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3 A STUDY OF DIELECTRIC AND ELECTRO-OPTICAL RESPONSE OF LIQUID CRYSTAL IN CONFINED SYSTEMS A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Hong Ding May, 1996

4 UMI Number: UMI Microform Copyright 1996, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103

5 Dissertation written by Hong Ding B.S., Sichuan University, 1985 M.S., Sichuan University, 1988 Ph.D., Kent State University, 1996 Approved by Co-Chairs, Doctoral Dissertation Committee Members, Doctoral Dissertation Committee Accepted by ) oaaj UJ- Chair, Department of Physics. j Dean, College of Arts and Sciences R ep rod uced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

6 TABLE OF CONTENTS ACKNOWLEGEMENT xii CHAPTER 1. INTRODUCTION: PHYSICS OF LIQUID CRYSTALS Introduction to liquid crystals Ferroelectric liquid crystals Order parameter o f LC and orientation The free energy o f liquid crystal systems 21 Reference PROPERTIES OF POLYMER DISPERSED LIQUID CRYSTAL (PDLC) Brief history o f PDLC Phase separation techniques Director configuration of PDLC droplet Optical and electric properties of PDLC 38 Reference DIELECTRIC SPECTROSCOPY Dielectric in an electric field Resonance and relaxation Relaxation processes in liquid crystal Relaxation processes in ferroelectric liquid crystals Dielectric properties of polymer Dielectric permittivities of heterogeneous systems 76 Reference ELECTRO-OPTICAL RESPONSE OF POLYMER DISPERSED LIQUID CRYSTAL (PDLC) Introduction Theoretical model 86 iii

7 4.2.1 Calculation of the free energy Calculation of the field free energy Switching fieid Relating the model to experiment Experiment Results and discussion Conclusions 125 Reference 125 DIELECTRIC RESPONSE OF LIQUID CRYSTAL IN CONFINED SYSTEMS Introduction Dielectric properties of nematic liquid crystal in pores Materials and sample preparation Experimental set-up Experiments Results and discussion Conclusion Influence o f confinement on dielectric properties of ferroelectric liquid crystal (FLC) Materials and set-up Results and discussion Conclusion 184 References 186 CONCLUSION 189 iv

8 LIST OF FIGURES Chapter 1 Figure 1.1 Figure 1.2 Figure 1.3 Figure 1.4 Figure 1.5 Figure 1.6 Figure 1.7 Figure 1.8 Figure 1.9 Schematic representation of the isotropic phase and nematic phase. The arrangement of molecules in the cholesteric mesophase. Schematic representation o f the smectic A and smectic C phase. (a) Schematic cross-section of the structure illustrating how the local layer polarization turns from layer to layer, (b) Illustration o f the linear coupling between tilt 9 and polarization P. The schematic temperature dependence of the order parameter for a nematic liquid crystal from Maier-Saupe mean field theory. a. Director fluctuation in A* phase, and its variation with temperature and (b) director fluctuation in the C* phase with constant tilt angle. Definition of coordinates and introduction of the order parameter and P. Schematic temperature dependence of the tilt angle 0 and polarization P at a second-order SmA* - SmC* transition. Physical distortion of the director field: (a) splay, (b) twist, (c) bend. Chapter 2 Figure 2.1 Illustration of the SIPS process. v

9 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Diagram illustrating the evolution of a PDLC material. through phase separation. Illustration of the TIPS process. Director configurations in a droplet of PDLC film a) radial b) axial c) bipolar d) toroidal. PDLC light shutter illustrating the opaque or scattering state with randomly oriented nematic liquid crystal droplets and the transparent state with the droplets aligned by an applied electric field. Chapter 3 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 Figure 3.7 Dielectric constants e, and e1 of nematic as a function o f temperature Schematically. Debye type relaxation for polar substances. Cole-Cole plot for the Cole-Cole equation at a = 0. Dispersion and loss curves for the Cole-Cole equation at a = 0.8 and 0.0 respectively. Relaxation processes of nematic liquid crystals. Schematic temperature dependence of dissipation peaks for the polymer. The self consistent field approximation for effective dielectric constant of heterogeneous system.

10 Chapter 4 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 4.7 Figure 4.8 Figure 4.9 Figure 4.10 Figure 4.11 Figure 4.12 Schematic representation of nematic director configuration of elliposoidal droplet in a PDLC film: (a) before switching, (b) after switching. The nematic unit director in cylindrical coordinates for a elliposoid droplet of PDLC at OFF state. The nematic director in cylindrical coordinates for a elliposoid droplet of PDLC at the ON state. Effective dielectric constant calculation. Consider a rotational ellipsoid a = b * c, a < c, a and a, are the short axis of inner and outer ellipsoid respectively. Switching field as a function of aspect ratio X for different droplet sizes. Contribution to switching field from the splay and bend elastic energy. The optical and capacitance method for the thickness measurement. Typical picture of the Scanning Electron Microscope for E7 and N65 PDLC film. Set-up for high voltage dielectric-optical measurement. The effective dielectric constant emof the PDLC film and dielectric constant ep of polymer binder as a function of temperature. Dissipation of N65&E7 mixture as a function of logarithmic frequency for different concentration of E7 at room temperature. Dissipation peaks in logarithemic frequency vs. concentration of liquid crystal E7 in the polymer. vii R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

11 Figure 4.13 Figure 4.14 Figure 4.15 Figure 4.16 Figure 4.17 The typical curves of capacitance and transmittance as a function of voltage for 50% N65&E7 PDLC film at temperature T 15 C. The dielectric constant as a function of temperature for E7 liquid crystal. Elastic constant Kn versus temperature for E7 liquid crystal. Switching field as a function of temperature for different aspect ratios at a fixed droplet size. Comparison between experimental and theoretical results on the switching field as a function temperature. Chapter 5 Figure 5.1 Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 Figure 5.6 Figure 5.7 Schematic picture of glass morphology, "worm holes": empty area pores and shadow area matrix. Experimental Set-up for the Dielectric measurement. The difference of dissipation for pore-size 100 A pore-size empty glass before the heating and after the heating at 450 C for 2 hours and measurement is taken at room temperature. Real Part of Permittivity of TL205 in Bulk sample at T=-10 C and T=+10 C. Imaginary part of permittivity of TL205 Bulk sample at T=-10 C and T=+10 C. The dependence of s real part of permittivity o f TL205 in the pores at T=-25 C. Relaxation time of TL205 in porous glasses and bulk as a function of temperature. viii

12 Figure 5.8 The Frequency shift of relaxation peaks of TL205 in porous glasses with respect to free state. Figure 5.9 Figure 5.10 Figure 5.11 Figure 5.12 Figure 5.13 Figure 5.14 Figure 5.15 Figure 5.16 Figure 5.17 Figure 5.18 Figure 5.19 Log(T) as a function of inverse of temperature for TL205 Bulk and the Activation Energy U= 0.74 (ev). Logarithmic relaxation time as a function of reciprocal o f temperature for TL205 in different pore. Activity energies for TL205 as a function of pore-size, for bulk sample U = 0.74 ev. The pore-size for saturation is about 1560 A. Potential Shapes in Anisotropic Phases. Activation Energies for TL205 LC in 500 A pores with the surface treatment and without the treatment. Relaxation times for TL205 in 500 A pores with surface treatment and without treatment. Dissipation frequency as a function of bias voltage for TL205 LC in 500 A pores at T=-15 C and dissipations are all same D = Cole-Cole Parameter as a Function of Pore-size, for Bulk sample a is Frequency dependence of real and imaginary permittivities for DOBAMBC in 1000 A pore at different temperature. The temperature dependence of soft mode dielectric strength and relaxation time for DOBAMBC in 1000A pores. The temperature dependence of soft mode dielectric strength and relaxation time for DOBAMBC in 100A pores. ix

13 Figure 5.20 Figure 5.21 Figure 5.22 Figure 5.23 Figure 5.24 Figure 5.25 Figure 5.26 Figure 5.27 Figure 5.28 Figure 5.29 The temperature dependence of soft mode relaxation frequency./j and the reciprocal of dielectric strength 1/Ae3 for DOBAMBC in 1000A pores. The temperature dependence of soft mode relaxation frequency/3 and the reciprocal of dielectric strength 1/As3for DOBAMBC in 100 A pores. The temperature dependence of the reciprocal of soft mode dielectric strength and the relaxation frequency in Sm C - Sm A phase for DOBAMBC in 1000A pores. The temperature dependence of the reciprocal of soft mode dielectric strength 1/Ae3 and relaxation frequency/3 for DOBAMBC in Sm C - Sm A phase at 100A pores. The temperature dependence of the soft mode rotational viscosity rje and nature logarithmic rotational viscosity for DOBAMBC in 1000A pores. The temperature dependence of soft mode rotational viscosity qe and nature logarithmic rotational viscosity for DOBAMBC in 100 A pores. The temperature dependence of Cole-Cole parameter for DOBAMBC in 1000A and 100A pores. The relaxation time t, of first dielectric relaxation as a function of temperature for DOBAMBC in 100 A and 1000 A pores. Dielectric strength Ae, of first relaxation as a function temperature for DOBAMBC in 100 A and 1000 A pores. Temperature dependence of first relaxation time t, of DOBAMBC in 100 A and 1000 A pores. x

14 Figure 5.30 Figure 5.31 Figure 5.32 Relaxation times x2 of DOBAMBC in 100 A and 1000 A pores as a function of temperature. The ln(xj as a function of reciprocal of temperature for DOBAMBC in 100 A and 1000 A pores. The dielectric strength As, as a function of temperature for DOBAMBC in 100 A and 1000 A pores.

15 ACKNOWLEDGMENTS I owe a great deal to my thesis advisors: Drs. Jack R. Kelly and J. William Doane for their guidance and supervision through these years. Jack deserves special thanks for always encouraging me and persistently keep me on the track with astute questions and observations. Jack's enthusiasm and guidance have made this thesis possible. He has taught me much about the process of doing research as well. I wish to thank various members of people at Physics Department and Liquid Crystal Institute. In particular, Professor A. Saupe, J. West, O. Lavrentovich, D. Allender, and M. Groom. Dr. Saupe has been an inspiration and a friend over several years. Thanks also to Professor Fouad Aliev at University of Puerto Rico, who opened his laboratories and gave of his time for many of experiments in this thesis. Next, I would like to express my special appreciation to Tom Buer for his caring and friendship which made the closing chapter of my life in graduate school vastly more endurable. Finally, to my family I extend my most heartfelt thanks for their encouragement and unwavering support which have been a constant source o f strength in my life. This work is dedicated to them, especially to my little boy Bill, without their love it never would have been completed. xii

16 CHAPTER ONE INTRODUCTION: PHYSICS OF LIQUID CRYSTALS Liquid crystals combine the properties of a solid and of an isotropic liquid. On one hand, they can flow like ordinary liquids; on the other hand, they are orientationally ordered, and exhibit anisotropic behavior as seen in their electrical, magnetic and optical properties. The dielectric properties of a liquid crystal depend on the molecular orientation. Molecular orientation by an applied field changes the dielectric properties of the liquid crystal system, and hence changes light propagation in liquid crystal. Therefore liquid crystals are promising materials for optical and display devices. In this chapter, we will review some physical properties of liquid crystals, and only focus on those aspects of liquid crystals which are essential for an understanding of the following chapters: what are liquid crystal materials? what are the primary thermotropic phases of liquid crystals (nematic, smectic A, smectic A*, smectic C, and smectic C*) What are the properties of each phase? And what physical quantities we can use to describe each phase?

17 1.1 Introduction to liquid crystals Liquid crystals are a state of matter intermediate between the solid crystalline phase and the isotropic liquid phase, and combine the properties o f a solid and isotropic liquid. They possess many of the mechanical properties of a liquid; for example, they can flow like an ordinary liquid. On the other hand, they are similar to crystals in that they exhibit anisotropy in their optical, electrical and magnetic properties. Liquid crystal mesophases can be observed in certain organic compounds, and are usually composed of elongated molecules. Liquid crystals can be divided into two groups: thermotropic and lyotropic, based on whether the phase behavior is induced thermally (thermotropic) or by the influence of solvents (lyotropic). When a substance which shows a thermotropic liquid crystalline phase is heated, the system may pass through one or more mesophases before it transforms from the crystal phase into the isotropic liquid. The melting point and the clearing point define the temperature range of the mesophases. In liquid crystalline mesophases, the molecules show some degree of orientational order (and in some cases partial translational order as well) even though a 3 -D crystal lattice does not exist. Therefore these phases are often called ordered fluid phases. In this dissertation we will focus our discussion on thermotropic liquid crystals. According to the molecular arrangement and ordering, thermotropic liquid crystals can be further classified into the following types: isotropic, nematic, cholesteric and smectic (however, the cholesteric is usually considered as a modified form of the nematic). On the temperature scale the liquid crystalline phases appear as following:

18 Crystalline solid Liquid crystalline phase Isotropic liquid smetic nematic(or Cholesteric) t melting point t clearing point The high temperature phase of a liquid crystal is called the isotropic phase; it is characterized by optical, electrical, and mechanical properties which are independent of orientation. In the isotropic phase the molecules possess neither orientational nor positional order, and the spatial average of the molecule orientational director a is zero, where a defined as the unit vector along axis o f the molecule. However, locally the molecules exhibit a certain degree of ordering and the two point correlation function <a(0)a(r)> is non-zero for sufficiently small r which is characterized by an isotropic correlation length E, (less than looa, also temperature dependent) and decays exponentially'. As the temperature decreases and the molecules begin to correlate their motion over longer length scales, a first order transition takes place at the isotropic-nematic transition temperature TS1, which brings the system into an anisotropic state nematic phase. Below r Nn, a certain degree of orientational long-range order exists in which the director n points, on the average, in some particular direction. The nematic phase is the least ordered liquid crystalline phase, being characterized by a high degree of long range

19 4 Isotropic Phase n(r) points along the direction of the long axis of the molecule at r Nematic Phase i m m m? m m m ^ 0(r) is the angular deviation of n(r) from n Figure 1.1 Schematic representation of the isotropic phase and nematic phase.

