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3 TEMPERATURE DEPENDENCE OF VISCO-ELASTIC PROPERTIES OF NEMATIC LIQUID CRYSTALS A dissertation submitted to Kent State University in partial fulfillment o f the requirements for the degree o f Doctor o f Philosophy by Mingji Cui August 2000

4 UMI N um ber Copyright 2000 by Cui, Mingji All rights reserved. UMI UMI Microform Copyright 2000 by Bell & Howell Information and Learning Company. Ail rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml

5 Dissertation written by Mingji Cui B.S., Jilin University, P. R. China, 1985 M.S., Jilin University, P. R. China, 1988 Ph.D., Kent State University, 2000 Approved by Chair, Doctoral Dissertation Committee Members, Doctoral Dissertation Committee Accepted by Chair, Department o f Physics Dean, College of Arts and Sciences u

6 TABLE OF CONTENTS LIST OF FIGURES... vi ACKNOWLEDGMENTS...xiii CHAPTER Page 1. INTRODUCTION Overview Introduction to Nematic Liquid Crystals Order Parameter in Nematic Liquid Crystals Anisotropy of Physical Properties in Nematic Liquid Crystals Frank Free Energy - Continuum Theory BACKGROUND Introduction Frederiks Transition Dynamics of Nematic Liquid Crystals Hydrodynamic Equations and Leslie Viscosity Coefficients Shear Flow and Miesowicz Viscosities Hydrodynamics of Twisted Nematic Devices Including Flow Microscopic Description of Viscosities o f NLCs OT Theory Orientational Fluctuation and Light Scattering Static Light Scattering From Orientational Fluctuations Dynamics of Orientational Fluctuation EXPERIMENTAL STUDIES OF TEMPERATURE DEPENDENCE OF REFRACTIVE INDICES, ORDER PARAMETER AND ELASTIC CONSTANTS Introduction Refractive Indices...61 iii

7 3.2.1 Experiment Results and Discussion Order Parameter Haller's Method Result and Discussion Elastic Constants K.,, and K The method for obtaining K,, and K Experimental Sample Preparation Determination of Cell Thickness Measurement o f Electric Field Dependence o f Capacitance Magnetic Splay Frederiks Transition Magnetic Bend Frederiks Transition Results and Discussion Results o f Kn and K Some Discussion About Elastic Constants DYNAMIC LIGHT SCATTERING EXPERIMENT Geometrical Configurations for Dynamic Light Scattering Experiment Geometry A Splay / Twist Geometry Geometry B Bend / Twist Geometry Geometry C Twist / Bend Geometry Electric Field Forced Dynamic Light Scattering (EFDLS) Photon Correlation Spectroscopy Experimental Details Sample Preparation Experimental Set-up Influence From Non-zero Pretilt Angle Multiple Scattering Strategy for Dynamic Light Scattering Measurement Geometry A Geometry B EFDLS Measurement Results and Discussion iv

8 4.4.1 Temperature Dependence of and the Ratios o f Elastic Constants Rotational Viscosity y, Experimental Result Influence of Order Parameter and Temperature on Rotational Viscosity Miesowicz Viscosities Leslie Viscosity Coefficients Simulation o f the Optical Response o f Twisted Nematic Device Including Flow SIMPLE SHEAR FLOW IN A FLAT CAPILLARY Introduction Theoretical Modeling Governing Equations Director Profile Effective Viscosity and Velocity Gradient Profiles Dissipation Calculation Experiment Sample Preparation Experimental Setup Observation Under Microscope Capacitance Measurement CONCLUSION REFERENCES v

9 LIST OF FIGURES Figure Page 1-1 A schematic representation o f nematic phase and isotropic phase A schematic representation o f the temperature dependence of nematic order parameter from Maier-Saupe mean field theory A schematic representation o f uniform director distortions occurring in a nematic Frederiks transitions under a magnetic field B: at B<Bth the molecules are parallel to the easy direction o f the alignment layers; at B>Blh the molecules in the middle layer rotate towards B Illustration of a fluid element which has a closed surface A, volume SV and mass density p Illustration of the director orientation with respect to the shear plane Illustration of the geometries for Miesowicz viscosities Illustration of the coordinates in dipole radiation Temperature dependence o f refractive indices o f 5CB using 632.8nm wavelength in both nematic and isotropic phases Temperature dependence o f refractive indices o f 5CB using 589.3nm wavelength in both nematic and isotropic phases Temperature dependence o f refractive indices and n o f ZLI-4792 using 632.8nm wavelength...66 vi

