STATISTICAL MECHANICS OF POLAR, BIAXIAL AND CHIRAL ORDER IN LIQUID CRYSTALS

Transcription

1 STATISTICAL MECHANICS OF POLAR, BIAXIAL AND CHIRAL ORDER IN LIQUID CRYSTALS A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Subas Dhakal August 2010

2 Dissertation written by Subas Dhakal M.A., Kent State University, 2006 Ph.D., Kent State University, 2010 Approved by Dr. Jonathan V. Selinger Dr. David W. Allender Dr. Elizabeth Mann Dr. John Portman Dr. Hiroshi Yokoyama Dr. Paul Sampson, Chair, Doctoral Dissertation Committee, Members, Doctoral Dissertation Committee,,,, Accepted by Dr. Bryon D. Anderson Dr. Timothy Moerland, Chair, Department of Physics, Dean, College of Arts and Sciences ii

3 TABLE OF CONTENTS LIST OF FIGURES vii ACKNOWLEDGMENTS xv 1 Introduction General Considerations Application of Liquid Crystals Experiments on Liquid Crystals Macroscopic and Microscopic Order Parameters Theoretical Studies on Liquid Crystals Landau-de Gennes Model for the Nematic Phase Maier-Saupe Theory for Nematic Computer Simulation in Liquid Crystals Scope of the Present Work BIBLIOGRAPHY Field Induced Phase Transition in SmCP phases of Bent-Core Liquid Crystals Review of the SmCP Phases of Bent-Core Liquid Crystals Lattice Monte Carlo Simulation for Polar, Chiral and Tilt Order iii

4 2.2.1 SmC A P A SmC A P F transition SmC A P A SmC S P F transition SmC S P A SmC S P F transition Phenomenological Free Energy Numerical Analysis of the Phenomenological Model SmC A P A SmC A P F transition SmC A P A SmC S P F transition : SmC S P A SmC S P F transition: Is Chirality an Independent Order Parameter? Summary BIBLIOGRAPHY Statistical Mechanics of Splay Flexoelectricity in Nematic Liquid Crystals Review of Flexoelectric Effect in a Nematic Splay flexoelectric Model Monte Carlo simulation Mean-field calculation Ising model General Mean-Field Calculation Conclusion BIBLIOGRAPHY iv

5 4 Giant Flexo-electricity in Bent Core Nematic Liquid Crystals: A Monte Carlo Simulation Study Introduction Interaction Potential for Bend Flexoelectric Effect Lattice Monte Carlo simulation Mean-Field Calculation of the Model Summary and Discussion BIBLIOGRAPHY Chirality and Biaxiality in Cholesteric Liquid Crystals Opening Remarks Dipole-Induced-Dipole Interaction Twisting Torque Between Two Chiral Molecules Monte Carlo Simulation for Chiral Molecules Mean-field Calculation for Biaxial Cholesteric Phenomenological Theory Summary and Discussion BIBLIOGRAPHY Concluding Remarks A Calculation of the Splay and Bend Vector From Site to Site A.1 Calculation of splay vector v

6 A.2 Energy term in the Splay Flexoelectric Model A.3 Calculation of the bend vector A.4 Energy term in Bend Flexoelectric Model B Self Adjusting Boundary Condition C Free Energy of the Biaxial Cholesteric Phase vi

7 LIST OF FIGURES 1.1 Examples of liquid crystalline molecule: (a) 5CB and (b) 8CB Different liquid crystal phases: (a) isotropic. (b) nematic. (c) cholesteric. (d) SmA. (e) SmC Schematic representation of molecular order in (a) nematic and (b) polar phase. The configurations are taken from lattice simulations of a model with a nematic and polar interaction Typical liquid crystal cell Schematic of a bent-core molecule Free energy and order parameter in Landau-de Gennes model for nematic Free energy and order parameter in Maier-Saupe theory for nematic Visual scheme of different SmCP phases: (a) SmC A P A. (b) SmC A P F. (c) SmC S P A. (d) SmC S P F Monte Carlo simulation results showing electric field dependence of order parameters: (a) chiral (b) polar (c) tilt and (d) polarization, for interaction strengths A = 0.6, B = 0.6, C = 0.6, C χ = 0.2, C p = 0.4, C θ = 0.5, g 1 = vii

8 2.3 Monte Carlo simulation results showing electric field dependence of order parameters: (a) chiral (b) polar (c) tilt and (d) polarization, for the interaction strengths A = 1.0, B = 0.6, C = 0.7 C χ = 0.4, C p = 0.2, C θ = 0.2, g 1 = Monte Carlo simulation results showing electric field dependence of order parameters: (a) chiral (b) polar (c) tilt and (d) polarization, for the interaction strengths A = 0.6, B = 0.6, C = 0.6 C χ = 0.2, C p = 0.4, C θ = 0.5, g 1 = Schematic representation of the electric field dependence of the order parameters: (a) polar order, showing antiferroelectric to ferroelectric transition (b) chiral order, showing chiral to antichiral transition. The results correspond to the values: a 1 = 1.0, a 2 = 1.0, c 0 = 0.2, c 1 = 0.4, c 2 = 0.3, g 1 = Schematic representation of the electric field dependence of the order parameters: (a) polar order, showing antiferroelectric to ferroelectric transition (b) tilt order, showing anticlinic to synclinic transition. The results correspond to the values: a 1 = 1.0, a 2 = 1.0, c 0 = 0.7, c 1 = 0.4, c 2 = 0.3, g 1 = viii

9 2.7 Schematic representation of the electric field dependence of the order parameters: (a) polar order, showing antiferroelectric to ferroelectric transition (b) chiral order, showing antichiral to chiral transition. The results correspond to the values: a 1 = 1.0, a 2 = 1.0, c 0 = 0.2, c 1 = 0.4, c 2 = 0.7, g 1 = Order parameters as a function of applied field for poor chiral model: (a) chiral order, showing chiral to antichiral transition (b) polar order, showing antiferroelectric to ferroelectric transition. The results correspond to the same values of the interaction parameters that has shown the preserved chirality [Fig. 2.6] for the model with well defined chiral order in a smectic layer Snapshots of the simulation results in the three phases: (a) Isotropic. (b) Nematic. (c) Polar. The software V Sim is used [21], and the color of each molecule represents the polar angle θ away from the z-axis.. 45 ix

10 3.2 Monte Carlo simulation results for the order parameters P 1, P 2, and θ as functions of temperature T, for different values of the interaction parameters chosen to show various types of transitions: (a) Zero-field results for A = 1.5, B = 0.09, C = 0.3, showing the isotropicnematic and nematic-polar transitions. (b) Zero-field results for A = 1.5, B = 0.4, C = 0.3, showing the direct isotropic-polar transition. (c) Phase diagram for zero field. (d) Simulation with applied electric field E = 0.06, for the same parameters as in part (a), showing the induced polar order and splay in the nematic phase Variation of splay as a function of temperature for several values of the applied electric field, using the same numerical parameters as in Fig. 3.2(a) Variation of splay as a function of temperature for several values of the interaction coefficient C, with a small field E = Numerical mean-field calculations for the Ising mapping, showing the splay and polar order as functions of temperature T, for parameters A = 1.5, B = 0.09, and C = 0.3, for zero and nonzero electric field.. 58 x

11 3.6 Numerical mean-field calculations of the order parameters P 1, P 2, and θ as functions of temperature T, for different values of the interaction parameters chosen to show various types of transitions: (a) Zero-field results for A = 1.5, B = 0.09, C = 0.3, showing the isotropic-nematic and nematic-polar transitions. (b) Zero-field results for A = 1.5, B = 0.4, C = 0.3, showing the direct isotropic-polar transition. (c) Phase diagram for zero field. (d) Simulation with applied electric field E = 0.06, for the same parameters as in part (a), showing the induced polar order and splay in the nematic phase Different bend configuration between neighboring molecules Visual scheme of different phases obtained in simulations: (a) Isotropic. (b) Uniaxial nematic. (c) Biaxial nematic. (d) Polar. The software V Sim is used [37], and the color of each molecule represents the polar angle θ away from the z-axis Monte Carlo simulation results for uniaxial (S), biaxial (V ) and polar (P) order parameters as functions of temperature T at zero field (E = 0.0) in units of interaction strength A: (a) I N U N B P transition for B 1 = 0.04 and B 2 = 0.4. (b) I N U P transition for B 1 = 0.14 and B 2 = 0.4. (c) I P transition for B 1 = 0.38 and B 2 = 0.4. (d) I N B P transition for B 1 = 0.03 and B 2 = xi

12 4.4 Complete phase diagram of the model at zero field (E = 0.0): (a) in terms of B 1 with B 2 = 0.4 and C = 0.4. (b) in terms of B 2 with B 1 = 0.03 and C = Monte Carlo simulation results for θ as a function of temperature T (in units of interaction strength A): (a) I N U N B P transition with B 1 = 0.04, B 2 = 0.4 and E = 0.04 for different values of C. (b) I N U P transition with B 1 = 0.14, B 2 = 0.4 and E = 0.04 for different values of C. (c) I N U N B P transition with B 1 = 0.04, B 2 = 0.4 and C = 0.4 for different values of E. (d) I N U P transition with B 1 = 0.14, B 2 = 0.4 and C = 0.4 for different values of E Schematic representation of the molecular order in the nematic phase showing bend in the long-axis. This is the snapshot from the Monte Carlo simulation just above the polar to nematic transition Mean-field results for the order parameters P 2, P 1 and θ as a function of T at zero field (E = 0), for different values of the interaction parameters chosen to show different kind of transitions: (a) B = 0.04 and C = 0.4. (b) B = 0.27 and C = Mean-field phase diagram as a function of B 2 and B Temperature dependence of bend in mean-field calculation: (a)n U N B P (b) N U P xii

13 5.1 (a) Achiral and (b) chiral biaxial molecular structures studied in this work Free uniaxial rotation between two chiral molecules Torque between two chiral molecules as function of molecular twist (χ) averaged over the biaxial angles Monte Carlo simulation results: (a) Uniaxial and biaxial order parameters as functions of temperature T, for achiral biaxial molecules with ellipsoid separation h = (b) Complete phase diagram for achiral biaxial molecules, in terms of h and T. (c) Uniaxial and biaxial order parameters as functions of T, for chiral biaxial molecules with h = 0.24 and molecular twist angle χ = (d) Boundary twist angle Φ as a function of T. For achiral (χ = 0) molecules, Φ is locked at π, indicating that the system is not twisted. For chiral (χ = 0.08) molecules, Φ is not a multiple of π, indicating that the system is twisted, and the cholesteric twist increases as T decreases. (e) Cholesteric phase of chiral molecules, showing the macroscopic twist Monte Carlo simulation results: (a) plot of biaxial correlation function in the cholesteric phase (T = 0.9) and (b) variation of pitch (P) with molecular chirality (χ) for different h xiii

14 5.6 Theoretical results for cholesteric twist as a function of temperature (in units of interaction strength A). Crosses represent numerical meanfield theory, while the solid line represents Landau theory. In simulations, twist from layer to layer is calculated from the boundary twist angle Φ and is represented by circles A.1 Different splay configuration of neighboring molecules on a lattice B.1 Self-determined boundary condition. Figure adopted from the reference xiv

15 ACKNOWLEDGMENTS In the past six years, I have been supported and assisted by a large number of people that ended my graduate carrier as a success. Foremost, I would like to express my deepest gratitude to my advisor, Dr. Jonathan V. Selinger. I am so fortunate to have an advisor who gave me the freedom to explore on my own, who always encourage me and at the same time the guidance to recover when my steps faltered. His patience, support, encouragement and dedication helped me overcome many difficult situation and finish this dissertation. I am dreaming whether I would be able to become a good advisor to my students as Jonathan has been to me. Dr. Robin Selinger insightful comments and advice at different stages of my research helped me a lot during this work. I am so grateful to her for teaching me the basics of computer simulation and giving input in each of the projects presented in this dissertation. I am also thankful to Dr. David W. Allender and Dr. Antal Jakli for many helpful discussions. I want to thank the other members of my dissertation committee: Dr. Elizabeth Mann, Dr. John Portman, Dr. Hiroshi Yokoyama, and Dr. Paul Sampson. My special thanks also goes to Cindy Miller and Loretta Hauser, Brenda Decker, Lynn Fagan, Dawn Miller and all the office staff, who were an immeasurable help all the time I was here. I am also thankful to all my friends, especially Badel Mbanga, Lena Lopatina, Jun xv

16 Geng, Vianney Gimenez, Parshu Gwyali, Sijan Baral, Bhuvan Joshi, Fanindra Bhatta, Sushil Dhakal, M.T Lama and Govinda Poudel for discussion, help and advices during my stay at Kent State University. Most importantly, none of this would have been possible without the love and patience of my family. My immediate family to whom this dissertation is dedicated to, has been a constant source of love, concern, support and strength all these years. I would like to express my heart-felt gratitude to my family. Finally, I appreciate the support from the following organizations and institutions: NSF that funded parts of the research discussed in this dissertation; the Ohio Supercomputing Center and KSU computer Science department for providing computational facilities. xvi

17 CHAPTER 1 Introduction 1.1 General Considerations For many years, it was believed that matter exists in three states: solid, liquid, and gas. This concept was proved to be wrong in 1888 after Reinitzer [1] found that some cholesterol derived organic compounds undergo a cascade of melting transitions on heating. In those pioneering experiments, he observed that a solid crystal melted into a cloudy liquid at C that persists up to C after which it transforms into transparent liquid. These intermediate phases between solid and liquid are called mesophases or liquid crystalline phases. An essential criterion for liquid crystalline phases to occur is that the molecule must be geometrically anisotropic, for example a rod or a disc. Thus, one can expect molecular structure to be an important factor to dictate the symmetry of a mesophase. Many optical properties of liquid crystals (LCs) may be improved by changing the shape of the constituent molecules. For a majority of LCs the building blocks are Figure 1.1: Examples of liquid crystalline molecule: (a) 5CB and (b) 8CB. 1

18 2 rod-shaped molecules and they are called calamitic LCs. Typical examples are shown in Fig The symmetry of a phase can be described in terms of different types of molecular order present in the system. The important types of order that are usually sufficient to describe mesophases are orientational and positional order. Positional order in a material can be realized regardless of the molecular shape; whereas, orientational order exists only if the constituent molecules are geometrically anisotropic. The center of mass of a molecule or aggregate of molecules in a crystal is fixed at regular points in space and the positional order in 3D is retained. In other words, crystals have long-range positional order of molecules in all dimensions in addition to the orientational order. Crystals that lack orientational order are called plastic crystals. In contrast, liquids lack both positional and orientational order. In between a solid crystal and liquid there may occur a large number of mesophases. Transitions among these mesophases can be initiated by a change in temperature or, in the case of mixtures, concentration, or both. The former class of materials is called thermotropic LCs, and the latter is called lyotropic LCs. Here, I will briefly describe the most common thermotropic LC phases found in rod-like, or calamitic, liquid crystals starting from the one at highest temperature. The phase which is less symmetric than an isotropic liquid (I) is nematic (N) liquid crystal phase. A schematic representation of the molecular ordering in the nematic phase is shown in Fig. 1.2(b). In the nematic phase, there is long-range orientational order and molecules tend

19 Figure 1.2: Different liquid crystal phases: (a) isotropic. (b) nematic. (c) cholesteric. (d) SmA. (e) SmC. 3

20 4 (a) (b) Figure 1.3: Schematic representation of molecular order in (a) nematic and (b) polar phase. The configurations are taken from lattice simulations of a model with a nematic and polar interaction. to align in parallel. The average orientation of the molecules is along a direction ˆn known as the nematic director. As there is no long-range correlation in the centers of mass of the molecules, nematic LCs are fluid and flow like ordinary liquids. Even for the nematic phase formed by polar molecules, the phase is indistinguishable with the transformation ˆn ˆn i.e., molecules are equally pointing up and down as shown in Fig The nematic phase described above is for a system of achiral (having mirror symmetry) molecules. For a nematic phase formed of chiral molecules (absence of mirror symmetry), the molecular director follows a helix on the macroscopic scale, whereas it is very similar to nematic locally. This kind of distortion is also found in cholesterol esters. Hence, this phase is also known as cholesteric or chiral nematic. The equilibrium structure of the cholesteric phase is shown in Fig. 1.2(c). Assuming that

21 z is the helical axis, description of the cholesteric director in space can be written as 5 n x = cos(q 0 z + φ) n y = sin(q 0 z + φ) (1.1) n z = 0, where φ is a constant. In this notation, the period of the cholesteric structure is L = π q 0, where q 0 is the pitch of the cholesteric phase and its sign determines the handedness of the helical structure. For the nematic phase, where the molecules are optically inactive or achiral q 0 = 0. Therefore, a cholesteric phase is nothing but a nematic phase for optically active or chiral molecules. On further reducing the temperature, translational order can develop in either one or two dimensions rather than in three dimensions, resulting in a wide variety of smectic phases. These phases are less symmetric than the nematic(n) phase but are more symmetric than a crystal. The next few paragraphs are devoted to describing common smectic phases for calamitic LCs. The primitive smectic phase in which the positional order is periodic in one dimension (1D) is SmA. The visual scheme of the molecular organization in SmA is previously shown in Fig. 1.2(d). In SmA, the molecules are stacked in layers and aligned parallel to the layer normal. However, there are no long-range correlations in the center of mass of the molecules within the layer. For the layer normal along z, the structure is invariant under the transformation ẑ ẑ. The symmetry of a SmA phase could be further reduced producing a SmC [see Fig. 1.2(e)]. The SmC phase is quite similar to SmA except that the molecules are tilted with respect to the layer

22 6 normal. The molecular tilt in SmC breaks the ẑ ẑ symmetry present in SmA and leaves the SmC structure optically, magnetically and electrically biaxial (a phase that is also anisotropic in the plane perpendicular to the primary director). The symmetry group describing the SmC phase is C 2h. As in SmA, the individual layers are fluid and there is an orientational order. Examples of less symmetric phases than SmC are B HEX, SmF, SmI, and SmL. These hexatic smectic phases are 2D fluids, similar to SmA and SmC, with bond-orientational order ( orientation of imaginary lines joining the centers of nearest neighbor molecules without a regular spacing along the line) within the layer. Further ordering leads to the crystal phase. Finally, a series of transitions among different mesophases could be observed as the temperature is increased. The equilibrium structure is determined by two competing factors-the energetic part which favors an ordered phase and the entropic part that favors randomness. The ordered phase is stable at a very low temperature. However, on increasing the temperature it gradually loses order until an isotropic phase is obtained. In experiments the usual order of stability is solid SmB SmC SmA nematic isotropic. This sequence of transitions is for a polymorphous material. For some other materials, one or more of the intermediate phases might be absent. The sequence of the transitions could then be obtained simply by omitting the phases that are absent.

