A Study in Pattern Formation: Electroconvection in Nematic Liquid Crystals

Transcription

1 UNIVERSITY OF CALIFORNIA Santa Barbara A Study in Pattern Formation: Electroconvection in Nematic Liquid Crystals A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in the subject of Physics by Michael Dennin August, 1995 Committee in charge: Professor David S. Cannell Professor Guenter Ahlers Professor James S. Langer

2 The dissertation of Michael Dennin is approved: August, 1995 ii

3 ACKNOWLEDGEMENTS As with any work that takes six years and over 200 pages to describe, there are a large number of people who helped make it possible. First, I would like to thank my advisors for introducing me to this fascinating subject and for seducing me away from theory into the lab where I have spent many enjoyable hours (as well as a few frustrating ones). I have had the pleasure of reaping the benefits of the dedication that David Cannell and Guenter Ahlers have to teaching and guiding their students. I am sure that the skills I have learned under their mentorship will serve me well in my future work. I would also like to thank Jim Langer for his time and effort as a member of my committee. In the local Santa Barbara area, I need to thank my golf partners Toby Falk, John Woodward, Art Bailey, and Will Nelson. This thesis may not have been possible without them getting their acts together and graduating, so that I could get off the golf course and into the lab. Also, my thanks to Andy Hays, John, and Mike Warren their companionship and support through the first year of graduate school was invaluable. As my first guide through the world of Taylor-Vortex flow, I wish to thank Ning Li for the many hours of interesting discussions on pattern formation. Of course, I must thank Steve Trainoff and Art Bailey to whom I owe copious amounts of chocalate for the patient fielding of my questions. I also wish to thank the community of St. Mark s parish for their wonderful support and friendship over the last seven years. On the international scene, I am in debt to Ingo Rehberg, Eberhard Bodenschatz, Stephen Morris and Barbara Frisken for the expertise they shared on the ins and outs of nematic liquid crystals. Without their knowledge and experience in the Black Arts of liquid crystal cell building these experiments would not have been possible. Also, in particular, I wish to thank Ingo for his patience in passing on his knowledge of electroconvection experiments. My conversations with Lorenz Kramer, Werner Pesch, and Helmut Brand contributed greatly to my understanding of the theoretical background of eleciii

4 troconvection and pattern formation, in general. Especially interesting were my interactions with Hermann Reicke and Martin Treiber, the two theorists to whom my experiments have so far most directly related. Finally, I must thank my family, not only will they kill me if I fail to mention them, but they truly have been a great support to me. I love them all dearly and could not have done this without them. Just for the record, that would include Mom, Dad, Elizabeth, Pete, David, Happy Cat, and, most recently, Chris. And, of course, I thank my wife, Jeni. She is the one who had to deal with most of my stress and worries over this piece of work. Her patience and understanding are without equal. Not, only do I thank her for herself, but in marrying her, I gained the wonderful support, love, and encouragement of her family as a welcome bonus. iv

5 VITA Date of Birth: December 10, A. B., Princeton University Research Assistant, Department of Physics, Princeton University Teaching Assistant, Department of Physics, University of California, Santa Barbara Graduate Student Researcher, Department of Physics, University of California, Santa Barbara Lecturer, Department of Physics, University of California, Santa Barbara. PUBLICATIONS 1. Origin of the Hopf bifurcation in electroconvection M. Dennin, M. Treiber, L. Kramer, G. Ahlers, D.S. Cannell, Phy. Rev. Lett., submitted. 2. Patterns in Electroconvection in the Nematic Liquid Crystal I52 M. Dennin, D.S. Cannell, G. Ahlers, Mol. Cryst. Liq. Cryst. 261, 377 (1995). 3. Measurement of Material Parameters of the Nematic Liquid Crystal I52 M. Dennin, G. Ahlers, D. S. Cannell, in Spatio-Temporal Patterns, ed. P. Palffy-Muhoray and P. Cladis (Addison-Wesley, 1994) p Measurement of a Short-Wavelength Instability in Taylor Vortex Flow M. Dennin, D. S. Cannell, G. Ahlers, Physical Review E, 49, 462 (1994). v

6 ABSTRACT A Study in Pattern Formation: Electroconvection in Nematic Liquid Crystals Michael Dennin I have studied fundamental issues in pattern formation using electroconvection in the nematic liquid crystal I52. An electroconvection cell consists of a nematic liquid crystal, which is doped with ionic impurities, confined between two glass plates which have transparent electrodes on their inner surfaces. The electrodes are used to apply an ac voltage perpendicular to the plates. By appropriate surface treatment of the glass plates, the average molecular alignment (the director) is made to be parallel to the plates. The nematic liquid crystals which are used for electroconvection typically have a negative dielectric anisotropy. As part of this thesis, I will provide a detailed prescription for the making of electroconvection cells. Above a critical applied voltage V c and below a critical applied frequency (the Lifshitz point), there is an instability to a spatially varying state with a wavevector at a nonzero angle Θ or π Θ to the director (oblique rolls, the two states are degenerate). The state can either be traveling (Hopf bifurcation) or stationary. The experimental results are divided into three sections. The first section presents experimental measurements of the critical voltage, the angle between the director and the wavevector of the pattern, and the traveling frequency of the pattern. Comparison of the experimental observations with a detailed linear stability analysis carried out by Martin Treiber and Lorenz Kramer is made, and the agreement between the theory and experiment is found to be good. The second section presents observations of states of spatio-temporal chaos which occur as the initial state above V c. I observe both states which are extended in space and localized states. These states involve the interaction of four modes: right- and left-traveling oblique rolls. The third section summarizes the wide variety of patterns which are obvi

7 served well above V c. I report on the existence of the observed patterns as a function of conductivity of the cell, applied voltage, and applied frequency. Also, I discuss the nature of the transitions between the different patterns. vii

8 Contents 1 Introduction 1 2 Theory Theoretical Tools Nematics Electroconvection Experimental Details Apparatus Shadowgraph Temperature Control Electronics Liquid Crystals Cell Construction Glass Preparation Alignment Sealing Filling Experimental Results General Methods Conductivity viii

9 4.3 Linear Behavior Weakly Non-linear Regime Nonlinear Results Open Issues and Future Directions 148 A Material Parameters 158 B Tables of data 171 C C-Code 197 D References 213 References ix

10 Chapter 1 Introduction One of the most challenging areas in the field of physics is the study of nonlinear, dissipative systems which are driven out of thermodynamic equilibrium by an external stress. Particularly dramatic and fascinating is the spontaneous appearance of spatial and temporal structures, or patterns, at a critical value R c of the applied stress, or control parameter R. Pattern formation, as this process is referred to, occurs in systems ranging in length scales from magnetic domains in garnet crystals to the distribution of galaxies and stars, and includes such diverse systems as the stripes and spots on animal coats, population distributions, cloud streets, and crystal formation. Given that patterns are ubiquitous in nature, the search for an underlying dynamics of patterns which can be applied to systems with different microscopic details is almost unavoidable. While this goal has certainly not been achieved, significant progress has been made in the understanding of patterns through a combination of theoretical developments and precision experiments 1. One way to appreciate the difficulties which arise in formulating a theoretical description of patterns is to briefly consider two ideas which are fundamental to the study of linear systems and thermodynamic equilibrium. We know 1 A recent review of the field of pattern formation is provided in Ref. [1]. For a more complete list of references on the phenomena discussed here, see the references in Ref. [1] 1

11 that any solution Ψ of a linear system can be written as a linear combination of the eigenfunctions F n with amplitudes a n, i.e. Ψ = a n F n. The F n are n often referred to as the normal modes, or simply modes, of the system. By definition, the eigenfunctions are independent, and their evolution is not dependent on the amplitudes of the other modes (the principle of superposition). One of the most common realizations of this idea is the use of Fourier modes, or plane wave solutions, for many problems in electrodynamics and quantum mechanics. The study of systems in thermodynamic equilibrium is simplified in many cases by the existence of a free energy. The state of the system is specified uniquely by the minimum of its free energy. These ideas of superposition and minimizing a free energy are essential elements of our intuitive understanding of linear and equilibrium systems. In contrast, spatially extended pattern forming systems are described by nonlinear partial differential equations (PDE) where the inherent nonlinearity destroys the principle of superposition. Even so, it is often useful to treat solutions of these nonlinear systems as a superposition of individual modes. For example, the eigenfunctions of a linearized version of the equations are often used to express the solution of the full equations. Unlike in the linear case, the evolution of each mode depends on the amplitudes of the other modes through the nonlinear couplings. This nonlinear evolution plays an essential role in the rich variety of pattern dynamics which are observed. For some special pattern forming systems, which are referred to as potential- or gradientflow, a generalized potential which is the equivalent of a free energy exists. For these situations, the description of the dynamics is simplified because the state of the system is specified by the minimum of the potential. However, pattern forming systems are inherently nonequilibrium, and it can not be assumed that a generalized potential exists. In fact, well known cases exist for which it is well established that no such potential exists. The absence of a potential often results in multiple states which are equally stable, and/or a strong history 2

12 dependence for the state of the system. The lack of superposition and general potentials means that much of our physical intuition is often wrong when applied to pattern formation. Despite the difficulties inherent in a theoretical study of patterns, a number of tools in the field of nonlinear dynamics have been developed which enable quantitative study of these systems[1]. First, the increase in computing power has made available numerical studies that just weren t possible in the past. In addition, a number of analytic tools have also proved useful, among these are linear stability analysis, amplitude equations[2], phase equations[3], and simple model equations. Linear stability analysis, as the name implies, is the analysis of a linearized version of the underlying PDEs. The goal of linear stability analysis is the study of the onset of patterns or other instabilities as one or more control parameters are varied. Amplitude equations are used to study the nonlinear interactions of the amplitudes of the relevant modes of the system. They can be derived as a perturbation of the full equations of motion near the onset of a pattern where the amplitudes are small. Phase equations consider the dynamics of large scale modulations of the wavevector of the pattern. Model equations are useful for numerical simulations and for separating the details of a system. For a review of these methods, see Ref. [1] and the references therein. I will briefly discuss linear stability analysis and amplitude equations in Sec Essential to the development of these analytic tools has been the ability to perform precision, quantitative experiments in systems for which the fundamental equations are well known. Precision experiments have both guided the theoretical formulation, as well as, provided quantitative tests of the theory. The solutions of the nonlinear PDEs which describe pattern forming systems depend on the boundary conditions. So, for quantitative tests of theoretically predicted solutions to be relevant, the experiments must maintain precise control over the boundaries. In addition, it is essentially impossible to find analytic solutions to PDEs except in very special cases. Precision experiments provide 3

13 a means of solving these equations over a wide parameter range by simply observing the dynamics of the patterns. Without control over the properties of the system, it is often difficult to distinguish dynamics which are related to the fundamental equations of the system from dynamics which are the result of dirt or poorly understood experimental conditions. Of course, to even consider a comparison of the experimental solutions to a given theory, knowledge of the fundamental equations for the system is needed. Because they meet the above criteria, fluid dynamical systems have played a large role in the advancement of our understanding of patterns[4]. The basic equations (Navier-stokes and additional conservation laws) are well established. The long history of experimental fluid dynamics has resulted in systems for which the boundary conditions and fluid properties are well defined and precisely controlled[5]. A number of quantitative measurement techniques have been developed. Another advantage of fluid dynamical systems is the natural existence of dimensionless control parameters. This allows experimental access to a wide parameter range by the use of appropriate geometries and/or fluids. Two classic examples of pattern forming systems for which precision experiments have been achieved are Taylor Vortex Flow (TVF)[6] and Rayleigh Bérnard convection (RBC)[7]. TVF occurs in a fluid which is confined between two concentric cylinders with one or both of the cylinders rotating. I will consider the case where the outer cylinder is fixed, and the inner cylinder is rotating. The fundamental equation is the Navier-Stokes equation, and the dimensionless control parameter is the Taylor number T = [2ri 2d3 /(r i + r 2 )](Ω/ν) 2 where r i (r o ) is the inner (outer) cylinder radius, d = r o r i, Ω is the rotation rate of the inner cylinder, and ν is the kinematic viscosity. Often the applied stress is described in terms of the Reynolds number R = (r o r i )r i Ω/ν because it is linear in the experimentally controlled parameter Ω. Below the critical value R c of the Reynolds number, the flow is laminar and axially uniform. For a fixed geometry and ν, R c sets the critical rotation rate Ω c. Reasonable values of Ω c are 4

14 achieved by adjusting the geometry and viscosity of the fluid. Above Ω c, pairs of counter-rotating rolls form along the axis of the cylinder. The resulting pattern looks like donuts which are stacked along the inner cylinder. This is an example of an effectively one dimensional system for which the pattern is characterized by its wavenumber. It is possible to make TVF apparatus[8, 9] with the spacing between the cylinders constant to better than 0.1% and the cylinders concentric to 0.1%. To minimize variations of the viscosity, the temperature of the fluid is kept constant to a few mk. Taking into account the various effects, variations of R c are on the order of 0.1%. In RBC, the fluid is confined between two parallel plates and is heated from below. The fundamental equations are Navier-Stokes for the velocity field, and equations for mass conservation and energy conservation. The control parameter is given by the Rayleigh number Ra = gα T d 3 /(νκ). Here g is the acceleration due to gravity, α is the thermal expansion coefficient, d the distance between the plates, κ is the thermal diffusivity, and T the temperature difference across the plates. For a given system, the geometrical factors and fluid properties are usually held constant, and T is varied. Below the critical temperature difference T c, heat is transported by conduction, and the system is spatially uniform. Above T c, the fluid begins to flow, and convection rolls form which have a width roughly equal to the distance between the plates. The patterns in RBC are intrinsically two dimensional, and for simple periodic patterns, the pattern is described by a wavevector (or wavevectors) which lies in a plane parallel to the plates. Depending on the details of the system, there is a great variety of possible patterns including straight rolls, hexagons[10], squares[11], and spirals[12]. The precision of RBC experiments is quite impressive[13]. The top and bottom plate can be made parallel to 0.5 µm. The lateral temperature variations along the top or bottom plate are typically a few mk while the temperature difference between the plates is of the order of a few degrees and is constant to ±0.1 mk. Combining the various effects, the convection apparatus in the 5

15 Santa Barbara group[13] have resolutions in the reduced control parameter ɛ = (R R c )/R c of 10 3 to A number of variations on RBC and TVF have been studied. RBC in one dimension is studied by confining the fluid to a narrow channel. RBC[14] in one dimension and TVF[15] have both been studied in the presence of an imposed flow. RBC in binary fluid mixtures[16, 17, 18] adds the concentration of one of the components of the fluid mixture as an additional variable. For binary fluid convection, the separation ratio Ψ is a measure of the response of the concentration field to a temperature gradient. For negative Ψ, the concentration behaves so as to stabilize the system against convection; whereas, for positive Ψ, convection occurs at a lower value of Ra than for a pure fluid. Finally, the effects of rotation about a vertical axis have been studied for RBC[19], and about an axis perpendicular to the cylinder axis for TVF[20]. There are, of course, many other examples of experiments in pattern formation, but I mention these two here to emphasize the level of precision which can be achieved in these systems. For this thesis, the focus is on a third fluid dynamical system: electroconvection (EC)[21] in nematic liquid crystal (NLC),[22]. EC in NLC is a paradigm for pattern formation in an anisotropic medium. NLC molecules have an inherent orientational order, but no positional order. The direction of the average molecular alignment is given by the director, which is a function of space and time. A NLC cell consists of the NLC confined between two glass plates which have been properly treated so as to produce spatially uniform alignment of the director which is parallel to the plates (planar alignment). Transparent conductors are evaporated onto the glass plates, and an ac voltage is applied perpendicular to the plates. The cells and their construction are described in detail in Sec For EC to occur, the NLC must be doped with ionic impurities, and the anisotropy in the dielectric constant ɛ a = ɛ ɛ must be negative or only slightly positive. Here ɛ (ɛ ) is the dielectric constant for electric fields parallel (perpendicular) to the director. Above a critical value V c of the applied 6

16 ac voltage, a pattern forms which consists of fluid flow in the form of convection rolls and a corresponding periodic spatial variation of the director. A great variety of spatio-temporal structures is observed, including rolls[23, 24], traveling waves[24, 25, 26, 27, 28], defect chaos[26, 29], and chaos at onset[28]. The details of the instability mechanism are discussed in Sec Another paradigm for pattern formation in an anisotropic medium is RBC in NLC with planar alignment[30]. For both RBC in NLC with planar alignment and EC, the inherent anisotropy introduces a new class of patterns: oblique rolls. Consider a pattern which consists of a set of parallel, straight rolls with a wavenumber q. In an isotropic medium the roll axis is equally likely to orient in any direction. The selection of a particular direction by the roll pattern breaks the underlying symmetry of the system. For a NLC sample with planar alignment, the director has already defined a direction. Therefore, a given periodic roll state has a well defined wavevector with components q and p parallel and perpendicular to the director, respectively. States with p = 0 have the roll axis perpendicular, or normal, to the director and are referred to as normal rolls. States with p 0 are referred to as oblique rolls and have a nonzero angle between their wavevector and the director. As a control parameter is varied, a point where the system goes from having p = 0 to p 0 is called a Lifshitz point[31]. Because the director is not a vector, the two states whose wavevectors have the same magnitude and form an angle θ and π θ with respect to the director are degenerate. These states are referred to as the zag and zig oblique roll states. As I will show, the interaction of the two degenerate zig and zag roll states leads to a number of interesting patterns. The inherent anisotropy of these systems results in a greatly increased complexity of the fundamental equations. A theoretical description of RBC convection in NLC requires the usual equations for RBC with additional equations describing the director field. A description of EC requires equations for the velocity field, the director field, the charge density, and the electric fields. The anisotropic nature of the NLC translates into material parameters which 7

17 are tensor quantities. The thermal conductivity, electrical conductivity, dielectric constant, index of refraction, and viscosity all are tensor quantities. The director has elastic properties which introduce three elastic constants. This list is provided here to highlight the complexity of the systems. The details of these tensor quantities and NLC are discussed in Chapter 2. This increased complexity serves as an ideal test of the tools developed in the study of RBC and TVF. While some discrepancies still exist, the quantitative agreement between the predictions for RBC in NLC and the experimental studies is excellent[30, 32]. EC has proven to be a more difficult system, but the agreement between theory and experiment has been quite striking. A detailed linear stability analysis[33, 34] of the model for EC originally introduced by Helfrich[35] and to which a number of authors have contributed (see Ref. [36] and the references therein) quantitatively predicts both V c and the initial wavevector of the pattern. However, this model fails to predict the traveling-roll states (Hopf bifurcation) which are observed experimentally[24, 25, 26, 27, 28] and the experimentally observed backward bifurcation[37], i.e. discontinuous transition. The discrepancies between the theoretical predictions and the experimental observations provided one of three reasons for studying EC in new NLC s. A number of possible reasons for the discrepancy had been suggested, including imperfect director alignment and conductivity effects[33, 34, 38]. I surveyed EC in a number of new NLC s with the goal of studying these proposed effects. As I will report (Sec. 4.2), the standard treatment of the conductivity of the EC cells as a frequency independent parameter was found to be inadequate. The recently introduced weak-electrolyte model (WEM)[39] extended the previous theoretical work by including the dissociation-recombination of the ionic impurities and the effects of this on the conductivity. This model provides the first detailed description of the Hopf bifurcation. Section 2.3 is a review of this model, and Sec. 4.3 presents the experimental evidence for the WEM. This is an example of the close interaction between experiment and theory which is 8

18 still an essential element of the study of patterns. The second motivation for studying new NLC s was provided by experiments in EC which were the first direct observations of the effects of thermal noise below R c [37]. Even though fluctuations have a negative growth rate below R c, it is possible to observe fluctuating patterns for R < R c which are generated by a noise source which couples in to the system with the right frequency and wavelength. Even in the absence of experimental noise sources, thermal noise is always present in any system, and because it is white noise, always has components which can couple to the system. Generally, the effects of thermal noise are too small to be observed directly in RBC and TVF, but recent work in RBC in compressed gases has also been able to observe fluctuating patterns driven by thermal noise[40]. For both of these systems, the initial transition is backward, i.e. to a finite amplitude. One would like to study the effects of noise in a system with a forward bifurcation where the amplitude of the pattern grows continuously from zero at onset. Given that the detailed calculations[33, 34] for EC predicted a forward bifurcation, it was hoped that the right NLC could be found for which the transition was forward. Then, it would be possible to study the effects of noise on a forward bifurcation. Most of the drawbacks of EC are related to the difficulties involved in making EC cells, and these difficulties provided the final motivation for studying new materials. Precision experiments in EC are often hampered by poor alignment of the director and chemistry, in particular electrochemistry, in the cell which results in unwanted impurities. These difficulties, along with suggested methods for dealing with them, are discussed in detail in Sec The other difficulty with EC as a model system is the lack of detailed knowledge of the material parameters. As outlined in Sec. 2.3, quantitative comparison with theory requires the knowledge of at least 14 material parameters. There is one single-component NLC, MBBA (see Sec. 3.2 for details), for which ɛ a < 0 and for which the necessary parameters have been measured. Unfortunately, this material is also highly unstable making it less than ideal for precision experi- 9

19 ments. Therefore, the third and final reason for studying EC in new materials: finding a suitable replacement for MBBA for quantitative experiments. The lack of knowledge regarding the material parameters severely limited the usefulness of most of the new materials I was able to obtain. However, one material, I52 (see Sec. 3.2 for details), is being marketed as a benchmark material[41], and as such, a number of material parameters had been measured. The work on I52 has proven useful for comparisons to the WEM, and has revealed a wealth of interesting pattern dynamics. In addition, I extended the knowledge of the material parameters by studying the Frederiks transition in I52 which is discussed in Appendix A. As such, the results from EC in I52 are the focus of this thesis. I studied EC in I52 as a function of the applied voltage, the applied frequency, and the sample conductivity. A rich variety of patterns with many intriguing aspects (see Secs. 4.4 and 4.5) were observed. Particularly interesting are the states of spatio-temporal chaos and localized states observed near onset (Sec. 4.4). Neither of these classes of behavior occur in linear systems which are in thermodynamic equilibrium. Nor do they occur in systems with a generalized potential. Spatio-temporal chaos refers to variations of the pattern that are aperiodic in time and space and results from the inherent nonlinearities in the system[1]. I will report here on a system which has a continuous oscillatory instability leading directly to spatio-temporal chaos which in principle can be compared quantitatively to coupled complex Ginzburg Landau amplitude equations[42]. This has not been possible in other chaotic systems[1] because the primary bifurcation is backwards, because chaos occurs only after secondary bifurcations, or because the system size accessible to experiments is too small. Localized states consist of regions of the system which exhibit a pattern coexisting with regions of the uniform base state. This is not possible in a system governed by a generalized potential (e.g. the free energy of an equilibrium system), except at very special points in parameter space where there 10

20 are two equal minima of the potential. I will show that the localized states in this system are stable over a wide range of parameters. Localized states in one dimension have been studied extensively in the context of RBC in binary fluids[43, 44, 45, 46, 47, 48, 49] but stable localized states in two dimensions have not been observed[18]. The confirmation of the linear predictions of the WEM [39](see Sec. 4.3) presents the unique possibility of a quantitative theoretical description of these states in terms of coupled amplitude equations (see discussion in Chapter 5). Finally, two general advantages of EC are the cell size and the intrinsic time scales. The EC cells have thicknesses d in the range of 10 µm to 100 µm. This makes it possible to study EC systems with aspect ratios l/d, where l is a typical lateral dimension, on the order of By comparison, the limitations of RBC and TVF systems are such that the aspect ratios are typically of the order 10 to 20, and 100 is considered extremely large. The large aspect ratios of EC are significantly closer to the theoretical idealization of a system of infinite lateral extent[1]. In addition to achieving extremely large aspect ratios, the ability to control the size of the electrodes allows one to make systems of extremely small spatial extent, and to make effectively one dimensional systems. This control over the system size is potentially useful for the study of spatio-temporal chaos 2. The combination of thin cells and an electrical driving force results in intrinsic time scales for EC which are typically of the order 10 3 to 10 1 s (see Sec. 2.3). This is significantly faster than the typical RBC experiment which has time scales of the order of minutes. Many patterns display interesting temporal, as well as spatial, behavior. For these patterns, the relatively short times scales of EC can simplify the measurement of the temporal dynamics of these patterns by shortening the time required to compile reasonable statistics. 2 For a nice discussion of the role of system size in studies of spatio-temporal chaos, see Ref [1]. The possible uses for this particular system are discussed in Chapter 5. 11

21 Chapter 2 Theory Section 4.3 presents a quantitative comparison between the experimental measurements of the Hopf frequency and the predictions resulting from a linear stability analysis of the weak electrolyte model (WEM)[39]. In principle, it is possible to derive amplitude equations from the WEM which can be quantitatively compared with the results of Sec In addition, the design of the apparatus is dependent on a number of the physical properties of nematic liquid crystals. This chapter provides the necessary background for both an understanding of the WEM and the experimental details. Section 2.1 is an introduction to the techniques of linear stability analysis and amplitude equations. Section 2.2 is a review of the physical properties of nematic liquid crystals which are relevant to the design of the apparatus and the models of EC. Because the WEM is an extension of the model of EC first introduced by Helfrich[35], Sec. 2.3 has a discussion of the Carr-Helfrich model of EC[36], and it concludes with a discussion of the changes incorporated into the WEM[39]. None of the theoretical work described in this section is the result of my own work. The material in Sec. 2.2 is based mainly on Ref. [1] and the references therein. The discussion of NLC properties is based on Ref. [22]. The description of the WEM in Sec. 2.3 is a result of extensive discussions with Martin Treiber, and the actual theoretical work was carried 12