10 3-4 Temperature dependence o f refractive indices and n o f ZLI-4792 using 589.3nm wavelength Temperature dependence o f order parameter for 5CB calculated from refractive index data: circle 632.8nm; square 589.3nm Temperature dependence o f order parameter for ZLI-4792 calculated from refractive index data: circle 589.3nm; square 632.8nm Influence o f the three fitting parameters on the reduced capacitance field dependence Transmitted spectra o f a homogeneous planar cell filled with ZLI-4792: circle calculated; solid line measurement data Transmitted spectra o f a homeotropic cell filled with ZLI-4792 at two incident angles: circle calculated; solid line measurement data Measurement curve o f electric field dependence o f capacitance normalized to the zero field capacitance for 5CB at various temperatures Measurement curve o f electric field dependence o f capacitance normalized to the zero field capacitance for ZLI-4792 at various temperatures Measurement curve o f capacitance vs. magnetic field for 5CB at 26 C: magnetic splay Frederiks transition Measurement curve o f capacitance vs. magnetic field for ZLI-4792 at 30 C: magnetic splay Frederiks transition Measurement curve o f intensity vs. magnetic field for 5CB at 25 C: magnetic bend Frederiks transition Measurement curve o f intensity vs. magnetic field for ZLI-4792 at 30 C: magnetic bend Frederiks transition...94 vii

11 3-16 Fitting the measurement data of capacitance dependence on the electric field for 5CB at 25 C: circle measurement data; solid line fitted data Fitting the measurement data of capacitance dependence on the electric field for ZLI-4792 at 20 C: circle measurement data; solid line fitted data Temperature dependence o f dielectric constants for 5CB obtained from capacitance measurements Temperature dependence o f dielectric constants for ZLI-4792 obtained from capacitance measurements Temperature dependence o f K,, for 5CB obtained from fitting the capacitance measurement. Results are compared with literature values Temperature dependence of K33 for 5CB obtained from fitting the capacitance measurement and the Frederiks transition measurement separately. Results are compared with literature values Temperature dependence of elastic constants Ku and K33 obtained from fitting the capacitance measurement for ZLI Results are compared with EM data K,, and K33 dependence on the square of order parameter S for 5CB. Dashed and solid lines are linear fits to the data K,, and K33 dependence on the square o f order parameter S for ZLI Dashed and solid lines are linear fits to the data Summary of geometry A in dynamic light scattering Ratio of geometric factors as a function o f scattering angle in geometry A calculated with 5CB data at 23 C and 632.8nm wavelength Summary of geometry B in dynamic light scattering viii

12 4-4 Ratio of q components as a function of scattering angle in geometry B calculated with 5CB data at 23 C and 632.8nm wavelength Summary of geometry C in dynamic light scattering Ratio of q components as a function of scattering angle in geometry C calculated with 5CB data at 23 C and 632.8nm wavelength Intensity autocorrelation function o f 5CB at 27 C measured at 0bb=2O in geometry B Set-up for dynamic light scattering experiment Temperature dependence o f the influence of pretilt angle a on the magic angle in geometry B using 5CB data Temperature dependence o f K,,/r sptay for 5CB obtained from geometry A measurement Temperature dependence o f K,,/tlsp(ay for ZLI-4792 obtained from geometry A measurement Temperature dependence o f K33/ribcn(1 for 5CB obtained from geometry B measurement Temperature dependence o f for ZLI-4792 obtained from geometry B measurement Result o f fitting geometry B measurement data to Eq. (4.32) to obtain A ^ ^ o ij2 using 5CB data at 27 C Typical measurement result in EFDLS: measured at 0bb= 15 and 24 C for 5CB Temperature dependence o f elastic constants of 5CB. K,, and K3J are obtained in Chapter ix

13 4-17 Temperature dependence o f elastic constants o f ZLI K,, and K33 are obtained in Chapter 3. Results are compared with EM data Dependence o f obtained from dynamic light scattering measurement on the square o f order parameter S for 5CB and a linear fit to the data Dependence o f K ^ obtained from dynamic light scattering measurement on the square o f order parameter S for ZLI-4792 and a linear fit to the data Temperature dependence of elastic constant ratios for 5CB Temperature dependence of elastic constant ratios for ZLI Comparison o f the temperature dependence of y, and r(,pta> for 5CB obtained from dynamic light scattering measurements Comparison o f the temperature dependence of y, and q ^. for ZLI-4792 obtained from dynamic light scattering measurements Logarithm plot o f the scaled rotational viscosity vs. inverse o f temperature for 5CB and a linear fit to the data Logarithm plot o f the scaled rotational viscosity vs. inverse o f temperature for ZLI-4792 and a linear fit to the data at low and high temperature regions separately Temperature dependence o f Miesowicz viscosities obtained from dynamic light scattering measurements and the viscosity in the isotropic phase qjso measured using viscometric method for 5CB Temperature dependence o f Miesowicz viscosities obtained from dynamic light scattering measurements and the viscosity in the isotropic phase r Lv) measured using viscometric method for ZLI In (q,)~l/t plots in the nematic and isotropic phases for 5CB x

