Biaxial Nematic Liquid Crystals – Theory, Simulation and Experiment

GEOFFREY R. LUCKHURST School of Chemistry, University of Southampton, UK TIMOTHY J. SLUCKIN School of Mathematics, University of Southampton, UK

About the Editors xiii List of Contributors xv Preface xvii 1 Introduction 1 Geoffrey R. Luckhurst and Timothy J. Sluckin 1.1 Historical Background 1 1.2 Freiser Theory 3 1.3 Nematic Order Parameters 4 1.4 Nematic Tensor Order Parameters 5 1.5 Theoretical Phase Diagrams 6 1.6 Landau de Gennes Theory 9 1.7 Computer Simulation 10 1.8 Other Theoretical Issues 11 1.9 Applications 12 1.10 Characterisation 12 1.11 Lyotropic and Colloidal Systems 14 1.12 Molecular Design 15 References 19 2 Biaxial Nematics: Order Parameters and Distribution Functions 25 Geoffrey R. Luckhurst 2.1 Introduction 25 2.2 The Cartesian Language 26 2.2.1 Order Parameters 26 2.2.2 Molecular Symmetry 28 2.2.3 Measurement 30 2.3 The Spherical Tensor Language 31 2.3.1 The Order Parameters of Biaxial Molecules in a Uniaxial Phase 31 2.3.2 Molecular Symmetry 33 2.3.3 Measurement 33 2.4 Extension to Biaxial Nematics 35 2.4.1 Orientational Order Parameters 35 2.4.2 Systems with D2h Point Group Symmetry 36 2.4.3 Measurement of the Order Parameters 37 2.4.4 Systems with C2h Point Group Symmetry and Their Order Parameters 38 2.4.5 Systems with C2h Point Group Symmetry: The Cartesian Language 39 2.5 Fourth-Rank Order Parameters 42 2.6 The Singlet Orientational Distribution Function 44 2.7 Appendices 47 2.7.1 Point Group Symmetry and the Associated Symmetry Operations 47 2.7.2 Legendre Polynomials, Modified Spherical Harmonics and Wigner Rotation Matrices 48 Acknowledgements 51 References 51 3 Molecular Field Theory 55 Epifanio G. Virga 3.1 Introduction 55 3.2 General Mathematical Theory 57 3.2.1 Two-Particle Hamiltonian 57 3.2.2 Ensemble Potentials 62 3.2.3 Molecular Field Approximation 65 3.2.4 Variational Principles 69 3.2.5 Local Stability Criterion 71 3.3 Non-Polar Molecules 74 3.3.1 Quadrupolar Hamiltonians 74 3.3.2 Phase Transitions 80 3.3.3 Universal Phase Diagram 87 3.3.4 Steric Effects 91 3.4 Polar Molecules 99 3.4.1 Dipolar Fluids 100 3.4.2 Dipolar Hamiltonian 102 3.4.3 Condensed Polar Phases 107 References 112 4 Hard Particle Theories 117 Andrew J. Masters 4.1 Introduction 117 4.2 Theoretical Approaches 119 4.3 Board-Like Models 122 4.4 Bent-Core Models 124 4.5 Rod Plate Mixtures 125 4.6 Conclusions and Speculations 128 Acknowledgements 129 References 129 5 Landau Theory of Nematic Phases 133 Lech Longa 5.1 Introduction 133 5.2 Symmetry of Biaxial Nematics and Primary Order Parameters 134 5.3 Landau Expansion 136 5.3.1 Generic NU I Phase Transition 136 5.3.2 Generic NB NU and NB I Phase Transitions 138 5.3.3 Role of Coupling between Nematic Order Parameters 141 5.3.4 Landau de Gennes Expansion in Terms of the Alignment Tensor 145 5.4 Conclusion 149 Acknowledgements 149 References 149 6 Computer Simulations of Biaxial Nematics 153 Roberto Berardi and Claudio Zannoni 6.1 Introduction 153 6.2 Order Parameters 156 6.3 Model Potentials and Applications 159 6.3.1 Lattice Models 159 6.3.2 Atomistic Models 162 6.3.3 Molecular Models 163 6.4 Conclusion 171 Acknowledgements 173 6.5 Appendices 173 6.5.1 Quaternions 173 6.5.2 Angular Momentum Operator 174 6.5.3 Kinematic and Dynamic Equations of Rotational Motion 175 6.5.4 Propagator/Integrator of Rotational Equations of Motion 176 6.5.5 Gradient of the Biaxial Gay Berne Potential 176 6.5.6 Torques of the Biaxial Gay Berne Potential 177 References 178 7 Continuum Theory of Biaxial Nematic Liquid Crystals 185 Iain W. Stewart 7.1 Introduction 185 7.2 Continuum Model and Energies 186 7.2.1 The Elastic Energy 187 7.2.2 The Magnetic and Electric Energies 187 7.2.3 The Total Energy 189 7.3 Dynamic Equations 189 7.3.1 Balance Laws 190 7.3.2 The Viscous Stress 192 7.3.3 The Dynamic Equations 194 7.3.4 Euler Angle Description 195 7.3.5 A Simple Shear Flow 196 7.4 Equilibrium Equations 198 7.4.1 The Equilibrium Equations 199 7.4.2 Alignment Induced by a Magnetic Field 200 7.5 Conclusion 202 References 202 8 The Alignment of Biaxial Nematics 205 Demetri