23 7 Figure 1.4: Typical liquid crystal cell. 1.2 Application of Liquid Crystals At this point it is worthwhile to mention the applications of liquid crystals. The most important application of liquid crystals is in displays. In this section, I describe the basic working mechanism of liquid crystal displays (LCDs). The schematic diagram of a LCD is shown in Figure 1.4. It consists of a liquid crystal cell placed between two glass plates. The orientational order of the liquid crystal inside the cell could be determined by surface treatment (surfactant layers). The arrangement is kept between a pair of polarizing films at the outer surfaces. This arrangement might or might not allow light to pass through the second polarizing film near the reflector. It depends on the order of liquid crystalline medium placed between the plates. For example, in the case of twisted nematic material inside the cell (with 90 0 twist from top to bottom, that could be achieved by surface alignment), a state of polarization of light traveling through such a medium gets rotated by Since the direction of

24 8 the polarizing film on the outside of the bottom piece of glass is perpendicular to the top polarizing film, the light passes through this film and the reflector at the bottom appears silver color. Upon applying an electric field across the indium-tin oxide (ITO) films, the twist in the director field is destroyed since the applied field tends to align the liquid crystal in its direction. In this state, light is extinguished by the polarizer on the bottom piece of the glass and the cell appears dark. This describes the basic working mechanism of the LCDs. However, there are several factors to be considered for a commercial display. First, it is the material itself. It must be stable over a long period of time and it must show liquid crystalinity at all temperatures at which the device is likely to be used. Second, the dielectric and mechanical properties of these materials should allow it to be switched at high speed. Third, the design of LCDs must consider the sharpness of the display response as they are turned on and off. Finally, minimizing the power consumption is another important issue. Therefore, investigation of new materials to improve these properties for the advances in the field of LC technology is at the frontier of liquid crystal research at the moment. 1.3 Experiments on Liquid Crystals In this section, I present some of the recent experiments on liquid crystals which motivates my research projects. One major area of interest is on ferroelectric liquid crystals because of their potential applications in LCDs. In most of the ferroelectric LCDs, chirality at the molecular level is exploited to develop a ferroelectric order

25 9 Figure 1.5: Schematic of a bent-core molecule in the system. In recent experiments, Link et al. [2 6] demonstrated a different mechanism to realize macroscopic chirality in a system of achiral bent-core molecules. An example of a typical bent-core molecule is shown in Fig Because of the packing constraint, the organization of these molecules in a smectic layer develops a polar order in the layer plane. If the molecules are tilted with respect to the layer normal, the combination of the tilt and polar order breaks the symmetry of the layer giving a chiral structure. This unique feature of bent-cores has attracted more researchers for experimental and theoretical studies. It has been observed in the experiments that one could induce phase transitions between chiral and antichiral structures with an electric field. The reversibility of these states with an applied field implies the possibility of making light shutters [2,6] from antiferroelectric liquid crystals of bent-core molecules. One of the themes of this dissertation is to develop a theoretical model to understand these chiral switching experiments. Another thrust in experimental liquid crystal science is to improve power consumption in LCDs. Low power consuming LCDs relying on the flexoelectric effect (coupling between the director distortion and the electric polarization in the nematic)

26 10 has been sought for a long time. Since its discovery in 1969 by Meyer [8], the flexoelectric effect has generated a tremendous interest because of its possible applications in strain gauges, transducers, micro power generators and electro-optical devices. For calamitic liquid crystals, the splay and bend flexoelectric coefficients are in the range of 3-20 pc/m. However, in recent experiments, Harden et al. [9, 10] found that bent-core liquid crystals have a surprisingly large bend flexoelectric coefficient, up to 35nC/m, roughly three orders of magnitude larger than the typical value. The question to be answered with regard to this problem is why is the behavior so different from that of calamitic liquid crystals? Another area of interest is about experiments on some biological polymers. Several bio-polymers, for example DNA, protein -α helices and rod like viruses can develop a liquid crystalline order at very high concentration. The most common is the cholesteric phase. It was found in these experiments that the cholesteric pitch is very large (and hence small twist) as compared to the molecular twist. In experiments on rod like fd and M13 viruses [11], the pitch of the so formed cholesteric phase is so large that it is difficult to measure. A few of the questions that arise are: what determines the cholesteric pitch and what is the relation between the twist at the microscopic and macroscopic length scales? One of the focal points of my research is to answer these fundamental questions about the cholesteric phase by a simple molecular model.

27 Macroscopic and Microscopic Order Parameters In microscopic theories of nematic LCs, the constituent molecules are treated as rods where the molecular axis is uniquely defined. The order of the phase could be determined by taking a statistical average of the molecular orientation with respect to the nematic director. The requirement for the order parameter (S) is that S = 0 in the isotropic phase and S 0 in the nematic phase. The quantity satisfying these criteria is the second Legendre polynomial S = P 2 (cosθ) = 3 cos2 θ 1. (1.2) 2 This is essentially an adequate definition to measure the order when the molecules are highly elongated. But when the molecular shape departs significantly from rods (for example broad shape), this microscopic definition [Eq. (1.2)] of the order parameter is no longer valid. In that case, one should rely on macroscopic responses of the system in an external field. These might be magnetic, electric or optical responses. To an experimentalist, it means the measurement of diamagnetic susceptibility, dielectric constants or refractive indices respectively. The tensor order parameter could be expressed in terms of the diamagnetic susceptibility [13] as Q ij = 2( χ) , where χ is the difference between susceptibility for a field parallel to the director and for a field perpendicular to the director. This clearly vanishes in the isotropic

28 12 phase but is not zero in the nematic phase and is a proper order parameter. Hereafter, we will call it a macroscopic order parameter. A relation between macroscopic and microscopic order parameters can be derived as explained in reference [13]. The relation is Q ij = P 2 (cosθ) Theoretical Studies on Liquid Crystals Of many theories, two analytical approaches to study the phase transition and various properties of the ordered phases are phenomenological theory and meanfield theory. In phenomenological theory, it is assumed that the order parameter is small near the phase transition and the free energy of the system is expanded in terms of the order parameters. This theory yields a wealth of information about the I N phase transition in a liquid crystal and is still being used to understand the equilibrium thermodynamics of a system. On the other hand, mean-field theory is an approximation which assumes that the force experienced by a molecule is the same as the force experienced by others in its surrounding. It means there are no spatial fluctuations in the order parameter. Therefore, mean-field theory becomes an exact theory when the range of the interaction becomes infinite. Because it is mathematically simple and has the ability to explain the I N transition, it is one of the first approaches taken to predict the phase diagram of a new system. Therefore, these two methods will certainly be the approaches used to understand the problems

29 in subsequent chapters. Here, I present examples of these theories to describe the nematic phase and the I N phase transition Landau-de Gennes Model for the Nematic Phase One analytical approach to understand the I N transition is a simple Landautype theory as first developed by de Gennes. In this theory, the detailed nature of molecular interactions is ignored. The free energy is expanded in terms of the order tensor (Q) as F = F 0 + A(T)Tr(Q 2 ) + B(T)Tr(Q 3 ) + C(T)Tr(Q 4 ), (1.3) where A(T), B(T) and C(T) are constants depending on temperature and material properties. F 0 is the free energy of the isotropic phase. For simplicity, here we assume that B(T) and C(T) are independent of temperature, as is typical for landau theories. Then, the free energy in terms of the scalar order parameter (S) with rescaled coefficients is F = F 0 + a(t) 2 S2 + b 3 S3 + c 4 S4. (1.4) Absence of a linear term in S ensures that S = 0 is a solution to df ds = 0, i.e., an isotropic phase (S = 0) is the minimum of F at high temperature. As usual, a(t) = a(t T 0 ), where T 0 is a temperature in the vicinity of I N. Though, we can get some insight into the phase transition from a plot of F as a function of S and T, it is imperative to carry out some analytical calculations first. At a glance, for the minimum of the free energy to be at finite value of S, clearly, we

30 14 need c > 0. The condition for the extremum of the free energy is df ds = 0 = a(t)s + bs2 + cs 3. (1.5) Solutions to Eq. (1.5) are S = 0 and S = b± b 2 4ac 2c. Here S = 0 and S 0 corresponds to the isotropic and nematic phases respectively. The temperature (T NI ) at which the I N takes place can be calculated by equating the free energy of the isotropic phase (S = 0) with that of the nematic phase(s 0) i.e., F(S = 0, T = T NI ) = F(S 0, T = T NI ) a(t)s bs3 3 + cs4 4 = 0. (1.6) Solving Eqs. (1.5) and (1.6), we have S(T NI ) = 0, S(T NI ) = 2b 3c. (1.7) Substituting the non -zero value of the order parameter in Eq. (1.6), we obtain T NI = T 0 + 2b2 9a 0 c. We can check whether these points correspond to the minimum of the free energy by evaluating the second derivative. Since, 2 F 2 S T=T NI,S=S(T NI ) = 2b2 9a 0 c > 0, these points essentially corresponds to the minimum of the free energy. The temperature T 0 may be inferred as the super-cooling limit of the isotropic phase. Alternatively, the isotropic phase exists down to the temperature of T 0, however the nematic phase has the lower energy at that temperature. Similarly, we can estimate the super heating limit of the nematic phase. Let us assume that the nematic phase exist up to the

31 15 4 x F/Nk B T 2 s 0.4 Nematic Isotropic 4 T >T NI T =T NI 0.2 T <T NI S (a) T (b) Figure 1.6: Free energy and order parameter in Landau-de Gennes model for nematic. point of T. At this point, F S = 0 and 2 F 2 S = 0. Solving these equations, we get T = T NI + b2 9a 0 c. (1.8) The variation of free energy with order parameter at different temperatures is shown in Fig. 1.6(a). Variation of the order parameter with temperature is shown in Fig. 1.6(b). These plots clearly show that as the system is cooled down from the high temperature isotropic phase, it transforms into the ordered phase at T = T NI. As a final remark, the Landau-de Gennes model for the nematic phase could be extended for smectic and cholesteric phases but is beyond the scope of this chapter Maier-Saupe Theory for Nematic Among different molecular theories of liquid crystals, the preeminent one is the original work of Maier and Saupe [14], more commonly known as Maier-Saupe theory.

32 16 In their initial work, it was assumed that molecules are cylindrically symmetric and it is a reasonable approximation for the majority of liquid crystal molecules forming nematic phase. In this theory, van der Waals dispersion interactions between pairs of molecules are assumed to dominate the energy of the system. The Maier-Saupe theory starts with a simple expression for the interaction energy that depends only on the orientation between the molecules. The interaction energy is U(Ω i, Ω j ) = A V 2P 2(cosθ ij ), (1.9) where V is the molar volume and θ ij is the angle between the long axes of molecules i and j. P 2 (cos θ ij ) is the second order Legendre polynomial and A is a constant independent of pressure, volume and temperature. In a mean-field calculation, it is assumed that a molecule is embedded in a sea of many molecules and each one experiences the same force as the others. The mean potential experienced by a molecule is u i = U(Ω i, Ω j ) j A V 2 P 2(cosθ i )P 2 (cosθ j ) j = A V 2P 2(cos θ i )S. (1.10) From the one body potential u i, we can construct a distribution function that gives the probability of a molecule being oriented at an angle θ i with respect to the nematic director. The single particle distribution function is f(θ) = exp( u i /k B T) 1 exp( u 0 i/k B T)d(cosθ i ), (1.11)

33 F/Nk B T T >T NI T =T NI s Nematic Isotropic T <T NI S (a) T (b) Figure 1.7: Free energy and order parameter in Maier-Saupe theory for nematic where k B is the Boltzmann constant and, T is the absolute temperature. In this construction, the internal energy per mole is the averaged value of the single particle potential in Eq. (1.10) and could be written as where α = U = N u iexp( u i /k B T)d(cosθ i ) 1 0 exp( u i/k B T)d(cosθ i ) = Nk BTαS 2, (1.12) 2 A V 2 k B. The entropy of the system is T S e = Nk B [αs(2s + 1) 2 ln 1 0 exp( 3αS cos2 θ i )d(cosθ i ) ], (1.13) 2 where the symbol S e is used to differentiate entropy with the order parameter(the convention is that entropy be denoted by S). The Helmholtz free energy of the system

34 18 is F = E TS e = Nk B T [ αs(s + 1) 2 ln 1 0 exp( 3αS cos2 θ i )d(cosθ i ) ], (1.14) 2 The plot of free energy as a function of the order parameter (S) at different temperatures evaluated from Eq. (1.14) is illustrated in Fig. 1.7(a). Clearly, for T > T NI, there is only one minimum in the free energy corresponding to S=0 and represents an isotropic phase. For T = T NI, there are two minima, one at S=0 and other at S 0. Finally, for T < T NI, there is only one minimum at S 0 which corresponds to an ordered phase. Though, the Maier-Saupe theory was able to explain most of the properties of the nematic phase, there are a few issues to be considered. The first one is that this theory doesn t take into account the long wavelength orientational fluctuations that could diminish the effective order parameter. Secondly, the assumption that the molecule is a rigid rod is an oversimplification. Real molecules contains flexible endchains that could affect the ordering process. Despite these limitations, mean field theories have been found to be capable of describing the qualitative behavior of many different co-operative phenomena. Therefore, it is essentially a subject of discussion in this dissertation.

35 Computer Simulation in Liquid Crystals The structure-property relationship in mesophases could be investigated through the synthesis of a series of similar mesogenic units. This, however, can be time consuming and may involve many difficult and expensive syntheses and characterizations. In these circumstances, simulations can play an important role that might suggest chemists should synthesize an alternative structure. In this section, I present an overview of simulation techniques appropriate for liquid crystals. A major challenge in these simulations stems from the properties of LC systems that involve different time and length scales and non-linear responses with external perturbation. As a consequence, a range of computer models are being used in these studies. One of the simplest approaches is the lattice simulation. The prototype for these models is the original Lebwohl-Lasher model [19]. In this model, each molecule is represented by 3D spins fixed at lattice points of a simple cubic lattice. The interaction is limited to the nearest neighbors and is written as V ij = A(ˆn i ˆn j ) 2, (1.15) where A is the strength of nematic interaction. This model has been successful in explaining most of the properties of the nematic phase and the I N phase transition. The second type of model falls in the category of hard core repulsion. The potential could be written as r ij < r 0 v ij = 0. (1.16) In these simulations, temperature is a redundant variable and the phase transition is

36 20 mainly governed by the density of the system. Another purely repulsive model for the mesogens has been used in many simulations [15 17]. It is the Lennard-Jones potential shifted and truncated at the minimum, also known as the WCA potential [18] [( ) 12 ( ) 6 ] σ 4ɛ ij σ ij r ij ij r ij + 1 r 4 ij < r 0 v ij = 0 r ij > r 0. (1.17) In these simulations, the mesogenic units are represented by a collection of spheres. In recent years, significant research effort has been directed toward the development of the force field with atomistic details. In these methods, the functional form of the force field that can be used to model assemblies of atoms and/or molecules is: E ff = k l 2 (l i l eq ) 2 + k θ 2 (θ i θ eq ) 2 + i i torsions N N 1 [( ) 12 ( ) 6 ] σij σij N N 1 + 4ɛ ij + r ij r ij i=1 j=i+1 i=1 j=i+1 V n [1 + cos(nω γ)] 2 q i q j 4πɛ 0 r ij (1.18) Here, the first two terms represent the intra-molecular interactions and accounts for bond stretching, bond angle bending respectively. The third term is a torsional potential that accounts the energy cost for bond rotations. The last term represents the electrostatic interaction. The non-bonded interactions are modeled by a Lennard- Jones 12-6 potential. Among these models, a particular choice depends on the system and the properties to be studied. As the molecular structure greatly determines the macroscopic behavior of mesophases, a slight change in molecular configuration might influence the

37 21 macroscopic behavior (optical, electric, magnetic, and structural) of LCs significantly. These changes in LCs take place at different time and length scales. For example, the changes in intra-molecular configuration (bond-vibration, bond-stretching, rotation etc.) occur at the time scale of picoseconds and length scale of nanometers, whereas the macroscopic variables change at the time scale of milliseconds and length scale of millimeters. The wide span of different liquid crystal phenomena on the temporal and spatial scales makes the liquid crystal simulation much more complicated. To study fine-structure details, a choice might be the microscopic model as presented in Eq. (1.18). The disadvantage is that it is computationally expensive and the size of the system that could be studied might be small. In that case, the result for macroscopic properties might be less accurate. On the other hand, one can use a coarse-grained model where a group of atoms is represented by an interaction site. In the case of a high degree of coarse-graining where the molecular details are ignored and the molecules are represented by a single interaction site, the simulations are computationally efficient and one could work out a complete phase diagram in a reasonable amount of computer time. This approximation might be useful if one is interested in the macroscopic behavior of a system without fine-structural details and it is employed for most of the work that I will present in this dissertation. In subsequent chapters, I present an interaction potential that defines the microscopic energetics of the simulated system. In these models, a molecule is described by a single or multiple interaction sites. The interaction between the molecules is restricted to two body interaction and quantum mechanical effects are ignored. As

38 we are interested in statistical evolution of a large collection of molecules, our choice for these simulations is Monte Carlo methods Scope of the Present Work Simulations and analytical calculations have been carried out to study different liquid crystal phenomena. The first one is about chirality switching in the SmCP phase of bent-core liquid crystals. The details are presented in Chapter 2. The results of the studies on the flexoelectric effect of nematic liquid crystals are presented in Chapters 3 and 4. Chapter 3 presents the splay flexoelectric effect in a nematic phase of pear-like molecules while the bend flexoelectricity in bent-core liquid crystals is presented in Chapter 4. In Chapter 5, a molecular model with dispersion interactions for chiral molecules is presented. The approach to this problem is three fold: computer simulation, meanfield calculation and a phenomenological theory. Finally, Chapter 6 summarizes these studies.