20 \\ WV \../ n X Figure 1.2 The arrangement of molecules in the cholesteric mesophase.

21 6 orientational order as shown in Figure 1.1, but no long range translational order. The molecules of a nematic liquid crystal tend to be parallel on the average, to some common direction. A unit vector in this preferred direction is called the nematic director n. The parallel and perpendicular components of the macroscopic properties of the nematic with respect to the director exhibit different values. If the values of two components perpendicular to each other as well as to the director are the same (complete rotational symmetry around the director n), then the nematic is called a uniaxial phase, Otherwise, it is called a biaxial phase2. Thermotropic nematic liquid crystals are usually uniaxial. A distorted form of the nematic phase is the cholesteric mesophase, which is caused by chiral molecules. The cholesteric phase is similar to the nematic phase in having long range orientational order and no long range translational order as shown in Figure 1.2. It differs from the nematic phase in that the cholesteric director varies with a helical form throughout the medium with a spatial period L-PI2=kI \ q01 where P is the pitch and q0 is the wave vector. The sign of q0 distinguishes between left and right helices and its magnitude determines the spatial periodicity. In fact, a nematic can be viewed as a cholesteric o f infinite pitch. If the temperature is lowered further from the nematic phase, the system can enter another ordered phase in which a certain amount of translational order is introduced, this is the smectic phase. The smectic phase has not only long range orientational order but also partial long range translational order. As many as eight smectic phases have been identified3. Here we

22 7 just give a few examples of relevance in later sections. In the smectic A phase the molecules are aligned perpendicular to the layers to form a one-dimension periodic structure with no long range translational order within the layer. This can be considered as a two dimensional fluid with an orientational order. Thus, the layers are individually fluid and inter-layer diffusion can occur, although with somewhat lower probability. The layer thickness, determined from x-ray scattering data, is essentially identical to the full molecular length in most cases. At thermal equilibrium the smectic A phase is optically uniaxial due to the infinite-fold rotational symmetry about an axis parallel to the layer normal. A schematic representation of smectic A order is shown in Figure 1.3a. The major characteristics o f smectic A phase are as follows: (a) A layered structure (with layer thickness close to the full length of the constituent molecules for the ordinary smectic A phase). There is a quasi long range order perpendicular to the layer. (b) Inside each layer, the centers of mass show no long range order, each layer is a two dimensional liquid. (c) The system is optically uniaxial, the optical axis being the normal OZ to the plane of the layer. (d) The directions Z and -Z are equivalent. The symmetry of the smectic A phase is Dm, which means the phase can not be distinguished for chiral and racemate materials. The requirement of constant interlayer

23 8 Smectic A Phase y 1 (a) Smectic C Phase IMMUllULl (b) Layer normal A Qrp n Layer Figure 1.3 Schematic representation of the smectic A and smectic C phase.

24 9 spacing imposes the condition curl(ri)=0 for all macroscopic deformations of a perfect smectic. Therefore the helix structure, which has curl(n)=qn^0 is forbidden. Strictly speaking, a smectic A made of chiral molecules should be labeled SA*, and considered as a phase different from the standard smectic A, with symmetry. Indeed, the macroscopic properties of SA and SA* are not equivalent: in the SA* phase, rotatory power and electro-clinic effect exist but not in SA. As the temperature is lowered still further from smectic A, another phase transition may take place in which the molecules retain their layered structure but undergo a tilt 6 with respect to the layer normal, this is called the smectic C phase. The projection of the average molecular long axis director in the layer plane is a 2D vector called the C director. Smectic C order is depicted in Figure 1.3b. X-ray scattering data from several smectic C phases indicate a layer thickness significantly less than the molecular length. This has been interpreted as evidence for a uniform tilting of the molecular axes with respect to the layer normal. The fact that the smectic C phase is optically biaxial is further evidence in support of a tilt angle. Tilt angles of up to 45 have been observed and in some materials the tilt angle has been found to be temperature dependent. The structure of a smectic C is defined as follows: (a) Each layer is still a two dimensional liquid. (b) The material is optically biaxial, the symmetry is specified by the point group

25 10 (a> \\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\\\\\\ /////////////////77777T7 ////////////////////// lllliiiiiniiiiiiiiiilii \\\\\\m iu \v \l\\\l\\ij\'\' \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ w \ \ \ \ \ \ \ Figure 1.4 (a) Schematic cross-section of the structure illustrating how the local layer polarization turns from layer to layer, (b) Illustration of the linear coupling between tilt 0 and polarization P.

26 II The stability of smectic A is governed by an elastic constant which keeps the long molecular axis parallel to the smectic layer normal. However, due to the thermal energy the director fluctuates, giving locally an instantaneous tilt angle between the director and smectic layer normal as shown in Figure 1.6a. When the temperature approaches the A-C transition temperature Tc, the elastic constant controlling the tilt fluctuation gets soft. Thus the fluctuation amplitude increases drastically and its susceptibility diverges at Tc. In the C phase, the tilt angle increases with decreasing temperature. Deep in the C phase, tilt becomes more stable against thermal fluctuations. So the soft mode can be seen close to C - A phase transition, but is suppressed as the temperature moves away from the transition in either direction as shown in Figure 1.6a. 1.2 Ferroelectric Liquid Crystals The smectic C phase produced by molecule of pure chirality is the smectic C phase. Because of the chirality of the molecules, the mirror plane o f symmetry in the Sm C, defined by the molecular director and the normal to smectic layer is broken. The symmetry of the system reduces to C, allowing a net local polarization to exist as shown in Figure 1.4. Ferroelectricity in liquid crystal, i.e. the existence of a spontaneous electric dipole moment in Sm C* phase, was predicted by Meyer et al in In the smectic C* phase molecule orientational director makes an angle with the smectic layer normal, and it often R ep rod uced with perm ission of the copyright ow ner. Further reproduction prohibited without perm ission.

27 12 precesses with a finite phase angle cp from one layer to another resulting in a helical structure with helical axis parallel to the normal to the smectic layer. Ferroelectric liquid crystals have following general properties: (a) The ground state of the chiral smectic C is a helicoidal structure in which the molecules within one layer are tilted uniformly but the direction of tilt precesses around the normal to the smectic layers as one goes from one layer to another to form a helix. (b) A polarization P exists in the plane of the layers which scales with director tilt angle 0 for small 0 (for large 0 higher order terms need to be considered). The precession angle for the tilt is the azimuthal angle (p. The pitch of the helix is typically a few microns4-5, so that in a thick sample, the quantity <0exp(i(p)> averages to zero. So does the net value of <P>, since the magnitude of polarization is proportional to 0exp(icp). To study the polarization and other properties of ferroelectricity, the helix must often be unwound by external fields ( either electric or magnetic field) to exhibit a net polarization. In the Fig. 1.4 for 0 > 0 we have chosen P to point into the paper. If we rotate it around the cone to the opposite position, corresponding to a change of sign in 0, then we see that P has reversed its direction, now pointing out of paper. Thus, a change of sign in 0 corresponds to a sign reversal of P, and from this it follows that, at least for small 0, P and 0 must be linearly related, or

28 13 P = Pokxn or P = PosinGG = PqQQ ( 1. 1 ) where k is layer normal direction and n is molecular direction. We can only expect a linear relation between P and 0 to hold true over a limited interval. The convention now adopted for the sign of P can be stated extremely simply: start from the smectic A* situation with the director n parallel to the layer normal k. Then let n tilt by 9 corresponding to the C* phase. Then the polarization P(~ 0), is said to be positive if it follows the direction of a right-handed screw when we rotate n out from k, i.e., in the 6 direction (direction kxn) as show in Figure 1.4b. The sign of P has been found negative in most synthesized molecules so far. 1.3 Order parameter of LC and orientation In order to describe the orientational order of a nematic liquid crystal, one should take into account two aspects: (1) the local preferred direction n, and (2) the degree of orientational order. Therefore the orientational properties of liquid crystals are described by a second rank tensor Q with Cartesian components 0 X] (i,j=x,y,z). For a molecule of arbitrary shape, the element of the tensor order parameter Ot] can be written in the generalized form6 Q,j = - < 3 a,a; - 5tJ > ( 1.2 )

29 14 Here we define a unit vector a(r) to describe the orientation of the symmetry axis of a molecule at position site r, and ij=x, y, z refer to the space-fixed axes. 5n is the Kronecker delta, and the brackets < > denote a volume average. The tensor Ot] is symmetric and traceless. It vanishes in the isotropic phase and thus serves as an order parameter. In an ordered nematic phase, Q usually has uniaxial symmetry. The symmetry axis is defined by the eignvector of Oip i.e. n(r), corresponding to the only non-degenerate eignvalue. By choosing a proper coordinate system, i.e., the principal axis frame, Qt] can be expressed in a diagonal form1 Q= S2 2 o 0 0 S33 (1.3) where 533 = - (5U + 5,. ). In the uniaxial nematic case, 5U= S2Z. By choosing the z-axis parallel to the nematic director, 0 has the form Q= - f f (1.4) where 5 is the scalar order parameter, which describes the degree of the orientational order of the molecules. 5 is defined as7 5 = ^(3 < cos20 > - 1) (15)

30 S (order parameter) j i j Temperature (K) Figure 1.5 The schematic temperature dependence of the order parameter for a nematic liquid crystal from Maier-Saupe mean field theory.

31 16 where 9 is the angle between the direction of molecule symmetry axis and the nematic director. Clearly, S = 1 for the fully oriented nematic phase and S = 0 for the randomly distributed isotropic phase. The scalar order parameter S is a function of the temperature. The schematically temperature dependence of order parameter S is shown schematically in Figure 1.5. In general, the nematic director n and order parameter S are spatially varying quantities. The molecular ordering at every spatial point r is characterized by a director n(r) pointing along the local axis of uniaxial symmetry and by a quantity S(r) giving the local orientational order of the molecules. Thus the tensor order parameter Q(J can be written as Qij(x) = T ^3n,(r)rt/r) - 5;;), i j = x,y,z (1.5) The directions n(r) and -n(r) correspond to physically equivalent states due to symmetry, but Otj and - 0 tj correspond to physically different states. In the smectic A phase, besides the order parameter above, we can introduce another order parameter which describes the periodic layer structure, namely the center of mass density function along the layer normal, p(z). In the majority of cases, the smectic C phase appears when cooling an A phase. In such cases, the transition can be continuous and as a first approximation can be described by a single order parameter 0 characterizing the appearance of the molecular tilt at the

32 17 tilt fluctuation instantaneous tilt angle c o CO 3 o 3 53 C* phase A* phase (a) (p large fluctuation small fluctuation spontaneous tilt angle (b) Figure 1.6 (a) Director fluctuation in A* phase, and its variation with temperature and (b) director fluctuation in the C* phase with constant tilt angle.

33 18 smectic layer Figure 1.7 Definition o f coordinates and introduction of the order parameter ^ and P.

34 19 transition temperature Tc. On entering the C phase the system of molecules must tilt. By symmetry, there are infinitely many tilt planes, and evidently we have the case of a continuous degeneracy in the sense that if all molecules would tilt in the same direction, given by the azimuthal angle cp, the chosen value of cp would not affect the free energy. The complete order parameter thus has to have two components, reflecting both the magnitude of the tilt 0 and its direction (p in space and can conveniently be written in complex form : = 0e,(p (1.6) Figures 1.6b and 1.7 illustrate the two-component order parameter ^=0e"* describing the smectic A-C transition. The phase variable cp is a so-called gauge variable and is fundamentally different from 0. The latter is a "hard" variable with relatively small fluctuations around its thermodynamically determined value <0> (its changes are connected to compression or dilation of the smectic layer, thus requiring a considerable elastic energy), whereas the phase angle cp has no thermodynamically predetermined value at all. The result is that we find large thermal fluctuations in cp around the cone for long wavelengths compared to a molecular scale; indeed, this mode gives rise strong scattering of visible light. The very easily excitable cone motion, sometimes called the spin mode or the Goldstone mode, is also the motion that can most easily be induced by an applied electric field E in the case of the ferroelectric C* phase where E couples to P. The tilt or

35 2 0 P(T) 0(T) Temperature (K) Figure 1.8 Schematic temperature dependence of the tilt angle 9 and polarization P at a second-order SmA* - SmC* transition.

