Tensors for Physics

14.5 Additional Formulas Involving Projectors 271 This expression is also traceless, notice thatetrμutrμ=eμ⊥u⊥μandeμ‖uμ‖+e⊥μu⊥μ= ...

Chapter 15 Liquid Crystals and Other Anisotropic Fluids Abstract This chapter deals with equilibrium properties of liquid crysta ...

274 15 Liquid Crystals and Other Anisotropic Fluids 15.1 Remarks on Nomenclature and Notations. Liquid crystals are substances w ...

15.1 Remarks on Nomenclature and Notations 275 Fig. 15.1Cartoon of the orientation of molecules in the nematic phase, as shown i ...

276 15 Liquid Crystals and Other Anisotropic Fluids years, the research dealing with blue phases was considered as a rather exot ...

15.2 Isotropic↔Nematic Phase Transition 277 15.2 Isotropic$Nematic Phase Transition. 15.2.1 Order Parameter Tensor. The existenc ...

278 15 Liquid Crystals and Other Anisotropic Fluids The alignment tensor is uniaxial fora 1 =0, with its symmetry axis parallel ...

15.2 Isotropic↔Nematic Phase Transition 279 whereP 2 (x)=^32 (x^2 −^13 )is the second Legendre polynomial. Frequently, the quant ...

280 15 Liquid Crystals and Other Anisotropic Fluids or direct computation from (15.11) leads to ΦμνLdG≡ ∂ΦLdG ∂aμν =Aaμν−B √ 6 a ...

15.2 Isotropic↔Nematic Phase Transition 281 has to be obeyed at equilibrium, these relations imply a=ani≡ 2 B 3 C , Ani=A 0 ( 1 ...

282 15 Liquid Crystals and Other Anisotropic Fluids Clearly,aeq∗ =1forθ=1. The nematic state is metastable in the range 1<θ&l ...

15.2 Isotropic↔Nematic Phase Transition 283 15.2.3 Maier-Saupe Mean Field Theory. Although Maier and Saupe [80] followed a diffe ...

284 15 Liquid Crystals and Other Anisotropic Fluids 0.5 0 0.5 1 1.5 2 2.5 3 a 0.5 0 0.5 1 1.5 2 F Fig. 15.4 The Maier-Saupe grap ...

15.3 Elastic Behavior of Nematics 285 15.3.1 Director Elasticity, Frank Coefficients Standard nematic liquid crystals, in therma ...

286 15 Liquid Crystals and Other Anisotropic Fluids For the derivation of this expression from (15.27), the condition (15.28) eq ...

15.3 Elastic Behavior of Nematics 287 has to be a minimum. This impliesδF= ∫ f(n+δn)d^3 r− ∫ f(n)d^3 r=0. The changeδnmust conse ...

288 15 Liquid Crystals and Other Anisotropic Fluids The free energy density has a minimum whenα=q 0 z. The quantityP= 2 π/q 0 is ...

15.3 Elastic Behavior of Nematics 289 Clearly, fork 1 =0, all three elasticity coefficients are equal, corresponding the isotrop ...

290 15 Liquid Crystals and Other Anisotropic Fluids The local orientation of liquid crystals, as observed optically via its bire ...

15.4 Cubatics and Tetradics 291 andcubatics. Here, as in [92, 93], the term ‘tetradic’ is meant as an abbreviation for ‘tetrahed ...