14 4-29 In (rij-l/t plots o f Miesowicz viscosity data in the nematic and isotropic phases for ZLI Solid and dashed lines are linear fits to the data Temperature dependence o f Leslie viscosity coefficients for 5CB obtained from dynamic light scattering measurements Temperature dependence o f Leslie viscosity coefficients for ZLI-4792 obtained from dynamic light scattering measurements Transmission vs. time o f a TN device for different viewing directions. Applied voltage is 5 V. Long dashes are the model and short dashes are the experiment. Figures on the left are tum-on and those on the right for turn-off Transmission vs. time o f a TN device for different viewing directions. Applied voltage is 8 V. Long dashes are the model and short dashes are the experiment. Figures on the left are tum-on and those on the right for turn-off Illustration o f geometry of a flat capillary and definition o f coordinates. xz is the shear plane Simulation o f director profile o f splay-mode in a flat capillary under different average flow velocity without external field Simulation o f director profile o f bend-mode in a flat capillary under different average flow velocity without external field Simulation o f effective viscosity profile in a flat capillary at V = 1.08x10-1m/s a\g 5-5 Simulation o f velocity gradient profile in a flat capillary at Vavg= 1.08x10"* m/s xi

15 5-6 Capillary preparation: (a) top view o f the capillary; (b) illustration of the surface treatment Illustration of experimental setup for Poiseuille flow study Reduced capacitance as a function of average flow velocity in a flat capillary for 5CB at room temperature xii

16 ACKNOWLEDGMENTS This dissertation marks the completion o f the long journey in my education. I would like to take this opportunity to thank those people who have helped me achieve this goal at Kent State. First, I would like to express my special thanks to my research advisor, Dr. Jack Kelly, for his constant guidance, encouragement, patience, and financial support throughout this project. Under his guidance, I have gained much in my professional skills and capability to face future challenges in my career. For this, I will be indebted forever. I am grateful to the Physics Department and faculty for giving me the education and financial support during the first two years of my study. I would like to thank the members and former members o f our research group: Tatiana Sergan, Weimin Liu, Yimin Ji, Syed Jamal and Hong Ding, for their help and useful discussions. I also would like to thank Merrill Groom for helping me in many situations. I wish to thank the office staff of the Liquid Crystal Institute and the Physics Department and many, who have helped me in many ways. Finally, my greatest thanks goes to my parents for their unconditional love, encouragement, and support throughout the entire development of my education and all aspects o f my life. I humbly dedicate this dissertation to my beloved parents. xiu

17 CHAPTER 1 INTRODUCTION 1.1 Overview The most notable application o f liquid crystal (LC) materials is in information display, which utilizes their anisotropic electro-optical properties. Many liquid crystal display (LCD) devices with various designs and applications have been created over the past 30 years. Mainstream work is still focused on improving the static performance: viewing angle, contrast ratio, and chromaticity (color). Due in large part to mathematical complexity, the dynamics in LCDs has been given much less theoretical emphasis in spite of its importance in designing and optimizing LCD devices, particularly with the introduction of video rate LCDs. During the middle of 70's, Berreman1 and Doom2 showed that the optical "bounce" observed in twisted nematic (TN) devices after a high voltage is removed is due to the transverse flow (backflow) in the middle layers of the display. Their work demonstrated the influence of the so-called Leslie viscosities on the electro-optical response o f a TN device. Numerical modeling of the electro-optical performance of LCDs, which allows for the quick characterization and optimization of proposed displays without the expense and time needed to build many prototypes, has become a high-tech tool in the display 1

18 design. Accuracy in simulating real device performance relies on a good understanding of the physics involved and reliable information on the material parameters such as indices of refraction. With the demand for video-rate displays and the advent of dynamic driving schemes, computer modeling o f dynamics of LCDs becomes more desirable. Today, most computer modeling of display dynamics still relies on the approximation of zero flow. One of the major problems when including flow effects is the unavailability of the six Leslie viscosity coefficients for many liquid crystal materials, especially those used for commercial LCDs3. There has been only qualitative success in describing the high-field dynamic response o f TN devices. Computer modeling of the display dynamics relies on the theoretical understanding of the hydrodynamics of nematic liquid crystals (NLCs). It is much more complicated than that of isotropic fluids due to the coupling between the director orientation and flow. Developing a full description of this phenomenon has been a big challenge to theoretical physicists. A macroscopic theory, based on classical mechanics, has been developed by Ericksen4, Leslie5 and Parodi6 (ELP theory). In their theory, six Leslie viscosity coefficients, a ; (five of them are independent), were introduced to describe the Theological properties of uniaxial NLCs. This theory had been tested in numerous experiments and proven to be the most successful theory of the hydrodynamics of NLCs. As seen from the above discussion, the Theological properties of LC materials are very important in the dynamic response o f LCDs. These properties strongly depend on

19 3 temperature, nematic order parameter and molecular properties. However, a macroscopic theory gives no information regarding either these dependencies or the origins of the viscosity coefficients. It is necessary to build a molecular theory of the nematic which can account for the Leslie viscosity coefficients at the molecular level7"8. The pioneering work in this area was done by Diogo and Martins9 (DM theory), who developed a microscopic theory for describing the rotational viscosity. They assumed that the Leslie viscosity coefficients are proportional to a characteristic relaxation time, which is related to the probability of overcoming a potential barrier during the molecular reorientation. DM theory fits well to some experiments, but it involves four free parameters that may lead to unreliable fitting. Also, expressions for the six Leslie viscosity coefficients are not available in this theory. A more consistent description o f microscopic theory was given by Kuzuu and Doi10"" (KD theory). They obtained microscopic expressions for all the six Leslie viscosity coefficients. However in their theory, a contribution which is called the 'tumbling parameter1was not expressed in terms of molecular model parameters. This parameter appears to be very important since it determines the additional temperature dependence of rotational viscosity. An expression for this parameter was later given by Archer and Larson12 and Kroger and Sellers13 separately based on the molecular theory. It depends on the order parameters and molecular aspect ratio. An article reviewing various microscopic theories including KD theory has been presented by Kroger and Sellers13.

