than the ordinary one.

ble *all-fiber* photonic devices, such as filters and *all-optical* switches.

The use of liquid crystals in flat panel electronic displays offers several advantages over the traditional ones wherein cathode-ray tubes (CRTs) are implemented. Though varieties of liq‐ uid crystal panels are available in the market, in Chapter 11, J. W. Huang makes an attempt to touch upon the design and fabrication of ultrashort throw projection systems for home cinema and display systems in automobiles. The author discusses the technique of liquid crystal on silicon for generating high-quality images. The emphasis is on the enhancement in viewing angle through reducing the throw ratio.

Another newly ventured application of liquid crystals would be in the pharmaceutical indus‐ try. Yuji Yamashita in Chapter 12 describes functional media comprised of unstable colloid dispersion systems using liquid crystals. The author specifically mentions the emulsification technology using liquid crystals constructed by self-assembly of several molecules and, also, the dispersion of liquid crystals, which would find prominent medicinal applications.

Finally, the book—*Liquid Crystals*: *Recent Advancements in Fundamental and Device Technologies* —overall highlights various research-related aspects that liquid crystal media have come across in recent years. Pioneering scientists from different countries contributed their re‐ search results; these are put in 11 different chapters. The editor hopes the book to be useful for novice as well as expert researchers—the former group of readers would remain abreast of recent research advancements in the relevant area, whereas the latter kind would be fueled to plan for new research ventures in the liquid crystal arena.

> **Prof. Pankaj Kumar Choudhury** Universiti Kebangsaan, Malaysia

**Chapter 1**

**Provisional chapter**

**Introductory Chapter: Whither Liquid Crystals?**

**Introductory Chapter: Whither Liquid Crystals?**

DOI: 10.5772/intechopen.74413

"*Whither liquid crystals?*"—a very general question that every researcher falling on the relevant research directives would ask about. A veteran researcher may also attempt to seek the answer to be abreast of the recent research developments in the area. Amazing physical and chemical properties of liquid crystals draw great attention of the research and development (R&D) community. These essentially make them indispensable for several technological applications, namely sensing [1, 2], communication systems [3], lasing actions [4, 5], flat panel

Though there are many different kinds of liquid crystals, nematic, smectic, and cholesteric exist as the three widely accepted phases (of these crystals) [9]. In short, the nematic version is characterized by molecules having no positional order, but aligned along the director with thread-like molecular formations. In the smectic phase, molecules have positional order only in one dimension, thereby having restricted movement within the planes. The cholesteric state is a kind of nematic phase wherein the molecular orientation undergoes helical rotation about the director. These may exist naturally or can be synthesized as well [10, 11]. Certain liquid-crystalline phases are abundant in living organisms, for example, proteins and cell membranes. Technologically developed liquid crystals are used for liquid crystal display (LCD) applications [6]. Apart from the LCD panels, there are host of other avenues where the

Liquid crystals exhibit chirality and possess very high electro-optic coefficient [12]. Chiral objects have the property to discriminate between the left-handed and right-handed electromagnetic

displays [6], holography [7], and nanotechnology-enabled medicinal needs [8].

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Pankaj Kumar Choudhury

**1. Introduction**

Pankaj Kumar Choudhury

http://dx.doi.org/10.5772/intechopen.74413

synthesized versions of liquid crystals are used.

**2. Liquid crystal properties**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

### **Introductory Chapter: Whither Liquid Crystals? Introductory Chapter: Whither Liquid Crystals?**

DOI: 10.5772/intechopen.74413

Pankaj Kumar Choudhury Pankaj Kumar Choudhury

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74413

### **1. Introduction**

"*Whither liquid crystals?*"—a very general question that every researcher falling on the relevant research directives would ask about. A veteran researcher may also attempt to seek the answer to be abreast of the recent research developments in the area. Amazing physical and chemical properties of liquid crystals draw great attention of the research and development (R&D) community. These essentially make them indispensable for several technological applications, namely sensing [1, 2], communication systems [3], lasing actions [4, 5], flat panel displays [6], holography [7], and nanotechnology-enabled medicinal needs [8].

Though there are many different kinds of liquid crystals, nematic, smectic, and cholesteric exist as the three widely accepted phases (of these crystals) [9]. In short, the nematic version is characterized by molecules having no positional order, but aligned along the director with thread-like molecular formations. In the smectic phase, molecules have positional order only in one dimension, thereby having restricted movement within the planes. The cholesteric state is a kind of nematic phase wherein the molecular orientation undergoes helical rotation about the director. These may exist naturally or can be synthesized as well [10, 11]. Certain liquid-crystalline phases are abundant in living organisms, for example, proteins and cell membranes. Technologically developed liquid crystals are used for liquid crystal display (LCD) applications [6]. Apart from the LCD panels, there are host of other avenues where the synthesized versions of liquid crystals are used.

## **2. Liquid crystal properties**

Liquid crystals exhibit chirality and possess very high electro-optic coefficient [12]. Chiral objects have the property to discriminate between the left-handed and right-handed electromagnetic

fields [13]. These *optically active* mediums are classified into the categories of isotropic and structurally chiral ones. The isotropic chiral molecules can be formed by randomly dispersed, randomly oriented, electrically small, handed inclusions in an isotropic achiral host medium. On the other hand, the structurally chiral molecules, such as those of chiral nematic liquid crystals, are randomly positioned and exhibit helicoidal kind of orientation. One may exemplify biological structures of plants and animals, such as cholesterols, which represent chiral molecules. The director of cholesteric liquid crystal molecules exhibits periodic helical structure depending on the chirality of molecules, and may be altered due to external conditions—the feature that has great potential in technological applications [14]. For example, the changes in the helix (formed by the rotation of director) pitch due to chiral dopants would modify the phase of liquid crystals.

As stated before, the *optical activity* of liquid crystals opens up varieties of avenues. Under high external electrical fields, an optical material would exhibit nonlinear characteristics, that is, the refractive index of medium may not vary linearly with the field [20]. Liquid crystals, being optically anisotropic mediums, possess the birefringence property that remains of great potential in optics-based applications [21, 22]. The increase of birefringence happens owing to the nonlinear phenomenon that liquid crystals also exhibit. In fact, liquid crystals are characterized by extremely high optical nonlinearity. Some of the featured nonlinear phenomena would be self-phase modulation, four-wave mixing, stimulated Brillouin scattering, optical bistability, and so on. [23].

Introductory Chapter: Whither Liquid Crystals? http://dx.doi.org/10.5772/intechopen.74413 3

Optical fibers with radially anisotropic liquid crystals have been greatly dealt with in the literature [24–26]. These have been much attractive owing to the fairly high optical anisotropic properties of liquid crystals—the feature that attracted the R&D community to introduce varieties of liquid crystal-based optical fibers in respect of geometry as well as material distributions. These include fibers of circular [25] and elliptical [26] cross-sections, and also, those with the loading of conducting helical structures, in order to achieve control over the dispersion characteristics [27–29]. It has been reported before that the radially anisotropic kind of nematic liquid crystal-loaded fibers become highly sensitive, and would be of potential for evanescent wave-based sensing applications [24, 25]. Indeed, the use of such fibers in chemi-

cal sensors would be one of the great avenues that liquid crystal mediums open up.

Photonic crystals are known to exhibit band gaps, that is, the range of frequencies (or wavelengths) for which the propagation of waves remains forbidden [30–32]. Microstructured dielectric mediums can be engineered to exhibit such an excellent feature, which has been of great technological use in many optics-related applications [33]. Apart from dielectrics, liquid crystals may also be utilized due to the fact that the physical and/or chemical features of these are highly dependent on the externally applied fields. Varying thermal ambience also plays a great role in manipulating the behavior of liquid crystals, which essentially happens owing to the effects on the birefringence property. Technically speaking, thermal and electrical tuning of liquid crystals would alter the spectral characteristics. As such, photonic crystals, infiltrated with liquid crystal mediums, would exhibit tunable band gap features, which would

Apart from the aforementioned applications of liquid crystals, there are many other varieties of usages that these materials offer [34]. Some of these include optical recording mediums, lasers,

**4. Liquid crystal-based fibers**

**5. Liquid crystals for band gap features**

be greatly interesting in optics-based needs.

**6. Other miscellaneous applications and summary**

### **3. Application-oriented R&D**

The unique properties of liquid crystals fueled scientists to invent new applications. Continuous research and development determine these mediums to gain increasingly important industrial and techno-scientific usages, and become vital in modern technological advancements. It is true that the research on liquid crystals tremendously bloomed after the invention of LCD panels [15]. Though the use of liquid crystals in flat panel electronic displays offers several advantages over the traditional ones, wherein cathode ray tubes (CRTs) are implemented [6], the LCDs have the drawback of having limited viewing angle, and also, higher manufacturing cost. However, these parameters have now become less significant with the advances in research, which becomes evident from the multitude of other applications of liquid crystals. This is primarily because external perturbations would introduce significant alterations in the macroscopic properties of liquid crystals [12, 13]. As an example, the chirality of liquid crystals allows these to acquire selective reflectance property, which can even be modified in the presence of external electric field. As such, these would be of great use in optical filters and imaging [16, 17] applications. Apart from these, the property of temperature dependence also makes liquid crystals to acquire the selectivity of reflection spectrum—the phenomenon that can be harnessed for devising temperature sensors [18].

Since the temperature plays a determining role to alter the refractive index values of liquid crystals, the birefringence property of these [19] allows splitting of light waves into the *slow* and *fast* components—the phenomenon which remains highly temperature dependent. A relatively higher temperature would induce a strong birefringence characteristic in certain form of liquid crystals, which would result due to higher temperature of the ambience. As such, a variation in temperature would introduce alterations in the phase difference between the incoming and outgoing light waves, thereby determining the polarization state of light.

As the properties of liquid crystals are affected by electric field, these mediums can be used to sense the field strength. Similarly, magnetic field also has effects on the properties of liquid crystals owing to the moving electric charges (magnetic dipoles are generated by the electrons moving around the nucleus in atoms). An externally applied magnetic field would make the liquid crystal molecules to align accordingly.

As stated before, the *optical activity* of liquid crystals opens up varieties of avenues. Under high external electrical fields, an optical material would exhibit nonlinear characteristics, that is, the refractive index of medium may not vary linearly with the field [20]. Liquid crystals, being optically anisotropic mediums, possess the birefringence property that remains of great potential in optics-based applications [21, 22]. The increase of birefringence happens owing to the nonlinear phenomenon that liquid crystals also exhibit. In fact, liquid crystals are characterized by extremely high optical nonlinearity. Some of the featured nonlinear phenomena would be self-phase modulation, four-wave mixing, stimulated Brillouin scattering, optical bistability, and so on. [23].

### **4. Liquid crystal-based fibers**

fields [13]. These *optically active* mediums are classified into the categories of isotropic and structurally chiral ones. The isotropic chiral molecules can be formed by randomly dispersed, randomly oriented, electrically small, handed inclusions in an isotropic achiral host medium. On the other hand, the structurally chiral molecules, such as those of chiral nematic liquid crystals, are randomly positioned and exhibit helicoidal kind of orientation. One may exemplify biological structures of plants and animals, such as cholesterols, which represent chiral molecules. The director of cholesteric liquid crystal molecules exhibits periodic helical structure depending on the chirality of molecules, and may be altered due to external conditions—the feature that has great potential in technological applications [14]. For example, the changes in the helix (formed by the rotation of director) pitch due to chiral dopants would modify the phase of liquid crystals.

2 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

The unique properties of liquid crystals fueled scientists to invent new applications. Continuous research and development determine these mediums to gain increasingly important industrial and techno-scientific usages, and become vital in modern technological advancements. It is true that the research on liquid crystals tremendously bloomed after the invention of LCD panels [15]. Though the use of liquid crystals in flat panel electronic displays offers several advantages over the traditional ones, wherein cathode ray tubes (CRTs) are implemented [6], the LCDs have the drawback of having limited viewing angle, and also, higher manufacturing cost. However, these parameters have now become less significant with the advances in research, which becomes evident from the multitude of other applications of liquid crystals. This is primarily because external perturbations would introduce significant alterations in the macroscopic properties of liquid crystals [12, 13]. As an example, the chirality of liquid crystals allows these to acquire selective reflectance property, which can even be modified in the presence of external electric field. As such, these would be of great use in optical filters and imaging [16, 17] applications. Apart from these, the property of temperature dependence also makes liquid crystals to acquire the selectivity of reflection spectrum—the phenomenon that

Since the temperature plays a determining role to alter the refractive index values of liquid crystals, the birefringence property of these [19] allows splitting of light waves into the *slow* and *fast* components—the phenomenon which remains highly temperature dependent. A relatively higher temperature would induce a strong birefringence characteristic in certain form of liquid crystals, which would result due to higher temperature of the ambience. As such, a variation in temperature would introduce alterations in the phase difference between the incoming and outgoing light waves, thereby determining the polarization state

As the properties of liquid crystals are affected by electric field, these mediums can be used to sense the field strength. Similarly, magnetic field also has effects on the properties of liquid crystals owing to the moving electric charges (magnetic dipoles are generated by the electrons moving around the nucleus in atoms). An externally applied magnetic field would make the

**3. Application-oriented R&D**

can be harnessed for devising temperature sensors [18].

liquid crystal molecules to align accordingly.

of light.

Optical fibers with radially anisotropic liquid crystals have been greatly dealt with in the literature [24–26]. These have been much attractive owing to the fairly high optical anisotropic properties of liquid crystals—the feature that attracted the R&D community to introduce varieties of liquid crystal-based optical fibers in respect of geometry as well as material distributions. These include fibers of circular [25] and elliptical [26] cross-sections, and also, those with the loading of conducting helical structures, in order to achieve control over the dispersion characteristics [27–29]. It has been reported before that the radially anisotropic kind of nematic liquid crystal-loaded fibers become highly sensitive, and would be of potential for evanescent wave-based sensing applications [24, 25]. Indeed, the use of such fibers in chemical sensors would be one of the great avenues that liquid crystal mediums open up.

### **5. Liquid crystals for band gap features**

Photonic crystals are known to exhibit band gaps, that is, the range of frequencies (or wavelengths) for which the propagation of waves remains forbidden [30–32]. Microstructured dielectric mediums can be engineered to exhibit such an excellent feature, which has been of great technological use in many optics-related applications [33]. Apart from dielectrics, liquid crystals may also be utilized due to the fact that the physical and/or chemical features of these are highly dependent on the externally applied fields. Varying thermal ambience also plays a great role in manipulating the behavior of liquid crystals, which essentially happens owing to the effects on the birefringence property. Technically speaking, thermal and electrical tuning of liquid crystals would alter the spectral characteristics. As such, photonic crystals, infiltrated with liquid crystal mediums, would exhibit tunable band gap features, which would be greatly interesting in optics-based needs.

### **6. Other miscellaneous applications and summary**

Apart from the aforementioned applications of liquid crystals, there are many other varieties of usages that these materials offer [34]. Some of these include optical recording mediums, lasers, light modulators, and so on. Also, the area of biomedical applications is no more untouched on the exploitation of liquid crystals. For example, certain forms of liquid crystal polymers can act as nonviral vectors in gene therapy, transfecting DNA to the nucleus cell. Furthermore, functional mediums composed of specific colloid dispersion systems using liquid crystals would be greatly useful in pharmaceutical industry.

[8] Lin Y-H. Liquid crystals for bio-medical applications. In: Lee C-C, editor. The Current Trends of Optics and Photonics. Vol. 129. Topics in Applied Physics. Dordrecht:

Introductory Chapter: Whither Liquid Crystals? http://dx.doi.org/10.5772/intechopen.74413 5

[9] Choudhury PK, editor. New Developments in Liquid Crystals and Applications. New York:

[10] Collings PJ. Liquid Crystals: Nature's Delicate Phase of Matter. Princeton: Princeton

[11] de Gennes PG, Prost J. The Physics of Liquid Crystals. Clarendon Press: Oxford; 1993

[12] Wu S-T, Efron U. Optical properties of thin nematic liquid crystal cells. Applied Physics

[13] Robbie K, Brett MJ, Lakhtakia A. Chiral sculptured thin films. Nature. 1996;**384**:616-617

[15] Castellano JA. Liquid Gold: The Story of Liquid Crystal Displays and the Creation of an

[16] Gebhart SC, Stokes DL, Vo-Dinh T, Mahadevan-Jansen A. Instrumentation considerations in spectral imaging for tissue demarcation: comparing three methods of spectral

[17] Levenson RM, Lynch DT, Kobayashi H, Backer JM, Backer MV. Multiplexing with mul-

[19] Madsen LA, Dingemans TJ, Nakata M, Samulski ET. Thermotropic biaxial nematic liq-

[20] Choudhury PK, Singh ON. Electromagnetic materials. In: Chang K, editor. Encyclopedia

[21] Green M, Madden SJ. Low loss nematic liquid crystal cored fiber waveguides. Applied

[22] Lin H, Muhoray PP, Lee MA. Liquid crystalline cores for optical fibers. Molecular

[24] Choudhury PK, Soon WK. On the transmission by liquid crystal tapered optical fibers.

[25] Choudhury PK. Evanescent field enhancement in liquid crystal optical fibers – A field characteristics based analysis. Advances in Condensed Matter Physics. 2013;**2013**:

resolution. Proceedings of SPIE. 2005;**5694**:41-52. DOI: 10.1117/12.611351

tispectral imaging: from mice to microscopy. ILAR Journal. 2008;**49**:78-88

[14] Moreno I. Liquid crystals for photonics. Optical Engineering. 2011;**50**:081201-081201

Springer; 2014

University Press; 1990

Letters. 1986;**48**:624-636

Optics. 1989;**28**:5202-5203

Optik. 2011;**122**:1061-1068

504868-1-504868-9

Industry. Singapore: World Scientific; 2005

[18] Plimpton RG. Pool Thermometer. U.S. Patent 4738549. 1988

uid crystals. Physical Review Letters. 2004;**92**:145505-1-145505-4

[23] Agrawal GP. Fiber Optic Communication Systems. Wiley: New York; 2012

of RF and Microwave Engineering. New York: Wiley; 2005

Crystals and Liquid Crystals. 1991;**204**:189-200

Nova; 2013

In summary, the book delineates several important advances occurring at the forefront of liquid crystal research. These are in terms of the development of fundamental theories as well as the exploitation of liquid crystals in inventing new devices. The subject matter of the book primarily focuses on the aspects of (i) varieties of liquid crystal polymer syntheses and their stability, (ii) physical and optical properties of complex liquid-crystalline states, and (iii) device applications of liquid crystals. The editor hopes that the topics included will be greatly useful for the R&D workers at universities and industries. The researchers use the book as springboard for their own thoughts in varieties of ways the different forms of liquid crystals can be exploited.

### **Author details**

Pankaj Kumar Choudhury

Address all correspondence to: pankaj@ukm.edu.my

Universiti Kebangsaan Malaysia, Bangi, Selangor, Malaysia

### **References**


[8] Lin Y-H. Liquid crystals for bio-medical applications. In: Lee C-C, editor. The Current Trends of Optics and Photonics. Vol. 129. Topics in Applied Physics. Dordrecht: Springer; 2014

light modulators, and so on. Also, the area of biomedical applications is no more untouched on the exploitation of liquid crystals. For example, certain forms of liquid crystal polymers can act as nonviral vectors in gene therapy, transfecting DNA to the nucleus cell. Furthermore, functional mediums composed of specific colloid dispersion systems using liquid crystals would

In summary, the book delineates several important advances occurring at the forefront of liquid crystal research. These are in terms of the development of fundamental theories as well as the exploitation of liquid crystals in inventing new devices. The subject matter of the book primarily focuses on the aspects of (i) varieties of liquid crystal polymer syntheses and their stability, (ii) physical and optical properties of complex liquid-crystalline states, and (iii) device applications of liquid crystals. The editor hopes that the topics included will be greatly useful for the R&D workers at universities and industries. The researchers use the book as springboard for their own thoughts in varieties of ways the different forms of liquid crystals can be exploited.

[2] Choudhury PK. Liquid crystal optical fibers for sensing applications. Proceedings of

[3] Wolinski TR, Lesiak P, Dabrowski R, Kedzierski J, Nowinowski E. Polarization mode dispersion in an elliptical liquid crystal core fiber. Molecular Crystals and Liquid

[4] Dolgaleva K, Wei SKH, Lukishova SG, Chen SH, Schwertz K, Boyd RW. Enhanced laser performance of cholesteric liquid crystals doped with oligofluorene dye. Journal of the

[6] O'Mara WC. Liquid Crystal Flat Panel Displays. New York: Van Nostrand Reinhold;

[7] Berenberg VA, Venediktov VY. Liquid crystal valves as dynamic holographic correctors. In: Choudhury PK, editor. New Developments in Liquid Crystals and Applications.

Optical Society of America B: Optical Physics. 2009;**25**:1496-1504

[5] Coles H, Morris S. Liquid-crystal lasers. Nature Photonics. 2010;**4**:676-685

be greatly useful in pharmaceutical industry.

4 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

Address all correspondence to: pankaj@ukm.edu.my

Universiti Kebangsaan Malaysia, Bangi, Selangor, Malaysia

[1] Khoo I-C. Liquid Crystals. Wiley: New York; 1994

SPIE. 2013;**8818**:88180E-1-88180E-10

Crystals. 2004;**421**:175-186

New York: Nova; 2013

**Author details**

**References**

1993

Pankaj Kumar Choudhury


[26] Moghaddas S, Choudhury PK, Ibrahim A-BMA. TE and TM mode power transmission through liquid crystal clad elliptical guides. Optik. 2017;**145**:113-120

**Chapter 2**

**Provisional chapter**

**Synthetic Cationic Cholesteric Liquid Crystal Polymers**

We report the synthesis of six multifunctional cationic cholesteric liquid crystals polyesters functionalized with choline, amine, and amide groups to obtain new chemical formulations involving macromolecular features with new properties added to those of precursor chiral cholesteric polyesters. They are designed as PTOBDME-choline

by NMR. Thermal behavior is studied by thermogravimetry (TG) and differential scanning calorimetry (DSC), showing all the polymers endothermic transition from crystal phase to liquid crystal mesophase. Chirality is determined by optical rotatory dispersion (ORD). The cationic cholesteric liquid crystal polymers described here have proved to act

**Keywords:** cholesteric LC, cationic polymers, chiral polyesters, synthesis, NMR, DSC

Cholesteric liquid crystal polyesters have received much attention in the last few years for their interesting chemical, optical, mechanical, and biological properties. Due to their anisotropic formulation and amphiphilic nature, their molecules are able to self-associate and/or aggregate in blocks to form species with supramolecular ordered structure, which presents

Two cholesteric liquid crystal polyesters, named PTOBDME and PTOBEE in **Figure 1**, were obtained by polycondensation reaction. Although only racemic materials were used in their synthesis, a cholesteric, chiral morphology, theoretically unexpected, was found. Evidence of this was obtained when a white solid, recrystallized, as the second fraction, from toluene mother

)n─C5

)n─C5

N)n. Structural characterization is performed

H13N]; PTOBDME-ammonium

H13N]; PTOBUME-amide

H13N]; PTOBEE-choline [(C26H20O8

N)n; and PTOBEE-amide (C26H19O9

H13N]; PTOBEE-ammonium [(C26H20O8

as nonviral vectors in gene therapy, transfecting DNA to the nucleus cell.

**Synthetic Cationic Cholesteric Liquid Crystal Polymers**

DOI: 10.5772/intechopen.70995

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

Mercedes Pérez Méndez

**Abstract**

[(C34H36O8

[(C34H36O8

(C33H33O9

**1. Introduction**

desirable material properties.

) <sup>n</sup>─C5

) <sup>n</sup>─C5

Mercedes Pérez Méndez

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70995


### **Synthetic Cationic Cholesteric Liquid Crystal Polymers Synthetic Cationic Cholesteric Liquid Crystal Polymers**

DOI: 10.5772/intechopen.70995

Mercedes Pérez Méndez Mercedes Pérez Méndez

[26] Moghaddas S, Choudhury PK, Ibrahim A-BMA. TE and TM mode power transmission

[27] Ghasemi M, Choudhury PK. Propagation through complex structured liquid crystal

[28] Ghasemi M, Choudhury PK. Conducting tape helix loaded radially anisotropic liquid crystal clad optical fiber. Journal of Nanophotonics. 2015;**9**:093592-1-0093592-15

[29] Ghasemi M, Choudhury PK. On the conducting sheath double-helix loaded liquid crystal optical fibers. Journal of Electromagnetic Waves and Applications. 2015;**29**:1580-1592

[30] Alkeskjold TT, Scolari L, Noordegraaf D, Laegsgaard J, Weirich J, Wei L, Tartarini G, Bassi P, Gauza S, Wu S-T, Bjarklev A. Biased liquid crystal infiltrated photonic bandgap

[31] Ibrahim A-BMA, Choudhury PK, Alias MSB. Analytical design of photonic band-gap

[32] Ibrahim A-BMA, Choudhury PK, Alias MSB. On the analytical investigation of fields and power patterns in coaxial omniguiding Bragg fibers. Optik. 2006;**117**:33-39

[33] Choudhury PK. Electromagnetics of Micro- and Nanostructured Guides – Pathways to

[34] Chandrasekhar S. Liquid Crystals. Cambridge University Press: Cambridge; 1992

through liquid crystal clad elliptical guides. Optik. 2017;**145**:113-120

6 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

optical fibers. Journal of Nanophotonics. 2014;**8**:083997-1-083997-13

fiber. Optical and Quantum Electronics. 2007;**39**:1009-1019

Nanophotonics. Malaysia: UKM Press; 2013

fibers and their dispersion characteristics. Optik. 2005;**116**:169-174

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70995

#### **Abstract**

We report the synthesis of six multifunctional cationic cholesteric liquid crystals polyesters functionalized with choline, amine, and amide groups to obtain new chemical formulations involving macromolecular features with new properties added to those of precursor chiral cholesteric polyesters. They are designed as PTOBDME-choline [(C34H36O8 ) <sup>n</sup>─C5 H13N]; PTOBEE-choline [(C26H20O8 ) <sup>n</sup>─C5 H13N]; PTOBDME-ammonium [(C34H36O8 ) <sup>n</sup>─C5 H13N]; PTOBEE-ammonium [(C26H20O8 )n─C5 H13N]; PTOBUME-amide (C33H33O9 N)n; and PTOBEE-amide (C26H19O9 N)n. Structural characterization is performed by NMR. Thermal behavior is studied by thermogravimetry (TG) and differential scanning calorimetry (DSC), showing all the polymers endothermic transition from crystal phase to liquid crystal mesophase. Chirality is determined by optical rotatory dispersion (ORD). The cationic cholesteric liquid crystal polymers described here have proved to act as nonviral vectors in gene therapy, transfecting DNA to the nucleus cell.

**Keywords:** cholesteric LC, cationic polymers, chiral polyesters, synthesis, NMR, DSC

### **1. Introduction**

Cholesteric liquid crystal polyesters have received much attention in the last few years for their interesting chemical, optical, mechanical, and biological properties. Due to their anisotropic formulation and amphiphilic nature, their molecules are able to self-associate and/or aggregate in blocks to form species with supramolecular ordered structure, which presents desirable material properties.

Two cholesteric liquid crystal polyesters, named PTOBDME and PTOBEE in **Figure 1**, were obtained by polycondensation reaction. Although only racemic materials were used in their synthesis, a cholesteric, chiral morphology, theoretically unexpected, was found. Evidence of this was obtained when a white solid, recrystallized, as the second fraction, from toluene mother

**2.1. Synthesis of cholesteric PTOBDME-choline [(C34H36O8**

**)n**─**C5**

Poly[oxy(1,2-dodecane)oxycarbonyl-1,4-phenylene-oxy-1,4-terephthaloyl-oxy-1,4-phenylene-carbonyl]-oxy-*N*, *N*, *N*-trimethylethan-1-ammonium (Choline) chloride, **II** in **Figure 2**, was obtained through polycondensation reaction between: 4,4′-(terephthaloyldioxydibenzoic

**Figure 2.** Synthetic process of cholesteric liquid-crystalline PTOBDME-choline (m = 9) (II) and PTOBEE-choline (m = 1) (III). Monomeric units are indicated, together with aliphatic end groups and choline aromatic end groups. The

C\*), respectively. Torsion angles *ϕ*, along (12C─11C) and (4

C─<sup>3</sup>

C) bonds,

asterisks indicate the chiral centers (12C\*) and (4

respectively, are shown.

**H13N]**

Synthetic Cationic Cholesteric Liquid Crystal Polymers http://dx.doi.org/10.5772/intechopen.70995 9

**Figure 1.** Monomeric unit of cholesteric liquid-crystalline PTOBEE (m = 1) and PTOBDME (m = 9). The three different zones of the monomer: *mesogen*, *spacer,* and *flexible side chain* are indicated. The asterisk indicates the chiral center. Torsion angle *ϕ*, along the bond containing the asymmetric carbon atom, is indicated. Aromatic-end acid and aliphaticend alcoholic groups are also specified.

liquor after the filtration of the polymer, was identified as ─PTOBDME, with [α]<sup>25</sup> <sup>589</sup> = −1.43 (1.538g/100 ml, toluene) [1, 2] and ─PTOBEE, with a value of [α]<sup>25</sup> <sup>589</sup> = −2.33 (0.0056 mol/l, toluene) [3], respectively. The synthetic method [4], based on the previously reported by Bilibin [5], leads to obtain two or more fractions of different kinetic rates, with different enantiomeric excess. Not always, the enantiomer in excess is the same.

We are interested in the molecular design and chemical modifications of these multifunctional cholesteric liquid crystals to obtain new chemical formulations involving macromolecular features with new properties added. Our main interest being to introduce cationic charge, hence favoring the creation of hydrogen bonds, through intra and intermolecular interactions, giving secondary structures with long-range supramolecular order, and enabling to interact with molecules of interest, such as biological molecules (lipids, DNA, and oligonucleotides) and metal surfaces. The functional groups selected to be introduced at the end of the main chains were Choline [─CH2 ─CH2 ─N─(CH3 )3 ] and ammonium [─CH2 ─CH2 ─CH2 ─NH─ (CH3 )2 ] and amide groups (─CONH2 ) at the end of the lateral hydrophobic chains.

The new synthetized cationic polymers reported here have proved to be able to interact with negatively charged DNA, forming polyplexes, which are able to condense and successfully transfect the new DNA into the nucleus cell, protecting it from damage during the transfection process, acting as nonviral vectors in Gene Therapy [6, 7]. Besides, they are sensitive to pH changes, acting as polycationic efficient transfection agents possessing substantial buffering capacity below physiological pH. These vectors have shown to deliver genes as well as oligonucleotides, both *in vitro* and *in vivo*, by protecting DNA from inactivation by blood components. Their efficiency relies on extensive endosome swelling and rupture that provides an escape mechanism for the polycation/DNA complexes [8].

### **2. Materials**

The new cholesteric liquid crystal polymers so designed have been synthesized as follows: PTOBDME-choline [(C34H36O8 )n─C5 H13N]; PTOBEE-choline [(C26H20O8 )n─C5 H13N]; PTOBDME-ammonium [(C34H36O8 )n─C5 H13N]; PTOBEE-ammonium [(C26H20O8 )n─C5 H13N]; PTOBUME-amide [(C33H33O9 N)n; and PTOBEE-amide (C26H19O9 N)n.

#### **2.1. Synthesis of cholesteric PTOBDME-choline [(C34H36O8 )n**─**C5 H13N]**

liquor after the filtration of the polymer, was identified as ─PTOBDME, with [α]<sup>25</sup>

ene) [3], respectively. The synthetic method [4], based on the previously reported by Bilibin [5], leads to obtain two or more fractions of different kinetic rates, with different enantiomeric

**Figure 1.** Monomeric unit of cholesteric liquid-crystalline PTOBEE (m = 1) and PTOBDME (m = 9). The three different zones of the monomer: *mesogen*, *spacer,* and *flexible side chain* are indicated. The asterisk indicates the chiral center. Torsion angle *ϕ*, along the bond containing the asymmetric carbon atom, is indicated. Aromatic-end acid and aliphatic-

We are interested in the molecular design and chemical modifications of these multifunctional cholesteric liquid crystals to obtain new chemical formulations involving macromolecular features with new properties added. Our main interest being to introduce cationic charge, hence favoring the creation of hydrogen bonds, through intra and intermolecular interactions, giving secondary structures with long-range supramolecular order, and enabling to interact with molecules of interest, such as biological molecules (lipids, DNA, and oligonucleotides) and metal surfaces. The functional groups selected to be introduced at the end of the

─N─(CH3

)3

The new synthetized cationic polymers reported here have proved to be able to interact with negatively charged DNA, forming polyplexes, which are able to condense and successfully transfect the new DNA into the nucleus cell, protecting it from damage during the transfection process, acting as nonviral vectors in Gene Therapy [6, 7]. Besides, they are sensitive to pH changes, acting as polycationic efficient transfection agents possessing substantial buffering capacity below physiological pH. These vectors have shown to deliver genes as well as oligonucleotides, both *in vitro* and *in vivo*, by protecting DNA from inactivation by blood components. Their efficiency relies on extensive endosome swelling and rupture that provides an

The new cholesteric liquid crystal polymers so designed have been synthesized as fol-

N)n; and PTOBEE-amide (C26H19O9

)n─C5

)n─C5

] and ammonium [─CH2

H13N]; PTOBEE-choline [(C26H20O8

N)n.

H13N]; PTOBEE-ammonium [(C26H20O8

) at the end of the lateral hydrophobic chains.

(1.538g/100 ml, toluene) [1, 2] and ─PTOBEE, with a value of [α]<sup>25</sup>

8 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

─CH2

excess. Not always, the enantiomer in excess is the same.

escape mechanism for the polycation/DNA complexes [8].

main chains were Choline [─CH2

end alcoholic groups are also specified.

] and amide groups (─CONH2

lows: PTOBDME-choline [(C34H36O8

PTOBDME-ammonium [(C34H36O8

PTOBUME-amide [(C33H33O9

(CH3 )2

**2. Materials**

<sup>589</sup> = −1.43

<sup>589</sup> = −2.33 (0.0056 mol/l, tolu-

─CH2

─CH2

)n─C5

)n─C5

H13N];

H13N];

─NH─

Poly[oxy(1,2-dodecane)oxycarbonyl-1,4-phenylene-oxy-1,4-terephthaloyl-oxy-1,4-phenylene-carbonyl]-oxy-*N*, *N*, *N*-trimethylethan-1-ammonium (Choline) chloride, **II** in **Figure 2**, was obtained through polycondensation reaction between: 4,4′-(terephthaloyldioxydibenzoic

**Figure 2.** Synthetic process of cholesteric liquid-crystalline PTOBDME-choline (m = 9) (II) and PTOBEE-choline (m = 1) (III). Monomeric units are indicated, together with aliphatic end groups and choline aromatic end groups. The asterisks indicate the chiral centers (12C\*) and (4 C\*), respectively. Torsion angles *ϕ*, along (12C─11C) and (4 C─<sup>3</sup> C) bonds, respectively, are shown.

chloride) **TOBC**, **I** in **Figure 2**, the racemic mixture of DL-1,2-dodecanediol, and choline chloride. Notation similar to precursor cholesteric liquid crystal PTOBDME [1, 2] is used.

*2.4.1. Preparation of PTOBMDE-ammonium and PTOBEE-ammonium*

cal preparation of PTOBDME-ammonium chloride is shown.

PTOBDME, which was filtered, washed with ethanol, and vacuum dried.

**2.5. Synthesis of cholesteric PTOBUME-amide [(C33H33O9**

*2.5.1. Preparation of undec-10-enoyl chloride (II in Figure 4)*

solution generated by reaction between 100 g of ClHN4

*2.5.2. Preparation of undec-10-enamide*

in a bath of dry ice/acetone. The NH3

A NH3

than with precursor cholesteric liquid crystal PTOBDME, **Figure 1**.

tion was used directly to prepare undec-10-enamide (**III** in **Figure 4**).

PTOBDME-ammonium chloride, **II** in **Figure 3**, and PTOBEE-ammonium chloride, **III** in **Figure 3**, were obtained through polycondensation reaction between 4 and 4′-(terephthaloyldioxydibenzoic chloride) **TOBC**, **I** in **Figure 3**, and the racemic mixture of **DL-1,2-dodecanediol** and **DL-1,2-butanediol,** respectively, and then reaction with 3-Dimethylamino-1-propanol. Notation of cholesteric liquid crystal PTOBDME and PTOBEE precursors is used. Next, a typi-

Into a flask of 50 ml, TOBC (0.0079 mol) and 1,2-dodecanediol (0.0079 mol) from Flucka Chemie GmBH (Buchs, Switzerland) and diphenyl oxide (19.7 ml) of from Sigma-Aldrich Chemie GmBH (Steinheim, Germany) were mixed, while the system was purged with stream of dry nitrogen from Praxair (Madrid, Spain), for 30 min at room temperature. Then, while maintaining the gas current, the flask was transferred to a bath at 200°C for 2 hours; since the liberation of HCl is still observed, the temperature of the bath was descended to 160°C, the polycondensation was stopped and was not observed HCl formation. 3-Dimethylamino-1-propanol (0.2 ml, 0.00156 mol) was added to the reaction mix, and the liberation of HCl returned again. After 2 hours, the reaction finished. The result of the polycondensation reaction was poured into 200 ml of toluene from Merck KGaA (Darmstadt, Germany), decanting

Poly[oxy(1,2-undecan-11-amidyl)-oxycarbonyl-1,4-phenylene-oxy-1,4-terephthaloyl-oxy-1,4-phenylene-carbonyl], **VII** in **Figure 4**, was obtained through polycondensation reaction between 4 and 4′-(terephthaloyldioxydibenzoic chloride) **TOBC** and the racemic mixture of **DL-10,11-dihydroxyundecanemide** (**V** in **Figure 4**) [11–15]. Similar notation has been used

To a stirred solution of 0.118 mol of undec-10-enoic acid in 100 ml of toluene, at 25°C, 0.078 mol of oxalyl chloride was added during 30 minutes. The solution was stirred for 30 minutes after emission of HCl gas had completed. The mixture reaction was concentrated to about half the initial volume by using a vacuum pump equipped with a sodium hydroxide trap. This solu-

gas stream was used to purge the stirred solution of undec-10-enoyl chloride, cooled

of NaOH solved in 50 ml of water at 10°C. The reflux condenser and a NaOH trap were connected between the ammonia solution and the mixture reaction to prevent moisture. After 30 minutes of reaction, when a white solid had precipitated and HCl gas emission was not

stream was produced by boiling to reflux ammonia

solved into 300 ml of H2

O and 76 g

**N)n**

Synthetic Cationic Cholesteric Liquid Crystal Polymers http://dx.doi.org/10.5772/intechopen.70995 11

#### **2.2. Synthesis of cholesteric PTOBEE-choline [(C26H20O8 ) <sup>n</sup>**─**C5 H13N]**

The structure of Poly[oxy(1,2-butane)oxycarbonyl-1,4-phenylene-oxy-1,4-terephthaloyl-oxy-1,4-phenylene-carbonyl]-oxy-*N*,*N*,*N*-trimethylethan-1-aminium (choline) chloride is shown in **III** of **Figure 2**. The polycondensation included DL-1,2-butanediol. Notation similar to precursor cholesteric liquid crystal PTOBEE [3, 4] is used.

### *2.2.1. Preparation of PTOBDME-choline and PTOBEE-choline*

The dichloride, TOBC, was obtained by reaction between thionyl chloride and 4,4′-(terephthaloyldioxydibenzoic) acid (TOBA), previously synthesized from terephthaloyl chloride and 4-hydroxybenzoic acid [5].

The polycondensation reaction between TOBC and the racemic mixture, the corresponding glycol, takes place in presence of 1/7 equimolecular choline chloride. The preparation of these compounds was performed on melting due to the insolubility of choline chloride in the solvents used in the synthesis of PTOBDME or PTOBEE precursors, diphenyl oxide, or chloronaphthalene.

A mixture of 0.0054 mol of the glycol, either DL-1,2-dodecanediol or DL-1,2-butanediol, from Flucka Chemie GmBH (Buchs, Switzerland) and 0.000775 mol of choline chloride from Sigma-Aldrich Chemie GmBH (Steinheim, Germany) were placed into a flask of 50 ml contained in a bath with a high-temperature transfer agent, while a current of dry nitrogen from Praxair (Madrid, Spain) was used to purge the system at room temperature and then maintained in the rest of the reaction. The mixture was stirred and heated to 110°C to whole dissolution of the choline chloride into diol. The bath was cooled to 80°C, and 0.0062 mol of TOBC was added; this temperature was maintained for 15 minutes. The bath was heated up to 190°C, the mixture was melted, and emission of HCl was observed. After 60 minutes, 15 ml of chloronaphthalene from Sigma-Aldrich Chemie GmBH (Steinheim, Germany) was added. The reaction mix was maintained into the solvent stirring at 190°C for 150 minutes. Then, it was poured into 150 ml of toluene from Merck KGaA (Darmstadt, Germany), decanting PTOBDME-choline or PTOBEE-choline, respectively, which was filtered, washed with ethanol, and vacuum dried.

#### **2.3. Synthesis of cholesteric PTOBDME-ammonium [(C34H36O8 ) <sup>n</sup>**─**C5 H13N]**

The structure of Poly[oxy (1,2-dodecane)-oxy-carbonyl-1,4-phenylene-oxy-1,4-terephthaloyloxy-1,4-phenylene-carbonyl]-oxy-3-dimethyl amine-1-propyl choride is shown in **II** of **Figure 3**.

#### **2.4. Synthesis of Cholesteric PTOBEE-ammonium [(C26H20O8 )n**─**C5 H13N]**

The structure of Poly[oxy(1,2-butane)oxycarbonyl-1,4-phenylene-oxy-1,4-terephthaloyl -oxy-1,4-phenylene-carbonyl]-oxy-3-dimethylamine-1-propyl choride is shown in **III** of **Figure 3**.

### *2.4.1. Preparation of PTOBMDE-ammonium and PTOBEE-ammonium*

chloride) **TOBC**, **I** in **Figure 2**, the racemic mixture of DL-1,2-dodecanediol, and choline chlo-

The structure of Poly[oxy(1,2-butane)oxycarbonyl-1,4-phenylene-oxy-1,4-terephthaloyl-oxy-1,4-phenylene-carbonyl]-oxy-*N*,*N*,*N*-trimethylethan-1-aminium (choline) chloride is shown in **III** of **Figure 2**. The polycondensation included DL-1,2-butanediol. Notation similar to pre-

The dichloride, TOBC, was obtained by reaction between thionyl chloride and 4,4′-(terephthaloyldioxydibenzoic) acid (TOBA), previously synthesized from terephthaloyl chloride and

The polycondensation reaction between TOBC and the racemic mixture, the corresponding glycol, takes place in presence of 1/7 equimolecular choline chloride. The preparation of these compounds was performed on melting due to the insolubility of choline chloride in the solvents used in the synthesis of PTOBDME or PTOBEE precursors, diphenyl oxide, or

A mixture of 0.0054 mol of the glycol, either DL-1,2-dodecanediol or DL-1,2-butanediol, from Flucka Chemie GmBH (Buchs, Switzerland) and 0.000775 mol of choline chloride from Sigma-Aldrich Chemie GmBH (Steinheim, Germany) were placed into a flask of 50 ml contained in a bath with a high-temperature transfer agent, while a current of dry nitrogen from Praxair (Madrid, Spain) was used to purge the system at room temperature and then maintained in the rest of the reaction. The mixture was stirred and heated to 110°C to whole dissolution of the choline chloride into diol. The bath was cooled to 80°C, and 0.0062 mol of TOBC was added; this temperature was maintained for 15 minutes. The bath was heated up to 190°C, the mixture was melted, and emission of HCl was observed. After 60 minutes, 15 ml of chloronaphthalene from Sigma-Aldrich Chemie GmBH (Steinheim, Germany) was added. The reaction mix was maintained into the solvent stirring at 190°C for 150 minutes. Then, it was poured into 150 ml of toluene from Merck KGaA (Darmstadt, Germany), decanting PTOBDME-choline or PTOBEE-choline, respectively, which was filtered, washed with etha-

The structure of Poly[oxy (1,2-dodecane)-oxy-carbonyl-1,4-phenylene-oxy-1,4-terephthaloyloxy-1,4-phenylene-carbonyl]-oxy-3-dimethyl amine-1-propyl choride is shown in **II** of **Figure 3**.

The structure of Poly[oxy(1,2-butane)oxycarbonyl-1,4-phenylene-oxy-1,4-terephthaloyl -oxy-1,4-phenylene-carbonyl]-oxy-3-dimethylamine-1-propyl choride is shown in **III** of **Figure 3**.

**) <sup>n</sup>**─**C5**

**H13N]**

**)n**─**C5**

**)n**─**C5**

**H13N]**

**H13N]**

ride. Notation similar to precursor cholesteric liquid crystal PTOBDME [1, 2] is used.

**2.2. Synthesis of cholesteric PTOBEE-choline [(C26H20O8**

10 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

cursor cholesteric liquid crystal PTOBEE [3, 4] is used.

4-hydroxybenzoic acid [5].

chloronaphthalene.

nol, and vacuum dried.

*2.2.1. Preparation of PTOBDME-choline and PTOBEE-choline*

**2.3. Synthesis of cholesteric PTOBDME-ammonium [(C34H36O8**

**2.4. Synthesis of Cholesteric PTOBEE-ammonium [(C26H20O8**

PTOBDME-ammonium chloride, **II** in **Figure 3**, and PTOBEE-ammonium chloride, **III** in **Figure 3**, were obtained through polycondensation reaction between 4 and 4′-(terephthaloyldioxydibenzoic chloride) **TOBC**, **I** in **Figure 3**, and the racemic mixture of **DL-1,2-dodecanediol** and **DL-1,2-butanediol,** respectively, and then reaction with 3-Dimethylamino-1-propanol. Notation of cholesteric liquid crystal PTOBDME and PTOBEE precursors is used. Next, a typical preparation of PTOBDME-ammonium chloride is shown.

Into a flask of 50 ml, TOBC (0.0079 mol) and 1,2-dodecanediol (0.0079 mol) from Flucka Chemie GmBH (Buchs, Switzerland) and diphenyl oxide (19.7 ml) of from Sigma-Aldrich Chemie GmBH (Steinheim, Germany) were mixed, while the system was purged with stream of dry nitrogen from Praxair (Madrid, Spain), for 30 min at room temperature. Then, while maintaining the gas current, the flask was transferred to a bath at 200°C for 2 hours; since the liberation of HCl is still observed, the temperature of the bath was descended to 160°C, the polycondensation was stopped and was not observed HCl formation. 3-Dimethylamino-1-propanol (0.2 ml, 0.00156 mol) was added to the reaction mix, and the liberation of HCl returned again. After 2 hours, the reaction finished. The result of the polycondensation reaction was poured into 200 ml of toluene from Merck KGaA (Darmstadt, Germany), decanting PTOBDME, which was filtered, washed with ethanol, and vacuum dried.

#### **2.5. Synthesis of cholesteric PTOBUME-amide [(C33H33O9 N)n**

Poly[oxy(1,2-undecan-11-amidyl)-oxycarbonyl-1,4-phenylene-oxy-1,4-terephthaloyl-oxy-1,4-phenylene-carbonyl], **VII** in **Figure 4**, was obtained through polycondensation reaction between 4 and 4′-(terephthaloyldioxydibenzoic chloride) **TOBC** and the racemic mixture of **DL-10,11-dihydroxyundecanemide** (**V** in **Figure 4**) [11–15]. Similar notation has been used than with precursor cholesteric liquid crystal PTOBDME, **Figure 1**.

### *2.5.1. Preparation of undec-10-enoyl chloride (II in Figure 4)*

To a stirred solution of 0.118 mol of undec-10-enoic acid in 100 ml of toluene, at 25°C, 0.078 mol of oxalyl chloride was added during 30 minutes. The solution was stirred for 30 minutes after emission of HCl gas had completed. The mixture reaction was concentrated to about half the initial volume by using a vacuum pump equipped with a sodium hydroxide trap. This solution was used directly to prepare undec-10-enamide (**III** in **Figure 4**).

### *2.5.2. Preparation of undec-10-enamide*

A NH3 gas stream was used to purge the stirred solution of undec-10-enoyl chloride, cooled in a bath of dry ice/acetone. The NH3 stream was produced by boiling to reflux ammonia solution generated by reaction between 100 g of ClHN4 solved into 300 ml of H2 O and 76 g of NaOH solved in 50 ml of water at 10°C. The reflux condenser and a NaOH trap were connected between the ammonia solution and the mixture reaction to prevent moisture. After 30 minutes of reaction, when a white solid had precipitated and HCl gas emission was not

three times with additional dichloromethane. The combined dichloromethane extracts were washed with brine, dried with anhydrous sodium sulfate, and concentrated to a solid on a rotatory evaporator [12–15]. The solid was recrystallized in a mix chloroform/hexane (1:1) to

(bs, 1H,) *(5.13)*, δ 4.96, (dd, 1H J = 7.2 Hz) *(4.88)*, δ 2.21, (t, 2H, J = 7.6 Hz) *(2.34)*; δ 2.02 *(2.13)*, (m, 2H), δ 1.62 (t, 2H, J = 7.4 Hz) *(1.53)*, δ 1.33-1.28, overlapped (10H) *(1.33, 1.30, 1.30, 1.30, 1.29)*.

*(38.7)*, 34.2 *(33.9*), 29.4 *(29.7, 29.7, 29.6, 28.9, 28.6)* and 25.4 *(25.3)*. HRMS m/z calc. For

was added, and then, 5.2 ml of water was added carefully. The resultant thick mixture was strongly stirred, while a solution of 40.6 g of oxone in 158 ml of water was added dropwise during 45 min. The reaction was monitored by thin layer chromatography (TLC) using a mix of ethyl acetate/hexane 2:1. After the reaction was complete, the acetone was removed by evaporation. The remaining solution was acidified with HCl 10% to pH 2 at 10°C and followed rapid extraction with 250 ml of dichloromethane. The aqueous phase was washed three times with additional dichloromethane. The combined organic phase was washed with brine, dried with anhydrous sodium sulfate, and concentrated to a white solid on a rotatory evapo-

δ 2.46, (dd, 1H J = 5.0 Hz), δ 2.21, (t, 2H, J = 7.6 Hz); δ 1.62 (t, 2H, J = 7.4 Hz), δ 1.51, (m, 2H), δ

**C**═), 38.7 (1C) (─H2

Na + [M + Na]+; found.

The previously obtained 10-11 epoxy undecanamide was stirred during 8 hours at 60°C in aqueous HCl 10%. The reaction was monitored by TLC using a mix of ethyl acetate/hexane 2:1. An oil, not miscible with water, was obtained. The mixture reaction was extracted with dichloromethane, and the aqueous phase was washed three times with additionally dichloromethane. The combined organic phase was washed with brine, dried with anhydrous sodium sulfate, and concentrated to a yellow oil on a rotatory evaporator. Yield (75%) (**V** in **Figure 4**).

δ 3.48, (dd, 1H J = 2.21, (t, 2H, J = 7.6 Hz); δ 1.62 (t, 2H, J = 7.4 Hz);), δ 1.51, (m, 2H), δ 1.44 (m,

100 MHz, δ; (ppm)): HRMS m/z calc. For C11H23NO3

To a stirred solution of undec-10-enamide (7 g;) in 108.4 ml of acetone, NaHCO3

300 MHz, δ; (ppm)): δ 7.19, (dd, 2H) *(7.03)*, δ 5.80, (m, 1H) *(5.82)*, δ 5.32-5.22,

100 MHz, δ; (ppm)): 175.2 *(173.6)*, 139.6 *(139.1)*, 114.5 *(115.7)*, 36.3

300 MHz, δ; (ppm)): δ 5.44, (bs, 2H), δ 2.90, (m, 1H), δ 2.74, (dd, 1H J = 4.6 Hz),

300 MHz, δ; (ppm)): δ 5.44, (bs, 2H), δ 3.78, (m, 1H), δ 3.60, (dd, 1H J = 11.0 Hz),

**C**─CONH2

100 MHz, δ; (ppm)): 173.8 (1C) (**C**ONH2

Synthetic Cationic Cholesteric Liquid Crystal Polymers http://dx.doi.org/10.5772/intechopen.70995

H3

O+ [M + H3

O] + 236.2;

), 29.7 (3C), 28.7 (2C), 25.3

(26.4 g)

13

),

give pure undec-10-enamide—yield (80%) and melting point 87°C (**III** in **Figure 4**).

1

H NMR (CDCl3

13C NMR (CDCl3

rator (**IV** in **Figure 4**).

H NMR (CDCl3

1

1

H NMR (CDCl3

13C NMR (CDCl3

found 236.2.

2H), δ 1.33-1.28, (bs, 8H);

In tilted numbers are the calculated shifts.

C11H21NONa + [M + Na] + 206.2; found 206.2.

*2.5.3. Preparation of 10-11 epoxy undecanamide*

1.44 (m, 2H), δ 1.33-1.28, (bs, 8H); 13C NMR (CDCl3

*2.5.4. Preparation of 10-11 of dihydroxyundecanamide*

137.6 (1C) (═**C**H─C), 115.7 (1C) (H2

(1C); HRMS m/z calc. For C11H21NO2

**Figure 3.** Synthetic process of cholesteric liquid-crystalline PTOBDME-ammonium (II), and PTOBEE-ammonium (III). Monomeric units are indicated, together with aliphatic end groups and ammonium aromatic end groups. The asterisks indicate the chiral centers (4 C\*) and (12C\*), respectively. Torsion angles *ϕ*, along (4 C─<sup>3</sup> C) and (12C─11C) bonds, are shown.

observed, the reaction flask was allowed to warm to room temperature. The mixture reaction was concentrated to a residue on a rotatory evaporator. The solid was partitioned between 10% aqueous sodium hydroxide and dichloromethane, and the aqueous phase was washed three times with additional dichloromethane. The combined dichloromethane extracts were washed with brine, dried with anhydrous sodium sulfate, and concentrated to a solid on a rotatory evaporator [12–15]. The solid was recrystallized in a mix chloroform/hexane (1:1) to give pure undec-10-enamide—yield (80%) and melting point 87°C (**III** in **Figure 4**).

1 H NMR (CDCl3 300 MHz, δ; (ppm)): δ 7.19, (dd, 2H) *(7.03)*, δ 5.80, (m, 1H) *(5.82)*, δ 5.32-5.22, (bs, 1H,) *(5.13)*, δ 4.96, (dd, 1H J = 7.2 Hz) *(4.88)*, δ 2.21, (t, 2H, J = 7.6 Hz) *(2.34)*; δ 2.02 *(2.13)*, (m, 2H), δ 1.62 (t, 2H, J = 7.4 Hz) *(1.53)*, δ 1.33-1.28, overlapped (10H) *(1.33, 1.30, 1.30, 1.30, 1.29)*. In tilted numbers are the calculated shifts.

13C NMR (CDCl3 100 MHz, δ; (ppm)): 175.2 *(173.6)*, 139.6 *(139.1)*, 114.5 *(115.7)*, 36.3 *(38.7)*, 34.2 *(33.9*), 29.4 *(29.7, 29.7, 29.6, 28.9, 28.6)* and 25.4 *(25.3)*. HRMS m/z calc. For C11H21NONa + [M + Na] + 206.2; found 206.2.

### *2.5.3. Preparation of 10-11 epoxy undecanamide*

To a stirred solution of undec-10-enamide (7 g;) in 108.4 ml of acetone, NaHCO3 (26.4 g) was added, and then, 5.2 ml of water was added carefully. The resultant thick mixture was strongly stirred, while a solution of 40.6 g of oxone in 158 ml of water was added dropwise during 45 min. The reaction was monitored by thin layer chromatography (TLC) using a mix of ethyl acetate/hexane 2:1. After the reaction was complete, the acetone was removed by evaporation. The remaining solution was acidified with HCl 10% to pH 2 at 10°C and followed rapid extraction with 250 ml of dichloromethane. The aqueous phase was washed three times with additional dichloromethane. The combined organic phase was washed with brine, dried with anhydrous sodium sulfate, and concentrated to a white solid on a rotatory evaporator (**IV** in **Figure 4**).

1 H NMR (CDCl3 300 MHz, δ; (ppm)): δ 5.44, (bs, 2H), δ 2.90, (m, 1H), δ 2.74, (dd, 1H J = 4.6 Hz), δ 2.46, (dd, 1H J = 5.0 Hz), δ 2.21, (t, 2H, J = 7.6 Hz); δ 1.62 (t, 2H, J = 7.4 Hz), δ 1.51, (m, 2H), δ 1.44 (m, 2H), δ 1.33-1.28, (bs, 8H); 13C NMR (CDCl3 100 MHz, δ; (ppm)): 173.8 (1C) (**C**ONH2 ), 137.6 (1C) (═**C**H─C), 115.7 (1C) (H2 **C**═), 38.7 (1C) (─H2 **C**─CONH2 ), 29.7 (3C), 28.7 (2C), 25.3 (1C); HRMS m/z calc. For C11H21NO2 Na + [M + Na]+; found.

#### *2.5.4. Preparation of 10-11 of dihydroxyundecanamide*

observed, the reaction flask was allowed to warm to room temperature. The mixture reaction was concentrated to a residue on a rotatory evaporator. The solid was partitioned between 10% aqueous sodium hydroxide and dichloromethane, and the aqueous phase was washed

**Figure 3.** Synthetic process of cholesteric liquid-crystalline PTOBDME-ammonium (II), and PTOBEE-ammonium (III). Monomeric units are indicated, together with aliphatic end groups and ammonium aromatic end groups. The asterisks

C─<sup>3</sup>

C) and (12C─11C) bonds, are shown.

C\*) and (12C\*), respectively. Torsion angles *ϕ*, along (4

12 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

indicate the chiral centers (4

The previously obtained 10-11 epoxy undecanamide was stirred during 8 hours at 60°C in aqueous HCl 10%. The reaction was monitored by TLC using a mix of ethyl acetate/hexane 2:1. An oil, not miscible with water, was obtained. The mixture reaction was extracted with dichloromethane, and the aqueous phase was washed three times with additionally dichloromethane. The combined organic phase was washed with brine, dried with anhydrous sodium sulfate, and concentrated to a yellow oil on a rotatory evaporator. Yield (75%) (**V** in **Figure 4**).

1 H NMR (CDCl3 300 MHz, δ; (ppm)): δ 5.44, (bs, 2H), δ 3.78, (m, 1H), δ 3.60, (dd, 1H J = 11.0 Hz), δ 3.48, (dd, 1H J = 2.21, (t, 2H, J = 7.6 Hz); δ 1.62 (t, 2H, J = 7.4 Hz);), δ 1.51, (m, 2H), δ 1.44 (m, 2H), δ 1.33-1.28, (bs, 8H);

13C NMR (CDCl3 100 MHz, δ; (ppm)): HRMS m/z calc. For C11H23NO3 H3 O+ [M + H3 O] + 236.2; found 236.2.

*2.5.6. Preparation of PTOBUME-amide.*

which was filtered, washed with ethanol, and vacuum dried.

**Figure 5.** Synthetic method of cholesteric liquid-crystalline PTOBEE-amide.

Yield first fraction 3.0 g (38.5%); yield first and second fraction 0.1 g (40.0%).

A mixture of TOBC (5.5 g; 0.012 mol), 10-11 of dihydroxyundecanamide (2.7 g; 0.012 mol) in 3 ml of diphenyl oxide from Sigma-Aldrich Chemie GmBH (Steinheim, Germany) was purged with dry nitrogen from Praxair (Madrid, Spain) for 25min at room temperature. Then, while maintaining the gas stream, the flask was transferred to a bath containing a high-temperature heat-transfer agent. The polycondensation was carried out for 360 minutes at 200°C. The reaction gets completed when emission of HCl had finished. The reaction mixture was poured into 300 ml of toluene from Merck KGaA (Darmstadt, Germany), decanting PTOBUME-amide. After 12 hours, it was filtered, washed with ethanol, and vacuum dried. After 3 weeks, a second fraction of polymer was precipitated of the toluene mother liquors,

Synthetic Cationic Cholesteric Liquid Crystal Polymers http://dx.doi.org/10.5772/intechopen.70995 15

**Figure 4.** Synthetic method of cholesteric liquid-crystalline PTOBUME-amide. The asterisk indicates the chiral center ( 11C\*). Torsion angle *ϕ*, along 10C─11C bond.

#### *2.5.5. Preparation of TOBC*

In the course of 20 minutes, 20 g TOBA were added to 350 ml thionyl chloride from Sigma-Aldrich Chemie GmBH (Steinheim, Germany), while stirring rapidly at room temperature (**VI** in **Figure 4**).

The solution was boiled with the reflux condenser. When the emission of HCl had finished and most of the sediment had dissolved, the hot solution was filtered and cooled down to 0°C for a day. The obtained product that separated out was filtered, vacuum dried, and recrystallized in chloroform, from SDS Votre Partenaire Chimie (Peypin, France).

Yield: 14 g (60%).

### *2.5.6. Preparation of PTOBUME-amide.*

A mixture of TOBC (5.5 g; 0.012 mol), 10-11 of dihydroxyundecanamide (2.7 g; 0.012 mol) in 3 ml of diphenyl oxide from Sigma-Aldrich Chemie GmBH (Steinheim, Germany) was purged with dry nitrogen from Praxair (Madrid, Spain) for 25min at room temperature. Then, while maintaining the gas stream, the flask was transferred to a bath containing a high-temperature heat-transfer agent. The polycondensation was carried out for 360 minutes at 200°C. The reaction gets completed when emission of HCl had finished. The reaction mixture was poured into 300 ml of toluene from Merck KGaA (Darmstadt, Germany), decanting PTOBUME-amide. After 12 hours, it was filtered, washed with ethanol, and vacuum dried. After 3 weeks, a second fraction of polymer was precipitated of the toluene mother liquors, which was filtered, washed with ethanol, and vacuum dried.

Yield first fraction 3.0 g (38.5%); yield first and second fraction 0.1 g (40.0%).

**Figure 5.** Synthetic method of cholesteric liquid-crystalline PTOBEE-amide.

*2.5.5. Preparation of TOBC*

11C\*). Torsion angle *ϕ*, along 10C─11C bond.

14 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

(**VI** in **Figure 4**).

(

Yield: 14 g (60%).

In the course of 20 minutes, 20 g TOBA were added to 350 ml thionyl chloride from Sigma-Aldrich Chemie GmBH (Steinheim, Germany), while stirring rapidly at room temperature

**Figure 4.** Synthetic method of cholesteric liquid-crystalline PTOBUME-amide. The asterisk indicates the chiral center

The solution was boiled with the reflux condenser. When the emission of HCl had finished and most of the sediment had dissolved, the hot solution was filtered and cooled down to 0°C for a day. The obtained product that separated out was filtered, vacuum dried, and recrystal-

lized in chloroform, from SDS Votre Partenaire Chimie (Peypin, France).

#### **2.6. Synthesis of Cholesteric PTOBEE-amide (C26H19O9 N)n**

Poly[oxy(1,2-butan-4-amidyl)-oxycarbonyl-1,4-phenylene-oxy-1,4-terephthaloyl-oxy-1,4 phenylene-carbonyl], **VI** in **Figure 5**, was obtained through poly-condensation reaction between 4 and 4′-(terephthaloyldioxydibenzoic chloride) **TOBC** and the racemic mixture of **DL-3,4-dihydroxybutanamide** (**IV** in **Figure 5**). The same notation has been used with precursor cholesteric liquid crystal PTOBEE, **Figure 1**.

Spectroscopy, through-space correlation method), HSQC (Heteronuclear Single-Quantum Correlation spectroscopy), and HMBC (Heteronuclear Multiple Bond Correlation) for correlations between carbons and protons that are separated by two, three, and sometimes four bonds,

The experiments were performed in a Bruker 300 MHz NMR spectrometer and VARIAN 400

Thermal stability was studied by Thermogravimetry on a Mettler TA4000-TG50 at heating rate of 10°C/min with nitrogen purge between 30 and 600°C. Thermal behavior was determined by differential scanning calorimetry (DSC) in a Mettler TA4000/DSC30/TC11 calorimeter, with series of heating/cooling cycles in a temperature range between 0 and 230°C.

The optical activity of the polymers was measured as optical rotatory dispersion (ORD) at 25°C in DMSO from Scharlau Chemie, in a Perkin Elmer 241 MC polarimeter with wavelengths: λNa = 589 nm, slit = 5 mm, integration time = 50 s; λHg = 574 nm, slit = 14 mm, integration time = 50s; λHg = 546 nm, slit = 30 mm, integration time = 50 s; λHg = 435 nm, slit = 5 mm,

H and 13C-NMR chemical shifts, in DMSO-d6

and the end groups of Polyester PTOBDME-choline, is given in **Table 1**. All the spectra have been analyzed and interpreted the help of MestReNova [9]. The predicted theoretical values, also in **Table 1**, have been calculated by ChemDraw [10]. Similar notations as those assigned

trum, corresponding to *the mesogen*, including aromatic protons between 11.0–7.00 ppm, *the spacer* where methylene and methine protons directly attached to oxygen atoms are observed, with signals between 6 and 3 ppm, and *the flexible side chain* formed by aliphatic protons between 2 and 0.8 ppm. The main feature of the proton spectrum is the presence of higher

bonded to 11C atom, allocated in α position with respect to the asymmetric carbon atom 12C\*. For that reason, they are diastereotopic and their 1H-NMR signals, usually indistinguishable,

with precursor cholesteric liquid crystal polyesters PTOBDME [1, 2] have been used.

Considering the monomer structure, three zones can be differentiated in the <sup>1</sup>

number of peaks than those expected for the monomeric unit. Hydrogen atoms Ha

split in two easily differentiated. The same effect is observed for Hd and He

C.

integration time = 50s; λHg = 365 nm, slit = 2.5 mm, integration time = 50 s.

**4. Structural characterization by NMR**

**4.1. Structural characterization of PTOBDME-choline**

shifts were calculated from the formula with ChemDraw Professional, v.15.1.0.144 [10].

and CDCl3

Synthetic Cationic Cholesteric Liquid Crystal Polymers http://dx.doi.org/10.5772/intechopen.70995

H chemical shifts were referenced to the residual solvent signal

) relative to tetramethylsilane (TMS). All the spectra were processed

, from Merck KGaA

17

H and 13C-NMR chemical

, of the monomeric unit

H-NMR spec-

and Hb

, bonded to 10C, and

are

in conjugated systems. Direct one-bond correlations being suppressed.

and 500 MHz spectrometers. The solvents used were DMSO-d6

and analyzed with MestReNova v.11.0.4 software [9]. Predicted 1

(Darmstadt, Germany), at 25°C. 1

at δ = 2.50 ppm (DMSO-d<sup>6</sup>

**3.2. Thermal behavior**

**3.3. The optical activity**

The designation of the 1

for Hf

and Hg

, both bonded to 9

### *2.6.1. Preparation of 3,4-dihydroxybutanoic acid*

To a stirred mixture of 10 g of but-3-enoic acid solved in 130 ml acetone, 34 g. NaHCO3 in 65 ml mili-Q water was added carefully. The resultant mixture was strongly stirred, while a solution of 51.1 g oxone in 200 ml of water was added dropwise during 120 min. The reaction was monitored by thin layer chromatography (TLC) using a mix of ethyl acetate/diethyl ether 1:1. After the reaction was complete, the acetone was removed by evaporation. The remaining solution was acidified with HCl 10% to pH 2 at 10°C and followed of rapid extraction with 250 ml of ethyl acetate. The aqueous phase was washed three times with additional ethyl acetate. The combined organic phase was washed with brine, dried with anhydrous sodium sulfate, and concentrated to a white solid on a rotatory evaporator (**III** in **Figure 5**).

### *2.6.2. Preparation of 4-hydroxydihydrofuran-2(3H)-one*

To 3,4-dihydroxybutanoic acid into a flask equipped with a Dean Stark adapter filled with a toluene column finally connected to a refrigerant, 0.5 ml trifluoroacetic acid was added in 100 ml toluene heating to 110°C, mixing for 3 hours. The reaction product was removed with ethanol, washed in ethyl acetate, and dried. The reaction was monitored by thin layer chromatography (TLC) using a mix of ethyl acetate/diethyl ether 1:1 (**IV** in **Figure 5**).

#### *2.6.3. Preparation of 3,4-dihydroxybutanamide*

To the 4-hydroxydihydrofuran-2(3*H*)-one, 100 ml NH3 33% was added stirring at 70°C with reflux for 12 h. The reaction product was removed with ethanol, filtered, and washed with water several times (**V** in **Figure 5**).

### *2.6.4. Preparation of PTOBEE-amide*

In a three-neck round-bottom flask, 0.2 g 3,4-dihydroxybutanamide was added dropwise to 1 g TOBC solved in 100 ml 1,1,2,2-Tetrachloroethane. The reaction was stirred at 90°C for 20 hours. The reaction product was filtered, washed in 50 ml ethanol, 100 ml water, 200 ml NaHCO3 (10%), 200 ml HCl (5%), 300 ml water, and 200 ml ethanol, and dried.

### **3. Characterization techniques**

### **3.1. Conventional NMR techniques**

The obtained polymers are characterized by 1 H-NMR, 13C-NMR, COSY (Homonuclear Correlation Spectroscopy), TOCSY (Total Correlation Spectroscopy), NOESY (Nuclear Overhauser Effect Spectroscopy, through-space correlation method), HSQC (Heteronuclear Single-Quantum Correlation spectroscopy), and HMBC (Heteronuclear Multiple Bond Correlation) for correlations between carbons and protons that are separated by two, three, and sometimes four bonds, in conjugated systems. Direct one-bond correlations being suppressed.

The experiments were performed in a Bruker 300 MHz NMR spectrometer and VARIAN 400 and 500 MHz spectrometers. The solvents used were DMSO-d6 and CDCl3 , from Merck KGaA (Darmstadt, Germany), at 25°C. 1 H chemical shifts were referenced to the residual solvent signal at δ = 2.50 ppm (DMSO-d<sup>6</sup> ) relative to tetramethylsilane (TMS). All the spectra were processed and analyzed with MestReNova v.11.0.4 software [9]. Predicted 1 H and 13C-NMR chemical shifts were calculated from the formula with ChemDraw Professional, v.15.1.0.144 [10].

### **3.2. Thermal behavior**

**2.6. Synthesis of Cholesteric PTOBEE-amide (C26H19O9**

16 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

cursor cholesteric liquid crystal PTOBEE, **Figure 1**.

*2.6.2. Preparation of 4-hydroxydihydrofuran-2(3H)-one*

*2.6.3. Preparation of 3,4-dihydroxybutanamide*

water several times (**V** in **Figure 5**).

*2.6.4. Preparation of PTOBEE-amide*

**3. Characterization techniques**

**3.1. Conventional NMR techniques**

The obtained polymers are characterized by 1

NaHCO3

To the 4-hydroxydihydrofuran-2(3*H*)-one, 100 ml NH3

*2.6.1. Preparation of 3,4-dihydroxybutanoic acid*

**N)n**

in

33% was added stirring at 70°C with

H-NMR, 13C-NMR, COSY (Homonuclear Correlation

Poly[oxy(1,2-butan-4-amidyl)-oxycarbonyl-1,4-phenylene-oxy-1,4-terephthaloyl-oxy-1,4 phenylene-carbonyl], **VI** in **Figure 5**, was obtained through poly-condensation reaction between 4 and 4′-(terephthaloyldioxydibenzoic chloride) **TOBC** and the racemic mixture of **DL-3,4-dihydroxybutanamide** (**IV** in **Figure 5**). The same notation has been used with pre-

To a stirred mixture of 10 g of but-3-enoic acid solved in 130 ml acetone, 34 g. NaHCO3

65 ml mili-Q water was added carefully. The resultant mixture was strongly stirred, while a solution of 51.1 g oxone in 200 ml of water was added dropwise during 120 min. The reaction was monitored by thin layer chromatography (TLC) using a mix of ethyl acetate/diethyl ether 1:1. After the reaction was complete, the acetone was removed by evaporation. The remaining solution was acidified with HCl 10% to pH 2 at 10°C and followed of rapid extraction with 250 ml of ethyl acetate. The aqueous phase was washed three times with additional ethyl acetate. The combined organic phase was washed with brine, dried with anhydrous sodium sulfate, and concentrated to a white solid on a rotatory evaporator (**III** in **Figure 5**).

To 3,4-dihydroxybutanoic acid into a flask equipped with a Dean Stark adapter filled with a toluene column finally connected to a refrigerant, 0.5 ml trifluoroacetic acid was added in 100 ml toluene heating to 110°C, mixing for 3 hours. The reaction product was removed with ethanol, washed in ethyl acetate, and dried. The reaction was monitored by thin layer chroma-

reflux for 12 h. The reaction product was removed with ethanol, filtered, and washed with

In a three-neck round-bottom flask, 0.2 g 3,4-dihydroxybutanamide was added dropwise to 1 g TOBC solved in 100 ml 1,1,2,2-Tetrachloroethane. The reaction was stirred at 90°C for 20 hours. The reaction product was filtered, washed in 50 ml ethanol, 100 ml water, 200 ml

Spectroscopy), TOCSY (Total Correlation Spectroscopy), NOESY (Nuclear Overhauser Effect

(10%), 200 ml HCl (5%), 300 ml water, and 200 ml ethanol, and dried.

tography (TLC) using a mix of ethyl acetate/diethyl ether 1:1 (**IV** in **Figure 5**).

Thermal stability was studied by Thermogravimetry on a Mettler TA4000-TG50 at heating rate of 10°C/min with nitrogen purge between 30 and 600°C. Thermal behavior was determined by differential scanning calorimetry (DSC) in a Mettler TA4000/DSC30/TC11 calorimeter, with series of heating/cooling cycles in a temperature range between 0 and 230°C.

### **3.3. The optical activity**

The optical activity of the polymers was measured as optical rotatory dispersion (ORD) at 25°C in DMSO from Scharlau Chemie, in a Perkin Elmer 241 MC polarimeter with wavelengths: λNa = 589 nm, slit = 5 mm, integration time = 50 s; λHg = 574 nm, slit = 14 mm, integration time = 50s; λHg = 546 nm, slit = 30 mm, integration time = 50 s; λHg = 435 nm, slit = 5 mm, integration time = 50s; λHg = 365 nm, slit = 2.5 mm, integration time = 50 s.

### **4. Structural characterization by NMR**

### **4.1. Structural characterization of PTOBDME-choline**

The designation of the 1 H and 13C-NMR chemical shifts, in DMSO-d6 , of the monomeric unit and the end groups of Polyester PTOBDME-choline, is given in **Table 1**. All the spectra have been analyzed and interpreted the help of MestReNova [9]. The predicted theoretical values, also in **Table 1**, have been calculated by ChemDraw [10]. Similar notations as those assigned with precursor cholesteric liquid crystal polyesters PTOBDME [1, 2] have been used.

Considering the monomer structure, three zones can be differentiated in the <sup>1</sup> H-NMR spectrum, corresponding to *the mesogen*, including aromatic protons between 11.0–7.00 ppm, *the spacer* where methylene and methine protons directly attached to oxygen atoms are observed, with signals between 6 and 3 ppm, and *the flexible side chain* formed by aliphatic protons between 2 and 0.8 ppm. The main feature of the proton spectrum is the presence of higher number of peaks than those expected for the monomeric unit. Hydrogen atoms Ha and Hb are bonded to 11C atom, allocated in α position with respect to the asymmetric carbon atom 12C\*. For that reason, they are diastereotopic and their 1H-NMR signals, usually indistinguishable, split in two easily differentiated. The same effect is observed for Hd and He , bonded to 10C, and for Hf and Hg , both bonded to 9 C.


In the aromatic zone singlet at 8.36 ppm belongs to 20H and doublets at 7.50 and 8.08 ppm are assigned to 16'H and 15'H, respectively, and doublets at 7.55 and 8.15 ppm to 16H and 15H; similar assignation was previously carried out in precursor PTOBDME [1]. In the spacer zone, multiplet

'Hg

they showed COSY correlations and were related with signals at 1.77 ppm (Hd) and 1.35 ppm

and they were assigned through TOCSY correlations observed for signal at 4.52 ppm (not

") and (Hb

Choline end group showed in **Table 2**, two set of signals probably due to conformational equilibrium: Multiplets assigned to 21H (4.54 ppm), 22H (3.33 ppm), and 23H (2.74 ppm), correlated in COSY, and another set was multiplets 21'H (4.76 ppm), 22'H (3.85 ppm) and a singlet

The HSQC experiment allowed the direct allocation of carbon atoms linked to hydrogens,

'), with the multiplet at 4.39 ppm (Hc

They are also related with Hf" (1.53 ppm) and Hg"(1.44 ppm) by the same experiment.

'. These peaks presented correlation signals in COSY and were related with other

" (4.39 ppm) by COSY experiments, and they were assigned to Hd" and He

' (4.63 ppm) and Hb

**shifts**

**H calc 13C calc DMSO DMSO <sup>1</sup>**

*4.56* 4.75

*3.32* 3.84

2.74 3.22

H and 13C-NMR chemical shifts (ppm), in DMSO-d6, for the -N, N, N-trimethylethan- 1-ammonium

"; and correlation between carbon 21C at (62.6 ppm)

**H(ppm) Atom 13C(ppm)**

58.8 58.5

42.5 52.7

13C 13'C

14C 14'C

15C 15'C

and double doublets at 3.95 and 3.89 ppm to Ha

', and the double doublets at 4.63 and 4.52 ppm correspond

' (1.45 ppm) by TOCSY experiment. Multiplet

Synthetic Cationic Cholesteric Liquid Crystal Polymers http://dx.doi.org/10.5772/intechopen.70995

' (4.52 ppm) and with 21H (4.54 ppm) (in **Table 2**),

", Hb

"). Signals at 1.83 and 1.45 ppm assigned

'(5.45 ppm). Peaks at 1.77 ppm (Hd He

(5.26 ppm). Signals at 1.92, 1.78 ppm are

and Hb

", and Hc

') and 4.52 (Hb

HNMR experiment. The

' (4.52 ppm); correlation

**Calc. chemical shifts**

55.0 3.70 66.5

**H calc 13C calc**

4.69 58.1

3.30 54.4

") and confirmed by HSQC. By

, and

19

" due to

') and

)

".

at 5.45 ppm is interpreted due to Hc

at 5.26 ppm was assigned to Hc

aliphatic end group are overlapped with Hb

carbon 11"C (67.8 ppm) with 4.52 ppm (Ha

', Hg

between 11"C (67.8 ppm) and Ha

Hg

' (1.83 ppm) and Hf

) respectively by TOCSY experiment. The peaks assigned to Ha

this method, carbon C11' (65.7 ppm) was correlated with signals at 4.63 (Ha

') and correlated by COSY with Hc

) are correlated by COSY with Hc

confirming the assignation of the proton signals overlapped in the <sup>1</sup>

**PTOBDME-choline PTOBEE-choline Observed chemical shifts Calc. chemical shifts Observed chemical** 

**H(ppm) Atom 13C(ppm) Atom 1**

" and Hb

4.69 58.1 13H

3.70 66.5 14H

3.30 54.4 15H

13'H

14'H

15'H

(Choline) oxy benzoate hydrochloride end group, in Polyester PTOBDME-choline and Polyester PTOBEE-choline, and

correlation of carbon atom 11'C (65.7 ppm) with Ha

and 21H at (4.54 ppm), are observed in **Table 2**.

**DMSO DMSO <sup>1</sup>**

62.6 58.8

55.6 64.0

43.2 53.0

21C 21'C

22C 22'C

23C 23'C

to Ha

(Hf Hg

' and Hb

aliphatic signals Hd'He

observed for 4.63 ppm, Ha

to (Hd') and (Hf

related with Hc

23'H (3.21 ppm).

**Atom 1**

21H, 21'H

22H, 22'H

23H, 23'H

4.54, 4.76

3.33 3.85

2.74 3.21

**Table 2.** Observed 1

theoretical calculated values.

and 1.35 ppm (Hf

The symbol ( ' ) and without it ( ) distinguish the two independent system of the repeating unit, the symbol ( " ) is used to mark signals due to the aliphatic end group. \* Signal of 4 C to 8 C at 28.8 ppm is a multiplet from 28.9 to 28.6.

**Table 1.** <sup>1</sup> H and 13C-NMR chemical shifts (ppm) observed and calculated for the repeating unit and the aliphatic end group polyester PTOBDME-choline.

The presence of two independent 1 H-NMR sets of signals are observed in the spectrum, one marked with ( ' ) and the other without it ( ). They are attributed to two conformers *gg* and *gt* of the spacer within the repeating unit respectively. The same effect has been reported for PTOBDME and PTOBEE, and accordingly, similar nomenclature is used to identify the signals. A third set of signals, marked with ( " ), is assigned to the aliphatic end group.

In the aromatic zone singlet at 8.36 ppm belongs to 20H and doublets at 7.50 and 8.08 ppm are assigned to 16'H and 15'H, respectively, and doublets at 7.55 and 8.15 ppm to 16H and 15H; similar assignation was previously carried out in precursor PTOBDME [1]. In the spacer zone, multiplet at 5.45 ppm is interpreted due to Hc ', and the double doublets at 4.63 and 4.52 ppm correspond to Ha ' and Hb '. These peaks presented correlation signals in COSY and were related with other aliphatic signals Hd'He ' (1.83 ppm) and Hf 'Hg ' (1.45 ppm) by TOCSY experiment. Multiplet at 5.26 ppm was assigned to Hc and double doublets at 3.95 and 3.89 ppm to Ha and Hb , and they showed COSY correlations and were related with signals at 1.77 ppm (Hd) and 1.35 ppm (Hf Hg ) respectively by TOCSY experiment. The peaks assigned to Ha ", Hb ", and Hc " due to aliphatic end group are overlapped with Hb ' (4.52 ppm) and with 21H (4.54 ppm) (in **Table 2**), and they were assigned through TOCSY correlations observed for signal at 4.52 ppm (not observed for 4.63 ppm, Ha '), with the multiplet at 4.39 ppm (Hc ") and confirmed by HSQC. By this method, carbon C11' (65.7 ppm) was correlated with signals at 4.63 (Ha ') and 4.52 (Hb ') and carbon 11"C (67.8 ppm) with 4.52 ppm (Ha ") and (Hb "). Signals at 1.83 and 1.45 ppm assigned to (Hd') and (Hf ', Hg ') and correlated by COSY with Hc '(5.45 ppm). Peaks at 1.77 ppm (Hd He ) and 1.35 ppm (Hf Hg ) are correlated by COSY with Hc (5.26 ppm). Signals at 1.92, 1.78 ppm are related with Hc " (4.39 ppm) by COSY experiments, and they were assigned to Hd" and He ". They are also related with Hf" (1.53 ppm) and Hg"(1.44 ppm) by the same experiment.

Choline end group showed in **Table 2**, two set of signals probably due to conformational equilibrium: Multiplets assigned to 21H (4.54 ppm), 22H (3.33 ppm), and 23H (2.74 ppm), correlated in COSY, and another set was multiplets 21'H (4.76 ppm), 22'H (3.85 ppm) and a singlet 23'H (3.21 ppm).

The HSQC experiment allowed the direct allocation of carbon atoms linked to hydrogens, confirming the assignation of the proton signals overlapped in the <sup>1</sup> HNMR experiment. The correlation of carbon atom 11'C (65.7 ppm) with Ha ' (4.63 ppm) and Hb ' (4.52 ppm); correlation between 11"C (67.8 ppm) and Ha " and Hb "; and correlation between carbon 21C at (62.6 ppm) and 21H at (4.54 ppm), are observed in **Table 2**.


**Table 2.** Observed 1 H and 13C-NMR chemical shifts (ppm), in DMSO-d6, for the -N, N, N-trimethylethan- 1-ammonium (Choline) oxy benzoate hydrochloride end group, in Polyester PTOBDME-choline and Polyester PTOBEE-choline, and theoretical calculated values.

The presence of two independent 1

group polyester PTOBDME-choline.

to mark signals due to the aliphatic end group. \*

**Atom <sup>1</sup>**

Hc

Hc

Ha 'Hb

Ha "Hb

Hf ', Hg

Hf ", Hg

8

7

6

5

4

3

2

1

**Table 1.** <sup>1</sup>

Hd",He

' 1.45 <sup>9</sup>

" 1.53, 1.44 <sup>9</sup>

'H 1.24 <sup>8</sup>

'H 1.24 <sup>7</sup>

'H 1.24 <sup>6</sup>

'H 1.24 <sup>5</sup>

'H 1.24 <sup>4</sup>

'H 1.24 <sup>3</sup>

'H 1.24 <sup>2</sup>

'H 0.85 <sup>1</sup>

H-NMR sets of signals are observed in the spectrum, one

marked with ( ' ) and the other without it ( ). They are attributed to two conformers *gg* and *gt* of the spacer within the repeating unit respectively. The same effect has been reported for PTOBDME and PTOBEE, and accordingly, similar nomenclature is used to identify the signals. A third set of signals, marked with ( " ), is assigned to the aliphatic end group.

**Set of signal of system ( ' ) and ( " ) Set of signal of system without apostrophe ( ) Calculated chemical** 

20'H 8.36 20'C 130.4 20H 8.36 20C 130.4 8.04 130.2

16'H 7.50 16'C 122.1 16H 7.55 16C 122.2 7.26 121.5 15'H 8.08 15'C 130.8 15H 8.15 15C 131.0 8.13, 8.11 130.3

' 5.45 12'C 72.3 Hc 5.26 12C 73.7 4.55 70.3

" 4.39 12"C 60.0 3.81 70.8

" 4.52 11"C 67.8 4.53, 4.28 70.8

" 1.92 1.78 10"C 1.44 34.0

, Hg 1.35 <sup>9</sup>

H 1.24 <sup>8</sup>

H 1.24 <sup>7</sup>

H 1.24 <sup>6</sup>

H 1.24 <sup>5</sup>

H 1.24 <sup>4</sup>

H 1.24 <sup>3</sup>

H 1.24 <sup>2</sup>

H 0.85 <sup>1</sup>

The symbol ( ' ) and without it ( ) distinguish the two independent system of the repeating unit, the symbol ( " ) is used

C to 8

H and 13C-NMR chemical shifts (ppm) observed and calculated for the repeating unit and the aliphatic end

Signal of 4

Hd' 1.83 10'C 30.0 Hd 1.77 10C 31.2 1.71 30.7

**H(ppm) Atom 13C(ppm) <sup>1</sup>**

, Hb 3.95, 3.89 11C 46.4 4.80, 4.55 67.5

C 24.4 1.29 23.3

C 28.9 m\* 1.29 29.6

C 28.9 m\* 1.29 29.6

C 28.9 m\* 1.26 29.6

C 28.9 m\* 1.26 29.6

C 28.6 m\* 1.26 29.3

C 31.3 1.26 31.8

C 22.1 1.26 22.7

C 14.0 0.86 14.1

C at 28.8 ppm is a multiplet from 28.9 to 28.6.

19'C 133.3 19C 133.3 135.4 18'C 163.5 18C 163.4 165.2 17'C 153.9 17C 154.4 155.6

14'C 128.8 14C 127.6 126.9 13'C 166.7 13C 164.7 165.9

"C 25.6 1.29 23.1

**H(ppm) Atom 13C(ppm) Atom <sup>1</sup>**

18 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

' 4.63, 4.52 11'C 65.7 Ha

'C 24.6 Hf

'C 28.9 m\* <sup>8</sup>

'C 28.9 m\* <sup>7</sup>

'C 28.9 m\* <sup>6</sup>

'C 28.9 m\* <sup>5</sup>

'C 28.6 m\* <sup>4</sup>

'C 31.3 <sup>3</sup>

'C 22.1 <sup>2</sup>

'C 14.0 <sup>1</sup>

**shift**

**H(ppm) 13C(ppm)**

### **4.2. Structural characterization of PTOBEE-choline**

The assignment of the 1 H and 13C-NMR chemical shifts, in CDCl3 and DMSO-d6 , of the monomeric unit and the end groups of PTOBEE-choline are given in **Table 3**, with the predicted values calculated by ChemDraw Professional [10]. Similar notations as those designated for precursor cholesteric liquid crystal PTOBEE [4] have been used.

with Ha

carbons. Ha

Hc

Hc

'. At 4.15, a weak multiplet is assigned to Hc

1.09 ppm is due to 1'H, with TOCSY correlation with Hc

at 4.75 and 3.84 ppm, respectively, singlet 15'H at 3.22 ppm.

