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Nonseparable interactions and orientational phenomena in liquid crystals Gingras, Michel 1990

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NONSEPARABLE INTERACTIONS AND  ORIENTATIONAL  P H E N O M E N A IN LIQUID C R Y S T A L S By Michel J.P. Gingras B . Sc. (Physics) Universite Laval (1983) M . Sc. (Physics) Universite Laval (1985) A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF PHILOSOPHY  in T H E FACULTY O F G R A D U A T E STUDIES PHYSICS  We accept this thesis as conforming to the required standard  T H E UNIVERSITY O F BRITISH COLUMBIA  May 1990 © Michel J.P. Gingras, 1990  In  presenting  degree at the  this  thesis  in  University of  partial  fulfilment  of  of  department  this thesis for or  by  his  or  requirements  British Columbia, I agree that the  freely available for reference and study. I further copying  the  representatives.  an advanced  Library shall make  it  agree that permission for extensive  scholarly purposes may be her  for  It  is  granted  by the  understood  that  head of copying  my or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department The University of British Columbia Vancouver, Canada  DE-6 (2/88)  Abstract  Intermolecular interactions are anisotropic since they depend on the orientations of the interacting molecules. They are responsible for the orientational phase transitions observed in molecular systems. The effects of nonseparable interactions, which depend on both the molecular and intermolecular vector orientations, are investigated using MonteCarlo simulations. It is found that for systems with quenched positional disorder, the nonseparability of the pair potential leads to orientational frustration. Expanding the pair potential in terms of its rotational invariants allows one to isolate random field terms whose presence are shown to be at the origin of the non-frustrated patches observed in such disordered systems. For liquid crystals, where translational and rotational degrees of freedom are in thermal equilibrium, Monte-Carlo results show that an ordering of the molecular orientations induces an ordering of the nearest-neighbor bond orientations only when the pair potential is nonseparable. These results suggest that it is necessary to reinterpret the mean-field theory of nematic liquid crystals as an approximation which neglects the presence of bond ordering in these systems. As well, they unravel the microscopic origin of the phenomenological coupling between bond and molecular orientations invoked in Ginzburg-Landau theories of liquid crystals. The nonseparability of the intermolecular pair potential leads to a coupling between the nearest-neighbor bond orientational field and the molecular orientational field. It is shown that such coupling introduces new and interesting effects in liquid crystalline systems where interactions between the molecular and nearest-neighbor bond orientations are usually ignored.  u  Table of Contents  Abstract  ii  List of Tables  viii  List of Figures  ix  Acknowledgements  xiv  1 Introduction to Liquid Crystals 1.1  Liquid Crystals  1  1.2  Separable and Nonseparable Interactions  5  1.3  Nematic Phase  8  1.4  Smectic Fluid Phases  11  1.4.1  Smectic A  12  1.4.2  Smectic C  15  1.4.3  Two-Dimensional Melting and Classification of Smectic Phases . .  19  1.5 2  1  Outline of the Thesis  31  Mean-Field Theory of the Isotropic-Nematic Phase Transition  34  2.1  Mean-Field Description of the Isotropic-Nematic Phase Transition . . . .  34  2.1.1  Mean-Field Theory of the Lebowhl-Lasher Lattice Model  35  2.1.2  Mean-Field Theory for a True Liquid Crystal Model - The MaierSaupe Model  2.2  39  Mean-Field Theory and Nonseparability iii  44  2.2.1  Macroscopic Manifestations of the Nonseparability  44  2.2.2  Mean-Field Approximation and Nonseparability  49  2.2.3  The Anisotropic van der Waals Potential - A Simple Nonseparable Model  2.3 3  51  Towards a Computer Experiment  54  Computer Simulations I  58  3.1  Computer Simulations, Monte Carlo and Metropolis Algorithm  58  3.1.1  A n Impetus for Computer Simulations  58  3.1.2  What is a Computer Simulation?  60  3.1.3  Molecular Dynamics  61  3.1.4  Monte Carlo Simulation and Metropolis Algorithm  62  3.2  3.3  3.4  Lattice Systems  65  3.2.1  Two-Dimensional Lattices  66  3.2.2  Three-Dimensional Lattices  68  3.2.3  Discussion of the Results for Lattice Systems  70  Quenched Positional Disorder  72  3.3.1  Random Parking Systems  72  3.3.2  Ground States of Random Parked Systems  73  3.3.3  Microscopic Origin of the Frustrated Islands  76  Annealed Positional Disorder: Preliminary Results on 2D Liquid Crystal Models  3.5  82  3.4.1  Why Study 2D Liquid Crystalline Models?  82  3.4.2  Pair Potentials  83  3.4.3  Results  85  Formation of Chain-Like Structures  iv  87  3.6 4  5  6  Discussion  95  Computer Simulations II  97  4.1  Induced Bond Orientational Ordering in Smectic Liquid Crystal Films . .  98  4.2  Computational Technique  103  4.3  Thermodynamic Quantities  105  4.4  Local Structure for the Separable and Nonseparable Systems  109  4.5  Correlation Functions  112  4.6  Defect Strutures in the Nonseparable System  122  4.7  Liquid-Like Behavior of the Nonseparable System  123  4.8  System-Size Dependence of Q  125  4.9  Summary  127  B O F - M O F Coupling and Director Fluctuations in 2D Liquid Crystals 131 5.1  Nature of the Ordering in 2D Liquid Crystals  132  5.2  B O F - M O F Coupling Model  135  5.3  Results  138  5.4  Discussion  143  Freedericksz Transition in Tilted Hexatics  145  6.1  Response of a Liquid Crystal Material to an External Field  145  6.2  Model  147  6.3  Results  149  6.3.1  Infinite B O F Elastic Constant  151  6.3.2  Finite B O F Elastic Constant  154  6.3.3  Numerical Investigation of the Euler-Lagrange Equations  161  6.3.4  Application to a Well Characterized Hexatic Material — 80SI . .  163  v  6.4 "7  9  165  Microscopic Model for the Sm-I - Sm-F Transition  167  7.1  The Smectic-I - Smectic-F Transition  168  7.2  Model for the I-F Transition and Intermolecular Pair Potential  169  7.2.1  Preliminaries  169  7.2.2  Interactions between Tilted Molecules  171  7.3 8  Discussion  Results and Discussion  176  Viscous Fingering  182  8.1  Importance of Anisotropy in Problems of Pattern Formation  182  8.2  Anisotropic Viscosities and Viscous Fingering in a Radial Geometry . . .  184  8.3  Discussion  187  Conclusion and Avenues for Future Work  188  9.1  Conclusion  188  9.2  Avenues for Future Work  190  Appendices  192  A Noise and the Linear Stability Analysis of Viscous Fingering  192  A.l  Introduction  193  A.2 The Model  196  A.3 Limit of validity of the linear regime  200  A.4 Number of fingers in the linear regime  204  A.5 Final Remarks  208  A.6 Acknowledgement  208  A.7 References  210  vi  212  Appendices B Breakdown of the Linear Stability Analysis of Laplacian Growth B.l  212  Amplitude-wavelength Relationship at the Breakdown of the Linear Regime213  B.2 Breakdown of the Linear Regime and Morphological Phase diagram . . .  215  B.3 References  218  Bibliography  220  vii  List of Tables  Summary of the K N T H Y theory  viii  L i s t of Figures  1.1  Nematogen p-azoxyanisole (PAA) and its steric model  5  1.2  Separable vs nonseparable interactions  7  1.3  Schematic representation of the orientational order in a nematic phase . .  9  1.4  Schematic arrangement of the molecules in a smectic-A phase  13  1.5  Rotational averaging of the pair-potential in a smectic-A phase  14  1.6  Schematic arrangement of the molecules in a smectic-C phase  16  1.7  Detailed geometry of a smectic-C phase  17  1.8  Planar arrangement of two interacting particles ' 1 ' and '2'  18  1.9  Spin vortex  21  1.10 Local hexatic structure  25  1.11 Long-range bond order and short-range positional order  26  1.12 Schematic representation of the tilt orientation in tilted hexatic smectic phases  29  1.13 Classification of liquid crystal phases according their degree of ordering .  30  2.1  Interactions between ordered domains in a nematic phase  46  2.2  Splay, bend and twist deformations in nematics  47  2.3  X-ray diffraction pattern for a nematic liquid crystal  49  2.4  Dependence of the van der Waals potential on the orientation of intermolecular vector  55  3.1  Construction and evolution of a scientific model  59  3.2  Schematic representation of a system with periodic boundary conditions .  66  ix  3.3  Lowest energy configurations for two-dimensional lattices  67  3.4  Lowest energy configuration of the face centered cubic lattice  70  3.5  Typical sequential adsorption pattern of discs in a plane  74  3.6  Radial pair distribution function for the 2D random parking problem at the jamming coverage density  75  3.7  Superposition of three ground state configurations for the 2D random system 77  3.8  Slices of thickness d of the ground state for a ZD random parking configuration  3.9  78  Superposition of 20 orientational ground state configurations for different values of the random  field  79  3.10 Snapshots of particle configurations for the separable system at p = 0.888  86  3.11 Order parameter Q as a function of temperature for the separable potential at density p = 0.888  87  3.12 Snapshots of particles configurations for the nonseparable system at p = 0.888 3.13 Oij dependence of r  88 for the nonseparable model with J = 1/2  90  3.14 Oij dependence of V for the nonseparable model with J = 1/2  91  3.15 Snapshots of chains configurations at T = 4.8, 4.2, 3.6 and 2.6  92  m  m  4.1  Schematic representation of a freely suspended liquid crystal  film  99  4.2  Order parameter Q vs temperature for the nonseparable system at p = 1.00107  4.3  Average configurational energy per particle vs temperature for the nonseparable system at p = 1.00  107  4.4  Pressure vs temperature for the nonseparable system at p = 1.00  108  4.5  Configurational specific heat at constant volume vs temperature for the nonseparable system  108  x  4.6  Snapshots of the configuration of the particles interacting via the nonseparable potential close to the isotropic-nematic transition  4.7  110  Snapshots of the configuration of the particles interacting via the separable potential close to the isotropic-nematic transition  Ill  4.8  Snapshots of lowest energy bonds network  113  4.9  Radial pair distribution function go(r) for the nonseparable system  . . .  114  4.10 Large distance behavior of go(r) for the nonseparable system .  115  4.11 Large distance behavior of go(r) for the separable system  115  4.12 Molecular orientational pair correlation function g2(r) for the nonseparable system  116  4.13 First nearest-neighbor bond orientational correlation function C2(r) for the nonseparable system  118  4.14 Third nearest-neighbor bond orientational correlation function c ±(r) for 2  the nonseparable system  119  4.15 Molecular orientational pair correlation function g2(f) for the separable system  120  4.16 Bond orientational correlation function for the separable system  121  4.17 Typical M O F defect structures for the nonseparable system  124  4.18 Time dependence of the M O F order Q for two different system sizes . . .  126  4.19 Subsystem size dependence of the M O F order Q  127  5.1  Theoretical system size dependence of the M O F fluctuations in a nonseparable system  5.2  139  Theoretical system size dependence of the M O F order parameter Q in a nonseparable system  5.3  139  System size dependence of (<fi ) for a separable system 2  xi  141  5.4  System size dependence of Q for a hard-needle  fluid  142  5.5  System size dependence of Q for a hard-ellipse  fluid  142  6.1  Freedericksz transition in a tilted hexatic phase with an infinite B O F elastic constant  6.2  153  Magnetic field dependence of the M O F distortion for a Freedericksz transition in tilted hexatics with finite B O F elastic constant  6.3  156  Magnetic field dependence of the B O F distortion for a Freedericksz transition in tilted hexatics with finite B O F elastic constant  6.4  157  Magnetic field dependence of the free energy for a Freedericksz transition in tilted hexatics with finite B O F elastic constant  6.5  158  Phase diagram for Freedericksz transition in tilted hexatics with finite B O F elastic constant  6.6  160  Comparison of 'exact' and SVA solution for a Freedericksz transition in tilted hexatics with finite B O F elastic constant  6.7  162  Period doubling route to chaos for iterative solution of Euler-Lagrange equations  7.1  163  Schematic arrangement of atomic species on interacting tilted smectic molecules  7.2  A.l  175  Phase diagram of close-packed ellipses interacting with distributed sources of isotropic van der Waals centers  178  Formation of fingers in the linear regime  201  A.2 Injection of air into a hele-Shaw cell filled with glycerine  202  A.3 Reduced breakdown radius X-/XQ versus reduced pressure £ for x = 0.1 0  and e = 1 0  203  - 5  xii  A.4 Fourier spectra of the amplitudes of the perturbations e (t) m  204  A.5 Average number of fingers (Nf) observed at the breakdown of the linear regime ( i = i _ ) versus the reduced pressure £  205  A . 6 Difference between the azymuthal number of fastest growing mode and average number of fingers observed at breakdown of linear regime versus noise level B. l  207  The velocities of a planar front moving with velocity v and perturbed by a sine wave of amplitude Ak(t)  214  B.2 Morphological phase diagram for viscous fingering in a radial Hele-Shaw cell217  xiii  Acknowledgements  I first want to thank Dr. Birger Bergersen, my thesis advisor, for his ready availability and for the large degree of freedom that he entrusted to me. In particular, I acknowledge a conversation in which he suggested I study van der Waals interactions in a random parking system. This study led, as a by-product, to other investigations, the results of which constitute a large fraction of the work presented in this thesis. I would like to thank my friends and collaborators, Peter Holdsworth and Zoltan Racz for all the interesting discussions about physics and life I had with them in the past five years. Their unfailing interest in physics has provided me with a continuous motivation in my work. I acknowledge Ian Affleck, Simon Fortin and Raymond Laflamme for helpful discussions. I am obliged to the Natural Sciences and Engineering Research Council, the Fond pour les Chercheurs et l'Aide a la Recherche, and the Physics Department at U B C for their pecuniary support. Finally, I would like thank my wife Joanna for carefully editing the manuscript for both grammar and content. More importantly, I am grateful for her moral support in periods of frustration and disillusion.  xiv  "The only people for me are the mad ones, the ones who are mad to live, mad to talk, mad to be saved, desirous of everything at the same time, the ones who never yawn or say a common-place thing, but burn, burn, burn like fabulous yellow roman candles..."  Jack Kerouac  XV  Chapter 1  Introduction to Liquid Crystals  This chapter is a short introduction to liquid crystals and is intended for the reader who is not familiar with the field. It is hoped that it will make this work as self-contained as possible and help the reader to better assimilate the material presented in the following chapters. Sections 1.1 and 1.2 introduce the topic of liquid crystals and the notion of nonseparable interactions we refer to in the title of the thesis. The Sections 1.3 and 1.4 contain a description of the nematics and smectics, the two main classes of liquid crystals that are discussed in this work. A n outline of the thesis is presented in Section 1.5.  1.1  Liquid Crystals  b  The everyday world we live in consists of three phases of matter: gas, liquid and solid. This picture is, however, not quite complete: some organic materials do not show a single phase transition from a liquid to a solid phase upon cooling, but rather one or several extra transitions leading to new phases. Both microscopic and macroscopic properties of these phases are intermediate between those of a liquid and a crystalline solid. For this reason, they are called mesomorphic phases (meso: middle and morph: form). The name liquid crystals was coined by researchers who found it more descriptive, and the  1  Chapter 1. Introduction to Liquid Crystals  2  two words are used synonymously . 1  In order to understand the origin of the various liquid crystal phases, it is useful to recall the distinction between an ordinary (isotropic) liquid and a crystalline solid. In a crystal, the centers of mass of the molecules are regularly positioned on a threedimensional lattice while no such long-range positional order exists in a liquid. mechanical properties of these two phases are most different: unlike a solid, a liquid does not support a shear force and flows easily. At the microscopic level, a crystal is recognized by its X-ray diffraction pattern which reveals sharp Bragg reflections characteristic of the regular lattice. The X-ray pattern of an ordinary isotropic liquid, on the other hand, shows only a faint circular ring in reciprocal space with a radius of the order of the inverse of the average intermolecular spacing. As well, the orientational degrees of freedom are completely disordered in an isotropic molecular fluid, as can be deduced from nuclear magnetic resonance ( N M R ) studies . Hence, mesophases can be obtained in two distinct ways: 2  1. By imposing positional ordering in one or two rather than three dimensions. A l though such proposition may sound surprising, this phenomenon happens frequently in liquid crystalline materials.  In most cases, one has positional order  in only one direction: such systems, called smectics can be viewed as a set of twodimensional (2D) liquid layers stacked on top of each other with a well-defined spacing.  See references [1, 2, 3, 4] for a review of the physics ofliquid crystals. The notion of orientational order or disorder is meaningless for atomicfluid(e.g. He,Ar,Kr,Xe) since these can be viewed as 'spherical objects'. 1  2  The  3  Chapter 1. Introduction to Liquid Crystals  2. B y allowing, in the case of nonspherical molecules, the orientational degrees of freedom to order at a temperature other than the melting temperature. Orientational transitions can occur in a crystal, in a liquid or even in a smectic phase. There are three different scenarios which can lead to orientational phase transitions: • Several crystals exhibit a transition from a low-temperature strongly orientationally ordered phase to higher temperature phase where the molecules commute freely between equivalent orientations. This high-temperature phase is positionally ordered (crystalline solid), but orientationally disordered. One often refers to these systems as plastic crystals. Solid hydrogen ( # 2 ) is n a  example in which such an orientational transition occurs. • Some mixed crystalline materials like AB —AMi- , where A and B are atomic x  x  species, M a molecular species and x the concentration of the B atoms, exhibit some kind of orientational phase transition. Unlike in plastic crystals, the molecular species M is orientationally disordered in both the high and lowtemperature phases. One observes, however, a dramatic change in the molecular reorientation response times as the system is cooled to low temperature. It appears as if the individual molecular orientational degrees of freedom are frozen out as the system is cooled below the so-called 'freezing temperature'. These systems are called molecular orientational glasses. KBr — K(C N)\x  and 7V  2)I  —  Ari_ are examples of systems where such glassy transitions have x  been extensively studied in the past years. • A very large number of organic substances show a mesophase where the molecules are oriented on average along a common direction, but without having the long-range positional order characteristic of crystals. These anisotropic liquids are called nematics.  Above some critical temperature, the nematic  x  Chapter 1. Introduction to Liquid Crystals  4  phase undergoes a first order transition to an isotropic liquid phase. The term liquid crystals applies to both nematics and smectics. It is now generally accepted that liquid crystalline phases are observed when the constitutive structures consist of elongated (strongly anisotropically steric) objects. The most common of these structures are: • Small Organic Molecules: A classical example is illustrated in Figure 1.1.  The  simplest way to induce a transition in these systems is by varying the temperature (although the effect of pressure is also studied). Hence, these systems are often refered to as  thermotropic.  • Long Helical Rods: The deoxyribonucleic acid (DNA) (TMV)  belong to this category.  and the tobacco mosaic  For these systems, called lyotropic,  virus  the phase  transition is most easily induced by changing the concentration of rods rather than the temperature. • Associated Structures: Typical examples in this category are found in the soapwater systems. A system of aliphatic chains with a polar head immersed in water cluster together in a way to minimize (maximize) the contact between the chain (polar head) and the water. The resulting objects give rise to associations and to the mesomorphic behavior observed in the present category. • Polymer Liquid Crystals: Long polymer molecules can also give rise to liquid crystalline behavior. The molecules themselves can exhibit nematic ordering or their alkyl side-chains can lead to smectic-like ordering. The situation is completely different for plastic crystals and molecular orientational glasses where the particle undergoing the orientational phase transition is usually nearly  Chapter 1. Introduction to Liquid Crystals  5  20A  5A  Steric Model of PAA  Figure 1.1: Nematogen p-azoxyanisole (PAA) and its steric model.  spherical in shape. For this reason, the fields of liquid crystals, plastic crystals and orientational glasses are usually regarded as being distinct and only vaguely related; in this thesis, we concentrate mainly on liquid crystals. We shall, however, illustrate that the thermodynamic behavior of orientational glasses may have the same microscopic origin, at the level of the intermolecular pair-potential, as some interesting orientational phenomena observed in exotic smectic liquid crystals.  1.2  Separable and Nonseparable Interactions  In the last section, we described orientational phase transitions in various systems. It is clear that the interactions between the particles must depend on their relative orientations in order to lead to such transitions. It turns out that a great deal of the physics involved in liquid crystal systems can be learned by considering pair potentials that depend only on the relative orientations between the molecules and not on the orientation of the  6  Chapter 1. Introduction to Liquid Crystals  intermolecular vector fij. A 'model' potential of this form is said to be separable. A n example of a separable interaction is the well known isotropic Heisenberg model of a ferromagnet: HHeisenberg = ~  J(\fi ~ rj\)S(fj) • S{fj) ,  (1.1)  «.J  where the sum is taken over the sites i and j of some lattice. The strength of the interaction between the spins S(ri) and S(rj), J(|f,- — Fj\), depends only on on the distance, \fi — fj\, separating them, but not on the direction of r,j. The most general case, where the interaction depends on the orientation of f,j, is said to be nonseparable. A picture is worth 1,000 words (Figure 1.2). The dipole-dipole potential is a common example of a nonseparable interaction: HdipolaT  = -3-(pi ' Pi ~ 3pi • rtjrij • Pj)  r«j •  where />, and pj are the strengths of the dipoles and  (1.2)  is a unit vector parallel to fij.  For example, two dipoles parallel to each other and to the vector, f{j, connecting their center of mass have an energy of —2piPjrjf while their energy is PiPjrjf when they are mutally parallel but perpendicular to rij. A system of particles interacting via a separable potential has a complete rotational symmetry of both the molecular variables and the intermolecular vector. A separable system is said to have an SO(3)xSO(3) symmetry. For a nonseparable potential, terms of the form r^ • pi lower or break the 'original' SO(3)xSO(3) symmetry of the system. For example, the dipolar Hamiltonian above has only an SO(3) symmetry. Later in the thesis, will discuss at length a nonseparable potential, V , which is given by: NS  V  V  NS  NS  oc Hl  polar  .  (1.3)  is invariant under an inversion of the molecules:  V (puPj) = V (-pi,pj) = V (pi,-pj) . NS  NS  NS  (1.4)  Chapter 1. Introduction to Liquid Crystals  7  a) Separable Interaction  b) Nonseparable Interaction ff(NS).  1  #<" >: 5  c) Nonseparable Interaction  Figure 1.2: Separable vs nonseparable interactions. In a Heisenberg ferromagnet, the interaction energy is invariant under any arbitrary rotation of the spin for fixed rij, as shown in a). In the case of a nonseparable interaction, the interaction energy is not invariant under any rotation of the particles as in b). It remains, however, invariant under a roation of both the intermolecular vector and molecular orientation as shown in c).  Chapter 1. Introduction to Liquid Crystals  8  This nonseparable system has an extra discrete (Ising-like) symmetry which is not present in the original  Hdipoiar',  it has an SO(3)xZ(2) symmetry.  Liquid crystal molecules are complex objects (see Figure 1.1) which interact via anisotropic repulsive forces (steric effects), dipolar, quadrupolar and van der Waals forces; all these interactions are nonseparable. Hence, one expects the full microscopic pair potential to be nonseparable.  In this work, we investigate what effects are introduced  by this nonseparability of the pair potential. We examine whether the presence of the nonseparability of the pair potential introduces new physical effects which are absent in separable models. In Chapter 2, we show that this question arises naturally when considering the mean-field theory of the isotropic-nematic phase transition for systems with nonseparable interactions. In the following two sections we review the basic and most important properties of nematic and smectic liquid crystals. 1.3  Nematic Phase  The name nematic comes from the Greek vn\ia which means thread and refers to threadlike defects commonly observed in these systems. A schematic representation of the orientational order present in a nematic phase is shown in Figure 1.3. The nematic phase is a fluid in which the molecules have long-range orientational order but only short-range positional order, as revealed by the absence of Bragg peaks in their X-ray diffraction pattern. In fact, nematics flow like ordinary fluid with a viscosity of approximately 0.1 Poise (by comparison, water has a viscosity of about 0.01 Poise at room temperature). The microscopic origin of this phase is most easily understood by simply considering  9  Chapter 1. Introduction to Liquid Crystals  N  Figure 1.3: Schematic representation of the orientational order in a nematic phase. A l though the molecules carry a dipolar moment, assumed parallel to the molecular axis as illustrated by the arrows, the system organizes itself in such a way that as many dipoles are 'up' as are 'down'. The molecules point, on average, along the director N.  the steric constraints that occur when filling space with asymmetric objects (see Figure 1.1). These constraints are introduced as the density of the system is increased, either by lowering the temperature or increasing the pressure. A system of asymmetric objects is able to fill the space more efficiently by allowing the build-up of some shortrange orientational order where the rod-like objects are, on average, parallel to each other over some microscopic length scale. Theoretical arguments as well as computer simulations show that, for objects with large enough eccentricity, the system undergoes an orientational phase transition to a fluid phase with long-range orientational order as the density is increased above some critical density. This is the isotropic to nematic phase transition. In the nematic phase, the molecules point on average along a common direction which is called the director and identified by the unit vector N. Furthermore, although individual the molecules are biaxial ('brick-like' and not truly cylindrically symmetric), there are very few cases of thermotropic nematics where there is not complete rotational  Chapter 1. Introduction to Liquid Crystals  10  symmetry around the axis N: nematics are uniaxial and not biaxial anisotropic fluids. This anisotropy reflects itself in several bulk tensorial properties such as magnetic and electric susceptibility, optical birefringence, and anisotropic viscosity coefficients. The direction of N is arbitrary in the lab frame but, in practice, small forces, such as scratches on the cell holding the fluid, imposes its orientation. This is analogous to the Heisenberg ferromagnet in which all the spins tend to be parallel to the total magnetic moment, but where the energy of the system is independent of the average orientation of the moment. One of the most important feature of nematics is that the states N and — N are indistinguishable. Although a single molecule can carry a sizeable electic dipolar moment (0(10*) Debye), the system organizes itself in such a way that as many dipoles are 'up' as are 'down' and is therefore not ferroelectric (see Figure 1.3). To summarize, the main features of the nematic phase are: 1. Short-range positional order. 2. Flows like an ordinary fluid. 3. Long-range orientational order along the direction N. 4. The direction of N is arbitrary in the lab frame. 5. Full rotational symmetry around N (uniaxial phase). 6. The states N and — N are indistinguishable. 7. Displays bulk anisotropic properties. To characterize the amount of orientational order in a nematic sample, one introduces an order parameter Q which is 0 in the isotropic phase and 1 in a perfectly oriented system. Assuming, temporarily, that the orientation of N is known, a convenient order  11  Chapter 1. Introduction to Liquid Crystals  parameter can be defined as:  where () means that a thermal average is taken, n,- is a unit vector along the molecular axis of molecule i. N  is the number of molecules in the sample, and P2(x) = \(% ~ 1)> x2  m  is the second Legendre polynomial. One cannot use the simple average ((hi • TV)) in a nematic phase since it is always 0 due to property 6 above. We shall study the nature and theoretical description of the orientational order in nematics in further detail in the following chapter. There exists a helical version of the nematics called the cholesteric phase which is similar to the ordinary nematic phase and lacks long-range positional order. However, N is not constant in space and the preferred direction is helical with N(r) = (N , N , N ) = x  v  z  (cos(fc z), sin(koz), 0). The phase is periodic along z, and since the states N and —TV are 0  again indistinguishable, the spatial period is equal to half the pitch of the helix. The pitch is typically of the order of 3000A and measurable by optical methods. In this work we are concerned only with the ordinary nematics.  1.4  Smectic Fluid Phases <  The name smectic (from the Greek cr unknot = soap) was given to the materials belonging to this family because their mechanical properties are reminiscent of soaps. There exist various phases in the smectic family: smectic-A, smectic-B, smectic-C, . . . , smectic-K, smectic-L. The reason for these rather mundane names is that the phases were labeled in their order of discovery without attempt to classify them as new phases were found. Some of these phases, initially identified as smectics, were later shown to be three-dimensional crystals. We will often refer to a specific smectic-X phase by its abbreviation sm-X.  12  Chapter 1. Introduction to Liquid Crystals  In this section we restrict ourselves to the true smectic liquid crystalline phases; these are the smectics A , B H , C, I, F and L . Such phases can, from a naive structural point of view, be described as a one-dimensional stacking of 2D sheets of liquid along the z direction. The interlayer spacing is well denned and can be measured by X-ray diffraction experiments. Liquid crystalline smectics are thus more ordered than nematics. In fact, for a given material, the smectic phases almost always occur at temperatures below the nematic phase, although there exist some materials which exhibit one or several re-entrant or less ordered phases as the temperature is reduced. For example the following progression of phases may be observed as the temperature is decreased: isotropic—mematic—•smA —^nematic—•sm-A... —* crystal. In this example, the second nematic phase is said to be re-entrant since it has less macroscopic order than the preceding sm-A phase. 1.4.1  Smectic A  A schematic representation of the order present in a smectic-A phase is shown in Figure 1.4. The main features of this phase are: 1. Well defined interlayer spacing d close to the full molecular length / . 3  2. Short-range positional order within a layer (quasi-2D isotropic fluid). 3. The states +z and — z are equivalent. 4. The system is uniaxial and there is full rotational symmetry of the director N around z. Let  us comment on feature 2 above.  In a well ordered sm-A phase the average  A  molecular orientation, TV, is strongly pinned in the direction of the layer normal, z, and I n some sm-A materials, individual layers are actually bilayers where the molecules are interlaced and the distance between layers is somewhat larger than a single molecule. 3  Chapter 1. Introduction to Liquid Crystals.  13  d  Figure 1.4: Schematic arrangement of the molecules in a smectic-A phase. The molecules are, on average, parallel to the layer normal, z. The distance, d, between two consecutive layers is of the order of the full length of a single molecule /. The positional correlations within a layer are liquid-like.  14  Chapter 1. Introduction to Liquid Crystals  x-axis view  y-axis view  z-axis view  rapid rotation around z  Figure 1.5: Rotational averaging of the pair-potential in a smectic-A phase  individual rotation around the long molecular axis occurs on a short time scale compared to the in-plane motion time scale. Hence, one recognizes that a smectic-A phase consists of a stack of quasi-2£) isotropic liquid layers. To illustrate further, consider molecules that look like elongated bricks and exhibit a sm-A phase where the long side of the brick is parallel to z. Their rapid rotation about z averages out the true three-dimensional pair potential and 'transforms' the brick-like molecule into a cylinder-like molecule (see 4  Figure 1.5). The effective, or averaged, intralayer pair-potential is therefore isotropic within the layer (hard disk in the present example) and independent of the orientation, <f>, of the molecule around z. This observation will be very useful when the time comes to classify the various smectic phases. I n the present example, the pair-potential is purely repulsive and corresponds to the shape of the hard object. 4  15  Chapter 1. Introduction to Liquid Crystals  The presence of the layers in the sm-A phase imply a modulation of the mass density p(z). One can expand p(z) in a Fourier series:  P(z) = po + ^=(/>« exp{iqz) + c.c.) + • • • ,  (1.6  where po is average mass density and q = 2ir/d. p is the order parameter of the smq  A phase; its amplitude and phase describe, respectively, the degree of layering and the displacement of the layers . 5  1.4.2  Smectic C  A schematic representation of the order present in a smectic-C phase is shown in Figure 1.6. The main features of this phase are: 1. Well defined interlayer spacing with thickness given by the z projection of the molecular length: d « /cos(u;), where u is the tilt angle measured from the layer normal i . See Figure 1.6. 2. Short-range positional order within a layer. A sm-C phase can be seen as a quasi-2£> anisotropic fluid or a quasi-2Z) liquid crystal system . 3. The in-plane projection of the tilt makes an angle <j> with respect to an arbitrary in-plane direction x perpendicular to z. n is a unit vector parallel to the in-plane projection of the director N; we refer to it as the 2D director. See Figure 1.7. 4. The system is biaxial. The second feature of the sm-C phase deserves some attention. In a well ordered sm-C phase, where the tilt angle w fluctuates very little, it is possible to 'integrate' out the true 5  D u e to the  c o u p l i n g between layers displacement a n d director  o n l y q u a s i - l o n g - r a n g e p o s i t i o n a l o r d e r a l o n g z, namic limit  [4].  a n d the  fluctuations,  order parameter, p, q  the  s m - A phase  vanishes in the  exhibits  thermody-  Chapter 1. Introduction to Liquid Crystals  17  Figure 1.7: Detailed geometry of a smectic-C phase. The director N makes an angle u with respect to the layer normal and its in-plane projection (unit vector n) has orientation <j> with respect to the x axis.  Chapter 1. Introduction to Liquid Crystals  18  Figure 1.8: Planar arrangement of two interacting particles ' 1 ' and '2' separated by f . The orientations of hi, h and fu are measured with respect to the fixed axis x. 12  2  pair potential along the z direction and obtain an effective 2D pair-potential, Vij, between particles i and j. In the most general case, this effective potential is nonseparable and depends on fjj, fa and <j>j, the intermolecular vector between molecule i and j and the in-plane orientation of these molecules respectively (see Figure 1.8):  Vij = Vij{rij;hi,hj) .  (1.7)  For example, consider a single sm-C liquid crystal layer where the individual molecule is, as a rough approximation, modeled by a cylinder. In a well ordered sm-C phase, where all the molecules have the same w and and  the equivalent 2 D object of a tilted cylinder  is an ellipse. In such a case, one can then consider a simpler 2D fluid of perfectly oriented ellipses of eccentricity 1/ cos(u;) and disregard the original three-dimensional problem. A smectic-C phase is thus a realization of a quasi-2D liquid crystal system which, in the  Chapter 1. Introduction to Liquid Crystals  19  case where only one layer can be stabilized in an experiment, becomes a true 2D liquid crystal. Since (u) ^ 0 and (e''*> ^ 0 in a sm-C phase, one introduces the following complex order parameter for the sm-C phase: $ = «e'* ,  (1.8)  where u and <f> are defined in Figure 1.7. A sm-C phase exhibits long-range order in the azimuthal angle <f>. A l l the molecules have, on average, the same ut and <f> in a single infinitely large sm-C domain. We have given a brief description of the sm-A and sm-C phases. However, we mentioned earlier that there exist other smectic fluid phases: sm-Bn, sm-I, sm-F and sm-L . 6  The liquid-like nature of these phases was demonstrated via X-ray diffraction experiments at the end of the 1970s. It was, at that time, difficult to understand what differentiated these phases from the 'simpler' sm-A and sm-C phases: one needed to introduce new order parameters different than p and q  Defining a 'good' order parameter in a given  phase transition problem however requires a great deal of insight into the physics involved. The reward is often great since one can build phenomenological theories once the relevant order parameters have been identified. As discussed in the next section, it is the development of the theory of dislocation-mediated melting of 2D solids which, by introducing the idea of bond-orientational order, has led to a natural classification of the smectic fluid phases.  1.4.3  Two-Dimensional Melting and Classification of Smectic Phases  This section is rather long, but contains a material to which we shall continually refer in the next chapters. We have tried to put the emphasis on the physical picture by keeping 6  The smectics B, D, E, G, H and J are believed to be true three-dimensional solids.  Chapter 1. Introduction to Liquid Crystals  20  the mathematical burden to a minimum. Almost all the microscopic devices that result from the microtechnological revolution are made of thin layers of materials. Such devices are nearly two-dimensional and they may have provided the impetus that led researchers to investigate the behavior of various physical processes in 2D systems in the early 1970s. The melting of a 2D isotropic (triangular) solid is an example of such a process (see the review by Strandburg [5]). Before explaining 2D melting itself, we present a simple example which reveals most of the physics involved in 2D systems. Kosterlitz-Thouless Transition Two-dimensional systems, with short-range interactions and which have an ordered phase with broken continuous symmetry, lack true long-range order at any non-zero temperature. The continuous symmetry implies that at least one branch of elementary excitations has the property that its energy approaches zero continuously at very long wavelength. Consider, for example, a simple ferromagnet with classical spins S (\S\ = 1) lying in the xy plane of a square lattice (xy model) and interacting via the following nearest-neighbor Hamiltonian: H  xy  = -JllSi  • S, = - j £ c o s ( < f c - h)  (1.9)  where J is a positive constant with units of energy and sets the strength of the interaction, —*  and fa is the in-plane orientation of the spin 5,-. The sum is restricted to the nearestneighbors i and j. At low temperature, we expect a slowly spatially varying orientation of the spins (spin waves) and (fc - ^ ) < 1. In this regime, one has cos(<^j — <£,•) « 1 — |((^- — t  <f>j) . One finds that this quadratic or elastic Hamiltonian leads to a power-law 2  decay of the spin-spin correlation function: g  (r) = (5(r) • S{0)) = (cos(^r) - <f>{0)) cx r " " ^  (1.10)  Chapter 1. Introduction to Liquid Crystals  21  Figure 1.9: Spin vortex. where r)(T) « hsTfJ\ T is the temperature, JCB is the Boltzmann constant, and a is the lattice constant. Hence, the thermally excited spin waves destroy the ordering in the xy system. If there was true long-range order in this system, g(r) would decay towards a positive constant equal to the square of the magnetization at large distance. However, we know that at high enough temperature, g(r) must decay exponentially to zero. The question of interest is then: what is the mechanism which converts the power law to exponential decay of g[r) as the temperature is raised? As shown above, spin waves have little to do with any phase transition in this system. Another mechanism must be responsible for driving the system to short-range order (exponential decay to zero of  Kosterlitz and Thouless suggested taking into account the effects of vortices in the system (see Figure 1.9). The energy increase due to the spins a distance r from the core of the vortex is, in the large core approximation,  Integrating over the area of the system, one finds that the total energy of a vortex, E , v  22  Chapter 1. Introduction to Liquid Crystals  in a system of size L and lattice spacing a, is:  l Ja r  a  The entropy of such a vortex is simply the logarithm of the number of states (sites) in which the vortex can be: ks l n ( £ / a ) . A t a temperature T, the total free energy for this 2  free vortex is:  F = E -TS v  v  = irJ\n(-)- k T\n(L/a) a  2  B  .  (1.12)  Below T (ksT = 7T«//2/:B) vortices are rare while they abound above it. c  c  This simple description neglects interactions between vortices of opposite vorticity. Both mean-field [6] and renormalization-group [7, 8] treatments of such vortex-vortex interactions lead to a precise and elegant interpretation of the Kosterlitz-Thouless ( K T ) transition. These show that, at low temperature, vortices of opposite vorticity attract each other and form tightly bound neutral pairs which screen the effective interaction between other pairs of bound vortices of opposite vorticity. As the temperature is raised, more pairs are excited and the mean radius of a pair (distance between vortices within a pair) increases; the bound pairs are less efficient in screening the pair-pair interaction. Above some critical temperature, T , the screening becomes too small and a rapid c  increase in the number of pairs, which then dissociate into free vortices, occurs. The transition is thus analogous to a 2D dielectric breakdown where the vortices play the role of lines of charge perpendicular to the plane of the lattice. In fact, the analogy is not coincidental since the vortex-vortex Hamiltonian is just that of a 2D two-component neutral Coulomb gas. In the Coulomb gas problem, the signature of the transition is the divergence of the 'dielectric constant' of the system. In the xy problem, as the temperature approaches the critical temperature from below, the effective (renormalized) elastic constant, decreases and eventually vanishes at T , where the system becomes unable to rec  act to long wavelength elastic distortions. The unbinding transition of vortices drives the  23  Chapter 1. Introduction to Liquid Crystals  spin-spin correlation function, g(r), to be short-ranged above T . K T transitions occur in c  systems where the low temperature phase would be expected to have long-range order, which breaks a continuous Abelian group, if it was not of the underlying fluctuations which destroy the long-range order. Although now fifteen years old, the detailed predictions of the K T theory are still intensively investigated. The debate is fueled by contradictory results coming from various Monte Carlo simulations. 2 D Melting and Bond Order Parameter In a 2D solid, the elastic vibrations (phonons) are the fluctuations (excitations) which destroy the ordering; they are analogous to the spin waves in the xy model. In an X-ray diffraction experiment, one does not observe true Bragg peaks (^-functions centered on the reciprocal lattice vectors of the 2D crystal) but rather peaks whose intensity falls off — #  as a power-law in the vicinity of a reciprocal lattice vector G. The amount of positional order is measured by g^(R), the density-density correlation function:  g (R) = (exp{iG.[u(R)-u(0)}}) , s  (1-13)  where the positions of the atoms are given by f = R-ru(R). u(R) is the atomic displacement field (fluctuation around the equilibrium position R). Using the classical elastic Hamiltonian, one finds that g<«{R) behaves as:  g {R) = R~« ^ T  Q  .  (1.14)  In a liquid phase, g^R) = e~ ^^/R, where ( is the temperature dependent posiR  tional correlation length. Hence, the thermal fluctuations destroy the long-range positional order of the solid phase, but are not sufficiently 'violent' to make the correlation functions decay exponentially with distance as in the case of a liquid. B y analogy to the  Chapter 1. Introduction to Liquid Crystals  24  xy model, the question which arises here is: how does a 2D solid melt or, what is the mechanism which converts quasi-long-range order (power-law decay of gg) to short-range order (exponential decay of g^jl Kosterlitz and Thouless [6] extended their predictions for the xy model to the 2D melting problem. The defect in this case is not a vortex as in the xy model but a dislocation, which can be viewed as an extra row of atoms stuck partway into the crystal. In the K T theory, it is the unbinding of these dislocation pairs which drive the melting of the 2D solid. In this case, as in the xy model, the elastic response of the system vanishes at the transition. The system thus becomes unable to react to a shear stress and becomes a liquid. The unbinding of dislocations in the K T theory of 2D melting drives the system from a solid phase with quasi-long-range order in the density-density correlation g@(f) —*  to a liquid phase with short-range order in gg(R). We have argued that phonons destroy the long-range positional order of the solid phase and transform it to quasi-long-range (Equation 1.14). This description is incomplete as pointed out by Mermin [9]. A 2D solid phase has quasi-long-range positional order, but exhibits true long-range bond orientational order. The sixfold symmetry of the triangular solid is not lost at low temperature. In other words, the nearest-neighbor bond correlation function decays to a non-zero constant at large distance in the solid phase. Consider Figure 1.10.  Assuming that the orientation of the bonds between two  nearest-neighbors of a central particle at f is known to be 0(f), one can define a sixfold bond correlation function, ce(r),as: ce(r) = (cos{6c?(r) - 66(0)}) .  (1.15)  Chapter 1. Introduction to Liquid Crystals  25  Figure 1.10: Local hexatic structure with bonds joining a central atom to its six nearest-neighbors. The angle between the two nearest-neighbors of the central particle is  0. The presence of long-range bond orientational order in the 2D solid phase can be mathematically stated as follows: lim ci (r) = const . olid  r—too  (1.16)  °  In a liquid phase, c$(r) is short-ranged and decays exponentially with distance: # (r) a -  ,  u,d  (1.17)  r  where cj6 is the bond correlation length. Halperin and Nelson [10, 11, 12] recognized that the melting process proposed by Kosterlitz and Thouless did not reveal anything about the behavior of the c$(r) at the melting transition. Halperin and Nelson came up with the interesting suggestion that the melting of a 2D triangular lattice could proceed in two steps (two-step melting) and lead to a novel phase of matter observable between the solid and liquid phase.  The  intermediate phase was predicted to exhibit exponential decay of the density-density correlation function, ^ ( r ) , as in an ordinary liquid, but power-law decay of the bond correlation, ce(r), as for quasi-long-range bond orientational order. This intermediate  26  Chapter 1. Introduction to Liquid Crystals  Figure 1.11: Long-range bond order and short-range positional order. The dots represent periodic triangular lattice points. phase was called  due to its sixfold nature. The hexatic phase is analogous to the  hexatic  xy system described above, where the vector connecting nearest-neighbors plays the role —*  of the spin vector S. Figure 1.11 illustrates how the positions of the particles can be disordered but still maintain sixfold bond orientational correlation. Although the molecules (open discs) at fi and r*2 form perfectly symmetric sixfold structures, the centers of mass of the two patches are positionally uncorrelated, as illustrated by the offset of the discs at fi with respect to the dots at r . According to Halperin and Nelson, the above two transitions 2  are both K T transitions. We shall not present the details of these two K T transitions or describe the defects associated with each of them here. The interested reader is invited to refer to the review by Strandburg [5]. The modern theory of defect-mediated melting of a 2D solid is referred to as the Kosterlitz-Thouless-Halperin-Nelson-Young ( K T H N Y ) theory . 7  7  Young  [13]  presented by  p e r f o r m e d a detailed s t u d y of the  problem  Table 1.1 contains a  a n d showed that s o m e of the  H a l p e r i n a n d Nelson in their orginial paper were incorrect.  numerical  values  27  Chapter 1. ^Introduction to Liquid Crystals  Phase  9rX ) r  ce(r)  solid  quasi-long-range  long-range  hexatic  short-range  quasi-long-range  liquid  short-range  short-range  Table 1.1: Summary of the K N T H Y theory. Upon heating, the solid shows a melting transition to a hexatic phase which further converts, at still higher temperature, to an ordinary liquid. summary of the K T H N Y theory. It is worth mentioning that a large amount of experimental work as well as computer simulations have been devoted to investigating the predictions of the K T H N Y theory, principally to identify the existence of a hexatic phase. It is, we believe, fair to say that the detailed predictions of the theory have not been entirely confirmed. Let us summarize the main points we have raised thus far: • Two-dimensional systems with continuous symmetry lack true long-range order and exhibit quasi-long-range order; the xy model and a two-dimensional solid are examples of such systems. • B y extending the work of Kosterlitz and Thouless, Halperin and Nelson proposed that a 2D solid need not melt to a liquid phase via a single transition. They postulated the existence of an intermediate hexatic phase between the solid and liquid phase.  Chapter I. • Introduction to Liquid Crystals  r  28  • The hexatic phase has short-range positional order like a liquid but shows quasilong-range nearest-neighbor bond orientational order. Classification of Smectic Phases Recall the problem that we raised at the end of Section 1.4.2 regarding the need to introduce new order parameters in order to classify the sm-P>H, sm-I, sm-F and sm-L phases. Inspired by the new ideas of 2D melting, Birgeneau and Litster [14] suggested in 1978 that some exotic smectic phases might be three-dimensional (3D) realizations of the bond ordered hexatic phase. In their view, each hexatic smectic layer is an almost independent two-dimensional hexatic phase.  The role of the weak three dimensional  couplings is to transform the quasi-long-range nature of the bond order in the hexatic phase to true long-range order. From this perspective, the SHI-BH is a 3D hexatic phase, hence the subscript H in sm-Bn. It can also be thought of as a bond orientationally ordered version of the sm-A. The SITL-BH has short-range positional order, as do ordinary fluid, while the sm-B is a crystalline phase with true long-range positional order and which exhibit Bragg peaks in X-ray diffraction patterns.  The sm-I, sm-F and sm-L  phases are the hexatic versions of the sm-C, and tilted 'cousins' of the sm-Bii. The inplane projection of the tilt is along, halfway between, or anywhere between two nearestneighbor in the sm-I, sm-F and sm-L respectively (see Figure 1.12). The notion of bond orientational order and the relative orientation of the in-plane projection of the molecular tilt with respect to the nearest-neighbor bond turn out to be the order parameters that are needed to understand what differentiates various smectic phases from each other. The current classification of the smectic liquid crystal phases is presented in Figure 1.13. The isotropic and nematic phases have also been included for completeness. As described earlier, a sm-A phase can be seen as a quasi-two-dimensional liquid which  29  Chapter 1. 'Introduction to Liquid Crystals  Smectic-I  Smectic-F ^  ^ ^  ^  ^ ^  ^  Figure 1.12: Schematic representation of the tilt orientation in tilted hexatic smectic phases. The figure illustrates one smectic layer, with the layer normal perpendicular to the page, and with the arrows indicate the in-plane orientation of the tilt. The tilt is along (0°), halfway (30°) and at an arbitrary orientation between the nearest-neighbors in the smectic I, F and L respectively.  exhibits short-range hexatic order. This is because the free energy of a 2 D isotropic liquid is minimized by allowing the build-up of some triangular-like patches of molecules. This is easy to visualize in the case of hard discs where the constraints of filling the space at a given density is best realized with a closely-packed configuration (local triangular structure).  As a material undergoes a sm-A—•sm-Bn transition, the bond correlation  function, c$(r), acquires quasi-long-range order and the amount of correlation of the triangular patches extends very far in the sample. In a 2D anisotropic fluid, one notes that the tilted bond-ordered version of the sm-Bn, the sm-I, sm-F and sm-L phases, will not be perfectly hexatic since the tilt imposes an underlying twofold distortion to the bond order. Again, this is best seen by considering the packing of ellipses of small eccentricity. In this case, the packing is almost perfectly hexagonal, but with a small twofold (orthorhombic) distortion. A t the experimental level,  30  Chapter 1. Introduction to Liquid Crystals  Disordered  Isotropic  Molecular Orientational Order  Nematic  Smectic-A  Smectic-Bn  Smectic-C  Smectic-I,F,L  Positional Order Normal to Layer  L A Y E R E D  Bond Order within Layer  P H A S E S  I N C R E A S I N G 0 R D E R  Tilted Smectics Figure 1.13: Classification of liquid crystal phases according their degree of ordering. For example, the smectic-A phase displays both molecular orientational order and positional order normal to the layer, but not the bond orientational order exhibited by the hexatic phases located just under it in the sequence.  Chapter 1.. Introduction to Liquid Crystals  31  : the importance of the distortion must be examined on a case-by-case basis. . F i n a l l y , it is important to note that we have described the possible sequence of phases that can occur as the temperature of a liquid crystalline material is lowered. In most cases, a given substance will not exhibit the full sequence of transitions described in Figure 1.13. It might, for example, go directly from an isotropic phase to a sm-A phase, avoiding the nematic phase, and then solidify upon further cooling.  1.5  Outline of the Thesis  The rest of this thesis is organized as follows. In Chapter 2, we present an overview of the mean-field theory of nematic liquid crystals. We show that the mean-field approximation for a true liquid crystal model neglects correlations between the molecular orientation and the orientation of the intermolecular vector. In practice, the mean-field aproximation transforms an originally nonseparable intermolecular pair potential into an effective separable pair potential. This implies that, within a mean-field approximation, a nonseparable liquid crystal model behaves very similarly to a separable model. Hence, we are led to ask ourselves if there exist interesting nonseparable effects which are neglected by the mean-field approximation of nematic liquid crystals and which cannot be explained by the mean-field improvements usually considered for lattice models of nematic liquid crystals. In Chapter 3 we use a nonseparable anisotropic van der Waals potential to investigate, using Monte Carlo simulations, the importance of quenched positional disorder in nonseparable systems. The computer simulation results show that systems with quenched positional disorder cannot be used to model nematic liquid crystal when the pair potential is nonseparable.  We present arguments which suggest that such system are much  more closely related to molecular orientational glasses than to liquid crystals. This is  Chapter L Introduction to Liquid Crystals  32  not the case for separable models of nematic liquid crystals which can be modelled.using perfect Bravais lattices or systems with quenched positional disorder. The conclusion "that translational degrees of freedom must be in thermal equilibrium with the rotational degrees of freedom in a nonseparable liquid crystal model leads us to perform a Monte Carlo simulation where the particles are allowed to translate and rotate. We find that the modified van der Waals potential we use in this Monte Carlo simulation exhibits a nematic to isotropic phase transition and a considerable amount of nearest-neighbor bond structure in the nematic phase. We show in Chapter 4 that the build-up of the nearest-neighbor bond structure in the nematic phase exhibited by the van der Waals system can be explained in terms of the phenomenon of induced bond orientational ordering observed in real smectic-C liquid crystal systems. These results allow us to reinterpret the mean-field theory of nematics as an approximation which neglects the phenomenon of induced bond orientational ordering in a nonseparable system. Furthermore, the same results provide a microscopic origin of the coupling between the nearest-neighbor bond orientational field (BOF) and the molecular orientational field (MOF) invoked in Ginzburg-Landau theories of smectic liquid crystals. The question which then arises is if the presence of induced bond orientational order in nonseparable liquid crystal models introduces new effects which are commonly neglected in nonseparable systems. We show in Chapter 5 that the finite-size scaling behaviour of two-dimensional nonseparable liquid crystal models can be strongly modified when such B O F ordering exists. In Chapter 6, we investigate how the response of liquid crystal phases to an external magnetic field is modified when there exists an underlying B O F ordering. In particular, we consider the problem of the Freederickzs transition in tilted hexatic smectic films which display such B O F ordering. We find that, due to this B O F M O F coupling, the Freederickzs transition in these systems can be considerably different  Chapter 1. Introduction to Liquid Crystals  33  than for ordinary liquid crystals. In Chapter 7, we take a different approach and investigate a problem where the M O F introduces interesting behavior for hexatic phases with predominant B O F ordering. In particular, the problem of the smectic-I to smectic-F is considered and a preliminary microscopic model is proposed. In Chapter 8, we investigate whether macroscopic and 'liquid-like' properties of the nonseparability in nematic liquid crystals could introduce interesting and important effects in a problem where the role of anisotropy has proven to be most important. As an example of such a macroscopic manifestation of the nonseparability, we consider the problem of viscous fingering in a system with anisotropic viscosities and investigate whether our previous results obtained for the problem of viscous fingering in isotropic systems are modified. We conclude with a discussion of possible avenues for extending the work presented in this thesis.  Chapter 2  Mean-Field Theory of the Isotropic-Nematic Phase Transition  The mean-field theory of the isotropic-nematic transition is a simple method for solving the thermodynamic problem of microscopic models of the isotropic-nematic transition. To introduce the subject, we review the mean-field treatment of a lattice model for the nematic phase. We then introduce a proper description of the mean-field approximation for a true liquid crystal model which includes translational degrees of freedom.  This  latter approach suggests that it is possible to find different potentials which, at the meanfield level of approximation, have the same thermodynamic behavior. In particular, we present a separable and a nonseparable model which are equivalent within a mean-field approximation. This observation leads us to question how the thermodynamic behavior of separable and nonseparable potentials compare when one goes beyond the mean-field approximation. Finally, we present some arguments which motivated us in investigating this problem via computer experiments.  2.1  Mean-Field Description of the Isotropic-Nematic Phase Transition  For a given system, the motion of the particles in phase space is described by a set of variables such as the position, orientation and conjugated momenta of each particle. In any real system, the particles interact with each other, and their interactions are described  34  35 Chapter 2. PMeari-Field Theory of the Isotropic-Nematic Phase Transition  by a coupling of their respective dynamical variables. The various couplings that can be constructed and allowed by symmetry are the building blocks of the Hamiltonian which describes the interactions present in the system. The mean-field approximation is often the starting point for investigating the thermodynamic behavior of a system of interacting particles. Technically, the mean-field approximation consists of transforming the interacting part of the Hamiltonian into an effective Hamiltonian which describes the interaction of a single particle with a ficticious field. This field is a function of the thermodynamical average of the various dynamical variables involved in the interactions. B y requiring that the free energy of the system is minimized with respect to all variations of these thermodynamical averages, one obtains a set of transcendental equations which allows the evaluation of the mean-field in a selfconsistent manner. This last step leads, most of the time, to a numerical problem which must be solved on a computer. We now illustrate how the mean-field approximation is carried out for a simple lattice model of the isotropic-nematic transition in liquid crystals.  2.1.1  Mean-Field Theory of the Lebowhl-Lasher Lattice Model  In its simplest formulation, the Lebowhl-Lasher [15, 16] model consists of an ensemble of classical rotors with orientation n,-, sitting on a simple cubic lattice and interacting with the nearest-neighbor Hamiltonian  HLL = -J2 £  <«j>  (2.1)  Ptfa-nj)  where J is positive and sets the strength of the interaction, and P is the second Legendre 2  2  polynomial, ^ ( z ) = | ( 3 x — 1). Notice that this Hamiltonian has nematic or inversion 2  symmetry: HLi,(ni,hj) = HLL{—The  above interaction favors an average parallel  orientation of the rotors which is independent of the orientation of the axis of the lattice; this is the low temperature phase. A t sufficiently high temperature, the free energy is  !  36 Chapter 2. Mean-Field Theory of the Isotropic-Nematic Phase Transition  minimized by having a random orientation of the rotors. The low temperature phase is thus a 'nematic-like' phase, and the orientationally disordered high temperature phase is the 'isotropic-like' phase. We will show that, within a mean-field calculation, a first order transition separates these two phases. The Lebowhl-Lasher model is thus believed to grasp the essential physics of the isotropic-nematic transition. It is generally believed that the coupling between positional and orientational degrees of freedom is weak and does not play an important role in the isotropic-nematic transition. Hence, a lattice model in which the center of mass of the molecules is fixed, would seem to be a reasonable starting point for studying the problem of the isotropic-nematic transition. Indeed, a considerable amount of work has been devoted to this model in the past twenty years or so ( 1 ,  15-17,25-33)  The remainder of this subsection presents a brief derivation of the mean-field theory of the Lebowhl-Lasher model since we will often refer to these results. We start by defining the matrices CiaP  where a, /3  = x,y,z,  = -(3n,- n,7j  —  a  (2.2)  &ap)  and rij is a Cartesian component of n, while a  is the Kronecker  6p a  delta. Using the o $ matrices, the above Hamiltonian can be rewritten as Q  2j Hu  =  3  2  T~ 2 6  2  O- 0<7jf3 iQ  a  .  (2.3)  <i,j> a,P=l  The mean-field theory assumes that fluctuations in the elements of o p are small or, a  technically, by imposing the following approximation: (cTicf} ~ {^iap)) {o-j p a  {o-jap)) « 0 .  (2.4)  The brackets indicate that a thermal average has been performed. The thermodynamic properties of the systems are obtained by calculating Z , the  ^  r  Chapter 2. Mean-Field Theory of the Isotropic-Nematic Phase Transition 37  partition function:  Z =  exp(-H /k T) .  JUdQi  LL  B  (2.5)  $7, are the polar orientations of n, with respect to some fixed axis in the lab frame. The free energy, F, of the system is given by  (2-6)  F = -k T\n(Z) . B  The mean-field equations are obtained by requiring that F is minimized by all variations in the average elements of Q p = a  (c.a/j).  B y analogy to the Weiss molecular field  theory of magnetism, there will always be a solution, Q p = 0, for all a and /3. These cora  respond to the isotropic high temperature phase. A t low temperature, nonzero solutions appear, and these correspond to the 'nematic-like' phase. These nonzero solutions will not be unique, due to the overall rotational symmetry of the system. A n infinitesimal external field would be required to dictate the orientation of broken symmetry in the nematic phase. However, since Q p is real and symmetric, there exists a frame in which a  it is diagonal. Let be 0 and <f> the polar angles of the average orientation of the molecules in the lab frame. Then sin(0) cos(</>) , sin(0) sin(</>) and cos(0) . We define the following new variables  P= 9 = P =  (p) and  Q =  (<?> •  Chapter 2. Mean-Field Theory of the Isotropic-Nematic Phase'Transition38  After some algebra, we rewrite the matrix Q 0 as a  /  0  -(Q-P)/2 0  Qa0= ^  0 \  -(Q + P)/2 0  0  0  Q )  If we choose the director to be along the z axis, we have P = 0 and, defining p = cos(#), 1  we can write F as  jf^exp JJ: = -k T\n ( 4x B  [^{(3^-l)Q-Q }]) 2  (2.7) where r is the number of nearest-neighbors in the lattice. It is now a simple matter of setting dF/dQ equal to 0 and solving numerically for Q. One finds a first order transition and a jump in the nematic order parameter Q from 0 in the isotropic phase to Q « 0.42 at the transition. Technically, the transition is first order since there are terms with odd power in Q in the Taylor expansion of F. Physically, the definition of the order parameter, Q = (|{3cos (0) — 1}), implies that there will be 2  such odd terms in F since the state with +Q does not correspond to the same state or orientation of the molecules as that with — Q. From the mean-field calculation, one finds that the difference between the thermodynamic isotropic-nematic transition temperature, T^i, and the limit of stability of a supercooled isotropic phase, T", is of the order of 0.17V/. Experimentally, this difference is rather of the order of O.OITWJ. There has been a long debate over this discrepancy in (17,25-33)  the liquid crystal literature  since it was not clear if the difference between exper-  imental results and the theoretical calculation was due to the mean-field approximation \t c a n b e s h o w n t h a t t h e  l  without an external  field.  phase the  l o w e s t f r e e e n e r g y p h a s e is u n i a x i a l ( P  =  0)  in the  present  model  Chapter 2. Mean-Field Theory of the Isotropic-Nematk Phase Transition  or if the Lebowhl-Lasher model could reproduce a difference T^i — T* of the order of 0.017V/. Recent and extensive Monte Carlo simulations [17] indicate that the LebowhlLasher model is indeed able to reproduce a supercooling temperature range of 0.01T;v/. These Monte Carlo results suggest that the supercooling range of O.ITAT/ found in the mean-field treatment of the Lebowhl-Lasher lattice model is entirely due to the neglect of orientational fluctuations. Notwithstanding the success of the simple Lebowhl-Lasher model, one would like to go beyond a lattice model and investigate the thermodynamic behavior of true nematic liquid crystal models where translation of the molecules is allowed. The mean-field theory of such models is the subject of the next section.  2.1.2  Mean-Field Theory for a True Liquid Crystal Model - The MaierSaupe Model  We have shown in the last section how the mean-field calculation for a lattice model of the isotropic-nematic transition is carried out. We now show how such mean-field theory can be constructed for a 'true' nematic liquid crystal model. We shall see that the interpretation of the mean-field approximation in the latter case is not as straightforward as in the Lebowhl-Lasher model. This difficulty in interpreting the mean-field approximation for a true liquid crystal model will lead us to ask a set of questions which differ considerably from the usual questions raised in mean-field calculations of thermodynamical lattice systems. Consider the i/-particle distribution function for a system of N particles interacting with some pair potential V^-: P<")M 9  ^''""^  \ _ /o 2\" {  m  > (N-v)\  // P { ~ f t E i j n]  • • • dn dr  v  ex  N  SfexT,{-PZ Vi]}dSll...dn dr ...dr itj  N  1  N  v+l  ...dr  N  '  (2.8)  •  3  40 'Chapter 2. 'Mean-Field Theory of the Isotropic-Nematic Phase Transition  where p = 1/ksT. The position of particle i is r,-, and fi, is a short notation for the three 1  Euler angles describing its orientation. The i/-particle distribution is the conditional probability density that the particles ( 1 . . . v) have a state with  ...  f\... r„). Of  particular interest are the one- and two-particle distribution functions:  ^ - ^'<i:l:^;^;lS:::St::::t:=^-"-> ™ and  U  '  ~ V**^  ]  1)  ffexp{-PZ ,V }dn ...dn dr ...dr i  ll  1  N  1  N  (2.10) =  p(f i ii )p(r 2Mgi.f ]n n ) f  f  1  where we have defined f  12  (2.ii)  f  13  1  u  2  — f — r\. The 1-particle density, p(fi,Clj), is the probability 2  density that particle i is at a position r, in the sample with orientation fi,-. The pair distribution function, g(r\,flj; f ,(l ), is a conditional probability that a particle 1 at f\ 2  2  will have an orientation O i when there is a particle 2 at f with orientation Q, . 2  2  From here on, we restrict ourselves to the case where the particles have complete rotational symmetry (Coo) around their main molecular axis whose orientation is denoted by the unit vector n,. In an isotropic phase, p(fi, n,) simply reduces to the number density which is independent of r; and hi. In a nematic phase, we have /?(f,,n,) = pf(fii). The probability density is uniform in space, but is a function of the orientation of the molecular axis, hi, with respect to the director (see Chapter 1). The problem consists of calculating f(hi). Taking the derivative of Equation 2.9 with respect to n, , the a component of the )0(  unit vector n,-, we find:  d  n  =  pjdrijjdllj  —  f{nj)g{rij;ni,nj)  (2.12)  41 Chapter 2. Mean-Field Theory of the Isotropic-Nematic Phase Transition  with f(hi)  =  exp{-/?e(n,)} .  We have introduced here the quantity  known as the pseudo-potential . 2  e(hi),  In Equation 2.13, the number density, p, and the pair potential, Vij, are known quantities which are input to the model. The pseudo-potential,  e(hi),  is the quantity to  be calculated. Physically, e(n,) measures the energy of a particle having some orientation n, with respect to the average orientation of the rest of the molecules in the sample, hence the name pseudo-potential. The pair distribution deserves some attention. Clearly, to calculate anything from 2.13, one needs to have some information about how the particles are located with respect to each other. Trying to carry the analysis without assuming that something is known about  would be the same as trying to find a transition temperature for a  g(fij\hi,hj)  ferromagnet problem without knowing on what kind of lattice the spins are sitting on. Well above the isotropic-nematic transition temperature, one assumes that the 'isotropic liquid problem' has been solved and the isotropic part of the pair distribution function, 9o°{ ri})i  n  a  s  D e e n  calculated. As the transition temperature is approached from above,  one can write: 9ij  =  9{rij]  hi,  hj)  =  g "  0  ^ )  + A$(ry;  hi,  hj)  (2.13)  where Ap(f,j; hi, hj) is the change in gij occuring due to the build-up of molecular orientational correlations close to the transition. In the spirit of the mean-field approximation, one neglects this increase in the correlations (or orientational fluctuations) and approximates  by the already known 5o" ( «i) >  r  :  9ij * giFto) 2  Equation  2.13  corresponds to  a B B G K Y  equation  [18].  •  (2.14)  42 Chapter 2. - Mean-Field Theory of the Isotropic-Nematic Phase Transition  ...Thus, in the present context, the mean-field approximation amounts to neglecting correlations between the relative orientation of the molecules as well as the correlations between the molecular orientations and the orientation of the intermolecular vector, fij. Practically, the second step is equivalent to assuming that the distribution of the centers of mass of the molecules is spherically symmetric with respect to a reference molecule at the origin. This approximation allows us to integrate Equation 2.13 with respect to n e(n.) = const + p J (f r J dty f(hj)Vij(fij; 3  tJ  hi, h )g (r ) 0  :  (2.15)  , so  j  t > a  tJ  where const is a constant of integration which has to be determined. One can rewrite Equation 2.15 in the following way: e(n.) = const + p J r d r 2  t i  ^ ( r ^ ) J dVlj f(hj) Jj  n,-, hj) sin(0,j) d 0 j j d $  tj  (2.16) where the intermolecular vector, fij, has orientation 0,j and  In the mean-field  approximation, the full anisotropic pair potential is averaged over all the orientations of fij, and the resulting spatially averaged pair potential depends only on the relative orientations of the molecules; this is an important point. One finally obtains: e(hi)  =  const + p J dQ,j f(hj)Vij(hi • hj)  =  j r jdrijg"\rij)  (2.17)  where Vij(hi-hj)  2  j j Vij(rij\hi,hj)sm(Qij)dQijd$ij  .  If one assumes that Kj(n,- • hj) is invariant under the change n,- —* —hi, which guaranties that all mesophases predicted within this description have the nematic symmetry (see Section 1.3), then one can expand Vij in terms of Legendre polynomials of even order, P2 : n  e(hi) = const + p / dSljf(hj) ^ -^2nP2n(n. • hj) . J  n=0  (2.18)  Chapter 2. Mean-Field Theory of the Isotropic-Nematic Phase Transition  The Maier-Saupe [19, 20] approximation consist of keeping only the terms with n = 0 •and n = 1 in Equation 2.18. The anisotropic term, J , is assumed positive and stabilizes, 2  at sufficiently low temperature, a configuration with the molecules on average parallel to each other. The result, 2.18, could be interpreted as a theoretical justification for studying a lattice model where the particles interact with their nearest neighbors with an anisotropic potential of the form — J P2{hi • hj) (Section 2.1.1). From Equation 2.18, 2  the mean-field treatment of the Maier-Saupe model follows the steps outlined above for the Lebowhl-Lasher model and we do not repeat it. Energy Scales in the Maier-Saupe Approximation Within the Maier-Saupe approximation, there is only one energy scale which is set by the strength of the average anisotropic interaction J . «7 varies from one potential, VJj, 2  2  to another, but since there is only one energy scale, all thermodynamic properties are functions of the single parameter pJi/ksT. This feature disappears if higher order terms, such as Pi(hi • hj), which may be present in the original pair potential, Vij, are taken into account in Equation 2.18. Reconsider the mean-field approximation 2.14 for two different anisotropic potentials V#> and V&K Assume that these two potentials can be chosen such that their average over Qij and  gives only an isotropic term and a P term without any other higher order 2  terms. In such a case, there is no approximation made by neglecting terms proportional to P  2 n  with n > 1 in Equation 2.18 since they are simply absent. The integration over <Prij  cannot generate such higher order terms since the mean-field approximation has already been made in 2.14 before carying the integration over dO,j and d$ij. That is, only the isotropic part of g<j, go°(rij), which is independent of hi and hj, has been retained.  .  Chapter 2. Mean-Field Theory of the Isotropic-Nematic Phase Transition 44  The only difference between the potentials V-j and V-j , is that their J s are differ2  ent . We thus expect, according to the preceding discussion, that the behavior of these 3  two systems would be identical as a function of their reduced parameter p^/hsT. This argument holds only within a mean-field treatment. Hence, any major difference occuring in the thermodynamic behavior of the original potentials V-p and VJp can be immediately traced-back to the mean-field approximation, where the replacement g,j « <7"°(«'j) r  is made. One sees that the mean-field approximation presented here is much more intricate than in the simplest Lebowhl-Lasher model since it is imbedded in approximation 2.14, which is clearly not as transparent as approximation 2.4. As in any other phase transition problem, one would like to understand the importance of the mean-field approximation in the context of the isotropic-nematic transition within a true liquid crystal model and not a lattice model. The approach we have taken to investigate this question is the subject of the next section.  2.2  Mean-Field Theory and Nonseparability  2.2.1  Macroscopic Manifestations of the Nonseparability  For spherically symmetric particles, like argon, the pair potential is isotropic since it depends only on the distance separating the centers of mass of two interacting particles. The Coulomb interaction between point charges and the usual van der Waals interaction, which decay as the inverse sixth power of the distance, are examples of such isotropic interactions. Practically, only atoms which have an electronic angular momentum of L = 0 can 3  The  do not  Jo's  are irrelevant i n the  present  discussion since they just shift the  average energy scale  p l a y a n y r o l e i n a n i s o t r o p i c - n e m a t i c t r a n s i t i o n a t c o n s t a n t d e n s i t y . T h i s is t r u e as l o n g as  is l a r g e e n o u g h t o s t a b i l i z e a n e m a t i c l i q u i d p h a s e a t a t e m p e r a t u r e a b o v e t h e s o l i d p h a s e .  and  J2/J0  45  ^'Chapter 2.- Mean-Field Theory of the Isotropic-Nematic.Phase Transition  be regarded as spherical. For any other particle, a nonspherical distribution of the electronic charge introduces anisotropic corrections to the otherwise isotropic potential. The importance of such corrections vary from one particles to another. Assuming an effective isotropic pair potential for a molecular system in its liquid phase (eg. liquid nitrogen) can be a fairly valid starting point, as long as the liquid phase exhibits isotropic bulk properties. Clearly, anisotropic liquid phases, such as liquid crystals, must interact via anisotropic forces, which depend on the relative orientation of the particles, in order to lead to the observed mesomorphic behavior. The Lebowhl-Lasher model and the Maier-Saupe model are among the simplest anisotropic models one may investigate to study the isotropic to nematic phase transition. However, both the starting potential in the Lebowhl-Lasher model 2.1 and the effective pair potential  in the Maier-Saupe model 2.18, are sep-  arable. They depend only on the relative orientation of the particles, and not on the orientation of the intermolecular vector r . A natural question one may ask at this point tJ  is: Do nonseparable interactions have to be considered at all in microscopic nematic liquid crystal models, and if so, why?  Elastic Constants of a Nematic Material As we will now show, the answer to the above question is that all macroscopic properties which depend on the interaction between large domains of highly ordered nematic molecules and on the relative spatial orientation of the ordered domains, would be isotropic if the true intermolecular potential was separable . 4  A  A  A  Consider three domains D j , D and D , their respective directors Ni, N and iV , and 3  2  4  the  The  word  domain  in  a  boundaries between two  ferromagnetism. within ± 4 5 °  system  with  continuous  regions w i t h different  W e loosely use  the  symmetry  is,  o r i e n t a t i o n are  formally not  sharp,  w o r d d o m a i n to refer to a r e g i o n w h e r e the  w i t h respect to s o m e location i n the  3  2  sample.  