20 4 They showed that these theories can be formally expressed in a similar format and related to the Ericksen-Leslie theory4"5. Following the idea of KD theory of considering an average of a "microscopic stress tensor" with an appropriate nonequilibrium distribution function, Osipov and Terentjev14' 15 proposed another microscopic theory (OT theory) which gives similar expressions for the six Leslie viscosity coefficients to those given by the KD theory. Although it was pointed out16 that the OT theory and the KD theory are equivalent as far as viscosity coefficients are concerned, the OT theory did go one step further and gave the expression for the microscopic friction coefficient which is not available in the KD theory. This microscopic friction coefficient describes the torque acting on the molecule when the local molecular angular velocity deviates from the average velocity. This microscopic friction coefficient is temperature dependent and should be taken into account in the final expressions when the temperature dependence of the Leslie viscosity coefficients is studied. Among17 the various existing microscopic theories o f NLCs, a general feature for the Leslie viscosity coefficients, far from the transition temperature, is an exponential Arrhenius temperature dependence, as observed for isotropic liquids, with comparable activation energies for all the viscosity coefficients. This temperature dependence is superposed by an individual dependence on the order parameter whose influence dominates in the neighborhood o f the clearing point. According to most theories, the

21 5 dependence o f Leslie viscosity coefficients on the order parameter S and temperature T are separable, i.e. a- =/[S(T)] g,(d 0-1) where g, is related to a relaxation process in the nematic liquid crystals and f, is a function of order parameter. Several simplifications have been introduced in developing these microscopic theories. Their validity and the results of the theories need to be verified. The experimental temperature dependence o f the viscosities is an excellent measurement for testing the predictions of these microscopic theories. However, experimental data on the Leslie viscosities as a function of temperature are rarely reported due to experimental difficulties18. Some measurements o f the shear viscosities as a function of temperature can be found19' and more studies on the rotational viscosity are available. The agreement between the experimental results and the theoretical predictions is not satisfactory'. A major reason is that the reliability of measurement results has often suffered from systematic errors leading to large differences between results of different researchers. Therefore more accurate measurements of the temperature dependence of viscosities are needed for testing these microscopic theories. For isotropic fluids, the viscosity is defined as the ratio o f shear stress ct (shear force per unit area) and shear rate dv/dx, i.e. q=o/(dv/dx). For nematic liquid crystals, the

22 6 viscosity is represented by the six Leslie viscosity coefficients. These viscosity coefficients can be positive or negative. They cannot be measured directly. However, effective viscosities, whose definition is similar to the isotropic viscosity, can be experimentally determined given the nematic director orientation, the flow velocity direction and the velocity gradient direction. These effective viscosities are combinations of the Leslie viscosity coefficients. A number of methods for experimental determination of the nematic viscosities have been developed. In the following, we will briefly introduce some of these methods: (a) Mechanical methods movement o f a plate (couette flow): In this method32'33, the liquid crystal is confined between two parallel plates. One plate is moved at a constant speed while a fixed orientation between the nematic director and the direction of plate motion is achieved by a strong magnetic field. Three effective viscosities can be determined by measuring the force needed to keep the plate moving given different nematic director orientations with respect to the plate movement. flat capillary (poiseuille flow): Liquids flow through a capillary with a rectangular cross-section ' by applying a constant flow rate or a constant pressure drop across the capillary. It is possible to obtain a complete set of viscosity coefficients when a magnetic field is used to control the director orientation with respect to the directions of flow and velocity gradient. Flow

23 7 alignment can be monitored by changes in birefringence, capacitance, or dielectric constants. torsional shear flow. In this method19 35'36, the shear is produced between two circular plates by rotating one of the plates with a low angular velocity. An external field (magnetic or electric) can be used to influence the director orientation. Values of some Leslie viscosity coefficients can be extracted from this experiment. (b) Dynamic light scattering (DLS) One o f the characteristic features of NLCs is a strong scattering of visible light which originates from thermally induced fluctuations of the nematic director. The visco-elastic ratio, i.e. the ratio of a viscosity coefficient and an elastic constant can be obtained from the scattered intensity correlation function22-37~40. In principle, it is possible, by using suitable fitting procedures, to determine all the Leslie viscosity coefficients from the angular dependence of the light scattering under different geometries. (c) Attenuation o f ultrasonic shear waves When a high frequency ultrasonic shear wave propagates across a solid-nematic liquid crystal interface41, shear is induced in a very small thickness of the nematic (near the solid). Two effective viscosities, depending on the orientation o f the molecules with respect to the polarization o f the wave, can be determined by measuring the reflection coefficients o f the wave at the interface.