22 out by Martin Treiber and Lorenz Kramer[39]. I will denote vectors in bold-face b. When not otherwise specified, x refers to the position vector in three dimensions with coordinates x i given by (x, y, z), and the components b i of a general vector will refer to the same Cartesian coordinate system. With this notation, j = / x j. For vector expressions written in components, I will use the Einstein summation convention, the Kronecker delta function δ ij and the fully anti-symmetric tensor ɛ ijk for writing cross products. Because of the central role of the dielectric tensor ɛ ij, there is some chance of confusion. However, if it is not clear from context, the number of indices distinguishes the dielectric tensor from ɛ ijk. 2.1 Theoretical Tools In general, analytic solutions to nonlinear partial differential equations (PDE) are rare. Therefore, in order to understand the behavior involved in pattern formation, a number of methods have been developed. One option is to consider a limit for which the solutions can be calculated perturbatively. Linear stability analysis and amplitude equations are two such techniques. In linear stability analysis, infinitesimal perturbations to a known solution of the full nonlinear equations are considered. The full equations are expanded in terms of the perturbations, and only terms linear in the perturbation are considered. The stability of the known solution is determined by the growth (unstable) or decay (stable) of the perturbations. Amplitude equations represent a perturbative description of the dynamics near a linear instability. This section provides a definition of the essential elements of linear stability analysis and amplitude equations used in the rest of the thesis and follows the development of Ref. [1]. For a more detailed description, see Ref. [1] and the references therein. I will consider a general system of nonlinear (PDE) for the state U of the 13

23 system given by: t U = G[U, x U,..., R]. (2.1) Here R is the control parameter for the system. For EC, U includes the velocity field, the director field, and the ionic impurity concentrations, and G[U] includes the Navier-Stokes equation, the conservation of angular momentum, the equations of electrostatics, and charge conservation. For this case, R is the square of the applied voltage. It is assumed that the spatially uniform base state U 0 is known and that U 0 is stable for 0 R < R c. Here R c is the critical value of R for which U 0 first becomes unstable. For example, in EC this state consists of v = 0 and planar alignment of the director. To determine R c, the state U = U 0 + δu is considered where δu is an infinitesimal perturbation. For simplicity, I will consider the case of pattern formation in one dimension and U 0 = 0. The system is assumed to be spatially infinite so that the perturbation can be expanded in Fourier modes δu = u (o) q i exp(iq i x+λ i t). If we consider an individual Fourier mode with wavenumber q, the perturbed solution becomes U = U 0 + δu = u (o) exp(iqx + λt) with u (o) infinitesimal. This solution is plugged into Eq. 2.1, and the result is expanded to first order in u (o), t [u (o) exp(iqx + λt)] = D[u (o) exp(iqx + λt)], (2.2) where D is the linear operator which results from expanding Eq Then, Equation 2.2 is solved for the eigenvalues λ α (q, R) of D. The linear stability of U o is entirely determined by the largest λ α, denoted as λ(q, R). If for all q and a given value of R, Re λ(q, R) < 0 (Re denotes the real part), then all perturbations u (o) exp(iqx + λt) decay exponentially in time. (Because λ(q, R) is the largest eigenvalue, if the above statement is true for λ(q, R), then it must be true for all λ α (q, R).) Because this is a linear analysis, this situation corresponds to the state U 0 being linearly stable against all perturbation of the form δu u (o) q i exp(iq i x+λ i t). If for some q and R, Re λ(q, R) > 0, then the corresponding perturbation grows exponentially, 14

24 and U 0 is said to be linearly unstable to perturbations with a wavenumber q. These perturbations are often referred to as the unstable modes. As defined above, R c corresponds to the lowest value of R for which there is some value of q q c such that Re λ(q c, R c ) = 0. It should be emphasized that there is nothing special about the assumptions of U 0 = 0 and one dimension. In fact, the system doesn t even have to be a pattern forming system. Linear stability analysis is applicable to the stability of any known solution of a nonlinear set of equations. The procedure outlined above is still followed, only now U 0 0 is carried along as part of the calculation. The defining characteristics are the assumption of infinitesimal perturbations and the resulting linearization of the equations. Another option is to do nonlinear stability analysis which considers the stability of the known solution to finite perturbations. The generalization to two-dimensional instabilities with a wavevector q is also straightforward. An additional feature is that there are often two or more degenerate wavevectors for which Re λ(q c, R c ) first becomes zero. For example, in the case of RBC in an isotropic fluid, there is an infinite degeneracy as the wavevectors with magnitude q c and arbitrary orientation are equivalent. For EC, the two wavevectors which form an angle θ and π θ with respect to the director are degenerate. Introducing the reduced control parameter ɛ = (R R c )/R c, (2.3) the behavior of λ(q) for three different values of ɛ is plotted in Fig This shows the three possibilities as a function of ɛ. For ɛ < 0, Re λ(q, ɛ) < 0 for all q, and the uniform state is stable. At ɛ = 0, the uniform state becomes unstable to a mode with wavenumber q c. For ɛ > 0, there is a band of wavenumbers against which the pattern is unstable. Reference [1] discusses three types of transition based on the value of q c. I will only consider the type I transitions for which q c 0. The imaginary part of λ (Im λ) determines if the transition 15

25 is stationary, Im λ = 0, or oscillatory, Im λ 0. (An oscillatory transition is often referred to as a Hopf bifurcation. Bifurcations and bifurcation theory will be considered briefly in the discussion of amplitude equations.) In the case of the oscillatory transition, the Im λ is referred to as the Hopf frequency and denoted by ω. growth rate (arb. units) ε > 0 ε < 0 ε = 0-2 q (arb. units) Figure 2.1: The figure illustrates the three possibilities for λ(q, ɛ). The dotted line is for ɛ > 0 for which there is a range of q such that λ(q, ɛ) > 0. For ɛ = 0 (dashed line), a single q value, q c, exists such that λ(q c, ɛ = 0) = 0. For ɛ < 0 (dashed dotted line), all q have λ(q, ɛ) < 0. Another way to view the results of the linear stability analysis is the marginal stability curve defined by Re λ(q, ɛ) = 0. An example of the marginal stability curve for TVF (a one dimensional case)[50] is shown in Fig The minimum of the marginal stability curve is the point (q c, R c ). For a given value of ɛ, the uniform state is stable against perturbations with wavenumbers q that are located in regions outside the curve; whereas, perturbations with wavenumbers q located inside the curve grow exponentially. The exponential growth of the unstable modes can not continue forever, as the nonlinearities re- 16

26 sult in saturation of the growth and interactions between the unstable modes. Amplitude equations are one method used to describe the dynamics of the system near the point (q c, R c ) where ɛ is small. This is referred to as the weakly nonlinear regime, or weakly nonlinear analysis ε reduced wavenumber (q - q c ) / q c Figure 2.2: The solid line represents the marginal stability curve calculated for TVF for a system with a ratio of the inner to outer cylinder radius of 0.5. The uniform state is stable against perturbations outside the marginal stability curve (region 2). Perturbations inside the marginal stability curve (region 1) grow exponentially. For a stationary transition, the simplest saturated state U(x, t) is a periodic state with a single wavenumber q o and a uniform amplitude a o, U(x, t) = ψ(y, z)a o exp(iq o x). I am still assuming U 0 = 0 and treating the case of one dimensional patterns. Here ψ(y, z) is the eigenvector of the linearized equations and contains the dependence of the solution on y and z. For a Hopf bifurcation, the simplest state is a traveling wave of uniform amplitude with a frequency ω, U(x, t) = ψ(y, z)a o exp[i(q o x ωt)]. Generally, the amplitude of the q o mode has a slow spatial and temporal modulation due to interactions with the band of unstable modes. For the stationary transition, one writes 17

27 this state of the system as U(x, t) = [ψ(y, z) A(x, t)e iqox + cc] + O(ɛ), (2.4) where cc refers to complex conjugate. The amplitude A(x, t) is assumed to be complex with A(x, t) = A(x, t) exp[iφ(x, t)]. A constant φ(x, t) corresponds to a spatial translation of the state, and φ(x, t)/ x gives the local variation of the wavenumber away from q o. The equation governing the dynamics of A(x, t) is referred to as an amplitude equation. There are a number of quantitative methods for deriving amplitude equations from the full equations of motion, but one of the strengths of amplitude equations is that the form of the equation can be written down based on the symmetries of the problem. For the one dimensional case considered here, when the system is invariant under spatial translations and A(x, t) A(x, t), the amplitude equation to lowest nonlinear order is τ o t A(x, t) = ɛa(x, t) + ξo 2 2 x A(x, t) g A(x, t) 2 A(x, t). (2.5) The details of the full equations are contained entirely in the constants τ o, ξ o, q o, and g. As mentioned, there are methods for deriving the coefficients, but in the absence of known fundamental equations, they can be measured experimentally. The amplitude equations provide a natural connection to bifurcation theory which is worth briefly discussing as the terms forward and backward bifurcation are prevelant in this thesis. For this discussion, I will consider the case where A(x, t) = A(t) is spatially uniform and real. Equation 2.5 reduces to τ o t A(t) = ɛa(t) ga(t) 3. (2.6) In Figs. 2.3a and 2.3b, the steady state ( t A(t) = 0) solutions for A are plotted as a function of ɛ for the two cases of g > 0 and g < 0. The solid lines represent a stable solution and the dashed lines represent an unstable solution. In the language of bifurcation theory both of these transitions are considered 18

28 amplitude (arb. units) amplitude (arb. units) amplitude (arb. units) amplitude (arb. units) (a) (b) ε ε (c) (d) 0 0 ε 0 0 ε Figure 2.3: The steady state solutions of A as a function of ɛ for 4 standard bifurcations. The solid lines represent stable solutions, and the dashed lines represent unstable solutions. Notice, at ɛ = 0, the solution A = 0 (which is the uniform state) always becomes unstable. (a) is a forward pitchfork bifurcation. (b) is a backward pitchfork bifurcation. (c) shows the backward pitchfork bifurcation with a stabilizing quintic term added to Eq (d) is a backward transcritical bifurcation corresponding to a quadratic term being added to Eq For both (c) and (d), the dashed-dotted lines illustrate the jump in amplitude at ɛ = 0 and the hysteresis upon decreasing ɛ. pitchfork bifurcations because of the shape of the A 0 solution. This is the result of the A A symmetry which precludes a quadratic term in Eq 2.6. As ɛ is varied, if the solution goes continuously from one stable branch to another, as in Fig. 2.3a, the bifurcation is considered to be supercritical or forward. If there is a loss of stability, as in Fig. 2.3b, the bifurcation is subcritical or backward. In a real system for which g < 0, a stabilizing quintic 19

29 term is generally added to Eq. 2.6, and one has the behavior shown in Fig. 2.3c of a finite jump in A at onset (ɛ = 0). The other common bifurcation in pattern forming systems occurs when the A A symmetry is broken. Then one generically has a quadratic term in Eq The behavior of A for this case is shown in Fig. 2.3d. This is referred to as a transcritical bifurcation, and depending on the sign of the quadratic term will be subcritical or supercritical. Figure 2.3d shows an example of a backward transcritical bifurcation. As mentioned in the discussion of linear stability analysis, bifurcations are also stationary or oscillatory, and oscillatory bifurcations are referred to as Hopf bifurcations. The initial transitions in pattern forming systems are generally characterized by their bifurcation class. Linear stability is sufficient for determining if the bifurcation is a stationary or Hopf bifurcation. However, it is the nonlinear coefficients of the amplitude equation which determines if the bifurcation is forward or backward, and by definition, these can only be determined by doing the weakly nonlinear calculation. Recall that the two unexplained features of EC are the Hopf bifurcation, a linear property of the system, and the backward bifurcation, a nonlinear property of the system. Whether or not the bifurcation is forward or backward has a significant impact on the possibility of making quantitative experimental studies of the solutions of the amplitude equations. For the forward case, because the amplitude grows continuously from zero, there is a well defined range of ɛ for which the the amplitude equation is a valid perturbation expansion of the full equations. For the case of a backward bifurcation, because of the jump in amplitude at onset, the initial amplitude might be too large for a perturbation expansion to be valid. At this point in a discussion of pattern formation, it is natural to make an analogy between pattern formation and thermodynamic phase transitions. Equation 2.5 is identical to the Ginzburg-Landau equation introduced to describe continuous phase transitions, in particular the superconducting transi- 20

30 tion[51]. The forward bifurcation is analogous to a second order phase transition. The amplitude of the pattern grows continuously from zero and there is no hysteresis upon decreasing the control parameter ɛ. The backward bifurcation is analogous to a first order phase transition. The amplitude has a finite jump at onset, and the system shows hysteresis as ɛ is decreased (see Fig. 2.3). However, with respect to pattern formation Eq. 2.6 is a special case. It is derivable from a potential of the same form as the free energy used for the phase transition applications. As discussed in the introduction, the dynamics of most pattern forming systems are not described by the minimization of a generalized potential. The weakly nonlinear dynamics are still described by a amplitude equation, often called a generalized Ginzburg-Landau equation, but the amplitude equation is generally not derivable from a potential. The one dimensional complex Ginzburg-Landau equations (CGL) is one of the paradigms of generalized amplitude equations which is not derivable from a potential. The equation has the same form as Eq. 2.5, only now the coefficients are complex, as well as the amplitude. The CGL is the appropriate amplitude equation when the system has a Hopf bifurcation. Of particular interest for this thesis is the fact that the CGL in both one[52, 53, 54, 55, 56] and two dimensions[57] has solutions which exhibit spatio-temporal chaos and localized states. I will show in Sec. 4.4 that EC in I52 exhibits both of these interesting behaviors. In Sec. 4.3, I show that the initial transition is a Hopf bifurcation and described by the WEM. Therefore, in principle, the correct CGL for EC I52 can be derived, and its solutions can be compared with the experimental observations. In the case of EC where the initial transition is a Hopf bifurcation, the initial state involves an interaction of two modes. In this case, one has an amplitude equation for each mode of interest with nonlinear couplings between the equations. An example of such a coupling is a Hopf bifurcation in one dimension where U(x, t) is the superposition of right and left traveling waves U(x, t) = (ψ(y, z) [A r (x, t)e i(qox ωt) + A l (x, t)e i(qox+ωt) + cc) + O(ɛ). (2.7) 21

31 For a forward bifurcation, one has two coupled complex Ginzburg-Landau equations (CGL) τ o ( t A r + s o x A r ) = ɛa r + ξ 2 o(1 + c 1 ) 2 xa r g(1 ic 3 ) A r 2 A r g 1 (1 ic 2 ) A l 2 A r, τ o ( t A l + s o x A l ) = ɛa l + ξ 2 o (1 + c 1) 2 x A l (2.8) g(1 ic 3 ) A l 2 A l g 1 (1 ic 2 ) A r 2 A l. Here s o is the linear group speed, the complex nature of the coefficients is written out explicitly, and the coupling is given by the last term in each equation. For the case of EC studied here, the initial transition is to traveling oblique rolls, so there are actually four modes, i.e. the two degenerate oblique roll modes each can be right- or left-traveling. 2.2 Nematics Liquid crystals (LC) are a thermodynamic phase of matter which possesses both fluid like properties (they flow) and crystal like properties (long range order)[22]. For this work, the most important LC phase is the nematic liquid crystal (NLC), or simply nematics. NLC posses orientational order in the form of an average alignment of the molecules, but the molecules have no positional order. The direction of the average alignment of the molecules is referred to as the director. Upon heating a NLC, at a critical temperature (the clearing point), there is a transition to an isotropic fluid for which the orientational order is not present. Upon cooling, a NLC will often have a number of transitions to LC phases which posses various degrees of positional order (Smectic A, Smectic C, etc.), and eventually, the NLC will crystallize. I will focus exclusively on the nematic phase, and for a discussion of the other phases, see for instance Ref. [22] and the references therein. 22

32 This section focuses on the physics of NLC which are relevant to an understanding of EC and the experimental details: the anisotropy of the material parameters, the elastic properties of NLC, and the nature of the viscosity tensor. I will consider the hydrodynamic limit where the dynamics of fluid elements are considered. Fluid elements represent an average over a region large enough to contain a sufficient number of individual molecules so that a continuum treatment is valid, but small enough to be treated as a point on the scale of the entire fluid. The detailed molecular interactions will not be considered except so far as they result in an average alignment of the molecules. The director is represented as a unit vector n in the direction of the average molecular alignment where states with n and n are equivalent. The equivalence of n and n is based on the observation that the molecular alignment does not distinguish right from left. For example, even in a NLC with a permanent dipole, on average, the number of molecules with the dipole aligned in a given direction along the director is equal to the number of molecules aligned in the opposite direction. The director field is a function of space and time, and its dynamics must be included in a description of the NLC. Finally, NLC have a cylindrical symmetry, so the axes perpendicular to the director are equivalent. The material properties of a NLC are generally anisotropic and are represented by tensors. Because of the cylindrical symmetry of the NLC, the tensors are uniaxial with their two principle axes perpendicular and parallel to the director. The properties considered here are the magnetic susceptibility χ, the dielectric constant ɛ, the index of refraction, and the electrical and thermal conductivities σ and λ, and they can be written in the general form: b ij = b δ ij + b a n i n j, (2.9) where the n i are the components of the director. For a given material property b, b is the principle value perpendicular to the director, b is the principle value parallel to the director, and b a = b b is the anisotropy of the ma- 23

33 terial parameter. For example, ɛ a is the anisotropy of the dielectric constant, or the dielectric anisotropy. The anisotropy of the index of refraction is responsible for the ability to image EC patterns by the shadowgraph technique. This is discussed in Sec The viscosity is also a tensor, but because it involves the velocity and director field, it is not uniaxial. I will discuss the viscosity separately. Because of the anisotropy of χ and ɛ, the director can be aligned by an external magnetic or electric field. In the absence of other forces, for χ a > 0 (ɛ a > 0), the director aligns parallel to the applied magnetic (electric) field, and for χ a < 0 (ɛ a < 0), the director aligns perpendicular to the applied magnetic (electric) field. (In general, χ a > 0, but ɛ a < 0 or ɛ a > 0.) One can see this either by directly computing the torques on the director, or by considering the free energy density F. As an example, consider the case of a magnetic field H at an arbitrary angle to the director n. The field produces a magnetization M = χ H + χ a (H n)n which contributes a term to F : F = M dh = 1 2 χ H χ a(n H) 2. (2.10) For χ a > 0, the free energy is minimized for n and H parallel (n H a maximum), and for χ a < 0, it is minimized for n and H perpendicular (n H = 0). The alignment of the director by external fields is not a statement about the degree to which the individual molecules are aligned. The degree of molecular ordering is determined entirely by the distance from the nematic-isotropic transition as a function of temperature, and is not affected by the field strengths considered here. The experiments reported on here were conducted at temperatures well away from the clearing point where the ordering remains roughly constant. Therefore, unless specified otherwise, alignment will refer to alignment of the director and not of the individual molecules. Any description of EC is necessarily complicated, but two essential elements are the elastic and viscous properties of NLC. The elastic behavior is best 24

34 described in terms of a free energy density F. The three basic types of elastic deformations are splay, twist and bend. Examples of these three basic types of deformations are given in Fig The elastic contribution F d to F in terms (a) (b) (c) Figure 2.4: The solid lines in the three figures represent the director orientation. (a) example of a pure splay (S) deformation. (b) example of a pure bend (B) deformation. (c) example of a pure twist (T) deformation. The length of the lines in (c) represent the director at an angle to the plane of the paper. of these three types of deformations is given by: F d = 1 2 [K 11( n) 2 + K 22 [n ( n)] 2 + K 33 n ( n) 2 ), (2.11) where K 11, K 22 and K 33 are referred to as the splay, twist and bend elastic constants, respectively[58]. Generally, these constants are on the order of N with the twist elastic constant the smallest. The total free energy density F contains additional contributions from external magnetic and electric fields, F m = 1 2 χ a(n H) 2, (2.12) and F e = 1 2 ɛ oɛ a (n E) 2, (2.13) respectively. Here I have not included the contributions to F which are independent of n, and ɛ o is the dielectric constant of the vacuum. 25

35 The equilibrium condition for the bulk is that the free energy F = F dx is a minimum with respect to variations of n subject to the constrain n 2 = 1. Following Ref. [22], the equilibrium condition is found using the Euler- Lagrange equations with Lagrange multipliers. One finds [ ] δf j δf = λ(x)n i, (2.14) δ( j n i ) δn i where δ is used to represent a functional derivative. Here λ(x) is an arbitrary function of x (the required Lagrange multiplier). It is useful to define a new vector function h with components [ ] δf h i = j δ( j n i ) which is referred to as the molecular field. δf δn i (2.15) In terms of h, the equilibrium condition is [h i + λ(x)n i ] = 0, i.e. h and n must be parallel. Since the cross product of two parallel vectors is zero, a necessary but not sufficient condition for equilibrium, independent of λ(x), is n h = 0 (2.16) I will use the notation h d, h m, h e to refer to the contribution to h from the elastic contribution to the free energy density F d, the magnetic contribution F m, and the electric contribution F e, respectively. For EC, it is more useful to discuss distortions of the director in terms of torques rather than free energies. The molecular field h provides a natural way to connect the free energy density with the torque per unit volume Γ. One finds Γ = n h. (2.17) (For the rest of this section and Sec. 2.3, when I use torque Γ, I will be refering to the torque per unit volume.) This can be derived using the variational principle, but the easiest way to see this is to consider a specific example. For the equilibrium configuration of a NLC subjected to a uniform magnetic field 26

36 H and no electric field, the torques generated by the elastic distortion must cancel the magnetic torques. I will show that this comes out naturally from the equilibrium condition given in Eq. 2.16, and the association of n h with the total torque follows. For an arbitrary angle between n and H, the magnetization M = χ H + χ a (H n)n produces a magnetic torque Γ m = M H = χ a (n H) n H. (2.18) As discussed, the magnetic field contribution to the free energy is The equilibrium condition (Eq. 2.16) gives F m = 1 2 χ a(n H) 2 (2.19) n h d + n h m = 0 n h d = χ a (n H) n H = Γ m. (2.20) Because Eq represents the condition for equilibrium, n h d must be the torque generated by the elastic deformations which balances the magnetic torque Γ m. Equation 2.17 is a convenient formulation of the torque because, as I will show below, the viscous torque is naturally written in this form as well. One application of the elastic torque is the alignment of NLC in the absence of external fields[59]. For thin enough samples, if the orientation of the director is fixed at the surface of the sample, the elastic torques generate a unique orientation in the bulk. This is an essential feature of the construction of LC displays and electroconvection cells. The techniques of alignment are discussed in Sec The hydrodynamics of NLC are governed by the Navier Stokes equation ρ( t v i + (v )v i ) = j t ji + f i, (2.21) 27

37 where ρ is the density, t ji are the components of the stress energy tensor, v i are the components of the velocity, and f i are body forces. The viscous contribution to the stress tensor σ ij can be written[60, 61] as σ ij = α 1 [n i (A kl n l )n j ] + α 2 [n i N j ] + α 3 [n j N i ] + α 4 [A ij ] + α 5 [n i n k A kj ] + α 6 [n j n k A ki ]. (2.22) Here A ij = 1 ( 2 iv j + j v i ), N = dn/dt ω n, and ω = 1 v, with 2 dn/dt = n/ t + (v )n. The α i range from the order of 1 to 10 2 cp, and are known as the Leslie coefficients[61]. There are other formulations of σ ij, but this is the one which is usually used in the theory of EC. There are only five independent α i because there is the Onsager relation α 6 α 5 = α 2 + α 3. There are three special cases of director orientation and fluid velocity for which the viscous contribution to Eq can be reduced to the usual form for an isotropic fluid of η 2 v. The three cases are shown schematically in Fig In all three cases, the flow consists of a single nonzero velocity component v i which has a single nonzero derivative j v i where j i. The η s are linear combinations of the α s. For the director aligned perpendicular to both v i and x j, For the director aligned parallel to v i, For the director aligned parallel to x j, η η a = α 4 /2. (2.23) η η b = (α 4 + α 3 + α 6 )/2. (2.24) η η c = (α 4 + α 5 α 2 )/2. (2.25) This is the case that is relevant for the simple description of the onset of EC given in Sec

38 (a) (b) (c) x i x j Figure 2.5: Three special orientations of the director and flow for which a shear viscosity η is defined. The arrows represent the direction of the velocity. (a) The orientation for measuring η a where the director is coming out of the plane of the paper (the solid dot). (b) The orientation for measuring η b where the director orientation is given by the ellipsoid. (c) The orientation for measuring η c where the director is given by the ellipsoid. The other two important viscosities are the rotational viscosities γ 1 = α 3 α 2 and γ 2 = α 6 α 5 = α 2 + α 3. The γ i are the linear combinations of the α s which appear in the expression for the viscous torque Γ v on the director field due to viscous stresses. The total torque Γ is equal to the rate of change of the angular momentum L, Γ = L = d dt dx (x ρv). (2.26) Using Eq to substitute for ρ d v, the viscous contribution to the torque, dt Γ v is just the antisymmetric part of the viscous stress tensor: Γ i = ɛ ijk σ jk = ɛ ijk n j (γ 1 N k + γ 2 n l A lk ), (2.27) or in vector notation, Γ v = n (γ 1 N + γ 2 n A). (2.28) Two important examples of the viscous torque are illustrated in Fig For both cases, the coordinate system is as shown, with the y axis chosen to give the usual right handed coordinate system. Also, v z (x) is the only nonzero 29

39 (a) α 3 < 0 α 3 > 0 z (b) α 2 > 0 x α 2 < 0 Figure 2.6: (a) The torque on the director (show by the ellipsoid) for the case of the director aligned parallel to the velocity component (v z ) and perpendicular to the shear x v z 0. The direction of the torque depends on the sign of α 3. The velocity field is show schematically by the two solid straight lines, and the initial direction of motion of the director is given by the curved dashed lines for the two possible signs of α 3. (b) The case when the director is parallel to the shear and perpendicular to the direction of the velocity. Here the sign of α 2 is relevant. The coordinate system is the same for both figures and the y axis is such that it is a usual right handed coordinate system. component of v with x v z > 0. For both cases, there is only a y component of the torque. For the situation shown in Fig. 2.6a, Eq gives Γ y = α 3 x v z, and for the situation shown in Fig. 2.6b, Eq gives Γ y = α 2 x v z. The sign of α 3 or α 2 determines the direction of rotation for the director as shown in Fig For rod-like NLC, α 2 < 0, but α 3 can be either positive or negative. The situation in Fig. 2.6b is the case that is considered in the simple model of EC discussed in Sec Two main results of this section are used in the description of EC: the conservation of momentum given by the Navier-Stokes equations (Eq. 2.21) and the conservation of angular momentum given by the balance of torques, 30