36 21 "soft" mode, is "hard" to excite in comparison with the cone mode, except at the A-C transition. For T * TAC, the two motions can be considered as essentially independent of each other; tilt mode is important for T %TAC, cone mode for T < TAC. Although, in a typical solid ferroelectric, polarization is the natural order parameter, in a LC the primary transition is in the tilting of the molecules, and P is only a secondary effect of this tilting. Thus, the tilt 0 is the primary order parameter and polarization is a secondary order parameter. Figure 1.8 shows the temperature dependence of order parameters 0 and P at a second-order SmA* - SmC* transition. 1.4 The free energy o f liquid crystal systems In an ideal nematic single crystal, the molecules are aligned on average along one common direction n. However, in most practical cases, because of thermal fluctuations, the limited surface of the sample, or external fields, this ideal conformation will not exist. For a uniaxial nematic liquid crystal, the orientational order is described by a director field n and order parameter S. In an equilibrium state, the system has minimum free energy. However, when the system is perturbed by some external factors such as an external field or various surface constraints, the free energy associated with the distortion in the director field will be added to the system. If the distance of significant variations of motion is large compared to the molecular scale, one can describe deformations of nematic by a continuum theory. Such a theory was first enunciated by Oseen and Zocher9, and perfected

37 2 2 by Frank10. In Frank theory 5 is assumed to be a constant. The contribution to the free energy density due to the distortion of the director field can be written in vector notation for an arbitrary deformation as9: F= ^ { K X(V n)2 + ^ 2(n (V x n))2 + ^ 3(n x (V x n))2} (1.6) where, K2, and K2 are the Frank elastic constants which depend on S and temperature T. The K{, K2, and correspond to splay, twist, and bend deformations respectively as shown in Figure 1.9. In 1937 Landau1112 proposed to describe second order phase transition phenomena near the transition point, where the order parameter is small, in terms of an expression of the free energy density in a power series in the order parameter. De Gennes1generalized Landau theory to include the first-order nematic-isotropic transition, where the order parameter is not small. The de Gennes theory is qualitatively correct, but it provides a satisfactory description of many properties of the system at the phase transition. The expansion of the Landau-de Gennes free energy in powers of the tensor order parameter On for a uniform uniaxial nematic is given by (in the absence of external fields)1 f= fo(t ) + ^A O u(r)qj,(r) + i Q v(r)oyfc(r) fa(r) + ^Q 0?(r)0 (r)]2 + 0(Q 5) (1.7)

38 23 Twist Figure 1.9 Physical distortion o f the director field: (a) splay, (b) twist, (c) bend.

39 2 4 where f 0(T) represents the free energy density of the isotropic phase; A=a0(T - T*) where P is a temperature slightly below the nematic-isotropic transition temperature Tc and aq>0; B, C are coefficients weakly dependent temperature T. i, j, k =1, 2, 3 denote the components along the three orthogonal axes of the coordinate system. For bulk nematics, by substituting equation (1.5) for Q ^r) into equation (1.7) we obtain13 F = f- M T ) = \ a 0( T - r ) ^ ( r ) + ± 5 S 3(r) + S \ r) (1.8) We can extend these ideas to the free energy of the smectic C phase. If we consider the tilt angle 0 as the order parameter for the smectic C phase, we can write the smectic C free energy density F in terms of a Landau expansion in powers of 0. This expansion can not contain odd powers of 0 because in the absence of any internal structure along the smectic layer the free energy must be independent of the sign of the tilt (±0). Hence, we may write, F = Fo + ±a02 + ^>e4 ++c (1.9) Here, for a second-order A-C transition, the general case, b is greater than zero and only weakly depends on temperature, whereas a has to change sign in order to allow the transition. Thus, with a=a(t-tc) as the simplest choice,

40 25 F = F 0 + i a ( T - r c)02 + i be4 + ±cq (1.10) A first-order /1-C transition occurs if b < 0 (Above discussion of phase transition is also applied to nematic Landau coefficients). As explained previously in the smectic C* phase, there are two order parameters c, and P. is a two-component tilt vector order parameter =(,, ^ as shown in Figure 1.9. The free energy density must therefore be expanded in both order parameters, subject to the restriction that only the different powers or power combinations which are invariant to the symmetry operations of the C* phase be retained. We will introduce a bilinear term, i.e., a term P, to take into account coupling between order parameters. =( 1, 2) related to our earlier order parameter as = 0e'<p = (0 cos (p, 0 sin cp) = ( ^ 1. ^ 2) ( 111) In the following, the helix axis is taken to be along the z-direction with the smectic layers parallel to the xy-plane. If we neglect, for simplicity, any in-plane variations in the director n(x, y, z), thus only derivatives dldz have to be considered. The free energy density can be written as ( 1. 12)

41 2 6 where e is a generalized permittivity, assumed positive and constant. The term C(PX^2 - Py,) takes into account the fact that the coupling between P and 6 is chiral in character (without chirality, finite tilt will not result in a polarization) and is a piezoelectric coefficient. This quantity changes sign when we change from a right- to a left-handed reference frame, which means that the optical antipode of a certain C* compound will have a polarization of opposite sign. The term preceded by the proportionality factor A has the same symmetry properties, and responsible for helicoidal structure. Both C and A vanish for nonchiral or racemic materials. The Frank elastic modulus is denoted by corresponds to a twist, and p is called the flexoelectric coefficient. We will come back to this Classic-Landau free energy expression in chapter three to discuss some thermodynamic properties and dynamic modes of ferroelectric liquid crystals. In order to determine the behavior of liquid crystals in the presence of fields, one has to consider the free energy associated with the external fields. In a static electric field, the free energy density1 for any system is given by Fe = ±jd E = -yso op a- p (113) where D is electric displacement and E is the applied electric field. Here "+" corresponds to constant charge and is for constant potential. Similarly, the free energy density in the presence of a magnetic field in SI unit is F m = - j B H = PoPaptfatfp (1.14)

42 2 7 B is the magnetic induction, H is the applied magnetic field and (i is the magnetic permeability and a > 1 for paramagnetic substances; p. < 1 for diamagnetic. R eference; 1. P.G. de Gennes, The physics of liquid crystals, Clarendon press, Oxford L. J. Yu and A. Saupe, Phys. Rev. Lett., 45, 1000(1980). 3. H. Sackmann and D. Demus, Fortschr, Chem. Forsch., 12,349(1969). 4. R.B. Meyer, L. Liebert, L. Strzeleki, and P. Keller, J. Phys. (Paris). Lett., 36X69(1975) 5. P. Martinot-Lagarde, J. Phys. (Paris) Lett., 38 L I7(1977). 6. A. Saupe, Z. Naturforsch. 19a, 161(1964). 7 E.F. Gramsbergen, L. Longa, and W. H. De Jeu, Phys. Rep. 135, 195(1986). 8. J.W. Goodby and R. Blinc, etc. Ferroelectric Liquid Crystal: Principles, Properties and Applications, Gordon and Breach Science Publishers, C. Oseen, Trans. Faraday Soc., 29, 883(1933). 10. F.C. Frank, Disc. Faraday Soc., 25, 19(1958). 11. L. D. Landau, On the Theory of Phase Transition, Part I and Part II, collected papers o f L. D. Landau, edited by D. ter Haar, Gordon and Breach, Science Pulishers, N.Y., 2nd Edition, 193(1967).

43 L.D. Landau and E.M. Lifshitz, "Electrodynamics o f Continous Media", Pergamon Press, NY, (1960). 13. E.B. Priestley, P.J. Wojtowicz, and P. Sheng, Introduction to Liquid Crystal, 1979.

44 CHAPTER TWO PROPERTIES OF POLYMER DISPERSED LIQUID CRYSTAL (PDLC) Polymer dispersed liquid crystals (PDLC) form a relatively new class of a wide variety of materials used in many types o f displays, switchable windows and other light shutter devices1'2, and especially for fabrication of large scale flexible displays. The use of PDLC films overcomes the two major problems normally encountered in display technology: liquid crystal fluidity and need for light polarizers. In this chapter we give a brief introduction to PDLCs about their history, techniques of phase separation, director configurations inside a PDLC droplet, and electro-optical properties of PDLC film. This serves as a background to chapter four, where we will focus on the detailed electro-optical properties and switching mechanism of PDLC films. 2.1 Brief history of PDLC In a patent application published in 1976, Hilsum describes what is perhaps the first light shutter device in which a nematic liquid is dispersed with a second medium to induce light scattering that can be electrically controlled3. He patented a device with 29

45 3 0 dispersions of glass spheres in a liquid crystal material. By electrically controlling the birefringence of the liquid crystal he was able to match the refractive index of the glass and liquid crystal. A similar principle was used by a group at Bell Labs where liquid crystal was filled into microporous filters. In 1982 Craighead et al, published a device where the second medium is a polymer4. They made use of a microporous filter, filling the micron-size pores with a nematic liquid crystal of positive dielectric anisotropy and sandwiching the film between ITO coated substrates. However the contrast ratio, which is a measure of how opaque the device is in the off state to how transparent it is in the on state, was poor for both devices and they were never commercialized. In 1983 Fergason succeeded in developing a film with micron size nematic liquid crystal droplets dispersed in a polymer matrix. Initially PVA (Poly Vinyl Alcohol)\ a water soluble polymer, is mixed with water to form a homogeneous solution. An emulsion of liquid crystal and the homogeneous polymer solution is then made. The water is evaporated, resulting in the forming of liquid crystal droplets in the PVA matrix film. In 1984 phase separation methods were developed at Kent State University to make such films. The process begins with a homogeneous solution of polymer or a prepolymer and a low molecular weight liquid crystal. Phase separation is then induced thermally, through polymerization or by solvent evaporation6. The results in the formation of droplets which grow in size until the polymer solidifies. Since these phase separation

46 31 procedures can be applied to a broad range of polymers including thermoplastics, thermoset polymer and UV-curable polymers, a wide variety of systems can be developed. 2.2 Phase separation techniques The phase separation techniques used to fabricate PDLC films can be classified into three main types: the SIPS (Solvent Induced Phase Separation), the PEPS (Polymerization Induced Phase Separation) and TIPS (Thermally Induced Phase Separation) processes6. The SIPS process is illustrated in Figure 2.1, a low molecular weight liquid crystal and a prepolymer are dissolved in a common solvent forming a homogeneous solution represented by point I. As the solvent evaporates the solution crosses the miscibility gap At this point labeled M, the liquid crystal becomes immiscible and phase separation occurs as liquid crystal droplets begin to form. The droplets continue to grow until the gelation of the polymer occurs with two phases finally reaching their equilibrium concentrations at points A and B. If the process is not quasistatic, the equilibrium concentration (A and B) are reached via point F. In this process the droplet size and droplet density can be controlled by the rate of evaporation. The PIPS process is a time dependent process. The different stages are illustrated in Figure 2.2. The liquid crystal is dissolved in a prepolymer and curing agent. The polymerization process is then induced thermally or photochemically (UV radiation). As

47 POLYMER Homogeneous Solution Miscibility Gap SOLVENT LIQUID CRYSTAL Figure 2.1 Illustration of the SIPS process.

48 mix i immiscibility I gelation homogeneous droplet droplet solution formation purification i final set i Time Figure 2.2 Diagram illustrating the evolution of a PDLC material through phase separation.

49 3 4 time increases the polymerization process results in the liquid crystal becoming increasingly immiscible in the polymer. After a certain period of time phase separation occurs and the droplets are formed. The droplet grow in size as the liquid crystal continues to phase separate out of the polymer. The growth of the droplets ceases with the gelation of the polymer. The polymerization process continues until the final cure. The rate of polymerization is controlled by the cure temperature for thermal process and light intensity for photopolymerization. The droplet size is controlled by the rate of polymerization and other factors such as the solubility of the liquid crystal, diffusion coefficients and the rate o f the chemical reaction. The TIPS process is useful for thermoplastics which melt below their decomposition temperature. In this process a binary mixture of polymer and liquid crystal forms a homogeneous solution at elevated temperatures. A typical path for the phase separation process is shown in Fig 2.3. Starting at point I, as the temperature decreases the miscibility gap is crossed at point M. Phase separation occurs and droplets begin to form. If the process is quaistatic, two phase will eventually form and have concentrations represented by points A and B. The droplets continue to grow until the polymer hardens. If the process is not quasistatic, the equilibrium concentrations represented by points A and B can also be reached via point P through rapid cooling. As in the PIPS case, the solution may also include other chemical agents to adjust the elecro-optical performance of the display, adhesion to the substrate, etc. Cooling the homogeneous solution into the miscibility gap causes phase separation of the liquid crystal. The droplet size is controlled

50 <L> u. 3 2 Homogeneous Solution Miscibility Gap Concentration Figure 2.3 Illustration of the TIPS process.