24 8 These effective viscosities are very similar to Miesowicz viscosities whose definition will be introduced in Chapter 2. (d) Rotating field method When a strong uniform magnetic field is applied to a cylindrical nematic sample, the director will tend to align parallel to the field. If the field is slowly rotated43-*4, the director will follow the field but with a certain phase lag. This phase lag is such that the frictional torque and the magnetic torque balance. This allows a determination of the rotational viscosity, which is the viscosity related to the rotation o f the nematic director. (e) Relaxation methods In relaxation methods42, a non-equilibrium orientation of the director is produced when an external field (magnetic or electric), which is stronger than the threshold field for the Frederiks transition, is suddenly removed. The time constant for the relaxation to the equilibrium is then determined. The relaxation process can be followed by optical detection, by measurements of the dielectric constant, or by measurement of the torque acting on the sample in a magnetic field. The rotational viscosity and two effective viscosities corresponding to splay or bend nematic director deformation can be obtained. Among the above methods, we chose two to use for determination of the Leslie viscosities: dynamic light scattering and poiseuille flow. Dynamic light scattering was chosen based on several considerations: i) this method is sensitive to visco-elastic ratios

25 9 and the results are accurate; ii) the consumption of LC materials can be kept extremely low; iii) the experimental set-up is relatively simple and some of the Leslie viscosity coefficients can be easily determined. The flat capillary method is the only method that has been used to obtain a complete set of the viscosity coefficients18. In order to get a complete set of viscosity coefficients, a strong magnetic field is needed which is not available in our lab facility. However we chose this method to explore the situation without any external field, to obtain information on the relatively small but important viscosity coefficient ctj, which is not easily accessible by dynamic light scattering. In this dissertation, we will experimentally study the visco-elastic properties of NLCs as a function of temperature in hope o f better understanding the hydrodynamic behaviors of NLCs. The writing is organized as follows: in the remaining sections o f this chapter, we will give a basic description of NLCs relevant to this research. In chapter 2, we introduce the theoretical background which is the basis for this dissertation. We will introduce both the macroscopic and microscopic theories for describing the hydrodynamics in nematics as well as the fundamental theories for the experiments we use in later chapters. In chapter 3, we present measurement results for some physical properties of NLCs including refractive indices and elastic constants whose data will be needed in the dynamic light scattering experiment. In chapter 4, we will describe using a dynamic light scattering experiment to obtain the visco-elastic ratios. Using the elastic constant data from Chapter 3, we extract several Leslie viscosity coefficients. With the measured viscosity data, we simulate the

26 10 dynamic response of a TN device using the Ericksen-Leslie equations including flow and compare with the experimental result. Also in this chapter, we will study the temperature dependence of the rotational viscosity and compare the experimental result with the prediction of microscopic theory14. In chapter 5, we present a study o f the flow behavior of NLCs in a flat capillary. We use a new technique, capacitance measurement, to obtain the 'tumbling parameter1from which the Leslie viscosity coefficient a, is determined. In the following sections, we are going to give an introduction to nematic liquid crystals. We will introduce some concepts regarding nematic liquid crystals such as nematic order parameter, anisotropic properties, and Frank elastic free energy. The viscous and elastic properties of the nematic liquid crystals, which are the subjects of this dissertation, are discussed in the Chapter Introduction to Nematic Liquid Crystals The liquid crystalline phase is an intermediate phase between the solid crystalline and the isotropic liquid phases. Material of liquid crystalline exhibits some degree of fluidity comparable to that of an ordinary liquid. It flows and will take the shape of its container. On the other hand, it possesses some degree o f translational and/or orientational order at dimensions much larger than the molecular level and therefore exhibits anisotropy in thermal, mechanical, optical and electromagnetic properties typically observed in a crystal. The term, liquid crystal, is used for a compound in its liquid crystalline phase. The transition between phases may be induced by a temperature

27 11 change or by a variation o f the concentration o f one component in the mixture. Compounds with these phase behaviors are known as thermotropic liquid crystal and lyotropic liquid crystal, respectively. Most of the thermotropic liquid crystals are composed of rod-like molecules. With a change of temperature, a thermotropic liquid crystal may pass through one or more liquid crystalline phases (mesophases) before it is transformed from a crystal into an isotropic liquid. These mesophases are classified according to the molecular arrangement and ordering47. The nematic phase is the least ordered liquid crystalline phase in which the center of mass of the molecules has no long-range order, but the orientation of the molecules has long-range order as shown in Fig Molecules in the nematic phase tend to align to a common direction. A unit vector n along this direction is denoted as the nematic director. For an infinite nematic the direction of n is degenerate in space; in practice, the degeneracy is removed by either an external field or the orienting influence of the walls of the container (anchoring force). A nematic is said to have homogeneous alignment if the nematic director n is parallel to the surface of its container, and to have homeotropic alignment if n is perpendicular to the surface of its container. Physical properties, such as the dielectric constant or magnetic susceptibility, exhibit different values along the n direction than along any other directions. A nematic phase is uniaxial if there is a complete rotational symmetry about the director n; otherwise, it is biaxial. This dissertation considers exclusively the uniaxial nematic liquid crystals.

