Pattern Formation in Liquid Crystals

In the last 20 years the study of nonlinear nonequilibrium phenomena in spa­ tially extended systems, with particular emphasis on pattern-forming phenomena, has been one of the very active areas in physics, exhibiting interesting ramifi­ cations into other sciences. During this time the study of the "classic" systems, like Rayleigh-Benard convection and Taylor vortex flow in simple fluids, has also been supplemented by the study of more complex systems. Here liquid crystals have played, and are still playing, a major role. One might say that liquid crystals provide just the right amount and right kind of complexity. They are full of non­ linearities and give rise to new symmetry classes, which are sometimes actually simpler to deal with qualitatively, but they still allow a quantitative description of experiments in many cases. In fact one of the attractions of the field is the close contact between experimentalists and theorists. Hydrodynamic instabilities in liquid crystals had already experienced a period of intense study in the late 1960s and early 1970s, but at that time neither the ex­ perimental and theoretical tools nor the concepts had been developed sufficiently far to address the questions that have since been found to be of particular interest. The renewed interest is also evidenced by the fact that a new series of workshops has evolved. The first one took place in 1989 in Bayreuth and united participants from almost all groups working in pattern formation in liquid crystals.

1 Introduction to Pattern Formation in Nonequilibrium Systems.- 1.1 General Remarks.- 1.2 A Simple Model.- 1.3 Pattern Formation in Liquid Crystals.- 1.3.1 Transient Patterns in the Freédericksz Transition.- 1.3.2 Patterns in Rotating Magnetic and Electric Fields.- References.- 2 Hydrodynamics and Electrohydrodynamics of Liquid Crystals.- 2.1 Introduction.- 2.2 Symmetries and Broken Symmetries.- 2.2.1 Conservation Laws.- 2.2.2 Broken Symmetries.- 2.2.3 Slowly Relaxing Variables.- 2.3 Statics.- 2.3.1 Thermodynamics.- 2.3.2 Energy and Thermodynamic Forces.- 2.4 Dynamics.- 2.4.1 Reversible Currents.- 2.4.2 Irreversible Currents.- 2.5 Electrohydrodynamics.- 2.5.1 External Fields.- 2.5.2 Statics and Dynamics.- 2.6 Additions to Nematodynamics.- 2.6.1 Fluctuating Forces.- 2.6.2 Biaxial Nematics.- 2.6.3 Order Parameter Variable.- 2.6.4 Side-Chain Polymers.- 2.6.5 Nonlinearities and Higher-Order Gradient Terms.- 2.7 Director-Type Degrees of Freedom.- 2.7.1 Smectic A Liquid Crystals.- 2.7.2 Cholesteric Liquid Crystals.- 2.7.3 Smectic C, C*, CM, and CM*Liquid Crystals.- 2.7.4 Smectic F, I, and L Liquid Crystals.- References.- 3 General Mathematical Description of Pattern-Forming Instabilities.- 3.1 Introductory Remarks.- 3.2 Linear Analysis.- 3.3 The Landau Equation.- 3.4 The Ginzburg―Landau Equations.- 3.4.1 Derivation.- 3.4.2 Application of the Ginzburg―Landau Equations.- 3.5 Extended Weakly Nonlinear Analysis.- 3.5.1 Derivation of Order Parameter Equations.- 3.6 From Order Parameter to Amplitude Equations.- 3.6.1 Derivation of Coupled Amplitude Equations.- 3.7 Concluding Remarks.- 3.7.1 Swift―Hohenberg Equation.