Another correlation with overlapped signal at 4.53 ppm Ha

attached to hydrogens matching the calculated model.

spectrum of PTOBDME-ammonium, in DMSO-d6

and Hb

', Hg

"(1.51 ppm) and Hg

", Hb

Hg

**4.3. Structural characterization of PTOBDME-ammonium**

', Ha

', and Hb

nals of COSY and are related with other aliphatic signals: Hd', He

", and Hc

(4.38 ppm), according to TOCSY correlations observed for signal Hb

(1.42 ppm) by TOCSY. Multiplet at 5.26 ppm is assigned to Hc

(5.26 ppm). The signal at 1.33 ppm cannot be observed in the 1

respectively . 23 H at 3.28 ppm was overlapped with signal of H2

and Hb

" and Hb

. The weak triplet at 1.13 ppm corresponded with 1

' (4.53 ppm) showed correlation with carbon 3

"Hb

(5.25 ppm). The overlapped signal at 1.90 ppm is identified as Hd', with cross signal with

'. A very weak COSY cross signal between 4.15 and Hd"(1.88 ppm) is observed. Triplet at

PTOBDME-choline, the choline end group shows, in **Table 2**, two set of signals due to conformational equilibrium. Multiplets 13H, 14H are observed at 4.56 and 3.32 ppm, respectively, and 15H at 2.74 ppm, correlated in COSY experiments, and another set was 13'H and 14'H multiplets

HSQC experiment was performed to determine the chemical shift of carbons bonded to the assigned hydrogen. The complex signal at 4.53 in proton presented several correlations with

C, and 4

The structure of PTOBDME-ammonium, as depicted **II** in **Figure 3**, is confirmed by <sup>1</sup>

13C-NMR, with the chemical shifts given in **Table 4**. In the aromatic zone of the 1

due to the low concentration. 13C-NMR experiment allowed the assignation of the carbons not

doublets at 7.50 and 8.07 ppm are assigned to 16'H and 15'H, respectively, and doublets at 7.57 and 8.13 ppm to 16H and 15H. In the spacer zone where methylene and methines attached to oxygen are observed, a multiplet at 5.45 ppm, and the double doublets at 4.64 ppm,

C, 2

signal 4.53 ppm, indicating the presence of Ha

'(4.60 ppm) and Hb

(67.8 ppm). Signals corresponding to 1

4.48 ppm correspond to Hc

and 3.88 ppm, to Ha

observed for Ha

lapping with CH2

related with Hc

related with Hf

(1.77 ppm) and Hf

aliphatic end group, Ha

') and 1.42 ppm (Hf

relation. Signals at 1.77 ppm (Hd He

He

He

3.74 ppm were identified as H<sup>a</sup>

with TOCSY correlation with Hc

", it presented COSY correlation with

Synthetic Cationic Cholesteric Liquid Crystal Polymers http://dx.doi.org/10.5772/intechopen.70995

". The two double doublets at 3.76 and

', while triplet at (1.03 ppm) is 1

" was observed with carbon 3

C of aliphatic end group were not observed

, a singlet at 8.36 ppm belongs to 20H and

' (1.81 ppm) and Hf

and double doublets at 3.94

' (4.48 ppm) and with 21H

) show TOCSY correlation with Hc

H sprectrum due to the over-

O of the deuterated solvent,

' (4.48 ppm) and not

'), also COSY cor-

" and are also

', respectively; these signals present correlation sig-

, they show COSY correlations and are related with signals Hd

(1.33 ppm), in the TOCSY experiment. In the set of signals due to

" are overlapped with Hb

') have TOCSY correlation with 5.45 ppm (Hc

) and 1.33 ppm (Hf

"(4.38 ppm) by COSY and TOCSY, are assigned to Hd" and He

Ammonium end group shows 23H, 22H, 24H multiplets at 3.28 ppm, 2.17 ppm, and 2.79 ppm,

' (4.63 ppm) and confirmed by HSQC. Aliphatic signals at 1.81 ppm (Hd',

Hg

, but it was clearly observed in TOCSY 2D. Signals at 1.91 and 1.78 ppm,

"(1.42 ppm) by the same experiment.

H,

21

"C

H and

', Hg '

H-NMR

"H. As in

'C (65.6 ppm).

and presented the expected COSY correlation with

In the 1 H-NMR experiment in CDCl3, observed chemical shifts are 12H singlet at (8.34 ppm), 8 H doublet at (7.34 ppm), 7 H doublet at (8.16 ppm), 8 'H doublet at (7.36 ppm) and 7 'H doublet at (8.18 ppm). Multiplets at 5.46 and at 5.25ppm are interpreted as Hc ' and Hc , respectively. The double doublet at 4.60 ppm is assigned to Ha ' and correlates in COSY with Hc ' signal. An overlapped signal at 4.53 ppm is identified as H<sup>b</sup> ', with COSY and TOCSY cross signal


**Table 3.** Observed and calculated 1 H and 13C-NMR chemical shifts (ppm) for polyester PTOBEE-choline in DMSO-d6 and CDCl3 . Repeating unit and the aliphatic end group.

with Ha '. At 4.15, a weak multiplet is assigned to Hc ", it presented COSY correlation with signal 4.53 ppm, indicating the presence of Ha " and Hb ". The two double doublets at 3.76 and 3.74 ppm were identified as H<sup>a</sup> and Hb and presented the expected COSY correlation with Hc (5.25 ppm). The overlapped signal at 1.90 ppm is identified as Hd', with cross signal with Hc '. A very weak COSY cross signal between 4.15 and Hd"(1.88 ppm) is observed. Triplet at 1.09 ppm is due to 1'H, with TOCSY correlation with Hc ', while triplet at (1.03 ppm) is 1 H, with TOCSY correlation with Hc . The weak triplet at 1.13 ppm corresponded with 1 "H. As in PTOBDME-choline, the choline end group shows, in **Table 2**, two set of signals due to conformational equilibrium. Multiplets 13H, 14H are observed at 4.56 and 3.32 ppm, respectively, and 15H at 2.74 ppm, correlated in COSY experiments, and another set was 13'H and 14'H multiplets at 4.75 and 3.84 ppm, respectively, singlet 15'H at 3.22 ppm.

HSQC experiment was performed to determine the chemical shift of carbons bonded to the assigned hydrogen. The complex signal at 4.53 in proton presented several correlations with carbons. Ha '(4.60 ppm) and Hb ' (4.53 ppm) showed correlation with carbon 3 'C (65.6 ppm). Another correlation with overlapped signal at 4.53 ppm Ha "Hb " was observed with carbon 3 "C (67.8 ppm). Signals corresponding to 1 C, 2 C, and 4 C of aliphatic end group were not observed due to the low concentration. 13C-NMR experiment allowed the assignation of the carbons not attached to hydrogens matching the calculated model.

### **4.3. Structural characterization of PTOBDME-ammonium**

**4.2. Structural characterization of PTOBEE-choline**

20 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

The double doublet at 4.60 ppm is assigned to Ha

**H(ppm) Atom 13C(ppm) Atom <sup>1</sup>**

'C 121.7 121.7 <sup>8</sup>

'C 131.5 130.6 <sup>7</sup>

'C 65.6 64.8 Ha

'C 9.8 9.2 <sup>1</sup>

9

8

7

6

5

3

2

. Repeating unit and the aliphatic end group.

An overlapped signal at 4.53 ppm is identified as H<sup>b</sup>

precursor cholesteric liquid crystal PTOBEE [4] have been used.

H and 13C-NMR chemical shifts, in CDCl3

**Set of signal of system (') and (") Set of signal of system without apostrophe ( ) Calculated** 

**CDCl3 DMSO CDCl3 DMSO CDCl3 DMSO DMSO CDCl3 DMSO <sup>1</sup>**

'C 154.5 154.2 <sup>9</sup>

'C 128.3 128.7 <sup>6</sup>

'C 165.4 166.4 <sup>5</sup>

'C 73.8 73.1 Hc 5.25 5.20 <sup>4</sup>

'C 24.4 23.0 Hd 1.86 1.81 <sup>2</sup>

, Hb

3.76, 3.74

12'H 8.34 8.35 12'C 130.4 130.1 12H 8.34 8.35 12C 130.4 130.0 8.04 130.2

H 7.34 7.53,

H 8.16 8.11,

11'C 133.8 133.2 11C 132.2 131.6 135.4 10'C 163.6 163.3 10C 163.6 163.3 165.2

7.51

8.09

"C 61.6\* 3.81 73.0

"C \* 1.48 26.8

"C 8.73\* 0.96 7.6

H and 13C-NMR chemical shifts (ppm) for polyester PTOBEE-choline in DMSO-d6

3.94, 3.91

"C 67.8 67.1 4.53,

H 1.03 0.95 <sup>1</sup>

8

7

3

H doublet at (8.16 ppm), 8

at (8.18 ppm). Multiplets at 5.46 and at 5.25ppm are interpreted as Hc

meric unit and the end groups of PTOBEE-choline are given in **Table 3**, with the predicted values calculated by ChemDraw Professional [10]. Similar notations as those designated for

H-NMR experiment in CDCl3, observed chemical shifts are 12H singlet at (8.34 ppm),

and DMSO-d6

' and Hc

', with COSY and TOCSY cross signal

C 154.5 153.8 155.6

C 121.7 122.0 7.26 121.5

C 127.9 127.3 126.9

C 165.4 164.7 165.9

C 75.1 74.4 4.55 72.5

C 24.4 24.2 1.75 23.5

C 9.6 9.8 0.96 7.8

C 45.23 45.6 4.80,

C 131.5 130.6 8.13,

'H doublet at (7.36 ppm) and 7

' and correlates in COSY with Hc

**H(ppm) Atom 13C(ppm) Atom Atom**

, of the mono-

'H doublet

' signal.

, respectively.

**chemical shifts**

**H 13C**

8.11

4.55

4.28

130.3

67.2

70.5

and

The assignment of the 1

H doublet at (7.34 ppm), 7

In the 1

**Atom <sup>1</sup>**

8

7

Hc

Hc

Ha ', Hb '

Ha ", Hb "

Hd", He "

1

1

CDCl3

'H 7.36 7.48,

'H 8.18 8.08,

7.50

8.06

' 5.46 5.38 <sup>4</sup>

" 4.15 4.36 <sup>4</sup>

4.63, 4.52

1.96, 1.80

4.53 4.52 \* <sup>3</sup>

Hd' 1.90 1.86 <sup>2</sup>

'H 1.09 1.02 <sup>1</sup>

"H 1.13 1.04\* <sup>1</sup>

**Table 3.** Observed and calculated 1

1.88, 1.13

\*Overlapped signal

4.60, 4.53

8

The structure of PTOBDME-ammonium, as depicted **II** in **Figure 3**, is confirmed by <sup>1</sup> H and 13C-NMR, with the chemical shifts given in **Table 4**. In the aromatic zone of the 1 H-NMR spectrum of PTOBDME-ammonium, in DMSO-d6 , a singlet at 8.36 ppm belongs to 20H and doublets at 7.50 and 8.07 ppm are assigned to 16'H and 15'H, respectively, and doublets at 7.57 and 8.13 ppm to 16H and 15H. In the spacer zone where methylene and methines attached to oxygen are observed, a multiplet at 5.45 ppm, and the double doublets at 4.64 ppm, 4.48 ppm correspond to Hc ', Ha ', and Hb ', respectively; these signals present correlation signals of COSY and are related with other aliphatic signals: Hd', He ' (1.81 ppm) and Hf ', Hg ' (1.42 ppm) by TOCSY. Multiplet at 5.26 ppm is assigned to Hc and double doublets at 3.94 and 3.88 ppm, to Ha and Hb , they show COSY correlations and are related with signals Hd He (1.77 ppm) and Hf Hg (1.33 ppm), in the TOCSY experiment. In the set of signals due to aliphatic end group, Ha ", Hb ", and Hc " are overlapped with Hb ' (4.48 ppm) and with 21H (4.38 ppm), according to TOCSY correlations observed for signal Hb ' (4.48 ppm) and not observed for Ha ' (4.63 ppm) and confirmed by HSQC. Aliphatic signals at 1.81 ppm (Hd', He ') and 1.42 ppm (Hf ', Hg ') have TOCSY correlation with 5.45 ppm (Hc '), also COSY correlation. Signals at 1.77 ppm (Hd He ) and 1.33 ppm (Hf Hg ) show TOCSY correlation with Hc (5.26 ppm). The signal at 1.33 ppm cannot be observed in the 1 H sprectrum due to the overlapping with CH2 , but it was clearly observed in TOCSY 2D. Signals at 1.91 and 1.78 ppm, related with Hc "(4.38 ppm) by COSY and TOCSY, are assigned to Hd" and He " and are also related with Hf "(1.51 ppm) and Hg "(1.42 ppm) by the same experiment.

Ammonium end group shows 23H, 22H, 24H multiplets at 3.28 ppm, 2.17 ppm, and 2.79 ppm, respectively . 23 H at 3.28 ppm was overlapped with signal of H2 O of the deuterated solvent,


correlations with 3.28 ppm (21H). The polymer holds positive charge due to ammonium pro-

The HSQC experiment confirmed the direct assignation of carbon atom 11'C (66.0 ppm) linked to

at 12"C (60.4 ppm) and another with 21H linked to carbon atom 21C (62.3 ppm). The correlations

ester PTOBEE-ammonium chloride and calculated for the repeating unit and the aliphatic

' and 5.25 ppm is Hc

' because of its shape and the COSY correlation with Hc

and presented the expected COSY correlation with Hc

', while triplet at 1.03 ppm with TOCSY correlation with Hc

(2.81 ppm), 15H multiplet (3.25 ppm), 14H multiplet (2.18 ppm), and 13H multiplet overlapped at (4.39 ppm) but presented COSY correlations with 2.18 ppm and TOCSY correlations with 3.25 ppm. The compound is positively charged, with the ammonium proton 17H observed at

. HSQC experiment exhibits several correlations of the complex proton signal at 4.53 ppm, car-

" and 3

lapped signal of proton 13H (4.39 ppm), within the ammonium end group, and 13C (61.9 ppm)

The structures of undec-10-enamide, 10-11-epoxy-undecanamide, and 10,11-dihydroxyunde-

NMR, COSY and HSQC, obtained in VARIAN 400 and 500 MHz spectrometers, also at room

The structure of PTOBUME-amide, **VII** in **Figure 4**, has also been confirmed by <sup>1</sup>

at 25°C in a Bruker 300 MHz NMR spectrometer. Chemical shifts and Mass spec-

" and 1.88 ppm (Hd"). Triplet signal at 1.09 ppm with TOCSY correlation with Hc

".

' (4.53) correlates with 3

' exhibits correlation with 11"C (67.8 ppm),

Synthetic Cationic Cholesteric Liquid Crystal Polymers http://dx.doi.org/10.5772/intechopen.70995

"(1.42 ppm) confirmed the previous assignation.

H and 13C-NMR chemical shifts (ppm) observed of poly-

H doublet and 7'H doublet, and 7

", and this signal presented COSY correlation with Hb

". The two double doublets at 3.76 and 3.74 ppm are

'. A very weak COSY cross signal is observed

are observed at (**Table 5**): 16H singlet

"C (67.8 ppm). Correlation between the over-

'C at (65.6 ppm). Another

H-NMR, 13C-NMR, registered in

H-NMR, 13C-

H-NMR experiment, in CDCl3, peaks observed at 8.34, 7.36, 7.34, 8.18, and

', by COSY and TOCSY cross signal with Ha

" (4.38 ppm), one with carbon atom

"(1.78 ppm), and carbon atom 9

. The double doublet at 4.60 ppm is

"C

23

H doublet,

'. A weak

',

'

'. An overlapped

. The overlapped

was assigned

' (4.48 ppm). Signal Hb

". Two cross signal are observed for Hc

of carbon atom 10"C (33.6 ppm) with Hd"(1.91 ppm) and He

"(1.51 ppm) and Hg

**4.4. Structural characterization of PTOBEE-ammonium**

" and Hb

and 13.2 ppm in CDCl3

'(4.60 ppm) Hb

. The weak triplet at 1.13 ppm corresponds to He

Signals of proton ammonium end group in DMSO-d6

**4.5. Structural characterization of PTOBUME-amide**

canamide (**III**, **IV** and **V** in **Figure 4**) were confirmed by <sup>1</sup>

8.16 ppm are assigned to 12H singlet, 8'H doublet, 8

ton 25H observed at 10.33 ppm.

" and Hb

**Table 6** shows the assignation of 1

respectively. Peak at 5.46 ppm is Hc

signal at 4.53 ppm is attributed to H<sup>b</sup>

multiplet at 4.15 ppm is assigned to Hc

and Hb

signal at 1.90 ppm is Hd' correlated with Hc

indicating the presence of Ha

' (4.63 ppm) and Hb

protons Ha

linked to Ha

(25.6 ppm) with Hf

end group. In the 1

interpreted as Ha

identified as H<sup>a</sup>

is interpreted as He

10.3 ppm in DMSO-d6

bon atoms. Double doublet Ha

is observed.

correlation is observed between proton Ha

trometry results are given in Section 2.3.

between Hc

to He

in CDCl3

DMSO-d6

The symbol (') and with no apostrophe ( ) distinguish the two independent systems of the repeating unit, the symbol (") is used to mark the signals of the aliphatic end group.\* Signal of 13C at 28.8 ppm is a multiplet from 28.9 to 28.7.

**Table 4.** <sup>1</sup> H and 13C-NMR chemical shifts (ppm) observed and calculated for the repeating unit of polyester PTOBDMEammonium chloride and the aliphatic end group.

and it was assigned due to the COSY and TOCSY correlations with signals at 4.38 ppm (21H) and at 2.17 ppm (22H) and HSQC correlation with 23C at 54.0 ppm. 21H signal was overlapped at 4.38 ppm, and it was identified by COSY correlations with 2.17 ppm (22H) and TOCSY correlations with 3.28 ppm (21H). The polymer holds positive charge due to ammonium proton 25H observed at 10.33 ppm.

The HSQC experiment confirmed the direct assignation of carbon atom 11'C (66.0 ppm) linked to protons Ha ' (4.63 ppm) and Hb ' (4.48 ppm). Signal Hb ' exhibits correlation with 11"C (67.8 ppm), linked to Ha " and Hb ". Two cross signal are observed for Hc " (4.38 ppm), one with carbon atom at 12"C (60.4 ppm) and another with 21H linked to carbon atom 21C (62.3 ppm). The correlations of carbon atom 10"C (33.6 ppm) with Hd"(1.91 ppm) and He "(1.78 ppm), and carbon atom 9 "C (25.6 ppm) with Hf "(1.51 ppm) and Hg "(1.42 ppm) confirmed the previous assignation.

### **4.4. Structural characterization of PTOBEE-ammonium**

**Set of signal of system (') and (") Set of signal of system without** 

22 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

**H (ppm Atom 13C(ppm) Atom <sup>1</sup>**

' 4.64, 4.48 11'C 66.0 Ha

'C 24.6 Hf

'C 28.6 m\* <sup>8</sup>

'C 28.6 m\* <sup>7</sup>

'C 28.6 m\* <sup>6</sup>

'C 28.5 m\* <sup>5</sup>

'C 28.1 m\* <sup>4</sup>

'C 31.3 <sup>3</sup>

'C 22.1 <sup>2</sup>

'C 14.0 <sup>1</sup>

is used to mark the signals of the aliphatic end group.\*

ammonium chloride and the aliphatic end group.

**Atom <sup>1</sup>**

Hc

Hc

Ha ', Hb

Ha ", Hb "

Hd', He

Hd", He "

Hf ', Hg

Hf ", Hg "

8

7

6

5

4

3

2

1

**Table 4.** <sup>1</sup>

' 1.42 <sup>9</sup>

'H 1.22 <sup>8</sup>

'H 1.22 <sup>7</sup>

'H 1.22 <sup>6</sup>

'H 1.22 <sup>5</sup>

'H 1.22 <sup>4</sup>

'H 1.22 <sup>3</sup>

'H 1.22 <sup>2</sup>

'H 0.84 <sup>1</sup>

1.51;1.42 <sup>9</sup>

**apostrophe ( )**

20'H 8.36 20'C 130.4 20H 8.36 20C 130.4 8.04 130.2

16'H 7.50 16'C 122.1 16H 7.57 16C 122.4 7.26 121.5 15'H 8.07 15'C 131.0 15H 8.13 15C 131.0 8.13;8.11 130.3

' 5.45 12'C 72.6 Hc 5.26 12C 73.4 4.55 70.4

" 4.38 12"C 60.4 3.81 70.9

, Hb 3.94, 3.88

4.48 11"C 67.8 4.53, 4.28 70.9

' 1.81 10'C 30.3 Hd, He 1.77 10C 31.3 1.71 30.8

1.91 1.78 10'C 33.6 1.44 34.1

, Hg 1.33 <sup>9</sup>

H 1.22 <sup>8</sup>

H 1.22 <sup>7</sup>

H 1.22 <sup>6</sup>

H 1.22 <sup>5</sup>

H 1.22 <sup>4</sup>

H 1.22 <sup>3</sup>

H 1.22 <sup>2</sup>

H 0.84 <sup>1</sup>

The symbol (') and with no apostrophe ( ) distinguish the two independent systems of the repeating unit, the symbol (")

and it was assigned due to the COSY and TOCSY correlations with signals at 4.38 ppm (21H) and at 2.17 ppm (22H) and HSQC correlation with 23C at 54.0 ppm. 21H signal was overlapped at 4.38 ppm, and it was identified by COSY correlations with 2.17 ppm (22H) and TOCSY

H and 13C-NMR chemical shifts (ppm) observed and calculated for the repeating unit of polyester PTOBDME-

"C 25.3 1.29 23.2

**H(ppm) Atom 13C(ppm) <sup>1</sup>**

19'C 133.3 19C 133.3 135.5 18'C 163.5 18C 163.4 165.2 17'C 153.8 17C 154.3 155.7

14'C 128.9 14C 127.6 127.0 13'C 166.6 13C 164.8 166.0

11C 46.4 4.80, 4.55 67.6

C 24.3 1.29 23.4

C 28.6 m\* 1.29 29.7

C 28.6 m\* 1.29 29.7

C 28.6 m\* 1.26 29.7

C 28.5 m\* 1.26 29.7

C 28.1 m\* 1.26 29.4

C 31.3 1.26 31.9

C 22.1 1.26 22.8

C 14.0 0.86 14.1

Signal of 13C at 28.8 ppm is a multiplet from 28.9 to 28.7.

**Calculated chemical shift**

**H(ppm) 13C(ppm)**

**Table 6** shows the assignation of 1 H and 13C-NMR chemical shifts (ppm) observed of polyester PTOBEE-ammonium chloride and calculated for the repeating unit and the aliphatic end group. In the 1 H-NMR experiment, in CDCl3, peaks observed at 8.34, 7.36, 7.34, 8.18, and 8.16 ppm are assigned to 12H singlet, 8'H doublet, 8 H doublet and 7'H doublet, and 7 H doublet, respectively. Peak at 5.46 ppm is Hc ' and 5.25 ppm is Hc . The double doublet at 4.60 ppm is interpreted as Ha ' because of its shape and the COSY correlation with Hc '. An overlapped signal at 4.53 ppm is attributed to H<sup>b</sup> ', by COSY and TOCSY cross signal with Ha '. A weak multiplet at 4.15 ppm is assigned to Hc ", and this signal presented COSY correlation with Hb ', indicating the presence of Ha " and Hb ". The two double doublets at 3.76 and 3.74 ppm are identified as H<sup>a</sup> and Hb and presented the expected COSY correlation with Hc . The overlapped signal at 1.90 ppm is Hd' correlated with Hc '. A very weak COSY cross signal is observed between Hc " and 1.88 ppm (Hd"). Triplet signal at 1.09 ppm with TOCSY correlation with Hc ' is interpreted as He ', while triplet at 1.03 ppm with TOCSY correlation with Hc was assigned to He . The weak triplet at 1.13 ppm corresponds to He ".

Signals of proton ammonium end group in DMSO-d6 are observed at (**Table 5**): 16H singlet (2.81 ppm), 15H multiplet (3.25 ppm), 14H multiplet (2.18 ppm), and 13H multiplet overlapped at (4.39 ppm) but presented COSY correlations with 2.18 ppm and TOCSY correlations with 3.25 ppm. The compound is positively charged, with the ammonium proton 17H observed at 10.3 ppm in DMSO-d6 and 13.2 ppm in CDCl3 .

HSQC experiment exhibits several correlations of the complex proton signal at 4.53 ppm, carbon atoms. Double doublet Ha '(4.60 ppm) Hb ' (4.53) correlates with 3 'C at (65.6 ppm). Another correlation is observed between proton Ha " and 3 "C (67.8 ppm). Correlation between the overlapped signal of proton 13H (4.39 ppm), within the ammonium end group, and 13C (61.9 ppm) in CDCl3 is observed.

### **4.5. Structural characterization of PTOBUME-amide**

The structures of undec-10-enamide, 10-11-epoxy-undecanamide, and 10,11-dihydroxyundecanamide (**III**, **IV** and **V** in **Figure 4**) were confirmed by <sup>1</sup> H-NMR, 13C-NMR, registered in DMSO-d6 at 25°C in a Bruker 300 MHz NMR spectrometer. Chemical shifts and Mass spectrometry results are given in Section 2.3.

The structure of PTOBUME-amide, **VII** in **Figure 4**, has also been confirmed by <sup>1</sup> H-NMR, 13C-NMR, COSY and HSQC, obtained in VARIAN 400 and 500 MHz spectrometers, also at room


temperature. The solvent used were DMSO-d6

**4.6. Structural characterization of PTOBEE-amide**

ues predicted by ChemDraw Professional, v. 15.1.0.144.

**H Atom 13C Atom <sup>1</sup>**

temperature. The solvent used was DMSO-d6

' 4.95, 4.30 10'C Ha

' 1.74 <sup>9</sup>

' 1.37 <sup>8</sup>

'H 1.22 <sup>7</sup>

'H 1.22 <sup>6</sup>

'H 1.22 <sup>5</sup>

'H 1.22 <sup>4</sup>

'H 1.52 <sup>3</sup>

'H 2.28 <sup>2</sup>

' 7.0 <sup>1</sup>

**Atom <sup>1</sup>**

Hc

Ha ', Hb

Hf ', Hg

7

6

5

4

3

2

NH2

Hc

Hd', He

and CDCl3

**H Atom 13C Atom <sup>1</sup>**

18'C 18C *133.7* 18C 135.5 17'C 17C *163.7* 17C 165.2 16'C 16C *154.7* 16C 153.7

13'C 13C *128.2* 13C 126.9 12'C 12C *165.7* 12C 165.9

10C Ha

C Hf

C <sup>7</sup>

C <sup>6</sup>

C <sup>5</sup>

C <sup>4</sup>

C <sup>3</sup>

C <sup>2</sup>

Germany). The spectra were processed and analyzed with the help of MestReNova 11.0.4 [9]. The chemical shifts are given in **Table 7**. Theoretical values predicted by ChemDraw

NMR, COSY and HSQC, obtained in VARIAN 400 and 500 MHz spectrometers, at room

experimental chemical shifts analyzed from the spectra are given in **Table 8**. Theoretical val-

Professional, v. 15.1.0.144. Tilted values are chemical shifts registered in CDCl3

The structure of PTOBEE-amide, **VI** in **Figure 5**, has also been confirmed by <sup>1</sup>

**System (') System without apostrophe ( ) Theoretical chemical shifts**

19'H 8.36 19'C 19H 8.36 19C *130.4* 19H 8.04 19C 130.2

15'H 8.07 15'C 15H 8.07 15C *121.6* 15H 7.52 15C 121.5 14'H 7.50 14'C 14H 7.50 14C *131.3* 14H 8.18 14C 130.3

' 5.76 11'C Hc 5.45 11C Hc 5.16 11C 70.3

1.55 <sup>9</sup>

1.22 <sup>8</sup>

H 1.22 <sup>7</sup>

H 1.22 <sup>6</sup>

H 1.22 <sup>5</sup>

H 1.22 <sup>4</sup>

H 1.52 <sup>3</sup>

H 2.28 <sup>2</sup>

Experimental signals "end group" Theoretical chemical shifts end

'C NH2 7.0 <sup>1</sup>

4.23, 4.18

, Hb

He

, Hg

'C Hd,

'C Hf

'C <sup>7</sup>

'C <sup>6</sup>

'C <sup>5</sup>

'C <sup>4</sup>

'C <sup>3</sup>

'C <sup>2</sup>

" 4.18 11"C Hc

from Merck KGaA (Darmstadt,

Synthetic Cationic Cholesteric Liquid Crystal Polymers http://dx.doi.org/10.5772/intechopen.70995

from Merck KGaA (Darmstadt, Germany). The

,Hb 4.78, 4.53

, Hg 1.25 <sup>8</sup>

H 1.25 <sup>7</sup>

H 1.26 <sup>6</sup>

H 1.30 <sup>5</sup>

H 1.30 <sup>4</sup>

H 1.53 <sup>3</sup>

H 2.34 <sup>2</sup>

" 4.57 11"C 73.6

C Hd, He 1.67 <sup>9</sup>

C NH2 7.03 <sup>1</sup>

group

.

**H Atom 13C**

10C 66.0

C 30.7

C 23.3

C 29.6

C 29.6

C 28.9

C 28.6

C 25.3

C 38.7

C 173.6

H-NMR, 13C-

25

**Table 5.** <sup>1</sup> H and 13C-NMR chemical shifts (ppm) observed for the 3-dimethylamine-1-propyl benzoate hydrochloride end group in polyester PTOBDME-ammonium and polyester PTOBEE-ammonium, and calculated values.


**Table 6.** <sup>1</sup> H and 13C-NMR chemical shifts (ppm) observed for polyester PTOBEE-ammonium chloride, both the repeating unit and the aliphatic end group, in DMSO-d6 and in CDCl3 , and calculated values.

temperature. The solvent used were DMSO-d6 and CDCl3 from Merck KGaA (Darmstadt, Germany). The spectra were processed and analyzed with the help of MestReNova 11.0.4 [9]. The chemical shifts are given in **Table 7**. Theoretical values predicted by ChemDraw Professional, v. 15.1.0.144. Tilted values are chemical shifts registered in CDCl3 .

### **4.6. Structural characterization of PTOBEE-amide**

**Set of signal of system (') and (") Set of signal of system without apostrophe ( ) Calculated** 

H and 13C-NMR chemical shifts (ppm) observed for the 3-dimethylamine-1-propyl benzoate hydrochloride end

**Atom 13C**

21H 4.38 21C 62.3 4.25 63.0 13H 4.39 4.49 13C 61.3 61.9 4.25 63.0 22H 2.17 22C 23.8 2.19 22.5 14H 2.18 2.42 14C 23.1 24.5 2.19 22.5 23H 3.28 23C 54.0 3.24 55.2 15H 3.25 3.14 15C 53.8 55.5 3.24 55.2 24H 2.79 24C 42.5 2.90 45.0 16H 2.81 2.82 16C 41.9 43.1 2.90 45.0

**DMSO DMSO DMSO CDCl3 DMSO CDCl3**

25H 10.33 7.0 17H 10.33 13.2 7.0

group in polyester PTOBDME-ammonium and polyester PTOBEE-ammonium, and calculated values.

**(ppm)**

**1 H (ppm)**

**CDCl3 DMSO CDCl3 DMSO CDCl3 DMSO CDCl3 DMSO <sup>1</sup>**

'C 154.5 <sup>9</sup>

'C 128.3 <sup>6</sup>

'C 165.4 <sup>5</sup>

'C 73.8 73.02 Hc 5.25 5.20 <sup>4</sup>

'C 24.4 23.06 Hd 1.90 1.82 <sup>2</sup>

'C 9.8 8.73 He 1.03 0.96 <sup>1</sup>

, Hb

3.76, 3.74

12'H 8.34 8.36 12'C 130.4 129.9 12H 8.34 8.36 12C 130.4 129.9 8.04 130.2

11'C 133.8 11C 132.2 135.4 10'C 163.6 10C 163.6 165.2

"C \* 3.81 73.0

3.96, 3.92

"C \* 1.48 26.8

"C 8.73\* 0.96 7.6

, and calculated values.

H and 13C-NMR chemical shifts (ppm) observed for polyester PTOBEE-ammonium chloride, both the repeating

"C 67.8 \* 4.53,

3

H 7.34 7.53 <sup>8</sup>

H 8.16 8.11 <sup>7</sup>

**H(ppm) Atom 13C(ppm) Atom <sup>1</sup>**

**PTOBDME-Ammonium PTOBEE-Ammonium**

24 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

**shifts**

**13C (ppm)**

**1 H (ppm)**

**Observed chemical shifts Calc. chemical** 

**(ppm)**

**Atom 13C**

'C 121.7 121.7 <sup>8</sup>

'C 131.5 130.4 <sup>7</sup>

'C 65.6 64.6 Ha

9

6

5

3

2

unit and the aliphatic end group, in DMSO-d6 and in CDCl3

**Atom <sup>1</sup>**

**Table 5.** <sup>1</sup>

**Atom <sup>1</sup> H (ppm)**

8

7

Hc

Hc

Ha ', Hb '

Ha ", Hb "

Hd", He "

1

1

**Table 6.** <sup>1</sup>

'H 7.36 7.53 <sup>8</sup>

'H 8.18 8.13 <sup>7</sup>

' 5.46 5.39 <sup>4</sup>

" 4.15 4.40 <sup>4</sup>

4.64 4.53

1.90, 1.86\*

4.53 4.48\* <sup>3</sup>

Hd' 1.90 1.86 <sup>2</sup>

'H 1.09 1.04 <sup>1</sup>

"H 1.13 1.06\* <sup>1</sup>

4.60, 4.53

1.88, 1.13

\*Overlapped signal.

**chemical shift**

**shifts**

**13C (ppm)**

**1 H (ppm)**

**13C (ppm)**

**H 13C**

8.11

4.55

4.28

130.3

67.2

70.5

**H(ppm) Atom 13C(ppm) Atom Atom**

**Observed chemical shifts Calc. chemical** 

**Atom <sup>1</sup> H (ppm)**

C 154.5 155.6

C 121.9 121.7 7.26 121.5

C 127.9 126.9

C 165.4 165.9

C 75.1 74.4 4.55 72.5

C 24.4 24.1 1.75 23.5

C 9.6 8.38 0.96 7.8

C 45.23 45.6 4.80,

C 131.5 130.4 8.13,

The structure of PTOBEE-amide, **VI** in **Figure 5**, has also been confirmed by <sup>1</sup> H-NMR, 13C-NMR, COSY and HSQC, obtained in VARIAN 400 and 500 MHz spectrometers, at room temperature. The solvent used was DMSO-d6 from Merck KGaA (Darmstadt, Germany). The experimental chemical shifts analyzed from the spectra are given in **Table 8**. Theoretical values predicted by ChemDraw Professional, v. 15.1.0.144.



**5. Thermal stability and differential scanning calorimetry (DSC)**

466°C in PTOBEE-choline.

(c) DSC of PTOBEE-choline, both at 10°C/min.

PTOBEE. First heating run of the original sample. All at 10°C/min.

The presence of choline group at the end of polymer chains causes in PTOBDME-choline a decrease in the thermal stability range compared to precursor PTOBDME. A 5% weight loss is observed for PTOBDME-choline at 230°C, while PTOBDME loses 5% weight at about 280°C. The thermal stability of PTOBEE-Choline is similar to that of polyester PTOBEE. PTOBEE-choline has 5% weight loss at 281°C, and PTOBEE at 280°C (see **Figures 6** and **7**). In the thermal stability curve of PTOBDME-choline, the first degradation step observed at 230°C is followed by two other weight loss step at 280 and 448°C. Two decomposition steps are observed at 280 and

Synthetic Cationic Cholesteric Liquid Crystal Polymers http://dx.doi.org/10.5772/intechopen.70995 27

In the DSC experiment of PTOBDME-choline, performed at 10°C/min, **Figure 6(b)**, a glass transition can be observed at 58.2°C, in the first heating run, and a weak endothermic peak at 99.5°C is interpreted as due to the first order transition from crystal phase to liquid crystal state. An exothermic peak at 171.2°C is also observed which is not explained, but the beginning of a second endothermic peak at 200°C can be attributed to fusion to the isotropic. In the cooling process, two exothermic peaks at 155°C and at 175 are observed, probably associated to crystal formation. In the second heating, a very broad endothermic peak at 100.2°C is

**Figure 6.** (a) Thermogravimetry of PTOBDME-choline and PTOBEE-choline; (b) DSC analysis of PTOBDME-choline and

**Figure 7.** (I) Thermogravimetry of precursor PTOBDME; (II) DSC analysis of PTOBDME; (III) DSC of PTOBEE: (a) first heating process of the original sample, (b) subsequent cooling down, (c) second heating process; and (d) DSC of (─)

observed again associated to the transition to liquid crystal mesophase.

**Table 7.** <sup>1</sup> H and 13C-NMR chemical shifts (ppm) observed and calculated for chiral Polyesteramide PTOBUME-amide.


**Table 8.** <sup>1</sup> H and 13C-NMR chemical shifts (ppm) observed and calculated for chiral Polyesteramide PTOBEE-amide.

### **5. Thermal stability and differential scanning calorimetry (DSC)**

**System (') System without apostrophe ( ) Theoretical chemical shifts**

**H Atom 13C Atom <sup>1</sup>**

"C Hd", He

"C Hf

"C <sup>7</sup>

"C <sup>6</sup>

"C <sup>5</sup>

"C <sup>4</sup>

"C <sup>3</sup>

"C <sup>2</sup>

H and 13C-NMR chemical shifts (ppm) observed and calculated for chiral Polyesteramide PTOBUME-amide.

**H Atom 13C Atom <sup>1</sup>**

C 132.35 <sup>8</sup>

C 123.06 <sup>7</sup>

C Ha

C <sup>9</sup>

C <sup>6</sup>

C <sup>5</sup>

C Hc 4.56 <sup>4</sup>

C Hd, He 2.46 <sup>2</sup>

C NH2 7.03 <sup>1</sup>

H 8.18 <sup>8</sup>

H 7.52 <sup>7</sup>

,Hb 4.78, 4.53

11'C 11C 11C 135.4 10'C 10C 10C 165.2

C**H**<sup>2</sup> 4.36 **C**H2 *61.2* C**H**<sup>2</sup> 4.29 **C**H2 60.9 C**H**<sup>3</sup> 1.35 **C**H3 *14.3* C**H**<sup>3</sup> 1.30 **C**H3 14.1

**System (') System without apostrophe ( ) Theoretical chemical shifts**

H 8.08 <sup>8</sup>

H 7.45 <sup>7</sup>

,Hb 4.32, 4.22

3

H and 13C-NMR chemical shifts (ppm) observed and calculated for chiral Polyesteramide PTOBEE-amide.

12'H 8.32 12'C 131.36 12H 8.32 12C 131.36 12H 8.04 12C 130.2

"C NH2

", Hb

", Hg

" 3.86, 3.80

" 1.67 <sup>9</sup>

" 1.25 <sup>8</sup>

"H 1.25 <sup>7</sup>

"H 1.26 <sup>6</sup>

"H 1.30 <sup>5</sup>

"H 1.30 <sup>4</sup>

"H 1.53 <sup>3</sup>

"H 2.34 <sup>2</sup>

" 7.03 <sup>1</sup>

**H Atom 13C**

10"C 64.3

"C 30.5

"C 25.6

"C 29.6

"C 29.6

"C 28.9

"C 28.6

"C 25.3

"C 38.7

"C 173.6

**H Atom 13C**

C 155.7

C 121.5

C 130.3

C 126.9

C 165.9

C 70.2

C 66.5

C 38.9

C 173.6

3

**H Atom 13C Atom <sup>1</sup>**

26 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

" 1.62 <sup>9</sup>

" 1.22 <sup>8</sup>

"H 1.22 <sup>7</sup>

"H 1.22 <sup>6</sup>

"H 1.22 <sup>5</sup>

"H 1.22 <sup>4</sup>

"H 1.52 <sup>3</sup>

"H 2.28 <sup>2</sup>

" 10.7 <sup>1</sup>

**H Atom 13C Atom <sup>1</sup>**

'C 132.35 <sup>8</sup>

'C 123.06 <sup>7</sup>

'C Ha

'C <sup>9</sup>

'C <sup>6</sup>

'C <sup>5</sup>

'C Hc 5.43 <sup>4</sup>

'C Hd, He 1.05 <sup>2</sup>

'C 17.47 NH2 7.61 <sup>1</sup>

9

6

5

3

2

" 10"C Ha

**Atom <sup>1</sup>**

Ha ", Hb

Hf ", Hg

7

6

5

4

3

2

NH2

**Table 7.** <sup>1</sup>

**Atom <sup>1</sup>**

8

7

Hc

Ha ', Hb '

Hd', He '

NH2

**Table 8.** <sup>1</sup>

'H 8.08 <sup>8</sup>

'H 7.45 <sup>7</sup>

' 5.67 <sup>4</sup>

4.63, 4.49

3.22, 3.10

' 7.61 <sup>1</sup>

Hd", He

The presence of choline group at the end of polymer chains causes in PTOBDME-choline a decrease in the thermal stability range compared to precursor PTOBDME. A 5% weight loss is observed for PTOBDME-choline at 230°C, while PTOBDME loses 5% weight at about 280°C. The thermal stability of PTOBEE-Choline is similar to that of polyester PTOBEE. PTOBEE-choline has 5% weight loss at 281°C, and PTOBEE at 280°C (see **Figures 6** and **7**). In the thermal stability curve of PTOBDME-choline, the first degradation step observed at 230°C is followed by two other weight loss step at 280 and 448°C. Two decomposition steps are observed at 280 and 466°C in PTOBEE-choline.

In the DSC experiment of PTOBDME-choline, performed at 10°C/min, **Figure 6(b)**, a glass transition can be observed at 58.2°C, in the first heating run, and a weak endothermic peak at 99.5°C is interpreted as due to the first order transition from crystal phase to liquid crystal state. An exothermic peak at 171.2°C is also observed which is not explained, but the beginning of a second endothermic peak at 200°C can be attributed to fusion to the isotropic. In the cooling process, two exothermic peaks at 155°C and at 175 are observed, probably associated to crystal formation. In the second heating, a very broad endothermic peak at 100.2°C is observed again associated to the transition to liquid crystal mesophase.

**Figure 6.** (a) Thermogravimetry of PTOBDME-choline and PTOBEE-choline; (b) DSC analysis of PTOBDME-choline and (c) DSC of PTOBEE-choline, both at 10°C/min.

**Figure 7.** (I) Thermogravimetry of precursor PTOBDME; (II) DSC analysis of PTOBDME; (III) DSC of PTOBEE: (a) first heating process of the original sample, (b) subsequent cooling down, (c) second heating process; and (d) DSC of (─) PTOBEE. First heating run of the original sample. All at 10°C/min.

In the DSC experiment of PTOBDME-ammonium chloride, at 10°C/min, **Figure 8**-**II**, a very broad exothermic peak centered at 96.8°C, is observed in the first heating, associated to low enthalpy value, which can be attributed to crystal to crystal transitions, involving molecular reordering between crystalline phases. An endothermic peak at 146.9°C is interpreted due to the transition to liquid crystal mesophase; finally, an exothermic peak at 186.8°C is observed. In the cooling run, very weak exothermic peaks at 154.4 and at 104.1°C were observed due crystallization process. In the second heating, a broad exothermic peak centered at 75.2°C, an endothermic peak at 149.1°C, and finally, an exothermic peak at 179.8°C

Synthetic Cationic Cholesteric Liquid Crystal Polymers http://dx.doi.org/10.5772/intechopen.70995 29

The DSC experiment of PTOBEE ammonium choride, at 10°C/min, **Figure 8**-**III**, shows in the first heating run a broad exothermic peak centered at 69.1°C, and a very strong endothermic peak at 146.2°C due to the fusion transition from crystalline phase to liquid crystal mesophase, and finally, a weak endothermic peak at 173.3°C, perhaps due to a partial fusion to isotropic. During the cooling, an exothermic peak appeared at 166°C would correspond to a crystallization from the mesophase state, and in the second heating, the broad exothermic peak observed in the first heating was observed to higher temperature centered at 114.8°C; the

The thermogravimetric curve and the DSC analysis of PTOBUME-amide are given in **Figure 9**. At 265°C, it loses 5% weight. At 340°C, a first decomposition step begins, followed by another three at 400, 450, and 510°C. In the first heating of the DSC, an endothermic peak is observed at 160°C interpreted as the transition to the mesophase state. In the cooling run, several week exothermic peaks could be associated to crystal formation

As in the polyester precursors PTOBEE-ammonium chloride and PTOBDME-ammonium chloride presented an unexpected optical activity and chiral morphology, although they were synthesized starting from equimolar quantities of TOBC and the racemic mixture of the corresponding glycol. The obtained chirality has been evaluated by optical rotatory dispersion, in **Figure 10**, the values of optical activity are given as [α]25°C, at different wavelengths. **Table 9**

In the optical characterization of precursor cholesteric liquid crystal polyesters [1, 3], even an increase of chirality was observed for a second fraction of the polymer, obtained by precipitation, after days of reaction of the liquors mother with respect to the initial first fraction of the polymer. The optical activity of PTOBDME-choline, PTOBEE-choline, PTOBUME-amide and PTOBEE-amide, has not been studied at the end of the present article but will be reported in

two endothermic peaks were again observed at 147.6 and 170.1°C.

**6.1. Optical activity of PTOBDME-ammonium and PTOBEE-ammonium**

were observed again.

processes.

the future.

**6. Optical characterization**

shows the measured values.

**Figure 8.** (I) Thermogravimetric curve of PTOBDME-ammonium and PTOBEE-ammonium; (II) DSC analysis of PTOBDME-ammonium; (III) and PTOBEE-ammonium chloride. All at 10°C/min of.

**Figure 9.** (a) Thermogravimetric curve of PTOBUME-amide; (b) DSC analysis of PTOBUME-amide.

In the DSC experiment of PTOBEE-choline at (10°C/min), **Figure 6(c)**, a glass transition can be observed at 60°C, and an endothermic peak at 130.2°C is attributed to the transition crystal to liquid crystal. A decreasing of baseline from 183.7°C to the end of heating was also observed in the first heating run due to a nonconcluded endothermic process or to the beginning of degradation to the polymer. A broad exothermic peak observed the cooling around 145°C would correspond to a crystallization from the mesophase state. In the second heating, only two glass transitions can be observed at 65 and 85°C.

The presence of ammonium chloride group at the end in the polymer chains, in **Figure 8**-**I**, produces a decrease of the thermal stability range compared to precursor polyesters. At 278°C, PTOBEE-ammonium chloride loses 10% weight and PTOBDME-ammonium chloride at 260° C, while precursor PTOBDME and PTOBEE at 310°C. In the thermal stability curve of the ammonium-polymers, the first degradation step observed at 228 and 230°C, respectively, was not observed in PTOBEE and PTOBDME. The two next inflexion points at 310 and 311°C and 466 and 471°C were equivalents to the observed in the precursor polyesters, which would indicate the same type of decomposition to principal core of the chain.

In the DSC experiment of PTOBDME-ammonium chloride, at 10°C/min, **Figure 8**-**II**, a very broad exothermic peak centered at 96.8°C, is observed in the first heating, associated to low enthalpy value, which can be attributed to crystal to crystal transitions, involving molecular reordering between crystalline phases. An endothermic peak at 146.9°C is interpreted due to the transition to liquid crystal mesophase; finally, an exothermic peak at 186.8°C is observed. In the cooling run, very weak exothermic peaks at 154.4 and at 104.1°C were observed due crystallization process. In the second heating, a broad exothermic peak centered at 75.2°C, an endothermic peak at 149.1°C, and finally, an exothermic peak at 179.8°C were observed again.

The DSC experiment of PTOBEE ammonium choride, at 10°C/min, **Figure 8**-**III**, shows in the first heating run a broad exothermic peak centered at 69.1°C, and a very strong endothermic peak at 146.2°C due to the fusion transition from crystalline phase to liquid crystal mesophase, and finally, a weak endothermic peak at 173.3°C, perhaps due to a partial fusion to isotropic. During the cooling, an exothermic peak appeared at 166°C would correspond to a crystallization from the mesophase state, and in the second heating, the broad exothermic peak observed in the first heating was observed to higher temperature centered at 114.8°C; the two endothermic peaks were again observed at 147.6 and 170.1°C.

The thermogravimetric curve and the DSC analysis of PTOBUME-amide are given in **Figure 9**. At 265°C, it loses 5% weight. At 340°C, a first decomposition step begins, followed by another three at 400, 450, and 510°C. In the first heating of the DSC, an endothermic peak is observed at 160°C interpreted as the transition to the mesophase state. In the cooling run, several week exothermic peaks could be associated to crystal formation processes.

### **6. Optical characterization**

In the DSC experiment of PTOBEE-choline at (10°C/min), **Figure 6(c)**, a glass transition can be observed at 60°C, and an endothermic peak at 130.2°C is attributed to the transition crystal to liquid crystal. A decreasing of baseline from 183.7°C to the end of heating was also observed in the first heating run due to a nonconcluded endothermic process or to the beginning of degradation to the polymer. A broad exothermic peak observed the cooling around 145°C would correspond to a crystallization from the mesophase state. In the second heating, only

**Figure 9.** (a) Thermogravimetric curve of PTOBUME-amide; (b) DSC analysis of PTOBUME-amide.

**Figure 8.** (I) Thermogravimetric curve of PTOBDME-ammonium and PTOBEE-ammonium; (II) DSC analysis of

PTOBDME-ammonium; (III) and PTOBEE-ammonium chloride. All at 10°C/min of.

28 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

The presence of ammonium chloride group at the end in the polymer chains, in **Figure 8**-**I**, produces a decrease of the thermal stability range compared to precursor polyesters. At 278°C, PTOBEE-ammonium chloride loses 10% weight and PTOBDME-ammonium chloride at 260° C, while precursor PTOBDME and PTOBEE at 310°C. In the thermal stability curve of the ammonium-polymers, the first degradation step observed at 228 and 230°C, respectively, was not observed in PTOBEE and PTOBDME. The two next inflexion points at 310 and 311°C and 466 and 471°C were equivalents to the observed in the precursor polyesters, which would

two glass transitions can be observed at 65 and 85°C.

indicate the same type of decomposition to principal core of the chain.

### **6.1. Optical activity of PTOBDME-ammonium and PTOBEE-ammonium**

As in the polyester precursors PTOBEE-ammonium chloride and PTOBDME-ammonium chloride presented an unexpected optical activity and chiral morphology, although they were synthesized starting from equimolar quantities of TOBC and the racemic mixture of the corresponding glycol. The obtained chirality has been evaluated by optical rotatory dispersion, in **Figure 10**, the values of optical activity are given as [α]25°C, at different wavelengths. **Table 9** shows the measured values.

In the optical characterization of precursor cholesteric liquid crystal polyesters [1, 3], even an increase of chirality was observed for a second fraction of the polymer, obtained by precipitation, after days of reaction of the liquors mother with respect to the initial first fraction of the polymer. The optical activity of PTOBDME-choline, PTOBEE-choline, PTOBUME-amide and PTOBEE-amide, has not been studied at the end of the present article but will be reported in the future.

in the spacer, along the copolymer backbone, with two possible helical screw sense of the polymer chain and in all the studied polymers. Chirality in racemic PTOBDME was proposed to be due to the kinetic resolution of a preferable helical diastereomer, such as S*gt*, with respect to the possible four forms, while the R/S ratio of asymmetric carbon atoms remained 50:50.

Synthetic Cationic Cholesteric Liquid Crystal Polymers http://dx.doi.org/10.5772/intechopen.70995 31

The presence of choline group or ammonium chloride groups at the end of polymer chains causes in precursor polyesters a decrease in their thermal stability range. PTOBDME-choline losses 5% weight at 230°C (PTOBDME at 280°C). The thermal stability of PTOBEE-choline is

At 260°C, PTOBDME-ammonium loses 10% weight and PTOBEE-ammonium at 278°C (pre-

All the synthetized cationic liquid crystal polymers show in DSC an endothermic peak assigned to the first order transition from crystalline phase to liquid crystal mesophase: PTOBDME-choline at 99.5°C; PTOBEE-choline at 130.2°C; PTOBDME-ammonium at 146.9°C;

At 265°C, PTOBUME-amide loses 5% weight. At 340°C, it has a first decomposition step, followed by another three at 400, 450, and 510°C. In the DSC first heating, it shows the endother-

Optical ORD values are provided for the second fractions of PTOBDME-ammonium and

The author thanks Dr. Javier Sanguino Otero for his valuable help during the development of this Project. She also thanks the financial support obtained in the Project "Nuevos vectores no virales basados en polímero cristal-líquido colestérico (PCLC) y su uso para transfección

[1] Pérez Méndez M, Sanguino Otero J. Cholesteric Liquid-Crystal Copolyester, Poly[oxycarbonyl-1,4-phenylene-oxy-1,4 terephthaloyl-oxy-1,4-phenylene-carbonyloxy

similar to that precursor PTOBEE, with 5% weight loss at 281°C.

cursor PTOBDME and PTOBEE at 310°C).

and PTOBEE-ammonium at 146.2°C.

PTOBEE-ammonium.

**Acknowledgements**

génica". PTR1995-0760-OP.

Mercedes Pérez Méndez

**Author details**

**References**

mic peak due to the mesophase transition at 160°C.

Address all correspondence to: perezmendez@ictp.csic.es

Instituto de Ciencia y Tecnología de Polímeros (ICTP), CSIC, Madrid, Spain

**Figure 10.** Optical activity of PTOBDME-ammonium chloride and PTOBEE-ammonium choride. Expressed as [α]25°C in DMSO-d6 at different wavelengths.


**Table 9.** Optical activity of PTOBDME-ammonium and PTOBEE-ammonium, expressed by optical rotatory dispersion.

### **7. Conclusions**

The synthetic methods of six new multifunctional cationic cholesteric liquid crystal polymers designed as PTOBDME-choline [(C34H36O8 ) <sup>n</sup>─C5 H13N]; PTOBEE-choline [(C26H20O8 ) <sup>n</sup>─C5 H13N]; PTOBDME-ammonium [(C34H36O8 ) <sup>n</sup>─C5 H13N]; PTOBEE-ammonium [(C26H20O8 ) <sup>n</sup>─C5 H13N]; PTOBUME-amide (C33H33O9 N)n and PTOBEE-amide (C26H19O9 N)n are given and their characterization by 1 H, 13C-NMR, COSY, and HSQC is reported.

The NMR analysis let us to conclude that the enantiomeric polymer chains present stereo regular head-tail, isotactic structure, explained in terms of the higher reactivity of the primary hydroxyl group in the glycol, with respect to the secondary one, through the polycondensation reaction.

According to our previous experience, each enantiomer, with two independent sets of signals observed by 1 H and 13C-NMR, differentiated with apostrophe (') and without it ( ), could be attributed to two diastereomeric conformers: *gg* and *gt*, related with two possible staggered conformations, of the torsion along the chemical bond containing the asymmetric carbon atom in the spacer, along the copolymer backbone, with two possible helical screw sense of the polymer chain and in all the studied polymers. Chirality in racemic PTOBDME was proposed to be due to the kinetic resolution of a preferable helical diastereomer, such as S*gt*, with respect to the possible four forms, while the R/S ratio of asymmetric carbon atoms remained 50:50.

The presence of choline group or ammonium chloride groups at the end of polymer chains causes in precursor polyesters a decrease in their thermal stability range. PTOBDME-choline losses 5% weight at 230°C (PTOBDME at 280°C). The thermal stability of PTOBEE-choline is similar to that precursor PTOBEE, with 5% weight loss at 281°C.

At 260°C, PTOBDME-ammonium loses 10% weight and PTOBEE-ammonium at 278°C (precursor PTOBDME and PTOBEE at 310°C).

All the synthetized cationic liquid crystal polymers show in DSC an endothermic peak assigned to the first order transition from crystalline phase to liquid crystal mesophase: PTOBDME-choline at 99.5°C; PTOBEE-choline at 130.2°C; PTOBDME-ammonium at 146.9°C; and PTOBEE-ammonium at 146.2°C.

At 265°C, PTOBUME-amide loses 5% weight. At 340°C, it has a first decomposition step, followed by another three at 400, 450, and 510°C. In the DSC first heating, it shows the endothermic peak due to the mesophase transition at 160°C.

Optical ORD values are provided for the second fractions of PTOBDME-ammonium and PTOBEE-ammonium.

### **Acknowledgements**

The author thanks Dr. Javier Sanguino Otero for his valuable help during the development of this Project. She also thanks the financial support obtained in the Project "Nuevos vectores no virales basados en polímero cristal-líquido colestérico (PCLC) y su uso para transfección génica". PTR1995-0760-OP.

### **Author details**

**7. Conclusions**

DMSO-d6

terization by 1

tion reaction.

observed by 1

designed as PTOBDME-choline [(C34H36O8

PTOBDME-ammonium [(C34H36O8

at different wavelengths.

30 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

PTOBUME-amide (C33H33O9

The synthetic methods of six new multifunctional cationic cholesteric liquid crystal polymers

**Table 9.** Optical activity of PTOBDME-ammonium and PTOBEE-ammonium, expressed by optical rotatory dispersion.

**Polymers (0.2 g/100 ml in DMSO) Hg (365 nm) Hg (435 nm) Hg (546 nm) Hg (578 nm) Na (589 nm)** PTOBEE-ammonium chloride +12.44° +5.75° +2.85° +3.44° +5.75° PTOBDME-ammonium chloride +9.00 +5.06 −0.55 −1.15 −4.04

**Figure 10.** Optical activity of PTOBDME-ammonium chloride and PTOBEE-ammonium choride. Expressed as [α]25°C in

N)n and PTOBEE-amide (C26H19O9

The NMR analysis let us to conclude that the enantiomeric polymer chains present stereo regular head-tail, isotactic structure, explained in terms of the higher reactivity of the primary hydroxyl group in the glycol, with respect to the secondary one, through the polycondensa-

According to our previous experience, each enantiomer, with two independent sets of signals

attributed to two diastereomeric conformers: *gg* and *gt*, related with two possible staggered conformations, of the torsion along the chemical bond containing the asymmetric carbon atom

H and 13C-NMR, differentiated with apostrophe (') and without it ( ), could be

H13N]; PTOBEE-choline [(C26H20O8

H13N]; PTOBEE-ammonium [(C26H20O8

) <sup>n</sup>─C5

) <sup>n</sup>─C5

N)n are given and their charac-

H13N];

H13N];

) <sup>n</sup>─C5

) <sup>n</sup>─C5

H, 13C-NMR, COSY, and HSQC is reported.

Mercedes Pérez Méndez

Address all correspondence to: perezmendez@ictp.csic.es

Instituto de Ciencia y Tecnología de Polímeros (ICTP), CSIC, Madrid, Spain

### **References**

[1] Pérez Méndez M, Sanguino Otero J. Cholesteric Liquid-Crystal Copolyester, Poly[oxycarbonyl-1,4-phenylene-oxy-1,4 terephthaloyl-oxy-1,4-phenylene-carbonyloxy (1,2-dodecane)] [C34H36O8 ] <sup>n</sup>, Synthesized from Racemic Materials: Kinetics, Structure and Optical Characterization. International Journal of Engineering Research and Applications (IJERA). July 2015;**5**(7, Part-2):48-62. ISSN: 2248-9622, http://www.ijera. com/papers/Vol5\_issue7/Part%20-%202/H57024862.pdf

[11] Knapp S, Levorse AT. Synthesis and Reactions of Iodo Lactams. Journal of Organic

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[12] Rani S, Vankar YD. An efficient one step dihydroxylation of 1,2-glycals with oxone in

[13] Corey PF, Ward FE. Buffered potassium peroxymonosulfate-acetone epoxidation of .Alpha.,.Beta.-unsaturated acids. The Journal of Organic Chemistry. 1986;**51**(10):1925-1926

[14] Zhu W, Ford WT. Oxidation of alkenes with aqueous potassium peroxymonosulfate and no organic solvent. The Journal of Organic Chemistry. 1991;**56**(25):7022-7026

[15] Ella-Menye JR, Sharma V, Wang G. New synthesis of chiral 1,3-oxazinan-2-ones from carbohydrate derivatives. The Journal of Organic Chemistry. 2005;**70**(2):463-469

Chemistry; **53**:4006-4014

acetone. Tetrahedron Letters. 2003;**44**:907-909


[11] Knapp S, Levorse AT. Synthesis and Reactions of Iodo Lactams. Journal of Organic Chemistry; **53**:4006-4014

(1,2-dodecane)] [C34H36O8

11/IJ16M1031.pdf

Engineering B. 2013;**3**(2):104-115

]

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com/papers/Vol5\_issue7/Part%20-%202/H57024862.pdf

of Macromolecular Science Part B Physics. 2001;**40**:553-576

Conditions. Macromolecules. 2003;**36**:8049-8055

JP2001513827-W, AU739076-B, DE69824182-E

properties in solution. Macromolecular Symposia. 1989;**29**:9-23

Santiago de Compostela, Spain, www.mestrelab.com, 2017

<sup>n</sup>, Synthesized from Racemic Materials: Kinetics, Structure

and Optical Characterization. International Journal of Engineering Research and Applications (IJERA). July 2015;**5**(7, Part-2):48-62. ISSN: 2248-9622, http://www.ijera.

[2] Fayos J, Sanchez-Cortes S, Marco C, Pérez-Méndez M. Journal of Macromolecular Science Part B Physics.Conformational analysis and molecular modeling of cholesteric liquid-crystal polyesters based on XRD, Raman and transition thermal analysis. Journal

[3] Perez-Mendez M, Marsal R, Garrido L, Martin-Pastor M. Self-Association and Stereoselectivity in a Chiral Liquid-Crystal Colesteric Polymer Formed under Achiral

[4] (a) Pérez-Méndez M, Marco C. New synthesis, thermal properties and texture of cholesteric poly[ethyl ethylene 4,4'-(terephthaloyldioxy)dibenzoate]. Acta Polymerica. 1997;**48**:502-506. (b) Pérez-Méndez M, Marco Rocha C. Process for obtaining cholesteric liquid crystals by stereoselective recrystallization. Patents: EP1004650-A, WO9831771-A, WO9831771-A1, AU9854863-A, ES2125818-A1, US6165382-A, MX9906732-A1,

[5] (a) Bilibin AY, Ten'kovtsev AV, Piraner ON, Skorokhodov SS. Synthesis of highmolecular weight liquid crystal polyesters based on a polycondensation mesogenic monomer. Polymer Science U.S.S.R. 1984;**26**(12):2882-2890. (b) Bilibin AY, Skorokhodov SS. Rational path of the synthesis of liquid-crystalline highmolecular weight polyesters and their

[6] Pérez Méndez M, Rodríguez Martínez D, Fayos Alcañíz J. Structure of non-viral vectors based on cholesteric liquid-crystal polymers by SAXS. International Journal of Advancement in Engineering Technology, Management and Applied Science (IJAETMAS). 2016;**03**(11):27-41. ISSN 2349-3224, http://www.ijaetmas.com/wp-content/uploads/2016/

[7] Pérez Méndez M, Hammouda B. SAXS and SANS investigation of synthetic cholesteric liquid-crystal polymers for biomedical applications. Journal of Materials Science and

[8] Pérez Méndez M, Rodriguez Martinez D, King SM. pH-induced size changes in solutions of cholesteric liquid crystal polymers studied by SANS. Journal of Physics: Conference Series. Dynamics of Molecules and Materials-II. 2014;**554**(012011):1-11. IOP Publishing

[9] (a) Cobas JC, Sardina FJ, Concepts in Magenetic Resonance. Nuclear magnetic resonance data processing. MestRe-C: A software package for desktop computers. Concepts in Magenetic Resonance. 2003;**19**:80-96; (b) MestReNova 11.0.4, Mestrelab Research SL,

[10] ChemDraw Professional, Version 15.1.0.144 (PerkinElmer Informatics 1985-2016)


**Chapter 3**

Provisional chapter

**From a Chiral Molecule to Blue Phases**

From a Chiral Molecule to Blue Phases

Chiral molecules play an important role in a wide range from biological structures of plants and animals to chemical systems and liquid crystal display technologies. These molecules were used in different research fields due to their opaqueness and iridescent colors changes as a function of the variation in temperature after their discovery by Lehman in 1889. The iridescent colors and different optical textures of cholesterol make it attractive for the new study field of cholesteric liquid crystals. The direction of the cholesteric liquid crystals generates a periodic helical structure depending on the chirality of molecules. This helical structure might be right or left handed configuration and it is very sensitive to the external conditions, such as chiral dopant concentration and temperature. The variation in a helical structure, which was induced by these external conditions, had a great attraction for the scientists working on the chirality in liquid crystals and their applications. This chapter will provide a general introduction not only about the chirality in nature and its application in liquid crystals, especially in blue phases but also about the trends in the stabilization of blue phases and the investigation of their electro-optical properties for advanced applications in

DOI: 10.5772/intechopen.70555

Keywords: chiral molecule, chirality, cholesteric liquid crystal, blue phase liquid crystal,

Chiral molecules play an important role in a wide range from biological structures of plants and animals to chemical systems and liquid crystal display technologies. The chiral molecules as liquid crystals in the form of cholesterol in the biological substances were discovered by the biologist Friedel in 1922 [1, 2]. After cholesterol was extracted from plants, cholesteryl esters were obtained by treating cholesterol with fatty acids and Reinitzer observed that cholesterol has two different melting points during heating from the crystalline phase or upon cooling

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70555

display, photonic devices.

1. Introduction

Bragg reflection, polymer stabilization

Emine Kemiklioglu

Emine Kemiklioglu

Abstract

Provisional chapter

## **From a Chiral Molecule to Blue Phases** From a Chiral Molecule to Blue Phases

### Emine Kemiklioglu

Additional information is available at the end of the chapter Emine Kemiklioglu

http://dx.doi.org/10.5772/intechopen.70555 Additional information is available at the end of the chapter

### Abstract

Chiral molecules play an important role in a wide range from biological structures of plants and animals to chemical systems and liquid crystal display technologies. These molecules were used in different research fields due to their opaqueness and iridescent colors changes as a function of the variation in temperature after their discovery by Lehman in 1889. The iridescent colors and different optical textures of cholesterol make it attractive for the new study field of cholesteric liquid crystals. The direction of the cholesteric liquid crystals generates a periodic helical structure depending on the chirality of molecules. This helical structure might be right or left handed configuration and it is very sensitive to the external conditions, such as chiral dopant concentration and temperature. The variation in a helical structure, which was induced by these external conditions, had a great attraction for the scientists working on the chirality in liquid crystals and their applications. This chapter will provide a general introduction not only about the chirality in nature and its application in liquid crystals, especially in blue phases but also about the trends in the stabilization of blue phases and the investigation of their electro-optical properties for advanced applications in display, photonic devices.

DOI: 10.5772/intechopen.70555

Keywords: chiral molecule, chirality, cholesteric liquid crystal, blue phase liquid crystal, Bragg reflection, polymer stabilization

### 1. Introduction

Chiral molecules play an important role in a wide range from biological structures of plants and animals to chemical systems and liquid crystal display technologies. The chiral molecules as liquid crystals in the form of cholesterol in the biological substances were discovered by the biologist Friedel in 1922 [1, 2]. After cholesterol was extracted from plants, cholesteryl esters were obtained by treating cholesterol with fatty acids and Reinitzer observed that cholesterol has two different melting points during heating from the crystalline phase or upon cooling

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

from the isotropic liquid [3]. The cholesteryl esters were examined with the help of a polarizing optical microscope by chemist Reinitzer in 1888 [3], biologists Planer in 1861 [4] and physicist Lehman in 1889 [5] and they noticed that these esters showed an opaqueness and iridescent colors with the changes in temperature. Therefore, the structure and optical behavior of cholesterol were explained with the contribution of biologist, physicist, chemist and the iridescent colors and different optical textures of cholesterol make it attractive for the new study field of cholesteric liquid crystals. The cholesteric liquid crystals are generally derivatives of the cholesterol which exhibit in organic compounds of elongated molecules (like nematic liquid crystal) without mirror symmetry [6–8]. They can be formed using pure chiral compounds or the mixture of the chiral and achiral compounds [9–13]. The direction of the cholesteric liquid crystals generates a periodic helical structure depending on the chirality of molecules. This helical structure might be right or left handed configuration and it is very sensitive to the external conditions, such as chiral dopant concentration and temperature [14]. The variation in a helical structure, which was induced by these external conditions, had a great attraction for the scientists working on the chirality in liquid crystals and their applications.

superimposed over its own mirror image. The main feature that gives rise to chirality at a molecular scale is the presence of an asymmetric ally handed carbon atom. A chiral molecule typically has a carbon atom in the center of the molecule surrounded by four different substituents and these molecules cannot be transformed into their mirror image by rotations [39–41]. Moreover, all these different groups are not in the same plane, although the positions of these groups form the corners of a tetrahedron with a central carbon atom (Figure 1). The configuration of these molecules can be classified into two different group, such as 'R'(for Latin rectus, right) or 'S'(for Latin sinister, left). In that case, handedness can be defined as right hand or left hand for a chiral molecule [42]. In order to define handedness of the molecule, the four groups are arranged in a priority list. Groups with the higher atomic number take precedence of the groups of lower atomic number. Each chiral center is labeled as R or S related to the priority of the substituents of the molecule based on their atomic numbers. To determine the handedness of the molecule, first the chiral center is determined according to the lowest priority of the four substituents. If the priority of the other three substituents decreases in clockwise direction, it is called R (right handed), if it decreases in counterclockwise direction, it is called S (left handed). Table 1 summarizes a list for the enantiomers of chiral materials. Moreover, an enantiomer can be named by its ability to rotate the plane of plane-polarized light (+/). The enantiomer is labeled (+), if it rotates light in a clockwise direction. If it rotates the light counterclockwise direction, it is labeled as (). Liquid crystals may have multiple chiral centers with handedness and configuration. Moreover, the chirality of atoms can be detected by optical experiments [43], which shows that the broken mirror symmetry in stable

From a Chiral Molecule to Blue Phases http://dx.doi.org/10.5772/intechopen.70555 37

Chiral molecules may give rise to an intrinsic helical structure of the director in liquid crystals inducing chirality [44–49]. The liquid crystal state is a mesophase between solid and liquid which is characterized by the alignment of rod-like molecules which has two aromatic rings

Figure 1. Simulation of a chiral molecule which cannot be superimposed with its mirror image [53].

atoms as a function of absorption of light.

Moreover, recent studies showed that the reduction of the pitch of the helical structure of the cholesteric liquid crystal by adding chiral dopants generates different phases, such as blue phase [15, 16]. Blue phases are mesophases with double-twisted cylinders of cholesteric liquid crystals and they come into existence in a self-organized three-dimensional (3D) structures in the narrow temperature range between the cholesteric and isotropic phases [3–17]. Recent studies showed the trends in the stabilization of blue phase in order to expand its narrow temperature range using different stabilization methods, such as photopolymerization [18–22], nanoparticles doping [23–27], polymer-modified carbon nanotubes (CNTs) [28, 29]. Specifically, the temperature range of blue phase was broadened up to 60 K by using a polymerized polymer network, called as the polymer-stabilized blue phase (PSBP) [18] whereas blue phase was stabilized over a range of about 50C by using a mixture nematic bimesogenic liquid crystals [30]. Furthermore, blue phase has some advantageous in the display applications due to its outstanding electro-optical properties. Blue phases have field-induced birefringence (Kerr effect) and their response time is in the level of submillisecond. Additionally, blue phases do not need any surface modification which leads simplicity in the fabrication process and they have wide and symmetric viewing angle.

This chapter will be focused on the stabilization and electro-optical properties of blue phases and their potentials for advanced applications in display as well as photonic devices [18–22, 31, 32]. The chapter concludes with the studies related to the recent novel studies on the encapsulation of blue phases [33], the stabilization of the encapsulated blue phases [34] and polymerizationinduced polymer-stabilized blue phase [35–37].

### 2. Cholesteric liquid crystals

### 2.1. Chirality

The word of chirality originates from Greek as a meaning of hand and chirality was discovered by Lord Kelvin in 1894 [38]. It was described as a property of a molecule that cannot be

superimposed over its own mirror image. The main feature that gives rise to chirality at a molecular scale is the presence of an asymmetric ally handed carbon atom. A chiral molecule typically has a carbon atom in the center of the molecule surrounded by four different substituents and these molecules cannot be transformed into their mirror image by rotations [39–41]. Moreover, all these different groups are not in the same plane, although the positions of these groups form the corners of a tetrahedron with a central carbon atom (Figure 1). The configuration of these molecules can be classified into two different group, such as 'R'(for Latin rectus, right) or 'S'(for Latin sinister, left). In that case, handedness can be defined as right hand or left hand for a chiral molecule [42]. In order to define handedness of the molecule, the four groups are arranged in a priority list. Groups with the higher atomic number take precedence of the groups of lower atomic number. Each chiral center is labeled as R or S related to the priority of the substituents of the molecule based on their atomic numbers. To determine the handedness of the molecule, first the chiral center is determined according to the lowest priority of the four substituents. If the priority of the other three substituents decreases in clockwise direction, it is called R (right handed), if it decreases in counterclockwise direction, it is called S (left handed). Table 1 summarizes a list for the enantiomers of chiral materials. Moreover, an enantiomer can be named by its ability to rotate the plane of plane-polarized light (+/). The enantiomer is labeled (+), if it rotates light in a clockwise direction. If it rotates the light counterclockwise direction, it is labeled as (). Liquid crystals may have multiple chiral centers with handedness and configuration. Moreover, the chirality of atoms can be detected by optical experiments [43], which shows that the broken mirror symmetry in stable atoms as a function of absorption of light.

from the isotropic liquid [3]. The cholesteryl esters were examined with the help of a polarizing optical microscope by chemist Reinitzer in 1888 [3], biologists Planer in 1861 [4] and physicist Lehman in 1889 [5] and they noticed that these esters showed an opaqueness and iridescent colors with the changes in temperature. Therefore, the structure and optical behavior of cholesterol were explained with the contribution of biologist, physicist, chemist and the iridescent colors and different optical textures of cholesterol make it attractive for the new study field of cholesteric liquid crystals. The cholesteric liquid crystals are generally derivatives of the cholesterol which exhibit in organic compounds of elongated molecules (like nematic liquid crystal) without mirror symmetry [6–8]. They can be formed using pure chiral compounds or the mixture of the chiral and achiral compounds [9–13]. The direction of the cholesteric liquid crystals generates a periodic helical structure depending on the chirality of molecules. This helical structure might be right or left handed configuration and it is very sensitive to the external conditions, such as chiral dopant concentration and temperature [14]. The variation in a helical structure, which was induced by these external conditions, had a great attraction for

36 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

the scientists working on the chirality in liquid crystals and their applications.

Moreover, recent studies showed that the reduction of the pitch of the helical structure of the cholesteric liquid crystal by adding chiral dopants generates different phases, such as blue phase [15, 16]. Blue phases are mesophases with double-twisted cylinders of cholesteric liquid crystals and they come into existence in a self-organized three-dimensional (3D) structures in the narrow temperature range between the cholesteric and isotropic phases [3–17]. Recent studies showed the trends in the stabilization of blue phase in order to expand its narrow temperature range using different stabilization methods, such as photopolymerization [18–22], nanoparticles doping [23–27], polymer-modified carbon nanotubes (CNTs) [28, 29]. Specifically, the temperature range of blue phase was broadened up to 60 K by using a polymerized polymer network, called as the polymer-stabilized blue phase (PSBP) [18] whereas blue phase was stabilized over a range of about 50C by using a mixture nematic bimesogenic liquid crystals [30]. Furthermore, blue phase has some advantageous in the display applications due to its outstanding electro-optical properties. Blue phases have field-induced birefringence (Kerr effect) and their response time is in the level of submillisecond. Additionally, blue phases do not need any surface modification which leads simplicity in the fabrication process and they have wide and symmetric viewing angle.

This chapter will be focused on the stabilization and electro-optical properties of blue phases and their potentials for advanced applications in display as well as photonic devices [18–22, 31, 32]. The chapter concludes with the studies related to the recent novel studies on the encapsulation of blue phases [33], the stabilization of the encapsulated blue phases [34] and polymerization-

The word of chirality originates from Greek as a meaning of hand and chirality was discovered by Lord Kelvin in 1894 [38]. It was described as a property of a molecule that cannot be

induced polymer-stabilized blue phase [35–37].

2. Cholesteric liquid crystals

2.1. Chirality

Chiral molecules may give rise to an intrinsic helical structure of the director in liquid crystals inducing chirality [44–49]. The liquid crystal state is a mesophase between solid and liquid which is characterized by the alignment of rod-like molecules which has two aromatic rings

Figure 1. Simulation of a chiral molecule which cannot be superimposed with its mirror image [53].

with the aliphatic chains. These rod-like shaped molecules are usually formed liquid crystal materials by aligning along a certain direction and this certain direction forms a helical structure with the addition of chiral molecules. In that case, the liquid crystal phase is called chiral nematic (cholesteric) phase which is one of the several additional phases in the temperature range between the crystalline and the isotropic liquid state (Figure 2). Moreover, chirality can be induced in smectic and columnar phases which are a quasi-long range positional order in less than three dimensions. In contrast to thermotropic liquid crystals, chiral columnar liquid crystals are formed by amphiphilic molecules in lyotropic liquid crystals. However, these amphiphilic molecules can be arranged in an anisotropic structure as in thermotropic liquid crystals, when the concentration of these molecules in a solvent is enough. Recently, Takezoe et al. induced a molecular chirality in the bent-core molecules which do not have molecular handednesses. They successfully induced a chirality in these achiral molecules based on the

Cloramphenicol It is an antibiotic and it works to treat serious

Levomethamphetamine Active ingredients in over-the-counter nasal

infections induced by certain bacteria

From a Chiral Molecule to Blue Phases http://dx.doi.org/10.5772/intechopen.70555 39

decongestants

Name of an enantiomer Chemical structure of an enantiomer Properties of enantiomers (S) Ibuprofen The over-the-counter painkiller

(R) Thalidomid Sedative and antinausea

(S) Thalidomid Teratogen

packing of the bent-core molecules [15, 17].

Table 1. Enantiomers and their properties.

Table 1. Enantiomers and their properties.

Name of an enantiomer Chemical structure of an enantiomer Properties of enantiomers

(R) Limonene Orange smell

38 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

(S) Limonene Lemon smell

(S) Asparagine Sweet taste

(R) Asparagine Bitter taste

(S) Carvone Caraway flavor

(R) Carvone Spearmint flavor

Pseudoephedrine Active ingredients in over-the-counter nasal

decongestants

with the aliphatic chains. These rod-like shaped molecules are usually formed liquid crystal materials by aligning along a certain direction and this certain direction forms a helical structure with the addition of chiral molecules. In that case, the liquid crystal phase is called chiral nematic (cholesteric) phase which is one of the several additional phases in the temperature range between the crystalline and the isotropic liquid state (Figure 2). Moreover, chirality can be induced in smectic and columnar phases which are a quasi-long range positional order in less than three dimensions. In contrast to thermotropic liquid crystals, chiral columnar liquid crystals are formed by amphiphilic molecules in lyotropic liquid crystals. However, these amphiphilic molecules can be arranged in an anisotropic structure as in thermotropic liquid crystals, when the concentration of these molecules in a solvent is enough. Recently, Takezoe et al. induced a molecular chirality in the bent-core molecules which do not have molecular handednesses. They successfully induced a chirality in these achiral molecules based on the packing of the bent-core molecules [15, 17].

Figure 2. Phase sequence in thermotropic liquid crystals.

### 2.2. Cholesteric liquid crystals

The cholesteric phase is a mesophase which exhibits between the smectic and isotropic phases in thermotropic liquid crystals. Chiral nematic liquid crystals are a type of liquid crystal which has a helical structure based on the molecular chirality of its components (Figure 3). This phase can be formed using the chiral dopants in an achiral nematic forms new chiral materials with specific helical pitches (Figure 4) [50–52].

Pitch plays an important role in the reflection of the wavelength of the incident light, as a result of the periodic structure of cholesteric liquid crystals [53]. Cholesteric liquid crystals have the ability to reflect a handedness of circularly polarized light when the pitch has the same wavelength of visible light [53]. The light will be circularly reflected if it is the same handedness as that of the cholesteric liquid crystal, whereas it will be circularly transmitted with opposite handedness as that of the cholesteric liquid crystal [44]. This selective reflection of circularly polarized light exhibits an iridescent color depending on the angular deviation. This property of selective refraction may practically be used in the application of liquid crystals, such as thermometers,

Figure 5. An illustration of cholesteric liquid crystal with a pitch p. Pitch plays an important role in the reflection of the

Furthermore, chirality in liquid crystals can be described related to inverse of the pitch of the material and a shorter helical pitch has a higher chirality. The normalized reciprocal of the

where p is helical pitch in microns and c is concentration of chiral dopant in the cholesteric

) is described as the helical twisting power (HTP) of a molecule and it can be defined

HTP ¼ 1=c ∗ p (1)

From a Chiral Molecule to Blue Phases http://dx.doi.org/10.5772/intechopen.70555 41

polarizing mirrors, refractive electro-optic displays and optical storage [53–58].

wavelength of the incident light, as a result of the periodic structure of cholesteric liquid crystals [53].

as the chiral dopant's ability to induce helicity in the molecule [17].

Figure 4. Formation of helical chiral nematic phase using chiral dopant.

pitch (p�<sup>1</sup>

liquid crystal mixture.

Cholesteric liquid crystals arrange within layer without any positional ordering in the layer whereas the director axis rotates with the layers as shown in Figure 5. The rotation of the director axis is periodic and its full rotation of 360 is called the pitch, p. The value of the pitch may change as a function of enantiomeric excess in an ideal mixture containing chiral and racemic.

Figure 3. Simulation of formation of the helical structure of a chiral nematic (cholesteric) phase.

Figure 4. Formation of helical chiral nematic phase using chiral dopant.

2.2. Cholesteric liquid crystals

specific helical pitches (Figure 4) [50–52].

Figure 2. Phase sequence in thermotropic liquid crystals.

40 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

The cholesteric phase is a mesophase which exhibits between the smectic and isotropic phases in thermotropic liquid crystals. Chiral nematic liquid crystals are a type of liquid crystal which has a helical structure based on the molecular chirality of its components (Figure 3). This phase can be formed using the chiral dopants in an achiral nematic forms new chiral materials with

Cholesteric liquid crystals arrange within layer without any positional ordering in the layer whereas the director axis rotates with the layers as shown in Figure 5. The rotation of the director axis is periodic and its full rotation of 360 is called the pitch, p. The value of the pitch may change as a function of enantiomeric excess in an ideal mixture containing chiral and racemic.

Figure 3. Simulation of formation of the helical structure of a chiral nematic (cholesteric) phase.

Figure 5. An illustration of cholesteric liquid crystal with a pitch p. Pitch plays an important role in the reflection of the wavelength of the incident light, as a result of the periodic structure of cholesteric liquid crystals [53].

Pitch plays an important role in the reflection of the wavelength of the incident light, as a result of the periodic structure of cholesteric liquid crystals [53]. Cholesteric liquid crystals have the ability to reflect a handedness of circularly polarized light when the pitch has the same wavelength of visible light [53]. The light will be circularly reflected if it is the same handedness as that of the cholesteric liquid crystal, whereas it will be circularly transmitted with opposite handedness as that of the cholesteric liquid crystal [44]. This selective reflection of circularly polarized light exhibits an iridescent color depending on the angular deviation. This property of selective refraction may practically be used in the application of liquid crystals, such as thermometers, polarizing mirrors, refractive electro-optic displays and optical storage [53–58].

Furthermore, chirality in liquid crystals can be described related to inverse of the pitch of the material and a shorter helical pitch has a higher chirality. The normalized reciprocal of the pitch (p�<sup>1</sup> ) is described as the helical twisting power (HTP) of a molecule and it can be defined as the chiral dopant's ability to induce helicity in the molecule [17].

$$\text{HTP} = 1/\text{c} \ast p \tag{1}$$

where p is helical pitch in microns and c is concentration of chiral dopant in the cholesteric liquid crystal mixture.

However, qo is the pitch in the ground state given by Eq. (2) and it is induced in nematic liquid crystal where there are long-range distortions

$$\mathbf{q}\_{\rm o} = 2\pi/\mathbf{p} \tag{2}$$

[61, 62]. Recently, it became possible to observe BPs in wide temperature range via the stabilization method. The thermodynamic stability of BPs which were composed of chiral nematic liquid crystal with a low chirality have been predicted using Landau theory [63] and it was proved that the instability of cholesteric phase increases at the temperature near the transition point depending on the increment in the planar helix structure. Alternatively, the temperature of the blue phase liquid crystal can be determined by the help of Meiboom's defect model depending on the Oseen-Frank elasticity equation [64]. The presence of the defect lines is essential for entity of the lattice structure in blue phase liquid crystals and the energy cost of the defects should be low enough to stabilize the entire phase for narrow range temperatures. Moreover, the free

where Fdiscl is the total free energy per unit length of the disclination, Fel is the elastic energy related the defect, Fsurf is the free energy at the disclination surface and Fint is the energy related to melting of area to the isotropic core. For blue phase double-twist cylinder lattices, the free energy calculations of Meiboom et al. comprise Fcore as the only temperature-

Fcore ¼ αð Þ Tiso � T πRo

where Tiso is the isotropic transition temperature, Ro is the defect core radius size and the difference in free energies of the isotropic and ordered phases at temperature T is represented

The surface energy at the interface between core and cholesteric is characterized by a surface

Fsurf can be turned into a surface integral and it is negligible and ignored, since surface terms do not scale competitively with the bulk terms. In that case the interior surface of the

Fdiscl ¼ Fcore þ FintFsurf þ Fel, (4)

Finterface ¼ 2σπRo (6)

<sup>2</sup> (5)

From a Chiral Molecule to Blue Phases http://dx.doi.org/10.5772/intechopen.70555 43

energy per unit length for the disclination line in BPs can be described as in Eq. (4):

dependent term:

Figure 7. Blue phase structures in (a) BPI and (b) BPII.

by α(Tiso�T).

tension, σ (Eq. (6));

and the pitch introduces in a scalar quantity of the free energy of cholesteric phases [44]:

$$F = \frac{1}{2} \left[ K\_{11} (\nabla.n)^2 + K\_{22} (n.\nabla \mathbf{x}n + q\_o)^2 + K\_{33} (n\mathbf{x} \nabla \mathbf{x}n)^2 \right] \tag{3}$$

where n is the director, K11 is splay elastic constant, K22 twist elastic constant and K33 is bend elastic constant. When the chirality of a material is high enough, in other words the pitch of the molecule is around 100 nm, another phase becomes energetically favorable, which is called blue phase with self-organized three-dimensional double twist structure [17].

