PHYSICAL ACOUSTICS: PROPERTIES OF GASES, LIQUIDS, AND SOLUTIONS
PART A: OVERVIEW ChapterAISome history 7 I1 Georges Friedel and liquid crystals 7 I2 The discovery of birefringence in fluid biological substances by Buffon, Virchow, and Mettenheimer: lyotropic liquid crystals 9 I3 Observation of the surprising behavior of cholesteryl esters by Planer and Reinitzer: thermotropic liquid crystals 12 I4 Fliessende Kristalle or “the flowing crystals” of Otto Lehmann 14 ChapterAIIModern classification of liquid crystals 17 II1 The terminology introduced by Georges Friedel 17 II1a) Polymorphism and mesomorphic states 17 II1b) Nematic and cholesteric phases 18 II1c) Smectic phases 19 II2 Modern definition of mesophases; broken symmetries; short- and long-distance order 20 II3 Classification of smectic phases 24 II4 Classification of columnar phases 29 II5 Chiral smectic phases 31 VI ChapterAIIIMesogenic anatomy 35 III1 Thermotropic liquid crystals 35 III1a) Hybrid molecular form and molecular frustration 36 III1b) Importance of small structural details 40 III1c) Eutectic mixtures 40 III1d) Precursors and “relatives” of cyanobiphenyls 43 III1e) General architecture of smectogenic and nematogenic products: calamitic molecules 46 III1f) Other examples of mesomorphic molecules: discoidal molecules 47 III2 Lyotropic liquid crystals 51 III3 Liquid crystal diblock copolymers 59 III4 Colloidal liquid crystals 60 PART B: MESOPHASES WITH AN ORIENTATIONAL ORDER ChapterBIStructure and dielectric properties of the nematic phase 65 I1 Quadrupolar order parameter 66 I1a) Uniaxial nematics 66 I1b) Biaxial nematics 71 I2 The uniaxial nematic-isotropic liquid phase transition 71 I2a) Landau-de Gennes theory 72 I2b) Determination of the Landau coefficients: the Cotton-Mouton and Kerr effects 74 I3 The uniaxial nematic-biaxial nematic phase transition 78 I4 Low-frequency dielectric properties 80 I4a) Potential energy of a molecule in an electric field 80 I4b) Permanent and induced dipoles of an isolated molecule: role of the molecular symmetry 81 I4c) Absence of spontaneous polarization and Flexo-electricity in the nematic phase 83 I4d) Dielectric anisotropy 87 I5 Optical properties 92 I5a) Preliminary experiment: double refraction of a nematic prism 92 I5b) Ordinary and extraordinary rays: eigenmodes 93 VII I5c) Calculation of the extraordinary ray trajectory: generalization of the Fermat principle (Grandjean 1919) 97 I5d) Eigenmodes of a twisted nematic 102 I5e) Polarized light contrast 105 I5f) Conoscopy 108 ChapterBII Nematoelasticity: Frederiks transition and light scattering 117 II1 Grupp experiment 119 II2 Frank-Oseen free energy 120 II3 Free energy minimization: molecular field and elastic torques 126 II4 Interpretation of the Grupp experiment 128 II5 Magnetic field action 131 II5a) Physical interpretation of the bulk torque 131 II5b) Molecular magnetic field 132 II5c) Magnetic coherence length 133 II5d) Frederiks instability in magnetic field 135 II5e) Application to the determination of the elastic constants 138 II6 Action of an electric field: displays 141 II6a) Employing an electric field 141 II6b) Setting up a 16-pixel display 143 II7 Elastic light scattering and the determination of the Frank constants 147 II8 Nonlinear optics 154 II8a) Optical Frederiks transition 154 II8b) Nonlinear analysis 159 II8c) Autofocusing of a laser beam 163 Appendix 1: Calculating the scattering cross-section 168 Appendix 2: Free energy expression in the Fourier space 171 ChapterBIIINematodynamics and flow instabilities 173 III1 Preliminary observations illustrating some fundamental differences between a nematic and an ordinary liquid 174 III1a) Frederiks instability and rotational viscosity 174 III1b) Scintillation and orientation fluctuations of the director 176 III1c) The Wahl and Fischer experiment: influence of shearing on the director orientation 178 III1d) “Backflow” effect 182 VIII III1e) Viscosities under shear: the experiments of Miesowicz and Gähwiller 183 III2 Equations of the linear nematodynamics 185 