39 BIBLIOGRAPHY [1] F. Reinitzer, Monatsch Chem., 9, 421 (1888). [2] G. Heppke, A. Jakli, and S. Rauch, Phys. Rev. E 60, 5575 (1999). [3] G. Pelzl, S. Diele, A. Jakli, Ch. Lischka, I. Wirth, and W. Weissflog, Liq. Cryst. 26, 135 (1999). [4] T. Niori, T. Sekine, J. Watanabe, T. Furukawa, and H. Takezoe, J. Mater. Chem. 6, 1231 (1996). [5] D. R. Link, G. Natale, R. Shao, J. E. Maclennan, N. A. Clark, E. Korblova, and D. M. Walba, Science 278, 1924 (1997). [6] A. Jakli, S. Rauch, D. Ltzsch, and G. Heppke, Phys. Rev. E 57, 6737 (1998). [7] J. V. Selinger, Phys. Rev. Lett. 90, (2003). [8] R. B. Meyer, Phys. Rev. Lett. 22, 918 (1969). [9] J. Harden, B. Mbanga, N. Eber, K. Fodor-Csorba, S. Sprunt, J. T. Gleeson, and A. Jakli, Phys. Rev. Lett. 97, (2006). [10] J. Harden, R. Teeling, J. T. Gleeson, S. Sprunt, and A. Jakli, Phys. Rev. E 78, (2008). [11] Z. Dogic and S. Fraden, Phys. Rev. Lett. 78, 2417 (1997). [12] A.B. Harris, R.D. Kamien, T.C. Lubensky, Phys. Rev. Lett. 78, 1476 (1997). [13] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Oxford Science, Oxford, 1993). [14] W. Maier and A. Saupe, Z. Naturforsch. A 14A, 882 (1959). [15] J. Xu, R. Selinger, J. Selinger, B. Ratna, and R. Shashidhar, Physical Review E 60,5584 (1999). [16] J. Xu, R. L. B. Selinger, J. V. Selinger, and R. Shashidhar, J. Chem. Phys. 115, 4333 (2001). [17] M. A. Glaser, R. Malzbender, N. A. Clark, and D. M. Walba, J. Phys.: Condens. Matter 6, A261 (1994). [18] J. D. Weeks, D. Chandler and H. C. Anderson, J. Chem. Phys (1971). [19] P. Lebwohl and G. Lasher, Phys. Rev. A 6, 426 (1972). 23

40 CHAPTER 2 Field Induced Phase Transition in SmCP phases of Bent-Core Liquid Crystals. In this chapter, studies on chiral switching in smectic phases of bent-core liquid crystals are presented. The phenomenon is investigated with a lattice Monte Carlo simulation and a phenomenological theory. Both methods show that the antiferroelectric (AFE) ground state switches into a ferroelectric state (FE) at sufficiently large electric field. In this transition, the chirality of the phase could change or not. In the case of weak inter-layer chiral coupling, anticlinic order of molecules from layer to layer could remain through the transition -thus changing the chirality of the phase. However, for a large value of chiral coupling, the anticlinic state could change to a synclinic state in the transition -thus preserving the chirality of the phase. 2.1 Review of the SmCP Phases of Bent-Core Liquid Crystals In recent years, ferroelectric liquid crystals have been of great interest because of their potential applications in display technology. Most ferroelectric LCDs rely on the chirality of mesophases that originates at the molecular level. However, in recent experiments, Link et al. [1,2] have shown that macroscopic chirality could be realized in a system of achiral bent-core molecules as well. Though similar molecules have been known for a long time, these observations have generated tremendous interest in ferroelectricity and chirality in achiral bent-core liquid crystals. 24

41 Figure 2.1: Visual scheme of different SmCP phases: (a) SmC A P A. (b) SmC A P F. (c) SmC S P A. (d) SmC S P F. 25

42 26 A bent-core molecule has the symmetry of arrow and bow as shown in Fig The smectic phases of these molecules have polar order in the layer plane due to close packing. If the molecules are tilted with respect to the layer normal, the combination of tilt and polar order breaks the mirror symmetry of the layers giving a chiral structure, or handedness to each layer. The phases with a tilt and polar order are denoted by SmCP. In this notation, Sm stands for smectic phase, and the letters C and P refer to the state of tilt and polarization respectively. A structure is said to synclinic or anticlinic if the molecules in successive layers tilt in the same or opposite direction. Similarly, a structure is said to be ferroelectric or antiferroelectric according to whether the molecular dipoles in successive layers point in the same or opposite direction. As the tilt and polar order defines the handedness of a layer, the macroscopic phases could be chiral or antichiral. A structure is said to be chiral if the neighboring layers have the same handedness. However, the macroscopic phase can have alternating layer chirality and is called racemic [1] or antichiral [3]. In this dissertation, I will use the term antichiral as proposed by Selinger [3] to emphasize the alternation of right and left handed domains. The possible phases with different combinations of tilt and polarization are: SmC S P F (synclinic and ferroelectric), SmC A P A (anticlinic and antiferroelectric), SmC S P A (synclinic and antiferroelectric) and SmC A P F (anticlinic and ferroelectric). The first two are the chiral states, and the other two are antichiral states. Molecular order in these phases are shown in Fig One can induce a phase transition from an antiferroelectric (AFE) to a ferroelectric

43 27 state (FE) with an electric field. The reversibility of these two states over the field implies the possibility of making light shutters from antiferroelectric bent-core liquid crystals [5, 6]. To a theoretical physicist, an eminent question is the mechanism for these chiral switching experiments. In an effort to understand the molecular configuration in SmCP phases and their structural changes in an external field, several theoretical models have been proposed [3, 4]. These models considered different symmetry breaking instabilities in smectic layers. In reference [3], only chiral order was considered as the symmetry of the system. In another reference [4], the coupling between polar and tilt order was considered. The aim of this study is to consider an explicit coupling between all three orders: chirality, polarity and tilt. 2.2 Lattice Monte Carlo Simulation for Polar, Chiral and Tilt Order In order to study chiral switching in SmCP phases of bent-core liquid crystals, we construct a lattice model considering three symmetry breaking instabilities - chiral order (χ), polar order (P) and tilt order (θ). Here, we assume that the molecular order at a lattice site i is represented by three Ising spins corresponding to chirality (χ i ), polarity (P i ) and tilt (θ i ). If the layer normal to the smectic planes is along the

44 28 z direction, the lattice Hamiltonian is H = A x,y χ i χ j B x,y P i P j C x,y θ i θ j C χ χ i χ j + C p P i P j + C θ θ i θ j z z z g 1 χ i P i θ i E P i. (2.1) i i In this model, the first three terms represent intra-layer coupling of the different orders. The positive values of constants A, B and C ensure a well defined order within a layer. The next three terms accounts for inter-layer coupling. Note that in this Hamiltonian the coupling between different orders enters as the product χp θ. The combination χp θ is symmetric under reflection. It means changing the sign of either P or θ, but not both simultaneously, changes the chirality of the system. E is an applied electric field acting on the polar order (P). The resulting phase diagram for this model is a consequence of subtle interplay between different coupling interactions. For example, a larger value of C χ favors homogeneous chiral structure from layer to layer and increases the probability that the system will be in a chiral state. Our first approach to the problem is a lattice Monte Carlo simulation. In these simulations, each molecule is fixed at a lattice site of a simple cubic lattice. It has periodic boundaries in all directions. There are three Ising spins associated with each molecule- χ i = ±1, P = ±1 and θ = ±1. In each Monte Carlo step, a lattice site is chosen randomly, a randomly chosen Ising spin (χ, P, θ) is flipped, and the change in energy H is calculated. The usual Metropolis algorithm was used for lattice updates. If H < 0 the move is accepted, and if H > 0 the move is accepted

45 29 with probability exp( H/k B T). Starting from a zero field, the system is subjected to a gradually increasing external electric field in steps of E = The final configuration at each field is taken as the initial configuration for the next higher field. The order parameters are O i = O 1 O 2 for odd layer for even layer, (2.2) where O = χ, P, θ and are averaged values of the spin variables within a layer. The order parameters are time averaged during the production cycle. Monte Carlo simulations have shown three types of phase transition on increasing the electric field: (i) SmC A P A SmC A P F (ii) SmC A P A SmC S P F and (iii) SmC S P A SmC S P F. I present a complete description of each type of transition in the following subsections SmC A P A SmC A P F transition Plot of order parameters as a function of electric field for a small inter-layer chiral coupling (C χ = 0.2) is shown in Fig The ground state at zero field is SmC A P A. On increasing the field, at the AFE FE transition, the anticlinic order from layer to layer is retained. The system thus displays a first-order transition from chiral to antichiral structure. Fig. 2.2(d) shows the electric field dependence of polarization, which is P 1 +P 2. It is clearly seen that polarization increases almost linearly for small value of the field as in the experiments [5, 6]. This indeed indicates that the antiferroelectric order between smectic layers is distorted even at very low field.

46 χ P χ 2 Chiral Antichiral 0.5 AFE FE 0 P E (a) θ 1 θ 2 Anticlinic (c) E Polarization(P) E (b) E E (d) E Figure 2.2: Monte Carlo simulation results showing electric field dependence of order parameters: (a) chiral (b) polar (c) tilt and (d) polarization, for interaction strengths A = 0.6, B = 0.6, C = 0.6, C χ = 0.2, C p = 0.4, C θ = 0.5, g 1 = 1.0.

47 E (a) Anticlinic Chiral χ 1 χ 2 Synclinic θ 1 θ (c) E Polarization(P) AFE FE P 1 P E (b) E E (d) E Figure 2.3: Monte Carlo simulation results showing electric field dependence of order parameters: (a) chiral (b) polar (c) tilt and (d) polarization, for the interaction strengths A = 1.0, B = 0.6, C = 0.7 C χ = 0.4, C p = 0.2, C θ = 0.2, g 1 = SmC A P A SmC S P F transition The variation of order parameters with the field for a stronger inter-layer chiral coupling (C χ = 0.7) is summarized in Fig The ground state at zero field is still SmC A P A. On increasing the field, the tilt order also changes discontinuously at AFE FE transition. In contrast to the previous case, the anticlinic order changes to the synclinic order. This altogether leads to a first-order phase transition where the chirality is preserved. These observations are in qualitative agreement with experiments [1,5,6]. The electric field dependence of spontaneous polarization is illustrated in Fig. 2.3 (d). However, this dependence is markedly different from that of SmC A P A

48 χ P Antichiral χ 2 Chiral AFE P 2 FE E (a) E (b) Synclinic θ 1 θ 2 Polarization(P) 0.5 E E (c) E (d) E Figure 2.4: Monte Carlo simulation results showing electric field dependence of order parameters: (a) chiral (b) polar (c) tilt and (d) polarization, for the interaction strengths A = 0.6, B = 0.6, C = 0.6 C χ = 0.2, C p = 0.4, C θ = 0.5, g 1 = 1.0. SmC A P F transition. It is clearly seen that there is considerable hysteresis in the polarization-field curve. However, we did not observe hysteresis in the polarizationfield curve in similar simulations at higher temperatures. In view of these results, it seems conceivable that at the low temperature, as expected, the number of metastable states with mixed chiral and antichiral domains is very large- hence, the simulated system might have been trapped in one of these states.

49 SmC S P A SmC S P F transition In this model, homogeneous tilt between smectic layers could be obtained at zero fields by reversing the sign of the inter-layer tilt coupling. For C θ = 0.5, the ground state at zero-field is SmC S P A. On increasing the field, the AFE FE transition is accompanied by a transition from antichiral to the chiral state. Therefore, the phase remains synclinic for the entire range of the field studied. These variations are shown in Fig The field dependence of polarization is very similar to that of the SmC A P A SmC A P F transition. 2.3 Phenomenological Free Energy In this section, I present another approach to understand the problem. In this method, the free energy is constructed based on symmetry considerations. It consists of the following- terms for intra-layer ordering, terms with inter-layer ordering, coupling between different orders and a field term. Here, we take the average of different orders over each layer to get an effective order parameter for that layer. Then, the free energy is expressed in terms of effective layer order parameters summed over all layers. The free energy of a smectic phase in terms of chiral, polar and tilt order is F = A 1 i B 1 χ 2 i 2 + A χ 4 i 2 4 C χ i i C 1 i P 2 i 2 + B 2 θ 2 i 2 + C 2 i G 1 χ i P i θ i E i i i χ i χ i+1 i P 4 i 4 + C p θ 4 i 4 + C θ P i P i+1 i θ i θ i+1 i P i, (2.3)

50 34 where the sum is over all smectic layers and A 1, A 2, B 1, B 2, C 1, C 2, C χ,c p, C θ and G 1 are phenomenological constants. E is an applied electric field. The phenomenological constants depend on material properties. They are generally temperature dependent. The distinct values of these constants imply the discrimination of one interaction over other. For example, the positive signs of C χ, C p and C θ favors homochiral (same handedness), anti-ferroelectric and anti-clinic order from layer to layer respectively. The interaction constant G 1 represents coupling of all three orders and is more subtle. To describe the phase diagram quantitatively, we use the same order parameter definition as in Eq. (2.2). The parameter space was reduced by rescaling the free energy and order parameters. The free energy in reduced parameter space is f = χ2 1 + χ χ4 1 + χ a 2(θ θ2 2 ) 2 a 1(P P 2 2 ) 2 + a 2(θ θ4 2 ) 4 + a 1(P P 4 2 ) 4 c 0 χ 1 χ 2 + c 1 P 1 P 2 + c 2 θ 1 θ 2 g 1 (P 1 χ 1 θ 1 + P 2 χ 2 θ 2 ) E(P 1 + P 2 ), (2.4) where a 1, a 2, c 0, c 1, c 2, g 1 and E are rescaled coefficients. 2.4 Numerical Analysis of the Phenomenological Model In this section, the numerical analysis of the phenomenological model presented in Sec. 2.3 is carried out. The field dependence of order parameters are calculated from numerical minimization (using conjugate gradient method [7]) of the free energy in Eq. (2.4). On exploring the phase diagram for the model, three types of phase transition have been found as in simulations: (i) SmC A P A SmC A P F (ii) SmC A P A SmC S P F and (iii) SmC S P A SmC S P F.

51 35 P 1,P P (a) E χ 1,χ χ (b) E P 2 χ 1 AFE FE Chiral Antichiral Figure 2.5: Schematic representation of the electric field dependence of the order parameters: (a) polar order, showing antiferroelectric to ferroelectric transition (b) chiral order, showing chiral to antichiral transition. The results correspond to the values: a 1 = 1.0, a 2 = 1.0, c 0 = 0.2, c 1 = 0.4, c 2 = 0.3, g 1 = SmC A P A SmC A P F transition Numerical minimization is performed using a 1 = 1.0, a 2 = 1.0, c 0 = 0.2, c 1 = 0.4, c 2 = 0.3, g 1 = 0.4. The results for the order parameters are reported in Fig We see that the chirality of the system changes in the transition. At this transition, we have χ 1 = χ χ 2 = χ P 1 = P P 2 = P χ 1 = χ χ 2 = χ P 1 = P P 2 = P (2.5) θ 1 = θ θ 2 = θ θ 1 = θ θ 2 = θ. The necessary conditions to be satisfied at this transition are F CA P A = F CA P F F CA P A P = 0 F CA P F χ = 0. (2.6)

52 36 P 1,P P (a) E θ 1,θ θ (b) E P 2 θ 1 AFE FE Anticlinic Synclinic Figure 2.6: Schematic representation of the electric field dependence of the order parameters: (a) polar order, showing antiferroelectric to ferroelectric transition (b) tilt order, showing anticlinic to synclinic transition. The results correspond to the values: a 1 = 1.0, a 2 = 1.0, c 0 = 0.7, c 1 = 0.4, c 2 = 0.3, g 1 = 0.4 Assuming that E th be the field at the transition, evaluating these equations with the values in Eq. (2.5), we get E Th P c P P 2 c χ χ 2 = 0 (2.7) a 1 P c P P + a 1 P 3 g 1 θχ = 0 (2.8) g 1 Pθ χ + c χ χ + 2χ 3 = 0. (2.9) The value of the field and order parameters at the transition can be estimated solving these equations with an approximation that the tilt is uniform (θ 1) near the transition. The solutions to these equations for the same set of interaction parameters as in Fig. 2.5 are: E Th = 0.72, χ = 1.12 and P = These values agrees with those obtained in numerical minimization [see Fig. 2.5].