### 3. Blue phases

Blue phases were first observed in 1888 by Reinitzer who noticed a brief hazy blue color that exhibited in the narrow temperature range between the chiral nematic (cholesteric) and the isotropic phases [17]. Blue phases are locally isotropic fluids. Moreover, the molecules are selforganized and complex three-dimensional (3D) structures and characterized by crystallographic space group symmetry in this kind of liquid crystal phase. The blue phases are generated by double-twisted cylinders separated by defect lines (Figure 6). Effectively, blue phase is classified by the network of the defect line and three network states are known as BPI, II and III as a function of increasing temperature. The Bravais lattice is body-centered and simple cubic for BPI and BPII, respectively, as shown in Figure 7 [59]. The BPI and BPII have soft, frequently coagulating platelet-small domains in a size of micrometer to submillimeter. The lattice constant which is around 100 nm depends on the radius of double-twisted helix and photonic band. This constant is mostly in the blue wavelength range and has the same order of magnitude as the cholesteric pitch. Additionally, the BPIII is called 'blue fog' since it has a cloudy and an amorphous appearance.

Moreover, Bragg scattering of BP which is the characteristics of the selective reflections of BPs have been comprehensively investigated since 1980 [59, 60]. Because of the exhibition of the BPs in a narrow temperature range, studies on BPs have been a challenge to the experimentalists

Figure 6. The simulation of a double twist cylinder structure of blue phase.

Figure 7. Blue phase structures in (a) BPI and (b) BPII.

However, qo is the pitch in the ground state given by Eq. (2) and it is induced in nematic liquid

and the pitch introduces in a scalar quantity of the free energy of cholesteric phases [44]:

<sup>2</sup> <sup>þ</sup> <sup>K</sup><sup>22</sup> <sup>n</sup>:∇xn <sup>þ</sup> qo

where n is the director, K11 is splay elastic constant, K22 twist elastic constant and K33 is bend elastic constant. When the chirality of a material is high enough, in other words the pitch of the molecule is around 100 nm, another phase becomes energetically favorable, which is called

Blue phases were first observed in 1888 by Reinitzer who noticed a brief hazy blue color that exhibited in the narrow temperature range between the chiral nematic (cholesteric) and the isotropic phases [17]. Blue phases are locally isotropic fluids. Moreover, the molecules are selforganized and complex three-dimensional (3D) structures and characterized by crystallographic space group symmetry in this kind of liquid crystal phase. The blue phases are generated by double-twisted cylinders separated by defect lines (Figure 6). Effectively, blue phase is classified by the network of the defect line and three network states are known as BPI, II and III as a function of increasing temperature. The Bravais lattice is body-centered and simple cubic for BPI and BPII, respectively, as shown in Figure 7 [59]. The BPI and BPII have soft, frequently coagulating platelet-small domains in a size of micrometer to submillimeter. The lattice constant which is around 100 nm depends on the radius of double-twisted helix and photonic band. This constant is mostly in the blue wavelength range and has the same order of magnitude as the cholesteric pitch. Additionally, the BPIII is called 'blue fog' since it has a

Moreover, Bragg scattering of BP which is the characteristics of the selective reflections of BPs have been comprehensively investigated since 1980 [59, 60]. Because of the exhibition of the BPs in a narrow temperature range, studies on BPs have been a challenge to the experimentalists

<sup>2</sup> h i

� �<sup>2</sup> <sup>þ</sup> <sup>K</sup>33ð Þ nx∇xn

qo ¼ 2π=p (2)

(3)

crystal where there are long-range distortions

<sup>F</sup> <sup>¼</sup> <sup>1</sup> 2

cloudy and an amorphous appearance.

Figure 6. The simulation of a double twist cylinder structure of blue phase.

3. Blue phases

K11ð Þ ∇:n

42 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

blue phase with self-organized three-dimensional double twist structure [17].

[61, 62]. Recently, it became possible to observe BPs in wide temperature range via the stabilization method. The thermodynamic stability of BPs which were composed of chiral nematic liquid crystal with a low chirality have been predicted using Landau theory [63] and it was proved that the instability of cholesteric phase increases at the temperature near the transition point depending on the increment in the planar helix structure. Alternatively, the temperature of the blue phase liquid crystal can be determined by the help of Meiboom's defect model depending on the Oseen-Frank elasticity equation [64]. The presence of the defect lines is essential for entity of the lattice structure in blue phase liquid crystals and the energy cost of the defects should be low enough to stabilize the entire phase for narrow range temperatures. Moreover, the free energy per unit length for the disclination line in BPs can be described as in Eq. (4):

$$\mathbf{F}\_{\text{discl}} = \mathbf{F}\_{\text{core}} + \mathbf{F}\_{\text{int}}\mathbf{F}\_{\text{surf}} + \mathbf{F}\_{\text{el}\nu} \tag{4}$$

where Fdiscl is the total free energy per unit length of the disclination, Fel is the elastic energy related the defect, Fsurf is the free energy at the disclination surface and Fint is the energy related to melting of area to the isotropic core. For blue phase double-twist cylinder lattices, the free energy calculations of Meiboom et al. comprise Fcore as the only temperaturedependent term:

$$\mathbf{F}\_{\text{core}} = \mathbf{a} (\mathbf{T}\_{\text{iso}} - \mathbf{T}) \,\, \pi \mathbf{R}\_{\text{o}} \,\, ^2 \tag{5}$$

where Tiso is the isotropic transition temperature, Ro is the defect core radius size and the difference in free energies of the isotropic and ordered phases at temperature T is represented by α(Tiso�T).

The surface energy at the interface between core and cholesteric is characterized by a surface tension, σ (Eq. (6));

$$\mathbf{F}\_{\text{interface}} = 2\sigma\pi\mathbf{R}\_0 \tag{6}$$

Fsurf can be turned into a surface integral and it is negligible and ignored, since surface terms do not scale competitively with the bulk terms. In that case the interior surface of the disclination must be taken into consideration and the solution covers the energy per unit length of the disclination line (Eq. (7)):

$$\mathbf{F}\_{\text{surf}} = -\pi (\mathbf{K}\_{22} + \mathbf{K}\_{24}) = -\pi \mathbf{K} \tag{7}$$

successfully polymerization of BPs in a wide temperature range [71] which is more than 50C [30]. Recently, Coles' group reported a study about the stabilization benefitted from the flexoelectric coupling between polar order and curvature of the director. Unfortunately, the report does not address the question of whether BPI appears at 16.5C on heating from the smectic phase. Therefore, thermodynamic stability of BPI were not clearly indicated [53, 72, 73]. Alternatively, Yoshizawa et al. [74] successfully manage to extent the temperature range of the BPs more than 10C using chiral T-shaped compounds. Yelamaggad et al. [75] were able to

From a Chiral Molecule to Blue Phases http://dx.doi.org/10.5772/intechopen.70555 45

Wang et al. [76] both introduced BPs in a wide temperature range using ZnS nanoparticles and showed the stability of the cubic structures against the electrical field. Recent studies on BPs with an broadened temperature range make them more attractive for applications because of some specific electro-optical (E-O) properties of BPs, such as fast response time [31], wide viewing angle and also any surface treatments are not necessary for the BPs. Moreover, Kemiklioglu et al. stabilized the cholesteric blue phases using polymerizable silicon-based nanoparticles to expand the temperature range of BPS. They showed that these polymerizable nanoparticles help to modify the interfacial properties of disclination cores broadening the blue phase temperature range and also the polymer concentration plays an important role in the thermodynamic stability of modulated liquid crystal blue phases. They also reported inorganic polymer leads to significant reduction in the switching voltage from about 140 to 40 V in corresponding device as a

stabilize the BP more than 20C using chemically linked bent core molecules.

result of the low surface energy property of the inorganic polymers [72, 73, 77, 20].

Nanoparticles [63, 64, 30, 72, 73] and polymer-modified carbon nanotubes (CNTs) [67] are emerging as new classes of nanoscaled materials and have become the subjects of extensive research because of their potential in improving the mechanical, electrical and thermal features of composite materials. Recently, with new approaches, such as doping MgO [68], ZnS [69] and CNTs into LCs, it has been possible to overcome the limitations of the transition temperature

Carbon nanotubes are not only anisotropic but also metallic or semiconducting nanoparticles based on the diameter and helicity of the carbon rings [78]. Moreover, they categorized into two different morphologies namely, single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes (MWCNTs). Moreover, the typical length of SWCNT which changes from submicron to microns is an important parameter for the determination of tensile strength of SWCNTs since they show exceptional tensile strength depending on their high aspect ratio and rigidity. In addition, the diameter in the range from 0.5 to 2 nm leads a high aspect ratio of tubes [53, 79]. Besides, MWCNTs show the similar electronic behaviors with those of SWCNTs

Carbon nanotubes have became an important research topic for the liquid crystal scientists after their discovery by Iijima in 1991 [80] because of the extraordinary electrical properties and strong interactions of the CNTs with the mesogenic units of liquid crystals [81]. Recently, different groups have reported studies on the alignment and characterization of CNTs in

3.2. Carbon nanotubes doping for the stabilization of blue phases

range and physical properties of LCs [53].

due to weak coupling between cylinders of them.

Fel is the elastic energy where K is the elastic constant and Rmax is the radius of the double twist cylinder and Ro is the defect core radius.

$$F\_{el} = \frac{1}{4}\pi K \ln\left(\frac{R\_{max}}{R\_o}\right) \tag{8}$$

According to Eq. (8), one parameter in this equation must minimize the energy of cost of the disclination line in BPs to expand the BP temperature range. It is expected to move the isotropic particles, such as nanoparticles or monomers, towards isotropic areas of liquid crystals in order to minimize the core energy. The addition of these nanoparticles into an isotropic phase of sample and cooling to the BP give rise to an aggregation of these nanoparticles in the defect lines. These nanoparticles will interrupt any inclination towards orientational order inside the core when temperatures decreased into the blue phase. However, the surface energy at the interface between core and cholesteric was assumed zero during the energy minimization of the system [53].

#### 3.1. Stabilization of blue phases

Blue phase liquid crystals have a great potential for various applications due to their electrooptical properties, such as fast response time, wide and symmetric viewing angle and lack of requirement of any surface alignment layer. However, BPs have limited usage in the practical applications because of their narrow temperature range [18, 65, 66]. Recently, two independently reported methods to expand the BP temperature range have a great attraction to blue phase materials, which have become a hot topic of comprehensive research in exploiting applications in new optics, photonics and information displays based on the outstanding electro-optical properties of BPLCs. The first reported approach uses a tiny amount of monomer for polymerization and it has been reported polymer stabilization which helps to expand the BP temperature range to more than 60 K including room temperature with an ultrafast response time [18]. Kikuchi et al. [67–69] developed a technique to extent the BP temperature range with a polymerized polymer network, denoted as the polymer-stabilized blue phase (PSBP). The synchrotron small angle X-ray scattering measurements exhibited that polymers are selectively concentrated in the disclination cores and a remarkably unique accumulation structure in the PSBP [70]. This result evidently conforms the mechanism of the stabilizing effect of BPI originating from the immobilization of the disclination in the blue phase by polymers. The first method proposed that the polymer network which is concentrated not only in the isotropic defect core but also in the disclination core of BP causes an increment in the temperature range of BP. Therefore, cross-linked network of the polymer which was produced by the process of in-situ polymerization blocked the molecular reorientation of liquid crystal directors [18]. The latter approach reported the usage of the nematic bimesogenic liquid crystal mixtures to stabilize the defect structures in the blue phase. This method provided a successfully polymerization of BPs in a wide temperature range [71] which is more than 50C [30]. Recently, Coles' group reported a study about the stabilization benefitted from the flexoelectric coupling between polar order and curvature of the director. Unfortunately, the report does not address the question of whether BPI appears at 16.5C on heating from the smectic phase. Therefore, thermodynamic stability of BPI were not clearly indicated [53, 72, 73].

Alternatively, Yoshizawa et al. [74] successfully manage to extent the temperature range of the BPs more than 10C using chiral T-shaped compounds. Yelamaggad et al. [75] were able to stabilize the BP more than 20C using chemically linked bent core molecules.

Wang et al. [76] both introduced BPs in a wide temperature range using ZnS nanoparticles and showed the stability of the cubic structures against the electrical field. Recent studies on BPs with an broadened temperature range make them more attractive for applications because of some specific electro-optical (E-O) properties of BPs, such as fast response time [31], wide viewing angle and also any surface treatments are not necessary for the BPs. Moreover, Kemiklioglu et al. stabilized the cholesteric blue phases using polymerizable silicon-based nanoparticles to expand the temperature range of BPS. They showed that these polymerizable nanoparticles help to modify the interfacial properties of disclination cores broadening the blue phase temperature range and also the polymer concentration plays an important role in the thermodynamic stability of modulated liquid crystal blue phases. They also reported inorganic polymer leads to significant reduction in the switching voltage from about 140 to 40 V in corresponding device as a result of the low surface energy property of the inorganic polymers [72, 73, 77, 20].

### 3.2. Carbon nanotubes doping for the stabilization of blue phases

disclination must be taken into consideration and the solution covers the energy per unit

Fel is the elastic energy where K is the elastic constant and Rmax is the radius of the double twist

According to Eq. (8), one parameter in this equation must minimize the energy of cost of the disclination line in BPs to expand the BP temperature range. It is expected to move the isotropic particles, such as nanoparticles or monomers, towards isotropic areas of liquid crystals in order to minimize the core energy. The addition of these nanoparticles into an isotropic phase of sample and cooling to the BP give rise to an aggregation of these nanoparticles in the defect lines. These nanoparticles will interrupt any inclination towards orientational order inside the core when temperatures decreased into the blue phase. However, the surface energy at the interface between core and cholesteric was assumed zero during the

Blue phase liquid crystals have a great potential for various applications due to their electrooptical properties, such as fast response time, wide and symmetric viewing angle and lack of requirement of any surface alignment layer. However, BPs have limited usage in the practical applications because of their narrow temperature range [18, 65, 66]. Recently, two independently reported methods to expand the BP temperature range have a great attraction to blue phase materials, which have become a hot topic of comprehensive research in exploiting applications in new optics, photonics and information displays based on the outstanding electro-optical properties of BPLCs. The first reported approach uses a tiny amount of monomer for polymerization and it has been reported polymer stabilization which helps to expand the BP temperature range to more than 60 K including room temperature with an ultrafast response time [18]. Kikuchi et al. [67–69] developed a technique to extent the BP temperature range with a polymerized polymer network, denoted as the polymer-stabilized blue phase (PSBP). The synchrotron small angle X-ray scattering measurements exhibited that polymers are selectively concentrated in the disclination cores and a remarkably unique accumulation structure in the PSBP [70]. This result evidently conforms the mechanism of the stabilizing effect of BPI originating from the immobilization of the disclination in the blue phase by polymers. The first method proposed that the polymer network which is concentrated not only in the isotropic defect core but also in the disclination core of BP causes an increment in the temperature range of BP. Therefore, cross-linked network of the polymer which was produced by the process of in-situ polymerization blocked the molecular reorientation of liquid crystal directors [18]. The latter approach reported the usage of the nematic bimesogenic liquid crystal mixtures to stabilize the defect structures in the blue phase. This method provided a

<sup>π</sup>Kln Rmax Ro 

Fel <sup>¼</sup> <sup>1</sup> 4

Fsurf ¼ �πð Þ¼� K22 þ K24 πK (7)

(8)

length of the disclination line (Eq. (7)):

44 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

cylinder and Ro is the defect core radius.

energy minimization of the system [53].

3.1. Stabilization of blue phases

Nanoparticles [63, 64, 30, 72, 73] and polymer-modified carbon nanotubes (CNTs) [67] are emerging as new classes of nanoscaled materials and have become the subjects of extensive research because of their potential in improving the mechanical, electrical and thermal features of composite materials. Recently, with new approaches, such as doping MgO [68], ZnS [69] and CNTs into LCs, it has been possible to overcome the limitations of the transition temperature range and physical properties of LCs [53].

Carbon nanotubes are not only anisotropic but also metallic or semiconducting nanoparticles based on the diameter and helicity of the carbon rings [78]. Moreover, they categorized into two different morphologies namely, single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes (MWCNTs). Moreover, the typical length of SWCNT which changes from submicron to microns is an important parameter for the determination of tensile strength of SWCNTs since they show exceptional tensile strength depending on their high aspect ratio and rigidity. In addition, the diameter in the range from 0.5 to 2 nm leads a high aspect ratio of tubes [53, 79]. Besides, MWCNTs show the similar electronic behaviors with those of SWCNTs due to weak coupling between cylinders of them.

Carbon nanotubes have became an important research topic for the liquid crystal scientists after their discovery by Iijima in 1991 [80] because of the extraordinary electrical properties and strong interactions of the CNTs with the mesogenic units of liquid crystals [81]. Recently, different groups have reported studies on the alignment and characterization of CNTs in nematic liquid crystals [82–86] as well as on the dielectric [87–89] and electro-optical properties [90, 91] of CNTs [93]. Different textures of CNTs were observed when the nematic LC droplets were embedded in a polymer matrix medium [53, 80, 92]. However, carbon nanotubes have been studied in blue phase liquid crystals to expand the temperature range of the blue phases depending on thermal stability of carbon nanotubes. Another group studied CNT-doped polymer-stabilized blue phase (PSBP) LC. The CNT-doped PSBP mixtures exhibit a good thermal stability in a wide BP temperature range which is more than 40C. They reported that BP temperature range and electro-optical properties, such as Kerr constant, switching voltages and response times of the PSBP LCs were able to improved when a mixture of monomer and BPLCs doped with CNT [93].

experiments. This low surface energy leads a significant decrement in the switching voltage from 140 to 40 V [19, 20, 72, 73]. The significant reduction in the switching voltage and widening of blue phase temperature range are useful for new electro-optical applications [22, 53]. Besides, Coles et al. reported that they managed to expand the blue phase temperature range to over 50�C by using the nematic bimesogenic liquid crystal mixtures to stabilize the defect structures of the blue phase [30]. However, a three-dimensional polymer network is formed by the reaction of benzoyl radicals with the double bonds of the diacrylate monomer through a chain reaction [53, 101]. Furthermore, molecular mobility of the network polymers obtained in the PSBP affects the stability of PSBP [53, 102]. Additionally, the electro-optic properties can be improved due to the variation of the flexibility of the molecule, the length of the rigid core and the polymerizable functional group of reactive monomer [53, 103]. All these studies showed that either thermally polymerizable [101] or photopolymerizable [18, 19, 32, 100–105] monomers can be used in the polymer stabilization of BP liquid crystals and PSBP liquid crystals have some advantages [31, 104–111]. PSBP liquid crystals become an attractive material as a next generation display technology [109–111] since, these materials have submillisecond response time, and wide viewing angle and also they do not need any surface alignment [53]. On the other hand, high operating voltage, and a low-contrast ratio due to residual birefringence and hysteresis [107, 108] are some disadvantages of PSBP liquid crystals which limits the wide-spread applications of them. There are two common approaches in order to overcome the issue of high operating voltage, there are two common approaches. One of these approaches is using a small electrode gap to produce a strong electric field [53, 110, 111]. The second approach is electric-fieldinduced birefringence known as the Kerr effect [53, 71]. Kerr effect is an electro-optical behavior of blue phases. Blue phase liquid crystals not have birefringence in the absence of the electric field and Kerr effect can be explained as the induced birefringence occurrence in the presence of the electrical field. PSBP liquid crystal with large Kerr constant was reported by Kikuchi et al.

[107] and Wu et al. [111, 112–114]. Kerr effect can be calculated using Eq. (9)

gence is linearly proportional to E<sup>2</sup>

3.4. Polymer dispersion of blue phases

valid only in the low field region [53, 111, 112].

where K is the Kerr constant, λ is the probe wavelength, and E (= V/l, where V is the applied voltage and l is the distance between electrodes) is the applied electric field. Induced birefrin-

Polymer-dispersed liquid crystals (PDLCs) have become the topic of considerable interest during the last decades, because of their potential applications such a smart windows, flexible displays, projection displays and holographic gratings [115–119]. The PDLC films have been widely studied as a candidate for the large area display because of the simplification of the preparation process and because their light transmittance is higher than conventional LCs in the absence of polarizer by the reason of their light scattering nature [120–126]. PDLC films are a mixed phase of micron-sized liquid crystal droplets, which are randomly dispersed inside a polymer matrix [127]. In general, the polymer weight concentration is between 30 and 60%

<sup>Δ</sup>n Eð Þ¼ <sup>λ</sup>KE<sup>2</sup> (9)

From a Chiral Molecule to Blue Phases http://dx.doi.org/10.5772/intechopen.70555 47

, where E is the electric field and this linear relationship is

Moreover, the electro-optical behaviors of liquid crystals were studied as a function of the addition of carbon nanotubes. Several liquid crystal textures were observed in the CNT-doped liquid crystal mixture based on the field-induced movement of CNTs inside nematic liquid crystal by applying a high electric field [94–96]. Furthermore, it has also been demonstrated that the rising time of CNT-doped nematic liquid crystals leads a decrement in the threshold voltage both of the twisted nematic and in-plane switching cells with the dispersion of a very small concentration of CNT dopant [90, 91, 97, 98]. The addition of CNTs in the optical controlled birefringence cells lead to a fast response time due to the increment in anchoring energy of the alignment layer by CNT doping [53, 99].

#### 3.3. Polymer stabilization of blue phases

Although BPLCs have the potential for various applications because of their electro-optical properties, such as fast response time, wide and symmetric viewing angle, the narrow temperature range of BPs is still one of the main limitations for their practical applications [18, 65, 66]. Therefore, stabilization of blue phase (PSBP) liquid crystals via polymerization have been studied commonly for two decades [18, 19, 32, 100–105] because of their great potential for use in display devices or as an optical modulator.

Recently, many studies on the increment in the BP temperature range via polymer stabilization methods have drawn attention to blue phase materials, which have thereafter become a subject of extensive research in exploiting applications in new optics, photonics and information displays because of the outstanding electro-optical properties of BPLCs [53]. One of these studies include a method which uses a small amount of polymer for polymerization that is phaseseparated to the defects of the blue phase based on the concept of the polymer-stabilized liquid crystal the orientation of liquid crystal directors can be stabilized by a crosslinked network dispersed in a liquid crystal [18]. Moreover, the polymer network plays a fundamental role in the increment of temperature range, causing the thermodynamic stabilization of BP [18, 19, 32, 100]. Furthermore, another study explored that the stabilization of cholesteric blue phases using polymerizable silicon-based nanoparticles to modify the interfacial properties of disclination cores and broaden the blue phase temperature range. This study showed that the polymer concentration has an important effect on the thermodynamic stability behaviors of modulated liquid crystal blue phases. There was a significant reduction in the switching voltage of the device as a result of the low surface energy property of the inorganic polymer used in the experiments. This low surface energy leads a significant decrement in the switching voltage from 140 to 40 V [19, 20, 72, 73]. The significant reduction in the switching voltage and widening of blue phase temperature range are useful for new electro-optical applications [22, 53]. Besides, Coles et al. reported that they managed to expand the blue phase temperature range to over 50�C by using the nematic bimesogenic liquid crystal mixtures to stabilize the defect structures of the blue phase [30]. However, a three-dimensional polymer network is formed by the reaction of benzoyl radicals with the double bonds of the diacrylate monomer through a chain reaction [53, 101]. Furthermore, molecular mobility of the network polymers obtained in the PSBP affects the stability of PSBP [53, 102]. Additionally, the electro-optic properties can be improved due to the variation of the flexibility of the molecule, the length of the rigid core and the polymerizable functional group of reactive monomer [53, 103]. All these studies showed that either thermally polymerizable [101] or photopolymerizable [18, 19, 32, 100–105] monomers can be used in the polymer stabilization of BP liquid crystals and PSBP liquid crystals have some advantages [31, 104–111]. PSBP liquid crystals become an attractive material as a next generation display technology [109–111] since, these materials have submillisecond response time, and wide viewing angle and also they do not need any surface alignment [53]. On the other hand, high operating voltage, and a low-contrast ratio due to residual birefringence and hysteresis [107, 108] are some disadvantages of PSBP liquid crystals which limits the wide-spread applications of them. There are two common approaches in order to overcome the issue of high operating voltage, there are two common approaches. One of these approaches is using a small electrode gap to produce a strong electric field [53, 110, 111]. The second approach is electric-fieldinduced birefringence known as the Kerr effect [53, 71]. Kerr effect is an electro-optical behavior of blue phases. Blue phase liquid crystals not have birefringence in the absence of the electric field and Kerr effect can be explained as the induced birefringence occurrence in the presence of the electrical field. PSBP liquid crystal with large Kerr constant was reported by Kikuchi et al. [107] and Wu et al. [111, 112–114]. Kerr effect can be calculated using Eq. (9)

$$
\Delta n(E) = \lambda KE^2 \tag{9}
$$

where K is the Kerr constant, λ is the probe wavelength, and E (= V/l, where V is the applied voltage and l is the distance between electrodes) is the applied electric field. Induced birefringence is linearly proportional to E<sup>2</sup> , where E is the electric field and this linear relationship is valid only in the low field region [53, 111, 112].

#### 3.4. Polymer dispersion of blue phases

nematic liquid crystals [82–86] as well as on the dielectric [87–89] and electro-optical properties [90, 91] of CNTs [93]. Different textures of CNTs were observed when the nematic LC droplets were embedded in a polymer matrix medium [53, 80, 92]. However, carbon nanotubes have been studied in blue phase liquid crystals to expand the temperature range of the blue phases depending on thermal stability of carbon nanotubes. Another group studied CNT-doped polymer-stabilized blue phase (PSBP) LC. The CNT-doped PSBP mixtures exhibit a good thermal stability in a wide BP temperature range which is more than 40C. They reported that BP temperature range and electro-optical properties, such as Kerr constant, switching voltages and response times of the PSBP LCs were able to improved when a mixture of monomer and

Moreover, the electro-optical behaviors of liquid crystals were studied as a function of the addition of carbon nanotubes. Several liquid crystal textures were observed in the CNT-doped liquid crystal mixture based on the field-induced movement of CNTs inside nematic liquid crystal by applying a high electric field [94–96]. Furthermore, it has also been demonstrated that the rising time of CNT-doped nematic liquid crystals leads a decrement in the threshold voltage both of the twisted nematic and in-plane switching cells with the dispersion of a very small concentration of CNT dopant [90, 91, 97, 98]. The addition of CNTs in the optical controlled birefringence cells lead to a fast response time due to the increment in anchoring

Although BPLCs have the potential for various applications because of their electro-optical properties, such as fast response time, wide and symmetric viewing angle, the narrow temperature range of BPs is still one of the main limitations for their practical applications [18, 65, 66]. Therefore, stabilization of blue phase (PSBP) liquid crystals via polymerization have been studied commonly for two decades [18, 19, 32, 100–105] because of their great potential for

Recently, many studies on the increment in the BP temperature range via polymer stabilization methods have drawn attention to blue phase materials, which have thereafter become a subject of extensive research in exploiting applications in new optics, photonics and information displays because of the outstanding electro-optical properties of BPLCs [53]. One of these studies include a method which uses a small amount of polymer for polymerization that is phaseseparated to the defects of the blue phase based on the concept of the polymer-stabilized liquid crystal the orientation of liquid crystal directors can be stabilized by a crosslinked network dispersed in a liquid crystal [18]. Moreover, the polymer network plays a fundamental role in the increment of temperature range, causing the thermodynamic stabilization of BP [18, 19, 32, 100]. Furthermore, another study explored that the stabilization of cholesteric blue phases using polymerizable silicon-based nanoparticles to modify the interfacial properties of disclination cores and broaden the blue phase temperature range. This study showed that the polymer concentration has an important effect on the thermodynamic stability behaviors of modulated liquid crystal blue phases. There was a significant reduction in the switching voltage of the device as a result of the low surface energy property of the inorganic polymer used in the

BPLCs doped with CNT [93].

energy of the alignment layer by CNT doping [53, 99].

46 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

3.3. Polymer stabilization of blue phases

use in display devices or as an optical modulator.

Polymer-dispersed liquid crystals (PDLCs) have become the topic of considerable interest during the last decades, because of their potential applications such a smart windows, flexible displays, projection displays and holographic gratings [115–119]. The PDLC films have been widely studied as a candidate for the large area display because of the simplification of the preparation process and because their light transmittance is higher than conventional LCs in the absence of polarizer by the reason of their light scattering nature [120–126]. PDLC films are a mixed phase of micron-sized liquid crystal droplets, which are randomly dispersed inside a polymer matrix [127]. In general, the polymer weight concentration is between 30 and 60% [127]. However, it was known four different methods for the fabrication of PDLCs. The first one of these four methods is encapsulation (emulsification) which includes the liquid crystal inside an aqueous solution of film-forming polymer. The second one is called the solventinduced phase separation (SIPS) which is used to dissolve the liquid crystal and thermoplastic polymer and forms a single phase by evaporating the solvent at a certain rate. The third method is known as thermally induced phase separation (TIPS) which includes the heating of the liquid crystal and thermoplastic polymer to obtain a single phase. After heating procedure, the liquid crystal phase separates into droplets with the cooling of mixture at a controlled rate. The fourth method is polymerization-induced phase separation (PIPS). This method contains the liquid crystal that is dissolved into the monomer [93] and uses ultraviolet radiation to initiate the free radical polymerization of monomers [127]. One of the main advantages of this method is the possibility to form a composite directly between two glass substrates coated with indium-tin-oxide (ITO) without any requirement of laminating procedure. The above methods produce a wide size distribution of liquid crystal domain size [127]. PDLCs are operated based on the micron-sized LC droplet dispersion inside the polymer matrix and the scattering performance of the PDLC film is determined by the LC droplet size. The operation principle of the PDLC films—electrically switchable between light scattering and transparent states or vice versa depends on the refractive indices matching between guest and host materials [128, 129]. The PDLC films normally seems milky since the randomly orientation of LC molecules inside the droplets causes a scattered light at zero voltage. As a function of an applying a voltage across the PDLC film, the LC directors align in the direction parallel to the applied field. Due to matching in indices of refraction between polymer and liquid crystal molecules under the electric field, PDLC film becomes transparent at normal viewing direction. Additionally, H-PDLC, which is the another type of PDLC, includes liquid crystal droplets smaller than that of PDLC [130] and they are staged in varying planes in accordance with the polymer. There are two modes of H-PDLCs, which are called transmissive and reflective. In the transmissive mode, diffraction occurs by an applying voltage and light is reflected in the absence of electric field. In the reflective mode, light is reflected in the absence of electric field, with the applying voltage it transmits through the display.

known as polymerization-induced phase separation (PIPS) containing the liquid crystal, monomer and a small amount of catalyst. After exposing the prepolymer mixture to an external stimulus, for example, light or heat, the monomer gels into a polymer matrix and

From a Chiral Molecule to Blue Phases http://dx.doi.org/10.5772/intechopen.70555 49

The operation principle of the PDLC films based on the electrical switching between light scattering and transparent states due to index matching between guest and host materials [115, 132, 127]. The PDLC films normally appear milky and scatter incident ambient light because the LC molecules orient randomly inside of the droplets in the absence of voltage. With the applying a voltage across the PDLC film, the LC directors align in the direction parallel to the applied field. As a result of index matching between polymer and LC molecules in the presence of the electric field, the PDLC film becomes transparent when viewed along the normal direction. These PDLC films have significant advantages for electro-optical device applications, since PDLCs do not require any polarizers also PDLCs have the property of high light transmittance [127]. A number of reports have appeared recently suggested application areas for PDLCs ranging from switchable light modulators [71, 133], smart windows [127], information displays [134] and holographically formed optical elements and devices [127–135]. The electro-optic properties of PDLC devices, such as displays and smart windows can be improved by using BPLCs. The polymer dispersed or encapsulated blue phase liquid crystal films have many advantages when compared to that of polymer dispersed or encapsulated nematic liquid crystals [33–35, 53]. One of these advantages of BPLCs is field-induced birefringence due to their submillisecond response time, which is at least one order of magnitude faster than the present nematic LC-based displays [53]. BPLCs do not require any surface alignment layer; thus, the device fabrication process is greatly simplified [53]. Another significant advantage of BPLCs is their wide and symmetric viewing angle due to the fact that their 'voltage off' state is optically isotropic and the 'voltage on' state forms multidomain structures [53, 108, 109]. Moreover, BPLC can be a substantial candidate for polymer encapsulated LC films due to their fast switching properties [53]. Due to all these advantages of BPLCs, polymer encapsulated blue phase liquid crystal films are a strong candidate for the next generation of displays and spatial light modulators due to their optical properties [22, 31, 32] and also these

films have potential for advanced applications in displays and photonic devices [53].

Encapsulation is one of the major methods used in the fabrication of PDLC films [53]. This method includes emulsion-based PDLC films which are formed of small liquid crystal droplets inside the aqueous solution of water soluble polymer [136, 137] or a colloidal suspension of a water insoluble polymer [130, 138]. Moreover, the energy is required to form the encapsulated droplets. This required energy which generally arises from the chemical potential of components or from mechanical devices break-up the droplets. The emulsion system is obtained by high shear; for example, by ultrasonication or high-pressure homogenizers and the rate of solidification and polymer solubility play a role in the yield. Microspheres formed by rapid solidification of the polymer may give a higher yield due to encapsulation of some of the soluble fractions in the matrix [53, 140, 141]. Besides, the emulsion is coated on a conductive

3.5. Polymer encapsulated blue phase liquid crystals

liquid crystal phase separates into droplets.

Polymer-dispersed liquid crystals (PDLCs) which are a class of important electro-optical (E-O) materials comprise of dispersions of micron-sized LC droplets inside a polymer matrix which were discovered by Fergason [127, 130]. The PDLC films were fabricated by using different methods, such as solvent evaporation, thermal induction or polymerization-induced phase separation [131]. The first method used in the preparation of PDLC films is the encapsulation (emulsification) of the liquid crystal inside an aqueous solution of film-forming polymer [130]. After water evaporated at a certain rate to induce phase separation, the film is laminated between two conductive electrode coated substrates. The second method which is called the solvent-induced phase separation (SIPS) includes solvent which is used to dissolve the liquid crystal and thermoplastic polymer and create a single phase. The certain solvent evaporation rate induces the phase separation. The third method is the thermally induced phase separation (TIPS). This method uses liquid crystal and thermoplastic polymer. These two phases are heated to obtain a melting and then mixed to form a single phase. Liquid crystal phase separates into droplets with cooling of the mixture at a controlled rate. The fourth method is known as polymerization-induced phase separation (PIPS) containing the liquid crystal, monomer and a small amount of catalyst. After exposing the prepolymer mixture to an external stimulus, for example, light or heat, the monomer gels into a polymer matrix and liquid crystal phase separates into droplets.

The operation principle of the PDLC films based on the electrical switching between light scattering and transparent states due to index matching between guest and host materials [115, 132, 127]. The PDLC films normally appear milky and scatter incident ambient light because the LC molecules orient randomly inside of the droplets in the absence of voltage. With the applying a voltage across the PDLC film, the LC directors align in the direction parallel to the applied field. As a result of index matching between polymer and LC molecules in the presence of the electric field, the PDLC film becomes transparent when viewed along the normal direction. These PDLC films have significant advantages for electro-optical device applications, since PDLCs do not require any polarizers also PDLCs have the property of high light transmittance [127]. A number of reports have appeared recently suggested application areas for PDLCs ranging from switchable light modulators [71, 133], smart windows [127], information displays [134] and holographically formed optical elements and devices [127–135].

The electro-optic properties of PDLC devices, such as displays and smart windows can be improved by using BPLCs. The polymer dispersed or encapsulated blue phase liquid crystal films have many advantages when compared to that of polymer dispersed or encapsulated nematic liquid crystals [33–35, 53]. One of these advantages of BPLCs is field-induced birefringence due to their submillisecond response time, which is at least one order of magnitude faster than the present nematic LC-based displays [53]. BPLCs do not require any surface alignment layer; thus, the device fabrication process is greatly simplified [53]. Another significant advantage of BPLCs is their wide and symmetric viewing angle due to the fact that their 'voltage off' state is optically isotropic and the 'voltage on' state forms multidomain structures [53, 108, 109]. Moreover, BPLC can be a substantial candidate for polymer encapsulated LC films due to their fast switching properties [53]. Due to all these advantages of BPLCs, polymer encapsulated blue phase liquid crystal films are a strong candidate for the next generation of displays and spatial light modulators due to their optical properties [22, 31, 32] and also these films have potential for advanced applications in displays and photonic devices [53].

### 3.5. Polymer encapsulated blue phase liquid crystals

[127]. However, it was known four different methods for the fabrication of PDLCs. The first one of these four methods is encapsulation (emulsification) which includes the liquid crystal inside an aqueous solution of film-forming polymer. The second one is called the solventinduced phase separation (SIPS) which is used to dissolve the liquid crystal and thermoplastic polymer and forms a single phase by evaporating the solvent at a certain rate. The third method is known as thermally induced phase separation (TIPS) which includes the heating of the liquid crystal and thermoplastic polymer to obtain a single phase. After heating procedure, the liquid crystal phase separates into droplets with the cooling of mixture at a controlled rate. The fourth method is polymerization-induced phase separation (PIPS). This method contains the liquid crystal that is dissolved into the monomer [93] and uses ultraviolet radiation to initiate the free radical polymerization of monomers [127]. One of the main advantages of this method is the possibility to form a composite directly between two glass substrates coated with indium-tin-oxide (ITO) without any requirement of laminating procedure. The above methods produce a wide size distribution of liquid crystal domain size [127]. PDLCs are operated based on the micron-sized LC droplet dispersion inside the polymer matrix and the scattering performance of the PDLC film is determined by the LC droplet size. The operation principle of the PDLC films—electrically switchable between light scattering and transparent states or vice versa depends on the refractive indices matching between guest and host materials [128, 129]. The PDLC films normally seems milky since the randomly orientation of LC molecules inside the droplets causes a scattered light at zero voltage. As a function of an applying a voltage across the PDLC film, the LC directors align in the direction parallel to the applied field. Due to matching in indices of refraction between polymer and liquid crystal molecules under the electric field, PDLC film becomes transparent at normal viewing direction. Additionally, H-PDLC, which is the another type of PDLC, includes liquid crystal droplets smaller than that of PDLC [130] and they are staged in varying planes in accordance with the polymer. There are two modes of H-PDLCs, which are called transmissive and reflective. In the transmissive mode, diffraction occurs by an applying voltage and light is reflected in the absence of electric field. In the reflective mode, light is reflected in the absence of electric field,

48 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

Polymer-dispersed liquid crystals (PDLCs) which are a class of important electro-optical (E-O) materials comprise of dispersions of micron-sized LC droplets inside a polymer matrix which were discovered by Fergason [127, 130]. The PDLC films were fabricated by using different methods, such as solvent evaporation, thermal induction or polymerization-induced phase separation [131]. The first method used in the preparation of PDLC films is the encapsulation (emulsification) of the liquid crystal inside an aqueous solution of film-forming polymer [130]. After water evaporated at a certain rate to induce phase separation, the film is laminated between two conductive electrode coated substrates. The second method which is called the solvent-induced phase separation (SIPS) includes solvent which is used to dissolve the liquid crystal and thermoplastic polymer and create a single phase. The certain solvent evaporation rate induces the phase separation. The third method is the thermally induced phase separation (TIPS). This method uses liquid crystal and thermoplastic polymer. These two phases are heated to obtain a melting and then mixed to form a single phase. Liquid crystal phase separates into droplets with cooling of the mixture at a controlled rate. The fourth method is

with the applying voltage it transmits through the display.

Encapsulation is one of the major methods used in the fabrication of PDLC films [53]. This method includes emulsion-based PDLC films which are formed of small liquid crystal droplets inside the aqueous solution of water soluble polymer [136, 137] or a colloidal suspension of a water insoluble polymer [130, 138]. Moreover, the energy is required to form the encapsulated droplets. This required energy which generally arises from the chemical potential of components or from mechanical devices break-up the droplets. The emulsion system is obtained by high shear; for example, by ultrasonication or high-pressure homogenizers and the rate of solidification and polymer solubility play a role in the yield. Microspheres formed by rapid solidification of the polymer may give a higher yield due to encapsulation of some of the soluble fractions in the matrix [53, 140, 141]. Besides, the emulsion is coated on a conductive substrate before allowing the water to evaporate to produce thin polymer films containing liquid crystal droplets dispersed in a matrix. Evaporation process is one of the key factors for the droplet shape deformation which affects the alignment of LC inside the film cavities, which has a significant effect on the physical properties of PDLC films. Droplets become spherical or oblate in the polymer film [71, 120, 127, 132–145] depending on the evaporation process. However, the size distribution of the liquid crystal droplets in the emulsion can be modified by the preparation process and materials used to produce the emulsion, for example, the stirring time and speed, viscosities of polymer and liquid crystal. With increase in time of mixing in an ultrasonic cleaner, the droplet size of emulsion decreases [53, 146]. Moreover, surfactant type and concentration are the other key factors which affect the size, stability, and polydispersity of the droplets. The droplet size and polydispersity index decreased with increase in surfactant concentration [53]. The size and size distribution of encapsulated LC droplets can have a significant effect on the electro-optical properties of the films. Large area applicability of the emulsion system enhances the range of useful applications of PDLCs, ranging from switchable light modulators [71, 133], smart Windows [127] and information displays [134], as well as holographically formed optical elements and devices [135, 137, 142–144]. In PDLC systems, LC droplets are dispersed in a polymer film and these LCs can be oriented in the polymer droplets leading a switching from scattering to transparent states or vice versa with an applying electric field. There is a mismatching of refractive indices in the field-off state and these refractive indices of LC and the polymer match in the field-on state [53, 115, 127, 132–134, 146]. This phenomenon gives the electro-optical performance of the corresponding device. This electro-optical performance of displays and smart windows can be improved by replacing the nematic LCs with BPLCs, and the dispersed or encapsulated BPLCs leads a development in the original optical and E-O properties with an external electical field [53].

composite materials of polymer-dispersed blue phase (PDBP) LCs were studied by combining PDLC films and BPLCs using two preparation methods, that is, polymer encapsulation and polymer stabilization in order to fabricate polymer encapsulated–polymer-stabilized bluephase (PEPSBP) LC droplets[33]. Encapsulated droplets were stabilized via the polymerization of reactive monomers after they were produced in a polyvinyl alcohol solution by emulsification. It was reported that polymer stabilized droplets caused an expansion of the BP temperature range from 53 to below 0�C. Moreover, this study concluded that low switching voltage and fast response time based on the decrement in the interfacial energy of polymer encapsulated and stabilized BPLC droplets. Furthermore, stabilization and positions of droplets in the

From a Chiral Molecule to Blue Phases http://dx.doi.org/10.5772/intechopen.70555 51

Kemiklioglu et al. firstly demonstrated the polymer-dispersed blue-phase liquid-crystal films between two indium-tin-oxide-coated conductive substrates by switching between light scat-

They experimentally investigated the photoinitiator effect on the electro-optical properties of the polymer-dispersed blue phase liquid crystals as well as the ratio between the crosslinking agent and the monomer. They showed that the increasing monomer concentration reduces the switching voltage of the corresponding device. Moreover, the increment of the monomer concentration in the polymer-dispersed blue phase liquid crystal samples leads an increment in the contrast ratio. All these significant improvements in the electro-optical properties of polymer-dispersed blue-phase liquid crystal devices are promising for new electro-optical

Department of Bioengineering, Faculty of Engineering, Manisa Celal Bayar University, Manisa,

[2] Streyer L. Chapter 10, Introduction to Biological Membranes Biochemistry. New York:

[5] Lehmann O. Uber flissende Kristalle. The Journal of Physical Chemistry. 1889;4:462

tering and transparent states with applying an electric field across the films [35].

aqueous phase have a great attraction for the researchers [142–146].

Address all correspondence to: emine.kemiklioglu@cbu.edu.tr

[1] Friedel G. Annales de Physique. 1922;18:273

[3] Reinitzer F. Monatshefte fuer Chemie. 1888;9:421

[4] Planer P. Liebigs Annalen. 1861;118:25

W. H. Freeman; 1975

applications [35].

Author details

Emine Kemiklioglu

Turkey

References

A recent study has appeared recently demonstrating polymer-encapsulated blue phase (PEBP) liquid crystal films were prepared via solvent evaporation-induced phase separation of a mixture of blue phase liquid crystal (BPLC) and polymer latex [33]. It was observed that the PEBP films induced the birefringence between crossed polarizers at low switching voltage and with fast response time. PEBP samples generated considerably large Kerr constants, in the range of 1.83\*10<sup>8</sup> –20\*108 V�<sup>2</sup> m (at 633 nm), which are about 10 times higher than those of the reported PSBPs [139]. Therefore, PEBP liquid crystal films are strong candidates for nextgeneration displays as a result of the outstanding E-O properties of blue phases [33].

#### 3.6. Polymer stabilization of polymer encapsulated blue phase liquid crystals

A novel report has appeared recently suggesting the stabilization of encapsulated BPLC droplets is useful technique to expand their wide temperature range and improve the E-O properties of PDLCs using BPLCs [33]. Besides, polymer stabilization is one of the most effective methods to expand the thermal stability of BPLCs. However, liquid crystal/polymer composites can be classified into two distinct groups as polymer-dispersed liquid crystal (PDLC) and polymer-stabilized liquid crystal (PSLC). Both PDLC and PSLC methods are usually operated between a transparent state and an opaque state [53]. In the PDLC systems, droplets of liquid crystal are dispersed in a polymer film, which can be switched from scattering state to transparent state or vice versa with an applied electric field. In the reported study, composite materials of polymer-dispersed blue phase (PDBP) LCs were studied by combining PDLC films and BPLCs using two preparation methods, that is, polymer encapsulation and polymer stabilization in order to fabricate polymer encapsulated–polymer-stabilized bluephase (PEPSBP) LC droplets[33]. Encapsulated droplets were stabilized via the polymerization of reactive monomers after they were produced in a polyvinyl alcohol solution by emulsification. It was reported that polymer stabilized droplets caused an expansion of the BP temperature range from 53 to below 0�C. Moreover, this study concluded that low switching voltage and fast response time based on the decrement in the interfacial energy of polymer encapsulated and stabilized BPLC droplets. Furthermore, stabilization and positions of droplets in the aqueous phase have a great attraction for the researchers [142–146].

Kemiklioglu et al. firstly demonstrated the polymer-dispersed blue-phase liquid-crystal films between two indium-tin-oxide-coated conductive substrates by switching between light scattering and transparent states with applying an electric field across the films [35].

They experimentally investigated the photoinitiator effect on the electro-optical properties of the polymer-dispersed blue phase liquid crystals as well as the ratio between the crosslinking agent and the monomer. They showed that the increasing monomer concentration reduces the switching voltage of the corresponding device. Moreover, the increment of the monomer concentration in the polymer-dispersed blue phase liquid crystal samples leads an increment in the contrast ratio. All these significant improvements in the electro-optical properties of polymer-dispersed blue-phase liquid crystal devices are promising for new electro-optical applications [35].

### Author details

substrate before allowing the water to evaporate to produce thin polymer films containing liquid crystal droplets dispersed in a matrix. Evaporation process is one of the key factors for the droplet shape deformation which affects the alignment of LC inside the film cavities, which has a significant effect on the physical properties of PDLC films. Droplets become spherical or oblate in the polymer film [71, 120, 127, 132–145] depending on the evaporation process. However, the size distribution of the liquid crystal droplets in the emulsion can be modified by the preparation process and materials used to produce the emulsion, for example, the stirring time and speed, viscosities of polymer and liquid crystal. With increase in time of mixing in an ultrasonic cleaner, the droplet size of emulsion decreases [53, 146]. Moreover, surfactant type and concentration are the other key factors which affect the size, stability, and polydispersity of the droplets. The droplet size and polydispersity index decreased with increase in surfactant concentration [53]. The size and size distribution of encapsulated LC droplets can have a significant effect on the electro-optical properties of the films. Large area applicability of the emulsion system enhances the range of useful applications of PDLCs, ranging from switchable light modulators [71, 133], smart Windows [127] and information displays [134], as well as holographically formed optical elements and devices [135, 137, 142–144]. In PDLC systems, LC droplets are dispersed in a polymer film and these LCs can be oriented in the polymer droplets leading a switching from scattering to transparent states or vice versa with an applying electric field. There is a mismatching of refractive indices in the field-off state and these refractive indices of LC and the polymer match in the field-on state [53, 115, 127, 132–134, 146]. This phenomenon gives the electro-optical performance of the corresponding device. This electro-optical performance of displays and smart windows can be improved by replacing the nematic LCs with BPLCs, and the dispersed or encapsulated BPLCs leads a development in the

50 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

original optical and E-O properties with an external electical field [53].

range of 1.83\*10<sup>8</sup>

A recent study has appeared recently demonstrating polymer-encapsulated blue phase (PEBP) liquid crystal films were prepared via solvent evaporation-induced phase separation of a mixture of blue phase liquid crystal (BPLC) and polymer latex [33]. It was observed that the PEBP films induced the birefringence between crossed polarizers at low switching voltage and with fast response time. PEBP samples generated considerably large Kerr constants, in the

reported PSBPs [139]. Therefore, PEBP liquid crystal films are strong candidates for next-

A novel report has appeared recently suggesting the stabilization of encapsulated BPLC droplets is useful technique to expand their wide temperature range and improve the E-O properties of PDLCs using BPLCs [33]. Besides, polymer stabilization is one of the most effective methods to expand the thermal stability of BPLCs. However, liquid crystal/polymer composites can be classified into two distinct groups as polymer-dispersed liquid crystal (PDLC) and polymer-stabilized liquid crystal (PSLC). Both PDLC and PSLC methods are usually operated between a transparent state and an opaque state [53]. In the PDLC systems, droplets of liquid crystal are dispersed in a polymer film, which can be switched from scattering state to transparent state or vice versa with an applied electric field. In the reported study,

generation displays as a result of the outstanding E-O properties of blue phases [33].

3.6. Polymer stabilization of polymer encapsulated blue phase liquid crystals

–20\*108 V�<sup>2</sup> m (at 633 nm), which are about 10 times higher than those of the

### Emine Kemiklioglu

Address all correspondence to: emine.kemiklioglu@cbu.edu.tr

Department of Bioengineering, Faculty of Engineering, Manisa Celal Bayar University, Manisa, Turkey

### References


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**Chapter 4**

Provisional chapter

**Phase Transition effect on the Parametric Instability of**

DOI: 10.5772/intechopen.70240

Phase Transition effect on the Parametric Instability of

We review advances in the last few years on the study of the Faraday instability onset on thermotropic liquid crystals of nematic and smectic A types under external magnetic fields which have been investigated with a linear stability theory. Especially, we show that thermal phase transition effects on nematics of finite thickness samples produce an enhanced response to the instability as a function of the frequency of Shaker's movement. The linear stability theory has successfully been used before to study dynamical processes on surfaces of complex fluids. Consequently, in Section 1, we show its extension to the study of the instability in the nematics, which set the theoretical framework for its further application to smectics or other anisotropic fluids such as lyotropic liquid crystals. We present the dispersion relationships of both liquids and its dependence on interfacial elastic parameters governing the surface elastic responses to external perturbations, to the sample size and their bulk viscosities. Finally, we point out the importance of following both experimental and theoretical analysis of various effects that needs to be incorporated into this model for the quantitative understanding of the

hydrodynamics behavior of surface phenomena in liquid crystals.

transition, nonlinear waves, complex fluids

Keywords: liquid crystals, parametric instability, surface hydrodynamics, phase

The Faraday wave instability emerges as a macroscopic nonlinear behavior of the dynamics at interfaces of different liquids and vapor [1–22]. It appears when the vessel containing the liquid is vibrated vertically with a given acceleration until the quiescent equilibrium interface develops unstable surface waves. It has been observed in Newtonian fluids [6], but their most interesting realizations occur in complex fluids where their viscoelastic responses are present due to different time scales associated with the molecular relaxation processes [8, 16]. Therefore,

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Liquid Crystals**

Liquid Crystals

Abstract

1. Introduction

Martin Hernández Contreras

Martin Hernández Contreras

http://dx.doi.org/10.5772/intechopen.70240

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter


Provisional chapter

### **Phase Transition effect on the Parametric Instability of Liquid Crystals** Phase Transition effect on the Parametric Instability of Liquid Crystals

DOI: 10.5772/intechopen.70240

Martin Hernández Contreras

Additional information is available at the end of the chapter Martin Hernández Contreras

http://dx.doi.org/10.5772/intechopen.70240 Additional information is available at the end of the chapter

### Abstract

[142] Qi J, Crawford GP. Displays. 2004;25:177

58 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

Abbott NL. Soft Matter. 2016;12:8781

2015;112:13195

Materials. 2015;27:6892

[143] Wang X, Bukusoglu E, Abbott NL. Chemistry of Materials. 2017;29:53

[144] Bukusoglu E, Wang X, Zhou Y, Martínez-González JA, Rahimi M, Wang Q, de Pablo JJ,

[145] Martínez-González JA, Zhou Y, Rahimi M, Bukusoglu E, Abbott, NL, de Pablo JJ. PNAS.

[146] Bukusoglu E, Wang X, Martínez-González JA, de Pablo JJ, Abbott NL. Advanced

We review advances in the last few years on the study of the Faraday instability onset on thermotropic liquid crystals of nematic and smectic A types under external magnetic fields which have been investigated with a linear stability theory. Especially, we show that thermal phase transition effects on nematics of finite thickness samples produce an enhanced response to the instability as a function of the frequency of Shaker's movement. The linear stability theory has successfully been used before to study dynamical processes on surfaces of complex fluids. Consequently, in Section 1, we show its extension to the study of the instability in the nematics, which set the theoretical framework for its further application to smectics or other anisotropic fluids such as lyotropic liquid crystals. We present the dispersion relationships of both liquids and its dependence on interfacial elastic parameters governing the surface elastic responses to external perturbations, to the sample size and their bulk viscosities. Finally, we point out the importance of following both experimental and theoretical analysis of various effects that needs to be incorporated into this model for the quantitative understanding of the hydrodynamics behavior of surface phenomena in liquid crystals.

Keywords: liquid crystals, parametric instability, surface hydrodynamics, phase transition, nonlinear waves, complex fluids

### 1. Introduction

The Faraday wave instability emerges as a macroscopic nonlinear behavior of the dynamics at interfaces of different liquids and vapor [1–22]. It appears when the vessel containing the liquid is vibrated vertically with a given acceleration until the quiescent equilibrium interface develops unstable surface waves. It has been observed in Newtonian fluids [6], but their most interesting realizations occur in complex fluids where their viscoelastic responses are present due to different time scales associated with the molecular relaxation processes [8, 16]. Therefore,

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

it is important to determine how the onset of the Faraday instability is determined by the underlying bulk fluid elasticity. The basic study of this phenomenon poses challenges to hydrodynamic and statistical theories that need to be adapted or developed to explain the onset of the instability. The description of the Faraday instability has recently motivated the advance of new experiments [23–26] and theoretical approaches for anisotropic fluids such as liquid crystals [27–40]. Also, it seems that its comprehensive experimental and theoretical investigation may have a major impact on the development of sensor technologies based on interfaces with biological or chemical components of practical interest. Absorbed molecules at the interface provide a surface coverage with absorbed molecular species that can present activity. Their impact on the instability has not well been understood yet. Liquid crystals remain as an ideal complex fluid where a controlled fine tuning of the cohesive energy of the absorbed molecules of interest and the nematogen's bulk average orientations imposed by surface treatment of their anchoring energy [41–46] can be experimentally reached. Reports on birefringent experiments on a lyotropic suspension of fd virus describe the effect of bulk microrheology on the surface wave [23]. The hysteresis of the wave amplitude under the harmonic external driving acceleration, and how the imposed perturbation shear lowers the viscosity for increasing driving impulse were observed. Such a rheological response of the liquid crystal led to a hydrodynamically induced transition from isotropic to nematic phase change, which produces the formation of patches at the deformed crest of the interface. In this chapter, we review recent work on the Faraday instability on thermotropic liquid crystals, and the effect that a thermal phase transition experienced by a nematic liquid crystal toward its isotropic state has on the instability onset. Thus, we review our understanding of thermal phase transitions that produce enhanced response on dynamical properties at the interface of thermotropic liquid crystals. The liquid crystal is subjected to vertical vibrations of the container which induce hydrodynamics instability on its surface. Temperature variations produce phase changes on liquid crystals [47]. We present our results on the liquid crystal phase change effect in the dynamics of the Faraday wave instability [39]. We focus our discussion on this coupled phenomenon on a hydrodynamic level of description based on the Navier-Stokes equation for the field velocity response of the liquid crystal. Our presentation incorporates the constitutive equation for taking into account properly the heat transfer into the liquid crystal which drives the phase transition. Also, the significant effects of various elastic parameters such as surface tension, bending modulus, and interfacial elasticity of the interface on the sustained wave are discussed. Further discussion shows how those elastic parameters determine the onset of the hydrodynamic instability through the critical acceleration of the surface wave, which is temperature dependent when the liquid crystal experiences a phase change. To set the theoretical framework, in Section 2 the phenomenological free energy of layers and surface deformations and its dependence on the elastic parameters are introduced. In Section 2.1, the hydrodynamic level of description of a model nematic liquid crystal is made. This section includes the mean field viscous shear stress tensor of the liquid crystal and the corresponding boundary conditions. In Section 2.2, we discuss the case of model nematic with nematogens aligned perpendicular to an external magnetic field but parallel to the surface. Such a model represents an isotropic liquid crystal case. We further present an analysis of the critical acceleration as a function of temperature variation from nematic up to the phase transition to the isotropic liquid crystal phase in Section 2.3. In Section 2.4, we analyze the dispersion relationship as a function of all elastic parameters

for a specific temperature. We then present in Section 2.5 the dispersion relation of an isotropic liquid. In Section 3, we discuss the occurrence of a parametric instability in smectic A liquids. In Sections 3.1 and 3.2, the finite thickness layer dispersion relationships for sustained Faraday surface waves in two configurations of the director on the magnetic field that orient the nematogens are provided. In Section 4, we discuss the experimental results in the literature on the phase transition effect on Faraday waves due to changes in particle concentration in a lyotropic liquid crystal of fd virus. In Section 5, a conclusion paragraph is provided. Finally,

We consider a finite thickness layer of depth L of nematic fluid with its molecules been oriented parallel to the liquid-air interface in the X-axis direction by an external magnetic field

The vector position giving the local elastic response of the interface has components ζ(x, y,t) and ξ(x, y,t) to a normal and in-plane perturbations. Those deformation fields are determined by a surface tension γ and interfacial shear elasticity ε produced by adsorbed surfactants at the interface which lower the surface tension. Perturbations of the interface produce coupling of vertical and lateral deformation whose strength is given by a parameter λ. A splay module K determines the curvature distortion of the nematogen's layers deformation, and a compressibility modulus B yields its compression. The magnetic field orients the nematogens as they are characterized by a magnetic susceptibility χa. Thus, the elastic-free energy of layers distortion

there is a list of the most relevant and updated list of references.

2. Thermotropic nematic liquid crystal layers

H as shown in Figure 1.

is given by [48]

<sup>F</sup>bulk <sup>¼</sup> <sup>1</sup>

2 ð d<sup>2</sup> r

Fsurface <sup>¼</sup> <sup>1</sup>

2 ð

> 8 < :

the shear normal to the surface <sup>γ</sup> <sup>=</sup> <sup>γ</sup><sup>0</sup> � <sup>π</sup> <sup>+</sup> <sup>i</sup>ωγ<sup>0</sup>

<sup>d</sup><sup>3</sup> r Bð Þ <sup>∂</sup>xu

�2λð Þ <sup>∂</sup>x<sup>ξ</sup> <sup>∂</sup><sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>K</sup> <sup>∂</sup><sup>2</sup>

<sup>x</sup><sup>ζ</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup> <sup>y</sup><sup>ζ</sup> � �<sup>2</sup>

whereas the interface elastic-free energy may be written approximately as [49]

yu <sup>þ</sup> <sup>∂</sup><sup>2</sup> zu � �<sup>2</sup>

<sup>γ</sup> ð Þ <sup>∂</sup>x<sup>ζ</sup> <sup>2</sup> <sup>þ</sup> <sup>∂</sup>y<sup>ζ</sup> � �<sup>2</sup> � � <sup>þ</sup> <sup>ε</sup> ð Þ <sup>∂</sup>x<sup>ξ</sup> <sup>2</sup> <sup>þ</sup> <sup>∂</sup>y<sup>ξ</sup> � �<sup>2</sup> � � <sup>þ</sup> <sup>κ</sup><sup>0</sup> <sup>∂</sup><sup>2</sup>

The molecular field u(x, y, z,t) takes into account the bulk elastic deformation of the stack of layers and which is caused by the acceleration, thermal fluctuations, and for smectics, also due to the movement (permeation) of molecules between layers u\_ � vx ¼ λph, where λp, h is the permeation length and molecular field, respectively, ∂<sup>β</sup> ≔ ∂/∂β, β = x , y, z. If we consider the Faraday wave on a liquid crystal that supports a monolayer, then the surface elastic response is affected by surface tension, dilational, and coupling modulus as given in Eq. (2). Crilly et al. [50] have shown using photon correlation spectroscopy experiments that the thermotropic phase transition on monoglyceride monolayers that originate from the little molecular area fluctuations can be accurately detected. They found that two surface elastic moduli can explain

<sup>þ</sup> <sup>χ</sup>aH<sup>2</sup> <sup>∂</sup>yu � �<sup>2</sup> <sup>þ</sup> ð Þ <sup>∂</sup>zu

Phase Transition effect on the Parametric Instability of Liquid Crystals

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61

and the other governs dilational distortion in

<sup>x</sup><sup>ζ</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup> <sup>y</sup><sup>ζ</sup> � �<sup>2</sup> 9 = ;: (2)

<sup>2</sup> n o h i : (1)

for a specific temperature. We then present in Section 2.5 the dispersion relation of an isotropic liquid. In Section 3, we discuss the occurrence of a parametric instability in smectic A liquids. In Sections 3.1 and 3.2, the finite thickness layer dispersion relationships for sustained Faraday surface waves in two configurations of the director on the magnetic field that orient the nematogens are provided. In Section 4, we discuss the experimental results in the literature on the phase transition effect on Faraday waves due to changes in particle concentration in a lyotropic liquid crystal of fd virus. In Section 5, a conclusion paragraph is provided. Finally, there is a list of the most relevant and updated list of references.

### 2. Thermotropic nematic liquid crystal layers

it is important to determine how the onset of the Faraday instability is determined by the underlying bulk fluid elasticity. The basic study of this phenomenon poses challenges to hydrodynamic and statistical theories that need to be adapted or developed to explain the onset of the instability. The description of the Faraday instability has recently motivated the advance of new experiments [23–26] and theoretical approaches for anisotropic fluids such as liquid crystals [27–40]. Also, it seems that its comprehensive experimental and theoretical investigation may have a major impact on the development of sensor technologies based on interfaces with biological or chemical components of practical interest. Absorbed molecules at the interface provide a surface coverage with absorbed molecular species that can present activity. Their impact on the instability has not well been understood yet. Liquid crystals remain as an ideal complex fluid where a controlled fine tuning of the cohesive energy of the absorbed molecules of interest and the nematogen's bulk average orientations imposed by surface treatment of their anchoring energy [41–46] can be experimentally reached. Reports on birefringent experiments on a lyotropic suspension of fd virus describe the effect of bulk microrheology on the surface wave [23]. The hysteresis of the wave amplitude under the harmonic external driving acceleration, and how the imposed perturbation shear lowers the viscosity for increasing driving impulse were observed. Such a rheological response of the liquid crystal led to a hydrodynamically induced transition from isotropic to nematic phase change, which produces the formation of patches at the deformed crest of the interface. In this chapter, we review recent work on the Faraday instability on thermotropic liquid crystals, and the effect that a thermal phase transition experienced by a nematic liquid crystal toward its isotropic state has on the instability onset. Thus, we review our understanding of thermal phase transitions that produce enhanced response on dynamical properties at the interface of thermotropic liquid crystals. The liquid crystal is subjected to vertical vibrations of the container which induce hydrodynamics instability on its surface. Temperature variations produce phase changes on liquid crystals [47]. We present our results on the liquid crystal phase change effect in the dynamics of the Faraday wave instability [39]. We focus our discussion on this coupled phenomenon on a hydrodynamic level of description based on the Navier-Stokes equation for the field velocity response of the liquid crystal. Our presentation incorporates the constitutive equation for taking into account properly the heat transfer into the liquid crystal which drives the phase transition. Also, the significant effects of various elastic parameters such as surface tension, bending modulus, and interfacial elasticity of the interface on the sustained wave are discussed. Further discussion shows how those elastic parameters determine the onset of the hydrodynamic instability through the critical acceleration of the surface wave, which is temperature dependent when the liquid crystal experiences a phase change. To set the theoretical framework, in Section 2 the phenomenological free energy of layers and surface deformations and its dependence on the elastic parameters are introduced. In Section 2.1, the hydrodynamic level of description of a model nematic liquid crystal is made. This section includes the mean field viscous shear stress tensor of the liquid crystal and the corresponding boundary conditions. In Section 2.2, we discuss the case of model nematic with nematogens aligned perpendicular to an external magnetic field but parallel to the surface. Such a model represents an isotropic liquid crystal case. We further present an analysis of the critical acceleration as a function of temperature variation from nematic up to the phase transition to the isotropic liquid crystal phase in Section 2.3. In Section 2.4, we analyze the dispersion relationship as a function of all elastic parameters

60 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

We consider a finite thickness layer of depth L of nematic fluid with its molecules been oriented parallel to the liquid-air interface in the X-axis direction by an external magnetic field H as shown in Figure 1.

The vector position giving the local elastic response of the interface has components ζ(x, y,t) and ξ(x, y,t) to a normal and in-plane perturbations. Those deformation fields are determined by a surface tension γ and interfacial shear elasticity ε produced by adsorbed surfactants at the interface which lower the surface tension. Perturbations of the interface produce coupling of vertical and lateral deformation whose strength is given by a parameter λ. A splay module K determines the curvature distortion of the nematogen's layers deformation, and a compressibility modulus B yields its compression. The magnetic field orients the nematogens as they are characterized by a magnetic susceptibility χa. Thus, the elastic-free energy of layers distortion is given by [48]

$$F\_{\text{bulk}} = \frac{1}{2} \left\{ d^3 \, r \left\{ \mathcal{B} (\partial\_x u)^2 + K \left( \partial\_y u + \partial\_z u \right)^2 + \chi\_d H^2 \left[ \left( \partial\_y u \right)^2 + \left( \partial\_z u \right)^2 \right] \right\}. \tag{1}$$

whereas the interface elastic-free energy may be written approximately as [49]

$$F\_{\text{surface}} = \frac{1}{2} \left\{ d^2 \, r \left\{ \begin{aligned} \chi & \left( \left( \partial\_{\mathbf{x}} \zeta \right)^2 + \left( \partial\_{\mathbf{y}} \zeta \right)^2 \right) + \varepsilon \left( \left( \partial\_{\mathbf{x}} \zeta \right)^2 + \left( \partial\_{\mathbf{y}} \zeta \right)^2 \right) + \kappa' \left( \partial\_{\mathbf{x}}^2 \zeta + \partial\_{\mathbf{y}}^2 \zeta \right)^2 \right\} \\ - 2\lambda (\partial\_{\mathbf{x}} \zeta) \left( \partial\_{\mathbf{x}}^2 \zeta + \partial\_{\mathbf{y}}^2 \zeta \right)^2 \end{aligned} \right\}. \tag{2}$$

The molecular field u(x, y, z,t) takes into account the bulk elastic deformation of the stack of layers and which is caused by the acceleration, thermal fluctuations, and for smectics, also due to the movement (permeation) of molecules between layers u\_ � vx ¼ λph, where λp, h is the permeation length and molecular field, respectively, ∂<sup>β</sup> ≔ ∂/∂β, β = x , y, z. If we consider the Faraday wave on a liquid crystal that supports a monolayer, then the surface elastic response is affected by surface tension, dilational, and coupling modulus as given in Eq. (2). Crilly et al. [50] have shown using photon correlation spectroscopy experiments that the thermotropic phase transition on monoglyceride monolayers that originate from the little molecular area fluctuations can be accurately detected. They found that two surface elastic moduli can explain the shear normal to the surface <sup>γ</sup> <sup>=</sup> <sup>γ</sup><sup>0</sup> � <sup>π</sup> <sup>+</sup> <sup>i</sup>ωγ<sup>0</sup> and the other governs dilational distortion in

Figure 1. Interface of nematic liquid crystal layer of thickness L and air. A magnetic field H orients the nematogens parallel to the wave vector k which lies along the X-axis. Gravity modulation acts in the Z-axis direction. A vector position with components (ξ, ζ ) designates the inplane and normal elastic deformation of the interface, respectively.

elastic properties of GOM monolayer and their variation as a function of temperature as in Figures 1–3 of paper [50], when the monolayer is deposited on surfaces of liquid crystals, are still lacking, and their complete experimental measurement as a function of temperature would be quite valuable. Such experimental studies would pave the way for models which can be extended to understand thermotropic phase changes on monolayers and their impact on liquid crystal surface dynamics. Presently, one can resort to several useful models that study the surface hydrodynamics such as parametric instability, thermal capillary waves on polymer solutions and gels [13, 52], and coarse-grained effective field theories [27] and atomistic simulation techniques [37] and lattice hydrodynamics [17]. These methods have been demonstrated to provide a rich description of the effects of phase transition at interfaces of liquid crystals. From the experimental side, there is an accumulated knowledge of prototype systems that are now well characterized with tested experimental techniques. Those studies provide accurate information on static (elastic parameters [50, 53]). Also, the time-dependent properties (power spectrum of the intensity of scattered light) at interfaces of isotropic and simple liquids supporting surfactant monolayers [54] have been reported. Those studies need to be extended to include anisotropic liquids. An example of two such comprehensive experimental studies was performed by Langevin [54] in the 1970s where a series of systematic experiments on the variation of viscous and elastic parameters of some liquid crystals

function of the nematic-isotropic phase transition temperature. These values were interpolated from the experimental

0

Phase Transition effect on the Parametric Instability of Liquid Crystals

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63

/η<sup>3</sup> of MBBA as a

Figure 2. From bottom up are the interpolated values of viscosities η3, surface tension γ, viscosity η

studies of [54].

the interfacial plane ε = ε<sup>0</sup> + iωε<sup>0</sup> . Both are complex quantities, with i <sup>2</sup> <sup>=</sup> � 1 and <sup>γ</sup> 0 , ε 0 viscous coefficients of a monolayer of surfactants with pressure π. γ<sup>0</sup> , ε<sup>0</sup> are unperturbed values of elastic parameters without the monolayer. The imaginary parts are dissipative processes. However, Crilly et al. [50] showed that only the real parts of these moduli determine completely the thermotropic phase change of the surfactant monolayer, thus, our expression in Eq. (2) is valid to incorporate surface viscoelastic properties. The third elastic parameter λ that couples normal and tangential elasticity seems not to be detected experimenlly yet. The glycerol monooleate (GMO) monolayer has a phase transition temperature about 15.5� Cwhere a lipid undergos an all-trans state at low temperature to a gauche conformation of the lipid (pointing out the chain-melting transition) for temperatures greater than 15.5� C. We note, however, that the liquid crystal methoxy benzylidine butyl aniline (MBBA) has a transition temperature of 45� C [51]; therefore, one might expect a theory prediction for the effect of thermotropic phase changes of lipid monolayers on the Faraday waves to occur prior to the phase transition in the nematic-isotropic liquid. Such a study is possible to perform either experimentally and theoretically. From the experimental viewpoint, Crilly et al. [50] measured the elastic properties of the monolayer and provided results similar to those of Figures 2 and 3 (see Figure 1 of [50]), both well below and above the transition temperature as it is required in the modeling approach that is presented in this review. However, the determination of the

Phase Transition effect on the Parametric Instability of Liquid Crystals http://dx.doi.org/10.5772/intechopen.70240 63

Figure 2. From bottom up are the interpolated values of viscosities η3, surface tension γ, viscosity η 0 /η<sup>3</sup> of MBBA as a function of the nematic-isotropic phase transition temperature. These values were interpolated from the experimental studies of [54].

the interfacial plane ε = ε<sup>0</sup> + iωε<sup>0</sup>

temperature of 45�

. Both are complex quantities, with i

coefficients of a monolayer of surfactants with pressure π. γ<sup>0</sup> , ε<sup>0</sup> are unperturbed values of elastic parameters without the monolayer. The imaginary parts are dissipative processes. However, Crilly et al. [50] showed that only the real parts of these moduli determine completely the thermotropic phase change of the surfactant monolayer, thus, our expression in Eq. (2) is valid to incorporate surface viscoelastic properties. The third elastic parameter λ that couples normal and tangential elasticity seems not to be detected experimenlly yet. The

Figure 1. Interface of nematic liquid crystal layer of thickness L and air. A magnetic field H orients the nematogens parallel to the wave vector k which lies along the X-axis. Gravity modulation acts in the Z-axis direction. A vector position

with components (ξ, ζ ) designates the inplane and normal elastic deformation of the interface, respectively.

62 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

a lipid undergos an all-trans state at low temperature to a gauche conformation of the lipid

however, that the liquid crystal methoxy benzylidine butyl aniline (MBBA) has a transition

thermotropic phase changes of lipid monolayers on the Faraday waves to occur prior to the phase transition in the nematic-isotropic liquid. Such a study is possible to perform either experimentally and theoretically. From the experimental viewpoint, Crilly et al. [50] measured the elastic properties of the monolayer and provided results similar to those of Figures 2 and 3 (see Figure 1 of [50]), both well below and above the transition temperature as it is required in the modeling approach that is presented in this review. However, the determination of the

C [51]; therefore, one might expect a theory prediction for the effect of

glycerol monooleate (GMO) monolayer has a phase transition temperature about 15.5�

(pointing out the chain-melting transition) for temperatures greater than 15.5�

<sup>2</sup> <sup>=</sup> � 1 and <sup>γ</sup>

0 , ε 0 viscous

Cwhere

C. We note,

elastic properties of GOM monolayer and their variation as a function of temperature as in Figures 1–3 of paper [50], when the monolayer is deposited on surfaces of liquid crystals, are still lacking, and their complete experimental measurement as a function of temperature would be quite valuable. Such experimental studies would pave the way for models which can be extended to understand thermotropic phase changes on monolayers and their impact on liquid crystal surface dynamics. Presently, one can resort to several useful models that study the surface hydrodynamics such as parametric instability, thermal capillary waves on polymer solutions and gels [13, 52], and coarse-grained effective field theories [27] and atomistic simulation techniques [37] and lattice hydrodynamics [17]. These methods have been demonstrated to provide a rich description of the effects of phase transition at interfaces of liquid crystals. From the experimental side, there is an accumulated knowledge of prototype systems that are now well characterized with tested experimental techniques. Those studies provide accurate information on static (elastic parameters [50, 53]). Also, the time-dependent properties (power spectrum of the intensity of scattered light) at interfaces of isotropic and simple liquids supporting surfactant monolayers [54] have been reported. Those studies need to be extended to include anisotropic liquids. An example of two such comprehensive experimental studies was performed by Langevin [54] in the 1970s where a series of systematic experiments on the variation of viscous and elastic parameters of some liquid crystals

The experimental elucidation of thermotropic phase changes in monolayers of lipids deposited on top of liquid crystals and its impact on interface-laden of liquid crystals is an open topic to be investigated yet. A possible program for that task would involve the determination of the variation of the surface area of mono- or bilayers of surfactants like GMO which may be supported in isotropic, nematic, or smectic phases of liquid crystals. Moreover, the surface tension and dilational modulus versus temperature in an ample range around the critical temperature have been measured. Experimental techniques of photon correlation spectroscopy and surface quasielastic light scattering have been used in this context for isotropic fluids. Moreover, the entropy of formation of the film (see Figure 2 in Ref. [50] for temperature gradient of surface tension), which is related to the Marangoni effect of diffusion of molecules in the surface, should be measured correspondingly. Such experiments might observe the phase transition in the film and also determine the surface viscosities ignored in our model of Eq. (2). It is expected that the viscosities will help in clarifying the nature of the monolayer phases [54–56]. A comprehensive experimental determination of the bulk Leslie viscosities of the most known liquid crystals in an ample interval of temperatures near their critical phase transition is scarce [53]. Their values can lead to the prediction of enhanced macroscopic hydrodynamic response on the parametric surface dynamics of liquid crystals [39, 40], or polymer melts [57]. A similar experimental program may be set up for studying lyotropic liquid crystals to determine the viscosities and elastic parameters as a function of temperature. Those thermodynamic parameters enter as inputs in models like the one developed in this section. As an alternative to experimental measurements of bulk and interfacial viscoelastic properties of liquid crystals, one can also resort to molecular dynamics simulation. This computational technique allows the determination of the bulk and surface elastic parameters for certain molecular models of liquid crystals where the nematogens are under pairwise forces of the Weeks-Chandler-Andersen type, and the particles of ellipsoidal shape constitute the colloidal suspensions. From the practical point of view, it is known that the viscoelastic behavior of films of surfactants has very practical consequences, for instance, the electrical properties of GOM films depend on the formation of pores arising from significant fluctuations

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in the area near the phase transition [37, 38].

2.1. Parallel magnetic field to wave vector and along the X-axis

In this section, we present the hydrodynamic description of the velocity field of the center of mass of an infinitesimal volume element containing nematogens and its corresponding boundary conditions on a fluid-air interface [39]. We consider a nematic layer in contact with air, which has an equilibrium interface that is located at position z = 0 and has a depth L and infinite lateral extension in order to avoid viscous boundary layer effects due to the wall of the container and also beneth the surface as it is shown in Figure 1. This model can be generalized to liquid-liquid interfaces that separate two phases of a liquid crystal. To ignore the finite size effect of the vessel, we assume that it has an infinite lateral extension. The fluid is subjected to vertical cosine-like vibration with an acceleration g(t) = g a cos(ωt), where a is the acceleration of the shaker in a reference frame fixed to the container. ω is the angular frequency of the oscillation and g being the gravitational acceleration. Because the director of

Figure 3. Interpolation of the entropy of deformation that contributes to Marangoni flow of nematogens during the thermal phase transition of MBBA from nematic to isotropic phase as a function of transition temperature. These values were interpolated from the experimental studies of [59].

experiencing thermotropic phase change were reported, and the thorough experiments by Earnshaw [55] on surfaces of water supporting monolayers of surfactants. These researches provided tabulated comprehensive data of the elastic parameters and viscous properties on bulk and in the plane of the interface in a wide range of temperatures and not only at a single specific temperature of interest. Our phenomenological description of the elastic behavior of the interface given in Eq. (2) relies on models developed for polymer solution surfaces with absorbed non-active surfactants [49]. They are simple models that do not incorporate the anchoring energy of nematogens on the liquid crystal side toward the lipid monolayer and its strength of variation. A more reliable model that takes into account surface elasticity and the modification of anchoring energy of nematogens due to absorption of lipids at surfaces was developed in [34–36], and the application of this method to study theoretically the main surface modes of thermal waves at surfactant-laden liquid-liquid crystal interfaces was made in [33]. In reference [33], it was predicted that the thermal wave has a dispersion law that involves energy dissipation through anisotropic coefficients due to compression of the surfactant layer. And a new relaxation mode of the director appears due to the boundary condition of anchoring at the interface in a perpendicular direction to the plane.

The experimental elucidation of thermotropic phase changes in monolayers of lipids deposited on top of liquid crystals and its impact on interface-laden of liquid crystals is an open topic to be investigated yet. A possible program for that task would involve the determination of the variation of the surface area of mono- or bilayers of surfactants like GMO which may be supported in isotropic, nematic, or smectic phases of liquid crystals. Moreover, the surface tension and dilational modulus versus temperature in an ample range around the critical temperature have been measured. Experimental techniques of photon correlation spectroscopy and surface quasielastic light scattering have been used in this context for isotropic fluids. Moreover, the entropy of formation of the film (see Figure 2 in Ref. [50] for temperature gradient of surface tension), which is related to the Marangoni effect of diffusion of molecules in the surface, should be measured correspondingly. Such experiments might observe the phase transition in the film and also determine the surface viscosities ignored in our model of Eq. (2). It is expected that the viscosities will help in clarifying the nature of the monolayer phases [54–56]. A comprehensive experimental determination of the bulk Leslie viscosities of the most known liquid crystals in an ample interval of temperatures near their critical phase transition is scarce [53]. Their values can lead to the prediction of enhanced macroscopic hydrodynamic response on the parametric surface dynamics of liquid crystals [39, 40], or polymer melts [57]. A similar experimental program may be set up for studying lyotropic liquid crystals to determine the viscosities and elastic parameters as a function of temperature. Those thermodynamic parameters enter as inputs in models like the one developed in this section. As an alternative to experimental measurements of bulk and interfacial viscoelastic properties of liquid crystals, one can also resort to molecular dynamics simulation. This computational technique allows the determination of the bulk and surface elastic parameters for certain molecular models of liquid crystals where the nematogens are under pairwise forces of the Weeks-Chandler-Andersen type, and the particles of ellipsoidal shape constitute the colloidal suspensions. From the practical point of view, it is known that the viscoelastic behavior of films of surfactants has very practical consequences, for instance, the electrical properties of GOM films depend on the formation of pores arising from significant fluctuations in the area near the phase transition [37, 38].

#### 2.1. Parallel magnetic field to wave vector and along the X-axis

experiencing thermotropic phase change were reported, and the thorough experiments by Earnshaw [55] on surfaces of water supporting monolayers of surfactants. These researches provided tabulated comprehensive data of the elastic parameters and viscous properties on bulk and in the plane of the interface in a wide range of temperatures and not only at a single specific temperature of interest. Our phenomenological description of the elastic behavior of the interface given in Eq. (2) relies on models developed for polymer solution surfaces with absorbed non-active surfactants [49]. They are simple models that do not incorporate the anchoring energy of nematogens on the liquid crystal side toward the lipid monolayer and its strength of variation. A more reliable model that takes into account surface elasticity and the modification of anchoring energy of nematogens due to absorption of lipids at surfaces was developed in [34–36], and the application of this method to study theoretically the main surface modes of thermal waves at surfactant-laden liquid-liquid crystal interfaces was made in [33]. In reference [33], it was predicted that the thermal wave has a dispersion law that involves energy dissipation through anisotropic coefficients due to compression of the surfactant layer. And a new relaxation mode of the director appears due to the boundary condition

Figure 3. Interpolation of the entropy of deformation that contributes to Marangoni flow of nematogens during the thermal phase transition of MBBA from nematic to isotropic phase as a function of transition temperature. These values

of anchoring at the interface in a perpendicular direction to the plane.

were interpolated from the experimental studies of [59].