51 3 6 by the rate of cooling and depends upon a number of material parameters, which include viscosity, chemical potential, etc. In general, larger concentrations of liquid crystals are required for these films as compared to phase separation by polymerization. 2.3 Director configurations of PDLC droplet Confining the liquid crystal in a droplet results in a particular director configuration. This configuration depends on a number of factors: how the molecules are anchored at the droplet wall, droplet size and shape, elastic constants of the liquid crystal and the direction and magnitude of any applied electric or magnetic field. Minimization of the free energy o f the droplet determines the director configuration in a droplet. Four director configurations have been observed7'9 as shown in Fig 2.4 Radial and axial configurations are observed when the molecules have perpendicular (homotropic) anchoring. Bipolar and toroidal configurations are observed when the molecules have tangential (homogeneous) anchoring. With the application of an electric or magnetic field, either a configuration transformation or reorientation can occur. An example of a configuration transformation is the radial to axial transformation9. A sufficiently high field will transform the radial configuration to axial configuration with the symmetry axis of the axial configuration aligning along the field direction. Configuration reorientation is often observed in bipolar droplets. The application of a field rotates the symmetry axis of the

52 3 7 (a) (b) (c) (d) Figure 2.4 Director configurations in a droplet of PDLC film a) radial b) axial c) bipolar d) toroidal

53 3 8 bipolar director configuration into the field direction without changing the configuration substantially. Typically the configuration o f the droplet in most devices is bipolar because they give the best scattering. For the PDLC systems studied in this work the configuration in the droplet is bipolar. 2.4 Optical and electric properties o f PDLC Figure 2.3 shows schematically how a PDLC film works. A PDLC display operates on the principle of electrically controlled light scattering. The nematic liquid crystal in the droplet is optically uniaxial with the optical axis parallel to the nematic director. It is birefringent with an ordinary refractive index n0 for light polarized perpendicular to the nematic director, and an extraordinary refractive index ne corresponding to the polarization parallel to the nematic director. The polymer matrix is typically isotropic with a single refractive index nm. In making the film the polymer refractive index nm is closely matched to the ordinary refractive index of the liquid crystal n0. In addition, a liquid crystal with positive dielectric anisotropy is often used so that the nematic symmetry axis aligns with field direction. The refractive index that incident light probes is dependent upon the director configuration in the droplet and angle between the optical axis (nematic director) and the direction of the incident light. For light polarized in the plane defined by the nematic

54 3 9 : t *... I - 'V ' * # 0 Is 4 OFF STATE ON STATE o Incident s Scattered t transitted Figure 2.5 PDLC light shutter illustrating the opaque or scattering state with randomly oriented nematic liquid crystal droplets and the transparent state with the droplets aligned by an applied electric field.

55 4 0 director n and the incident wavevector k, the refractive index at angle 4>(determined from coaj) = n»k) is10 (2. 1) In the absence of a field the average nematic director in a droplet varies from droplet to droplet and there is a wide distribution of n(<j>). The incident light probes a range of refractive indices from ndto ne and light is strongly scattered causing the film to appear opaque. With application of an electric field E the nematic symmetry axis reorients parallel to the field. The incident light probes a single refractive index (the ordinary index): «(<j>) = fio * n m (2.2) and the scattering is minimized, causing the film to appear transparent. The electro-optical properties of PDLC films are controlled by the types of materials used, the droplet morphology and the method of the film construction. Desirable properties include high clarity and transmission of the film on the ON and OFF states, low driving voltage, low power consumption, fast switching times and high film resistance. All these properties are related to each other, so it is usually not possible to change them independently.

56 41 In order to quantify the optical performance of a the PDLC film, we introduce the clarity and transmission The clarity is a measure of the sharpness of an image viewed through a film, and the transmission is a measure of the efficiency o f the light passage through the film. The transmission through the film is defined as the intensity of light transmitted by a film divided by the incident light intensity. The clarity is defined as the intensity of the light transmitted unscattered divided by the total light transmitted. It can be measured with a haze meter or with an integrating sphere. The clarity of a PDLC film in the ON state depends on the match of n0 and nm. The closer the match, the clearer the film in the ON state. This is usually achieved by precisely adjusting the refractive index of the matrix nm. On the other hand, liquid crystal dissolved in the binder of a PDLC varies in its refractive index12. Also, the effective n0 of the droplet is not precisely equal to ndof the bulk liquid crystal because the alignment is not parallel throughout the droplet. The OFF state clarity and transmission are determined by the size and density of the droplet and birefringence of the liquid crystal. Maximum scattering and therefore minimum transmission and clarity are achieved when the droplet size and spacing is on the order of the wavelength of light. Highly birefringent liquid crystals offer the largest mismatch of the refractive indices in the OFF state. Thicker films are also more scattering; however they also reduce the clarity in the ON state, and require higher switching fields. The refractive index match of the liquid crystal and the polymer is also temperature dependent. Because n0 tends to increase with temperature while nm tends to decrease, it

57 4 2 is usually not possible to have an exact match over the entire operating temperature range of the film13. One of the important parameters associated with a PDLC film is its driving voltage. For a perfectly spherical droplet, no elastic distortion is required to align a bipolar droplet with an electric field. In practice the droplets in PDLC materials are never perfectly spherical and the random orientation of bipolar droplets in a PDLC film is caused by a distribution in the shapes and orientations of slightly elongated droplets. The driving voltage is dependent on a variety of factors, such as dielectric properties, director configuration, droplet shape etc. We will discuss this in more detail in Chapter 4. References: 1. J.W. Doane, A. Golemme, J.L. West, J.B. Whitehead, Jr., and B.G. Wu, Mol. Liq. Cryst (1988). 2. Paul S. Drzaic, J. Appl. Phys. 60, 22142(1986). 3. Hilsum, U.K. Patent 1,442,360, July14, H.G. Craighaed, J. Cheng, and S. Hackwood, Appl. Phys. Lett 40, 22(1982) 5. James L. Fergason, SID Digest of Technical Papers 16, 68(1985) 6. J.L. West, Mol. liq. Cryst. 157, 427(1988). 7. A. Golemme, S.Zumer, D.W. Allender, and J.W. Doane, Phys. Rev. Lett. 61, 2937(1988)

58 43 8. P.Drzaic, Mol. Cryst. Liq. Cryst 154, 289(1987) 9. J.H.Erdmann, S.Zmer, J.W.Doane, Phys. Rev. Lett. 64, 1907(1990) 10. B. Bahadur, Liquid crystals applications and uses, Vol 1, World Scientific 11. G.P. Montgomery, Jr. and N. A Vaz, Applied Optics 26, 738(1987). 12. AM. Lacker, J.D. Margerum, E. Ramos, S T. Wu and K.C. Lim, Proc. SPEE958, 73(1988) 13. N.A. Vaz and G.P. Montgomery, Jr., J. Appl. Phys. 62, 3161(1987)

59 CHAPTER THREE DIELECTRIC SPECTROSCOPY For a dielectric, one of the most important consequences of the imposition of an external electric field is induced polarization. Dielectric spectroscopy is based on the interaction of an electromagnetic field with the electric dipole moments of a material and is an effective method to study molecular systems. In this chapter, we will review some basic dielectric properties of materials; basic theory of dielectric relaxation processes of liquid crystals; and the dielectric response of heterogeneous systems. This will be an introduction for chapters four and five where we study the dielectric response of some composite systems containing liquid crystal. 3.1 Dielectrics in an electric field In non-conducting condensed materials (insulators), the constituent molecules may have permanent dipole moments on an atomic scale. In addition to permanent dipole moments, charges can be spatially separated over microscopic distances resulting in induced dipoles due to the presence of an external electric field. When a material is brought into an external electric field, for instance between the plates of a capacitor, every 44

60 45 portion of the material is subjected to an internal field which for the linear dielectrics is proportional to the external electric field. In a conducting material, charge carriers such as electrons in metals or ions in the liquid will migrate over large distances (on an atomic scale); equilibrium is not be reached until the total field strength has become zero at all points in the material. In the case of insulators (dielectrics), however, only very small displacements of charges occur. When an electric field is applied, the electric forces acting upon the charges brings about a small displacement of the electrons relative to the nuclei. Furthermore electric field tends to orient the permanent dipoles. In both cases the electric field gives rise to a dipole density; the electric field polarizes the dielectric. a) Static electric fields: Dielectrics may be broadly divided into "non-polar material" and "polar material". In non-polar materials, when the molecules are placed in an external electric field the positive and negative charges experience electric forces tending to move them apart in the direction of the external field. The distance is very small ( " m) since the displacement is limited by restoring forces which increase with increasing displacement. The centers of positive and negative charges no longer coincide and the molecules are said to be polarized. The dipoles so formed are known as induced dipoles since when the field is removed the charges resume their normal distribution and the dipoles disappear.

61 4 6 In polar dielectrics the molecules, which are normally composed of two or more different atoms, have dipole moments even in the absence of an electric field. Normally these molecular dipoles are randomly oriented throughout the material owing to thermal agitation, so that the average dipole moment over any macroscopic volume element is zero. In the presence of an externally applied field the molecules tend to orient themselves in the direction o f the field. In static case, for a linear, isotropic dielectric, the time independent polarization is related to the electric field by1 P = XE (3.1) where the % is the dielectric susceptibility which depends on the temperature, pressure, chemical composition. The polarization P is related to the electric field E and electric displacement D by D = e 0E + P (3.2) Using equation 3.2 may be re-written as D =eosre = ee (3.3) where s = 1+x and is called the dielectric constant which provides the link between the macroscopic and atomic theory of dielectrics. The s0= 8.85 x 10'12 C^/Nm2 is the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

62 4 7 permittivity of free space. P and E are parallel if the medium is isotropic, i.e, has the same properties in all directions. For non-isotropic dielectrics e becomes a tensor, in its principal axis system it can be given in the form: u 0 0 //= 0 E S 33 (3.4) If we go down further into the basic microscopic concepts of the dielectric theory, the total polarization has several contributions: electronic, ionic, and orientational polarizatioa We can write the total dipole moment per molecule by adding the three polarizabilities2: P = (a e + a, + a 0)E (3.5) This equation is known as the Langevin-Debye formula. a e is the electronic polarizability and a, the ionic polarizability of the molecules. a o is the contribution due to orientation of the molecules to the applied field. For non-interacting dipoles it is a Q=p2 I2KT. p is average dipole moment of each molecule in the direction of the field. Another important component of the polarization is interfacial or space charge polarizatioa This usually arises from the presence of electrons or ions capable of migrating over distances of macroscopic magnitude. Interfacial polarization is of particular importance in heterogeneous or multiphase materials. Due to the differences in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

63 48 electrical conductivity of the phases present, charges move through the more conducting phases and build up on the surfaces that separate them from the more resistive phases. Effectively it will be apparent as an increase in the average moment of the molecules given by P = a se, where the a, is the interfacial or space charge polarizability. Interfacial polarization is of importance in practical dielectric systems (our PDLC film is one example). It is also referred to quite often as Maxwell-Wagner polarization. In summary, the total polarization in any material is made up four components according to the nature of the charge displaced. The average polarizability per molecule a is the sum of the individual polarizabilities ( a e + a, + a 0 + a s). b) Static dielectric properties of a nematic liquid crystal A liquid crystal molecule usually has a permanent dipole, which causes the dielectric properties o f the liquid crystal to be strongly frequency and temperature dependent. For uniaxial nematic liquid crystals, the dielectric tensor s can be diagonalized with eignvalues e and e,, where e ; and e refer to the dielectric constants for polarization parallel and perpendicular to the nematic director n, respectively. In general, the dielectric tensor can be written asj e,; = s 5, ; + T i< 2,; ( 3. 6 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

64 4 9 where e = (e ; + 2e. )/3, t\ = 2(e,: - e, )/S is a material constant and S is the scalar part of the vector order parameter. In uniaxial nematics, equation (3.6) becomes etj = Ex5,; + Asn,rij (3.7) where Ae = Si i - e is dielectric anisotropy and n0 n] are the components of the director n. Maier and Meier4 extended Onsager s theory of isotropic dielectrics to the nematic phase. For a molecule with permanent dipole moment (i inclined at angle 3 with respect to the long axis, the equations for the principal components of the dielectric permittivity tensor in the low frequency range are eu = l+(nhf/.0){a + ^ a as+ F - - [ 1 - ( 1-3 cos2p)^]} (3.8) Sx = 1 + (NhF/e0){a - as + F ~ - ^. 1+ ^-(1-3 co s^ )^ } j J (3.9) where N is the Avogadro number, p the density, M the molecular weight. h = 3s/(2s+ 1) (3.10) is the cavity factor and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