53 SmC A P A SmC S P F transition : In analogy with simulations, upon making the inter-layer chiral coupling stronger, we observed SmC A P A SmC S P F phase transition. The numerical minimization results are depicted in Fig In this transition, we have χ 1 = χ χ 2 = χ P 1 = P P 2 = P χ 1 = χ χ 2 = χ P 1 = P P 2 = P (2.10) θ 1 = θ θ 2 = θ θ 1 = θ θ 2 = θ. The following conditions must be satisfied at the transition F CA P A = F CS P F F CA P A P = 0 F CA P A θ = 0. (2.11) If E = E Th at the transition, evaluating these equations with values in Eq. (2.10), we get 2E Th P 2c p P 2 2c θ θ 2 = 0 2a 1 P 2c p P + 2a 1 P 3 2g 1 θχ = 0 2a 2 θ 2c θ θ + 2a 2 θ 3 2g 1 Pχ = 0. (2.12) These equations are solved for the same set of the interaction parameters in Fig The results are: E Th = 0.92, χ = 1.31 and P = These estimates agree with the numerical minimization results [see Fig. 2.6].

54 38 P 1,P P (a) E χ 1,χ χ Antichiral (b) E P 2 χ 1 AFE FE Chiral Figure 2.7: Schematic representation of the electric field dependence of the order parameters: (a) polar order, showing antiferroelectric to ferroelectric transition (b) chiral order, showing antichiral to chiral transition. The results correspond to the values: a 1 = 1.0, a 2 = 1.0, c 0 = 0.2, c 1 = 0.4, c 2 = 0.7, g 1 = SmC S P A SmC S P F transition: Fig. 2.7 shows the field dependence of order parameters obtained in numerical minimization of the free energy for c 2 = 0.7. At this transition, we have χ 1 = χ χ 2 = χ P 1 = P P 2 = P χ 1 = χ χ 2 = χ P 1 = P P 2 = P (2.13) θ 1 = θ θ 2 = θ θ 1 = θ θ 2 = θ. The following conditions must satisfied at the transition F CS P A = F CS P F F CS P A P = 0 (2.14) F CS P A χ = 0. Evaluating these equations with values in Eq. (2.13), we get

55 39 2E th P 2C p P 2 + 2C χ χ 2 = 0 2a 1 P 2C p P + 2a 1 P 3 2gθχ = 0 2gPθ 2χ + 2C χ χ + 2χ 3 = 0. (2.15) These equations are solved for the same set of interaction parameters as in Fig The solutions are: E Th = 0.335, χ = 1.13 and P = This is another illustration that this numerical prediction exactly agrees with minimization results [Fig. 2.7]]. 2.5 Is Chirality an Independent Order Parameter? One important question that remains to answer with regard to chirality of a phase is whether the system has independent spontaneous chiral order, or whether chirality is just a consequence of the combination of tilt and polarity. If there is independent spontaneous chiral order, then we expect that the free energy should have minima at positive and negative values of χ. This free energy can be represented as χ 2 + χ 4, as discussed in the previous sections. By contrast, if chirality is just induced by the combination of tilt and polarity, then we expect that the free energy should have a minimum at χ = 0, which is slightly perturbed by the coupling with tilt and polarity. This free energy can be represented as +χ 2. In this section, our goal is to test whether the chirality is an independent order parameter. The free energy [Eq. 2.4] is modified to represent a poor chiral order from layer to layer. In this case, the intra-layer chiral term in the free energy is represented by χ2 1 +χ2 2 2 without quartic terms. A phase diagram showing the electric field dependence of order parameters

56 χ 1,χ Chiral χ 1 Antichiral (a) E 1 P χ 1 2 P 1,P AFE FE P (b) E Figure 2.8: Order parameters as a function of applied field for poor chiral model: (a) chiral order, showing chiral to antichiral transition (b) polar order, showing antiferroelectric to ferroelectric transition. The results correspond to the same values of the interaction parameters that has shown the preserved chirality [Fig. 2.6] for the model with well defined chiral order in a smectic layer. for the same interaction parameters as in Fig. 2.6 is shown in Fig It is clearly seen that the chirality of a system changes during AFE FE transition with a poor chiral order in a layer. This result doesnot agree with the experiments. Therefore, we might conclude that the chirality of a system is not just induced by tilt and polarity, but it is an independent order parameter. 2.6 Summary The main focus of this chapter has been on electric field induced phase transitions among different SmCP phases of bent-core liquid crystals. Experiments have shown that the chirality of the system may be preserved or may change at the AFE FE transition. To understand these experiments, a lattice model for SmCP phases of bent-core liquid crystals is developed. The model considers an explicit coupling between three symmetry breaking instabilities: chirality, polarity and tilt. The unique

57 41 feature of the model is that the coupling term (χpθ) between different types of order is invariant under reflection. In this construction, changing the sign of one of them induces a discontinuous change in another one, while the third is kept fixed. Through this model, the phase behavior is studied as a function of applied field. This model shows that SmCP phase of achiral bent-core molecules exhibits two antiferroelectric structures-the SmC A P A and SmC S P A. The preference of a system over these states depends on the inter-layer interactions. It has been found that at sufficiently large value of the field, antiferroelectric (AFE) state changes into ferroelectric state (FE). Simulation of the model have shown three types of the transitions: (i) SmC A P A SmC A P F, (ii) SmC A P A SmC S P F and (iii) SmC S P A SmC S P F. In the first and third transition, the chirality of the system changes at the transition. However, in the second case, the chirality of the system is preserved at AFE FE transition. Therefore, this simple model could explain the observed chiral switching in experiments [1, 2, 5, 6] on bent-core liquid crystals. Similar calculation that represents a poor chiral order within a smectic layer does not agree with experiments. Therefore, we conclude that the chirality is not just induced by tilt and polarity, but rather the system must have spontaneous chiral order.

58 BIBLIOGRAPHY [1] D. R. Link, G. Natale, R. Shao, J. E. Maclennan, N. A. Clark, E. Korblova, and D. M. Walba, Science 278, 1924 (1997). [2] T. Niori, T. Sekine, J. Watanabe, T. Furukawa, and H. Takezoe, J. Mater. Chem. 6, 1231 (1996). [3] J. V. Selinger, Phys. Rev. Lett. 90, (2003). [4] A. Roy, N. V. Madhusudana, P. Toledano, and A. M. Figueiredo Neto, Phys. Rev. Lett. 82, 1466 (1999) [5] A. Jakli, S. Rauch, D. Ltzsch, and G. Heppke, Phys. Rev. E 57, 6737 (1998). [6] G. Heppke, A. Jakli, S. Rauch, and H. Sawade, Phys. Rev. E 60, 5575 (1999). [7] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes (in C): The Art of Scientific Computing (Cambridge University Press, Cambridge, 1992). 42

59 CHAPTER 3 Statistical Mechanics of Splay Flexoelectricity in Nematic Liquid Crystals This Chapter presents studies of the splay flexoelectric effect in nematic liquid crystals with a lattice model. In this model, each lattice site has a spin representing the local molecular orientation, and the interaction between neighboring spins represents pear-shaped molecules with shape polarity. Simulations have shown that there is a large splay flexoelectric effect, which diverges in the vicinity of a nematic to polar transition. These results clearly indicate that flexoelectricity can be a statistical phenomenon associated with the onset of polar order. The bulk of this chapter is published in Physical Review E [1]. 3.1 Review of Flexoelectric Effect in a Nematic Flexoelectricity is a coupling between elastic deformation and electrostatic polarization in a liquid crystalline medium. In general, a splay or bend deformation of the nematic director leads to an electrostatic polarization, which can be observed as a macroscopic dipole moment of the liquid crystal. Conversely, an applied electric field induces an electrostatic polarization, which leads to a combination of splay and bend distortions in the nematic director. Since its discovery in 1969 by Meyer [2], the flexoelectric effect has drawn great interest because of its possible applications [3, 4] in strain gauges, transducers, actuators, micro power generator and electro-optical devices. 43

60 44 There have been many experimental and theoretical studies to determine the flexoelectric coefficients of nematic liquid crystals [5 19], using a range of different approaches. For typical calamitic (rod-shaped) liquid crystals, the splay and bend flexoelectric coefficients are in the range of 3 20 pc/m. However, in recent experiments, Harden et al. [3, 4] found that bent-core liquid crystals have a surprisingly large bend flexoelectric coefficient, up to 35 nc/m, roughly three orders of magnitude larger than the typical value. With this large bend flexoelectric coefficient, bent-core liquid crystals may be practical materials for converting mechanical into electrical energy. For theoretical physics, a key question is how to explain the large flexoelectric effect found in bent-core nematic liquid crystals, so that it can be exploited for technological applications. Our conjecture is that the large flexoelectric effect is a statistical phenomenon associated with a nearby polar phase. Near a polar phase, a nematic liquid crystal is on the verge of developing spontaneous polar order, and hence any deformation of the director should induce a large polar response. To test this conjecture, we would like to build a model with nematic and polar phases, and determine the behavior of the flexoelectric effect as a function of temperature above the nematic-polar transition. In this chapter, we begin the study by investigating the splay flexoelectric effect in a system of uniaxial pear-shaped molecules. In a subsequent chapter, we will investigate the more complex case of bend flexoelectricity in bent-core liquid crystals, as in the experiments. To study the splay flexoelectric effect, we generalize the Lebwohl-Lasher lattice

61 45 (a) (b) (c) Figure 3.1: Snapshots of the simulation results in the three phases: (a) Isotropic. (b) Nematic. (c) Polar. The software V Sim is used [21], and the color of each molecule represents the polar angle θ away from the z-axis. model of nematic liquid crystals [20]. In the original Lebwohl-Lasher model, each lattice site i has a spin n i, which represents the local nematic director, with the symmetry n i n i. In our generalization, the spins represent the orientations of pear-shaped molecules, which do not have that symmetry. For that reason, the interaction between neighboring spins includes three terms one term favoring nematic order, another term favoring polar order, and a third term that couples polar order with splay of the nematic director. With this interaction, we find a phase diagram with isotropic, nematic, and polar phases, as illustrated in the snapshots of Fig The nematic phase has a flexoelectric effect, which increases as the system approaches the polar phase. Thus, this calculation demonstrates explicitly that the flexoelectric effect can be a collective, statistical phenomenon, which is strongest near the transition to a phase with spontaneous polar order. The plan of this chapter is as follows. In Sec. 3.2, I set up the theoretical framework leading to the model interaction and discuss the relevant order parameters.

62 46 In Sec. 3.3, Monte Carlo simulation methods and results are presented. In Sec. 3.4, I present a mean-field theory for the model, and compare with simulation results. Finally, in Sec. 3.5, I discuss and summarize the conclusions of this study. 3.2 Splay flexoelectric Model In this study, our goal is to simulate the splay flexoelectric effect in a system of uniaxial pear-shaped molecules. For these simulations, we construct a lattice model that can represent both nematic and polar order. In this model, the local molecular orientation at lattice site i is represented by a unit vector ˆn i. If the system has nematic order, then the molecular orientations tend to be aligned along a preferred axis; i.e. there is a nonzero order parameter P 2 (ˆn i ˆd), where ˆd is the overall director and P 2 is the second Legendre polynomial. If the system has polar order, then the molecular orientations tend to point in a particular direction; i.e. there is a nonzero order parameter P 1 (ˆn i ˆd), where P 1 is the first Legendre polynomial. Note that the system can have nematic order without polar order, but it cannot have polar order without nematic order. The lattice Hamiltonian must have four terms: one term that favors nematic order, one term that favors polar order, one term that gives a coupling between polar order and an applied electric field, and a final term that gives a coupling between polar order and splay of the nematic director. The term favoring nematic order can be written simply as A(ˆn i ˆn j ) 2, summed over all pairs of neighboring sites i and j, as in the Lebwohl-Lasher model [20]. The term favoring polar order can be written

63 47 even more simply as B(ˆn i ˆn j ), again summed over all pairs of neighboring sites i and j, as in the Heisenberg model of magnetism. The coupling between polar order and an applied electric field can be written as E ˆn i, summed over i. The coupling between polar order and nematic splay is somewhat more subtle. For this coupling we need a lattice expression for the local splay between neighboring sites i and j. Our expression for the local splay should depend only on the nematic director, and hence it should be invariant under the transformation ˆn ˆn. We cannot describe splay by the scalar ˆn, because it is not invariant under that transformation. Rather, we must describe splay by the vector ˆn( ˆn), which has the correct symmetry. In the following calculation, we let Latin letters refer to lattice sites and Greek letters refer to directions. On a continuum basis, the splay vector ˆn( ˆn) can be written in terms of the local nematic order tensor Q αβ (r), or equivalently in terms of the dyad n α (r)n β (r), as n α β n β = 1 2[ β (n α n β ) + (n α n γ ) β (n β n γ ) (n β n γ ) β (n α n γ ) ]. (3.1) Hence, a lattice approximation to the splay vector between sites i and j can be written as [n α β n β ] ij = 1 2 [ r ijβ (n jα n jβ n iα n iβ ) (3.2) + n iαn iγ + n jα n jγ r ijβ (n jβ n jγ n iβ n iγ ) 2 n ] iβn iγ + n jβ n jγ r ijβ (n jα n jγ n iα n iγ ), 2

64 where ˆr ij = (r j r i )/ r j r i is the unit vector from site i to j on the lattice. After some algebra, this expression simplifies to 48 [ˆn( ˆn)] ij = 1 2[ˆnj (ˆr ij ˆn j ) ˆn i (ˆr ij ˆn i ) +ˆn i (ˆn i ˆn j )(ˆr ij ˆn j ) (3.3) ˆn j (ˆn i ˆn j )(ˆr ij ˆn i ) ]. Note that this expression is invariant under the transformations ˆn i ˆn i, ˆn j ˆn j, and i j. Now that we have found an expression for the local splay vector, we can couple it with the local polar order. The coupling term in the lattice Hamiltonian can be written as the dot product of the splay between sites i and j with the average polar order on these sites, V int = C[ˆn( ˆn)] ij ˆn i + ˆn j. (3.4) 2 Simplifying with the use of Eq. (3.3), the coupling term between splay and polar order in the Hamiltonian is ( ) ˆni ˆn j V int = C ˆr ij (ˆn j ˆn i ). (3.5) 2 Combining all these terms, our final expression for the lattice Hamiltonian is H = i,j [ A(ˆn i ˆn j ) 2 + B(ˆn i ˆn j ) (3.6) ( ) ˆni ˆn j +C ˆr ij (ˆn j ˆn i )] 2 i E ˆn i. To compare with previous theoretical work, we note that other investigators have studied models with both nematic and polar order; for example, see Refs. [22 24].