64 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

In this section, we present the hydrodynamic description of the velocity field of the center of mass of an infinitesimal volume element containing nematogens and its corresponding boundary conditions on a fluid-air interface [39]. We consider a nematic layer in contact with air, which has an equilibrium interface that is located at position z = 0 and has a depth L and infinite lateral extension in order to avoid viscous boundary layer effects due to the wall of the container and also beneth the surface as it is shown in Figure 1. This model can be generalized to liquid-liquid interfaces that separate two phases of a liquid crystal. To ignore the finite size effect of the vessel, we assume that it has an infinite lateral extension. The fluid is subjected to vertical cosine-like vibration with an acceleration g(t) = g a cos(ωt), where a is the acceleration of the shaker in a reference frame fixed to the container. ω is the angular frequency of the oscillation and g being the gravitational acceleration. Because the director of the nematogens remains fixed by the magnetic field, their time variation is neglected in the hydrodynamic equations. Under these conditions, a spatial ripple at the interface is generated which propagates symmetrically in the X- and Y-axis directions. Therefore, it is sufficient to consider that the surface wave propagates with wave vector k in the x-coordinate only and independently of coordinate y. The static magnetic field H keeps the director n=(1,0,0) of the nematogens in the X-axis direction oriented. Langevin calculated that thermal fluctuations produce periodic distortions of the nematic director with frequencies of strength ωundulation = (Kk<sup>2</sup> <sup>+</sup> <sup>χ</sup>aH � <sup>H</sup>)/<sup>η</sup> <sup>≈</sup> 9.9 Hz for MBBA nematic liquid with viscosity <sup>η</sup>. Both the frequency of the mechanical excitation and that of inertial effects ωinertia = η/ρL<sup>2</sup> = 10<sup>6</sup> ωundulation are larger than the elastic frequency of director variations. ρ is the density of the nematic. Thus, the nematic director dynamics is neglected in the hydrodynamic description of the generated surface wave. Therefore, the governing equation of the fluid velocity v about the quiescent state of rest is provided by the linearized Navier-Stokes equation which uses a reference frame that moves attached to the container [58]

$$
\rho \frac{\partial V}{\partial t} = \nabla \cdot \sigma \tag{3}
$$

Normal interface displacements are balanced by the elastic force obtained from Eq. (2)

<sup>x</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup> y � �<sup>ζ</sup> <sup>þ</sup> <sup>λ</sup> <sup>∂</sup><sup>2</sup>

The tangential forces at the interface result from the Marangoni instability due to surface tension variation with temperature, and the in-plane elastic deformation which are obtained

In this last equation, elongational deformation ε and the coupling with normal deformation through the elastic parameter λ are included. This is the case when one needs to study interfaces that support mono- and bilayers of surface-active surfactants. Because the normal displacement is small compared with the wavelength, one can approximate the velocity field with the rate of surface deformations through the components of the vector that locates the

In the last equality, κ is the thermal conductivity of a thermally insolated surface with fixed

In the application that follows below, we do not consider the effect of the coefficients ε and λ and will be made zero. From the first two conditions of Eq. (7) and with help of Eq. (8), we obtain

dT <sup>∂</sup><sup>2</sup>

� �∇<sup>2</sup> <sup>∂</sup>x∂zVx � <sup>∂</sup><sup>2</sup>

Now taking the divergence with the gradient operator ∇⊥ ≔ (∂x, ∂y) of the Navier-Stokes

We now replace this expression in the z component of the total stress tensor σ and take its

<sup>x</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup> y � �<sup>ξ</sup> <sup>þ</sup> <sup>λ</sup> <sup>∂</sup><sup>2</sup>

<sup>x</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup> y

<sup>∂</sup>t<sup>ζ</sup> <sup>¼</sup> Vz, <sup>∂</sup>t<sup>ξ</sup> <sup>¼</sup> Vx at <sup>z</sup> <sup>¼</sup> <sup>0</sup>, <sup>κ</sup>∂zT <sup>¼</sup> <sup>0</sup>: (10)

<sup>x</sup>Vz � � <sup>þ</sup> <sup>η</sup>

� � ∇<sup>2</sup>

V ¼ 0 (11)

<sup>∂</sup>zVz <sup>¼</sup> <sup>0</sup>: (12)

0 ∂2

<sup>⊥</sup> <sup>þ</sup> <sup>∇</sup><sup>2</sup> � �∂xVx <sup>þ</sup> <sup>η</sup>

<sup>2</sup> <sup>=</sup> � 1 with the result (from now on, we will not use <sup>u</sup><sup>~</sup> for

<sup>x</sup>½ � T � Aζ (13)

<sup>x</sup>∂x∂zVx: (14)

<sup>x</sup>∂xVx (15)

0 ∂2

<sup>x</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup> y � �ζ, <sup>σ</sup><sup>r</sup>

Phase Transition effect on the Parametric Instability of Liquid Crystals

� �ξ: (8)

http://dx.doi.org/10.5772/intechopen.70240

yz ¼ 0: (9)

67

σr

from Eq. (2) as

σr

xz <sup>¼</sup> <sup>f</sup> <sup>x</sup> <sup>¼</sup> <sup>d</sup><sup>γ</sup>

interface positions as follows (see Figure 1):

moreover, there is no penetration of the wall

<sup>ρ</sup>∂<sup>t</sup> � <sup>η</sup>2∇<sup>2</sup> � �∇<sup>2</sup>

<sup>⊥</sup><sup>p</sup> <sup>¼</sup> <sup>ρ</sup>∂<sup>t</sup> � <sup>η</sup>2∇<sup>2</sup> � <sup>η</sup><sup>3</sup> � <sup>η</sup><sup>2</sup>

d2 re<sup>i</sup>k∙<sup>r</sup> u, i

any transformed function but just simply u)

∇2

Fourier transform <sup>u</sup><sup>~</sup> <sup>¼</sup> <sup>Ð</sup>

(Eq. (3)) moreover, using the divergence of Eq. (4) yields

zz <sup>¼</sup> <sup>f</sup> <sup>z</sup> <sup>¼</sup> <sup>γ</sup> <sup>∂</sup><sup>2</sup>

dT ½ �� <sup>∂</sup>xT � <sup>A</sup>∂x<sup>ζ</sup> <sup>ε</sup> <sup>∂</sup><sup>2</sup>

flux. At the botton of the container z = � L, there is no slip of fluid

<sup>η</sup><sup>3</sup> <sup>∇</sup><sup>2</sup>

<sup>⊥</sup> � <sup>∂</sup><sup>2</sup> z � �Vz <sup>¼</sup> <sup>d</sup><sup>γ</sup>

Vz ¼ η<sup>3</sup> � η<sup>2</sup>

� �∂<sup>2</sup>

From the second identity of Eq. (7), and from Eq. (8) together with Eq. (9), we obtain

x � �∂zVz <sup>þ</sup> <sup>η</sup><sup>3</sup> � <sup>η</sup><sup>2</sup>

The total stress tensor of the liquid <sup>σ</sup> ¼ �p<sup>I</sup> <sup>þ</sup> <sup>σ</sup><sup>0</sup> <sup>þ</sup> <sup>σ</sup><sup>r</sup> � <sup>ρ</sup>g tð Þbezbez has dissipative contributions σ 0 due to the bulk viscous response and an elastic part σ<sup>r</sup> given by the different elastic parameters in the free energy (Eq. (2)). Here, the unit vector <sup>b</sup>ez is directed along the <sup>Z</sup>-axis, and the hydrostatic fluid pressure is p. The unit matrix I<sup>β</sup> , <sup>δ</sup> = 1 if β = δ and zero otherwise. For nematic liquids

$$
\sigma'\_{\vec{\eta}} = \eta' n\_i n\_j V\_{l0} n\_l n\_0 + 2\eta\_2 V\_{\vec{\eta}} + 2\left(\eta\_3 - \eta\_2\right) \left(n\_i n\_l V\_{l\vec{\eta}} + n\_j n\_l V\_{l\vec{\iota}}\right) \tag{4}
$$

where the shear viscosities η2, η3, and η 0 are defined in terms of the Leslie coefficients as <sup>η</sup><sup>3</sup> <sup>¼</sup> <sup>α</sup>2þα<sup>5</sup> <sup>2</sup> � α2γ2= 2γ<sup>1</sup> � �, <sup>η</sup><sup>3</sup> <sup>¼</sup> <sup>α</sup><sup>4</sup> <sup>2</sup> , <sup>η</sup><sup>0</sup> <sup>¼</sup> <sup>α</sup><sup>1</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> <sup>2</sup>=γ1, where γ<sup>1</sup> = α<sup>3</sup> � α2, and γ<sup>2</sup> = α<sup>3</sup> + α2. The strain rate Vij = (∂iVj + ∂jVi)/2 has components i , j , l , o = x , y, z. For frequencies that are much less than the first sound frequency of the liquid, it holds the incompressible condition

$$\nabla \cdot V = \mathbf{0}.\tag{5}$$

In the experiment, a constant and vertical gradient of temperature that produces Marangoni flow with the rate of change of local temperature T variation given by the linearized heat diffusion equation [39] is applied from top to the bottom of the vessel

$$\frac{\partial T}{\partial t} = AV\_z + \alpha \left(\partial\_x^2 T + \partial\_z^2 T\right) \tag{6}$$

where A = � 3 � C/mm is the temperature gradient per unit length with the heating occurring from the air side. At the bottom solid wall T = 0� C. The balance of the normal and tangential shear stresses at the interface z = 0 defines the boundary conditions to be

$$
\sigma\_{zz} = 0, \sigma\_{xz} = 0, \sigma\_{yz} = 0 \tag{7}
$$

Normal interface displacements are balanced by the elastic force obtained from Eq. (2)

$$
\sigma\_{zz}^r = f\_z = \mathcal{V} \left(\partial\_x^2 + \partial\_y^2\right) \zeta + \lambda \left(\partial\_x^2 + \partial\_y^2\right) \xi. \tag{8}
$$

The tangential forces at the interface result from the Marangoni instability due to surface tension variation with temperature, and the in-plane elastic deformation which are obtained from Eq. (2) as

$$
\sigma\_{xz}^{\prime} = f\_x = \frac{d\gamma}{dT} [\partial\_x T - A \partial\_x \zeta] - \varepsilon \left(\partial\_x^2 + \partial\_y^2\right)\xi + \lambda \left(\partial\_x^2 + \partial\_y^2\right)\zeta,\\
\sigma\_{yz}^{\prime} = 0. \tag{9}
$$

In this last equation, elongational deformation ε and the coupling with normal deformation through the elastic parameter λ are included. This is the case when one needs to study interfaces that support mono- and bilayers of surface-active surfactants. Because the normal displacement is small compared with the wavelength, one can approximate the velocity field with the rate of surface deformations through the components of the vector that locates the interface positions as follows (see Figure 1):

$$
\partial\_t \zeta = V\_z, \quad \partial\_t \xi = V\_x \text{ at } z = 0, \kappa \partial\_z T = 0. \tag{10}
$$

In the last equality, κ is the thermal conductivity of a thermally insolated surface with fixed flux. At the botton of the container z = � L, there is no slip of fluid

$$V = 0\tag{11}$$

moreover, there is no penetration of the wall

the nematogens remains fixed by the magnetic field, their time variation is neglected in the hydrodynamic equations. Under these conditions, a spatial ripple at the interface is generated which propagates symmetrically in the X- and Y-axis directions. Therefore, it is sufficient to consider that the surface wave propagates with wave vector k in the x-coordinate only and independently of coordinate y. The static magnetic field H keeps the director n=(1,0,0) of the nematogens in the X-axis direction oriented. Langevin calculated that thermal fluctuations produce periodic distortions of the nematic director with frequencies of strength ωundulation = (Kk<sup>2</sup> <sup>+</sup> <sup>χ</sup>aH � <sup>H</sup>)/<sup>η</sup> <sup>≈</sup> 9.9 Hz for MBBA nematic liquid with viscosity <sup>η</sup>. Both the frequency of the

the elastic frequency of director variations. ρ is the density of the nematic. Thus, the nematic director dynamics is neglected in the hydrodynamic description of the generated surface wave. Therefore, the governing equation of the fluid velocity v about the quiescent state of rest is provided by the linearized Navier-Stokes equation which uses a reference frame that moves

The total stress tensor of the liquid <sup>σ</sup> ¼ �p<sup>I</sup> <sup>þ</sup> <sup>σ</sup><sup>0</sup> <sup>þ</sup> <sup>σ</sup><sup>r</sup> � <sup>ρ</sup>g tð Þbezbez has dissipative contributions

due to the bulk viscous response and an elastic part σ<sup>r</sup> given by the different elastic parameters in the free energy (Eq. (2)). Here, the unit vector <sup>b</sup>ez is directed along the <sup>Z</sup>-axis, and the hydrostatic fluid pressure is p. The unit matrix I<sup>β</sup> , <sup>δ</sup> = 1 if β = δ and zero otherwise. For

ρ ∂V

ninjVl0nln<sup>0</sup> þ 2η2Vij þ 2 η<sup>3</sup> � η<sup>2</sup>

0

rate Vij = (∂iVj + ∂jVi)/2 has components i , j , l , o = x , y, z. For frequencies that are much less than

In the experiment, a constant and vertical gradient of temperature that produces Marangoni flow with the rate of change of local temperature T variation given by the linearized heat

<sup>x</sup><sup>T</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup>

C/mm is the temperature gradient per unit length with the heating occurring

<sup>∂</sup><sup>t</sup> <sup>¼</sup> AVz <sup>þ</sup> <sup>α</sup> <sup>∂</sup><sup>2</sup>

<sup>2</sup> , <sup>η</sup><sup>0</sup> <sup>¼</sup> <sup>α</sup><sup>1</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup>

the first sound frequency of the liquid, it holds the incompressible condition

diffusion equation [39] is applied from top to the bottom of the vessel

∂T

shear stresses at the interface z = 0 defines the boundary conditions to be

ωundulation are larger than

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>∇</sup> � <sup>σ</sup> (3)

are defined in terms of the Leslie coefficients as

<sup>2</sup>=γ1, where γ<sup>1</sup> = α<sup>3</sup> � α2, and γ<sup>2</sup> = α<sup>3</sup> + α2. The strain

∇ � V ¼ 0: (5)

<sup>z</sup><sup>T</sup> � � (6)

C. The balance of the normal and tangential

σzz ¼ 0, σxz ¼ 0, σyz ¼ 0 (7)

� � (4)

� � ninlVlj <sup>þ</sup> njnlVli

mechanical excitation and that of inertial effects ωinertia = η/ρL<sup>2</sup> = 10<sup>6</sup>

66 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

attached to the container [58]

σ0 ij ¼ η 0

<sup>2</sup> � α2γ2= 2γ<sup>1</sup>

where the shear viscosities η2, η3, and η

� �, <sup>η</sup><sup>3</sup> <sup>¼</sup> <sup>α</sup><sup>4</sup>

from the air side. At the bottom solid wall T = 0�

σ 0

nematic liquids

<sup>η</sup><sup>3</sup> <sup>¼</sup> <sup>α</sup>2þα<sup>5</sup>

where A = � 3

�

$$
\partial\_z V\_z = 0.\tag{12}
$$

In the application that follows below, we do not consider the effect of the coefficients ε and λ and will be made zero. From the first two conditions of Eq. (7) and with help of Eq. (8), we obtain

$$
\eta\_3 \left[ \nabla\_\perp^2 - \partial\_z^2 \right] V\_z = \frac{d\mathcal{V}}{dT} \partial\_\mathbf{x}^2 [T - A\zeta] \tag{13}
$$

From the second identity of Eq. (7), and from Eq. (8) together with Eq. (9), we obtain

$$
\nabla \left[ \rho \partial\_t - \eta\_2 \nabla^2 \right] \nabla^2 V\_z = \left( \eta\_3 - \eta\_2 \right) \nabla^2 \left[ \partial\_x \partial\_z V\_x - \partial\_x^2 V\_z \right] + \eta' \partial\_x^2 \partial\_{\bar{x}} \partial\_{\bar{z}} V\_{\bar{x}}.\tag{14}
$$

Now taking the divergence with the gradient operator ∇⊥ ≔ (∂x, ∂y) of the Navier-Stokes (Eq. (3)) moreover, using the divergence of Eq. (4) yields

$$\nabla^2\_\perp p = \left[\rho \partial\_t - \eta\_2 \nabla^2 - \left(\eta\_3 - \eta\_2\right) \partial\_x^2\right] \partial\_z V\_z + \left(\eta\_3 - \eta\_2\right) \left[\nabla\_\perp^2 + \nabla^2\right] \partial\_x V\_x + \eta' \partial\_x^2 \partial\_x V\_x \tag{15}$$

We now replace this expression in the z component of the total stress tensor σ and take its Fourier transform <sup>u</sup><sup>~</sup> <sup>¼</sup> <sup>Ð</sup> d2 re<sup>i</sup>k∙<sup>r</sup> u, i <sup>2</sup> <sup>=</sup> � 1 with the result (from now on, we will not use <sup>u</sup><sup>~</sup> for any transformed function but just simply u)

$$\frac{1}{2}\left[\partial\_t + (\nu\_3 + 2\nu\_2)\dot{\mathbf{k}}^2 - \nu\_2 \partial\_z^2\right] \partial\_{\mathbf{z}} V\_{\mathbf{z}} = -(\nu\_3 - \nu\_2) \left[-2\dot{\mathbf{z}}^3 + \dot{\mathbf{z}}\partial\_z^2\right] V\_{\mathbf{z}} + \nu' \dot{\mathbf{z}} \mathbf{k}^3 V\_{\mathbf{z}} - g(t) \mathbf{k}^2 \zeta - \frac{\mathcal{V}}{\rho} \mathbf{k}^4 \zeta. \tag{16}$$

As a consequence, the Fourier transformed form of Eq. (13) is

$$
\nu\_3 \left[ k^2 + \partial\_z^2 \right] V\_z = \frac{k^2}{\rho} \frac{d\gamma}{dT} [T - A\zeta]\_\prime \tag{17}
$$

moreover, the capillary frequency w<sup>2</sup>

A<sup>1</sup> ¼ �m<sup>2</sup> k

B<sup>1</sup> ¼ �m<sup>1</sup> k

Pn ¼ � <sup>ζ</sup>nμ<sup>n</sup> m<sup>2</sup> <sup>1</sup> � <sup>m</sup><sup>2</sup> 2

Rn <sup>¼</sup> <sup>ζ</sup>nμ<sup>n</sup> m2 <sup>1</sup> � <sup>m</sup><sup>2</sup> 2

k <sup>2</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup>

<sup>0</sup> � <sup>μ</sup><sup>n</sup>

<sup>n</sup> <sup>k</sup>; <sup>μ</sup><sup>n</sup> <sup>¼</sup> <sup>s</sup> <sup>þ</sup> <sup>i</sup>ð Þ <sup>α</sup><sup>r</sup> <sup>þ</sup> <sup>n</sup> <sup>ω</sup> � �,

k mð Þ <sup>1</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> <sup>α</sup>

μn � � 8 >>>><

>>>>:

M<sup>∞</sup> <sup>n</sup> <sup>¼</sup> <sup>2</sup> <sup>k</sup> <sup>w</sup><sup>2</sup>

0

BBBB@

� 2 k D<sup>∞</sup> k <sup>2</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup> 1 � � � <sup>m</sup><sup>2</sup> <sup>k</sup><sup>4</sup> <sup>A</sup>

<sup>2</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup> 2 � � � <sup>m</sup><sup>1</sup> <sup>k</sup>

<sup>1</sup> <sup>þ</sup> <sup>k</sup><sup>4</sup> <sup>A</sup>

μnη<sup>3</sup>

dγ dT <sup>þ</sup> <sup>1</sup> � �

<sup>k</sup><sup>2</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup>

<sup>k</sup><sup>2</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup>

� �

� � α

ð Þ m<sup>1</sup> coshð Þ Lm<sup>1</sup> coshð Þ� Lm<sup>2</sup> m<sup>2</sup> sinhð Þ Lm<sup>1</sup> sinhð Þ Lm<sup>2</sup> ,

<sup>1</sup> þ k

Eq. (22) is the first of our most significant results. It permits the calculation of the wave amplitude modes ζ<sup>n</sup> as dictated by the viscoelastic properties of the nematic liquid and the hydrodynamic fluid velocity. It does not depend on adjustable free parameters and includes the Marangoni number Ma = (A/μnη3)dγ/dT that takes into account the thermal instability induced by the heating process applied to the nematic layer. All the material parameters that appear in Eq. (22) have already been reported by the experimental work of other authors [58] and that we are going to use further in the subsequent text. In contrast for a system of semi-infinite thickness, we obtained

<sup>2</sup> þ k

<sup>4</sup> A μnη<sup>3</sup>

<sup>2</sup> þ k

<sup>0</sup>≔gk <sup>þ</sup> <sup>γ</sup>k<sup>3</sup>

Qn <sup>¼</sup> <sup>ζ</sup>nμnA<sup>1</sup> m2 <sup>1</sup> � <sup>m</sup><sup>2</sup> 2 � � <sup>1</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup>

dγ dT <sup>þ</sup> <sup>1</sup> � �

� �

� � α

ð Þ m<sup>2</sup> coshð Þ Lm<sup>1</sup> coshð Þ� Lm<sup>2</sup> m<sup>1</sup> sinhð Þ Lm<sup>1</sup> sinhð Þ Lm<sup>2</sup> ,

de≔ � m<sup>2</sup> coshð Þ Lm<sup>2</sup> sinhð Þþ Lm<sup>1</sup> m<sup>1</sup> coshð Þ Lm<sup>1</sup> sinhð Þ Lm<sup>2</sup> ,

<sup>4</sup> A μnη<sup>3</sup>

> þ k 2 m2 1

<sup>4</sup> A μnη<sup>3</sup>

<sup>4</sup> A μnη<sup>3</sup>

μ<sup>n</sup> þ k <sup>2</sup> <sup>3</sup>ν<sup>3</sup> <sup>þ</sup> <sup>ν</sup> � � � �<sup>0</sup>

�ν3k <sup>2</sup> m<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup>

<sup>þ</sup> <sup>α</sup> ð Þ <sup>m</sup><sup>1</sup> � <sup>m</sup><sup>2</sup> <sup>2</sup>

Sn <sup>¼</sup> <sup>ζ</sup>nμnB<sup>1</sup> m2 <sup>1</sup> � <sup>m</sup><sup>2</sup> 2 � � <sup>1</sup> <sup>þ</sup> <sup>k</sup>

<sup>ρ</sup> : The coefficients are

α μn � �de

� � α

μn

2α μn � �de

dγ dT <sup>þ</sup> <sup>1</sup> � �

� � α

μn

dγ dT

dγ dT

þ k 2 m2 1

Phase Transition effect on the Parametric Instability of Liquid Crystals

þ k 2 m2 2

α=μ<sup>n</sup> � �

� �

<sup>1</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> <sup>α</sup> μn

α=μ<sup>n</sup> � �

� �

k <sup>2</sup> � <sup>m</sup>1m<sup>2</sup> � �

> 1m<sup>2</sup> 2

> > <sup>2</sup> � <sup>ν</sup>3m<sup>4</sup> 2

<sup>1</sup> � <sup>ν</sup>3m<sup>4</sup> 1 9 >>>>=

1

CCCCA

(25)

>>>>;

<sup>2</sup> <sup>þ</sup> <sup>B</sup>3m<sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>B</sup>3m<sup>2</sup>

� �

� � " #

<sup>2</sup> þ m1m<sup>2</sup> � � � <sup>ν</sup>3m<sup>2</sup>

<sup>A</sup><sup>2</sup> <sup>m</sup>1m<sup>2</sup> <sup>þ</sup> <sup>ν</sup>3m1m<sup>3</sup>

<sup>þ</sup>B<sup>2</sup> <sup>m</sup>1m<sup>2</sup> <sup>þ</sup> <sup>ν</sup>3m2m<sup>3</sup>

1 þ k <sup>2</sup> α μn

�

�

μn þ

http://dx.doi.org/10.5772/intechopen.70240

69

μn þ

(24)

dγ dT <sup>þ</sup> <sup>1</sup> � �

μnη<sup>3</sup>

þ k 2 m2 2

whereas Eq. (14) takes the form

$$\left[\partial\_{l} - \nu\_{2}\left(\partial\_{z}^{2} - k^{2}\right)\right] \left(\partial\_{z}^{2} - k^{2}\right) V\_{z} = \left(\nu\_{3} - \nu\_{2}\right) \left(-\partial\_{z}^{2} + k^{2}\right) \left[ik\partial\_{z}V\_{x} + k^{2}V\_{z}\right] + \nu^{\prime}ik^{3}\partial\_{z}V\_{x\prime} \tag{18}$$

With <sup>υ</sup>j<sup>≔</sup> <sup>η</sup><sup>j</sup> <sup>ρ</sup> , j ¼ 2, 3, ν<sup>0</sup> ≔η<sup>0</sup> =ρ. Similarly, Eq. (6) becomes

$$\frac{\partial T}{\partial t} = AV\_z + \alpha \left(\partial\_z^2 T - k^2 T\right),\tag{19}$$

The acceleration g(t) is a function with period 2π/ω, and according to Floquet theory the solution of Eqs. (5) and (16)–(19) is the superposition <sup>ζ</sup>ðÞ¼ <sup>t</sup> <sup>P</sup><sup>∞</sup> <sup>n</sup>¼�<sup>∞</sup> <sup>ζ</sup><sup>n</sup> exp <sup>μ</sup>n<sup>t</sup> � � [6]

where the modes μ<sup>n</sup> = s + i(n + αr)ω. The quantities ζ<sup>n</sup> ,s, α<sup>r</sup> are real numbers. There are two types of waves Harmonic with frequency equaling the external forcing frequency and determined by α<sup>r</sup> = 0,with ζ<sup>n</sup> equaling its complex conjugate and subharmonic with α<sup>r</sup> = 1/2, with <sup>ζ</sup><sup>n</sup> <sup>¼</sup> <sup>ζ</sup><sup>∗</sup> <sup>n</sup>�<sup>1</sup>. The velocity Vz also has a similar Floquet expansion that is substituted in Eqs. (5) and (18) providing the equation for each component of the velocity amplitude Vzn

$$\left[\partial\_z^4 + b\_n \partial\_z^2 + c\_n\right] V\_{zn}(z) = 0,\tag{20}$$

where bn ¼ � <sup>μ</sup><sup>n</sup> <sup>ν</sup><sup>3</sup> þ k <sup>3</sup> <sup>2</sup> <sup>þ</sup> <sup>ν</sup> 0 ν3 h i � � , cn <sup>¼</sup> <sup>k</sup><sup>2</sup> <sup>μ</sup><sup>n</sup> <sup>ν</sup><sup>3</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> h i: A trial solution of Eq. (20) in the form vzn(z) ~ e <sup>m</sup>(k)<sup>z</sup> ,C = m<sup>2</sup> simplifies Eq. (20) to C<sup>2</sup> + bnC + cn = 0. We denote the two possible modes Cj≔m<sup>2</sup> <sup>j</sup> , j ¼ 1, 2 which lead to the solution of Eq. (20) in the form

$$V\_{zn}(z) = P\_n \cosh(zm\_1) + Q\_n \sinh(zm\_1) + R\_n \cosh(zm\_2) + S\_n \sinh(zm\_2). \tag{21}$$

The coefficients Pn , Qn , Rn , Sn in this solution are derived by the substitution of Eq. (21) in Eqs. (10)–(12) (we recall ξ = λ = 0 here). Finally, the use of Eq. (21) in Eq. (16) yields the surface amplitude of deformation ζ<sup>n</sup>

$$M\_n \zeta\_n = a(\zeta\_{n-1} + \zeta\_{n+1}) \tag{22}$$

with

$$M\_n = \frac{2}{k} \left\{ \begin{aligned} w\_0^2 + \frac{m\_1}{k} \left[ \mu\_n + k^2 \left( 3\nu\_3 + \nu' \right) - \nu\_3 m\_1^2 \right] \frac{Q\_n}{\zeta\_n} + \\ + \frac{m\_2}{k} \left[ \mu\_n + k^2 \left( 3\nu\_3 + \nu' \right) - \nu\_3 m\_2^2 \right] \frac{S\_n}{\zeta\_n} \end{aligned} \right\} \equiv \frac{2}{k} D\_{n\nu} \tag{23}$$

moreover, the capillary frequency w<sup>2</sup> <sup>0</sup>≔gk <sup>þ</sup> <sup>γ</sup>k<sup>3</sup> <sup>ρ</sup> : The coefficients are

<sup>∂</sup><sup>t</sup> <sup>þ</sup> ð Þ <sup>ν</sup><sup>3</sup> <sup>þ</sup> <sup>2</sup>ν<sup>2</sup> <sup>k</sup>

whereas Eq. (14) takes the form

<sup>ρ</sup> , j ¼ 2, 3, ν<sup>0</sup>

<sup>∂</sup><sup>t</sup> � <sup>ν</sup><sup>2</sup> <sup>∂</sup><sup>2</sup> <sup>z</sup> � <sup>k</sup><sup>2</sup> � � � � <sup>∂</sup><sup>2</sup>

With <sup>υ</sup>j<sup>≔</sup> <sup>η</sup><sup>j</sup>

<sup>ζ</sup><sup>n</sup> <sup>¼</sup> <sup>ζ</sup><sup>∗</sup>

~ e

Cj≔m<sup>2</sup>

with

where bn ¼ � <sup>μ</sup><sup>n</sup>

<sup>ν</sup><sup>3</sup> þ k

amplitude of deformation ζ<sup>n</sup>

Mn <sup>¼</sup> <sup>2</sup> k

w2 <sup>0</sup> þ m<sup>1</sup>

8 >><

>>:

þ m<sup>2</sup>

<sup>3</sup> <sup>2</sup> <sup>þ</sup> <sup>ν</sup> 0 ν3

h i � �

<sup>2</sup> � <sup>ν</sup>2∂<sup>2</sup> z � �∂zVz ¼ �ð Þ� <sup>ν</sup><sup>3</sup> � <sup>ν</sup><sup>2</sup> <sup>2</sup>ik<sup>3</sup> <sup>þ</sup> ik∂<sup>2</sup>

As a consequence, the Fourier transformed form of Eq. (13) is

68 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

<sup>z</sup> � k

≔η<sup>0</sup>

<sup>ν</sup><sup>3</sup> <sup>k</sup><sup>2</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup> z � �Vz <sup>¼</sup> <sup>k</sup>

<sup>2</sup> � �Vz <sup>¼</sup> ð Þ� <sup>ν</sup><sup>3</sup> � <sup>ν</sup><sup>2</sup> <sup>∂</sup><sup>2</sup>

=ρ. Similarly, Eq. (6) becomes

<sup>∂</sup><sup>t</sup> <sup>¼</sup> AVz <sup>þ</sup> <sup>α</sup> <sup>∂</sup><sup>2</sup>

and (18) providing the equation for each component of the velocity amplitude Vzn

∂4 <sup>z</sup> <sup>þ</sup> bn∂<sup>2</sup>

, cn <sup>¼</sup> <sup>k</sup><sup>2</sup> <sup>μ</sup><sup>n</sup>

<sup>j</sup> , j ¼ 1, 2 which lead to the solution of Eq. (20) in the form

<sup>k</sup> <sup>μ</sup><sup>n</sup> <sup>þ</sup> <sup>k</sup>

<sup>k</sup> <sup>μ</sup><sup>n</sup> <sup>þ</sup> <sup>k</sup>

The acceleration g(t) is a function with period 2π/ω, and according to Floquet theory the

where the modes μ<sup>n</sup> = s + i(n + αr)ω. The quantities ζ<sup>n</sup> ,s, α<sup>r</sup> are real numbers. There are two types of waves Harmonic with frequency equaling the external forcing frequency and determined by α<sup>r</sup> = 0,with ζ<sup>n</sup> equaling its complex conjugate and subharmonic with α<sup>r</sup> = 1/2, with

<sup>z</sup> þ cn

<sup>m</sup>(k)<sup>z</sup> ,C = m<sup>2</sup> simplifies Eq. (20) to C<sup>2</sup> + bnC + cn = 0. We denote the two possible modes

The coefficients Pn , Qn , Rn , Sn in this solution are derived by the substitution of Eq. (21) in Eqs. (10)–(12) (we recall ξ = λ = 0 here). Finally, the use of Eq. (21) in Eq. (16) yields the surface

> <sup>2</sup> <sup>3</sup>ν<sup>3</sup> <sup>þ</sup> <sup>ν</sup> <sup>0</sup> � �

h i Sn

<sup>2</sup> <sup>3</sup>ν<sup>3</sup> <sup>þ</sup> <sup>ν</sup> <sup>0</sup> � �

h i Qn

Vznð Þ¼ z Pn coshð Þþ zm<sup>1</sup> Qn sinhð Þþ zm<sup>1</sup> Rn coshð Þþ zm<sup>2</sup> Sn sinhð Þ zm<sup>2</sup> : (21)

� <sup>ν</sup>3m<sup>2</sup> 1

ζn

� <sup>ν</sup>3m<sup>2</sup> 2

<sup>ν</sup><sup>3</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> h i

<sup>n</sup>�<sup>1</sup>. The velocity Vz also has a similar Floquet expansion that is substituted in Eqs. (5)

∂T

solution of Eqs. (5) and (16)–(19) is the superposition <sup>ζ</sup>ðÞ¼ <sup>t</sup> <sup>P</sup><sup>∞</sup>

z � �Vx <sup>þ</sup> <sup>ν</sup>

> 2 ρ dγ

<sup>z</sup> þ k <sup>2</sup> � � ik∂zVx <sup>þ</sup> <sup>k</sup>

<sup>z</sup><sup>T</sup> � <sup>k</sup><sup>2</sup>

0 ik3

Vx � g tð Þk

dT ½ � <sup>T</sup> � <sup>A</sup><sup>ζ</sup> , (17)

2 Vz � � <sup>þ</sup> <sup>ν</sup>

T � �, (19)

<sup>n</sup>¼�<sup>∞</sup> <sup>ζ</sup><sup>n</sup> exp <sup>μ</sup>n<sup>t</sup> � � [6]

� �Vznð Þ¼ <sup>z</sup> <sup>0</sup>, (20)

Mnζ<sup>n</sup> ¼ að Þ ζ<sup>n</sup>�<sup>1</sup> þ ζ<sup>n</sup>þ<sup>1</sup> (22)

9 >>=

>>;

� 2 k

Dn, (23)

ζn þ

: A trial solution of Eq. (20) in the form vzn(z)

2 <sup>ζ</sup> � <sup>γ</sup> ρ k 4 ζ: (16)

0 ik3 ∂zVx, (18)

$$Q\_n = \frac{\zeta\_n \mu\_n A\_1}{\left(m\_1^2 - m\_2^2\right)\left(1 + \frac{k^2 \alpha}{\mu\_n}\right) dc}$$

$$A\_1 = -m\_2(k^2 + m\_1^2) - m\_2 \left[k^4 \left(\frac{A}{\mu\_n \eta\_3} \frac{d\gamma}{dT} + 1\right) + k^2 m\_1^2\right] \frac{\alpha}{\mu\_n} + 1$$

$$\left\{k^2 + m\_2^2 + \left[k^4 \left(\frac{A}{\mu\_n \eta\_3} \frac{d\gamma}{dT} + 1\right) + k^2 m\_2^2\right] \frac{\alpha}{\mu\_n}\right\} .$$

$$\left(m\_2 \cosh(Lm\_1) \cosh(Lm\_2) - m\_1 \sinh(Lm\_1) \sinh(Lm\_2)\right)\_l$$

$$S\_n = \frac{\zeta\_n \mu\_n B\_1}{\left(m\_1^2 - m\_2^2\right) \left(1 + \frac{k^2 \alpha}{\mu\_n}\right) d\sigma}$$

de≔ � m<sup>2</sup> coshð Þ Lm<sup>2</sup> sinhð Þþ Lm<sup>1</sup> m<sup>1</sup> coshð Þ Lm<sup>1</sup> sinhð Þ Lm<sup>2</sup> ,

$$\begin{aligned} B\_1 &= -m\_1 \left( k^2 + m\_2^2 \right) - m\_1 \left[ k^4 \left( \frac{A}{\mu\_n \eta\_3} \frac{d\gamma}{dT} + 1 \right) + k^2 m\_2^2 \right] \frac{\alpha}{\mu\_n} + \\\ &\left\{ k^2 + m\_1^2 + \left[ k^4 \left( \frac{A}{\mu\_n \eta\_3} \frac{d\gamma}{dT} + 1 \right) + k^2 m\_1^2 \right] \frac{\alpha}{\mu\_n} \right\} .\end{aligned}$$
 
$$\begin{aligned} (\text{we, } \cosh(I \,\text{w.v.}) \cosh(I \,\text{w.v.})) \quad \text{we, } \sinh(I \,\text{w.v.}) \sinh(I \,\text{w.v.})) \end{aligned}$$

$$(m\_1 \cosh(Lm\_1)\cosh(Lm\_2) - m\_2 \sinh(Lm\_1)\sinh(Lm\_2))\_\nu$$

$$\begin{split} P\_n &= -\frac{\zeta\_n \mu\_n}{m\_1^2 - m\_2^2} \left[ k^2 + m\_2^2 + k^4 \frac{A}{\mu\_n \eta\_3} \frac{d\gamma}{dT} \frac{\left(\alpha/\mu\_n\right)}{\left(1 + k^2 \frac{\alpha}{\mu\_n}\right)} \right], \\ R\_n &= \frac{\zeta\_n \mu\_n}{m\_1^2 - m\_2^2} \left[ k^2 + m\_1^2 + k^4 \frac{A}{\mu\_n \eta\_3} \frac{d\gamma}{dT} \frac{\left(\alpha/\mu\_n\right)}{\left(1 + k^2 \frac{\alpha}{\mu\_n}\right)} \right], \end{split} \tag{24}$$

Eq. (22) is the first of our most significant results. It permits the calculation of the wave amplitude modes ζ<sup>n</sup> as dictated by the viscoelastic properties of the nematic liquid and the hydrodynamic fluid velocity. It does not depend on adjustable free parameters and includes the Marangoni number Ma = (A/μnη3)dγ/dT that takes into account the thermal instability induced by the heating process applied to the nematic layer. All the material parameters that appear in Eq. (22) have already been reported by the experimental work of other authors [58] and that we are going to use further in the subsequent text. In contrast for a system of semi-infinite thickness, we obtained

$$\begin{aligned} \mathcal{M}\_n^\alpha &= \frac{2}{k} \left( w\_0^2 - \frac{\mu\_n}{k(m\_1 + m\_2) \left( 1 + k^2 \frac{\alpha}{\mu\_n} \right)} \left\{ \begin{array}{c} \left[ \mu\_n + k^2 (3\gamma\_3 + \nu') \right] \left( k^2 - m\_1 m\_2 \right) \\ - \nu\_3 k^2 \left( m\_1^2 + m\_2^2 + m\_1 m\_2 \right) - \nu\_3 m\_1^2 m\_2^2 \\ + \frac{\alpha}{(m\_1 - m\_2)^2} \left[ \begin{array}{c} A\_2 \left( m\_1 m\_2 + \nu\_3 m\_1 m\_2^2 + B\_3 m\_2^2 - \nu\_3 m\_1^4 \right) \\ + B\_2 \left( m\_1 m\_2 + \nu\_3 m\_2 m\_1^3 + B\_3 m\_1^2 - \nu\_3 m\_1^4 \right) \end{array} \right] \right\} \end{aligned} \tag{25}$$
  $\equiv \frac{2}{k} D\_n^\alpha (k, \mu\_n = s + i(a\_r + n)\alpha),$ 

where the coefficients

$$A\_2 = k^4 \left(\frac{A}{\mu\_\pi \eta\_3} \frac{d\gamma}{dT} + 1\right) + k^2 m\_{1\prime}^2\\B\_2 = k^4 \left(\frac{A}{\mu\_\pi \eta\_3} \frac{d\gamma}{dT} + 1\right) + k^2 m\_2^2 \text{ and }\\B\_3 = \mu\_n + (3\nu\_3 + \nu^\prime)k^2.$$

#### 2.2. Magnetic field in the Y-axis direction with wave vector along the X-axis

In this section, we consider a nematic liquid layer of thickness L with the average director of nematogens n=(0,1,0) oriented in the Y-axis direction by the magnetic field and the wave propagates with wave number k in the X-axis. Thus, there is no coupling of the director with the flux. The hydrodynamic Navier-Stokes equation is the same as an isotropic liquid with a single viscosity η ≔ η<sup>3</sup> = η2. The components of the viscous stress tensor in Eq. (4) now read [53]

$$
\sigma'\_{xx} = 2\eta \partial\_x V\_{x\prime} \sigma'\_{zz} = 2\eta \partial\_z V\_{z\prime} \sigma'\_{zx} = \eta (\partial\_x V\_z + \partial\_z V\_x). \tag{26}
$$

Moreover, the forces normal to the interface and in the plane are, respectively, the same as in Eqs. (8)–(9) of Section 4. The boundary conditions are the same as in Eqs. (10)–(12), and the heat diffusion equation (Eq. (13)) is still valid. Consequently, the same method of section a results in the following eigenvalue equation for the mode amplitude of the wave:

$$M\_n^i \zeta\_n = a(\zeta\_{n-1} + \zeta\_{n+1}),\tag{27}$$

critical parameters are obtained in separate plots of the driving acceleration a for the onset of Faraday waves versus wave number. The lowest value of a = ac in the lower branch in that plot for a given k yields their critical values, so we need to first see this plot, the so-called instability tongue because of its shape, to know what it is the corresponding minimum value of a and associated wave number. One of such plots is given in Figure 4 of the neutral stability curve. This picture was obtained using Eq. (27) for a semi-infinite medium of nematic and tempera-

data for MBBA <sup>η</sup><sup>3</sup> = 0.0163 Pa s, <sup>γ</sup> = 0.03853 N/m, <sup>ρ</sup> = 1.03881 � 103 kg/m3 [58]. The main wave

For the pure nematic state, we first solved numerically Eqs. (22)–(24) with s =0, α<sup>r</sup> = 1/2 for n = 22 modes using the real materials data of the nematic MBBA obtained from interpolations of the experimental data which are provided in Figures 2 [58] and 3 [59]. Experiments with laser light scattering from the interface of MBBA performed by Langevin [54, 58] show that the

entropy dγ/dT as the only necessary data to characterize fully the phase transition. MBBA has

disappears. This was confirmed experimentally by Langevin who found that in the isotropic state of the liquid the intensity of scattered light is well characterized by a single bulk viscosity

was the same and equal to the constant value <sup>ρ</sup> = 1.03881 � <sup>10</sup>�<sup>3</sup> kg/m3 as required by the

Figure 4. Neutral stability curve of Faraday waves of subharmonic type for a semi-infinite layer of nematic in contact with an air interface. The use was made of the real material parameters of MBBA η<sup>3</sup> = 0.0163 Pa s, γ = 0.03853 N/m,

C from the phase transition.

, and temperature TNI � <sup>T</sup> = 3�

and the same elastic parameters above. In both cases, the density of the liquid crystal

0

C from the nematic-isotropic transition temperature and the experimental

, η3, the surface tension α and the interface structural

Phase Transition effect on the Parametric Instability of Liquid Crystals

http://dx.doi.org/10.5772/intechopen.70240

71

C, and above this temperature the anisotropy of the viscosity

ture of TNI � <sup>T</sup> = 3�

η = η<sup>3</sup> = η 0

<sup>ρ</sup> = 1.03881 � 103 kg/m<sup>3</sup>

sustained by the interface is of subharmonic type.

nematic state has two viscosities η

the critical temperature of Tc ≈ 45�

with

$$M\_n^i = \frac{2}{k} \left\{ w\_0^2 + \nu \left( q\_n^2 + k^2 \right) \frac{Q\_n^i}{\zeta\_n} \right\} + 4\nu q\_n \frac{S\_n^i}{\zeta\_n},\tag{28}$$

and

$$Q\_{n}^{j} = \zeta\_{n} \left\{ \begin{aligned} &\nu \eta\_{n} k^{2} \left\{ -2 \left( 1 + k^{2} \frac{\alpha}{\mu\_{n}} \right) + \frac{A}{\nu \eta} \frac{d\gamma}{dT} \frac{1}{k^{2} - q\_{n}^{2}} \left[ 1 - q\_{n}^{2} \frac{\nu}{\mu\_{n}} + (\nu + a) \frac{k^{2}}{\mu\_{n}} \right] \right\} + \\ &\nu \left\{ - \left( k^{2} + q\_{n}^{2} \right) \left( 1 + k^{2} \frac{\alpha}{\mu\_{n}} \right) + \frac{A}{\nu \eta} \frac{d\gamma}{dT} \frac{k^{2}}{k^{2} - q\_{n}^{2}} \left[ 1 - q\_{n}^{2} \frac{\nu}{\mu\_{n}} + (\nu + a) \frac{k^{2}}{\mu\_{n}} \right] \right\} \\ &\times \left[ -q\_{n} \cosh(kL) \cosh(q\_{n}L) + k \sinh(kL) \sinh(q\_{n}L) \right] \\ &\times \left\{ \left( 1 + k^{2} \frac{\alpha}{\mu\_{n}} \right) \left[ q\_{n} \sinh(kL) \cosh(q\_{n}L) - k \cosh(kL) \sinh(q\_{n}L) \right] \right\}^{-1} \end{aligned} \tag{29}$$

#### 2.3. Thermal phase transition effect on surface dynamics

In this section, we study the critical acceleration, and wave number of the Faraday wave at the interface of nematic liquid crystal and air as a function of temperature. We use the real material parameters reported in the literature of nematic methoxy benzylidine butyl aniline liquid crystal that experiences a thermal nematic-isotropic phase transition [58]. We now explain how we calculated the wave properties just mentioned. These properties ac , kc need to be determined in the nematic phase and then in the isotropic phase of the liquid. Notice that the critical parameters are obtained in separate plots of the driving acceleration a for the onset of Faraday waves versus wave number. The lowest value of a = ac in the lower branch in that plot for a given k yields their critical values, so we need to first see this plot, the so-called instability tongue because of its shape, to know what it is the corresponding minimum value of a and associated wave number. One of such plots is given in Figure 4 of the neutral stability curve.

where the coefficients

þ k 2 m2 <sup>1</sup>, B<sup>2</sup> ¼ k

σ0

xx <sup>¼</sup> <sup>2</sup>η∂xVx, <sup>σ</sup><sup>0</sup>

Mi <sup>n</sup> <sup>¼</sup> <sup>2</sup> <sup>k</sup> <sup>w</sup><sup>2</sup>

<sup>ν</sup>qnk<sup>2</sup> �2 1 <sup>þ</sup> <sup>k</sup><sup>2</sup> <sup>α</sup>

<sup>2</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup> n � � <sup>1</sup> <sup>þ</sup> <sup>k</sup>

2.3. Thermal phase transition effect on surface dynamics

μn � �

ν � k

� <sup>1</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> <sup>α</sup>

0

BBBBBBBB@

4 A μnη<sup>3</sup> dγ dT þ 1 � �

70 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

2.2. Magnetic field in the Y-axis direction with wave vector along the X-axis

þ k 2 m2

In this section, we consider a nematic liquid layer of thickness L with the average director of nematogens n=(0,1,0) oriented in the Y-axis direction by the magnetic field and the wave propagates with wave number k in the X-axis. Thus, there is no coupling of the director with the flux. The hydrodynamic Navier-Stokes equation is the same as an isotropic liquid with a single viscosity η ≔ η<sup>3</sup> = η2. The components of the viscous stress tensor in Eq. (4) now read [53]

zz <sup>¼</sup> <sup>2</sup>η∂zVz, <sup>σ</sup><sup>0</sup>

results in the following eigenvalue equation for the mode amplitude of the wave:

<sup>0</sup> <sup>þ</sup> <sup>ν</sup> <sup>q</sup><sup>2</sup>

<sup>n</sup> þ k <sup>2</sup> � � <sup>Q</sup><sup>i</sup>

> 1 k <sup>2</sup> � <sup>q</sup><sup>2</sup> n

( ) � �

k <sup>2</sup> � <sup>q</sup><sup>2</sup> n

qn sinhð Þ kL cosh qnL � � � <sup>k</sup> coshð Þ kL sinh qnL n o � � � � �<sup>1</sup>

In this section, we study the critical acceleration, and wave number of the Faraday wave at the interface of nematic liquid crystal and air as a function of temperature. We use the real material parameters reported in the literature of nematic methoxy benzylidine butyl aniline liquid crystal that experiences a thermal nematic-isotropic phase transition [58]. We now explain how we calculated the wave properties just mentioned. These properties ac , kc need to be determined in the nematic phase and then in the isotropic phase of the liquid. Notice that the

( ) � �

k 2

� �

n ζn

þ 4νqn

<sup>1</sup> � <sup>q</sup><sup>2</sup> n ν μn

> <sup>1</sup> � <sup>q</sup><sup>2</sup> n ν μn

Mi

μn

� �

þ A νη dγ dT

> þ A νη dγ dT

� �qn coshð Þ kL cosh qnL � � <sup>þ</sup> <sup>k</sup> sinhð Þ kL sinh qnL � � � �

<sup>2</sup> α μn

� �

Moreover, the forces normal to the interface and in the plane are, respectively, the same as in Eqs. (8)–(9) of Section 4. The boundary conditions are the same as in Eqs. (10)–(12), and the heat diffusion equation (Eq. (13)) is still valid. Consequently, the same method of section a

<sup>2</sup> and B<sup>3</sup> = μ<sup>n</sup> + (3ν<sup>3</sup> + ν')k

<sup>n</sup>ζ<sup>n</sup> ¼ að Þ ζ<sup>n</sup>�<sup>1</sup> þ ζ<sup>n</sup>þ<sup>1</sup> , (27)

Si n ζn

þ ð Þ ν þ α

k 2 μn

þ ð Þ ν þ α

2 .

zx <sup>¼</sup> <sup>η</sup>ð Þ <sup>∂</sup>xVz <sup>þ</sup> <sup>∂</sup>zVx : (26)

, (28)

þ

1

CCCCCCCCA

(29)

k 2 μn

4 A μnη<sup>3</sup> dγ dT þ 1 � �

A<sup>2</sup> ¼ k

with

and

Qi <sup>n</sup> ¼ ζ<sup>n</sup> This picture was obtained using Eq. (27) for a semi-infinite medium of nematic and temperature of TNI � <sup>T</sup> = 3� C from the nematic-isotropic transition temperature and the experimental data for MBBA <sup>η</sup><sup>3</sup> = 0.0163 Pa s, <sup>γ</sup> = 0.03853 N/m, <sup>ρ</sup> = 1.03881 � 103 kg/m3 [58]. The main wave sustained by the interface is of subharmonic type.

For the pure nematic state, we first solved numerically Eqs. (22)–(24) with s =0, α<sup>r</sup> = 1/2 for n = 22 modes using the real materials data of the nematic MBBA obtained from interpolations of the experimental data which are provided in Figures 2 [58] and 3 [59]. Experiments with laser light scattering from the interface of MBBA performed by Langevin [54, 58] show that the nematic state has two viscosities η 0 , η3, the surface tension α and the interface structural entropy dγ/dT as the only necessary data to characterize fully the phase transition. MBBA has the critical temperature of Tc ≈ 45� C, and above this temperature the anisotropy of the viscosity disappears. This was confirmed experimentally by Langevin who found that in the isotropic state of the liquid the intensity of scattered light is well characterized by a single bulk viscosity η = η<sup>3</sup> = η 0 and the same elastic parameters above. In both cases, the density of the liquid crystal was the same and equal to the constant value <sup>ρ</sup> = 1.03881 � <sup>10</sup>�<sup>3</sup> kg/m3 as required by the

Figure 4. Neutral stability curve of Faraday waves of subharmonic type for a semi-infinite layer of nematic in contact with an air interface. The use was made of the real material parameters of MBBA η<sup>3</sup> = 0.0163 Pa s, γ = 0.03853 N/m, <sup>ρ</sup> = 1.03881 � 103 kg/m<sup>3</sup> , and temperature TNI � <sup>T</sup> = 3� C from the phase transition.

incompressibility condition (Eq. (5)). We used a liquid layer of thickness L = 4.5 mm. Thus, for the isotropic phase we used the hydrodynamic equations which are similar to those of a simple liquid derived in this section. The critical parameters were numerically calculated from Eqs. (27)–(29) for the isotropic branch. The result of such a procedure is depicted in Figure 5.

Figure 5 shows the transition of the main sustained waves which are of subharmonic type, from low up to high temperatures across the critical temperature. One can observe that there is a significant variation of (ac , kc) due to their discontinuous behavior at the critical temperature of phase transition. Figure 5a and b depict kc, and Figure 5c yields ac, versus the nematicisotropic transition temperature T Tc at two values of the external frequency ω. For ω = 20π Hz, the plots of Figure 5a and c with symbol Ο correspond to the inclusion of Marangoni flow. For a higher frequency ω = 40π Hz we used symbol ● in Figure 5b and c and Marangoni flow is included too. However, in Figure 5 the plots with symbol ⋆ do not include Marangoni flow. We used the viscosities and surface tension in all the ranges of temperature variations as shown in Figures 2 and 3. When the Marangoni number is included Ma = (A/μnη3)dγ/dT in Eqs. (22)–(24) and (27)–(29), the critical acceleration and wave number (ac, kc) roughly coincide with their values when Ma = 0 is neglected. The same results are obtained if the anistotropy of the thermal diffusivity is taken as either α<sup>∥</sup> or α⊥; therefore in the plot of Figure 5 only α<sup>∥</sup> for the nematic side and αiso for the isotropic branch are used. On the other hand, from Figures 2 and 3 we can observe that the vertically applied temperature gradient of A = 3 C/mm on top of MBBA produces major changes in the magnitude of viscosities, surface tension, and on thermal diffusivity. However, the single parameter, the gradient of surface tension <sup>d</sup><sup>γ</sup> dT , which is related to Marangoni flow, does not affect the Faraday wave. This phenomenon occurs for instance at ω = 40Hz in Figure 5. We note from Figure 5 that (ac , kc) display the same discontinuous behavior as the viscous and elastic parameters do across the thermal phase transition temperature. However, a different mechanism for discontinuity of these critical parameters was measured experimentally by Huber et al. [57] during the surface freezing of a polymer monolayer made of tretraconazole melt. They found that a decrease of temperature leads to the formation of a monolayer at the interface of polymer-air that changed the surface tension but without an observable quantitative change of bulk viscosity. As a consequence, a high flow velocity gradient close to the surface at a characteristic temperature during the cooling down process from the high temperature regime is produced. Such an effect was accurately confirmed with a Faraday wave calculation by those authors and it is similar to our results here. We note that an exact match of the experimentally measured power spectrum of scattered light by thermal fluctuations of the nematic-air interface with a theoretical calculation ignoring Marangoni flow and for constant temperature suggests that a monolayer of nematogens does not form at the interface of MBBA. Unlike the MBBA liquid, the critical parameters of the Faraday instability at the interface of the liquid-vapor of CO<sup>2</sup> [6] do not display discontinuous behavior as in the liquid crystal study.

#### 2.4. Dispersion relation of an MBBA liquid layer with the inclusion of Marangoni flow

In Figure 6, the dispersion relation of the real MBBA (Figure 6a) together with an ideal model of a nematic (Figure 6b and c) as calculated from Eqs. (22)–(24). The real MBBA liquid (Figure 6a) presents a minimum of ac ≈ 0.455 g at oscillaton frequency ω = 55π Hz for a layer thickness of L = 4.5 mm. In this picture, the Marangoni flow was included, whereas the ideal model of nematic has a higher viscosity η<sup>3</sup> = 0.163 Pa than the real one, and its thickness is

and 3.

Figure 5. Calculated critical parameters of the Faraday wave during a thermal phase transition of nematic MBBA. In Figure 5, the critical wave number kc (5a, 5b) and acceleration ac (5c) as a function of the transition temperature are plotted. Two cases of frequency of excitation were used, ω = 20π Hz with symbol Ο, (5a, 5c), ω = 40π Hz symbol ● and white stars (5b, 5c). The nematic layer depth is L = 4.5 mm. Symbols Ο and ● include Marangoni flow, and it is neglected in the plot with a star symbol. The material parameters used across the thermal phase transition are those given in Figures 2

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incompressibility condition (Eq. (5)). We used a liquid layer of thickness L = 4.5 mm. Thus, for the isotropic phase we used the hydrodynamic equations which are similar to those of a simple liquid derived in this section. The critical parameters were numerically calculated from Eqs. (27)–(29) for the isotropic branch. The result of such a procedure is depicted in Figure 5. Figure 5 shows the transition of the main sustained waves which are of subharmonic type, from low up to high temperatures across the critical temperature. One can observe that there is a significant variation of (ac , kc) due to their discontinuous behavior at the critical temperature of phase transition. Figure 5a and b depict kc, and Figure 5c yields ac, versus the nematicisotropic transition temperature T Tc at two values of the external frequency ω. For ω = 20π Hz, the plots of Figure 5a and c with symbol Ο correspond to the inclusion of Marangoni flow. For a higher frequency ω = 40π Hz we used symbol ● in Figure 5b and c and Marangoni flow is included too. However, in Figure 5 the plots with symbol ⋆ do not include Marangoni flow. We used the viscosities and surface tension in all the ranges of temperature variations as shown in Figures 2 and 3. When the Marangoni number is included Ma = (A/μnη3)dγ/dT in Eqs. (22)–(24) and (27)–(29), the critical acceleration and wave number (ac, kc) roughly coincide with their values when Ma = 0 is neglected. The same results are obtained if the anistotropy of the thermal diffusivity is taken as either α<sup>∥</sup> or α⊥; therefore in the plot of Figure 5 only α<sup>∥</sup> for the nematic side and αiso for the isotropic branch are used. On the other hand, from Figures 2

72 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

and 3 we can observe that the vertically applied temperature gradient of A = 3

behavior as in the liquid crystal study.

thermal diffusivity. However, the single parameter, the gradient of surface tension <sup>d</sup><sup>γ</sup>

of MBBA produces major changes in the magnitude of viscosities, surface tension, and on

is related to Marangoni flow, does not affect the Faraday wave. This phenomenon occurs for instance at ω = 40Hz in Figure 5. We note from Figure 5 that (ac , kc) display the same discontinuous behavior as the viscous and elastic parameters do across the thermal phase transition temperature. However, a different mechanism for discontinuity of these critical parameters was measured experimentally by Huber et al. [57] during the surface freezing of a polymer monolayer made of tretraconazole melt. They found that a decrease of temperature leads to the formation of a monolayer at the interface of polymer-air that changed the surface tension but without an observable quantitative change of bulk viscosity. As a consequence, a high flow velocity gradient close to the surface at a characteristic temperature during the cooling down process from the high temperature regime is produced. Such an effect was accurately confirmed with a Faraday wave calculation by those authors and it is similar to our results here. We note that an exact match of the experimentally measured power spectrum of scattered light by thermal fluctuations of the nematic-air interface with a theoretical calculation ignoring Marangoni flow and for constant temperature suggests that a monolayer of nematogens does not form at the interface of MBBA. Unlike the MBBA liquid, the critical parameters of the Faraday instability at the interface of the liquid-vapor of CO<sup>2</sup> [6] do not display discontinuous

2.4. Dispersion relation of an MBBA liquid layer with the inclusion of Marangoni flow

In Figure 6, the dispersion relation of the real MBBA (Figure 6a) together with an ideal model of a nematic (Figure 6b and c) as calculated from Eqs. (22)–(24). The real MBBA liquid

C/mm on top

dT , which

Figure 5. Calculated critical parameters of the Faraday wave during a thermal phase transition of nematic MBBA. In Figure 5, the critical wave number kc (5a, 5b) and acceleration ac (5c) as a function of the transition temperature are plotted. Two cases of frequency of excitation were used, ω = 20π Hz with symbol Ο, (5a, 5c), ω = 40π Hz symbol ● and white stars (5b, 5c). The nematic layer depth is L = 4.5 mm. Symbols Ο and ● include Marangoni flow, and it is neglected in the plot with a star symbol. The material parameters used across the thermal phase transition are those given in Figures 2 and 3.

(Figure 6a) presents a minimum of ac ≈ 0.455 g at oscillaton frequency ω = 55π Hz for a layer thickness of L = 4.5 mm. In this picture, the Marangoni flow was included, whereas the ideal model of nematic has a higher viscosity η<sup>3</sup> = 0.163 Pa than the real one, and its thickness is

Figure 6. Dispersion relation of subharmonic waves of nematic MBBA at the fixed transition temperature TNI � <sup>T</sup> = 3� C for layer thickness of L = 4.5 mm (6a) and L = 2.5mm for (6b, 6c). Inset in Figure 6a depicts the critical acceleration calculated using the real parameters of MBBA and inclusion of Marangoni flow was made. Pictures in Figure 6b and c are for a model of nematic with 10 times the viscosity of the real nematic and neglect Marangoni flow and its critical acceleration is given in 6c. The left and right vertical lines in the insets of panels b and c depict the critical accelerations ac <sup>g</sup> ¼ 38:12, 51:13 which occur at wave vectors kc = 1.9284 , 0.6312 mm¯1 and frequencies ω = 135π , 215π Hz, respectively.

Figure 7. Dispersion relation of subharmonic waves of the isotropic nematic for low nematic depths L = 4.5 , 2.5 mm, Figure 7a and b, respectively. Other parameters are as in Figure 6. Figure 7a includes Marangoni number and uses real data of MBBA, whereas Figure 7b is for a model nematic with 10 times the real viscosity with no Marangoni flux. The

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insets correspond to the critical acceleration versus frequency.

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Figure 7. Dispersion relation of subharmonic waves of the isotropic nematic for low nematic depths L = 4.5 , 2.5 mm, Figure 7a and b, respectively. Other parameters are as in Figure 6. Figure 7a includes Marangoni number and uses real data of MBBA, whereas Figure 7b is for a model nematic with 10 times the real viscosity with no Marangoni flux. The insets correspond to the critical acceleration versus frequency.

Figure 6. Dispersion relation of subharmonic waves of nematic MBBA at the fixed transition temperature TNI � <sup>T</sup> = 3�

74 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

<sup>g</sup> ¼ 38:12, 51:13 which occur at wave vectors kc = 1.9284 , 0.6312 mm¯1 and frequencies ω = 135π , 215π Hz, respectively.

ac

for layer thickness of L = 4.5 mm (6a) and L = 2.5mm for (6b, 6c). Inset in Figure 6a depicts the critical acceleration calculated using the real parameters of MBBA and inclusion of Marangoni flow was made. Pictures in Figure 6b and c are for a model of nematic with 10 times the viscosity of the real nematic and neglect Marangoni flow and its critical acceleration is given in 6c. The left and right vertical lines in the insets of panels b and c depict the critical accelerations

C

almost half the size of the real MBBA L = 2.5 mm without Marangoni instability. The ideal model shows a monotonously increasing critical acceleration versus frequency (Figure 6b and c). A similar minimum value of ac was detected in silicon oil and water system. From Figure 6b and c, it can be seen that the harmonic type wave length λ<sup>c</sup> versus frequency shows a bifurcation. First, λ<sup>c</sup> decreases continuously as the frequency starts growing until a value ω ≈ 165π Hz is reached and suddenly the wave length makes a jump to a new value of the same magnitude λ<sup>c</sup> = 12.12 mm; it has at the starting value of the stimulus excitation ω ≈ 20π Hz. The inset in Figure 6b shows this transition in a regime of critical acceleration depicted with the small vertical lines. The discontinuities in wave length have been documented also to appear in viscoelastic fluids, in Newtonian liquids, and for silicon water oil systems.

#### 2.5. Dispersion relation of isotropic liquid crystal layer with the inclusion of Marangoni flow

In this case, the dispersion relation is calculated from Eqs. (27)–(29). Figure 7 presents the dispersion relation, whereas the insets depict the curve of the critical acceleration as a function of frequency for two systems; one with a layer thickness of L = 4.5 mm, Figure 7a that includes Marangoni flow, and the second one for L = 2.5 mm with Ma = 0 as shown in Figure 7b. The same pattern of behavior as in Figure 6 is obtained.

### 3. Thermotropic smectic A liquid crystal layers

### 3.1. Smectic order parameter, wave vector, and magnetic field directed along the X-axis direction

In this section, we discuss the Faraday instability in smectic A liquid crystal layers [40]. We consider a smectic liquid crystal of average thickness L and infinite lateral extension in contact with a vapor. The nematogens are oriented by an external magnetic field in the X-axis as depicted in Figure 8.

The stack of layers deformation is given by Eq. (1), whereas the elastic response of the interface is given by Eq. (2). The governing hydrodynamic equations of the velocity, viscous stress tensor, and the boundary conditions were reported in Refs. [40, 53]. Following their use and with the help of the methods of Section 2, we derived the following recursive equation of the amplitude of deformation ζn:

$$M\_n \zeta\_n = a(\zeta\_{n-1} + \zeta\_{n+1}) \tag{30}$$

<sup>A</sup><sup>1</sup> <sup>¼</sup> <sup>ζ</sup>nieL ffiffiffi

cn <sup>¼</sup> <sup>μ</sup><sup>2</sup>

n λpKν<sup>3</sup>

dn <sup>¼</sup> <sup>μ</sup><sup>2</sup>

þ k

n λpKν<sup>3</sup>

2kRn

S1 p

ffiffiffiffiffi S1 p �

> ffiffiffiffiffi S1 <sup>p</sup> ð�<sup>1</sup> <sup>þ</sup> eL ffiffiffiffi

Rn <sup>¼</sup> <sup>λ</sup>pBK<sup>2</sup> <sup>þ</sup> <sup>λ</sup>pKS<sup>2</sup>

<sup>2</sup> μ<sup>n</sup> λ2 <sup>1</sup>ν<sup>3</sup>

<sup>þ</sup> <sup>μ</sup><sup>n</sup> ξ2 <sup>H</sup>ν<sup>3</sup> þ 2μ<sup>n</sup> <sup>λ</sup>pK <sup>þ</sup>

> <sup>þ</sup> <sup>μ</sup><sup>n</sup> λpK

!

<sup>λ</sup><sup>p</sup> <sup>¼</sup> <sup>1</sup> ρυ3κ<sup>s</sup>

<sup>B</sup><sup>1</sup> <sup>¼</sup> <sup>ζ</sup>ni kRn

> <sup>2</sup> μ<sup>n</sup> λ2 <sup>1</sup>ν<sup>3</sup>

> > þ k

� <sup>1</sup> <sup>þ</sup> cothð<sup>L</sup> ffiffiffiffiffi

parallel to the interface surface. The velocity flow given by the wave vector k is parallel to the external field.

S1 p Þ ½μnS1e

!

<sup>1</sup> � <sup>λ</sup>pχaH<sup>2</sup>

þ k 4 λ2 1

<sup>2</sup> , <sup>υ</sup><sup>0</sup> <sup>¼</sup> <sup>η</sup><sup>0</sup>

In this eigenvalue equation, we ignored the elongational elasticity and coupling between inplane and normal elastic deformations. A plot of the Faraday stability curve is given in Figure 9

ν0 μn λpKν<sup>3</sup> þ κs 2 λ2 1

S1 <sup>p</sup> <sup>Þ</sup> �

Figure 8. Model of the deformation of a smectic A liquid crystal layer of thickness L due to the external perturbation. The liquid crystal has a free interface with air. The nematogens are oriented by a magnetic field H in the X-axis direction and

> <sup>L</sup> ffiffiffiffi S1 p

½μnS<sup>1</sup> þ k

� k 2 ð�1 þ e

, <sup>ξ</sup><sup>H</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffi K=χ<sup>a</sup>

o

<sup>ρ</sup> , <sup>κ</sup><sup>s</sup> <sup>≈</sup> <sup>100</sup> <sup>A</sup>:

S<sup>1</sup> þ μn, S<sup>1</sup> ¼ k

2 ð�1 þ e

<sup>L</sup> ffiffiffi S1 p ÞRn�

!

K=B p ,

<sup>L</sup> ffiffiffi S1 p ÞRn�

2 dn=cn,

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<sup>þ</sup> <sup>k</sup><sup>4</sup> <sup>1</sup> ξ2 H þ 2 λ2 1 þ ν0 ν3λ<sup>2</sup> 1

<sup>p</sup> <sup>=</sup>H, <sup>λ</sup><sup>1</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffi

$$M\_{\boldsymbol{n}} = \frac{2}{k} \left\{ w\_0^2 + \left\{ \frac{\left[Bk^2 + K\mathcal{S}\_1^2 + K\mathcal{k}^2\mathcal{S}\_1 + \chi\_\boldsymbol{n} H^2\mathcal{S}\_1\right]}{\rho} + \\ \left[\mu\_\pi + \lambda\_\mathbb{P} Bk^2 + \lambda\_\mathbb{P} \{K\mathcal{S}\_1^2 - \chi\_\boldsymbol{n} H^2\mathcal{S}\_1\}\right] \left[\mu\_\pi + k^2 \left(3\nu\_\mathbb{P} + \nu^\prime\right) - \nu\_\mathbb{S}\mathbf{S}\_1\right]} \right\} \frac{(A\_1 + B\_1)}{i\zeta\_n} \right\},$$

where

almost half the size of the real MBBA L = 2.5 mm without Marangoni instability. The ideal model shows a monotonously increasing critical acceleration versus frequency (Figure 6b and c). A similar minimum value of ac was detected in silicon oil and water system. From Figure 6b and c, it can be seen that the harmonic type wave length λ<sup>c</sup> versus frequency shows a bifurcation. First, λ<sup>c</sup> decreases continuously as the frequency starts growing until a value ω ≈ 165π Hz is reached and suddenly the wave length makes a jump to a new value of the same magnitude λ<sup>c</sup> = 12.12 mm; it has at the starting value of the stimulus excitation ω ≈ 20π Hz. The inset in Figure 6b shows this transition in a regime of critical acceleration depicted with the small vertical lines. The discontinuities in wave length have been documented also to

appear in viscoelastic fluids, in Newtonian liquids, and for silicon water oil systems.

same pattern of behavior as in Figure 6 is obtained.

3. Thermotropic smectic A liquid crystal layers

76 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

flow

direction

Mn <sup>¼</sup> <sup>2</sup>

where

<sup>k</sup> <sup>w</sup><sup>2</sup> <sup>0</sup> þ

>:

8 ><

depicted in Figure 8.

amplitude of deformation ζn:

8 ><

>:

Bk<sup>2</sup> <sup>þ</sup> KS<sup>2</sup>

<sup>1</sup> <sup>þ</sup> Kk<sup>2</sup>

<sup>μ</sup><sup>n</sup> <sup>þ</sup> <sup>λ</sup>pBk<sup>2</sup> <sup>þ</sup> <sup>λ</sup><sup>p</sup> KS<sup>2</sup>

� � ρ

<sup>S</sup><sup>1</sup> <sup>þ</sup> <sup>χ</sup>aH<sup>2</sup>

� � � � <sup>μ</sup><sup>n</sup> <sup>þ</sup> <sup>k</sup>

2.5. Dispersion relation of isotropic liquid crystal layer with the inclusion of Marangoni

3.1. Smectic order parameter, wave vector, and magnetic field directed along the X-axis

In this section, we discuss the Faraday instability in smectic A liquid crystal layers [40]. We consider a smectic liquid crystal of average thickness L and infinite lateral extension in contact with a vapor. The nematogens are oriented by an external magnetic field in the X-axis as

The stack of layers deformation is given by Eq. (1), whereas the elastic response of the interface is given by Eq. (2). The governing hydrodynamic equations of the velocity, viscous stress tensor, and the boundary conditions were reported in Refs. [40, 53]. Following their use and with the help of the methods of Section 2, we derived the following recursive equation of the

S1

<sup>1</sup> � <sup>χ</sup>aH<sup>2</sup>

þ

S1

Mnζ<sup>n</sup> ¼ að Þ ζ<sup>n</sup>�<sup>1</sup> þ ζ<sup>n</sup>þ<sup>1</sup> (30)

<sup>2</sup> <sup>3</sup>ν<sup>3</sup> <sup>þ</sup> <sup>ν</sup> � �<sup>0</sup>

� �

� ν3S<sup>1</sup>

9 >=

ð Þ A<sup>1</sup> þ B<sup>1</sup> iζ<sup>n</sup>

9 >=

>; ,

>;

In this case, the dispersion relation is calculated from Eqs. (27)–(29). Figure 7 presents the dispersion relation, whereas the insets depict the curve of the critical acceleration as a function of frequency for two systems; one with a layer thickness of L = 4.5 mm, Figure 7a that includes Marangoni flow, and the second one for L = 2.5 mm with Ma = 0 as shown in Figure 7b. The

Figure 8. Model of the deformation of a smectic A liquid crystal layer of thickness L due to the external perturbation. The liquid crystal has a free interface with air. The nematogens are oriented by a magnetic field H in the X-axis direction and parallel to the interface surface. The velocity flow given by the wave vector k is parallel to the external field.

<sup>A</sup><sup>1</sup> <sup>¼</sup> <sup>ζ</sup>nieL ffiffiffi S1 p 2kRn ffiffiffiffiffi S1 p � � <sup>1</sup> <sup>þ</sup> cothð<sup>L</sup> ffiffiffiffiffi S1 <sup>p</sup> <sup>Þ</sup> � ½μnS<sup>1</sup> þ k 2 ð�1 þ e <sup>L</sup> ffiffiffi S1 p ÞRn� <sup>B</sup><sup>1</sup> <sup>¼</sup> <sup>ζ</sup>ni kRn ffiffiffiffiffi S1 <sup>p</sup> ð�<sup>1</sup> <sup>þ</sup> eL ffiffiffiffi S1 p Þ ½μnS1e <sup>L</sup> ffiffiffiffi S1 p � k 2 ð�1 þ e <sup>L</sup> ffiffiffi S1 p ÞRn� Rn <sup>¼</sup> <sup>λ</sup>pBK<sup>2</sup> <sup>þ</sup> <sup>λ</sup>pKS<sup>2</sup> <sup>1</sup> � <sup>λ</sup>pχaH<sup>2</sup> S<sup>1</sup> þ μn, S<sup>1</sup> ¼ k 2 dn=cn, cn <sup>¼</sup> <sup>μ</sup><sup>2</sup> n λpKν<sup>3</sup> þ k <sup>2</sup> μ<sup>n</sup> λ2 <sup>1</sup>ν<sup>3</sup> <sup>þ</sup> <sup>μ</sup><sup>n</sup> ξ2 <sup>H</sup>ν<sup>3</sup> þ 2μ<sup>n</sup> <sup>λ</sup>pK <sup>þ</sup> ν0 μn λpKν<sup>3</sup> þ κs 2 λ2 1 ! <sup>þ</sup> <sup>k</sup><sup>4</sup> <sup>1</sup> ξ2 H þ 2 λ2 1 þ ν0 ν3λ<sup>2</sup> 1 ! dn <sup>¼</sup> <sup>μ</sup><sup>2</sup> n λpKν<sup>3</sup> þ k <sup>2</sup> μ<sup>n</sup> λ2 <sup>1</sup>ν<sup>3</sup> <sup>þ</sup> <sup>μ</sup><sup>n</sup> λpK ! þ k4 λ2 1 , <sup>ξ</sup><sup>H</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffi K=χ<sup>a</sup> <sup>p</sup> <sup>=</sup>H, <sup>λ</sup><sup>1</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffi K=B p , <sup>λ</sup><sup>p</sup> <sup>¼</sup> <sup>1</sup> ρυ3κ<sup>s</sup> <sup>2</sup> , <sup>υ</sup><sup>0</sup> <sup>¼</sup> <sup>η</sup><sup>0</sup> <sup>ρ</sup> , <sup>κ</sup><sup>s</sup> <sup>≈</sup> <sup>100</sup> <sup>A</sup>: o

In this eigenvalue equation, we ignored the elongational elasticity and coupling between inplane and normal elastic deformations. A plot of the Faraday stability curve is given in Figure 9

Figure 9. Plot of the acceleration versus wave number response of the surface wave sustained at the interface of a smectic A liquid-air for two frequencies and fixed room temperature. The configuration of the nematogens is given in Figure 8. The material parameters are provided in Section 3.1 and are typical of smectic A.

where the driving acceleration is plotted regarding the wave number. This picture shows us that parametric instability can be generated in shallow layers of smectic liquid crystals when it is excited with low frequencies. The magnitude of the normalized acceleration a/g falls well in the range of resolution of accessible experimental techniques that have been used before to

waves are depicted with ∘. This plot and that of Figure 9 show that small depth Sm A liquid crystal layers can sustain

Figure 10. Neutral stability curve of Faraday wave at the interface of Sm A and air. There is no applied magnetic field. The nematogens are oriented in layers parallel to the interface and the flow velocity is in the X-axis direction. The material

Phase Transition effect on the Parametric Instability of Liquid Crystals

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, whereas harmonic

parameters used are given in the main text. Note that subharmonic waves are plotted with symbol <sup>∗</sup>

parametric waves for excitation frequencies of typical experiments.

Figure 10. Neutral stability curve of Faraday wave at the interface of Sm A and air. There is no applied magnetic field. The nematogens are oriented in layers parallel to the interface and the flow velocity is in the X-axis direction. The material parameters used are given in the main text. Note that subharmonic waves are plotted with symbol <sup>∗</sup> , whereas harmonic waves are depicted with ∘. This plot and that of Figure 9 show that small depth Sm A liquid crystal layers can sustain parametric waves for excitation frequencies of typical experiments.

where the driving acceleration is plotted regarding the wave number. This picture shows us that parametric instability can be generated in shallow layers of smectic liquid crystals when it is excited with low frequencies. The magnitude of the normalized acceleration a/g falls well in the range of resolution of accessible experimental techniques that have been used before to

Figure 9. Plot of the acceleration versus wave number response of the surface wave sustained at the interface of a smectic A liquid-air for two frequencies and fixed room temperature. The configuration of the nematogens is given in Figure 8.

The material parameters are provided in Section 3.1 and are typical of smectic A.

78 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

study other fluids. The material parameters used to make this plot correspond to typical smectic liquids: K= 10�<sup>11</sup> N, B = 10<sup>6</sup> N/m<sup>2</sup> , L = 10�<sup>4</sup> m, λ<sup>p</sup> = 10�<sup>14</sup> m<sup>4</sup> /Ns, ρ = 103 kg/m3 , η<sup>3</sup> = 1 P, η<sup>3</sup> = η2, η 0 = 10�<sup>2</sup> η3, χ<sup>a</sup> = 10�<sup>8</sup> kg/(m s<sup>2</sup> G), H = 3000 G, the magnetic field that keeps fixed the orientation of nematogens, and γ = 0.033 N/m. In particular, Figure 9a points out that the excitation frequency a/g ≈ 0 for the low frequency ω = 6π Hz, that is, infinitesimal accelerations, can excite subharmonic waves in a similar manner as it occurs in inviscid ideal fluids.

viscosity, and order parameter of a model of rod-like suspension of particles in the isotropic phase [60]. A recent complete numerical simulation of the hydrodynamic equations governing the Faraday waves was developed by Perinet et al. [61] for a system formed by two immiscible fluids with the supporting fluid forming a shallow layer smaller than its boundary layer. These authors confirmed a hysteresis of the amplitude of the surface deformation as a function of the driving acceleration. They conclude that the wave amplitude bifurcates into two different waves. The hysteresis of the lower amplitude wave is attributed to a change of the shear stress in the fluid that results from variations in the fluid flow that produce a balance of hydrostatic

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Unlike lyotropic liquids, a successful continuum mean field model for thermotropic nematics has allowed the description of the rheology of the bulk of confined layers of nematics under oscillatory shear. Such a model has predicted shear thinning of the viscosity as a function of the oscillation frequency of the imposed shear. The extension of this study to understand the shear thinning effect on viscosity in the experiments of Ref. [23] seems feasible. A different perspective is obtained with computer simulations as those made by Germano et al. [38]. These researches have shown through molecular dynamics simulations on a molecular model of bulk ellipsoids with pair interaction of Weeks-Chandler-Andersen type as they say "nematic fluids may adopt inhomogeneous steady states under shear flow" [38]. Thus, shear flow modifies the molecular ordering in the liquid crystal producing changes in macroscopic viscosities like shear thinning and thickening and shear banding similarly as that found by Ballesta in their experiment of fd virus suspension under parametric instability. However, those simulations cannot be compared directly with the experiments of Ballesta [23]. Germano et al. [38] were interested in the capillary waves spectrum at the interface formed during the transition from nematic to isotropic. However, they did compare their simulations with a theory of Landau-De Gennes type for the free energy of the interface which incorporates an average director parallel to the interface. They found isotropic capillary waves that propagate at long wavelengths governed by the macroscopic surface tension. At short wavelength, however, the surface tension becomes anisotropic and depends on the wave vector. In a recent series of papers by Popa-Nita et al. [27–31], they developed a Landau-De Gennes theory to describe the capillary waves originating from thermal fluctuations, and at the interface of a ternary mixture of liquid crystal, colloid, and impurities. They considered both homeotropic (perpendicular to the interface) and also the variation of the nematic director. As in the Germano et al. method, Popa-Nita uses a free energy of the liquid crystal that predicts the bulk phase diagram, and additionally a Cahn-Hilliard equation was incorporated for taking into account the diffusion of impurities and colloids inside the liquid crystal. For such a mixture, they predicted the surface tension to decrease with the presence of colloids, whereas the impurities enhance its strength. Also, the temperature of the bulk phase transition is lowered on the pure liquid crystal nematic-isotropic transition temperature. The interfaces so formed experience thermal fluctuations. With the help of this approach, Popa-Nita were able to find that there are two regions of propagating capillary waves as it was also observed by Germano et al. in their simulation work. In the first region of long wavelengths, there is dissipation produced by shear flow and the ternary mixture behaves like an isotropic fluid which can be described by a single effective bulk viscosity. The hydrodynamic equations of the velocity field underlying that dispersion relation depends on the respective viscosity for each of the two phases formed which are

and lubrication stresses.

#### 3.2. Smectic A layers parallel to the surface and no magnetic field

In this case, we derived the modes of the surface amplitude of deformation ζ<sup>n</sup> that satisfies the eigenvalue Eqs. (27)–(29) with the function

$$M\_n = \frac{2}{k} \left\{ \pi \nu\_0^2 + \left( \frac{\mathcal{B}k\sqrt{S\_1}}{\rho} + \frac{\sqrt{S\_1}}{k} \mu\_n \left[ \mu\_n + k^2 \left( \Im \nu\_3 + \nu\_1' \right) - \nu\_3 S\_1 \right] \right) \coth\left( L\sqrt{S\_1} \right) \right\}.\tag{31}$$

The curly brackets in the above expression for Mn with the value n = 0 yield the well now dispersion relationship of thermal capillary waves reported by other authors. In Figure 10, we depict the stability curve of external acceleration versus wave number at a fixed frequency of oscillation for the given material parameters; K = 10�<sup>10</sup> N, B = 104 N/m<sup>2</sup> , layer thicknesses L = 0.03 , 0.05 m, frequency ω = 18π Hz, λ<sup>p</sup> = 10�<sup>14</sup> m4 /(N s), ρ = 103 kg/m3 , and viscosities and surface tension as in Section 3.1 earlier. We can observe that parametric surface waves also can evolve in this configuration of smectic A liquid. In a forthcoming manuscript, we will evaluate this parametric instability when the smectic phase can develop from the nematic phase as a thermal transition.

### 4. Lyotropic liquid crystal

In the previous section, we investigated how the bulk microstructure of the liquid crystal can modify the parametric instability through a thermal phase change. Now, we describe phase changes produced by particle volume fraction variations. Using birefringent measurements, Ballesta et al. [23] demonstrated a hydrodynamic phase change from isotropic to the nematic ordering of particles in a colloidal suspension. For the first time in the Faraday instability that occurs at the interface of air-colloidal suspension made of fd virus, they found that Faraday waves induce local nematic ordering of the nematogens in the wave crest when the colloid concentration is increased and close to the isotropic-nematic critical concentration. Such regions of nematic ordering become more permanent as the concentration is raised, and finally large areas of stable nematic patches that follow the wave flow are developed. This phenomenon was interpreted as a change in the local viscosity from its unperturbed value and decreases as a function of shear generated by the surface movement, which may reach high values of 100 Hz. A consequence of such shear thinning of viscosity is the appearance of hysteresis in the amplitude of the normal direction of the surface deformation as a function of the driving acceleration. Their analysis of the hysteretic behavior of the wave amplitude required them to use the Cross model of viscosity for bulk Newtonian fluids. Whereas for interpreting the intensity of observed birefringent experiments, it was necessary to use the viscosity, and order parameter of a model of rod-like suspension of particles in the isotropic phase [60]. A recent complete numerical simulation of the hydrodynamic equations governing the Faraday waves was developed by Perinet et al. [61] for a system formed by two immiscible fluids with the supporting fluid forming a shallow layer smaller than its boundary layer. These authors confirmed a hysteresis of the amplitude of the surface deformation as a function of the driving acceleration. They conclude that the wave amplitude bifurcates into two different waves. The hysteresis of the lower amplitude wave is attributed to a change of the shear stress in the fluid that results from variations in the fluid flow that produce a balance of hydrostatic and lubrication stresses.

study other fluids. The material parameters used to make this plot correspond to typical

orientation of nematogens, and γ = 0.033 N/m. In particular, Figure 9a points out that the excitation frequency a/g ≈ 0 for the low frequency ω = 6π Hz, that is, infinitesimal accelerations,

In this case, we derived the modes of the surface amplitude of deformation ζ<sup>n</sup> that satisfies the

� � h i

The curly brackets in the above expression for Mn with the value n = 0 yield the well now dispersion relationship of thermal capillary waves reported by other authors. In Figure 10, we depict the stability curve of external acceleration versus wave number at a fixed frequency of

surface tension as in Section 3.1 earlier. We can observe that parametric surface waves also can evolve in this configuration of smectic A liquid. In a forthcoming manuscript, we will evaluate this parametric instability when the smectic phase can develop from the nematic phase as a

In the previous section, we investigated how the bulk microstructure of the liquid crystal can modify the parametric instability through a thermal phase change. Now, we describe phase changes produced by particle volume fraction variations. Using birefringent measurements, Ballesta et al. [23] demonstrated a hydrodynamic phase change from isotropic to the nematic ordering of particles in a colloidal suspension. For the first time in the Faraday instability that occurs at the interface of air-colloidal suspension made of fd virus, they found that Faraday waves induce local nematic ordering of the nematogens in the wave crest when the colloid concentration is increased and close to the isotropic-nematic critical concentration. Such regions of nematic ordering become more permanent as the concentration is raised, and finally large areas of stable nematic patches that follow the wave flow are developed. This phenomenon was interpreted as a change in the local viscosity from its unperturbed value and decreases as a function of shear generated by the surface movement, which may reach high values of 100 Hz. A consequence of such shear thinning of viscosity is the appearance of hysteresis in the amplitude of the normal direction of the surface deformation as a function of the driving acceleration. Their analysis of the hysteretic behavior of the wave amplitude required them to use the Cross model of viscosity for bulk Newtonian fluids. Whereas for interpreting the intensity of observed birefringent experiments, it was necessary to use the

can excite subharmonic waves in a similar manner as it occurs in inviscid ideal fluids.

<sup>k</sup> <sup>μ</sup><sup>n</sup> <sup>μ</sup><sup>n</sup> <sup>þ</sup> <sup>k</sup>

3.2. Smectic A layers parallel to the surface and no magnetic field

ffiffiffiffiffi S1 p

oscillation for the given material parameters; K = 10�<sup>10</sup> N, B = 104 N/m<sup>2</sup>

Bk ffiffiffiffiffi S1 p ρ þ

80 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

L = 0.03 , 0.05 m, frequency ω = 18π Hz, λ<sup>p</sup> = 10�<sup>14</sup> m4

m, λ<sup>p</sup> = 10�<sup>14</sup> m<sup>4</sup>

η3, χ<sup>a</sup> = 10�<sup>8</sup> kg/(m s<sup>2</sup> G), H = 3000 G, the magnetic field that keeps fixed the

<sup>2</sup> <sup>3</sup>ν<sup>3</sup> <sup>þ</sup> <sup>ν</sup> <sup>0</sup> � �

� � � � <sup>p</sup> : (31)

� ν3S<sup>1</sup>

/(N s), ρ = 103 kg/m3

/Ns, ρ = 103 kg/m3

coth L ffiffiffiffiffi S1

, layer thicknesses

, and viscosities and

, η<sup>3</sup> = 1 P,

, L = 10�<sup>4</sup>

smectic liquids: K= 10�<sup>11</sup> N, B = 10<sup>6</sup> N/m<sup>2</sup>

eigenvalue Eqs. (27)–(29) with the function

<sup>k</sup> <sup>w</sup><sup>2</sup> <sup>0</sup> þ

Mn <sup>¼</sup> <sup>2</sup>

thermal transition.

4. Lyotropic liquid crystal

η<sup>3</sup> = η2, η 0 = 10�<sup>2</sup>

> Unlike lyotropic liquids, a successful continuum mean field model for thermotropic nematics has allowed the description of the rheology of the bulk of confined layers of nematics under oscillatory shear. Such a model has predicted shear thinning of the viscosity as a function of the oscillation frequency of the imposed shear. The extension of this study to understand the shear thinning effect on viscosity in the experiments of Ref. [23] seems feasible. A different perspective is obtained with computer simulations as those made by Germano et al. [38]. These researches have shown through molecular dynamics simulations on a molecular model of bulk ellipsoids with pair interaction of Weeks-Chandler-Andersen type as they say "nematic fluids may adopt inhomogeneous steady states under shear flow" [38]. Thus, shear flow modifies the molecular ordering in the liquid crystal producing changes in macroscopic viscosities like shear thinning and thickening and shear banding similarly as that found by Ballesta in their experiment of fd virus suspension under parametric instability. However, those simulations cannot be compared directly with the experiments of Ballesta [23]. Germano et al. [38] were interested in the capillary waves spectrum at the interface formed during the transition from nematic to isotropic. However, they did compare their simulations with a theory of Landau-De Gennes type for the free energy of the interface which incorporates an average director parallel to the interface. They found isotropic capillary waves that propagate at long wavelengths governed by the macroscopic surface tension. At short wavelength, however, the surface tension becomes anisotropic and depends on the wave vector. In a recent series of papers by Popa-Nita et al. [27–31], they developed a Landau-De Gennes theory to describe the capillary waves originating from thermal fluctuations, and at the interface of a ternary mixture of liquid crystal, colloid, and impurities. They considered both homeotropic (perpendicular to the interface) and also the variation of the nematic director. As in the Germano et al. method, Popa-Nita uses a free energy of the liquid crystal that predicts the bulk phase diagram, and additionally a Cahn-Hilliard equation was incorporated for taking into account the diffusion of impurities and colloids inside the liquid crystal. For such a mixture, they predicted the surface tension to decrease with the presence of colloids, whereas the impurities enhance its strength. Also, the temperature of the bulk phase transition is lowered on the pure liquid crystal nematic-isotropic transition temperature. The interfaces so formed experience thermal fluctuations. With the help of this approach, Popa-Nita were able to find that there are two regions of propagating capillary waves as it was also observed by Germano et al. in their simulation work. In the first region of long wavelengths, there is dissipation produced by shear flow and the ternary mixture behaves like an isotropic fluid which can be described by a single effective bulk viscosity. The hydrodynamic equations of the velocity field underlying that dispersion relation depends on the respective viscosity for each of the two phases formed which are

separated by a sharp interface. With appropriate boundary conditions on each thermodynamic phase, the dispersion relation of the capillary waves was predicted. The generated wave depends on the average effective constant surface tension of the nematic and isotropic interface. The propagating ripple depends on one viscosity and the compression and bending modulus of the surface. The second region corresponds to a diffuse gap of particles close to the interface and corresponds to low values of wavenumber. This wave is dominated by the relaxation of the order parameter and the surface tension which is dependent on the density variation within the diffuse zone and the inhomogeneous distribution of nematogens inside it. The boundary conditions consist of the matching of the velocity field inside the diffuse zone with that from the bulk isotropic and nematic regions. The theoretical model of Popa-Nita [27– 31] might be useful to study Faraday waves in interfaces of phase-separated regions of liquid crystals as a function of the concentration of particles. Presently, the phase transition on bulk phases of liquid crystals constitutes a large body of knowledge [62], but its effect on the dynamical responses of parametric waves on the interfaces so formed in the transition is still an open subject of research.

References

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PhysRevLett.81.4384

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### 5. Conclusion

We reviewed recent results underlying the hydrodynamics description of Faraday waves under a thermal phase transition in thermotropic nematic liquid crystals. The numerical evaluation of the effect of phase change on the critical acceleration at the onset of the instability points out its pertinent experimental observation with birefringence or surface light-scattering techniques. Consequently, other liquid crystals can be studied with this theoretical approach. In Section 4, one such experimental example of a lyotropic liquid crystal of fd virus was mentioned. Also, a correction to the conceptual framework of Sections 2 and 3 to include the effect of variations of volume fraction of particles that can lead to a phase transition can be considered in this case.

### Acknowledgements

The author acknowledges the General Coordination of Information and Communications Technologies (CGSTIC) at CINVESTAV for providing HPC resources on the Hybrid Supercomputer "Xiuhcoatl," which has contributed to the research results reported in this paper.