speaking, as  in the  incorrect  since  Ising m o d e l  d i r e c t o r f i e l d is  of  constant  46 Chapter 2. Mean-Field Theory of the Isotropic-Nematic Phase Transition  J  c i?12  \  fa  -* •^13  J  \  Figure 2.1: Interactions between ordered domains in a nematic phase. The square blocks represent domains of highly oriented nematic molecules. The director TV, indicates the average orientation of the molecules within a domain i and varies slowly going from A to Dj.  their relative spatial orientations i 2 and i? 3 (Figure 2.1). Due to thermal fluctuations, 1 3  2  long wavelength fluctuations of the director,  5  which are analogous to spin waves in  a Heisenberg ferromagnet, will be excited. One would expect, by analogy with the Heisenberg ferromagnet, that this distortion of the director field increases the energy of the system. Experimentally [2, 3], one can measure the energy required to locally distort the director field while keeping the director in a different orientation in a nearby region. One observes that the energy required to distort the orientation of N  2  with  respect to N\ is not the same as the energy needed to distort the orientation of TVjj with respect to N\. Hence, unlike for the Heisenberg ferromagnet, nematic liquid crystals are characterized by three elastic constants associated with splay, bend and twist distortions 5  B y long wavelength we mean long compared to a typical intermolecular separation.  47 Chapter 2.tMean-Field Theory of the Isotropic-Nematic Phase Transition Glass  ^f^-^^ Nematic &  ----- — - _ C l a s s < ^ i ) ' V l | j ' l  S p l a y Deformation  Bend D e f o r m a t i o n  Twist Deformation  Figure 2.2: Splay, bend and twist deformations in nematics. The thin lines illustrate higly ordered domains. The glass plates can be treated in such a way that they impose a constant orientation of the director on the plates are shown by the thick double arrows. of the director field. These three types of deformation are illustrated in Figure 2.2. The experimental difference between the elastic constants associated with the above distortions is a macroscopic manifestation of the nonseparability of the pair potential. A system of particles interacting with an isotropic potential would have equal splay, bend and twist elastic constants [21] since the interaction energy of two molecules i and j does not depend on the orientation of the intermolecular vector, fij, nor does the energy required to rotate one molecule with respect to the other while keeping r,j fixed. This example shows that a nonseparable pair potential, although it is formally anisotropic since it depends on hi and hj, it is isotropic with respect to the spatial variable fij. A spatially isotropic anisotropic pair potential is the simplest definition one can give to a separable potential. Another macroscopic property of a liquid of particles interacting with a separable  48 * Chapter 2\ MeanrField Theory of the Isotropic-Nematic Phase Transition  potential is that it would have isotropic viscosity coefficients. Again, this does not correspond to the experimental evidence, where different viscosity coefficients are measured depending on the direction of the fluid flow with respect to a director maintained aligned with either surface treatment or with a strong magnetic field [2, 3]. X-Ray Diffraction Pattern of Nematics For a separable potential, the distance of approach of two particles with a given relative orientation is independent of their angle of approach. From this observation, one immediately concludes that in both the isotropic and nematic phase, the X-ray diffraction pattern should display a perfectly circular ring of uniform intensity. Consider an attractive anisotropic separable potential. In a mean-field point of view, the anisotropic part of the potential vanishes above the transition since all the (Pin) in Equation 2.18 are zero in the isotropic phase. Further, due to packing constraints, there is a large probability that two molecules are separated by a distance, r " ° , in the isotropic phase. Hence, in an X-ray diffraction pattern of an isotropic liquid crystal material, one expects to see a perfectly circular ring of uniform intensity at a momentum transfer q ° « l / r " ° far above the isotropic-nematic transition. ,s  The ring is diffuse  and has a radial width in reciprocical space of l / £ , the positional correlation length (see Figure 2.3a). In the nematic phase, each P 2 n component of the pseudo-potential acquires a non-zero value and the mean strength of the attractive potential (isotropic + anisotropic part) increases. Hence, the minimum distance of approach decreases slightly and one should again observe a perfectly circular ring of uniform intensity, but at a momentum transfer q  nematic  ta l/ ™  matic  r  with q  nematic  > q . This is not what is observed iao  experimentally [23, 22, 24] as shown in Figure 2.3b. As discussed in Chapter 1 (see Figure 1.1), the nematics are elongated molecules. The shape of the molecule will prevent nearby molecules from getting as close along the long  Chapter 2. Mean-Field Theory of the Isotropic-Nematic Phase Transition 49  (a)  (b)  Figure 2.3: X-ray diffraction pattern for a nematic liquid crystal both above (a) and below the transition temperature (b). Photographs taken form reference [22]. molecular axis as perpendicular to it. One thus observes two rings modulated in intensity: one along the direction of N at q « 1//, and one in the direction perpendicular to N at q « 1 jr where / and r are, respectively, a typical molecular length and molecular diameter. 0  0  Hence, the fact that the X-ray diffraction pattern of nematic liquid crystals do not exhibit a ring of uniform intensity in g-space can be interpreted as another manifestation of the nonseparability of the intermolecular pair potential.  2.2.2  Mean-Field Approximation and Nonseparability  We have argued in the first section of this chapter that the Maier-Saupe mean-field theory neglects orientational fluctuations and, as a by-product, averages out Vij over d 0  t J  and $ij leading to an effective separable potential Vij(hi • hj). This second step makes the interpretation of the Maier-Saupe mean-field approximation much less transparent than that of the simple Lebowhl-Lasher model. The question we then asked is if the nonseparability of the true pair potential is observable at a macroscopic level. In the  50 Chapter 2. Mean-Field Theory of the Isotropic-Nematic Phase Transition  previous subsection, we gave examples, taken from experimentally observed facts, which demonstrate that the pair potentials in nematic liquid crystals are nonseparable and manifest themselves at the macroscopic level. It could be argued that such manifestations of the nonseparability are not fundamentally interesting since they do not seem to play an important role in the isotropic-nematic transition. The important questions which arise at this point and that we try to answer in this thesis are: • Do separable models really grasp the essential physics of the isotropic-nematic transition? • Do nonseparable effects alter the thermodynamic behavior of the system close to its isotropic-transition, or is the nonseparability of the true pair potential just a cumbersome technical detail? If the answer to these two questions was yes, it would mean that there are problems with a description of the isotropic-nematic transition in terms of separable models. Further, it would show indirectly that the mean-field approximation 2.14 is missing 'something' which is beyond the improvements that are brought in by the improved mean-field theories devised for the Lebowhl-Lasher model [1, 25, 26, 27, 28, 29, 30, 31, 32, 17, 33]. Two preliminary questions must be answered before we can tackle the two previous questions: • A t what level or on what footing can we compare the thermodynamic behavior of separable and nonseparable potentials ? • What type(s) of pair potential should be used ?  51 Chapter 2. MeantField Theory of the Isotropic-Nematic Phase Transition:.  2.2.3  The Anisotropic van der Waals Potential - A Simple Nonseparable Model  We have shown in Equation 2.14 that the Maier-Saupe approximation averages out a nonseparable potential over the orientations of the intermolecular vector f,j. We shall start from there and make use of the properties of the Maier-Saupe approximation we have described in 2.1.2 to come up with a simple nonseparable model for the pair potential. We need a potential whose spatial average reduces to a separable potential of the form: V&* = Vo(l^-l) + V (\f \)P (h, • hj) 2  without higher order P  2n  tj  2  (2.19)  terms.  As already illustrated by our description of the X-ray diffraction pattern, the presence of an isotropic-nematic transition in systems of hard objects depends on the degree of anisotropic of their shape, hence their nonseparability. This implies that we will not be able to investigate purely steric models such as hard ellipsoids [34, 35] or hard spherocylinders [36, 37, 38]. In order to study the approximation 2.14, we will have to investigate systems with soft anisotropic potentials. There is a prejudice against renouncing the investigation of hard steric objects in the problem of the isotropic-nematic transition in liquid crystals. Since the pioneering work of Onsager [39], there has been a strong consensus in the liquid crystal community that anisotropically repulsive forces (steric effects) may be the major forces responsible for stabilizing the nematic phase in liquid crystal materials. Furthermore, it was generally accepted until about two years ago that attractive forces had to be present to stabilize a smectic phase [40, 41, 42]. There is now rapidly increasing evidence from computer simulations that hard objects, such as hard spherocylinders, can exhibit a smectic-A phase without the presence of any attractive forces [36, 37, 38]. Hence, attractive forces have recently attracted little attention regarding the problem of phase transition in liquid  52 Chapter 2. Mean-Field Theory of the Isotropic-Nematic Phase Transition  crystals. However, we will show in this thesis that attractive forces are important in the physics of liquid crystals. In particular, we will find that several interesting mesomorphic behaviors in some liquid crystals systems are much more easily explained and investigated using soft potentials than with anisotropically steric models. Some realistic intermolecular pair potentials have been proposed [43, 44] and investigated in computer simulations [45, 46, 47]. These are soft potentials and, in contrast to purely steric contact potentials as in spherocylinders, contain information on both the anisotropic repulsive (steric) and anisotropic attractive part of the potential. Unfortunately, such realistic potentials are complicated and it is useful to investigate whether a simpler model can exhibit the essential features of nematic order. A t a more technical level, these potentials [43, 44] do not satisfy our criterion of reducing to the simple form 2.19. One possibility is to describe the isotropic to nematic transition by purely anisotropic attractive interactions.  Assuming that the anisotropy of the pair potential is due to  the molecular polarizability, the resulting van der Waals interaction [48] gives rise to an anisotropic part of the pair potential for molecules of cylindrical symmetry of the form: VvDw{fij\hi,hj)  —  - J  (hi • hj — 3hi • e.-je.-j • hj) 2 .  (2.20)  Here e,j is a unit vector parallel to f,j, the intermolecular vector which connects the center of mass of two molecules. The length scale is set by r , which is a measure of the 0  effective molecular diameter, hi and hj are unit vectors parallel to the symmetry axis of the molecules and J is the strength of the interaction which has units of energy. This potential can be derived from second order perturbation theory [48]. We shall not repeat the derivation here, but rather present a more heuristic approach. Nematics do not necessarily have a permanent electric dipole moment, but at any moment a fluctuation  53 Chapter ^.i-Mean^-Field Theory of the Isotropic-Nematic Phase Transition  in the electronic charge distribution can occur which creates a spontaneous dipole. As described in Chapter 1, most of the nematogens have quite a large length to breadth ratio as shown in Figure 1.1. One can thus expect the molecule to have an anisotropic molecular polarizability. The electric field, E(fij) produced by a spontaneous dipole moment, p, of orientation n,- at a relative position f,j (f,j = e,jr,j) from a molecule i is given by E{fij) oc \  (hi - 3n, • e.j-e.j) .  (2.21)  7*Y.  —*  If a molecule, j, sits at fij, it gets polarized by the spontaneous field, E, created by i. The induced moment, pj, along hj, is pj  =  xnjhj-E(fij)  oc — (hi • hj — 3n, • e,je,j • hj) hj  (2.22)  where x is the molecular polarizability. The potential energy, VVDW, of the two dipoles is VVDW  — -pj • E(fij)  =  —  y—g-^J  (hi • hj — 3n,- • e,je,j • hj)  2  (2.23)  where we have introduced the constant of proportionality Jf\. This is potential 2.20. Since the potential 2.20 is so simple, it is useful to investigate its validity as a model for a thermotropic nematic liquid crystal. Indeed, this potential was first introduced in the context of nematics by Maier and Saupe [19, 20] and is the starting point of the improved mean-field theory of Ypma and Vertogen [25]. It is easy to show that 2.20 satisfies the requirement 2.19: VVDW  =  JJ VVDW sin(Q )  = 4TTJ  h;  dSijd^j  fe)  (2.24)  54 Chapter 2. Mean-Field Theory of the Isotropic-Nematic Phase Transition  If the molecules are lying completely into the xy plane, the unit vectors n, and e;j can be written as n,- = { cos (<^,), sin (<£,)} and e,j = {cos(8ij,sin(0;j)},  where </>•, and  the orientations of hi and e,j with respect to a fixed axis x (see Figure 1.8). dimensions, the rotational invariant with nematic symmetry  6  are In two  equivalent to P2(hi • hj) is  2(n,- • hj) — 1 = cos(2<i>, — 2(f>j). Performing the following integration: 2  (2.25) we obtain the two-dimensional version of the Maier-Saupe approximation. It is instructive to look at the dependence of potential 2.20 on the orientation of e,j for various relative orientations of the molecules i and j (Figure 2.4). Consider the orientations giving rise to an extremum in energy for a pair of molecules interacting via potential 2.20 for a given separation r^-. The lowest energy configuration has n , hj and t  parallel, with Vynw = —4 in units of J^(r /rij) . The configuration in which 6  0  but hi and hj ± r,j, is a saddle point with VVDW = —1> while the energy has an absolute maximum when all three vectors are mutually perpendicular. We shall now describe the approach we have taken to investigate potential 2.20 and its mean-field versions (Equations 2.24 and 2.25 respectively).  2.3  Towards a Computer Experiment  Several approaches can be taken to investigate systems of particles interacting with anisotropic potentials and exhibiting mesomorphic phases: 1. Lattice models with anisotropic potentials 6  are  A g a i n we zero.  assume a  fluid  p h a s e w h e r e t h e r e is n e m a t i c s y m m e t r y a n d w h e r e t h e  a v e r a g e s o f cos(0,j)  Chapter 2.  Mean-Field Theory of the Isotropic-Nematic Phase Transition  55  25.0 -i  0  Figure 2.4: Dependence of the van der Waals potential, in units of J", on the orientation of intermolecular vector 0, for fixed distance r,j, and for various relative molecular orientation \<f>\. The following planar orientations are assumed: n, = (0,1), hj = (sin(c^),cos(</>)) and e,j = (sin(0),cos(0)). The solid curves are for positive <j> while the dashed ones are for <p < 0. For clarity, each curve is shifted by a positive value of +4 from the nearest lower one.  56 . Chapter 2. Mean-Field Theory of the Isotropic-Nematic Phase Transition  • Theoretical investigations • Computer simulations 2. True liquid models with anisotropic potentials • Theoretical investigations • Computer simulations. There have been several theoretical and computer simulation investigations of lattice models the isotropic-nematic transition with separable potentials in the past ten years. A brief review of these results was given in Section 2.1.1. We shall illustrate in the next chapter that results for nonseparable lattice models cannot be meaningfully interpreted in the context of an isotropic-nematic transition. We are interested in the implication of the mean-field approximation within a liquid description (Equation 2.14) of the isotropic-nematic transition. Hence, we are interested in the second class of approaches above. In recent years, the technique of density functional calculations coupled to numerical evaluations of the Ornstein-Zernike and approximate closure relations [49, 50, 51] have proved to be very good at reproducing the computer simulation results of various model systems [34, 35, 36, 37, 38, 52]. Unfortunately, with a few exceptions [53], this approach does not allow investigation of the nematic phase. Finding a way to describe the nematic phase using such techniques is a very difficult problem, and is the subject of current and intense study. For all the above reasons, and since we felt there was a need to increase our understanding of the effects induced by nonseparable anisotropic potentials in liquid crystals, we decided to investigate, using computer simulations, the thermodynamic behavior of systems of particles interacting with the van der Waals potential 2.20. The results of  Chapter 2. Mean-Field Theory of the Isotropic-Nematic Phase Transition  these computer experiments are presented in the next two chapters.  Chapter 3  Computer Simulations I: Quenched and Annealed Positional Disorder in Systems with Nonseparable Interactions  In this chapter, we first present the motives for using computer simulations, and a short overview of common computer simulation methods used to study statistical thermodynamic problems. We then use one of these methods, the Monte Carlo technique, to investigate the behavior of the van der Waals potential (Equation 2.20) for lattice systems and systems with quenched positional disorder.  The results obtained for these  systems show that the translational degrees of freedom must be annealed (in thermal equilibrium) with the orientational degrees of freedom in order to lead to nematic-like orientational ordering. These observations led us to perform preliminary Monte Carlo simulations of systems where both translation and rotation of the particles are allowed.  3.1 3.1.1  Computer Simulations, Monte Carlo and Metropolis Algorithm A n Impetus for Computer Simulations  Observations are often made in the laboratory which are not immediately explained by existing theories or are perhaps not expected (Figure 3.1, step 1). In response to this, the theoretician may propose a model to explain what is observed. The model is put into  58  Chapter 3. Computer Simulations I  -j if ok' v f not ok  59 Exper i m e n t a l results  Analysis of the r e s u l t s  4  4\  f  Theoret i c a l Model  Set o f Approximations  g  | |  ,  A n a l y s i s of the r e s u l t s  Analytical results  Computer Simulation  1  Figure 3.1: Construction and evolution of a scientific model. See text for details. mathematical terms to facilitate comparison with the physical system (step 2). Effects such as nonlinearities, lack of symmetry or a large number of degrees of freedom prevalent in any realistic model are very difficult, if not impossible, to deal with analytically. In the past, arbitrary simplifications had to be made before the model could be compared with the physical system (step 3 and 4). If the experimental results and theoretical predictions were in agreement with each other, the initial model was thought to be appropriate to describe most of the physics going on in the experiments (step 5). However, this is not what happens most of the time; theoretical predictions and experimental results are often in qualitative agreement at the most. In such circumstances one is left to decide if the disagreement comes from the introduction of an oversimplified model at step 2 or from the approximations used at step 3. This is clearly not a simple task. W i t h the advent of powerful computers, it is now often possible to solve the model 'exactly' (step 6). This gives to the theoretician a way to study the limit of validity of the approximations he relied on in order to obtain some analytical results (step 7). Once a  Chapter 3.  Computer Simulations I  60  given model, as well as its set of approximations, are fully understood, it becomes easier to decide how well this model can describe the physical system under study. Computer simulation is adding a new dimension to scientific investigation and establishing a role of equal importance with the traditional approaches of experiment and theory. 3.1.2  What is a Computer Simulation?  Computer simulations have been used for the last three decades as an extension of laboratory and theoretical work, and are becoming especially useful now with the advent of powerful computers. In the field of phase transitions in statistical mechanics problems, simulations are used to calculate statistical properties of the system such as energy, order parameters and correlation functions. They are particularly applicable to the field of liquid crystals because phase transitions predicted by various model systems are easily simulated by computer. There exist two popular computer simulation techniques used to investigate the thermodynamic behavior of a given model. These are the Monte Carlo method and the molecular dynamics technique. In the context of the present work, a computer simulation consists of N particles interacting through some known potential and each particle is allocated a set of arbitrary initial co-ordinates (position and orientation). The Monte-Carlo method generates a random displacement for each particle, and the move is accepted or rejected depending on its new energy and a certain probability function. No time scale is involved, and the sequence of configurations is entirely probabilistic. The molecular dynamics method uses the known potential to calculate forces and torques, and each particle is moved by integrating its equation of motion over some small time increment. Unlike the Monte Carlo method, the molecular dynamics allows the study real time-dependent phenomena in various systems.  Chapter 3. Computer Simulations I  3.1.3  61  Molecular Dynamics  The molecular dynamics method is based on the microcanonical ensemble rather than on the canonical ensemble: the total energy of the system is conserved during the length of the simulation and the temperature is a fluctuating quantity. The ergodicity assumption is crucial for the applicability of this method. The ergodicity assumption states that during any significant time interval the phase point, XN = ( n • • • nv; Q i . . . fi;v), spends equal time in all regions of the constant energy surface. If the ergodicity assumption is applicable to the system under study, one can use a time averaging procedure rather than an averaging over the states of the system. The molecular dynamics equations are the Newton's equation for each individual particle in the system. The positions and orientations of the particles are obtained by integrating Newton's equation forward in time using some finite difference algorithm. The criterion that the total energy of the system must be conserved in this method can be used to check the accuracy of the algorithm used to integrate the equations of motion. The translational temperature of the system, T , is not a fixed quantity and must be tT  obtained form the average squared velocities: (3.1)  (3.2)  M is the duration of the molecular dynamics run after disregarding some initial transient. v (j) is the center of mass velocity of particle p, of mass m , at step j. A similar equation u  M  can be used to calculate the rotational temperature by using a time averaging of the individual angular velocities. In this work we are primarily interested in the problem of phase transitions in liquid crystal models, transitions which are induced by a change of the temperature and we  Chapter 3. Computer Simulations I  62  are not concerned with time-dependent phenomena . For these two reasons, we have 1  preferred to use a Monte Carlo approach using the standard Metropolis [54] algorithm, rather than a molecular dynamics simulation. 3.1.4  Monte Carlo Simulation and Metropolis Algorithm  We are interested in the thermal average of various functions f(Xw):  }exp{-pV (X )}dX N  N  N  Ideally, one would replace the integral above by a sum over the M configurations generated in the Monte Carlo procedure:  ,f(Y  u  Ej=r exp{-/3^Q-)}/(i)) to,  Ej=i  exp{-/0VjvU)}  where VN(J) is the total potential energy of the configuration, j, and M t is the total to  number of configurations allowed by the system. Because M t is large, even in a very small system, any attempt to calculate (/) from to  the above prescription is a hopeless task. One may hope that a much smaller number of states, M <C M t, could be used to give an adequate estimate of (/): to  Ei=i  exp{-0Vjvt7)}  The problem with the last approach is that randomly selected states are very improbable and contribute very little to the sums. It is essential to introduce some form of importance sampling which removes this bias and where configurations are selected according to a prescribed probability distribution V(j).  Then, when averaging over the M configu-  rations, a weight must be assigned to each state which eliminates the bias in the selection x  A s we  s h a l l see,  t h i s is n o t  system under study in thermal  to say  t h a t one  equilibrium!  does not  h a v e to often w a i t a very l o n g t i m e to get  the  63  Chapter 3. Computer Simulations I  process:  Monte Carlo simulation is almost synonymous with the sampling scheme proposed by Metropolis et al. [54] with the choice:  The average (/) is then obtained from: i=M  1  (/) * T7 £  j=i  m  /0')  (3-8)  The Markov process allows us to generate the states, j, of the system with the correct probability. Consider a single-step transition probability, pj,,-, that the system is in state i at time t and in state j at time t + 1. The transition probability matrix, p,,,-, is time independent and its elements satisfy the conditions M  EPi..  =  (-)  1  3 9  Pj,.- > 0 .  (3.10)  The system is started in a state i at time t = 0. The probability that the system is in 0  state j at time t, after M Markov steps, is:  Pj,k  =  £  Pi.«M-lP«M-l>'M-2  • • •P»2.»'lP»l.»'o •  (3-11)  Furthermore, the limit lim pW M-*oo  =  H  (3.12)  j  J  exists for all j and z and is independent of the initial starting state z . In a statistical 0  problem, the IIj's are known: ^  =  •  (3.13)  64  ' Chapter 3.' Computer Simulations I  Notice that the:transition steps obey the following equality  = Ek ^ g f " •  (3-14)  0  We find that after taking the limit M —* oo, n; = I>;.*ri* •  (3.15)  k  One  is free to choose any form of transition probability pj i as long as they lead to the t  correct limiting distribution 3.13.  Most often, the unnessarily strong condition of detailed  balance is imposed:  .  p U= ,U iij  j  Pj  i  i  (3.16)  In this case, a simple choice for the P,,,- is Phi  = -ij.  Pi.i =  if n,->n.-  (3.17)  < v (i))  (v (j) N  N  JVocc  T^TT  ^  n,<n,-  (V (j) N  (3.18)  > V (i)) N  where NJ£) is the number of states accessible from state i, which does not have to be C  known. In practice, the system is in a given configuration, say i. A new configuration (positions and orientations) of the particles is generated. cepted if VN(J) < VJV(I).  The new configuration is ac-  If V^v(jf) > V^v(i), the state is accepted with probability  Pj i = exp{—(3 [VN(J) — V}v(*)]}. This last step is done by comparing pj i with a random t  number, R, uniformly generated in the interval  t  [0,1]. If R  < pj,,-, the state is accepted.  If R > pj,,-, the new state j is rejected and the old state i must be taken into account again in the averaging procedure. This process is repeated until averages are acceptable, or until either computing time or funds, are exhausted.  Chapter 3. Computer Simulations I  65  Periodic Boundary Conditions Both the molecular dynamics and Monte Carlo methods are subject to the limitation of very small sample sizes: in most published works, N is of the order of a hundred to a few thousand particles, which is extremely small by macroscopic standards. Periodic boundary conditions are almost invariably imposed in order to minimize surface effects and to simulate more closely the properties of an infinite system. If a particle leaves the cell, it is brought back in via the opposite boundary. This is most easily visualized with the aid of Figure 3.2 in which nine identical cells are shown, but only the central one is the system of interest. As a particle moves out of the central cell it also moves out of one of the eight surrounding cells and back into the central cell. One runs into problems with either free or periodic boundary conditions when the correlation length of the system reaches a value of the order of one half the size of the system. We have used periodic boundary conditions in all the simulations we have performed.  3.2  Lattice Systems  Before investigating positionally disordered systems, we shall first see what the ground states exhibited by potential 2.20 on some simple Bravais lattices are. These results will prove to be useful in interpreting the results of simulations in systems with either quenched or annealed positional disorder. More precisely, they will give a strong indication that local structure is vitally important if nematic orientational ordering is to exist in a system of particles interacting with a nonseparable potential such as the van der Waals potential. It is clear from our description of the extremum in energy for the van der Waals potential (Section 2.2.3) that if the ground state of the system has nematic order (i.e. hi • hj = ± 1 for all i , j at T = 0), some of the bonds will not be in their lowest energy  66  Chapter 3. Computer Simulations I  T  •  •  •f  •  •—  •  •—  •  •—  •r  •  •—  •—  • —  •  • * •  •-  Figure 3.2: Schematic representation of a system with periodic boundary conditions. The central box (dark boundaries) represent the true system while its eight neighbors are a replica of the central box. The dots represent the particles moving with some velocity in the direction shown by the vector. configuration.  There is thus a possibility that nematic order will not be favored due  to this frustration. In the present section we illustrate this observation by a number of examples.  3.2.1  Two-Dimensional Lattices  We first consider the case where the long axis is constrained to lie in the xy plane since it is the easiest to visualize. The ground state for a system of particles interacting via 2.25 has nematic order irrespective of the lattice, but what about potentials such as 2.20 which depend on the angle between  and n, or hj? In Figure 3.3 we show the lowest energy  configurations for a few two-dimensional lattices where we employed the potential 2.20 with the particles constrained to lie in the lattice plane. The ground states were found b y zero temperature Monte Carlo simulations on systems of the order of 150 particles, and with periodic boundary conditions. Typically, we used 500 Monte Carlo steps per  Chapter 3. Computer Simulations I  Y YY / YY  /  \  ^  \  /  \  N  67  i  i  /  \  /  i  \  I  /  \  I  I  (c)  Figure 3.3: Lowest energy configurations for two-dimensional lattices, a) Square lattice, b) Triangular lattice and c) Honeycomb lattice.  Chapter 3. Computer Simulations I  68  particle (MCS) with either random or ordered starting configurations. For the lattices shown in Figure 3.3 no ambiguities arose, and we are confident that we have found the ground state configurations. The square lattice has a ground state with parallel alignment of the n,s when only nearest neighbor interactions are included, but there is no continuous orientational symmetry; the director is pinned along one of the square axes (< 10 > direction). If more neighbors are included, the ground state changes and is shown in Figure 3.3a. If the lattice is sufficiently stretched into a rectangular lattice, the parallel orientation reappears with the alignment parallel to the short axis. The potential 2.20 with the molecules constrained to lie in the plane is similar, but not identical, to the quadrupolar interaction between dumbell molecules. A lattice gas model employing quadrupolar interaction has been applied to the adsorption of diatomic molecules such as H2, N2 and O2 on graphite [55]. In the case of the triangular lattice with nearest neighbor interactions, the "railroad track" ground state of Figure 3.3b has two sublattices with different energy per particle for the "rails" and the "tressels". This is in complete analogy with the "herringbones" of Harris et al. [55]. The "pinwheel" ground state of the honeycomb lattice of Figure 3.3c is a linear superposition of three equivalent railroad tracks just as in the case considered by Harris et al. . 2  3.2.2  Three-Dimensional Lattices  We have examined the lowest energy configurations of a number of three-dimensional lattices in which the particles interact via 2.20, and have found some interesting and highly frustrated configurations. In analogy with the square lattice, the ground state of a simple cubic lattice with nearest neighbor interactions has all the hi pinned in a 2  a  The  fact that the  zero temperature  correctly.  g r o u n d state in Figure  Monte  3.3c  C a r l o simulation gave  a n d its e n e r g y was us  good  p r e d i c t e d before we  confidence  that  our  program  f o u n d it was  with  working  Chapter 3~ Computer Simulations I  69  < 100 > direction . Humphries et al. [56] found a transition to an ordered phase of this • , type i n a Monte Carlo simulation where the potential 2.20 was considered as a model of a liquid crystal, but they did not try other lattice structures and therefore did not notice that their result was atypical. For lattice structures other than the simple cubic, it was necessary to cool the system slowly to T = 0 rather than quenching it from infinite temperature.  If quenched, the  system would invariably be trapped in a metastable state. The symmetry of the lattice produces local minima in phase space, from which there is no escape without thermal fluctuations. We will refer to the lowest energy state found as the 'ground state', but we cannot say for certain that this is the true ground state. The ground states that we have found are, for all but the simple cubic lattice, lower in energy than any "nematic" configuration with the n, aligned parallel to each other. In Figure 3.4 we show our ground state configuration for a face centered cubic lattice (f.c.c). lattice. The figure shows a < 111 > plane. The structure in the plane above is the same, but translated by a basis vector. The arrow on each rotor is pointing up from the plane, while the length is proportional to the projection in the plane. We observe a configuration which is similar to the pinwheel structure of Figure 3.3b. The rotors at the centre of the wheel have a declination of 15° with respect to the normal to the plane. We measure angles for the projection onto the plane relative to the in-plane basis vector by which successive planes are translated. The in-plane angle of the centre is then 150°. The rotors on the wheel have declinations that alternate between approximately 65° and 55°, and the in-plane angles are 150° ± 120° and 100° ± 120° respectively. The energy per site for the hexagonal closed packed lattice was found to be the same as for the f.c.c, and its structure is very similar. The simulation for the f.c.c lattice was carried out for 4 of the 6 x 6 layers shown in Figure 3.4, with periodic boundary conditions. The calculation was repeated for a  Chapter 3. Computer Simulations I  70  Figure 3.4: Lowest energy configuration of the face centered cubic lattice. This figure shows a < 111 > plane. The length of each rotor is proportional to its projection in the plane. The direction out of the plane is indicated by an arrow. 4 x 4 x 4 structure with an identical outcome (this is, of course, no guarantee that we have found the exact ground state). It was usually sufficient to have 1000-2000 MCS  per  temperature. The particles were selected at random from the full 47r solid angle at high temperatures. At lower temperatures only small changes in orientation were attempted. In this way we were able to keep the acceptance ratio in the 30%-40% range until the onset of a high symmetry, low energy configuration. 3.2.3  Discussion of the Results for Lattice Systems  We have shown that lattice systems may display complicated structures when the pairpotential is nonseparable.  Furthermore, it is clear that lattice models with particles  interacting via nonseparable potentials are not suitable for studying the isotropic-nematic transition. The reasons are twofold:  71  Chapter 3. Computer Simulations I  • In a nonseparable system, the presence of an underlying lattice breaks the continu•ous symmetry. The discrete symmetry introduced by the lattice is non-existent in a real liquid. Hence, lattice models with nonseparable potentials are inappropriate for the isotropic to nematic transition . 3  • One way to get around the above argument about lattice models with nonseparable potentials would be to argue that nematics are dense fluids with a rather considerable short-range positional correlation and the underlying lattice could mimic this short-range structure. If this sort of argument should be taken with a grain of salt for 3D systems, it is, in most cases, totally invalid for two-dimensional (2D) lattices. Due to their continuous symmetry, 2D liquid crystals are expected to exhibit quasi-long-range molecular orientational order (see Section 1.4). The presence of an underlying lattice in a nonseparable system will, in most cases [8], destroy this behavior and impose true long-range orientational order. These two problems do not arise in lattice models with separable potentials since the underlying lattice does not break the continuous symmetry. Clearly, positional disorder is essential in order to investigate nonseparable models of nematics. Recall the mean-field approximation (Equation 2.14) which led to the spatial averaging of Vij in Equation 2.16. It is very tempting to reinterpret the approximation 2.14 as an averaging of all the possible short-range discrete symmetries allowed by the short-range positional order present in the liquid phase. As described earlier, the approximation 2.14 requires an input which is assumed to be known a priori: the radial distribution function <7o- These observations led us to ask the following questions: 3  A s  we  shall discuss in C h a p t e r  between tilted hexatic smectic  7, s u c h l a t t i c e m o d e l s m i g h t n o t  phases.  b e so b a d for m o d e l l i n g t r a n s i t i o n s  Chapter 3. Computer Simulations I  72  • Assume, for the moment, that the particles are allowed to rotate, but not to translate, and are randomly distributed such that their center of mass pair distribution function, go, is liquid-like (decays rapidly to a value of 1). How would the orientational degrees of freedom in such a system behave as a function of temperature? • Is it possible that quenched positional disorder is enough to get rid of the problems encountered with lattice models of nonseparable potentials, restore the spatial averaging invoked in Equation 2.15 and allow the system to exhibit a static nematic-like phase? These are the questions we shall investigate in the following section.  3.3  Quenched Positional Disorder  By quenched positional disorder we mean that the translation of the molecules occurs on a time scale which is much longer than their rotational time scale, and the centers of mass of the particles do not have long-range positional order. The two degrees of freedom are not in thermal equilibrium in such a case. Systems where the translational degrees of freedom are said to be quenched are experimentally realized with molecules adsorbed on a substrate where the substrate potential is much larger than the interaction between the adsorbed molecules. In our studies of quenched systems, we take the relaxation time for translational motion to be infinitely long and consider only the rotation of the particles.  3.3.1  Random Parking Systems  As mentioned previously, we would like to have a quenched positionally disordered systems with a liquid-like radial density distribution function go. Hence, as a first step in understanding how the system behaves when one introduces the positional disorder  Chapter 3. Computer Simulations I  73  characteristic of a liquid, we carry out Monte Carlo simulations on rotors interacting via the anisotropic potential 2.20. The rotors are placed at the center of sequentially parked circles of diameter d in two dimensions, and spheres in three dimensions. It is interesting to test the hypothesis that coupling between spatial and orientational correlations in the liquid does not play an essential role in nematic ordering [42]. We therefore wish to represent the liquid by a model containing as few positional correlations as possible and the sequential absorption (random parking) model would seem to satisfy this criterion (see Hinrichsen [57] and references therein). As the name indicates, random parking consists of sequential placement of objects (circles, squares, spheres, etc.) randomly and without overlap. Once an object is parked, its position remains fixed. As the volume is filled, it becomes more and more difficult to park a new object. Eventually, the time required to park a new object diverges and the system is said to have reached the jamming limit. At this limit, the total volume occupied by the parked objects, divided by the initially accessible volume is called the jamming density. The jamming density is 54.7% for circles adsorbed onto a plane [58] and approximately 34% for spheres randomly parked in a cube [59]. This is significantly smaller than the closely packed density of 90.7% in two dimensions and 74% in three dimensions. Our simulations were performed at a covering density of 1% to 2% below the jamming density.  