J. Photinos 8.1 Introduction 205 8.2 Alignment by an External Electric or Magnetic Field 206 8.3 Surface Alignment 208 8.3.1 Macroscopic Description 208 8.3.2 Molecular Scale Description 210 8.4 Flow Alignment 210 8.5 Lower Symmetry Biaxial Nematics and Hierarchical Domain Structures 211 Acknowledgements 212 References 212 9 Applications 215 Paul D. Brimicombe 9.1 Introduction 215 9.1.1 Materials Considerations 215 9.1.2 Surface Alignment 216 9.2 Thin-Film Electro-Optic Devices 217 9.2.1 Minor-Director In-Plane Switching Devices 218 9.2.2 Electric Field-Induced Biaxiality Effects 220 9.2.3 Planar Biaxial Nematic Devices 221 9.2.4 Twist Effects in Biaxial Nematics and Biaxial Pi-Cells 222 9.2.5 Bistable Biaxial Nematic Devices 223 9.2.6 Spontaneous Chirality Effects 224 9.3 Non-Device Applications of Biaxial Nematic Liquid Crystals 225 9.3.1 Optical Compensation Films 225 9.4 Conclusion 225 References 226 10 Characterisation 229 10.1 Textures of Nematic Liquid Crystals 230 Ingo Dierking 10.1.1 Polarising Microscopy 230 10.1.2 Simple Liquid Crystal Optics 230 10.1.3 Optical Biaxiality 232 10.1.4 Textures 234 References 240 10.2 Refractive Index Studies 242 Antonio J. Palangana 10.2.1 Introduction 242 10.2.2 Optical Indicatrix 242 10.2.3 Optical Conoscopy 244 10.2.4 Results 246 10.2.5 Acknowledgements 250 References 250 10.3 Orientational Order Parameters of Nematic Liquid Crystals Determined by Infrared and Raman Spectroscopy 251 Jagdish K. Vij and Antoni Kocot 10.3.1 Introduction 252 10.3.2 Polarised IR Spectroscopy 252 10.3.3 Scalar Order Parameters of a Second-Rank Tensor 252 10.3.4 IR Absorbance Components 254 10.3.5 Experimental Method 256 10.3.6 Results for the Order Parameters for the Tetrapodes 256 10.3.7 Discussion of the Order Parameters 258 10.3.8 Raman Spectroscopy 259 10.3.9 Comparisons of IR and Raman Spectroscopy for Determining Order Parameters 263 References 264 10.4 NMR Spectroscopy 265 Louis A. Madsen 10.4.1 Introduction: NMR Basics, Advantages and Limitations 265 10.4.2 Probing Orientational Order 266 10.4.3 Creating a Director Distribution to Observe Biaxiality 267 10.4.4 Spectral Analysis Considerations: Fitting and Rotational Modulations 268 10.4.5 Incorporating Deuterium: Direct Mesogen Labelling Versus Probe Solutes 270 10.4.6 Powder Spectra and Monodomain Spectra: Examples 271 10.4.7 Alternative and Emerging Methods 272 References 274 10.5 Structural Studies of Biaxial Nematics: X-Ray and Neutron Scattering 276 Patrick Davidson 10.5.1 Introduction 276 10.5.2 Theoretical Considerations 276 10.5.3 Experimental Details 279 10.5.4 Specificities of the Scattering by Different Kinds of Biaxial Nematics 280 References 283 11 Lyotropic Systems 285 Antonio M. Figueiredo Neto and Yves Galerne 11.1 Introduction 285 11.2 Phase Diagrams 286 11.3 The Potassium Laurate Decanol Water Mixture: A Working Example 287 11.4 The Intrinsically Biaxial Micelles Model 294 11.5 Theoretical Reconstruction of the Lyotropic Nematic Phase Diagram: a Landau-Like Approach 298 11.6 Conclusions 302 Acknowledgements 302 References 302 12 Colloidal Systems 305 Gert Jan Vroege 12.1 Introduction 305 12.2 Onsager Theory and Extensions 306 12.3 Special Features of Colloids and Colloidal Liquid Crystals 307 12.4 Biaxiality in Mixtures of Rods and Plates 308 12.5 Particles with Inherent Biaxial Shape 311 12.6 Concluding remarks 315 References 316 13 Thermotropic Systems: Biaxial Nematic Polymers 319 Anke Hoffmann, Felicitas Brommel, and Heino Finkelmann 13.1 Introduction 319 13.2 Main-Chain Liquid Crystal Polymers 321 13.3 Side-Chain Liquid Crystal Polymers 321 13.4 Comparison of Attachment Geometries Influence of Molecular Dynamics and Molecular Shape 327 13.5 Conclusion 330 References 330 14 Low Molar Mass Thermotropic Systems 333 Matthias Lehmann 14.1 Preamble 333 14.2 Introduction and General Considerations 333 14.3 Single Component 336 14.3.1 Biaxial Board-Shaped Mesogens 336 14.3.2 V-Shaped Nematogens 338 14.3.3 Multipodes 350 14.4 Mixtures 354 14.5 Concluding Remarks 360 References 360 15 Final Remarks 369 Geoffrey R. Luckhurst and Timothy J. Sluckin References 373 Index 375