### Author details

Martin Hernández Contreras

Address all correspondence to: marther@fis.cinvestev.mx

Physics Department, Center for Research and Advanced Studies of the National Polytechnic Institute, CD Mexico, Mexico

### References

separated by a sharp interface. With appropriate boundary conditions on each thermodynamic phase, the dispersion relation of the capillary waves was predicted. The generated wave depends on the average effective constant surface tension of the nematic and isotropic interface. The propagating ripple depends on one viscosity and the compression and bending modulus of the surface. The second region corresponds to a diffuse gap of particles close to the interface and corresponds to low values of wavenumber. This wave is dominated by the relaxation of the order parameter and the surface tension which is dependent on the density variation within the diffuse zone and the inhomogeneous distribution of nematogens inside it. The boundary conditions consist of the matching of the velocity field inside the diffuse zone with that from the bulk isotropic and nematic regions. The theoretical model of Popa-Nita [27– 31] might be useful to study Faraday waves in interfaces of phase-separated regions of liquid crystals as a function of the concentration of particles. Presently, the phase transition on bulk phases of liquid crystals constitutes a large body of knowledge [62], but its effect on the dynamical responses of parametric waves on the interfaces so formed in the transition is still

82 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

We reviewed recent results underlying the hydrodynamics description of Faraday waves under a thermal phase transition in thermotropic nematic liquid crystals. The numerical evaluation of the effect of phase change on the critical acceleration at the onset of the instability points out its pertinent experimental observation with birefringence or surface light-scattering techniques. Consequently, other liquid crystals can be studied with this theoretical approach. In Section 4, one such experimental example of a lyotropic liquid crystal of fd virus was mentioned. Also, a correction to the conceptual framework of Sections 2 and 3 to include the effect of variations of volume fraction of particles that can lead to a phase transition can be considered in this case.

The author acknowledges the General Coordination of Information and Communications Technologies (CGSTIC) at CINVESTAV for providing HPC resources on the Hybrid Supercomputer "Xiuhcoatl," which has contributed to the research results reported in this paper.

Physics Department, Center for Research and Advanced Studies of the National Polytechnic

an open subject of research.

Acknowledgements

Author details

Martin Hernández Contreras

Institute, CD Mexico, Mexico

Address all correspondence to: marther@fis.cinvestev.mx

5. Conclusion


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jphys:01972003301010100


**Chapter 5**

**Provisional chapter**

**Lyotropic Liquid Crystals Incorporated with Different**

Liquid crystals (LCs) are considered as the "fourth state of matter," which can display properties between crystals and isotropic liquids. LCs can be classified into lyotropic liquid crystals (LLCs) and thermotropic liquid crystals (TLCs), among which LLCs are a kind of self-assemblies formed by amphiphile molecules in a given solvent within certain concentration ranges. The structures and properties of LLCs can be tuned by the incorporation of various kinds of additives, which represents an interesting and novel route for realizing functional composites. This review focuses on recent progress on LLCsbased materials assembled with diverse additives including carbon nanotubes, graphene, graphene oxide, and biomolecules. The thermal stability and mechanical strength of the host LLCs can be greatly improved after the guests are incorporated. In addition, new functions such as conductivity, photothermal effect, and bioactivity can be introduced by the incorporation of the guests, which significantly widens the applications of LLCs-based hybrids in nanotechnology, electrochemistry, drug delivery, and life

**Keywords:** liquid crystals, amphiphilic molecule, carbon nanotubes, graphene oxide,

Liquid crystals (LCs), including lyotropic liquid crystals (LLCs) and thermotropic liquid crystals (TLCs), are an intermediate state between isotropic liquid and ordered crystal [1]. They normally show the anisotropic physical properties because of the long orientational order of the molecular self-assemble aggregates. Thus, LCs are currently of great significance in nanotechnology to act as templates to devise, arrange, or even synthesize interesting composites due to the intrinsic self-assembly speciality [2–9]. Among them, LLCs were a special

**Lyotropic Liquid Crystals Incorporated with Different** 

DOI: 10.5772/intechopen.70392

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

**Kinds of Carbon Nanomaterials or Biomolecules**

**Kinds of Carbon Nanomaterials or Biomolecules**

Zhaohua Song, Yanzhao Yang and Xia Xin

Additional information is available at the end of the chapter

Zhaohua Song, Yanzhao Yang and Xia Xin

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70392

**Abstract**

science.

**1. Introduction**

biomolecule

**Provisional chapter**

### **Lyotropic Liquid Crystals Incorporated with Different Kinds of Carbon Nanomaterials or Biomolecules Kinds of Carbon Nanomaterials or Biomolecules**

**Lyotropic Liquid Crystals Incorporated with Different** 

DOI: 10.5772/intechopen.70392

Zhaohua Song, Yanzhao Yang and Xia Xin Additional information is available at the end of the chapter

Zhaohua Song, Yanzhao Yang and Xia Xin

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70392

#### **Abstract**

Liquid crystals (LCs) are considered as the "fourth state of matter," which can display properties between crystals and isotropic liquids. LCs can be classified into lyotropic liquid crystals (LLCs) and thermotropic liquid crystals (TLCs), among which LLCs are a kind of self-assemblies formed by amphiphile molecules in a given solvent within certain concentration ranges. The structures and properties of LLCs can be tuned by the incorporation of various kinds of additives, which represents an interesting and novel route for realizing functional composites. This review focuses on recent progress on LLCsbased materials assembled with diverse additives including carbon nanotubes, graphene, graphene oxide, and biomolecules. The thermal stability and mechanical strength of the host LLCs can be greatly improved after the guests are incorporated. In addition, new functions such as conductivity, photothermal effect, and bioactivity can be introduced by the incorporation of the guests, which significantly widens the applications of LLCs-based hybrids in nanotechnology, electrochemistry, drug delivery, and life science.

**Keywords:** liquid crystals, amphiphilic molecule, carbon nanotubes, graphene oxide, biomolecule

### **1. Introduction**

Liquid crystals (LCs), including lyotropic liquid crystals (LLCs) and thermotropic liquid crystals (TLCs), are an intermediate state between isotropic liquid and ordered crystal [1]. They normally show the anisotropic physical properties because of the long orientational order of the molecular self-assemble aggregates. Thus, LCs are currently of great significance in nanotechnology to act as templates to devise, arrange, or even synthesize interesting composites due to the intrinsic self-assembly speciality [2–9]. Among them, LLCs were a special

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

kind of self-assemblies of amphiphilic molecules within the scope of the long-range ordered arrangement, and their unique functions cannot be replaced by other materials. For example, the LLCs can be used as mediums for various organic and inorganic reactions that can be carried out in the hydrophilic or hydrophobic domains in confined spaces of those mesophases [10–13]. Moreover, the lamellar, hexagonal and cubic phases of LLCs can also be used as soft templates to control the structure and morphology of desired composites [14]. The Bi, PdS nanoparticles can also been synthesized by the LLCs [15]. A large variety of functional materials possessing mechanical or processing properties, biocompatibility, and so on, could be produced in this way. The LLCs have been the ideal self-organizing templates of carbon nanotubes, graphene oxide, biomolecules, and a large number of functional nanoparticles, because of their alignment regulated by the ordered matrix of LLCs [3, 16]. The preparation of LLCs-based hybrid materials has been proven to be an efficient approach to prepare ordered functional composites.

A modified method for the introduction of the CNTs into LLCs is accompanied by rigorous sonication and heating, which is complicated and harmful for the LLC matrix. The heating process may also induce the desorption of the amphiphilic molecules from the surface of the CNTs, leading to the aggregation of the CNTs [26]. To overcome these problems, Xin and coworkers developed a nondestructive strategy to incorporate SWNTs into LLCs utilizing the spontaneous phase separation between a nonionic surfactant (n-dodecyl hexaoxyethylene

phase separation, an upper phase contains hexagonal LLCs incorporated by SWNTs together with a bottom isotropic phase consisting of PEG (**Figure 1A** and **B**). The type of the LLC phase (hexagonal or lamellar phase) could be regulated by varying the ratio of PEG and C12E<sup>6</sup>

The quality of SWNTs/LLC composites was characterized by polarized microscopy (POM) observations (**Figure 1C** and **D**) and small-angle X-ray scattering (SAXS) measurements (**Figure 1E** and **F**). The d-spacing of the upper hexagonal phase was improved with increasing amount of SWNTs, which was consistent with the results obtained from POM observations.

**Figure 1.** SWNTs embedded in the upper hexagonal phase formed by C12E<sup>6</sup>

Corresponding polarized micrographs of the upper C12E<sup>6</sup>

SWNTs, and water [27].

by PEG 20,000 (20 wt%). The percent of SWNTs in the upper phase was calculated to be (A) 0 wt% and (B) 0.25 wt%.

SWNTs. (E) Small-angle diffraction rings of sample B and (F) SAXS results of SWNTs/LLC composites. (G) Schematic

representation of the phase separation process in the four-component mixture of the surfactant C12E<sup>6</sup>

) and a hydrophilic polymer (poly(ethyiene glycol), (PEG)) [27]. After

Lyotropic Liquid Crystals Incorporated with Different Kinds of Carbon Nanomaterials or...

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.

91

(10 wt%) after phase separation induced

, PEG 20,000,

hexagonal phases without (C) and with (D) incorporated

glycol monoether, C12E<sup>6</sup>

In the present chapter, a deep review on structure and property of LLCs-based materials is presented. Emphasis will be put on the hybrids composed of LLCs of amphiphilic molecules incorporated with different kinds of additives including carbon nanotubes, graphene oxide, and biomolecules.

### **2. Carbon nanotubes/LLC hybrid materials**

Carbon nanotubes (CNTs) possess wide potential applications in physics, chemistry, and material and life sciences due to their distinct cylindrical π-conjugated structure and remarkable mechanical, electronic, optical, and thermal properties [17, 18]. The orienting and aligning of CNTs in CNTs-based composites is crucial for the advancement of their applications. Great efforts have been paid to disperse single-walled carbon nanotubes (SWNTs) into aqueous solutions by noncovalent method typically with surfactants, polymers, and polyelectrolytes [19, 20]. One of the efficient methods to develop aligned CNTs is to make use of the order and fluidity of LLCs [21]. The incorporation of CNTs in LLCs has achieved well-dispersed and uniformly-aligned CNTs, which are two key prerequisites during the application of CNTs.

Studies about the dispersion and alignment of CNTs in LLCs have been reported [4, 16, 22–24]. For example, Lagerwall et al. presented an approach of using self-organized lyotropic nematic LC to align and disperse SWNTs [21]. This work also pointed out the possibility of using other LLCs with higher orientational orders such as lamellar and hexagonal phases to align CNTs. However, the above LC composites must be presented in a test cell to avoid the fluiding of the materials. Okano et al. proposed an approach of using sulfonated polyaramide to prepare ordered SWCNTs/LLCs film. The fabrication process of this system is simple (LLCs in water) and low-cost, endowing the broad versatility performance of LLC polymer film with SWCNTs [25]. These characteristic properties also demonstrated that the LLC/SWNTs systems took an advantage over the TLC/SWCNTs systems.

Although using LLCs to align CNTs is a promising method to promote the application of CNTs, the direct incorporation of SWNTs into the LLCs with high viscosity is usually difficult. A modified method for the introduction of the CNTs into LLCs is accompanied by rigorous sonication and heating, which is complicated and harmful for the LLC matrix. The heating process may also induce the desorption of the amphiphilic molecules from the surface of the CNTs, leading to the aggregation of the CNTs [26]. To overcome these problems, Xin and coworkers developed a nondestructive strategy to incorporate SWNTs into LLCs utilizing the spontaneous phase separation between a nonionic surfactant (n-dodecyl hexaoxyethylene glycol monoether, C12E<sup>6</sup> ) and a hydrophilic polymer (poly(ethyiene glycol), (PEG)) [27]. After phase separation, an upper phase contains hexagonal LLCs incorporated by SWNTs together with a bottom isotropic phase consisting of PEG (**Figure 1A** and **B**). The type of the LLC phase (hexagonal or lamellar phase) could be regulated by varying the ratio of PEG and C12E<sup>6</sup> . The quality of SWNTs/LLC composites was characterized by polarized microscopy (POM) observations (**Figure 1C** and **D**) and small-angle X-ray scattering (SAXS) measurements (**Figure 1E** and **F**). The d-spacing of the upper hexagonal phase was improved with increasing amount of SWNTs, which was consistent with the results obtained from POM observations.

kind of self-assemblies of amphiphilic molecules within the scope of the long-range ordered arrangement, and their unique functions cannot be replaced by other materials. For example, the LLCs can be used as mediums for various organic and inorganic reactions that can be carried out in the hydrophilic or hydrophobic domains in confined spaces of those mesophases [10–13]. Moreover, the lamellar, hexagonal and cubic phases of LLCs can also be used as soft templates to control the structure and morphology of desired composites [14]. The Bi, PdS nanoparticles can also been synthesized by the LLCs [15]. A large variety of functional materials possessing mechanical or processing properties, biocompatibility, and so on, could be produced in this way. The LLCs have been the ideal self-organizing templates of carbon nanotubes, graphene oxide, biomolecules, and a large number of functional nanoparticles, because of their alignment regulated by the ordered matrix of LLCs [3, 16]. The preparation of LLCs-based hybrid materials has been proven to be an efficient approach to prepare

In the present chapter, a deep review on structure and property of LLCs-based materials is presented. Emphasis will be put on the hybrids composed of LLCs of amphiphilic molecules incorporated with different kinds of additives including carbon nanotubes, graphene oxide,

Carbon nanotubes (CNTs) possess wide potential applications in physics, chemistry, and material and life sciences due to their distinct cylindrical π-conjugated structure and remarkable mechanical, electronic, optical, and thermal properties [17, 18]. The orienting and aligning of CNTs in CNTs-based composites is crucial for the advancement of their applications. Great efforts have been paid to disperse single-walled carbon nanotubes (SWNTs) into aqueous solutions by noncovalent method typically with surfactants, polymers, and polyelectrolytes [19, 20]. One of the efficient methods to develop aligned CNTs is to make use of the order and fluidity of LLCs [21]. The incorporation of CNTs in LLCs has achieved well-dispersed and uniformly-aligned CNTs, which are two key prerequisites during the application of CNTs. Studies about the dispersion and alignment of CNTs in LLCs have been reported [4, 16, 22–24]. For example, Lagerwall et al. presented an approach of using self-organized lyotropic nematic LC to align and disperse SWNTs [21]. This work also pointed out the possibility of using other LLCs with higher orientational orders such as lamellar and hexagonal phases to align CNTs. However, the above LC composites must be presented in a test cell to avoid the fluiding of the materials. Okano et al. proposed an approach of using sulfonated polyaramide to prepare ordered SWCNTs/LLCs film. The fabrication process of this system is simple (LLCs in water) and low-cost, endowing the broad versatility performance of LLC polymer film with SWCNTs [25]. These characteristic properties also demonstrated that the LLC/SWNTs

Although using LLCs to align CNTs is a promising method to promote the application of CNTs, the direct incorporation of SWNTs into the LLCs with high viscosity is usually difficult.

ordered functional composites.

**2. Carbon nanotubes/LLC hybrid materials**

90 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

systems took an advantage over the TLC/SWCNTs systems.

and biomolecules.

**Figure 1.** SWNTs embedded in the upper hexagonal phase formed by C12E<sup>6</sup> (10 wt%) after phase separation induced by PEG 20,000 (20 wt%). The percent of SWNTs in the upper phase was calculated to be (A) 0 wt% and (B) 0.25 wt%. Corresponding polarized micrographs of the upper C12E<sup>6</sup> hexagonal phases without (C) and with (D) incorporated SWNTs. (E) Small-angle diffraction rings of sample B and (F) SAXS results of SWNTs/LLC composites. (G) Schematic representation of the phase separation process in the four-component mixture of the surfactant C12E<sup>6</sup> , PEG 20,000, SWNTs, and water [27].

Next, the incorporation of CNTs into the LLC phase formed by n-dodecyl tetraethylene monoether (C12E<sup>4</sup> ) through phase separation at the presence of PEG with different molecular weights has been systematically studied (**Figure 2**) [28]. The LLC/CNTs hybrid material maintained the lamellar organizations of the host LLCs according to POM observations and SAXS measurements. The increase of the d-spacing of the LLC/CNTs hybrid with increasing concentration of incorporated CNTs indicated that the CNTs have been successfully integrated within the layer of lamellar LLCs. UV-vis and Raman spectra further confirmed that CNTs have been incorporated into the LLC phase, which also revealed the alignment of CNTs in the LLC matrix. The mechanical strength of the hybrid material has also been improved after the introduction of CNTs.

Furthermore, the LLCs formed by ionic surfactants such as sodium dodecyl sulfate (SDS) and cetyltrimethylammonium bromide (CTAB) have also been selected as the host matrix for SWNTs incorporation. In these cases, polyelectrolytes such as poly(sodium styrenesulfonate) (PSS) or poly(diallyldimethylammonium chloride) (PDADMAC) instead of PEG were utilized to induce phase separation (**Figure 3**) [29]. The final concentration of the SWNTs in the upper LLC phase was increased a few times compared with that in the initial aqueous dispersion. Thus, it can be concluded that incorporation of SWNTs into LLC by phase separation method provides a practical way to achieve highly concentrated SWNTs aqueous

dispersion. Moreover, it is surprising to find the stability of SWNTs/LLC hybrids fabricated from ionic surfactants is much better than those prepared from nonionic surfactants, which indicates that the SWNTs/LLC hybrids prepared by the combination of ionic surfactant/poly-

Corresponding polarized micrographs of the upper SWNTs/LLC hybrids. The weight percent of SDS and PSS to water are 10 wt%/40 wt% (D), 20 wt%/30 wt% (E), and 30 wt%/20 wt% (F). (G) SAXS results of the three typical samples of A–C. (H) SAXS results of one typical sample of SWNTs in CTAB/LLC as a function of temperature. The weight fractions

Lyotropic Liquid Crystals Incorporated with Different Kinds of Carbon Nanomaterials or...

CNTs possess hollow lumens with diameters of a few to hundred nanometers, which are an ideal geometry for drug transport and delivery. However, CNTs are observed to exhibit weak infrared emissions. For diagnosis, the nanotubes must be functionalized with spectroscopically characteristic fluorescent dyes [30]. Xin and coworkers fabricated a luminescent CNTs-based hybrid material by anchoring lanthanide complexes (Eu or Tb) onto the surfaces of multi-walled CNTs (MWNTs) (**Figure 4A**). UV–vis measurements was used to demonstrate the successful coupling of Eu(III) complexes to MWCNTs (**Figure 4B**). Then, they incorporated the lumines-

liquid crystal phase (**Figure 4C**). The introduction of the Eu-MWNTs merely induced the swol-

Cellulose nanocrystals (CNCs), obtained from hydrolysis of the cellulose, have attracted considerable attention due to their anisotropic properties and their self-assembly behavior [32]. Recently, Yuan and coworkers firstly fabricated the composite films with ordered oxidized CNTs (o-CNTs) using the lyotropic nematic liquid crystals (CNLCs), and the framework of the CNLCs can be retained in the final solid films (**Figure 5**) [33]. The randomly oriented o-CNTs in the aqueous dispersion were aligned orderly due to the confinement of the liquid crystal matrix of CNCs. The composite films were endowed with the anisotropic conductivity with the help of ordered arrangement of o-CNTs and the anisotropy of the CNLCs. Their work provides a new approach of fabricating LCs/CNTs material with application in sensors

len of the hexagonal lattice, and the luminescent property was retained (**Figure 4D**).

LLC phase with a phase separation method using PEG [31].

O system after phase separation. (D–F)

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93

was remained, and the Eu-MWNTs were ordered in the hexagonal

electrolyte may find potential applications at higher temperatures.

**Figure 3.** (A–C) Photographs of three typical samples of SWNTs/SDS/PSS/H<sup>2</sup>

of CTAB and PDADMAC to water are 20 and 15 wt%, respectively [29].

cent Eu-MWCNTs into the C12E<sup>6</sup>

The hexagonal phase of C12E<sup>6</sup>

and photoelectronics.

**Figure 2.** (A) CNTs embedded in the upper LLC phase formed by C12E<sup>4</sup> (10 wt%) after phase separation induced by PEG 20,000 (20 wt%). (B) Schematic illustrations of the states of LLC/CNTs composites as a function of the concentration of incorporated CNTs in the upper LLC phase: (a) 0, (b) 0.04 wt%, (c) 0.08 wt% and (d) 0.10 wt%. (C) SAXS results of LLC/ CNTs composites as a function of the concentration of incorporated CNTs in the upper LLC phase. (D) Raman spectra excited at 1064 nm of (a) pure C12E<sup>4</sup> LLC, (b) the raw CNTs, (c) 0.05 wt% CNTs dispersed in 10 mL of 0.1 wt% C12E<sup>4</sup> and (d) C12E<sup>4</sup> LLC/CNTs composites. (E) Rheological results for LLC/CNTs composites with increasing concentration of CNTs [28].

Lyotropic Liquid Crystals Incorporated with Different Kinds of Carbon Nanomaterials or... http://dx.doi.org/10.5772/intechopen.70392 93

Next, the incorporation of CNTs into the LLC phase formed by n-dodecyl tetraethylene

92 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

weights has been systematically studied (**Figure 2**) [28]. The LLC/CNTs hybrid material maintained the lamellar organizations of the host LLCs according to POM observations and SAXS measurements. The increase of the d-spacing of the LLC/CNTs hybrid with increasing concentration of incorporated CNTs indicated that the CNTs have been successfully integrated within the layer of lamellar LLCs. UV-vis and Raman spectra further confirmed that CNTs have been incorporated into the LLC phase, which also revealed the alignment of CNTs in the LLC matrix. The mechanical strength of the hybrid material has also been improved

Furthermore, the LLCs formed by ionic surfactants such as sodium dodecyl sulfate (SDS) and cetyltrimethylammonium bromide (CTAB) have also been selected as the host matrix for SWNTs incorporation. In these cases, polyelectrolytes such as poly(sodium styrenesulfonate) (PSS) or poly(diallyldimethylammonium chloride) (PDADMAC) instead of PEG were utilized to induce phase separation (**Figure 3**) [29]. The final concentration of the SWNTs in the upper LLC phase was increased a few times compared with that in the initial aqueous dispersion. Thus, it can be concluded that incorporation of SWNTs into LLC by phase separation method provides a practical way to achieve highly concentrated SWNTs aqueous

20,000 (20 wt%). (B) Schematic illustrations of the states of LLC/CNTs composites as a function of the concentration of incorporated CNTs in the upper LLC phase: (a) 0, (b) 0.04 wt%, (c) 0.08 wt% and (d) 0.10 wt%. (C) SAXS results of LLC/ CNTs composites as a function of the concentration of incorporated CNTs in the upper LLC phase. (D) Raman spectra

LLC/CNTs composites. (E) Rheological results for LLC/CNTs composites with increasing concentration

) through phase separation at the presence of PEG with different molecular

(10 wt%) after phase separation induced by PEG

LLC, (b) the raw CNTs, (c) 0.05 wt% CNTs dispersed in 10 mL of 0.1 wt% C12E<sup>4</sup>

monoether (C12E<sup>4</sup>

after the introduction of CNTs.

**Figure 2.** (A) CNTs embedded in the upper LLC phase formed by C12E<sup>4</sup>

excited at 1064 nm of (a) pure C12E<sup>4</sup>

and (d) C12E<sup>4</sup>

of CNTs [28].

**Figure 3.** (A–C) Photographs of three typical samples of SWNTs/SDS/PSS/H<sup>2</sup> O system after phase separation. (D–F) Corresponding polarized micrographs of the upper SWNTs/LLC hybrids. The weight percent of SDS and PSS to water are 10 wt%/40 wt% (D), 20 wt%/30 wt% (E), and 30 wt%/20 wt% (F). (G) SAXS results of the three typical samples of A–C. (H) SAXS results of one typical sample of SWNTs in CTAB/LLC as a function of temperature. The weight fractions of CTAB and PDADMAC to water are 20 and 15 wt%, respectively [29].

dispersion. Moreover, it is surprising to find the stability of SWNTs/LLC hybrids fabricated from ionic surfactants is much better than those prepared from nonionic surfactants, which indicates that the SWNTs/LLC hybrids prepared by the combination of ionic surfactant/polyelectrolyte may find potential applications at higher temperatures.

CNTs possess hollow lumens with diameters of a few to hundred nanometers, which are an ideal geometry for drug transport and delivery. However, CNTs are observed to exhibit weak infrared emissions. For diagnosis, the nanotubes must be functionalized with spectroscopically characteristic fluorescent dyes [30]. Xin and coworkers fabricated a luminescent CNTs-based hybrid material by anchoring lanthanide complexes (Eu or Tb) onto the surfaces of multi-walled CNTs (MWNTs) (**Figure 4A**). UV–vis measurements was used to demonstrate the successful coupling of Eu(III) complexes to MWCNTs (**Figure 4B**). Then, they incorporated the luminescent Eu-MWCNTs into the C12E<sup>6</sup> LLC phase with a phase separation method using PEG [31]. The hexagonal phase of C12E<sup>6</sup> was remained, and the Eu-MWNTs were ordered in the hexagonal liquid crystal phase (**Figure 4C**). The introduction of the Eu-MWNTs merely induced the swollen of the hexagonal lattice, and the luminescent property was retained (**Figure 4D**).

Cellulose nanocrystals (CNCs), obtained from hydrolysis of the cellulose, have attracted considerable attention due to their anisotropic properties and their self-assembly behavior [32]. Recently, Yuan and coworkers firstly fabricated the composite films with ordered oxidized CNTs (o-CNTs) using the lyotropic nematic liquid crystals (CNLCs), and the framework of the CNLCs can be retained in the final solid films (**Figure 5**) [33]. The randomly oriented o-CNTs in the aqueous dispersion were aligned orderly due to the confinement of the liquid crystal matrix of CNCs. The composite films were endowed with the anisotropic conductivity with the help of ordered arrangement of o-CNTs and the anisotropy of the CNLCs. Their work provides a new approach of fabricating LCs/CNTs material with application in sensors and photoelectronics.

**3. Graphene (or graphene oxide)/LLC hybrid materials**

to engineering high-performance nanocomposites.

Apart from one-dimensional (1D) CNTs, the two-dimensional (2D) carbon-based nanomaterials, i.e., graphene nanosheets, have also attracted a great deal of attention. Graphene, a single-layer and 2D carbon lattice, is one of the most promising materials with great potential applications due to its unique mechanical, quantum, and electrical properties [34–36]. The properties of superior dispersibility, stability, and processability in water also promote the exploration of graphene assemblies [37]. For example, Behabtu et al. reported the formation of LC phase of graphene at high concentrations (~20–30 mg ml−1) [38]. The graphite can be spontaneously exfoliated into single-layer graphene in chlorosulphonic acid without the need for surfactant stabilization and sonication. The LC phases are promising for functionalization, and for scalable manufacturing of nanocomposites, films, coatings, and highperformance fibers. The oxidized derivative of graphene, i.e., graphene oxide (GO), is a good hydrophilic and biocompatible material. Due to the presence of a variety of hydroxyl, epoxide, and carbonyl groups at their basal planes and edges, GO has been regarded as the most important substitute for graphene to form stable colloidal carbon-based composites in water and polar organic solvents [39–43]. Kim et al. prepared nematic LCs of exfoliated graphene oxide by a modified Hummer's method [44]. The ionic strength and pH are the influencing factors on the stability of the LCs. They also successfully tuned the macroscopic orientation of GO in LCs by applying a magnetic field (**Figure 6**). This method provided a viable route

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**Figure 6.** (A) Magnetic-field-induced alignment of graphene oxide liquid crystals. a) Shear-induced birefringent morphology formed after sample preparation. b) Nematic schlieren morphology formed about 3 h after sample preparation without any external field. c) Top: experimental scheme for magnetic field application; bottom: magneticfield-induced highly aligned liquid-crystal texture. (B) Mechanical-deformation-induced alignment of PAA/grapheme oxide composites. Water/PAA/graphene oxide three-component liquidcrystal mixtures: a) without and b) with crossed polarizers. c) Handdrawn gel composite fiber. The strong optical birefringence was caused by homogeneously dispersed, uniaxially oriented graphene oxide platelets. d) Highly aligned graphene oxide morphology along the fiber axis. e)

Randomly oriented graphene oxide morphology in the fiber cross section [44].

**Figure 4.** (A) Synthetic procedure for Eu(III) and Tb(III)-coupled MWNTs (Eu or Tb-MWNTs). (B) UV-vis spectra of Eu-MWNTs (a) and ligand-modified MWNTs (b) dispersed in 1 wt% SDS. (C) Typical SAXS results of the upper condensed phase formed by C12E<sup>6</sup> at the presence of PEG without (a) and with (b) Eu-MWNTs incorporated. The dotted lines are guides for the eyes to highlight the peaks. (D) A typical image obtained under a fluorescence microscope for the upper C12E<sup>6</sup> hexagonal phase incorporated with 0.1 wt% Eu-MWNTs [31].

**Figure 5.** The composite films with ordered CNTs using the chiral nematic liquid crystals (CNLCs) prepared by the cellulose nanocrystals (CNCs) [33].

### **3. Graphene (or graphene oxide)/LLC hybrid materials**

Apart from one-dimensional (1D) CNTs, the two-dimensional (2D) carbon-based nanomaterials, i.e., graphene nanosheets, have also attracted a great deal of attention. Graphene, a single-layer and 2D carbon lattice, is one of the most promising materials with great potential applications due to its unique mechanical, quantum, and electrical properties [34–36]. The properties of superior dispersibility, stability, and processability in water also promote the exploration of graphene assemblies [37]. For example, Behabtu et al. reported the formation of LC phase of graphene at high concentrations (~20–30 mg ml−1) [38]. The graphite can be spontaneously exfoliated into single-layer graphene in chlorosulphonic acid without the need for surfactant stabilization and sonication. The LC phases are promising for functionalization, and for scalable manufacturing of nanocomposites, films, coatings, and highperformance fibers. The oxidized derivative of graphene, i.e., graphene oxide (GO), is a good hydrophilic and biocompatible material. Due to the presence of a variety of hydroxyl, epoxide, and carbonyl groups at their basal planes and edges, GO has been regarded as the most important substitute for graphene to form stable colloidal carbon-based composites in water and polar organic solvents [39–43]. Kim et al. prepared nematic LCs of exfoliated graphene oxide by a modified Hummer's method [44]. The ionic strength and pH are the influencing factors on the stability of the LCs. They also successfully tuned the macroscopic orientation of GO in LCs by applying a magnetic field (**Figure 6**). This method provided a viable route to engineering high-performance nanocomposites.

**Figure 6.** (A) Magnetic-field-induced alignment of graphene oxide liquid crystals. a) Shear-induced birefringent morphology formed after sample preparation. b) Nematic schlieren morphology formed about 3 h after sample preparation without any external field. c) Top: experimental scheme for magnetic field application; bottom: magneticfield-induced highly aligned liquid-crystal texture. (B) Mechanical-deformation-induced alignment of PAA/grapheme oxide composites. Water/PAA/graphene oxide three-component liquidcrystal mixtures: a) without and b) with crossed polarizers. c) Handdrawn gel composite fiber. The strong optical birefringence was caused by homogeneously dispersed, uniaxially oriented graphene oxide platelets. d) Highly aligned graphene oxide morphology along the fiber axis. e) Randomly oriented graphene oxide morphology in the fiber cross section [44].

**Figure 5.** The composite films with ordered CNTs using the chiral nematic liquid crystals (CNLCs) prepared by

**Figure 4.** (A) Synthetic procedure for Eu(III) and Tb(III)-coupled MWNTs (Eu or Tb-MWNTs). (B) UV-vis spectra of Eu-MWNTs (a) and ligand-modified MWNTs (b) dispersed in 1 wt% SDS. (C) Typical SAXS results of the upper

lines are guides for the eyes to highlight the peaks. (D) A typical image obtained under a fluorescence microscope

hexagonal phase incorporated with 0.1 wt% Eu-MWNTs [31].

94 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

at the presence of PEG without (a) and with (b) Eu-MWNTs incorporated. The dotted

the cellulose nanocrystals (CNCs) [33].

condensed phase formed by C12E<sup>6</sup>

for the upper C12E<sup>6</sup>

LLC phases containing graphene and GO sheets have received considerable attention because of the new dimension to soft self-assembly science. A series of LLCs containing graphene and GO have been fabricated employing novel amphiphilic molecules, which achieved functional materials with enhanced properties, self-assembly, and alignment of graphene and GO [45, 46]. Pasquali et al. explored the lyotropic phase behavior of giant GO flakes in water with an order of magnitude higher than other works in 2011 [47]. Wallace and coworkers produced the GO LLC phase in a wide range of organic solvents (ethanol, acetone, tetrahydrofuran, and some other organic solvents) fully using the ultralarge GO sheets (**Figure 7A**) [48]. The GO LLC phase could be used to take the exploitation of organizing and aligning SWNTs through the addition of LC GO to the SWNT dispersions (**Figure 7B**). The presence of fine interband transitions of the UV/vis-near-IR spectra of the LC GO and LC GO-SWNT demonstrated that SWNT sizes have been preserved in the composite formulation (**Figure 7C**).

Xin and coworkers successfully incorporated graphene and GO into LLC matrix con-

and GO were all fully exfoliated and well-dispersed in aqueous solutions, indicating that the carbon nanosheets can be efficiently stabilized by the steric repulsions created by

structures. The strength of the composite could be enhanced by the addition of a small amount of well dispersed graphene and GO. A phase separation method was taken to demonstrate the difference between the interaction mechanisms of graphene and GO with C12E<sup>4</sup> LLC because of the different nature of graphene and GO. The schematic illustration of gra-

phene and GO facilitated the manipulation and processing of graphene and GO in nano-

The graphene (or GO)/LLC matrix could be facilely regulated by changing its composition, for example, the type of amphiphilic molecules. Following the above work, Xin and coworkers successfully incorporated GO into a hybrid LLC matrix constructed by the mixture of C12E<sup>4</sup> and 1-dodecyl-3-methylimidazolium bromide ionic liquid (C12mimBr) [50]. The GO was well-

its a hexagonal structure. GO could not only improve the mechanical properties, but also tune the phase state of hybrid LLC matrixes from lamellar to hexagonal state. The addition of large

indicated that the improved mechanical and electrical properties of the C12E<sup>4</sup>

observations and SAXS results, which indicated that GO/60 wt% C12E<sup>4</sup>

amounts of C12mimBr greatly increased the thermal stability of GO/C12E<sup>4</sup>

**Figure 8.** Graphene (A) and graphene oxide (B) incorporated into the LLC phase formed by 35 wt% C12E<sup>4</sup>

(H, I) LLC composites [49].

percent of graphene and GO is 0 (a), 0.03 (b), 0.09 (c), 0.15 (d), 0.20 (e) mg mL-1 in (A) and 0 (a), 0.1 (b), 0.5 (c), 1.0 (d) 1.5 (e) mg mL-1 in (B), respectively. (C, D) Typical TEM images of grapheme and GO dispersed in water. POM images

 (**Figure 8C** and **D**). According to the polarized optical microscope (POM) and smallangle X-ray scattering (SAXS) results, all of the composites including graphene–C12E<sup>4</sup>

(**Figure 8A and B**) [49]. Typical TEM images showed that graphene

Lyotropic Liquid Crystals Incorporated with Different Kinds of Carbon Nanomaterials or...

/C12mimBr hybrid LLC matrixes at room temperature according to POM

(**Figure 8H** and **I**) are characteristic of lamellar

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LLC phase is presented in **Figure 9**. The results

LLC by gra-

97

/10 wt% C12mimBr LLC

/C12mimBr LLC

. The weight

/60 wt% C12mimBr LLC composite exhib-

structed by C12E<sup>4</sup>

dispersed in the C12E<sup>4</sup>

for graphene–C12E<sup>4</sup>

(E–G) and GO–C12E<sup>4</sup>

(**Figure 8E**–**G**) and graphene oxide–C12E<sup>4</sup>

phene and GO incorporated into the C12E<sup>4</sup>

technology, electrochemical and biochemical areas.

composite is the lamellar phase and GO/10 wt% C12E<sup>4</sup>

C12E<sup>4</sup>

**Figure 7.** (A) Representative POM images of LC GO in various organic solvents at a concentration of 2.5 mg mL−1. (B) (a) Representative photograph of a flexible free-standing paper of LC GO-SWNT made by cast drying method. (b) SEM image of the cross section of as-cast dried LC GO-SWNT paper. (c) SEM image of the surface of the layer-by-layer composite, which is marked as region (i) in (b). Some of the SWNTs are laid on the surface of the paper (white arrow), while others are placed between layers of GO sheets (black arrow). Transparency of the monolayer/few layers of GO sheets allows observing tube sites in different layers. (d-f) Cross section of composite paper at different magnifications (marked as (ii) in (b)) confirmed the self-oriented nature of the composite as well as maintaining SWNTs debundled after the fabrication of composite. (C) POM micrographs of LC GO-SWNTs/CHP (a), (b) LC GO-SWNTs/DMF. (c) UV-visnear-IR spectra of SWNTs and LC GO dispersions before and after mixing together [48].

Xin and coworkers successfully incorporated graphene and GO into LLC matrix constructed by C12E<sup>4</sup> (**Figure 8A and B**) [49]. Typical TEM images showed that graphene and GO were all fully exfoliated and well-dispersed in aqueous solutions, indicating that the carbon nanosheets can be efficiently stabilized by the steric repulsions created by C12E<sup>4</sup> (**Figure 8C** and **D**). According to the polarized optical microscope (POM) and smallangle X-ray scattering (SAXS) results, all of the composites including graphene–C12E<sup>4</sup> (**Figure 8E**–**G**) and graphene oxide–C12E<sup>4</sup> (**Figure 8H** and **I**) are characteristic of lamellar structures. The strength of the composite could be enhanced by the addition of a small amount of well dispersed graphene and GO. A phase separation method was taken to demonstrate the difference between the interaction mechanisms of graphene and GO with C12E<sup>4</sup> LLC because of the different nature of graphene and GO. The schematic illustration of graphene and GO incorporated into the C12E<sup>4</sup> LLC phase is presented in **Figure 9**. The results indicated that the improved mechanical and electrical properties of the C12E<sup>4</sup> LLC by graphene and GO facilitated the manipulation and processing of graphene and GO in nanotechnology, electrochemical and biochemical areas.

LLC phases containing graphene and GO sheets have received considerable attention because of the new dimension to soft self-assembly science. A series of LLCs containing graphene and GO have been fabricated employing novel amphiphilic molecules, which achieved functional materials with enhanced properties, self-assembly, and alignment of graphene and GO [45, 46]. Pasquali et al. explored the lyotropic phase behavior of giant GO flakes in water with an order of magnitude higher than other works in 2011 [47]. Wallace and coworkers produced the GO LLC phase in a wide range of organic solvents (ethanol, acetone, tetrahydrofuran, and some other organic solvents) fully using the ultralarge GO sheets (**Figure 7A**) [48]. The GO LLC phase could be used to take the exploitation of organizing and aligning SWNTs through the addition of LC GO to the SWNT dispersions (**Figure 7B**). The presence of fine interband transitions of the UV/vis-near-IR spectra of the LC GO and LC GO-SWNT demonstrated that SWNT sizes have been preserved in the composite formula-

96 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

**Figure 7.** (A) Representative POM images of LC GO in various organic solvents at a concentration of 2.5 mg mL−1. (B) (a) Representative photograph of a flexible free-standing paper of LC GO-SWNT made by cast drying method. (b) SEM image of the cross section of as-cast dried LC GO-SWNT paper. (c) SEM image of the surface of the layer-by-layer composite, which is marked as region (i) in (b). Some of the SWNTs are laid on the surface of the paper (white arrow), while others are placed between layers of GO sheets (black arrow). Transparency of the monolayer/few layers of GO sheets allows observing tube sites in different layers. (d-f) Cross section of composite paper at different magnifications (marked as (ii) in (b)) confirmed the self-oriented nature of the composite as well as maintaining SWNTs debundled after the fabrication of composite. (C) POM micrographs of LC GO-SWNTs/CHP (a), (b) LC GO-SWNTs/DMF. (c) UV-vis-

near-IR spectra of SWNTs and LC GO dispersions before and after mixing together [48].

tion (**Figure 7C**).

The graphene (or GO)/LLC matrix could be facilely regulated by changing its composition, for example, the type of amphiphilic molecules. Following the above work, Xin and coworkers successfully incorporated GO into a hybrid LLC matrix constructed by the mixture of C12E<sup>4</sup> and 1-dodecyl-3-methylimidazolium bromide ionic liquid (C12mimBr) [50]. The GO was welldispersed in the C12E<sup>4</sup> /C12mimBr hybrid LLC matrixes at room temperature according to POM observations and SAXS results, which indicated that GO/60 wt% C12E<sup>4</sup> /10 wt% C12mimBr LLC composite is the lamellar phase and GO/10 wt% C12E<sup>4</sup> /60 wt% C12mimBr LLC composite exhibits a hexagonal structure. GO could not only improve the mechanical properties, but also tune the phase state of hybrid LLC matrixes from lamellar to hexagonal state. The addition of large amounts of C12mimBr greatly increased the thermal stability of GO/C12E<sup>4</sup> /C12mimBr LLC

**Figure 8.** Graphene (A) and graphene oxide (B) incorporated into the LLC phase formed by 35 wt% C12E<sup>4</sup> . The weight percent of graphene and GO is 0 (a), 0.03 (b), 0.09 (c), 0.15 (d), 0.20 (e) mg mL-1 in (A) and 0 (a), 0.1 (b), 0.5 (c), 1.0 (d) 1.5 (e) mg mL-1 in (B), respectively. (C, D) Typical TEM images of grapheme and GO dispersed in water. POM images for graphene–C12E<sup>4</sup> (E–G) and GO–C12E<sup>4</sup> (H, I) LLC composites [49].

and there was no obvious structural change during the heating or cooling process (**Figure 10**). **Figure 11** gives the schematic illustration of the formation of GO/C12E<sup>4</sup> /C12mimBr lamellar and hexagonal LLC composites.

**Figure 9.** Schematic illustration of graphene and GO incorporated into the C12E<sup>4</sup> LLC phase. Graphene incorporates into the hydrophobic layers of the C12E<sup>4</sup> LLC phase while GO tends to stay in the hydrophilic layers [49].

**4. Manipulation of LLC behavior of amphiphilic molecules by** 

Apart from varying the type of amphiphilic molecules, the structure and properties of LLC can also be tuned by changing the additives to the LLC composites. Much attention has been paid to study the effect of biomolecules to the amphiphilic molecules LLCs with new types of functions. The literatures have shown that LLCs with biomolecules are useful for biological sensing and NMR RDC analysis [51]. Clark et al. have reported the formation of LLC phases of double-stranded DNA and RNA oligomers in water [52, 53]. The formation of LLC by hydrated duplex DNA plays a crucial role in deciphering the structure of DNA and enables

/C12mimBr lamellar and hexagonal LLC composites [50].

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Amino acids are the basic units of proteins and peptides, which give the proteins and peptides specific molecular structure features [54]. Oligomers of the β-amino acids (β-peptide), a second class of biomolecules, can aggregate into diverse nanostructures [55, 56]. Gellman et al. investigated the effect of β-peptide modifications on the propensity of these helical molecules to form LLC in water [57]. The side chain of β-peptides displayed an important role to well-defined nanostructures and rules for creating LLC phases, which can also endow

**biomolecules**

the alignment of the DNA chains.

**Figure 11.** Schematic illustration of the formation of GO/C12E<sup>4</sup>

**Figure 10.** POM images of liquid crystal composites as the change of temperature, *c*GO = 0.3 mg mL−1, (A) 60 wt% C12E<sup>4</sup> /10 wt% C12mimBr and (B) 10 wt% C12E<sup>4</sup> /60 wt% C12mimBr. The temperature is (A<sup>1</sup> and B<sup>1</sup> ) 20°C, (A2) 80°C, (B2) 90°C and (A<sup>3</sup> and B<sup>3</sup> ) back to 20°C from high temperature. The SAXS results of LLC composites: 60 wt% C12E<sup>4</sup> /10 wt% C12mimBr (C<sup>1</sup> and C2 ); 10 wt% C12E<sup>4</sup> /60 wt% C12mimBr (D<sup>1</sup> and D<sup>2</sup> ). The concentration of GO is 0 mg mL−1 (C1 and D<sup>1</sup> ) and 0.3 mg mL−1 (C2 and D<sup>2</sup> ) [50].

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and there was no obvious structural change during the heating or cooling process (**Figure 10**).

**Figure 10.** POM images of liquid crystal composites as the change of temperature, *c*GO = 0.3 mg mL−1, (A) 60 wt%

/60 wt% C12mimBr (D<sup>1</sup>

**Figure 9.** Schematic illustration of graphene and GO incorporated into the C12E<sup>4</sup>

/60 wt% C12mimBr. The temperature is (A<sup>1</sup>

LLC phase while GO tends to stay in the hydrophilic layers [49].

) back to 20°C from high temperature. The SAXS results of LLC composites: 60 wt% C12E<sup>4</sup>

and D<sup>2</sup>

and B<sup>1</sup>

LLC phase. Graphene incorporates into

). The concentration of GO is 0 mg mL−1 (C1

) 20°C, (A2) 80°C, (B2)

/10 wt%

 and D<sup>1</sup> )

/C12mimBr lamellar

**Figure 11** gives the schematic illustration of the formation of GO/C12E<sup>4</sup>

98 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

and hexagonal LLC composites.

C12E<sup>4</sup>

90°C and (A<sup>3</sup>

C12mimBr (C<sup>1</sup>

and 0.3 mg mL−1 (C2

/10 wt% C12mimBr and (B) 10 wt% C12E<sup>4</sup>

and D<sup>2</sup>

); 10 wt% C12E<sup>4</sup>

) [50].

and B<sup>3</sup>

and C2

the hydrophobic layers of the C12E<sup>4</sup>

**Figure 11.** Schematic illustration of the formation of GO/C12E<sup>4</sup> /C12mimBr lamellar and hexagonal LLC composites [50].

### **4. Manipulation of LLC behavior of amphiphilic molecules by biomolecules**

Apart from varying the type of amphiphilic molecules, the structure and properties of LLC can also be tuned by changing the additives to the LLC composites. Much attention has been paid to study the effect of biomolecules to the amphiphilic molecules LLCs with new types of functions. The literatures have shown that LLCs with biomolecules are useful for biological sensing and NMR RDC analysis [51]. Clark et al. have reported the formation of LLC phases of double-stranded DNA and RNA oligomers in water [52, 53]. The formation of LLC by hydrated duplex DNA plays a crucial role in deciphering the structure of DNA and enables the alignment of the DNA chains.

Amino acids are the basic units of proteins and peptides, which give the proteins and peptides specific molecular structure features [54]. Oligomers of the β-amino acids (β-peptide), a second class of biomolecules, can aggregate into diverse nanostructures [55, 56]. Gellman et al. investigated the effect of β-peptide modifications on the propensity of these helical molecules to form LLC in water [57]. The side chain of β-peptides displayed an important role to well-defined nanostructures and rules for creating LLC phases, which can also endow

**Figure 12.** (A) Acylated β-peptides containing hydrocarbon acyl tails, Ac-1 and 5-13, and biological recognition groups, 14 and 15. Model for the multistate assembly of amphiphilic β-peptides, progressing of formation of nanofibers directly from monomeric β-peptides. (B) Optical micrographs of aqueous solutions of acylated β-peptides 7, 12, and 13 between crossed polarizers [57].

the LLCs useful properties (**Figure 12**). They also demonstrated that LLC formed by the modified β-peptides was useful as NMR alignment media to small organic molecules in aqueous solution and provided initial evidence for enantiodiscrimination [58].

The interactions between the surfactants and amino acids have also been investigated, which is not only of fundamental important in theoretics but also practical in industrial applications [59–62]. Except for traditional surfactants, long-chain alkyl ionic liquids are special kinds of amphiphilic surfactant molecules which can also form LLC. Xin and coworkers systematically studied the effects of alkaline amino acids L-Arginine (L-Arg) and L-Lysine (L-Lys) on the LLC behavior of C14mimBr. C14mimBr/L-Arg system remained the hexagonal phases and merely led to the variation of the mechanical strength (**Figure 13A–F**) [63]. L-Lys could induce a transition of the C14mimBr LLC phase to worm-like micelles (WLMs). The balance among electrostatic interaction, H-bond interaction, and hydrophobic interaction between amino acids, C14mimBr and water contributes to the phase transition. The schematic illustrations for phase transition introduced by the amino acids are shown in **Figure 13G**. These changes can contribute to a better understanding of the effect of the additives on the influence of the structure and macroscopic properties of LLCs.

**5. Conclusion and outlook**

and L-Lys [63].

In summary, LCs, with their fluidity as well as long-range organization, represent an interesting and novel route for realizing functional composites. In the ordered LLC phases, the molecules tend to align along a common direction, forming orientationally ordered macroscopic domains which can provide a way to control the orientation of guest materials. Thus, different kinds of additives including SWNTs, MWNTs, graphene, GO, and biomolecules can be incorporated into LLCs, which will induce various different properties of LLCs. The mechanical, electric, physicochemical properties of the hybrid LLC materials will be improved largely after the incorporation, which will open the door for the applications of these interesting

**Figure 13.** POM images (100×) for 38 wt% C14mimBr–amino acids mixtures with different water contents at room temperature. The amount of the amino acids is: (A) 0; (B) 1.0 wt% L-Lys; (C) 5.0 wt% L-Lys; (D) 2.5 wt% L-Arg; (E) 5 wt% L-Arg; and (F) 7.5 wt% L-Arg. (G) Schematic illustrations of the phase transition between LLCs and WLMs by the L-Arg

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hybrid materials in nanotechnology, electrochemical and biochemical areas.

**Figure 13.** POM images (100×) for 38 wt% C14mimBr–amino acids mixtures with different water contents at room temperature. The amount of the amino acids is: (A) 0; (B) 1.0 wt% L-Lys; (C) 5.0 wt% L-Lys; (D) 2.5 wt% L-Arg; (E) 5 wt% L-Arg; and (F) 7.5 wt% L-Arg. (G) Schematic illustrations of the phase transition between LLCs and WLMs by the L-Arg and L-Lys [63].

### **5. Conclusion and outlook**

the LLCs useful properties (**Figure 12**). They also demonstrated that LLC formed by the modified β-peptides was useful as NMR alignment media to small organic molecules in aqueous

**Figure 12.** (A) Acylated β-peptides containing hydrocarbon acyl tails, Ac-1 and 5-13, and biological recognition groups, 14 and 15. Model for the multistate assembly of amphiphilic β-peptides, progressing of formation of nanofibers directly from monomeric β-peptides. (B) Optical micrographs of aqueous solutions of acylated β-peptides 7, 12, and 13 between

The interactions between the surfactants and amino acids have also been investigated, which is not only of fundamental important in theoretics but also practical in industrial applications [59–62]. Except for traditional surfactants, long-chain alkyl ionic liquids are special kinds of amphiphilic surfactant molecules which can also form LLC. Xin and coworkers systematically studied the effects of alkaline amino acids L-Arginine (L-Arg) and L-Lysine (L-Lys) on the LLC behavior of C14mimBr. C14mimBr/L-Arg system remained the hexagonal phases and merely led to the variation of the mechanical strength (**Figure 13A–F**) [63]. L-Lys could induce a transition of the C14mimBr LLC phase to worm-like micelles (WLMs). The balance among electrostatic interaction, H-bond interaction, and hydrophobic interaction between amino acids, C14mimBr and water contributes to the phase transition. The schematic illustrations for phase transition introduced by the amino acids are shown in **Figure 13G**. These changes can contribute to a better understanding of the effect of the additives on the influence of the structure and macroscopic

solution and provided initial evidence for enantiodiscrimination [58].

100 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

properties of LLCs.

crossed polarizers [57].

In summary, LCs, with their fluidity as well as long-range organization, represent an interesting and novel route for realizing functional composites. In the ordered LLC phases, the molecules tend to align along a common direction, forming orientationally ordered macroscopic domains which can provide a way to control the orientation of guest materials. Thus, different kinds of additives including SWNTs, MWNTs, graphene, GO, and biomolecules can be incorporated into LLCs, which will induce various different properties of LLCs. The mechanical, electric, physicochemical properties of the hybrid LLC materials will be improved largely after the incorporation, which will open the door for the applications of these interesting hybrid materials in nanotechnology, electrochemical and biochemical areas.

### **Acknowledgements**

We gratefully acknowledge the financial support obtained from the National Natural Science Foundation of China (21476129, 21203109) andYoung Scholars Program of Shandong University (2016WLJH20).

[9] Kossyrev PA, Yin A, Cloutier SG, Cardimona DA, Huang DH, Alsing PM, Xu JM. Electric field tuning of plasmonic response of nanodot array in liquid crystal matrix. Nano

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### **Author details**

Zhaohua Song, Yanzhao Yang and Xia Xin\*

\*Address all correspondence to: xinx@sdu.edu.cn

National Engineering Technology Research Center for Colloidal Materials, Shandong University, Jinan, PR China

### **References**


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**Acknowledgements**

(2016WLJH20).

**Author details**

**References**

1997

University, Jinan, PR China

adma.200306196

Zhaohua Song, Yanzhao Yang and Xia Xin\*

364. DOI: 10.1002/adma.200600889

1255. DOI: 10.1021/cm020899e

DOI: 10.1021/bm049789m

\*Address all correspondence to: xinx@sdu.edu.cn

102 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

We gratefully acknowledge the financial support obtained from the National Natural Science Foundation of China (21476129, 21203109) andYoung Scholars Program of Shandong University

National Engineering Technology Research Center for Colloidal Materials, Shandong

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**Chapter 6**

Provisional chapter

**Electrical and Thermal Tuning of Band Structure and**

DOI: 10.5772/intechopen.70473

Electrical and Thermal Tuning of Band Structure and

We describe the main results previously studied concerning the thermal and electrical tuning of photonic band gap structures and the temperature-dependent defect modes in multilayer photonic liquid crystals using nematic liquid crystal slabs in a twisted configuration. In addition to this, we present new results regarding the electrical control of defect modes in such multilayer structures. In order to achieve this goal, we establish and solve numerically the equation governing the twisted nematic configurations under the action of the external electric field by assuming arbitrary anchoring conditions at the boundaries. After this, we write Maxwell's equations in a 4 4 matrix representation and, by using the matrix transfer technique, we obtain the transmittance and reflectance

Keywords: photonic band bap, electrical and thermal tuning, nematic liquid crystal,

Photonic crystals (PCs) are artificial structures with spatially periodic dielectric permittivity whose interesting optical properties have attracted the attention of the scientific community since the seminal works made by Yablonovitch [1] and John [2]. The most attractive attribute of these periodic structures is the existence of photonic band gaps (PBGs) in which the propagation of electromagnetic waves is prohibited for a specific wavelength range. In onedimensional PCs, this phenomenon is usually called Bragg reflection. Liquid crystals (LCs) are anisotropic intermediate phases between the solid and liquid states of matter that possess positional and orientational order just like those of the solid crystals, and they can flow as a

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Defect Modes in Multilayer Photonic Crystals**

Defect Modes in Multilayer Photonic Crystals

Carlos G. Avendaño, Daniel Martínez and

Carlos G. Avendaño, Daniel Martínez and

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

for incident circularly polarized waves.

multilayer structure, defect mode

http://dx.doi.org/10.5772/intechopen.70473

Ismael Molina

Ismael Molina

Abstract

1. Introduction

Provisional chapter

### **Electrical and Thermal Tuning of Band Structure and Defect Modes in Multilayer Photonic Crystals** Electrical and Thermal Tuning of Band Structure and

DOI: 10.5772/intechopen.70473

Defect Modes in Multilayer Photonic Crystals

Carlos G. Avendaño, Daniel Martínez and Ismael Molina Carlos G. Avendaño, Daniel Martínez and

Additional information is available at the end of the chapter Ismael Molina

http://dx.doi.org/10.5772/intechopen.70473 Additional information is available at the end of the chapter

#### Abstract

We describe the main results previously studied concerning the thermal and electrical tuning of photonic band gap structures and the temperature-dependent defect modes in multilayer photonic liquid crystals using nematic liquid crystal slabs in a twisted configuration. In addition to this, we present new results regarding the electrical control of defect modes in such multilayer structures. In order to achieve this goal, we establish and solve numerically the equation governing the twisted nematic configurations under the action of the external electric field by assuming arbitrary anchoring conditions at the boundaries. After this, we write Maxwell's equations in a 4 4 matrix representation and, by using the matrix transfer technique, we obtain the transmittance and reflectance for incident circularly polarized waves.

Keywords: photonic band bap, electrical and thermal tuning, nematic liquid crystal, multilayer structure, defect mode

### 1. Introduction

Photonic crystals (PCs) are artificial structures with spatially periodic dielectric permittivity whose interesting optical properties have attracted the attention of the scientific community since the seminal works made by Yablonovitch [1] and John [2]. The most attractive attribute of these periodic structures is the existence of photonic band gaps (PBGs) in which the propagation of electromagnetic waves is prohibited for a specific wavelength range. In onedimensional PCs, this phenomenon is usually called Bragg reflection. Liquid crystals (LCs) are anisotropic intermediate phases between the solid and liquid states of matter that possess positional and orientational order just like those of the solid crystals, and they can flow as a

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

conventional liquid [3]. For many decades, LCs have been used as optoelectronic substances on account of easy tunability of their properties under the stimuli of external agents as temperature, pressure and electromagnetic fields. This fact suggests the conception of new artificial structures by making a convenient combination of LCs with PCs, whose most prominent feature is the externally controlled PBG. First studies reported on this subject in threeand two-dimensional structures are attributed to Busch and John [4] and Leonard et al. [5], respectively. In [4], it was demonstrated the tunability of the PBG under the action of an external electric field meanwhile the temperature tuning when a nematic LC is infiltrated into the void regions of solid PCs is showed in [5].

the proximity of the defects (defect modes). Ozaki et al. [13] developed the first tunable PC/LC hybrid structure by using a planar aligned NLC as a defect layer sandwiched between two one-dimensional periodical multilayers (dielectric materials with high- and low-refractiveindex layers stacked alternatively) and demonstrated the electrical tuning of the defect modes. Thermal tunability of one-dimensional PC/LC cells was demonstrated by Arkhipkin et al. [14]. Electrical-dependent defect mode in PC/LC hybrid structures using a twisted nematic LC as defect layer was studied by Lin et al. [15] and Timofeev et al. [16]. Thermal tuning of defect

Electrical and Thermal Tuning of Band Structure and Defect Modes in Multilayer Photonic Crystals

http://dx.doi.org/10.5772/intechopen.70473

111

In this chapter, we describe the main results previously studied concerning the thermal and electrical tuning of PBG and the temperature-dependent defect modes in MPLCs using NLC slabs in a twisted configuration. In addition to this, we present new results regarding the electrical control of defect modes in MPLCs. In order to achieve this goal, we establish and solve numerically the equation governing the twisted nematic configurations under the action of the external electric field by assuming arbitrary anchoring conditions at the boundaries. After this, we write Maxwell's equations in a 4 � 4 matrix representation and, by using the matrix transfer technique, we obtain the transmittance and reflectance for incident circularly

As said above, we are focused on a 1D structure consisting in N NLC slabs in a twisted configuration alternated by N transparent isotropic dielectric films as it is illustrated in Figure 1 (a) and (b). For each of the NLC cells, the nematic is sandwiched between two dielectric layers in such a way that its director is aligned parallel in both frontiers. A twist is then imposed on the NLC by rotating an angle 2φt, one of the dielectric layers about its own normal direction. Because of the possibility of molecular reorientation under the influence of external stimuli,

with α(z) and φ(z), the polar (zenithal) and azimuthal angles made by n with the xy plane and the x-axis, respectively. For the present physical system and assuming small distortions in the

K2ð Þ n � ∇ � n

where the positive elastic moduli K1, K<sup>2</sup> and K<sup>3</sup> refer to splay, twist and bend bulk deformations, respectively. At this point, it is important to mention that it has experimentally found that when the nematic temperature is increasing to its transition temperature TNI (where the NLC becomes isotropic), a reduction of the values of the elastic moduli is induced [3, 17]. On the other hand, if we take into account the presence of an external electric field E, the interac-

nematic [17], the expression that describes the elastic energy density of the nematic is

tion of this field with the LC is described through the energy density <sup>f</sup> em ¼ � <sup>1</sup>

n � n½ �¼ αð Þz ;φð Þz ½ � cos αð Þz cosφð Þz ; cos αð Þz sinφð Þz ; sin αð Þz , (1)

K3ð Þ n � ∇ � n

2

, (2)

<sup>2</sup> Re <sup>E</sup> � <sup>D</sup><sup>∗</sup> f g,

modes in MPLCs using twisted nematic LC was recently shown in Ref. [8].

such as electromagnetic fields, the director n takes the general form

K1ð Þ ∇ � n

polarized waves.

2. Nematic-twisted configuration

<sup>f</sup> el <sup>¼</sup> <sup>1</sup> 2

Multilayer photonic liquid crystals (MPLCs) consisting of LCs alternated by transparent isotropic dielectric films have been previously studied. In Ref. [6], Ha et al. demonstrated experimentally simultaneous red, green and blue reflections (multiple PBGs) using the single-pitched polymeric cholesteric LC films. Later, Molina et al. [7] investigated the strong dependence of electric field on the PBG for incident waves of left- and right-circular polarization at arbitrary incidence angles using nematic liquid crystal (NLC) slabs in a twisted configuration. In a recent paper, Avendaño and Reyes [8] studied the optical band structure for reflectance and transmittance considering that the dielectric matrix of a similar one-dimensional photonic structure to that studied in Ref. [7] depends on temperature and wavelength. Twisted nematic LCs, where the molecular orientation exhibits a 90 twist, have proven technological advantages to control light flow. They have been used to switch effectively the pass of polarized light in nematic displays by means of a normally applied low-frequency electric field.

Surface anchoring plays an essential role in the science and technology of LCs. The structure of LCs in the bulk is different than that near the interface, and the boundary conditions established from this interface structure influence the behaviour of the LCs in the bulk. There are two cases of surface anchoring of particular interest. First, a strong anchoring case in which the molecules near the surface adopt a rigidly fixed orientation, and the anchoring energies are very large. Second, a weak anchoring case where the surface strengths are not strong enough to impose a well-defined molecular orientation at the interface, and the expression for the anchoring energy is some finite function that depends on the LC properties at the surface, the surface properties and the external fields (e.g., electric and magnetic fields) and temperature [9, 10]. Anchoring effects on the electrically controlled PBG in MPLCs were previously investigated by Avendaño [11]. They considered a generalization of the model studied in [7] for which arbitrary anchoring of the nematic at the boundaries is taken into account. They also found the nematic configuration versus the anchoring forces and the PBG under the action of a strong enough external field parallel to the periodicity axis, which is able to modify the configuration of the nematic-twisted LC in the whole material including at the boundaries of each nematic slab. Later, Avendaño and Martínez [12] theoretically exhibited that this system is able to produce an omnidirectional PBG that can be electrically controlled for circularly polarized incident waves. An omnidirectional PBG requires that there be no states in the given frequency range for propagation in any direction in the material for both polarizations, which implies the total reflectivity for all incident angles.

Resonant transmittance peaks in the PBG can be induced in PCs when defects are introduced in the periodic lattice. In this case, standing waves with a huge energy density are localized in the proximity of the defects (defect modes). Ozaki et al. [13] developed the first tunable PC/LC hybrid structure by using a planar aligned NLC as a defect layer sandwiched between two one-dimensional periodical multilayers (dielectric materials with high- and low-refractiveindex layers stacked alternatively) and demonstrated the electrical tuning of the defect modes. Thermal tunability of one-dimensional PC/LC cells was demonstrated by Arkhipkin et al. [14]. Electrical-dependent defect mode in PC/LC hybrid structures using a twisted nematic LC as defect layer was studied by Lin et al. [15] and Timofeev et al. [16]. Thermal tuning of defect modes in MPLCs using twisted nematic LC was recently shown in Ref. [8].

In this chapter, we describe the main results previously studied concerning the thermal and electrical tuning of PBG and the temperature-dependent defect modes in MPLCs using NLC slabs in a twisted configuration. In addition to this, we present new results regarding the electrical control of defect modes in MPLCs. In order to achieve this goal, we establish and solve numerically the equation governing the twisted nematic configurations under the action of the external electric field by assuming arbitrary anchoring conditions at the boundaries. After this, we write Maxwell's equations in a 4 � 4 matrix representation and, by using the matrix transfer technique, we obtain the transmittance and reflectance for incident circularly polarized waves.

### 2. Nematic-twisted configuration

conventional liquid [3]. For many decades, LCs have been used as optoelectronic substances on account of easy tunability of their properties under the stimuli of external agents as temperature, pressure and electromagnetic fields. This fact suggests the conception of new artificial structures by making a convenient combination of LCs with PCs, whose most prominent feature is the externally controlled PBG. First studies reported on this subject in threeand two-dimensional structures are attributed to Busch and John [4] and Leonard et al. [5], respectively. In [4], it was demonstrated the tunability of the PBG under the action of an external electric field meanwhile the temperature tuning when a nematic LC is infiltrated into

Multilayer photonic liquid crystals (MPLCs) consisting of LCs alternated by transparent isotropic dielectric films have been previously studied. In Ref. [6], Ha et al. demonstrated experimentally simultaneous red, green and blue reflections (multiple PBGs) using the single-pitched polymeric cholesteric LC films. Later, Molina et al. [7] investigated the strong dependence of electric field on the PBG for incident waves of left- and right-circular polarization at arbitrary incidence angles using nematic liquid crystal (NLC) slabs in a twisted configuration. In a recent paper, Avendaño and Reyes [8] studied the optical band structure for reflectance and transmittance considering that the dielectric matrix of a similar one-dimensional photonic structure to that studied in Ref. [7] depends on temperature and wavelength. Twisted nematic LCs, where the molecular orientation exhibits a 90 twist, have proven technological advantages to control light flow. They have been used to switch effectively the pass of polarized light in nematic

Surface anchoring plays an essential role in the science and technology of LCs. The structure of LCs in the bulk is different than that near the interface, and the boundary conditions established from this interface structure influence the behaviour of the LCs in the bulk. There are two cases of surface anchoring of particular interest. First, a strong anchoring case in which the molecules near the surface adopt a rigidly fixed orientation, and the anchoring energies are very large. Second, a weak anchoring case where the surface strengths are not strong enough to impose a well-defined molecular orientation at the interface, and the expression for the anchoring energy is some finite function that depends on the LC properties at the surface, the surface properties and the external fields (e.g., electric and magnetic fields) and temperature [9, 10]. Anchoring effects on the electrically controlled PBG in MPLCs were previously investigated by Avendaño [11]. They considered a generalization of the model studied in [7] for which arbitrary anchoring of the nematic at the boundaries is taken into account. They also found the nematic configuration versus the anchoring forces and the PBG under the action of a strong enough external field parallel to the periodicity axis, which is able to modify the configuration of the nematic-twisted LC in the whole material including at the boundaries of each nematic slab. Later, Avendaño and Martínez [12] theoretically exhibited that this system is able to produce an omnidirectional PBG that can be electrically controlled for circularly polarized incident waves. An omnidirectional PBG requires that there be no states in the given frequency range for propagation in any direction in the material for both polarizations, which implies the total reflectivity for all incident angles.

Resonant transmittance peaks in the PBG can be induced in PCs when defects are introduced in the periodic lattice. In this case, standing waves with a huge energy density are localized in

the void regions of solid PCs is showed in [5].

110 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

displays by means of a normally applied low-frequency electric field.

As said above, we are focused on a 1D structure consisting in N NLC slabs in a twisted configuration alternated by N transparent isotropic dielectric films as it is illustrated in Figure 1 (a) and (b). For each of the NLC cells, the nematic is sandwiched between two dielectric layers in such a way that its director is aligned parallel in both frontiers. A twist is then imposed on the NLC by rotating an angle 2φt, one of the dielectric layers about its own normal direction. Because of the possibility of molecular reorientation under the influence of external stimuli, such as electromagnetic fields, the director n takes the general form

$$\mathfrak{n} \equiv \mathfrak{n}[a(z), \varphi(z)] = [\cos a(z)\cos \varphi(z), \cos a(z)\sin \varphi(z), \sin a(z)]\tag{1}$$

with α(z) and φ(z), the polar (zenithal) and azimuthal angles made by n with the xy plane and the x-axis, respectively. For the present physical system and assuming small distortions in the nematic [17], the expression that describes the elastic energy density of the nematic is

$$f\_{cl} = \frac{1}{2}K\_1(\nabla \cdot \mathfrak{n})^2 + \frac{1}{2}K\_2(\mathfrak{n} \cdot \nabla \times \mathfrak{n})^2 + \frac{1}{2}K\_3(\mathfrak{n} \times \nabla \times \mathfrak{n})^2,\tag{2}$$

where the positive elastic moduli K1, K<sup>2</sup> and K<sup>3</sup> refer to splay, twist and bend bulk deformations, respectively. At this point, it is important to mention that it has experimentally found that when the nematic temperature is increasing to its transition temperature TNI (where the NLC becomes isotropic), a reduction of the values of the elastic moduli is induced [3, 17]. On the other hand, if we take into account the presence of an external electric field E, the interaction of this field with the LC is described through the energy density <sup>f</sup> em ¼ � <sup>1</sup> <sup>2</sup> Re <sup>E</sup> � <sup>D</sup><sup>∗</sup> f g,

where we have assumed that the nematic follows the constitutive relation D=ε0ε�E character-

with εa=ε∥�ε<sup>⊥</sup> the dielectric anisotropy of the medium and ε<sup>0</sup> the permittivity of free space. Here, δij is the Kronecker delta, ε<sup>⊥</sup> and ε<sup>∥</sup> denote the relative dielectric permittivity perpendicular and parallel to the nematic axis, respectively, and they are related to the ordinary no and

The study of confined nematic liquid crystals is strongly influenced by the physical properties of the boundary walls [18]. From the macroscopic-geometrical and microscopic interactions between the molecules of such surfaces and of the nematic, the alignment of the director n on the boundary surfaces, known as anchoring, can be completely determined. Once the anchoring conditions are established, the orientation of the NLC molecules at the substrate surface

Several methods and techniques for surface alignment have been developed [19, 20]. In the case of rubbed polymer films [21, 22], it has been observed that NLC molecules are strongly anchored at the surface, and the alignment is parallel to the grooves produced by the rubbing process. Also, the orientation of NLC molecules at the surfaces is preserved even if an external

On the other hand, photoalignment [23] and nanostructuring polymer surfaces [24] are contact-free methods where it is induced a surface ordering that causes an anchoring of controllable strength, which corresponds to a weak anchoring. For this anchoring condition, alignment of the NLC molecules before and after the application of external fields is different. Anchoring energy can be expressed in terms of the surface anchoring coefficients which are related to the interaction strength between the NLC and the wall substrate for the deviation of the easy axis along the correspondent directions. It is experimentally found that these coefficients are temperature dependent [18] and their values for specific NLCs can be obtained by using the dynamic light scattering [25]. Thus, if we write the director in terms of α(z) and φ(z), as in expression (1), the anchoring energy of each NLC slab can be expressed in terms of the

<sup>m</sup> <sup>þ</sup> <sup>W</sup><sup>φ</sup> cos <sup>2</sup>α<sup>L</sup>

<sup>m</sup> <sup>þ</sup> <sup>W</sup><sup>φ</sup> cos <sup>2</sup>α<sup>R</sup>

polar and azimuthal angles at the left (right) boundary of each NLC, respectively, and φ<sup>t</sup> the

Thus, strong anchoring conditions are achieved when the anchoring coefficients are sufficiently large and can be modelled by considering that Wα!∞ and Wφ!∞. In contrast, for weak anchoring conditions, it is taken that Wα!0 and Wφ!0. Another criterion to establish

<sup>m</sup> sin <sup>2</sup> φ<sup>L</sup>

<sup>m</sup> sin <sup>2</sup> φ<sup>R</sup>

<sup>m</sup> þ φ<sup>t</sup> 

<sup>m</sup> � φ<sup>t</sup>

<sup>m</sup> α<sup>R</sup> m and φ<sup>L</sup>

, (4)

<sup>m</sup> φ<sup>R</sup> m are the

<sup>o</sup> and <sup>ε</sup><sup>∥</sup> <sup>¼</sup> <sup>n</sup><sup>2</sup>

e .

Electrical and Thermal Tuning of Band Structure and Defect Modes in Multilayer Photonic Crystals

ε ¼ ε⊥δij þ εann, (3)

http://dx.doi.org/10.5772/intechopen.70473

113

ized by the uniaxial dielectric tensor

determined the director in the bulk.

extraordinary ne refractive indices by <sup>ε</sup><sup>⊥</sup> <sup>¼</sup> <sup>n</sup><sup>2</sup>

field (electric or magnetic) is applied and removed.

surface anchoring coefficients W<sup>α</sup> and W<sup>φ</sup> [26] as follows:

gL <sup>¼</sup> <sup>W</sup><sup>α</sup> sin <sup>2</sup>α<sup>L</sup>

gR <sup>¼</sup> <sup>W</sup><sup>α</sup> sin <sup>2</sup>α<sup>R</sup>

which is an extension of the Rapini-Papoular model [9] and where α<sup>L</sup>

twist angle. These anchoring coefficients are measured in energy per area units.

whether the anchoring is strong or weak is based on the extrapolation length [27].

Figure 1. (a) Schematic of a MPLC consisting of N nematic LC slabs in a twisted configuration alternated by N transparent homogeneous isotropic dielectric films with thicknesses d and h, respectively. (b) Schematic of the polar α and azimuthal φ angles made by the director n with the xy-plane and the x-axis, respectively, at the boundaries of each of nematic LC slabs; the twist angle is given by ϕ<sup>t</sup> and, at the middle of the slab, α=φ=0. (c) An obliquely incident electromagnetic field with wave vector k<sup>0</sup> impinges on the structure in the xz-plane and it makes an angle θ with respect to the z-axis. Here, aL and aR represent the amplitudes of left- and right-circularly polarized components of incident wave, respectively, and rL, rR, tL and tR correspond to those of the reflected and transmitted waves.

where we have assumed that the nematic follows the constitutive relation D=ε0ε�E characterized by the uniaxial dielectric tensor

$$
\varepsilon = \varepsilon\_\perp \delta\_{i\bar{j}} + \varepsilon\_a \mathfrak{m}\_\prime \tag{3}
$$

with εa=ε∥�ε<sup>⊥</sup> the dielectric anisotropy of the medium and ε<sup>0</sup> the permittivity of free space. Here, δij is the Kronecker delta, ε<sup>⊥</sup> and ε<sup>∥</sup> denote the relative dielectric permittivity perpendicular and parallel to the nematic axis, respectively, and they are related to the ordinary no and extraordinary ne refractive indices by <sup>ε</sup><sup>⊥</sup> <sup>¼</sup> <sup>n</sup><sup>2</sup> <sup>o</sup> and <sup>ε</sup><sup>∥</sup> <sup>¼</sup> <sup>n</sup><sup>2</sup> e .

The study of confined nematic liquid crystals is strongly influenced by the physical properties of the boundary walls [18]. From the macroscopic-geometrical and microscopic interactions between the molecules of such surfaces and of the nematic, the alignment of the director n on the boundary surfaces, known as anchoring, can be completely determined. Once the anchoring conditions are established, the orientation of the NLC molecules at the substrate surface determined the director in the bulk.

Several methods and techniques for surface alignment have been developed [19, 20]. In the case of rubbed polymer films [21, 22], it has been observed that NLC molecules are strongly anchored at the surface, and the alignment is parallel to the grooves produced by the rubbing process. Also, the orientation of NLC molecules at the surfaces is preserved even if an external field (electric or magnetic) is applied and removed.

On the other hand, photoalignment [23] and nanostructuring polymer surfaces [24] are contact-free methods where it is induced a surface ordering that causes an anchoring of controllable strength, which corresponds to a weak anchoring. For this anchoring condition, alignment of the NLC molecules before and after the application of external fields is different.

Anchoring energy can be expressed in terms of the surface anchoring coefficients which are related to the interaction strength between the NLC and the wall substrate for the deviation of the easy axis along the correspondent directions. It is experimentally found that these coefficients are temperature dependent [18] and their values for specific NLCs can be obtained by using the dynamic light scattering [25]. Thus, if we write the director in terms of α(z) and φ(z), as in expression (1), the anchoring energy of each NLC slab can be expressed in terms of the surface anchoring coefficients W<sup>α</sup> and W<sup>φ</sup> [26] as follows:

*x*

*z*

*tR*

*k*0

*z*

*z*

*tL*

*n*

**(d) (d)**

**0 d**

Figure 1. (a) Schematic of a MPLC consisting of N nematic LC slabs in a twisted configuration alternated by N transparent homogeneous isotropic dielectric films with thicknesses d and h, respectively. (b) Schematic of the polar α and azimuthal φ angles made by the director n with the xy-plane and the x-axis, respectively, at the boundaries of each of nematic LC slabs; the twist angle is given by ϕ<sup>t</sup> and, at the middle of the slab, α=φ=0. (c) An obliquely incident electromagnetic field with wave vector k<sup>0</sup> impinges on the structure in the xz-plane and it makes an angle θ with respect to the z-axis. Here, aL and aR represent the amplitudes of left- and right-circularly polarized components of incident wave,

**- <sup>t</sup>***n* **<sup>t</sup>**

**d/2**

*n*

**(0) (0)**

*Dielectric LC*

*1st-Layer Nth-Layer*

*y*

*rR*

*x*

*rL*

respectively, and rL, rR, tL and tR correspond to those of the reflected and transmitted waves.

*aL* 

*aR*

*y*

**(c)** *<sup>k</sup>*<sup>0</sup>

*k*0

*y*

**(b)**

**(a)**

*d*

*h*

*x*

112 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

$$\begin{aligned} \mathcal{g}^L &= \mathcal{W}\_a \sin^2 \alpha\_m^L + \mathcal{W}\_{\phi} \cos^2 \alpha\_m^L \sin^2 \left(\phi\_m^L + \phi\_t\right) \\ \mathcal{g}^R &= \mathcal{W}\_a \sin^2 \alpha\_m^R + \mathcal{W}\_{\phi} \cos^2 \alpha\_m^R \sin^2 \left(\phi\_m^R - \phi\_t\right) \end{aligned} \tag{4}$$

which is an extension of the Rapini-Papoular model [9] and where α<sup>L</sup> <sup>m</sup> α<sup>R</sup> m and φ<sup>L</sup> <sup>m</sup> φ<sup>R</sup> m are the polar and azimuthal angles at the left (right) boundary of each NLC, respectively, and φ<sup>t</sup> the twist angle. These anchoring coefficients are measured in energy per area units.

Thus, strong anchoring conditions are achieved when the anchoring coefficients are sufficiently large and can be modelled by considering that Wα!∞ and Wφ!∞. In contrast, for weak anchoring conditions, it is taken that Wα!0 and Wφ!0. Another criterion to establish whether the anchoring is strong or weak is based on the extrapolation length [27].

Thus, the equations governing the equilibrium configuration of the system are obtained by considering specific anchoring conditions and by minimizing the total free energy

$$F = \int\_{V} (f\_{el} + f\_{em})dV + \frac{1}{2} \int\_{S\_0} \mathbf{g}^L d\mathbf{S} + \frac{1}{2} \int\_{S\_d} \mathbf{g}^R d\mathbf{S},\tag{5}$$

same set of coupled equations given by (6) and (7) subjected to boundary conditions at each

<sup>1</sup> � <sup>Γ</sup> sin <sup>2</sup> <sup>φ</sup> <sup>þ</sup> <sup>φ</sup><sup>t</sup> � � � � sin 2α fð Þ α

<sup>1</sup> � <sup>Γ</sup> sin <sup>2</sup> <sup>φ</sup> � <sup>φ</sup><sup>t</sup> � � � � sin 2α fð Þ α

sin 2 φ þ φ<sup>t</sup>

sin 2 φ � φ<sup>t</sup>

The interaction between electromagnetic fields and matter is governed by Maxwell's equations and their corresponding constitutive equations. Optical propagation in layered media can be studied by conveniently writing Maxwell's equations in a 4�4 matrix representation. In this matrix representation, the boundary conditions of waves impinging on material can be imposed in such a way that the transfer and scattering matrix formalism to obtain the trans-

In systems where boundary conditions cannot be avoided, Maxwell's equations require the continuity of tangential components of electric E and magnetic H fields at the boundaries. In studying the optical properties of dielectric layers which are confined between parallel walls, it is useful to write the set of Maxwell's equations in a representation where only appears, at the same time, the transversal components of E and H (two components for E and two components for H). This formalism is frequently referred to as Marcuvitz-Schwinger representation [30]. If we consider that the optical properties of a multilayer structure depends only on spatial

ikxx�iω<sup>t</sup> <sup>¼</sup>

with ω the angular frequency of the propagating wave and kx the transversal component of the wave vector. Maxwell's equations, inside a nonmagnetic medium, can be written in the follow-

exð Þz eyð Þz hxð Þz hyð Þz 1

CCA e ikxx e �iωt

0

BB@

� � cos <sup>2</sup>α gð Þ α

Electrical and Thermal Tuning of Band Structure and Defect Modes in Multilayer Photonic Crystals

� � cos <sup>2</sup>α gð Þ α

� � � � z¼zL m

> � � � � z¼zR m

� � � � z¼zL m

> � � � � z¼zR m

, (11)

115

http://dx.doi.org/10.5772/intechopen.70473

, (12)

, (13)

, (14)

, (15)

layer:

dα dz � � � � z¼zL m

dα dz � � � � z¼zR m

> dφ dz � � � � z¼zL m

dφ dz � � � � z¼zR m

With Γα=1/γa,Γ=γφ/γa, γa=Wαd/K<sup>1</sup> and γφ=Wφd/K1.

3.1. 4 � 4 matrix representation

ing matrix form:

<sup>¼</sup> <sup>1</sup> d Γ<sup>α</sup>

¼ � <sup>1</sup> d Γ<sup>α</sup>

> <sup>¼</sup> <sup>Γ</sup> d Γ<sup>α</sup>

¼ � <sup>Γ</sup> d Γ<sup>α</sup>

3. Electromagnetic propagation in a layered medium

mittances and reflectances can be used in a natural way [28, 29].

variable z, we define the time-harmonic transversal four-vector

Ψð Þ¼ x; y; z ψð Þz e

that can be achieved by considering strong or weak anchoring conditions.

#### 2.1. Strong anchoring

Let us assume that the structure shown in Figure 1(a) is subjected to a DC electric field Edc= (0, 0,Edc) parallel to z-axis, and we consider that the orientation of the director at the surfaces of each nematic cell are fixed and given by α=0�, φ z<sup>L</sup> <sup>m</sup> <sup>¼</sup> ð Þ <sup>m</sup> � <sup>1</sup> ð Þ <sup>d</sup> <sup>þ</sup> <sup>h</sup> � � ¼ �φ<sup>t</sup> and <sup>φ</sup> zR <sup>m</sup> <sup>¼</sup> � ð Þ <sup>m</sup> � <sup>1</sup> ð Þþ <sup>d</sup> <sup>þ</sup> <sup>h</sup> <sup>d</sup>� ¼ <sup>φ</sup><sup>t</sup> for <sup>m</sup>=1,2,3, …,N. Here, zL <sup>m</sup> and zR <sup>m</sup> represent the positions of the left and right boundaries of the N nematic layers, respectively. Under these circumstances and by using a standard variational calculus procedure, the minimum free-energy condition δF=0 together with the restriction δn=0 at the surface of each slab generate the equations [12]

$$0 = f(a)\frac{d^2\alpha}{dz^2} + \frac{1}{2}\frac{df(a)}{da}\left(\frac{da}{dz}\right)^2 - \frac{1}{2}\frac{dg(a)}{da}\left(\frac{d\varphi}{dz}\right)^2 + \frac{1}{2}\left(\frac{\sigma}{d}\right)^2\sin 2\alpha,\tag{6}$$

$$0 = g(\alpha) \frac{d^2 \phi}{dz^2} + \frac{d g(\alpha)}{d \alpha} \frac{d \alpha}{dz} \frac{d \phi}{dz} \prime \tag{7}$$

where we have defined the dimensionless parameter <sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>ε</sup>0εaE<sup>2</sup> dc<sup>=</sup> <sup>K</sup>1=d<sup>2</sup> � � which represents the ratio between the electric and elastic energies. The functions f(α) and g(α) are defined as

$$\begin{aligned} f(a) &= \cos^2 \alpha + \frac{K\_3}{K\_1} \sin^2 \alpha \\ g(a) &= \left(\frac{K\_2}{K\_1} \cos^2 \alpha + \frac{K\_3}{K\_1} \sin^2 \alpha \right) \cos^2 \alpha. \end{aligned} \tag{8}$$

In absence of the dc electric field, the polar angle α(z)=0� for any value of z and, the solution of Eqs. (6) and (7) is simply

$$
\varphi\_m(z) = \frac{2\varphi\_t}{d}(z - (m - 1)(d + h)) - \varphi\_{t^\*} \tag{9}
$$

where φm(z) represents the configuration of the mth layer in the region (m�1)(d+h)≤ z ≤(m�1) (d+h)+d and the nematic director (1) is reduced to

$$\mathfrak{n} \equiv \mathfrak{n}[\varphi(z)] = [\cos \varphi(z), \sin \varphi(z), 0]. \tag{10}$$

#### 2.2. Weak anchoring

In this case, we consider a free-end-point variation for which the director orientation is affected by the existence of finite anchoring coefficients [12]. This minimization procedure leads to the same set of coupled equations given by (6) and (7) subjected to boundary conditions at each layer:

$$\left. \frac{d\alpha}{dz} \right|\_{z=z\_m^l} = \frac{1}{d} \frac{\left(1 - \Gamma \sin^2 \left(\varphi + \varphi\_t\right)\right) \sin 2\alpha}{f(\alpha)} \Big|\_{z=z\_m^l} \tag{11}$$

$$\left. \frac{da}{dz} \right|\_{z=z\_n^k} = -\frac{1}{d\,\Gamma\_\alpha} \frac{\left(1 - \Gamma \sin^2 \left(\varphi - \varphi\_t\right)\right) \sin 2\alpha}{f(a)} \Big|\_{z=z\_n^k}^{z} \tag{12}$$

$$\left. \frac{d\boldsymbol{\varrho}}{d\boldsymbol{z}} \right|\_{\boldsymbol{z} = \boldsymbol{z}\_{\rm m}^{l}} = \frac{\Gamma}{d} \frac{\sin 2\left(\boldsymbol{\varrho} + \boldsymbol{\varrho}\_{t}\right) \cos^{2} \boldsymbol{\alpha}}{\operatorname{g}(\boldsymbol{\alpha})} \Big|\_{\boldsymbol{z} = \boldsymbol{z}\_{\rm m}^{l}} \tag{13}$$

$$\left. \frac{d\boldsymbol{\varphi}}{d\boldsymbol{z}} \right|\_{\boldsymbol{z} = \boldsymbol{z}\_n^R} = -\frac{\Gamma}{d\,\Gamma\_\alpha} \frac{\sin 2(\boldsymbol{\varphi} - \boldsymbol{\varphi}\_t) \cos^2 \boldsymbol{a}}{\boldsymbol{g}(\boldsymbol{\alpha})} \Big|\_{\boldsymbol{z} = \boldsymbol{z}\_n^R} \tag{14}$$

With Γα=1/γa,Γ=γφ/γa, γa=Wαd/K<sup>1</sup> and γφ=Wφd/K1.

### 3. Electromagnetic propagation in a layered medium

The interaction between electromagnetic fields and matter is governed by Maxwell's equations and their corresponding constitutive equations. Optical propagation in layered media can be studied by conveniently writing Maxwell's equations in a 4�4 matrix representation. In this matrix representation, the boundary conditions of waves impinging on material can be imposed in such a way that the transfer and scattering matrix formalism to obtain the transmittances and reflectances can be used in a natural way [28, 29].