3.3.2  Ground States of Random Parked Systems  Two Dimensions Figure 3.5 shows a typical sequential adsorption pattern in two dimensions. Figure 3.6 shows the radial pair distribution function, go, of the 2D random parking problem at the jamming limit with the go of a hard disc system at the same density. Notice the small  Figure 3.5: Typical sequential adsorption pattern of discs in a plane. A typical infinite temperature configuration (fi = 0) with a rotor at center of each circle is shown.  -  Chapter 3. Computer Simulations I  75  7 6 5  3 2 1  n .5  i 1.0  l.S  i  i  2.0 p/d  2.5  i 3.0  Figure 3.6: Radial pair distribution function for the 2D random parking problem at the jamming coverage density of 54.7%. amount of short-range positional order in this system as illustrated by the small peak at r « 2d. This is to be compared with a hard disk fluid at the same density where the height of the peak at r « 2d is about 20% of the peak at r = d. Hence, there is much less positional correlation in a randomly parked configuration than in a 2D hard disc fluid. We parked 512 circles of diameter d onto a plane of size 28.5dx 28.5c?. This corresponds to a coverage of 49.5% . The potential was cut off at a distance of either 3.5d or 2.15c? (when the potential strength is 1% of the value at contact).  No noticable difference  between the results in the two cases was found. Both annealed and quenched simulations were performed. In the first case we initially decreased the temperature from fij = 0 in 14 intervals of fi J = 0.3. The system was then quenched to fij = oo. In the second case we quenched from fij = 0 to fij = oo directly. In both simulations we picked molecules at random for each Monte-Carlo step per particle ( M C S ) , with an average of 2500  ''Chapter 3. Computer Simulations I  - MCS  76  at each temperature. The initial and final configurations are shown in Figures 3.5  and 3.7. For the final configurations the results of three runs are superimposed The quenched and annealed systems look qualitatively the same and lack long-range nematiclike orientational order.  However, the zero temperature configurations present some  short-range order and look quite different from a totally disordered configuration (see Figure 3.5). A n interesting feature of the simulation is the occurence of regions where is always the same. We shall comment further on this point in Section 3.3.3 Three Dimensions In the three dimensional case, we randomly parked 512 spheres in a cube of size 10c? x 10c? x 10c?, giving an occupied volume of 26.8%. The potential was cut off at the distance 3.5c?. We performed the same kind of simulations as in two dimensions. The initial and final configurations of some typical slices cut into the cube are shown in Figure 3.8. Once again, no transition from a disordered to a nematic phase was observed. 3.3.3  Microscopic Origin of the Frustrated Islands  It is clear that local structure is important in determining the orientational behavior of a nonseparable system. If the molecules are fixed on a lattice, the nonseparability leads to a crystal field that breaks the continuous orientational symmetry [60, 61]. In an amorphous solid, where the translational degrees of freedom are frozen in a random configuration, the system cannot relax and the orientational degrees of freedom suffer random frustration. The lack of long-range nematic-like orientational order is due to the frustration caused by the quenched positional degrees of freedom. Figure 3.7 is made up qualitatively of two types of regions: the first, which we call field dominated, is a region where, although disordered, the particles always choose the same orientations, indicating the presence of  Chapter 3.  77  Computer Simulations I  1 1 ^ 1  \  \  /\LJ  /  / \  Figure 3.7: Superposition of three ground states (/3 = oo) ground state configurations for the two-dimensional random system of Figure 3.5 with the rotors interacting via 2.20.  Chapter 3. Computer Simulations I  78  Figure 3.8: Slices of thickness d of the ground state for a 3D random parking configuration. Only molecules whose centers lie within the slice are shown. Arrows indicate direction out of the plane. The two figures represent two different slices with two superimposed final configurations for each slice.  Chapter 3. Computer Simulations I  /=0.0  /=3.0  79  ;  /=1.5  /=6.0  Figure 3.9: Superposition of 20 orientational groundstate configurations for different values of the random field for a single random park of positions. The figure shows the ground states for the rotors interacting via the van der Waals potential with / = 0.0, 1.5, 3.0 and 6.0 respectively in Equation 3.19.  80  Chapter 3. Computer Simulations I  a field that dictates a unique ground state. The second is what we shall term a spin glass region, where there appear to be many ground state configurations of approximately the same energy, and the subsystem reaches a different one on each simulation. This interpretation becomes much clearer if we rewrite the potential 2.20 in terms of the angles 6{j, fa: VVDW  =  )* (10+  Nij(</>i,<l>j)  FittiAi)  ii  =  =  )  ij  + f{F  I  +  F }+gG ) j  ij  cos(2&-2^-) ,  = cos(2fa-0 )  Fj((j>j,9ij)  GiM^ipyAi)  =  rN  ,  cos(2<j)j — 6ij) and  cos(2c6,- + 2fa - 4%)  .  (3.19)  r\ = 1, / = 6 and g = 9 respectively. Written in this form we can clearly isolate which terms are responsible for the field dominated and the spin glass regions. The function Nij favors parallel orientation of interacting neighbors, independent of the orientation of the intermolecular vector. The function F, depends on the orientation of a single rotor, <f>i,  and the random bond angle,  Hence, it describes the effect of a random field on  particle i. The function G%j depends on the orientation of both rotors fa and <pj and the random phase  The random particle configuration leads to random bond frustration  and we interpret G„ as a spin glass term. Changing the relative size of the field and spin glass terms in the potential 3.19 alters the extent of the two regions in a superposition of ground states. Figure 3.9 shows the change in the structure of the ground state as a function of the strength, / , of the random field terms. Figure 3.9 shows the result of 20 zero temperature Monte Carlo simulations for this reduced potential. We observe that if the field terms are completely removed,  Chapter 3. Computer Simulations I  81  the areas where a unique ground state exists disappear completely and the figure is now completely frustrated. We thus propose that our amorphous nonseparable system is similar to a Ising spin glass problem (g / 0), in a random magnetic field ( / ^ 0), and with a small ferromagnetic coupling (77 ^ 0). The quenched bond angles, c?,j, break the continuous symmetry of the rotational degrees of freedom. Present opinion is that the lower critical dimensionality is larger than two for a lattice system with discrete orientations and nearest neighbor interactions [62] and it seems likely that the same will be true for our system. The effects of random fields on nearest neighbor spin glass models is not known in detail and much work must be done to parameterize such systems quantitatively [61, 63]. However in a recent mean field calculation, Holdsworth [64] has shown that the infinite range Ising model with a Gaussian random field has a stable spin glass phase below a finite critical field variance. A field variance line analagous to the de Almeida-Thouless [65] line for a constant field is predicted. The above system is analagous to a quadrupolar, or orientational glass such as paraortho hydrogen or a mixed crystal of K B r - K C N [66] for which there has been a long debate concerning the existence of a glass transition. The quadrupole-quadrupole interaction is nonseparable and can be expanded, in the same way, in field and bond terms.  Our  results for anisotropic amorphous solids show that orientational glass behaviour is the result of nonseparability in the anisotropic pair interaction when the translational degrees of freedom are quenched. We have recently investigated the spin glass aspect of the van der Waals potential at finite temperature [63]. We do not present the results here, since it would take us too far from the main subject we are interested in, namely orientational phenomena in liquid crystals. In the context of liquid crystal models, the main conclusion of this section is that the  Chapter 3. Computer Simulations I  82  translational and rotational degrees of freedom should be in thermal equilibrium in order for a nonseparable potential to lead to nematic-like orientational ordering behavior. In a liquid phase, the system is able to relax and find the minimum free energy for both degrees of freedom.  Our results on lattice systems and for amorphous solid already  suggest that, if a system interacting with the van der Waals potential ( 2.20) is to lead to liquid crystalline behavior, there has to be a strong and nontrivial coupling between the translational and rotational degrees of freedom. A preliminary investigation of a system of particles interacting with a modified van der Waals potential in a true liquid phase is presented in the next section.  3.4  Annealed Positional Disorder: Preliminary Results on 2D Liquid Crystal Models  3.4.1  Why Study 2D Liquid Crystalline Models?  We give here only the practical motivations for first studying a 2D fluid instead of a Z D one, and reserve the more theoretical ones for the next chapter. Firstly, for 2D fluids, we have only three degrees of freedom to consider, in contrast to five in three dimensions for molecules with cylindrical symmetry; this reduces the computing time considerably. Secondly, it is much easier to visualize the configuration of the particles in 2D than in 31? (compare, for example, Figures 3.7 and 3.8). Finally, there is a hope that some of the physics present in a 3D nonseparable system can be first observed and at least qualitatively understood, for a fraction of the computing cost, in a 2D system. It will, however, soon become apparent that one has to disregard, from theoretical arguments, the direct comparison of the results obtained for 2D nematic liquid crystals with those for Z D systems.  83  Chapter 3. Computer Simulations I  3.4.2  Pair Potentials  We would like to return to our initial investigation of approximation 2.14. To do this, we proceed as explained in the previous chapter and compare a nonseparable potential with a separable one. The nonseparable potential chosen is:  -Wt[2{hi.hj) -\\ 2  r~;  f * A O * A A A 12  J  —6  (3.20)  —47e[n,- • n,- — 3n, • e,je,j • rij\ where we have defined r,j =  \fij\/r  0  with  r  0  being an effective molecular diameter. Here  hi = (cos(<6,), sin(</>i)) is a unit vector in the direction of the molecular anisotropy, and iij  = (cos(0,-j), a'm(Oij))  is a unit vector in the direction of  fij.  The last term in 3.20 corresponds to the anisotropic Van der Waals interaction we have used in the previous two sections.  The other anisotropic term, proportional to  /3, and independant of e , is purely phenomenological. We introduce this term in the tJ  nonseparable potential to make sure that the ground state has nematic order and is not a modulated phase as observed in the section on the lattice models [60]. Since the particles are now free to move, a repulsive term is needed to stabilize the interaction at short distance. As discussed in Chapter 2, we have chosen to neglect steric anisotropy. Hence, a simple isotropic repulsive term, 4er~ , is added to prevent the particles of collapsing 12  towards each other. Finally, an attractive isotropic term, 4aer~- , is added, a, 6  and 7  are positive constants, and e is a measure of the strength of the interaction and has units of energy. In the spirit of the discussion presented in Sections 2.1.2 and 2.2.3, we choose a, /5 and 7 be 0, 2/5 and 4/5 respectively so that averaging V ^ over the orientation 0,j 5  of e,j leads to our separable potential: ^-r «-J\2(hi.hjY-l}r-f\ T  (3.21)  84  Chapter 3. Computer Simulations I  with J = 1/2. This is best seen if the potential 3.20 is expanded in terms of its rotational invariants:  T  \j  T  37e r  6. 9ft  2r  6  ij  CT  ij  [cos(2& - 28ij) + cos(2<f>j - 20^)] [cos(2^ + 2 ^ - . 4 ^ ) ]  •  (3-22)  Hence, the quantities a, 0, 7 and J ate not independent but are related by a  =  P =  1 J-l  T  .  (3.23)  Potential 3.21 corresponds to the two-dimensional Maier-Saupe model for nematic liquid crystals [19, 20]. This potential leads to a 2D nematic phase if J is larger than some critical value . At high temperature, in the isotropic phase, where (e >) w (e >) w 0 4  and  2lBi  4t0i  ((hi • hj) ) « 1/2, one recovers an effective Lennard-Jones pair potential from both 2  the separable and the nonseparable potential with the above particular choice of or, j3 and  7.  We note here that the potentials we used exhibit the symmetry Vij(fij-hi, hj) = Vij(rij] -hi,hj) = V^j(r„; n<, -hj)  ,  (3.24)  which ensures that the two systems can, in principle, exhibit a mesophase with nematic symmetry. We next consider the effects of translational motion. Monte Carlo simulations have been performed on these 2D systems (potentials 3.20 and 3.21) at various temperatures where the particles could move and rotate. A lattice version of this model has been studied via renormalization group by Albinet and Tremblay [67] and a Monte Carlo simulation with fixed particles on a triangular lattice has been performed by Denman et al. [68]. 4  85  Chapter 3. Computer Simulations I  3.4.3  Results  >. A : system size of 400 particles at a fixed density p = 0.888, (in units where lengths and temperatures are given in units of r„ and e/ks respectively) was investigated for both potentials 3.20 and 3.21. We performed between 2000 and 3000 MCS  for both  equilibration and production of data at each temperature. We first present the results for the separable potential 3.21. When the temperature is varied we find three different phases as illustrated in Figure 3.10. At high temperatures, as in Figure 3.10a, the system is isotropic. At T = 2.4, we observe the onset of nematic order, as shown in Figure 3.11, where we plot the nematic order parameter Q given by the average positive eigenvalue of the traceless 2 x 2 matrix [69]:  j i=N  Qcp = TT £ i V  n,  a  t=i  [2n nip - 6 p] . ia  a  (3.25)  is the a cartesian component of n,-. Q = 0 in an isotropic phase and Q = 1 in  a perfectly ordered (nematic) phase. The eigenvector associated with Q is called the director (see Chapter 1). Technically, one calculates Q from successive realizations of the particle configuration and one keeps a running average of the Qs. Another possibility would be to keep a running average of the elements of Q p and calculate Q from the a  resulting average. For systems with continuous symmetry, the first procedure is superior since the second technique would give a value for Q which tends to zero as the simulation time is increased. A snapshot taken in the middle of the nematic region is given in Figure 3.10b. When the system is cooled further, it crystallizes in a triangular lattice (Figure 3.10c). The empty regions are characteristic of the phase separation seen in a constant volume simulation. We next consider the case for the nonseparable potential, Equation 3.20. There is now a strong correlation between orientational and translational motion, and the low  Figure 3.10: Snapshots of particles configurations for the separable system at p = 0.888. (a) Isotropic phase at T = 4.0. (b) Nematic phase at T = 1.2. (c) Low temperature, T — 0.2, with the system freezing into a triangular lattice.  87  Chapter 3- Computer Simulations I 1-1  <H  1  1  1  1  1  1  05. 1 t5 2 25. 3 35.  1  4  •  4.5  TEMPERATURE Figure 3.11: Order parameter Q as a function of temperature for the separable system at density p = 0.888. temperature crystalline phase is rectangular, allowing parallel alignment of the rotors (see Section 3.2). In the liquid phase we see the build-up of strong short-range positional and orientational correlations. It is interesting to find that the only lattice structure that supports nematic order in its ground state is the one which is favored by the system once translational motion is allowed. Snapshots of configurations at different temperatures are shown in Figure 3.12. 3.5  Formation of Chain-Like Structures  One observes in Figure 3.12b, for the nonseparable potential 3.20 a build-up of bond orientational structure and the formation of chains whose axis lies in the direction of the molecular anisotropy n,-. As the temperature is reduced, the chains increase in length, become noticeably less flexible and register with respect to each other with the perpendicular distance between chains being somewhat longer than a link along a chain. In contrast, for the separable potential, there is no obvious evidence of build-up of  Chapter 3. Computer Simulations I  88  (c)  Figure 3.12: Snapshots of particles configurations for the nonseparable system at p = 0.888 for three different temperatures: (a) T = 4.0. (b) T = 2.0. (c) T = 0.2. A strong coupling between the translational and orientational degrees of freedom is observed. The system appears to freeze into a rectangular lattice which favors orientational ordering.  89  Chapter 3. Computer Simulations I  bond correlation in the nematic phase. The microscopic origin of this phenomenon can be understood by calculating the dependence of V^  s  on 0,j for a distance r (8ij) where m  (3.26) where we have used Equations 3.23. The potential depth, V , at r , is given by m  m  V V  m  = - l + J _ - l + {l_3cos (^)} 2  m  (3.27)  2  7  has an absolute mimimum at 0,j = 0, a local mimimum at # = 90°, and an absolute tJ  maximum at Oij = cos (v 3/3) « 55°. The dij dependence is shown in Figures 3.13 1  and 3.14 for r  m  and V  m  /  respectively, for several values of 7. First notice that the maxi-  mum at 55° disrupts the normal tendency of isotropic, and separable anisotropic potentials to form a local sixfold (60°) symmetric nearest-neighbor bond shell. B y examining Figure 3.14, we see why, in the nematic phase, where hi « \\hj and for a large value of 7, the large depth of the potential for 0,j = 0 will strongly favor the formation of weakly coupled chain-like structures. We have looked in detail at the formation of chains in the nonseparable system, but at a density p = 1.00 to avoid the phase separation observed in Figure 3.12. Snapshot configurations of the chains are shown in Figure 3.15 for four different temperatures. We define a chain as follows: take two neighboring particles and calculate the lowest two bond energies for each. If they share one of these bonds then this bond is a link in the chain. This is clearly not a unique definition, but is serves our purpose in analysing the chain structure appearing in the ordered phase. At high temperature, T = 4.8, there are only short chains, but as the temperature is reduced to 3.6, we observe a rapid build-up until, at T = 2.6, they span the entire system. The chains are reminiscent of those observed in dipolar magnetic systems in strong magnetic fields [70, 71, 72, 73]. In our case, the isotropic to nematic phase transition  Chapter 3.~iComputer Simulations I  Figure 3.13: 0,j dependence of r  TO  for the nonseparable model with J = 1/2.  90  Chapter 3.  Computer Simulations I  Figure 3.14: 0,j dependence of V for the nonseparable model with J — 1/2. m  91  Figure 3.15: Snapshots of chains configurations at T = 4.8, 4.2, 3.6 and 2.6. The criterion used for chain construction is explained in the text.  93  ^^Chapter 3: Computer Simulations I  induces a self-consistent orientational field that plays qualitatively the same role as an external field in dipolar magnetic systems. The potential energy,  of a particle i in  this field would be given in such a mean field argument by Ei cx -Qcos(2<f>i - 2 $ ) 0  (3.28)  where Q is the nematic order parameter and $ is the orientation of the director in the 0  lab frame. There is, however, a major theoretical difficulty in explaining the formation of chains with the above mechanism in a 2D system like the present one. As discussed in Chapter 1, most two-dimensional systems with continuous degrees of freedom lack long-range order at any non-zero temperature. This result follows from an elastic or continuum description of the fluctuations away from a perfectly ordered state. As in other systems, one finds that an elastic description of director or orientational fluctuations in two-dimensional liquid crystals [2] leads to the conclusion that true long-range orientational order is absent [74]. Practically, this means that Q vanishes in the thermodynamic limit and our interpretation in terms of an effective field as in Equation 3.28 would be meaningless for a very large sample. The observation that true long-range orientational ordering is forbidden in a 2D liquid crystalline phase raises new questions. If true long-range order does not exist, this means that our study of the mean-field approximation 2.14 in a 2D liquid phase is illdefined, as is our comparitive study of a separable and a nonseparable potential through the spatial averaging of e  5 tJ  .  This investigation is, nevertheless, interesting from a  theoretical point of view since Straley [75] has shown that the presence of long-range order in two-dimensional liquid crystals is rigorously excluded only if the intermolecular pair potential is short-range and separable. Straley's failure to rule out true long-range order in a general case opens the small possibility that our system could exhibit true 8  T h i s is w h y  r e s u l t s f o r 2D  we  s a i d earlier ( s e c t i o n 3.4.1) t h a t it w o u l d b e  s y s t e m s to t h o s e for 3 D  systems.  necessary to renounce directly  mapping  Chapter 3. Computer Simulations I  94  long-range orientational order. If this were the case, we would not have to give up our original interpretation of the chain formation in terms of a true rotational symmetry breaking in the liquid crystal phase. Evidence from computer simulations on nonseparable potentials is quite controversial and there is no consensus at present as to the nature of the ordering. Viellard-Baron [76] found an isotropic to nematic phase transition in a hard ellipse fluid which he interpreted as first order, but no attempt was made to distinguish between long-range and quasi-long-range order. Soft ellipses have been studied by Tobochnik and Chester [45]. They found strong evidence for long-range order in the molecular orientational field but were unable to study the system-size dependence of the M O F order parameter because of prohibitively long relaxation times. Frenkel and Eppenga [69] recently studied a twodimensional system of needle-like particles. They observed a transition to a high density phase that clearly exhibits quasi-long-range orientational order, with the orientional correlation function decaying as a power law with density dependent exponent. They also isolated defect pairs [8, 6, 7] and interpreted the transition in terms of an unbinding of these pairs (Kosterlitz-Thouless transition). Finally, they were unable to reproduce the results of Tobochnik and Chester [45] and their preliminary investigation of the hard ellipse fluid did not support a first order isotropic to nematic phase transition [76]. Very recently, Cuesta and Frenkel [77] have reported further results on the 2D hard ellipse fluid which suggest that this system has quasi-long-range order and not true long-range orientational order. Although the question regarding the nature of the ordering in 2D liquid crystalline systems remains open, a large number of experiments on nonseparable liquid crystals [78, 79, 80] have been very succesfully explained by continuum theories that exclude longrange order. Hence, it would be most surprising if any short-range nonseparable potential led to long-range orientational order in a fluid phase.  Chapter 3. Computer Simulations I  95  If we accept that it is highly probable that our nonseparable system with potential 3.20 would show quasi-long-range order in the large system limit, one is still left to explain the origin of the chain-like structures observed in that system. Indeed, it seems reasonable to assume that, even if the nonseparable system was to exhibit only quasi-long-range order, one would still observe the formation of strongly oriented nearest-neighbor bonds on some length scale, even for an infinite system.  3.6  Discussion  It has often been speculated that the van der Weals anisotropic potential we have studied in this chapter may lead to a nematic phase. We have focused our attention on spherical molecules to avoid a coupling between steric effects and the anisotropic attractive forces in which we are interested. We observed that different lattices have different ground state symmetries. Nematiclike ground states were found only for the sufficiently stretched rectangular lattices. We studied randomly parked circles and spheres in two and three dimensions. We did not observe a nematic-like ground state for these systems. One might have expected this result knowing that different lattices have different ground state symmetries. In addition, we have observed for quenched positionally disordered systems, the appearance of frustrated islands and traced back their origin to loose regions with small random fields which arise from the nonseparability of the pair potential. Preliminary Monte Carlo simulations with translational motion suggest that molecules interacting with the potential 3.20 exhibit strong correlations between translational and rotational degrees of freedom which lead to the formation chain-like structures similar to those observed in dipolar systems in strong magnetic fields. The probable existence of quasi-long-range order in such a system makes it difficult to explain the formation of  Chapter 3: Computer Simulations I  96  these chains within the framework of a self-consistent symmetry breaking field (mean-field scenario). Finding a satisfactory microscopic model for the formation of these nearestneighbor structures in this system is the subject of the next chapter.  Chapter 4  Computer Simulations II: Induced Bond Orientational Ordering in 2 D Liquid Crystals  In this chapter, we continue our computer simulation investigations of the orientational phenomena exhibited by the modified anisotropic van der Waals potential in a twodimensional liquid phase. In Chapter 3 we observed that the nonseparable van der Waals potential leads to a noticeable build-up of local positional structure.  Such a phenomenon is not observed  for the separable system. We argued that, from a rigourous theoretical point of view, the interpretation of this increase in positional correlation is not satisfactorily explained in terms of chain formation induced by a self-consistent molecular orientational field appearing below the isotropic-nematic transition. Another mechanism must be found in order to explain the build-up of the observed nearest-neighbor structures. Several authors have discussed the phenomenon of tilt-induced bond orientational ordering in smectic films from a theoretical point of view [81, 82, 83, 84]. This effect has now been observed experimentally in several systems [85, 80, 86, 87]. We will show that it is possible to reinterpret the increase of positional correlation in the nonseparable system as a manifestation of such induced bond orientational ordering. We investigate, using Monte Carlo simulations, the presence or absence of an induced 97  Chapter 4. Computer Simulations II  98  bond orientational order in 2D liquid crystal models. We find that the nonseparable anisotropic van der Waals potential exhibits a nematic phase where the ordering in the molecular director induces an ordering of the nearest-neighbor bond orientations with rectangular symmetry. To make a comparison with this behavior, we study a separable system that also exhibits a nematic phase. Unlike the nonseparable case, we find that the separable system exhibits only short-range order in the correlations of the nearestneighbor bond orientations. The phenomenological coupling between molecular and bond orientations present in Landau theories of liquid crystals [81, 82, 83, 84] plays an important role in explaining the phase transitions observed in smectic liquid crystal films. Our results lead us to conclude that these couplings have their microscopic origin in the nonseparability of the intermolecular pair potential.  4.1  Induced Bond Orientational Ordering in Smectic Liquid Crystal Films  A freely suspended smectic liquid crystal film is realized in the following way: a drop of liquid crystal material is put on a piece of glass and, using another piece of glass, the liquid is spread across a hole in the plate (see figure 4.1). Depending on the speed at which the liquid is spread across the hole, (of diameter w 10mm) film thicknesses of between 2 and 100 smectic layers can be stabilized and studied for several months [4]. Nelson and Halperin [81] realized that freely suspended smectic liquid crystals could provide particularly well suited systems to test the new ideas of 2D melting theories [10, 11, 12]. We present their ideas in this section since we will use them to explain the bond ordering observed for the nonseparable system studied in Chapter 3. Chapter 1 we described in various smectic phases. Let us consider the bond angle, 0(f), which describes the orientation of the sixfold symmetric nearest-neighbor shell (see  Chapter 4. Computer Simulations II  99  Figure 4.1: Schematic representation of a freely suspended liquid crystal film. Figure 1.10). Long wavelength spatial fluctuations of the nearest-neighbor bond orientation are described by the following Hamiltonian: =  L  -J  d*r K \Ve(r)\  (4.1)  2  B  KB is the elastic constant for distortion in 0(f), the bond orientational field (BOF). Below some critical temperature, defects in 0(f), which are similar to the vortices in the xy model, are rare and the correlation function, c$(r), has quasi-long-range order: ce(r) = (cos{6(0(r)-0(O)}) « r - "  ( T )  .  (4.2)  As the temperature is increased, defect pairs are created and reduce KB- Above the critical temperature, the defects unbind and ce(r) is driven to short-range order with exponential decay. This transition corresponds to the smectic-Bn —* smectic-A transition in a thin, freely suspended smectic film. If the molecules are tilted with respect to the layer normal, as in the smectic-C phase, one repeats the above argument and introduces a new term in H, which measures the  100  Chapter 4. Computer Simulations II  energy required to impose long wavelength distortion to  <J>(f),  the in-plane orientation of  the molecular tilt:  H kT  =  -Jd r  l  [K \v0(f)\  2  2  B  + K \v<t>(f)\ } 2  M  (4.3)  B  where <j>(f) is defined on the interval  [—7r,7r]. KM  is the Frank constant for fluctuations  in <f>(f), the molecular orientational field ( M O F ) . The unbinding of vortex pairs in <f>(f) corresponds to the smectic-C —» smectic-A or a smectic-C —• smectic-Bn transition in a 2D smectic film depending on whether 0(f) has short or long-range order. Notice that, up to this point in the present discussion, there is nothing that prevents a given liquid crystalline material from exhibiting both a smectic-Bn —• smectic-C transition, followed by a smectic-C —* smectic-A transition as the temperature is raised (see classification of smectic phases in Figure 1.13); that is two independent transitions, which are both expected to be of Kosterlitz-Thouless type [6, 7]. The important idea of Nelson and Halperin [81] was to recognize that the last Hamiltonian was not the most general, and that one should include a coupling between the fields 0(f) and <f>(f):  H kT  i  j  dr 2  [K \V0(f)\  2  B  + K \V<f>(f)\  2  M  + 2gV0(f) • V<f>(f)}  B  (4.4) The coupling term, h , arises because both 0(f) and e  <f>(f)  feel a sixfold symmetric potential  when rotated with the other field held fixed. A gradient cross-coupling term, proportional to <7, would be present in the most general situation. Let us explain the implications of the periodic coupling term. Assume that KM is larger than K . Hence, as the temperature is lowered, the field, B  <f>(f)i  acquires quasi-long-range order before 0(f), and ordering of <j>(f) occurs on a very  long length scale. On this length scale, <f>(f) picks-up a definite orientation. Once 4>(f) is  101  Chapter 4. Computer Simulations II  quasi-long-range ordered, not all the orientations available to 6(f) axe equivalent anymore within this region since one particular direction has been picked up by <p(f). Because of the he coupling term, some induced ordering in 6(f) sets in. In some sense, the ordering in <f>(f)  plays the same role on 6(f), via the h term, as does a magnetic field in a ferromagnet 6  above the Curie temperature. The induced ordering in 6(f) eliminates a possible smecticC —• smectic-B# transition at a lower temperature, although remnants of the transition may be observed if  is small. This is again similar to the ferromagnetic problem where  the presence of a magnetic field washes the transition away since the system has a finite magnetization at all temperature . 1  If KB is larger than KM, quasi-long-range order sets in for 6(f) at a higher temperature than for <f>(f). However, the ordering in 6(f) does not induce an ordering in <f>(f) for the simple reason that once 6(f) acquires quasi-long-range order, there are still six equivalent direction (the directions of the six nearest-neighbors) for <f>(f) to choose from, so <f>(f) can remain disordered. Bruinsma and Nelson [82] have shown that the above prediction of tilt induced B O F ordering in a thick (three-dimensional) system smectic-C film applies as well. These predictions for both thin [85] (2D) and thick [80] (3D) smectic-C films have recently been verified experimentally. Hence, the classification of smectic phases presented in Chapter 1 (Figure 1.13) is not totally correct: because of this mechanism of tilt-induced B O F ordering, a smectic-C phase is actually a hexatic phase since it has bond order orientational within the smectic layer! In a general case, one expects that more harmonics will be present in the sixfold periodic B O F - M O F coupling in H. In a recent work, Selinger and Nelson [83, 84] have l  A t  the  level of a L a n d a u theory, the  a t r a n s i t i o n f r o m a sm-C —» s m - I  o r sm-C  —» sm-F  sm-C  — • sm-Bjj  is a l l o w e d , t h o u g h m o s t s y s t e m s t h a t  undergo  t o a h e x a t i c p h a s e e x h i b i t a t r a n s i t i o n t o a t i l t e d h e x a t i c p h a s e s u c h as a transition.  sm-C  102  Chapter 4. Computer Simulations II  extended the above Hamiltonian for tilted smectic films:  H kT  =  I J d r [K \Ve(f)\ + K \V<f>(r)\ + 2gV9{r) • VcA(r)] 2  2  B  2  M  B  - £ > e « f d r cos{6n [6(f) - <f>(?)]} . 2  (4.5)  n=l  Using this Hamiltonian functional, they were able to explain both the first order smecticI —• smectic-F transition in a thermotropic material [88], and a sequence of two second order transitions smectic-I —• smectic-L —• smectic-F in a lyotropic liquid crystal [87]. There is a strong resemblance between a single smectic-C layer and our 2D van der Waals nematic liquid crystal model. Indeed, a well ordered smectic-C phase, where the tilt angle shows only small fluctuations, can be regarded as a realization of a 2D nematic fluid, if there are as many molecules 'up' as 'down' ((n • z = 0) . 2  The analogy between smectic-C and 2D nematics, as well as the success of the above Landau theory (Equation 4.5) in explaining the phenomenon of induced B O F ordering observed in smectic-C phases [85, 80, 88, 87], motivated us to investigate the presence of an induced B O F ordering in the nonseparable van der Waals system. We will show that induced bond orientational ordering is the mechanism we are looking for to explain the build-up of chain-like structures observed in the nonseparable system and described in Chapter 3. As a by-product, we shall gain some physical insight into the microscopic origin of the B O F - M O F couplings invoked in the Landau theories of tilted smectic systems. 2  R e c a l l i t e m 3 i n p a g e 15  a n d the  following discussion in page  18.  103  Chapter 4. Computer Simulations II  4.2  C o m p u t a t i o n a l Technique  For convenience, we rewrite here the modified van der Waals potential 3.20 we investigated in Chapter 3:  —4~ft[hi • hj — 3n • e.je.j • hj] r,^ 2  6  t  (4.6)  We have studied a nonseparable potential with a = 0, /? = 2/5 and 7 = 4 / 5 . The separable potential was obtained from 4.6 with a = 1.0, /3 = 1/2 and 7 = 0. The reader is invited to refer to Chapter 3 for a definition of the various parameters in Equation 4.6 and a description of how these two potentials are related to each other. We have investigated the thermodynamic behavior of systems of particles interacting via the above two potentials with a Monte Carlo simulation using the standard Metropolis algorithm at constant area, number density and temperature. A Monte Carlo step consists of an attempt to move and rotate simultaneously one particle in a cell with periodic boundary conditions. To make the program work as efficiently as possible, a local neighborhood of size 3.5r was recorded around each particle. The particles move 0  in their local environment for a small number of Monte Carlo steps per particle (MCS). After typically 5 MCS, the local environments are reconnected. In this way, the time taken for m MCS  is approximatively linear in the number of particles N. B y varying  the size of the attempted move in f and h space, we were able to keep an acceptance rate between 30 and 50% at all temperatures. The density p = 0.888 used in our preliminary investigation (previous chapter) proved to be too low, as phase separation occured at low temperature. For the results presented below we used a density of p = 1.00 and a system of 400 particles. The separable and  104  Chapter 4. Computer Simulations II  nonseparable potentials were cut-off at a distance of 2.50r . o  The results of Chapter 3, and an investigation of the ground states exhibited by the nonseparable potential showed that a rectangular lattice with the particles oriented along the shortest lattice vector ("nematic" ground state) is favored for a range of pressure and degree of nonseparability (ratio  For the values of 7 = 4/5 and 0 = 2/5,  we anticipate a rectangular solid with lattice vectors a « 0.91, b « 1.09 and with the molecules oriented along the a axis. The ground state bond energies are then — 13.00e and —1.44c along the a and the b axis respectively. Two simulations were started at T = 2.40, one from the ordered rectangular phase described above and one starting from a random configuration in both positions and orientations. After 120,000 MCS  for the random start and 20,000 MCS  for the ordered  one, the energy, pressure and nematic order parameter statistics became indistinguishable for the two runs. From T = 2.40, we heated the system to T = 4.8 and cooled it to T = 0.2. At each temperature the final configuration from the previous temperature was used as the starting configuration. Enough MCS  each temperature enough were  performed to re-establish the thermodynamic equilibrium state of the system. Typically, equilibration runs of up to 100,000 MCS data were 40,000 MCS MCS  were performed, while runs for production of  in both the solid and isotropic fluid phase and up to 200,000  in the nematic phase (500,000 MCS  at T = 3.4).  The nonseparable system proved to be extremely sluggish and huge relaxation times were observed throughout the liquid crystalline phase. Similar observations were made by Tobochnik and Chester [45] for the nonseparable potential they studied. Hence, we found it most efficient to collect our thermodynamic data each time the local particle environments were reconnected, that is every 5 MCS. B y comparison, for the separable system at the same density and away from the isotropic-nematic transition, we found that 10,000 to 20,000 MCS  for both equilibration and production runs were sufficient to  105  Chapter 4. Computer Simulations II  ensure good statistics in all the thermodynamic quantities we measured.  4.3  Thermodynamic Quantities  To characterize the amount of orientational ordering present in our systems, we reintroduce the order parameter Q, which is given by the positive eigenvalue of the 2 x 2 matrix [69]: j i=N Qa/3  =  "77 £  [ «'a »7? 