40 Γ = 0. Taking advantage of the similar forms of Eq and Eq. 2.17, the total torque Γ can be written as Γ = n S, (2.29) where S is defined 1 as S = h (γ 1 N + γ 2 n A). (2.30) 2.3 Electroconvection As discussed in the introduction, one of the major motivations for studying EC in a new NLC is the attempt to resolve the discrepancies between experimental observations and the predictions of the detailed analysis of EC presented in Ref. [33, 34]. The model used in Ref. [33, 34] is the full three dimensional version of the unidimensional model originally introduced by Helfrich[35]. I will refer to the set of equations and assumptions presented in Ref. [33, 34] as the standard model (SM). The mechanism which produces EC in the SM is referred to as the Carr-Helfrich mechanism[36]. The essential elements of the Carr-Helfrich mechanism are planar alignment of the director, an electric field applied perpendicular to the director, ionic impurities in the NLC to provide a nonzero conductivity σ, and a positive σ a = σ σ. In practice, an ac electric field is used to prevent charge accumulation at the boundaries, but for low frequencies the basic mechanism is the same in the ac case as in the dc case. Also, ɛ a = ɛ ɛ is generally taken to be negative, but it is possible for ɛ a to be slightly positive. If ɛ a is large and positive, the Frederiks transition dominates (see Appendix A). The weak-electrolyte model (WEM), considered in detail in Ref. [39], is an extension of the SM which addresses the origin of the observed Hopf bifurca- 1 Equation 2.29 is the same as Eq. (2.4) in Ref. [33]. But, because the definition of h follows Ref. [22], careful tracking of minus signs must be used when comparing the two equations. 31

41 tion. The essential difference between the SM and the WEM is the treatment of the ionic impurities and their resulting conductivity. Both models assume equal numbers of positive and negative ions, so the fluid is electrically neutral. Under the influence of the electric field, local variations of the charge density will develop[62] which provide a body force on the fluid. The difference between the two models is the treatment of the conductivity. In the SM, the conductivity of the sample is frequency independent, and the electric fields and currents are related by Ohm s Law. In the WEM, a dissociationrecombination reaction for the positive and negative ionic species is considered, and the resulting conductivity is an additional dynamical variable. Because the Carr-Helfrich mechanism is central to both the SM and the WEM, I will illustrate it using the unidimensional model of Helfrich. Then, I will discuss the additional effects included in the WEM which produce a Hopf bifurcation. The equations of the SM are the Navier-Stokes equation (Eq. 2.21) with the body force f i = ρ e (x)e i (x), the conservation of angular momentum (Eq. 2.29), incompressibility ( v = 0), the equations of electrostatics, [ɛ 0 ɛe(x)] = ρ e (x), E(x) = 0, (2.31) and charge conservation j(x) + t ρ e (x) = 0. (2.32) Here ρ e (x) is the charge density of the ionic impurities, E(x) is the applied electric field, ɛ is the dielectric tensor, ɛ 0 is the dielectric constant of the vacuum, and j(x) = σe(x) + ρ e (x)v (2.33) is the electric current. I have explicitly written the x dependence of ρ e to emphasize its role as a dynamical variable; whereas, the conductivity σ is a constant (tensor) parameter. Before discussing the unidimensional model introduced by Helfrich, some general comments about EC with an ac electric field are required. There are 32

42 two regimes of EC with an applied ac electric field (E(x, t) = E(x) cos(ωt)): the conduction regime and the dielectric regime. For low enough applied frequencies (the conduction regime), the charge density ρ e (x, t) is able to follow the applied field (i.e. ρ e (x, t) = ρ e (x) cos(ωt)), and the director and velocity are essentially time independent. For high frequencies (the dielectric regime), the director and velocity oscillate with the applied frequency Ω, and the horizontal variation of the charge density is essentially constant in time. The crossover between the two regimes is known as the cutoff frequency. The cutoff frequency is set by the two time scales of the SM[33]: a charge-relaxation time, τ q = ɛ 0 ɛ /σ, and a director-relaxation time, τ d = γ 1 d 2 /[(K 11 + K 33 )π 2 ]. The conduction regime corresponds roughly to the conditions Ωτ d 1 and τ d τ q. The cutoff frequency scales with the conductivity of the sample and ranges from a few Hz for very pure samples to khz. All of the work presented here, both the theory and experiments, has been on EC in the conduction regime. Because the charge follows the applied field, the description of the Carr-Helfrich mechanism for the dc case carries over naturally to the ac case. Also, for the cases where there is a Hopf bifurcation, the Hopf frequency ranges from 0.06 to 0.2 Hz. For comparison, in the dielectric regime, the director oscillates with the frequency of the applied voltage, which for my experiments ranges from 25 to 200 Hz. The situation considered by Helfrich[35] is shown schematically in Fig The director is assumed to be confined to the x-z plane, initially with planar (n = [1, 0, 0]) alignment in the x direction. A fluctuation of the director n = [1, 0, φ(x)] is considered, where φ(x) is assumed to be small. There is an applied dc electric field in the z-direction E z which is assumed to be constant. I will follow closely the treatment of Ref. [63] which makes the simplification of assuming ɛ a = 0. Helfrich s[35] original work accounts for the more general case of ɛ a 0. This necessitates including the dielectric torques in addition to the elastic and viscous torques. The quantitative result is changed, but the qualitative features of the instability are essentially the 33

43 z E z x φ α 2 < 0 Figure 2.7: A schematic diagram of the Carr-Helfrich mechanism. The director (shown by the ellipsoid) is assumed to have a small fluctuation (φ) away from its initially planar alignment (horizontal dashed line). A constant electric field E z is applied in the positive z-direction. The current induced in the x- direction produces a separation of positive and negative ionic impurities which then experience a force in the z-direction. The resulting shear flow is shown by the arrows. The situation corresponds to Fig. 2.6b with α 2 < 0, and generates a destabilizing torque. The viscous torque is balanced by the elastic torque induced by distorting the director. same. The relevant equations are the ones listed above for the SM. All of the variables are only considered to have variations in the x direction; hence, the name unidimensional. For the director orientated at an angle φ, E z produces a current in the x-direction given by I x = σ a E z φ(x) which results in a charge density ρ e (x) due to the separation of the positive and negative ions. The charge separation produces a field in the x-direction which generates an opposing current I x = 34

44 σ E x. For the steady state case, charge conservation gives x I x = 0, or σ x E x = σ a E z x φ (2.34) (Recall, everything is constrained to one dimension, and E z is constant to lowest order in φ.) Using Poisson s equation (ɛe) = ρ e, the space charge density ρ e (x) generated by E z is ρ e = ɛ 0 ɛ x E x = ɛ 0 ɛ σ a σ E z x φ (2.35) There is now a body force on the fluid of E z ρ e. The velocity field which results from the body force is equivalent to the example in Fig. 2.5c. Therefore, for steady state flow ( t v = 0), the Navier Stokes equation gives (assuming that the viscous terms dominate over the term (v )v) or 0 = η c 2 x v z + E z ρ e, (2.36) η c 2 x v z = E z ρ e = E 2 z ɛ 0ɛ σ a σ x φ, (2.37) with η c = (α 4 + α 5 α 2 )/2 as defined in Eq For both φ(x) and v z (x), only a single Fourier mode with wavenumber q is considered, so 2 xv z = iq x v z. Therefore, Eq becomes Recalling Eq and Fig. 2.6b, x v z iqη c x v z = E 2 zɛ 0 ɛ σ a σ x φ. (2.38) produces a torque on the director Γ v = (0, α 2 x v z, 0). For NLC used in EC, α 2 < 0, so the viscous torque is always destabilizing (see Fig. 2.6b). This torque is opposed by the elastic torque 2 Γ d. In the absence of magnetic fields and for ɛ a = 0, Γ d = (n h) = 2 The result for Γ d is not immediately obvious, but it is straight forward to work out from the elastic free energy, Eq. 2.11, and use Eq to compute the torque. If ɛ a 0, the additional torque due to the electric field is accounted for by including F e when calculating h. 35

45 (0, (K 33 2 xφ), 0) (see Eq ). As E z is increased, a critical value E c will be reached at which the total torque Γ = Γ d + Γ v = 0, which gives K 33 2 x φ = α 2 x v z. (2.39) Combining Eq and Eq. 2.38, using 2 xφ = iq x φ, and eliminating x φ gives E 2 c = K 33σ η c q 2 c ( α 2 )ɛ 0 ɛσ a, (2.40) where q c is the critical wavenumber. Experiments suggest q c scales with the cell thickness d, and in fact, the detailed linear stability analysis[33, 34] predicts this scaling with d. This is used to convert the expression for E c to one for V c = E c d. Taking q c π/d, V 2 c = πk 33σ η c ( α 2 )ɛ 0 ɛσ a. (2.41) Despite the fact that the value of V c derived from this simple model is too low, the three essential elements of the Carr-Helfrich mechanism are illustrated by this calculation. First, the generation of a charge density ρ e (x) by the anisotropic conductivity is essential. Second, from Eq. 2.35, the generation of the charge density requires fluctuations of the director with x φ 0. The fluctuations of the director are believed to be driven by thermal noise[37]. Finally, because of the d dependence of the critical field, the actual control parameter is the voltage squared, V 2. Even when the full equations of motion are considered, the mechanism remains essentially the same. A Hopf bifurcation is not predicted for any combination of the parameters. The addition of the conductivity as a dynamical variable in the WEM[39] provides a mechanism for the Hopf bifurcation. The WEM starts with the two species of ionic charge carriers[64] with charges ±e and couples them through a simple dissociation-recombination reaction[65]. The charges are considered to have number densities n + (x, t) and n (x, t), and constant, possibly different, mobility tensors µ ± with principal values perpendicular and parallel to the director, µ ± and µ±, respectively. 36

46 The µ ± (, ) are defined in terms of the ion s steady-state velocities in a quiescent fluid with an applied electric field, v ± (, ) = µ± (, ) E (, ). For simplicity, the ratio µ /µ is assumed to be the same for both species. The WEM expresses the total space-charge density ρ(x, t), which is the same as in the SM, as ρ(x, t) = e[n + (x, t) n (x, t)]. Unlike for the SM, the local conductivity tensor is an additional variable with components given by σ ij (x, t) = σ(x, t)[δ ij + n i n j (µ /µ 1)]. Here σ(x, t) = e[µ + n+ (x, t) + µ n (x, t)], The basic equations of the WEM are two equations which couple σ(x, t) and ρ(x, t) plus the SM equations for the director n and fluid velocity v. The new equations for σ(x, t) and ρ(x, t) are given in Ref. [39]. The charge density is coupled to n and v as in the SM, and σ(x, t) couples to n and v via the conductivity tensor and the ρ e (x)v part of the current. The actual boundary conditions in the experiment are unknown, but using the frequency dependence of the capacitance, I have experimentally determined that the thickness of the charged boundary layers at the electrodes is small compared to the thickness of the cell (see Sec. 4.3). In this limit, the model is insensitive to the details of the boundary conditions[39]. As discussed, the SM has two relevant time scales: τ q and τ d. The consideration of the individual ions introduces two new time scales: a recombination time τ rec = 1/(2k r n 0 ) for the dissociation-recombination reaction, and a migration time τ mig = d 2 /[π 2 (µ + +µ )V (0) ] for a charge to traverse the cell under an applied voltage V (0) which is of order the critical voltage for low external frequencies. Here k r is the recombination rate of the ions, n 0 is the equilibrium number density of either species of ion, and V (0) = π[(k 11 + K 33 )/(τ q σ a )] 1/2. The SM is recovered in the limit of τ rec /τ q 0 and τ mig /τ q. Reference [39] applied the WEM to the case of normal rolls, and ref. [42] applied the WEM to the more general case of oblique rolls. I will only consider here the final result. The calculation can be reduced to an equation for the amplitude of the local deviation of the conductivity from its equilibrium value A σ (t), and the critical SM mode A n (t) which includes the deformation of 37

47 the director. eliminated.) (The velocity field and charge density are both adiabatically A σ = λ σ (R)A σ α 2 Rσ a (eff) (σ τ d ) 1 A n, ) A n = Rσ 2 C 1+(βΩτ q) 2 Aσ + λ n (R)A n, ( σ a (eff) τ d (2.42) Here R = (V/V (0) ) 2 is the control parameter, and Ω is the angular frequency of the applied voltage. The diagonal coefficients λ σ (R) = τrec 1 1 τd (R α2 β)/[1+ (βωτ q ) 2 ] < 0 and λ n (R) = ɛ/τ0 SM are the growth rates of the σ mode and the SM amplitude, respectively. Here τ0 SM is the correlation time of the SM amplitude equation (O(τ d )), ɛ = R/R c 1 is the relative distance from threshold R c, and β = [(1 + ɛ a /ɛ )(q c d) 2 + (p c d) 2 + 1]/[(1 + σ a /σ )(q c d) 2 + (p c d) 2 + 1] The dimensionless quantity C contains only SM quantities[39] and is of order unity for our experiments[66]. The effective conductivity anisotropy σ a (eff) σ a [β ɛ a σ /(σ a ɛ )]/[1 + (βωτ q ) 2 ] is proportional to the charge produced by the Carr-Helfrich mechanism[36]. The important coupling between A σ and A n is proportional to α 2 = µ + µ γ 1π 2 /(σ a d 2 ). (2.43) In the limit of the SM, τ mig which gives α 2 0, and the two modes are decoupled. The other condition for recovering the SM, τ rec 0, gives λ σ (R) τ 1 rec which fixes the conductivity at its equilibrium value, A σ = 0. There are two main predictions of the WEM. For sufficiently high mobilities and small recombination rates, there is a nonzero Hopf frequency ω at threshold (the imaginary part of the growth rate of the 2x2 equations is nonzero for zero real part), ω = ω 1 (λ σ (R c )/ ω) 2, where ω = C ( ) 3 π Rc(K 11 +K 33 ) µ + d 1+(βΩτ q) 2 µ γ 1 σ a. (2.44) The condition for a nonzero Hopf frequency can be written λ σ (R c ) < ω. By making the appropriate substitutions, this sets an upper bound on the 38

48 combination ( τd τ q ) 1/2 τ mig τ rec (σ d 2 ) 3/2. (2.45) The result that the condition for a Hopf frequency scales with σ d 2 is consistent with experimental observations that the Hopf bifurcation tends to occur in thin cells and at low frequencies[26, 27]. Because both τ d /τ q and τ mig /τ rec scale as σ d 2, it is also expected that the regions of existence for the nonlinear patterns might scale with σ d 2 (see Sec. 4.4 and Sec. 4.5). There is an upward shift of ɛ relative to the SM prediction. For nonzero ω (λ σ (R c ) < ω), the shift is ɛ = Rc RSM c R SM c = τ SM 0 λ σ (R c ), (2.46) and for ω = 0 (λ σ (R c ) > ω), the shift is ɛ = ω 2 τ SM 0 /λ σ (R c ). (2.47) These conditions are equivalent when λ σ (R c ) = ω: the crossover from ω = 0 to a nonzero ω. Therefore, ɛ = ωτ SM 0 represents the maximum possible deviation between the SM and the WEM. The net result is that within the accuracy of the measured material parameters, the WEM predicts the SM values for V c. Linear stability analysis also predicts the wavevector of the pattern at onset, and in particular for EC, the angle Θ between the wavevector of the pattern and the director is computed. The WEM also recovers the SM prediction for Θ. Both the SM model prediction of V c and Θ are found to agree with experiments[33, 34]. In Sec. 4.3, I present the results of a quantitative calculation of the WEM provided by Martin Treiber and compare them to my experimental measurements. Returning briefly to Eq. 2.42, the stabilizing coupling of A σ and A n provides the mechanism of the Hopf bifurcation. From Eq. 2.42, the linear growth rate λ for the most unstable mode is λ = (λ σ + λ n )/2 ± [ (λ σ λ n ) 2 /4 B ] 1/2, (2.48) 39

49 where B > 0 is the product of the off diagonal coefficients in Eq. 2.42, both of which are positive. It is the negative cross-coupling that allows for the possibility of a nonzero Imλ. Had the coupling been destabilizing (one of the coefficients negative), then B < 0 and λ is always real. Physically, for large enough α 2, the growth of A n causes A σ to become negative. (Recall that λ σ (R) < 0 and A σ is the deviation from the equilibrium conductivity.) A negative A σ retards the growth of A n, stabilizing the director mode. When the A σ and A n have sufficiently different relaxation time scales, the feedback of A σ is out of phase with the growth of A n and establishes an oscillation. I will discuss this mechanism and its relation to Hopf bifurcations in other systems again in Chapter 5. 40

50 Chapter 3 Experimental Details There are three main elements to the experiment: the apparatus, the NLC s, and construction of the NLC cells. There are currently two EC apparatus in use in our labs. They follow the same general design except for the temperature regulation. The apparatus described here was designed for room temperature operation. The second apparatus is described in Ref. [67] and was designed to operate at temperatures up to 200 C. Provided here is a detailed description of the methods of cell construction. Included are comments on the methods which were found not to work which will help to guide future modifications to the cell design. In the section on the NLC s, I discuss the relative merits of four liquid crystals. Included is a discussion of the methods used to dope the various NLC s. 3.1 Apparatus The apparatus consists of three main parts, the imaging system, temperature control stage, and electronic controls, which are shown schematically in Fig The details of each part will be discussed separately. An expanded view of the imaging system and temperature control stage is shown in Fig The temperature control stage also serves as the cell holder. The electronics 41

51 computer video in to pceye board ac signal out from wavetec DIO lines on labmaster A/D on daughter board multi-meter external electronics box digital in to DAC ac in to DAC ac out to cell (Fig. 3.9) current to voltage conversion (Fig. 3.10) analog temperature control Figure 3.1: Schematic of the apparatus. are used for generating the applied ac voltage and measuring the conductivity of the cells Shadowgraph The shadowgraph technique[68, 69] is a well developed method of visualizing variations in the dielectric constant ɛ, or index of refraction n, of a fluid. Consider a RBC or EC cell which consists of fluid confined between two plates. We take the z-direction to be perpendicular to the plates, and the x direction is taken parallel to the wavevector of the pattern. There is a variation of ɛ or n associated with the pattern, and for many situations, geometric optics is sufficient to explain the shadowgraph signal produced by this variation. Because rays are bent toward regions of high n, light propagating in the z- 42

52 CCD Camera shadow graph system Nikon Camera Lens Temperature- Control Stage lower lens cell electronics in/out Translation Stage light source Figure 3.2: Schematic of the imaging system (shadowgraph), light source, and temperature control stage. direction through the cell is focused in a plane above the cell. This creates an image or shadow of the pattern in which the bright regions correspond to the regions of the fluid with a relatively large n and the dark regions to regions 43

53 with a relatively low n. (For NLC, there is an additional effect because of the inherent anisotropy of the material which will be discussed later.) Fermat s principle (minimization of the optical path) provides a quantitative calculation of the shadowgraph signal for the geometric optics limit[68]. This calculation clearly breaks down at the caustics, i.e. locations where the intensity is predicted to be infinite. For any physical system, diffraction effects prevent the intensity from actually diverging. The distance to the first caustic above the cell is referred to as the focal length of the pattern. A more complete calculation which includes diffraction effects has been carried out recently[69]. This physical optics approach recovers the predictions of geometric optics in the correct limit and provides a nice intuitive picture of the shadowgraph method in terms of diffraction. The horizontal variation of n generated by the pattern is equivalent to a phase grating and/or an amplitude grating. An incoming beam of light is diffracted by this grating, and the interference of the diffracted beams with the main beam generates the intensity pattern. I highlight here some of the relevant results of both the geometric[68] and physical optics[69] calculations for RBC and EC. The shadowgraph apparatus used in the EC experiments is a modification of the RBC, and the modifications are best understood by comparing the two systems. For the case of thermal convection in simple fluids, the variation of n is directly proportional to variations in the fluid s density induced by temperature variations. Therefore, quantitative measurements of the temperature field are possible using the shadowgraph technique. Because the thermal convection rolls consist of regions of hot upward-traveling fluid alternating with regions of cold-downward traveling fluid, the shadowgraph provides a qualitative image of the vertical average of the velocity field as well. The one to one correspondence between the temperature variation and the variation of n yields the nice result that the wavelength of the pattern in the shadowgraph image corresponds to the wavelength of the physical pattern. The case of NLC is more complicated. 44

54 As discussed in Sec. 2.2, NLC have a tensor ɛ and n with principle axes parallel and perpendicular to the director. Light polarized parallel to the director is called the extraordinary beam and has an index of refraction n e = n = ɛ. Light polarized perpendicular to the director is referred to as the ordinary beam and has an index of refraction n o = n = ɛ. In EC experiments, one uses light polarized in the direction of the undistorted director. When a pattern is present, the local director gains a z-component and makes an angle φ with respect to the horizontal. In this case, the light sees an effective index of refraction ñ = n o n e (n 2 o cos 2 (φ) + n 2 e sin 2 (φ)) 1/2. It is the horizontal variations in ñ which produces a diffraction grating. Because ñ depends on the square of cos(φ) and sin(φ), director distortions with an angle φ and π φ produce equivalent ñ. However, they represent half a wavelength in terms of the pattern (see Fig. 3.3). Therefore, unlike the case of RBC, the phase grating generated by the pattern in EC has half the wavelength of the pattern. Furthermore, the intensity of the shadowgraph signal due to the phase grating is proportional to the square of the director angle. Hence, in EC, the phase grating contribution to the shadowgraph signal is often referred to as the nonlinear or quadratic shadowgraph effect. As first pointed out by Rasenat, et. al.[68], there is a second contribution to the formation of a shadowgraph image in EC which is equivalent to an amplitude grating. This contribution is linear in the director angle and produces intensity variations which have the same wavelength as the pattern. This is usually referred to as the linear effect. As with the phase grating, there is a geometric optics explanation of this contribution. When φ is nonzero, the light traveling through the cell has a polarization component parallel and perpendicular to the director as shown in Fig 3.4. Because the velocities of these two components are not equal, the usual Hugyen s construction results in wavelets which are elongated into ellipsoids instead of the usual spherical wavelets. The envelope of the ellipsoids (shown by the dashed dotted line in Fig. 3.4) still corresponds to a plane wave which is essentially parallel to the incident wave. 45

55 (a) (b) φ modulation 0 x Figure 3.3: (a) schematically shows a director variation and the corresponding effect on a parallel beam of light. Notice, there are two effects represented by the solid and dashed rays above the cell. The dashed rays represent a pure focusing due to the phase grating effects. The solid lines represent a shift in the rays which corresponds to the amplitude grating effect. (b) shows the director variation (solid line) and the corresponding index of refraction variation (dashed line) which has half the wavelength for the situation shown in (a). However, the direction of energy propagation (the ray direction) is no longer normal to the wavefront. The ray direction is parallel to the line connecting the origin of the wavelet and the point of tangency with the planar wavefront (shown by the dashed line in Fig. 3.4). Because for most liquid crystals, n e is greater than n o, the component of polarization perpendicular to the director has the greater velocity which results in a shift of the rays along the director (Fig. 3.3). This produces a horizontal intensity variation of the light leaving the cell, but the direction of travel of the rays is unaltered (see Fig. 3.3). Because there is no focusing of the light, the resulting intensity modulation I/I, has a constant amplitude as a function of z in the geometrical optics limit[68]. This is not the case of course in reality. Here I use the expression for I/I given in Ref. [68], I/I 4φ ˆn (d/λ), (3.1) 1 + ˆn where ˆn = 1 (n e /n o ) 2, d is the cell thickness, and λ is the wavelength of the pattern. In contrast, the phase grating effect is a pure focusing effect within geometrical optics and produces a modulation of the light which has a zero 46

56 amplitude at the surface of the cell and reaches a maximum at the caustic. Therefore, in the geometrical optics limit, the image near the surface of the cell is dominated by the linear effects. In practice, for extremely strong patterns, even right near the surface of the cell the quadratic effects will often dominate the image. However, the various effects can be separated by Fourier analysis. k director director v e v o polarization direction Figure 3.4: The Hugyen s construction explanation of the shift of the optical rays as they travel through the EC cell. The initial polarization is shown in the lower left. The director is given by the solid line. The ellipsoid for n e > n o is shown with the corresponding ray direction as a dashed line. The dashed dotted line gives the wavefront formed by the wavelets. The physical optics calculation predicts similar amplitude and phase grating effects. As in geometric optic, the phase grating produces an intensity variation which is proportional to the square of the director variation and which has twice the wavenumber of the underlying pattern. The amplitude grating effect is linear in the director variation and has the wavelength of the underlying pattern. However, instead of being constant as a function of z, the amplitude of the intensity modulation due to the linear effect has a cos z dependence, where z = 0 at the top surface of the cell. The amplitude of the 47

57 quadratic effect has a sin z dependence. So, the linear effect still dominates when one focuses near the top of the cell. This behavior has been confirmed qualitatively, but quantitative comparisons with the physical optics calculation are still needed. Because of the different imaging properties of RBC patterns and EC, the principles behind the operation of the shadowgraph apparatus are quite different, even though the designs are essentially identical. The goal in both cases is to produce as faithful a representation of the pattern as possible. For RBC, there is no image at the cell surface. One must image the shadow which exists at some distance above the cell. For typical convection patterns, this requires imaging at a distance which is meters away from the cell. An objective lens produces an image of the intensity modulation which is either projected on a screen, or preferably, imaged by a camera/lens system similar to the one shown in Fig Notice, the objective lens creates the image of the shadow between itself and its focal plane. This is shown schematically in Fig shadow light cell objective lens focal plane image Figure 3.5: Top figure schematically shows the imaging method for RBC. Here the objective lens is used to move the shadow in from a plane near infinity to a plane within the lens focal length. The bottom figure shows the method used in EC. Here one wants to image the cell itself, not a shadow which exists at a large distance away from the cell. 48