### 3.1. 4 � 4 matrix representation

Thus, the equations governing the equilibrium configuration of the system are obtained by

1 2 ð

Let us assume that the structure shown in Figure 1(a) is subjected to a DC electric field Edc= (0, 0,Edc) parallel to z-axis, and we consider that the orientation of the director at the surfaces of

and right boundaries of the N nematic layers, respectively. Under these circumstances and by using a standard variational calculus procedure, the minimum free-energy condition δF=0 together with the restriction δn=0 at the surface of each slab generate the equations [12]

> � 1 2 dgð Þ α dα

the ratio between the electric and elastic energies. The functions f(α) and g(α) are defined as

K3 K1 sin <sup>2</sup> α

� �

In absence of the dc electric field, the polar angle α(z)=0� for any value of z and, the solution of

where φm(z) represents the configuration of the mth layer in the region (m�1)(d+h)≤ z ≤(m�1)

In this case, we consider a free-end-point variation for which the director orientation is affected by the existence of finite anchoring coefficients [12]. This minimization procedure leads to the

<sup>d</sup> <sup>ð</sup><sup>z</sup> � ð Þ <sup>m</sup> � <sup>1</sup> ð Þ <sup>d</sup> <sup>þ</sup> <sup>h</sup> Þ � <sup>φ</sup><sup>t</sup>

dα dz � �<sup>2</sup>

> d2 φ dz<sup>2</sup> <sup>þ</sup> dgð Þ <sup>α</sup> dα

0 ¼ gð Þ α

where we have defined the dimensionless parameter <sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>ε</sup>0εaE<sup>2</sup>

gð Þ¼ α

φmð Þ¼ z

(d+h)+d and the nematic director (1) is reduced to

<sup>f</sup>ð Þ¼ <sup>α</sup> cos <sup>2</sup><sup>α</sup> <sup>þ</sup>

K2 K1 cos <sup>2</sup> α þ K3 K1 sin <sup>2</sup> α

2φ<sup>t</sup>

S0

gLdS <sup>þ</sup> 1 2 ð

<sup>m</sup> and zR

dφ dz � �<sup>2</sup>

dα dz dφ þ 1 2 σ d � �<sup>2</sup>

cos <sup>2</sup>α:

n � n½ �¼ φð Þz ½ � cos φð Þz ; sin φð Þz ; 0 : (10)

Sd

gRdS, (5)

<sup>m</sup> <sup>¼</sup> �

(8)

<sup>m</sup> <sup>¼</sup> ð Þ <sup>m</sup> � <sup>1</sup> ð Þ <sup>d</sup> <sup>þ</sup> <sup>h</sup> � � ¼ �φ<sup>t</sup> and <sup>φ</sup> zR

<sup>m</sup> represent the positions of the left

dz , (7)

dc<sup>=</sup> <sup>K</sup>1=d<sup>2</sup> � � which represents

, (9)

sin 2α, (6)

considering specific anchoring conditions and by minimizing the total free energy

f el þ f em � �dV <sup>þ</sup>

that can be achieved by considering strong or weak anchoring conditions.

F ¼ ð

114 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

each nematic cell are fixed and given by α=0�, φ z<sup>L</sup>

ð Þ <sup>m</sup> � <sup>1</sup> ð Þþ <sup>d</sup> <sup>þ</sup> <sup>h</sup> <sup>d</sup>� ¼ <sup>φ</sup><sup>t</sup> for <sup>m</sup>=1,2,3, …,N. Here, zL

d2 α dz<sup>2</sup> <sup>þ</sup> 1 2 dfð Þ α dα

0 ¼ fð Þ α

2.1. Strong anchoring

Eqs. (6) and (7) is simply

2.2. Weak anchoring

V

In systems where boundary conditions cannot be avoided, Maxwell's equations require the continuity of tangential components of electric E and magnetic H fields at the boundaries. In studying the optical properties of dielectric layers which are confined between parallel walls, it is useful to write the set of Maxwell's equations in a representation where only appears, at the same time, the transversal components of E and H (two components for E and two components for H). This formalism is frequently referred to as Marcuvitz-Schwinger representation [30]. If we consider that the optical properties of a multilayer structure depends only on spatial variable z, we define the time-harmonic transversal four-vector

$$\Psi(x,y,z) = \Psi(z)e^{ik\_x x - i\omega t} = \begin{pmatrix} e\_x(z) \\ e\_y(z) \\ h\_x(z) \\ h\_y(z) \end{pmatrix} e^{ik\_x x} e^{-i\omega t},\tag{15}$$

with ω the angular frequency of the propagating wave and kx the transversal component of the wave vector. Maxwell's equations, inside a nonmagnetic medium, can be written in the following matrix form:

$$\frac{\partial \Psi(z)}{\partial z} = iA(z).\Psi(z),\tag{16}$$

This propagation matrix gives the right-side field amplitudes of the multilayer structure as

Electrical and Thermal Tuning of Band Structure and Defect Modes in Multilayer Photonic Crystals

http://dx.doi.org/10.5772/intechopen.70473

117

(iii) The scattering matrix S giving the output field as function of the incident one. The matrix S is defined through the relation αout=S�αin, where αin and αout are the amplitudes of the in-

To find out S, the field must expressed, in any one of the external media, as a superposition of

The relation ψ=T�α can be interpreted as a basis change in the four dimensional space of the state vectors ψ. The columns of T are the ψ vectors representing the four plane waves generated by the incident waves in the two external media (assumed as identical). The elements of vector α are the amplitudes of the four plane wave. The choice of the new basis could be different

Uff Ubf

bb <sup>U</sup>f b <sup>U</sup>bf <sup>U</sup>�<sup>1</sup>

bb <sup>U</sup>f b <sup>U</sup>�<sup>1</sup>

bb

bb !: (22)

Uð Þ¼ α

<sup>S</sup> <sup>¼</sup> <sup>U</sup>ff � <sup>U</sup>bf <sup>U</sup>�<sup>1</sup>

�U�<sup>1</sup>

where the symbols + and f (� and b) mean forward (backward) propagating waves.

the plane wave transmission and reflection from surfaces of multilayer structures.

ψð Þ¼ z<sup>0</sup> e

We point out that the methods of transfer and scattering matrices are very useful in studying

Differential equation (16) can be formally integrated over a certain distance z<sup>0</sup> of the medium

i Ð z0 <sup>0</sup> <sup>A</sup> <sup>z</sup><sup>0</sup> ð Þdz<sup>0</sup>

and by straight comparison of Eqs. (18) and (23), the transfer matrix U(0,z0) is defined as:

i Ð z0 <sup>0</sup> <sup>A</sup> <sup>z</sup><sup>0</sup> ð Þdz<sup>0</sup>

where plane waves are incident and reflected in the half-space z < 0, and plane waves are

Uð Þ¼ 0; z<sup>0</sup> e

<sup>ψ</sup> <sup>¼</sup> <sup>T</sup> � <sup>α</sup>; <sup>U</sup>αð Þ¼ <sup>0</sup>; <sup>z</sup><sup>0</sup> <sup>T</sup>�<sup>1</sup> � <sup>U</sup>ð Þ� <sup>0</sup>; <sup>z</sup><sup>0</sup> <sup>T</sup>, (20)

<sup>U</sup>f b <sup>U</sup>bb !, (21)

� ψð Þ0 , (23)

, (24)

(ii) For a specific value z=z0, the transfer matrix is defined as U(0,z0).

function of the left-side ones.

going and out-going waves.

<sup>1</sup> ; a<sup>þ</sup> <sup>2</sup> ; a� <sup>1</sup> ; a� 2

the scattering matrix writes:

transmitted on the half-space z > z0.

� �<sup>T</sup>

.

depending on the particular problem. By setting

planes waves by setting:

where α ¼ a<sup>þ</sup>

for which the 4�4 matrix A(z) is given by

$$\mathbf{A}(z) = \begin{pmatrix} -\frac{k\_x \varepsilon\_{zx}}{k\_0 \varepsilon\_{zx}} & -\frac{k\_x \varepsilon\_{zy}}{k\_0 \varepsilon\_{zx}} & 0 & 1 - \frac{k\_x^2}{k\_0^2 \varepsilon\_{zx}}\\ 0 & 0 & -1 & 0\\ -\varepsilon\_{yx} + \frac{\varepsilon\_{yz}\varepsilon\_{zx}}{\varepsilon\_{zx}} & \frac{k\_x^2}{k\_0^2} - \varepsilon\_{yy} + \frac{\varepsilon\_{yz}\varepsilon\_{zy}}{\varepsilon\_{zx}} & 0 & \frac{k\_x \varepsilon\_{yz}}{k\_0 \varepsilon\_{zx}}\\ \varepsilon\_{xx} - \frac{\varepsilon\_{xx}\varepsilon\_{zx}}{\varepsilon\_{zx}} & \varepsilon\_{xy} - \frac{\varepsilon\_{xz}\varepsilon\_{zy}}{\varepsilon\_{zx}} & 0 & -\frac{k\_x \varepsilon\_{xz}}{k\_0 \varepsilon\_{zx}} \end{pmatrix} \tag{17}$$

where εij (i,j=x,y,z) represents the elements of dielectric matrix in the structure, k0=2π/λ=ω/c is the wavenumber in free space, λ is the wavelength and c denotes the speed of light in vacuum. Also the fields e(z)=(ex(z), ey(z), ez(z)) and h(z)=(hx(z), hy(z), hz(z)) are related to the electric <sup>E</sup>(z) and magnetic <sup>H</sup>(z) fields by the following expressions <sup>e</sup>ð Þ� <sup>z</sup> <sup>Z</sup>�1=<sup>2</sup> <sup>0</sup> Eð Þz and <sup>h</sup>ð Þ� <sup>z</sup> <sup>Z</sup><sup>1</sup>=<sup>2</sup> <sup>0</sup> <sup>H</sup>ð Þ<sup>z</sup> , with <sup>Z</sup><sup>0</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffi μ0=E<sup>0</sup> p the impedance in vacuum, ε<sup>0</sup> and μ<sup>0</sup> the permittivity and permeability of free space, respectively.

For a homogeneous and isotropic dielectric medium, the matrix ε(z) is diagonal and independent of the position, whereas for a nematic slab, ε(z) depends on the local orientation of the principal axis of the liquid crystal molecules characterized by expression (3).

#### 3.2. Boundary condition

Let us consider a multilayer structure where each of the layers is confined between two planes, and the whole structure is surrounded by air. An electromagnetic wave impinging from the left side of the multilayer structure will propagate through the sample, and it will be transmitted and reflected outside the medium (see Figure 1 (c)).

The general solution of the differential equation (16) for electromagnetic waves propagating in homogeneous media is the superposition of four plane waves: two left-going and two rightgoing waves. With this in mind, we state the procedure to find the amplitudes of the transmitted and reflected waves in terms of incident ones (at plane z=0). This implies the definition of the following quantities [31]:

(i) The propagation matrix U(0,z) that is implicitly defined by the equations

$$
\psi(z) = \mathbf{U}(0, z). \psi(0), \qquad \mathbf{U}(0, 0) = \mathbf{1}, \tag{18}
$$

where 1 is the identity matrix and U(0,z) satisfies the same propagation equation (16) found for ψ:

$$\partial\_z \mathbf{U}(0, z) = \mathbf{i} \mathbf{A}(z) \mathbf{J} \mathbf{I}(0, z) \,. \tag{19}$$

This propagation matrix gives the right-side field amplitudes of the multilayer structure as function of the left-side ones.

(ii) For a specific value z=z0, the transfer matrix is defined as U(0,z0).

(iii) The scattering matrix S giving the output field as function of the incident one. The matrix S is defined through the relation αout=S�αin, where αin and αout are the amplitudes of the ingoing and out-going waves.

To find out S, the field must expressed, in any one of the external media, as a superposition of planes waves by setting:

$$
\psi = T \cdot \mathfrak{a}; \qquad \mathfrak{U}\_{\mathfrak{a}}(0, z\_0) = T^{-1} \cdot \mathfrak{U}(0, z\_0) \cdot T, \tag{20}
$$

where α ¼ a<sup>þ</sup> <sup>1</sup> ; a<sup>þ</sup> <sup>2</sup> ; a� <sup>1</sup> ; a� 2 � �<sup>T</sup> .

∂ψð Þz

� kxεzy k0εzz

� εyy þ

where εij (i,j=x,y,z) represents the elements of dielectric matrix in the structure, k0=2π/λ=ω/c is the wavenumber in free space, λ is the wavelength and c denotes the speed of light in vacuum. Also the fields e(z)=(ex(z), ey(z), ez(z)) and h(z)=(hx(z), hy(z), hz(z)) are related to the

For a homogeneous and isotropic dielectric medium, the matrix ε(z) is diagonal and independent of the position, whereas for a nematic slab, ε(z) depends on the local orientation of the

Let us consider a multilayer structure where each of the layers is confined between two planes, and the whole structure is surrounded by air. An electromagnetic wave impinging from the left side of the multilayer structure will propagate through the sample, and it will be transmit-

The general solution of the differential equation (16) for electromagnetic waves propagating in homogeneous media is the superposition of four plane waves: two left-going and two rightgoing waves. With this in mind, we state the procedure to find the amplitudes of the transmitted and reflected waves in terms of incident ones (at plane z=0). This implies the definition of

where 1 is the identity matrix and U(0,z) satisfies the same propagation equation (16) found for ψ:

<sup>ε</sup>xy � <sup>ε</sup>xzεzy εzz

0 0 �1 0

εyzεzy εzz

0

p the impedance in vacuum, ε<sup>0</sup> and μ<sup>0</sup> the permittivity and

ψð Þ¼ z Uð Þ 0; z :ψð Þ0 , Uð Þ¼ 0; 0 1, (18)

∂zUð Þ¼ 0; z iAð Þz :Uð Þ 0; z : (19)

� kxεzx k0εzz

<sup>ε</sup>xx � <sup>ε</sup>xzεzx εzz

μ0=E<sup>0</sup>

ted and reflected outside the medium (see Figure 1 (c)).

εyzεzx εzz

k2 x k2 0

electric <sup>E</sup>(z) and magnetic <sup>H</sup>(z) fields by the following expressions <sup>e</sup>ð Þ� <sup>z</sup> <sup>Z</sup>�1=<sup>2</sup>

principal axis of the liquid crystal molecules characterized by expression (3).

(i) The propagation matrix U(0,z) that is implicitly defined by the equations

�εyx þ

for which the 4�4 matrix A(z) is given by

0

116 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

BBBBBBBBBBB@

Að Þ¼ z

<sup>0</sup> <sup>H</sup>ð Þ<sup>z</sup> , with <sup>Z</sup><sup>0</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffi

permeability of free space, respectively.

3.2. Boundary condition

the following quantities [31]:

<sup>h</sup>ð Þ� <sup>z</sup> <sup>Z</sup><sup>1</sup>=<sup>2</sup>

<sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>i</sup>Að Þ<sup>z</sup> :ψð Þ<sup>z</sup> , (16)

0 1 � <sup>k</sup>

<sup>0</sup> � kxεxz k0εzz

2 x k2 <sup>0</sup>εzz 1

CCCCCCCCCCCA

(17)

<sup>0</sup> Eð Þz and

kxεyz k0εzz

> The relation ψ=T�α can be interpreted as a basis change in the four dimensional space of the state vectors ψ. The columns of T are the ψ vectors representing the four plane waves generated by the incident waves in the two external media (assumed as identical). The elements of vector α are the amplitudes of the four plane wave. The choice of the new basis could be different depending on the particular problem. By setting

$$\mathbf{U}(\mathbf{a}) = \begin{pmatrix} \mathbf{U}\_{\varnothing} & \mathbf{U}\_{b\circ} \\ \mathbf{U}\_{\varnothing} & \mathbf{U}\_{bb} \end{pmatrix}' \tag{21}$$

the scattering matrix writes:

$$\mathbf{S} = \begin{pmatrix} \mathbf{U}\_{\sharp \prime} - \mathbf{U}\_{b\circ} \mathbf{U}\_{bb}^{-1} \mathbf{U}\_{fb} & \mathbf{U}\_{b\circ} \mathbf{U}\_{bb}^{-1} \\ -\mathbf{U}\_{bb}^{-1} \mathbf{U}\_{fb} & \mathbf{U}\_{bb}^{-1} \end{pmatrix} . \tag{22}$$

where the symbols + and f (� and b) mean forward (backward) propagating waves.

We point out that the methods of transfer and scattering matrices are very useful in studying the plane wave transmission and reflection from surfaces of multilayer structures.

Differential equation (16) can be formally integrated over a certain distance z<sup>0</sup> of the medium

$$
\Psi(z\_0) = \varepsilon^{i \int\_0^{z\_0} A(z') dz'} \cdot \Psi(0),
\tag{23}
$$

and by straight comparison of Eqs. (18) and (23), the transfer matrix U(0,z0) is defined as:

$$\mathbf{U}(0, z\_0) = \boldsymbol{\varepsilon}^{i \int\_0^{z\_0} A(z') dz'},\tag{24}$$

where plane waves are incident and reflected in the half-space z < 0, and plane waves are transmitted on the half-space z > z0.

It can be seen immediately that the problem of finding U(0,z0) is reduced to find a method to integrate expression (24) on the whole multilayer structure. Because of the non-homogeneity of the medium proposed here, we consider it as broken up into many very thin parallel layers, each of them with homogeneous anisotropic optical parameters [32]. In this way, U(0,z0) is obtained by multiplying iteratively the matrix for each sublayer from z = 0 to z = z0.

#### 3.3. Transmission and reflection by multilayer structures

As said above, the general solution of the differential equation (16) for electromagnetic waves propagating in homogeneous media is the superposition of forward and backward propagating waves. The obliquely incident and reflected electromagnetic fields in free half-space z ≤ 0 (Figure 1 (c)), for an arbitrary polarization state which are solutions of equation (16), can be expressed as:

$$
\begin{pmatrix} \mathbf{e} \\ \mathbf{h} \end{pmatrix}\_{\text{inc}} = \begin{pmatrix} [a\_L(i\mathbf{u} - \mathbf{v}\_+) - a\_R(i\mathbf{u} + \mathbf{v}\_+)] \exp\left(i\mathbf{k}\_{0z}z\right) \\ -i[a\_L(i\mathbf{u} - \mathbf{v}\_+) + a\_R(i\mathbf{u} + \mathbf{v}\_+)] \exp\left(i\mathbf{k}\_{0z}z\right) \end{pmatrix} \tag{25}
$$

P ¼

transmitted amplitudes tL and tR (for z ≥ z0).

S ¼

Q<sup>1</sup> ¼

0

BBBB@

0

BBBB@

follows

where M=P�<sup>1</sup>

where

and

0

BBBB@

cos θ cos θ cos θ cos θ �ii i �i i cos θ �i cos θ i cos θ �i cos θ 1 1 �1 �1

By using Eqs. (23), (24) and (29), the problem of reflection-transmission can be established as

aR aL rR rL

�U(0,z0)�P and U(0,z0) are defined in (24). Notice that the matrix equation (31)

1

Electrical and Thermal Tuning of Band Structure and Defect Modes in Multilayer Photonic Crystals

0

BBB@

tR tL 0 0

1

CCCA <sup>¼</sup> <sup>M</sup> �

gives a set of coupled equations relating the amplitudes aL, aR, rL and rR (from z ≤ 0) to the

The scattering matrix S relates the amplitudes tL, tR, rL and rR to the known incident ampli-

CCCA <sup>¼</sup> <sup>S</sup> � aR

aL !

0

BBB@

tudes aL and aR. This relation can be expressed in terms of matrix M as [33]

0

BBB@

1

tRR tRL tLR tLL rRR rRL rLR rLL

1

CCCCA

, Q<sup>2</sup> ¼

Co-polarized coefficients have both subscripts identical meanwhile cross-polarized coefficients have different subscripts. The square of the amplitudes of t and r is the corresponding transmittance and reflectance, respectively; thus, TRR=|tRR|2 is the co-polarized transmittance corresponding to the transmission coefficient tRR, TRL=|tRL|2 is the cross-polarized transmittance

0

BBBB@

tR tL rR rL 1

1

CCCCA

: (30)

119

http://dx.doi.org/10.5772/intechopen.70473

CCCA, (31)

: (32)

CCCCA <sup>¼</sup> <sup>Q</sup><sup>1</sup> � MQ<sup>2</sup> ð Þ�<sup>1</sup> MQ<sup>1</sup> � <sup>Q</sup><sup>2</sup> ð Þ (33)

1

CCCCA

: (34)

and

$$
\begin{pmatrix} \mathbf{e} \\ \mathbf{h} \end{pmatrix}\_{\text{ref}} = \begin{pmatrix} -[r\_L(i\mathbf{u} - \mathbf{v}\_-) - r\_R(i\mathbf{u} + \mathbf{v}\_-)] \exp\left(-i k\_{0z} z\right) \\ i[r\_L(i\mathbf{u} - \mathbf{v}\_-) + r\_R(i\mathbf{u} + \mathbf{v}\_-)] \exp\left(-i k\_{0z} z\right) \end{pmatrix} \tag{26}
$$

where k0=(k0x, k0y, k0z)=k0(sinθ,0,cosθ) is the wave vector of the incident wave making an angle θ with respect to the z-axis, aL and aR represent the amplitudes of left- and right-circularly polarized (LCP and RCP) components of incident wave, respectively, and rL and rR correspond to those of the reflected wave (see Figure 1(c)). The unit vectors u and v are defined as

$$\mathbf{u} = \frac{\mathbf{u}\_y}{\sqrt{2}}, \qquad \mathbf{v}\_{\pm} = \frac{\mp \cos \theta \mathbf{u}\_x + \sin \theta \mathbf{u}\_z}{\sqrt{2}}, \tag{27}$$

with ux,uy,uz the triad of Cartesian unit vectors. In the region z ≥ z0, the transmitted electromagnetic field is

$$
\begin{pmatrix} \mathbf{e} \\ \mathbf{h} \end{pmatrix}\_{tr} = \begin{pmatrix} [t\_{\mathcal{L}}(i\mathbf{u} - \mathbf{v}\_{+}) - t\_{\mathcal{R}}(i\mathbf{u} + \mathbf{v}\_{+})] \exp\left(i\mathbf{k}\_{\mathbb{R}\mathbb{Z}}(z - \mathcal{N}(d+h) - h)\right) \\ -i[t\_{\mathcal{L}}(i\mathbf{u} - \mathbf{v}\_{+}) + t\_{\mathcal{R}}(i\mathbf{u} + \mathbf{v}\_{+})] \exp\left(i\mathbf{k}\_{\mathbb{R}\mathbb{Z}}(z - \mathcal{N}(d+h) - h)\right) \end{pmatrix},\tag{28}
$$

where tL and tR are the amplitudes of LCP and RCP components, respectively, of transmitted wave. As the tangential components of e and h must be continuous across the planes z=0 and z=z0, the boundary values ψ(0) and ψ(z0) can be fixed as:

$$
\Psi(0) = \frac{\mathbf{P}}{\sqrt{2}} \cdot \begin{pmatrix} a\_{\mathbb{R}} \\ a\_{\mathbb{L}} \\ r\_{\mathbb{R}} \\ r\_{\mathbb{L}} \end{pmatrix} \quad \text{and} \quad \Psi(z\_0) = \frac{\mathbf{P}}{\sqrt{2}} \cdot \begin{pmatrix} t\_{\mathbb{R}} \\ t\_{\mathbb{L}} \\ 0 \\ 0 \end{pmatrix} \tag{29}
$$

with

Electrical and Thermal Tuning of Band Structure and Defect Modes in Multilayer Photonic Crystals http://dx.doi.org/10.5772/intechopen.70473 119

$$\mathbf{P} = \begin{pmatrix} \cos \theta & \cos \theta & \cos \theta & \cos \theta \\ -i & i & i & -i \\ i \cos \theta & -i \cos \theta & i \cos \theta & -i \cos \theta \\ 1 & 1 & -1 & -1 \end{pmatrix} \tag{30}$$

By using Eqs. (23), (24) and (29), the problem of reflection-transmission can be established as follows

$$
\begin{pmatrix} t\_R \\ t\_L \\ 0 \\ 0 \end{pmatrix} = \mathbf{M} \cdot \begin{pmatrix} a\_R \\ a\_L \\ r\_R \\ r\_L \end{pmatrix}' \tag{31}
$$

where M=P�<sup>1</sup> �U(0,z0)�P and U(0,z0) are defined in (24). Notice that the matrix equation (31) gives a set of coupled equations relating the amplitudes aL, aR, rL and rR (from z ≤ 0) to the transmitted amplitudes tL and tR (for z ≥ z0).

The scattering matrix S relates the amplitudes tL, tR, rL and rR to the known incident amplitudes aL and aR. This relation can be expressed in terms of matrix M as [33]

$$
\begin{pmatrix} t\_R \\ t\_L \\ r\_R \\ r\_L \end{pmatrix} = \mathbf{S} \cdot \begin{pmatrix} a\_{\mathbb{R}} \\ a\_L \end{pmatrix}. \tag{32}
$$

where

(25)

, (26)

, (28)

CCA, (29)

p , (27)

It can be seen immediately that the problem of finding U(0,z0) is reduced to find a method to integrate expression (24) on the whole multilayer structure. Because of the non-homogeneity of the medium proposed here, we consider it as broken up into many very thin parallel layers, each of them with homogeneous anisotropic optical parameters [32]. In this way, U(0,z0) is

As said above, the general solution of the differential equation (16) for electromagnetic waves propagating in homogeneous media is the superposition of forward and backward propagating waves. The obliquely incident and reflected electromagnetic fields in free half-space z ≤ 0 (Figure 1 (c)), for an arbitrary polarization state which are solutions of equation (16), can be expressed as:

> <sup>¼</sup> ½ � aLð Þ� <sup>i</sup><sup>u</sup> � <sup>v</sup><sup>þ</sup> aRð Þ <sup>i</sup><sup>u</sup> <sup>þ</sup> <sup>v</sup><sup>þ</sup> exp ð Þ ik0zz �i a½ � <sup>L</sup>ð Þþ iu � v<sup>þ</sup> aRð Þ iu þ v<sup>þ</sup> exp ð Þ ik0zz � �

<sup>¼</sup> �½ � rLð Þ� <sup>i</sup><sup>u</sup> � <sup>v</sup>� rRð Þ <sup>i</sup><sup>u</sup> <sup>þ</sup> <sup>v</sup>� exp ð Þ �ik0zz i r½ � <sup>L</sup>ð Þþ iu � v� rRð Þ iu þ v� exp ð Þ �ik0zz � �

where k0=(k0x, k0y, k0z)=k0(sinθ,0,cosθ) is the wave vector of the incident wave making an angle θ with respect to the z-axis, aL and aR represent the amplitudes of left- and right-circularly polarized (LCP and RCP) components of incident wave, respectively, and rL and rR correspond to

<sup>p</sup> , <sup>v</sup>� <sup>¼</sup> <sup>∓</sup> cos <sup>θ</sup>u<sup>x</sup> <sup>þ</sup> sin <sup>θ</sup>u<sup>z</sup> ffiffiffi

with ux,uy,uz the triad of Cartesian unit vectors. In the region z ≥ z0, the transmitted electro-

<sup>¼</sup> ½ � tLð Þ� <sup>i</sup><sup>u</sup> � <sup>v</sup><sup>þ</sup> tRð Þ <sup>i</sup><sup>u</sup> <sup>þ</sup> <sup>v</sup><sup>þ</sup> exp ð Þ ik0zð Þ <sup>z</sup> � N dð Þ� <sup>þ</sup> <sup>h</sup> <sup>h</sup> �i t½ � <sup>L</sup>ð Þþ iu � v<sup>þ</sup> tRð Þ iu þ v<sup>þ</sup> exp ð Þ ik0zð Þ z � N dð Þ� þ h h � �

where tL and tR are the amplitudes of LCP and RCP components, respectively, of transmitted wave. As the tangential components of e and h must be continuous across the planes z=0 and z=z0, the

and ψð Þ¼ z<sup>0</sup>

P ffiffiffi 2 p � tR tL 0 0

1

0

BB@

2

those of the reflected wave (see Figure 1(c)). The unit vectors u and v are defined as

obtained by multiplying iteratively the matrix for each sublayer from z = 0 to z = z0.

3.3. Transmission and reflection by multilayer structures

118 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

e h !

e h !

and

magnetic field is

with

e h !

tr

boundary values ψ(0) and ψ(z0) can be fixed as:

<sup>ψ</sup>ð Þ¼ <sup>0</sup> <sup>P</sup>

ffiffiffi 2 p � aR aL rR rL 1

CCA

0

BB@

inc

ref

<sup>u</sup> <sup>¼</sup> <sup>u</sup><sup>y</sup> ffiffiffi 2

$$\mathbf{S} = \begin{pmatrix} t\_{RR} & t\_{RL} \\ t\_{LR} & t\_{LL} \\ r\_{RR} & r\_{RL} \\ r\_{LR} & r\_{LL} \end{pmatrix} = (\mathbf{Q}\_1 - \mathbf{M}\mathbf{Q}\_2)^{-1} (\mathbf{M}\mathbf{Q}\_1 - \mathbf{Q}\_2) \tag{33}$$

and

$$\mathbf{Q}\_1 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \quad \mathbf{Q}\_2 = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}. \tag{34}$$

Co-polarized coefficients have both subscripts identical meanwhile cross-polarized coefficients have different subscripts. The square of the amplitudes of t and r is the corresponding transmittance and reflectance, respectively; thus, TRR=|tRR|2 is the co-polarized transmittance corresponding to the transmission coefficient tRR, TRL=|tRL|2 is the cross-polarized transmittance corresponding to the transmission coefficient tRL, and so forth. In the absence of dissipation of energy inside the sample, the principle of conservation of energy must be satisfied from which we have that

$$T\_{RR} + T\_{LR} + R\_{RR} + R\_{LR} = 1 \qquad \text{and} \qquad T\_{RL} + T\_{LL} + R\_{RL} + R\_{LL} = 1. \tag{35}$$

4.1.1. Strong anchoring conditions

**0 0.2 0.4**

a straight line with slope 2φt=90

value σc, α=0

**w=z/d 0.6 0.8 <sup>1</sup> <sup>0</sup>**

.

**<sup>90</sup> <sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup>**

(a) (b)

Indeed, most of molecules tend to spread far from xz plane.

For strong anchoring conditions, the orientation of the nematic molecules at the walls of each NLC is specified in Section 2.1. The curves for α(z) and φ(z) are shown in Figure 2(a) and (b), respectively, as function of dimensionless variable w = z/d above the critical value σ<sup>c</sup> = 3.26. As it can be noticed in Figure 2(a), an increment in the electric field involves the augmentation in the polar angle α. Owing to the influence of the external field, the nematic molecules tend to be aligned parallel to it (z-axis). As expected, for σ < σc, α=0 for all values of w, which means that the director in this case is perpendicular to the z-axis. In Figure 2(b), we observe that for σ < σc, the curves for azimuthal angle φ are reduced to straight lines with slope equal to 2φt=90, that corresponds to the configuration of a pure twisted NLC. Above the critical value, the strong anchoring condition is really dominant on the parameter φ as the electric field increases.

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121

Figure 3 exhibits the co-polarized and cross-polarized transmittances and reflectances for LCP and RCP waves impinging normally on the structure as function of the dimensionless parameter d/λ for continuous values of the electric field above the critical value σ<sup>c</sup> (it is worth to mention that below this value, the not-shown curves are very similar to that of σ = σc). Note the strong influence of σ on the transmission and reflection spectra in enhancing and extinguishing bands. Indeed, Figure 3 clearly shows that for σ = σc, the curves for transmittances exhibit several stop bands of different widths in the plotted interval and, as σ increases, each stop band gets wider for co-polarized transmittances TRR and TLL. Also, cross-polarized transmittances TLR and TRL are totally absent for high enough values of the electric field. On the other hand, at the critical value, co-polarized reflectances RRR and RLL exhibit narrow-reflection bands with relatively high amplitudes and cross-polarized reflectances RLR and RRL show one-dominant high-reflection band. In this case, co-polarized reflectances reduce their band amplitudes practically to zero, and reflection bands of cross-polarized reflectances are highly enhanced for larger values of σ. These optical properties allow us to use this MPLC as an electrically shiftable universal rejection filter for incident RCP and LCP waves where, by

Figure 2. (a) Curves of the polar angle α as function of dimensionless variable w at different values of σ: σ=σc+0.005 (solid line), σ=3.5 (dashed line), σ=4.5 (dotted line), σ=8 (dot-dashed line) and σ=13.5 (large dashed line). Below the critical

. (b) Curves of the azimuthal angle φ at the same values of σ as in (a). Below the critical value σc, the curve is

**0 0.2 0.4 0.6 0.8 1**

**w z d**

**0 0.2 0.4 0.6 0.8 1**

Before ending this section, we mention that an alternative way to find the transmission and reflection coefficients is using the expressions given by (21) and (22). Also, the system of equations (31) can be solved numerically to find the scattering matrix.

### 4. Numerical results and discussion

In previous sections, we have presented in detail a general mathematical formalism to determine the reflectances and transmittances by multilayer structures. In this section, we apply this formalism to MPLCs using NLC slabs in a twisted configuration considering that circularly polarized light impinges on the structure in order to analyse the optical spectra and their dependence on external agents. In particular, we describe the main results previously studied concerning the thermal and electrical tuning of optical spectra and the temperature-dependent defect modes. In addition to this, we present new results regarding the electrical control of defect modes.

#### 4.1. Electrical tuning of band structure and defect mode

In this section, we present the influence of the electric field on the optical band structure and defect mode by considering arbitrary anchoring conditions at the boundaries. To this aim, the equilibrium configuration of each NLC layer as a function of σ is obtained by solving the second order differential equations (6) and (7) for α(z) and φ(z) subjected to the conditions expressed in Eqs. (11)–(14). Then, this configuration is substituted into Eq. (23) in order to obtain the transfer matrix M as function of σ for circularly polarized incident waves.

Numerical calculations were performed by considering a NLC phase 5CB for which K1=0.62� <sup>10</sup>�11N, <sup>K</sup>2=0.39�10�11N, <sup>K</sup>3=0.82�10�11<sup>N</sup> [17] and refractive indices at optical frequencies no <sup>¼</sup> ffiffiffiffiffi ε⊥ <sup>p</sup> <sup>¼</sup> <sup>1</sup>:53 and ne <sup>¼</sup> ffiffiffiffiffi ε∥ <sup>p</sup> <sup>¼</sup> <sup>1</sup>:717. The twist angle is taken 2φt=90�, and the homogeneous isotropic dielectric medium is zinc sulphide (ZnS) with refractive index nd=2.35. The MPLC consists of N=11 NLC layers alternating with N=11 dielectric slabs with the same thickness. Finally, we report our results parameterizing all the spatial variables by the NLC thickness d. In this way, the dimensionless thickness of each NLC cell is h' =d/d=1, whereas for each ZnS slab is h' =h/d=1, and so forth.

Due to the competition between orientation produced by influence of the external electric field and by surface anchoring effects, we expect a deformation in the NLC only above a certain critical value σc. This critical electric field is expected to be maximum for the case of strong anchoring conditions, whereas for the weak anchoring case, σ<sup>c</sup> will decrease as the surface forces get smaller [7, 11].

#### 4.1.1. Strong anchoring conditions

corresponding to the transmission coefficient tRL, and so forth. In the absence of dissipation of energy inside the sample, the principle of conservation of energy must be satisfied from which

Before ending this section, we mention that an alternative way to find the transmission and reflection coefficients is using the expressions given by (21) and (22). Also, the system of

In previous sections, we have presented in detail a general mathematical formalism to determine the reflectances and transmittances by multilayer structures. In this section, we apply this formalism to MPLCs using NLC slabs in a twisted configuration considering that circularly polarized light impinges on the structure in order to analyse the optical spectra and their dependence on external agents. In particular, we describe the main results previously studied concerning the thermal and electrical tuning of optical spectra and the temperature-dependent defect modes. In addition to this, we present new results regarding the electrical control of

In this section, we present the influence of the electric field on the optical band structure and defect mode by considering arbitrary anchoring conditions at the boundaries. To this aim, the equilibrium configuration of each NLC layer as a function of σ is obtained by solving the second order differential equations (6) and (7) for α(z) and φ(z) subjected to the conditions expressed in Eqs. (11)–(14). Then, this configuration is substituted into Eq. (23) in order to

Numerical calculations were performed by considering a NLC phase 5CB for which K1=0.62� <sup>10</sup>�11N, <sup>K</sup>2=0.39�10�11N, <sup>K</sup>3=0.82�10�11<sup>N</sup> [17] and refractive indices at optical frequencies

neous isotropic dielectric medium is zinc sulphide (ZnS) with refractive index nd=2.35. The MPLC consists of N=11 NLC layers alternating with N=11 dielectric slabs with the same thickness. Finally, we report our results parameterizing all the spatial variables by the NLC

Due to the competition between orientation produced by influence of the external electric field and by surface anchoring effects, we expect a deformation in the NLC only above a certain critical value σc. This critical electric field is expected to be maximum for the case of strong anchoring conditions, whereas for the weak anchoring case, σ<sup>c</sup> will decrease as the surface

<sup>p</sup> <sup>¼</sup> <sup>1</sup>:717. The twist angle is taken 2φt=90�, and the homoge-

=d/d=1, whereas for

obtain the transfer matrix M as function of σ for circularly polarized incident waves.

equations (31) can be solved numerically to find the scattering matrix.

120 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

4. Numerical results and discussion

4.1. Electrical tuning of band structure and defect mode

ε∥

=h/d=1, and so forth.

thickness d. In this way, the dimensionless thickness of each NLC cell is h'

TRR þ TLR þ RRR þ RLR ¼ 1 and TRL þ TLL þ RRL þ RLL ¼ 1: (35)

we have that

defect modes.

no <sup>¼</sup> ffiffiffiffiffi ε⊥

each ZnS slab is h'

forces get smaller [7, 11].

<sup>p</sup> <sup>¼</sup> <sup>1</sup>:53 and ne <sup>¼</sup> ffiffiffiffiffi

For strong anchoring conditions, the orientation of the nematic molecules at the walls of each NLC is specified in Section 2.1. The curves for α(z) and φ(z) are shown in Figure 2(a) and (b), respectively, as function of dimensionless variable w = z/d above the critical value σ<sup>c</sup> = 3.26. As it can be noticed in Figure 2(a), an increment in the electric field involves the augmentation in the polar angle α. Owing to the influence of the external field, the nematic molecules tend to be aligned parallel to it (z-axis). As expected, for σ < σc, α=0 for all values of w, which means that the director in this case is perpendicular to the z-axis. In Figure 2(b), we observe that for σ < σc, the curves for azimuthal angle φ are reduced to straight lines with slope equal to 2φt=90, that corresponds to the configuration of a pure twisted NLC. Above the critical value, the strong anchoring condition is really dominant on the parameter φ as the electric field increases. Indeed, most of molecules tend to spread far from xz plane.

Figure 3 exhibits the co-polarized and cross-polarized transmittances and reflectances for LCP and RCP waves impinging normally on the structure as function of the dimensionless parameter d/λ for continuous values of the electric field above the critical value σ<sup>c</sup> (it is worth to mention that below this value, the not-shown curves are very similar to that of σ = σc). Note the strong influence of σ on the transmission and reflection spectra in enhancing and extinguishing bands. Indeed, Figure 3 clearly shows that for σ = σc, the curves for transmittances exhibit several stop bands of different widths in the plotted interval and, as σ increases, each stop band gets wider for co-polarized transmittances TRR and TLL. Also, cross-polarized transmittances TLR and TRL are totally absent for high enough values of the electric field. On the other hand, at the critical value, co-polarized reflectances RRR and RLL exhibit narrow-reflection bands with relatively high amplitudes and cross-polarized reflectances RLR and RRL show one-dominant high-reflection band. In this case, co-polarized reflectances reduce their band amplitudes practically to zero, and reflection bands of cross-polarized reflectances are highly enhanced for larger values of σ. These optical properties allow us to use this MPLC as an electrically shiftable universal rejection filter for incident RCP and LCP waves where, by

Figure 2. (a) Curves of the polar angle α as function of dimensionless variable w at different values of σ: σ=σc+0.005 (solid line), σ=3.5 (dashed line), σ=4.5 (dotted line), σ=8 (dot-dashed line) and σ=13.5 (large dashed line). Below the critical value σc, α=0 . (b) Curves of the azimuthal angle φ at the same values of σ as in (a). Below the critical value σc, the curve is a straight line with slope 2φt=90 .

Figure 3. (a)–(h) Plots of co-polarized and cross-polarized reflectances and transmittances for LCP and RCP waves impinging normally on a MPLC as function of the dimensionless parameter d/λ and continuous values of σ within the interval σ<sup>c</sup> < σ < 13.

increasing the electric field, one can highly enhance the cross-polarized reflection bands and supress the co-polarized ones.

Nevertheless, as σ increases, most of the molecules tend to be aligned parallel to z-axis (see Figure 2(a)) and β!0�. These results show that the amplitude of defect mode and its position

Figure 4. (a) At normal incidence, plots of co-polarized transmittance TRR for LCP incident waves as function of the dimensionless parameter d/λ and continuous values of σ. (b) At normal incidence, cross-polarized reflectance RLR for LCP

Electrical and Thermal Tuning of Band Structure and Defect Modes in Multilayer Photonic Crystals

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It is experimentally found that for a LC phase 5CB, the polar anchoring γα is of the order of 101

and this value is one or two orders stronger than the azimuthal anchoring γφ [34]. Under these considerations, the values of the dimensionless anchoring parameters are taken as Γ=Γα=0.1. The curves for α(z) and φ(z) are shown in Figure 5(a) and (b), respectively, as function of dimensionless variable w=z/d above the critical value σc=2.86. In Figure 5(a), we can notice that, as σ augments, the values of α increase, getting a maximum at the middle of the cell. Because of the influence of external electric field, the polar angle at both borders enlarges by increasing σ highlighting the fact that even at the borders, the field is able to distort the configuration. Figure 5 (b) shows two interesting phenomena: (i) for σ < σc, the curves are reduced to straight lines with slope equal to 2φ0c, where φ0<sup>c</sup> represents the azimuthal angle adopted by the MPLC at the walls of each NLC cell for values of electric field below the critical field; (ii) above the critical value, most of the molecules tend to acquire an angle φt=�45� for 0< w < 0.5 and φ<sup>t</sup> = 45� for 0.5 < w<1. Figure 6 shows the co-polarized and cross-polarized transmittances and reflectances for LCP and RCP waves impinging normally on the structure as function of the dimensionless parameter d/λ for continuous values of the electric field above the critical value σ<sup>c</sup> (below this value, the not-shown curves are very similar to those corresponding to σ=σc). Although, the optical properties shown in Figure 6 are qualitatively similar to those of Figure 3 where strong anchoring conditions were considered, we notice that in the case of weak anchoring conditions, the behaviour of transmittances and reflectances in Figure 6 is enhanced in comparison with Figure 3. Because of the strong influence of the electric field on the molecular orientation

,

123

can be tuned by a DC electric field.

incident waves at the same values of σ as in (a).

4.1.2. Weak anchoring conditions

In [7], it is shown that for a fixed value of σ the band structure of the reflectances and transmittances are shifted towards smaller wavelength regions as the incident angle θ increases. This behaviour results from the fact that for plane electromagnetic waves propagating obliquely with respect to the layer interfaces, only the normal component of the wave vector is involved in the photonic band formation. Hence, as the incident angle augments, the relative position of the bands is moved towards smaller wavelengths.

If one of the layers possesses a different size compared with the remaining ones, this layer can act as a defect, and an optical defect mode can be induced. Here, we specifically consider that the middle NLC-ZnS stack of the MPLC has a different size compared with the remaining ones. We choose specific values dd=2d' and hd=2h' , where dd and hd are the dimensionless thicknesses of the NLC and ZnS defect layers, respectively. Figure 4(a) and (b) displays the defect mode induced in the photonic band of the co-polarized transmittance TRR and crosspolarized reflectance RLR, respectively, by LCP waves impinging normally on the MPLC. We notice that as the parameter σ increases two important facts occur: (i) two defect modes with small amplitude are induced within the first stop band (see Figure 3) which gradually merge into only one; the position of the defect mode possessing the largest wavelength moves toward regions of smaller wavelengths, keeping fixed the position of the other one and (ii) the amplitude of the defect modes gets larger. Physically, the origin of the defect mode is the phase change due to the variation in the optical path length caused by the defective medium. Once the defect mode is created at specific position, it can be controlled by inducing reorientation in the nematic molecules by means of an external electric field [7]. Indeed, since the refractive index of the LC depends on the angle β between the wave vector k of the electromagnetic wave in the LC and the local orientation of the director n, the refractive index (and the optical path length) can be changed by varying β. At normal incidence and σ < σc, β = 90 for all positions z.

Electrical and Thermal Tuning of Band Structure and Defect Modes in Multilayer Photonic Crystals http://dx.doi.org/10.5772/intechopen.70473 123

Figure 4. (a) At normal incidence, plots of co-polarized transmittance TRR for LCP incident waves as function of the dimensionless parameter d/λ and continuous values of σ. (b) At normal incidence, cross-polarized reflectance RLR for LCP incident waves at the same values of σ as in (a).

Nevertheless, as σ increases, most of the molecules tend to be aligned parallel to z-axis (see Figure 2(a)) and β!0�. These results show that the amplitude of defect mode and its position can be tuned by a DC electric field.

#### 4.1.2. Weak anchoring conditions

increasing the electric field, one can highly enhance the cross-polarized reflection bands and

Figure 3. (a)–(h) Plots of co-polarized and cross-polarized reflectances and transmittances for LCP and RCP waves impinging normally on a MPLC as function of the dimensionless parameter d/λ and continuous values of σ within the

In [7], it is shown that for a fixed value of σ the band structure of the reflectances and transmittances are shifted towards smaller wavelength regions as the incident angle θ increases. This behaviour results from the fact that for plane electromagnetic waves propagating obliquely with respect to the layer interfaces, only the normal component of the wave vector is involved in the photonic band formation. Hence, as the incident angle augments, the relative

If one of the layers possesses a different size compared with the remaining ones, this layer can act as a defect, and an optical defect mode can be induced. Here, we specifically consider that the middle NLC-ZnS stack of the MPLC has a different size compared with the remaining

thicknesses of the NLC and ZnS defect layers, respectively. Figure 4(a) and (b) displays the defect mode induced in the photonic band of the co-polarized transmittance TRR and crosspolarized reflectance RLR, respectively, by LCP waves impinging normally on the MPLC. We notice that as the parameter σ increases two important facts occur: (i) two defect modes with small amplitude are induced within the first stop band (see Figure 3) which gradually merge into only one; the position of the defect mode possessing the largest wavelength moves toward regions of smaller wavelengths, keeping fixed the position of the other one and (ii) the amplitude of the defect modes gets larger. Physically, the origin of the defect mode is the phase change due to the variation in the optical path length caused by the defective medium. Once the defect mode is created at specific position, it can be controlled by inducing reorientation in the nematic molecules by means of an external electric field [7]. Indeed, since the refractive index of the LC depends on the angle β between the wave vector k of the electromagnetic wave in the LC and the local orientation of the director n, the refractive index (and the optical path length) can be changed by varying β. At normal incidence and σ < σc, β = 90 for all positions z.

, where dd and hd are the dimensionless

position of the bands is moved towards smaller wavelengths.

122 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

ones. We choose specific values dd=2d' and hd=2h'

supress the co-polarized ones.

interval σ<sup>c</sup> < σ < 13.

It is experimentally found that for a LC phase 5CB, the polar anchoring γα is of the order of 101 , and this value is one or two orders stronger than the azimuthal anchoring γφ [34]. Under these considerations, the values of the dimensionless anchoring parameters are taken as Γ=Γα=0.1.

The curves for α(z) and φ(z) are shown in Figure 5(a) and (b), respectively, as function of dimensionless variable w=z/d above the critical value σc=2.86. In Figure 5(a), we can notice that, as σ augments, the values of α increase, getting a maximum at the middle of the cell. Because of the influence of external electric field, the polar angle at both borders enlarges by increasing σ highlighting the fact that even at the borders, the field is able to distort the configuration. Figure 5 (b) shows two interesting phenomena: (i) for σ < σc, the curves are reduced to straight lines with slope equal to 2φ0c, where φ0<sup>c</sup> represents the azimuthal angle adopted by the MPLC at the walls of each NLC cell for values of electric field below the critical field; (ii) above the critical value, most of the molecules tend to acquire an angle φt=�45� for 0< w < 0.5 and φ<sup>t</sup> = 45� for 0.5 < w<1.

Figure 6 shows the co-polarized and cross-polarized transmittances and reflectances for LCP and RCP waves impinging normally on the structure as function of the dimensionless parameter d/λ for continuous values of the electric field above the critical value σ<sup>c</sup> (below this value, the not-shown curves are very similar to those corresponding to σ=σc). Although, the optical properties shown in Figure 6 are qualitatively similar to those of Figure 3 where strong anchoring conditions were considered, we notice that in the case of weak anchoring conditions, the behaviour of transmittances and reflectances in Figure 6 is enhanced in comparison with Figure 3. Because of the strong influence of the electric field on the molecular orientation

Figure 5. (a) Curves of the polar angle α as function of dimensionless variable w at different values of σ: σ=σc+0.005 (solid line), σ=3.5 (dashed line), σ=4.5 (dotted line), σ=8 (dot-dashed line) and σ=13.5 (large dashed line). (b) Curves of the azimuthal angle φ at the same values of σ as in (a).

orientation for all values of z, including the walls of each NLC slab in the MPLC. This implies that the defect-mode amplitude gets larger for smaller values of σ in comparison to that of the

Figure 7. (a) At normal incidence, plots of co-polarized transmittance TRR for LCP incident waves as function of the dimensionless parameter d/λ and continuous values of σ. (b) At normal incidence, cross-polarized reflectance RLR for LCP incident waves at the same values of σ as in (a). Here, we consider weak anchoring conditions at the walls of each NLC.

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125

Here, we assume that the orientation of the director at the surfaces of each nematic cell is strongly anchored at the boundaries. In order to obtain the band structure, we apply the same mathematical procedure as depicted in Section 4.1, but in this case, we have to take into account that the director n is given by expressions (9) and (10) and the elements of dielectric tensor ε(z) depend on the wavelength and temperature [8]. By considering E7 LC mixture slabs and ZnS dielectric layers, it is found that in the interval of temperatures [15C,50C] and for RCP waves impinging normally, the position of the photonic bands of the co-polarized transmittance TRR and cross-polarized reflectance RLR (analogous to those of Figure 3 with σ=0) can be shifted from regions of small wavelengths toward regions of higher wavelengths by increasing the thickness d of the NLC layers. In addition to this, bandwidth increases for thicker layers, and new narrower transmission bands are created in regions of smaller wavelengths. In summary, the position, the width, and the number of bands augment as the thickness d is increased. Physically, when the magnitude of d gets larger, the optical path lengths increase, and hence, the wavelength zones of destructive or constructive interference are shifted towards higher wavelength regions. For constant thickness and temperature, it is observed that as the incident angle θ augments, the photonic bands undergo a shift towards smaller wavelengths and their widths get narrower. As said previously, this behaviour results from the fact that for plane electromagnetic waves propagating obliquely with respect to the layer interfaces, only the normal component of the wave vector is involved in the photonic band formation. Thus, as the incident angle augments the relative position of the bands moves towards smaller wavelengths, and the overall band is always closed up. On the other hand, for constant thickness and a fixed incident angle, as the temperature augments, the photonic

4.2. Temperature-dependent band structure and defect mode

strong anchoring case.

Figure 6. (a)–(h) Plots of co-polarized and cross-polarized reflectances and transmittances for LCP and RCP waves impinging normally on a MPLC as function of the dimensionless parameter d/λ and continuous values of σ within the interval σc<σ<13. Here, we consider weak anchoring conditions at the walls of each NLC.

for all values of z (including the walls of each cell), the alignment of most of the nematic molecules parallel to z-axis occurs at smaller values of electric field unlike for strong anchoring. Hence, the phenomenon of extinguishing and enhancing bands is present at smaller values of σ.

Now, we induce a defect mode in the photonic band structure by generating a defect in the MPLC in the same way as explained in Section 4.1.1. Figure 7(a) and (b) displays the defect mode induced in the photonic band of the co-polarized transmittance TRR and cross-polarized reflectance RLR, respectively, for LCP waves impinging normally on the MPLC. Similar to the case of strong anchoring conditions, we can observe that when the parameter σ augments the amplitude of the defect modes gets larger, and the position of the defect mode possessing the largest wavelength moves toward regions of smaller wavelengths, while the position of the other defect mode remains fixed. These facts are enhanced in comparison to those of strong anchoring assumptions because of the strong influence of the electric field on the molecular

Electrical and Thermal Tuning of Band Structure and Defect Modes in Multilayer Photonic Crystals http://dx.doi.org/10.5772/intechopen.70473 125

Figure 7. (a) At normal incidence, plots of co-polarized transmittance TRR for LCP incident waves as function of the dimensionless parameter d/λ and continuous values of σ. (b) At normal incidence, cross-polarized reflectance RLR for LCP incident waves at the same values of σ as in (a). Here, we consider weak anchoring conditions at the walls of each NLC.

orientation for all values of z, including the walls of each NLC slab in the MPLC. This implies that the defect-mode amplitude gets larger for smaller values of σ in comparison to that of the strong anchoring case.

#### 4.2. Temperature-dependent band structure and defect mode

for all values of z (including the walls of each cell), the alignment of most of the nematic molecules parallel to z-axis occurs at smaller values of electric field unlike for strong anchoring. Hence, the phenomenon of extinguishing and enhancing bands is present at smaller values of σ. Now, we induce a defect mode in the photonic band structure by generating a defect in the MPLC in the same way as explained in Section 4.1.1. Figure 7(a) and (b) displays the defect mode induced in the photonic band of the co-polarized transmittance TRR and cross-polarized reflectance RLR, respectively, for LCP waves impinging normally on the MPLC. Similar to the case of strong anchoring conditions, we can observe that when the parameter σ augments the amplitude of the defect modes gets larger, and the position of the defect mode possessing the largest wavelength moves toward regions of smaller wavelengths, while the position of the other defect mode remains fixed. These facts are enhanced in comparison to those of strong anchoring assumptions because of the strong influence of the electric field on the molecular

Figure 6. (a)–(h) Plots of co-polarized and cross-polarized reflectances and transmittances for LCP and RCP waves impinging normally on a MPLC as function of the dimensionless parameter d/λ and continuous values of σ within the

interval σc<σ<13. Here, we consider weak anchoring conditions at the walls of each NLC.

Figure 5. (a) Curves of the polar angle α as function of dimensionless variable w at different values of σ: σ=σc+0.005 (solid line), σ=3.5 (dashed line), σ=4.5 (dotted line), σ=8 (dot-dashed line) and σ=13.5 (large dashed line). (b) Curves of the

**<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> <sup>0</sup>**

azimuthal angle φ at the same values of σ as in (a).

**w z d**

**<sup>90</sup> <sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup>**

(a) (b)

124 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

α

**0 0.2 0.4 0.6 0.8 1**

**w z d**

**0 0.2 0.4 0.6 0.8 1**

Here, we assume that the orientation of the director at the surfaces of each nematic cell is strongly anchored at the boundaries. In order to obtain the band structure, we apply the same mathematical procedure as depicted in Section 4.1, but in this case, we have to take into account that the director n is given by expressions (9) and (10) and the elements of dielectric tensor ε(z) depend on the wavelength and temperature [8]. By considering E7 LC mixture slabs and ZnS dielectric layers, it is found that in the interval of temperatures [15C,50C] and for RCP waves impinging normally, the position of the photonic bands of the co-polarized transmittance TRR and cross-polarized reflectance RLR (analogous to those of Figure 3 with σ=0) can be shifted from regions of small wavelengths toward regions of higher wavelengths by increasing the thickness d of the NLC layers. In addition to this, bandwidth increases for thicker layers, and new narrower transmission bands are created in regions of smaller wavelengths. In summary, the position, the width, and the number of bands augment as the thickness d is increased. Physically, when the magnitude of d gets larger, the optical path lengths increase, and hence, the wavelength zones of destructive or constructive interference are shifted towards higher wavelength regions. For constant thickness and temperature, it is observed that as the incident angle θ augments, the photonic bands undergo a shift towards smaller wavelengths and their widths get narrower. As said previously, this behaviour results from the fact that for plane electromagnetic waves propagating obliquely with respect to the layer interfaces, only the normal component of the wave vector is involved in the photonic band formation. Thus, as the incident angle augments the relative position of the bands moves towards smaller wavelengths, and the overall band is always closed up. On the other hand, for constant thickness and a fixed incident angle, as the temperature augments, the photonic bands move towards the short-wavelength region. Physically, since the average refractive index of the liquid crystal decreases as the temperature gets larger, the optical path length diminishes, and thus, the wavelength regions where the waves are able to undergo constructive or destructive interference shift towards smaller wavelengths zones.

[3] de Gennes PG, Prost J. The Physics of Liquid Crystals. 2nd ed. Oxford: Oxford Science

Electrical and Thermal Tuning of Band Structure and Defect Modes in Multilayer Photonic Crystals

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127

[4] Busch K, John S. Liquid-crystal photonic-band-gap materials: The tunable electromagnetic vacuum. Physical Review Letters. 1999;83:967-970. DOI: 10.1103/PhysRevLett.

[5] Leonard SW, van Driel H, Toader O, John S, Busch K, Birner A, Gosele U. Tunable twodimensional photonic crystals using liquid crystal infiltration. Physical Review B.

[6] Ha NY, Ohtsuka Y, Jeong S, Nishimura S, Suzaki G, Takanishi Y, Ishikawa K, Takezoe H. Fabrication of a simultaneous red-green-blue reflector using single-pitched cholesteric

[7] Molina I, Reyes JA, Avendaño CG. Electrically controlled optical bandgap in a twisted photonic liquid crystal. Journal of Applied Physics. 2011;109:113510-113516. DOI: 10.1063/

[8] Avendaño C, Reyes A. Temperature-dependent optical band structure and defect mode in a one-dimensional photonic liquid crystal. Liquid Crystals. 2017;1-12. DOI: 10.1080/

[9] Rapini A, Papoular M. Distorsion d'une lamelle nématique sous champ magnétique conditions d'ancrage aux parois. Journal de Physique, Colloque. 1969;30:C4-54-C4-56.

[10] Sonin AA. The Surface Physics of Liquid Crystals. Amsterdam: Gordon & Breach; 1995.

[11] Avendaño CG, Molina I, Reyes JA. Anchoring effects on the electrically controlled optical band gap in twisted photonic liquid crystals. Liquid Crystals. 2013;40:172-184. DOI:

[12] Avendaño CG, Martinez D. Tunable omni-directional mirror based on one-dimensional photonic structure using twisted nematic liquid crystal: The anchoring effects. Applied

[13] Ozaki R, Matsui T, Ozaki M, Yoshino K. Electro-tunable defect mode in one-dimensional periodic structure containing nematic liquid crystal as a defect layer. Japanese Journal of

[14] Arkhipkin VG, Gunyakov VA, Myslivets SA, Gerasimov VP, Zyryanov VY, Vetrov SY, Shabanov VF. One-dimensional photonic crystals with a planar oriented nematic layer: Temperature and angular dependence of the spectra of defect modes. JETP. 2008;106:388-

[15] Lin Y-T, Chang W-Y, Wu C-Y, Zyryanov VY, Lee W. Optical properties of onedimensional photonic crystal with a twisted-nematic defect layer. Optics Express.

liquid crystals. Nature Materials. 2008;7:43-47. DOI: 10.1038/nmat2045

Publications; 1993. 597 p. ISBN: 978-0198517856

2000;61:R2389-R2392. DOI: 10.1103/Phys.RevB.61.R2389

83.967

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10.1080/02678292.2012.735706

398. DOI: 10.1134/S1063776108020179

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Optics. 2014;53:4683-4690. DOI: 10.1364/AO.53.004683

Applied Physics. 2002;41:L1482-L1484. DOI: 10.1143/JJAP.41.L1482

ISBN: 978-2881249952

In a similar way as demonstrated above, a defect mode can be induced by considering that the middle layer of the homogeneous and isotropic slabs (ZnS) has a different size compared with the remaining ones. If we consider normal incident RCP waves, for hd=1.991h' and temperature values in the interval [15C, 50C], the position of the defect mode induced in the photonic band of the co-polarized transmittance TRR and cross-polarized reflectance RLR (analogous to those of Figure 4 with σ=0) shifts from larger wavelengths toward smaller ones as the temperature gets increasing. Because the origin of the defect mode is the phase change due to the variation in the optical path length caused by the defective medium, the defect wavelength can be shifted towards smaller wavelength regions as the temperature is increased by taking into account that the average refractive index of the NLC decreases as temperature increases.

### 5. Conclusion

We presented a series of results concerning the thermal and electrical tuning of photonic band gaps and defect modes in multilayer photonic liquid crystals consisting of liquid crystal layers alternated by transparent isotropic dielectric films using nematic liquid crystal slabs in a twisted configuration. We exhibited that the position and width of the band gaps can be electrically and thermally controlled. When one of the homogeneous and isotropic slabs has a different size compared with the remaining ones, a defect mode is induced in the band structure whose wavelength can be tuned. Tuning of the transmission and reflection bands and the defect mode investigated here could be useful in the implementation of tunable optical filters and waveguides.

### Author details

Carlos G. Avendaño\*, Daniel Martínez and Ismael Molina

\*Address all correspondence to: caravelo2000@gmail.com

Autonomous University of Mexico City, Mexico City, Mexico

### References


[3] de Gennes PG, Prost J. The Physics of Liquid Crystals. 2nd ed. Oxford: Oxford Science Publications; 1993. 597 p. ISBN: 978-0198517856

bands move towards the short-wavelength region. Physically, since the average refractive index of the liquid crystal decreases as the temperature gets larger, the optical path length diminishes, and thus, the wavelength regions where the waves are able to undergo construc-

In a similar way as demonstrated above, a defect mode can be induced by considering that the middle layer of the homogeneous and isotropic slabs (ZnS) has a different size compared with

values in the interval [15C, 50C], the position of the defect mode induced in the photonic band of the co-polarized transmittance TRR and cross-polarized reflectance RLR (analogous to those of Figure 4 with σ=0) shifts from larger wavelengths toward smaller ones as the temperature gets increasing. Because the origin of the defect mode is the phase change due to the variation in the optical path length caused by the defective medium, the defect wavelength can be shifted towards smaller wavelength regions as the temperature is increased by taking into account that the average refractive index of the NLC decreases as temperature increases.

We presented a series of results concerning the thermal and electrical tuning of photonic band gaps and defect modes in multilayer photonic liquid crystals consisting of liquid crystal layers alternated by transparent isotropic dielectric films using nematic liquid crystal slabs in a twisted configuration. We exhibited that the position and width of the band gaps can be electrically and thermally controlled. When one of the homogeneous and isotropic slabs has a different size compared with the remaining ones, a defect mode is induced in the band structure whose wavelength can be tuned. Tuning of the transmission and reflection bands and the defect mode investigated here could be useful in the implementation of tunable optical filters and waveguides.

[1] Yablonovitch E. Inhibited spontaneous emission in solid-state physics and electronics. Physical Review Letters. 1987;58:2059-2062. DOI: 10.1103/PhysRevLett.58.2059

[2] John S. Strong localization of photons in certain disordered dielectric superlattices. Phys-

ical Review Letters. 1987;58:2486-2489. DOI: 10.1103/PhysRevLett.58.2486

and temperature

tive or destructive interference shift towards smaller wavelengths zones.

126 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

5. Conclusion

Author details

References

Carlos G. Avendaño\*, Daniel Martínez and Ismael Molina \*Address all correspondence to: caravelo2000@gmail.com

Autonomous University of Mexico City, Mexico City, Mexico

the remaining ones. If we consider normal incident RCP waves, for hd=1.991h'


[16] Timofeev IV, Lin Y-T, Gunyakov VA, Myslivets SA, Arkhipkin VG, Vetrov SY, Lee W, Zyryanov VY. Voltage-induce defect mode coupling in a one-dimensional photonic crystal with a twisted-nematic defect layer. Physical Review E. 2012;85:011705. DOI: 10.1103/ PhysRevE.85.011705

[30] Marcuvitz N, Schwinger J. On the representation of electric and magnetic field produced by currents and discontinuities in waveguides. Journal of Applied Physics. 1951;22:806-

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[31] Altman C, Suchy K. Reciprocity, Spatial Mapping and Time Reversal in Electromagnetics.

[32] Berreman DW, Scheffer TJ. Bragg reflection of light from single-domain cholesteric liquidcrystal films. Physical Reviews Letters. 1970;25:577-581. DOI: 10.1103/PhysRevLett. 25.902.4

[33] Avendaño CG, Ponti S, Reyes JA, Oldano C. Multiplet structure of the defect modes in 1D helical photonic crystals with twist defects. Journal of Physics A: Mathematical and

[34] Nastishin YA, Polak RD, Shiyanovskii SV, Bodnar VH, Lavrentovich OD. Nematic polar anchoring strength measured by electric field techniques. Journal of Applied Physics.

2nd ed. Heidelberg: Springer; 2011. DOI: 10.1007/978-94-007-1530-1

General. 2005;38:8821-8840. DOI: 10.1088/0305-4470/38/41/001

1999;86:4199-4213. DOI: 10.1063/1.371347

819. DOI: 10.1063/1.1700052


[30] Marcuvitz N, Schwinger J. On the representation of electric and magnetic field produced by currents and discontinuities in waveguides. Journal of Applied Physics. 1951;22:806- 819. DOI: 10.1063/1.1700052

[16] Timofeev IV, Lin Y-T, Gunyakov VA, Myslivets SA, Arkhipkin VG, Vetrov SY, Lee W, Zyryanov VY. Voltage-induce defect mode coupling in a one-dimensional photonic crystal with a twisted-nematic defect layer. Physical Review E. 2012;85:011705. DOI: 10.1103/

[17] Stewart IW. The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction. 1st ed. New York: Taylor & Francis Group; 2004. 351 p. ISBN: 978-

[18] Rasing T, Muševič I. Surfaces and Interfaces of Liquid Crystals. 1st ed. New York:

[19] Castellano JA. Surface anchoring of liquid crystal molecules on various substrates. Molecular Crystals and Liquid Crystals. 1983;94:33-41. DOI: 10.1080/00268948308084245 [20] Takatoh K, Hasegawa M, Koden M, Itoh N, Hasegawa R, Sakamoto M. Alignment Technologies and Applications of Liquid Crystals Devices. 1st ed. New York: Taylor &

[21] Rüetschi M, Grütter P, Fünfschilling J, Güntherodt H. Creation of liquid crystal waveguides with scanning force microscopy. Science. 1994;265:512-514. DOI: 10.1126/sci-

[22] Pidduck AJ, Haslam SD, Bryan-Brown GP, Bannister R, Kitely ID. Control of liquid crystal alignment by polyimide surface modification using atomic force microscopy.

[23] Gibbons WM, Shannon PJ, Sun S, Swtlin BJ. Surface-mediated alignment of nematic liquid crystals with polarized laser light. Nature. 1991;351:49-50. DOI: 10.1038/351049a0

[24] Chigrinov VG, Kozenkov VM, Kwok H-S. Photoalignement of Liquid Crystalline Materials: Physics and Applications. West Sussex: John Wiley & Sons; 2008. p. 248. DOI:

[25] Vilfan M, Mertelj A, Copic M. Dynamic light scattering measurements of azimuthal and zenithal anchoring of nematic liquid crystals. Physical Review E. 2002;65:041712-1-

[26] Zhao W, Wu C, Iwamoto M. Weak boundary anchoring, twisted nematic effect and homeotropic to twisted-planar transition. Physical Review E. 2002;65:031709. DOI:

[27] Baek S-I, Kim S-J, Kim J-H. Measurement of anchoring coefficient of homeotropically aligned nematic liquid crystal using a polarizing optical microscope in reflective mode.

[28] Chuang SL. Physics of Photonic Devices. 2nd ed. New Jersey: John Wiley & Sons; 2009. p. 840 [29] Hecht E, Zajac A. Óptica. 3rd ed. Madrid: Addison Wesley Iberoamericana; 2010. 722 p.

Springer; 2004. p. 298. DOI: 10.1007/978-3-662-10157-5

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Francis Inc.; 2005. p. 320. DOI: 10.1201/9781420023015

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0748408962

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041712-7. DOI: 10.1103/PhysRevE.65.041712

AIP Advances. 2015;5:097170. DOI: 10.1063/1.4931950


**Chapter 7**

Provisional chapter

**Nonlinear Optical Phenomena in Smectic A Liquid**

DOI: 10.5772/intechopen.70997

Liquid crystals (LC) are the materials characterized by extremely high optical nonlinearity. Their physical properties such as temperature, molecular orientation, density, and electronic structure can be easily perturbed by an applied optical field. In particular, in smectic A LC (SALC), there is a specific mechanism of the cubic optical nonlinearity determined by the smectic layer normal displacement. The physical processes related to this mechanism are characterized by a comparatively large cubic susceptibility, short time response, strong dependence on the optical wave polarization and propagation direction, resonant spectral form, low scattering losses as compared to other LC phases, and weak temperature dependence in the region far from the phase transition. We investigated theoretically the nonlinear optical phenomena caused by this type of the cubic nonlinearity in SALC. It has been shown that the light self-focusing, self-trapping, Brillouin-like stimulated light scattering (SLS), and four-wave mixing (FWM) related to the smectic layer normal displacement are strongly manifested in SALC. We obtained the exact analytical solutions in some cases and made the numerical evaluations of the basic parameters such as the optical

Keywords: smectic liquid crystals, second sound, nonlinear optics, cubic nonlinearity,

Liquid crystals (LC) are characterized by the physical properties intermediate between ordinary isotropic fluids and solids [1]. LC flow like liquids but also exhibit some properties of crystals [1, 2]. The various phases in which such materials can exist are called mesophases [1, 2]. The LC molecules are large, anisotropic, and complex [2]. Dielectric constants, elastic constants, viscosities,

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

stimulated scattering of light, four-wave mixing, surface plasmon polariton

Boris I. Lembrikov, David Ianetz and Yossef Ben Ezra

Nonlinear Optical Phenomena in Smectic

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.70997

beam width and SLS gain.

1. Introduction

Boris I. Lembrikov, David Ianetz and

**Crystals**

Yossef Ben Ezra

Abstract

A Liquid Crystals

#### **Nonlinear Optical Phenomena in Smectic A Liquid Crystals** Nonlinear Optical Phenomena in Smectic A Liquid Crystals

DOI: 10.5772/intechopen.70997

Boris I. Lembrikov, David Ianetz and Yossef Ben Ezra Boris I. Lembrikov, David Ianetz and

Additional information is available at the end of the chapter Yossef Ben Ezra

http://dx.doi.org/10.5772/intechopen.70997 Additional information is available at the end of the chapter

### Abstract

Liquid crystals (LC) are the materials characterized by extremely high optical nonlinearity. Their physical properties such as temperature, molecular orientation, density, and electronic structure can be easily perturbed by an applied optical field. In particular, in smectic A LC (SALC), there is a specific mechanism of the cubic optical nonlinearity determined by the smectic layer normal displacement. The physical processes related to this mechanism are characterized by a comparatively large cubic susceptibility, short time response, strong dependence on the optical wave polarization and propagation direction, resonant spectral form, low scattering losses as compared to other LC phases, and weak temperature dependence in the region far from the phase transition. We investigated theoretically the nonlinear optical phenomena caused by this type of the cubic nonlinearity in SALC. It has been shown that the light self-focusing, self-trapping, Brillouin-like stimulated light scattering (SLS), and four-wave mixing (FWM) related to the smectic layer normal displacement are strongly manifested in SALC. We obtained the exact analytical solutions in some cases and made the numerical evaluations of the basic parameters such as the optical beam width and SLS gain.

Keywords: smectic liquid crystals, second sound, nonlinear optics, cubic nonlinearity, stimulated scattering of light, four-wave mixing, surface plasmon polariton

### 1. Introduction

Liquid crystals (LC) are characterized by the physical properties intermediate between ordinary isotropic fluids and solids [1]. LC flow like liquids but also exhibit some properties of crystals [1, 2]. The various phases in which such materials can exist are called mesophases [1, 2]. The LC molecules are large, anisotropic, and complex [2]. Dielectric constants, elastic constants, viscosities,

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and eproduction in any medium, provided the original work is properly cited.

absorption spectra, transition temperatures, anisotropies, and optical nonlinearities of LC are determined by the structure of these molecules [1, 2]. There exist three different types of LC: lyotropic, polymeric, and thermotropic [1, 2]. Lyotropic LC are obtained when an appropriate concentration of a material is dissolved in a solvent [2]. They can demonstrate a one-, two-, or three-dimensional positional order [2]. Liquid crystalline polymers are built up by the joining together the rigid mesogenic monomers [2]. Thermotropic LC exhibit different mesophases depending on temperature [1, 2]. Typically, they consist of organic molecules elongated in one direction and represented as rigid rods [2]. There are two types of LC sample orientation with respect to the boundary: (i) a homeotropic orientation when the long molecular axes are perpendicular to the boundary and (ii) a planar orientation when the long molecular axes are parallel to the boundary [1].

waveguide is a thin film of LC with a thickness of about 1μm sandwiched between two glass slides of lower refracted index than LC [2]. Stimulated light scattering (SLS), self-phase modulation (SPM), self-focusing, spatial soliton formation, optical wave mixing, harmonic generation, optical phase conjugation, and other nonlinear optical effects in LC have been investigated [3]. NLC is the most useful and widely studied type of LC [2–4]. However, the practical integrated electro-optical applications of NLC are limited by their large losses of about 20 dB/cm and relatively slow responses [2]. The scattering losses in SALC are much lower, and they can be useful in nonlinear optical applications [2]. Recently, the LC applications in plasmonics attracted a wide interest due to the combination of the surface plasmon polaritons (SPP) strong electric

Nonlinear Optical Phenomena in Smectic A Liquid Crystals

http://dx.doi.org/10.5772/intechopen.70997

133

We investigated theoretically the nonlinear optical phenomena in SALC related to the specific mechanism of the cubic nonlinearity, which is determined by the smectic layer normal displacement u xð Þ ; y; z; t in the electric field of optical waves and SPP [5–13]. This mechanism combining the properties of the orientational and electrostrictive nonlinearities [2] occurs without the mass density change, strongly depends on the optical wave polarization and propagation direction, and has a resonant form of the frequency dependence. It is character-

The theoretical analysis of the nonlinear optical phenomena in SALC related to the layer displacement was based on the simultaneous solution of the Maxwell equations for the optical waves propagating in SALC and the equation of motion for the SALC layers in the electric field of these waves. We used the slowly varying amplitude approximation (SVAA) [14]. We investigated the following nonlinear optical effects in SALC based on the layer displacement nonlinearity: self-focusing and self-trapping, SLS, and four-wave mixing (FWM) [5–10]. We applied the developed theory of the nonlinear optical phenomena in SALC to the SPP interactions in SALC [11–13]. The SPP stimulated scattering in SALC and the metal/insulator/metal (MIM) plasmonic waveguide with the SALC core are theoretically studied [11–13]. The detailed calculations and complicated explicit analytical expressions can be found in Refs [5– 13]. In this chapter, we describe the general approach to the theoretical analysis of the

The chapter is constructed as follows. The equation of motion for the smectic layer normal displacement u xð Þ ; y; z; t in the electric field is derived in Section 2. The self-focusing and selftrapping of the optical wave in SALC are considered in Section 3. The SLS in SALC is investigated in Section 4. The FWM in SALC is analyzed in Section 5. The SPP interaction in

!

ð Þ x; y; z; t is

ized by a comparatively short response time similar to acousto-optic processes [2, 14].

nonlinear optical phenomena in SALC and present the main results.

SALC is discussed in Section 6. The conclusions are presented in Section 7.

The structure of the homeotropically oriented SALC in an external electric field E

The hydrodynamics of SALC is described by the following system of Eq. [1]

2. The smectic layer equation of motion

presented in Figure 1.

fields and the unique electro-optical properties of LC [4].

In this work, we consider only thermotropic LC, which are divided into three groups according to their symmetry: nematic LC (NLC), cholesteric LC (CLC), and smectic LC (SLC) [1, 2]. NLC are characterized by some long-range order in the direction of the molecular long axes, while the centers of gravity of the molecules do not have any long range order [1, 2]. The general direction of the molecules is defined by a unit vector function n ! ð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup>; <sup>t</sup> ; n! ð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup>; <sup>t</sup> <sup>¼</sup> 1 called director

[1, 2]. NLC molecules are centrosymmetric such that the n ! and � <sup>n</sup> ! directions are equivalent; NLC are optically uniaxial media with a comparatively large birefringence of about 0.2 [1, 2]. LC consisting of chiral molecules yield CLC phase with the helical structure [1, 2]. The molecule centers of gravity in CLC do not have a long range order like in NLC, while the direction of the molecular orientation rotates in space about the helical axis Z with a period of about 300 nm [1, 2]. The smectic LC (SLC) are characterized by the positional long range order in the direction of the elongated molecular axis and exhibit a layer structure [1, 2]. The layer thickness d ≈ 2nm is approximately equal to the length of the constituent molecule [1, 2]. SLC can be considered as natural nanostructures. Inside a layer the molecules form a two-dimensional liquid [1, 2]. The layers can easily move one along another because the elastic constant <sup>B</sup> <sup>≈</sup> <sup>10</sup><sup>6</sup> � 107 Jm�<sup>3</sup> related to the layer compression is two orders of magnitude less than the elastic constant related to the bulk compression [1]. There exist different phases of SLC: (i) smectic A LC (SALC) where the molecule long axes are perpendicular to the layer plane; (ii) smectic B LC with the in-layer hexagonal ordering of the molecules; (iii) smectic C LC where the molecules are tilted with respect to the layers; (iv) smectic C\* LC consisting of the chiral molecules and possessing the spontaneous polarization; (v) different exotic smectic mesophases [1]. In this work, we consider only SALC. The SALC layered structure can be described by the one-dimensional mass density wave characterized by the complex order parameter. The modulus of this order parameter describes the mass density and its phase is related to smectic layer displacement u xð Þ ; y; z; t along the direction perpendicular to the layer plane [1]. SALC is an optically uniaxial medium [2].

LC are highly nonlinear optical materials due to their complex physical structures, and their temperature, molecular orientation, mass density, electronic structure can be easily perturbed by an external optical field [2–4]. Almost all known nonlinear optical phenomena have been observed in LC in time scale range from picoseconds to hours, involving laser powers from 106 Watt to 10�<sup>9</sup> Watt, in different configurations such as bulk media, optical waveguides, optical resonators and cavities, and spatial light modulators [3]. For instance, a typical LC slab optical waveguide is a thin film of LC with a thickness of about 1μm sandwiched between two glass slides of lower refracted index than LC [2]. Stimulated light scattering (SLS), self-phase modulation (SPM), self-focusing, spatial soliton formation, optical wave mixing, harmonic generation, optical phase conjugation, and other nonlinear optical effects in LC have been investigated [3]. NLC is the most useful and widely studied type of LC [2–4]. However, the practical integrated electro-optical applications of NLC are limited by their large losses of about 20 dB/cm and relatively slow responses [2]. The scattering losses in SALC are much lower, and they can be useful in nonlinear optical applications [2]. Recently, the LC applications in plasmonics attracted a wide interest due to the combination of the surface plasmon polaritons (SPP) strong electric fields and the unique electro-optical properties of LC [4].

We investigated theoretically the nonlinear optical phenomena in SALC related to the specific mechanism of the cubic nonlinearity, which is determined by the smectic layer normal displacement u xð Þ ; y; z; t in the electric field of optical waves and SPP [5–13]. This mechanism combining the properties of the orientational and electrostrictive nonlinearities [2] occurs without the mass density change, strongly depends on the optical wave polarization and propagation direction, and has a resonant form of the frequency dependence. It is characterized by a comparatively short response time similar to acousto-optic processes [2, 14].

The theoretical analysis of the nonlinear optical phenomena in SALC related to the layer displacement was based on the simultaneous solution of the Maxwell equations for the optical waves propagating in SALC and the equation of motion for the SALC layers in the electric field of these waves. We used the slowly varying amplitude approximation (SVAA) [14]. We investigated the following nonlinear optical effects in SALC based on the layer displacement nonlinearity: self-focusing and self-trapping, SLS, and four-wave mixing (FWM) [5–10]. We applied the developed theory of the nonlinear optical phenomena in SALC to the SPP interactions in SALC [11–13]. The SPP stimulated scattering in SALC and the metal/insulator/metal (MIM) plasmonic waveguide with the SALC core are theoretically studied [11–13]. The detailed calculations and complicated explicit analytical expressions can be found in Refs [5– 13]. In this chapter, we describe the general approach to the theoretical analysis of the nonlinear optical phenomena in SALC and present the main results.

The chapter is constructed as follows. The equation of motion for the smectic layer normal displacement u xð Þ ; y; z; t in the electric field is derived in Section 2. The self-focusing and selftrapping of the optical wave in SALC are considered in Section 3. The SLS in SALC is investigated in Section 4. The FWM in SALC is analyzed in Section 5. The SPP interaction in SALC is discussed in Section 6. The conclusions are presented in Section 7.

### 2. The smectic layer equation of motion

absorption spectra, transition temperatures, anisotropies, and optical nonlinearities of LC are determined by the structure of these molecules [1, 2]. There exist three different types of LC: lyotropic, polymeric, and thermotropic [1, 2]. Lyotropic LC are obtained when an appropriate concentration of a material is dissolved in a solvent [2]. They can demonstrate a one-, two-, or three-dimensional positional order [2]. Liquid crystalline polymers are built up by the joining together the rigid mesogenic monomers [2]. Thermotropic LC exhibit different mesophases depending on temperature [1, 2]. Typically, they consist of organic molecules elongated in one direction and represented as rigid rods [2]. There are two types of LC sample orientation with respect to the boundary: (i) a homeotropic orientation when the long molecular axes are perpendicular to the boundary and (ii) a planar orientation when the long molecular axes are parallel to

In this work, we consider only thermotropic LC, which are divided into three groups according to their symmetry: nematic LC (NLC), cholesteric LC (CLC), and smectic LC (SLC) [1, 2]. NLC are characterized by some long-range order in the direction of the molecular long axes, while the centers of gravity of the molecules do not have any long range order [1, 2]. The general direction

NLC are optically uniaxial media with a comparatively large birefringence of about 0.2 [1, 2]. LC consisting of chiral molecules yield CLC phase with the helical structure [1, 2]. The molecule centers of gravity in CLC do not have a long range order like in NLC, while the direction of the molecular orientation rotates in space about the helical axis Z with a period of about 300 nm [1, 2]. The smectic LC (SLC) are characterized by the positional long range order in the direction of the elongated molecular axis and exhibit a layer structure [1, 2]. The layer thickness d ≈ 2nm is approximately equal to the length of the constituent molecule [1, 2]. SLC can be considered as natural nanostructures. Inside a layer the molecules form a two-dimensional liquid [1, 2]. The

to the layer compression is two orders of magnitude less than the elastic constant related to the bulk compression [1]. There exist different phases of SLC: (i) smectic A LC (SALC) where the molecule long axes are perpendicular to the layer plane; (ii) smectic B LC with the in-layer hexagonal ordering of the molecules; (iii) smectic C LC where the molecules are tilted with respect to the layers; (iv) smectic C\* LC consisting of the chiral molecules and possessing the spontaneous polarization; (v) different exotic smectic mesophases [1]. In this work, we consider only SALC. The SALC layered structure can be described by the one-dimensional mass density wave characterized by the complex order parameter. The modulus of this order parameter describes the mass density and its phase is related to smectic layer displacement u xð Þ ; y; z; t along the direction perpendicular to the layer plane [1]. SALC is an optically uniaxial medium [2].

LC are highly nonlinear optical materials due to their complex physical structures, and their temperature, molecular orientation, mass density, electronic structure can be easily perturbed by an external optical field [2–4]. Almost all known nonlinear optical phenomena have been observed in LC in time scale range from picoseconds to hours, involving laser powers from 106 Watt to 10�<sup>9</sup> Watt, in different configurations such as bulk media, optical waveguides, optical resonators and cavities, and spatial light modulators [3]. For instance, a typical LC slab optical

layers can easily move one along another because the elastic constant <sup>B</sup> <sup>≈</sup> <sup>10</sup><sup>6</sup> � 107

! ð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup>; <sup>t</sup> ; n! ð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup>; <sup>t</sup> 

! and � <sup>n</sup>

 

<sup>¼</sup> 1 called director

Jm�<sup>3</sup> related

! directions are equivalent;

the boundary [1].

of the molecules is defined by a unit vector function n

[1, 2]. NLC molecules are centrosymmetric such that the n

132 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

The structure of the homeotropically oriented SALC in an external electric field E ! ð Þ x; y; z; t is presented in Figure 1.

The hydrodynamics of SALC is described by the following system of Eq. [1]

Figure 1. Homeotropically oriented SALC in an external electric field E ! ð Þ x; y; z; t .

$$
\vec{\text{div}} \cdot \vec{v} = \mathbf{0} \tag{1}
$$

where K is the Frank elastic constant associated with the SALC purely orientational energy, ε<sup>0</sup> is the free space permittivity, and εik is the SALC permittivity tensor including the nonlinear

; εzz ¼ ε<sup>∥</sup> þ a<sup>∥</sup>

; εyz ¼ εzy ¼ �ε<sup>a</sup>

Here, ε⊥, ε<sup>∥</sup> are the diagonal components of the uniaxial SALC permittivity tensor perpendicular and parallel to the optical axis Z, respectively, and a<sup>⊥</sup> � 1; a<sup>∥</sup> � 1 are the phenomenological dimensionless coefficients. For the smectic layer displacement u xð Þ ; y; z; t depending on z, the purely orientational second term in the free energy density F (7) can be neglected. Indeed,

kS<sup>⊥</sup> is the in-plane component of the smectic layer displacement wave vector. The contribution of the first term containing the normal layer strain is dominant. We consider the smectic layer normal displacement with kSz 6¼ 0. Taking into account the assumptions mentioned above and combining Eqs. (1)–(9), we obtain the equation of motion for the smectic layer normal displace-

ð Þ <sup>α</sup><sup>4</sup> <sup>þ</sup> <sup>α</sup><sup>56</sup> <sup>∇</sup><sup>2</sup>

z

responsible for the decay of the smectic layer displacement are neglected, Eq. (10) coincides

<sup>2</sup> <sup>¼</sup> <sup>B</sup>∇<sup>2</sup> ⊥ ∂2 u

uncoupled acoustic modes: (i) the ordinary longitudinal sound wave caused by the mass density oscillations; (ii) SS wave caused by the smectic layer oscillations [1]. The SS propagation may be considered separately from ordinary sound since B is much less than the elastic constant of the mass density oscillations [1]. The SS dispersion relation corresponding to

> kS⊥kSz kS

Eq. (11) that SS is neither longitudinal, nor purely transverse, and it vanishes for the wave

; s<sup>0</sup> ¼

S, s<sup>0</sup> are SS frequency, wave vector and phase velocity, respectively. It is seen from

<sup>S</sup> perpendicular or parallel to the layer plane. SS represents the oscillations of the

� � ∂u

∇2

� � � � � � :

∂t

u=∂y2. If the external electric field is absent and the viscosity terms

∂ ∂x

!

ffiffiffi B r

s

<sup>þ</sup> <sup>B</sup>∇<sup>2</sup> ⊥ ∂2 u ∂z<sup>2</sup>

> ∂ ∂y

EyEz

<sup>∂</sup>z<sup>2</sup> : (10)

<sup>S</sup> in SALC, there exist two practically

ð Þþ ExEz

∂u ∂z ;

∂u ∂y

; ε<sup>a</sup> ¼ ε<sup>∥</sup> � ε<sup>⊥</sup>

Nonlinear Optical Phenomena in Smectic A Liquid Crystals

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(8)

135

(9)

(11)

<sup>S</sup><sup>⊥</sup> ≪ B where

∂u ∂z

∂u ∂x

for the typical values of <sup>B</sup> and <sup>K</sup> � <sup>10</sup>�<sup>11</sup><sup>N</sup> [1], the following inequality is valid: Kk<sup>2</sup>

terms related to the smectic layer strains. It is given by [1]

εxz ¼ εzx ¼ �ε<sup>a</sup>

ment u xð Þ ; y; z; t [5, 6, 10]

<sup>⊥</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup>

Eq. (11) has the form [1]

!