2n  n  f><*0]  _  (-) 4  7  »=1  As mentioned in Chapter 3, the eigenvector associated with this eigenvalue is the director (see Chapter 1). It is easy to show that the orientation of the director, $ , in the lab0  frame, is such that the quantity A  = 4 Iy I > o s { 2 ( < k - $ ) }  t=i  is maximized with respect to  0  (4-8)  and that A = Q. This gives another interpretation of  the director: it is the direction along which the average projection of the anisotropy axis, hi, is the largest. Figure 4.2 shows the temperature dependence of Q for the nonseparable potential. Even if the low temperature phase has only algebraic order, Q is meaningful for a finite size simulation and should decay to zero as the size of the system increases [69]. This point will be discussed in Section 4.8. There is evidence that an orientational phase transition occurs around T « 3.6. Figure 4.3 shows the average configurational energy per particle. One observes a rapid change in the energy at T « 3.6 as well as a discontinuity in the slope of this curve around T « 1.8. We also measured the pressure using the virial equation of state:  PA  = Nk T B  ^i • ViVg^&hithj)  (4.9)  106  Chapter 4. Computer Simulations II  This quantity gives no indication of a transition at T « 3.6 but exhibits a discontinuity in the slope around T « 1.8 which we interpret as the melting temperature (Figure 4.4). The configurational specific heat at constant volume, C , calculated from the fluctuv  ations is illustrated by the triangular symbols in Figure 4.5. We also calculated C from v  finite difference of the energy curve. The heat capacity, at a temperature T , midway between two neighboring temperatures T\ and T and separated by T\ — T = 0.20 on 2  2  the energy curve, is shown in Figure 4.5 by the square symbols. The two methods lead to similar results suggesting that fairly good thermal equilibrium has been reached. The heat capacity shows a sharp peak at T = 3.6, which, from our other thermodynamic data and the orientational correlation functions, appears to be at or very close to the phase transition. This is in contrast with the simple 2D xy model where a broad peak in the heat capacity occurs at a temperature about 15% above the transition. Similar behavior was observed for Q, E and P for the separable system. As in the nonseparable system, an orientational phase transition is observed, but at a temperature T = 2.8. For both systems, we interpret the high temperature transition as an isotropic-nematic phase transition. In the following sections, we present results that support this assertion. Two phase transitions were also observed for the separable system and the high temperature one was again interpreted as an isotropic-nematic transition. Most of our efforts and computer power were oriented towards the investigation of the isotropic-nematic phase transition for both potentials. It would also be interesting to study the nature of the melting of these two-dimensional anisotropic solids [89].  Chapter 4.  107  Computer Simulations II  1.00  O  E  D  0.80  A  0.60  H  o • •  a  • • •  CD  O  D  0-  0.40  O  0.20  o •• i  0.00  0.50  i i i i i i i i i i i i i i i i i i i | 1.50 2.50  I I  i  i i i i i i i i i i i i i i i i i 3.50 4.50  Temperature  (£/k ) B  Figure 4.2: Order parameter Q vs temperature T (in units of t/ks) for the nonseparable system at p = 1.00. - 4 . 0 0 -]  H  -6.00  to  cm • H  -8.00  CD  C -10.00  UJ  -12.00 H  — 14.00  |  I I I I I I I'l'l J I I I I I 1 ! M  0.00  1.00  I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I  2.00  3.00  4.00  5.00  Temperature ( e / k ) Figure 4.3: Average configurational energy per particle (in units of e) vs temperature (T) for the nonseparable system at p = 1.00. B  Chapter 4: Computer Simulations II  108  40.00 -,  ^ - N 30.00 b w  V  20.00  cn  CO cu  Cl-  io.oo  'f'T T T T T ' T T T T  0.00  0.50  | 1 1'T T"T I 'I T T ' | T T I 1 t 1 I I I f T ' T T T I  1.50  2.50  1 1 T T ) t 1 I 1 1  3.50  Temperature  4.50  (£/k ) B  Figure 4.4: Pressure vs temperature for the nonseparable system at p = 1.00.  2  c  Q.  6.00  -i  5.00  -  4.00  3.00  o  oo I  A O AP 2.00  A  -  O^*  1 00 -  T  0.50  A * 0  0  AQA ^  D  A  D  AP*  iTir iiii| ' ' iiiiiii| iiiiiiiii| ' ' ' i 1.50  2.50  3.50  Temperature (units of  | i i i ii  4.50 c/k ) 9  Figure 4.5: Configurational specific heat at constant volume C (in units of kg) vs temperature (T) for the nonseparable system. The triangles correspond to C obtained from the fluctuations in the energy while the squares correspond to C obtained from finite difference of the energy curve. v  v  v  Chapter 4. Computer Simulations II  4.4  109  L o c a l S t r u c t u r e for the Separable and Nonseparable Systems  By comparing the behavior of the thermodynamic quantities, it appears that the only major difference between the separable and the nonseparable systems is that the isotropicnematic transition occurs at a lower temperature (T=2.8) in the separable case.  A  snapshot of a particle configuration below the transition reveals, however, an important qualitative difference in the positional configurations of the two systems (Figures 4.6 and 4.7). In Figure 4.6 we show a snapshot of the nonseparable system just above and below the transition. As described in Chapter 3, we observe a rapid and considerable increase of bond orientational structure and the build-up of chains whose axis lies in the direction of the molecular anisotropy vector, h i . The nearest-neighbor bonds are illustrated in Figure 4.6 for several particles. The lowest energy configuration is with the chains registered parallel to each other so that many particles have well denned third and fourth nearest neighbors in a line perpendicular to the first and second neighbor pair, forming a local rectangular structure. In contrast, for the separable potential, there is no obvious evidence of a build-up of bond correlation. In Figure 4.7 we show the nearest-neighbor bonds for several particles; the most probable local structure is hexagonal, as in an isotropic fluid. The hexagonal structures illustrate the sixfold bond orientational field, which remains disordered below the isotropic-nematic transition. Snapshot configurations of the nearest neighbor bond network are shown in Figure 4.8 for four different temperatures. We have defined the network as follows: take two neighboring particles and calculate the lowest two bond energies for each. If they share one of these bonds then this is part of the network and a link is drawn between the two particles (solid line). Each pair of particles are treated in this way. The procedure is then  110  Chapter 4. „.Computer Simulations II  Figure 4.6: Snapshot of the configuration of the particles interacting via the nonseparable potential both above (T = 4.0, a) and below (T = 3.6, b) isotropic-nematic transition. The circle represents the particle and has diameter r ; the line indicates the orientation of the unit vector n,. The lines joining the centers of three particles illustrate the onset of order in the twofold bond orientational field at the isotropic-nematic phase transition. 0  Chapter:^i:-: Computer Simulations II  111  Figure 4.7: Snapshot of the configuration of the particles interacting via the separable potential both above (T = 3.2, a) and below (T = 2.8, b) the isotropic-nematic transition. The lines joining the centers of six neighboring particles illustrate the the sixfold bond orientational field.  Chapter 4.  112:  Computer Simulations II  repeated for the third and fourth lowest energy bonds (dashed line). We have purposely kept the first and second lowest energy bonds and the third and fourth lowest energy bonds separate, because we wish to show the development of the two independent twofold symmetries of the rectangular structure. The positions of the particles are indicated by a small circle. At T = 4.2, well above the isotropic-nematic transition temperature, the system is clearly isotropic and only short-range structure is visible. The figure for T = 3.6 shows the network very close to the nematic transition, where the lowest energy bonds form long chain structures along the direction of molecular orientation and the chains begin to register, forming rectangular plaquettes.  At T = 2.6, the development of the two  perpendicular twofold symmetries constituting rectangular bond orientational order is clear, and the network has become more ordered and noticeably less flexible. Finally we show a snapshot taken in the solid phase (T = 0.8), where the network is very well defined except for frozen in dislocations. One or two further bonds are, suprisingly missing. This is because the fourth lowest energy bond is occasionally that of a second nearest neighbor along the chain, or short axis of the rectangle (see Figure 3.15 for comparison.)  4.5  C o r r e l a t i o n Functions  For all correlation functions we calculated, a bin size of 0.05r„ was used. collected every 5 MCS  over a total run time of 20,000-50,000 MCS.  Data was  No smoothing of  the resulting correlation functions was performed. We show in Figure 4.9 the radial distribution function, go(r). A t high temperature (T = 4.2, solid curve) the shape of <7o(r) is characteristic of a liquid distribution, while at T = 2.6, well below the isotropic-nematic transition, the distribution function is still liquid-like but displays much more short-range structure (dashed-curve). This is unlike  114  Chapter 4. Computer Simulations II r  2.50 ;  2.oo :  1.50 -j o  i.oo :  0.50 :  0.00  —  i4 i i  0.00  1.00  i i i i i i i i i  r  i  i  iii  i i  2.00  i  i i  i  i  ii i ii  3.00  Figure 4.9: Radial pair distribution function go{r) vs separation (r) (in units of r ) for the nonseparable system. The solid and dashed curves are for T = 4.2 and T = 2.6 respectively. 0  the separable system whose c/ (r) is found to be essentially featureless for all temperatures 0  above the melting temperature. The large-distance behavior of g for both the separable 0  and nonseparable systems clearly shows that these are liquid phases since go decays quickly to its limiting liquid-like value of 1 (Figures 4.10 and 4.11.) In order to characterize the correlations in the molecular orientational field ( M O F ) , we measured the orientational correlation function, <7 (r), [45, 69] defined as: 2  g (r) = (cos{2^(r) - 2#0)}> 3  (4.10)  where <j>(r) is the angle, relative to a fixed axis, of the vector hi of some particle at a distance r from a reference particle at the origin. In a phase where there is true long-range order, the limit of c/ (r) when r —> oo is non zero and equal to the square of 2  the order parameter Q. Unfortunately, this criterion cannot be used to check for the existence of long-range order since the same result applies to a system with quasi longrange order; that is, the value of c/ (r) at the edge of a cell of size L is equal to the square 2  Chapter 4. Computer Simulations II  115  Figure 4.10: Large distance behavior of go(r) for the nonseparable system at T = 4.2 (solid curve) and T = 3.4 (dashed curve).  3.00 -  2.00 -  Figure 4.11: Large distance behavior of go(r) for the separable system at T = 2.8, which corresponds to the isotropic-nematic phase transition temperature.  • / • V? ' Chapter 4.' Computer Simulations II  . 116  0.95 n  0.75 -  v.  -0.05 • 111 0.000  1111111  T=4.0  2.00  4.00  r  6.00  I ' l l '  8.00  10.00  Figure 4.12: Molecular orientational pair correlation function gi(r) vs r for the nonseparable system. The long and short-dashed curves are exponential and power law fits respectively. of the order parameter observed for that system. Figure 4.12 shows the behavior of g (r) 2  around the isotropic-nematic transition. A n exponential decay of the envelope (longdashed curve) is observed for T = 4.0 and T = 3.8. The decay is a lot slower at T = 3.6 and consistent with a power law r~ with n « 0.27 ± 0.04 (short-dashed curve). This n  suggests that the two-dimensional nematic phase exhibits algebraic order as predicted by harmonic theories. It is, however, also consistent with an exponential decay to finite value (0.39) as would be the case for true long-range order. The long-dashed curve for T = 3.6 corresponds to such a fit with a correlation length £ = 3.11. Nelson and Halperin [81] have shown that algebraic order in the M O F of a smectic-C layer induces algebraic order in the sixfold bond orientational field (BOF). There exist some differences between Nelson and Halperin's model and our simulation. Their theory applies to a onefold symmetric M O F (due to the in-plane projection of the tilted molecules) coupled to a sixfold symmetric B O F (arising from the triangular short-range  Chapter 4. Computer Simulations II  117  positional order), whereas our simulation is concerned with the coupling of a twofold symmetric M O F (due to the h <-• —n invariance of the pair potential) with a twofold symmetric B O F (arising from the rectangular short-range positional order). These differences are, however, qualitatively irrelevant for the phenomenon we are interested in. That is, there cannot be ordering in the M O F of a nematic fluid phase without having ordering in a B O F with reflection symmetry. The similarities between our two-dimensional liquid crystal and the predictions of Nelson and Halperin allow us to reinterpret the chain formation we described in Chapter 3 as a manifestation of the rapid build-up of correlations in the twofold B O F . These observations led us to carry out a calculation of a twofold version of the hexatic order parameter introduced by Nelson and Halperin in the context of two-dimensional melting of an isotropic solid [5, 10, 11, 12, 90, 91]. We define the twofold B O F correlation c (r) as [92]: 2  c (r) = (* (r)*;(0)) 2  (4.11)  *2(*o = 5 i ; « p ( 2 t " M * o )  (- )  2  with  where ^ ( r )  1 S  *  n e  4  12  l ° l order parameter of the twofold B O F , and the sum in 4.12 is c a  taken over the bond angles 0,j of the two nearest-neighbors (see bond construction in Figure 4.6). Below the transition, these two neighbors are the most probable neighbors that form the two lowest energy links in the chain-like structures. Figure 4.13 illustrates this correlation function. The envelope of c (r) decays exponentially at T = 4.0 and 2  T = 3.8, but the decay is much slower at T = 3.6. The short-dashed curve is a power law fit (r~ ) with n = 0.39 ± 0.05, and the long-dashed curve is a fit to an exponential v  decay to finite value (0.15) with correlation length £ = 2.63 as would be the case if there was true long-range order in the B O F .  Chapter 4.  118  Computer Simulations II  0.55 -  0.35 -  -0.05 ' ' 0.00 0  11111111111 M  2.00  111  T=4.0 i I ' '  4 4.00  rI  I " I I I I I I I I I I I I) I I I I I I I  11111111111111111111111111  r  i  I  6.00  8.00  I  10.00  Figure 4.13: First nearest-neighbor bond orientational correlation function C2(r) vs r for the nonseparable system. The long and short-dashed curve are exponential and power law fits respectively. We have also measured an order parameter, $ 2 1 1 (^2J.(r)^2x( ))) u  f°  r t  n  e  a  n  a  the correlation function, c i(r) = 2  third and fourth nearest neighbors, which are on average per-  pendicular to the director (Figure 4.14). The same behavior as for c (r) was observed, 2  indicating that the transverse BOF is locked perpendicular to the MOF below the transition and that the chains are not free to slide parallel to each other. We found, however, that the transverse correlation function, C2i(r), has an amplitude smaller than c (r). 2  This is because bonds along the chain are much stronger than those perpendicular to it (see Section 4.2). An anisotropic liquid crystalline phase where quasi long-range order in the BOF is driven by the quasi-long-range ordering in the MOF is called a locked phase, and the theory of hexatic films predicts that the exponents  T\BOF  and T}MOF should be the same [81].  The discrepancy between the two exponents in our nonseparable system could be due to  Chapter 4. Computer Simulations II  119  0.35 - i  0.25 -  ol 0.15  u  :  :  0.05 -  -0.05 0.00  TTT  I I  I  2.00  I I I I I I I I I  I  4.00  I I I I I I I I I  I  6.00  I  8.00  10.00  r  Figure 4.14: Third nearest-neighbor bond orientational correlation function C 2 i ( r ) vs r for the nonseparable system. One observes that the isotropic-nematic transition induces a locking of the transverse bonds. finite-size effects and short-range structural details. Since the theory is based on a continuum approximation which neglects these effects, it would be interesting to investigate a much larger system in order to reduce the importance of finite-size effects, and perform a detailed study of the two correlation functions to check the theoretical predictions. The short-range structural details in the correlation functions can be removed by Fourier transforming the data, filtering the high frequency modes and transforming back to real space. See, for example, reference [93] where such technique has been successfully used. The separable system also exhibits an isotropic-nematic phase transition. In Figure 4.15 we show in the orientational correlation function, <72(r), both above and below the isotropic-nematic transition. The envelope of gi(r) decays exponentially for T = 3.2, but decays much more slowly for T = 2.8 and T = 3.0, and is consistent with a power law decay. Snapshots of the particle configuration (see Figure 4.7) reveal that the separable  Chapter 4. Computer Simulations II  120  0.80 -i  0.60 -  r  Figure 4.15: Molecular orientational pair correlation function <72( ) vs r for the separable system. The long and short-dashed curves are exponential and power law fits respectively. r  system has a B O F with sixfold symmetry as in an isotropic system [5, 10, 11, 12, 90, 91]. To reflect this sixfold symmetry, we replace the 2 in Equations 4.11 and 4.12 by a 6 (c6(r)) and take the summation over the first six nearest-neighbors (see Figure 4.7). This definition corresponds to the standard definition of the hexatic bond order parameter for isotropic two-dimensional fluids [90, 91]. Figure 4.16 shows the behavior of C6(r). The envelope of Ce(r) decays exponentially with distance both above (T = 3.2) and below (T = 2.8) the isotropic to nematic transition with correlation lengths of 2.20 and 3.30 respectively (long-dashed fits). The sixfold B O F of the separable system exhibits short-range order in the nematic phase. The increase in the correlation length is due to the build-up of bond correlations as one gets closer to the melting point and has nothing to do with any symmetry breaking coupling between the M O F and the B O F [81]. It thus appears that the separable system exhibits an anisotropic fluid phase with ordering in the M O F but exponentially decaying correlations in the B O F .  121  Chapter 4. Computer Simulations II  0.45 :  0.35 -.  0.25 -.  0.15  0.05 ;  I -0.05 0.00  T=3.2  •| I •! T I I T I I I I8.00 III  2.00  M4.00 I  6.00 I  r  1  It  I10.00 I I  1  I T I t TTTT T T I T | rj  Figure 4.16: Bond orientational correlation function c$(r) vs r for the separable system. The long and short-dashed curves are exponential and power law fits respectively. At low temperature the separable system freezes into a perfect triangular solid with nematic orientational order and with the director orientation independent of the lattice vectors. We have not studied the details of the melting transition, but it is tempting to invoke the two stage melting theory [5, 10, 11, 12] and then to interpret the isotropicnematic transition as a third Kosterlitz-Thouless transition [6, 7] involving the unbinding of vortices of charge 1/2 in the M O F [94]. One should be able to tune the transition so that algebraic order in the M O F appears at any point in the phase diagram by varying the size of the anisotropic constant J in Equation 3.21. The existence of a solid phase introduces a low temperature bound to the presumed quasi-long-range behavior of the nonseparable system. The solid phase displays true longrange order in the B O F [9, 10, 11, 12, 81, 89], and the presence of B O F - M O F symmetry breaking terms in the pair potential therefore implies true long-range order in the M O F at low temperature (see Equation 3.22). This is not the case for the separable system which displays a M O F with algebraic order at any non-zero temperature, even in the  1  122  Chapter 4. Computer Simulations II  solid phase.  4.6  Defect Strutures in the Nonseparable System  The theory of Nelson and Halperin [81] relies on the relevance of the defect density fugacity of the B O F , the M O F , and the B O F - M O F coupling. The isotropic to nematic transition would, according to their theory, be driven by the unbinding of M O F defect pairs of charge 1/2. In order to check if the isotropic-nematic transition for the nonseparable system is compatible with such an unbinding mechanism, it is useful to identify M O F defects (vortices in the director field). Clearly, there are several ways in which this can be achieved; we have chosen to proceed in a way similar to Frenkel and Eppenga [69] where the isotropic-nematic transition was interpreted in terms of the unbinding of director defects. We introduce at every point f a tensorial field, S p(f), defined as: a  S p(f) = a  E  ~  [2n.«(*)*tf(*) - 6 \ aP  (4.13)  where w(f — f , ) is a weight function chosen such that the nearby molecules dominate the field at f . We choose a Gaussian weight function: w(f — fi) — exp [—|f — f , | / c r J 2  2  (4-14)  with a = 1.0 in units of r . We then determine $ ( f ) , the orientation of the local 0  eigenvector of S p{r) (director), at a grid of points that form a square lattice with spacing a  S = 1.0 in units of r . Disclinations are identified by observing the change in $ ( f ) on 0  going clockwise around an elementary plaquette of grid points. A positive (negative) disclination of the nematic director corresponds to a net change in $ of +7r (—ir). The method we choose to calculate the director field suffers from some degree of arbitrariness, but it is still interesting to compare snapshots of defect structures for  123  Chapter 4. Computer Simulations II  various temperatures that have been obtained the same way, that is for a particular choice of w(r — fi), a and a. Some typical defect structures are illustrated in Figure 4.17. It is difficult to make a direct comparison between these results and the theory; it is, however, interesting to observe that, as predicted by theory, half-integer defects are rare and tightly bound below T = 3.4, but proliferate and unbind above that temperature, where all our other thermodynamic quantities and correlation functions indicate that the isotropic to nematic phase transition occurs.  4.7  Liquid-Like Behavior of the Nonseparable System  One could argue that the intermediate phase we observe for the nonseparable potential corresponds to a phase separated region (partially melted solid) since the simulation was performed at constant density and not at constant pressure. A melange of solid rectangular islands with long-range B O F and M O F , and isotropic fluid regions with short-range order in both fields could, presumably, exhibit properties analogous to a truly homogeneous anisotropic mesophase. Similar arguments [5] have cast very serious doubts upon the results obtained from constant-density computer simulations of the Lennard-Jones system which were originally interpreted in terms of the predicted hexatic phase [90, 91]. We argue that our results are not subject to this criticism. Firstly, notice that the large distance behavior of go(r) for the nematic phase of the nonseparable system (Figure 4.10) decays to its limiting value of 1, as expected for a liquid phase, within six molecular diameters. Monte Carlo trajectories over 10,000 MCS  in the intermediate phase did not show  isolated solid-like and fluid-like patches with regions of diffusion, as observed just above the melting temperature of the Lennard-Jones system [5, 91]. As well, our intermediate phase occurs over a relative temperature range ((T/vj — TMeit.)/TMeit.) that is over an  Chapter 4.  Computer Simulations II  124  T-3.6  T-4.0  +  - +  + •-  +  • +  +-  - •+  +-  T-3.4  T-3.5  +-  - ++-  +  +  Figure 4.17: Typical M O F defect structures for the nonseparable system. Positive and negative disclinations of the discretized director field are indicated by + and - respectively.  125  Chapter 4. Computer Simulations II  order of magnitude wider than the relative temperature range over which an intermediate anisotropic (hexatic-like) region is observed in the Lennard-Jones system [5, 90, 91]. Finally, the fact that we were able to reproduce the same statistics in the middle of the intermediate phase starting either from a random or an ordered configuration (Section 4.2) indicates that the nematic phase is not a superheated state. We thus believe that the intermediate phase we observe in the nonseparable system is a true mesophase and not an artifact due to an unusually wide anisotropic solid-isotropic liquid coexistence region.  4.8  System-Size Dependence of Q  One finds, using a simple elastic Hamiltonian describing only fluctuations in the M O F , that the order parameter Q decays as a simple power law with the size of the system, ruling out long-range order [74, 69]. To test this prediction for our nonseparable system, we have investigated the system size dependence of Q and present here our preliminary results. We point out that extremely long simulation runs are needed to obtain reliable statistics in Q once nematic ordering has set in the system. If we assume, as did Frenkel and Eppenga [69], that the typical relaxation time, r , scales as N , we find that, for the 16002  particle system we investigated at T = 3.4, T > 2 • 10 MCS 6  or 3.2 • 10 configurations 9  (see Section 4.2). This estimate indicates that reliable results on the size dependence of Q for our nonseparable system are very difficult to obtain with available computers and without using sophisticated cluster simulation algorithms . We are not aware of the ex3  istence of such algorithm for a disordered system and with an f-dependent Hamiltonian. Hence, if one looks at independent systems of different sizes, one observes very large 3  There  has  been  much  interest  recently  m o d e l [95],  for s y s t e m s w i t h a H a m i l t o n i a n  a n d for the  xy model  [97].  in  developping  fast  simulation  a l g o r i t h m s for  the  Potts  h a v i n g g l o b a l (either discrete or continuous) s y m m e t r y  [96]  Chapter 4. Computer Simulations II  126  0 . 7 4 -»  N=1600  O U - 0.44 -  o  \i  i '' ii i.,  :  1  N=100 0.34 0.00T-r  i i i I i  50.00  100.00  T—i—j—n—i—i—i—r—i—i—i—f-1—i—i—r-i—r  150.00  Time ( x 1 2 5 0 MCS)  Figure 4.18: Time dependence of the M O F order Q for two different system sizes. The data were accumulated at T = 3.4 over a total run time of about 190,000 MCS. The solid (dashed) curve is for a subsystem size of 1600 (100) particles. and uncorrelated fluctuations in Q which makes it difficult to extract any meaningful information. We have even observed that, for long periods of time (t > 50,000 MCS), a small system could have a Q smaller than the Q of a bigger system. B y taking a large system and splitting it into smaller subsystems one finds, however, that the fluctuations are correlated on different length scales (Figure 4.18). The technique described in the previous paragraph has been used in the past with some success [69]. We have used this method and studied subsystem sizes (N ) of 1/16, s  1/8, 1/4, 1/2 and 1/1 of a 1600-particle system at T = 3.4 and p = 1.00. The results are shown in Figure 4.19 where the dashed line corresponds to the best power law fit Q(N,) = ctNs . Clearly, the huge error bars prevent us from concluding anything P  definitive at this time. However, we mention that we have done a calculation using a simple model for B O F - M O F coupling 4  The  results of this calculation  are  4  presented  which shows that Q(N ) does not decay as a S  in t h e next  Chapter.  127  Chapter 4. Computer Simulations II 0 . 7 *i  D CL i_  d)  X>  o °-5 -l-i 100  1  -i——i  1—i—i  i i|  ,  1000  Area of subsystem  Figure 4.19: Subsystem size dependence of the M O F order parameter (Q(N,)) as a function of the subsystem area (in units of r ,). The dashed-line corresponds to the best straight line fit to the data. 2  simple power law at intermediate length scale, but in a way similar to the one seen in Figure 4.19 if the error bars are disregarded. This model is described in the following chapter. Finally, we note that Frenkel and Eppenga have also observed such positive curvature in In [Q(N )] vs \n(N ) for both the hard needle and hard ellipse fluids [69]. a  4.9  3  Summary  We have performed Monte Carlo simulations on a separable and a nonseparable pair potential and found that both exhibit an isotropic-nematic phase transition. Their structures are, however, extremely different. In the nonseparable case we observe a very strong coupling between the rotational and translational degrees of freedom leading to an orientationally ordered liquid crystalline phase, with a twofold bond orientational field (BOF) and an underlying rectangular symmetry. Both the molecular and bond orientational fields are ordered on a scale of  Chapter 4. Computer Simulations II  128  the size of the system. Furthermore, the nematic phase exhibited by our nonseparable system is clearly reminiscent of the locked smectic-C hexatic phase predicted by Nelson and Halperin's model of thin smectic films [81] where the B O F ordering is also driven by the ordering in the molecular orientations. Hence it is very appealing to interpret our mesophase as a rectangular analogue of this locked hexatic phase with quasi-long-range order. The separable system is much simpler. It behaves qualitatively like a planar spin system [6, 7, 98] displaying quasi-long-range order in the molecular orientations with short-range positional and bond orientational order. At low temperature it freezes into a perfect triangular lattice as for an isotropic system.  If the melting of this system  proceeds via a two stage melting [5] we expect that, due to the very weak coupling between the molecular anisotropy and the translational degrees of freedom, it would exhibit three Kosterlitz-Thouless transitions [6, 7, 8]; the phase transition in the molecular orientational field ( M O F ) driven by the unbinding of vortices of strength 1/2 [94] and occuring independently of the melting. Our results lead us to conclude that the coupling between M O F and B O F invoked in continuum models of liquid crystals [81, 82, 83, 84] has its microscopic origin in the intrinsic nonseparability of realistic intermolecular pair potentials. In the separable case there exists no such bare coupling and this is the reason why the B O F remains shortrange below the isotropic-nematic transition. We also suggest that the theory of Nelson and Halperin is a first approximation to a two-dimensional liquid crystal film where any twofold distortion is ignored and the sole effect of the nonseparability is to couple the M O F to an otherwise perfect sixfold symmetric B O F . One difference between our simulation and the theory is that our B O F has rectangular symmetry whereas it is triangular in the theory. The symmetry of the B O F depends on the values of the parameters a, 0 and 7 in Equation 4.6. In the weakly nonseparable  Chapter 4. Computer Simulations II  129  limit (small 7) we would expect the B O F to remain locked to the M O F and to have an almost perfect sixfold symmetry with a small underlying twofold distorsion. We note that X-ray scattering experiments performed in three dimensional nematics reveal the existence of an outer diffuse ring [23, 22, 24] modulated in intensity. The modulation is a manifestation of the uniaxial B O F whose long-range order is driven by a coupling to a M O F with long-range order [99] (see Chapter 2 and Figure 2.3). It would thus be very interesting to extend the work presented in this section to three-dimensional and other two-dimensional liquid crystal models where a nonseparable potential would, presumably, also lead to a coupling between the B O F and the M O F . Finally, we would like to emphasize a very important point. By comparing the behavior of separable and nonseparable system we have been able to draw a clear picture of what is implied by the mean-field approximation 2.14. By neglecting the correlations between molecular orientations and the orientation of the intermolecular vector, the mean-field theory for nematic liquid crystal transforms an initial microscopic nonseparable potential into an effectively separable one. By showing that separable systems do not exhibit induced bond orientational ordering, we have implicitly shown that the mean-field approximation (Equation 2.14) amounts to ignoring the phenomenon of induced bond orientational ordering in nematic liquid crystals. Having observed the phenomenon of induced bond orientational ordering, it is natural to ask if the B O F is just a boring slave field locked to the M O F , or if its coupling to the M O F introduces new and interesting effects otherwise absent when the presence of a B O F is ignored. This is the question we shall investigate in the following two chapters. Having argued in this chapter that an induced B O F ordering is a direct conseqence of the nonseparability of the pair potential for an ordered nematic liquid crystal, the above question regarding the importance of B O F - M O F coupling in liquid crystals can be regarded as a rephrasing of the two original questions we asked at the end of Section 2.2.2  Chapter 4.  Computer Simulations II  concerning the importance of nonseparability in the isotropic-nematic phase transition.  130  Chapter 5  B O F - M O F Coupling and Director Fluctuations in 2D Liquid Crystals  In the previous chapter, we showed how the ordering in the molecular orientational field (MOF)  induces an ordering of the bond orientational field (BOF) when the pair potential  is nonseparable.  It appears that the B O F would be a slave field, locked to the M O F ,  and not playing much of a role in the thermodynamic behavior of the system. It is the purpose of this chapter, and the following one, to show that the presence of a B O F - M O F coupling can significantly alter the thermodynamic behavior and the static field response of liquid crystalline systems with bond orientational ordering. In this chapter, we show that induced nearest-neighbor bond orientational ordering present in some two-dimensional (2D) liquid crystal systems modifies the usual logarithmic infrared divergence of director fluctuations. It is found that the presence of such an induced bond ordering introduces a crossover length scale between two different asymptotic regimes with fluctuations increasing logarithmically with the system size in both regimes. This suggests that, for liquid crystals where induced bond ordering exists, the interpretation of finite-size scaling results, in terms of a model that considers only fluctuations in the molecular orientation, is questionable [100].  131  Chapter 5. BOF-MOF Coupling and Director Fluctuations in 2D Liquid Crystals  5.1  Nature of the Ordering in 2D Liquid Crystals  It was argued in Chapter 1 that most (2D) systems with continuous symmetry lack longrange order at any non-zero temperature. In a seminal paper, Kosterlitz and Thouless [6] suggested that these systems have, however, a transition from a low temperature phase that exhibits quasi-long-range order to a high temperature disordered phase. In the low temperature phase, the relevant correlation functions decay as a power law rather than exponentially to zero with distance. We already mentioned in Chapter 3 that, as in other 2D systems with continuous symmetry [74], one finds that long-wavelength elastic distortions of the director field  1  destroy the long-range molecular orientational order in 2D liquid crystals [2]. This can be seen by considering the following elastic Hamiltonian which describes the collective fluctuations, <f>(f), away from the average molecular orientation:  ^  = \j  d  2  r  K  ^m\  (5.i)  2  where KM is an effective Frank elastic constant for distortion in the M O F . T is the temperature, and k the Boltzmann constant. B  The most general form for H would involve two bare elastic constants K  M  and  K  M  associated with distortions parallel and perpendicular to the director. Nelson and Pelcovits [101] have shown, however, that these two constants renormalize to the same value, KM, at large length scales. For smaller systems, where the distinction between K  M  K  M  and  is still meaningful, most of the predictions obtained from Equation 5.1 are still valid  provided one introduces an effective elastic constant *We  loosely  refer  orientation of the  here  to  the  molecules and  director do not  as  the  unit  vector  necessarily reserve the  KM in  = the  {KM^M) ^ ' 1  direction of  2  average  in-plane  d e n o m i n a t i o n to nematic liquid  the  crystals.  Chapter 5. BOF-MOF Coupling and Director Fluctuations in 2D Liquid Crystals  From the above elastic Hamiltonian, one finds that the mean-square of the fluctuations, (<f> ), diverges logarithmically with the size L of the system: 2  where r is a typical intermolecular spacing. Other quantities which are also often cal0  culated in computer simulations of 2D liquid crystals are the order parameters Q . 2  n  These measure the amount of orientational ordering present in the system and vanish as a power law with the system size: = (cos(n^)) = {Llr )- l * »  Q  n2  n  where n is an integer.  A  (5.3)  K  0  Although long-range orientational order is absent, the above  elastic description of director fluctuations can be shown to lead to power law decay of the orientational correlation functions [74]. Hence, 2D liquid crystals exhibit quasi-longrange orientational order and one expects the predictions of Kosterlitz and Thouless [6] to apply to these systems [94]. As the temperature is increased, topological defects (vortices) appear in tightly bound pairs [7, 8] and renormalize (reduce) KM (see Chapter 1). The predictions of the defectfree elastic theory remain valid in the low temperature phase and at very large length scales provided one uses the renormalized elastic constant  K  M  \  In contrast to the above predictions, Straley [75] has shown, using a different approach, that true long-range orientational order in 2D liquid crystals is rigorously excluded only if the pair potential, V^-, between molecule i and j is independent of the orientation, of the intermolecular vector f,j: V(fi ;<f> ,d> ) = V(\f \;{ l> -<i> }) . j  2  N o t i c e t h a t the  variable Q  we  i  j  ij  <  i  (5.4)  j  c a l c u l a t e d i n C h a p t e r s 3 a n d 4 is the  s a m e as Q  2  here.  Chapter 5. BOF-MOF Coupling and Director Fluctuations in 2D Liquid Crystals  <f>i is the orientation of molecule i with respect to a fixed axis, and |f,j | is the magnitude of f^. This is a separable potential. We have argued, however, that realistic intermolecular potentials are nonseparable. Although the existence of long-range orientational order in 2D liquid crystal systems of particles interacting via a short-range nonseparable potential remains an open theoretical possibility, the results of recent experiments in various nonseparable 2D liquid crystals [78, 79, 80, 88] have been very successfully explained by continuum theories that forbid long-range order. Hence, it would be most surprising if any 2D liquid crystals would exhibit true long-range molecular orientational order. One might think that computer simulations could help to resolve this issue. To demonstrate the lack of long-range order (or the existence of quasi-long-range order) with a Monte Carlo or molecular dynamics simulation, one must show that for the 2D liquid crystal model under investigation, the system size dependence of (<f> ) and Q 2  n  behaves  as predicted by the elastic theory (Equations 5.2 and 5.3). Unfortunately, the evidence from computer simulations on nonseparable liquid crystal models is most controversial and there is no concensus at present as to the nature of the ordering [45, 69, 76, 92, 102] (see Chapter 4 for a description of the various computer simulation results). In the previous chapter, we studied a system of particles interacting via an anisotopic van der Waals potential [60]. We found that this nonseparable potential exhibits a 2D nematic phase where the ordering in the molecular orientation induces an ordering of the orientation of the nearest-neighbor bonds with rectangular symmetry. Induced bond ordering has also been observed in various experiments on two [80, 86, 88] and three-dimensional [85, 87] tilted hexatic smectic films and explained in terms of the Landau theories [81, 82, 83, 84] which invoke phenomenological B O F - M O F couplings described in Chapter 4. The results from our computer simulation [92, 102] have led us to conclude that these couplings have their microscopic origin in the nonseparability of the intermolecular pair potential, and that induced bond ordering should be observable  Chapter 5. BOF-MOF Coupling and Director Fluctuations in 2D Liquid Crystals  in a large number of nonseparable 2D liquid crystal models. 5.2  B O F - M O F Coupling Model  Theoretically, it is now well established that to fully characterize a 2D isotropic fluid, one must introduce a sixfold or hexatic B O F describing the local orientation of the six nearest-neighbor bonds [5]. We have shown in Chapter 1 how successfully this concept was used in classifying the various smectic phases. In 2D liquid crystals, one has to consider all the relevant couplings between the B O F and the M O F that are allowed by the molecular and local bond symmetries. Following such a prescription, Nelson and Halperin [81], and more recently, Selinger and Nelson [83, 84], have investigated an effective Hamiltonian functional describing the coupling between the in-plane projection of the molecular tilt orientation of smectic-C molecules, (f>(f), with the orientation of a local nearest-neighbor B O F , 0(f), having sixfold symmetry. For convenience, we rewrite here this Hamiltonian described in Chapter 4: =  -Jd r  l  [K \V0(f)\ -rKM\^m\  2  2  B  oo  + ^(f)-V<f>(f)]  .  / d r cos{6n [0(f) - <f>(f)}} .  - £ h  (5.5)  2  6n  n=l  2  J  The sixfold periodic coupling term between bond and molecular orientations arises because either 0(f) or <j>(f) feels a sixfold symmetric potential when rotated with the other field held fixed. KM is the effective Frank elastic constant for fluctuations in the molecular orientation. KB is the stiffness constant for fluctuations in the bond orientation. A gradient cross-coupling term proportional to g is present in the most general situation. Nelson and Halperin [81] were the first to recognize that the presence of the h  6n  BOF-  M O F coupling terms induce bond ordering in a smectic-C film or, in other words, that a smectic-C phase is a hexatic phase [80, 82, 85, 88]. We refer the reader to Chapter 4 for  Chapter 5. BOF-MOF Coupling and Director Fluctuations in 2D Liquid Crystals  a description of this phenomenon. In such a phase, the correlation function C*_(r) = (exp{Qi [8.