58 For EC, one generally wants to image near the surface of the cell where the linear effect dominates. The intensity modulation of the light will have the same wavelength as the pattern, and the signal is linear in the director angle. Even if we wanted to image the shadow as in RBC, one still focuses close to the top of the cell because the focal length of the pattern for the case of EC is centimeters or less, not meters. Also, the wavelength of the patterns range from 10 to 100 µm, so high magnification and good resolution is required. In order to image the surface of the cell with sufficient magnification, most EC experiments use a commercial microscope. This limits one to objective lenses with a short focal length or very expensive microscopes. In our case, we desired a space of at least 5 cm above the cell for the temperature control stage. We found that the basic RBC shadowgraph design with a 5 cm focal length objective lens and the cell located about 6.5 cm away from the lens acts as a microscope/telescope with the required magnification (up to 30x) and resolution ( 4 µm based on the Rayleigh criteria). The shadowgraph apparatus used in the EC experiments is a modified version of the RBC shadowgraph tower[13]. In thermal convection, the bottom plate is generally opaque. Therefore, RBC is imaged by illuminating the sample from above and using a reflecting bottom plate. This requires that a light source, beam splitter, etc. be included as part of the shadowgraph tower. In EC, we do not have this limitation because the cells are transparent, and we can illuminate the sample from below. Figure 3.2 shows that the light source is a separate unit mounted below the sample, and the shadowgraph tower above the cell only requires an objective lens, camera lens and camera. The specifics of the system are as follows. The light source is a separate unit mounted below the cell. A parallel beam is generated by locating a point source at the focal point of a lens. In the initial design, the point source was made by mounting a 40 µm diameter pinhole in front of a high-power red LED. The plastic lens of the LED had been machined away so that the pinhole was located approximately 250 µm from the active element of the LED. The 49

59 apparatus now uses a LED coupled into a single mode optical fiber[72] with a diameter of 50 µm. The optical fiber is 2 m long with a 0.22 numerical aperture and SmA connectors on each end. The LED provides 660 nm light and is run above the rated current at roughly 100 ma [72]. The optical fiber setup provides more uniform illumination than the pinhole/led combination with essentially the same amount of light. The mount for the light source can hold either the new LED/fiber or old LED pinhole. The light is converted into a parallel beam by a 10 mm diameter achromatic lens with a 20 mm focal length. A dichroic sheet polarizer is placed between the light source and the liquid crystal cell and can be rotated with respect to the cell. Another dichroic sheet polarizer can be placed between the cell and the lens system to be used as an analyzer which is crossed with respect to the lower polarizer. The lens system used to image the cell from above consists of two lenses and a CCD camera[70] which are mounted in a 1.2 m high aluminum tube. The camera uses a 1.27 cm charge-coupled device (CCD) with 510 x 492 picture elements. The image is digitized in the computer using a PCEYE board[71] which produces an 8-bit gray scale. The image is divided pixel by pixel by an image of the cell without any pattern present. This is done to remove inhomogeneities in the lighting. The lower lens is a 20 mm diameter achromat with a 52.7 mm focal length. The lens is fixed in place 6.63 cm above the cell. The second lens is a Nikon 50 mm f/1.4 camera lens. The Nikon lens and CCD camera are mounted on separate movable carriages. The design of the carriages allows the relative position of the Nikon lens and CCD camera and the position of the Nikon lens and camera as a unit to be adjusted independently. The magnification is estimated using the thin lens formulas 1/f = 1/i + 1/o and M = i/o where f is the focal length of the lens, i is the image distance, o is the object distance, and M is the magnification. Using these formulas, the fixed achromat lens provides a magnification of approximately 4x. This also places the image of the first lens at a distance of 25 cm from 50

60 the lens. Because of the design of the carriages, the Nikon lens can only get within 6 cm of the camera element for a magnification of 0.2x. This gives a combined magnification of roughly 0.8x. With the maximum separation of the camera and lens, the magnification of the Nikon lens is approximately 5x, for a combined magnification of 20x. For accurate determination of the magnification, the aluminum cylinder containing the sample cell is mounted on an x-y translation stage equipped with micrometers accurate to 1 µm. To calibrate the magnification of the shadowgraph system, a reference object is imaged in two measured lateral positions. The shadowgraph apparatus functions as a microscope where the objective lens and eyepiece (in our case, the camera lens) can be moved relative to each other. By adjusting the location of the camera lens, one can image the focal plane of the objective lens. The image in the focal plane of the objective lens is the square of the modulus of the optical Fourier transform of the pattern (the power spectrum). This follows directly from a consideration of Fraunhofer diffraction. Two examples of images of the focal plane of the objective lens and the corresponding image of the pattern are shown in Fig The image on the left is for a relatively strong pattern, so the dominant peaks in the power spectrum are due to the quadratic effect. Because the underlying pattern is a superposition of degenerate oblique rolls, the quadratic dependence on the director variation produces eight peaks corresponding to the sums and differences of the fundamentals. Because of the bright central spot (the image of the light source) and the large intensity of the second harmonics, the fundamental peaks are barely visible in this image. The image on the right is for a weaker pattern. In this case, we only show the first quadrant of the focal plane, and by enhancing the contrast of the image of the focal plane we can pick out the fundamental spot. This highlights one of the limitations of using the focal plane; the trick of minimizing the quadratic term by focusing on or near the cell is not available. The other limitation is that unless one blocks off regions of the cell, the image in the focal plane is the power spectrum of the 51

61 entire cell, not just the region which is convecting. There are possible benefits of using the focal plane. For example, a number of well know image enhancing techniques exist which involve manipulations of the spots in the focal plane. Two examples are dark-field microscopy (blocking the central peak in the focal plane) and the Schlieren method (block half of the focal plane). a k y b k x c d Figure 3.6: The images (a) and (b) are of the focal plane of the objective lens in the shadowgraph apparatus. Image (a) has (k x = 0, k y = 0) in the center as shown, and image (b) has (k x = 0, k y = 0) in the lower left corner of the image. The image (c) and (d) are the corresponding patterns which are observed in the cell, respectively. The patterns consist of a superposition of degenerate rightand left-traveling zig and zag rolls. Image (c) is a relatively large amplitude pattern, and the (a) highlights the dominance of the nonlinear shadowgraph effect in the focal plane. Image (d) is for weaker convection, and (b) has been sufficiently enhanced that the spot corresponding to the fundamental is visible (the circled region in (b)). 52

62 3.1.2 Temperature Control The temperature-control stage is shown schematically in Fig. 3.7 and consisted of a cylinder of aluminum 6.78 cm high with a diameter of 9.78 cm wrapped with 0.64 cm of insulating foam. The aluminum cylinder consists of a top and bottom half which are screwed together. To allow for illuminating the cell from below and viewing it from above, there is a 1.40 cm diameter hole along the axis of the aluminum cylinder. The hole is closed at the top and bottom with glass windows. A cm wide and 2.54 cm high circular channel with an inner radius of 3.56 cm is located with its midplane at the midplane of the aluminum cylinder and surrounds the center of the cylinder. Water enters the channel on one side of the aluminum block, flows around the cell, and exits from the side on which it entered. (In Fig. 3.7, only the inflow connection is visible.) The water is used to maintain the temperature of the aluminum cylinder. The temperature of the apparatus is regulated with an analog control system. The cylinder s temperature is measured with a calibrated thermistor which is embedded in the aluminum just below the cell to the side of the inflow channel. The thermistor serves as one arm of a standard Wheatstone bridge. The operating temperature is set by a variable resistor which is also part of the bridge. The error signal from the bridge is input into an integrating amplifier, and the output of the amplifier is used to set the voltage across a wire heater embedded in the circulating water. The integrating amplifier provides both integral and proportional control of the temperature. The integral control eliminates long term drifts in temperature. Both the proportional gain and the integration time constant are set by resistors and capacitors in the amplifier. The gain and time constant can be adjusted by selecting one of 6 ranges. The wire heater has a resistance of 7 ohm. The power output from the heater ranges from 3.5 W to 60 W, depending on the operating temperature. The typical ohmic heating of the EC cell itself is negligible. The samples have 53

63 Insulation Aluminum Block CELL water thermistor Electronic connections Translation Stage Figure 3.7: Schematic of the temperature control stage. Main elements are the water channel, electronics feed-through, and thermistor locations. The aluminum block consists of two halves (divided at the horizontal white line) which are screwed together. In this view, the thermistor is going into the page, and there is another thermistor hole opposite the one shown here. The output for the water channel is located directly behind the input connection.there are four BNC connectors in line with the one shown for input/output of electronics. a resistance of roughly ohm and are typically run at 20 V for a power of 50 µw. When operating at temperatures above 25 C, the room is an adequate heat sink. For lower temperatures, it is necessary to pass the circulating water through a heat exchanger in which the cold side is maintained by a Neslab RTE-110 refrigerator. Currently, the upper limit to the operating temperature of the experiment is around 70 C. (Above this temperature, the tubing currently used in the circulating bath has an increased risk of leaking.) For higher temperature work, the apparatus described in Ref. [67] can be used. The temperature stability of the aluminum is measured by a second thermistor embedded in the aluminum just below the cell but next to the outflow of the circulating water. Temperature stability of ±5 mk (rms) for the aluminum is easy to achieve, and generally we operate with a temperature stability of ±1 mk (rms). Typical temperature records for two hours and for 14 hours are given in Fig

64 The thermistors in this apparatus were initially calibrated against a glass thermometer for which the absolute temperature can be off by degrees. Recently, the thermistors were recalibrated against a thermistor that was calibrated against a standard platinum thermometer. The results of this calibration are used in this thesis, and both calibrations are given in Appendix A temperature ( o C) time (min) temperature ( o C) time (hours) Figure 3.8: Two records of the temperature of the aluminum block. The top record is for a two hour time span with the temperature recorded every minute, and the bottom record is for a 14 hour run with the temperature measured every 10 minutes. In both cases, the temperature was measured to be (34.9 ± 0.2) C with a rms stability of ±0.001 K. 55

65 1K DI/O data K 7.5K 100K 15K 10K 75K 1 M Electronics The ac voltage signal was generated by a computer controlled synthesizer card[73]. The card was capable of generating arbitrary waveforms, but for these experiments, only sinusoidal waveforms were used. When changing the amplitude of the output waveform, the synthesizer card recomputed the entire waveform. In general, this resulted in a phase jump in the output waveform. To eliminate this problem and achieve higher resolution voltage steps, the output of the synthesizer card was used as the input to the circuit shown in Fig INA105 3 K 1 nf 100K D E F 10K 1 µf V out 10K input stage A B C 200 V dd output stage M selection stage V ss V ss 12-bit multiplying DAC Figure 3.9: Schematic of circuit used to divide voltage from quatech synthesizer card. The circuit is divided into four sections by the dashed boxes. The resistor and capacitor values are shown in Fig Unless labeled otherwise, the op amps are all LF

66 The first element of the input stage is a Burr-Brown INA105 differential amplifier which is being used as a unity gain difference amplifier. The output of the synthesizer card is a BNC cable with the inner conductor connected to the negative input (pin 2) of the INA105 and the shield connected to the positive input (pin 3) of the INA105. The INA105 is connected to ground through its pin 1. The output is at pin 6 and goes to a low pass filter with 1/(2πRC) = 53 KHz. This smoothes out the digital steps in the waveform generated by the synthesizer. Finally, the signal is amplified by a factor of 2. The second stage is used to select the input to the multiplying DAC. If the points D and F are connected, the switches AB and BC must be open. For this case, the full signal is used as the input to the multiplying DAC. If D and E are connected, then by connecting A and B or B and C either 10% or 1% of the full signal is used as the input to the DAC, respectively. For these two settings, higher resolution voltage steps are achieved while the overall voltage range is limited to ±10% or ±1% of the base signal. The central element of the circuit is the 12-bit multiplying DAC chip, Analog Devices AD7845. The input to the chip is the analog signal from the selection stage (pin 17) and a digital signal generated by the digital input/output (DIO) lines (pins 2-14) on a Labmaster card[74] in the computer. The output of the 12-bit multiplying DAC is the analog signal divided by a factor ranging from 0 to 4096 which is set by the digital input. The 12-bit multiplying DAC produces changes in the output without any phase jumps as the value of the digital input is changed. The final stage is the output stage. It contains an adder circuit for use when only 10% or 1% of the signal was the DAC input. In those cases, the output of the DAC is added back to the portion of the full signal which is at D and E. In the case when the full signal is the input to the DAC, this circuit serves as a unity gain amplifier. The last stage of the output is a high pass filter to remove any dc component. The signal from the circuit was amplified with a commercial power amp- 57

67 lifier[75]. The frequencies used for the electro-convection experiments ranged from 25 Hz to 2000 Hz and the voltages used ranged from 0 V to 85 V (all voltages quoted are root mean square values). The resistance R and the capacitance C of the cell were measured using the circuit shown in Fig Based on modeling the cell as a resistor and R f Cell C C f V in R Vout Figure 3.10: Schematic of circuit used to measure capacitance and resistance of cells. capacitor in parallel, we define 1/R(ω) = Re[I(ω)/V (ω)] (3.2) ωc(ω) = Im[I(ω)/V (ω)], (3.3) where I(ω) is the current through the cell in response to an applied voltage V (ω) = V in cos(ωt). The circuit shown in Fig converts the current I to a voltage which is digitized in the computer with an A/D converter. The phase and amplitude of the output voltage are extracted by fitting the digitized data to V o (ω) = V out cos(ωt + φ). The amplitude and phase of the output voltage are related to the resistance and capacitance of the cell by V out = R f R 1 + ω2 R 2 C ω2 R 2 f C2 f V in (3.4) φ = π + tan 1 ( ωrc ωr fc f 1 + ω 2 RCR f C f ) (3.5) 58

68 2 1.5 gain applied frequency (Hz) Figure 3.11: The circles are the measured values of the gain V out / V in for the circuit shown in Fig with the cell replaced by a resistor R i = ohm. The solid line is the fit to Eq The result of the fit is R f = ohm and R f C f = 5.95 x 10 4 s. Independent of the circuit, I measured R f = ohm using an ohm meter, and using a capacitance bridge, C f = pf. This gives R f C f = 5.96 x 10 4 s. In order to compute the resistance and capacitance of the cell we need to know R f and R f C f. Since we want to know their effective values in the circuit, we calibrated R f and R f C f by replacing the cell with a known resistor R i = ohm (measured using a digital multimeter). We then measured the gain V out / V in and fit to the function V out V in = R f R i [1 + (ωr f C f ) 2 ] 1/2 (3.6) with R f /R i and R f C f as the fit parameters. A typical measurement and the resulting fit is shown in Fig The accuracy of our measurements was checked by measuring the resistance and capacitance of the parallel combination of a known resistor and 59

69 capacitor similar in value to those of a filled cell. The measured value of the capacitance only varied by ±0.2% over a range of 20 V. The resistance measurements showed a monotonic decrease over the 20 V range of 2%. The results for the measurements as a function of frequency are shown in Fig and Fig One can see that this technique had difficulties at high frequencies. We computed the values of C and R from R = C = V in R f V out (cos φ + ωr f C f sin φ) (3.7) V out (sin φ ωr f C f cos φ) V in ωr f (3.8) To avoid oscillations in the op amp circuit, I used R f C f = 5.95 x 10 4 s. This meant that even at frequencies as low as 1000 Hz our technique was sensitive to any errors in the measured phase shift φ. The systematic error increased as the difference between the measured resistance and the test resistor R f increased. For the EC cells, I actually needed the conductivity σ = (d/a)(1/r) where d is the cell thickness and A is the area of the electrode. As d could only be determined to 10%, the error for the typical cell resistances (of the order 8 6 ohm, see Fig. 3.13) were not significant. There were cases where the cell resistance was closer to ohm, and with the chosen values of R f and C f, the errors at high frequency could be as large as 20%. This is shown in Fig where the measurement made with this circuit is compared with a measurement made using a capacitance bridge[76]. The 20% error is the result of an error of only rad in the determination of the phase of the output voltage relative to the input voltage. This error is probably due to nonideal op amp behavior. Even this error is not generally a problem for my measurements. The values chosen for R f and C f result in the circuit being least sensitive to the phase in the range of 50 Hz to 100 Hz. As shown in Fig. 3.14, the circuit does a good job of measuring even the ohm resistor in that frequency range. However, it is important to be aware of this limitation of the circuit when measuring the frequency as a 60

70 function of resistance for resistances significantly greater than ohm. 61

71 500 capacitance (pf) applied frequency (Hz) 10 capacitance (pf) applied frequency (Hz) capacitance (pf) applied frequency (Hz) Figure 3.12: The three figures show the measured capacitance as a function of frequency for a test capacitor (top), test resistor (middle), and the resistor and capacitor in parallel (bottom). The bottom figure also shows the capacitance for the parallel combination of resistor and capacitor which you would expect from the individually measured capacitances. (solid line) 62

72 resistance (10 6 ohm) applied frequency (Hz) resistance (10 6 ohm) applied frequency (Hz) resistance (10 6 ohm) applied frequency (Hz) Figure 3.13: The three figures show the measured resistance as a function of frequency for a test capacitor (top), test resistor (middle), and the resistor and capacitor in parallel (bottom). The bottom figure also shows the resistance for the parallel combination of resistor and capacitor which you would expect from the individually measured resistances. (solid line) 63

73 18 resistance (10 6 ohm) applied frequency (Hz) Figure 3.14: Measurement of a ohm resistor in parallel with a 465 pf capacitor. The open circles were measured using a capacitance bridge and lock-in amplifier. The closed circles were measured using the circuit of Fig

74 3.2 Liquid Crystals The standard NLC used for EC is 4-methoxybenzylidene-4 -butylaniline (MB- BA), and the main NLC studied in this thesis is 4-ethyl-2-fluoro-4 -[2-(trans- 4-pentylcyclohexyl)-ethyl]biphenyl (I52)[41]. For comparison, their chemical formulas are shown in Fig The other common NLC used in EC is a mixture known as Merck Phase V. Since this is a mixture, I have not used this particular NLC. The other two NLCs which I have studied are trans- 4n-pentyl(2/-fluor4/n-pentylphenyl)cyclohexane carboxylic acid ester (D55-F) and trans-4n-propylcyclohexane carboxylicacid-(4-n-propylencyclohexanol) ester (OS-33). These are also single component NLC s which show promising behavior for future study. Currently, they are difficult to obtain as the manufacturer, Merck Inc.[77], is limiting the purchase of single component NLCs. (a) O C N (b) Figure 3.15: (a) Chemical formula for MBBA. (b) Chemical formula for I52. F 65

75 The use of MBBA in most EC experiments is largely historical. MBBA was one of the first single-component, room-temperature NLC with ɛ a < 0. Its nematic range is rather small, 20 to 40 C. It was developed for use in displays which were based on the transition to the dynamic scattering mode[63]. The dynamic scattering mode occurs as a secondary bifurcation from the initial transition to EC. Being of interest for displays, a lot of information on the alignment properties, material parameters, and doping characteristics of MBBA were obtained. This information was useful for the original EC experiments. In particular, the knowledge of the material parameters was essential for quantitative comparison of experiment and theory. There are a number of standard dopants used with MBBA for which the conductivity as a function of concentration has been measured[63]. One of the most common is a solution of 0.01% tetrabutylammonium bromide in MBBA which provides conductivities of the order 10 7 ohm 1 m 1. Another important parameter for EC is ɛ a. For MBBA, ɛ a 0.5 where the exact value is slightly sample dependent. The absolute value of ɛ a is large enough that one generally observes normal rolls in EC in MBBA. There are definite drawbacks to using MBBA. First, it is a known health risk. One must be very careful to wear gloves and work under a hood at all times. Second, it is highly unstable. MBBA easily decomposes in the presence of water, and for this reason, it is difficult to obtain extremely pure MBBA. A cell made with MBBA will initially age as any trace water that is present in the cell (and often the epoxy used to seal the cell) reacts with the MBBA and changes the properties of the cell. The cells eventually achieve an equilibrium though the time required is often months to a year. I52 is essentially the opposite of MBBA. It is a relatively new NLC, so most of its material parameters are unknown[41]. It is nematic at room temperature, but it has a wide nematic range with a melting point of 24 C and a clearing point of C. It does have a smectic B phase in the range 13 to 24 C. In the nematic phase, ɛ a monotonically increases with increasing temperature from 66

76 0.05 to 0.07, going through zero around 60 C (see Appendix A). Because ɛ a 0, oblique rolls are the dominant pattern. Also, I52 was designed to be a benchmark chemical and unlike MBBA, is very stable in the presence of both light and water. I have found that some properties of I52 will change upon extreme heating, as discussed below. Because it is a nonpolar molecule, it is rather difficult to dope. Table 3.1 shows a list of attempted dopants 1 and the success or failure. Here failure corresponds to a conductivity < 10 9 ohm 1 m 1 for which the cutoff frequency falls below 10 Hz and EC is not observed. As you can see from the list, I 2 is the only dopant which was found to work. Also, the concentrations required are relatively large, O(2%); whereas, the resulting conductivities range from 10 9 to 10 8 ohm 1 m 1 and are one or two orders of magnitude smaller than is typical for MBBA samples. It takes roughly two weeks to a month before enough I 2 dissolves and dissociates in the I52 for a solution to achieve a useful conductivity at temperatures of roughly 30 C to 60 C. Because I 2 is highly volatile, the doped solutions generally remain useful for only 6 months before the loss of I 2 becomes too great. However, the relatively rapid evaporation of I 2 means that the final concentration of the solution is always unknown. Therefore, it is not unreasonable to recharge a solution that is no longer useful by adding more I 2. In fact, a cell design utilizing a reservoir of NLC allows for the placement of pellets of I 2 in the reservoir which would maintain a saturated solution. This would allow a given sample to be used for longer periods of time. The time it takes for the I52-I 2 solution to reach a useful conductivity can be decreased by heating the solution. However, one must be careful to maintain the temperature at or below the clearing point. Samples of I52 heated well above the clearing point for long periods of time underwent a permanent change in ɛ a. After such a heating, a value of ɛ a > +0.2 was found for the 1 I have to thank Floyd Klavetter at Uniax, Inc., Goleta, CA. for help in choosing possible dopants for I52. 67

77 Table 3.1: Dopants for I52. Chemical percent dopant success (yes/no) tetra-butyl ammonium bromide (TBAB) 0.01 no TBAB 0.03 no TBAB 0.05 no TBAB 0.1 no sodium dodecylbenzene sulfonate 1.0 no dodecylbenzene sulfonic acid no dodecylbenzene sulfonic acid 0.7 no dioctylsulfosuccinate, sodium salt 0.3 no Bis(ethylhexyl)hydrogen phosphate 0.1 no cetylpyridinium, bromide 0.1 no 3-nitrophenol 0.2 no TCNQ 0.5 no phenylenediamine (P1) 0.6 no tetracyanoethelyene (T1) 0.8 no mixture of P1 and T1 0.8 no FeCl no iodine (I 2 ) 1.0 no I yes entire nematic range. This effect was discovered while attempting to find a method of doping I52 using various salts. When the sample was kept at high enough temperatures to induce the change in ɛ a, solutions with TBAB had conductivities which approached 10 9 ohm 1 m 1. However, the change in ɛ a rendered this method of doping irrelevant for EC. The change in ɛ a was observed for pure I52 as well, so it does not seem to be an effect caused by the dopants themselves, only the temperature. This change in ɛ a was never 68

78 observed for samples kept at temperatures below the clearing point 2. The color of an I52-I 2 solution serves as an indication of the conductivity. The solutions initially are bright red. Solutions with a dark red to black color have a conductivity of roughly 5 x 10 9 to 2 x 10 8 ohm 1 m 1. As the sample continues to age, and the conductivity decreases, the color fades to a pale pink. Once the sample has reached the dark red color range, a more precise determination of the conductivity is made using a commercial cell from DisplayTech[78] before filling a homemade cell. For testing the conductivity of solutions, the commercials cells have the advantages that they require less than two hours to assemble, always fill well, and always align. Whereas, a homemade cell requires two days to make, doesn t always fill, and doesn t always align. The commercial cells are also smaller, so they require less NLC. Finally, the area of the electrode is better defined in the commercial cells than in the homemade cell. The electrode in the commercial cells covers a well-defined, limited region of the cell which only contains NLC. In the homemade cells, the electrode covers the entire slide and measurement of the geometrical factor is complicated by the sealant, gasket, and air bubbles. This is important when measuring conductivities which are defined as σ = (1/R)(d/A) where R is the measured resistance, d is the cell thickness, and A is the area of the electrode (see Sec ). The DisplayTech cells are 10 µm thick with an electrode that is 0.5 cm 0.5 cm in area. The cell is shown schematically in Fig The bottom glass plate of the cell extends out on one side (in Fig. 3.16, this extension is to the right) and has two conducting strips for the attachment of wires. The wires are connected using the same method described in Sec for the connection of wires to the homemade cells. The cells are filled using capillary action from one of the two holes located opposite each other. The cell is supported with one hole at the top, and a small drop of solution is placed on the hole. After 2 There have been reports by J.T. Gleeson that temperatures as low as 80 C can affect ɛ a, but I have not observed this. 69

79 Figure 3.16: Schematic drawing of the top view of a commercial cell. The gray shaded regions represent the central electrode and the two electrodes used for connecting wires to the cell. The dark circles represent the glass beads embedded in the UV epoxy which set the cell spacing. Notice, there are two filling channels, one at the top and one at the bottom of the picture. the cell has filled, the holes are wiped clean of any excess solution and sealed with 5 minute epoxy. The cells are then placed in the apparatus where the conductivity can be measured as a function of temperature. In addition to measurements of σ, the commercial cells can be used to determine if the ɛ a of an NLC is positive or negative. If ɛ a is positive, one will observe the Frederiks transition instead of EC when a large enough voltage is applied to the cell. The onset voltage for the Frederiks transition will give a measure of the magnitude of ɛ a when the elastic constants are known. Initial observation of any new NLC or NLC/dopant combinations are best done in the commercial cells. One quickly determines the alignment properties of a rubbed polyimide and the conductivity of the solution. Initial surveys of EC in new NLC are possible in the commercial cells. However, when observing EC in the commercial cells, it is important to know that the alignment layer has been assembled in the parallel orientation (see Fig in Sec ). This is believed to result in drifts of the EC patterns 70

80 which may be confused with a Hopf bifurcation. A characteristic of the drifting pattern is that it always travels in the same direction in the cell. Another drawback of the commercial cells for EC is the electrode design. Because there is a region of NLC outside of the conducting coatings (see Fig. 3.16), EC always starts at the edge of the conductor. The apparent cause is the fringing fields of the conductor which preferentially tilt the director away from its planar orientation. The director field is softer in this region and will tend to convect first. Finally, 10 µm is a very thin cell. The wavelength of the pattern is close to the limit of resolution of the shadowgraph, and the validity of many of the SM assumptions begins to break down for thicknesses at and below 10 µm. As mentioned, the NLC D55-F and OS-33 are candidate NLC which are of interest. They have ɛ a 1 which is a range of ɛ a for which normal rolls dominate. Both of these NLCs are easily doped with I 2, but being nonpolar molecules like I52, they are difficult to dope with any of the traditional (and many nontraditional) salts. There was limited success in achieving conductivities in the 10 8 ohm 1 m 1 range by heating solutions of either OS-33 or D55-F and tertrabutylammonium bromide for 1 month at temperatures well above the clearing point. So far, no adverse effects of the high temperatures have been observed as in I52, but a supply shortage has prevented any definite conclusions. The initial studies of D55-F and OS-33 have suggested the transition is forward and stationary. One definitely observes normal rolls. One possible drawback of these two chemicals is that their index of refraction anisotropy n is rather small. This means that the power in the shadowgraph images is lower for both D55-F and OS-33 than it is for I52 and MBBA (see Sec ). Detailed studies of EC were not carried out for D55-F and OS-33 largely due to the limits of I 2 doping and availability of the NLCs. A key element of the experiments was the ability to vary σ. In a given sample, σ was varied by approximately a factor of 2 by changing the tem- 71