65 5 0 Perm ittivities iso T em perature Figure 3.1 Dielectric constants e and e of nematic as a function of temperature Schematically. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

66 51 F= 1/(1- a f) (311) is the reaction field factor for a spherical cavity in an isotropic medium; Both o f them also are assumed to be remain equal to their isotropic values in Maier and Meier's extension. / = 47tATp(2e - 2)/3M (2s + 1) (3.12) where s is the mean dielectric constant, a the mean polarizability, a a the polarizability anisotropy, A/the molecular weight, S the order parameter. kbis the Boltzmann constant, and T is the temperature. Since S is temperature dependent, the dielectric components are temperature dependent. The temperature dependence of s for a nematic liquid crystal with positive dielectric anisotropy is shown schematically in Figure 3.1. The low frequency dielectric anisotropy of a molecule is determined by two factors: 1) the polarizability anisotropy a a from electronic and ionic contributions which for the elongated molecules of nematogenic compounds always makes a positive contribution (i.e, a larger contribution for the measuring field parallel to the long molecular axis) and 2) the dipole orientational contribution. The sign o f the latter contribution is positive if the net dipole moment of the molecule makes only a small angle with its long axis and is negative if the angle is large ("magic angle 3 ~ 57 ). We have the expression from equations (3.8) and (3.9) like (3-13) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

67 52 From the Equation 3.13 we can see that the dielectric anisotropy is directly proportional to the square of dipole moment. c) Dynamic electric fields: The dynamic response is most easily studied with the application of a sinusoidally varying electric field. The time dependence of the electric field strength is then given by5: E(r) = E c o sq ) / ( ) where E is the amplitude and co the angular frequency of the sinusoidal variation. For linear systems the time dependence of the dielectric displacement can also be described as sinusoidal with frequency o, but with a constant phase difference 5 with respect to the electric field5: D(0 = D cos(<o/-5) (3.15) where the D is the amplitude of the sinusoidal variation. When the frequency of the electric field is changed, the phase difference 8 changes. If we introduce a new notation according to. cos5(co) = e (co) / > (3.16) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

68 53 sin6(co) = s"((o)eqfdq then we obtain with the help of equations (3.16) and (3.17): D(/) = e,(o))e0cosffl/ + e,/(co)e0sin / (3.18) where s' and s" can be considered as a generalization of the dielectric constant for sinusoidally varying fields. When a dielectric material is subject to an alternating field the orientation of dipoles, and hence the polarization, will tend to reverse every time the polarity of the field changes. The component e" determines the loss of the energy in the dielectric and is called the loss factor. Equations (3.15) and (3.18) can be written in a more compact way by using a complex notation. In this notation the harmonic field is represented by: E(/) = E0e,at (3.19) Similarly the complex dielectric displacement can be written as: D(f) = D o e " ) (3.20) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

69 5 4 Comparing equations (3.19) and (3.20), we see that the relation between D(t) and E(t) can be written in the same form as the relation D=sE valid in the static case, by introducing a complex frequency-dependent dielectric constant e (ffl): leading to: D(/) = e (oo)e(/) (3.22) The ratio D / and quantities 5 and e* depend on frequency, but due to the superposition principle of electrodynamics, are independent o f the amplitude E of the applied field. Substituting Equation (3.19) into (3.22) and taking the real part of this equation, which must be equal to the expression for D(t) given by equation (3.18), we find that the real and imaginary part of e are equal to e' ( ) and -e"(<o), respectively, so that: e*(co) = e^to) - /e^ffl) (3.23) s* is called the complex dielectric constant. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

70 Resonance and relaxation Dielectric spectroscopy is the study of the dynamic interaction between an external electromagnetic field and the dipole moments o f a material. The electric permittivity or the susceptibility is the macroscopic manifestation of the polarization phenomena which take place on a microscopic or atomic scale and are essentially the formation and reorientation of dipoles within the dielectric material. Due to the fact that the dipole moments are coupled to the molecules, the dynamics of the dipole reorientation in an alternating electric field can provide some information about the individual or collective motion of various parts of the molecules. For example, dielectric properties give direct information about the orientation of liquid crystal molecules which have strong permanent and induced dipole moments. The dipole is the probe by which the molecular level motion can be studied dielectric spectroscopy. There are two different kinds of dielectric behavior in the time dependent regime: resonance and relaxation. These processes are different from each other by their origins. In a condensed system relaxation arises when the electric field changes too rapidly for permanent dipoles to follow the field. When the frequency is higher, the atomic and electronic dipoles induced by the external field will give rise to resonance. The permanent dipole moments in molecules arise from the distribution of the charges in the molecules in the absence of an external field. To a first approximation molecules can be considered as composed of atoms linked together by bonds between pairs of atoms6. In a nonmetallic solid, the atoms may be bonded with ionic, covalent, or Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

71 56 van der Waal's bonds7. The dipoles may be added vectorially for each molecule or for a group. In organic molecules most of the atoms are arranged symmetrically, so that most of bond dipole moments cancel each other. The net dipole of a group is the sum of the moments which are not compensated and usually a few per group. The appropriate group is determined by that portion o f a molecule which can considered as 'rigid'. When the permanent dipoles of a dielectric material are subjected to an alternating electric field, the orientation of the dipoles, and hence the polarization tends to reverse every time the polarity of the field changes. As long as the frequency remains low typically below about 1 MHz (although specific materials may have higher frequency limits), the polarization follows the alternation of the field without any significant lag. In this region the permittivity of the material remains independent o f the frequency. When the frequency is increased sufficiently, the permanent dipoles will no longer be able to rotate fast enough and their oscillations begin to lag behind those of the field and decrease in amplitude. When the frequency is further increased the dipoles can not follow the field and the contribution to the dielectric constant from this molecular process becomes vanishingly small. This usually occurs in frequency range of 10-10u Hz. Relaxation phenomena are associated with the frequency dependence of the orientation polarization. For frequencies below the infra-red the contribution of the ionic and electronic polarizations to the total polarization P are independent of the frequency and may be expressed as1 Pe + P, = P«= E0( oc - 1)E (3.24) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

72 5 7 where is the relative permittivity at frequencies which are too high for the permanent dipoles to follow and arises solely therefore from the electronic and ionic polarizations. Under static conditions P = P0 + P= (3-25) so that the orientational polarization PDis given by P0 = P-Poo = e o ( s - c c ) E (3.26) Now when a static field is applied to the dielectric, according to Debye8, P0 approaches its final value exponentially so that the orientational polarization at any instant time t after the application of the field id given by P0(t) = Pc(l - e _f x) (3.27) where t is called relaxation time of the dielectric medium; it determines the rate at which the polarization builds up. x is independent of the frequency but depends upon the temperature. In the case o f an alternating field E=E0e '" we have ^ P = i[8 (e -8 «)E -P 0«)l (3.28) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

73 58 > EW <o Cl e - e, Cm o W - cc a. CO u. o CO Frequency Figure 3.2 Debye type relaxation for polar substances. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

74 The steady-state solution o f equation 3.22 gives P*(/) = e- - - E 1 +y<ax (3.29) and the complex total polarization is therefore, P (/) = P0(r) + Pec (0 = e (^ " E + e«(e - e») (3.30) From equation 3.2 we have D 6qE + P(0 So E «+ ' e - e tt y'cox E (3.31) Comparing equation 3.31 with 3.22, the complex dielectric constant will become e*= + S Soc 1 + i(&t (3 32) Separating the real and imaginary parts we find e / (co) = e«+ S Soc 1 + C 0 2 X2 (3.33) i n \ (e - e*)a>t e"(<a) = ^ t-fit X z (3.34) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

75 cd "I Z -E z-z Figure 3.3 Cole-Cole plot for the Cole-Cole equation at a = 0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

76 \ e' x,, 6 - \ a=0.8 a= o io g ( f) Figure 3.4 Dispersion and loss curves for the Cole-Cole equation at a = 0.8 and 0.0 respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

77 6 2 These are known as the Debye8 relaxation equations. Figure 3.2 displays s' and e" as a function o f frequency for Debye relaxation. In 1941 K.S. Cole and R.H. Cole1suggested a graphical representation from which it is immediately clear whether the experimental points for e'(co) and s"(co) can be described by a single relaxation time, or if a distribution of relaxation times is necessary. This representation, generally called the Cole-Cole plot, is obtained by plotting the experimental values of s"( ) against those of e'(o>). From the equations (3.33) and (3.34) we have (6//)2 = ( s - e /)(e/ -e=c) (3.35) Figure 3.3 shows that imaginary part of permittivity versus the real part and is called Cole-Cole plot. In general, for a single relaxation time the Cole-Cole plot is a semi-circle. If there is a distribution of relaxation times then Cole-Cole plot gives a circular arc with origin below the real axis. The behavior of the orientational polarization of most condensed systems in a time-dependent field can, as a good approximation, be characterized with a distribution of relaxation times. One the most widely used empirical equation was given by K.S. Cole and R. H. Cole in 1941' e * (co) = «+ (e - e=c) ^ (3.36) 1 + ( / cot) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

78 6 3 a is the Cole-Cole parameter and it is readily seen that for a=0 this expression reduces to the equation for a single relaxation time, i.e. equation Figure 3.4 shows that the real and imaginary part of permittivity from equation (3.36) as a function of frequency with a=0.8 and a = 0. We can see when a*0 there is a distribution of relaxation times and the region o f relaxation is much broader than for the single relaxation time curve. More generally, there is a continuous distribution of relaxation time. In the frequency range corresponding to the characteristic times for the molecular reorientation. The complex dielectric constant can be written with5 G(ln x)d\n x (3.37) Here G(lnx) is distribution function with: J G(ln x)d\n x = 1 (3.38) If there is a polarization in the absence of an electric field, due to the occurrence of a field in the past, the decrease of the orientational polarization is independent of the history of the dielectric, and only depends on the value of the polarization at that instant, with which is proportional. Denoting the proportionality constant by 1/x, since it has the dimension of a reciprocal time. The distribution function G(lnx) reduces to a delta function. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

79 6 4 complex dielectric constant equation 3.37 will reduce to equation 3.32 which is single relaxation time process Debye situation. 3.3 Relaxation processes in liquid crystals In the Maier-Saupe3 mean field theory, the liquid crystal molecules in the nematic phase are considered to be long, rigid rod like molecules with strong permanent dipole moments which make an angle (B with their long axis as shown in the Figure 3.5. This permanent dipole moment can be decomposed into two components, one parallel and other perpendicular to the long axis of the molecule. The motion o f each component can be monitored by means of dielectric measurements in two distinct alignment textures: the planar alignment in which the molecules lay parallel to the substrate and the homotropic alignment in which the molecules lay perpendicular to the substrate. In each case the electric field is applied perpendicular to the substrate as shown in Figure 3.5. In this way it is possible to measure the relaxation processes of each component of the permanent dipole moment. There are two types of rotations which tend to align molecular dipole moment along the field direction: rotation around the short axis of molecules; and rotation around long axis of molecules as shown in Figure 3.5. However, the dipole themselves will not align parallel to filed because of the constraint of the nematic potential. The relaxation time o f the orientational dipole moment depends on the dielectric anisotropy. For a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

80 ^ Frequency Figure 3.5 Relaxation processes of nematic liquid crystals Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

81 66 substance with strong positive dielectric anisotropy, orientational dipole moment parallel to the director will have longer relaxation time which means relaxation takes place at low frequency because of strong hindering of this component about the transverse axis. On the other hand, the component of the orientational dipole moment perpendicular to the director will have shorter relaxation time which means at high frequencies. There may be additional dipole moments in parts of the molecules with their own rotational degrees of freedom which may give rise to a separate relaxation process. 3.4 Relaxation processes in ferroelectric liquid crystals When the molecules are subject to an alternating electric field perpendicular to the smectic layer normal, the director fluctuation in smectic can be divided into two parts. One due to the fluctuation o f the tilt and other due to the rotation of the director around the axis normal to the smectic layers. Fluctuation of the tilt is called the "soft mode" and the rotation o f the director about the layer normal is called the "Goldstone mode". The idea of "a soft mode" can be traced to the 1940's. Raman9 and Nedungadi had identified a mode in the Raman spectrum of quartz with anomalous temperature dependence in quartz when the a~p transition temperature is approached. The term "soft mode" arises from the fact that, by approaching the transition temperature, binding forces or the corresponding elastic constant soften against a certain type of mode of vibration. This mode is characterized by s0 -> cc and co -> O10. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

