Biaxial Nematic Liquid Crystals

GEOFFREY R. LUCKHURSTSchool of Chemistry, University of Southampton, UKTIMOTHY J. SLUCKINSchool of Mathematics, University of Southampton, UK

• About the Editors xiiiList of Contributors xvPreface xvii1 Introduction 1Geoffrey R. Luckhurst and Timothy J. Sluckin1.1 Historical Background 11.2 Freiser Theory 31.3 Nematic Order Parameters 41.4 Nematic Tensor Order Parameters 51.5 Theoretical Phase Diagrams 61.6 Landau–de Gennes Theory 91.7 Computer Simulation 101.8 Other Theoretical Issues 111.9 Applications 121.10 Characterisation 121.11 Lyotropic and Colloidal Systems 141.12 Molecular Design 15References 192 Biaxial Nematics: Order Parameters and Distribution Functions 25Geoffrey R. Luckhurst2.1 Introduction 252.2 The Cartesian Language 262.2.1 Order Parameters 262.2.2 Molecular Symmetry 282.2.3 Measurement 302.3 The Spherical Tensor Language 312.3.1 The Order Parameters of Biaxial Molecules in a Uniaxial Phase 312.3.2 Molecular Symmetry 332.3.3 Measurement 332.4 Extension to Biaxial Nematics 352.4.1 Orientational Order Parameters 352.4.2 Systems with D2h Point Group Symmetry 362.4.3 Measurement of the Order Parameters 372.4.4 Systems with C2h Point Group Symmetry and Their Order Parameters 382.4.5 Systems with C2h Point Group Symmetry: The Cartesian Language 392.5 Fourth-Rank Order Parameters 422.6 The Singlet Orientational Distribution Function 442.7 Appendices 472.7.1 Point Group Symmetry and the Associated Symmetry Operations 472.7.2 Legendre Polynomials, Modified Spherical Harmonics and Wigner Rotation Matrices 48Acknowledgements 51References 513 Molecular Field Theory 55Epifanio G. Virga3.1 Introduction 553.2 General Mathematical Theory 573.2.1 Two-Particle Hamiltonian 573.2.2 Ensemble Potentials 623.2.3 Molecular Field Approximation 653.2.4 Variational Principles 693.2.5 Local Stability Criterion 713.3 Non-Polar Molecules 743.3.1 Quadrupolar Hamiltonians 743.3.2 Phase Transitions 803.3.3 Universal Phase Diagram 873.3.4 Steric Effects 913.4 Polar Molecules 993.4.1 Dipolar Fluids 1003.4.2 Dipolar Hamiltonian 1023.4.3 Condensed Polar Phases 107References 1124 Hard Particle Theories 117Andrew J. Masters4.1 Introduction 1174.2 Theoretical Approaches 1194.3 Board-Like Models 1224.4 Bent-Core Models 1244.5 Rod–Plate Mixtures 1254.6 Conclusions and Speculations 128Acknowledgements 129References 1295 Landau Theory of Nematic Phases 133Lech Longa5.1 Introduction 1335.2 Symmetry of Biaxial Nematics and Primary Order Parameters 1345.3 Landau Expansion 1365.3.1 Generic NU –I Phase Transition 1365.3.2 Generic NB –NU and NB –I Phase Transitions 1385.3.3 Role of Coupling between Nematic Order Parameters 1415.3.4 Landau–de Gennes Expansion in Terms of the Alignment Tensor 1455.4 Conclusion 149Acknowledgements 149References 1496 Computer Simulations of Biaxial Nematics 153Roberto Berardi and Claudio Zannoni6.1 Introduction 1536.2 Order Parameters 1566.3 Model Potentials and Applications 1596.3.1 Lattice Models 1596.3.2 Atomistic Models 1626.3.3 Molecular Models 1636.4 Conclusion 171Acknowledgements 1736.5 Appendices 1736.5.1 Quaternions 1736.5.2 Angular Momentum Operator 1746.5.3 Kinematic and Dynamic Equations of Rotational Motion 1756.5.4 Propagator/Integrator of Rotational Equations of Motion 1766.5.5 Gradient of the Biaxial Gay–Berne Potential 1766.5.6 Torques of the Biaxial Gay–Berne Potential 177References 1787 Continuum Theory of Biaxial Nematic Liquid Crystals 185Iain W. Stewart7.1 Introduction 1857.2 Continuum Model and Energies 1867.2.1 The Elastic Energy 1877.2.2 The Magnetic and Electric Energies 1877.2.3 The Total Energy 1897.3 Dynamic Equations 1897.3.1 Balance Laws 1907.3.2 The Viscous Stress 1927.3.3 The Dynamic Equations 1947.3.4 Euler Angle Description 1957.3.5 A Simple Shear Flow 1967.4 Equilibrium Equations 1987.4.1 The Equilibrium Equations 1997.4.2 Alignment Induced by a Magnetic Field 2007.5 Conclusion 202References 2028 The Alignment of Biaxial Nematics 205Demetri J. Photinos8.1 Introduction 