81 perature over the range of 25 to 65 C. By varying the amount of dopant, I achieved an additional variation in σ of up to a factor of 10 from sample to sample. Because I varied σ for a given sample by changing temperature, it was necessary to understand the role of the other temperature dependent material parameters. A detailed discussion of the temperature dependence of the material parameters is given in Appendix A. 3.3 Cell Construction The standard EC cell consists of two glass plates separated by a spacer and sealed by an epoxy. The glass plates are coated with a transparent conductor and are surface treated so as to align the NLC. I will refer to these as sealed cells. The construction of these type of NLC cells remains more of an art form than a science because of the difficulties associated with aligning the NLC and the potential for chemistry between the NLC, the dopant, the sealant and the spacer. There are commercial cells available; however, they have limited uses and were discussed in detail in Sec I will address both methods which I have confirmed to work as well as ones which others have used but have not worked for me. The second category is included as alternatives for those people for whom my methods fail. The cells are designed to be sturdy and easy to handle. Depending on the NLC, they have a useful lifetime of anywhere from 6 months to years. The main drawback to this design is that both the spacer and the epoxy often react with either the NLC or the dopant. This results in cells which age. The better cells have a relatively short time constant for the aging process and reach an equilibrium state in a reasonable time. For many cells, the time constant is quite long and the critical voltage in such cells drifts in a roughly linear fashion (see Sec. 4.3). In general, the cells with a long time constant reach a prohibitively low conductivity before achieving equilibrium. In some situations, the aging can be solved by a better choice of epoxy and spacer. An 72

82 alternative is to change the cell design completely to an open design with adjustable top and bottom plates. The methods describe here are not 100% effective. For this reason, the equipment used in cell construction has been designed to make two cells at a time. There are four main steps to the building of a cell: preparation of the glass, alignment treatment, sealing, and filling. The steps up to and including sealing the cell will take about 6-8 hours plus an additional 6-10 hours curing time for the epoxy. Filling the cells takes anywhere from 1-3 hours. A list of required materials by step are giving in Table 3.2. I will discuss each step in the process separately Glass Preparation The first step in the construction of the cells is to acquire glass slides with indium-tin oxide (ITO) coatings. Other transparent conductors are possible, but ITO is the most common one and easy to obtain. Donnely[79] and Libbey- Owen[80] are two companies which provide free samples of ITO coated glass. The samples come in sheets of approximately 12 x 12 and are standard float glass. Donnelly provides glass which is 1 mm thick and has a resistance of 200 ohm per square. The glass from Libbey-Owen has a thickness of 2.5 mm and a resistance of 20 ohm per square. This thick glass is preferable because the resulting cells are more durable, and it is easier to cut than the thinner glass. Because of this, all glass-slide holders have been designed to hold the thicker slides. (All but the evaporation holders can actually accommodate slides of either thickness.) The cells are currently made from 1 x 1 pieces of the glass. After the glass has been cut, the slides must be cleaned. Practically everyone has their own favorite cleaning method, and I settled on a combination of methods that works well. The steps in the cleaning procedure are: 1. Wash your hands so that they are as clean as possible. Then, using your 73

83 hands, scrub the glass slides in a solution of Joy liquid detergent and water. Rinse well with tap water to remove the soap, and then rinse with the deionized water to remove the tap water. 2. Ultrasound the slides in a solution of 10% liquinox and 90% diluted ammonium hydroxide. (350 ml distilled water, 50 ml NH 4 OH, 40 ml liquinox) for min. 3. Rinse with tap water followed by a rinse with Milli-Q[81] filtered water. 4. Ultrasound for 20 min in spectroscopic grade acetone. 5. Ultrasound for 20 min in spectroscopic grade methanol. 6. Rinse in Milli-Q[81] filtered water. 7. Dry by blowing 99.99% pure N 2 gas across the slides under the Laminar Flow hood. A slide holder (typically Teflon) should be used for carrying the cleaned slides and placing them in the solvents. After cleaning the glass slides, the rest of the steps should be performed under the laminar flow hood to keep them clean. The only exception to this is steps involving evaporations. For transporting the cells to and from the evaporator, one should use a sealed container. There are microscope-slide holders which are ideal for this purpose. The next step in the preparation of the glass slides is evaporation. There are three different types of evaporations which are used in EC cells. Depending on the type of cell which is being made, all, none or some combination of the evaporations are used in the following order: uniform insulating layer, beaches, and alignment layer. I will discuss the general techniques of evaporation first, and then highlight the important features of each of the three types of evaporation. All evaporations are performed on the side of the glass which has the ITO coating. SiO is a convenient material to use for all three of the possible 74

84 evaporations. Because it is an insulator, it is ideal for both the insulating layer and the beaches. Its aligning properties are well known, and it can be used to obtain a number of different alignments. Also, evaporation of Si0 does not require extremely low pressures, so it it possible to accomplish it relatively quickly. One should be aware that even though one uses SiO as the source for evaporating, the resulting layer is actually SiO x, i.e. some combination of SiO and SiO 2. SiO evaporation requires the use of special boats[82]. SiO tends to sputter when evaporating, and the boats provide baffles which prevent large pieces of SiO sticking to your sample. I recommend using boats from the SM-series of the R.D. Mathis Co. [82]. The SiO should be in pieces of roughly 1 mm in diameter (this corresponds to a +10 mesh size) and not in powder form. When being used for the first time, the SiO needs to be heated to the point where it just starts to spark, and then the temperature should be lowered to just below this point. Let the SiO sit at this temperature for roughly 20 minutes. This allows contaminates to leave the SiO. Finally, let it cool under vacuum. After this treatment, the SiO can be taken in and out of the evaporator as much as necessary without repeating the procedure. Relatively high pressures are used for evaporation, only 10 4 torr with air bleeding. (This is the required pressure for alignment purposes. For other evaporations, the pressure is not critical.) The boat temperature should be around C, but will vary depending on the desired rate of evaporation. If one uses the R.D. Mathis SM-8 boats[82], this temperature range corresponds to a current range of Amps. For more information on evaporation techniques, R.D. Mathis provides excellent technical support. In general, the ITO slides already come with an insulating layer coating the conducting ITO. In the context of the WEM theory of EC, this is precisely the boundary conditions we desire: perfectly insulating boundaries, i.e. no charge flows between the NLC and the electrodes. This implies that the samples would have infinite dc resistance. Initial measurements of the conductivity 75

85 do suggest that the dc resistance is effectively infinite. Therefore, I have not evaporated any extra insulating layers onto the glass slides. In contrast to the insulating layer, the Si0 beaches were evaporated on all of the EC cells for which results are reported in this thesis. The beaches are used to isolate the region in which EC first occurs from the spacers used to set the thickness of the cell. This is accomplished by reducing the voltage across the NLC under the beaches. For studies of the Frederiks transition, the beaches are not desired because it is more important that the voltage be uniform everywhere than that the transition be isolated from the spacers. The beaches are constructed by evaporating a 1 µm thick layer of silicon monoxide (SiO) in 0.2 cm wide strips onto the glass slides. The strips form a square which separates the central 0.5 cm 0.5 cm from the outer edges of the cell. A rate of [?]/sec is used. This corresponds to Amps for the R.D. Mathis SM-8 boats, depending on the amount of SiO in the boat. The mask is made from 0.25 thick aluminum and is shown in Fig The (a) (b) "" ## $ $ %% && '' "" ## $ $ %% && "" ## $ $ %% && "" ## $ $ %% && Top view of mask side view of cell Figure 3.17: (a) shows a schematic top view of the mask where the shaded regions represent the aluminum. Notice that open sections are offset from from the center of the mask. This is necessary so that the beaches are aligned when the cell is assembled. (b) shows a side view of an assembled cell with the beaches, represented by the hashed boxes, aligned. mask is designed to evaporate both slides at once and to correctly position 76

86 the beaches so that they are aligned when the cell is constructed. There is a small section of each beach on which no SiO is evaporated. These regions are positioned so that they do not overlap when the cell is assembled. The asymmetry of the two beaches can be used to determine the direction of alignment when needed (see Sec ). Due to the drop in voltage across the SiO layer, the voltage drop across the NLC is smaller under the beaches than in the central 0.25 cm 2. This ensures that convection occurs first in the central region of the cell, and that it does not propagate in from the spacers. Figure 3.18 shows a Dektak scan of a typical beach edge and the resulting profile of the voltage drop across the liquid crystal. Notice that the x- and y-axes have very different scales, so that the beach is actually extremely gradual. Height (µm) Voltage (V rms ) Position, arb. origin (mm) Figure 3.18: Shown here is the result of a Dektak scan across one edge of the beach. The scan measures the layers relative height. Also shown is an estimate of the corresponding voltage drop across the NLC when 7 V is applied across the entire cell. The estimate assumes the cell and SiO layer act as dielectrics in series. If evaporation is being used for alignment, it is always the final evaporation so that the entire cell will be aligned. In particular, the region of the cell corresponding to the beaches must be aligned if they are to effectively suppress convection. Unaligned regions of NLC generally convect first which eliminates 77

87 the effectiveness of the beaches. All slides should have some type of alignment, and whenever possible, that alignment should be done by evaporation. For more details on alignment by evaporation, as well as by other methods, see Sec The final stage of slide preparation is the cleaning after evaporating. It was found that even when a cold trap was used, evaporation of material onto the slides without cleaning afterwards seriously hindered convection. The postevaporation cleaning also serves as a test of the pre-evaporation cleaning. If the slides are not well cleaned before evaporation, the SiO does not stick well to the surface. In this case, the post-evaporation cleaning will cause the SiO beaches to flake off, and such slides should be rejected. The cleaning consists of: 1. Ultrasound for 3 min in trichloroethylene. 2. Ultrasound for 5 min in spectroscopic grade acetone. 3. Rinse in Milli-Q[81] filtered water. 4. Dry by blowing 99.99% pure N 2 gas across the slides under the Laminar Flow hood. It is important to note that no alcohol is used in the post-evaporation cleaning. If for some reason, nothing is evaporated onto the glass, one must skip the alcohol step listed in the initial cleaning method, and this second cleaning is not needed. If alcohol is used to clean the glass slides either after the evaporation of material or when no evaporation is performed, EC is suppressed. This effect has been observed by other groups and is a classic example of the black magic that is sometimes needed to make a good EC cell. There is no fundamental understanding of what the cleaning with alcohol does to ruin EC. If SiO alignment has been used, the slides are ready to be sealed. Otherwise, after the evaporation of the beaches, the slides are ready for their alignment treatment. 78

88 3.3.2 Alignment A comprehensive source on aligning techniques is the reference by Jacques Cognard, Alignment of Nematic Liquid Crystals and their Mixtures[59]. It contains an extensive comparison of the various types of alignment, as well as explanations of how alignment works. I will focus here on the techniques that I have tried. As with evaporations, all aligning layers are applied to the side of the glass with the ITO coating. There are two general types of alignment, homeotropic and planar. Homeotropic alignment is when the director is perpendicular to the glass plates. For our purposes, this is useful for studies of the electric Frederiks transition in NLC with ɛ a < 0. Also, homeotropic alignment is required to study the behavior of EC following a Frederiks transition in NLC with ɛ a < 0. Uniform planar (often just called planar) alignment is when the director is parallel to the glass plates and in a uniform direction. This is the standard type of alignment which is used for EC. Each of these general types of alignment is further modified by the presence of any pretilt. In most display applications, it is required that the alignment not be perfectly perpendicular or parallel to the glass plates. A small nonzero angle between the director and the glass, referred to as pretilt, is required to break the symmetry of the state. This limits the number of defects which occur during the transitions. For basic studies of EC and Frederiks transition, we almost always desire zero pretilt. In the case of planar alignment, it is difficult to achieve exactly zero pretilt, but pretilts less than 2 are often referred to as zero for practical purposes. The reasons why a given combination of surface and NLC produces either homeotropic or planar alignment are not completely understood, but there are some general principles which appear to hold true. Whether one has homeotropic alignment or planar alignment depends strongly on the interaction between the surface and the NLC. For example, generally it is energetically 79

89 favorable for a NLC to sit with one end preferentially attached to a surfactant (homeotropic alignment). Whereas, with most polymers and inorganic surfaces, it is energetically favored for the NLC to be parallel to the surface (planar alignment). For uniform planar alignment, the molecules must also be forced to have a particular direction as well. This usually involves some method of establishing grooves on the surface. Because of the elastic cost when the director is perpendicular to the grooves, one obtains uniform alignment in the direction parallel to the grooves. Because of this added complication, uniform planar alignment is usually harder to achieve than homeotropic alignment. For homeotropic alignment, I have used a coating of lecithin (egg yolk). A 1% solution of lecithin in chloroform is recommended though the exact concentration is not critical. It is possible to buy precisely this mixture commercially [83]. The glass slides should be cleaned using the method described in Sec After cleaning, use the following procedure: 1. Have a petri dish, or other shallow dish, ready with chloroform for rinsing the slides. 2. Remove the lecithin/chloroform solution from its container using a syringe and needle. The bottle has a septum, so it is not necessary to remove the cap, and the lecithin will remained sealed. Occasionally, dry nitrogen should be flowed into the lecithin bottle using a syringe needle so as to maintain the pressure of nitrogen. 3. Holding one slide, place a few drops of lecithin on the slide. 4. Take the other slide and place it on top of the first slide. Press them together so that the lecithin completely wets the two surfaces, removing any bubbles. (Steps 3 and 4 must be done as quickly as possible, as the cholorform evaporates quickly) 80

90 5. Carefully separate the two slides and place them face up in the petri dish. 6. Remove them from the dish and place them in a holder with a loose cover. 7. Bake the slides for 30 min at 80 C. Visually check the slides after removal from the petri dish. If they appear extremely messy, they can be gently rinsed with chloroform from a spray bottle. During baking, the slides should be covered in some fashion to minimize the accumulation of dust. However, they shouldn t be in a airtight container as part of the heating process is the removal of the remaining chloroform. Currently, this procedure must be performed in the fume hood to avoid breathing the chloroform. This is not ideal as the slides are exposed to dust in the air which can lead to dirty cells. So far, this alignment technique has only been used to look at the Frederiks transition, and there has been no observation of problems due to dust. However, this must be kept in mind if one plans to study Frederiks to EC transitions. I have confirmed that this alignment method works for 5CB, 8CB, MBBA, and I52. If a new liquid crystal is used for which lecithin does not work, there are plenty of other methods available. The other homeotropic techniques are essentially the same as the above method, only the specific chemical and solvent vary. Because most planar samples have a small pretilt, there are two types of planar alignment: parallel and anti-parallel. Consider the slides before assembly with the polymer side up and the direction of rubbing is either to the right or the left. The director can either be at an angle of Θ or π Θ with respect to the plane of the glass slides. Now, fold the slides together for assembly as shown in Fig Parallel alignment refers to a cell in which the director has a pretilt of Θ on one plate and π Θ on the other. This results in a variation across the cell of the angle between the director and the plane of the 81

91 plates. For anti-parallel alignment, the director is at either an angle of Θ or π Θ on both glass slide. This results in a constant director orientation across the cell. These two possible planar alignments are illustrated in Fig (a) (b) (π Θ) (Θ) (π Θ) (π Θ) Figure 3.19: (a) Schematically shows the arrangement of the glass slides and the rubbing direction (given by arrows) for parallel alignment. Notice the director will splay as a function of position across the cell to match the boundary conditions at each plate. (b) Schematically shows the arrangement of slides and rubbing direction for anti-parallel alignment. In this case, the director is uniform across the cell. There are two main methods of achieving planar alignment[59]: obliqueangle evaporation and rubbed polymers. The oblique-angle evaporation is the preferred method. For oblique-angle evaporation, a thin layer of material is evaporated onto the glass slide with a nonzero angle between the slide and the direction of evaporation (see Fig. 3.20). Under the correct conditions, the particles already deposited on the slide shadow the particles which arrive later. This shadowing produces steps or ridges which run along the glass slides and provide the uniform direction for the director field. A number of materials can be used for the evaporation, but I have focused on SiO because of the wide range of pretilt angles which are possible. In most cases, one wants the pretilt to be as small as possible, and with SiO evaporation one can achieve samples with pretilts less than 1. Pretilts up to 30 are achieved in a reproducible 82

92 fashion by selecting the appropriate angle of evaporation. The exact value of the pretilt will depend on the NLC being used 3. This could be used to study the effect of a variation in the pretilt angle as a function of horizontal position on EC patterns[84]. SiO evaporated at an angle of 30 from the vertical with approximately 12 between the source and the substrate (see Fig. 3.20) will achieve a nominally zero pretilt for many NLC s. The general techniques for evaporation of SiO and slide preparation are described in Sec When using SiO for alignment, one wants a 200 to 500 [?] thick layer evaporated at a rate of 5 [?]/sec using a pressure of 10 4 torr. For evaporation, there is no rubbing direction, but there are still the two possible alignments: parallel and anti-parallel. In general, the two slides for a given cell are evaporated at the same time (see Fig. 3.20). The pretilt will be in the same relative direction for both slides. If the cell is assembled by holding the slides in the same relative orientation as when they were evaporated and then folding the slides together, the result is anti-parallel alignment. If one slide is first rotated by 180 and then the slides are folded together, the result is parallel alignment. The evaporation method was found to work very well with 5CB, 8CB, MBBA, D55-F, OS-33. One drawback is that it does not work for I52. I52 has a strong tendency to align perpendicular to the evaporated layer. Some effort was made to vary the evaporation parameters, but no values were found that worked well. MF 2 was tried as an evaporating material, but it also produced homeotropic alignment. Further work might reveal a combination of material and evaporation parameters that produced planar alignment for I52, but because the rubbed polymer method does work for I52, evaporation techniques were not pursued further. The most frustrating method of planar alignment is the rubbed-polymer technique. The number of prescriptions for obtaining alignment seem to be as 3 The values for the pretilt are from Ref. [59]. 83

93 (a) director alignment (b) θ substrate "ridges" (higly expanded view) source source Figure 3.20: Shown here is the standard arrangement for SiO evaporation for planar alignment. (a) shows the front view of the holder with the two glass slides (shaded regions). The alignment will be in the horizontal direction. (b) shows the side view and a magnified view of the ridges which result from the shadowing effect. The angle θ should be 30 to produce nominally 0 pretilt. To achieve nonzero pretilts, θ is varied. numerous as the number of people who make EC cells. This proliferation of techniques is probably due to the delicate balance between choice of polymer, rubbing material, and rubbing method. For rubbed polymers, the rubbing makes the required grooves in the surface. I will describe the method I have found to be most successful, and then offer variations that have worked for other people. The solution I recommend is a polyimide solution made from 1% poly(ether-imide)[85] in methylene chloride. This corresponds to g of polymer in 40 ml of solvent. If the polymer is found to streak upon spin-coating, a drop of mineral oil can be added to the 40 ml solution. The steps used to produce the aligning layer are: 1. Spin coat the slide with the polymer. 84

94 2. Let the slide sit for minutes. 3. Rub slides with a polishing cloth. To spin coat one of the glass slides, place the slide in the spinner and coat it with a thin layer of the polymer solution. Turn on the spinner as soon as the slides is coated to avoid significant evaporation of the solvent. For the cleanest possible films, the polymer solution should be applied to the glass slide using a syringe which has a 0.45 µm syringe filter. A spinning rate of 5000 rpm results in a film with a thickness of roughly 0.35 µm. The UCSB fluids lab spinner is calibrated so that a setting of 30 V on the variac corresponds to 5000 rpm. The film should be essentially transparent and appear smooth to the naked eye. If the film is excessively streaky or too thick, remove the film with methylene chloride and repeat the spinning procedure. Rubbing the polymer coating by hand appears to be the best method to achieve uniform alignment. A rubbing surface is constructed by wrapping an Extec synthetic velvet[86] polishing cloth around a 2 x4 board of length 6. A piece of plexiglass is clamped to the board to serve as a guide when rubbing. The slide is placed polymer side down on the cloth and rapidly moved along the surface 12 times. It is important to use a very light touch applying essentially no pressure to the slide. The strokes should be as smooth and as quick as possible with all of the strokes in the same direction. Remember to note the direction of rubbing for each slide so that the desired parallel or anti-parallel alignment can be achieved. One should be aware that during the rubbing, the slide will not always slide perfectly straight or smoothly. It may even catch on the plexiglass and swing quite far to the side. After all, you are doing this quickly and by hand. Based on my experience with aligning, these mistakes generally do not hurt the alignment and small imperfections in the rubbing may be essential to the aligning process. This conclusion is based in part on observations of the quality of alignment which is achieved by rubbing machines. The best 85

95 machine alignment was nominally uniform but had variations perpendicular to the rubbing direction, or streaks. The machine duplicated the rubbing by hand in all but two regards: the motion was always regular, and the speed of rubbing was slower. Slowing down the rubbing by hand to match the machine speed did not introduce streaks. This suggests that it is the very regular motion of the rubbing machine which generates the streaky alignment, and the more irregular motion of rubbing by hand eliminates the streaks. A wide variety of polymers have been successfully used by others. Two highly recommended solutions[87] are 1. polyvinal alcohol (PVA) with molecular weight 85, ,000: 60% distilled water, 40% propanol, few milligrams PVA. After spin coating, the slides need to be baked for 30 min at 80 C. 2. polyvinal formal (PVF): g PVF in 4 g chloroform. No baking needed. These can be spin coated using the same method as described above. There may be cases where another polymer in needed. For example, polyimides are known to have problems at high temperatures. In some commercial cells which used an unknown polyimide, the alignment of I52 changed from planar to homeotropic when heated above 70 C. The alignment of I52 using the poly(ether-imide) recommended here was at least stable to the maximum 65 C. Polyamides are good candidates for work at significantly higher temperatures. An alternate spin coating method is to have the substrate spinning and drop a few drops into the center. My experience with this method of spin coating is that it causes streaking in the film which can affect the alignment. There are a number of techniques and rubbing materials used to rub polymers to achieve uniform alignment. An alternate material is cloth (cotton) diapers. I have found that the diapers bought in Santa Barbara did not work that well. An alternate rubbing method is to wrap the rubbing material around a drill chuck inserted into a drill press with the drill running at high speed and 86

96 press the slide firmly against the cloth[88]. One group uses a Rayvel polishing cloth wrapped around a cylinder of 2 cm diameter, rotating at 1 to 10 Hz[89]. The slide moves under the cylinder at a rate of 3 mil/sec. It is possible that there was sufficient play in this rubbing machine compared to the one built by myself to produce the irregularities required for good alignment. The quality of the alignment can only be checked after the cell is constructed and filled with the NLC. The simplest way to check the alignment is to observe the sample between crossed-polarizers. As discussed in the Sec , NLC s are birefringent. Polarized light passing through an EC cell may have its polarization rotated depending on the relative orientation of the polarization direction and the director. If the light is propagating along the director axis, as is the case for a homeotropic cell with light propagating perpendicular to the glass plates, all polarizations of the light have the same index of refraction and the polarization does not rotate. Therefore, independent of the relative orientation of the polarizers and the sample, a cell with uniform homeotropic alignment between crossed polarizer will be uniformly dark. As discussed earlier, light which has an arbitrary angle between its polarization and the director is composed of an extraordinary ray and an ordinary ray. For light propagating perpendicular to the director, the two rays acquire a relative phase as they travel through the cell because of their differing velocities. The phase shift produces a rotation of the polarization of the light exiting the cell relative to the initial polarization. For the two special cases of light polarized parallel or perpendicular to the director, there is only one ray and the polarization is not rotated. Therefore, uniform planar alignment can be checked by placing the cell between crossed polarizers. The transmitted intensity is zero when the director is aligned with one of the polarizers and a maximum when the relative angle between the cell and polarizers is 45. If the sample is not uniform, there will be light and dark regions for all relative orientations of the cell and the polarizers. The alignment can be measure more quantitatively using conoscopy [90]. 87

97 A converging beam of light is passed through the cell. The ordinary and extraordinary rays interfere with each other to produce an interference pattern which can be quantitatively related to the director orientation. This method is a local measurement of the director and depends on the cell thickness. Because of our cell thicknesses, the current optical system would need to be modified to perform conoscopy Sealing Sealing the cells is a straightforward process but does require a judicious choice of materials. At this stage, one should have two glass slides with some or all of the following: an ITO coating, insulating layer, beaches, and alignment layer. A schematic of the assembled cell which shows all of the main parts is given in Fig The first step in sealing the cell is to make a spacer. In general, you want the spacer to form a gasket which separates the working fluid from the epoxy as shown in Fig In practice, one must leave a small fill channel in the gasket. For certain choices of material, it is possible to fill the cell even if there is no fill channel in the gasket. This is not recommended because cells filled by this method generally have a large number of bubbles. Mylar is the most common choice of spacer material due to its availability, inertness, and ease of use. It does not appear to react with most of the NLC used in EC. Two other materials which have been tried are mica and kapton. Kapton is as easy to work with as mylar but much harder to acquire. Mica is fairly easy to acquire, but if one wants cells with a thickness less than 75 µm, mica becomes difficult to work with. The main drawback of mylar is that it absorbs I 2. As discussed in Sec. 3.2, I 2 is used to dope I52. The absorption of I 2 by the mylar results in a drift in the conductivity of the sample. We tested mylar, kapton, Teflon, mica, and glass for their relative absorption I 2, and only glass did not show significant absorption. Glass spacers would be 88