III2a) Choice of the hydrodynamic variables 185 III2b) Surface torque field 186 III2c) Conservation equations for the mass and the momentum 187 III2d) Constructing the stress tensor 188 III2e) Equation of torque balance: application of the angular momentum theorem 192 III2f) Irreversible entropy production 195 III2g) Summary of the equations and description of the hydrodynamic modes 197 III3 Laminary Couette and Poiseuille flows 201 III3a) Simple shear under a strong magnetic field (Couette) and determining the Miesowicz viscosities 201 III3b) Poiseuille flow and “transverse” viscous stress 204 III4 Laminary flows and their stability 206 III4a) Reynolds and Ericksen numbers 206 III4b) Definition of the hydrodynamic torque 208 III4c) Origin of the hydrodynamic torque 210 III4d) Measuring the ?3 viscosity 211 III4e) Destabilization under continuous shear when ?3 > 0 in the (b) Miesowicz geometry (n // v and n ? grad v) 216 III4f) Destabilization under continuous shear for ?3 < 0 in the (a) Miesowicz geometry (n ? v and n ? grad v) 222 III4g) Destabilization under linear oscillating shear in the (a) Miesowicz geometry (n ? v and n ? grad v) 227 III4h) Elliptical shear: director precession 233 III4i) Instabilities under elliptic shear 239 III5 Convective instabilities of electrohydrodynamic origin 242 III5a) Theoretical model 243 III5b) Nonlinear regime 249 III6 Thermal instabilities 251 III6a) Qualitative analysis, role of the thermal conductivity anisotropy 251 III6b) Thermal conductivity anisotropy 253 III6c) Heat focusing 254 III6d) Calculating the instability threshold in planar geometry 255 III6e) Instabilities in homeotropic geometry 260 III6f) Experimental evidence 261 III6g) Thermal instabilities in the presence of a magnetic field 263 Appendix 1: “Derivation under the integral” theorem 270 Appendix 2: Rotational identity 272 IX Appendix 3: Calculation of the irreversible entropy production 273 Appendix 4: Energy dissipation and constitutive laws in the formalism of Leslie-Ericksen-Parodi 274 Appendix 5: The Rayleigh-Bénard instability in isotropic fluids 277 ChapterBIVDefects and textures in nematics 283 IV1 Polarizing microscope observations 284 IV2 The Volterra-de Gennes-Friedel process 289 IV2a) Symmetry elements of a uniaxial nematic 289 IV2b) Generating a disclination 289 IV3 Energy of a wedge planar line in isotropic elasticity 293 IV4 Continuous core model: Landau-Ginzburg-de Gennes free energy 296 IV5 Interaction energy between two parallel wedge lines 300 IV6 Dynamics of a planar wedge line: calculating the friction force 302 IV7 Wedge line stability: escape in the third dimension 311 IV7a) Topological stability 311 IV7b) Energetic stability (infinite medium) 313 IV7c) Disclinations in a confined medium (capillary tube) 316 IV8 Bloch and Ising walls induced by the Frederiks instability 323 IV8a) Experimental setup 323 IV8b) Theoretical predictions 324 IV8c) Comparison with experiment 332 ChapterBV Anchoring and anchoring transitions of nematics on solid surfaces 339 V1 Precursors 340 V2 On the notion of interface 341 V3 Interface symmetry and classification of the different types of anchoring 342 V4 Wetting and anchoring selection 348 V5 Anchoring transitions 354 V5a) Anchoring transitions on evaporated SiO layers 354 V5b) Anchoring transitions on muscovite mica 358 V5c) Anchoring transitions on gypsum 365 V6 Measuring the anchoring energy in the homeotropic case 366 ChapterBVI The nematic-isotropic liquid interface: static properties and directional growth instabilities 371 VI1 Anchoring angle, surface tension, and width of the nematic-isotropic interface 372 VI2 Landau-Ginzburg-de Gennes theory 379 VI3 Instabilities in confined geometry 384 VI4 Elastic correction to the Gibbs-Thomson relation 389 VI5 Directional growth of the nematic-isotropic liquid front 396 VI5a) Experimental observation of the Mullins and Sekerka instability 396 VI5b) Secondary instabilities in the rapid solidification regime and absolute restabilization 400 VI5c) Depinning of a disclination line 406