82 6 7 In the smectic A phase the molecules are aligned in a direction parallel to the layer normal. The stability of this structure is governed by an elastic constant which keeps the long molecular axis parallel to the smectic layer normal. However, due to thermal energy the director will fluctuate, giving locally an instantaneous tilt angle between the director and smectic layer normal as shown in Figure 1.5. When the temperature approaches the A-C transition temperature Tc, the elastic constant controlling the tilt fluctuation gets soft. Thus the fluctuation amplitude increases drastically and its susceptibility diverges at Tc. In the C* phase, the tilt angle increases with decreasing temperature. Deep in the C* phase, tilt becomes more stable against thermal fluctuations. So the soft mode can be seen close to C* - A phase transition. The concept of the Goldstone mode originates from the field of elementary particles and the concept has now a wide range of application in physics", e.g. in ferromagnetic systems, lattice dynamics, and superfluids. In a nematic liquid crystal, there are cooperative fluctuations of molecular orientation which decay infinitely slowly as its wave vector k -> 0. This is the Goldstone mode of the isotropic-nematic transition. The C* phase is characterized by its chirality, molecular tilt and sometimes the helical structure. The director makes an angle with the smectic layer normal, and it often precesses with a finite phase angle cp from one layer to another resulting in a helical structure with helical axis parallel to the normal to the smectic layer. If we let the director in a certain layer fluctuate locally as a result of thermal excitation, then this partial reorientation will propagate along the helical axis with a long wavelength 2. This type of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

83 68 fluctuation manifests itself as a distortion of the phase angle cp (the Goldstone mode in the C* phase). In the C* phase, both the Goldstone mode (cp fluctuation) and soft mode ( 0 fluctuation) are present. The soft mode relaxation is well separated from other relaxation processes in the smectic A phase and is therefore easily observable. But in the smectic C* phase molecular fluctuations are dominated by the Goldstone mode. Quite often the relaxation times of Goldstone mode and soft mode are too close to each other to be easily resolved. It is possible to suppress the Goldstone mode by applying a DC electric field parallel to the smectic layers. Since the Goldstone mode is directly related to the helical structure and the twist o f the permanent polarization, it will be destroyed by a DC field strong enough to unwind the helix. It is worth noting that the soft and Goldstone mode do also exist in the non-chiral smectic A and C phase. Let us go back to Landau theory equation 1.12 for the smectic phase to discuss some frequency and temperature dependence of the dielectric constant of ferroelectric liquid crystals near the smectic-c* smectic-^ phase transition. The helicoidal variation of the azimuthal angle ^can be written as vj/(r) = qz (3.39) where q = 27t/Z is wave vector and Z is the pitch which describes the helical periodicity of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

84 69 smectic C* phase, we obtain for our two order parameters (tilt and polarization in section 1.3)13. (S u 2) = (0 cos qz, 9 sin qz) (3.40) (Px, Py) = (-P sin qz, P cos qz) (3.41) Inserting these expressions into equation 1.12 immediately yields F = iflg2 + i& e4 + i p 2 - CP0 + ^33<72e 2 - A^e2 - vqp (3.42) To obtain the equilibrium values of P and q, we minimize the free energy with respect to P and q, obtaining ^o = X(W + Q 6 (3.43) (344) Equation (3.43) is the expected linear relation between P and 0 and, from equation (3.44), the wave vector is temperature independent. This Landau model incorporates the symmetry properties of the system and was believed to properly describe the thermodynamic properties of a system, with the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

85 7 0 exception of the critical region. However, it has been shown recently that some thermodynamic properties show qualitative disagreement with the predictions of this model. The most pronounced disagreement can be observed for the temperature dependence of the pitch of the helix, which slowly increases with increasing temperature in the Sm C* phase, reaches a maximum approximately one degree below Tc, and then sharply decreases to a finite value at Tc. For a proper description it is necessary to introduce of the higher-order terms into the Landau expansion in the tilt and polarization. The generalized Landau model for the free energy density describing the Sm A -Sm C* transition can then be expressed as following with help o f equations 3.40 and as follows V] ) 4 0 l +P l) +C(PA l - P, -kkpai+ +/*;)2 + s t) ( «1 ^ -ir (3.45) Here three additional terms have been added to the classical model. The H term represents the biquadratic coupling between the tilt and the transverse polarization. The y term has been added to stabilize the system, and d term describes the monotonic increase of the pitch with temperature at low temperatures. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

86 71 The ferroelectric Sm C* phase represents a spatially modulated structure which is characterized by two two-component order parameters the tilt and in-plane spontaneous polarization (equations 3.40 and 3.41). We thus expect four order parameter excitation modes: two high-frequency polarization modes and two modes of lower frequency connected to the reorientations of the director. The director reorientation modes are just the soft mode, and the Goldstone mode described above. The relaxation frequencies of these two modes are normally of the order of / s khz and f G ~ Hz, respectively. The dynamic equations of the system in the presence of a homogeneous time-dependence field E=E0eJ"' can be formulated as a set of torque equations: ^elastic + p vucouj _ q (3 4 5 ) Here, P la3uc is given by the Euler-Lagrange terms using equation (3.38), whereas r vucous =-r\d(8q)/dt, with r\ being a viscosity coefficient. Actually there are four viscosity coefficients corresponding to each of the modes. Since the eignmodes fall into two groups w ith/s, / G«/ ps, /pq, we can use the fact when we are studying the low-frequency director modes that the molecular rotation around the long axis is so fast that for each director configuration the polarization takes its corresponding equilibrium infinitely fast. Mathematically this can be expressed by putting rips and riro equal to zero. By solving the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

87 7 2 dynamic equations, we can obtain the relaxation frequency f Qand dielectric strength AeG for Goldstone mode: q2kn 2 k t \ g (3.47) (3.48) Here we have a renormalized elastic constant (3.49) where P0 and 0Oare polarization and tilt angle at equilibrium. For the temperature- and frequency- dependent dielectric constant of the chiral smectic -A phase, we can proceed in the same way except for some special considerations: first in the smectic-/! phase and P are zero in equilibrium; secondly only second order terms in, and P are considered (the dielectric response is by definition the response taken in the limit of zero electric field); putting the d/'dz = 0. The relaxation frequency / s and dielectric strength Aes of the soft mode are then given by (3.50) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

88 eoaes = s 2C2/[a(7 ~ 7c) + ( e p 2)<?o (3-51) where t\s is the viscosity coefficient of soft mode. Above we briefly gave some dynamic properties of ferroelectric liquid crystals, which we will use to discuss the experimental results for ferroelectric liquid crystal in confining geometries. 3.5 Dielectric Properties of Polymer In a PDLC system, the dielectric response is composed of contributions from both the liquid crystal and the polymer matrix. Here we discuss some basic dielectric properties of polymers. Polymers can be divided into two general classes14 based on their electrical response. The first class, the non-polar polymers, have no permanent dipole moment. They have a low dielectric constant and show a relatively small dielectric loss. The dielectric response has a very slight frequency dependence (especially at low frequency). Contribution to the dielectric constant is primarily due to the electronic and ionic polarization. The second class, the polar polymers, have a large permanent dipole moment. The dipole moment can reside on the main axis or in a side group. Therefore, in addition to electronic and ionic polarization, their is significant contribution to the dielectric constant Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

89 7 4 a T em perature Figure 3.6 Schematic temperature dependence of dissipation peaks for the polymer. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

90 75 from the orientation polarization. The dielectric response is also strongly frequency dependent. The vast majority of polymers fall into this class. Examples of common polar polymers are polystyrene and polymethyl methacylate(pmma). In polar polymers there may be more than one relaxation as a function of temperature at any given frequency15-16 as shown in Figure 3.6. These dissipation peaks are conventionally labeled with Greek letters, with the highest temperature peak labeled a, the next peak 3 etc. The a peak is usually associated with the glass transition temperature. The glass transition temperature of a polymer is the temperature at which the polymer goes from a glassy state to a more rubbery flexible state. As the temperature is increased beyond the glass transition temperature the main chain of the polymer becomes more mobile. This increasing motion decreases the ability of dipoles to align with the direction of the field and the polarization generally decreases above the glass transition. The secondary peaks are generally associated with motion of the dipole moments of side chain groups that are attached to the main chain of the polymer. The dispersions associated with the main chain and side chain group motion can also be observed if the temperature is kept constant and the frequency is varied. The a peak appears as the lowest frequency peak, with the secondary peaks occurring successively higher frequencies. These peaks move to higher frequency if the temperature is increased. In general these peaks have an Arrhenius temperature is dependence15. f p ~ Qxp(U/RT) (3.52) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

91 76 Here/p is the peak frequency, 7* is the absolute temperature, R is the universal gas constant and U is the activation energy in SI units. 3.6 Dielectric permittivities o f heterogeneous systems The expressions for permittivities of heterogeneous mixtures dates back to the beginning of the nineteenth century. A review of the earlier work is given by Van Beek16. A more recent survey is given by Hale17. Brown18 showed that the dielectric constant of a composite material depends not only on the dielectric constants of two phases and their volume fractions, but also on the composite geometry. Attempts to derive exact theoretical expressions of general applicability for the dielectric constants of multiple phase geometries have been futile. However, many approximate expressions have been derived and are discussed below. The dielectric constant e of a mixture of two phases must lie between certain limits whatever the geometry, and if some knowledge of the phase geometry is available, still better estimates can be derived for e. For a few special geometries exact solutions can be obtained. Most of the earlier expressions for the composite permittivity concentrated on dilute suspensions. The volume fraction of the dispersed particles is then low enough that the effect of neighboring particles can be neglected. Exact expressions have been calculated for dilute suspensions of a volume fraction v, of randomly oriented particles Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

92 7 7 (dielectric constant in a continuous matrix (dielectric constant e,) for certain composite geometries. For spheres the expression is the Raleigh mixture formula16. -e ~ -e-1 = v2;a ~ Sj (3.53) i At higher concentrations the effect of neighboring particles has to be taken into account. Approximate expressions have been calculated using different methods. The most well known expression is the Bottcher mixture formula16. e = si + 3v2ng-( L>- (3.54) Most of these expressions apply to mixtures of two phase materials, with the phases being indistinguishable with respect to which phase is dispersed and which is continuous. Thus one of the constraints on these expressions is that they have to be symmetric; inverting the phases so that the dispersed phase becomes the continuous phase and vice versa results gives the same value for s. However for the PDLC film, inverting the phases so that the liquid crystal is in the matrix and the polymer is in dispersed droplet would result in different physical properties. Therefore any expression that adequately describes the dielectric response of the film should be asymmetric. An expression that is asymmetric has been developed by Bruggeman16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

93 78 f E*D d \ - f E^Dcfa J a J B Figure 3.7 The self consistent field approximation for effective dielectric constant of heterogeneous system. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

94 79 <3-5 s > However, the expression is cubic and make it difficult to work with. An asymmetric expression that is relatively simpler in form has been calculated by Hashin and Shtrikman using an effective medium method with a self-consistent field approximation technique19. The original calculation was done for the composite magnetic permeability of two phase systems. However for reasons of mathematical analogy this expression also applies to the composite permittivity. The method is illustrated in Fig A potential \\iqis applied to a homogeneous system B of permittivity e. A sphere in this system is then replaced by a composite sphere of inner part of radius ri and permittivity e2 and a concentric shell of radius rb with permittivity e,. System A in Figure 3.7 macroscopically measured dielectric response will be the same if the energies of the two systems are equal. Equating the energy of the two configurations results in an expression between the composite permittivity e, the constituent permittivities e2, s, and the volume fraction (v2) of phase A. That is Energy config a = Energy configb (J 56) This condition is equivalent to requiring that there be no net polarization associated with the sphere and shell in configuration A. The composite permittivity of system e is then determined to be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

95 80 e = e, + 3v2si(e2.ei). - - (3.57) E2 + 2si - v 2(e2- e i ) The Hashin-Shtrikman model provides a good framework from which to calculate the composite permittivity of a PDLC film for spherical droplet. References: 1. B. K. P. Scaife, Principles of Dielectrics, Clarendoon Press, Oxfordl Dielectric Solids, a series edited by L. Jacob 3. P.G. de Gennes, The physics of liquid crystal, Clarendon press, Oxford S. Chandrasekhar, Liquid Crystal, Cambridge University Press 1977, C.J.F. Bottcher and P. Bordewijk, Theorey of Electric Polarization 6. P. Hedvig, Dielectric Spectroscopy of Polymer, by John Weily and Sons, New York (1977) 7. H. Walter, Electric Structure and Properties of Solids, by W.H. Freeman and Co. San Francisco (1980) 8. P. Debye, Phys, Z 13, 97(1912) 9. C. V. Raman and T. M. K. Nedungadi, Nature, 145, 147(1940) 10. F. Gouda, K. Skarp, and S.T. Lagerwall, Ferroelectrics, 113, , 1991 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