98 a nice option, but currently the thinnest glass available for spacers is 50 µm optical fibers. These are extremely difficult to work with and are obviously only useful for making 50 µm thick cells. One possibility is to copy commercial cells and to use glass beads embedded in an epoxy. But, this is not useful as a gasket which isolates the NLC from the epoxy. Pretreating the spacer by exposing it to I 2 is also a possibility, but some experimentation is needed to find a proper length of time for the pretreatment. The spacer needs to be cleaned, but remember NO ALCOHOL can be used in cleaning the spacer. This causes the same problems as cleaning the glass slides with alcohol (see Sec ). The spacers should only be washed with soap and water and dried with N 2. Figure 3.21 diagrams the cell assembly. Once the spacer is cut out, it is placed on one of the slides. If beaches are being used, the spacer should be centered on the beach. If there are no beaches, the spacer needs to be placed off center on the slide. The second slide is placed on top of the spacer, shifted relative to the bottom slide. There should be about 0.1 width of each glass which is uncovered and on opposite sides. This is for attachment of wire connectors. The fill channel of the spacer should be on one of the sides which is offset. This is the side which will be used for filling. The two slides are clamped together so that one has easy access to the fill hole. A small amount of glass wool is placed in the location where the fill hole will be to serve as a filter. Cells constructed with this filter are essentially free of visible dust in the interior of the cell. The glass wool is held in place by the sealant which is placed around the slides. The two sides used for filling are sealed first leaving a hole on each side for filling. The side which has the filter and fill channel should also have a small well made from the sealant to hold the NLC in the pre-filling stage (see Fig. 3.21). Now, adjust the clamps so that the non-filling sides are accessible and apply the sealant to the entire length of the remaining two sides taking special care to seal the corners. There are a number of choices for sealants. The three used most commonly 89

99 in this lab are Torr Seal[91], UV epoxy[92] and adhesive cement[93]. The adhesive cement is necessary for high temperature work. The Torr Seal is the easiest to use. It is known to react with MBBA, but a good gasket between the sealant and the NLC will limit the effects of this reaction (as well as the reactions of any of the other sealants with the NLC). The UV epoxy is the one used by much of the liquid crystal industry. All three sealants have been tested for their reactivity with MBBA and no noticeable difference was detected. One of the difficulties is that MBBA is so naturally unstable that it is hard to detect a difference. With I52, no difference has been observed between different sealants because the dominant effect on the aging of the cell is the absorption of the I 2 by the spacer. For UV epoxy the sample is cured under UV light[94] for 10 minutes with the clamps on the cell. After this, the clamps can be removed, but the epoxy should be cured for another 20 minutes under the light. Then, it should be put in the oven over night at 50 C. This last stage bonds the epoxy to the glass. Torr seal is cured at 55 C to 65 C for at least two hours. However, the best cells have been made with the Torr seal curing overnight. The cement is cured in hours at room temperature. It is a good idea after the epoxy is completely cured to use a mechanical pump and place the cell under a low pressure for roughly an hour before filing it. This ensures that the epoxy has bonded to the glass and allows some of the residue from the cleaning agents, epoxies, etc. to be removed. If the epoxy has not bonded to the glass, it will clearly bulge into the cell while pumping on the cell. This is done without any liquid crystal present. Whenever pumping on a NLC cell, it is necessary to use a cold trap as the pump oil can contaminate the cell. 90

100 3.3.4 Filling There are a number of different filling methods 4 The simplest method is to fill the cells under atmospheric pressure using capillary action. This is the quickest method and works fine except that it will always leave bubbles of air in the cell. The bubbles are a real problem when using I 2 doped NLCs as the air will absorb I 2. Also, depending on the final location of the bubbles, they can interfere with the EC. However, for the commercial cells, bubbles which result from this filling method are always far away from the electrodes. Therefore, this is the recommended method for filling commercial cells, especially when they are being used for conductivity measurements or preliminary measurements of EC. The best filling method is to use capillary action under a vacuum. In this case, the liquid crystal is placed in the well formed by the epoxy during the sealing stage (see Fig. 3.21). Typically, 2-3 drops from a pipet is plenty. The sample is held at an angle so that the NLC does not touch the fill hole. The entire sample and cell is pumped on with a mechanical pump for roughly 5 minutes. It is important to watch the solution closely during this time. I52 has an extremely large surface tension, and as bubbles form you need to make sure that the I52 does not run over onto the top glass surface. If the sample is not too bubbly, one can pump for longer than 5 minutes. A dessicator was fitted with a holder that allows two cells to be filled simultaneously and was used only for filling NLC cells. The sample was tilted by tilting the entire dessicator. Once the sample has been pumped on, seal off the vacuum. Keeping the sample under vacuum, tilt the slides forward so that the NLC contacts the fill hole. The sample will fill under capillary action. Once the cell has filled in this fashion, there still will be bubbles, but these are at a significantly lower 4 I have Ingo Rehberg and Eberhard Bodenschatz to thank for information on previously used filling methods. All of the methods described here had been used by them. 91

101 pressure than air pressure. Slowly release the vacuum, and the various bubbles will be dissolved into the NLC, and one will have a bubble free sample. A few warnings on filling. On occasion, the cell won t fill completely on the first try for various reasons. An attempt can be made to fill it again by adding some NLC to the fill hole and pumping on the sample. One should be aware that this moves around the material which is already in the cell. In the past, this tends to produce highly nonuniform convection in the cell. If one is lucky, the convection in samples in which this happens eventually becomes uniform. The final filling method is similar to the above method but less reliable. One uses only a single fill hole. A drop of liquid crystal is placed over the hole and the sample is pumped on. The air inside the cell bubbles out through the NLC and the cell reaches very low pressures. For this method, one should pump until there are essentially no more bubbles leaving the cell. This takes at least an hour for most cells. Then, one slowly releases the vacuum. Done slowly enough, the NLC is forced into the cell, and if the sample has been pumped on long enough, the resulting cell is free of bubbles. This method does not work well with I52 because of the high surface tension. I52 makes rather large bubbles, and when they burst, they spread the I52 all over the top of the cell. It will work for MBBA, but care needs to be taken to watch for extra large bubbles. The filling method is one reason to use torr seal over the UV epoxy as a sealant when making cells with I52. I52 wets the UV epoxy so well that it wicks right up into the cell before you can begin pumping on it. With Torr seal, the I52 remains in the well made from the epoxy. The other liquid crystals discussed in Sec. 3.2 did not have this problem with the UV epoxy. After the cells have been filled, the areas around the fill holes need to be wiped clean. Initially, they should be wiped only with DRY cotton swabs or tissue. NO SOLVENTS OR WATER SHOULD BE USED. Once all the NLC has been wiped off, they should be sealed using the Hardman 5 minute 92

102 epoxy[95]. This has been recommended by NLC display manufacturers as having essentially no reactions with the NLC as it cures. Once the 5 minute epoxy has been applied, the cell can be carefully cleaned with water or acetone to remove the remaining NLC residue. Having sealed the cells, all that remains is to attach wires with connectors to the electrodes. Use 28 gauge multi-strand wire. Solder the connectors onto the wire first, and then attach the wire to the electrodes of the cell. For connectors on the wire, use single pins from IC sockets with the plastic removed. (These connectors can be bought in breakaway strips for single pin use.) The connecters in the apparatus are female, so solder the wire which is fixed to the cell to the female end of the IC pin. Heat shrink tubing is used on the wire/connector for strain relief. The wire is attached to the electrodes with silver conducting paint[96]. A layer of 5 minutes epoxy is applied for extra strain relief. It is important to make sure the silver conducting paint has dried completing before applying the epoxy. You now have a finished NLC cell. 93

103 Table 3.2: Required materials for cell construction. Step material comments/ recommendations general needs Teflon-coated tweezers used to manipulate glass multi-meter scissors exacto knife laminar flow hood powder free gloves protective gloves evaporator cleaning acetone (spectroscopic grade) methanol grade) liquinox (spectroscopic ammonium hydroxide trichloroethylene high purity N 2 gas beakers Teflon slide carrier determining ITO side of glass for flow hood for handling NLC one per solvent holds slides in solvents 94

104 Step material comments/ recommendations evaporation SiO evaporation boat R. D. Mathis SM-series[82] +10 Mesh SiO sealed slide carrier (glass) biology storeroom evaporation mask slide holder for oblique and normal angles evaporator alignment syringe/ syringe filter spinner planar alignment polymer solution planar alignment rubbing material synthetic velvet lecithin homeotropic alignment chloroform homeotropic alignment shallow dish homeotropic alignment 95

105 Step material comments/ recommendations sealing spacer/gasket mylar sealant torr seal[91], UV epoxy[92] or adhesive cement[93] UV lamp[94] if using UV epoxy UV protection goggles clamps glass wool filling mechanical pump dessicator fit with two cell holders pipet disposable glass 5 minute epoxy Hardman epoxy[95] cotton swabs removal of NLC from slides 96

106 filling well glass wool *)*+*,*-*. *)*+*,*-*. *)*+*,*-*. *)*+*,*-*. *)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/ *)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/ *)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/ *)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/ *)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/ *)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/ *)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/ *)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/ *)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/ *)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/ *)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/ *)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/ *)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/ *)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/ *)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/ *)*+*,*-*.*/*0*1*2*(*)*+*,*-*.*/ Figure 3.21: Shown here is the standard assembled cell. The hashed region represents the beaches. The gray region is the spacer and the solid black is the sealant. The two glass slides are offset so as to be able to attach the electrodes. To fill the cells, two holes are left in the sealant along the sides which are offset. One of the fill holes has some glass wool placed in front of it as a filter. This is the hole that will be used for filling. In addition, there is an extra bead of epoxy (shown by a black line) extending out from the edge of the fill hole to form a well. 97

107 Chapter 4 Experimental Results 4.1 General Methods In the course of this work, well over 100 cells were made. The results presented here for EC using I52 are from a limited number of cells which are listed in Table 4.1. (The cells used in the Frederiks experiment are discussed separately in Appendix A.) For the rest of the thesis, I will refer to the cells by their label. The majority of the experiments used cells I5243 and I5263. For these two cells, EC was studied as a function of three control parameters: applied voltage Table 4.1: Summary of main cells. label date filled dopant % date doped thickness I5221 2/9/93 1.4% I 2 1/26/93 10 µm I /16/93 2.2% I 2 11/21/93 28 µm I5263 5/19/94 2.3% I 2 5/6/94 28 µm I5268 9/12/94 2.3% I 2 8/16/94 52 µm I /28/94 2.3% I 2 8/16/94 54 µm I /21/94 2.3% I 2 8/16/94 30 µm 98

108 V, applied frequency Ω, and conductivity of the sample σ. The cells I5221, I5268 and I5275 were used to make preliminary observations of the dependence of EC on cell thickness d. A separate reduced control parameter ɛ(σ, Ω) = (V/V c (σ, Ω)) 2 1 is defined for each value of σ and Ω. Here V c (σ, Ω) is the critical voltage for the onset of EC as a function of σ and Ω. For this work, all of the studies were done by varying V at fixed values of σ and Ω. So, V c (σ, Ω) and ɛ(σ, Ω) will be written as V c and as ɛ, respectively, with the σ and Ω dependence assumed. This is consistent with definitions used in the theoretical calculations where V c is computed for a given value of σ and Ω, and the weakly nonlinear analysis is done in terms of small ɛ = (V/V c ) 2 1. The patterns consisted of roll-like solutions (modes) which can be written as A n (x, t) cos(k n x ω n t), where n labels the modes. Here k n and ω n are the wavevector and the frequency of a given mode. The A n (x, t) corresponds to the complex amplitude described by the amplitude equations discussed in Sec. 2.1: A n (x, t) = A n (x, t) exp[φ(x, t)] where A n (x, t) is a slowly varying function, and φ(x, t) gives the deviations of k and ω from k n and ω n. Most commonly, the modes of interest were the four degenerate right- and left-traveling zig and zag oblique roll states. For these solutions, the k n and the ω n are all equal, and the angle Θ between k n and the director is θ for the zag states and π θ for the zig states. Choosing the x-axis parallel to the director, Θ = tan 1 p n /q n where q n and p n are the x and y components of k n. There were two main types of analysis used to study these patterns. The first method was a general use of the power spectrum of the digitized image to determine k n, ω n and the total amplitude A n of a given mode. (The details of the image digitization are given in Sec ) The images I l,m,t consist of discrete pixels where l, m are the x and y indices and t is the time index. I will use a continuous notation I(x, t) I l,m,t where it simplifies the expressions without introducing ambiguities. The wavevector k is taken to have components q and p parallel and perpendicular to the director, respec- 99

109 tively, and ω is the angular frequency of the pattern. For time series of images, the discrete power spectrum is S p,q,w = 1 N all l,m,t I l,m,t exp[2πi(pl/l + qm/m + wt/t )] 2. (4.1) Here L, M, T are the total number of x, y, and t points, respectively, with the total number of points N = L + M + T. For single spatial images, S p,q = 1 N all l,m I l,m exp[2πi(pl/l + qm/m)] 2. (4.2) Here N = L + M. Again, where convenient I will use continuous notation and write S(k, ω) S p,q,w and S(k) S p,q. By Parseval s theorem, the normalization is chosen so that S(k, ω) = all k,ω [I(x, t)] 2, i.e. the sum of the power spectrum is equal to the mean 1 N all x,t squared amplitude. If we consider only spatial images of the form I(x) = A n cos(k n x), then, [S(k n ) + S( k n )] 1/2 = A n / 2. So, for a given mode, we define A n 2([ k n S(k)]+[ k n S(k)]) 1/2, where the summation is over a small region around k n and k n. The summation is performed over a small region around the peak because the underlying pattern is generally not commensurate with the finite extent of I(x), and this results in a finite width for the peak at k n. Equation 3.1 converts the A n computed in this fashion from the shadowgraph signal to a value for the amplitude of the director variation in radians. In addition, one determines k n from k n ( S(k)k)/( S(k)) where the summation is over a limited region around the peak of interest. Equivalent definitions and methods are used to determine ω n, k n, and A n from the time series with the summation over ω in addition to k. The other analysis technique was used in the characterization of the observed spatio-temporal chaos for which the behavior of A n (x, t) is of particular interest. To extract A n (x, t) of a given mode, one demodulates the images by performing the space-time Fourier transform of a time series of images and setting the Fourier transform to zero everywhere except for a small region around 100

110 spatial Fourier transform zag k y k x inverse Fourier transform zig Figure 4.1: Example of a spatial demodulation into zig and zag rolls. the k n and ω n of interest. Taking the inverse Fourier transform of this modified function results in a complex function of space and time. The real part of this function corresponds to A n (x, t) cos[k n x ω n t + φ(x, t)]. The modulus of the function is just A n (x, t). One can also demodulate single spatial images to find the amplitude of the zig and zag rolls without distinguishing between right- and left-traveling states. An example of spatial demodulation which takes the real part of the inverse Fourier transform is shown in Fig Note, all of the images in this chapter have the director aligned horizontally (the 101

111 x-axis), and the gray-scaled plots of the power spectrum have (k x = 0, k y = 0) in the center of the image. In analyzing the shadowgraph images, there is a difficulty which arises for traveling rolls which is not present for stationary rolls. The effect is an artifact of the frame grabber[71] which is used in these experiments and predominately affects the quadratic terms in the shadowgraph signal (see Sec ). Figure 4.2 shows two images (left-hand side) which were taken at rather large values of ɛ where the quadratic effect dominates the shadowgraph signal 1. Also shown are their corresponding power spectra (right-hand side). The top image was taken at ɛ = 0.015, and the nonlinear peak corresponding to the direction perpendicular to the director is substantially larger than the peak in the parallel direction. The lower image is at ɛ = 0.070, and the two peaks have roughly equivalent amplitudes, as expected. For ɛ = the rolls are traveling and for ɛ = the rolls are stationary. Because the frame grabber[71] only reads single lines not frames, each image is composed of a number of frames. This smears out the rolls in the direction of travel, in this case, parallel to the director. Therefore, the relative strength of the peak perpendicular and parallel to the director is misleading for the case of the traveling rolls. One can see that the relative strengths are actually equal by observing the optical power spectrum of traveling rolls (for an example, see Fig. 3.6 in Sec ). There the frame grabber plays no roll in the computation of the power spectrum, and the peaks perpendicular and parallel to the director clearly have equal intensity. 1 Even when focusing on the cell (see Sec ), diffraction effects and the finite thickness of the cell result in the quadratic shadowgraph effect eventually dominating the image for high enough values of ɛ. 102

112 Figure 4.2: The top image (left) is a snapshot of a superposition of right and left traveling zig and zag rolls at ɛ = The director is aligned in the horizontal direction. In the spatial power spectrum of this image (right), the peaks corresponding to the sum of zig and zag rolls which gives a wavevector perpendicular to the director are clearly stronger than those parallel. (The peaks of interest are highlighted by circles.) The bottom image shows the superposition of stationary zig and zag rolls at ɛ = Here the peaks in the power spectrum have roughly equal amplitudes. Notice, for the case of ɛ = 0.015, the fundamental peaks are still visible. 103

113 4.2 Conductivity As discussed in Chapter 2, one of the outstanding problems in EC in NLC is the existence of traveling waves. The detailed linear stability analysis[33, 34] of the SM predicts that the initial bifurcation is to a stationary pattern; whereas, there are a number of experimental observations of traveling patterns[24, 25, 26, 27, 28, 37, 97]. One of the motivations for studying new liquid crystals is to acquire information which would be useful in guiding modifications of the SM. Two assumptions of the SM, perfect planar alignment with no pretilt and Ohmic conductivity (see Sec. 2.3), have been considered as possible weaknesses of the SM[33, 34, 38] which are related to its failure to predict the observed traveling waves. It would be interesting to study both of these assumptions. Measurements of a number of alignment techniques[59] reveal that the pretilts are less than 3, and for many applications this is effectively a zero pretilt. However, pretilts of less than 1 are rarely achieved, if at all. In order to quantify the effect of pretilts on EC, a reliable measurement of this generally small pretilt angle is necessary. In principle, by converting the shadowgraph light source from a parallel beam to a converging beam, the well known method of conoscopy could be used to measure the pretilt (see Sec ). However, for the typical cell thicknesses used in the EC studies, conoscopic measurements on cells of planar alignment do not have the required resolution to make accurate measurements of the pretilt. Furthermore, the recent theoretical work described in the Sec. 2.3, the WEM[39], makes a strong case for the conductivity σ as the primary source of a Hopf bifurcation. For these reasons, I have focused my studies on the conductivity of the I52 samples. The conductivity of the sample is due to equal numbers of positive and negative ionic impurities[64] which generally have different mobilities, undergo a dissociation-recombination reaction[65], and have a finite time for traversing the width of the cell. The WEM includes these additional effects into the SM (see Sec. 2.3 and Ref. [39]), and the resulting conductivity is frequency 104

114 dependent. The dopant used with I52 was 2% by weight I 2. In the simple picture, I 2 forms a charge transfer complex with the benzene ring present in I52. It is this complex which undergoes a dissociation-recombination reaction producing positively and negatively charged species. This multi-step process is clearly more complicated than the simple dissociation-recombination picture used in the WEM. I measured the frequency behavior of the conductivity of the I52 samples to check for deviations from the predictions of the WEM. The cell is treated as a resistor and capacitor in parallel with the resistance R and capacitance C defined by 1/R(ω) = Re[I(ω)/V (ω)] (4.3) ωc(ω) = Im[I(ω)/V (ω)], (4.4) where I(ω) is the current in response to an applied voltage V (ω) = V in cos(ωt). For the ohmic conductivity assumed by the SM, 1/R and C are independent of frequency. In the context of the WEM, C has a high frequency limit corresponding to the dielectric capacitance C d = ɛ o ɛa/d. At low frequencies, the free ions can generate boundary layers which result in an increase of C as Ω is decreased. This behavior is demonstrated by a typical measurement of C which is shown in Fig The high frequency limit is consistent with measurements of ɛ made in undoped samples. The boundary layer thickness depends on the details of the electrical boundary conditions at the two electrodes, but can be deduced from the low frequency rise of the capacitance. This is apparent for the case of insulating boundaries for which analytic expressions for C and R can be derived from the WEM[99]. Assuming a boundary layer thickness l b, ) (d/2l b ) C = C d ( (d/2l b ) 2 (Ωτ q ) 2 (4.5) R = R o [1 + (d/2l b ) 2 (Ωτ q ) 2 ]/(d2l b ) 2 (Ωτ q ) 2. (4.6) Here R o is a constant and represents the asymptotic high frequency value of R. 105

115 capacitance (pf) applied frequency (Hz) Figure 4.3: Capacitance of I5263 at T = 49 C as a function of frequency. Figure 4.4 shows a plot of the data from Fig. 4.3 and three curves computed using Eq Even though Eq. 4.5 captures the correct low and high frequency limits, it does not fit the detailed shape of the curve in the transition region. Despite this, one can use these curves to estimate an upper limit of the boundary layer thickness of l b < 0.04d 1 µm. One possible source of the discrepancy is that the boundaries are not perfect insulators. However, if this were true, there would be a finite dc resistance. Figure 4.5 shows a typical measurement of R using an applied voltage of 5 V rms as a function of frequency. I have observed no evidence of a finite dc resistance. Figure. 4.5 clearly shows the sharp rise in R in the low frequency limit which is consistent with the curve predicted by Eq. 4.5 given by the dashed line in Fig However, the high frequency limit of R does not behave as expected. One contribution to the total R of the NLC cell which is not included in the WEM is dielectric losses in the material itself. Figure 4.6 shows the measured R of a cell filled with undoped I52. The cell has the same dimensions and construction as cell I5263. For comparison, the R of a mica capacitor which is 106

116 900 capacitance (pf) applied frequency (Hz) Figure 4.4: The measured capacitance of I5263 at T = 49 C compared with three curves calculated using Eq The curves use a boundary layer thickness of 5.0 µm (solid line), 1.0 µm (dashed line), and 0.2 µm (dot-dashed line). known to be have relatively high dielectric losses and a polystyrene capacitor which is known to be relatively loss free are shown as well. (The data for the polystyrene capacitor does not exist for frequencies above 200 Hz because the resistance was too large to measure.) The loss for I52 is definitely larger than that for the mica capacitor, but it is consistent with other measurements of dielectric losses in NLC[63]. The rather large losses in NLC at these low frequencies are generally attributed to long relaxation times which result from the various intermolecular interactions which contribute to the nematic order. In addition to losses in the NLC itself, this measurement includes any losses inherent in the cell, for example, the mylar spacers. One can correct the resistance predicted by Eq. 4.5 by including the resistance of the undoped I52 cell as an additional resistor in parallel with the ionic contribution given by Eq The solid line in Fig. 4.5 is the result of fitting this effective resistance to the data with R o and l b as fit parameters. The resulting values are R o = ohm and l b = 0.9 µm. This result for 107

117 resistance (10 6 ohm) applied frequency (Hz) Figure 4.5: The measured resistance of I5263 at T = 49 C. The dashed line is computed using Eq. 4.5 with R o = ohm and a boundary layer thickness of 0.9 µm. The solid curve is a calculation of the resistance due to the resistance given by the dashed line in parallel with the measured resistance of an undoped cell (see Fig. 4.6). l b is consistent with the capacitance measurements. In addition to the frequency dependence, I have also measured a voltage dependence of R. This is shown in Fig The capacitance was found to be voltage independent. The voltage dependence of R can not be explained by the WEM. In summary, the WEM explains the high and low frequency limits of the frequency dependence of R and C. A more detailed description of the chemistry of the doping process and physics of the ionic species is needed to explain the voltage dependence of R and some of the details of the frequency dependence of R and C. However, the WEM represents a clear improvement over the SM assumption of a frequency and voltage independent σ. 108

118 resistance (10 6 ohm) applied frequency (Hz) Figure 4.6: The measured resistance of a polystyrene capacitor (open triangles), mica capacitor (open circles), undoped I52 cell at 44 C (solid circles), undoped I52 cell at 49 C (solid triangles). The solid lines are fits of the data to R = A/(f + f 0 ), where f is the applied frequency and f 0 = 70 Hz. 8.8 resistance (10 6 ohm) voltage (V rms ) Figure 4.7: The measured resistance the cell I5263 at 49 C and and applied frequency of 50 Hz as a function of applied voltage. 109

119 4.3 Linear Behavior As discussed in Sec. 2.3, the linear properties of EC in I52, V c, Θ and ωτ q, have been calculated within the context of a new model, the WEM, as a function of Ωτ q, σ and d (as well as the other material parameters). I will present the results of the measurement of these three quantities in four cells and compare the results with the calculations. The four cells used for the linear studies are I5221, I5243, I5263, and I5275 (see Table 4.1). Initial studies of EC were done using the cells I5221 and I5243. These were among the earliest cells made and quantitative data on the Hopf frequency was not acquired. The cell for which the most extensive quantitative work was carried out was I5263 with a thickness of 28 µm. Limited measurements were made in the thicker cell I5275 to study the d dependence of ω. Because the WEM predicts the linear values of ω and Θ, it was experimentally useful that the initial bifurcation is forward, i.e. continuous and non-hysteretic, in the parameter range of interest. The evidence for the forward bifurcation is presented in Sec To extract the linear values, ω and Θ were measured as a function of ɛ, and the results in the limit as ɛ 0 were compared with theory. The step size in ɛ was δɛ = with a waiting time of 300 s (400τ d ). For each value of ɛ, a time series of images and a separate single image were taken. The single images covered a square region 30 wavelengths on a side. The time series consisted of 64 images taken 2 seconds ( 2.5τ d ) apart with each image covering 6 wavelengths. From the single images, I averaged Θ of the two degenerate modes (the values of Θ for the two modes agreed to ±1 degree). I used the average ω of the four degenerate traveling rolls computed from a weighted average of the relevant peaks in S(k, ω) (the four frequencies differed by at most ±2%). I determined V c using the total power under the relevant pairs of peaks in S(k) and S(k, ω). I found that S(k) and S(k, ω) yielded the same results for V c. A meaningful test of the WEM requires knowledge of the following SM 110