65 49 Those models considered energy functions with a strong tendency toward nematic order, as represented by our A term, and a weak tendency toward polar order, as represented by our B term. However, those models did not consider the coupling between polar order and nematic splay, represented by our C term. This coupling is the key new feature of our lattice model, which enables it to describe flexoelectricity. At this point, we want to use the lattice Hamiltonian of Eq. (3.6) to calculate the nematic order parameter P 2, the polar order parameter P 1, and the average splay vector ˆn( ˆn) as functions of the parameters A, B, and C and the electric field E. In the following sections, we will do this calculation through Monte Carlo simulations and mean-field theory. 3.3 Monte Carlo simulation As a first step in exploring this model, we carry out Monte Carlo simulations of a system of pear-like molecules interacting with the lattice Hamiltonian of Eq. (3.6). In these simulations, we use a simple cubic lattice of size When an electric field is applied, it is in the z direction, so that the molecules tend to align along z, with splay in the x and y directions. The lattice has periodic boundary conditions in z, but free boundaries in x and y, so that it can form splay in those directions. (We have also done limited simulations with a face-centered-cubic lattice, and the results are consistent with the simulations presented here.) The usual Metropolis algorithm was used for lattice updates. In each Monte Carlo step, a lattice site is chosen randomly, its orientation is changed slightly, and

66 50 the change in energy E is calculated. If E < 0 the move is accepted, and if E > 0 the move is accepted with probability exp( E/k B T). Starting from the high-temperature isotropic phase, the system is cooled down slowly with temperature steps of T = The final configuration at each temperature is taken as the initial configuration for the next lower temperature. Typical runs take about 10 5 steps to come to equilibrium, while runs near phase transitions take about steps. The nematic and polar order parameters and the splay vector are calculated and time-averaged during the production cycle. The nematic order parameter P 2 is calculated by the usual method using the 3D nematic order tensor Q αβ = 1 N N i=1 ( 3 2 n iαn iβ 1 ) 2 δ αβ. (3.7) where α and β = x, y, z, and N is the total number of lattice sites. The largest eigenvalue of this order tensor corresponds to P 2. To calculate the polar order parameter P 1, we assume that polar order is oriented along the same axis as nematic order, as expected for uniaxial molecules. The eigenvector corresponding to the largest eigenvalue of the nematic order tensor Q αβ is the instantaneous director ˆd. Hence, the polar order parameter is calculated as the average dot product of the director with the molecular orientation, P 1 = 1 N N i=1 ˆd ˆn i. (3.8) The splay vector is calculated from Eq. (3.3), averaged over the four bonds in the (x, y) plane. The magnitude of this vector gives the average angle between the

67 51 molecular orientations on neighboring lattice sites. For that reason, we report this magnitude as θ. Figure 3.2 shows plots of the order parameters P 2, P 1, and θ as functions of temperature for several values of the interaction parameters. In Fig. 3.2(a), for a small polar coupling B and no applied electric field, we see an isotropic-nematic transition at high temperature followed by a nematic-polar transition at low temperature. At the isotropic-nematic transition, the nematic order parameter goes from zero to a nonzero value. Here the transition is rounded by finite-size effects; we would expect a sharp first-order transition for an infinite system. Throughout the nematic temperature range, the polar order parameter and splay are both zero. At the nematic-polar transition, the polar order parameter becomes nonzero, and this polar order induces an accompanying splay. The nematic order parameter decreases as the system moves into the polar phase, because the splayed molecular orientation partially averages out the alignment, as shown in the snapshot of Fig. 3.1(c). In Fig. 3.2(b), for a larger polar coupling B, we see a direct transition from the isotropic to the polar phase, with no intervening nematic phase. In this case, the nematic and polar order parameters both become nonzero at the same transition temperature. Once again, the polar order induces a splay, which inhibits the growth of the nematic order parameter. The simulation results for the phase diagram are shown in Fig. 3.2(c). In this phase diagram, the vertical axis shows temperature while the horizontal axis shows the polar coupling B for a constant nematic coupling A. For small B the phase diagram

68 P 2, P 1, θ P 2 P 1 θ P 2, P 1, θ P 2 P 1 θ (a) T (b) T Isotropic 0.8 T Nematic Polar P 2, P 1, θ P P 1 θ (c) B (d) T Figure 3.2: Monte Carlo simulation results for the order parameters P 1, P 2, and θ as functions of temperature T, for different values of the interaction parameters chosen to show various types of transitions: (a) Zero-field results for A = 1.5, B = 0.09, C = 0.3, showing the isotropic-nematic and nematic-polar transitions. (b) Zero-field results for A = 1.5, B = 0.4, C = 0.3, showing the direct isotropic-polar transition. (c) Phase diagram for zero field. (d) Simulation with applied electric field E = 0.06, for the same parameters as in part (a), showing the induced polar order and splay in the nematic phase.

69 53 shows isotropic, nematic, and polar phases, with a nematic range that decreases as B increases. At a sufficiently large value of B, the nematic phase disappears and there is a direct transition from isotropic to polar. Note that this phase diagram is quite similar to the phase diagram found in recent work on the 2D isotropic, tetratic, and nematic phases [25]. The question is now: What happens to the nematic phase when an electric field is applied? To answer this question, Fig. 3.2(d) shows the simulation results for the same parameters as Fig. 3.2(a), but in the presence of a small electric field. In the high-temperature isotropic phase, the field induces some polar and nematic order, but this effect is very small. However, in the nematic phase, the field induces a more substantial polar order, and that polar order induces a splay in the nematic director, i.e. a converse flexoelectric effect. Both the polar order and the splay are quite temperature-dependent, increasing as the system approaches the nematic-polar transition temperature, as would be expected for a divergent susceptibility above a second-order transition. The nematic-polar transition is rounded off by the applied field, and the polar order parameter and splay saturate in the low-temperature polar phase. To provide further insight into the effect of an applied electric field, Fig. 3.3 shows the splay as a function of temperature for several values of the field. Within the nematic phase, the splay increases as the electric field increases, as expected for the converse flexoelectric effect. This trend is reasonable because an increasing electric field enhances polar order. For small field the splay is quite sensitive to

70 E = 0.0 E = 0.03 E = 0.06 E = 0.09 θ Polar Nematic Isotropic T Figure 3.3: Variation of splay as a function of temperature for several values of the applied electric field, using the same numerical parameters as in Fig. 3.2(a). temperature, but for large field it becomes less temperature-dependent, as the induced polar order grows larger and approaches saturation. In the low-temperature polar phase, the splay shows the opposite trend with electric field; it now decreases as the field increases. This trend is reasonable because an increasing electric field cannot enhance the polar order, which is already saturated; it only aligns the direction of polar order. This alignment reduces the induced splay, since splay necessarily involves some misalignment of the molecular orientation. For comparison, Fig. 3.4 presents a plot of splay as a function of temperature for several values of the interaction coefficient C. This graph shows that the splay increases as C increases, over the full temperature range, in all phases. This result is reasonable because the coefficient C represents the flexoelectric coupling between polar order and splay. As a final point, note that the behavior presented here can only occur in the

71 θ C = 0.18 C = 0.24 C = 0.30 C = 0.36 C = 0.42 C = 0.50 Polar Nematic Isotropic T Figure 3.4: Variation of splay as a function of temperature for several values of the interaction coefficient C, with a small field E = limit of small splay θ π/n, where N is the system size. In the opposite limit θ π/n, the system is too large for one single splay from side to side. Instead, it must break up into modulated structures consisting of regions of splay separated by domain walls. These modulated structures might be splay stripes or even more complex two- or three-dimensional arrangements of splay cells [26]. We have observed such modulated structures in our simulations, but we have not explored them in detail because they are not likely to occur in experiments, where the magnitude of splay is generally small. 3.4 Mean-field calculation In this section we discuss two approximate analytic approaches to solve the problem. First, we map the interaction onto an Ising model, and use this Ising model to calculate the splay and polar order as functions of temperature and electric field. Second, we present a more general mean-field calculation with full rotational degrees

72 of freedom, and use it to calculate the full phase diagram with isotropic, nematic, and polar phases Ising model For a simple Ising-type model of the splay flexoelectric effect, we suppose that the system has well-defined nematic order, with variable amounts of splay and polar order. Consider a particular site i surrounded by six nearest neighbors on a cubic lattice. We suppose that site i has its director along the z-axis, as do the two neighbors above and below, while the four nearest neighbors in the xy-plane have directors that are splayed outward by a small angle θ. The polar order at any site j is represented by an Ising spin variable σ j = ±1, which indicates whether the molecular orientation is pointing up or down along the local director. Thus, the central site i has the molecular orientation ˆn i = σ i (0, 0, 1), while the six neighbors have the orientations ˆn +x = σ +x (sin θ, 0, cos θ), ˆn x = σ x ( sin θ, 0, cos θ), ˆn +y = σ +y (0, sin θ, cos θ), ˆn y = σ y (0, sin θ, cos θ), ˆn +z = σ +z (0, 0, 1), and ˆn z = σ z (0, 0, 1). We now substitute these expressions for the molecular orientations into the lattice Hamiltonian of Eq. (3.6). As usual in mean-field theory, we assume that all the neighbors of site i have polar order given by σ j = M. (This quantity is called P 1 in the other sections; here we use the symbol M to emphasize the analogy with the Ising magnetization.) The mean potential experienced by site i, expanded to second

73 57 order in the small splay θ, is then V mean = A(6 4( θ) 2 ) BMσ i (6 2( θ) 2 ) 2C θ(σ i + M) Eσ i. (3.9) Hence, the effective field acting on the Ising spin σ i is E eff = E + 6BM + 2C θ 2BM( θ) 2. (3.10) As a result, the polar order parameter must satisfy the self-consistency equation ( Eeff M = σ i = tanh k B T ( E + 6BM + 2C θ 2BM( θ) 2 = tanh k B T ) ). (3.11) Furthermore, minimization of the mean potential over the splay θ gives θ = CM 2A + BM2. (3.12) Solving Eqs. (3.11) and (3.12) simultaneously gives the equilibrium values of the splay θ and polar order M, as functions of electric field E, temperature T, and energetic parameters A, B, and C. To calculate the response to an electric field in the nematic phase, we assume that E, M, and θ are all small, and expand Eqs. (3.11) and (3.12) to linear order in these quantities. From these expansions we obtain M = θ = E k B T (6B + C 2 /A), (3.13) CE 2A[k B T (6B + C 2 /A)]. (3.14)

74 E = 0.0 E = E = 0.0 E = 0.06 θ M (a) T (b) T Figure 3.5: Numerical mean-field calculations for the Ising mapping, showing the splay and polar order as functions of temperature T, for parameters A = 1.5, B = 0.09, and C = 0.3, for zero and nonzero electric field. Note that Eq. (3.13) gives the polar order parameter induced by an applied electric field, while Eq. (3.14) gives the converse flexoelectric effect induced by the field. Both of these responses increase as the temperature decreases toward the secondorder nematic-polar transition at the temperature k B T NP = 6B + C2 A. (3.15) At this transition, they diverge as (T T NP ) γ, with critical exponent γ = 1, as expected for the susceptibility to an applied field, in mean-field theory for the Ising model. For a more precise calculation, we solve Eqs. (3.11) and (3.12) numerically as functions of temperature and field. The numerical results for splay θ and polar order M are shown in Fig As in the approximate analytic calculation above, we see a second-order nematic-polar transition. The high-temperature nematic phase has no polar order or splay without a field, but an applied field induces both of

75 59 these quantities. By contrast, the low-temperature polar phase has both spontaneous polar order and spontaneous splay, and they both increase moderately when a field is applied. Although the Ising model is successful in explaining some features of our Monte Carlo simulations, it is incomplete because it assumes perfect nematic order the molecules can have only two possible orientations, up and down. It cannot describe the behavior of the nematic order parameter as a function of temperature. For that reason, we proceed to a more general mean-field theory, in which each molecule has full rotational degrees of freedom General Mean-Field Calculation In mean-field theory, the free energy can be written as F = U TS = H + k B T log ρ, (3.16) averaged over the single-particle distribution function ρ. Thus, our goal is to express the single-particle distribution function in terms of some variational parameters, calculate the energetic and entropic terms in the free energy, and then minimize the free energy over those variational parameters. For this mean-field calculation, we write the molecular orientation at each lattice site in terms of the polar angle θ i and azimuthal angle φ i with respect to the local director. We assume the distribution function depends only on the polar angle θ i, and hence write ρ(θ i ) = exp[v 1 P 1 (cosθ i ) + v 2 P 2 (cos θ i )] π 0 exp[v 1P 1 (cosθ i ) + v 2 P 2 (cosθ i )]dθ i, (3.17)

76 P 2, P 1, θ P 2 P 1 θ P 2, P 1, θ P 2 P 1 θ (a) T (b) T Isotropic T Nematic Polar P 2, P 1, θ P 2 P θ (c) B (d) T Figure 3.6: Numerical mean-field calculations of the order parameters P 1, P 2, and θ as functions of temperature T, for different values of the interaction parameters chosen to show various types of transitions: (a) Zero-field results for A = 1.5, B = 0.09, C = 0.3, showing the isotropic-nematic and nematic-polar transitions. (b) Zero-field results for A = 1.5, B = 0.4, C = 0.3, showing the direct isotropic-polar transition. (c) Phase diagram for zero field. (d) Simulation with applied electric field E = 0.06, for the same parameters as in part (a), showing the induced polar order and splay in the nematic phase. where v 1 and v 2 are variational parameters. The order parameters are then P 1 = π 0 P 1(cos θ)ρ(θ)dθ and P 2 = π 0 P 2(cosθ)ρ(θ)dθ, and the partition function is Z = π 0 exp[v 1P 1 (cosθ) + v 2 P 2 (cosθ)]dθ. The entropic contribution to the free energy is therefore TS = k B T log ρ(θ i ) = k B T[v 1 P 1 + v 2 P 2 log(z)] (3.18) per lattice site.

77 61 As in the previous section, we suppose that site i has its director along the z-axis, as do the two neighbors above and below, while the four nearest neighbors in the xy-plane have directors that are splayed outward by a small angle θ. To calculate the average energy, we combine our distribution function of Eq. (3.17) with the lattice Hamiltonian of Eq. (3.6). After averaging over all the angles, neglecting terms involving the third-order Legendre polynomials, and expanding to second order in the small splay θ, we obtain H = A A P 2 [ 2 2 2( θ) 2] (3.19) B P 1 [ 2 3 ( θ) 2] [ ] ] P2 C P 1 θ E P 1 [1 ( θ) per lattice site. We now have an expression for the total free energy F = A A P 2 [ 2 2 2( θ) 2] B P 1 [ 2 3 ( θ) 2] [ ] ] P2 C P 1 θ E P 1 [1 ( θ) k B T[v 1 P 1 + v 2 P 2 log(z)] (3.20) per lattice site. In this expression, note that P 1, P 2, and Z are all determined by the parameters v 1 and v 2 in the distribution function. Hence, the free energy is a function of just three variational parameters: v 1, v 2, and θ. Thus, in the mean-field calculation, we must minimize the free energy numerically with respect to those three parameters. After this minimization, we can calculate the order parameters P 1 and

78 62 P 2, and hence determine whether the system is in an isotropic, nematic, or polar phase. Figure 3.6 shows the numerical mean-field results for the order parameters P 1 and P 2 and splay θ, as well as a complete phase diagram as a function of temperature T. For a small polar interaction B = 0.09 there are two transitions, first from the high-temperature isotropic phase ( P 1 = 0, P 2 = 0) to the intermediate nematic phase (( P 1 = 0, P 2 0), and then from the nematic phase to the lowtemperature polar phase ( P 1 0, P 2 0). The isotropic-nematic transition is first-order, while the nematic-polar transition is second-order. On increasing the polar interaction strength B, the polar phase becomes stable even at higher temperature. For B = 0.36, these two transitions merge into a single first-order transition directly from the high-temperature isotropic phase to the low-temperature polar phase. If E 0, the polarization and splay are nonzero even in the nematic phase and scale with the magnitude of the field. For nonzero field, the magnitude of the splay increases on reducing temperature and is enhanced greatly near the transition to the polar phase. Note that the numerical mean-field results of Fig. 3.6 are very similar to the Monte Carlo simulation results of Fig. 3.2, both in the overall phase diagram and in the splay and polar response to an electric field. This similarity demonstrates that the mean-field theory captures the essential physics of this model.

79 Conclusion In conclusion, we have developed a lattice model for splay flexoelectricity in a system of uniaxial pear-shaped molecules. This model predicts a phase diagram showing isotropic, nematic, and polar phases, and it further predicts a converse flexoelectric effect in the nematic phase. The converse flexoelectric effect is proportional to the applied electric field, and it increases dramatically as the temperature decreases toward the nematic-polar transition. Indeed, we can regard this effect as a susceptibility to an applied field, which diverges at the second-order transition to a polar phase. Thus, flexoelectricity is not just a molecular effect arising from the microscopic interaction of liquid crystals with a field. Rather, it can be a statistical effect associated with the response of correlated volumes of molecules, which increases as one approaches a polar phase. The recent experiments of Harden et al. [3, 4] have found an anomalously large bend (rather than splay) flexoelectric effect in systems of bent-core liquid crystals. We speculate that the same considerations discussed in this chapter can explain the large bend flexoelectric coefficient in those experiments. The bent-core liquid crystal should be close to a polar phase, with order in the transverse dipole moments of the molecules. As a result, there should be large correlated volumes of molecules, leading to a high susceptibility to an applied field, which induces both polar order and bend. The model for bend flexoelectricity will necessarily be more complex than the splay model, because the bent-core molecules are not uniaxial and hence their orientations must be characterized by two vectors and is the subject of Chapter 4.

80 BIBLIOGRAPHY [1] S. Dhakal and J. V. Selinger, Phys. Rev. E 81, (2009). [2] R. B. Meyer, Phys. Rev. Lett. 22, 918 (1969). [3] J. Harden, B. Mbanga, N. Eber, K. Fodor-Csorba, S. Sprunt, J. T. Gleeson, and A. Jakli, Phys. Rev. Lett. 97, (2006). [4] J. Harden, R. Teeling, J. T. Gleeson, S. Sprunt, and A. Jakli, Phys. Rev. E. 78, (2008). [5] J. P. Straley Phys. Rev. A 14, 1835 (1976). [6] M. A. Osipov, J. Phys. (Paris) Lett. 45, 823 (1984). [7] Y. Singh and U. P. Singh, Phys. Rev. A 39, 4254 (1989). [8] A. M. Somoza and P. Tarazona, Molec. Phys. 72, 911 (1991). [9] F. Biscarini, C. Zannoni, C. Chiccoli, and P. Pasini, Mol. Phys. 73, 439 (1991). [10] P. R. M. Murthy, V. A. Raghunathan, and N. V. Madhusudana, Liq. Cryst. 14, 483 (1993). [11] L. M. Blinov, Liq. Cryst. 24, 143 (1998). [12] J. Stelzer, R. Berardi, and C. Zannoni, Chem. Phys. Lett. 299, 9 (1999). [13] J. L. Billeter and R. A. Pelcovits, Liq. Cryst. 27, 1151 (2000). [14] L. M. Blinov, M. I. Barnik, M. Ozaki, N. M. Shtykov, and K. Yoshino, Phys. Rev. E 62, 8091 (2000). [15] R. Berardi, M. Ricci, and C. Zannoni, ChemPhysChem 2, 443 (2001). [16] C. Zannoni, J. Mater. Chem. 11, 2637 (2001). [17] A. Ferrarini, Phys. Rev. E 64, (2001). [18] A. Dewar and P. J. Camp, J. Chem. Phys. 123, (2005). [19] A. Kapanowski, Phys. Rev. E 75, (2007). [20] P. Lebwohl and G. Lasher, Phys. Rev. A 6, 426 (1972). [21] Sim/V Sim/index.en.html. 64

81 65 [22] T. J. Krieger and H. M. James, J. Chem. Phys. 22, 796 (1954). [23] D. H. Lee and G. Grinstein, Phys. Rev. Lett. 55, 541 (1985). [24] M. Campbell and L. Chayes, J. Phys. A 32, 8881 (1999). [25] J. Geng and J. V. Selinger, Phys. Rev. E 80, (2009). [26] For a survey of modulated structures in liquid crystals, see R. D. Kamien and J. V. Selinger, J. Phys.: Condens. Matter 13, R1 (2001).