28 12 n n(r) \ / /«I ' ' I 0(rj 0(r) is the angular deviation of n(r) from n Nematic phase \ \ l \ / /\ Isotropic phase Figure 1-1. A schematic representation o f nematic phase and isotropic phase.

29 Order Parameter in Nematic Liquid Crystals The orientational order in nematics is not nearly as perfect as in a solid; the molecules do not in general point along the director n but at any instant in time are distributed about this direction. The alignment state of the molecules is described by a distribution function giving the probability o f finding a molecule oriented at an angle 6 from the nematic director n. An order parameter5 0 ' 5 3 giving the average orientation of the molecules with respect to the director is used to describe the degree o f orientational order. This order parameter must be non-zero in the nematic phase and vanish in the isotropic liquid. A scalar order parameter S for the uniaxial nematic system was first proposed by Tsvetkov4 8 S=^<3cos20 ( r)- 1 > (1.2) where 0 (r) is the angle between the orientation o f each individual molecule and the nematic director n, and the brackets < > denote a spatial average over the entire sample. It gives S=1 for a fully aligned system (e.g. perfect crystal), and S=0 for completely random orientation (e.g. isotropic liquid). The order parameter of a nematic decreases as the temperature is increased. Dramatic changes occur while approaching TNI (or Tc used alternatively), the temperature

30 14 at which the transition between nematic phase and isotropic phase occurs. Typical values of S for a nematic liquid crystal lie between 0.3 and 0.8. The order parameter as a function of temperature is shown schematically in Fig. 1-2 based on the Maier-Saupe mean field theory49. This theory predicts that the order parameter at the nematic-isotropic transition is 0.44; the transition is first order. 1.4 Anisotropy o f Physical Properties in Nematic Liquid Crystals Orientational order existing in a nematic causes anisotropy in its physical properties. These anisotropies are highly temperature dependent since the degree of orientational order varies with temperature. Here, we discuss optical, electric and magnetic anisotropies. In nematic liquid crystals, light polarized along the director propagates at a different velocity than light polarized perpendicular to the director. Therefore, a uniaxial system has two principal refractive indices, nqand ne, observed for an "ordinary" ray and an "extraordinary" ray, respectively. Then the birefringence or optical anisotropy is given by An = n. - na (1.3) The optical dispersion, i.e. the dependence of refractive indices on wavelength, X, is given by Cauchy equations

31 C/ O Temperature Figure 1-2. A schematic representation o f the temperature dependence of nematic order parameter from Maier-Saupe mean field theory.

32 16 (1.4a) (14b) where is the refractive index extrapolated to infinite wavelength and A is a material specific coefficient. A non-conducting material may respond to external electric field E in two ways: atomic charges (nuclei and electrons) are driven apart slightly by the electric field and give rise to induced electric dipoles. If permanent dipoles are present, they will tend to align along the applied field. In both cases the electric field produces a polarization P. The polarization from permanent dipoles is typically much stronger than that from induced dipoles. In a linear, anisotropic medium, this polarization is proportional to the external field, but need not be parallel to it. The electric displacement D in such a medium is related to E by D = e e E (1-5) where e is called the dielectric tensor; ea is the permittivity o f the vacuum. For uniaxial nematic liquid crystals, this dielectric tensor can be diagonalized with eigenvalues e, and

33 17 el ( 2 -fold degenerate), the dielectric constants measured along and normal to the nematic director respectively. It can be written in dyad form as e = e.i + Aenn (1-6) where Ae=e.-ex is the dielectric anisotropy; Kis the second rank isotropic tensor. Most molecules o f nematic liquid crystals are diamagnetic. For a uniaxial system, the magnetic susceptibility has two different negative (diamagnetism) values of x and xx, measured with the magnetic field parallel and perpendicular to n, respectively. Therefore, by introducing a susceptibility tensor x, the magnetization M can be written in terms of the magnetic induction B as M = n 0 'x«b (1.7) where i0is the permeability o f the vacuum. The susceptibility tensor x is given by X = X -l + Axnn (1.8) where the magnetic anisotropy A x^rx j. is usually positive for a nematic. 1.5 Frank Free Energy - Continuum Theory In an ideal nematic, the molecules are aligned on average along one common direction n. However, in m ost practical cases, this ideal conformation will not be

34 18 compatible with the constraints exerted on the molecules by surface anchoring and external fields (magnetic, electric, etc.)- There will be some deformation of the alignment: the direction of n varies from place to place over macroscopic distances (typically a few microns). There are three possible types of uniform deformations in nematics: splay, twist and bend, as shown in Fig Based on the fact that the spatial variation of n is small on a molecular scale, a nematic may be pictured as being locally still uniaxial with a fixed order parameter S but with a variable preferred direction n(r). The curvature of n can be treated with a continuum theory disregarding the details o f the molecular structure. This theory was first suggested by Oseen5 4 and Zocher5 5 and extensively developed by Frank56. In this theory, the order parameter S is assumed to be unchanged by director distortions. Considering only the first derivatives of the director n, the contribution to the free energy density due to the distortion o f the director field is given by F, = n)! + ^ ( i i.( V x n))! + W n x (V x n))2}. (1.9) Each of the terms in Eq. (1.9) represents the free-energy density associated with the three basic types of deformations displayed in Fig Therefore, the coefficients K,,, K^and K3 3 are called the Frank elastic constants describing the forces necessary to obtain splay, twist and bend deformations, respectively. They are functions of order parameter S and temperature T.