120 parameters: the three elastic constants (K 11, K 22, K 33 ), six viscosities (α 1, α 2, α 3, α 4, α 5, α 6 ), σ, σ, ɛ, ɛ, and d. In addition, two new parameters µ + µ and λ σ(r c ) are needed. The requirement that the WEM recovers the correct SM prediction for V c and Θ sets an upper limit on λ σ (R c ) from λ σ (R c ) τ SM 0 1. The literature values for γ 1, η, and ɛ were used, and independent measurements of ɛ a, K 33, σ and d were made. This fixed eight of the SM model parameters using α 2 = α 3 γ 1 (definition of γ 1 ), α 5 = α 6 α 2 α 3 (Onsager relation), and α 4 = 2η. The remaining three viscosities (α 1, α 3, and α 6 ) and the remaining two elastic constants (K 11 and K 22 ) were fixed by fitting V c and Θ for a single and temperature and assuming the temperature dependence of the unknown parameters was similar to the known ones. The remaining SM parameter, σ a /σ, was determined by fitting the low frequency behavior of V c at each temperature. In all, six parameters were determined from the six V c curves and the six Θ curves. Further details and the results of the fits are given in Appendix A. The final unknown parameter, µ + µ, was fit using measured travelling frequencies ω. Figure 4.8 is a plot of V c for the cell I5263. For each T, I show V c as a function of the scaled applied frequency Ωτ q. These curves were measured with a resolution in ɛ of Figure 4.8b shows the comparison of theory and experiment for two typical curves. Figure 4.9 shows the onset curves from I5221 and I5243. These curves were only measured with a resolution in ɛ of In a number of the onset curves, there is a definite inflection point. One of the clearest examples of this behavior is shown in Fig. 4.9b. The inflection point[99] is always close to, but above, the Lifshitz point[31]. The Lifshitz point is the value of Ω at which Θ goes to zero. In the measurements using I5263, the Lifshitz point roughly coincided with the cutoff frequency, and this work focused on the conduction regime of EC. Therefore, for most of the runs using I5263, there is no data above the Lifshitz point, and the inflection point was not observed. By symmetry arguments, Θ will go to zero with a square root dependence 111

121 (a) applied voltage (V rms ) applied voltage (V rms ) (b) applied frequency (Ωτ q ) Figure 4.8: (a) Onset curves for 6 temperature in I5263. Circles, squares, up triangles, diamonds, down triangles, and open diamonds are for 29 C, 34 C, 39 C, 44 C, 49 C, and 59 C, respectively. (b) onset curves for 29 C and 44 C with the corresponding WEM predictions (solid lines). The values of the material parameters used in the theoretical calculations are given in Appendix A. on the relevant parameter at the Lifshitz point. For example, in thermal convection in NLC, the Lifshitz point occurs as a function of applied magnetic 112

122 applied voltage (V rms ) (a) applied voltage (V rms ) (b) Ωτ q Figure 4.9: Onset voltages V c. (a) The open symbols are from cell I5243 at T = 24 C (squares) and T = 44 C (circles). The solid symbols are from the cell I5221 at T = 44 C (squares) and T = 64 C (circles). (b) The data from I5221 at T = 44 C with a fit to V c = a+b(ωτ q ) 2 +c(ωτ q ) 4. Here a = 13.2 V rms, b = 3.24 V rms, and c = V rms. field, and the square root behavior was found to hold for all field values[30, 32]. However, the range of validity of the square root dependence will be system dependent. In EC, the relevant parameter is Ωτ q. Two typical results for Θ as a function of Ωτ q are given in Fig The two curves are for very different parameter sets. For both cases, the solid line gives the WEM prediction for Θ. Given the large uncertainty in the material parameters for I52, the agreement between theory and experiment is quite good. Figure 4.11 shows the measured Hopf frequency ω and the calculation of 113

123 40 (a) (b) Θ (degrees) Ωτ q Ωτ q Figure 4.10: (a) Angle Θ at onset for cell I5263 corresponding to the onset curve for T = 44 C shown in Fig Solid line represents the theoretical calculation. (b) Θ at onset for cell I5243 corresponding to the onset curve for T = 24 C shown in Fig Solid line represents the theoretical calculation. The values of the material parameters used in the theoretical calculations are given in Appendix A. ω as a function of temperature for each of the six values of the temperature. Both τ0 SM and 2π/ ω are O(1 s), so λ σ (R c ) τ0 SM 1 implies (λ σ (R c )/ ω) 2 1. Therefore, ω ω and is computed using Eq with only one adjustable parameter, µ + µ (see Appendix A), for each of the six temperatures. Recall, for any values of the parameters, the SM prediction ω = 0 can not be shown on the scale used in Fig Also, the shape of the theoretical curves is completely determined by the SM model parameters which are fixed from the measurements of V c and Θ. The precise values of µ + µ simply set the overall scale (which is definitely greater than zero) and are consistent with previous measurements of µ + µ [39]. There are three predictions of the WEM which are independent of the unknown material parameters and the fitting procedure I used. First, the sign of the curvature of ω as a function of Ω is determined by the sign of ɛ a, which is measured. For ɛ a < 0, ω increases as Ω is increased, and for 114

124 1.5 ω (sec -1 ) Ωτ q 0.6 ω (sec -1 ) Ωτ q Figure 4.11: Plotted here is the Hopf frequency ω as a function of Ωτ q for the six runs shown in Fig The six curves are shown in two separate plots in order that they may be plotted on two different scales. The top plot is for 29 C (circles), 34 C (squares) and 39 C (triangles). The lower plot is for 44 C (diamonds), 49 C (triangles), and 59 C (circles). The solid lines are the corresponding theory. It is important to note that ω = 0, the SM prediction, can not be shown on the scale used here. ɛ a > 0, ω decreases as Ω is decreased. Because of the drift in σ, I was able to measure ω for two values of σ at a fixed value of T and d. The WEM predicts that ω σ 1/2 d 3. Figure 4.12 shows that the data are consistent with the predicted σ dependence. The cell I5275 was used to check the thickness dependence. Measurements 115

125 (a) ω σ 1/2 (s -1 ohm -1/2 m -1/2 ) (b) (c) Ωτ q Figure 4.12: (a) ωσ 1/2 as a function of Ωτ q for 34 C in cell I5263 at σ = ohm 1 m 1 (open symbols) and σ = ohm 1 m 1 (closed symbols). (b) ωσ 1/2 as a function of Ωτ q for 44 C in cell I5263 at σ = ohm 1 m 1 (open symbols) and σ = ohm 1 m 1 (closed symbols). (c) ωσ 1/2 as a function of Ωτ q for 49 C in cell I5263 at σ = ohm 1 m 1 (open symbols) and σ = ohm 1 m 1 (closed symbols). of ω at the same temperature in cell I5263 and I5275 are shown in Fig Here I also scale by σ 1/2 as the conductivities were slightly different. The high frequency limit follows the correct scaling of d 3. In the thicker cell, ω goes to zero as Ω is decreased. Qualitatively, this behavior is predicted by the WEM, and is related to the value of τ rec. Because of the small value of ɛ a, the 116

126 1.2 1 ω σ 1/2 d 3 x Ωτ q Figure 4.13: Scaled ω for the two cells I5263 (triangles) and I5275 (circles) at 49 C. theory is extremely sensitive to the chosen values of τ rec, and because of the drift in σ, the experimental value of Ω at which ω went to zero. These measurements confirm that the WEM has captured the main features of the Hopf bifurcation for EC. I have shown experimentally that for most of the parameter regime which was considered, the linear state consists of four degenerate modes: right- and left-traveling zig and zag modes. In the next section, I will discuss the interactions of these modes in the weakly nonlinear case and the ramifications of this for further tests of the WEM. 4.4 Weakly Non-linear Regime Linear stability analysis is not sufficient to predict if a transition is forward or backward. Furthermore, the linear stability analysis does not predict which pattern will be selected just above onset. For example, for Ωτ q less than the Lifshitz point, there are two degenerate modes (or four in the case of a Hopf 117

127 bifurcation), and the state which is actually observed above ɛ = 0 is a result of nonlinear interactions between these modes. In this section, I will discuss the experimental observations of the weakly nonlinear regime. I will focus on the range of conductivity which was discussed in Sec. 4.3; however, I will mention some preliminary results from higher conductivities and thicker cells. I have shown that the WEM captures the correct linear behavior of EC in I52. In theory, it is possible to derive amplitude equations from the WEM which should quantitatively describe the patterns discussed in this section. As of now, this calculation has not been carried out, so comparison between experiment and theory is not yet possible. One difficulty in determining whether or not the transitions were forward or backward was the drift in the conductivity of the I52 cells as a function of time (see Sec ). However, the drift was slow enough that the resulting drift in V c was linear over relatively long time periods. To determine V c, the voltage was increased quasistatically with steps of ɛ = waiting for 15 minutes at each step. The amplitude of the pattern was measured using the power under the relevant peaks in S(k). I determined V c to be the value of V halfway between the step in voltage where the measured amplitude first becomes nonzero and the previous step. Typical results for V c as a function of time are shown in Fig The drift in the critical voltage in the various cells was roughly V/hr for a V c 15 V rms, which corresponds to a drift in ɛ of roughly /hr for the typical voltages applied to the cell. To determine if the transition is forward or backward, the voltage was both quasistatically increased and decreased with the same step size as described above, and the amplitude was determined using S(k). The data obtained from increasing V were used to check the drift of V c. Then, ɛ was computed for each step in voltage for both the increasing and decreasing directions using the computed V c (t). If the amplitude as a function of ɛ was either hysteretic, or more importantly, showed a finite jump in amplitude as a function of ɛ, the transition was considered to be backward. If the amplitude grew continu- 118

128 V c (V) resistance (10 6 ohm) time (hours) Figure 4.14: Top figure shows the drift in V c over a period of 10.5 days. The linear fit to V c as a function of time (solid line) gives V/hr. The corresponding drift in the cell s resistance is shown in the lower figure. ously from zero with no evidence of hysteresis, then within my resolution, the transition was considered to be forward. A typical measurement is shown in Fig for cell I5263. An example from cell I5243 which shows a smaller range of ɛ is given in Fig The most important features of the onset curves are that the amplitude exhibits no large jump in value at ɛ = 0 and that there is no measurable hysteresis upon decreasing ɛ. The value of the amplitude in radians is computed using Eq. 3.1, 119

129 amplitude (rad) ε Figure 4.15: A typical result of an onset measurement in the cell I5263 for σ = ohm 1 m 1. The open symbols (circles) are for increasing the voltage and the closed symbols (triangles) are for decreasing the voltage. and is only an estimate of the amplitude. For example, Eq. 3.1 does not include any diffraction effects in the shadowgraph signal which would decrease the measured amplitude. Figure 4.17 shows three images taken from the onset run shown in Fig Also shown are the corresponding power spectra. The images are of a limited region of the cell where convection first occurs, and they illustrate the continuous nature of the transition. These pictures should be compared with Fig Present in both measurements is a relatively large baseline. Comparison with the results of Ref. [37] reveal that the baseline is on the order of, but larger than, the fluctuating amplitudes due to thermal noise. The measurements shown in Figs and 4.16 used only a single spatial image. The result in Ref [37] were obtained from spatio-temporal structure functions which were computed using a large number of images. A study of the onset for this parameter range in I52 using the techniques of Ref. [37] is needed. This should reveal a smooth transition from fluctuating patterns driven by noise to the 120

130 0.03 amplitude (rad) ε Figure 4.16: A typical result of an onset measurement in the cell I5243 for σ = ohm 1 m 1. The open symbols (circles) are for increasing the voltage and the closed symbols (triangles) are for decreasing the voltage. deterministic spatio-temporal chaos. The chaos is apparent in these measurements in the fluctuations of the measured amplitude as a function of ɛ, but will be discussed in detail later. There was a measurable decrease of ω with increasing ɛ. Figure 4.18 shows an example of the dependence of the Hopf frequency for small values of ɛ. In Sec. 4.5, I will report on a secondary transition for which ω makes a finite jump to zero. The behavior of ω for small ɛ should be contained within the CGL description (see Sec. 2.1) for which the complex coefficients provide both a linear and a nonlinear frequency shift. The only observations of a discontinuous transition in cells with a thickness of 28 µm were for σ > ohm 1 m 1. This transition was also to a stationary state. However, these measurements used rather large steps in ɛ of the order 0.01 (see Sec. 4.5). From the WEM (see Sec. 2.3), it is believed that the value of the product σd 2 determines when the transition becomes stationary, and if this is so, it is likely that the value of σd 2 also determines 121

131 a d b e c f Figure 4.17: Three images and their corresponding power spectra from the onset run shown in Fig (a) Image from ɛ = and power spectrum (d). (b) Image from ɛ = and power spectrum (e). (c) Image from ɛ = and power spectrum (f). The power spectra in (d) and (e) are enhanced a factor of 25 more than the one in (f). All of the images were scaled the same. 122

132 0.4 ω (sec -1 ) ε Figure 4.18: Measurement of hopf freq ω as a function of ɛ in cell I5263 at 49 C, σ = ohm 1 m 1, and Ωτ q = when the transition is backward. Indeed, when working with the thicker cells, I observed a discontinuous transition to a stationary state at significantly lower values of σ which is consistent with this assumption. Figure 4.19 shows two images, one taken just below onset and the other taken just above onset, from the cell I5275 for σ = ohm 1 m 1. Shown here is a small region of the cell where EC first occurs. The pattern is stationary above onset, and the initially observed director variation has an amplitude of roughly 200 mrad. This represents a significant jump in amplitude and strongly suggests that the transition is backwards. The measured amplitudes of the director variation just above onset in this case are consistent with the results of the backward bifurcation measured in Ref. [37]. For the rest of this chapter, I will focus on the the parameter range σ < ohm 1 m 1. Figures 4.20a, 4.20b, 4.20c and 4.20d show four images from the cell I5278 and their corresponding power spectrum. The images 123

133 (a) (b) Figure 4.19: Images from onset run in I5275 which is considered to be backward. (a) was taken at V, and there is no evidence of convection. (b) was taken at V, which represents a step in ɛ of The power spectrum gives an estimate of 200 mrad for the director amplitude in (b). This represents a large jump in amplitude at onset (see for instance, Fig. 4.16). were taken at ɛ = (4.20a) and ɛ = (4.20b, 4.20c and 4.20d) with σ = 1.5, 1.5, 1.0, and ohm 1 m 1, respectively. All four images have been displayed using the same gray scale. Figures 4.20a and 4.20b are examples of a typical extended state which has a nonperiodic spatial and temporal variation of the amplitude of the four modes. The pattern in Fig. 4.20b is extremely weak, but the four peaks corresponding to the zig and zag modes can clearly be seen in the power spectrum. For σ = ohm 1 m 1, I observed temporary localization of the pattern and large increases in the magnitude of the amplitude. For the instance in time shown in Fig. 4.20c, the pattern in the left portion of the image has localized in the direction perpendicular to the director (the y-direction), but the extended state is still present on the right. This is apparent in the power spectrum where there are still the peaks which correspond to the zig and zag rolls, but now one sees an additional peak, highlighted by the dashed box, which is the signature of the localized structure. One should compare the relative amplitudes of the localized state and the extended state in Fig. 4.20c to the images in Figs. 4.20a and 4.20b. 124

134 For σ = ohm 1 m 1, I observed no evidence of an extended pattern in the image or the power spectrum. Only patterns localized in the y-direction were observed, as shown in Fig. 4.20d. The patterns at all three values of σ are examples of spatio-temporal chaos at onset where the amplitudes of the four modes vary aperiodically in space and time. A typical example of a spatial demodulation for the extended state found at σ = ohm 1 m 1 is shown in Fig Shown is the original superposition of zig and zag rolls and the amplitudes of the corresponding zag rolls. The spatial correlation length of the individual A n (x) is computed from the power spectrum of A n (x). I find a correlation length of 25λ along the director, and perpendicular to the director, I measure a correlation length of 20λ, where λ = 2π/ k. Currently, this is only a rough estimate of the spatial correlation lengths as our largest images only span a region of 60λ. I studied the spatially-local temporal behavior of the pattern A n (t), using time series of images with a spatial extent of roughly 4λ. I demodulated the time series to get the A n (x, t). For each A n (x, t), I computed A n (t) = An (x, t)dx. Figure 4.22 shows a 30 minute segment of A n (t) for the righttraveling zig and zag rolls. One observes periods of time where one of the modes dominates as well as times when the modes have approximately equal amplitudes. Computing the cross-correlation of pairs of the A n (t) from a 4 hour long time series, I find that the four modes are anti-correlated with each other. Estimating the correlation time of a single A n (t) from its power spectrum yields a value of roughly 1000τ d (τ d is O(1 s)) for all four modes. Recall that the traveling frequency ω is O(1 Hz). I observed this state for over 48 hours at ɛ = 0.005, and no localized states were observed. In addition, no localization was observed as a function of ɛ. The next transition, which occurred at ɛ = 0.1, was to an extended stationary state. These two measurements are an extremely limited initial characterization of the dynamics of this state. The time series measures the local amplitudes of all four degenerate modes separately. During time periods for which the 125

135 a k y k x b c d Figure 4.20: (a), (b), (c), and (d) show typical spatial patterns (left side) which exist for Ω/2π = 25 Hz at an ɛ = for (a) and an ɛ = for (b), (c) and (d) and σ = 1.5, , and ohm 1 m 1, respectively. Their corresponding power spectra are shown on the right side. All four patterns are from the cell I5278. (a) and (b) are examples of the extended state at two different values of ɛ. (c) shows the coexistence of a localized state with the extended state of lower mean amplitude. In the power spectrum, the dashed box is around the peak due to the localized structure and the dashed circle is around the peak due to the zag rolls. (d) shows the localized state. All four patterns are time dependent, and all four images are scaled the same. The power spectra for images (b), (c) and (d) were scaled by the same factor, but the power spectrum for (a) is scaled such that black represents a power 4x the power represented by black in the power spectra (b), (c) and (d). amplitudes of right- and left-traveling modes are equal, there are only two degenerate modes: zig and zag standing waves. However, there are equally long periods where the traveling modes have unequal amplitudes. During these pe- 126

136 Figure 4.21: The left-hand image is a snapshot from a time series at ɛ = 0.01, and σ = ohm 1 m 1 in a cell with d = 28 µm. It shows the superposition of the zig and zag rolls which exists throughout the cell. The right-hand image shows the corresponding amplitude of the zag rolls with the white regions corresponding to large amplitude. Amplitude (arb. units) time (min.) Figure 4.22: Plot of the temporal variation of the amplitude of the righttraveling zig (solid lines) and zag (dashed lines) rolls. The time series was taken at ɛ = 0.01 and σ = ohm 1 m

137 riods, the resulting superposition is not a standing wave; however, the local zig (zag) amplitude still varies periodically in time with the underlying traveling frequency. The single snapshots measure the amplitude of this superimposed right- and left-traveling zig (zag) state at a single instance in time. Because the spatial demodulation represents a single instance in time, there are a number of sources of the spatial variation of the amplitude shown in Fig Obviously, one could have local zig (zag) standing waves which were globally in phase but had a spatial variation of their amplitude. A standing wave with a spatially uniform amplitude and a spatially varying phase would also have an instantaneous spatial variation of the zig (zag) amplitude. Indeed, it is possible to imagine a great variety of combinations of variations in the relative phase and amplitudes of the four modes which would result in the spatial variation observed in Fig It is clear that for a more comprehensive understanding of this state of spatio-temporal chaos, one requires time series of images of large spatial extent from which the phase and amplitude of all four modes can be studied. One important question which can only be answered in this fashion is the role of defects in the pattern. Defects have been found to play an essential role in other forms of spatio-temporal chaos. But, to adequately study the defects in this system, one needs to look at the underlying four modes. Observations on the superposition of modes (which the current spatial measurements represent) can be misleading. For σ = ohm 1 m 1, I still observe an extended state which is similar in amplitude to the state observed for σ = ohm 1 m 1. However, the dynamics of the A n (x, t) is quantitatively different. There are bursts of large amplitude convection which occur chaotically in time and in random spatial locations. The bursts either result in structures which are highly localized in the y-direction (a width of roughly λ) or result in blobs of high amplitude convection. Figure 4.23 shows four instances in time from a series taken at ɛ = Figure 4.23a, 4.23b, 4.23c and 4.23d show the 128

138 a b c d Figure 4.23: Snapshots from a time series of images at ɛ = and σ = ohm 1 m 1 in a cell with d = 28 µm. (a) is t = 324 min, (b) is t = 340 min, (c) is t = 496 min, and (d) is t = 564 min, where t = 0 min corresponds to when the voltage was set to ɛ = extended state, a state which has localized in the central region, the extended state again, and a blob state, respectively. At a fixed value of ɛ, this behavior repeated in an apparently aperiodic fashion for the duration of our observations (the longest observation period was 48 hours). An important feature of this state was the long time scales. The state would often initially remain uniform for up to 3 hours. This would explain why the state was not apparent in the initial studies of cell I5263 at σ = ohm 1 m 1 where the studies as a function of ɛ involved waiting times of at most an hour. 129

139 For σ = ohm 1 m 1, we observed no evidence of any extended state. The only states which were observed (up to ɛ 0.1) were localized states (worms) having a constant width in the y-direction equal to the y- component of the wavelength of the zig and zag states observed at σ = ohm 1 m 1. The worms have a distribution of lengths in the x-direction. For low amplitudes, the worms appear to be stationary, or slowly drifting, and blink with the underlying traveling frequency. Above a critical, as of yet undetermined, value of the worm s amplitude, it stabilizes into either a left- or a right-traveling state of apparently constant amplitude. The state is composed of a superposition of right- or left-traveling zig and zag rolls, respectively, i.e. the worm travels in the opposite direction of the traveling waves which compose it (see Fig 4.25 and discussion there). Just above onset, worms grow and die throughout the cell with lifetimes on the order of τ d. The average number and length of the worms increases with ɛ. At high enough values of ɛ, worms are observed to extend across the cell and exist until they travel out of the region of observation or interact with other worms. The relatively long lifetimes of the worms near onset is consistent with the long correlation times observed in the extended state. The worms have a number of interesting interactions with each other. Figure 4.24 shows four images taken 1 minute apart at an ɛ = The worms are traveling in both directions and are arranged with a roughly fixed spacing in the y-direction. As ɛ is increased, the spacing between the worms decreases until the cell is filled with convection. One worm is highlighted by a white rectangle to emphasize its movement across the cell. During a head on collision of worms, the overlapped regions fluctuate widely in amplitude and spread in the direction perpendicular to the director alignment. The worms do not generally move in the y-direction. However, if the distance in the y-direction between two states is small enough, they are attracted to each other. Upon contact, they generally repel each other which results in a rapid undulation of the two states in the y-direction. Occasionally, the two worms at different 130

140 a b c d Figure 4.24: Snapshots from a time series at ɛ = 0.05 and σ = ohm 1 m 1 in a cell with d = 28 µm. The images are 1 minute apart. The white rectangle highlights a worm which is traveling to the the right. y-positions join upon contact to produce a bend in the resulting worm state. Eventually, the combined worm moves to one of the original y-positions. The worms appear to be a generic feature of oblique rolls for low σ and thin cells. We have observed worms in two different cells (I5263 and I5278) with a thickness of 28 µm. In addition, we have observed the worms in a cell of thickness d = 52 µm (I5268). In this case, the worms were observed at a value of σ = ohm 1 m 1. We believe the worms only occur below a critical value of σd 2. This expectation still needs to be confirmed, but it 131

141 would explain why the worms have not been previously observed 2. All of the previous work in EC which we are aware of has been in thicker cells or at a higher σ than we have studied here. An image of a typical isolated worm is shown in Fig The worm is traveling to the right while the rolls are traveling to the left. It appears that the worms always travel in the direction opposite the underlying traveling wave direction. There is a sharp increase of amplitude at the front of the worm and a long tail at the trailing edge. The worm is a superposition of zig and zag rolls, but nonlinear effects in the optics and the extreme localization in the direction perpendicular to the director combine to create the optical effect of normal rolls. The zig and zag nature of the underlying rolls is apparent in the weak amplitude convection which extends slightly in the y-direction around the leading edge of the state. It is also apparent when two worms interact and the state is temporarily delocalized and has formed a blob similar to the image in Fig Figure 4.25b shows an image of the same worm shown in Fig. 4.25a, but with the light polarized in the y-direction instead of the x-direction. The uniformity of the image shows that the underlying director distortions remain in the x-direction, as they are for the extended state. 4.5 Nonlinear Results The nonlinear patterns in this system show a rich variety of behavior as a function of the three control parameters V, Ωτ q, and σ. As discussed in Sec. 4.4, it is believed that the patterns also depend on d, and that the relevant control parameters are V, Ωτ q, and σd 2. For the nonlinear patterns, the majority of the detailed measurements are from the cell I5263, and I am unable to comment on the d dependence except to say that the general trends are consistent 2 There are previous reports of localized states for normal rolls[25], but these states did not travel, and from the report in Ref [25], do not appear to have the same wide range of lengths and existence 132

142 a b Figure 4.25: (a) Close up of an isolated worm in the cell I5268. (b) Image of the same region of the cell with the worm present, only the light is polarized perpendicular to the director alignment. This demonstrates the lack of director variation perpendicular to the original alignment direction. with this assumption. Even limiting the discussion to the three control parameters considered here, there is a rather large parameter space to study and I will focus on two sets of data. One set was taken at a fixed Ωτ q as a function of ɛ and σ (Fig. 4.28) and the other one at a fixed value of σ as a function of ɛ and Ωτ q (Fig. 4.29). The spatially extended patterns are labeled using three letters as given in Table 4.2. Worms is used in the same sense as Sec. 4.4 and refers to the localized states. This is the only pattern which does not follow the three letter scheme. For a given pattern, the letters in the name are determined right to left. I will discuss the naming process and use some examples for illustration. The details of where in parameter space the various patterns are found is discussed later, but Fig and Fig can be referred to if that information is desired. The third letter distinguishes between patterns which consist of a superposition of zig and zag rolls and patterns which consist of only a single set of rolls. By superposition, I mean specifically zig and zag rolls which coexist in the same spatial location. For the rest of this chapter, I will use coexisting rolls to refer to a pattern which consists of zig and zag rolls in separate spatial regions. A pattern which consists of a single set of zig rolls which coexist with a set of zag rolls is still considered a single set and is designated by an R. In 133