2058.2 Alignment by an External Electric or Magnetic Field 2068.3 Surface Alignment 2088.3.1 Macroscopic Description 2088.3.2 Molecular Scale Description 2108.4 Flow Alignment 2108.5 Lower Symmetry Biaxial Nematics and Hierarchical Domain Structures 211Acknowledgements 212References 2129 Applications 215Paul D. Brimicombe9.1 Introduction 2159.1.1 Materials Considerations 2159.1.2 Surface Alignment 2169.2 Thin-Film Electro-Optic Devices 2179.2.1 Minor-Director In-Plane Switching Devices 2189.2.2 Electric Field-Induced Biaxiality Effects 2209.2.3 Planar Biaxial Nematic Devices 2219.2.4 Twist Effects in Biaxial Nematics and Biaxial Pi-Cells 2229.2.5 Bistable Biaxial Nematic Devices 2239.2.6 Spontaneous Chirality Effects 2249.3 Non-Device Applications of Biaxial Nematic Liquid Crystals 2259.3.1 Optical Compensation Films 2259.4 Conclusion 225References 22610 Characterisation 22910.1 Textures of Nematic Liquid Crystals 230Ingo Dierking10.1.1 Polarising Microscopy 23010.1.2 Simple Liquid Crystal Optics 23010.1.3 Optical Biaxiality 23210.1.4 Textures 234References 24010.2 Refractive Index Studies 242Antonio J. Palangana10.2.1 Introduction 24210.2.2 Optical Indicatrix 24210.2.3 Optical Conoscopy 24410.2.4 Results 24610.2.5 Acknowledgements 250References 25010.3 Orientational Order Parameters of Nematic Liquid Crystals Determined by Infrared and Raman Spectroscopy 251Jagdish K. Vij and Antoni Kocot10.3.1 Introduction 25210.3.2 Polarised IR Spectroscopy 25210.3.3 Scalar Order Parameters of a Second-Rank Tensor 25210.3.4 IR Absorbance Components 25410.3.5 Experimental Method 25610.3.6 Results for the Order Parameters for the Tetrapodes 25610.3.7 Discussion of the Order Parameters 25810.3.8 Raman Spectroscopy 25910.3.9 Comparisons of IR and Raman Spectroscopy for Determining Order Parameters 263References 26410.4 NMR Spectroscopy 265Louis A. Madsen10.4.1 Introduction: NMR Basics, Advantages and Limitations 26510.4.2 Probing Orientational Order 26610.4.3 Creating a Director Distribution to Observe Biaxiality 26710.4.4 Spectral Analysis Considerations: Fitting and Rotational Modulations 26810.4.5 Incorporating Deuterium: Direct Mesogen Labelling Versus Probe Solutes 27010.4.6 Powder Spectra and Monodomain Spectra: Examples 27110.4.7 Alternative and Emerging Methods 272References 27410.5 Structural Studies of Biaxial Nematics: X-Ray and Neutron Scattering 276Patrick Davidson10.5.1 Introduction 27610.5.2 Theoretical Considerations 27610.5.3 Experimental Details 27910.5.4 Specificities of the Scattering by Different Kinds of Biaxial Nematics 280References 28311 Lyotropic Systems 285Antonio M. Figueiredo Neto and Yves Galerne11.1 Introduction 28511.2 Phase Diagrams 28611.3 The Potassium Laurate–Decanol–Water Mixture: A Working Example 28711.4 The Intrinsically Biaxial Micelles Model 29411.5 Theoretical Reconstruction of the Lyotropic Nematic Phase Diagram: a Landau-Like Approach 29811.6 Conclusions 302Acknowledgements 302References 30212 Colloidal Systems 305Gert Jan Vroege12.1 Introduction 30512.2 Onsager Theory and Extensions 30612.3 Special Features of Colloids and Colloidal Liquid Crystals 30712.4 Biaxiality in Mixtures of Rods and Plates 30812.5 Particles with Inherent Biaxial Shape 31112.6 Concluding remarks 315References 31613 Thermotropic Systems: Biaxial Nematic Polymers 319Anke Hoffmann, Felicitas Brömmel, and Heino Finkelmann13.1 Introduction 31913.2 Main-Chain Liquid Crystal Polymers 32113.3 Side-Chain Liquid Crystal Polymers 32113.4 Comparison of Attachment Geometries – Influence of Molecular Dynamics and Molecular Shape 32713.5 Conclusion 330References 33014 Low Molar Mass Thermotropic Systems 333Matthias Lehmann14.1 Preamble 33314.2 Introduction and General Considerations 33314.3 Single Component 33614.3.1 Biaxial Board-Shaped Mesogens 33614.3.2 V-Shaped Nematogens 33814.3.3 Multipodes 35014.4 Mixtures 35414.5 Concluding Remarks 360References 36015 Final Remarks 369Geoffrey R. Luckhurst and Timothy J. SluckinReferences 373Index 375