96 D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, W. A. Benjamin, Inc. (1975) 12. G. Durand and P H. Martinot-Lagrade, Ferroelectrics, 24, 89(1980) 13. J.W. Goodby, etc. Ferroelectric Liquid Crystal (principles, properties and applications, Gordon and Breach Science Publishers, T. Carlsson and B. Zeks, etc. Phys. Rev. A 42, 2(1990) 15. A. J. Curtis, Progress in Dielectrics, edited by J. B. Birks and J. H. Schulman (Wiley, New York, 1960) Vol. 2, p P. Hedvig, Dielectric Spectroscopy of Polymers, (Wiley, New York, 1977) 17. L. K. H van Keek, Progress in Dielectrics, edited by J. B. Birks, Vol. 7, Heywood, London, D. K. Hale, J. Mat. Sci, II, 2105(1976) 19. W. F. Brown, J. Chem. Phys. 23 (1955) Z. Hashin and S. Shtrikman, J. Appl. Phys. 33, 3125 (1962) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

97 CHAPTER FOUR ELECTRO-OPTICAL RESPONSE OF POLYMER DISPERSED LIQUID CRYSTAL The polymer dispersed liquid crystal (PDLC) is an inhomogenous system which consists of a low molecular weight liquid crystal dispersed in polymer matrix. A wide variety of droplet shapes, sizes and nematic director configurations inside the droplet are possible, depending on material properties of liquid crystals and polymers and their concentration. The PDLC films can be switched from an opaque state to a transparent state upon the application of an external field. The switching field of a PDLC film depends on a variety of factors: the dielectric properties of the liquid crystal and polymer, the interaction between the liquid crystal and polymer, the nematic director configuration inside the droplets, the droplet sizes and shapes, and the operating temperature. It is then of great interest to understand the factors determining the switching voltage to be able to choose materials having suitable properties and techniques of fabrication for a specific PDLC technological application. In this chapter we present a theoretical model to analytically calculate the switching field of a PDLC film. Our model is an extension of previous work1" In this extension, measured values of the elastic constants of the liquid crystal at different 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

98 83 temperature are used, as well as an effective dielectric constant which more accurately accounts for the effect of the field on a droplet. The theoretical calculation is compared with experimental data on the switching of a UV-cured PDLC. We are going to address the following issues: 1) Does the elastic deformation energy or surface anchoring play the primary role in the switching, 2) how does the shape of the droplet affect the switching field and 3) how does the switching field change with the operating temperature. 4.1 Introduction A PDLC film is a very complicated system. For different material properties of the liquid crystal and polymer we may have different droplet shape, size, and director configurations. In general, for relatively strong surface anchoring at the droplet wall, ordering throughout the droplet will correspond to a specific director configuration: as discussed in Chapter 3 there will be the bipolar configuration for tangential anchoring at the droplet wall and the radial and axial configurations for normal anchoring conditions. The competition between anchoring, elastic and field terms determines the director configuration in the droplet. Specifically, the resulting configuration must minimize the free energy given by: F = i J[ i (V n)2 + * 2(n 'V x n f+ ^ C n x V x n)2-d E]dv+Fs (4.1) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

99 8 4 The first three terms in the equation give the elastic free energy of the liquid crystal. The fourth term is the electric field contribution. The last term is the surface anchoring free energy which includes polar anchoring and azimuthal anchoring energy. In order to understand the switching mechanism of PDLC film, ideally one needs to solve the Frank free energy equation (4.1) for the director configuration. But in reality, it is almost impossible to do so since there is a distribution of droplet size, shape, and orientation and the free energy is dependent not only on the droplet geometry but also on the interaction between the polymer and liquid crystal. The problem has only been approached by approximate techniques. Previous models of the switching field of a PDLC film1-2 have assumed that the droplet is a slightly elongated ellipsoid with a bipolar director configuration (strong polar anchoring energy). The bipolar axis prefers to orient along the long axis of the cavity where both splay and bend deformation are minimized. These models also assume that the deformation free energy density inside the droplet is governed by the radius of curvature of the cavity and dominated by bend deformation energy. At the equator of the elongated bipolar droplet, near the wall, the elastic deformation is pure bend and the free energy density is K^/2R2, where K33 is the bend constant and the R is the radius of curvature at the equator. By balancing the electric and elastic torque, one finds that the switching voltage is: (4.2) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

100 85 where </0 is the thickness of the film, a and / are length of the semi-major and aspect ratio of the ellipsoidal droplet, emand eic are the dielectric constants of polymer matrix and liquid crystal respectively, and AT is an effective deformation elastic constant. More recently, Drzaic and Muller3 have suggested that elastic free energy is determined by the difference in free energy obtained from reorienting a nematic in a non-spherical cavity, in this case, the free energy is determined by the detailed direction configuration inside the droplet. An effective dielectric constant and a one elastic constant approximation were also used in this calculation. In all previous models, the droplet is assumed to possess strong polar surface anchoring; the azimuthal surface anchoring is neglected. In this case, there is an approximate inverse relationship between the field required to orient a nematic droplet and the droplet diameter. This relationship can in fact be easily extracted from dimensional arguments. The switching voltage must be proportional to ~ (1/a) (A7eoAe)1/2 (here K is the effective deformation constant and a is the semi-major axis of the droplet). This result is expected, since in a coherence model4 the elastic deformation energy density scales as A7^2, where K is an elastic constant and is a deformation length (or coherence length). Since the electric field energy density scales as se2, balancing the field and deformation energy densities gives the approximate relationship E - holds. What does this model predict about the temperature dependence of the switching field? According to the Landau-de Gennes model4, K ~ S1 (to lowest order in S) and As - S, where S is the order parameter of the liquid crystal. So from above we have V ~ S'2 for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

101 86 the switching voltage approximately. This means that the switching voltage decreases as temperature increases, and the curve of this function V(T) has negative curvature because of the dependence of S on temperature has negative curvature shown in Figure 1.5. This provides a qualitative picture of the temperature dependence of the switching voltage. However, this picture has a number of problems. First, the voltage is usually higher than the model predicts and also as we shall see, the curvature of the voltage as a function of temperature can be positive for some temperatures. This discrepancy between the simple calculation and experimental observation implies that factors other than droplet contribute to the threshold switching fields of the films. In this chapter, we will study the switching voltage from more complete theoretical and experimental perspectives. We begin by proposing and analyzing a theoretical model for the switching field of a PDLC film which incorporates the measured values of the elastic constants of nematic liquid crystal and the effective dielectric constant for the PDLC film both as a functions of temperature and droplet geometry. The possible switching mechanisms are discussed in light of a comparison between the experimental results and theoretical calculation. 4.2 Theoretical Model: In a PDLC film, the nematic liquid crystal molecules are confined in spheroidal droplets surrounded by a polymer binder. The director configuration of the nematic liquid crystal molecules in a droplet is determined by elastic torque and surface interactions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

102 8 7 Experimental evidence5 from light scattering measurements indicates that the director configuration inside the droplet for the UV-curable polymer liquid crystal Norland 65 and E7 system is of bipolar structure. Previous theoretical calculation3 shows that the droplet is at its lowest energy when the bipolar axis is aligned along a major axis of the spheroid; other orientations of the droplet symmetry axis increase the deformation energy within the droplet. In order to express the free energy inside the droplet analytically, we have to make some approximations for mathematical simplicity. We assume identical ellipsoidal shaped droplets with strong surface (polar) anchoring and a bipolar director configuration. Azimuthal anchoring is taken as negligible. The long axis of the ellipsoids is chosen to lie in the film plane so that in the absence of an applied field the director symmetry axis is in this plane as well. From the dielectric measurements3, for some systems like NCAP films, indicate that the ellipsoidal droplets tend to align parallel with the glass substrate with the bipolar axis in the same direction in the absence of external field. However, this is not true for the PDLC system. We do not calculate the director configuration but instead assume it has a form that is physically reasonable and analytically tractable. Specifically the director configuration in an ellipsoidal droplet is assumed to be of ellipsoidal shape with rotational symmetry axis around the droplet long axis in the absence of external field, as shown in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

103 88 y X (a) (b) Figure 4.1 Schematic representation of nematic director configuration of elliposoidal droplet in a PDLC film: (a) before switching, (b) after switching. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

104 89 Figure 4.1a. Upon application of an electric field above the switching field, the rotational symmetry axis no longer exists. But, the director configuration in a droplet is still of ellipsoidal shape, with the director aligned along the electric field as shown in Figure 4.1b. For this model the switching field is defined as the field for which the free energy of state (b) becomes less than state (a) Calculation of the free energy: For the model, in cylindrical coordinates (p, <j>, z), the nematic director configuration in droplets for zero field case (a) is shown in Figure 4.2 and given by n = (4.3) where a and c are the lengths of the semi-minor and semi-major axis, respectively, of an ellipsoidal droplet. By substituting the equation (4.3) into the Frank free energy equation (1.6), we obtain the corresponding elastic free energy. In cylindrical coordinates Eq. 1.6 becomes F = H { ^ (V n)2 +K2[n (V x n)]2 +K3[n x (V x n)]2 JprfpoWcp (4.4) V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

105 90 tan a = (cr-z2)/p; cos a = pz / [p2 z2+(c2-z2)]; sin a = (c2-z2)/[p2 z2+(c2-z2)] The unit tangent vector n in the Cartesian coordinates: n = (n*, ny, n j = (0, -cosa, sina) In the cylindrical corrdinates: n = (p, (p, z) = (-cosa, 0, sina) = (-pz / [p2z2+(cr-z2)]1/2, 0, (c2-z2) / [p2z2+(c2-z2)]l/2) Figure 4.2 The nematic unit director in cylindrical coordinates for a elliposoid droplet of PDLC at OFF state. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

106 91 Explicitly, 4^ 4p2z2C2 + p V C 4 p2z2+(c2- z 2)2 [p 2z2 +(C 2-22)2]2 [p 2z2+(C2- z 2)2] (4.5) [n (V x n)]2 = 0 (4.6) [n x (V x n)]2 = {n x -f-(wp) - 4-(wz) dz p/ op cp}: 213 = p2c4(c2 - z 2)2/[p 2z2 +(C2 - z2) 2] (4.7) The splay and bend free energy have following forms: Fspiay = \K i\(v n)2p dp dz dip = nki ] J (V n) Vp2 dz o o C Po (4.8) Fbend = (n x V x n)2p dp dz dip - kki j J (n x V x n)dp2dz 2 oo (4.9) Here p0 = crcl-z^c2). We will calculate F^lay and Fbrad separately. First of all, by substituting equation (4.5) into (4.8) we have Fspiay = nki { (/i + h +Iz)dz (4.10)

107 92 I\ =\ 4z2/[p 2z2 + (C2 - j 2)2]«ip = 4ln(C2-5 z 2)/(C2 - z 2) (4.11) Ii =J 4p2z2C2/Tp2z2 +(C 2 - z 2)2] dp2 o = ^ - [ ( C 2 - z2)/(c2-5z2) In (C2-5z2)/(C2 - z2)] (4.12) /3 =J p4z2c4/[p 2z2 + (C2 ~ z2)2] ^P2 01f 3 1 C2-5z2, 2(C ^ -z 2) (C2 - z 2)2 4 _ 2 C2 - z 2 C2-5 z 2 2(C2-8 z 2) 2 _ (4.13) Here 6 = (CF-cryC1. If we put equations (4.11), (4.12), and (4.13) into equation (4.10) and let z/c=x then equation (4.10) becomes FspUv^nKxC] {(2 + i ) 2ln^ -5 * 2 0. * -K 5-I) S x 2 2(1-5 x 2)2 x2(l -5 x 2)2 dx (4.14) After integrating each term, the F 1>yfree energy can be obtained as a series in 5=1 -cr/c2, Fsplay = Ttfi c ( In ±52 - ^ (54)) (4.15) The same procedure is applied to the bend free energy F ^. We have

108 93 F M = r f, c ( i - f 5 + ^ + 0(6*)) (4.16) Finally we have the total elastic free energy for the state (a) (OFF state): F a) = naki( + i + ^ln S ) + (4.17) where K xand K3 are elastic constants associated with splay and bend, and a and C are the length of the short axis and long axis of an ellipsoidal droplet respectively. For a spherical droplet, 5 = 0, and the free energy is that of a sphere. For the ON state as shown in Figure 4.3, we assume that the tangent vector along the ellipsoid at (x^y^zg) must pass through z-axis, i.e. the director is on the plane which is formed by the point (x^y^zo) and the z-axis. The equation of the plane is given by x y z x y z - a =y0x - x 0y = x - x q y - y 0 z-zo (4.18) The point (x^y,,^) satisfies the equation X n V n Zn (4.19)

109 9 4 x In the Cylindrical coordinates: n={p2/[p2z2+(c2-z2)2]1 0, -(a V y tp V + ^ - z 2)2} Figure 4.3 The nematic director in cylindrical coordinates for a elliposoid droplet of PDLC at the ON state. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

