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.