(f)  - MO)]}) ,  (5-6)  where 0-(f) = 9(f) — <f>(f), measures the relative orientation between the bonds and the in-plane projection of the molecular tilt, decays exponentially towards a nonzero value at large distance r. Such behavior is characteristic of a locked phase where the order in the nearest-neighbor orientation is induced by the molecular orientational ordering, and where Cg_(f) has long-range order. Most computer simulations on 2D nonseparable liquid crystals have been performed on nematic models [45, 60, 69, 76, 92, 102] where the field, <f>(f), has a periodicity of 7r instead of 2n as in the case of smectic-C molecules. We remind the reader that thin smectic-C films are analogs of the xy ferromagnet model described in Chapter 1: the pair potential does not display the h <-• —n symmetry as in the case of two-dimensional nematic models . The model of Nelson and Halperin [81] remains valid for these systems 3  and one concludes from it that a 2D nematic phase with quasi-long-range order in <f>(f) should induce ordering in a nearest-neighbor B O F with reflection symmetry. However, we showed in the previous chapter that this conclusion does not apply to separable liquid crystal models where there exists no bare B O F - M O F coupling [92, 102]. A modified version of Hamiltonian 5.5, which describes a locked phase in a nonseparable 2D liquid crystal is: ~  =  yd r[K \V9(^\ -rK \S7cl>(f)\ 2  2  B  M  h J <fV{0(r) - c6(r)}  2  t  2  + 2gV9(f).V<f>(f)} + (5.7)  where hi is an effective B O F - M O F coupling in the locked liquid crystalline phase. For simplicity, we take the case where the fields are locked parallel to each other. Also, we A s well as a nonseparable nematic model, Tobochnik and Chester [45] have also investigated a separable model of a 2D smectic-C liquid crystal. 3  Chapter 5. BOF-MOF Coupling and Director Fluctuations in 2D Liquid Crystals  only consider a highly ordered phase in which disclinations in 8(f) and vortices in <f>(f) are rare and can be ignored. Equation 5.7 simplifies upon defining the fields 0±(f) [81, 83, 84]: 6 (f) = aO(f) + P<f>(f) +  6 .(f) =  (5.8)  6(f)-<j>(f)  where  In terms of 0±(f), the diagonalized Hamiltonian reads: ^  = Jd r 2  [K+\W (f)\ + K.\V6.(f)\ + M K * ) ] 2  2  +  (5.10)  where K+ = K  B  +K  M  + 2g  and .  K. = ° "-? K  K  (5.11)  The transformation 5.8 is interesting since it diagonalizes 5.7 by expressing it as a free Hamiltonian in 0+ and a massive one for 0_ . Physically, this allows one to interpret 4  0+(f) and 0-(f) as the acoustic and optical modes of the system respectively [83, 84, 88]. The Hamiltonian 5.10 is quadratic, and fluctuations in the B O F and the M O F , as well as their order parameters, can easily be evaluated; the details of the calculations can be found in standard textbooks [74, 103]. The fluctuations (</>) are: 2  (W  ( l  ^  )  =  /(ft  with (IMfll ) = ^ 2  4  The  choice 8 =  » d (IMfll ) = ^ T 2  I  T  t  •  (5.13)  1 — /? i s c o n v e n i e n t s i n c e t h e J a c o b i a n o f t h e t r a n s f o r m a t i o n h a s a b s o l u t e v a l u e 1.  Chapter 5. BOF-MOF Coupling and Director Fluctuations in 2D Liquid Crystals  Integrating 5.12, one obtains:  1+h N~ + h  (5.14)  l  where we have defined  h,r  2  h=  (5.15)  with N = (L/r ) , the number of particles in the system. The order parameters Q 0  n  are:  -n o*/8irK_  Q  n  = (cos(n<£)) = / V - " / 8 ^ 2  +  1+ h N~ + h_  s  (5.16)  1  Notice that if h = 0 and g = 0, one recovers Equations 5.2 and 5.3 from Equations 5.12 and 5.16 respectively.  The averages (6 ) and (cos(n#)) are given by Equations 5.14 2  and 5.16 respectively, provided the change a —• /? is made.  5.3  Results  In a nematic system exhibiting a hexatic phase, the renormalized elastic constants, and K  M  \  K$  are equal to 12/T: and 8/ir respectively at the decoupled multicritical fixed  point [94, 81, 12]. This point is the location where the hexatic-isotropic (sm-B# - sm-A) and nematic-isotropic phase transition boundary meet in the K$ — K ^ phase diagram. M  Figure 5.1 shows the system size dependence of the fluctuations (c6 ) for various reduced 2  B O F - M O F coupling strengths and where we have set KB and KM to the above critical values, and g = 0. Figure 5.2 shows the system size dependence of the M O F order parameter Q (Q = Q ) for various reduced B O F - M O F coupling strengths and for the 2  same values of KB, KM, and g = 0 as in Figure 5.1. Clearly, the director fluctuations for a nonseparable system, where induced bond ordering is present, do not increase logarithmically with the system size. This is unlike what is predicted by Equation 5.3. A different choice for KB, KM and g would not give qualitatively different results.  Chapter 5. BOF-MOF Coupling and Director Fluctuations in 2D Liquid Crystals 0.06 -a 0.05 -.  0.04  -e-  o.o3 -. 0.02 0.01 0.00  Figure 5.1: System size dependence of the reduced molecular orientational fluctuations 4irK (<f> ) as a function of N for various strengths of the reduced B O F - M O F coupling parameter h. 2  +  1.000  0.995 -  0.990 -  Q 0.985 -  0.980  0.975 -  0.970 -j—i 1  ti-0.01 i 11 i n n — 10  H I — i  i  100  11  mil 1000  i—i 10000  mil 100000  I I I  N Figure 5.2: System size dependence of the M O F order parameter Q as a function of the size of the system, N, for various strengths of the reduced B O F - M O F coupling parameter h.  Chapter 5. BOF-MOF Coupling and Director Fluctuations in 2D Liquid Crystals  Two asymptotic regimes, linear in ln(A ), are observed. For small length scales (small r  TV), both the acoustic and optical modes are strongly fluctuating and (4> ) increases 2  rapidly with the system size. However, the optical mode must relax towards a constant value since the B O F - M O F locking interaction, hi, induces long-range order in C$_. Hence, one observes a reduction of (<f> ) for a crossover system size N of the order ~ ( £ / r ) 2  o p  c  ~ K-jh\r\ ~ h~ , where £ l  op  0  is the correlation length of the optical mode. We find that  for a reduced coupling strength h ~ 10~ — 1 0 , curvature in (</>) and in \n(Q ), as a 3  _1  2  n  function of ln(iV), is observable for system sizes that are typically investigated in finitesize scaling studies of nonseparable 2D liquid crystal models (iV ~ 10 —10 ) [45, 69,102]. 1  3  c  Let us briefly summarize the major points we have discussed in this chapter: • We assumed that a harmonic description of B O F and M O F fluctuations in 2D nonseparable liquid crystal models is valid. Hence, we find that such models should exhibit quasi-long-range order. • We assumed that in the anisotropic phase of a nonseparable system there is an induced ordering of the B O F , and Equations 5.14 and 5.16 are the ones that apply. • The predictions of Equations 5.2 and 5.3 apply only for separable systems, where induced bond ordering is absent. These predictions are quite unambiguous and we now test them against the finite-size scaling results that are available in the literature. As predicted, one finds that a linear dependence of (G4 ) vs \n(N) is observed in the 2  data for the separable system studied by Tobochnik and Chester [45] (Figure 5.3). The potential they used is very similar to the separable potential we used in Chapter 3 and in Chapter 4: V(r) =  4e/(r)[l - p cos{<f>i - fa)] with  2  Chapter 5. BOF-MOF Coupling and Director Fluctuations w±2D Liquid Crystals  —i  30  1  i  i  1  i_  4.0 50 6 0 7.0 80 In H  Figure 5.3: System size dependence of (4> ) for a separable system. The results are from reference [45]. fcsT/e = 1.6, pr] = 0.856 (p is the number density) and p. = 0.80. System sizes of N = 16, 64, 256, 1024 and 4096 particles were investigated. 2  r„  12  T V  6  /(O = (7) - (r f) r where r = |fj — r_,|, t > 0 and p. > 0. Frenkel and Eppenga [69] performed some scaling studies of Q for 2D hard-needle 2  and hard-ellipse fluids. Their results were presented in the form of tables; we reproduce them graphically in Figures 5.4 and 5.5. These two fluids are nonseparable systems and one might suspect that the presence of a B O F may invalidate the interpretation of the results in terms of the simple model (5.2 and 5.3). The behavior observed in Figures 5.4 and 5.5 is to be compared with that shown in Figure 5.2. Finally, we recall the results that we have obtained for the system size dependence study of Q (Q = Q2) for the nonseparable system (Figure 4.19). There also, a curvature in the data was observed. The presence of an induced bond ordering and of the resulting B O F - M O F coupling in the above nonseparable systems could be responsible for the  Chapter 5. BOF-MOF Coupling and Director Fluctuations in 2D Liquid Crystals  p=8.75  /o=7.50  p=7.00  2  4 5 6 7 8 9  100  J 4 5 6 7 8 9  2  3 4  1000  N u m b e r of P a r t i c l e s ( N ) Figure 5.4: System size dependence of Q for a hard-needle fluid. The results are from reference [69]. Calculations were done for systems of number density p = 8.75, 7.50 and 7.00. Systems with N = 50, 200, 800 and 3200 needles were studied.  Number of Particles (N) Figure 5.5: System size dependence of Q for a hard-ellipse fluid. The results are from reference [69]. The aspect ratio of the ellipses is 6.0 and the number density is p = 0.6p (po is the density of close packing). Systems with N = 46, 186, 744 and 2976 ellipses were investigated.  0  Chapter 5. BOF-MOF Coupling and Director Fluctuations in 2D Liquid Crystals  observed curvature. This interpretation is further substantiated by the fact that the data for a separable system [45] (Figure 5.3) do not show any curvature.  5.4  Discussion  We have shown that induced nearest-neighbor bond ordering in a nonseparable 2D liquid crystal modifies the usual logarithmic infrared divergence of director fluctuations. As a consequence, the power law decay of the order parameters with the system size is also modified. The presence of bond ordering introduces a crossover system size N and two c  different asymptotic logarithmic regimes. Our results suggest that the interpretation of finite-size scaling studies of 2D liquid crystals is more involved than predicted by a simple model that ignores B O F - M O F coupling and fluctuations in the orientation of the nearest-neighbor bonds. We propose that to clearly show quasi-long-range order, a computer simulation of a 2D nonseparable liquid crystal system should demonstrate whether induced bond ordering is present or not. When bond ordering exists, finite-size scaling investigations should be performed for system sizes well above the crossover system size N . However, in the case c  of weak B O F - M O F coupling, this crossover system size could be as large as N  c  ~ 10  3  particles, making the finite-size scaling study of director fluctuations very difficult with currently available computers. A word of caution is needed here.  The model we have presented is based on a  harmonic approximation, Equation 5.7, and applies only when slowly varying fluctuations of <f>(f) and 6(f) are present. In other words, it applies in a well-ordered liquid crystalline phase. In a finite system, especially of the size investigated in computer simulations, non-harmonic corrections will become important as the temperature is raised. Such non-harmonic corrections can manifest themselves in two ways: via interactions between  Chapter 5. BOF-MOF Coupling and Director Fluctuations in 2D Liquid Crystals  director fluctuations of different wavevectors , or via the presence of defects, like vortices 5  in the M O F (see Chapter 1). In the Kosterlitz-Thouless theory, the vortices are assumed to have a large core and the elastic deformation of the director field outside the core of the vortex is the only one that is considered. A renormalization calculation [7, 8], which applies to large systems (thermodynamic limit), shows that the main effect of the vortices is to reduce the effective elastic constants KM and KB- However, in a system size of 0(1O ) particles, the presence of even a small number of bound vortex pairs will have 3  a disastrous effects on the behavior of the orientational fields if the core of the vortices, where the harmonic approximation breaks down, is sufficiently large. These remarks apply to both separable and nonseparable systems. We expect that both B O F - M O F coupling effects and non-harmonic corrections will contribute to reducing the applicability of Equations 5.2 and 5.3 to finite-size studies of 2D liquid crystal models. We believe, however, that it would be a very difficult task to isolate these two effects in a computer experiment.  5  T h i s is a n a l o g o u s t o i n t e r a c t i o n s b e t w e e n s p i n - w a v e s i n a H e i s e n b e r g  ferromagnet.  Chapter 6  B O F - M O F Coupling and Response to an External Field: The Freedericksz Transition in Tilted Hexatic Liquid Crystals  In the previous chapter we showed that the coupling between the B O F and the M O F modifies the finite-size scaling behavior of nonseparable 2D liquid crystals. It is natural to ask if other properties of liquid crystals are modified when the presence of a B O F coupled to a M O F is considered. In this chapter, we investigate how the response of a tilted hexatic liquid crystal film to an external field is affected when the system exhibits bond orientational ordering as in smectic-C, smectic-I and smectic-F phases. We shall concentrate on effects induced by magnetic fields in order to clearly expose the new effects introduced by the presence of a B O F - M O F coupling and to avoid the mathematical burden and complications arising when electric field induced distortions are considered [104].  6.1  Response of a Liquid Crystal Material to an External Field  Liquid crystals are usually diamagnetic, and the diamagnetization is particularly strong in aromatic molecules. Consider Figure 1.1. When a magnetic field is applied normal to a benzene ring, a current of electrons builds up inside the ring and generates a magnetic moment which reduces the flux going through it; the lines of magnetic field are thus expelled and this raises the energy of the system. When the benzene ring is parallel 145  Chapter 6. Freedericksz Transition in Tilted Hexatics  146  to the field, the lines are practically undistorted and the magnetic energy density is not raised. Hence, the molecules tend to orient themselves in a way to minimize the magnetic energy increase, that is, with their benzene ring(s) parallel to the field. For most liquid crystal molecules, this corresponds to a favored orientation of the long molecular axis parallel to the field (see Figure 1.1). We mentioaned in Chapter 2 that samples of liquid crystal material can be oriented by chemically treating the walls of the container. A n external field can be applied and tuned such to be perpendicular to the undistorted orientation of the molecules. This introduces a competition between the orientation imposed by the walls and the one favored by the magnetic field. Hence, one sees that there is a possibility that a liquid crystal sample uniformly aligned between parallel plane surfaces can undergo a transition to an elastically deformed state when subject to an external field larger than some critical value. This transition was first observed by Freedericksz and Zolina [105], and has been the subject of considerable study in the past fifty years. In this chapter we are interested in the response of tilted hexatic liquid crystals to an external field. Tilted hexatic liquid crystals are characterized by the tilt angle of the molecules with respect to the layer normal, and by an ordered B O F which describes the orientation of their nearest-neighbor bond structure (see Chapter 1). The B O F does not couple directly to the external magnetic field, but interacts with the M O F which is directly coupled to the magnetic field. We will show that the presence of a B O F can drastically influence the field response of the M O F of a tilted smectic film. Until now, most experimental and theoretical work has concentrated on nematic liquid crystals (see Frisken [104] and references therein). There have been a few theoretical and experimental investigations of the Freedericksz transition in smectic-C phases [2, 106, 107, 108, 109], but we are not aware of any work that has addressed the question of the  147  Chapter 6. Fr6edericksz Transition in Tilted Hexatics  role played by an underlying induced B O F on the Freedericksz transition in this system . l  6.2  Model  Consider a tilted hexatic phase where the layers are parallel to the x — y plane with the layer normal along z (see Figure 1.7). We assume that the magnetic field, H, is applied in the y direction and that the boundary, at y = 0 and y = L are parallel to the x — z plane and treated in such a way that the director, N, is constrained to be in the orientation  u(y = 0) = u(y = L) = u> 4>{y = 0) = <f>(y = L) = 0  (6.1)  where the orientation of N(y) is N(y) = {sin(w) cos(<f>), sin(a>) sin(^), cos(u;)} .  (6.2)  u is the tilt angle and <f> is the azimuthal angle which indicates the orientation of the in-plane projection of the tilt. We shall assume that the system is far from a nematic-smectic-A transition, and that the layers are incompressible, hence the layer spacing and tilt angle are constant. The azimuthal angle is the soft mode of the system and its originial direction (<p = 0) can be modified by applying an external magnetic field. The magnetic field, parallel to y, induces a mixed bend-splay elastic distortion of the in-plane tilt projection, c6(y), whose M O F elastic free energy density F  F and K  h  M  M  - = \ [ dy K {V • Nf + K (N xVxN)  asi  M  K  - is given by [2]:  ast  M  b  M  M  2  (6.3)  are the splay and bend elastic constant respectively [2] and A is the area  of the cell walls parallel to the x — z plane. We also assume that a B O F is present in 1  R e c a l l that the  s m e c t i c - C p h a s e is  a tilted hexatic phase (Chapters 4 and  5).  148  Chapter 6. Freedericksz Transition in Tilted Hexatics  the x — y plane and write a simple phenomenological, one-elastic constant free energy density for B O F distortions, F  - (see Chapters 4 and 5):  last  B  2  ^  = 2/0  d  y  °  K  dO(y) dy  (6.4)  We assume that it is energetically favorable for the system to have a B O F parallel to the cell walls at y = 0 and at y = L. This can be understood in terms of the packing constraint of the molecular cores which arise at the solid-liquid interface. Henceforth, we set 0(0) = 9(L) = 0. We choose a B O F - M O F coupling analogous to Equation 5.5:  F -M  = Ah  B  2n  rl j dy cos [2n{<f>(y) - 9(y)}] Jo  where we have imposed inversion symmetry for the interaction F -MB  (6.5) h  —  2n  ^6^,3  corresponds to the original coupling proposed by Nelson and Halperin [81]. In a binary liquid crystal mixture of smectic-C materials, the B O F in the present description would play the role of the M O F of the other component, and the term h  2n  = h 6 ,i would 2  n  correspond to the coupling between the two M O F s . The magnetic free energy density, ^ jr^ag.  =  m a g  _M(A> . H)  2  - , is given by: - %± H -(N2  H)  (6.6)  Introducing a new reduced variable, u = iry/L, we rewrite the total free energy of the system as  fo =  2F  —  „,  + Xi#  2  „, 2A r 2h = — f du F(u)  +  2n  L  7T  = | -2 + —2 -h  2n  cos  W') ) dy  v  K  JO  ) + KB(^) 2  [cos(2n<£ - 2n9) - 1]  2  +  Y(COS(2<^) - 1)  149  Chapter 6. Fre'edericksz Transition in Tilted Hexatics  where W  = ( |,-X±)# sin (u;)  K  =  2  2  X  AK =  Ki =  sin (w) K  K -K 1  2  = (j) K sin\u )  2  2  2  b  M  J  K  B  = {j)K  B  (6.7)  and /o = 0 in the undistorted state. 6.3  Results  The state of the system which is realized is such that /o is minimized with respect to  (j>(u) and 0(u). The functions (j>(u) and 6(u) which minimize fo are the solutions of th Euler-Lagrange equations  *- -£ =0 du dq dq da  (6.8)  d  where q = 6 or <j> and q = dO/du or d<f>/du. This results in two coupled, nonlinear second order differential equations. Since the M O F and B O F are anchored at y = 0 and y = L, one has, by symmetry, the maximum distortion in the two fields at y =• L/2 in the elastically distorted state. Hence, the boundary conditions are  6(u = 0) = 4>(u = 0) =  dO_ = 0. du u = i r / 2 d4  du tl=7T/2  = 0.  Three cases are easily solved: 1. h  2n  = 0. In this case, the Euler-Lagrange equations are decoupled and the solution  corresponds to the usual Freedericksz transition where the presence of a B O F is ignored. One finds that for W < W , the state of the system is with no distortion c  in either the B O F or the M O F . For W > W , the angle of maximum distortion in c  Chapter 6. Freedericksz Transition in Tilted Hexatics  150  the M O F at y = L/2, </> , increases continuously from zero while the B O F remains m  undistorted (0(y) = 0); this is the usual second order Freedericksz transition [106, 107, 2, 108, 109]. We shall not discuss this situation any further. 2. hin = oo. Here the B O F is locked to the M O F since it would take an infinite amount of energy for the system not to have 9(y) — <j>(y) for all y. This problem reduces to the previous one with a second order Freedericksz transition since the net result of the infinite B O F - M O F locking is just to increase K —• K + KB3. The limit K  B  —• oo is interesting and corresponds, as a first approximation, to a  system which is deep in the tilted hexatic phase (eg. sm-I or sm-F). In this case, one has 6(y) = 0 for all y since it would cost an infinite amount of B O F elastic energy if it were otherwise. A first integral of the resulting differential equation for <j>(y) can be found and the final solution expressed in terms of an elliptical integral. Although exact, this approach is not very instructive at this point of the discussion since we have very little feeling of what the physical phenomena involved in the Freedericksz transition in tilted smectics are. Hence, we shall, instead, first solve the problem by using the approximate ansatz described below. The general case is usually refered to as a two point boundary value problem. In the present situation this turns out to be a difficult problem to solve and one must resort to a numerical investigation using an iterative procedure [110]. As in item 3 above, such a solution, even when available, is of little use if one does not have some physical insight into the problem. A slowly varying approximate (SVA) solution of the form  <t>(u) = <f> sm(u) m  0(u) = e sm(u) m  (6.9)  Chapter 6. Freedericksz Transition in Tilted Hexatics  151  leads to a better understanding of the problem than the an 'exact' numerical solution. This ansatz fulfills the above boundary conditions. <f> and 6 are the 'order parameters' m  m  of the problem. They are both zero in the undistorted state, and acquire a non-zero value in the elastically distorted phase when the effective field W is increased beyond some critical value W . Using the above ansatz for 4>(u) and 6(u), Equation 6.7 is integrated c  over du to give  / = ^  +^ ^ +^  2 n  [W ) m  + J (2^ )] + 2  m  W — \M24>m) - 1] - 2h2n [J (2n{<£ - 6 }) - 1] 0  TO  m  (6.10)  where J is the ordinary Bessel function of order p. 6.3.1 Infinite B O F Elastic Constant p  Consider case 3 above where lim KB —* oo and 6 = 0. It is instructive to construct a m  Landau free energy by expanding fo in powers of <p . Up to fourth order, we find m  /  0  =  (Ki-W  +A n ^ ^  + iW-AK-An^n^JS  In the absence of a B O F - M O F coupling (h  (6.11)  = 0), we recover the results of previous  2n  calculations [106, 107, 2, 108, 109] where the Freedericksz transition, in the geometry considered here, was found to be second order when the presence of an intrinsic (sm-I, sm-F) or induced B O F (sm-C) is ignored. We find, however, that the B O F - M O F coupling can drive the Freedericksz transition first order provided that An h (n - 1) > K 2  There is a tricritical point at h  2n  .  2  2n  2  (6.12)  = K {4n (n — l ) } . The state of the system, within 2  - 1  2  2  the S V A approximation, is given by dfo/d<j> = 0. We have solved this equation with m  Chapter 6. Freedericksz Transition in Tilted Hexatics  152  KB — oo. Figure 6.1 shows some typical results for <f> as a function of applied field, W, m  for various hexatic B O F - M O F couplings h (see Chapters 4 and 5). 6  It is interesting to note that first order Freedericksz transitions have been observed in systems where two (static and/or optical) fields are applied, where feedback is present, and in systems where the alignment of the molecules at the boundaries is tilted. In these examples, the appearance of a first order Freedericksz transition is due to the presence of another aligning field which competes with the external applied field. In the present case, it is the B O F anchored at the boundaries which plays the role of the 'internal' aligning field. The tricritical point is found by setting an equality sign in Equation 6.12.  One  cannot have a first order Freedericksz transition if n = 1 since K > 0 by definition. As 2  mentioned earlier, this situation corresponds to a simple model of a binary liquid crystal mixture where the component which plays the role of the B O F has zero anisotropic magnetic susceptibility. The location of the line of second order transition, hence the limit of stability of the uniform state, and the location of the tricritical point, are predicted exactly by the above SVA. At the first order phase transition boundary, the system experiences a jump in <f> . m  The results of the SVA are inaccurate as the jump in <f> becomes large. As mentioned m  above, this problem could be solved in terms of elliptical integrals and one could compare the exact solution with the above Landau expansion or with the full SVA solution (Equation 6.10). This would give some indication of the range of validity of both the Landau expansion and the full SVA ansatz. This type of approach has been investigated by Frisken [104] for systems in crossed magnetic and electric fields. It was found that, although the Landau expansion and the full SVA solution are inaccurate deep in the first order regime, the full solution does not introduce any new or interesting physics which is not already predicted by the approximate solutions. Since we are not interested  153  Chapter 6. Freedericksz Transition in Tilted Hexatics  0  Figure 6.1: Freedericksz transition in tilted hexatic phase with an infinite B O F elastic constant. The squares, triangles, crosses and stars are for h = 0, 10~ , 5 x 10~ and 10~ respectively. We have set AK = 0 and w and h are measured in units of K\. First (1st) and second (2nd) order transitions occur at the locations marked by the arrows. The minimum free energy state of the system follows the solid line upon an increase of the magnetic field w. 3  6  2  6  3  154  Chapter 6. Freedericksz Transition in Tilted Hexatics  in investigating the location of the first order phase transition boundary, we shall not pursue the infinite KB limit any further, but look instead at the more realistic case with  K  ±00.  B  6.3.2  Finite B O F Elastic Constant  We showed earlier that the infinite h  2n  case is trivial since the the B O F and the M O F  must remain locked together (0(y) = <t>{y)). Due to this locking mechanism, the B O F and the M O F become indistinguishable and one recovers the original Freedericksz transition problem for a single effective M O F in an external magnetic field for which the Freedericksz transition is second order. We found in the previous section that if the limit KB —* 00 is taken first, the transition is first order for all h  2n  of h  2n  larger than h , where h 2n  2n  is the value  at the tricritical point. These observations lead us to suspect that there might be,  for finite KB, a reentrant second order transition regime on the phase boundary as h  2n  is increased beyond a value h  2n  = h . 2n  To qualitatively illustrate this effect we perform an expansion to first order in h /K . 2N  B  The effective M O F free-energy is, to first order in h\JKB, given by  W  1 - 1  (6.13) Expanding 6.13 up to fourth order in <f> , we find m  fo =  (Ki-W + 4n h -l6n hlJK )<f> j2 2  4  2n  (W-AK-4n h 4  +ten«h JK )<j> j8. 2  2n  +  2  B  4  B  (6.14)  The limit of stability of the undistorted state with <f> — 6 = 0 is, up to order h /(KB), m  m  2n  155  Chapter 6. Freedericksz Transition in Tilted Hexatics  given by  ' _ An h 2  W = e  1  (6.15)  2n  K +4n h 2  2n  KB  This is to be compared with the exact solution obtained by locating the change in sign of one of the eigenvalues of the quadratic expansion of fa  K  + An h 2  B  2n  We find that, up to fourth order in c6 , one has a first order transition if the coefficient m  of the fourth order is negative at W  c  or when  An h (n - 1 ) 4 - 1 6 ^ ^ ( 1 - An ) > K 2  2  .  2  2  2n  (6.17)  KB  Compare 6.17 with the infinite KB result (Equation 6.12). For h  2n  smaller than h% , one n  has a second order transition; this is the situation described earlier. For a B O F - M O F coupling with h  2n  larger than h , the M O F and the B O F are locked together and one 2n  recovers the second order Freedericksz transition as expected for strongly locked B O F M O F . For couplings with h  2n  < h  2n  < h , there exists a first order transition regime 2n  provided that  h  2n  and h  2n  are found by setting an equality sign in Equation 6.17.  Because of the approximation h /KB 2N  <C 1, the location of h% and h% is inaccurately  predicted and one must resort to a numerical investigation of Equation 6.10 via the full SVA solution. These results are presented in Figures 6.2, 6.3 and 6.4 which show the behavior of <f> , 6 and fo respectively, for various values of h& above h% as a function of m  m  the applied field W. The curves with the crosses represent all the solutions that we found by solving the equations  dfo _  df  0  Chapter 6. Freedericksz Transition in Tilted Hexatics  156  Figure 6.2: Magnetic field dependence of the MOF distortion, <f> , for a Freedericksz transition in tilted hexatics with finite BOF elastic constant. The values of /i6 were set to 0.1, 0.2 and 0.5 for the figures from top to bottom respectively. We have chosen AK = 0 and K = K. W and he are measured in units of K\. The crosses show all the states which are free energy extrema. First (1st) and second (2nd) order transitions occur at the locations marked by the arrows. The minimum free energy state of the system follows the solid line upon an increase of the magnetic field W. m  B  Chapter 6. Freedericksz Transition in Tilted Hexatics  157  Figure 6.3: Magnetic field dependence of the BOF distortion, 0 , for a Freedericksz transition in tilted hexatics with finite BOF elastic constant. Refer to Figure 6.2 for details. m  158  Chapter 6. Freedericksz Transition in Tilted Hexatics  — 1.50 111111i11 111' i ' i ' i ' ' i ' 111111111 II 11111111 i I I i ' 1.00 2.00 3.00 4.00 5.00  W  Figure 6.4: Magnetic field dependence of the free energy measured in units of K\ for a Freedericksz transition in tilted hexatics with finite B O F elastic constant. The various branches correspond to local extrema of fo which are observed in Figures 6.2 and 6.3 for h = 0.5. 6  The solid line shows the field dependence of the state with absolute minimum of the free energy / . The W — h phase diagram obtained by solving Equation 6.19 (3) numerically 0  6  is shown in Figure 6.5 and where we have set K = 1. For small applied field (small W ) , the system is in the undistorted state with <j> — 6 = 0. As anticipated, we find m  m  that as W is increased, the system initially undergoes either a second (solid boundary) or a first order Freedericksz transition (short-dashed boundary between point a and /3) depending on the value of he. Interestingly, we find that the phase diagram is much richer than expected. One observes that if he > h$ or if /t§ < he < ^6>  a  subsequent  first order transition order occurs if W is increased beyond some /^-dependent threshold (long-dashed boundary). The point c is a critical point as shown in the inset of Fig. 1. Since the two long-dashed curves meet the short-dashed one with a continuous slope at h% and Ag, one can regard these three first order phase transition boundaries as being a single first order boundary which extends from he = h% to he = oo.  Chapter 6. Freedericksz Transition in Tilted Hexatics  159  The subsequent first order transitions (long-dashed curves) correspond to reorientation transitions of the bond field with respect to the tilt field. As W is increased sufficiently far beyond the second order phase transition boundary (solid curve), the bond and tilt fields undergo a large elastic distortion. Since the bond field does not interact directly with the magnetic field, the system increases its bond elastic free energy by keeping a small angle between the tilt and bond orientational fields. At sufficiently large W, the magnetic interaction Tm becomes saturated and, since the bond distortion is considerable, the bond field 'springs back' to a small 0 , hence reducing its bond elasm  tic free energy. This is possible since the tilt can then reorient itself closer to one of the other sixfold symmetry axis of the bond field and the system does not increase its tilt-bond coupling free energy during the process. This interpretation is susbtantiated by observing, in Fig. 2, the behavior of (f> and 0 as a function of W for he — 0.5. Similar m  m  behavior for <f> and 0 is observed in the region h% < h$ < h%. m  m  The isolated first order Freedericksz transition boundary (short-dashed curve) is a remnant of the first order regime predicted above for h& >  in the infinite-reapproxi-  mation. Indeed, the dependence of <f> on W across this first order Freedericksz transition m  boundary for « = 1 is very similar of the one observed for h& > h when re = o o . 6  The structure of the phase diagram depends strongly on the value of re. A Sreis increased, the points h% and h% move to smaller and larger values respectively, and the length of the hi — h% branch decreases rapidly until it vanishes for re « 1.25. In smectic-C films, the hexatic bond ordering is entirely due to the molecular tilt and one has re 1. As K is decreased, the points h% and h$ move to larger and smaller values respectively, until the first order Freedericksz transition regimes in-between the two second order branches disappear. This occurs for re « 0.607. However, the first order tilt-bond reorientation transition above the second order Freedericksz transition is still present with c moving to smaller values of hi as re is decreased further.  160  Freedericksz Transition in Tilted Hexatics Chapter 6.  Figure 6.5: Phase diagram for Freedericksz transition in tilted hexatics withfiniteB O F elastic constant. Continuous lines correspond to second order phase transitions while the dashed line is afirstorder phase transition boundary. We have set AK = 0 and K — K and W and h are measured in units of K\. The values of h at alpha, 0 and c are hg, hi and h%. The values of h%, h% and h are 0.009, 0.153 and 0.007 respectively. (Note the logarithmic scale for h .) B  6  6  c  6  6  161  Chapter 6. Freedericksz Transition in Tilted Hexatics  6.3.3  N u m e r i c a l Investigation of the E u l e r - L a g r a n g e E q u a t i o n s  We briefly present some results which compare the SVA solution with a full numerical solution of the Euler-Lagrange Equation 6.8 which, for AK = 0 read - •• W K<f>+ — sin(2c6) - 2nh  2n  sm{2n<f> - 2n6)  K 6 + 2nh sm(2n<t>-2n6) B  =  =  2n  0  0 .  (6.20)  We have attempted to solve these coupled equations using the following Fourier series expansion: oo  <f>(u)  =  ^A(q)sm{{2q  + l)u}  oo  6(u)  =  £ £ ( ( ? ) sin{(2c/ + l)u}  .  (6.21)  9=0  We calculated the Fourier coefficients using an iterative procedure W [ -—- / du sin(gu) sin{2w >J Ai(k) sin((2g — l)u)} — litq Jo g>—i 2t  KAi+i(q)  =  /  ""9  K B (q) B  w  =  / ^o  —*r\  du sin( u) sin{2n T, {A,{k) - B,(k)} sin((2c/ - l)u)} 9  / = 1  sm(qu)sm{2nYi{Mk)-Bi(k)}sm((2q'-l)u)}  du  (6.22) where the index / in A/(c/) and B/(c/) means the / and B(l) correspond to <f> and 6 m  m  t h  iterate, The Fourier coefficients A(l)  respectively. It is easy to show that for A(q) = B(q)  = 0 for all q > 0, one recovers the algebraic equations obtained from Equations 6.19. This technique was found to converge rapidly for not-too-large values of W and h . 6  Figure 6.6 shows a comparison of the full numerical solution (square symbols) using the above iterative procedure with the SVA solution (crosses) for AK  = 0, K  B  = K and  Chapter 6. Freedericksz Transition in Tilted Hexatics  162  55.00 •  50.00 -  45.00 -  40.00 '•'5  1.25  , j  5  , ' . 4 5 l ' . 5 5 "  "  w  Figure 6.6: Comparison of 'exact' and SVA solution for a Freedericksz transition in tilted hexatics with finite B O F elastic constant. We have set AK = 0, KB = K and he = 0.02. W and h are measured in units of K \ . See text for details. Q  h = 0.02 (in units of K\). A first order transition is observed for W « 1.35 (dashed line). 6  The amplitudes A(3) and A(5) were found to be smaller than 1° for 1.35 < W < 1.60 so, clearly, the SVA solution is quite satisfactory. For large values of W or h , we found numerical instabilites which invalidate the use e  of the above iterative procedure. For example, at fixed /i6 (^6 = 0.02) and small W (W < 1.65), we could find an unambiguous and unique solution to the Euler-Lagrange equations. We observed, upon an increase of W, a doubling sequence in the number of solutions (1,2,4,8) up to W w 2.38 where the iterative procedure was found not to converge. We interpret this behavior as the well-known period doubling route to chaos (see Figure 6.7). The iterative technique we used is therefore inappropriate and one would have to devise a more subtle numerical technique in order to find a full solution of the EulerLagrange equations beyond the SVA approximation. It is often possible to find converging  163  Chapter 6. Freedericksz Transition in Tilted Hexatics  100.00 -i  20.00 :  o.oo  1.30  1.70  w  [ i i i i i i i i2.50 i i ii  0 2.10  Figure 6.7: Period doubling route to chaos for iterative solution of Euler-Lagrange equations with h = 0.02, AK = 0 and K = K. 6  B  iterative solutions of a given set of equations by rewriting the iterative map in a different form. We have applied this approach to Equations 6.20. Unfortunately, we could only find one stable fixed point corresponding to the undistorted state with <f> = 0. Nevertheless, m  we believe that the main features of the system are explained by only considering the SVA solution and we shall not pursue the present numerical study of Equations 6.20 any further. 6.3.4  Application to a Well Characterized Hexatic Material — 80SI  We conclude this chapter by investigating whether the behavior predicted for the Freedericksz transition in tilted hexatics is experimentally accessible. The important parameter to calculate is the B O F - M O F coupling, he. We are aware of only one material, abbreviated as 80SI, in which he has been determined [111]. 80SI exhibits a smectic-C to smectic-I  164  Chapter 6. Freedericksz Transition in Tilted Hexatics  (hexatic) phase transition. Using a phenomenologic parametric equation of state, Garland et al. [ I l l ] have estimated the value of A to be 6  h  (80si)  w  1 0  _  5 r c  w  i  t  h  j,^ ^  3  5  Q  (6.23)  R  This gives a value h  « 5 x 10~ J/molecule .  (6.24)  26  SOSI) 6  Assuming a molecular weight of 300(7 and a density of the order of lg/cm , we get a 3  B O F - M O F coupling free energy density of the order of h  &OSI) 6  « 5 x 10" J/molecule 26  W ^ m « 100J/m 3  3  .  (6.25)  Typical values for molecular Frank elastic constants are [2] K  M  « 10  - 6  - 10- dyne « 10" 7  - 10" A^ .  11  12  (6.26)  Finally, we find that the quantity we are interested in, h^/Ki, is « 10 - 100 (I//xm)  2  (6.27)  K\ where the length of the cell, L, is expressed in pm. In two-dimensional tilted hexatics, the bond and tilt elastic constants are  72kBT/ir  and 2A:BT/TT respectively at the decoupled fixed point [81]. For 3D hexatics, we also expect K = KB/KT  > 1. Hence, for definitiveness, we fixe « = 36. Assuming a tilt angle  u « 20° and typical magnetic anisotropic susceptibility [2], we find that for a thickness L « lpm, the system is at or very close to the interesting region near 0 with a magnetic field strength, H, of the order of 1 to 10 Tesla. It was assumed in the above calculations and numerical estimates that a strong anchoring of tilt and bond fields prevailed. Previous experiments have shown that it is possible to realize a strong anchoring of the molecular tilt in smectic-C liquid crystals  Chapter 6. Freedericksz Transition in Tilted Hexatics  165  for the planar geometry we consider [109]. However, to impose strong anchoring on the bond field, interfaces with bond orientational order at y = 0 and y = L are required. Monocrystalline boundaries could be used to impose strong anchoring of a well defined bond orientation. Instead of the planar geometry studied here, one could consider the ./V—geometry, first discussed by Rapini [107] and later by Meirovitz [108], in which the anchoring of the B O F and the B O F is at z = 0 and z = L. This geometry would allow to take advantage of the fact that free interfaces of suspended smectic liquid crystal films tend to favor more ordered tilted smectic phases [4, 112, 113]. For example, the formation of hexatic smectic-I layers at the free interfaces of smecticC films has been observed [112]. Further, recent experiments have shown that the rotational viscosity in thin smectic-I films is much larger than in the smectic-C phase [78, 88]. A n experiment in which a smectic-C film is slowly cooled (annealed) in a strong magnetic field would lead to free interfacial smectic-I layers with inner smectic-C layers with both well defined tilt and bond orientation. At the same time as the annealing field is turned off, a field applied parallel to the layers could induce a dynamical Freedericksz transition of the inner layers before the outer smectic-I layers could respond to the field which induces the dynamical Freedericksz transition.  