Here, ΩS, k

vectors k !

.Here, ∇<sup>2</sup>

�r∇<sup>2</sup> <sup>∂</sup><sup>2</sup> u ∂t

<sup>u</sup>=∂x<sup>2</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup>

¼ ε0 <sup>2</sup> <sup>∇</sup><sup>2</sup> ⊥ ∂ ∂z

<sup>2</sup> <sup>þ</sup> <sup>α</sup>1∇<sup>2</sup> ⊥ ∂2 ∂z<sup>2</sup> þ 1 2

with the equation of the so-called second sound (SS) [1]

Generally, for an arbitrary direction of the wave vector k

a<sup>⊥</sup> E<sup>2</sup>

<sup>x</sup> <sup>þ</sup> <sup>E</sup><sup>2</sup> y � � <sup>þ</sup> <sup>a</sup>∥E<sup>2</sup>

� � � <sup>2</sup>ε<sup>a</sup>

<sup>r</sup>∇<sup>2</sup> <sup>∂</sup><sup>2</sup> u ∂t

Ω<sup>S</sup> ¼ s<sup>0</sup>

εxx ¼ εyy ¼ ε<sup>⊥</sup> þ a<sup>⊥</sup>

$$
\rho \frac{\partial \sigma\_i}{\partial t} = -\frac{\partial \Pi}{\partial \mathbf{x}\_i} + \Lambda\_i + \frac{\partial \sigma'\_{ik}}{\partial \mathbf{x}\_k} \tag{2}
$$

$$
\Lambda\_i = -\frac{\delta F}{\delta u\_i} \tag{3}
$$

$$
\sigma'\_{\rm ik} = \alpha\_0 \delta\_{\rm ik} A\_{\rm ll} + \alpha\_1 \delta\_{\rm iz} A\_{zz} + \alpha\_4 A\_{\rm ik} + \alpha\_{56} (\delta\_{\rm iz} A\_{zk} + \delta\_{\rm kz} A\_{\rm zi}) + \alpha\_7 \delta\_{\rm iz} \delta\_{\rm kz} A\_{\rm ll} \tag{4}
$$

$$A\_{ik} = \frac{1}{2} \left( \frac{\partial v\_i}{\partial \mathbf{x}\_k} + \frac{\partial v\_k}{\partial \mathbf{x}\_i} \right) \tag{5}$$

$$
\upsilon\_z = \frac{\partial u}{\partial t} \tag{6}
$$

Here, v ! is the hydrodynamic velocity, r is the mass density, Π is the pressure, Λ ! is the generalized force density, σ<sup>0</sup> ik is the viscous stress tensor, α<sup>i</sup> are the viscosity Leslie coefficients, δik ¼ 1, i ¼ k; δik ¼ 0, i 6¼ k, and F is the free energy density of SALC. Typically, SALC is supposed to be incompressible liquid according to Eq. (1) [1]. For this reason, we assume that the pressure Π ¼ 0 and the SALC-free energy density F do not depend on the bulk compression [1]. The normal layer displacement u xð Þ ; y; z; t by definition has only one component along the Z axis. In such a case, the generalized force density Λ ! has only the Z component according to Eq. (3): Λ ! ¼ ð Þ 0; 0; Λ<sup>z</sup> . Eq. (6) is specific for SALC since it determines the condition of the smectic layer continuity [1]. The SALC free energy density F in the presence of the external electric field E ! ð Þ x; y; z; t has the form [1]

$$F = \frac{1}{2}B\left(\frac{\partial u}{\partial z}\right)^2 + \frac{1}{2}K\left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right)^2 - \frac{1}{2}\varepsilon\_0 \varepsilon\_{ik} E\_i E\_k \tag{7}$$

where K is the Frank elastic constant associated with the SALC purely orientational energy, ε<sup>0</sup> is the free space permittivity, and εik is the SALC permittivity tensor including the nonlinear terms related to the smectic layer strains. It is given by [1]

$$\begin{aligned} \varepsilon\_{\text{xx}} = \varepsilon\_{yy} &= \varepsilon\_{\perp} + a\_{\perp} \frac{\partial u}{\partial z}; \varepsilon\_{zz} = \varepsilon\_{\parallel} + a\_{\parallel} \frac{\partial u}{\partial z}; \\ \varepsilon\_{\text{xz}} = \varepsilon\_{\text{zz}} &= -\varepsilon\_{a} \frac{\partial u}{\partial x}; \varepsilon\_{yz} = \varepsilon\_{zy} = -\varepsilon\_{a} \frac{\partial u}{\partial y}; \varepsilon\_{a} = \varepsilon\_{\parallel} - \varepsilon\_{\perp} \end{aligned} \tag{8}$$

Here, ε⊥, ε<sup>∥</sup> are the diagonal components of the uniaxial SALC permittivity tensor perpendicular and parallel to the optical axis Z, respectively, and a<sup>⊥</sup> � 1; a<sup>∥</sup> � 1 are the phenomenological dimensionless coefficients. For the smectic layer displacement u xð Þ ; y; z; t depending on z, the purely orientational second term in the free energy density F (7) can be neglected. Indeed, for the typical values of <sup>B</sup> and <sup>K</sup> � <sup>10</sup>�<sup>11</sup><sup>N</sup> [1], the following inequality is valid: Kk<sup>2</sup> <sup>S</sup><sup>⊥</sup> ≪ B where kS<sup>⊥</sup> is the in-plane component of the smectic layer displacement wave vector. The contribution of the first term containing the normal layer strain is dominant. We consider the smectic layer normal displacement with kSz 6¼ 0. Taking into account the assumptions mentioned above and combining Eqs. (1)–(9), we obtain the equation of motion for the smectic layer normal displacement u xð Þ ; y; z; t [5, 6, 10]

div v

<sup>Λ</sup><sup>i</sup> ¼ � <sup>δ</sup><sup>F</sup> δui

> ∂vi ∂xk þ ∂vk ∂xi

vz <sup>¼</sup> <sup>∂</sup><sup>u</sup>

! is the hydrodynamic velocity, r is the mass density, Π is the pressure, Λ

δik ¼ 1, i ¼ k; δik ¼ 0, i 6¼ k, and F is the free energy density of SALC. Typically, SALC is supposed to be incompressible liquid according to Eq. (1) [1]. For this reason, we assume that the pressure Π ¼ 0 and the SALC-free energy density F do not depend on the bulk compression [1]. The normal layer displacement u xð Þ ; y; z; t by definition has only one component along the

smectic layer continuity [1]. The SALC free energy density F in the presence of the external

þ 1 2 <sup>K</sup> <sup>∂</sup><sup>2</sup> u ∂x<sup>2</sup> þ !

¼ ð Þ 0; 0; Λ<sup>z</sup> . Eq. (6) is specific for SALC since it determines the condition of the

∂2 u ∂y<sup>2</sup> <sup>2</sup>

� 1 2

Aik <sup>¼</sup> <sup>1</sup> 2 þ Λ<sup>i</sup> þ

∂σ<sup>0</sup> ik ∂xk

!

ð Þ x; y; z; t .

ik ¼ α0δikAll þ α1δizAzz þ α4Aik þ α56ðδizAzk þ δkzAziÞ þ α7δizδkzAll (4)

ik is the viscous stress tensor, α<sup>i</sup> are the viscosity Leslie coefficients,

r ∂vi <sup>∂</sup><sup>t</sup> ¼ � <sup>∂</sup><sup>Π</sup> ∂xi

Figure 1. Homeotropically oriented SALC in an external electric field E

134 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

σ0

generalized force density, σ<sup>0</sup>

!

Z axis. In such a case, the generalized force density Λ

ð Þ x; y; z; t has the form [1]

<sup>F</sup> <sup>¼</sup> <sup>1</sup> 2 <sup>B</sup> <sup>∂</sup><sup>u</sup> ∂z <sup>2</sup>

Here, v

Eq. (3): Λ !

electric field E

!¼ <sup>0</sup> (1)

<sup>∂</sup><sup>t</sup> (6)

has only the Z component according to

ε0εikEiEk (7)

(2)

(3)

(5)

is the

!

$$\begin{split} & -\rho \nabla^2 \frac{\partial^2 u}{\partial t^2} + \left[ \alpha\_1 \nabla\_\perp^2 \frac{\partial^2}{\partial z^2} + \frac{1}{2} (\alpha\_4 + \alpha\_{56}) \nabla^2 \nabla^2 \right] \frac{\partial u}{\partial t} + B \nabla\_\perp^2 \frac{\partial^2 u}{\partial z^2} \\ & = \frac{\varepsilon\_0}{2} \nabla\_\perp^2 \Big[ \frac{\partial}{\partial z} \left( a\_\perp \left( E\_x^2 + E\_y^2 \right) + a\_\parallel E\_z^2 \right) - 2 \varepsilon\_d \left( \frac{\partial}{\partial x} (E\_x E\_z) + \frac{\partial}{\partial y} (E\_y E\_z) \right) \Big]. \end{split} \tag{9}$$

.Here, ∇<sup>2</sup> <sup>⊥</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup> <sup>u</sup>=∂x<sup>2</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup> u=∂y2. If the external electric field is absent and the viscosity terms responsible for the decay of the smectic layer displacement are neglected, Eq. (10) coincides with the equation of the so-called second sound (SS) [1]

$$
\rho \nabla^2 \frac{\partial^2 u}{\partial t^2} = B \nabla\_\perp^2 \frac{\partial^2 u}{\partial z^2} \,. \tag{10}
$$

Generally, for an arbitrary direction of the wave vector k ! <sup>S</sup> in SALC, there exist two practically uncoupled acoustic modes: (i) the ordinary longitudinal sound wave caused by the mass density oscillations; (ii) SS wave caused by the smectic layer oscillations [1]. The SS propagation may be considered separately from ordinary sound since B is much less than the elastic constant of the mass density oscillations [1]. The SS dispersion relation corresponding to Eq. (11) has the form [1]

$$\mathbf{L}\Omega\_{\ $} = \mathbf{s}\_0 \frac{k\_{\$ } k\_{\ $} \mathbf{s}\_z}{k\_{\$ }}; \mathbf{s}\_0 = \sqrt{\frac{\mathcal{B}}{\rho}}\tag{11}$$

Here, ΩS, k ! S, s<sup>0</sup> are SS frequency, wave vector and phase velocity, respectively. It is seen from Eq. (11) that SS is neither longitudinal, nor purely transverse, and it vanishes for the wave vectors k ! <sup>S</sup> perpendicular or parallel to the layer plane. SS represents the oscillations of the SALC complex order parameter phase [1]. If we take into account the viscosity terms in Eq. (10), then we can obtain the SS relaxation time τ<sup>S</sup> given by

$$\tau\_S = 2\rho \left[ \alpha\_1 \frac{\left(k\_{\rm Sx}^2 + k\_{\rm Sy}^2\right) k\_{\rm Sz}^2}{k\_{\rm S}^2} + \frac{1}{2} (\alpha\_4 + \alpha\_{56}) k\_{\rm S}^2 \right]^{-1} \tag{12}$$

SS has been observed experimentally [15–17].

### 3. Self-focusing and self-trapping of optical beams in SALC

We first consider the self-action effects of the optical waves propagating in an anisotropic inhomogeneous nonlinear medium. The light beam propagation through a nonlinear medium is accompanied by the intensity-dependent phase shift on the wavefront of the beam [2]. Selffocusing of light results from the wavefront distortion inflicted on the beam by itself while propagating in a nonlinear medium [14]. In such a case, the field-induced refractive change Δn has the form <sup>Δ</sup><sup>n</sup> <sup>¼</sup> <sup>n</sup>2j j <sup>E</sup> <sup>2</sup> where <sup>n</sup><sup>2</sup> <sup>¼</sup> const [14]. A light beam with a finite cross section also diffracts [14]. At a certain optical power level, the beam self-focusing and diffraction can be balanced in such a way that the beam propagates in the nonlinear medium with a plane wavefront and a constant transverse intensity profile [1]. This phenomenon is called selftrapping of an optical beam [1]. The optical wave propagation in a nonlinear medium is described by the following wave equation for the electric field E ! ð Þ x; y; z; t [14]

$$
\vec{\alpha}\,\,\text{curl}\,\text{curl}\,\,\vec{E} + \mu\_0 \frac{\partial^2 \vec{D}}{\partial t^2} = -\mu\_0 \frac{\partial^2 \vec{D}}{\partial t^2} \tag{13}
$$

The corresponding linear electric induction vectors D

Figure 2. Propagation direction and polarization of the ordinary wave E

!<sup>L</sup>

<sup>e</sup> ¼ ε<sup>0</sup> a !

> k 2 <sup>o</sup> ¼ ε<sup>⊥</sup>

<sup>x</sup>ε⊥eex þ a !

dispersion relations for the ordinary and extraordinary waves, respectively [7, 18]

ω2 c2 ; k 2 ex ε∥ þ k2 ez ε⊥

In the linear approximation, substituting Eqs. (14)–(16) into the wave Eq. (13) we obtain the

It should be noted that the ordinary and extraordinary beams in the uniaxial medium propa-

We consider separately the self-focusing and self-trapping of the ordinary and extraordinary beams [7]. We start with the analysis of the slab-shaped ordinary beam with the dimension in the Y direction much greater than in the incidence XZ plane. In such a case, the dependence on the coordinate y may be neglected [7, 9]. Substituting expression (14) into the equation of

; DNL

Expression (18) shows that the nonlinearity related to the smectic layer normal strain is the

<sup>o</sup> ¼ ε0a<sup>⊥</sup>

∂u ∂z

<sup>B</sup> j j Ao <sup>2</sup>

<sup>e</sup> and D !<sup>L</sup>

with respect to the Z axis [18]. Here, θ<sup>1</sup> is the angle between k

!

DL

θ<sup>e</sup> ¼ arctan ε⊥=ε<sup>∥</sup>

motion (9), we obtain [7, 9]

the Z axis.

oy ¼ ε0ε⊥Ey; D

gate in different directions and the vectors E

tanθ<sup>1</sup>

wave propagates in the direction of the beam vector s

∂u <sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>ε</sup>0a<sup>⊥</sup>

Kerr-type nonlinearity [14]. We introduce now the coordinates x<sup>0</sup>

the ordinary beam propagation direction, respectively [7, 9]

!<sup>L</sup> <sup>o</sup> , D !<sup>L</sup>

!

¼ ω2

! ⊥E !

<sup>e</sup> are given by [7, 18]

<sup>o</sup> and extraordinary wave E

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137

<sup>c</sup><sup>2</sup> (17)

! <sup>e</sup> in SALC.

<sup>e</sup> are not parallel [18]. The extraordinary

<sup>e</sup>, which is determined by the angle

Eo (18)

; z<sup>0</sup> ð Þ parallel and normal to

! <sup>e</sup> and

<sup>z</sup>ε∥eez Aeexp½i kð Þ exx <sup>þ</sup> kezz � <sup>ω</sup><sup>t</sup> � þ <sup>c</sup>:c: (16)

Here, μ<sup>0</sup> is the free space permeability, D !<sup>L</sup> and D !NL are the linear and nonlinear parts of the electric induction. In SALC as a uniaxial medium two waves with the same frequency ω can propagate: an ordinary wave with the wave vector k ! <sup>o</sup> and an extraordinary one with the wave vector k ! <sup>e</sup> [2, 18]. Taking into account the SALC symmetry, we can choose the xz plane as a propagation plane. Then, the ordinary wave is polarized along the Y axis, and its electric field is given by

$$E\_{oy} = A\_o \exp[i(k\_{ox}\mathbf{x} + k\_{ox}\mathbf{z} - \omega t)] + \text{c.c.}\tag{14}$$

The extraordinary wave is polarized in the XZ plane having a component along the optical axis Z. The electric field of the extraordinary wave has the form

$$
\overrightarrow{E}\_{\varepsilon} = \overrightarrow{e}\_{\varepsilon} A\_{\varepsilon} \exp[i(k\_{\varepsilon x} \mathbf{x} + k\_{\varepsilon z} \mathbf{z} - \omega t)] + \varepsilon \,\mathrm{c}.\tag{15}
$$

Here, e ! <sup>e</sup> ¼ a ! xeex þ a ! zeez is the polarization unit vector of the extraordinary wave, a ! x, <sup>z</sup> are the unit vectors of the X, Z axes, and c.c. stands for complex conjugate. The propagation direction and polarization of the ordinary and extraordinary waves in SALC are shown in Figure 2.

Figure 2. Propagation direction and polarization of the ordinary wave E ! <sup>o</sup> and extraordinary wave E ! <sup>e</sup> in SALC.

The corresponding linear electric induction vectors D !<sup>L</sup> <sup>o</sup> , D !<sup>L</sup> <sup>e</sup> are given by [7, 18]

SALC complex order parameter phase [1]. If we take into account the viscosity terms in

k 2 Sz

We first consider the self-action effects of the optical waves propagating in an anisotropic inhomogeneous nonlinear medium. The light beam propagation through a nonlinear medium is accompanied by the intensity-dependent phase shift on the wavefront of the beam [2]. Selffocusing of light results from the wavefront distortion inflicted on the beam by itself while propagating in a nonlinear medium [14]. In such a case, the field-induced refractive change Δn has the form <sup>Δ</sup><sup>n</sup> <sup>¼</sup> <sup>n</sup>2j j <sup>E</sup> <sup>2</sup> where <sup>n</sup><sup>2</sup> <sup>¼</sup> const [14]. A light beam with a finite cross section also diffracts [14]. At a certain optical power level, the beam self-focusing and diffraction can be balanced in such a way that the beam propagates in the nonlinear medium with a plane wavefront and a constant transverse intensity profile [1]. This phenomenon is called selftrapping of an optical beam [1]. The optical wave propagation in a nonlinear medium is

þ 1 2

ð Þ α<sup>4</sup> þ α<sup>56</sup> k

2 S

!

∂2 D ! NL ∂t

Eoy ¼ Aoexp½i kð Þ oxx þ kozz � ωt � þ c:c: (14)

eAeexp½i kð Þ exx þ kezz � ωt � þ c:c: (15)

ð Þ x; y; z; t [14]

are the linear and nonlinear parts of the

<sup>o</sup> and an extraordinary one with the wave

<sup>2</sup> (13)

!

x, <sup>z</sup> are the

3 5

�1

(12)

Eq. (10), then we can obtain the SS relaxation time τ<sup>S</sup> given by

136 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

2 4

described by the following wave equation for the electric field E

axis Z. The electric field of the extraordinary wave has the form

E ! <sup>e</sup> ¼ e !

Here, μ<sup>0</sup> is the free space permeability, D

vector k !

Here, e ! <sup>e</sup> ¼ a ! xeex þ a !

is given by

propagate: an ordinary wave with the wave vector k

curlcurl E!

þμ<sup>0</sup> ∂2 D ! L ∂t

!<sup>L</sup>

and D !NL

electric induction. In SALC as a uniaxial medium two waves with the same frequency ω can

propagation plane. Then, the ordinary wave is polarized along the Y axis, and its electric field

The extraordinary wave is polarized in the XZ plane having a component along the optical

unit vectors of the X, Z axes, and c.c. stands for complex conjugate. The propagation direction and polarization of the ordinary and extraordinary waves in SALC are shown in Figure 2.

<sup>2</sup> ¼ �μ<sup>0</sup>

!

<sup>e</sup> [2, 18]. Taking into account the SALC symmetry, we can choose the xz plane as a

zeez is the polarization unit vector of the extraordinary wave, a

k 2 Sx þ k 2 Sy � �

3. Self-focusing and self-trapping of optical beams in SALC

k 2 S

τ<sup>S</sup> ¼ 2r α<sup>1</sup>

SS has been observed experimentally [15–17].

$$D\_{ay}^{L} = \varepsilon\_{0}\varepsilon\_{\perp}\underline{E}\_{y}; \overrightarrow{D}\_{\varepsilon}^{L} = \varepsilon\_{0}\left(\overrightarrow{a}\_{x}\varepsilon\_{\perp}\underline{e}\_{ex} + \overrightarrow{a}\_{z}\varepsilon\_{\parallel}\underline{e}\_{\varepsilon z}\right)A\_{\epsilon}\exp[\mathrm{i}(k\_{cx}x + k\_{cz}z - \omega t)] + \text{c.c.}\tag{16}$$

In the linear approximation, substituting Eqs. (14)–(16) into the wave Eq. (13) we obtain the dispersion relations for the ordinary and extraordinary waves, respectively [7, 18]

$$k\_o^2 = \varepsilon\_\perp \frac{\omega^2}{c^2}; \frac{k\_{ex}^2}{\varepsilon\_\parallel} + \frac{k\_{ez}^2}{\varepsilon\_\perp} = \frac{\omega^2}{c^2} \tag{17}$$

It should be noted that the ordinary and extraordinary beams in the uniaxial medium propagate in different directions and the vectors E ! <sup>e</sup> and D !<sup>L</sup> <sup>e</sup> are not parallel [18]. The extraordinary wave propagates in the direction of the beam vector s ! ⊥E ! <sup>e</sup>, which is determined by the angle θ<sup>e</sup> ¼ arctan ε⊥=ε<sup>∥</sup> tanθ<sup>1</sup> with respect to the Z axis [18]. Here, θ<sup>1</sup> is the angle between k ! <sup>e</sup> and the Z axis.

We consider separately the self-focusing and self-trapping of the ordinary and extraordinary beams [7]. We start with the analysis of the slab-shaped ordinary beam with the dimension in the Y direction much greater than in the incidence XZ plane. In such a case, the dependence on the coordinate y may be neglected [7, 9]. Substituting expression (14) into the equation of motion (9), we obtain [7, 9]

$$\frac{\partial \mu}{\partial z} = \frac{\varepsilon\_0 a\_\perp}{B} |A\_o|^2; D\_o^{\text{NL}} = \varepsilon\_0 a\_\perp \frac{\partial \mu}{\partial z} E\_o \tag{18}$$

Expression (18) shows that the nonlinearity related to the smectic layer normal strain is the Kerr-type nonlinearity [14]. We introduce now the coordinates x<sup>0</sup> ; z<sup>0</sup> ð Þ parallel and normal to the ordinary beam propagation direction, respectively [7, 9]

$$\mathbf{x}' = \mathbf{x}\sin\theta\_o + \mathbf{z}\cos\theta\_o; \mathbf{z}' = -\mathbf{x}\cos\theta\_o + \mathbf{z}\sin\theta\_o \tag{19}$$

Here, θ<sup>o</sup> is the angle between k ! <sup>o</sup> and the Z axis. We use the SVAA for the ordinary beam amplitude Ao [14]

$$\left|\frac{\partial^2 A\_o}{\partial \mathbf{x}'^2}\right| \ll \left|k\_o \frac{\partial A\_o}{\partial \mathbf{x}'}\right| \sim \left|\frac{\partial^2 A\_o}{\partial \mathbf{z}'^2}\right|\tag{20}$$

We are interested in the spatially localized solutions with the following boundary conditions [7, 9]

$$\lim\_{z' \to \infty} |A\_o(z')| = 0; \frac{\partial |A\_o(z')|}{\partial z'} \Big|\_{z'=0} = 0; |A\_o(z'=0)| = |A\_o|\_{\max} \tag{21}$$

Then, substituting expressions (14), (18), (19) and the first ones of Eq. (16), (17) into Eq. (13) and taking into account the SVAA conditions (20), we obtain the truncated equation for the SVA Ao x<sup>0</sup> ; z<sup>0</sup> ð Þ, which has the form [7]

$$i\frac{\partial A\_o}{\partial \mathbf{x}'} + \frac{1}{2k\_o} \frac{\partial^2 A\_o}{\partial \mathbf{z}'^2} + \frac{\omega^2}{c^2} \frac{\varepsilon\_0 a\_\perp^2}{2Bk\_o} |A\_o|^2 A\_o = 0 \tag{22}$$

he <sup>¼</sup> <sup>a</sup>⊥e<sup>2</sup>

θ<sup>o</sup> ¼ π=6.

� 105

ex <sup>þ</sup> <sup>a</sup>∥e<sup>2</sup>

soliton is given by [7, 10]

samples with a thickness of 10�<sup>4</sup>

The cubic susceptibility of SALC χð Þ<sup>3</sup>

ez � �sinθ<sup>e</sup> <sup>þ</sup> <sup>2</sup>εaeexeezcosθe, le<sup>∥</sup> <sup>¼</sup> ke∥ð Þ <sup>1</sup> <sup>þ</sup> ð Þ <sup>ε</sup>a=ε<sup>⊥</sup> sinθ<sup>e</sup> �<sup>1</sup>

we ¼

tor component parallel to the beam vector. The width we of the extraordinary beam spatial

Figure 3. The self-trapped ordinary beam normalized intensity for the maximum amplitude j j Ao max <sup>¼</sup> 105V=<sup>m</sup> and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

eezsinθ<sup>e</sup>

m have been demonstrated experimentally [2, 17].

ð Þ sinθ<sup>e</sup> c

SALC related to the smectic layer compression is larger than χð Þ<sup>3</sup>

<sup>p</sup> he Ae j jmax<sup>ω</sup> (25)

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139

; Amax ¼ A zð Þ<sup>0</sup> (26)

εa ε⊥

s � �

ffiffiffiffi ε0

<sup>V</sup>=m, <sup>ω</sup> � 1015s�<sup>1</sup> and small angle <sup>θ</sup><sup>e</sup> the spatial soliton width is wo,e � <sup>10</sup>�<sup>4</sup>

solution represents a bright surface wave with the amplitude A zð Þ given by [7]

A zð Þ¼ <sup>A</sup>max cosh <sup>z</sup> � <sup>z</sup><sup>0</sup>

For the typical values of the SALC parameters [1, 2], the optical beam electric field Ao,e j jmax

The optical wave self-trapping can occur also at the interface between the linear medium in the region z < 0 with the permittivity ε<sup>s</sup> and the SALC cladding ð Þ z > 0 . For the light wave Ey ¼ A zð Þexpi kð Þ ox � ωt propagating along the interface parallel to the X axis, the self-trapped

> wo � � � � �<sup>1</sup>

related to the Kerr nonlinearity in organic liquids [14], but it is much less than the giant orientational nonlinearity (GON) in NLC [2]. However, the optical beam intensity in SALC may be much greater than in NLC, which are extremely sensitive to the strong optical fields [2]. In such cases, the approach based on the purely orientational mechanism of the optical nonlinearity is invalid.

2B 1 þ

, and ke<sup>∥</sup> is the wave vec-

m [7]. SALC

Eq. (22) is the nonlinear Schrodinger equation (NSE) [19]. The coefficient of the last term in the left-hand side (LHS) of Eq. (22) is positive definite ω<sup>2</sup>ε<sup>2</sup> 0a2 <sup>⊥</sup>= 4c<sup>2</sup>Bko � � > 0, which corresponds to the stationary two-dimensional self-focusing of the light beam. The solution of Eq. (22) with the boundary conditions (21) has the form [7]

$$A\_o(\mathbf{x}', \mathbf{z}') = |A\_o|\_{\text{max}} \exp\left(i\frac{\varepsilon\_0 a\_\perp^2 |A\_o|\_{\text{max}}^2}{4B\varepsilon\_\perp} k\_o \mathbf{x}'\right) \left[\cosh\left(\frac{\sqrt{\varepsilon\_0} a\_\perp |A\_o|\_{\text{max}}}{\sqrt{2B\varepsilon\_\perp}} k\_o \mathbf{z}'\right)\right]^{-1} \tag{23}$$

The self-trapped beam (23) is the so-called spatial soliton with the width wo <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi 2Bε<sup>⊥</sup> p ffiffiffiffi ε0 <sup>p</sup> <sup>a</sup><sup>⊥</sup> � j j Ao maxkoÞ �<sup>1</sup> [7].

The self-trapped ordinary beam normalized intensity spatial distribution is shown in Figure 3.

The self-trapping of the extraordinary beam (15) can be realized only when the anisotropy angle ð Þ <sup>θ</sup><sup>1</sup> � <sup>θ</sup><sup>e</sup> is small enough: tanð Þ <sup>θ</sup><sup>1</sup> � <sup>θ</sup><sup>e</sup> <sup>≪</sup> ð Þ kewo �<sup>1</sup> [7]. For the typical values of <sup>ε</sup>⊥, <sup>ε</sup><sup>∥</sup> [2], the following condition is valid: 0 ≤ tanð Þ θ<sup>1</sup> � θ<sup>e</sup> ≤ 0:12, and the self-trapping condition for the extraordinary beam can be satisfied [7]. Then, using the procedure described above for the ordinary beam, we obtain the spatial soliton of the extraordinary beam. It has the form [7, 10]

$$A\_e = |A\_e|\_{\max} \exp\left[i\frac{\varepsilon\_0 h\_e^2 |A\_e|\_{\max}^2 \omega^2}{4Bl\_{\varepsilon \parallel} \left(1 + \frac{\varepsilon\_d}{\varepsilon\_\perp} c\_{\varepsilon \bar{z}} \sin \theta\_e\right) c^2 \sin^2 \theta\_e} \right] \left[\cosh\left(\frac{z''}{w\_e}\right)\right]^{-1} \tag{24}$$

Here, x<sup>00</sup> ¼ xsinθ<sup>e</sup> þ zcosθe; z<sup>00</sup> ¼ �xcosθ<sup>e</sup> þ zsinθ<sup>e</sup> are the coordinates parallel and perpendicular to the beam vector, respectively,

x<sup>0</sup> ¼ xsinθ<sup>o</sup> þ zcosθo; z<sup>0</sup> ¼ �xcosθ<sup>o</sup> þ zsinθ<sup>o</sup> (19)

� � � �

<sup>o</sup> and the Z axis. We use the SVAA for the ordinary beam

¼ 0; Ao z<sup>0</sup> j j¼ ð Þ ¼ 0 j j Ao max (21)

Ao ¼ 0 (22)

� � > 0, which corresponds to

(20)

(23)

ε0 <sup>p</sup> <sup>a</sup><sup>⊥</sup> �

(24)

2Bε<sup>⊥</sup> p ffiffiffiffi

Here, θ<sup>o</sup> is the angle between k

lim z0

; z<sup>0</sup> ð Þ, which has the form [7]

Ao x<sup>0</sup>

�<sup>1</sup> [7].

!<sup>∞</sup> Ao <sup>z</sup><sup>0</sup> j j¼ ð Þ <sup>0</sup>;

i ∂Ao ∂x<sup>0</sup> þ

left-hand side (LHS) of Eq. (22) is positive definite ω<sup>2</sup>ε<sup>2</sup>

the boundary conditions (21) has the form [7]

; z<sup>0</sup> ð Þ¼ j j Ao maxexp i

Ae <sup>¼</sup> Ae j jmaxexp <sup>i</sup> <sup>ε</sup>0h<sup>2</sup>

ular to the beam vector, respectively,

4Ble<sup>∥</sup> 1 þ

amplitude Ao [14]

Ao x<sup>0</sup>

j j Ao maxkoÞ

!

138 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

∂2 Ao ∂x0<sup>2</sup> � � � �

� � � � ≪ ko

∂ Ao z<sup>0</sup> j j ð Þ ∂z<sup>0</sup>

1 2ko ∂2 Ao <sup>∂</sup>z0<sup>2</sup> <sup>þ</sup>

ε0a<sup>2</sup> <sup>⊥</sup>j j Ao <sup>2</sup> max 4Bε<sup>⊥</sup>

� � � �

∂Ao ∂x<sup>0</sup>

We are interested in the spatially localized solutions with the following boundary conditions [7, 9]

Then, substituting expressions (14), (18), (19) and the first ones of Eq. (16), (17) into Eq. (13) and taking into account the SVAA conditions (20), we obtain the truncated equation for the SVA

> ω2 c2

Eq. (22) is the nonlinear Schrodinger equation (NSE) [19]. The coefficient of the last term in the

the stationary two-dimensional self-focusing of the light beam. The solution of Eq. (22) with

!

The self-trapped beam (23) is the so-called spatial soliton with the width wo <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi

The self-trapped ordinary beam normalized intensity spatial distribution is shown in Figure 3. The self-trapping of the extraordinary beam (15) can be realized only when the anisotropy angle ð Þ <sup>θ</sup><sup>1</sup> � <sup>θ</sup><sup>e</sup> is small enough: tanð Þ <sup>θ</sup><sup>1</sup> � <sup>θ</sup><sup>e</sup> <sup>≪</sup> ð Þ kewo �<sup>1</sup> [7]. For the typical values of <sup>ε</sup>⊥, <sup>ε</sup><sup>∥</sup> [2], the following condition is valid: 0 ≤ tanð Þ θ<sup>1</sup> � θ<sup>e</sup> ≤ 0:12, and the self-trapping condition for the extraordinary beam can be satisfied [7]. Then, using the procedure described above for the ordinary beam, we obtain the spatial soliton of the extraordinary beam. It has the form [7, 10]

> <sup>e</sup> Ae j j<sup>2</sup> maxω<sup>2</sup>

eezsinθ<sup>e</sup> � �

Here, x<sup>00</sup> ¼ xsinθ<sup>e</sup> þ zcosθe; z<sup>00</sup> ¼ �xcosθ<sup>e</sup> þ zsinθ<sup>e</sup> are the coordinates parallel and perpendic-

c2sin<sup>2</sup>θ<sup>e</sup>

x00

cosh <sup>z</sup><sup>00</sup> we � � � � �<sup>1</sup>

εa ε⊥ kox<sup>0</sup>

ε0a<sup>2</sup> ⊥ 2Bko

j j Ao <sup>2</sup>

0a2

cosh

<sup>⊥</sup>= 4c<sup>2</sup>Bko

ffiffiffiffi ε0 <sup>p</sup> <sup>a</sup>⊥j j Ao max ffiffiffiffiffiffiffiffiffiffi 2Bε<sup>⊥</sup> p koz<sup>0</sup> � � � � �<sup>1</sup>

� � � � z0 ¼0 � � � � � ∂2 Ao ∂z0<sup>2</sup> � � � �

Figure 3. The self-trapped ordinary beam normalized intensity for the maximum amplitude j j Ao max <sup>¼</sup> 105V=<sup>m</sup> and θ<sup>o</sup> ¼ π=6.

he <sup>¼</sup> <sup>a</sup>⊥e<sup>2</sup> ex <sup>þ</sup> <sup>a</sup>∥e<sup>2</sup> ez � �sinθ<sup>e</sup> <sup>þ</sup> <sup>2</sup>εaeexeezcosθe, le<sup>∥</sup> <sup>¼</sup> ke∥ð Þ <sup>1</sup> <sup>þ</sup> ð Þ <sup>ε</sup>a=ε<sup>⊥</sup> sinθ<sup>e</sup> �<sup>1</sup> , and ke<sup>∥</sup> is the wave vector component parallel to the beam vector. The width we of the extraordinary beam spatial soliton is given by [7, 10]

$$w\_{\varepsilon} = \frac{\sqrt{2B \left(1 + \frac{\varepsilon\_d}{\varepsilon\_\perp} e\_{\varepsilon x} \sin \theta\_\varepsilon \right)} (\sin \theta\_\varepsilon) c}{\sqrt{\varepsilon\_0} h\_\varepsilon |A\_\varepsilon|\_{\text{max}} \omega} \tag{25}$$

For the typical values of the SALC parameters [1, 2], the optical beam electric field Ao,e j jmax � 105 <sup>V</sup>=m, <sup>ω</sup> � 1015s�<sup>1</sup> and small angle <sup>θ</sup><sup>e</sup> the spatial soliton width is wo,e � <sup>10</sup>�<sup>4</sup> m [7]. SALC samples with a thickness of 10�<sup>4</sup> m have been demonstrated experimentally [2, 17].

The optical wave self-trapping can occur also at the interface between the linear medium in the region z < 0 with the permittivity ε<sup>s</sup> and the SALC cladding ð Þ z > 0 . For the light wave Ey ¼ A zð Þexpi kð Þ ox � ωt propagating along the interface parallel to the X axis, the self-trapped solution represents a bright surface wave with the amplitude A zð Þ given by [7]

$$A(z) = A\_{\text{max}} \left[ \cosh \left( \frac{z - z\_0}{w\_o} \right) \right]^{-1}; A\_{\text{max}} = A(z\_0) \tag{26}$$

The cubic susceptibility of SALC χð Þ<sup>3</sup> SALC related to the smectic layer compression is larger than χð Þ<sup>3</sup> related to the Kerr nonlinearity in organic liquids [14], but it is much less than the giant orientational nonlinearity (GON) in NLC [2]. However, the optical beam intensity in SALC may be much greater than in NLC, which are extremely sensitive to the strong optical fields [2]. In such cases, the approach based on the purely orientational mechanism of the optical nonlinearity is invalid.

### 4. Stimulated light scattering (SLS) in SALC

SLS is a process of parametric coupling between light waves and the material excitations of the medium [14]. We consider the SLS in SALC related to the smectic layer normal displacement and SS excited by the interfering optical waves [5, 6, 8–10]. We have taken into account the combined effect of SALC layered structure and anisotropy. It should be noted that SS propagates in SALC without the change of the mass density in such a way that the SS wave and the ordinary sound wave are decoupled [1].

In general case when the coupled optical waves have arbitrary polarizations and propagation directions, each optical wave in SALC ð Þ z > 0 splits into the extraordinary and ordinary ones with the same frequency and different wave vectors due to the strong anisotropy of SALC [6, 10, 18]. The polarizations of these waves are shown in Figure 4. The XZ plane is chosen to be the propagation plane of the waves E !o,e <sup>1</sup> . In such a case, the extraordinary wave E !e <sup>1</sup> is polarized in the XZ plane, while the ordinary wave E !o <sup>1</sup> is parallel to the Y axis [18]. The ordinary wave E !o 2 is polarized in the XY plane perpendicular to the optical Z axis, and the extraordinary wave E !e 2 possesses a three-dimensional polarization vector e !e <sup>2</sup> [18]. The wave vectors k !o <sup>1</sup>,<sup>2</sup> and k !e <sup>1</sup> of these waves satisfy the dispersion relations (17) while the three-dimensional wave vector k !e 2 satisfies the dispersion relation ke 2x � �<sup>2</sup> <sup>þ</sup> ke 2y � �<sup>2</sup> � �ε�<sup>1</sup> <sup>∥</sup> þ k e 2z � �<sup>2</sup> ε�<sup>1</sup> <sup>⊥</sup> ¼ ð Þ ω2=c <sup>2</sup> [18]. The fundamental ordinary and extraordinary waves have the form, respectively

$$\begin{aligned} \overline{E}\_1^{o,\epsilon} &= \overline{\vec{e}}\_1^{o,\epsilon} \left\{ A\_1^{o,\epsilon}(\mathbf{z}) \text{expi} \left[ \left( \overline{\vec{k}}\_1^{o,\epsilon} \cdot \overrightarrow{\vec{r}} \right) - w\_1 t \right] + c.c. \right\} \\ \overline{E}\_2^{o,\epsilon} &= \overline{\vec{e}}\_2^{o,\epsilon} \left\{ A\_2^{o,\epsilon}(\mathbf{z}) \text{expi} \left[ \left( \overline{\vec{k}}\_2^{o,\epsilon} \cdot \overrightarrow{\vec{r}} \right) - w\_2 t \right] + c.c. \right\} \end{aligned} \tag{27}$$

Here,Δk ! <sup>1</sup> ¼ k !e <sup>1</sup> � k !o <sup>2</sup>;Δk ! <sup>2</sup> ¼ k !e <sup>1</sup> � k !e <sup>2</sup>;Δk ! <sup>3</sup> ¼ k !o <sup>1</sup> � k !o <sup>2</sup>;Δk ! <sup>4</sup> ¼ k !o <sup>1</sup> � k !e 2;

<sup>h</sup><sup>1</sup> <sup>¼</sup> <sup>a</sup>⊥Δk1ze<sup>e</sup>

<sup>h</sup><sup>2</sup> <sup>¼</sup> <sup>a</sup>⊥Δk2ze<sup>e</sup>

<sup>h</sup><sup>4</sup> <sup>¼</sup> <sup>a</sup>⊥Δk4ze<sup>e</sup>

1xe<sup>o</sup>

1xe<sup>e</sup>

<sup>2</sup><sup>x</sup> � <sup>ε</sup><sup>a</sup> <sup>Δ</sup>k1xe<sup>e</sup>

<sup>2</sup><sup>x</sup> <sup>þ</sup> <sup>a</sup>∥Δk2ze<sup>e</sup>

! j � � <sup>¼</sup> ð Þ <sup>Δ</sup><sup>ω</sup>

> Δkj<sup>⊥</sup> � �<sup>2</sup> <sup>Δ</sup>kjz � �<sup>2</sup> Δkj � �<sup>2</sup> þ

<sup>2</sup><sup>y</sup> � <sup>ε</sup>aΔk4ye<sup>e</sup>

Gj Δω;Δk

ω<sup>2</sup> by the other pair of optical waves E

εN

Nb xz ¼ Nb zx ¼ �ε<sup>a</sup>

<sup>Γ</sup><sup>j</sup> <sup>¼</sup> <sup>1</sup> r α1

permittivity tensor (8) ε<sup>N</sup>

1ze<sup>o</sup>

<sup>2</sup>z; <sup>M</sup><sup>1</sup> <sup>¼</sup> Ae

1zee

Figure 4. The polarizations of the fundamental ordinary waves E

<sup>2</sup><sup>x</sup> <sup>þ</sup> <sup>Δ</sup>k1ye<sup>e</sup>

<sup>2</sup><sup>z</sup> � <sup>ε</sup><sup>a</sup> <sup>Δ</sup>k2<sup>x</sup> <sup>e</sup><sup>e</sup>

<sup>1</sup> <sup>A</sup><sup>o</sup> 2 � �<sup>∗</sup>

h i,

The parametric amplification of the fundamental optical waves E

1ze<sup>o</sup> 2y

<sup>2</sup> <sup>þ</sup> <sup>i</sup>ΔωΓ<sup>j</sup> � <sup>Ω</sup><sup>2</sup>

1 2

" #

!o,e

ik ¼ Nb iku xð Þ ; y; z; t ; Nb xx ¼ Nb yy ¼ a<sup>⊥</sup>

∂ ∂x

1xe<sup>e</sup> <sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>e</sup><sup>e</sup> 1zee 2x � � <sup>þ</sup> <sup>Δ</sup>k2ye<sup>e</sup>

; M<sup>2</sup> <sup>¼</sup> <sup>A</sup><sup>e</sup>

j

ð Þ α<sup>4</sup> þ α<sup>56</sup> Δkj

SLS on the light-induced smectic layer dynamic grating (28) [6, 10]. It is actually the Stokes SLS [14]. The fundamental optical waves also create Stokes and anti-Stokes small harmonics with the combination frequencies and wave vectors. The analysis of SLS in SALC is based on the simultaneous solution of the smectic layer equation of motion (9), the wave Eq. (13) for ordinary waves (14) and extraordinary waves (15) with the permittivity tensor (8). The nonlinear part of the

<sup>1</sup> <sup>A</sup><sup>e</sup> 2 � �<sup>∗</sup> <sup>M</sup><sup>3</sup> <sup>¼</sup> <sup>A</sup><sup>o</sup>

!o

� �<sup>2</sup>

ik in the three-dimensional case can be written as follows [6]

; Nb yz ¼ Nb zy ¼ �ε<sup>a</sup>

∂ ∂z

> ∂ ∂y

1zee 2y

<sup>1</sup>, <sup>2</sup> and extraordinary waves E

<sup>1</sup> <sup>A</sup><sup>o</sup> 2 � �<sup>∗</sup>

!o,e

<sup>1</sup> with the higher frequency ω<sup>1</sup> occurs in SALC due to the

; Nb xy ¼ Nb yx ¼ 0;

; Nb zz ¼ a<sup>∥</sup>

∂ ∂z 2y;

, and

(29)

(30)

<sup>1</sup> <sup>A</sup><sup>e</sup> 2 � �<sup>∗</sup>

<sup>1</sup>, <sup>2</sup> in SALC (z > 0).

<sup>2</sup> with the lower frequency

, M<sup>4</sup> <sup>¼</sup> <sup>A</sup><sup>o</sup>

!e

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141

h i; <sup>h</sup><sup>3</sup> <sup>¼</sup> <sup>a</sup>⊥Δk3ze<sup>o</sup>

; Ω<sup>2</sup> <sup>j</sup> ¼ s 2 0 Δkj<sup>⊥</sup> � �<sup>2</sup> <sup>Δ</sup>kjz � �<sup>2</sup> Δkj � �<sup>2</sup>

Here, ω<sup>1</sup> > ω<sup>2</sup> and Δω ¼ ω<sup>1</sup> � ω<sup>2</sup> ≪ ω1. Each pair of the waves (27) has the same frequency, and for this reason, we define the nonlinear mixing of these waves as partially frequency degenerate FWM [6]. We assume that the complex amplitudes Ao,e <sup>1</sup>, <sup>2</sup>ð Þ¼ <sup>z</sup> Ao,e <sup>1</sup>,2ð Þz � � � � � �expiγo,e <sup>1</sup>, <sup>2</sup>ð Þz are slowly varying along the optical axis Z: ∂<sup>2</sup> Ao,e 1,2=∂z<sup>2</sup> � � � � � � <sup>≪</sup> <sup>k</sup> o,e <sup>1</sup>, <sup>2</sup>z∂Ao,e <sup>1</sup>,2=∂z � � � � � �. As a result, the nonlinear two-wave mixing analyzed in Ref. [5] transforms into a partially degenerate FWM [6, 10]. We substitute the waves (27) into equation of motion (9). The interfering optical waves (27) with close frequencies ω1,<sup>2</sup> create a dynamic grating of the smectic layer normal displacement u xð Þ ; y; z; t consisting of four propagating harmonics with the same frequency and different wave vectors. It has the form [6]

$$u(\mathbf{x}, y, z, t) = \frac{i\varepsilon\_0}{\rho} \sum\_{j=1}^{4} \frac{\left(\Delta k\_{j\perp}\right)^2 h\_j M\_j}{\left(\Delta k\_j\right)^2 G\_j \left(\Delta \omega, \Delta \vec{k}\_j\right)} \exp[\left(\Delta \vec{k}\_j \cdot \vec{r}\right) - \Delta \omega t] + c.c.\tag{28}$$

4. Stimulated light scattering (SLS) in SALC

140 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

ordinary sound wave are decoupled [1].

the propagation plane of the waves E

satisfies the dispersion relation ke

in the XZ plane, while the ordinary wave E

possesses a three-dimensional polarization vector e

E !o,e <sup>1</sup> ¼ e !o,e <sup>1</sup> <sup>A</sup>o,e

E !o,e <sup>2</sup> ¼ e !o,e <sup>2</sup> <sup>A</sup>o,e

are slowly varying along the optical axis Z: ∂<sup>2</sup>

iε<sup>0</sup> r X 4

j¼1

Δkj � �<sup>2</sup>

Δkj<sup>⊥</sup> � �<sup>2</sup>

ent wave vectors. It has the form [6]

u xð Þ¼ ; y; z; t

SLS is a process of parametric coupling between light waves and the material excitations of the medium [14]. We consider the SLS in SALC related to the smectic layer normal displacement and SS excited by the interfering optical waves [5, 6, 8–10]. We have taken into account the combined effect of SALC layered structure and anisotropy. It should be noted that SS propagates in SALC without the change of the mass density in such a way that the SS wave and the

In general case when the coupled optical waves have arbitrary polarizations and propagation directions, each optical wave in SALC ð Þ z > 0 splits into the extraordinary and ordinary ones with the same frequency and different wave vectors due to the strong anisotropy of SALC [6, 10, 18]. The polarizations of these waves are shown in Figure 4. The XZ plane is chosen to be

is polarized in the XY plane perpendicular to the optical Z axis, and the extraordinary wave E

these waves satisfy the dispersion relations (17) while the three-dimensional wave vector k

2y

!e

ε�<sup>1</sup> <sup>∥</sup> þ k e 2z � �<sup>2</sup> ε�<sup>1</sup>

� �

� �

!o,e <sup>1</sup> � r ! � �

!o,e <sup>2</sup> � r ! � �

Here, ω<sup>1</sup> > ω<sup>2</sup> and Δω ¼ ω<sup>1</sup> � ω<sup>2</sup> ≪ ω1. Each pair of the waves (27) has the same frequency, and for this reason, we define the nonlinear mixing of these waves as partially frequency

> � � �

nonlinear two-wave mixing analyzed in Ref. [5] transforms into a partially degenerate FWM [6, 10]. We substitute the waves (27) into equation of motion (9). The interfering optical waves (27) with close frequencies ω1,<sup>2</sup> create a dynamic grating of the smectic layer normal displacement u xð Þ ; y; z; t consisting of four propagating harmonics with the same frequency and differ-

hjMj

! j � � exp<sup>i</sup> <sup>Δ</sup><sup>k</sup>

Gj Δω;Δk

� �

� �

Ao,e 1,2=∂z<sup>2</sup>

<sup>1</sup> . In such a case, the extraordinary wave E

� ω1t

� ω2t

� � � <sup>≪</sup> <sup>k</sup> o,e <sup>1</sup>, <sup>2</sup>z∂Ao,e <sup>1</sup>,2=∂z

� � �

! <sup>j</sup>� r ! � �

h i

<sup>1</sup> is parallel to the Y axis [18]. The ordinary wave E

<sup>2</sup> [18]. The wave vectors k

þ c:c:

þ c:c:

<sup>1</sup>, <sup>2</sup>ð Þ¼ <sup>z</sup> Ao,e

� Δωt

� � �

� � <sup>1</sup>,2ð Þz

� � �expiγo,e <sup>1</sup>, <sup>2</sup>ð Þz

�. As a result, the

þ c:c: (28)

<sup>⊥</sup> ¼ ð Þ ω2=c

!e

!o

<sup>1</sup> is polarized

<sup>1</sup>,<sup>2</sup> and k

<sup>2</sup> [18]. The funda-

!o 2

!e 2

!e 2

(27)

!e <sup>1</sup> of

!o,e

2x � �<sup>2</sup> <sup>þ</sup> ke

mental ordinary and extraordinary waves have the form, respectively

degenerate FWM [6]. We assume that the complex amplitudes Ao,e

!o

� �<sup>2</sup> � �

<sup>1</sup> ð Þz expi k

<sup>2</sup> ð Þz expi k

Figure 4. The polarizations of the fundamental ordinary waves E !o <sup>1</sup>, <sup>2</sup> and extraordinary waves E !e <sup>1</sup>, <sup>2</sup> in SALC (z > 0).

Here,Δk ! <sup>1</sup> ¼ k !e <sup>1</sup> � k !o <sup>2</sup>;Δk ! <sup>2</sup> ¼ k !e <sup>1</sup> � k !e <sup>2</sup>;Δk ! <sup>3</sup> ¼ k !o <sup>1</sup> � k !o <sup>2</sup>;Δk ! <sup>4</sup> ¼ k !o <sup>1</sup> � k !e 2; <sup>h</sup><sup>1</sup> <sup>¼</sup> <sup>a</sup>⊥Δk1ze<sup>e</sup> 1xe<sup>o</sup> <sup>2</sup><sup>x</sup> � <sup>ε</sup><sup>a</sup> <sup>Δ</sup>k1xe<sup>e</sup> 1ze<sup>o</sup> <sup>2</sup><sup>x</sup> <sup>þ</sup> <sup>Δ</sup>k1ye<sup>e</sup> 1ze<sup>o</sup> 2y h i, <sup>h</sup><sup>2</sup> <sup>¼</sup> <sup>a</sup>⊥Δk2ze<sup>e</sup> 1xe<sup>e</sup> <sup>2</sup><sup>x</sup> <sup>þ</sup> <sup>a</sup>∥Δk2ze<sup>e</sup> 1zee <sup>2</sup><sup>z</sup> � <sup>ε</sup><sup>a</sup> <sup>Δ</sup>k2<sup>x</sup> <sup>e</sup><sup>e</sup> 1xe<sup>e</sup> <sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>e</sup><sup>e</sup> 1zee 2x � � <sup>þ</sup> <sup>Δ</sup>k2ye<sup>e</sup> 1zee 2y h i; <sup>h</sup><sup>3</sup> <sup>¼</sup> <sup>a</sup>⊥Δk3ze<sup>o</sup> 2y; <sup>h</sup><sup>4</sup> <sup>¼</sup> <sup>a</sup>⊥Δk4ze<sup>e</sup> <sup>2</sup><sup>y</sup> � <sup>ε</sup>aΔk4ye<sup>e</sup> <sup>2</sup>z; <sup>M</sup><sup>1</sup> <sup>¼</sup> Ae <sup>1</sup> <sup>A</sup><sup>o</sup> 2 � �<sup>∗</sup> ; M<sup>2</sup> <sup>¼</sup> <sup>A</sup><sup>e</sup> <sup>1</sup> <sup>A</sup><sup>e</sup> 2 � �<sup>∗</sup> <sup>M</sup><sup>3</sup> <sup>¼</sup> <sup>A</sup><sup>o</sup> <sup>1</sup> <sup>A</sup><sup>o</sup> 2 � �<sup>∗</sup> , M<sup>4</sup> <sup>¼</sup> <sup>A</sup><sup>o</sup> <sup>1</sup> <sup>A</sup><sup>e</sup> 2 � �<sup>∗</sup> , and Gj Δω;Δk ! j � � <sup>¼</sup> ð Þ <sup>Δ</sup><sup>ω</sup> <sup>2</sup> <sup>þ</sup> <sup>i</sup>ΔωΓ<sup>j</sup> � <sup>Ω</sup><sup>2</sup> j <sup>Γ</sup><sup>j</sup> <sup>¼</sup> <sup>1</sup> r α1 Δkj<sup>⊥</sup> � �<sup>2</sup> <sup>Δ</sup>kjz � �<sup>2</sup> Δkj � �<sup>2</sup> þ 1 2 ð Þ α<sup>4</sup> þ α<sup>56</sup> Δkj � �<sup>2</sup> " #; Ω<sup>2</sup> <sup>j</sup> ¼ s 2 0 Δkj<sup>⊥</sup> � �<sup>2</sup> <sup>Δ</sup>kjz � �<sup>2</sup> Δkj � �<sup>2</sup> (29)

The parametric amplification of the fundamental optical waves E !o,e <sup>2</sup> with the lower frequency ω<sup>2</sup> by the other pair of optical waves E !o,e <sup>1</sup> with the higher frequency ω<sup>1</sup> occurs in SALC due to the SLS on the light-induced smectic layer dynamic grating (28) [6, 10]. It is actually the Stokes SLS [14]. The fundamental optical waves also create Stokes and anti-Stokes small harmonics with the combination frequencies and wave vectors. The analysis of SLS in SALC is based on the simultaneous solution of the smectic layer equation of motion (9), the wave Eq. (13) for ordinary waves (14) and extraordinary waves (15) with the permittivity tensor (8). The nonlinear part of the permittivity tensor (8) ε<sup>N</sup> ik in the three-dimensional case can be written as follows [6]

$$\begin{aligned} \varepsilon\_{ik}^{N} = \widehat{N}\_{ik} u(x, y, z, t); \widehat{N}\_{xx} = \widehat{N}\_{yy} = a\_{\perp} \frac{\partial}{\partial z}; \widehat{N}\_{xy} = \widehat{N}\_{yx} = 0;\\ \widehat{N}\_{xz} = \widehat{N}\_{zx} = -\varepsilon\_{a} \frac{\partial}{\partial x}; \widehat{N}\_{yz} = \widehat{N}\_{zy} = -\varepsilon\_{a} \frac{\partial}{\partial y}; \widehat{N}\_{zz} = a\_{\parallel} \frac{\partial}{\partial z} \end{aligned} \tag{30}$$

Combining Eqs. (27)–(30), we obtain the nonlinear part of the electric induction, or the nonlinear polarization DNL <sup>i</sup> <sup>¼</sup> <sup>ε</sup>0ε<sup>N</sup> ikEk [6]. This nonlinear polarization consists of two types of terms: (i) four harmonics, which are phase-matched with fundamental waves (27); (ii) all other terms with the combination frequencies and wave vectors, which give rise to the small scattered Stokes and anti-Stokes harmonics similar to the Brillouin scattering [6, 10, 14]. The combination of the anisotropy and nonlinearity also results in the creation of the small additional components of the waves E !o,e <sup>1</sup> and E !o <sup>2</sup> polarized in the XZ plane and XY plane, respectively [6].

ω1 c � ��<sup>2</sup>

l o <sup>1</sup> <sup>A</sup><sup>o</sup> 1 � � � � <sup>2</sup> <sup>þ</sup> <sup>l</sup> e <sup>1</sup> Ae 1 � � � �

wo,e <sup>1</sup> <sup>¼</sup> <sup>w</sup>o,e

> wo,e <sup>2</sup> <sup>¼</sup> wo,e

> > γo,e <sup>1</sup> � <sup>γ</sup>o,e

> > γo,e <sup>2</sup> � <sup>γ</sup>o,e

Here, the dimensionless variables are given by [6]:

wo,e <sup>1</sup>,<sup>2</sup> <sup>¼</sup> <sup>1</sup> I0

Cj <sup>¼</sup> <sup>ω</sup>1ω<sup>2</sup> c2

<sup>2</sup> h i <sup>þ</sup>

ω2 c � ��<sup>2</sup>

ðz

0

0

2 ðz

2 ðz

0

� � � 2 ; w<sup>o</sup> <sup>1</sup> <sup>þ</sup> <sup>w</sup><sup>e</sup>

0

8 < :

The solution of the system of Eqs. (31)–(34) can be written in the integral form [6, 10]

8 < :

<sup>1</sup> ð Þ0 exp �

<sup>2</sup> ð Þ<sup>0</sup> exp <sup>ð</sup><sup>z</sup>

<sup>1</sup> ð Þ¼� <sup>0</sup> <sup>1</sup>

<sup>2</sup> ð Þ¼� <sup>0</sup> <sup>1</sup>

l o,e <sup>1</sup>,<sup>2</sup> Ao,e 1,2 � � �

β<sup>j</sup> ¼ CjΓjΔω, δ<sup>j</sup> ¼ Cj ð Þ Δω

� �<sup>2</sup>

parametric amplification process is dominant while XPM can be neglected.

components with other polarizations are small in such a way that w<sup>e</sup>

the first approximation, the normalized intensities w<sup>e</sup>

form [6, 10]

� �<sup>2</sup> ; d<sup>1</sup> <sup>¼</sup> <sup>l</sup>

ω1,<sup>2</sup> c � ��<sup>2</sup>

� �<sup>2</sup> ε0I<sup>0</sup> Δkj<sup>⊥</sup>

r Gj � � � � 2 dj Δkj

Comparison of Eq. (36), (37), and (40) shows that for <sup>z</sup> ! <sup>∞</sup> <sup>w</sup>o,e

l o <sup>2</sup> Ao 2 � � � � <sup>2</sup> <sup>þ</sup> <sup>l</sup> e <sup>2</sup> <sup>A</sup><sup>e</sup> 2 � � � �

<sup>β</sup>3,2wo,e

<sup>β</sup>3,2wo,e

δ3, <sup>2</sup>wo,e

δ3, <sup>2</sup>wo,e

<sup>2</sup> � <sup>Ω</sup><sup>2</sup> j h i, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>, <sup>4</sup>,

> e 1l o <sup>2</sup>, d<sup>2</sup> ¼ l e 1l e <sup>2</sup>, d<sup>3</sup> ¼ l o 1l o <sup>2</sup>, d<sup>4</sup> ¼ l o 1l e

the pumping waves with the larger frequency ω<sup>1</sup> are depleted, the signal waves with smaller frequency ω<sup>2</sup> < ω<sup>1</sup> are amplified with the saturation at the sufficiently large distances, and the system is stable. The gain terms β<sup>j</sup> reach their maximal values close to the SS resonance condition Δω ≈ Ωj, which can be satisfied for Δω � ω1s0=c ≪ ω<sup>1</sup> [6]. In such a case, β<sup>j</sup> ≫ δj, the

In general case, the exact analytical solution of Eqs. (31)–(34) is hardly possible. However, the explicit expressions for the coupled wave SVA have been obtained when both waves are propagating in the same XZ plane [5, 6]. For instance, assume that the pumping extraordinary wave with the frequency ω<sup>1</sup> > ω<sup>2</sup> is mainly polarized in the XZ plane, the signal ordinary wave with the frequency ω<sup>2</sup> is mainly polarized along the Y axis, and the intensities of the

<sup>1</sup>, wo

<sup>2</sup> <sup>þ</sup> <sup>β</sup>4,1we, <sup>o</sup> 2

� �dz<sup>0</sup>

<sup>1</sup> <sup>þ</sup> <sup>β</sup>1,4we, <sup>o</sup> 1

<sup>2</sup> <sup>þ</sup> <sup>δ</sup>4,1we, <sup>o</sup> 2

<sup>1</sup> <sup>þ</sup> <sup>δ</sup>1,4we, <sup>o</sup> 1

<sup>1</sup> <sup>þ</sup> <sup>w</sup><sup>o</sup>

<sup>2</sup> <sup>þ</sup> <sup>w</sup><sup>e</sup>

<sup>1</sup> ! 0 and <sup>w</sup><sup>o</sup>

<sup>1</sup> ≫ w<sup>o</sup> 1; w<sup>o</sup> <sup>2</sup> ≫ w<sup>e</sup>

<sup>2</sup> of the main components have the

� �dz<sup>0</sup>

<sup>2</sup> h i <sup>¼</sup> const <sup>¼</sup> <sup>I</sup><sup>0</sup> (35)

Nonlinear Optical Phenomena in Smectic A Liquid Crystals

http://dx.doi.org/10.5772/intechopen.70997

9 = ;

9 = ;

� �dz<sup>0</sup> (38)

� �dz<sup>0</sup> (39)

<sup>2</sup> ¼ 1 (40)

<sup>2</sup> <sup>þ</sup> <sup>w</sup><sup>e</sup>

<sup>2</sup> (41)

<sup>2</sup> ! 1. Hence,

<sup>2</sup>. Then, in

(36)

143

(37)

We start with the analysis of the parametric coupling among the waves (27). Substituting expressions (27)–(30) and the phase-matched part of D !NL into wave Eq. (13), taking into account SVAA for the complex amplitudes Ao,e <sup>1</sup>,2ð Þz , and separating the real and imaginary parts, we obtain the truncated equations for the magnitudes Ao,e <sup>1</sup>,2ð Þz � � � � � � and phases <sup>γ</sup>o,e <sup>1</sup>, <sup>2</sup>ð Þz of these SVA [6]

$$\begin{split} &2l\_{1}^{\alpha,\varepsilon} \frac{\partial |A\_{1}^{\alpha,\varepsilon}|}{\partial z} = -\left(\frac{\omega\_{1}}{c}\right)^{2} \\ &\times \frac{\varepsilon\_{0} \Delta \omega}{\rho} \left\{ \frac{h\_{3,2}^{2} (\Delta k\_{3,2,1})^{2} \Gamma\_{3,2}}{|G\_{3,2}|^{2} (\Delta k\_{3,2})^{2}} |A\_{2}^{\alpha,\varepsilon}|^{2} + \frac{h\_{4,1}^{2} (\Delta k\_{4,11})^{2} \Gamma\_{4,1}}{|G\_{4,1}|^{2} (\Delta k\_{4,1})^{2}} |A\_{2}^{\varepsilon,\nu}|^{2} \right\} |A\_{1}^{\alpha,\varepsilon}| \\ &\times \partial |A\_{2}^{\alpha,\varepsilon}| \\ &\times \partial |A\_{2}^{\alpha,\varepsilon}|^{2} \end{split} \tag{31}$$

$$\begin{split} \mathcal{Q}\_{2}^{\boldsymbol{\rho},\boldsymbol{\epsilon}} \frac{\partial \left| A\_{2}^{\boldsymbol{\alpha},\boldsymbol{\epsilon}} \right|}{\partial \boldsymbol{z}} &= \left( \frac{\omega\_{2}}{\boldsymbol{c}} \right)^{2} \\ \times \frac{\varepsilon\_{0} \Delta \boldsymbol{\omega}}{\rho} \Bigg{{} \left\{ \frac{h\_{3,2}^{2} (\Delta k\_{3,2})^{2} \Gamma\_{3,2}}{\left| G\_{3,2} \right|^{2} (\Delta k\_{3,2})^{2}} \left| A\_{1}^{\boldsymbol{\alpha},\boldsymbol{\epsilon}} \right|^{2} + \frac{h\_{1,4}^{2} (\Delta k\_{1,4})^{2} \Gamma\_{1,4}}{\left| G\_{1,4} \right|^{2} (\Delta k\_{1,4})^{2}} \left| A\_{1}^{\boldsymbol{\alpha},\boldsymbol{\epsilon}} \right|^{2} \right\} \Bigg{{} \left| A\_{2}^{\boldsymbol{\alpha},\boldsymbol{\epsilon}} \right|} \end{split} \tag{32}$$

$$\begin{split} &2h\_{1}^{\alpha,\varepsilon} \frac{\partial \mathcal{V}\_{1}^{\phi,\varepsilon}}{\partial \boldsymbol{z}} = -\left(\frac{\alpha \boldsymbol{v}\_{1}}{c}\right)^{2} \frac{\varepsilon\_{0}}{\rho} \\ & \times \left\{ \frac{h\_{3,2}^{2} \left(\Delta k\_{3,2,1}\right)^{2} \left[\left(\Delta \boldsymbol{\omega}\right)^{2} - \Omega\_{3,2}^{2}\right]}{\left|\left(\boldsymbol{G}\_{3,2}\right)^{2} \left(\Delta k\_{3,2}\right)^{2}\right|} \left|A\_{2}^{\alpha,\varepsilon}\right|^{2} + \frac{h\_{4,1}^{2} \left(\Delta k\_{4,1}\right)^{2} \left[\left(\Delta \boldsymbol{\omega}\right)^{2} - \Omega\_{4,1}^{2}\right]}{\left|\left(\boldsymbol{G}\_{4,1}\right)^{2} \left(\Delta k\_{4,1}\right)^{2}} \left|A\_{2}^{\varepsilon,\nu}\right|^{2} \right\} \end{split} \tag{33}$$

$$\begin{split} &2h\_{2}^{\theta,\varepsilon}\frac{\partial\mathcal{Y}\_{2}^{\theta,\varepsilon}}{\partial\varepsilon} = -\left(\frac{\alpha\nu\_{2}}{c}\right)^{2}\frac{\varepsilon\_{0}}{\rho} \\ & \times \left\{ \frac{h\_{3,2}^{2}\left(\Delta k\_{3,2,1}\right)^{2}\left[\left(\Delta\omega\right)^{2} - \Omega\_{3,2}^{2}\right]}{\left|G\_{3,2}\right|^{2}\left(\Delta k\_{3,2}\right)^{2}} \left|A\_{1}^{\theta,\varepsilon}\right|^{2} + \frac{h\_{1,4}^{2}\left(\Delta k\_{1,4}\right)^{2}\left[\left(\Delta\omega\right)^{2} - \Omega\_{1,4}^{2}\right]}{\left|G\_{1,4}\right|^{2}\left(\Delta k\_{1,4}\right)^{2}} \left|A\_{1}^{\varepsilon,\vartheta}\right|^{2} \right\} \end{split} \tag{34}$$

Here, l o <sup>1</sup>, <sup>2</sup> ¼ k o <sup>1</sup>, <sup>2</sup>z; l<sup>e</sup> <sup>1</sup>, <sup>2</sup> ¼ k e <sup>1</sup>, <sup>2</sup><sup>z</sup> 1 � e e <sup>1</sup>, <sup>2</sup><sup>z</sup> k !e <sup>1</sup>,<sup>2</sup> � e !e 1,2 � � <sup>k</sup><sup>e</sup> 1,2z � ��<sup>1</sup> � �.

Eqs. (31) and (32) describe the parametric energy exchange between the fundamental waves, Eqs. (33) and (34) describe the cross-phase modulation (XPM) of these waves [6]. Combining Eqs. (31) and (32), we obtain the Manley-Rowe relation, which expresses the conservation of the total photon number [14]. In our case, it has the form [6]

Nonlinear Optical Phenomena in Smectic A Liquid Crystals http://dx.doi.org/10.5772/intechopen.70997 143

$$\left(\frac{\omega\_1}{\mathcal{L}}\right)^{-2} \left[l\_1^o \left|A\_1^o\right|^2 + l\_1^e \left|A\_1^e\right|^2\right] + \left(\frac{\omega\_2}{\mathcal{L}}\right)^{-2} \left[l\_2^o \left|A\_2^o\right|^2 + l\_2^e \left|A\_2^e\right|^2\right] = const = I\_0 \tag{35}$$

The solution of the system of Eqs. (31)–(34) can be written in the integral form [6, 10]

$$w\_1^{o, \epsilon} = w\_1^{o, \epsilon}(0) \exp\left\{-\int\_0^z \left(\beta\_{3, 2} w\_2^{o, \epsilon} + \beta\_{4, 1} w\_2^{\epsilon, o}\right) dz'\right\} \tag{36}$$

$$w\_2^{\rho,\epsilon} = w\_2^{\rho,\epsilon}(0) \exp\left\{ \int\_0^z \left( \beta\_{3,2} w\_1^{\rho,\epsilon} + \beta\_{1,4} w\_1^{\epsilon,\imath} \right) dz' \right\} \tag{37}$$

$$
\gamma\_1^{o,\epsilon} - \gamma\_1^{o,\epsilon}(0) = -\frac{1}{2} \prod\_{0}^{z} (\delta\_{3,2} w\_2^{o,\epsilon} + \delta\_{4,1} w\_2^{\epsilon,o}) dz' \tag{38}
$$

$$
\gamma\_2^{\rho,\epsilon} - \gamma\_2^{\rho,\epsilon}(0) = -\frac{1}{2} \Bigg| \begin{pmatrix} \delta\_{3,2} w\_1^{\rho,\epsilon} + \delta\_{1,4} w\_1^{\epsilon,\rho} \end{pmatrix} dz' \tag{39}
$$

Here, the dimensionless variables are given by [6]:

Combining Eqs. (27)–(30), we obtain the nonlinear part of the electric induction, or the

terms: (i) four harmonics, which are phase-matched with fundamental waves (27); (ii) all other terms with the combination frequencies and wave vectors, which give rise to the small scattered Stokes and anti-Stokes harmonics similar to the Brillouin scattering [6, 10, 14]. The combination of the anisotropy and nonlinearity also results in the creation of the small addi-

We start with the analysis of the parametric coupling among the waves (27). Substituting

ikEk [6]. This nonlinear polarization consists of two types of

!NL

<sup>4</sup>, <sup>1</sup>ð Þ Δk4, <sup>1</sup><sup>⊥</sup>

<sup>1</sup>, <sup>4</sup>ð Þ Δk1, <sup>4</sup><sup>⊥</sup>

<sup>4</sup>, <sup>1</sup>ð Þ Δk4, <sup>1</sup><sup>⊥</sup>

<sup>1</sup>, <sup>4</sup>ð Þ Δk1, <sup>4</sup><sup>⊥</sup>

ke 1,2z j j G4, <sup>1</sup> 2 ð Þ Δk4, <sup>1</sup>

j j G1, <sup>4</sup> 2 ð Þ Δk1, <sup>4</sup>

.

j j G4, <sup>1</sup> 2 ð Þ Δk4,<sup>1</sup>

j j G1, <sup>4</sup> 2 ð Þ Δk1,<sup>4</sup>

( )

( )

<sup>2</sup> polarized in the XZ plane and XY plane, respec-

<sup>1</sup>,2ð Þz , and separating the real and imaginary

� �

<sup>1</sup>,2ð Þz

<sup>2</sup> <sup>A</sup>e, <sup>o</sup> 2 � � � � 2

<sup>2</sup> <sup>A</sup>e, <sup>o</sup> 1 � � � � 2

<sup>2</sup> ð Þ <sup>Δ</sup><sup>ω</sup>

<sup>2</sup> ð Þ <sup>Δ</sup><sup>ω</sup>

<sup>2</sup> � <sup>Ω</sup><sup>2</sup> 4,1

<sup>2</sup> � <sup>Ω</sup><sup>2</sup> 1,4

h i

<sup>2</sup> <sup>A</sup>e, <sup>o</sup> 2 � � � � 2

<sup>2</sup> <sup>A</sup>e, <sup>o</sup> 1 � � � � 2

9 = ;

9 = ;

h i

� � �

2 Γ4, <sup>1</sup>

2 Γ1, <sup>4</sup>

into wave Eq. (13), taking into

Ao,e 1 � � � �

Ao,e 2 � � � �

� and phases <sup>γ</sup>o,e

<sup>1</sup>, <sup>2</sup>ð Þz of

(31)

(32)

(33)

(34)

nonlinear polarization DNL

tively [6].

these SVA [6]

tional components of the waves E

2l o,e 1

� ε0Δω r

2l o,e 2

� ε0Δω r

∂γo,e 1 <sup>∂</sup><sup>z</sup> ¼ � <sup>ω</sup><sup>1</sup>

∂γo,e 2 <sup>∂</sup><sup>z</sup> ¼ � <sup>ω</sup><sup>2</sup>

h2

8 < : <sup>3</sup>, <sup>2</sup>ð Þ Δk3, <sup>2</sup><sup>⊥</sup>

<sup>3</sup>, <sup>2</sup>ð Þ Δk3, <sup>2</sup><sup>⊥</sup>

j j G3, <sup>2</sup> 2 ð Þ Δk3,<sup>2</sup>

j j G3, <sup>2</sup> 2 ð Þ Δk3,<sup>2</sup>

<sup>1</sup>, <sup>2</sup> ¼ k e <sup>1</sup>, <sup>2</sup><sup>z</sup> 1 � e

h2

8 < :

2l o,e 1

�

2l o,e 2

�

Here, l o <sup>1</sup>, <sup>2</sup> ¼ k o <sup>1</sup>, <sup>2</sup>z; l<sup>e</sup> <sup>i</sup> <sup>¼</sup> <sup>ε</sup>0ε<sup>N</sup>

142 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

expressions (27)–(30) and the phase-matched part of D

account SVAA for the complex amplitudes Ao,e

∂ Ao,e 1 � � � � <sup>∂</sup><sup>z</sup> ¼ � <sup>ω</sup><sup>1</sup>

∂ Ao,e 2 � � � � <sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup> c � �<sup>2</sup>

h2

h2

c � �<sup>2</sup> ε<sup>0</sup> r

c � �<sup>2</sup> ε<sup>0</sup> r

<sup>2</sup> ð Þ <sup>Δ</sup><sup>ω</sup>

!o,e <sup>1</sup> and E !o

parts, we obtain the truncated equations for the magnitudes Ao,e

c � �<sup>2</sup>

> 2 Γ3, <sup>2</sup>

> 2 Γ3, <sup>2</sup>

<sup>2</sup> � <sup>Ω</sup><sup>2</sup> 3, 2

<sup>2</sup> � <sup>Ω</sup><sup>2</sup> 3, 2

> e <sup>1</sup>, <sup>2</sup><sup>z</sup> k !e <sup>1</sup>,<sup>2</sup> � e !e 1,2 � �

the total photon number [14]. In our case, it has the form [6]

h i

<sup>2</sup> Ao,e 2 � � � � 2 þ h2

<sup>2</sup> Ao,e 1 � � � � 2 þ h2

� ��<sup>1</sup> � �

Eqs. (31) and (32) describe the parametric energy exchange between the fundamental waves, Eqs. (33) and (34) describe the cross-phase modulation (XPM) of these waves [6]. Combining Eqs. (31) and (32), we obtain the Manley-Rowe relation, which expresses the conservation of

h i

<sup>2</sup> Ao,e 2 � � � � 2 þ h2

<sup>2</sup> Ao,e 1 � � � � 2 þ h2

<sup>3</sup>,2ð Þ Δk3, <sup>2</sup><sup>⊥</sup>

<sup>3</sup>,2ð Þ Δk3, <sup>2</sup><sup>⊥</sup>

j j G3, <sup>2</sup> 2 ð Þ Δk3, <sup>2</sup>

j j G3, <sup>2</sup> 2 ð Þ Δk3, <sup>2</sup>

<sup>2</sup> ð Þ <sup>Δ</sup><sup>ω</sup>

$$w\_{1,2}^{o,\epsilon} = \frac{1}{I\_0} \left(\frac{\omega\_{1,2}}{c}\right)^{-2} l\_{1,2}^{o,\epsilon} \left| A\_{1,2}^{o,\epsilon} \right|^2; w\_1^o + w\_1^{\epsilon} + w\_2^o + w\_2^{\epsilon} = 1 \tag{40}$$

$$\beta\_j = \mathbf{C}\_j \Gamma\_j \Delta \boldsymbol{\omega}\_j \, \delta\_j = \mathbf{C}\_j \left[ (\Delta \boldsymbol{\omega})^2 - \Omega\_j^2 \right]; j = 1, 2, 3, 4, \tag{41}$$

$$\mathbf{C}\_j = \left(\frac{\omega\_{1} \omega\_2}{c^2}\right)^2 \frac{\epsilon\_0 \boldsymbol{l}\_0 (\Delta k\_{j\perp})^2}{\rho |\mathbf{C}\_j|^2 d\_j (\Delta k\_j)^2}; \boldsymbol{d}\_1 = \mathbf{f}\_1^{\epsilon} \mathbf{f}\_2^{o}, \boldsymbol{d}\_2 = \mathbf{f}\_1^{\epsilon} \mathbf{f}\_{2^\prime}^{\epsilon}, \boldsymbol{d}\_3 = \mathbf{f}\_1^{\rho} \mathbf{f}\_2^{o}, \boldsymbol{d}\_4 = \mathbf{f}\_1^{\rho} \mathbf{f}\_2^{\epsilon} \tag{41}$$

Comparison of Eq. (36), (37), and (40) shows that for <sup>z</sup> ! <sup>∞</sup> <sup>w</sup>o,e <sup>1</sup> ! 0 and <sup>w</sup><sup>o</sup> <sup>2</sup> <sup>þ</sup> <sup>w</sup><sup>e</sup> <sup>2</sup> ! 1. Hence, the pumping waves with the larger frequency ω<sup>1</sup> are depleted, the signal waves with smaller frequency ω<sup>2</sup> < ω<sup>1</sup> are amplified with the saturation at the sufficiently large distances, and the system is stable. The gain terms β<sup>j</sup> reach their maximal values close to the SS resonance condition Δω ≈ Ωj, which can be satisfied for Δω � ω1s0=c ≪ ω<sup>1</sup> [6]. In such a case, β<sup>j</sup> ≫ δj, the parametric amplification process is dominant while XPM can be neglected.

In general case, the exact analytical solution of Eqs. (31)–(34) is hardly possible. However, the explicit expressions for the coupled wave SVA have been obtained when both waves are propagating in the same XZ plane [5, 6]. For instance, assume that the pumping extraordinary wave with the frequency ω<sup>1</sup> > ω<sup>2</sup> is mainly polarized in the XZ plane, the signal ordinary wave with the frequency ω<sup>2</sup> is mainly polarized along the Y axis, and the intensities of the components with other polarizations are small in such a way that w<sup>e</sup> <sup>1</sup> ≫ w<sup>o</sup> 1; w<sup>o</sup> <sup>2</sup> ≫ w<sup>e</sup> <sup>2</sup>. Then, in the first approximation, the normalized intensities w<sup>e</sup> <sup>1</sup>, wo <sup>2</sup> of the main components have the form [6, 10]

$$w\_1^\varepsilon = \frac{1}{2} J\_1 \left[ 1 - \tanh\left(\eta - \eta\_0\right) \right]; w\_2^\rho = \frac{1}{2} J\_1 \left[ 1 + \tanh\left(\eta - \eta\_0\right) \right] \tag{42}$$

The Brillouin-like SLS also results in the generation of six Stokes small harmonics with the frequency ð Þ ω<sup>2</sup> � Δω and combination wave vectors, six anti-Stokes small harmonics with the frequency ð Þ ω<sup>1</sup> þ Δω and combination wave vectors, and eight small harmonics with the funda-

Consider now the nondegenerate FWM in SALC [8–10]. Assume that four coupled fundamental optical waves have different close frequencies ω<sup>n</sup> such that Δωmn ¼ ω<sup>m</sup> � ω<sup>n</sup> � s0ωn=c ≪ ωn. For the sake of definiteness, we suppose that ω<sup>1</sup> < ω<sup>2</sup> < ω<sup>3</sup> < ω4. These fundamental waves

> <sup>m</sup>� r ! � � � <sup>ω</sup><sup>t</sup> h i <sup>þ</sup> <sup>c</sup>:c:

The interfering waves (44) create a dynamic grating of the smectic layer normal displacement of the type (28), but this time each harmonic has a different frequency Δωmn ¼ ω<sup>m</sup> � ωn, m, n ¼ 1, …, 4. We discuss two cases: (i) all waves (44) are polarized in the directions perpendicular to the propagation plane and propagate as ordinary waves; (ii) all waves (44) are polarized in the propagation plane and behave as extraordinary waves [8–10]. Using the SVAA and the theory developed in the previous section, we obtain the truncated equations for the slowly varying magnitudes j j Amð Þz and phases γmð Þz similar to Eqs. (31)–(34). The analysis of these equations shows that the wave with the lowest frequency ω<sup>1</sup> is amplified up to the saturation level determined by the integral of motion I<sup>0</sup> similar to the one from Eq. (35) [8–10]

Here, the factors lm are defined above for the ordinary or extraordinary wave, respectively. Three other waves with the higher frequencies ω2, <sup>3</sup>, <sup>4</sup> undergo the depletion like the pumping

analytical solution of the type (42) and (43) has been obtained for the case when the pumping

In the special case when some ordinary optical waves (44) have perpendicular polarizations

vanish [8–10]. In the case of the extraordinary wave mixing, the polarization-decoupled FWM is impossible because of the SALC anisotropy. If the electric field of the signal ordinary wave

<sup>1</sup> is perpendicular to the fields of all other waves than this wave propagates though SALC

!

n o, m <sup>¼</sup> <sup>1</sup>,…, <sup>4</sup> (44)

j j Am <sup>2</sup> <sup>¼</sup> const (45)

Nonlinear Optical Phenomena in Smectic A Liquid Crystals

http://dx.doi.org/10.5772/intechopen.70997

145

<sup>2</sup>, <sup>3</sup>, <sup>4</sup> is accompanied by the XPM with the rapidly

<sup>1</sup> saturates at large distances. The

mn ¼ a⊥Δkmnz e

!

! <sup>m</sup><sup>⊥</sup> � e ! n⊥ � �

<sup>2</sup>,<sup>3</sup> with the

!

<sup>1</sup> are much stronger than the idler waves E

<sup>n</sup><sup>⊥</sup> the polarization-decoupled FWM is possible [8–10]. Such waves do not excite

mental frequencies ω1, <sup>2</sup> and combination wave vectors [6].

<sup>m</sup> Amð Þ<sup>z</sup> expi k!

<sup>I</sup><sup>0</sup> <sup>¼</sup> <sup>X</sup> 4

m¼1 lm ω<sup>m</sup> c � ��<sup>2</sup>

5. The nondegenerate FWM in SALC

E ! <sup>m</sup> ¼ e !

waves [8–10]. The depletion of the waves E

<sup>4</sup> and the signal wave E

intermediate frequencies ω2,<sup>3</sup> [8–10].

wave E !

vectors e ! <sup>m</sup>⊥⊥e !

E !

increasing phases while the phase of the signal wave E

!

the dynamic grating since the corresponding coupling constants ho

have the form [8–10]

Here, w<sup>e</sup> <sup>1</sup> <sup>þ</sup> <sup>w</sup><sup>o</sup> <sup>2</sup> <sup>¼</sup> <sup>J</sup><sup>1</sup> <sup>¼</sup> const <sup>¼</sup> we <sup>1</sup>ð Þþ <sup>0</sup> wo <sup>2</sup>ð Þ0 , η ¼ β1J1z=2. It is seen from Eq. (42) that for η ! ∞ we <sup>1</sup> ! <sup>0</sup>; wo <sup>2</sup> ! J1, and the crossing point z<sup>0</sup> ¼ β1J<sup>1</sup> �<sup>1</sup> ln w<sup>e</sup> <sup>1</sup>ð Þ<sup>0</sup> <sup>=</sup>w<sup>o</sup> <sup>2</sup>ð Þ<sup>0</sup> exists only for we <sup>1</sup>ð Þ<sup>0</sup> <sup>=</sup>w<sup>o</sup> <sup>2</sup>ð Þ<sup>0</sup> <sup>&</sup>gt; 1. The coordinate dependence of the normalized intensities we <sup>1</sup>, wo <sup>2</sup> is presented in Figure 5. The numerical estimations show that for the typical values of SALC parameters [1–3] in the resonant case the coupling constant per unit optical intensity β1=Popt � ð � 0:01 10Þcm=MW [6]. For the optical intensity Popt � 106 � 107 Wcm�<sup>2</sup> , the SLS gain <sup>β</sup>1max � 102 cm�<sup>1</sup> , which is at least an order of magnitude larger than the gain at Brillouin SLS in isotropic organic liquids [14]. Such optical intensities are feasible [20, 21].

The explicit expressions of the small component intensities w<sup>o</sup> <sup>1</sup> and we <sup>2</sup> can be obtained in the second approximation. They have the form [6]

$$\begin{aligned} w\_1^{\rho} &= w\_1^{\rho}(0) \left\{ \frac{\exp(-\eta) \cosh \left( \eta\_0 \right)}{\cosh \left( \eta - \eta\_0 \right)} \right\}^{\beta\_3/\beta\_1} \\\ w\_2^{\epsilon} &= w\_2^{\epsilon}(0) \left\{ \frac{\exp(\eta) \cosh \left( \eta\_0 \right)}{\cosh \left( \eta - \eta\_0 \right)} \right\}^{\beta\_2/\beta\_1} \end{aligned} \tag{43}$$

It is easy to see from Eq. (43) that for <sup>η</sup> ! <sup>∞</sup> <sup>w</sup><sup>o</sup> <sup>1</sup> ! 0 and <sup>w</sup><sup>e</sup> <sup>2</sup> ! we <sup>2</sup>ð Þ<sup>0</sup> <sup>1</sup> <sup>þ</sup> <sup>w</sup><sup>e</sup> <sup>1</sup>ð Þ<sup>0</sup> <sup>=</sup>w<sup>o</sup> <sup>2</sup>ð Þ<sup>0</sup> <sup>β</sup>2=β<sup>1</sup> ¼ const.

The evaluation of the phases γo,e <sup>1</sup>,<sup>2</sup> shows that the pumping wave phases <sup>γ</sup>o, <sup>e</sup> <sup>1</sup> rapidly increase that results in the fast oscillations of the depleted amplitudes Ao,e <sup>1</sup> ð Þ<sup>z</sup> [6]. The phases <sup>γ</sup>o, <sup>e</sup> <sup>2</sup> of the signal waves tend to the constant values at sufficiently large η [6].