82 CHAPTER 4 Giant Flexo-electricity in Bent Core Nematic Liquid Crystals: A Monte Carlo Simulation Study. The focus of this chapter is on the bend flexoelectric effect in a system of bent-core molecules. A new method to calculate the bend flexoelectric coefficient is presented. The model for splay flexoelectricity presented in Chapter 3 is refined to incorporate the bent geometry of molecules. Using this model the flexoelectric response of bent-core liquid crystals is studied with a lattice simulation followed by a mean-field calculation. 4.1 Introduction One of the key issues in experimental liquid crystal science is the switching time of LCDs. The quest for fast switching LCDs relying on the flexoelectric effect in twisted nematic or in nematic phase has been surging since its introduction in 1969 by Meyer [1]. As postulated by Meyer, imposing a deformation in the director field in a system of anisotropic objects with permanent dipole moment might induce a spontaneous polarization. The general form of induced polarization density due to imposed splay and bend deformation of the nematic director ˆn is P = e sˆn( ˆn) + e bˆn ( ˆn), (4.1) 66

83 67 where e s and e b are the splay and bend flexoelectric coefficients respectively. Alternatively, an applied electric field ( E) could induce local polarization in such a system parallel to ( E), resulting in a macroscopic deformation in the director field because of the packing constraints. This phenomenon is known as the converse flexoelectric effect and is the subject of this chapter. Though the simplest picture of liquid crystal molecules as rods [2 4] has been successful in explaining most features of the uniaxial nematic phase, real mesogenic molecules have a symmetry lower than that of rods. One can expect less symmetric phases from the molecules of reduced symmetry. Considering lath-like molecules, Freiser [5] was able to predict a biaxial nematic phase that has non-zero orientational order of molecules along a second macroscopic direction. It is anisotropic in a plane perpendicular to the primary director. In such an anisotropic medium, a beam of linearly polarized light can maintain its state of polarization unchanged in two different directions. The anticipated quicker response of the short axis of molecules to an applied field has triggered research on the biaxial nematic phase at the moment. On the experimental front, although there are strong arguments for the existence of a thermotropic biaxial nematic phase as revealed by NMR [6] and X-ray [7] studies, the observed biaxiality lacks unanimous explanation [8 10]. Freiser s prediction, supported by other theoretical works [11 15] have paved a way to look for a candidate molecule with reduced symmetry to control the orientational order of liquid crystalline phases. A new classes of compounds- bent-cores (V-shaped) - has been synthesized [16,17]

84 68 to study structure-property relationships. The notable features of these compounds as revealed from the studies to date are -the formation of chiral ferroelectric and antiferroelectric smectic phases [18, 19], and the formation of biaxial nematic phases [6, 7], both having potential use in LCDs. Therefore, studies on bent-core liquid crystals has been surging both experimentally and theoretically. Keeping aside the debate about the biaxial nematic phase for the moment our goal in this project is to study the flexoelectric response of a nematic phase of bent-core molecules. There have been many experimental and theoretical studies using a range of approaches to understand the flexoelectric responses of nematic liquid crystals [20 35]. It has been found that the flexoelectric response of rod-shaped molecules is smaller than for other shapes and is in the range of 3-20 pc/m. Recently, Harden et al. [20,21] demonstrated that bent-core liquid crystals have a surprisingly large bend flexoelectric coefficient that is three orders of magnitude larger than for typical calamitic liquid crystals ( 35 nc/m). With this distinct feature, bent-core liquid crystals may be practical materials for converting mechanical energy into electrical energy. The purpose of this project is (i) to develop a lattice model for bend flexoelectricity and simulate it, (ii) to study the variation of bend flexoelectric coefficient with temperature and electric field, and (iii) give a possible explanation of the observed giant flexoelectricity in experiments.

85 69 Figure 4.1: Different bend configuration between neighboring molecules. 4.2 Interaction Potential for Bend Flexoelectric Effect The candidate molecules to study bend flexoelectric effect are bent-cores with a transverse dipole. The orientation of a bent-core molecule could be described by two vectors or three Euler angles. Here, the following notation is used- ˆn represents the long-axis of the molecule, and ˆb represents the short-axis [see Fig. 4.1]. As these molecules are less symmetric than rod-like molecules, we can expect phases of reduced symmetry compared to the nematic phases of rod-like molecules. The interaction energy of such molecules consists of five terms: a term that favors uniaxial order, a term that favors biaxial order, a term that favors polar interaction, a coupling term between polar order and an applied electric field, and a final term that couples polar order and bend deformation. The term favoring uniaxial nematic order can be written in terms of the longaxis as A(ˆn i ˆn j ) 2, summed over all pairs of neighboring sites i and j, as in the Lebwohl-Lasher model [36]. The term favoring polar order can be written in terms of the short-axis as B 1 (ˆb i ˆb j ), again summed over all pairs of neighboring sites i and j. The term favoring biaxial nematic order is similar to the uniaxial term and

86 can be written as B 2 (ˆb i ˆb j ) 2, summed over all pairs of neighboring sites i and j. The coupling between polar order and an applied electric field is E ˆb i, summed over i. Finally, the coupling between polar order and bend is somewhat more subtle. For this coupling, we need a lattice expression for the local bend between neighboring sites i and j. It should be invariant under the transformation ˆn ˆn as illustrated in Fig The bend vector is defined as ˆn ( ˆn); and is calculated from the difference in Q tensor between neighboring sites as in the Ref. [22] and the detail of the calculation is presented in Appendix A. Hence, a lattice approximation to the bend vector between sites i and j can be written as [ˆn ( ˆn)] ij = ˆn i(ˆr ij ˆn i ) 2 ˆn j(ˆr ij ˆn j ) 2 + ˆn i(ˆn i ˆn j )(ˆr ij ˆn j ) 2 70 ˆn j(ˆn i ˆn j )(ˆr ij ˆn i ). (4.2) 2 A relationship between the magnitude of bend [Eq. (4.2)] and induced polarization would enable one to estimate the bend flexoelectric coefficient (e b ) using Eq. (4.1). Knowing the explicit expression for bend vector between two molecules [Eq. (4.2)], we can calculate the possible interaction that favors bend deformation between them by taking the dot product of the bend vector with an average polarization between neighboring sites. The bend interaction energy is V bend = [ˆn ( ˆn)] ˆb i + ˆb j 2 = (ˆb j ˆn i ) [ˆr ij {ˆn i + ˆn j (ˆn i ˆn j )}] 4 (ˆb i ˆn j ) [ˆr ij {ˆn j + ˆn i (ˆn i ˆn j )}], (4.3) 4

87 71 where the interaction is between the nearest neighbors and r ij = r i r j with centers of molecules at positions r i and r j. Therefore, an interaction potential with a stable polar phase is V min = A(ˆn i ˆn j ) 2 B 1 (ˆb i ˆb j ) B 2 (ˆb i ˆb j ) 2, (4.4) where the first term favors the alignment of the long-axes (ˆn) of the molecules at site i anf j. The second term (B 1 ) accounts for the polar interaction and B 2 term favors the alignment of the side-wise axis (ˆb). On top of these interactions, an electric field is applied along the direction of the dipole to study the converse flexoelectric effect. The field term is V field = E 0 (ẑ ˆb i ). (4.5) Then, the total interaction for our bend flexoelectric model is the sum of all these terms: V ij = V min + CV bend + V field. (4.6) For B 1 = 0, C = 0 and E 0 = 0, this model is similar to Straley s model for lath-like molecules [11]. The interaction parameters (A, B 1, B 2 and C) are related to molecular properties. For example, the magnitude of B 1 is related to the strength of the dipole moment. On the other hand, B 2 is related to an opening angle of bent-core molecules. For bent-core molecules with an opening angle 180 0, the biaxiality parameter B 2 is very small and molecules behave as rigid rods. The magnitude of constant C reflects both molecular anisotropy and dipole moment.

88 72 (a) (b) (c) (d) Figure 4.2: Visual scheme of different phases obtained in simulations: (a) Isotropic. (b) Uniaxial nematic. (c) Biaxial nematic. (d) Polar. The software V Sim is used [37], and the color of each molecule represents the polar angle θ away from the z-axis.

89 Lattice Monte Carlo simulation As a first step in exploring this model, Monte Carlo simulations of a system of bent-core molecules interacting as in Eq. (4.6) is carried out. In these simulations, we use a simple cubic lattice of size When an electric field is applied, it is in the Z direction, so that the molecular dipoles (or the short axis) tend to align along Z, with bend perpendicular to it. The lattice has periodic boundary conditions in Z, but free boundaries in X and Y, so that it can form bend in one of those directions. The usual Metropolis algorithm was used for lattice updates. In each Monte Carlo step, a lattice site is chosen randomly, its orientation is changed slightly, and the change in energy E is calculated. If E < 0 the move is accepted, and if E > 0 the move is accepted with probability exp( E/k B T). Starting from the high-temperature isotropic phase, the system is cooled down slowly with temperature steps of T = The final configuration at each temperature is taken as the initial configuration for the next lower temperature. Typical runs take about 10 5 steps to come to equilibrium, while runs near phase transitions take about steps. The observables are time-averaged during the production cycle. Fig. 4.2 shows the molecular orders in a polar (P), biaxial nematic (N B ), uniaxial nematic (N U ) and isotropic (I) phase obtained in simulations. The symmetries of these phases could be described quantitatively in terms of order parameters. For a system of biaxial molecules, the orientational order parameters can be parameterized in terms of the Euler angles [11]. The relevant order parameters for a system of bent core molecules are [12,14]

90 74 S = 3 2 cos2 θ 1 2 T = 3 2 sin2 θ cos 2ψ U = 3 2 sin2 θ cos 2φ V = 1 2 (1 + cos2 θ) cos 2θ cos 2ψ cos θ sin 2φ sin2ψ. (4.7) Alternatively, the orientational order of a rigid biaxial molecule in a biaxial phase can be characterized at the second rank level by the Cartesian supermatrix with elements [13] S AB ab = 3l aal bb δ aa δ bb (4.8) 2 Here the subscripts a and b denotes the molecular axes, A and B are the laboratory axes, l aa is the direction cosine between the molecular axis a and laboratory axis b. The order parameters of biaxial molecules in biaxial phase can be expressed as [23], S = S ZZ zz U = Sxx ZZ SZZ yy T = S XX zz V = 1 3 [(SXX xx S Y Y zz Sxx Y Y ) (Syy XX Syy Y Y )] (4.9) In this definition, S measures the ordering of the molecular z axis with the laboratory Z axis and is the usual nematic order parameter. U measures the difference in the ordering for the molecular z axis with respect to the laboratory X and Y. Conversely, T indicates the difference in the ordering of the laboratory Z axis with respect to the molecular x and y axes. Finally, V measures the differences in the ordering of the molecular x and y axes with respect to the laboratory X and Y axes. These order

91 75 parameters are calculated as follows [15,23]: the Q -tensor is calculated for all molecular axes (ˆn, ˆb,ĉ). The largest eigen value of these matrices represents the magnitude of ordering of the primary molecular axis (z). The eigen vector corresponding to that eigen value is the laboratory Z axis. Similarly, the second largest eigen value is taken to identify the secondary molecular ordering axis. The corresponding eigen vector is the laboratory Y axis. The remaining molecular axis is taken as x and the corresponding laboratory axis is perpendicular to Y and Z. Identifying the molecular and laboratory axes the calculation of the order parameters is straight forward in terms of the direction cosines of two axes as expressed above. Of these four parameters, S and U are non-zero in both uniaxial and biaxial phases while T and V vanishes in the uniaxial phase and are non-zero in the biaxial phase. In the limit of high orientational order, S and V tends to 1. In contrast, both U and T tend to zero. We can define polar order parameter (P) for the macroscopic phases formed by polar molecules as an average orientation of the short axis of all molecules. Plots of order parameters as a function of temperature for different values of interaction strength is shown in Fig 4.3. Though, all the order parameters (S, T, U, V, P) were measured, only the uniaxial (S), biaxial (V ) and polar (P) orders are reported. As is clearly seen from the graphs, Monte Carlo simulations have shown four types of phase transitions depending upon the relative interaction strengths. For small B 1 and B 2, there is a cascade of transitions i.e., I N U N B P. Such a transition takes place for molecules with weak dipole (small B 1 ) and molecular biaxiality (small B 2 ). For a slightly larger value of B 1, a direct transition from

92 76 S,V,P S V P P N N I B U 0.2 P N I U S,V,P S V P S,V,P (a) T P (c) T I S V P S,V,P (b) T P N B I S V P (d) T Figure 4.3: Monte Carlo simulation results for uniaxial (S), biaxial (V ) and polar (P) order parameters as functions of temperature T at zero field (E = 0.0) in units of interaction strength A: (a) I N U N B P transition for B 1 = 0.04 and B 2 = 0.4. (b) I N U P transition for B 1 = 0.14 and B 2 = 0.4. (c) I P transition for B 1 = 0.38 and B 2 = 0.4. (d) I N B P transition for B 1 = 0.03 and B 2 = 0.95.

93 Isotropic Isotropic N U 0.6 T T N U N B 0.2 N B Polar B (a) 1 Polar (b) B 2 Figure 4.4: Complete phase diagram of the model at zero field (E = 0.0): (a) in terms of B 1 with B 2 = 0.4 and C = 0.4. (b) in terms of B 2 with B 1 = 0.03 and C = 0.4. uniaxial nematic to polar phase takes place i.e., I N U P. With even stronger B 1, direct transition from isotropic to polar phase takes place i.e., I P. Finally, if the molecules are strongly biaxial (large B 2 ), the intermediate uniaxial nematic phase might be absent i.e., I N B P. From these observations, we can infer that the stability of a biaxial nematic phase can be changed with dipole moment (B 1 ) and molecular biaxiality (B 2 ). In order to understand how different phases are separated in parameter space, a series of similar simulations have been carried out. In these simulations, keeping other variables fixed, we observe the phase behavior as a function of interaction parameters B 1 and B 2. Figure 4.4 shows the full simulated phase diagrams. The magnitude of bend/bond as calculated from Eq. (4.2) obtained in simulations without applied field (E = 0) for I N U N B P and I N U P transitions, is shown by circles in Fig. 4.5 (c) and (d). This clearly shows a non-zero bend in

94 78 the polar phase and is as expected since the model interaction involves a term that couples bend with polar order. However, the magnitude of bend/bond is negligible in both nematic phases and is undefined in the isotropic phase. One last point about the second rank order parameter worthwhile to mention here is about the reduction in the uniaxial order parameter (S) [see Fig. 4.3] in the polar phase. This is reasonable as the order parameters are computed globally. Up to now, I have described the behavior of the model without an applied field. The question is now: what happens to the nematic phases (both uniaxial and biaxial) when an electric field is applied? We are specifically interested in flexoelectric responses around N U P and N U N B P transitions. To answer these questions, Figs. 4.5 (c) and (d) shows the simulation results for bend/bond for different values of the field. It is seen from these plots that bend ( θ) is negligible in the isotropic phase for all values of the field studied. On increasing the field, there is substantial amount of bend in both uniaxial and biaxial nematic phases. A key feature of these studies is that the bend/bond diverges near the transition to the polar phase. This behavior is similar in both types of transitions- a direct transition from uniaxial nematic, and a transition from biaxial nematic to the polar phase. To provide further insight into the effect of the interaction parameter C, Figs. 4.5(a) and (b) shows bend as a function of temperature for different values of C at a constant electric field. From these studies, we see that the amount of bend in nematic phases increases with C and is reasonable as it couples the bend deformation with polar order. As a final point, the behavior presented here is for small C. For

95 79 θ C=0.2 C=0.3 C=0.4 C=0.5 θ C=0.2 C=0.3 C=0.4 C= P N B N U I (a) T 0.01 P N U I (b) T θ E=0.0 E=0.04 E=0.08 E=0.12 θ E=0.0 E=0.04 E=0.08 E= P N N B U I 0.01 P N I U (c) T (d) T Figure 4.5: Monte Carlo simulation results for θ as a function of temperature T (in units of interaction strength A): (a) I N U N B P transition with B 1 = 0.04, B 2 = 0.4 and E = 0.04 for different values of C. (b) I N U P transition with B 1 = 0.14, B 2 = 0.4 and E = 0.04 for different values of C. (c) I N U N B P transition with B 1 = 0.04, B 2 = 0.4 and C = 0.4 for different values of E. (d) I N U P transition with B 1 = 0.14, B 2 = 0.4 and C = 0.4 for different values of E.