6.4  Discussion  In this chapter we have investigated the response of tilted hexatic materials to an applied magnetic field, concentrating on the Freedericksz transition. We have found that the presence of B O F ordering can drastically modify the behavior of the response of the system under an applied field. Rough estimates suggest that some tilted hexatic materials could exhibit the phenomena described in this chapter.  166  Chapter 6. Freedericksz Transition in Tilted Hexatics  The B O F - M O F coupling strength is material-dependent and the interesting parts of our phase diagram in Figure 6.5 could be investigated in experiments performed with materials with a different h . For example, previous X-ray diffraction experiments [114] e  in a material abbreviated as 70.7 showed no induced B O F ordering in a smectic-C phase, hence a smaller he coupling than for 80SI. Experimental results from Freedericksz transitions in tilted hexatic would be of considerable interest since they would provide a new experimental way to determine important parameters, such as KT, K  B  help to characterize these systems.  and h , and 6  Chapter 7  Towards a Microscopic Model for the Smectic-I - Smectic-F Transition  In Chapter 4 we found that a strong nonseparability of the pair potential can induce a considerable amount of bond orientational ordering in a liquid crystal phase with an ordered M O F . We argued that the presence of induced B O F ordering is an intrinsic feature of liquid crystal phases where the particles interact via nonseparable potentials. Although it appeared, at first sight, that the induced B O F in these systems behaves as a locked slave field, we showed in the previous two chapters that the presence of a B O F can significantly modify the finite-size scaling behavior of 2D liquid crystals (Chapter 5) and the magnetic field response of tilted smectic liquid crystals (Chapter 6). Our approach, in Chapters 5 and 6, was to investigate the effects of a B O F on common properties of liquid crystalline systems where the presence of such bond ordering is usually neglected. In this chapter, we do the reverse: we investigate whether the presence of a M O F coupled to the B O F introduces interesting effects in systems with predominant B O F ordering and in which the M O F would seem to play an unimportant role. As an example of this type of investigation, we propose a microscopic model for the smectic-I - smectic-F (I-F) transition in a freely suspended hexatic liquid crystal film. We suggest that the transition between these two hexatic phases can be easily explained by assuming that it is driven entirely by an increase in the molecular tilt angle, which 167  Chapter 7. Microscopic Model for the Sm-I - Sm-F Transition  168  plays the role of the M O F in the present problem. We present ground state calculations which show that our model exhibits two types of zero temperature close-packed phases, which we call 'I-like' and 'F-like' phases, depending on the favored molecular tilt direction with respect to the nearest-neighbor bond. We refer to them as 'I-like' and 'F-like' phases since they are solid phases, and not true liquid crystalline phases. Assuming that the tilt angle increases upon a decrease of temperature, our results provide an explanation for the experimental observation that the smectic-F phases always occurs at a lower temperature than the smectic-I phase [4]. As well, we argue that the transition exhibited by our model is not confined to a solid phase and that its presence could be investigated for a true liquid crystalline phase with a computer simulation.  7.1  The Smectic-I - Smectic-F Transition  Before deriving our microscopic model for the I-F transition, we present a brief description of this transition. Some of the hexatic smectics that have recently been studied are tilted phases [85, 80, 88] (see Chapter 1 for description of hexatic smectic phases). These can be characterized by the relative orientation of the in-plane projection of the tilt with respect to the nearestneighbor bond (see Figure 1.12). The in-plane molecular projection can be along the nearest-neigbor bond as in the smectic-I phase (0°), or halfway between nearest-neigbor bonds as in the smectic-F phase (30°). In all cases where both phases are present, the smectic-F phase is the most ordered one [4] since it occurs at a lower temperature. This is a feature that any microscopic model of the I-F transition should be able to reproduce. In a recent experiment, Dierker and Pindak [88] observed a first order I-F transition in a thermotropic liquid crystal material they studied. Using a Landau theory, Selinger  169  Chapter 7. Microscopic Model for the Sm-I - Sm-F Transition  and Nelson [83, 84] were able to explain this first order I-F transition. This is the model we described in Equation 4.5 and in Equation 5.5. For a strongly first order I-F transition, one can neglect the fluctuations in <f>(r) and 0(f) and consider only the periodic coupling. The phase of the system is such that the B O F - M O F coupling energy, VB-M{0 — 4>),  minimized with respect to the B O F - M O F  13  offset (6 — <f>). Selinger and Nelson considered only the first two harmonics with h and e  hu in Equation 4.5: V -M(0-<j>) = -hecos{6(6-<f>)}-h cos{12(e-<f>)} B  Taking h  i2  12  .  (7.1)  > 0, they find that for h > 0, the minimum energy state is with 6 — <f> = 0 e  which is a smectic-I phase. For h  i2  > 0 and h < 0, the minimum energy state is with 6  $ — (ff = 30" which is a smectic-F phase. A first order phase transition separates the two states at h& = 0. Unfortunately, the microscopic origin of the phenomenological coupling between the M O F , <f>, and the B O F , 0, invoked in Equation 7.1 is not known. In particular, one would like to find a microscopic model which exhibits a I-F transition and for which the couplings he and h\ can be evaluated and check if a I-F transition does indeed 2  occur when he changes sign while h\ remains positive. This would allow a direct check 2  of Selinger and Nelson's model and largely improve our understanding of the hexatic smectic phases.  7.2 7.2.1  Model for the I-F Transition and Intermolecular Pair Potential Preliminaries  It is difficult to construct microscopic models for systems as complicated as hexatic liquid crystals. The presence of quasi (in 2D) or true (in 3D) long-range bond orientational  Chapter 7. Microscopic Model for the Sm-I - Sm-F Transition  170  order is a feature not easily incorporated into a microscopic model. In fact, we are not aware of any successful attempts in that direction. The I-F transition can be considered as a 'sort o f structural transition between two phases that both have the same very long distance behavior of their bond orientational correlation function. Since the main feature of this transition is a reorientation of the tilted M O F (see Figure 1.12), it seems plausible that the microscopic explanation for the transition lies in the competition between different nonseparable interactions present in the system. This is the point of view we shall adopt here. We indicated in Chapter 3 that one should not use lattice models with nonseparable interactions to model nematic liquid crystals since the underlying lattice breaks the continous rotational symmetry of the high temperature isotropic-like phase. In a hexatic phase, the system exhibits bond ordering and often has a positional correlation length larger than 20 to 50 molecular diameters [4]. The presence of such 'long' short-range positional order in these systems suggests that a lattice model for these systems might be a valid starting point for modelling the I-F transition. This provides a motivation for considering a model for the I-F transition where the particles are arranged on a perfectly regular lattice. Selinger and Nelson [84] have recently proposed such lattice model as a method to calculate the couplings constants appearing in their Landau theory for the I-F transition. In Chapter 2 we found that a large amount of physical insight can be gained by first considering the ground state orientational behavior of system of particles interacting via nonseparable potentials. This is also the approach we shall adopt in this chapter where we investigate a simple nonseparable lattice model which exhibits both an 'I-like' and an 'F-like' phase, depending on the value of some parameters to be defined below. We shall then argue that such a microscopic model may, in some circumstances, exhibit the equivalent I-F transition, but in a true liquid phase.  171  Chapter 7. Microscopic Model for the Sm-I - Sm-F Transition  At a more practical level, this lattice-model type of approach is useful since ground state calculations are orders of magnitude faster and cheaper to perform than Monte Carlo or molecular dynamics simulations, and can quickly reveal a potentially interesting behavior of the system in its liquid crystalline phase. 7.2.2  Interactions between Tilted Molecules  Steric Interactions The molecules are tilted with repect to the layer normal in both the smectic-I and the smectic-F phases (see Figure 1.7). We assume that we are far from the smectic-A —• smectic-C transition [4] and that the fluctuations of the tilt angle, a;, are negligible. Hence, all the molecules have the same UJ. Since we are trying to understand the I-F transition with the simplest model possible, we take for granted that we have a tilted phase and do not question the microscopic origin of the tilt . From now on, the tilt 1  angle, u>, will be considered an independent variable which, in a true liquid crystal, is a measure of the temperature of the system. The motivation for this assumption is that, in real liquid crystal systems, the tilt increases as the temperature is lowered and going from the smectic-C phase to the smectic-I phase [4, 116]. Furthermore, we shall neglect the coupling between smectic layers and consider a single liquid layer of tilted rods. Assuming a single layer is an uncontrolled approximation and its importance should be carefully investigated in later studies since it has been experimentally demonstrated that, for a given material, the smectic-I phase is not present in thick smectic films [117] First, we take a simple case where individual molecules can be modeled by hard 1  Once  such  sophisticated origin of the  'given-tilt' m o d e l  model  similar  to  for  the  tilt in tilted smectics.  the one  I - F transition has introduced  by  been devised,  McMillan  [115]  one  which  could introduce proposes  a  a  more  microscopic  172  Chapter 7. Microscopic Model for the Sm-F Sm-I Transition -  sphere-cylinders [36, 37, 38]. Since the tilt angle does not fluctuate, the effective interaction between the spherocylinders reduces to the interaction between their in-plane cross-section; the spherocylinders have become hard ellipses [76, 77]. The length of the major and minor axis of the ellipses are 2a/cos(u>) and 2a respectively, where a is the radius of the spherocylinder. The aspect ratio, k, is defined as k = l/cos(u>). This effective two-dimensional liquid crystal system is similar to a smectic-C phase, and one expects that it will exhibit induced bond orientational ordering [85, 80, 81, 83, 84, 82]. A perfectly oriented hard ellipse liquid crystalline phase is also realized by uniformly increasing (rescaling), in some direction, the diameter of hard discs in a dense fluid phase. This direction becomes the major axis of the ellipses and corresponds to the orientation of the tilt in the smectic plane. In a dense hard disc fluid phase, each disc has, on average, six nearest-neighbor bonds at about every 60°. We call this anisotropic phase 'I-like' or 'F-like' depending on whether tilt lies along or halfway between the original nearest-neighbor bonds of the original hard discs fluid . 2  Recall, however, that for our model of the I-F transition, we assume that the above liquid crystalline phase can be modelled as close-packed lattice model of tilted rods. This anisotopic solid is obtained by distorting a perfect close-packed lattice of hard-discs in the direction of the in-plane projection of the tilt. Having started with a perfect triangular lattice of hard-discs, the resulting distorted lattice is the closest-packed lattice one can build-up with ellipses, whose packing fraction, and free-energy, is independent of the direction of the distortion (orientation of the ellipses). Hence, ground state calculations show that 'I-like' and 'F-like' lattices have the same free energy . Other forces must be 3  taken into account in order to lift the degeneracy between all the equivalent hard-ellipse 2  The  presence  structure of the 3  of  a  tilt  original  T h e s e results m a y  introduces  a  small  t r i a n g u l a r l a t t i c e [85,  twofold  distortion  which  alters  the  sixfold  symmetric  80].  indicate that i n d u c e d b o n d orientational order w o u l d be s m a l l in a liquid  phase of uniformly oriented hard  fluid  ellipses.  crystal  173  Chapter 7. Microscopic Model for the Sm-I - Sm-F Transition  configurations in the solid phase and induce either an 'I-like' or 'F-like' phase. A t t r a c t i v e Interactions - P a r t I The obvious candidates which might removes the degeneracy between all the above closepacked lattice of ellipses are isotropic attractive forces. We have calculated the ground states of a system of hard ellipses interacting with each other via an isotropic attractive van der Waals interaction of the form: Vij  = -r"  6  .  (7.2)  The distance, rjj, between the centers of mass of the ellipse i and ellipse j, is measured in units of the original triangular lattice spacing, a. The ellipses are uniformly aligned on a close packed lattice, and the total energy of the system is calculated as a function of the orientation of the ellipses with respect to the lattice orientation. We find that the interaction 7.2 lifts the degeneracy between all the equivalent close-packed lattices of hard ellipses and, independently of the tilt angle (u> > 0), the resulting ground state is 'F-like', with the major axis of the ellipse oriented halfway (30°) between two nearest-neighbors. These results suggest that an orientationally ordered 2D hard ellipse fluid interacting via attractive van der Waals forces would have a nearest-neighbor bond structure strongly reminiscent of a smectic-F phase, and not an 'F-like' solid phase, provided parameters such as temperature, pressure and aspect ratio of the ellipse are suitably chosen.  It  would be interesting to investigate this prediction via computer simulations. As well, these results suggest that competing interactions must be present if a 'I-like' phase is to exist.  174  Chapter 7. Microscopic Model for the Sm-I - Sm-F Transition  Attractive Interactions - Part II In the calculation above, we have assumed that the global effect of the attractive intermolecular van der Waals forces can be taken into account by introducing an effective source of van der Waals interactions at the center of each molecule (ellipse). We now improve this approximation by considering the presence of van der Waals centers of interaction (atoms) away from the geometrical center of the cylindrically symmetric molecule. This introduces an attractive nonseparable interaction in our model which already contains a sterically repulsive part (the hard ellipse part), and an isotropic attractive one (the — r~f term in Equation 7.2). These three competing interactions will prove to be sufficient to lead to both 'I-like' and 'F-like' ground states. We introduce an atom of type 'o', located at the geometric center of the the cylindrical molecule and on the symmetry axis (see Figure 7.1). As well, we introduce four other atoms, two of type ' a ' and two of type '/?'. The 'a-atoms' are located on the long molecular axis, parallel to n, and their position is ±r\\ with respect to atom 'o'. The ' d i atoms' are located on the transverse, or short molecular axis, h±, which is perpendicular to h, and their position is i r j . with respect to atom 'o'. The purpose of the 'o', ' a ' and '/?' atoms is to introduce several energy scales in the problem. Assuming that all atoms interact via isotropic van der Waals forces, as in Equation 7.2, we find that the total intermolecular pair potential, Vij, is given by:  (|f,j + 2 r | | n | | | ~ + \fij - 2r||7i|j|~ ) - t  -t  6  aa  -2e  0Q  (\rij  + niniil"6  6  + |r - - H i ^ t l l " ) ~ 6  0  2  c  (|r + 2r h \~ + | r - 2r±n±\~ ) 6  m  ° 0  ti  ±  9  ti  ±  + ^ - d '  +  6  ~  ^ " l l "  6  )  -2e 0 (\fij + (r||ft|| - r nx)|" + |f - - (r\\h\\ - r h )\~ ) 6  x  a  -2e  a(3  6  tJ  L  x  (\fij + (ryriH + r n i ) | " + | r - (r\\n\\-r r h )\~ ) 6  x  e  ti  r  ±  .  (7.3)  Chapter 7. Microscopic Model for the Sm-I Sm-F Transition  a)  175  b)  Figure 7.1: Schematic arrangement of atomic species on interacting tilted smectic molecules. Figure a) shows the long molecular axis, n, which makes an angle u with the normal of the smectic layer, z. h± is a unit vector perpendicular to h, and b is the in-plane projection of n and is in the direction of the major axis of the ellipse. Figure b) shows the locations of the van der Waals centers. The atomic species 'o' is at the geometrical center of the molecule while the atoms ' a ' and '/?' are located at a distance ry and r± from the o-atom, on the n and h± axis respectively.  Chapter 7. Microscopic Model for the Sm-F Sm-I -Transition  176  (•op, tea, (-00 and c 3 are, respectively, the strengths of the o - o, o - a, o — 0, Q/  a — a, 0 — /3 and a — j3 van der Waals interactions. 7.3  R e s u l t s and D i s c u s s i o n  It is instructive to expand 7.3 up to second order in r||/r,j and rj_/r,j. This shows that, at least up to that order, the anisotropic correction has the lowest energy for a configuration with two ellipses parallel to their intermolecular vector. Such anisotropic terms tends to favor an 'I-like' phase:  Vij = - [e + 4 (e + e p 4- e + epp)] rjf 00  oa  0  aa  -3[t + 2t + 2e 0]r ^8sin (u>)(e,- • S) - l ) r " .  8  -Z[t + 2e + 2t p]r\(8 cos (a;)(e, • bf - l ) r "  8  2  oa  aa  2  2  j  O  2  o0  pp  a  j  .  (7.4)  iij is a unit vector parallel to f,j, and b is a unit vector parallel to the in-plane projection of the molecular tilt and indicates the orientation of the major axis of the ellipse (see Figure 7.1). At this point we can already make two observations from Equation 7.4: • The intermolecular potential, Vij, is strongly attractive for e,j parallel to b and and favors an 'I-like' configuration. Hence one should be able to observe a competition between these terms and the isotropic o — o interaction which favors an 'F-like' state. • The contribution to Vij coming from the term proportional to rjj (second term) is much less important than the one coming from the term proportional to r  2  (third  term). For small tilt angle, the effect of the nonseparable e,j • b interaction in the second term is considerably reduced since it is multiplied by sin (u>). For large ry, 2  the ' a ' atoms are far from the V and '/?' atoms and the r  - 6  dependence in the full  Chapter 7. Microscopic Model for the Sm-I - Sm-F Transition  177  potential 7.3 kills its contribution. Finally, for large tilt angle, the sin (a>) term 2  may play some role. We found, however, that for the case we have concentrated on with all the t \> interactions being equal, the eccentricity of the ellipses is such a  that, when this happens, the system has already been driven to the 'F-like' phase by the o — o interactions. The ground states exhibited by potential V;J in Equation 7.3 were obtained in the following way: an arbitrary frame, x' — y', with the x' axis having an orientation <j> with respect to the x axis of a perfect triangular lattice (x — y frame) is chosen. The coordinates, ( i , j ) , of each lattice point in the x — y frame are calculated, and once their projection along the x' axis is known, it is increased by a factor k > 1. This produces a close packed lattice of hard ellipses oriented along x', at an angle <f> with the x-axis and with major (x ) and minor (y ) axis of length ka and a respectively. The energy of the 1  1  system (Equation 7.3) is then calculated as a function of <j> € [0,30°], for fixed k, ry and r±. Once the lowest energy state is found, the aspect ratio, k, is slightly increased, and the procedure is repeated. We present the results for a case where all the e„f, interactions are set equal. Figure 7.2 shows the r j. — OJ phase diagram we have obtained with the full interaction 7.4 and where the lattice sum was carried up to (i,j) = ( ± 8 , ± 8 ) . A different cut-off range does not alter the results significantly. r  x  is measured in units of the molecular radius, which is  half the lattice spacing of the original triangular lattice. The distance ry was set to 2 in the same units, but as discussed above, the results for ry = 1 and 3 were found to be very similar to the case with ry = 2. For small tilt angle (high temperature), the ground state is 'I-like' with the ellipses (molecular tilt) oriented along the nearest-neighbor bond of the original triangular lattice. At larger tilt (low temperature), the system favors an 'F-like' state and a 'first order'  Chapter 7. Microscopic Model for the Sm-I - Sm-F Transition  178  35-,  — Low  30-  '.P-like  25-  1  (  T  CD  O  Z5  20-  D <D CL  15-  £ '/-like  10-  CD 1  5-  1  0.00  ' '  i i ' M  0.10  I I i i i i i i i i i i  0.20  i ii  High  i i i i i i i i i i i i i  0.30  0.40  Figure 7.2: Phase diagram of close-packed ellipses interacting with distributed sources of isotropic van der Waals centers. The tilt angle, u>, is expressed in degrees, and r± is measured in units of the molecular radius. The dashed-curve corresponds to a 'first order' phase transition boundary between the 'I-like' and the 'F-like' ground states. The orientations of the tilt with respect to the nearest-neighbor bond of the original perfect triangular lattice is illustrated by the orientation of ellipses at the top right of the phase diagram, but where k has been exaggerated for clarity.  Chapter 7. Microscopic Model for the Sm-I - Sm-F Transition  179  phase transition separates the two states. Two points which deserve particular attention are discussed in the following two subsections. 'First Order' I-F Phase Transition Boundary In Figure 7.2, we refer to the boundary separating the 'I-like' phase from the 'F-like' phase as a first order phase transition boundary. This categorization is not totally adequate as we now explain. Since the smectic-I and the smectic-F phase have the same symmetry, one generally expects to find a first order transition separating these two phases. In most systems which exhibit a first order phase transition from one state to another, there exists a free energy barrier separating the two phases. This is not what we found when calculating the I-F boundary illustrated in Figure 7.2. We first explain what one would normally expect to occur close to the I-F boundary in a phase diagram such as the one in Figure 7.2. In the 'I-like' phase, but very close to the I-F boundary, one would expect an absolute energy minimum for \9 — <f>\ = 9- = 0°, a local minimum at 9- = 30°, and an absolute maximum for some 0- between 0° and 30°. As the system crosses the boundary, the state with 9- = 30° would become the absolute minimum, the state with 9- = 0° would then become the absolute maximum, and there would still be an energy barrier separating the two states. We found, instead, that in the 'I-like' phase, below the dashed-line, the energy is minimum for 9- = 0°  4  and increases monotonically ously up to 9- = 30°, with no  energy barrier between the state with 9- = 0° from the state with 0_ = 30°. This behavior persists until the boundary is reached. On the phase boundary, and beyond (larger w), we find that the energy is now a maximum for 6- = 0°, decreases monotonously up to 9- = 30°, with still no energy barrier separating the two states. The origin of this behavior lies in the discontinous 'stretching-unstretching-stretching' 4  B y  s y m m e t r y , t h e s t a t e w i t h 3 0 ° — \b\ h a s  the  s a m e e n e r g y as t h e s t a t e w i t h 3 0 °  +  Chapter 7. Microscopic Model for the Sm-I - Sm-F Transition  180  of the triangular lattice which we use to find the lowest energy state of the system. Due to the impenetrability of the hard-ellipses, we cannot perform a continuous transformation (rotation of the triangular lattice) from the state with 0_ = 0° to the state with 0_ = 30°. Although the procedure we use to cross the I-F boundary does not correspond to the path a real system follows when undergoing an I-F transition, the location of the I-F boundary we find is still meaningful, since we do find the lowest energy state of the system for a given u. Since the transformation we used to find the energy of the states with 0_ € [0°,30°] is not continuous, we cannot evaluate the h& and hi coefficients the B O F - M O F interac2  tion 7.1. This brings up the second point.  Success of the Present M o d e l and Suggested Improvements The inability to evalate h& and hi2 is a unfortunate drawback of our hard-ellipse + van der Waals interactions model. However, this model has allowed us to identify mechanisms which can lead to an I-F transition. Furthermore, assuming that the tilt angle increases with a decrease of temperature, our model provides an explanation for why the F phase occurs at a lower temperature than the I phase. A t a more pedagological level, our model could be considered as one of the few microscopic models for a liquid crystal phase in which competition between steric forces, isotropic attractive forces and anisotropic attractive forces is necessary to explain the transition of interest. Identifying some mechanisms which drive a I-F transition is a useful first step. The same kind of approach as the one outlined in this chapter could be taken with softellipses where off-center van der Waals sources of interaction would also be introduced. 5  5  Soft-ellipse m o d e l s have been p r o p o s e d i n the  p u t e r s i m u l a t i o n s [45,  46,  47].  c o n t e x t o f n e m a t i c s [43,  44]  and investigated in  com-  Chapter 7. Microscopic Model for the Sm-I - Sm-F Transition  181  Since they are not impenetrable, soft-ellipses would allow a continuous rotation from the state with 0_ = 0° to the state with 0_ = 30°. One could then calculate the coefficients of the Fourier series for the B O F - M O F interaction, such as the one described in Equation 7.1. Using such a microscopic model exhibiting an 'I-like' to an 'F-like' transition would provide a first check of the Landau theory of Selinger and Nelson [83, 84]. If the behavior of soft-ellipses with off-center van der Waals sources in the solid phase did exhibit an I-F transition, it would be interesting to investigate that system in a truly liquid phase.  Chapter 8  Nonseparability and Anisotropic Viscosities: Application to Viscous Fingering  In this thesis we have concentrated on the strong coupling between positional and molecular orientational degrees of freedom which originates from the nonseparability of the pair potential. We did, however, discuss in Chapter 2 that there exist more 'common' manifestations of the nonseparability than those we have investigated. For example, the difference in the splay, bend and twist elastic constants or the anisotropy of the viscosity coefficients are such common manifestations. We would like to conclude our study of nonseparable effects in liquid crystals by looking at more macroscopic 'liquid-like properties' of the nonseparability. To do so, we ask ourselves whether the nonseparability-induced anisotropy in bulk properties of liquid crystals introduces interesting effects in some phenomena where the presence of anisotropy plays a crucial role.  8.1  Importance of Anisotropy in Problems of Pattern Formation  Several problems of pattern formation within the Laplacian growth mechanism have been found to exhibit drastically different behavior when anisotropy is present then when it is not. The following examples are illustrative. 182  183  Chapter 8. Viscous Fingering  • Computer simulation of diffusion-limited aggregation ( D L A ) of large clusters of particles reveal that the anisotropy imposed by the underlying square lattice, on which the particles random walk, introduces a noticible fourfold symmetry to the D L A cluster [118]. In this case, one observes structures strongly reminiscent of the dendrites observed in crystal growth [119]. • It has recently been shown that the presence of crystalline anisotropy plays a fundamental role in selecting the growth velocity of a parabolic crystalline dendrite [120, 119]. • Finally, it has recently observed that the formation of viscous fingers in a HeleShaw filled with a liquid crystal material behave quite differently depending on the temperature at which the experiment is performed [121, 122, 123] For some range of temperatures, stable parabolic fingers are formed and do not tip-split as happens in experiments with isotropic fluids [124, 122]. The above examples clearly illustrate that anisotropy may play an important role in the Laplacian growth problem We have recently studied the range of applicability of the linear stability analysis in the problem of viscous fingering for isotropic fluids [125]. The results we obtained in [125] led us to propose a morphological phase diagram for the problem of viscous fingering using results from a linear stability analysis [126]. Knowing the importance of the role played by the anisotropy in the problem of pattern formation, we felt it was a worthwhile exercise to extend our previous results [125, 126] to a case where anisotropy in the viscosity is taken into account. A brief report of our results on the importance of 1  In  all the  system  we  described,  s o l u t i o n of w h i c h determines the growth.  one  has  local normal  to  solve a  Laplace equation on  g r o w t h velocity of the  a moving  interface, hence the  boundary,  name  the  Laplacian  Chapter 8.  184  Viscous Fingering  anisotropic viscosities in the problem of viscous fingering for nematic liquid crystals is presented in this chapter.  8.2  Anisotropic Viscosities and Viscous Fingering in a Radial Geometry  This Chapter is a continuation of the work presented in appendices A and B and, as the notations and experiments considered here are the same, it is suggested that the non-experts refer to these appendices prior to reading Chapter 8. Consider a Hele-Shaw cell of radius Rc, with two parallel plates separated by a distance b. The fluid between the two plates is assumed to be a liquid crystal material in its nematic phase. A i r at constant pressure is injected into the cell through a hole at the center of one of the plates which pushes the fluid away, leaving behind a cicular air bubble. The plates are treated in such a way that the orientation of the director is normal to the plates before the injection of air (homeotropic alignment). In most cases the alignment imposed by the surface treatment is weak compared to the viscous torque [2] produced by the fluid flow. The torque reorients the director in a radial direction in front of the moving air-fluid interface. The planar orientation of the director field is the origin of the anisotropy in this system. The viscosity coefficient along the director, in the radial direction, is u , while it is u$ in the azimuthal direction . If inertial terms are neglected 2  T  in the Navier-Stokes equation, one derives a depth-averaged equation of motion for the velocity field v of the fluid: b  2  + ^ ~ p. v e + ugv eg = — e— dr r 06 T  T  r  e  r  (8.1)  where P is the pressure field. Assuming that the fluid is incompressible, one finds that 2  S i n c e we are w o r k i n g i n c y l i n d r i c a l c o o r d i n a t e s , we s h o u l d a d o p t the c o n v e n t i o n w h e r e the  azimuthal  a n g l e i s <j>. W e h a v e p r e f e r r e d t o u s e 0 i n o r d e r t o k e e p t h e s a m e n o t a t i o n a s i n A p p e n d i x A , E q u a t i o n  A.6.  Chapter 8. Viscous Fingering  185  P obeys the following equation:  a a P  1 d dP  - - T - r — +  —  .  2  2  -  —  =  0  •  . (8.2)  a = p /pe and is typically of the order of 1. In the case of an isotropic liquid, p = pe, r  r  and P obeys Laplace's equation. Notice that all the effects of the anisotropy of the liquid crystal material have been lumped into the variable a. We extend our earlier work [125] and present here a linear stability analysis using the boundary condition with kinetic correction. The motivation for this is to check whether kinetic effects coupled to anisotropy in the viscosity introduces extra effects which are not present otherwise. Hence, we use the boundary equation A.3 rather than A.4. After finding the solution for a uniform bubble, we carry a linear stability analysis following the steps described in Equations A.6 to A . 12. The general solution of Equation 8.2 is  P(r, 6) = A ln(r) + B + £ (C r  + D r~ ) exp(im9) .  am  (8.3)  am  m  m  m>l  As in Appendice A , we seek a linear perturbative solution for the perimeter of the bubble in the form  R (6,t) = R{t) 1 + 2 rn{t) COS [me + <p )  (8.4)  C  b  m  m=l  where c is the relative amplitude of the pertubations at the air- fluid interface. m  One can show that the evolution of the noise amplitude, e , for the azimuthal mode m  m is given by the following set of equations: e fe m  9m  -R /R 0  m  0  l-i/  [l +  _  ^ - ( ^ ) ( ^ ) ]  -  2+ Q  +  _  (7.6)^ ( f r - l )  m  12  ^ _ ( ir) x - h\n{l/xyR, 2  l  1  -  (8  1  2  ,  1 7 / 7  '  5)  (8-6)  Chapter 8.  186  V i s c o u s Fingering  R (t) is the average increasing circular radius of the bubble, 0  x = R (t)/R , 0  c  £ =  4P i? /(7ra), and 7 = 2/3 in the viscous fingering problem [122, 125]. 0  c  It is worthwhile noticing that the influence of anisotropy in the present problem manifests itself within a linear stability analysis. This is to be compared to the problem of dendritic growth [120] where the effect of anisotropy is introduced within an anisotropic surface tension correction. Anisotropic corrections in the surface tension enters in the problem only when the growth occurs away from the local interfacial normal which, to first order, is in a constant direction during the growth process. Hence, we have found that surface tension anisotropy does not come into consideration within a linear study. The first mode to become unstable is m = 2, with all the modes m > 2 being stable. One notices that a circular interface is stable if  a < 1/2;  or u /u > 2 . e  (8.7)  r  The same anisotropy effect has been invoked to explain the absence of tip splitting in viscous fingering experiments in nematic liquid crystals [122]. In [125] we investigated the behavior of Equation 8.6 with no anisotropy (a = 1) and no kinetic term {v — 0). Typically, the range of experimental parameters is such that £x ^> 1 and m* « O(10 ), where m* is the fastest growing unstable mode. Hence, if x  a > 1/2, and the system is in the unstable regime, one can approximate m — l/(£x — 1) 2  Let us assume that m can be taken as a continuous variable. This is not a bad approximation for m* > 10 [125]. By redefining a m = m and a £ = £, one recovers the growth 2  equation previously obtained by Buka and Palffy-Muhoray [122] for viscous fingering in an isotropic fluid. Hence, for a > 2 and in the unstable regime, the present anistropy does not introduce any new interesting physics in the problem of viscous fingering. As well, the morphological phase diagram obtained in appendix B would also apply in the  187  Chapter 8. Viscous Fingering  present anisotropic case if a new variable, s, was defined as s = s/ct = 2  8.3  xa/(Aa P R ). 2  0  0  Discussion  The role of anisotropy has proven to be most important in the problem of Laplacian growth. Nematic liquid crystals are anisotropic systems in which the amount of anisotropy can be easily tuned by varying the temperature. At a microscopic level, the anisotropy of the viscosity coefficient in nematics arises from the nonseparability of the intermolecular interactions. We have extended our previous analysis of the breakdown of a linear stability analysis in the viscous fingering problem by including anisotropy in the viscosity. We find that, unlike a case where anisotropy is included only in the surface tension, a [120], the anisotropy in the viscosity coefficients manifests itself in a linear regime. We find, however, that this correction is unimportant in the unstable regime since the viscosity ratio, a, can be scaled out of the problem by redefining the fastest growing mode, m*, and the effective driving force, £.  Chapter 9  Conclusion and Avenues for Future Work  9.1  Conclusion  In this thesis, we investigated some effects that nonseparable interactions can have on the orientational behavior exhibited by liquid crystalline systems. In particular: • We showed that the standard mean-field theory of nematic liquid crystals transforms an initial nonseparable pair potential, via a spatial averaging over the orientations of the intermolecular vector, into an effective separable potential. We then argued that the mean-field theory of nematic liquid crystals differs considerably from the ordinary mean-field approximation for lattice systems.  This led us to  re-examine, using Monte Carlo simulations, the physical meaning of the mean-field approximation in the context of nematic liquid crystals. • We showed that the spatial averaging invoked in the mean-field theory of nematics must be dynamical and not static in order to lead to 'nematic-like' orientational ordering in a nonseparable system. We found that a quenched positionally disordered system, with a spherically symmetric center of mass distribution, leads to 'glassy-like' behavior in a system of particles interacting via nonseparable forces.  188  er 9. Conclusion and Avenues for Future Work  189  We found that a system of particles interacting via a nonseparable anisotropic van der Waals potential, where translation and rotation are allowed, displayed a noticeable increase of nearest-neigbor bond orientational correlation below the isotropic-nematic transition. This behavior was not observed for an 'equivalent' separable model. We showed that this phenomenon could be explained by invoking the phenomenon of induced bond orientational ordering observed in smectic-C films. Our Monte Carlo results show that the nonseparability of the pair potential leads to induced bond orientational order in a nematic phase. As a by-product, they show that the coupling between the bond orientational field (BOF) and the molecular orientational field (MOF) invoked in Ginzburg-Landau theory of liquid crystals has its microscopic origin in the nonseparability of the pair potential. The difference in behavior between the separable and nonseparable potentials we investigated led us to reinterpret the mean-field theory of nematic liquid crystals as an approximation which neglects the phenomenon of induced bond orientational order in these systems. We showed that the presence of an underlying ordered B O F in nonseparable twodimensional nematic liquid crystal models invalidates the interpretation of finitesize scaling analysis of these systems in terms of a simple model which neglects the presence of such B O F . We presented results which show that the elastic response (Freederickzs transition) of tilted smectic materials subject to an external field can be substantially modified when the presence of a coupling between the B O F and the M O F in these systems is considered. In particular, we found that the sixfold symmetric B O F in a tilted hexatic phase can drive the Freederickzs transition first order if the B O F - M O F  Chapter 9. Conclusion and Avenues for Future Work  190  coupling and the strength of the applied magnetic field are appropriately chosen. • The coupling between the B O F and the M O F can induce 'structural' transitions in tilted hexatic liquid crystals. A preliminary microscopic model of such transition, the smectic-I to smectic-F transition, was proposed. This model has allowed us to isolate some microscopic parameters which are responsible for driving a transition from a smectic-I to a smectic-F. More importantly, we were able to argue, assuming that the tilt angle increases upon a decrease in temperature, why the smectic-F always occur at a lower temperature than the smectic-I phase. • The anisotropy of the viscosity coefficients in nematics is also a manifestation of the nonseparability of the pair potential.  We investigated whether anisotropic  viscosities could modify our previous results obtained for the problem of viscous fingering in isotropic fluids. We found that, at least in the linear regime, the anisotropy in the viscosity coefficients can be scaled-out and does not introduce new effects in the problem of viscous fingering.  9.2  Avenues for Future Work  This section briefly present some possible extensions to the work presented in this thesis. • In the context of orientational glasses, it would be instructive to extend our investigations of two-dimensional van der Waals and quadrupolar glasses [63] to threedimensional systems to see if a true thermodynamical glass transition is observed. As well, the importance of the random symmetry breaking fields in such a threedimensional system could be studied. • In Chapter 4, we investigated the modified nonseparable anisotropic van der Waals potential with the particular choice: J = 1/2, a = 0, /? = 2/5 and 7 = 4/5.  