Figure 5. The dependence of the normalized pumping and signal intensities we 1=J1; w<sup>o</sup> <sup>2</sup>=J<sup>1</sup> on the dimensionless coordinate η for the pumping-to-signal ratio we <sup>1</sup>ð Þ<sup>0</sup> <sup>=</sup>wo <sup>2</sup>ð Þ¼ 0 1:5; 5 (curves 1 and 2, respectively).

The Brillouin-like SLS also results in the generation of six Stokes small harmonics with the frequency ð Þ ω<sup>2</sup> � Δω and combination wave vectors, six anti-Stokes small harmonics with the frequency ð Þ ω<sup>1</sup> þ Δω and combination wave vectors, and eight small harmonics with the fundamental frequencies ω1, <sup>2</sup> and combination wave vectors [6].

### 5. The nondegenerate FWM in SALC

we <sup>1</sup> <sup>¼</sup> <sup>1</sup> 2

<sup>2</sup> <sup>¼</sup> <sup>J</sup><sup>1</sup> <sup>¼</sup> const <sup>¼</sup> we

[6]. For the optical intensity Popt � 106 � 107

Such optical intensities are feasible [20, 21].

second approximation. They have the form [6]

It is easy to see from Eq. (43) that for <sup>η</sup> ! <sup>∞</sup> <sup>w</sup><sup>o</sup>

The evaluation of the phases γo,e

η for the pumping-to-signal ratio we

Here, w<sup>e</sup>

¼ const.

<sup>1</sup> ! <sup>0</sup>; wo

we

we <sup>1</sup>ð Þ<sup>0</sup> <sup>=</sup>w<sup>o</sup>

<sup>1</sup> <sup>þ</sup> <sup>w</sup><sup>o</sup>

J<sup>1</sup> 1 � tanh η � η<sup>0</sup> ; w<sup>o</sup>

<sup>1</sup>ð Þþ <sup>0</sup> wo

<sup>2</sup>ð Þ<sup>0</sup> <sup>&</sup>gt; 1. The coordinate dependence of the normalized intensities we

in Figure 5. The numerical estimations show that for the typical values of SALC parameters [1–3] in the resonant case the coupling constant per unit optical intensity β1=Popt � ð � 0:01 10Þcm=MW

Wcm�<sup>2</sup>

least an order of magnitude larger than the gain at Brillouin SLS in isotropic organic liquids [14].

<sup>1</sup>ð Þ<sup>0</sup> expð Þ �<sup>η</sup> cosh <sup>η</sup>ð Þ<sup>0</sup> cosh <sup>η</sup>�<sup>η</sup> ð Þ<sup>0</sup> <sup>β</sup>3=β<sup>1</sup>

<sup>2</sup>ð Þ<sup>0</sup> expð Þ <sup>η</sup> cosh <sup>η</sup>ð Þ<sup>0</sup> cosh <sup>η</sup>�<sup>η</sup> ð Þ<sup>0</sup> <sup>β</sup>2=β<sup>1</sup>

<sup>1</sup> ! 0 and <sup>w</sup><sup>e</sup>

<sup>1</sup>,<sup>2</sup> shows that the pumping wave phases <sup>γ</sup>o, <sup>e</sup>

<sup>2</sup> ! J1, and the crossing point z<sup>0</sup> ¼ β1J<sup>1</sup>

144 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

The explicit expressions of the small component intensities w<sup>o</sup>

wo <sup>1</sup> <sup>¼</sup> wo

we <sup>2</sup> <sup>¼</sup> we

that results in the fast oscillations of the depleted amplitudes Ao,e

signal waves tend to the constant values at sufficiently large η [6].

Figure 5. The dependence of the normalized pumping and signal intensities we

<sup>1</sup>ð Þ<sup>0</sup> <sup>=</sup>wo

�<sup>1</sup>

J<sup>1</sup> 1 þ tanh η � η<sup>0</sup>

ln w<sup>e</sup>

, the SLS gain <sup>β</sup>1max � 102

<sup>1</sup> and we

<sup>2</sup> ! we

1=J1; w<sup>o</sup>

<sup>2</sup>ð Þ¼ 0 1:5; 5 (curves 1 and 2, respectively).

<sup>2</sup>ð Þ<sup>0</sup> <sup>1</sup> <sup>þ</sup> <sup>w</sup><sup>e</sup>

<sup>1</sup> ð Þ<sup>z</sup> [6]. The phases <sup>γ</sup>o, <sup>e</sup>

<sup>2</sup>=J<sup>1</sup> on the dimensionless coordinate

<sup>2</sup>ð Þ0 , η ¼ β1J1z=2. It is seen from Eq. (42) that for η ! ∞

<sup>1</sup>ð Þ<sup>0</sup> <sup>=</sup>w<sup>o</sup>

(42)

<sup>2</sup>ð Þ<sup>0</sup> exists only for

<sup>1</sup>, wo

cm�<sup>1</sup>

<sup>2</sup> can be obtained in the

<sup>1</sup>ð Þ<sup>0</sup> <sup>=</sup>w<sup>o</sup> <sup>2</sup>ð Þ<sup>0</sup> <sup>β</sup>2=β<sup>1</sup>

<sup>1</sup> rapidly increase

<sup>2</sup> of the

<sup>2</sup> is presented

, which is at

(43)

Consider now the nondegenerate FWM in SALC [8–10]. Assume that four coupled fundamental optical waves have different close frequencies ω<sup>n</sup> such that Δωmn ¼ ω<sup>m</sup> � ω<sup>n</sup> � s0ωn=c ≪ ωn. For the sake of definiteness, we suppose that ω<sup>1</sup> < ω<sup>2</sup> < ω<sup>3</sup> < ω4. These fundamental waves have the form [8–10]

$$\overrightarrow{E}\_m = \overrightarrow{e}\_m \left\{ A\_m(z) \exp[\left(\overrightarrow{k}\_m \cdot \overrightarrow{r}\right) - \omega t] \right. \\ \left. + c.c. \right\}, m = 1, \ldots, 4 \tag{44}$$

The interfering waves (44) create a dynamic grating of the smectic layer normal displacement of the type (28), but this time each harmonic has a different frequency Δωmn ¼ ω<sup>m</sup> � ωn, m, n ¼ 1, …, 4. We discuss two cases: (i) all waves (44) are polarized in the directions perpendicular to the propagation plane and propagate as ordinary waves; (ii) all waves (44) are polarized in the propagation plane and behave as extraordinary waves [8–10]. Using the SVAA and the theory developed in the previous section, we obtain the truncated equations for the slowly varying magnitudes j j Amð Þz and phases γmð Þz similar to Eqs. (31)–(34). The analysis of these equations shows that the wave with the lowest frequency ω<sup>1</sup> is amplified up to the saturation level determined by the integral of motion I<sup>0</sup> similar to the one from Eq. (35) [8–10]

$$I\_0 = \sum\_{m=1}^{4} I\_m \left(\frac{\alpha\_m}{c}\right)^{-2} |A\_m|^2 = const \tag{45}$$

Here, the factors lm are defined above for the ordinary or extraordinary wave, respectively. Three other waves with the higher frequencies ω2, <sup>3</sup>, <sup>4</sup> undergo the depletion like the pumping waves [8–10]. The depletion of the waves E ! <sup>2</sup>, <sup>3</sup>, <sup>4</sup> is accompanied by the XPM with the rapidly increasing phases while the phase of the signal wave E ! <sup>1</sup> saturates at large distances. The analytical solution of the type (42) and (43) has been obtained for the case when the pumping wave E ! <sup>4</sup> and the signal wave E ! <sup>1</sup> are much stronger than the idler waves E ! <sup>2</sup>,<sup>3</sup> with the intermediate frequencies ω2,<sup>3</sup> [8–10].

In the special case when some ordinary optical waves (44) have perpendicular polarizations vectors e ! <sup>m</sup>⊥⊥e ! <sup>n</sup><sup>⊥</sup> the polarization-decoupled FWM is possible [8–10]. Such waves do not excite the dynamic grating since the corresponding coupling constants ho mn ¼ a⊥Δkmnz e ! <sup>m</sup><sup>⊥</sup> � e ! n⊥ � � vanish [8–10]. In the case of the extraordinary wave mixing, the polarization-decoupled FWM is impossible because of the SALC anisotropy. If the electric field of the signal ordinary wave E ! <sup>1</sup> is perpendicular to the fields of all other waves than this wave propagates though SALC without any change of its SVA: Ao <sup>1</sup> ¼ const. If E !o <sup>1</sup>⊥E !o <sup>2</sup>,<sup>3</sup> and E !o <sup>1</sup>∥E !o 4, then FWM is divided in two separate two-wave mixing processes between the waves E !o <sup>1</sup>, <sup>4</sup> and the waves E !o <sup>2</sup>,<sup>3</sup> with the solutions similar to solution (42) [8].

SALC parameters and for the pumping wave intensity of about 100 MWcm�<sup>2</sup> [9], which is feasible [20, 21]. OPC in the homeotropically oriented SALC film with the thickness of 250 μm

fundamental waves (46) give rise to 12 doubly degenerate combination harmonics of the type.

� �t h i and 12 harmonics of the type <sup>A</sup><sup>2</sup>

6. Nonlinear interaction of surface plasmon polaritons (SPP) in SALC

are chosen to be perpendicular and parallel to the interface z ¼ 0, respectively.

Integration of strongly nonlinear LC with plasmonic structures and metamaterials would enable active switching and tuning operations with low threshold [4]. LC may be also used in reconfigurable metamaterials for tuning the resonant frequency, the transmission/ reflection coefficient, and the refractive index [23]. Combination of metamaterials and active plasmonic structures with NLC has been investigated [4, 23]. In this section, we discuss the nonlinear optical effects caused by the SPP mixing in SALC, which is characterized by low losses and a strong nonlinearity related to the smectic layer normal displacement without the change of the mass density [11–13]. Consider the interface z ¼ 0 between a homeotropically oriented SALC ð Þ z > 0 and a metal ð Þ z < 0 shown in Figure 6 [11, 12]. The SALC optical Z axis and the X axis

SPP from the metal penetrate into SALC. The permittivity of the metal εmð Þ ω determined by

plasma frequency, n<sup>0</sup> is the free electron density in the metal, e, m are the electron charge and mass, respectively, ω, τ are the SPP angular frequency and lifetime, respectively [24, 25]. The efficient SLS in SALC takes place for the counter-propagating SPP with close frequencies

Figure 6. The counter-propagating SPP at the interface between a metal ð Þ z < 0 and a homeotropically oriented SALC

<sup>p</sup>ω�<sup>1</sup>ð Þ <sup>ω</sup> <sup>þ</sup> ð Þ <sup>i</sup>=<sup>τ</sup> �<sup>1</sup> where <sup>ω</sup><sup>p</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

!NL

, which are not phase matched to the

http://dx.doi.org/10.5772/intechopen.70997

Nonlinear Optical Phenomena in Smectic A Liquid Crystals

mA<sup>∗</sup> <sup>n</sup>exp 147

<sup>n</sup>0e<sup>2</sup>=ð Þ <sup>ε</sup>0<sup>m</sup> <sup>p</sup> is the

had been demonstrated experimentally [20].

<sup>m</sup> þ k ! <sup>p</sup> � k ! n

� �� <sup>r</sup>

h i [8–10].

the Drude model is given by <sup>ε</sup>mð Þ¼ <sup>ω</sup> <sup>1</sup> � <sup>ω</sup><sup>2</sup>

! <sup>n</sup>� r ! � � <sup>þ</sup> <sup>ω</sup>nt

AmApA<sup>∗</sup>

i 2 k ! <sup>m</sup>� r ! � � � <sup>ω</sup>mt � � � <sup>k</sup>

ð Þ z > 0 .

nexpi k!

The components of the nonlinear electric induction D

! � � � <sup>ω</sup><sup>m</sup> <sup>þ</sup> <sup>ω</sup><sup>p</sup> � <sup>ω</sup><sup>n</sup>

In the important case when the pumping wave E ! <sup>4</sup> and the signal wave E ! <sup>1</sup> are much stronger than the idler waves E ! <sup>2</sup>,3, the approximate solution can be obtained similarly to the solution (42) and (43) in the case of SLS [8–10]. It has been shown that this solution is stable in the case of FWM [8].

In the particular case when the fundamental waves (44) are counter-propagating, the phase conjugation is possible as a result of the nondegenerate FWM in SALC [8–10]. Optical phase conjugation (OPC) is the wavefront reversion property of a backward propagating optical wave with respect to a forward propagating wave [22]. The optical waves are phase conjugated to each other if their complex amplitudes are conjugated with respect to their phase factors [22]. Typically, OPC results from nonlinear optical processes such as FWM and SLS [20]. LC are commonly used for FWM and OPC [22].

Suppose that the waves E ! <sup>1</sup>,<sup>4</sup> are phase-conjugate while the waves E ! <sup>2</sup>,<sup>3</sup> are forward-going and backward-going pumping waves, which have the form [8]

$$\begin{aligned} \overrightarrow{E}\_1 &= \overrightarrow{e}\_1 \left\{ A\_1 \text{exp} \overrightarrow{i} \left[ \left( \overrightarrow{k} \cdot \overrightarrow{r} \right) + \omega \mathbf{1} \mathbf{t} \right] + c.c. \right\} \\ \overrightarrow{E}\_2 &= \overrightarrow{e}\_2 \left\{ A\_2 \text{exp} \overrightarrow{i} \left[ \left( \overrightarrow{k} \cdot \overrightarrow{r} \right) - \omega \mathbf{1} \right] + c.c. \right\} \\ \overrightarrow{E}\_3 &= \overrightarrow{e}\_3 \left\{ A\_3 \text{exp} \overrightarrow{i} \left[ \left( \overrightarrow{k} \cdot \overrightarrow{r} \right) + \left( \Delta \, \overrightarrow{k} \cdot \overrightarrow{r} \right) + \omega \mathbf{3} \mathbf{t} \right] + c.c. \right\} \\ \overrightarrow{E}\_4 &= \overrightarrow{e}\_4 \left\{ A\_4 \text{exp} \overrightarrow{i} \left[ \left( \overrightarrow{k} \cdot \overrightarrow{r} \right) - \omega \mathbf{1} \right] + c.c. \right\} \end{aligned} \tag{46}$$

Here, Δ k ! ¼ Δk ! <sup>32</sup> is the wave vector mismatch of the FWM process. In the case of OPC caused by SLS the frequency balance condition between the waves with the same vectors is necessary. We assume that ω<sup>3</sup> � ω<sup>1</sup> ¼ ω<sup>4</sup> � ω2. Suppose that the pumping waves E ! <sup>2</sup>,<sup>3</sup> are much stronger than the probe wave E ! <sup>4</sup> and the phase-conjugate wave E ! <sup>1</sup> propagating in the negative direction as it is seen from Eq. (46). In such a case, using the constant pumping approximation (CPA) [14] where A2,<sup>3</sup> ¼ const, we obtain the following solution for the probe wave and the phase-conjugate wave SVA A1, <sup>4</sup> [8–10]

$$A\_{1,4} = A\_{01,4} \exp\left[gr \pm \frac{\dot{\mathbf{r}}}{2} \left(\Delta \stackrel{\rightarrow}{k} \cdot \stackrel{\rightarrow}{r}\right)\right] \tag{47}$$

Analysis of the truncated equations for A1,<sup>4</sup> shows that there exists the solution with the gain Reg < 0 corresponding to the amplification of the phase-conjugate wave E ! <sup>1</sup> [8–10]. Such a case can be characterized as a kind of the Brillouin-enhanced FWM (BEFWM) based on the optical nonlinearity related to the smectic layer normal displacement [8–10]. Numerical estimations show that the amplification of the phase-conjugate wave E ! <sup>1</sup> is possible for the typical values of SALC parameters and for the pumping wave intensity of about 100 MWcm�<sup>2</sup> [9], which is feasible [20, 21]. OPC in the homeotropically oriented SALC film with the thickness of 250 μm had been demonstrated experimentally [20].

without any change of its SVA: Ao

solutions similar to solution (42) [8].

than the idler waves E

Suppose that the waves E

of FWM [8].

Here, Δ k ! ¼ Δk !

than the probe wave E

<sup>1</sup> ¼ const. If E

separate two-wave mixing processes between the waves E

146 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

In the important case when the pumping wave E

LC are commonly used for FWM and OPC [22].

E ! <sup>1</sup> ¼ e !

E ! <sup>2</sup> ¼ e !

E ! <sup>3</sup> ¼ e !

E ! <sup>4</sup> ¼ e !

!

phase-conjugate wave SVA A1, <sup>4</sup> [8–10]

!

backward-going pumping waves, which have the form [8]

<sup>1</sup> <sup>A</sup>1expi k!

<sup>2</sup> <sup>A</sup>2expi k!

<sup>3</sup> <sup>A</sup>3expi k!

<sup>4</sup> <sup>A</sup>4expi k!

We assume that ω<sup>3</sup> � ω<sup>1</sup> ¼ ω<sup>4</sup> � ω2. Suppose that the pumping waves E

<sup>4</sup> and the phase-conjugate wave E

A1,<sup>4</sup> ¼ A01,4exp gr �

Reg < 0 corresponding to the amplification of the phase-conjugate wave E

show that the amplification of the phase-conjugate wave E

!

!o <sup>1</sup>⊥E !o

!

(42) and (43) in the case of SLS [8–10]. It has been shown that this solution is stable in the case

In the particular case when the fundamental waves (44) are counter-propagating, the phase conjugation is possible as a result of the nondegenerate FWM in SALC [8–10]. Optical phase conjugation (OPC) is the wavefront reversion property of a backward propagating optical wave with respect to a forward propagating wave [22]. The optical waves are phase conjugated to each other if their complex amplitudes are conjugated with respect to their phase factors [22]. Typically, OPC results from nonlinear optical processes such as FWM and SLS [20].

<sup>1</sup>,<sup>4</sup> are phase-conjugate while the waves E

þ ω1t h i

� ω2t h i

> þ Δ k ! � r ! � �

� ω4t h i

h i

n o

þ c:c:

þ c:c:

þ c:c:

!

<sup>32</sup> is the wave vector mismatch of the FWM process. In the case of OPC caused

þ ω3t

<sup>4</sup>� r ! � �

<sup>2</sup>� r ! � �

<sup>2</sup>� r ! � �

<sup>4</sup>� r ! � �

n o

n o

n o

by SLS the frequency balance condition between the waves with the same vectors is necessary.

tion as it is seen from Eq. (46). In such a case, using the constant pumping approximation (CPA) [14] where A2,<sup>3</sup> ¼ const, we obtain the following solution for the probe wave and the

Analysis of the truncated equations for A1,<sup>4</sup> shows that there exists the solution with the gain

can be characterized as a kind of the Brillouin-enhanced FWM (BEFWM) based on the optical nonlinearity related to the smectic layer normal displacement [8–10]. Numerical estimations

i <sup>2</sup> <sup>Δ</sup> <sup>k</sup> ! � r

! � � � �

!

<sup>2</sup>,<sup>3</sup> and E !o <sup>1</sup>∥E !o

!o

<sup>4</sup> and the signal wave E

!

þ c:c:

!

!

<sup>1</sup> is possible for the typical values of

<sup>1</sup> propagating in the negative direc-

<sup>2</sup>,3, the approximate solution can be obtained similarly to the solution

4, then FWM is divided in two

!o

<sup>2</sup>,<sup>3</sup> are forward-going and

<sup>2</sup>,<sup>3</sup> are much stronger

<sup>1</sup> [8–10]. Such a case

(46)

(47)

<sup>1</sup> are much stronger

<sup>2</sup>,<sup>3</sup> with the

<sup>1</sup>, <sup>4</sup> and the waves E

!

The components of the nonlinear electric induction D !NL , which are not phase matched to the fundamental waves (46) give rise to 12 doubly degenerate combination harmonics of the type.

$$\begin{split} & \left[ A\_m A\_p A\_n^\* \text{expi} \right] \left[ \left( \left( \overrightarrow{k}\_m + \overrightarrow{k}\_p - \overrightarrow{k}\_n \right) \cdot \overrightarrow{r} \right) - \left( \omega\_m + \omega\_p - \omega\_n \right) t \right] \text{ and 12 harmonics of the type } A\_m^2 A\_n^\* \text{expi} \\ & \text{s.t.} \left[ 2\left( \left( \overrightarrow{k}\_m \cdot \overrightarrow{r} \right) - \omega\_m t \right) - \left( \overrightarrow{k}\_n \cdot \overrightarrow{r} \right) + \omega\_n t \right] \text{[S-10].} \end{split} $$

### 6. Nonlinear interaction of surface plasmon polaritons (SPP) in SALC

Integration of strongly nonlinear LC with plasmonic structures and metamaterials would enable active switching and tuning operations with low threshold [4]. LC may be also used in reconfigurable metamaterials for tuning the resonant frequency, the transmission/ reflection coefficient, and the refractive index [23]. Combination of metamaterials and active plasmonic structures with NLC has been investigated [4, 23]. In this section, we discuss the nonlinear optical effects caused by the SPP mixing in SALC, which is characterized by low losses and a strong nonlinearity related to the smectic layer normal displacement without the change of the mass density [11–13]. Consider the interface z ¼ 0 between a homeotropically oriented SALC ð Þ z > 0 and a metal ð Þ z < 0 shown in Figure 6 [11, 12]. The SALC optical Z axis and the X axis are chosen to be perpendicular and parallel to the interface z ¼ 0, respectively.

SPP from the metal penetrate into SALC. The permittivity of the metal εmð Þ ω determined by the Drude model is given by <sup>ε</sup>mð Þ¼ <sup>ω</sup> <sup>1</sup> � <sup>ω</sup><sup>2</sup> <sup>p</sup>ω�<sup>1</sup>ð Þ <sup>ω</sup> <sup>þ</sup> ð Þ <sup>i</sup>=<sup>τ</sup> �<sup>1</sup> where <sup>ω</sup><sup>p</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>n</sup>0e<sup>2</sup>=ð Þ <sup>ε</sup>0<sup>m</sup> <sup>p</sup> is the plasma frequency, n<sup>0</sup> is the free electron density in the metal, e, m are the electron charge and mass, respectively, ω, τ are the SPP angular frequency and lifetime, respectively [24, 25]. The efficient SLS in SALC takes place for the counter-propagating SPP with close frequencies

Figure 6. The counter-propagating SPP at the interface between a metal ð Þ z < 0 and a homeotropically oriented SALC ð Þ z > 0 .

ω<sup>1</sup> > ω<sup>2</sup> such that Δω ¼ ω<sup>1</sup> � ω<sup>2</sup> ≪ ω<sup>1</sup> [12]. The spatially localized electric fields of these SPP in SALC have the form [24, 25]

$$\overrightarrow{E}\_{1,2} = \frac{1}{2} \left\{ \overrightarrow{e}\_{1,2} A\_{1,2}(\mathbf{x}, t) \exp\left[ \pm i k\_{\mathbf{x}} (\mathbf{x} \pm d) - k\_{z}^{S} \mathbf{z} - i \omega\_{1,2} t \right] + c.c. \right\} \tag{48}$$

grating (50) can be characterized as an enhanced Rayleigh wave of SS [26]. Analysis of

frequency difference <sup>Δ</sup><sup>ω</sup> � <sup>10</sup><sup>7</sup> � 108 � �s�1, and the spontaneous SS surface wave can be

to the localized grating (50) is essentially complex. For the typical values of the SALC parameters, SPP in silver, <sup>ω</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup>:<sup>4</sup> � 1015s�<sup>1</sup> and <sup>Δ</sup><sup>ω</sup> � 107 � <sup>10</sup><sup>8</sup> � �s�<sup>1</sup> the numerical estimations

We substitute the localized layer displacement u xð Þ ; z; t (50) into the expression of the SALC

(48), and by using the standard procedure, we obtain from Eq. (13) the truncated equations for the SPP SVA A1, <sup>2</sup>ðÞ¼ t j j A1,2ð Þt expiγ1,2ð Þt . The dependence of SVA A1, <sup>2</sup> on the x coordinate can be neglected in the central part of the dynamic grating (50) for the distances of several SPP wavelengths [12]. Integrating the SPP electric field and nonlinear electric induction over z ∈½ � 0; ∞ , we obtain the following truncated equations for the normalized SPP intensities

tude than the cubic susceptibilities of some organic liquids and solid materials [27].

nonlinear permittivity (30), evaluate the nonlinear part of the electric induction DNL

∂I1, <sup>2</sup>

hbIm G kx; kS

<sup>6</sup><sup>r</sup> <sup>ε</sup>⊥j j ex <sup>2</sup> <sup>þ</sup> <sup>ε</sup>∥j j ez <sup>2</sup> � � ð Þ 2Rekx <sup>2</sup> � 2Re<sup>k</sup>

<sup>z</sup> �a⊥j j <sup>e</sup>1<sup>x</sup> <sup>2</sup> <sup>þ</sup> <sup>a</sup>∥j j <sup>e</sup>1<sup>z</sup> <sup>2</sup> � � <sup>þ</sup> <sup>4</sup>εað Þ Rekx Im <sup>e</sup>1ze<sup>∗</sup>

I1ð Þ0

S

� � <sup>10</sup>�<sup>20</sup> � <sup>10</sup>�<sup>19</sup> � �m<sup>2</sup>=V<sup>2</sup> [11], which is larger by one-two orders of magni-

� � <sup>¼</sup> <sup>I</sup><sup>0</sup> <sup>¼</sup> const is the integral of motion obtained from the Manley-

<sup>z</sup> ;Δ<sup>ω</sup> � � � � <sup>I</sup>0ω1ω2expð Þ �<sup>2</sup>

<sup>I</sup>1ð Þþ <sup>0</sup> ½ � <sup>1</sup> � <sup>I</sup>1ð Þ<sup>0</sup> expð Þ �gt ; I2ðÞ¼ <sup>t</sup> ½ � <sup>1</sup> � <sup>I</sup>1ð Þ<sup>0</sup> expð Þ �gt

Expressions (55) show that the energy exchange between SPP takes place. In the limiting case t ! ∞, we obtain: I1ðÞ!t 1; I2ðÞ!t 0 [12]. The time dependence of the normalized SPP intensities I1ð Þt , I2ð Þt (55) is presented in Figure 7. The phases γ1,2ð Þ0 of the SPP SVA have the

S z � �<sup>2</sup> h i G kx; kS

1x

<sup>z</sup> ;Δ<sup>ω</sup> � � <sup>¼</sup> 0 cannot be achieved for the

http://dx.doi.org/10.5772/intechopen.70997

Nonlinear Optical Phenomena in Smectic A Liquid Crystals

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>∓</sup> gI1I<sup>2</sup> (53)

<sup>z</sup> ;Δ<sup>ω</sup> � � � � �

�

� �. Solution of Eq. (53) has the

<sup>I</sup>1ð Þþ <sup>0</sup> ½ � <sup>1</sup> � <sup>I</sup>1ð Þ<sup>0</sup> expð Þ �gt (55)

<sup>2</sup> > 0 (54)

ijklð Þ Δω ; i, j, k, l ¼ x, z related

!

for SPP

149

<sup>z</sup> ;Δ<sup>ω</sup> � � (52) shows that the resonant case ReG kx; <sup>k</sup>

neglected [12]. The cubic susceptibility of the SALC-metal system χð Þ<sup>3</sup>

G kx; kS

yield χð Þ<sup>3</sup> xxxx � � �

I1,<sup>2</sup> ¼ j j A1, <sup>2</sup>

Here, j j A<sup>1</sup>

2 =ω1, <sup>2</sup> � �<sup>I</sup>

2 =ω<sup>1</sup> � � <sup>þ</sup> j j <sup>A</sup><sup>2</sup>

Here, b ¼ �2Rek

form [12]

form

�1 0 .

> 2 =ω<sup>2</sup>

Rowe relation [14], and the gain g has the form [12]

<sup>g</sup> <sup>¼</sup> <sup>ε</sup>0ð Þ 2Rekx <sup>2</sup>

It is easy to see from Eq. (55) that I1ð Þþ t I2ðÞ¼ t 1.

S

I1ðÞ¼ t

� � � <sup>≈</sup> <sup>χ</sup>ð Þ<sup>3</sup> zzzz � � �

� �

The SPP are polarized as transverse magnetic (TM) waves with the electric field components Ex, <sup>z</sup> and the magnetic field component Hy [23, 24]. In an optically uniaxial SALC, SPP propagate as extraordinary waves [18]. The numerical estimations show that for the optical frequency range and the small frequency difference <sup>Δ</sup><sup>ω</sup> <sup>≈</sup> <sup>10</sup>�<sup>7</sup> � <sup>10</sup>�<sup>5</sup> � �ω<sup>1</sup> the SPP1,2 wave vectors are practically equal [11, 12]. They have the form [11, 12]

$$\begin{aligned} k\_z^S &= \sqrt{k\_x^2(\varepsilon\_\perp/\varepsilon\_\parallel) - \omega\_1^2 \varepsilon\_\perp/c^2} \\ k\_x &= (\omega\_1/c)\sqrt{\varepsilon\_m(\omega\_1)[1 - (\varepsilon\_m(\omega\_1)/\varepsilon\_\perp)]\left[1 - \left(\varepsilon\_m^2(\omega\_1)/\left(\varepsilon\_\perp \varepsilon\_\parallel\right)\right)\right]^{-1}} \end{aligned} \tag{49}$$

Numerical estimations show that for the typical values of ω1, <sup>2</sup>, ωp, τ the following relations are valid: Rek<sup>S</sup> <sup>z</sup> ≫ Imk S <sup>z</sup> , Rekx ≫ Imkx [11, 12, 24, 25]. For the optical wavelength λopt ≈ ð Þ 0:6 � 1:33 μm, the SPP propagation length Lx and the wavelength λ<sup>s</sup> are given by, respectively: Lx <sup>¼</sup> ð Þ Imkx �<sup>1</sup> <sup>≈</sup> ð Þ <sup>84</sup> � <sup>550</sup> <sup>μ</sup>m, <sup>λ</sup><sup>s</sup> <sup>¼</sup> <sup>2</sup>π=ð Þ Rekx <sup>≈</sup> ð Þ <sup>0</sup>:<sup>33</sup> � <sup>0</sup>:<sup>77</sup> <sup>μ</sup><sup>m</sup> <sup>≪</sup> Lx [12]. The SPP localization length Lz ¼ Rek S z � ��<sup>1</sup> � <sup>10</sup>�<sup>6</sup> m belongs to the subwavelength scale: Imk S <sup>z</sup> � <sup>10</sup>m�<sup>1</sup> <sup>≪</sup> Re<sup>k</sup> S <sup>z</sup> and can be neglected [12].

Substituting the SPP fields (48) into the smectic layer equation of motion (9), we obtain the dynamic grating u xð Þ ; z; t given by [11, 12]

$$u(\mathbf{x}, z, t) = 0.5 \left[ \mathcal{U} \exp \left\{ i(2 \text{Re} k\_x) \mathbf{x} - 2 (\text{Im} k\_x) d - \left( 2 \text{Re} k\_z^S \right) z - i \Delta \omega t \right\} + c.c. \right] \tag{50}$$

Here,

$$\mathcal{U} = -\frac{\varepsilon\_0 (2\text{Re}\mathbf{k}\_x)^2 h A\_1(\mathbf{x}, t) A\_2^\*(\mathbf{x}, t)}{\rho \left[ - (2\text{Re}\mathbf{k}\_x)^2 + \left( 2\text{Re}\mathbf{k}\_z^S \right)^2 \right] \mathcal{G} \left( k\_x, k\_z^S, \Delta\omega \right)} \tag{51}$$

$$\begin{split} h &= -\left(2\text{Re}\mathbf{k}\_z^\circ\right)\left(-a\_\perp|e\_\mathbf{1}|^2 + a\_\parallel|e\_\mathbf{1}|^2\right) - 4\varepsilon\_d (2\text{Re}\mathbf{k}\_x)\text{Im}\left(e\_\mathbf{1}e\_\mathbf{1}^s\right); \\ G\left(k\_x, k\_z^\circ, \Delta\omega\right) &= \left(\Delta\omega\right)^2 - \frac{B\left(2\text{Re}\mathbf{k}\_x\right)^2\left(2\text{Re}\mathbf{k}\_z^\circ\right)^2}{\rho\left[-\left(2\text{Re}\mathbf{k}\_x\right)^2 + \left(2\text{Re}\mathbf{k}\_z^\circ\right)^2\right]} \\ &- i\frac{\Delta\omega}{\rho}\left[-a\_1\frac{\left(2\text{Re}\mathbf{k}\_x\right)^2\left(2\text{Re}\mathbf{k}\_z^\circ\right)^2}{\left[-\left(2\text{Re}\mathbf{k}\_x\right)^2 + \left(2\text{Re}\mathbf{k}\_z^\circ\right)^2\right]} + \frac{\left(a\_4 + a\_5\delta\right)\left[-\left(2\text{Re}\mathbf{k}\_x\right)^2 + \left(2\text{Re}\mathbf{k}\_z^\circ\right)^2\right]}{2}\right] \end{split} \tag{52}$$

Unlike the dynamic grating (28) created by the interfering optical waves, the grating (50) caused by SPP is spatially localized both in the X and in the Z directions [11, 12]. The localized grating (50) can be characterized as an enhanced Rayleigh wave of SS [26]. Analysis of G kx; kS <sup>z</sup> ;Δ<sup>ω</sup> � � (52) shows that the resonant case ReG kx; <sup>k</sup> S <sup>z</sup> ;Δ<sup>ω</sup> � � <sup>¼</sup> 0 cannot be achieved for the frequency difference <sup>Δ</sup><sup>ω</sup> � <sup>10</sup><sup>7</sup> � 108 � �s�1, and the spontaneous SS surface wave can be neglected [12]. The cubic susceptibility of the SALC-metal system χð Þ<sup>3</sup> ijklð Þ Δω ; i, j, k, l ¼ x, z related to the localized grating (50) is essentially complex. For the typical values of the SALC parameters, SPP in silver, <sup>ω</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup>:<sup>4</sup> � 1015s�<sup>1</sup> and <sup>Δ</sup><sup>ω</sup> � 107 � <sup>10</sup><sup>8</sup> � �s�<sup>1</sup> the numerical estimations yield χð Þ<sup>3</sup> xxxx � � � � � � <sup>≈</sup> <sup>χ</sup>ð Þ<sup>3</sup> zzzz � � � � � � � <sup>10</sup>�<sup>20</sup> � <sup>10</sup>�<sup>19</sup> � �m<sup>2</sup>=V<sup>2</sup> [11], which is larger by one-two orders of magnitude than the cubic susceptibilities of some organic liquids and solid materials [27].

We substitute the localized layer displacement u xð Þ ; z; t (50) into the expression of the SALC nonlinear permittivity (30), evaluate the nonlinear part of the electric induction DNL ! for SPP (48), and by using the standard procedure, we obtain from Eq. (13) the truncated equations for the SPP SVA A1, <sup>2</sup>ðÞ¼ t j j A1,2ð Þt expiγ1,2ð Þt . The dependence of SVA A1, <sup>2</sup> on the x coordinate can be neglected in the central part of the dynamic grating (50) for the distances of several SPP wavelengths [12]. Integrating the SPP electric field and nonlinear electric induction over z ∈½ � 0; ∞ , we obtain the following truncated equations for the normalized SPP intensities I1,<sup>2</sup> ¼ j j A1, <sup>2</sup> 2 =ω1, <sup>2</sup> � �<sup>I</sup> �1 0 .

$$\frac{\partial I\_{1,2}}{\partial t} = \mp g I\_1 I\_2 \tag{53}$$

Here, j j A<sup>1</sup> 2 =ω<sup>1</sup> � � <sup>þ</sup> j j <sup>A</sup><sup>2</sup> 2 =ω<sup>2</sup> � � <sup>¼</sup> <sup>I</sup><sup>0</sup> <sup>¼</sup> const is the integral of motion obtained from the Manley-Rowe relation [14], and the gain g has the form [12]

$$\log = \frac{\varepsilon\_0 (2 \text{Re} \mathbf{k}\_x)^2 hb \text{Im} \left\{ G \left( \mathbf{k}\_x, \mathbf{k}\_z^S, \Delta \omega \right) \right\} I\_0 \omega\_1 \omega\_2 \exp(-2)}{6 \rho \left( \varepsilon\_\perp |\varepsilon\_x|^2 + \varepsilon\_\parallel |\varepsilon\_z|^2 \right) \left[ (2 \text{Re} \mathbf{k}\_x)^2 - \left( 2 \text{Re} \mathbf{k}\_z^S \right)^2 \right] \left[ G \left( \mathbf{k}\_x, \mathbf{k}\_z^S, \Delta \omega \right) \right]^2} > 0 \tag{54}$$

Here, b ¼ �2Rek S <sup>z</sup> �a⊥j j <sup>e</sup>1<sup>x</sup> <sup>2</sup> <sup>þ</sup> <sup>a</sup>∥j j <sup>e</sup>1<sup>z</sup> <sup>2</sup> � � <sup>þ</sup> <sup>4</sup>εað Þ Rekx Im <sup>e</sup>1ze<sup>∗</sup> 1x � �. Solution of Eq. (53) has the form [12]

$$I\_1(t) = \frac{I\_1(0)}{I\_1(0) + [1 - I\_1(0)] \exp(-gt)} ; I\_2(t) = \frac{[1 - I\_1(0)] \exp(-gt)}{I\_1(0) + [1 - I\_1(0)] \exp(-gt)}\tag{55}$$

It is easy to see from Eq. (55) that I1ð Þþ t I2ðÞ¼ t 1.

ω<sup>1</sup> > ω<sup>2</sup> such that Δω ¼ ω<sup>1</sup> � ω<sup>2</sup> ≪ ω<sup>1</sup> [12]. The spatially localized electric fields of these SPP in

The SPP are polarized as transverse magnetic (TM) waves with the electric field components Ex, <sup>z</sup> and the magnetic field component Hy [23, 24]. In an optically uniaxial SALC, SPP propagate as extraordinary waves [18]. The numerical estimations show that for the optical frequency range and the small frequency difference <sup>Δ</sup><sup>ω</sup> <sup>≈</sup> <sup>10</sup>�<sup>7</sup> � <sup>10</sup>�<sup>5</sup> � �ω<sup>1</sup> the SPP1,2 wave vectors are practically

n o

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>z</sup> , Rekx ≫ Imkx [11, 12, 24, 25]. For the optical wavelength λopt ≈ ð Þ 0:6 � 1:33 μm,

<sup>m</sup>ð Þ ω<sup>1</sup> = ε⊥ε<sup>∥</sup>

S z � �<sup>z</sup> � <sup>i</sup>Δω<sup>t</sup> � � <sup>þ</sup> <sup>c</sup>:c: � � (50)

> 1x � �;

ð Þ� <sup>α</sup><sup>4</sup> <sup>þ</sup> <sup>α</sup><sup>56</sup> ð Þ 2Rekx <sup>2</sup> <sup>þ</sup> 2Re<sup>k</sup>

2

� �<sup>2</sup> h i

<sup>2</sup>ð Þ x; t

G kx; k S S

<sup>z</sup> � <sup>10</sup>m�<sup>1</sup> <sup>≪</sup> Re<sup>k</sup>

<sup>z</sup> ;Δ<sup>ω</sup> � � (51)

S z

3 5 (52)

S <sup>z</sup> and

� � � � � � �<sup>1</sup> <sup>q</sup> (49)

S <sup>z</sup> <sup>z</sup> � <sup>i</sup>ω1,2<sup>t</sup> � � <sup>þ</sup> <sup>c</sup>:c:

(48)

<sup>1</sup>,2A1,2ð Þ x; t exp �ikxð Þ� x � d k

SALC have the form [24, 25]

E ! <sup>1</sup>, <sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> <sup>e</sup> !

equal [11, 12]. They have the form [11, 12]

k2 <sup>x</sup> ε⊥=ε<sup>∥</sup> � � � <sup>ω</sup><sup>2</sup>

q

kx ¼ ð Þ ω1=c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

148 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

1ε⊥=c<sup>2</sup>

εmð Þ ω<sup>1</sup> ½ � 1 � ð Þ εmð Þ ω<sup>1</sup> =ε<sup>⊥</sup> 1 � ε<sup>2</sup>

Numerical estimations show that for the typical values of ω1, <sup>2</sup>, ωp, τ the following relations are

the SPP propagation length Lx and the wavelength λ<sup>s</sup> are given by, respectively: Lx <sup>¼</sup> ð Þ Imkx �<sup>1</sup> <sup>≈</sup> ð Þ <sup>84</sup> � <sup>550</sup> <sup>μ</sup>m, <sup>λ</sup><sup>s</sup> <sup>¼</sup> <sup>2</sup>π=ð Þ Rekx <sup>≈</sup> ð Þ <sup>0</sup>:<sup>33</sup> � <sup>0</sup>:<sup>77</sup> <sup>μ</sup><sup>m</sup> <sup>≪</sup> Lx [12]. The SPP localization

Substituting the SPP fields (48) into the smectic layer equation of motion (9), we obtain the

u xð Þ¼ ; z; t 0:5 U exp ið Þ 2Rekx x � 2 Imð Þ kx d � 2Rek

<sup>U</sup> ¼ � <sup>ε</sup>0ð Þ 2Rekx <sup>2</sup>

� � �a⊥j j <sup>e</sup>1<sup>x</sup> <sup>2</sup> <sup>þ</sup> <sup>a</sup>∥j j <sup>e</sup>1<sup>z</sup> <sup>2</sup> � �

ð Þ 2Rekx <sup>2</sup> 2Re<sup>k</sup>

�ð Þ 2Rekx <sup>2</sup> <sup>þ</sup> 2Re<sup>k</sup>

� �<sup>2</sup> h i <sup>þ</sup>

<sup>r</sup> �ð Þ 2Rekx <sup>2</sup> <sup>þ</sup> 2Re<sup>k</sup>

<sup>2</sup> � <sup>B</sup>ð Þ 2Rekx <sup>2</sup> 2Rek<sup>S</sup>

S z

S z � �<sup>2</sup>

<sup>r</sup> �ð Þ 2Rekx <sup>2</sup> <sup>þ</sup> 2Re<sup>k</sup>

� �<sup>2</sup> h i

Unlike the dynamic grating (28) created by the interfering optical waves, the grating (50) caused by SPP is spatially localized both in the X and in the Z directions [11, 12]. The localized

� �<sup>2</sup> h i

m belongs to the subwavelength scale: Imk

hA1ð Þ <sup>x</sup>; <sup>t</sup> <sup>A</sup><sup>∗</sup>

S z

� <sup>4</sup>εað Þ 2Rekx Im <sup>e</sup>1ze<sup>∗</sup>

S z

z � �<sup>2</sup>

kS <sup>z</sup> ¼

S z � ��<sup>1</sup> � <sup>10</sup>�<sup>6</sup>

<sup>h</sup> ¼ � 2Rek<sup>S</sup>

2 4

G kx; kS

�i Δω <sup>r</sup> �α<sup>1</sup>

z

<sup>z</sup> ;Δ<sup>ω</sup> � � <sup>¼</sup> ð Þ <sup>Δ</sup><sup>ω</sup>

dynamic grating u xð Þ ; z; t given by [11, 12]

<sup>z</sup> ≫ Imk S

valid: Rek<sup>S</sup>

Here,

length Lz ¼ Rek

can be neglected [12].

Expressions (55) show that the energy exchange between SPP takes place. In the limiting case t ! ∞, we obtain: I1ðÞ!t 1; I2ðÞ!t 0 [12]. The time dependence of the normalized SPP intensities I1ð Þt , I2ð Þt (55) is presented in Figure 7. The phases γ1,2ð Þ0 of the SPP SVA have the form

$$\gamma\_1(t) - \gamma\_1(0) = \frac{\text{Re}\{G(k\_x, k\_z^S, \Delta\omega)\}}{2\text{Im}\{G(k\_x, k\_z^S, \Delta\omega)\}} \ln\left[ (1 - I\_1(0))\exp(-gt) + I\_1(0) \right] \tag{56}$$

and large nonlinearity [28]. Photonic components based on plasmonic waveguides with NLC core have been theoretically investigated in a number of articles (see, for example, [28–31] and

We consider the nonlinear optical processes in an MIM waveguide with the SALC core [13]. The structure of such a waveguide is shown in Figure 8. SPP propagating in the metal claddings and in SALC core are TM waves [24, 25]. The SPP electric and magnetic fields in

<sup>1</sup>,2ð Þ x; z; t , E

yH1, 20exp ∓k<sup>m</sup>

<sup>z</sup> <sup>z</sup> � � <sup>þ</sup> <sup>B</sup>exp �kS

! z kx ωε0ε<sup>∥</sup>

<sup>z</sup> <sup>z</sup> � � � � ( )

expression (49) and the SPP wave number in the metallic claddings is given by k

<sup>z</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

SPP fields (58)–(61) at the interface z ¼ d between the metallic cladding and the SALC core, we

SAð Þ x; z; t have the form, respectively [24]

xE1,2x<sup>0</sup> þ a ! zE1,2z<sup>0</sup> h iexp <sup>∓</sup> km

<sup>y</sup> Aexp kS

<sup>z</sup> <sup>z</sup> � � � <sup>B</sup>exp �k<sup>S</sup> <sup>z</sup> <sup>z</sup> � � � � � <sup>a</sup>

S

obtain the dispersion relation for the MIM modes [13, 24]

Figure 8. The MIM waveguide with the homeotropically oriented SALC as a core.

1 2 a ! !

<sup>1</sup>,2ð Þ x; z; t , and in the SALC core j j z ≤ d

Nonlinear Optical Phenomena in Smectic A Liquid Crystals

http://dx.doi.org/10.5772/intechopen.70997

<sup>z</sup> <sup>z</sup> <sup>þ</sup> ikxx � <sup>i</sup>ω<sup>t</sup> � � <sup>þ</sup> <sup>c</sup>:c:, zj j <sup>&</sup>gt; <sup>d</sup> (58)

<sup>z</sup> <sup>z</sup> � � � � expð Þþ ikxx � <sup>i</sup>ω<sup>t</sup> <sup>c</sup>:c:, zj j <sup>≤</sup> <sup>d</sup> (60)

Aexp kS

<sup>z</sup> of SPP in SALC in the linear approximation is determined by

[24]. Using the boundary conditions for the tangential components of the

<sup>z</sup> <sup>z</sup> <sup>þ</sup> ikxx � <sup>i</sup>ω<sup>t</sup> � � <sup>þ</sup> <sup>c</sup>:c:, zj j <sup>&</sup>gt; <sup>d</sup> (59)

<sup>z</sup> <sup>z</sup> � � <sup>þ</sup> <sup>B</sup>exp �kS

(61)

151

m

references therein).

SAð Þ x; z; t , E

E !

¼ 1 <sup>2</sup> <sup>a</sup> ! x k S z iωε0ε<sup>⊥</sup>

k2

q

!

E !

> H !

SAð Þ x; z; t

H !

the metallic claddings <sup>z</sup> <sup>&</sup>gt; d; z <sup>&</sup>lt; �d H!

<sup>1</sup>,2ð Þ¼ x; z; t

1 <sup>2</sup> <sup>a</sup> !

> 1 2 a !

Aexp k S

H !

<sup>1</sup>, <sup>2</sup>ð Þ¼ x; z; t

SAð Þ¼ x; z; t

�expi kð Þþ xx � ωt c:c:; zj j ≤ d

The complex wave number k

<sup>x</sup> � εmð Þ ω ω<sup>2</sup>=c<sup>2</sup>

$$\gamma\_2(t) - \gamma\_2(0) = -\frac{\text{Re}\{\mathcal{G}(k\_x, k\_z^S, \Delta\omega)\}}{2\text{Im}\{\mathcal{G}(k\_x, k\_z^S, \Delta\omega)\}}\ln\left[I\_1(0)\exp(\mathcal{g}t) + 1 - I\_1(0)\right] \tag{57}$$

It is easy to see from Eqs. (56) and (57) that for t ! ∞ the phase γ1ð Þt of the amplified SPP I<sup>1</sup> tends to a constant value <sup>γ</sup>1ð Þ� <sup>t</sup> <sup>γ</sup>1ð Þ!<sup>0</sup> Re G kx;k<sup>S</sup> <sup>z</sup> f g ð Þ ;Δ<sup>ω</sup> 2Im G kx;k<sup>S</sup> <sup>z</sup> f g ð Þ ;Δ<sup>ω</sup> ln½ � <sup>I</sup>1ð Þ<sup>0</sup> , while the phase of the depleted SPP I<sup>2</sup> γ2ð Þ� t γ2ð Þ0 for large time intervals such that gt ≫ 1 takes the form <sup>γ</sup>2ð Þ� <sup>t</sup> <sup>γ</sup>2ð Þ!� <sup>0</sup> Re G kx;kS <sup>z</sup> f g ð Þ ;Δ<sup>ω</sup> 2Im G kx;k<sup>S</sup> <sup>z</sup> f g ð Þ ;Δ<sup>ω</sup> gt and <sup>γ</sup>2ðÞ!� <sup>t</sup> <sup>∞</sup> for <sup>t</sup> ! <sup>∞</sup>. The SVA of the depleted SPP <sup>I</sup><sup>2</sup> undergoes strong XPM and rapidly oscillates in the time domain. The results (55)–(57) show that the Rayleigh stimulated scattering [27] of SPP on the smectic layer normal displacement localized grating is accompanied by XPM and the parametric energy exchange between SPP [12]. The rise time of the amplified SPP is about 1 � 2μs as it is seen from Figure 7. It is much faster than the thermal response time τ<sup>R</sup> ¼ 100 μs and the purely orientational response time ≈ 25 ms in NLC [4]. Numerical estimations show that for the SPP electric field of 107 V=m the rise time of about 10 ns can be achieved, which is much less than the Brillouin relaxation time τ<sup>B</sup> ≈ 200 ns [4, 12].

Structures consisting of alternative conducting and dielectric thin films are capable of guiding SPP light waves [24, 25]. Each single interface can sustain bound SPP. When the distance between adjacent interfaces is comparable or smaller than the SPP localization length Lz <sup>¼</sup> Rek<sup>S</sup> z �<sup>1</sup> , the coupled modes occur due to the interaction between SPP [24]. The following specific three-layer guiding systems can be considered: (i) an insulator/metal/insulator (IMI) heterostructure where a thin metallic layer is sandwiched between two infinitely thick dielectric claddings; (ii) a metal/insulator/metal (MIM) heterostructure where a thin dielectric core layer is sandwiched between two infinitely thick metallic claddings [24]. LC can be used as a tunable cladding material or as the guiding core material due to their excellent electro-optic properties

Figure 7. The temporal dependence of the SPP normalized intensities <sup>I</sup>1, <sup>2</sup>ð Þ<sup>t</sup> for the input electric field of 106V=<sup>m</sup> and optical wavelengths λopt<sup>1</sup> ¼ 0:6μm (curves 1) and λopt<sup>1</sup> ¼ 1:33μm (curve 2).

and large nonlinearity [28]. Photonic components based on plasmonic waveguides with NLC core have been theoretically investigated in a number of articles (see, for example, [28–31] and references therein).

γ1ð Þ� t γ1ð Þ¼ 0

<sup>γ</sup>2ð Þ� <sup>t</sup> <sup>γ</sup>2ð Þ!� <sup>0</sup> Re G kx;<sup>k</sup>

τ<sup>B</sup> ≈ 200 ns [4, 12].

z �<sup>1</sup>

Lz <sup>¼</sup> Rek<sup>S</sup>

γ2ð Þ� t γ2ð Þ¼� 0

tends to a constant value <sup>γ</sup>1ð Þ� <sup>t</sup> <sup>γ</sup>1ð Þ!<sup>0</sup> Re G kx;k<sup>S</sup>

S <sup>z</sup> f g ð Þ ;Δ<sup>ω</sup> 2Im G kx;k<sup>S</sup>

Re G kx; k<sup>S</sup> <sup>z</sup> ; <sup>Δ</sup><sup>ω</sup>

150 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

S

It is easy to see from Eqs. (56) and (57) that for t ! ∞ the phase γ1ð Þt of the amplified SPP I<sup>1</sup>

depleted SPP I<sup>2</sup> γ2ð Þ� t γ2ð Þ0 for large time intervals such that gt ≫ 1 takes the form

undergoes strong XPM and rapidly oscillates in the time domain. The results (55)–(57) show that the Rayleigh stimulated scattering [27] of SPP on the smectic layer normal displacement localized grating is accompanied by XPM and the parametric energy exchange between SPP [12]. The rise time of the amplified SPP is about 1 � 2μs as it is seen from Figure 7. It is much faster than the thermal response time τ<sup>R</sup> ¼ 100 μs and the purely orientational response time

rise time of about 10 ns can be achieved, which is much less than the Brillouin relaxation time

Structures consisting of alternative conducting and dielectric thin films are capable of guiding SPP light waves [24, 25]. Each single interface can sustain bound SPP. When the distance between adjacent interfaces is comparable or smaller than the SPP localization length

specific three-layer guiding systems can be considered: (i) an insulator/metal/insulator (IMI) heterostructure where a thin metallic layer is sandwiched between two infinitely thick dielectric claddings; (ii) a metal/insulator/metal (MIM) heterostructure where a thin dielectric core layer is sandwiched between two infinitely thick metallic claddings [24]. LC can be used as a tunable cladding material or as the guiding core material due to their excellent electro-optic properties

Figure 7. The temporal dependence of the SPP normalized intensities <sup>I</sup>1, <sup>2</sup>ð Þ<sup>t</sup> for the input electric field of 106V=<sup>m</sup> and

optical wavelengths λopt<sup>1</sup> ¼ 0:6μm (curves 1) and λopt<sup>1</sup> ¼ 1:33μm (curve 2).

, the coupled modes occur due to the interaction between SPP [24]. The following

≈ 25 ms in NLC [4]. Numerical estimations show that for the SPP electric field of 107

<sup>z</sup> f g ð Þ ;Δ<sup>ω</sup> 2Im G kx;k<sup>S</sup>

<sup>z</sup> f g ð Þ ;Δ<sup>ω</sup> gt and <sup>γ</sup>2ðÞ!� <sup>t</sup> <sup>∞</sup> for <sup>t</sup> ! <sup>∞</sup>. The SVA of the depleted SPP <sup>I</sup><sup>2</sup>

Re G kx; k<sup>S</sup> <sup>z</sup> ; <sup>Δ</sup><sup>ω</sup>

2Im G kx; kS

<sup>z</sup> ;Δ<sup>ω</sup> ln 1ð Þ � <sup>I</sup>1ð Þ<sup>0</sup> expð Þþ �gt <sup>I</sup>1ð Þ<sup>0</sup> (56)

<sup>z</sup> ;Δ<sup>ω</sup> ln <sup>I</sup>1ð Þ<sup>0</sup> expð Þþ gt <sup>1</sup> � <sup>I</sup>1ð Þ<sup>0</sup> (57)

<sup>z</sup> f g ð Þ ;Δ<sup>ω</sup> ln½ � <sup>I</sup>1ð Þ<sup>0</sup> , while the phase of the

V=m the

2Im G kx; k

We consider the nonlinear optical processes in an MIM waveguide with the SALC core [13]. The structure of such a waveguide is shown in Figure 8. SPP propagating in the metal claddings and in SALC core are TM waves [24, 25]. The SPP electric and magnetic fields in the metallic claddings <sup>z</sup> <sup>&</sup>gt; d; z <sup>&</sup>lt; �d H! <sup>1</sup>,2ð Þ x; z; t , E ! <sup>1</sup>,2ð Þ x; z; t , and in the SALC core j j z ≤ d H ! SAð Þ x; z; t , E ! SAð Þ x; z; t have the form, respectively [24]

$$\overrightarrow{H}\_{1,2}(\mathbf{x}, z, t) = \frac{1}{2}\overrightarrow{a}\_y H\_{1,20} \exp\left(\mp k\_z''' \mathbf{z} + i\mathbf{k}\_x \mathbf{x} - i\omega t\right) + c.c., |\mathbf{z}| > d\tag{58}$$

$$\overrightarrow{E}\_{1,2}(\mathbf{x}, z, t) = \frac{1}{2} \left[ \overrightarrow{a}\_x E\_{1,2x0} + \overrightarrow{a}\_z E\_{1,2x0} \right] \exp\left( \mp k\_z''' \mathbf{z} + i \mathbf{k}\_x \mathbf{x} - i \omega t \right) + c.c. \left| z \right| > d \tag{59}$$

$$\overrightarrow{H}\_{\text{SA}}(\mathbf{x}, z, t) = \frac{1}{2}\overrightarrow{a}\_y \left[ A \exp(k\_z^{\text{S}} z) + B \exp(-k\_z^{\text{S}} z) \right] \exp(i k\_x \mathbf{x} - i \omega t) + c.c., |z| \le d \tag{60}$$

$$\begin{aligned} &\stackrel{\cdot}{E}\_{SA}(\mathbf{x},z,t) \\ &=\frac{1}{2}\left\{ \stackrel{\cdot}{a}\_{x}\frac{k\_z^S}{i\omega\varepsilon\_0\varepsilon\_\perp} \left[ A\exp\left(k\_z^S z\right) - B\exp\left(-k\_z^S z\right) \right] - \stackrel{\cdot}{a}\_z \frac{k\_x}{i\omega\varepsilon\_0\varepsilon\_\parallel} \left[ A\exp\left(k\_z^S z\right) + B\exp\left(-k\_z^S z\right) \right] \right\} \\ &\times \exp(k\_x x - \omega t) + c.c.; |z| \le d \end{aligned} \tag{61}$$

The complex wave number k S <sup>z</sup> of SPP in SALC in the linear approximation is determined by expression (49) and the SPP wave number in the metallic claddings is given by k m <sup>z</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 <sup>x</sup> � εmð Þ ω ω<sup>2</sup>=c<sup>2</sup> q [24]. Using the boundary conditions for the tangential components of the SPP fields (58)–(61) at the interface z ¼ d between the metallic cladding and the SALC core, we obtain the dispersion relation for the MIM modes [13, 24]

Figure 8. The MIM waveguide with the homeotropically oriented SALC as a core.

$$\exp\left(-4k\_z^S d\right) = \left(\frac{k\_z^m}{\varepsilon\_r(\omega)} + \frac{k\_z^S}{\varepsilon\_\perp}\right)^2 \left(\frac{k\_z^m}{\varepsilon\_r(\omega)} - \frac{k\_z^S}{\varepsilon\_\perp}\right)^{-2} \tag{62}$$

D !NL SA ¼ ε<sup>0</sup>

> �2i ∂E<sup>0</sup> ∂t E∗

¼ ω

F<sup>1</sup> k<sup>S</sup>

F<sup>2</sup> k<sup>S</sup>

<sup>þ</sup><sup>8</sup> <sup>a</sup><sup>2</sup>

<sup>þ</sup><sup>8</sup> <sup>1</sup>

4

<sup>z</sup> ; kx; <sup>d</sup> � � <sup>¼</sup> <sup>ε</sup><sup>⊥</sup> <sup>1</sup> <sup>þ</sup>

<sup>z</sup> ; kx; <sup>d</sup> � � <sup>¼</sup> <sup>a</sup><sup>⊥</sup> <sup>þ</sup> <sup>a</sup><sup>∥</sup>

ε2 <sup>⊥</sup>j j kx <sup>2</sup> ε2 <sup>∥</sup> <sup>k</sup><sup>S</sup> j j <sup>z</sup> 2 � �<sup>2</sup> !sinh 2 Rek<sup>S</sup>

> ε2 <sup>⊥</sup>j j kx <sup>2</sup> ε2 <sup>∥</sup> kS z � � � � 2

j j E<sup>0</sup>

enhanced by a large geometric factor F<sup>2</sup> k

102 � 104 for 2<sup>d</sup> <sup>¼</sup> <sup>1</sup>μ<sup>m</sup> and Re<sup>k</sup>

<sup>⊥</sup> � a<sup>∥</sup>

<sup>2</sup> <sup>a</sup><sup>⊥</sup> <sup>þ</sup> <sup>a</sup><sup>∥</sup>

The solution of Eq. (67) has the form

0 @

At the distances <sup>x</sup> <sup>≪</sup> Lx <sup>¼</sup> ð Þ Imkx �<sup>1</sup>

ε0j j E<sup>0</sup> 4

expiwð Þt .

Here,

∂u ∂z E<sup>0</sup> a !

xa⊥cosh kS

<sup>0</sup>ε<sup>⊥</sup> cosh <sup>k</sup><sup>S</sup> <sup>z</sup> <sup>z</sup> � � � � �

> ∂j j E<sup>0</sup> 2 <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>0</sup>;

<sup>z</sup> <sup>z</sup> � � � <sup>a</sup> ! zia<sup>∥</sup> kxε<sup>⊥</sup> kS <sup>z</sup> ε<sup>∥</sup>

> � 2 <sup>þ</sup> j j kx <sup>2</sup> ε⊥

<sup>B</sup> exp½ � �2 Imð Þ kx <sup>x</sup> <sup>a</sup><sup>⊥</sup> cosh kS

k S z � � � � 2 ε∥

obtain the following equations for the magnitude j j E tð Þ and phase wð Þt of the SPP SVA.

∂w <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>ω</sup>

<sup>ε</sup>⊥j j kx <sup>2</sup> ε<sup>∥</sup> k<sup>S</sup> z � � � � 2

!

ε2 <sup>⊥</sup>j j kx <sup>2</sup> ε2 <sup>∥</sup> <sup>k</sup><sup>S</sup> j j <sup>z</sup> 2

� �<sup>2</sup>

1 A

S

2

2 <sup>2</sup>

2 4

" #

Substituting the SPP electric field (63) and nonlinear polarization (65) into Eq. (13) and separating linear and nonlinear parts, we obtain the truncated equation for SVA E0ðÞ¼ t j j E0ð Þt

> sinh k S <sup>z</sup> <sup>z</sup> � � � � �

> > � 2 þ a<sup>∥</sup>

<sup>z</sup> <sup>z</sup> � � � � �

ð Þ Imkx x� ≈ 1 [13]. The dispersion effects can be neglected because the dispersion length LD ≫ Lx [13]. Integrating both sides of Eq. (66) over z ∈½ � �d; d and separating real and imaginary parts we

> ε0j j E<sup>0</sup> 2

F<sup>2</sup> kS <sup>z</sup> ; kx; <sup>d</sup> � �

F<sup>1</sup> kS

� �<sup>d</sup> � � <sup>þ</sup> <sup>1</sup> � <sup>ε</sup>⊥j j kx <sup>2</sup>

ε<sup>∥</sup> kS z � � � � 2

<sup>z</sup> ; kx; <sup>d</sup> � � ( )<sup>t</sup> (69)

<sup>z</sup> ; kx; <sup>d</sup> � �, which can achieve a value of

!

16B

sinh 2 RekS

sinh 4 RekS

ε2 <sup>⊥</sup>j j kx <sup>2</sup> ε2 ∥ k S z � � � � 2

Eq. (69) shows that the strong SPM of the even SPP mode in the MIM wave guide occurs. It is

<sup>z</sup> ; kx; <sup>d</sup> � �=F<sup>1</sup> <sup>k</sup><sup>S</sup>

S

z � �d � �

þ a<sup>⊥</sup> � a<sup>∥</sup>

0 @

<sup>2</sup> <sup>¼</sup> const; <sup>w</sup>ðÞ¼ <sup>t</sup> <sup>ω</sup>

z

z � �d � �

> 1 A

ε0j j E<sup>0</sup> 2

16B

3 5 Rek S z � �d

F<sup>2</sup> k S <sup>z</sup> ; kx; <sup>d</sup> � �

F<sup>1</sup> k S

<sup>z</sup> � 106 � <sup>3</sup> � 106 � �m�<sup>1</sup>, Rekx � <sup>5</sup> � 106 � <sup>10</sup><sup>7</sup> � �m�1.

" #

� 2

> j j kx <sup>2</sup> ε2 ⊥

sinh kS <sup>z</sup> <sup>z</sup> � � � � �

Nonlinear Optical Phenomena in Smectic A Liquid Crystals

http://dx.doi.org/10.5772/intechopen.70997

� 2

<sup>z</sup> ; kx; <sup>d</sup> � � (67)

2 RekS z � �d

3 5

<sup>2</sup> (66)

153

(68)

kS z � � � � 2 ε2 ∥

, the SVA dependence on x can be neglected since exp½�2

sinh k<sup>S</sup> <sup>z</sup> <sup>z</sup> � � " #exp½ �þ i kð Þ xx � <sup>ω</sup><sup>t</sup> <sup>c</sup>:c: (65)

Numerical estimations show that for the typical values of the SPP frequency ω, the plasma frequency ωp, the SPP lifetime τ mentioned above, and the MIM thickness 2d ¼ 1μm the following relationships take place:

Rek S <sup>z</sup> � <sup>10</sup><sup>6</sup> m�<sup>1</sup> ≫ Imk S <sup>z</sup> � <sup>10</sup><sup>4</sup> <sup>m</sup>�<sup>1</sup>, Rekx � 107 <sup>m</sup>�<sup>1</sup> <sup>≫</sup> Imkx � 103 m�1. The SPP wavelength in the Z direction is given by 2π Imk<sup>S</sup> z � ��<sup>1</sup> � <sup>102</sup> μm and can be neglected inside the MIM waveguide core with the thickness of 2d � 1μm. Then, a single localized TM can exist in the MIM waveguide according to the dispersion relation (62). The even TM mode has the form [13]

$$\overrightarrow{E}\_{SA} = E\_0 \left[ \overrightarrow{a}\_x \cosh(k\_z^S z) - \overrightarrow{a}\_z i \frac{k\_x \varepsilon\_\perp}{k\_z^S \varepsilon\_\parallel} \sinh(k\_z^S z) \right] \exp[i(k\_x x - \omega t)] + c.c.\tag{63}$$

The distribution of the TM even mode normalized intensity E ! SA � � � � � � 2 =j j E<sup>0</sup> <sup>2</sup> in the MIM waveguide core is presented in Figure 9. It is seen from Figure 9 that the intensity is filling the MIM waveguide core due to the overlapping of SPP inserted from the metallic claddings z < �d; z > d. Substituting the SPP field (63) into equation of motion (9), we obtain the smectic layer normal strain [13].

$$\frac{\partial \mathbf{u}}{\partial \mathbf{z}} = \frac{\varepsilon\_0 |E\_0|^2}{B} \exp[-2(\text{Im} k\_x)\mathbf{x}] \left\{ a\_\perp \left| \cosh(k\_z^\mathcal{S} z) \right|^2 + a\_\parallel \frac{\left| k\_x \right|^2 \varepsilon\_\perp^2}{\left| k\_z^\mathcal{S} \right|^2 \varepsilon\_\parallel^2} \left| \sinh(k\_z^\mathcal{S} z) \right|^2 \right\} \tag{64}$$

The nonlinear polarization in the SALC core caused by the smectic layer strain (64) has the form [13]

Figure 9. Distribution of the SPP normalized intensity E ! SA � � � � � � 2 <sup>=</sup>j j <sup>E</sup><sup>0</sup> <sup>2</sup> in the MIM waveguide core (arbitrary units).

Nonlinear Optical Phenomena in Smectic A Liquid Crystals http://dx.doi.org/10.5772/intechopen.70997 153

$$\overrightarrow{D}\_{SA}^{NL} = \varepsilon\_0 \frac{\partial \mathfrak{u}}{\partial \mathbf{z}} \mathbb{E}\_0 \left[ \overrightarrow{a}\_x a\_\perp \cosh(k\_z^\mathrm{S} z) - \overrightarrow{a}\_z i a\_\parallel \frac{k\_x \varepsilon\_\perp}{k\_z^\mathrm{S} \varepsilon\_\parallel} \sinh(k\_z^\mathrm{S} z) \right] \exp[i(k\_x \mathbf{x} - \omega t)] + c.c. \tag{65}$$

Substituting the SPP electric field (63) and nonlinear polarization (65) into Eq. (13) and separating linear and nonlinear parts, we obtain the truncated equation for SVA E0ðÞ¼ t j j E0ð Þt expiwð Þt .

$$\begin{split} & -2i \frac{\partial E\_0}{\partial t} E\_0^\* \varepsilon\_\perp \left[ \left| \cosh(k\_z^S z) \right|^2 + \frac{|k\_x|^2 \varepsilon\_\perp}{|k\_z^S|^2 \varepsilon\_\parallel} \left| \sinh(k\_z^S z) \right|^2 \right] \\ &= \omega \frac{\varepsilon\_0 |E\_0|^4}{B} \exp[-2(\text{Im} k\_x)\mathbf{x}] \left[ a\_\perp \left| \cosh(k\_z^S z) \right|^2 + a\_\parallel \frac{|k\_x|^2 \varepsilon\_\perp^2}{|k\_z^S|^2 \varepsilon\_\parallel^2} \left| \sinh(k\_z^S z) \right|^2 \right] \end{split} \tag{66}$$

At the distances <sup>x</sup> <sup>≪</sup> Lx <sup>¼</sup> ð Þ Imkx �<sup>1</sup> , the SVA dependence on x can be neglected since exp½�2 ð Þ Imkx x� ≈ 1 [13]. The dispersion effects can be neglected because the dispersion length LD ≫ Lx [13]. Integrating both sides of Eq. (66) over z ∈½ � �d; d and separating real and imaginary parts we obtain the following equations for the magnitude j j E tð Þ and phase wð Þt of the SPP SVA.

$$\frac{\left|\partial \left|E\_{0}\right|\right|^{2}}{\partial t} = 0; \frac{\partial \varphi}{\partial t} = \omega \frac{\varepsilon\_{0} \left|E\_{0}\right|^{2} F\_{2}\left(k\_{z}^{S}, k\_{x}, d\right)}{16B} \tag{67}$$

Here,

exp �4kS

152 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

following relationships take place:

S <sup>z</sup> � <sup>10</sup><sup>4</sup>

SA ¼ E<sup>0</sup> a ! <sup>x</sup>cosh k S <sup>z</sup> <sup>z</sup> � � � <sup>a</sup> ! zi kxε<sup>⊥</sup> kS <sup>z</sup> ε<sup>∥</sup>

2

Figure 9. Distribution of the SPP normalized intensity E

m�<sup>1</sup> ≫ Imk

Z direction is given by 2π Imk<sup>S</sup>

E !

layer normal strain [13].

form [13]

∂u <sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>ε</sup>0j j <sup>E</sup><sup>0</sup>

Rek S <sup>z</sup> � <sup>10</sup><sup>6</sup> <sup>z</sup> <sup>d</sup> � � <sup>¼</sup> <sup>k</sup>

<sup>m</sup>�<sup>1</sup>, Rekx � 107

z � ��<sup>1</sup> � <sup>102</sup>

The distribution of the TM even mode normalized intensity E

<sup>B</sup> exp½ � �2 Imð Þ kx <sup>x</sup> <sup>a</sup><sup>⊥</sup> cosh <sup>k</sup>

m z εrð Þ ω þ kS z ε⊥

!<sup>2</sup>

Numerical estimations show that for the typical values of the SPP frequency ω, the plasma frequency ωp, the SPP lifetime τ mentioned above, and the MIM thickness 2d ¼ 1μm the

core with the thickness of 2d � 1μm. Then, a single localized TM can exist in the MIM wave-

guide core is presented in Figure 9. It is seen from Figure 9 that the intensity is filling the MIM waveguide core due to the overlapping of SPP inserted from the metallic claddings z < �d; z > d. Substituting the SPP field (63) into equation of motion (9), we obtain the smectic

The nonlinear polarization in the SALC core caused by the smectic layer strain (64) has the

! SA � � � � � � 2

guide according to the dispersion relation (62). The even TM mode has the form [13]

<sup>z</sup> <sup>z</sup> � � " #

8 < : <sup>m</sup>�<sup>1</sup> <sup>≫</sup> Imkx � 103

sinh k S

S <sup>z</sup> <sup>z</sup> � � � � �

� 2 þ a<sup>∥</sup>

km z <sup>ε</sup>rð Þ <sup>ω</sup> � kS z ε⊥

!�<sup>2</sup>

(62)

m�1. The SPP wavelength in the

exp½ �þ i kð Þ xx � ωt c:c: (63)

<sup>2</sup> in the MIM wave-

μm and can be neglected inside the MIM waveguide

! SA � � �

j j kx <sup>2</sup> ε2 ⊥

sinh k S <sup>z</sup> <sup>z</sup> � � � � �

<sup>=</sup>j j <sup>E</sup><sup>0</sup> <sup>2</sup> in the MIM waveguide core (arbitrary units).

� 2 9 = ;

(64)

k S z � � � � 2 ε2 ∥

� � � 2 =j j E<sup>0</sup>

F<sup>1</sup> k<sup>S</sup> <sup>z</sup> ; kx; <sup>d</sup> � � <sup>¼</sup> <sup>ε</sup><sup>⊥</sup> <sup>1</sup> <sup>þ</sup> <sup>ε</sup>⊥j j kx <sup>2</sup> ε<sup>∥</sup> k<sup>S</sup> z � � � � 2 !sinh 2 RekS z � �<sup>d</sup> � � <sup>þ</sup> <sup>1</sup> � <sup>ε</sup>⊥j j kx <sup>2</sup> ε<sup>∥</sup> kS z � � � � 2 !2 RekS z � �d " # F<sup>2</sup> k<sup>S</sup> <sup>z</sup> ; kx; <sup>d</sup> � � <sup>¼</sup> <sup>a</sup><sup>⊥</sup> <sup>þ</sup> <sup>a</sup><sup>∥</sup> ε2 <sup>⊥</sup>j j kx <sup>2</sup> ε2 <sup>∥</sup> <sup>k</sup><sup>S</sup> j j <sup>z</sup> 2 � �<sup>2</sup> sinh 4 RekS z � �d � � <sup>þ</sup><sup>8</sup> <sup>a</sup><sup>2</sup> <sup>⊥</sup> � a<sup>∥</sup> ε2 <sup>⊥</sup>j j kx <sup>2</sup> ε2 <sup>∥</sup> <sup>k</sup><sup>S</sup> j j <sup>z</sup> 2 � �<sup>2</sup> !sinh 2 Rek<sup>S</sup> z � �d � � <sup>þ</sup><sup>8</sup> <sup>1</sup> <sup>2</sup> <sup>a</sup><sup>⊥</sup> <sup>þ</sup> <sup>a</sup><sup>∥</sup> ε2 <sup>⊥</sup>j j kx <sup>2</sup> ε2 <sup>∥</sup> kS z � � � � 2 0 @ 1 A 2 þ a<sup>⊥</sup> � a<sup>∥</sup> ε2 <sup>⊥</sup>j j kx <sup>2</sup> ε2 <sup>∥</sup> kS z � � � � 2 0 @ 1 A 2 <sup>2</sup> 4 3 5 Rek S z � �d (68)

The solution of Eq. (67) has the form

$$\left|E\_{0}\right|^{2} = \text{const}; \varphi(t) = \left\{\omega \frac{\varepsilon\_{0}|E\_{0}|^{2}}{16B} \frac{F\_{2}\left(k\_{z}^{S}, k\_{x}, d\right)}{F\_{1}\left(k\_{z}^{S}, k\_{x}, d\right)}\right\} t \tag{69}$$

Eq. (69) shows that the strong SPM of the even SPP mode in the MIM wave guide occurs. It is enhanced by a large geometric factor F<sup>2</sup> k S <sup>z</sup> ; kx; <sup>d</sup> � �=F<sup>1</sup> <sup>k</sup><sup>S</sup> <sup>z</sup> ; kx; <sup>d</sup> � �, which can achieve a value of 102 � 104 for 2<sup>d</sup> <sup>¼</sup> <sup>1</sup>μ<sup>m</sup> and Re<sup>k</sup> S <sup>z</sup> � 106 � <sup>3</sup> � 106 � �m�<sup>1</sup>, Rekx � <sup>5</sup> � 106 � <sup>10</sup><sup>7</sup> � �m�1.

### 7. Conclusions

In SALC, there exists a specific mechanism of the optical nonlinearity related to the normal displacement u xð Þ ; y; z; t of smectic layers. This mechanism combines the properties of the orientational mechanism typical for LC and of the electrostrictive mechanism. In particular, the smectic layer oscillations occur without the mass density change. Under the resonant condition (11), the SS acoustic wave propagates in SALC in the direction oblique to the layer plane. The cubic nonlinearity related to this mechanism is characterized by a strong anisotropy, a short time response, a weak temperature dependence, a resonant frequency dependence, and a strong dependence on the optical wave polarization and propagation direction. The cubic susceptibility related to the smectic layer displacement is larger than the Kerr type susceptibility in ordinary organic liquids. It should be noted that the nonlinear optics of NLC has been mainly studied. However, SALC are promising candidates for nonlinear optical applications due to their low losses and higher degree of the long range order.

LC applications in nanophotonics, plasmonics, and metamaterials attracted a wide interest due to the combination of LC large nonlinearity and strong localized electric fields of SPP. Until now, NLC applications in nanophotonics and plasmonics have been investigated. We studied theoretically the nonlinear optical processes at the interface of a metal and a homeotropically oriented SALC. In such a case, SPP penetrating into SALC create the spatially localized surface dynamic grating of smectic layer normal displacement. We have shown that for optical frequencies of

cubic susceptibility may be one to two orders of magnitude larger than the cubic susceptibility of isotropic organic liquids. We solved the wave Eq. (13) for the counter-propagating SPP in SALC with the spatially localized nonlinear polarization and obtained the explicit expressions (55)–(57) for the magnitudes and phases of the coupled SPP SVA. It has been shown that the Rayleigh stimulated scattering of SPP on the surface smectic layer oscillations occurs. The rise time of the amplified SPP of about 10 ns can be achieved, which is much faster than the Brillouin relaxation

The plasmonic waveguides with NLC for nanophotonic and plasmonic have been theoretically investigated. We proposed an MIM waveguide with an SALC core. We evaluated the electric field of the strongly localized SPP even mode, the smectic layer normal strain and the nonlinear polarization in the MIM core. The evaluation of the SPP SVA shows that the strong SPM process takes place. The SPM is enhanced by the geometric factor caused by the strong

Department of Electrical Engineering and Electronics, Holon Institute of Technology (HIT),

[1] De Gennes PG, Prost J. The Physics of Liquid Crystals. 2nd ed. New York, USA: Oxford

[2] Khoo I-C. Liquid Crystals. 2nd ed. Hoboken, New Jersey, USA: Wiley; 2007. 368 p. DOI:

[3] Khoo IC. Nonlinear optics of liquid crystalline materials. Physics Reports. 2009;471:221-

[4] Khoo IC. Nonlinear optics, active plasmonics and metamaterials with liquid crystals. Progress in Quantum Electronics. 2014;38(2):77-117. DOI: 10.1016/j.pquantelec.2014.03.

1

Nonlinear Optical Phenomena in Smectic A Liquid Crystals

http://dx.doi.org/10.5772/intechopen.70997

, the SALC-metal system

155

<sup>1</sup> and coupled SPP frequency difference of about 108 s

about 1015 s

constant in NLC.

Author details

Holon, Israel

References

001

SPP localization in the MIM core.

ISBN: 978-0-471-75153-3

267. DOI: 10.1016/j.physrep.2009.01.001

Boris I. Lembrikov\*, David Ianetz and Yossef Ben Ezra

Univeristy Press; 1993. 597 p. DOI: ISBN: 978-0198517856

\*Address all correspondence to: borisle@hit.ac.il

We derived the equation of motion (9) of the smectic layer displacement u xð Þ ; y; z; t in the electric field of optical waves. We investigated theoretically the nonlinear optical phenomena in SALC based on this specific mechanism. We solved simultaneously the equation of motion (9) and the wave Eq. (13) for the optical waves including the nonlinear polarization. The solution was based on the SVAA.

In an optically uniaxial SALC, an ordinary wave and an extraordinary one can propagate. Both the ordinary and extraordinary optical beams propagating in SALC undergo self-focusing and self-trapping and form spatial solitons. The optical wave self-trapping can occur at the interface between the linear medium and SALC. We obtained the analytical solutions for the SVA of the self-trapped beams.

SLS of two arbitrary polarized optical waves in SALC transforms into the partially frequency degenerate FWM because each optical wave splits into the ordinary and extraordinary waves. The coupled optical waves create a dynamic grating of the smectic layer normal displacement u xð Þ ; y; z; t and undergo the parametric energy exchange and XPM. The signal optical waves with the lower frequency are amplified up to a saturation level determined by the Manley-Rowe relation, while the pumping optical waves with higher frequency are depleted. It has been shown that the system of the coupled optical waves and the dynamic grating is stable. The analytical expressions for the magnitudes and phases of SVA have been obtained in the limiting case when the waves are mainly polarized either perpendicular to the propagation plane, or in it. The SLS gain coefficient is significantly larger than the one in the case of the Brillouin SLS in isotropic organic liquids. The SLS in SALC also results in the generation of the Stokes and anti-Stokes harmonics with the combination wave vectors.

The nondegenerate FWM in SALC results in the amplification of the signal optical wave with the lowest frequency and depletion of three other waves with higher frequencies. The polarizationdecoupled FWM may take place when the polarizations of some optical waves are perpendicular to one another. If the coupled optical waves are counter propagating and their frequencies satisfy the balance conditions typical for OPC process then BEFWM takes place accompanied by the amplification of the phase-conjugate wave. The spectrum of the scattered harmonics consists of 24 Stokes and anti-Stokes terms with combination frequencies and wave vectors.

LC applications in nanophotonics, plasmonics, and metamaterials attracted a wide interest due to the combination of LC large nonlinearity and strong localized electric fields of SPP. Until now, NLC applications in nanophotonics and plasmonics have been investigated. We studied theoretically the nonlinear optical processes at the interface of a metal and a homeotropically oriented SALC. In such a case, SPP penetrating into SALC create the spatially localized surface dynamic grating of smectic layer normal displacement. We have shown that for optical frequencies of about 1015 s <sup>1</sup> and coupled SPP frequency difference of about 108 s 1 , the SALC-metal system cubic susceptibility may be one to two orders of magnitude larger than the cubic susceptibility of isotropic organic liquids. We solved the wave Eq. (13) for the counter-propagating SPP in SALC with the spatially localized nonlinear polarization and obtained the explicit expressions (55)–(57) for the magnitudes and phases of the coupled SPP SVA. It has been shown that the Rayleigh stimulated scattering of SPP on the surface smectic layer oscillations occurs. The rise time of the amplified SPP of about 10 ns can be achieved, which is much faster than the Brillouin relaxation constant in NLC.

The plasmonic waveguides with NLC for nanophotonic and plasmonic have been theoretically investigated. We proposed an MIM waveguide with an SALC core. We evaluated the electric field of the strongly localized SPP even mode, the smectic layer normal strain and the nonlinear polarization in the MIM core. The evaluation of the SPP SVA shows that the strong SPM process takes place. The SPM is enhanced by the geometric factor caused by the strong SPP localization in the MIM core.

### Author details

7. Conclusions

solution was based on the SVAA.

the self-trapped beams.

In SALC, there exists a specific mechanism of the optical nonlinearity related to the normal displacement u xð Þ ; y; z; t of smectic layers. This mechanism combines the properties of the orientational mechanism typical for LC and of the electrostrictive mechanism. In particular, the smectic layer oscillations occur without the mass density change. Under the resonant condition (11), the SS acoustic wave propagates in SALC in the direction oblique to the layer plane. The cubic nonlinearity related to this mechanism is characterized by a strong anisotropy, a short time response, a weak temperature dependence, a resonant frequency dependence, and a strong dependence on the optical wave polarization and propagation direction. The cubic susceptibility related to the smectic layer displacement is larger than the Kerr type susceptibility in ordinary organic liquids. It should be noted that the nonlinear optics of NLC has been mainly studied. However, SALC are promising candidates for nonlinear optical applications

We derived the equation of motion (9) of the smectic layer displacement u xð Þ ; y; z; t in the electric field of optical waves. We investigated theoretically the nonlinear optical phenomena in SALC based on this specific mechanism. We solved simultaneously the equation of motion (9) and the wave Eq. (13) for the optical waves including the nonlinear polarization. The

In an optically uniaxial SALC, an ordinary wave and an extraordinary one can propagate. Both the ordinary and extraordinary optical beams propagating in SALC undergo self-focusing and self-trapping and form spatial solitons. The optical wave self-trapping can occur at the interface between the linear medium and SALC. We obtained the analytical solutions for the SVA of

SLS of two arbitrary polarized optical waves in SALC transforms into the partially frequency degenerate FWM because each optical wave splits into the ordinary and extraordinary waves. The coupled optical waves create a dynamic grating of the smectic layer normal displacement u xð Þ ; y; z; t and undergo the parametric energy exchange and XPM. The signal optical waves with the lower frequency are amplified up to a saturation level determined by the Manley-Rowe relation, while the pumping optical waves with higher frequency are depleted. It has been shown that the system of the coupled optical waves and the dynamic grating is stable. The analytical expressions for the magnitudes and phases of SVA have been obtained in the limiting case when the waves are mainly polarized either perpendicular to the propagation plane, or in it. The SLS gain coefficient is significantly larger than the one in the case of the Brillouin SLS in isotropic organic liquids. The SLS in SALC also results in the generation of the

The nondegenerate FWM in SALC results in the amplification of the signal optical wave with the lowest frequency and depletion of three other waves with higher frequencies. The polarizationdecoupled FWM may take place when the polarizations of some optical waves are perpendicular to one another. If the coupled optical waves are counter propagating and their frequencies satisfy the balance conditions typical for OPC process then BEFWM takes place accompanied by the amplification of the phase-conjugate wave. The spectrum of the scattered harmonics consists of

due to their low losses and higher degree of the long range order.

154 Liquid Crystals - Recent Advancements in Fundamental and Device Technologies

Stokes and anti-Stokes harmonics with the combination wave vectors.

24 Stokes and anti-Stokes terms with combination frequencies and wave vectors.

Boris I. Lembrikov\*, David Ianetz and Yossef Ben Ezra

\*Address all correspondence to: borisle@hit.ac.il

Department of Electrical Engineering and Electronics, Holon Institute of Technology (HIT), Holon, Israel

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[19] Newell AC. Solitons in Mathematics and Physics. 1st ed. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics; 1985. 244 p. DOI: ISBN: 978-08-9871-1967

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**Chapter 8**

Provisional chapter

**Temperature Effects on Liquid Crystal Nonlinearity**

DOI: 10.5772/intechopen.70414

Temperature Effects on Liquid Crystal Nonlinearity

The effect of temperature variation on nonlinear refractive indices of several types of liquid crystal (LC) compounds has been reported. Five samples have been investigated: two pure components (E7, MLC 6241-000) and three mixtures are obtained by mixing the previous two in different proportions. Birefringence, the average refractive index and the temperature gradients of refractive indices of the LCs are determined. The variations in refractive indices and birefringence were fitted theoretically using the modified Vuks equation. Excellent agreement is obtained between the fitted values and experimental data. Finally, the bistability of nonlinear refractive indices with temperature of liquid crystal (LC) compounds has been studied. The bistability of liquid crystals based on temperature is clearly observed for all samples. Also, the extraordinary refractive index has larger bistability than the ordinary refractive index. The measurements are performed at 1550 nm wavelength using wedged cell refractometer method.

Keywords: extraordinary, refractive index, liquid crystal, birefringence, order parameter

Liquid crystals exhibit optical anisotropy or birefringence (Δn). This is an essential physical property of liquid crystals and is a key element in how they are implemented in the display [1, 2], photonic devices [3], communications signal processing [4], and beam steering [5].

When light propagates through anisotropic media such as liquid crystals, it will be divided into two rays which travel through the material at different velocities, and therefore have different refractive indices, the ordinary index (no), and extraordinary index (ne), and the difference is called as birefringence or double refraction (Δn=ne � no). Depending on the

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

values of ne and no, birefringence can be positive or negative [6, 7].

Lamees Abdulkaeem Al-Qurainy and

Lamees Abdulkaeem Al-Qurainy and

http://dx.doi.org/10.5772/intechopen.70414

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Kais A.M. Al Naimee

Kais A.M. Al Naimee

Abstract

1. Introduction

Provisional chapter