110 95 where b is the range of (0, a) and c' is in the range of (0, c). Then we have the director configuration of droplet for case (b): n = f W W f W W f W ^ ) <t dx dz dx (4.20) generally. The relation between Cartesian and cylindrical coordinates is i = pocoscp - <psmcp (4.21) j = posincp + cpcos (4.22) k = k0 (4.23) So we have the nematic director in the cylindrical coordinates as: n = [ r, 0, - J p 2z2 +(a2 - z 2)2 a 2 - z 2 J p 2z 2 +{a2 - z 2f (4.24) If we apply all the procedures for the OFF state to ON state, we obtain the expression for the elastic free energy for the state (b): ^. = B * l ( 2 + l + hl I 6>+ ^ ) +

111 96 ^ 3( i + i 5 + -^ ) +0(S4) (4.25) Calculation o f the field free energy: Upon the application of an electric field, one needs to include the free energy of the film associated with the electric field, which includes the free energy both inside and outside the droplets. This is due to the strong depolarization fields inside and outside the droplets in the presence of an electric field6. We use an effective medium approach7 to obtain the free energy of the film. The basic idea of the method was already discussed in Chapter 3. In order to calculate the effective dielectric constant, we need to start from the Poisson equation for the potential V (s V<j>) = p (4.26) Here <{>is the potential and p is the free charge in system. For a homogeneous medium in an electrostatic field and free of charge, equation (4.26) reduces to Laplace's equation V2<J> = 0 (4.27) Considering the droplet shape, we will use ellipsoidal (,, r\, 0 coordinates defined as follows8:

112 is the equation o f an ellipsoid whose semi-principal axes are of length a, b, c. Then, a2+, b2 +4 cz+4 (4.29) a2+r\ b2 +r\ cz+q = 1, (-c2> r > -62) (4.30) = 1, (-b 2 >C,> - a 2) (4.31) are the equations respectively of an ellipsoid, a pyperboloid of one sheet, and a hyperboloid of two sheets, all confocal with the ellipsoid (equation 4.28). Through each point of space there will pass just one surface of each kind, and to each point there will correspond a unique set of values for, r\, Q, these are the ellipsoidal coordinates. In these ellipsoidal coordinates, Laplace's equation (4.27) reduces to with hv h2, and h3 defined as h =1 (5-tQ( -Q 2l ( +a2)( + i>2)($+ c2)

113 98 / ^ 0 < $ < 4 ' a 2 I x ^ l c n. polymer Effective medium e. ((>3 Figure 4.4 Effective dielectric constant calculation. Consider a rotational ellipsoid a = b * c, a < c; a and a2 are the short axis of inner and outer ellipsoid respectively.

114 99 ~t1/2 (n -Q O i- 5 ) (T\+a2)(T]+b2)(T]+c2) (4.33) (C-UXC-n) L ft+ a X C + ^X C + e2). 1/2 For convenience we introduce the abbreviation R s = J (s + a 2)(s + b2)(s + c2) (4.34) In the effective medium approach, we need to calculate the perturbation of a uniform, parallel field due to a dielectric ellipsoid. According to Figure 4.4, there are three regions. The potentials which satisfy Laplace's equation (4.27) have the following forms8: <1)i=C3/'1(4)F2(ti)F3(Q,- a 2 < < 0 (4.35) h = C2F 1( ^ 2(t1)F3(Q + C4F 1( ^ 2(ti)F(0 { oo ds, 0 < ^ < a\ - a 2 (4.36) { (s + a2)rs <()3 = C. F K ^ to ^ Q + C 5F, G ) / ' 2( t1) F '3 ( Q { oo ds \ (s+ a2)rs (4.37) where F(s) = (s + a)xa and = [(s+orxs+zrxs+c2)]1/2 = (s+orxts+c2)]1'2 The variable s has replaced t, under the integral to avoid confusion with the lower limit. Notice that the second term in <j>3is the contribution from a dipole moment. The boundary conditions are:

115 100 At = 0: 4>i = <t>2 (4.38).h i 1 54>1 ' 1 a h - 62 [hi % J =o (4.39) At ^ = a 2- a1 <t>2 = <t>3 (4.40) 1 54)2 *c 1 54)3 [ ^ % Ji~a\-a G.Ai _ (4.41) From the boundary conditions we have: (4.42) C 2 + C 4 \ Ci + Cs J (4.43) E1C3 2a = S2 OC -f f 2 a T la J C4 (4.44) Ci+Cs J ds [s+a'^rs (4.45) By combining these four equations, we obtain

116 101 r 282T l-r 0(e - S 2XI -T A -r IB) 5 2e(l - TA + W ) - R 0B (e - e2)(l -T A + T B )- 2e2 TB 1 K } with T = o^cc&j-s, )/2e2 and = a{(a{-<r+c)m. A and B are defined as follows: A = (4.47) J (s + a2)rs B= T (4.48) J (s+cr2)/?* From the requirement mentioned in Chapter 3 that the net dipole moment in the effective medium is zero, the coefficient of dipole moment term in <t>3, C5 is equal to zero. Substituting A, B, T, and R0 into equation (4.46), we have Rote ~ e2)(l ~ + 2 ^ - (g-2g~j 2 = 0 (4.49) with X defined as a^-o X= f ---- ~ - r - (4.50) 0 (s + a 2)Rs

117 102 Expanding equation (4.49) into a series in 8 s (c2-cr)lcr as before, we obtain the equation for e. Setting the ratio of the long axis and short axis of the droplet and that of the polymer shell to be the same (c/a=c2/«2), we obtain the dielectric constant of the effective medium as follows: eda ] + 3ufc8pe ( l ) + epe((35 + y52) e(p5+y52) (4.51) with a = e l c + 2&p - v Lc ( s l c - sp) r - 1 K, * 3 (a V p-?'2vtc+tovlc'jt; (4.52) T - i 35 " 2VLC + f f e ) " ^ V' c f e ) + 5VLCf e. Here, a2 is the length of the short axis of the ellipsoidal polymer shell, ep is the dielectric constant of polymer matrix which may contain a certain percentage of the liquid crystal, and elc is the film-averaged dielectric constant of liquid crystal inside the droplet along the direction of the applied field, E6:

118 103 zic =< e«efc «e = e x + j ( l + IS ^ A e (4.53) where e is unit vector along electric the field E, e, is the perpendicular dielectric constant of the liquid crystal, As is the dielectric anisotropy, and s = elc - ep. Sd is the droplet order parameter which describes the degree of orientational order of the nematic director inside droplet and is defined as6 Sd =< i(3 (Nrf n(r))2 - l) >«, (4.54) where Nd is the droplet director and n(r) is nematic director. The physical significance of the droplet director Nd is that it is parallel to the direction of average orientation of the nematic director n(r). We assume it is a constant and Sd * 0.7 for bipolar droplet, (Sd = 0 for radial droplet). Sf is the order parameter of the PDLC film and is defined as6: j - f / V e. N,W B = i - f (? * -L-) - dq. 47tJ 4 yj(h2 _ I)2 + 4A(e L) 1 3(/t2 + 1) 3(3h2 + 1) I// h 2 32A3 \h-l (4.55) where PJx) is the second Legendre polynomial, L is the cavity symmetry axis, and h = ERdfi QAs/K)!z is dimensionless applied field (reduced field). vlc is the volume fraction of liquid crystal in droplets in the PDLC film.

119 104 If we apply the electric field to the film, there will be another term in the free energy density of a PDLC film which can be written as (4.56) where E0 is an applied field Switching field The switching field is defined as the field that makes the total free energies for the two state (a) and (b) the same. For simplicity, the effect of field induced director reorientation will be neglected. Balancing the free energy o f the two states gives rise to the switching field of a PDLC film (4.57) where X is the aspect ratio of the long axis and the short axis of a droplet, i.e. X=c/a. According to this model, the switching field of a PDLC film depends on film order parameter S{, the elastic constants of liquid crystal Kn and AT33, the dielectric constant of liquid crystal and polymer, droplet size and shape, liquid crystal volume fraction in the film, and the operating temperature.

120 R = 0.8 pm 2.0 R = 1.0 pm R = 1.2 pm > ) <E= ra I 1.0 +>* 0.5 I o.o( A spect ratio k Figure 4.5 Switching field as a function of aspect ratio X for different droplet sizes.

121 106 bend Aspect ratio X. Figure 4.6 Contribution to switching field from the splay and bend elastic energy. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

122 107 Figure 4.5 shows switching field as a function of aspect ratio for droplets of radii 0.8, 1, and 1.2 pm at 20 C, respectively. Deformed droplets are assumed to have the same volume as a spherical droplet with the given radius. The switching field increases with aspect ratio X, and decreases as droplet size increases. To first order approximation, the switching field is proportional to the droplet shape factor (5/A.a2)1' This results is different from previous work1, where the aspect ratio does not appears in their expression. Different elastic constant are used in our calculation (AT K2, and Kz). The contributions to switching field from the splay and bend elastic energy at 20 0 C are show in Figure 4.6. The sum of the splay and the bend energy difference between state (a) and state (b), i.e. ( F ^ - F ^ ) + (Fbend<b)-Fbendfr)), determine the switching field. Splay plays a more important role than bend, and the relative importance of the splay increases as droplet aspect ratio increases. From Eq. (4.57), one can determine the relative importance of the splay and bend elastic free energy Relating the model to experiment We need to decide how we can relate the theoretical results from equation 4.50 to the experimental results for PDLC switching. We can study the switching by monitoring either the capacitance or transmittance versus voltage. Typical capacitance and transmission curves are shown in Figure There are some questions: which point should we use (e.g. 10% ON or 90% ON) to determine the switching voltage. Also the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

123 108 transmittance is as dependent on the cell thickness as it is on the director configuration of liquid crystal (or director configuration of droplet for PDLC). Let us take a look at the film order parameter S{ first. If we plot the film order parameter S( (equation 4.54) as a function of applied field (reduced field h) it will be a step function for our model because we assume that there are only two directions for the droplet director: either aligned along the substrate for the OFF state or parallel with the field for the ON state. When the electric field is applied to the film, the direction of droplet director will turn 90 degrees at some threshold voltage. In reality, there is a distribution of droplet director orientations. The function S{ is not a step-function any more, but the inflection point for S{, plotted versus applied voltage corresponds to the threshold field for the "step-function" case. This suggests that we should determine the switching voltage by using the inflection points of physical quantities which are directly related to the film order parameter S{ versus applied voltage. From equation (4.53), we can see that the capacitance is proportional to the film order parameter. So the dielectric data, capacitance vs. applied voltage, can give a direct fingerprint o f the switching mechanism i.e. director orientation inside the droplet. For the transmittance, if we ignore contributions from multiple scattering, the intensity I(z) of the beam after traversing a distance z in the sample is6 I(z) = / 0exp(-p</ Oj z) (4.58) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

124 109 where pd is the number density of droplets and o$ is the droplet scattering cross-section, averaged over the sample. For a droplet size comparable to the wavelength of light a, is proportional to film order parameter Sf, with C, and r\ defined as c=k ]-i. p i Ex E x En (4.60) *1 = E ll ~ S x 3sp (4.61) Here k is wavevector and d is an average droplet dimension. Then the logarithm of the transmittance is ln(t) - S(. Consequently for determining the switching field of PDLC, we should use data from either capacitance or logarithm of the transmittance to compare with the model. It can easily be shown from equation (4.58) that the inflection point is bigger for the transmittance than the logarithm of it. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

125 no 4.3 Experiment PDLC samples were made with a 50% eutectic liquid crystal mixture (E7) and 50% ultraviolet curable Norland photopolymer (Norland 65). The E7 and prepolymer mixture was sandwiched between two glass plates with 0.75 in. x 0.75 in. patterned ITO electrodes. 15 pm glass fiber spacers were used to control the film thickness. In order to obtain the exact thickness of a sample, we measured the thickness of the empty cell using both optical and capacitance method as shown in Figure 4.7. The thickness was 13.3±0.5 pm. The droplet sizes for this type of PDLC film are controlled by the UV light intensity applied during the curing process. We measured droplet size and shape with a Scanning Electron Microscope. With the UV intensity of 1.9 mw/cnr in our experiment, we obtained a PDLC film with a droplet radius around 1.1pm, and an aspect ratio k ranging from 1 to 1.5 from droplet to droplet. Typical values for k are from 1.1 to 1.2 as obtained from SEM pictures (Fgure 4.8). The experiment set-up is shown in Figure 4.9. The sample was mounted on a computer-controlled heating/cooling stage which is also connected to a circulating temperature bath (PolyScience 5900). The temperature was controlled to within ±0.1 C. The heating stage has a window which permits a laser beam (HeNe at 633 run) to pass through for the transmittance measurement. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

126 Ill (A) EMPTY CELL (B) Schlumberger LCR Meter CELL Figure 4.7 The optical and capacitance method for the thickness measurement (A) The optical method for thickness measurement, d = Xm/[2(cos9, - cosg,)] (m: number of intensity change); (B) The capacitance method for the thickness measurement, d = sjs/c (C: capacitance; S: area; d. thickness). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

127 112 Figure 4.8 Typical picture of the Scanning Electron Microscope for E7 and N65 PDLC film. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.