Chapter 9. Conclusion and Avenues for Future Work  191  This choice of parameters led to a liquid crystalline phase with induced bond orientational order with rectangular symmetry. We expects this rectangular liquid crystalline phase to convert into a hexatic phase with a slight twofold distortion upon reducing 7 , while keeping the value J = 1/2. We find that the above transformation occurs in the ground state via a first order transition from a distorted triangular lattice (large pressure, small 7) to a rectangular lattice (low pressure, large 7 ) . It would be interesting to verify if this first order transition is observable in the liquid crystalline phase. If the transition were observed, the Fourier coefficients of the B O F - M O F coupling could be evaluated and their behavior close to the transition compared with a Landau theory of the type proposed by Selinger and Nelson for the problem of transition among tilted hexatics [83, 84]. This would constitute the first direct microscopic test of this type of phenomenological Landau theory for liquid crystals with bond orientational ordering. • It would be interesting to extend the calculations of the Freederickzs transition in tilted hexatics. In particular, a direct numerical investigation of the Euler-Lagrange equations and its comparison with the slowly varying approximation (SVA) would be useful. • Finally, another line of study would be to improve the microscopic model smectic-I to smectic-F presented in this thesis. In particular, it would be interesting to repeat the same calculations, but for soft-ellipses. As explained in Chapter 7, soft-ellipses would allow a continuous transformation from the 'I-like' state to the 'F-like' state. This would provide another check of the theory of Selinger-Nelson for transitions among tilted hexatics.  Appendix A  Noise and the Linear Stability Analysis of Viscous Fingering  M.J.P. Gingras* and Z. Racz  Institute for Theoretical Physics, Eotvos University Budapest 1088, Puskin u. 5-7, Hungary  A linear stability analysis of the Saffman-Taylor instability in a radial Hele-Shaw cell is presented. It is shown that the limit of validity of the linear stability analysis can be estimated without having to solve the problem to higher orders. The formation of fingers initiated by the presence of white noise is also examined. We find that, depending on the amplitude of the noise, the number of fingers that is observed as the limit of the linear regime is reached may differ significantly from the number of fingers predicted by the fastest growing mode.  Paper appeared in Physical Review A , 40, 5960 (1989).  192  Appendix A. Noise and the Linear Stability Analysis of Viscous Fingering  A.l  Introduction  Pattern formation in systems driven far from equilibrium has attracted much attention recently. One of the most studied processes is the so called diffusion-controlled growth which produces characteristic morphology in dendritic solidification , viscous fingering 1  2  and diffusion-limited aggregation ( D L A ) . The detailed mechanism of pattern formation 3 x  in the above examples is not quite understood but it is suspected that apart from the instabilities inherent in moving diffusive fronts, noise may play an important role. The following examples are illustrative. D L A is often considered as the noisy limit of diffusioncontrolled growth processes' . Viscous fingering resembles D L A if the fingering occurs in a 1  porous media . The sidebranching activity observed in dendritic growth might arise from 5  the presence of noise at the tip of the dendrite , although other mechanisms may also 6  be responsible . Finally, the wobbles of the Saffman-Taylor fingers and the seemingly 1,7  random patterns in viscous fingering may also be produced by the presence of noise . 8  Unfortunately, the sources of noise in diffusion-controlled growth problems are usually not known. Noise may be produced by thermal fluctuations or by the presence of impurities in case of dendritic growth . In two-dimensional viscous fingering , on the 1  2  other hand, irregularities in either the glass plates or in the aperture where the less viscous fluid enters may be the origin of randomness. Another source of noise, which may play a role in the pattern formation in liquid crystals is the anisotropy of ordered domains . The above examples have been selected to illustrate that noise may appear in 9  the mathematical description of the corresponding phenomena through i) initial conditions, ii) boundary conditions and iii) the dynamical equations. A systematic study of the above possibilities should, in principle, contribute to our understanding of unstable growth processes. x  T h e reader s h o u l d note t h a t the  p r i a t e references a p p e a r at the  end  reference n u m b e r s i n the of the  appendix a n d not  present a p p e n d i x are l o c a l a n d the in the  main  bibliography.  appro-  19  Appendix A. Noise and the Linear Stability Analysis of Viscous Fingering  Investigation of the sources of noise is hampered by the fact that diffusion-controlled growth problems are highly nonlinear and nonlocal making the experimental regimes rather inaccessible to calculations. Various schemes can be used to bridge the gap. For example, noise can be applied locally at the tip of the growing dendrite , the relevant 6  randomness may be assumed to be present in the initial conditions  10,11  or noise may be  completely forgotten by using linearization procedures in conjunction with the assumption that the fastest growing mode determines the p a t t e r n  12-15  . These phenomenological  approaches are surprisingly successful in explaning experimentally observed facts. We believe, however, that the reason for their success is yet to be understood. One would expect that investigating the earliest stage of growth of a given diffusioncontrolled growth process and comparing the results with the predictions of a linear theory would provide important clues about the origin of the noise. Unfortunately, there are few experiments where such early linear regime has been clearly identified and investigated. Experiments performed in a linear geometry indicate that the local perturbations caused by the sidewalls initiate the pattern formation . The problems 16  introduced by the sidewall interference effects make the interpretation of experimental results obtained in the linear geometry very difficult.  These problems are eliminated  in the circular geometry and an example where direct comparison between the linear theory and experiment may be possible is the viscous fingering experiment performed by Buka and Palffy-Muhoray . In this experiment, a stable circular interface is prepared 15  as an initial condition in a radial Hele-Shaw cell . Then a stepwise increase in pressure 12  is used to make the system unstable and the growth of perturbations on the interface is monitored. As a quantitative measure of the state of the system, the 'number of fingers' (number of throughs in the perturbation) is counted. Since the amplitudes of the perturbations are small at the initial stages of growth, Buka and Palffy-Muhoray assume that this regime may be described by using the linearized equations of motion. Making  19  195 Appendix A. Noise and the Linear Stability Analysis of Viscous Fingering  then the additional assumption that the fastest growing mode determines the number of fingers, and using the surface tension as a fitting parameter, they obtain good agreement between the observed and calculated numbers of fingers. The above experiment is conceptually simple and its analysis contains the ingredients used in the mathematical description of later stages of growth ' . Thus, we believe it is 12  13  worth examining the assumptions underlying the analysis. The questions which naturally arise are as follows: i) What is the extent of the regime where linearized equations can be used? ii) Does the fastest growing mode determine the number of fingers? Question ii) actually addresses the problem of the initial amplitude of various unstable modes and thus brings up the last questions: iii) What is the role of noise in selecting the number of fingers? Can we deduce the noise amplitude from the experiment? In this paper we address the above questions. In Section II, we present the model and derive the equations which govern the evolution of the unstable modes in the linear regime. A visibility parameter which measures how much structure is present at the interface is also introduced. Then, in Section III, we establish a criterion for the limit of validity of the linear regime using the fact that no inward motion of the interface is observed experimentally. Next (Section IV), the problem of the connection between the fastest growing mode and the number of fingers is examined. We find that the number of fingers is well approximated by the fastest growing mode only if the amplitude of the noise initiating the formation of patterns is sufficiently large. In particular, we deduce that the initial roughness of the bubble perimeter in Buka and Palffy-Muhoray's experiment should be on a lengthscale of 1 0 c m . _3  15  196 Appendix A.' Noise and the Linear Stability Analysis of Viscous Fingering  A.2  The Model  Pattern formation in hydrodynamics is a common phenomenon, and the Saffman- Taylor problem in which two fluids move in a narrow space between two parallel plates (Hele17  Shaw cell) is a frequently studied example. In this paper we restrict ourselves to the radial version of the Saffman-Taylor p r o b l e m  12,13  '  15  and a Hele-Shaw cell of radius R is c  considered. The plate separation is b and the plates are assumed to be horizontal so that the gravity does not enter the problem. A i r is injected into the cell at constant pressure through a hole located at the center of one plate. If inertial terms are neglected, the velocity field, v, of the fluid parallel to the plates is given by  where u is the viscosity of the fluid and P is the pressure field. Equation A . l constitutes Darcy's approximation. It is derived from the Navier-Stokes equation by assuming a parabolic flow profile with vanishing velocity at the plates and by averaging the velocity over the vertical direction. If the fluid is assumed incompressible, the pressure field obeys Laplace's equation (A.2)  V P = 0, 2  and the boundary conditions to Equation A . 2 are as follows. The applied air pressure in the bubble and the pressure at the outer radius of the cell are constant with P being 0  the difference between the two. The pressure jump across the air fluid interface, 6P\ , a  is given by a Gibbs-Thompson type relation which can be derived from an asymptotic analysis of the full set of hydrodynamical equations  18  (A.3) Here a is the surface tension, K is the curvature in the plane of the plates and the capillary number is given by Ca = uv /o~, where v is the normal velocity of the interface. n  n  197 Appendix A. Noise and the Linear Stability Analysis of Viscous Fingering  This boundary condition has been much discussed recently. Comparisons of experiments  19  with numerical solutions the Saffman-Taylor problem seem to indicate that the 20  wetting layer which produces both the kinetic term and the 7r/4 coefficient in the curvature term may be important. Nevertheless, experimental r e s u l t s  12-15,21  have been success-  fully analyzed by changing the exponent of C a , by ignoring the 7r/4 coefficient ' 1 3  12 13,15  or  by leaving out the kinetic term altogether ' ' . Thus we feel that in our initial studies 12  15  21  of the effect of noise it is adequate to neglect the kinetic term and use a mathematically simpler boundary condition  22  7T  (A.4)  6P\ = -*K . a  Equations A . l , A.2 and A.4 have a solution corresponding to a circular bubble of radius R(t). The time evolution of R(t) is given by  dR dt  b Po12pR 2  1  7TC  4R HRc/R)  (A.5)  '  One can see that the bubble expands if the pressure difference Po is larger than the pressure due to the surface tension TTO~/(4R).  The question of linear stability of the  expanding bubble has been the subject of several studies ' ' . 12  13  15  In these studies, a  perturbed circle of radius Ri,(6,t) is assumed  R ($,t) = R(t) 1+2 b  £ m  m=l  W  C O S  + Vm)  (A.6)  where 0 is the azymuthal angle, while e and <p are the relative amplitude and phase m  m  of the mode of wavenumber m. Then, using a linearization procedure, one obtains the equations governing the time evolution of the perturbations (•m/tn  x/x  2  = -2 + m 1 +  F (x) = m  m - 1, . .  (1 + x (1 - x  2 m  2 m  ) )  tx-1  r In (x) F (x) m  (A.7)  198 Appendix A. Noise-and the Linear Stability Analysis of Viscous Fingering  where x = R(t)/R and £ =  Typically, P ss lkPa, R  4P R /(Trcr).  c  0  Q  c  c  « 10cm and  a « 10~ Pa • m so £ « 10 — 10 and x « 1 0 . It is, therefore, a good approximation 2  3  to say that (x >• 1 and x  4  - 1  <C 1 for the modes that play some role in the formation of  2 m  patterns at the interface. Using these approximations, Equation A.7 is integrated and we obtain «m(t) = em(0)exp(r ) .  (A.8)  m  Here T  m  is given by  ^  T  m ( m — 1)" l + ln(x) = (m - 2) In (—) - ^ — i  .  m  , . x.  2  1 -f ln(x )' 0  x  (A.9)  0  and xo = x(t = 0). Two numbers of importance can be derived from Equations A.7. There exists a fastest growing mode (rn = m*) that maximizes the growth velocity in Equation A.7. It has often been assumed  12,13  '  15  that the number of fingers that are observed at the air-fluid  interface is given by m*(x). Another quantity which, we feel, may be a better measure of the number of fingers is the azymuthal number, m = m/(x), of the largest-amplitude mode. These two numbers which will be discussed in Section IV are obtained from the following equations m  1/2  £x - 1  (A.10)  = v^[ln(l/x) + 1 3m/ - 1 1 + ln(x) _ 1 + ln(xo) 2  — ln(e (0))| m  m = m (  =  i  x  (A.11)  XQ  ln(f) .  (A.12)  XQ  The linear regime is of course valid only for a short initial period (t < t i) beyond which n  nonlinear effects take over. We will discuss in the next Section how the limit of validity of the linear regime can be estimated. Assuming that t i is known, there still remains n  the question whether the pattern formed in the linear regime can be observed in a given  199 Appendix A. Noise and the Linear Stability Analysis of Viscous Fingering  experiment. Thus it is useful to introduce a visibility coefficient, V , that gives a measure of the deviation from the circular form:  V = / Jo 2  —  R {6,t)-R(t) b  2TT  4E.  ,(0)exp(2r ) m  (A.13)  One can detect the presence of structures (fingers) only if the resolution of the apparatus that is used to monitor the interface is greater than or equal to V. This is in complete analogy with the study of fringes in interferometry. Finally, to complete the description of the model, we must discuss the origin of noise in the system in order to estimate the amplitude of the initial values of the perturbations (c (0)). The sources of randomness may, of course, be rather diverse. They may come, for m  example, from the irregularities in the interplane spacing (b). These irregularities could be mathematically described by introducing a multiplicative noise term in Equation A . l . Also, Equation A.2 would have to be modified and, furthermore, the pressure jump at the interface (Equation A.3) would be influenced. Similarly, modifications would be caused by thermal fluctuations or by fluctuations of the air pressure inside the bubble. Equation A.7 does not contain any information about the above sources of noise since it was derived by assuming a uniform interplane spacing as well as constant pressure and temperature. Some of the noise effects, however, survive through the initial conditions e (0). In the present work, we omit the dynamical sources of noise and assume that TO  the effect of noise can be described through the initial amplitudes of the perturbations at the circular interface. The motivation for this assumption comes from the fact that the initial disturbances are exponentially amplified and one might expect that the role of later perturbations will be less significant. Some support for this assumption comes from the studies of diffusion controlled growth where D L A type structures were obtained by introducing randomness only in the initial conditions ' . 10 11  A n estimate of the lower limit for |e (0) | can be obtained in the following way. Assume m  200 Appendix A. Noise and the Linear Stability Analysis of Viscous Fingering  that fluctuations on the initial bubble occur on all length scales and that their amplitude is of the same order of magnitude as the lengthscale on which irregularities in the plate spacing occur. The size of irregularities for optical quality glass plates cannot be larger than (A/10) where (A) is the wavelength of the visible light (A « 500nm). Thus we have  23  A/10 « e (0)R(0) for all m and consequently m  e ( ) = m  m  0  K  )  c  » —^—» l0xoRc  ' " , » lO" 10-0.1 - 5 - 10" m 5  1  0  7  m  (A.14)  5  2  v  ;  For the purpose of the discussion in the following sections, we shall call the noise producing these equal amplitude perturbations white noise. Another possible source of perturbation is the thermal noise. The amplitude of the fluctuations in this case can be estimated by assuming equipartition of energy.  The  change in the energy of the system due to fluctuations of the surface, S, resulting from the mode m, is given by SE = a8S w k T where, up to second order in e , 6S can be m  B  written as SS «  As a result we find  •KbRm 2e 2n(Q)/2.  1  "<°> " m  2k T B  £  10~  "—  6  .  ( A  where we used 6 fa lOpm, R fa 1cm, a fa 10~ Pa • m and T = 30QK. 2  '  . 1 5 )  Since the  thermal noise is at least an order of magnitude smaller than the noise coming from the irregularities of the plates, we shall ignore its presence. It should be noted, however, that in case of small a (almost miscible fluids) the thermal fluctuations may play an important role. We shall assume in the following that other types of noise would also have a white noise spectrum and thus would only change the value of e in Equation A.14.  A.3  Limit of validity of the linear regime  The range of validity of a linear stability type calculation can, in principle, be checked by evaluating second and higher order terms in a perturbative expansion. Unfortunately,  Appendix A. Noise and the Linear Stability Analysis of Viscous Fingering  Figure A . l : Formation of fingers in the linear regime The values of the parameters are x = R(0)/R = 0.1, £ = 7000, and e = 1 0 . The interfaces are drawn at equidistant values of x(t) (6x = 0.015). -5  0  c  the solution of the equations beyond the first order is very involved for pattern formation in diffusive growth processes . Thus, one would like to find a criterion that establishes 24  the limit of validity of the linear regime without recourse to higher order terms. In order to see how such a criterion may be obtained, we consider the evolution of a bubble using the linearized equations of motion A.8, A.9. We choose typical values for both the initial reduced radius of the bubble x = R(0)/R = 0.1 and the reduced pressure ( = 7000. 0  c  The amplitude of perturbations on the interface are set somewhat arbitrarily to the lower limit (e = 10" ) calculated in Equation A.14 and the phases <p are choosen randomly 5  m  from the interval [0, 2TT]. The results shown in Figure A . l were obtained by drawing the successive surfaces at equidistant values of x(t) (6x = 0.015). Comparing Figure A . l with an adaptation of a multiple exposure photograph taken during the injection of air into a Hele-Shaw cell filled with glycerin  12  (Figure A.2), one can see some resemblance  for small radii. It can also be observed, however, that in contrast to the experiments  12  2  Appendix A. Noise and the Linear Stability Analysis of Viscous Fingering  Figure A.2: Adaptation of Figure 3 of reference 12. The original figure corresponds to a multiple exposure photograph taken during the injection of air into a Hele-Shaw cell filled with glycerine. (Figure A.2) the base of the fingers in Figure A . l starts to move inwards as the radius becomes larger. This behaviour is an artifact of the linear approximation which should not be present in the exact solution of the equations. Thus the time, t-, when the radial velocity becomes negative for the first time gives an upper limit for the period ( i / < £_) n  in which the linear description is valid. The quantity of experimental interest is actually not <_ but the breakdown radius x_ = x(z_) at which the linear stability analysis ceases to be applicable. We have numerically studied how x_ varies as a function of the reduced pressure £ at fixed c = 1 0  - 5  and xo = 0.1. Since the initial phases (y> ) are random, there are some fluctuations in m  the values of x_. The results shown by the solid curve in Figure A.3 were obtained by averaging over 500 growth processes. The breakdown radius may also be calculated by assuming that the evolution of the interface is determined by the mode with the largest  2  Appendix A. Noise and the Linear Stability Analysis of Viscous Fingering ion  8-  x  4-  z-t 0  1 2000  1 — 1 4000 8000 t  1 8000  1 10000  1 OOOO  Figure A.3: Reduced breakdown radius x _ / x versus reduced pressure £ for x = 0.1 and c = 10~ . The solid curve describes the numerical results averaged over 500 different realizations of the random phases <p . The dashed curve was obtained by considering only the mode with the largest amplitude. 0  0  5  m  amplitude (m/) (dashed curve in Figure A.3). This approximation leads to a fair agreement with the results obtained from the numerical solution. The partial disagreement can be understood from the Fourier spectra of the interface during the growth (Figure A.4). One finds that the spectrum is not particularly peaked at m/; the modes mj ± 1 have an amplitude that is not less than 85% of the maximum amplitude. These satellite modes contribute to the r.m.s. inward velocity of the base of the fingers and thus they reduce the breakdown radius compared to what is predicted by considering the largest-amplitude mode alone. We note here that the large-amplitude modes have relative amplitudes of the order of 10~ at x_. It has been observed in Ref.13 that predicting the morphological transition 2  from D L A to dense branching morphology was beyond the scope of the linear theory and would require a full nonlinear study; our results suggest that such nonlinear effects are already important at the early stages of patterning and they should, in principle, be  203  Appendix A. Noise and the Linear Stability Analysis of Viscous Fingering o-i  Mode #  Figure A.4: Fourier spectra of the amplitudes of the perturbations e (t). The solid line corresponds to time t = 0 and each successive curve, going upward, was plotted at equidistant values of x(t) (6x = 0.015). The parameters are the same as in Figure A . l . m  taken into account in perturbative calculations.  A.4  N u m b e r of fingers i n the linear regime  A quantity which is observed experimentally is the number of fingers.  The number  of fingers at the breakdown of the linear regime (Nj) can be obtained numerically by counting the number of maxima in  t). The results from an average over 500 growth  simulations are shown by the solid curve (squares) in Figure A.5. Assuming that the growth is entirely controlled by the mode with the largest amplitude (m/), one obtains the dashed curve (crosses) shown in Figure A.5. It is of interest to compare these results with the azymuthal number of the fastest growing mode at the breakdown radius (m*(a:_)) and also with the initial fastest growing mode (m*(xo)). The results are shown in Figure A.5 by the chaindashed (triangles) and dotted (diamonds) curves for m*(i_) and m*(xo) respectively. The agreement with the numerical results is  20  Figure A.5: Average number of fingers (Nf) observed at the breakdown of the linear regime (x = x_) versus the reduced pressure £. The parameters are the same as in Figure A . l . The solid curve was obtained by averaging the numerical results over 500 different realizations of the random phases ip . The dashed (crosses) curve corresponds to the azymuthal number of the largest-amplitude mode m/ evaluated at the breakdown radius x_. The chaindashed (triangles) and dotted (diamonds) curves correspond to the azymuthal number of the fastest growing mode m* evaluated at x_ and x respectively. m  0  20  Appendix A. Noise and the Linear Stability Analysis of Viscous Fingering  good only for m/. The small discrepancy between m/ and Nf is again due to the role played by the satellite modes in the growth of the interface. Figure A.5 was obtained using a noise amplitude e = 1 0 . In this case the breakdown - 5  radius (x_) is significantly larger than the initial radius (xo). This implies that during the growth m* and m/ which are initially equal but change at different rates have time to diverge from each other. For an increased initial noise level the limit of validy of the linear regime is reached earlier. One then expects that the difference between m* and Nf is smaller and, consequently, the hypothesis that the fastest growing mode determines the observed number of fingers works better. Indeed, we find (Figure A.6 that both m*(x_) — Nf and Nf — m*(x ) decreases with increasing c, and for the range of parameters 0  £ which are typical in experiments, the difference m*(x_) — Nf vanishes for e > 10~ . 3  We shall now try to use the above results to extract the noise amplitude in Buka and Palffy-Muhoray's experiment. In their setup the radius of the initial bubble is 0.5 or 1cm and they use a relatively large pressure difference so that £ « (1 — 4) • 10 . 4  The change in the radius by the time the pattern is observed is of the order i.e.  x/xo w 1.1.  25  of 10%  The amplitude of the interface modulation is of the order of the  change in the radius, thus the visibility coefficient (equation 12) has a value of V « 0.1. Assuming that the observations can be accounted for by the linear theory and that the noise initiating the pattern formation is a white noise (e (0) = e), we can use equation m  (12) to estimate the amplitude of the initial roughness of the interface ei?(0). The result is eR(0) = (2 ± 1) • 10~ cm where the uncertainty comes from the fact that neither x / x 3  nor V is known precisely and, consequently, we obtain different values for e for the various points which can be examined in Figure 2 of reference 15. Provided the initial roughness  is caused by the variations in the plate separation (6), the above result means that the lengthscale on which the fluctuations in b occur is of the order of 2 • 10~ cm. Note that 3  the above calculation does not provide an estimate for the amplitude of the variations in  0  207 Appendix A. Noise and the Linear Stability Analysis of Viscous Fingering  tin l  »7: 5I  3-|  TV,  'E 1-  >r—=*-  -1-3-  g -4.5  -4  * -3.5  Noise level  9  A  § n  A  o  •  m'lx,) - N,  —r-3  -2.5  -2  Oog^fe])  Figure A.6: Difference between the azymuthal number of the fastest growing mode and the average number of fingers (Nf) observed at the breakdown of the linear regime (x = x-) versus the noise level e. The initial condition is x = 0.1. The squares, triangles and crosses correspond to values of reduced pressure £ = 5000,10000, and 15000 respectively. The upper part of the figure corresponds to the difference between the fastest growing mode at the average breakdown radius (m*(x_)) and Nf. The lower part of the figure corresponds to the difference between the initial fastest growing mode (m*(xo)) and Nf. The dashed lines are linear fits and are guides to the eye. 0  Appendix A. Noise and the Linear Stability Analysis of Viscous Fingering  b, it gives only the lengthscale of these variations. Having estimated e, one may ask whether this estimate is consistent with the assumption that the pattern is formed in the linear regime. We find that for most of the experimental points one has just reached the limit of validity of the linear theory. The observation that the limit of the linear regime must have been reached is also clear from the fact that the size of the fingers is of the same order of magnitude as the increment in the radius. Another consequence of the ei?(0) = 2 • 1 0 c m result is that the number of _3  fingers is well accounted by m* (see Figure A.6). and  This provides a justification for Buka  Palffy-Muhoray's assumptions that the linear theory can be used and that their  fingering experiment can be analysed in terms of m*.  A.5  Final Remarks  Although noise is believed to play an important role in most systems where pattern formation occurs, a quantitative description is lacking. In this paper, we have shown how  the properties of noise in growth processes may be investigated by studying an  initial linear regime. It is hoped that the criterion we introduced to describe the range of validity of linear theories will be helpful in designing experiments where the linear regime can  A.6  be investigated and the noise extracted.  Acknowledgement  The  authors would like to thank B . Bergersen, A. Buka, G.M. Homsy, P. Palffy-Muhoray,  and  M . Plischke for helpful discussions and D.A. Reinelt for providing us with a preprint  of his results prior to publication. One of us (M.G.) would like to acknowlege a scholarship from N S E R C and F C A R as well the financial support and the hospitality of the members of the Institute of Theoretical Physics at the Eotvos University. This research  208  Appendix A. Noise and the Linear Stability Analysis of Viscous Fingering  was supported by the Hungarian Academy of Sciences through Grants No. O T K A 819 and 425.  2  Appendix A. Noise and the Linear Stability Analysis of Viscous Fingering  A.7  References  * On leave from Department of Physics,University of British Columbia, Vancouver, B . C . , Canada V 6 T 2A6. 1. J.S.Langer; Rev.Mod.Phys., 5 2 , 1 (1980). 2. G.M.Homsy; Ann.Rev.Fluid Mech., 1 9 , 271 (1981). 3. T.A.Witten and L.M.Sander; Phys.Rev.Lett., 47, 1400 (1981). 4. P.Meakin, R.C.Ball, P.Ramanlal and L.M.Sander; Phys.Rev.A, 35 5233 (1987). P.Meakin, Phys.Rev.A, 36 332 (1987). 5. K.J.Maloy, J.Feder and J.Jossang; Phys.Rev.Lett. 55, 2681 (1985). 6. R.Pieters and J.S.Langer; Phys.Rev.Lett., 56, 1948 (1987). 7. Y.Saito, G.Goldbeck-Wood and H.Muller-Krumbhaar; Phys.Rev.Lett., 58, 1541 (1987). O.Martin and N . D . Goldenfeld, Phys.Rev.A, 35, 1382 (1987). 8. G . L i , D.A.Kessler and L.M.Sander; Phys.Rev.A, 34, 3535 (1986). Y.Couder, N.Gerard and M.Rabaud; Phys.Rev.A, 34, 5175 (1986). Y.Couder, P.Cardoso, D.Dupuy, P.Tavernier, and W.Thorn; Europhys.Lett., 2, 437 (1986). A deterministic mechanism for the sidebranching activity in viscous fingering is discussed in J.S.Langer; Science, 243, 1150 (1989). 9. S.Arora, A.Buka, P.PalfFy-Muhoray and Z.Racz;Europhys.Lett., 7, 43 (1988). 10. L.M.Sander, P.Ramanlal, and E.Ben-Jacob; Phys.Rev.A, 32, 3160 (1985). 11. D.A.Kessler, J.Koplik, and H.Levine; Phys.Rev.A, 30, 3161 (1984).  21  Appendix A. Noise and the Linear Stability Analysis of Viscous Fingering  12. L.Paterson; J.Fluid Mech., 113, 513 (1981). 13. E.Ben-Jacob, G.Deutscher, P.Garik, N.D.Goldenfeld and Y.Lareah; Phys.Rev.Lett., 57, 1903 (1986). 14. L.Schwartz, Phys. Fluids; 29, 3086 (1986). 15. M.W.DiFrancesco and J.V.Maher; Phys.Rev.A, 39, 4709 (1989). 16. A.Buka and P.Palffy-Muhoray; Phys.Rev.A, 36,1527 (1987). 17. P.G.Saffman and G.I.Taylor; Froc.R.Soc. London Ser.A, 245, 312 (1958). 18. F.P.Bretherton; J.Fluid.Mech., 10,166 (1961). C.W.Park and G.M.Homsy; J.Fluid Mech., 139, 291, (1984). D.A.Reinelt; J.Fluid.Mech., 183, 219 (1987). 19. P.Tabelling and A.Libchaber; Phys.Rev.A, 33, 794 (1986). 20. S.Sarkar and D. Jasnow; Phys.Rev.A, 35,4900 (1987). 21. C.W.Park and G.M.Homsy; J.Fluid Mech., 141, 275 (1984); Corrigendum: 144, 468 (1984). 22. The term 2o~/b can be incorporated in the overall pressure drop PQ. 23. The quantity e (0) is, within the above description, analogous to the roughness m  factor introduced in the study of turbulent flow in pipes. See for example B.S. Massey; Mechanics of Fluids, (Van Norstrand Reinhold,U.K., 1983).  24. M . Kerszberg; Phys.Rev.B, 27, 6796 (1983). Ibid.,28, 247 (1983). 25. Private communication by P.Palffy-Muhoray.  211  Appendix B  Stability Analysis of Laplacian Growth: Breakdown of a Linear Regime  M.J.P. Gingras and Z. Racz*  Department of Physics, University of British Columbia Vancouver, B . C . Canada V 6 T 2A6  It is shown that the limit of validity of a linear stability analysis for diffusive-growth type problems such as viscous fingering and crystal growth can be estimated without solving the problem to higher orders. As an application, we derive a rule often used by experimentalists, namely that the linear regime breaks down when the amplitude and the wavelength of the perturbations are approximately equal. Assuming that noisy (DLAtype) growth occurs when no linear regime exists at all, a morphological phase diagram is drawn for viscous fingering.  To appear in the Proceedings of the 2nd Woodward Conference, San Jose, November 17-18, 1989.  212  Appendix B. Breakdown of the Linear Stability Analysis of Laplacian Growth :  B.l  Amplitude-wavelength Relationship at the Breakdown of the  Linear  . Regime Stability analysis is a standard method in the theory of pattern formation. Typically, homogeneous or highly-symmetric solutions of a given problem can be found easily and their stability can be examined by linearizing the equations of motion with respect to small perturbations around the known solutions. This approach has been effectively used in determining both the limit of stability of a given state and the properties of the new state which appears at the onset of instability . Unfortunately, a drawback to this 1 1  approach is that unless the problem is solved to higher order in the perturbations, the range of applicability of the linear analysis is unknown. This problem becomes crucial when the pattern formation is not a static phenomenon but results from the loss of stability as the system evolves towards an unknown state at a much later time. Examples are the initial roughening in solidification instabilities  2,3  or  the first stages of viscous fingering . The difficulty in comparing the experimentally 4-8  observed patterns with the results of a linear stability analysis is that a macroscopic pattern grows out of microscopic fluctuations in a finite time t\ after the system has lost its stability at to = 0. Thus, a direct comparison is possible only if the linear approximation is valid in the time interval t\. Information about the validity of the linear analysis can be obtained, in principle, by taking into account higher order terms in a perturbative expansion. Such calculations are rather involved, however, and they are rarely carried out . Recently, we suggested a way out of this situation for irreversible 9  10  growth processes and used our estimate of the validity of the linear regime to extract the noise amplitude in a viscous fingering experiment . Here we show that our method 6  can be used to derive a 'visual' criterion for the breakdown of the linear regime and, 1  The  reader s h o u l d note t h a t the  p r i a t e references a p p e a r at  the  end  reference n u m b e r s in the of the  appendix and  not  present a p p e n d i x are l o c a l a n d the i n the  main  bibliography.  appro-  Appendix B. Breakdown of the Linear Stability Analysis of Laplacian 214Growth  v-v  k  Figure B . l : The velocities of a planar front moving with velocity v and perturbed by a sine wave of amplitude Ak(t) are in the range of [v + Vk,v — Vk] where Vk = Ak- The linear regime breaks down when v = Vkfurthermore, assuming that the absence of linear regime is related to the onset of 'noisy growth', we propose a morphological phase diagram for viscous fingering. The basic ingredient in our estimate of the validity of the linear regime is the observation that the normal velocity of a moving surface never changes sign in a process which is equivalent to an irreversible aggregation of particles. On the other hand, if a perturbation of wavenumber k and amplitude Ak(t) is added to an unstable interface moving with a velocity v then there are places on the perturbed surface which move with velocity v — Vk = v — Ak(t) (see Figure B . l ) . A linear stability analysis yields Ak(t) = u>kAk(t) and consequently an exponentially growing amplitude Ak(t) = Ak(0)exp(u>kt). Since generally v and u>k are constant or slowly varying functions of time, it is clear that the linear theory predicts an unphysical change of sign in the normal velocity of the surface at a time i _ which can be calculated from the following condition:  v = u .A >{t-) k  k  (B.l)  Appendix B. Breakdown of the Linear Stability Analysis,ol Laplacian Growth  where k* is the wavenumber of the largest amplitude mode at time t-. Within the linear stability analysis, k* is usually assumed to be of the order of the wavenumber of the fastest growing mode and is thus determined as the value which maximizes  For  diffusive growth problems, Uk can be rather generally written as  u = vk[l-f(k,{p})] .  (B.2)  k  Here the term vk results from the instability inherent in moving diffusive interfaces while 2  f(k, {p}) describes the stabilizing forces such as the surface tension or kinetic effects and {p} is a notation for the control parameters. The function f(k, {/>}) is smooth in all the examples studied so far (in the simplest case f(k, {p}) « k ), and the maximum of ix>k is 2  well approximated by u v w vk* .  (B.3)  Using this estimate, we arrive at the first result of our paper by rewriting the condition for the breakdown of the linear regime (1) in the following form:  k*A >(t.) » 1 .  (B.4)  k  The above condition means that the amplitude of the dominating perturbation at the breakdown of the linear regime is of the order of the characteristic wavelength of the pattern. It should be noted that experimentalists have been using Equation B.4 as a rule of thumb for estimating the extent of the linear regime . 7  B.2  Breakdown of the Linear Regime and Morphological Phase diagram  The central idea here is that the value of t- which is found from Equation B . l depends on the initial amplitudes, Ak(0), of the perturbations.  For large enough Ak{0), the  result is <_ = 0, i.e. no linear regime exists at all. Assuming that pattern formation  Appendix B. Breakdown of the Linear Stability Analysis of Laplacian Growth  is qualitatively different for t- = 0 and for t- > 0, one may draw a phase boundary separating the two morphologicalrphases. The remaining problems are how to find the Ak(0)-s needed for the calculation and how to characterize the morphological phases. The initial amplitudes Ak(Q) are determined by the fluctuations (noise) present in the system. In general, it is not known which are the relevant fluctuations but there are a few cases, such as viscous fingering and crystal growth between closely placed parallel plates, where the noise is believed to come from the irregularities on the p l a t e s  11-13  . In  these cases, ^ ( 0 ) is of the order of the characteristic wavelength (e) of the irregularities on the plates . Thus we have Ak(0) = t and the phase boundary is obtained from the 10  following equation  (B-5)  v = u)k*e .  As an application of this equation, we studied viscous fingering in a radial Hele-Shaw cell  5  and used the simplest approximation in which the only stabilizing effect is the surface tension (o~). In this case, the control parameters are the noise amplitude (c) and the ratio of the surface tension to the excess pressure (p) driving the growth of the bubble (s = o~/p). The phase diagram we find is displayed in Figure B.2. For small driving force (s > s = 1/6), one finds that a circular bubble is stable c  against small perturbations. As s is decreased below s , the bubble becomes unstable c  and this unstable growth can be specified further by examining the existence of a 'linearly' unstable regime. The dividing line between the presence or absence of a linear regime is obtained by solving equation (5) with the result shown by the solid line in Figure B.2. We call the region above the solid line 'noisy growth' regime. The reason for this name is that, due to the large noise amplitude, no linear regime exists in this region. Since it is known from experiments '  11 12  that large noise results diffusion-limited-aggregation (DLA)  type growth in both viscous fingering and crystal growth, we speculate that this region  '  Appendix B. Breakdown of the Linear Stability Analysis of Laplacian 217Growth  0.0  10"  4  0.05  0.15  0.25  Figure B.2: Morphological phase diagram for viscous fingering in a radial Hele-Shaw cell. Both the noise amplitude (c) and the ratio of the surface tension to the excess pressure driving the growth of the bubble (s = c/p) are measured in units of the radius of the bubble.  Appendix B. Breakdown of the Linear Stability Analysis of Laplacian Growth  should be identified as a D L A regime. Another reason for the use of the expression ' D L A regime' is that viscous fingering in the zero-surface-tension limit (5 = 0) yields D L A structures . 14  There exists a linear regime below the solid line in Figure B.2. The morphology in this region is not obvious to us but it is a distinct possibility that the structures characterizing this regime belong to the much discussed  15  dense branching morphology ( D B M ) . 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