35 19 Twist Bend Figure 1-3. A schematic representation of uniform director distortions occurring in a nematic.

36 20 Fd is the elastic distortion energy. Under the presence o f an external field (electric, magnetic, etc.), the nematic director tends to align with the field. This will add energy to the system. If an electric field E is applied, the free energy density associated with this field is given by F, = ± e 0 Ae(n E ) 2 ( 1. 10) where "+" corresponds to a system under the influence of fixed charges and to a system with conductors at fixed potentials. Similarly, the distortion free energy density in the presence of a magnetic field B is given by Fm= - i p 0'AX(n.B )2 ( 1. 11) In the equilibrium state, nematic molecules choose the orientation in such a way that a minimum total free energy, associated with the elastic distortions, external fields, etc., is reached. Before ending this chapter, we briefly mention the LC materials used in this dissertation. Two LC materials, 4-cyano-4'-pentyl-biphenyl (5CB) and ZLI-4792, are used throughout this dissertation. 5CB was chosen because it has been relatively well studied. Reference data on many o f its physical properties can be easily found in the literature for comparison. It is a single component system with a nematic temperature range o f

37 21 24~-35 C. ZLI-4792 has been used in commercial LCDs for many years. It is a mixture of fluorinated molecules with a wide nematic range (-20 C to 92 C). This material has been used previously to study the dynamic response of twisted nematic devices. However, little data exists on the visco-elastic behavior or physical properties of this material, especially the temperature dependence of these physical properties.

38 CHAPTER 2 BACKGROUND 2. 1 Introduction In this chapter, we introduce the theoretical basis for the experimental work presented in subsequent chapters. First we introduce the field-induced Frederiks transition and derive the expressions for its threshold field. Later, in Chapter 3, we will use these expressions to experimentally determine the Frank elastic constants. Then we introduce the macroscopic description (ELP theoiy) for the hydrodynamics of nematic liquid crystals. We give the governing equations of nematodynamics and introduce the concept of Leslie viscosity coefficients. The Leslie viscosity coefficients are our interest in this dissertation. The knowledge in this section will be used in our experimental work in Chapter 5: study o f the flow behavior of nematic liquid crystals in a flat capillary. Following the macroscopic theory, we present a microscopic theory o f the nematic viscosities (OT theory). Finally, we will discuss nematic director fluctuations and light scattering by them. These are the basis for our dynamic light scattering experiments described in Chapter Frederiks Transition 22

39 23 We consider a nematic sample, with positive dielectric and magnetic anisotropy, homogeneously aligned between two glass substrates. The nematic easy direction is imposed by alignment layers on the two interior glass surfaces. An external field (magnetic or electric) is applied normal to the substrates. For small field, the surface alignment still determines the nematic conformation throughout the sample. At sufficiently high field, molecules are reoriented nearly along the external field, except those within a thin transition region near each surface where the reorientation is restrained by the surface alignment. The Frederiks transition occurs at a critical field, the threshold of the transition, where the nematic director starts to deform from the initial orientation to align with the applied field. The existence of the Frederiks transition is predicted by the continuum theory, and the threshold field is determined by the nematic elastic constants. Therefore, it provides a way of obtaining experimentally the elastic constants. We now derive the expression for the threshold field using the continuum theory. Three possible experimental geometries for the Frederiks transition under a magnetic field are given in Figure 2-1. The director pattern is developed into splay, twist and bend deformations respectively after the magnetic field exceeds the threshold B^. The first case in Figure 2-1, i.e. the magnetic splay Frederiks transition, is chosen as an example for the derivation. The result is easily generalized for the other cases.

40 Twist Bend B<B* B>Blh Figure 2-1. Frederiks transitions under a magnetic field B: at B<Bth the molecules are parallel to the easy direction of the alignment layers; at B>Bth the molecules in the middle layer rotate towards B.

41 25 A magnetic field is applied perpendicular to the initial director orientation, along the +z axis. 6 (z) is defined as the angle between the director orientation at z and its initial uniform orientation along the x-direction. Therefore, we can write the director field and the magnetic field as n = (cos 0,0, sin 0) and B = (0,0, B) (2. 1) According to the continuum theory introduced in Section 1.5 of Chapter 1, the free energy density o f the nematic is given by F, = ^K (l-k sin e )(d ) -jrsm e (2.2) where and 4 has unit o f length and is called the magnetic coherence length*5. At any given field, the director pattern 0(z) can be obtained by minimizing the total free energy of the nematic. We use variational calculus to determine the director profile which minimizes the total free energy. This leads to the E-L equation