- 3.7.2 Phase Equations.- References.- 4 Flow Instabilities in Nematics.- 4.1 Introduction.- 4.2 Continuous Description of Nematics and Viscometry.- 4.2.1 Nematohydrodynamics.- 4.2.2 Viscometry.- 4.2.3 Apparent Non-Newtonian Behavior and Flow Alignment.- 4.2.4 Anisotropy of Viscous Forces.- 4.2.5 Viscous Relaxation of the Orientation, Flow, and the Ericksen Number.- 4.3 Stability Analysis and Basic Mechanisms.- 4.3.1 Stability Analysis.- 4.3.2 The Pieranski―Guyon Mechanism.- 4.4 Shear Flow Instabilities with the Director Perpendicular to the Shear Plane.- 4.4.1 Simple Shear Flow.- 4.4.2 Alternating Shear Flows.- 4.4.3 Poiseuille Flow.- 4.5 Flow Instabilities with the Director Initially Parallel to the Shear Plane.- 4.6 Elliptical Shear Instability in Homeotropic Configuration.- 4.6.1 Experimental Results.- 4.6.2 Theoretical Account.- 4.7 Further Developments.- Appendix A: Linear stability problem when the director is perpendicular to the shear plane.- Appendix B: Elliptical Shear Equations.- References.- 5 Experiments on Thermally Driven Convection.- 5.1 Introduction.- 5.1.1 Instability Mechanisms.- 5.1.2 Stability Analysis.- 5.1.3 Pattern Formation.- 5.1.4 Materials.- 5.2 Planar Alignment and a Horizontal Magnetic Field.- 5.2.1 Introductory Remarks.- 5.2.2 Theoretical Predictions.- 5.2.3 Experimental Results.- 5.3 Homeotropic Alignment and a Vertical Magnetic Field.- 5.3.1 General Remarks.- 5.3.2 Heating from Below.- 5.3.3 Heating from Above.- 5.4 Two-Phase Convection.- 5.4.1 Theoretical Predictions.- 5.4.2 Experimental results.- Appendix A: Experimental Methods.- Appendix B: Physical Properties of 5CB.- References.- 6 Electrohydrodynamic Instabilities in Nematic Liquid Crystals.- 6.1 Introduction.- 6.1.1 General Considerations.- 6.1.2 Theoretical preliminaries.- 6.2 Planar alignment: linear theory.- 6.2.1 Conduction regime.- 6.2.2 Dielectric regime.- 6.3 Planar alignment: nonlinear theory.- 6.3.1 Results of Ginzburg―Landau Equation (GLE).- 6.3.2 Beyond the GLE.- 6.4 Homeotropic alignment.- 6.4.1 Case C.- 6.4.2 Case F.- 6.5 Concluding remarks.- References.- 7 Mesophase Growth.- 7.1 Introduction.- 7.2 The Mullins―Sekerka Instability.- 7.2.1 Undercooled Pure Material.- 7.2.2 Thin Layer of a Binary Alloy in a Temperature Gradient.- 7.3 Directional Growth Experiments.- 7.3.1 The Initial Instability.- 7.3.2 Secondary Instabilities.- 7.3.3 Tertiary and Higher Instabilities.- 7.3.4 Other Experimental Systems.- 7.4 Free-Growth Experiments.- 7.4.1 Microscopic-Solvability Theory.- 7.4.2 Dendritic Growth at the Discotic-Isotropic Interface.- 7.4.3 Dendritic Growth in Other Mesophase Systems.- 7.5 Prospects.- References.- 8 Viscous Fingering.- 8.1 Introduction.- 8.2 Theoretical Background.- 8.3 Experiments.- 8.3.1 Isotropic Systems.- 8.3.2 Anisotropic Systems.- 8.4 Concluding Remarks.- References.- 9 Thermal Fluctuations in Pattern Forming Instabilities.- 9.1 Introduction.- 9.2 Macroscopic Stochastic Equations for Thermal Noise.- 9.3 Stochastic Amplitude Equations.- 9.4 Theoretical Results.- 9.4.1 Rayleigh―Bénard Convection.- 9.4.2 Taylor―Couette Flow.- 9.4.3 Planar Electrohydrodynamic Convection.- 9.5 Experimental Results.- 9.6 Discussion.- References.