143 Table 4.2: Naming scheme for patterns in I52. First Position Second Position Third Position letter meaning letter meaning letter meaning T traveling N normal S superposition of rolls S stationary O oblique R single set of rolls general, oblique roll patterns which consist of a single set of rolls always have large regions of zig coexisting with large regions of zag. As defined here, the distinction between superposition and coexisting is made by a visual inspection of the pattern, and not by looking at the power spectrum, as the power spectrum does not reveal information about spatial locations. Figure 4.26 shows an example of a state which would have an S for the third letter 3 and one which would have an R, and their corresponding power spectra are shown. The second letter is decided upon differently for the superposition states and the single roll states. For the single roll state, oblique and normal have the definition given in Chapter 1 and refer to the angle between the wavevector of the pattern and the orientation of the director. States with a wavevector parallel to the director are normal ; otherwise, the pattern is oblique. If the pattern is a superposition of two rolls, oblique and normal refers to the orientation of the resulting grid pattern. This is best determined by the spatial power spectrum. If the two superimposed modes are degenerate, the pattern is considered normal (i.e., the last two letters are NS ). An example image and its power spectrum is given in Fig If the two modes are not degenerate, the pattern is considered to be oblique (i.e., the last two letters are OS ). One can see from the example image in Fig that the grid 3 The superimposed patterns are visually similar to the grid patterns of Ref [98]. But, Ref [98] does not provide power spectra, and the patterns were generated by a combined ac and dc voltage, so the connection with the patterns observed here is not clear. 134

144 a b c d Figure 4.26: (a) is an example of a pattern which consists of a single set of oblique rolls. In this case, there are coexisting zig and zag rolls with a grain boundary between them. (b) is the power spectrum of image (a) and it contains peaks corresponding to both sets of roll. (c) is an example of a superposition of oblique rolls. (d) is the power spectrum of image (c) and it also contains peaks which correspond to both sets of rolls. Image (c) shows the characteristic grid-pattern of a state which is the superposition of two oblique modes. pattern of an OS pattern is rotated with respect to the axis defined by the director, and the power spectrum clearly shows that the superimposed wavevectors are not degenerate. The first letter is determined from a time series of images and denotes whether or not the peaks in the power spectra S(q, ω) (see Sec. 4.1) are at ω = 0 ( S ) or at ω 0 ( T ). For the case of patterns labeled by T/S, there were nonzero peaks at both ω = 0 and ω 0, and as is discussed later, 135

145 a b c d Figure 4.27: (a) is an example of a superposition of degenerate zig and zag rolls and would be denoted with a second letter N. (b) is the power spectrum of image (a) and it shows that the two wavevectors form the same angle with respect to the director. (c) is an example of a superposition of nondegenerate oblique rolls and would be denoted with a second letter O. (d) is the power spectrum of image (c) and it shows that the two wavevectors form different angles with respect to the director. it has not been completely determined whether or not this is the result of a coexistence of or a superposition of traveling and stationary modes. I will now discuss the various transitions shown in Fig and Fig For the range of σ studied in the cell I5263 (see Fig. 4.28), the initial transition is a forward Hopf bifurcation to either the worms or TNS as discussed in Sec 4.4. From the limited measurements in I5243 (the x s in Fig. 4.28), the initial transition at high values of σ appeared to be backward and to a stationary state. However, these measurements were taken with relatively large steps in ɛ of 0.01, and clearly, the measurement of the initial transition must be done with a resolution in ɛ which is better than For example, in the 136

146 cell I5263 at 25 C, I observed an initial transition to traveling rolls with a transition to a stationary pattern at a value of ɛ of only In this case, σ = ohm 1 m 1 and Ωτ q = At this temperature, the value of ɛ at which the transition from traveling to stationary rolls occured was dependent on the applied frequency. There have also been reports of an initial Hopf bifurcation with a transition to a stationary state at values of ɛ < 0.01 for EC in other NLC 4. However, in the thicker cell I5275, the initial transition was observed to be a transition to the SOR state with a resolution in ɛ of as discussed in Sec This is consistent with the expectation that the patterns depend on the combination σd 2 and that higher values of σ and thicker cells result in a backward bifurcation to stationary rolls as observed in Ref. [37] and predicted by the WEM. Further quantitative work is required to determine the value of σ (or σd 2 ) at which the initial transition switches from a Hopf bifurcation to a stationary bifurcation. As noted in Fig. 4.28, the majority of this study was carried out by varying the operating temperature. This allowed for a reproducible increase/decrease of σ with increasing/decreasing T. As discussed in Appendix A, many other material parameters, including ɛ a which changes sign, are temperature dependent. My observations suggest that T does not affect the qualitative nature of the observed patterns for a given σ and d. The measurements in I5243 at T = 44 C and the measurement in I5263 at T = 49 C exhibit the same nonlinear patterns as observed for the cell I5263 at T = 59 C. Also, after the cell had aged for 6 months, the worm states and the TNS states reported in Sec. 4.4 were studied in the temperature range of 49 C to 64 C as opposed to T < 54 C which was used for the initial observations reported in Fig However, the quantitative aspects of the pattern, i.e. ɛ value of the transitions, traveling frequency and angle of the oblique rolls, does exhibit some dependence on the other material parameters. Ideally, future studies will be carried 4 L. Kramer, private communication 137

147 0.2 (T/S)NS SOS ε 0.1 SNS SOR 0 worms T TNS conductivity (10-8 ohm -1 m -1 ) Figure 4.28: Observed patterns as a function of ɛ and σ at Ωτ q = Here σ was varied by changing the temperature T, as indicated by the arrow. The points connected by lines (open circles) were taken using cell I5263 and a temperature range of 44 C to 59 C. The crosses are from the cell I5263 and T = 49 C (The point corresponding to T = 49 C from the open circle data is at σ = ohm 1 m 1 ). The x s are from a run in the cell I5243 where T = 44 C. The solid vertical line corresponds to σ = ohm 1 m 1, which is the value of σ used in Fig The dashed vertical line is discussed in the text. out as a function of d at a fixed temperature using a cell of variable thickness. I will present in detail data for two values of σ from Fig to illustrate the main transitions which are observed. The first is for σ = ohm 1 m 1 and is given by the vertical dashed line in Fig Figure 4.30 shows images of the three patterns which are observed along this line: TNS, SNS, (T/S)NS. Also shown are the spatial power spectra of the images. The initial pattern (TNS) consists purely of traveling rolls and undergoes a 138

148 0.2 (T/S)OS ε (T/S)NS SOS SOR SNS TNS 0.15 SOS ε SOR Ωτ q Figure 4.29: Top figure is the observed patterns as a function of ɛ and Ωτ q at σ = ohm 1 m 1 for the cell I5263. The solid vertical line corresponds to the value of Ωτ q = 1.34 which was used in Fig The bottom curve is the observed patterns as a function of ɛ and Ωτ q in the cell I5243 for the value of σ plotted in Fig Here the horizontal dashed line is at ɛ = 0.01 which was the ɛ step size used to measure the boundaries. secondary transition to a stationary state. What is particularly intriguing is that the next transition is to a state with both stationary and traveling components. Figure 4.31 is a plot of S(k, ω) (which is a three dimensional quantity depending on p, q, and ω) for four values of ɛ. As I can only plot two axes, 139

149 a b c Figure 4.30: The left hand column are three images taken along the dashed line of Fig Image (a) is for ɛ = (b) is ɛ = 0.066, and (c) is for ɛ = On the right is shown the power spectra for the corresponding left-hand image. 140

150 I show a section of the full S(k, ω). One axis corresponds to the entire ω axis. The other axis is a slice along the direction in k-space which corresponds to the wavevector of the zag rolls. A number of features of the transition are highlighted by this plot. First, for the length of time series used here, S(k, ω) for ɛ = shows that the right and left traveling waves had unequal amplitudes. This was discussed in Sec At ɛ = 0.056, the transition to a stationary state has occurred, and there is only a peak at ω = 0. At ɛ = 0.081, one finds that the traveling modes reappear with a relatively small amplitude. By ɛ = 0.086, there is a clear superposition of traveling and stationary modes. The S(k, ω) for ɛ = is suggestive of a continuous transition. However, one must also consider the spatial extent of the traveling and stationary components of the patterns. Ideally, demodulated time series of images covering many correlation lengths should be studied; whereas, because of previously discussed limitations, the images used to compute these S(k, ω) contained only 8 wavelengths. Initial demodulations of the pattern using these images of small spatial extent, and observations of movies of the pattern, strongly suggest that the traveling rolls and stationary patterns are not superimposed throughout space. It is clear from real time observations that localized regions of purely stationary patterns exist. However, these regions do not remain purely stationary modes. Regions of traveling modes move through the cell in an apparently chaotic fashion. One can get a feel for this from image (c) in Fig In this image, there are regions where the zig or zag rolls appear to dominate (at the very least the image is fuzzier). For example, across the top of the image, the individual oblique rolls are more apparent. However, the right-hand edge, halfway down the image, looks more like the image (b) in Fig This corresponds to a region which was observed to be stationary. What is not so clear is the nature of the regions with traveling rolls. The results of the demodulations of the current S(k, ω) suggest that the stationary mode is superimposed with the traveling mode in some regions of space. But, the images are much too small to say anything conclusive. It is a very intriguing 141

151 ε = ε = ω (sec 1 ) k ω (sec 1 ) k ε = ε = ω (sec 1 ) k ω (sec 1 ) k Figure 4.31: S(k, ω) for four values of ɛ taken from along the dashed line of Fig thought that this pattern consists of a stationary background across which localized traveling wave packets are moving. In this case, the small amplitude in the power spectrum could be the result of localized traveling modes. Clearly, the interaction of the stationary and traveling components of this pattern and the overall dynamics of this state are interesting aspects of this system. The traveling frequency corresponding to the dominant peaks in S(k, ω) are plotted in Fig. 4.32a. The transition to a stationary pattern at ɛ = occurs via a finite jump in the frequency. Similarly, at ɛ = 0.08, there is again a finite jump in frequency to a value approximately equal to the value of ω which existed below ɛ = The corresponding behavior of the ampli- 142

152 0.6 ω (sec -1 ) amplitude (arb. units) ε Figure 4.32: (a) frequency ω of pattern along the dashed line of Fig (b) amplitude of the fundamental (circles) and the harmonic (triangles) of the shadowgraph signal measured from the power spectrum of images taken along the dashed line of Fig tude as a function of ɛ for both the fundamental and the second harmonic of the shadowgraph signal is given in Fig. 4.32b and 4.32c. There is an obvious decrease in the amplitude of the power spectrum when the pattern becomes stationary. Also, at the point where ω makes a finite jump, the behavior of the fundamental peak and the second harmonic are quite different. A quantitative explanation of this behavior should be possible when the physical optics calculation has been completed for EC, but currently, there is no explanation. The transition in the cell I5243 from SOR to SOS has none of the complications of traveling modes but has some equally intriguing new features. Figure 4.33 shows six typical images of a small section of the cell. I want to emphasize here that there were some difficulties studying this state associated 143

153 a b c d e f Figure 4.33: Six images of oblique rolls for the run in cell I5243 shown in Fig The images were taken at (a) ɛ = 0.014, (b) ɛ = 0.052, (c) ɛ = 0.067, (d) ɛ = 0.082, (e) ɛ = and (f) ɛ = with the properties of the shadowgraph images of EC in NLC. The transition from SOR to SOS occurred at high enough values of ɛ that the images were usually dominated by the second harmonics. This is clear in the power spectrum of the images from Fig which are shown in Fig Here the dominant peaks in the power spectrum are all due to the quadratic effects of the shadowgraph, and the linear peaks are barely visible, if at all. For the case of SOR, both the zig and zag rolls exist in the cell separated by grain boundaries. An example of a grain boundary is given in Fig. 4.33b. What is clearly shown by the power spectra in Fig. 4.34f is that the SOS consist of a superposition of one set of the original zig or zag rolls and another set of oblique rolls whose wavevector is roughly perpendicular to the original set. For comparison, consider the power spectrum of the grain boundary which is a superposition of degenerate zig and zag rolls (Fig. 4.34c) and the power spectrum of the SOS state (Fig. 4.34f). There are a number of details of the SOR/SOS transition which still need to be measured. One issue is the angle between the original set of rolls and the new set which grows. Typically, the angle is around 90 degrees, but there 144

154 a c 115 d f Figure 4.34: Power spectra of four of the images in Fig The letters correspond to the letters in Fig The lines in (c) and (f) highlight the angles between the two modes which are present in the images. exists a range of angles near 90 which are allowed. I have observed angles as large as 110 degrees as shown by the state pictured in Fig (Here the fundamental peaks in the power spectrum were clearly visible.) Another issue is the dependence of the angle Θ of the initial zig or zag roll on ɛ. For small values of Θ near onset, Θ increased continuously as a function of ɛ and plateaud around 35. The details of the behavior depended strongly on σ and Ωτ q. I have observed no discontinuity of Θ as ɛ is increased through the SOR/SOS transition, but this needs to be studied more systematically. By jumping from a value of ɛ below the SOR/SOS transition to a value of ɛ which is above the transition, I observe the transition occurring first in 145

155 108 Figure 4.35: Image from cell I5263 at ɛ = 0.17 and 57.1 C with a σ = 1.56 ohm 1 m 1, and the power spectrum of the image. The two lines in the power spectrum are drawn through the fundamental peaks. The angles between the two sets of rolls is 108. the middle of regions of zig and zag rolls and not from the grain boundaries. It still remains to be determined whether or not the transition is forward or backward, and what hysteresis, if any, is present. Some images (for example Fig. 4.33d) suggest that the transition occurs due to an undulation instability in the rolls. Calculations of such instabilities from amplitude equations exist, so there exists the real possibility of phenomological description of this transition using amplitude equations. It is clear from Fig and Fig that there are a number of interesting transitions which exist as a function of σ and Ωτ q that were not studied here because these studies all were done as a function of ɛ. For example, the region around the point where SOS, SOR, SNS, and (T/S)NS meet in Fig involves the interaction of a number of very different types of patterns. Essentially all possible combinations of traveling, coexisting, superposition, degenerate and nondegenerate meet at a single point. This should lead to some extremely interesting dynamics. Another interesting transition which is found in Fig is the spontaneous break up of the (T/S)NS state into regions of differently oriented (T/S)OS. In the SOR/SOS transition, each region had an initial orientation defined by 146

156 the existing zig or zag rolls. In the observed (T/S)NS to (T/S)OS transition, the initial state is a superposition of the zig and zag rolls. Therefore, each region had to decide, by some mechanism, whether to retain the zig or zag roll state which formed the (T/S)NS state, and then a set of oblique rolls which was roughly at an angle of 90 to the chosen original mode would grow. In addition, the pattern still had both a traveling and a stationary component. This is just one more example of the richness of patterns which occur as a function of ɛ, Ωτ q and σ for this system. In the future, a cell design which incorporates the ability to easily adjust d should vastly increase the ability to quantitatively study this rich system of interesting patterns, as well as test the postulated dependence on σd

157 Chapter 5 Open Issues and Future Directions In the introduction, I gave three reasons for studying EC in NLC other than MBBA and Phase V: resolution of discrepancies between theory and experiment, measurements of thermal noise, and finding a replacement for MBBA as the standard NLC used in EC. In addition, the study of the NLC I52 has revealed a rich landscape of patterns to be explored. I will first discuss the degree to which the search of NLC achieved each of the three goals, and then, I will discuss possible directions to take with the fascinating system of EC in I52. There were two issues to be resolved between theory and experiment: the Hopf bifurcation and the backward bifurcation. In Sec. 4.3, I showed that the WEM[39] does an excellent job of predicting the Hopf bifurcation in the limit of thin cells and low conductivity. Based on experiments from MBBA[68] and my limited work using thick cells, the initial bifurcation in EC is stationary for thick cells with a high enough conductivity. Also, there is an intermediate range of thickness and conductivity for which the Hopf bifurcation exists for Ω close to the cutoff frequency and goes to zero as Ω is decreased. The WEM qualitatively predicts both of these behaviors[39], but quantitative experiments 148

158 are needed. As mentioned in Sec. 2.3, one of the nice features of the WEM is that the mechanism for producing the Hopf bifurcation is similar to the mechanisms of other pattern forming systems which exhibit Hopf bifurcations. The charge density (which is proportional to A n for τ q /τ d 1) drives the instability via the electrical volume force, but the coupling of A n to A σ provides an additional stabilizing mechanism. This stabilizing mechanism is responsible for the Hopf bifurcation provided that the time scale for the feedback is sufficiently slow. This interplay between a primary instability mechanism and a slower stabilizing mechanism also occurs in RBC in binary fluids and RBC in NLC with homeotropic alignment. In RBC in a binary fluid with a negative separation ratio, the concentration field is the slow field which provides the stabilizing mechanism that counteracts the buoyancy-driven instability[100]. In thermal convection in a homeotropically aligned NLC, the director field is the slow field which stabilizes the buoyancy-driven instability[101]. The main unresolved issue with the linear predictions[42] of the WEM is the value of τ rec. The limits set by my experiments require τ rec > 10 s for I 2 doped I52. Compared to typical numbers from the literature, τ rec 0.05 s[102], this is a very slow time. However, I 2 doped I52 forms a charge transfer complex and then undergoes a dissociation-recombination reaction. Since this is a multistep process, it is expected that the recombination times would be longer. In addition, I52 is a nonpolar molecule which is essentially impossible to dope, and the reported measurements have been on more polar liquid crystals which are relatively easy to dope. For example, a solution of 0.01% TBBA in MBBA produces a conductivity of 10 7 ohm 1 m 1. For I52 with concentrations of up to 0.1% TBBA, the conductivity was less than 10 9 ohm 1 m 1, and it took 2% I 2 to eventually achieve conductivities of 10 8 ohm 1 m 1!! This represents a difference of 3 to 4 orders of magnitude between I52 and MBBA in their ability to produce dissociation for a typical organic salt. This implies extremely small values for the rate constants of the dissociation-recombination 149

159 reactions in I52 and correspondingly large values for τ rec. Of course, τ rec should be measured for I52 doped with I 2. As discussed in Sec. 3.2, the large uncertainty in the final concentration of I 2 makes these measurements difficult. One possibility is to determine the concentration of I 2 using optical absorbance. As predicted by the weakly nonlinear analysis[33, 34] of the SM, I observed initial transitions which were forward for EC in I52. However, the range for which I observed a forward bifurcation corresponds with the range for which the WEM, not the SM, correctly describes the linear behavior[42]. Therefore, the correct theory to compare to is a weakly nonlinear analysis of the WEM. This remains to be done. In addition to predicting the forward bifurcation observed in I52, it remains to be seen if the WEM can provide an explanation for the observed backward bifurcation in MBBA[37]. These observations were from a cell 13 µm thick, but at a higher conductivity than my studies in I52. Also, my initial studies of EC in I52 in thick cells (see results for I5268, Sec. 4.4) suggest that the bifurcation becomes backward for I52 as well. Thick cells and high conductivity are the limits for which the WEM recovers the SM. Since the SM was unable to explain the backward bifurcation, this argues that the WEM will also be unable to explain the backward bifurcation. One intriguing possibility is suggested by the data of Fig Figure 4.28 shows the secondary instability to stationary rolls approaching the initial bifurcation as the conductivity is increased. At this transition, there is a large jump in the amplitude of the pattern. Both in I52 and in Merck Phase V, this secondary transition has been observed to occur at rather small values of ɛ, ɛ It may be that the value of ɛ for which the secondary transition occurs approaches zero, but the initial transition is always forward as predicted by the SM. At some point, the region of the forward transition might not be resolvable experimentally, and an initially backward transition would be observed. These ideas are clearly speculative, but definitely the behavior in Fig and Ref. [103] is suggestive. For now, the evidence for the backward bifurcation in Ref. [37] is very convincing and remains a fact that theory must 150

160 explain. As a system for studying fluctuations below onset, I52 is definitely promising. The initial transition is forward as desired. The existence of the initial transition to spatio-temporal chaos and localized states would make a study of the role of noise below onset in this system particularly interesting. Below V c the pattern is linear, and the individual modes do not interact. Presumably, it is the interaction of the modes which produces the spatio-temporal chaos and the localization, and at the very least, the nonlinearities are essential. It would be interesting to measure the transition from noise driven fluctuations to the deterministic spatio-temporal chaos or a localized state. The search of NLC revealed two other possible candidates for the study of thermal noise. Both D55-f and OS-33 appear to have forward bifurcations to normal rolls. For these NLC, the initial transition is also stationary. There are two difficulties with these materials. First, they are difficult to obtain. Second, the anisotropy in the index of refraction is relatively small for these materials which produces a decrease in the shadowgraph signal. The final goal was to find a replacement for MBBA. In this regard, I52 is a strong candidate. The material is reported to be highly stable, and a number of the essential material parameters have been measured. I have measured some additional parameters using the bend electric Frederiks transition. Measurements of the other two elastic constants are possible using different Frederiks configurations. The large nematic range in temperature and the variation of ɛ a with temperature allow for the measurement of EC for a range of material parameters using a single NLC. From the results of Sec. 4.4 and Sec. 4.5, one is presented with a potentially overwhelming number of interesting patterns to study. In the nonlinear regime, three problems stand out. There is the transition from SNR to SOS and the similar transition from (T/S)NS to (T/S)OS. For both of these systems, there are a number of features of the instability which require further study. Among these, the range of angles between the two nondegenerate, su- 151

161 perimposed modes needs to be studied. Also, whether or not the second mode grows continuously from zero amplitude or comes in with a finite amplitude needs to be determined. For the (T/S)NS to (T/S)OS transition, there is the additional issue of the dynamics by which the pattern separates into regions of different orientation. Another challenge is the actual nature of the (T/S)NS state. So far, I only have the most cursory characterization of the state. As far as I know, there is no other example of a pattern forming system for which there is a state which is a superposition of a traveling mode and a stationary mode. The first thing which needs to be determined is the extent to which the two modes are superimposed, or merely coexisting. This will be relatively easy to determine with improvements to the image taking capability. An interesting probe of this state would be to modulate the drive frequency at twice the traveling frequency. It has been shown that such a modulation will stabilize a traveling state into a standing wave[103]. Given the existence of the traveling modes, it would be interesting to study the effect of such a stabilizing modulation on this pattern. Finally, there is the interesting point where the four patterns SNS, (T/S)- NS, SOS, SOR meet as a function of the control parameters. The dynamics in this region should be quite fascinating and might be described by a phenomenological amplitude equation. All three phenomena just discussed occur at ɛ > 0.1 where the applicability of amplitude equations is limited. I will close with some thoughts on the spatio-temporal chaos and localized states which occur in the weakly nonlinear regime. For the spatio-temporal chaos and localized states, it is hoped that the appropriate CGL will be derived from the WEM, so that quantitative comparison between theory and experiment is possible. There are a number of examples of spatio-temporal chaos, but there are two in particular for which comparison to solutions of appropriate CGL have been made: dispersive chaos in 152

162 RBC in binary fluid mixtures[104], and defect turbulence in EC in NLC[26]. Even though both of these phenomena are qualitatively described by CGL amplitude equations[57], there are difficulties with quantitative comparisons between theory and experiment because of the relatively large amplitudes of the patterns. For RBC in binary fluids, the chaos occurs near onset, but the initial transition is a backward bifurcation (see Sec. 2.1). For the defect chaos, the chaos occurs after a secondary instability for which the amplitudes are already large. One of the great advantages of the chaotic states reported on here is that they occur as the result of a forward bifurcation. Therefore, it is expected that the patterns can be studied for a range of ɛ where the CGL are strictly valid. Even without a quantitative derivation, many of the coefficients in the CGL could be measured experimentally, and the coupled CGL could be used as a phenomenological model. There are a number of possible probes of the extended state of spatiotemporal chaos. First, the system size can be varied. For a small enough system, the spatial degrees of freedom can be eliminated. A number of pattern forming systems exhibit dynamical chaos when the system size is small enough[1], but one would expect that without the spatial coupling, the pattern could become regular. By varying the system size, the effects of the spatial coupling on the dynamics could be studied. Reference [105] serves as a possible guide for what to expect in a small enough system for which the spatial degrees of freedom are not relevant. Reference [105] explores the solutions of a set of coupled amplitude equations appropriate to a system of superimposed, traveling oblique rolls with no spatial variation of the amplitude. A number of regular solutions are found, as well as at least one chaotic solution. A particularly intriguing result is the existence of an alternating roll state which is the superposition of two standing waves that are one quarter of a cycle out of phase in time. The states of extended chaos reported on here display local regions which temporarily contain zig and zag standing waves that are out of phase with each other. A time 153

163 a b c d e f g h Figure 5.1: Time series of images from cell I5243. The applied frequency was 85 Hz, σ = ohm 1 m 1, and ɛ = The images were taken roughly 1 s apart. series from such a region is shown in Fig For comparison, I show Fig. 2 from Ref. [105] in Fig This shows the results of a numerical simulation of the alternating roll solution to the amplitude equations used in Ref. [105]. When comparing Fig. 5.1 to Fig. 5.2, it is important to remember that the alternating-roll states observed in the experiment (Fig. 5.1) are not stable, but exist as part of the state of spatio-temporal chaos. However, this comparison does suggest that equations of the form used in Ref. [105] with the appropriate spatial gradient terms would correctly describe the dynamics of the system. And, conversely, that a system of small enough spatial extent would be described by the equations used in Ref.[105]. Another probe of the spatio-temporal chaos is a modulation of the applied frequency at twice the traveling frequency. This is known to stabilize standing wave states from traveling wave states in EC[103]. This could potentially phase lock the system into two standing wave modes: zig and zag. The amplitude of the modes might still exhibit spatio-temporal chaos, but the overall system would be relatively simpler as it would only consist of two instead of four modes. 154

164 Figure 5.2: Examples of the alternating roll solution from Ref. [105], Fig. 2. One of the most striking features of the spatio-temporal chaos in this system is the connection between the extended states and the localized states (see Sec. 4.4). The localized states are examples of spatio-temporal chaos in that their lengths, number density, and possibly even their velocity and amplitude, vary in time and space. The apparently continuous transition as a function of conductivity from a spatially extended state, to a state which is occasionally localized, to a purely localized state suggests that the localized states represent a limit of the extended state of spatio-temporal chaos in which the correlation length perpendicular to the director reaches a minimum. I use apparently continuous because I only have three data points at this time. The existence of this transition raises a number of interesting questions. The intermediate range is similar in some respects to the dispersive chaos observed in RBC in binary fluid convection[104]. Both systems exhibit bursts of localized convection which occur randomly in space and time. Given that the mechanism for producing the traveling waves is similar in both cases, it is worth exploring the possible connections between these two states. In addition to the similar bursting states, the possible connection between the worms and observations of localized states in RBC in binary-fluid 155