42 where 9' is the first derivative of 0(z). Inserting Fd into Eq. (2.4), the E-L equation becomes 2(1 - K sin 20)0" - K sin 20 0* + :sin 20 = 0 (2.5) where 0" is the second derivative o f 0(z). Near the transition, 9(z) is very small. Therefore, Eq. (2.5) can be linearized as 0,/ + C"9 = O. (2.6) Finding the solution for this equation, which satisfies the boundary conditions: 0=0 at the substrate interfaces (strong anchoring is assumed), we obtain that the magnetic coherence length at the threshold is ^=<1/71, where d is the thickness of the nematic sample. From Eq. (2.3), the magnetic threshold field is given by B*=d JW c2-7 ) This formula can also be generalized to the other two geometries in Figure 2-1. Therefore we obtain

43 27 (i = l, 2, 3). (2.8) Similar to magnetic transition, the threshold voltage for electric Frederiks transition can be obtained by substituting (2.9) into Eq. (2.8). Experimentally the transition can be detected by a change in physical properties. For example, the capacitance as a function o f magnetic field is measured if capacitance is chosen to monitor the splay Frederiks transition induced by magnetic field. Below the transition, the capacitance remains constant when the field is increased. Above the transition, the capacitance increases with the field. The sharpness of the capacitance versus magnetic field curve at the transition depends on how good the alignment is. The transition threshold field can be obtained by linearly extrapolating the curve just below and above the transition and taking their intersection as the transition point. In Chapter 3, we are going to use this technique to obtain the Frank elastic constants. As we see in this technique, the external field needs to be applied exactly either perpendicular or parallel to the initial director orientation depending on positive or negative anisotropy. If this condition is not fulfilled, which is not easy to fulfill experimentally, there is no sharp transition. This makes the extrapolation to obtain the threshold field difficult and leads to

44 28 a too small value for the threshold field31. Therefore in Chapter 3, we will introduce another technique, fitting the electric field dependence of capacitance measurement to the continuum theory, to obtain the elastic constants. 2.3 Dynamics of Nematic Liquid Crystals A nematic liquid crystal flows very much like a conventional organic liquid consisting of similar molecules. However, the flow mechanics is more complex and more difficult to study experimentally due to the coupling between the translational motion and orientational motion of the molecules. In a nematic, a velocity gradient exerts a viscous torque on the director, and conversely, a local rotation of the director may induce a flow. Two approaches have been used to describe the coupling between the orientation and flow in nematics. A macroscopic approach, based on classical mechanics in which the fluid is regarded as a continuous medium, has been developed by Ericksen4, Leslie5 and Parodi6 (usually referred to as ELP). This theory is considered to be the most successful theory to describe flow phenomena in nematic liquid crystals. In this section, we are going to introduce the hydrodynamic equations in this theory. However, the macroscopic theory gives no information regarding the physical origin of viscosity coefficients. Therefore microscopic approaches have been used by different groups7 ' 1 4 to study the nematic hydrodynamics at the molecular level. In Section 2.4, we will introduce a microscopic theory proposed by Osipov and Terentjev1 4 (referred to as OT theory).

45 Hydrodynamic Equations and Leslie Viscosity Coefficients In continuum theory the fluid is regarded as a continuous medium. Various fluid properties such as density, velocity, director orientation, and temperature are continuous functions o f position and time. On this basis it is possible to establish fluid motion equations which are independent o f the molecular structure. The moving state of a nematic fluid can be described by the velocity field v(r, t) and the director field n(r, t), together with two thermodynamic quantities pertaining to the fluid, the pressure p(r, t) and the density p(r, t). All these quantities refer to the fluid at a given point r = (x,y,z) and time t. One of the fundamental laws in fluid mechanics is the law of conservation of mass. As shown in Fig. 2-2, in the fluid we choose a closed surface A which encloses a volume SV entirely occupied by fluid, p is the density of the fluid. The mass conservation law says that the mass change inside the volume 5V is equal to the mass flowing across the surface A. This leads to (2. 10) which is called the continuity equation. A fluid is said to be incompressible if its density is a constant. Therefore the continuity equation for an incompressible fluid is

46 30 5V Figure 2-2. Illustration of a fluid element which has a closed surface A, volume SV and mass density p.

47 31 V v = 0 or -r-2- = 0 dxa (2.11) where \ a (a=l,2,3) refer to Cartesian variables x, y, z. In this dissertation we will treat the nematic as an incompressible fluid. The fluid equation o f motion is governed by Newton s second law. In fluid dynamics, it is given by the Navier-Stokes equation, i.e. 3 1 (2. 12) where the forces on the right-hand side are in unit volume. There are two categories of forces in the total force acting on the fluid. The first category is the body force; it is proportional to the volume o f the fluid. It comes from the presence of external field (e.g. gravitational, electric or magnetic field) and is given by fbin Eq. (2.12). The second category is the surface force. By meaning, it is the force exerted on the surface of a fluid element, which is given by the second term on the right-hand side of Eq. (2.12). The surface force contains normal and shearing forces. There is always a normal surface force -Vp exerted on the surface o f a fluid element due to pressure p. There are also normal and shearing forces exerted on the surface of a fluid element due to the resistance to dilation, compression and shearing. In Eq. (2.12), o is a second-rank stress tensor, whose