96 80 Figure 4.6: Schematic representation of the molecular order in the nematic phase showing bend in the long-axis. This is the snapshot from the Monte Carlo simulation just above the polar to nematic transition. large value of C, the system size might be too large for a single bend from side to side. Instead, it might form some modulated structures consisting of region of bend separated by domain walls. Finally, Fig. 4.6 shows molecular order in the uniaxial nematic phase showing substantial amount of bend going from one end to another. This is the simulation result at T = This temperature corresponds to the phase point just above uniaxial nematic to polar (N U P) transition (T U P 0.32) for a small field. 4.4 Mean-Field Calculation of the Model In this section, I present mean-field calculation to solve the problem. It is based on Maier-Saupe theory for the nematic. In mean-field theory, the free energy can be

97 81 written as F = U TS = H + k B T log ρ, (4.10) averaged over the single-particle distribution function ρ. Thus, our goal is to express the single-particle distribution function in terms of some variational parameters, calculate the energetic and entropic terms in the free energy, and then minimize the free energy over those variational parameters. For these calculations, we assume that the bend is along the direction of the applied field (E = E 0ˆx). We suppose that site i has its long axis along the Z axis as do the four neighbors in XY plane. Let θ be the magnitude of bend from one site to another. Thus, the central site has the long axis a ˆn i = (0, 0, 1) while the six neighbors have the orientations ˆn +x = (0, 0, 1), ˆn x = (0, 0, 1), ˆn +y = (0, 0, 1), ˆn y = (0, 0, 1), ˆn +z = (sin θ, 0, cos θ) and ˆn z = ( sin θ, 0, cos θ). We further assume that the short axis of the molecules lies in the XY plane and its orientation is (cos θ i, sin θ i, 0), with θ i being angle made by ˆb i with X-axis. Considering these assumptions, the distribution function depends only on angle θ i, and is written as ρ(θ i ) = exp[v 1 cosθ i + v 2 cos 2θ i ] π 0 exp[v 1 cosθ i + v 2 cos 2θ i ]dθ i, (4.11) where v 1 and v 2 are variational parameters. The order parameters are then P 1 = 2π 0 cos θρ(θ)dθ and P 2 = 2π 0 cos 2θρ(θ)dθ. In this notation, P 1 is the polar order parameter and P 2 is the biaxial order parameter. Here, we assume a perfect uniaxial order through the entire range of temperature studied. The partition function is Z = 2π 0 exp[v 1 cosθ + v 2 cos 2θ]dθ.

98 82 The interaction energy in Eq. (4.6) is averaged with the distribution function in Eq. (4.11). On expanding to second order in small bend θ, the internal energy of the system is U = 4A + B 2 θ 2 3B 1 P B 2P B 1P1 2 θ 2 CP 1 θ B 2P 2 θ B 2P 2 2 θ2 EP 1. (4.12) The entropic contribution to the free energy is therefore TS = k B T log ρ(θ i ) = k B T[v 1 P 1 + v 2 P 2 log(z)], (4.13) per lattice site. The expression for total free energy is F = 4A + B 2 θ 2 3B 1 P B 2P B 1P1 2 θ2 CP 1 θ B 2P 2 θ B 2P 2 2 θ2 EP 1 +k B T[v 1 P 1 + v 2 P 2 log(z)]. (4.14) The problem to be solved is then the following: for a set of interaction parameters A, B 1, B 2 and C, one needs to determine the values of v 1 and v 2 and θ that minimizes the free energy, F = U TS. From the values of v 1 and v 2, we can calculate the order parameters P 1 and P 2. The free energy in Eq. (4.14) was numerically minimized with Mathematica for different values of interaction parameters. Examples of the variation of order parameters with temperature is shown in Fig It is clearly seen that for small B 1 and B 2, there is a transition from N U N B followed by a transition from N B P at low temperature [see Fig. 4.7(a)]. For slightly larger B 1, the line of N U N B and

99 83 P 2, P 1, θ θ P 1 P 2 P 2, P 1, θ P N N B U 0.2 P N U θ P 1 P (a) T (b) T Figure 4.7: Mean-field results for the order parameters P 2, P 1 and θ as a function of T at zero field (E = 0), for different values of the interaction parameters chosen to show different kind of transitions: (a) B = 0.04 and C = 0.4. (b) B = 0.27 and C = 0.4. the N U P transition line coalesces into a single line, and there is a direct transition from the N U to P phase [see Fig. 4.7(b)]. A complete phase diagram as a function of B 1 and B 2 is shown in Fig. 4.8 and is quite similar to the simulation results. For moderate values of B 1 and B 2, there is cascade of transitions as N U N B P. Starting from the low temperature polar phase, on increasing the temperature there is a transition to a biaxial nematic phase followed by another transition to the uniaxial nematic phase. Let T U B be the temperature at which transition takes place from biaxial to uniaxial nematic phase. On this line, 2 F 2 v 2 = 0 v 1 0 and v 2 1 (4.15) Evaluating and solving these equations, the temperature at which the uniaxial to biaxial nematic transition takes place is: T U B = 3B 2 2. Along the phase boundary separating the polar and the biaxial nematic phase,

100 N U 0.4 N U 0.8 T T N B P 0.1 P N B B B 1 Figure 4.8: Mean-field phase diagram as a function of B 2 and B 1 the following conditions must be satisfied, 2 F 2 v 1 = 0 2 F 2 v 2 = 0 v 1 1 andv 2 1 (4.16) The phase boundary shown in Fig. 4.8 was numerically evaluated from the above equations in combination with the free energy [Eq. (4.14)]. For larger values of B 1, on increasing the temperature there is a direct transition to the uniaxial nematic phase at some temperature (T U P ). The conditions for the minimum of the free energy are: F v 1 = 0 2 F 2 v 1 = 0 F v 2 = 0 and 2 F 2 v 2 = 0 (4.17) At this temperature both variational parameters are small i.e. v 1, v 2 1. These equations were solved simultaneously to estimate the phase boundary between the

101 85 θ P N 0.02 P N B N U U θ (a) T (b) T Figure 4.9: Temperature dependence of bend in mean-field calculation: (a)n U N B P (b) N U P. uniaxial nematic phase and the polar phase. Fig. 4.9 shows the schematic variation of the bend/bond with temperature obtained from numerical minimization of the mean-field free energy. Note that this is the result for an applied field, E = The graphs corresponds to two different types of transition: (a)n U N B P and (b)n U P. In both cases, it is seen that the bend flexoelectricity is significant in nematic phases and increases dramatically near the transition to the polar phase. Note that the mean-field results are very similar to the simulation results both in variation of order parameters and phase diagrams. This indeed indicates that the mean-field calculation captures the essential physics of this model. The only difference in two methods is in the transition temperatures. It is expected that mean-field theory has an exaggerated tendency toward the ordered phase.

102 Summary and Discussion One issue with regard to theoretical studies on the flexoelectric effect is to calculate the magnitude of flexoelectric coefficients. We believe that we have shown a new method to calculate it from the difference in the Q tensor between the sites. This is one of the key features of this study. From the coupling of bend vector with polarization, a term in the interaction energy that favors bend deformation is computed. Through the model, the flexoelectric response of bent-core liquid crystals is studied as a function of applied field and temperature. Monte Carlo simulations have shown four stable phases at different temperatures: isotropic, uniaxial nematic, biaxial nematic and polar. The converse flexoelectric effect is proportional to an applied electric field and increases dramatically near the transition to the polar phase. We can consider this effect as a susceptibility to an applied field, which diverges at the second order transition to the polar phase. Therefore, flexoelectricity is not just a molecular effect but could be a statistical effect associated with the response of correlated volumes of molecules, which increases as one approaches the polar phase. Therefore, this model could explain the observed giant flexoelectricity [20,21] in bent-core liquid crystals.

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105 CHAPTER 5 Chirality and Biaxiality in Cholesteric Liquid Crystals Chiral liquid crystals commonly form a cholesteric phase, in which the molecular director is twisted into a helix. In this chapter, I present various theoretical approaches to understand some fundamental questions about cholesteric phases. 5.1 Opening Remarks In liquid crystals, there is a close connection between chirality, asymmetry under reflection, and biaxiality, orientational order in the plane perpendicular to the director. In the 1970s, Priest and Lubensky [1] recognized that a cholesteric liquid crystal must have some slight biaxial order because of the difference between the directions along and perpendicular to the helical axis. Brand and Pleiner [2] showed theoretically that chirality can smear out the transition between uniaxial and biaxial phases, and Kroin et al. [3] confirmed this smearing experimentally in lyotropic liquid crystals. Later, Harris, Kamien, and Lubensky [4] developed a microscopic model of molecules interacting through classical central-force interatomic potentials, and found that cholesteric twist can only form if there are at least short-range biaxial correlations between molecules. By comparison, in a system with quantum dispersive interactions, cholesteric twist can form even without such correlations [5]. In recent years, there has been a resurgence of interest in biaxial liquid crystals driven in part by experimental reports of the discovery of a biaxial nematic phase in 89

106 90 thermotropic liquid crystals [6, 7], and in part by prospects for using biaxial liquid crystals for fast-switching display devices [8]. For that reason, it is now important to re-examine the interplay between chirality and biaxiality in liquid crystals. The key issue is: How is the cholesteric pitch affected by biaxial order either by long-range biaxial order or by short-range biaxial correlations? In this chapter, we investigate this issue through three theoretical approaches: (1) Monte Carlo simulations of a lattice model for chiral molecules interacting via anisotropic van der Waals forces. (2) Mean-field theory for the same lattice model. (3) Landau theory based on symmetry-allowed couplings between twist and biaxial order. Through all three approaches, we calculate the cholesteric twist as a function of molecular chirality, molecular biaxiality, and temperature. These calculations show that chirality acts as an effective field on the biaxial order, which changes the secondorder uniaxial-biaxial transition into a rapid but nonsingular evolution. Conversely, biaxial order enhances the cholesteric twist, i.e. reduces the pitch, so that the pitch greatly decreases in the low-temperature, highly biaxial state. The calculations also allow us to reconsider the relationship between twist and short-range biaxial correlations. Based on this theoretical work, we discuss opportunities for experimental studies of chiral biaxial liquid crystals. 5.2 Dipole-Induced-Dipole Interaction For the simulations, we need a model molecular structure that can exhibit biaxial order with or without chirality. Inspired by van der Meer et al. [9], we consider

107 91 (a) (b) Figure 5.1: (a) Achiral and (b) chiral biaxial molecular structures studied in this work. a structure with two ellipsoids arranged rigidly in the shape of the letter H. Each ellipsoid represents an extended, anisotropic charge distribution within the molecule. If the two ellipsoids are parallel, as in Fig. 5.1(a), this is an achiral biaxial structure, with a biaxiality characterized by the separation h. By contrast, if the ellipsoids are twisted about the central connector in the opposite directions, as in Fig. 5.1(b), this is a chiral biaxial structure. The twist angle (χ) of each ellipsoid from the parallel configuration determines the chirality of the structure. The interaction between any two ellipsoids on neighboring molecules is the van der Waals dipole-induced-dipole interaction. Hence, the total interaction between two molecules i and j is the sum of four pairwise interactions among the constituent ellipsoids, U ij = A α,β=1,2 (ê iα ê jβ ) 2, (5.1) r 6 iα,jβ where ê iα is the orientation of ellipsoid α on molecule i, and r iα,jβ = r jβ r iα is the center-to-center distance between two interacting ellipsoids iα and jβ. 5.3 Twisting Torque Between Two Chiral Molecules As a first test to this model, the torque between two molecules is calculated in the limit of free uniaxial rotation (i.e., the biaxial correlations are ignored) [see

108 92 Figure 5.2: Free uniaxial rotation between two chiral molecules. Fig. 5.2]. Mathematically, it means averaging out the interaction potential over the biaxial angles and calculating the derivative of the average energy. For a quantitative description, the molecular axes are written in terms of Euler s angles. For a molecule at site i, the molecular axes are: ˆn i (θ i, φ i ), ˆb i (θ i, φ i, ψ i ), and ĉ i = ˆn i ˆb i. Similarly for a molecule at site j, the molecular axes are: ˆn j (θ j, φ j ), ˆb j (θ j, φ j, ψ j ), and ĉ j = ˆn j ˆb j. Now, with this construction the interaction energy between two molecules is calculated for a general form of the potential V ij = J (ˆn iα ˆn jβ ) 2 f( r jβ r iα ). This potential is expanded in a Taylor series with respect to the ratio of intra-molecular size divided by intermolecular spacing( h r ij ) and then averaged over biaxial angles, ψ i and ψ j. The average interaction energy for θ i = θ j = π 2 is V ij [ ] cos = f[1] 2 φ ij (1 + 3 cos 2χ) 2 + (1 + 3 cos 2 χ) sin 2 χ 4 + h 4 f [1](1 + 3 cos2χ) sin 2χ sin 2φ ij, (5.2) where φ ij = φ j φ i. Surprisingly, but as predicted for such a quantum interaction [5], the effective chiral interaction between the molecular directors survives the rotational

109 93 Torque(τ) Molecular twist(χ) Figure 5.3: Torque between two chiral molecules as function of molecular twist (χ) averaged over the biaxial angles. averaging. This interaction gives an effective torque on parallel molecules, which favors a cholesteric twist. The average torque between two molecules is τ = V ij φ j φi =φ j = h 2 (1 + 3 cos 2χ) sin 2χf [1]. (5.3) sin 2χ The normalized torque is, τ = (1 + 3 cos 2χ) and depends on the extent of 2 molecular chirality [see Fig. 5.3]. This simple calculation indicates that the twisting torque is non-zero and the cholesteric phase can twist even without the short-ranged biaxial correlations for the dispersion induced interaction. This point is justified by other approaches in the preceding section. 5.4 Monte Carlo Simulation for Chiral Molecules In this Section, our goal is to simulate the cholesteric phase formed by chiral molecules that resemble the structure of a twisted dumb-bell as shown in Fig. 5.1(b).

110 94 Through these simulations, the macroscopic pitch is calculated as a function of molecular chirality, molecular biaxiality and temperature. First, I will discuss the simulation technique and then the phase diagrams obtained in the simulation. Initially, we perform Monte Carlo simulations of achiral biaxial molecules. We simulate a simple cubic lattice of size , with a molecule centered on each lattice site. In each Monte Carlo step, a molecule is randomly selected and its orientation is changed, following the standard Metropolis algorithm. The uniaxial order parameter S and biaxial order parameter V are calculated as described by Bates and Luckhurst [10]. The order parameters are defined as: S = 3 2 cos2 θ 1 2 T = 3 2 sin2 θ cos 2ψ U = 3 2 sin2 θ cos 2φ V = 1 2 (1 + cos2 θ) cos 2θ cos 2ψ cos θ sin 2φ sin2ψ. (5.4) Figure 5.4(a) shows a sample plot of the order parameters for h = It is clearly seen that the sequence of phases on lowering the temperature is: isotropic, uniaxial nematic and biaxial nematic. For small ellipsoid separation h, we find a first-order transition from isotropic (I) to uniaxial nematic (N u ), followed by a second-order transition to biaxial nematic (N b ) at lower temperature. The temperature range of biaxial nematic phase increases with h, as would be expected for broader molecules. Fig. 5.4(b) shows the full simulated phase diagram. We now use the same approach to simulate chiral biaxial molecules. In this system,

111 I s,v s v T 3 2 N U 0.2 N B N U I 1 N B (a) T (b) h (c) s,v N B N U I T s v (d) Φ N B N U I χ=0.0 χ= T (e) Figure 5.4: Monte Carlo simulation results: (a) Uniaxial and biaxial order parameters as functions of temperature T, for achiral biaxial molecules with ellipsoid separation h = (b) Complete phase diagram for achiral biaxial molecules, in terms of h and T. (c) Uniaxial and biaxial order parameters as functions of T, for chiral biaxial molecules with h = 0.24 and molecular twist angle χ = (d) Boundary twist angle Φ as a function of T. For achiral (χ = 0) molecules, Φ is locked at π, indicating that the system is not twisted. For chiral (χ = 0.08) molecules, Φ is not a multiple of π, indicating that the system is twisted, and the cholesteric twist increases as T decreases. (e) Cholesteric phase of chiral molecules, showing the macroscopic twist

112 96 we expect molecular chirality to induce a cholesteric twist. This twist is generally not consistent with periodic boundary conditions. Hence, we use self-adjusting twisted boundary conditions in the z-direction, following the method of Memmer [11]. In this method, the boundary twist angle Φ from the top to bottom of the cell is a free simulation variable, determined by the Monte Carlo process. In an untwisted system, Φ must be a multiple of π. Hence, the deviation of Φ from a multiple of π is a measure of the twist across the system, i.e. the inverse pitch. With this method, the simulation forms a cholesteric phase over a wide temperature range. A sample configuration showing the molecular orientations along the z-axis is shown in Fig. 5.4(e). Using these simulations, we determine the uniaxial and biaxial order parameters for systems of chiral molecules. Figure 5.4(c) shows S and V as functions of temperature T for ellipsoid separation h = 0.24 and molecular twist angle χ = At T = 3.4 there is a first-order transition from isotropic to cholesteric, as seen from the jump in S. In the cholesteric phase there is a slight nonzero value of V, as expected from Ref. [1]. As T decreases further, V gradually increases toward its maximum value of 1. There is no phase transition between uniaxial and biaxial, but only a nonsingular increase in V. Apparently the chirality acts as an effective field on the biaxial order, which smears out the N u -N b transition. We also determine the boundary twist angle Φ as a function of T, as shown in Fig. 5.4(d). For achiral molecules, the boundary twist angle is locked at Φ = π,

