On the Elastic Theory of Uniaxial Liquid Crystals

Abstract

Simple derivations of the elastic theory of liquid crystals with uniaxial molecular order are given using a molecular Cauchy‐like approach and a more general phenomenological approach. In a second‐order approximation, both methods lead to the same expression for the elastic energy density, except for the existence of one “Cauchy relation” between the elastic constants in the molecular case. Our phenomenological equation contains nine elastic constants which is two more than in Frank's earlier derivation. Our molecular equation is in agreement with Oseen's original expression. Differences between the molecular and the phenomenological approach and the justification of neglecting terms which do not contribute to the Euler–Lagrange differential equation are discussed. Higher than second‐order terms are given for cholesteric liquids. In addition, the consequence of curvature‐induced electric polarization for the equilibrium structure of liquid crystals is discussed.