113 97 (a) Biaxial Correlations Distance (b) 2π/P h = 0.24 h = Molecular twist(χ) Figure 5.5: Monte Carlo simulation results: (a) plot of biaxial correlation function in the cholesteric phase (T = 0.9) and (b) variation of pitch (P) with molecular chirality (χ) for different h. indicating that the system is in a uniform nematic phase, either uniaxial or biaxial. By contrast, for chiral molecules with χ = 0.08, the results for Φ show a twisted cholesteric phase. The cholesteric twist is substantial just below the isotropiccholesteric transition, although the biaxial order parameter is very small there. The twist increases further as T decreases, and becomes largest in the temperature range that would be the biaxial phase for an achiral system. Thus, we see that the cholesteric twist and the biaxial order increase together, reinforcing each other, as T decreases. In addition to the long-range biaxial order, we measure the short-range biaxial correlations as a function of distance between nearby lattice sites. Mathematically, the biaxial correlations are expressed as g( r ij ) = 3 2 (ˆb i ˆb j ) 2 + (ĉ i ĉ j ) 2 (ˆb i ĉ j ) 2 (ĉ i ˆb j ) 2. (5.5) The biaxial correlation at a cholesteric temperature (T =0.9) is reported in Fig. 5.5(a). Through most of the cholesteric temperature range, these biaxial correlations are very

114 98 small; they do not become noticeable until slightly above the achiral N u -N b transition temperature. The dependence of the cholesteric pitch on molecular parameters χ and h is shown in Fig. 5.5(b). This variation clearly indicates that the macroscopic twist increases non-linearly with molecular chirality. In addition, the macroscopic twist increases with molecular biaxiality. 5.5 Mean-field Calculation for Biaxial Cholesteric To compare with the simulations, we construct a Maier-Saupe-type mean-field theory for the same lattice model. Here, we assume the system has perfect order of the long axes of the molecules, but variable biaxial order and variable cholesteric twist. Suppose that site i has its long axis along the x-direction, as do the four neighbors in the xy-plane, while the two neighbors in the z-direction have long axes twisted about the z-axis. The long axis at site i is ˆn i = (1, 0, 0), while the long axes of the neighbors are ˆn ±x = (1, 0, 0), ˆn ±y = (1, 0, 0), and ˆn ±z = R z (± θ)(1, 0, 0), where θ is the cholesteric twist from one layer to the next and R z ( θ) is the rotation operator about the z-axis. At each site, the molecular short axis must be in the plane perpendicular to the long axis. Hence, the short axis at site i is ˆb i = (0, sin φ i, cosφ i ), while the short axes of the neighbors are ˆb ±x = (0, sinφ ±x, cosφ ±x ), ˆb ±y = (0, sin φ ±y, cosφ ±y ), and ˆb ±z = R z (± θ)(0, sin φ ±z, cosφ ±z ), where the local angle φ represents the azimuthal angle of the short axis. We now construct a distribution function for the local azimuthal angle φ, which

115 can be written as ρ(φ) = 99 exp(c cos 2φ) 2π 0 dφ exp(c cos 2φ), (5.6) where C is a variational parameter representing the effective biaxial potential. It is related to the biaxial order parameter by V = 2π 0 cos(2φ)ρ(φ)dφ. With this distribution function, the mean-field free energy per site can be written as the sum of energetic and entropic terms, F = H + k B T log ρ, (5.7) where H is the average interaction energy of Eq. (5.1) between site i and its six neighbors. This free energy depends on two variational parameters (effective biaxial potential C and cholesteric twist θ), two molecular parameters (ellipsoid separation h and molecular chirality χ), and temperature T (in units of interaction strength A). We numerically minimize the free energy over the variational parameters for each set of molecular parameters and temperature, to find the biaxial order and twist. The numerical mean-field results are consistent with the Monte Carlo simulations. For achiral biaxial molecules, the system has a uniaxial phase with V = 0 at high temperature. As a critical temperature, it undergoes a second-order transition to a biaxial phase with V 0, and the biaxial order parameter increases as a power law as T decreases. This achiral system is untwisted, with θ = 0 for all T. By contrast, for chiral biaxial molecules, the system has a high-temperature cholesteric phase with a small but nonzero value of V. As T decreases, V increases gradually, without any phase transition between uniaxial and biaxial. The cholesteric twist θ is substantial

116 100 θ Mean field theory Landau theory Monte Carlo N B N U I T Figure 5.6: Theoretical results for cholesteric twist as a function of temperature (in units of interaction strength A). Crosses represent numerical mean-field theory, while the solid line represents Landau theory. In simulations, twist from layer to layer is calculated from the boundary twist angle Φ and is represented by circles. at high temperature, even when V is small, and it increases further as T decreases, as shown by crosses in Fig Thus, as in the simulations, we see that the chirality acts as a field that induces biaxial order and smears out the N u -N b transition, and conversely, the biaxial order increases the cholesteric twist. 5.6 Phenomenological Theory For further insight into the relationship between cholesteric twist and biaxial order, we construct a Landau theory for a chiral biaxial liquid crystal. The orientation of a biaxial molecule could be described in terms of three mutually perpendicular axes. Here, the following notation is used the long axis of a molecule is represented by ˆn, the short axis by ˆb, and ĉ refers to the third axis perpendicular to both ˆn and ˆb [see Fig. 5.2]. As in the mean-field theory above, we suppose the system has perfect uniaxial

117 101 order along the local axis ˆn(r), but variable biaxial order. The biaxial order can be described by the tensor B ij = V (b i b j c i c j ), where V is the magnitude of the order, and ˆb(r) and ĉ(r) are the two principal axes orthogonal to ˆn(r). The free energy can then be expanded in B ij and in gradients of ˆn(r), to obtain F = 1 2 K( in j )( i n j ) Kq 0 ɛ ijk n i j n k r(t T UB)Tr(B 2 ) str(b4 ) ttr(b6 ) uɛ ijk B jl n i l n k wɛ ijk B jl n i k n l (5.8) In this expression, the first line is the Frank free energy for director gradients in a chiral liquid crystal, the second line is a power series expansion in B ij, and the third line is a pair of chiral couplings between B ij and director gradients. If we now assume a cholesteric modulation of the form ˆn = (cosqz, sin qz, 0), ˆb = (0, 0, 1), and ĉ = ( sin qz, cosqz, 0), with an arbitrary twist wave vector q, the free energy simplifies to F = 1 2 Kq2 Kq 0 q r(t T UB)V sv tv 6 (u + w)v q. (5.9) In the limit of high temperature, where biaxial order is small and the s and t terms are negligible, we minimize this free energy over V and q to obtain V (u + w)q 0 r(t T UB ), q q 0 + (u + w)2 q 0 Kr(T T UB ). (5.10a) (5.10b) Equations (5.10) demonstrate that the twist q acts as a field on the biaxial order V, and conversely, the biaxial order increases the twist, and hence reduces the pitch.

118 102 Instead of treating the Landau coefficients as purely phenomenological parameters, we can derive them from the lattice model presented previously, by expanding the free energy of Eq. (5.7) in powers of biaxial order and twist. The coefficients in terms of the molecular parameter h, χ and temperature (T) are K 2 Kq 0 = 11A Ah2 + 39Ah4 + 3A cos 2χ Ah2 cos 2χ Ah4 cos 2χ A cos 4χ Ah2 cos 4χ Ah4 cos 4χ] = 6Ah sin 2χ + 18Ah 3 sin 2χ + 9Ah sin 4χ + 45Ah 3 sin 4χ r(t T UB ) 2 = 9A 2 15Ah Ah T + 6A cos 2χ +15Ah 2 cos 2χ Ah4 cos 2χ 3 2 A cos 4χ 45 4 Ah2 cos 4χ Ah4 cos 4χ Ah4 sin 2 χ 3 8 Ah4 cos 2χ sin 2 χ s 4 = 203Ah4 + T Ah4 cos 2χ Ah4 cos 4χ 1 16 Ah4 sin 2 χ Ah4 cos 2χ sin 2 χ 7Ah 4 sin 4 χ t 6 = 1015Ah4 + 5T Ah4 cos 2χ Ah4 cos 4χ 3 64 Ah4 sin 2 χ Ah4 cos 2χ sin 2 χ 39 8 Ah4 sin 4 χ u + w = 14Ah 3 sin 2χ + 12Ah sin 4χ + 105Ah 3 sin 4χ. Results of this calculation are shown by the solid line in Fig The predictions of Landau theory are consistent with simulation and mean-field results, except at low temperature where biaxial order is large and series expansion is unreliable.

119 Summary and Discussion It is interesting to compare our results with Ref. [4], which argued that short-range biaxial correlations are a key factor in determining cholesteric twist in systems with classical central-force interactions. We also find an important connection between biaxiality and cholesteric twist, but it differs from their argument in two ways: (a) Our model shows some twist even in the limit of no biaxiality. This result does not contradict Ref. [4], because our system does not have central-force interactions; it is consistent with Ref. [5] for quantum dispersive interactions. However, it draws attention to the fact that most liquid crystals have quantum dispersive interactions, while central-force interactions are unusual. (To be sure, quantum dispersive interactions are derived from fluctuating microscopic central-force interactions among electrons, and these electrons might have some biaxial correlations. However, such correlations would be difficult to observe in either experiments or simulations; normal observations average over the fluctuations.) (b) Our results show there is not a cause-and-effect relationship between biaxiality and cholesteric twist; rather, there is a mutually reinforcing interaction between them. Twist acts as a field on biaxial order, and conversely, biaxial order helps to increase twist. As temperature decreases, biaxiality and twist increase together. The theory presented here has implications for experiments on biaxial liquid crystals, either thermotropic or lyotropic. It should be possible to choose an achiral host that has a uniaxial-biaxial transition, and add a chiral dopant. The theory predicts that the dopant will induce a small twist (large pitch) in the uniaxial phase, but

120 104 the twist will increase (pitch will decrease) as the uniaxial-biaxial transition is approached. For low dopant concentration the twist will diverge as (T T UB ) 1, while for larger concentration the divergence will be more rounded. At the same time, the chiral dopant will smear out the uniaxial-biaxial transition. The chirality-induced rounding of the uniaxial-biaxial transition has been observed in lyotropics [3], but has not yet been investigated in thermotropics. Moreover, to our knowledge, no experiments have yet examined the cholesteric twist around the uniaxial-biaxial transition in either thermotropics or lyotropics. This should be a promising area for experimental research, to further characterize the close relationship between chirality and biaxiality.

121 BIBLIOGRAPHY [1] R. Priest and T. C. Lubensky, Phys. Rev. A 9, 99 (1974). [2] H. R. Brand and H. Pleiner, J. Physique Lett. 46, L-711 (1985). [3] T. Kroin, A. M. Figueiredo Neto, L. Liébert, and Y. Galerne, Phys. Rev. A 40, 4647 (1989). [4] A. B. Harris, R. D. Kamien, and T. C. Lubensky, Phys. Rev. Lett. 78, 1476 (1997); Rev. Mod. Phys. 71, 1745 (1999). [5] S. A. Issaenko, A. B. Harris, and T. C. Lubensky, Phys. Rev. E 60, 578 (1999). [6] B. R. Acharya, A. Primak, and S. Kumar, Phys. Rev. Lett. 92, (2004). [7] L. A. Madsen, T. J. Dingemans, M. Nakata, and E. T. Samulski, Phys. Rev. Lett. 92, (2004). [8] S. Kumar, U. S. Patent 7,604,850 (2009). [9] B. W. van der Meer, G. Vertogen, A. J. Dekker, and J. G. J. Ypma, J. Chem. Phys. 65, 3935 (1976). [10] M. A. Bates and G. R. Luckhurst, Phys. Rev. E 72, (2005). [11] R. Memmer, J. Chem. Phys. 114, 8210 (2001). 105

122 CHAPTER 6 Concluding Remarks In this dissertation questions about recent experiments on liquid crystal science are answered with computer simulations and analytical calculations. However, these stories are far from complete. A variety of questions remain to be explored and more avenues are open for further investigations. In this chapter, I summarize the main results of my work. Smectic phases of achiral bent-core molecules comprise a very active field of research in liquid crystal science. The polar packing of these molecules in combination with a molecular tilt with respect to the layer normal induces chiral structures which are not present for typical calamatic liquid crystals. The interplay of different orders - polar, tilt and chiral- leads to several interesting structures that depend strongly on the electric field. In an effort to understand recent experiments that demonstrated chiral switching in smectic phases of bent-core liquid crystals in an applied field, a lattice model considering an explicit coupling between three symmetry breaking instabilities was developed. The model revealed a rich variety of phases depending on various interaction strengths as described in Chapter 2. In experiments these variations could be realized by interdigitation of smectic layers or by making stronger dipoles or by segregating a solvent that localizes at the layer interface and suppresses the out of layer fluctuations. Monte Carlo simulations have shown that the SmCP phase of 106

123 107 achiral bent-core molecules exhibits two antiferroelectric structures -the SmC A P A and SmC S P A at zero electric field. The preference of the system between these two states depends on various interaction strengths. The simulation showed that the antiferroelectric state could be changed to the ferrolectric state by an electric field. During this transition, the chirality of the system could either remain or be destroyed as in the experiments. Despite success in explaining the observed behavior qualitatively, it has been difficult to compare the simulation results directly with experiments. Therefore, a model that incorporates fine-molecular details is highly desirable and will be the subject of future study. Additionally, the unique feature in the polarization-field curve that shows hysteresis remains an open area for further investigations. Another thrust of these studies is on the flexoelectric effect in nematic liquid crystals. The question addressed in this dissertation is why the flexoelectric response of the nematic phase of bent-core molecules is so large in comparison to that for calamitic liquid crystals. One issue in regard to the theoretical studies of the flexoelectric effect is to calculate the magnitude of the flexoelectric coefficients. A remarkable feature of this study is a new method to calculate flexoelectric coefficients from the difference in Q -tensor between the neighboring molecules. In Chapter 3, a lattice model to study splay flexoelectric effect is presented. In this model, each lattice site has a spin representing the local molecular orientation, and the interaction between neighboring spins represents pear-shaped molecules with shape polarity. Through, these model the flexoelectric responses as a function of interaction

124 108 parameters, temperature, and applied electric field is studied. The resulting phase diagram has three phases: isotropic, nematic, and polar. This model predicts that, there is a large splay flexoelectric effect in the nematic phase, which diverges as the system approaches the transition to the polar phase. In Chapter 4, the splay flexoelectric model is extended to study the bend flexoelectricity in a system of bent-core liquid crystals. From the coupling of the symmetry permuted bend vector with polarization, a term in the interaction energy that favors bend deformation from site to site is computed. Through this model, the flexoelectric response of bent-core liquid crystals as a function of applied field and temperature is studied. Monte Carlo simulations have shown four stable phases on reducing the temperature: isotropic, uniaxial nematic, biaxial nematic and polar. The converse flexoelectric effect is proportional to an applied electric field and it increases dramatically near the transition to the polar phase. We can consider this effect as a susceptibility to an applied field, which diverges at the second order transition to the polar phase. Therefore, flexoelectricity is not just a molecular effect but could be a statistical effect associated with the response of correlated volumes of molecules, which increases as one approaches the polar phase. Despite these achievements, this model also has some limitations. As the model ignores the molecular details, a direct comparison of the simulation results with experiments was somewhat restricted. Secondly, for large value of the bend interaction parameter C, the amount of bend across the system might exceed 180 0, and hence the system could form stripes in bend patterns. Therefore, a refinement of the model

125 109 to address these issues could be an area of interest for future. In addition, though the mean-field calculation captures the qualitative behavior in the simulation, several approximations and assumptions are made to simplify the problem. For example, in these calculations, a perfect nematic order through the entire temperature range is assumed. Consequently, these calculations were unable to reproduce the whole phase diagram. A more general mean-field calculation for this model could be along the line of future works as well. One of the contributions of this dissertation is in understanding some fundamental questions about the cholesteric phase of liquid crystals. An outstanding problem on this topic is to determine the pitch of the cholesteric helix in terms of microscopic parameters, and to explain the pitch that is so much larger than the molecular length scales. We believe that we have achieved the goal for a special case of molecules interacting via the dispersion forces. Through the simulation of the lattice model it is seen that the cholesteric twist decreases with increase in the temperature. It is also found that the biaxial correlations enhance the twist but are not required for a twist with this fluctuation-induced interaction. The simulation results are consistent with mean-field calculations for this model. One important question that remains is whether the short range biaxial correlations are required to have a macroscopic twist for molecules interacting via central force potential. Initial simulation with central force interactions hinted that upon cooling from the isotropic phase, the system transforms to the crystal phase without any intermediate mesophases for a simple Lennard-Jones type interaction and deserve further investigation.

126 APPENDIX A Calculation of the Splay and Bend Vector From Site to Site In this appendix, the full calculation of splay and bend vector starting from Q tensor is presented. The splay and the bend interaction terms in the lattice Hamiltonian is derived from these vectors. A.1 Calculation of splay vector Fig. A.1 shows different configuration with the same magnitude of splay between neighboring molecules on a lattice. It implies that the magnitude of splay must be invariant on interchanging the center of masses and on reversing the orientation of the molecules. Mathematically, amount of splay must respect the following symmetry operations: i j or ˆn i ˆn i and ˆn j ˆn j. This is not possible with a scalar ( n) because it s linear in n. However, the quantity ˆn( ˆn) respect these symmetry operations and represents splay vector. The lattice version of the splay vector could Figure A.1: Different splay configuration of neighboring molecules on a lattice. 110