Macroscopic constitutive equations for liquid crystals induced by their mesoscopic orientation distribution

Abstract

By starting out with mesoscopic orientational balance equations for each orientational component of a liquid crystal which is described as a formal mixture, a set of independent macroscopic variables forming the state space is induced. This set includes a second-order tensorial measure of alignment, called the alignment tensor, and its derivatives. In terms of these state space variables constitutive equations are proposed by exploiting the dissipation inequality due to Coleman and Noll. The results are compared to those of Ericksen and Leslie, who described the alignment in a liquid crystal with only a macroscopic unit director field d( x, t) indicating the “mean orientation” of the media. In a recent paper Ericksen introduced beside the macroscopic director an additional scalar order parameter S( x,t) and its derivatives (Maier-Saupe theory) which turns out to be the uniaxial case in the alignment tensor formulation. Also in this case the restrictions to the constitutive equations caused by the dissipation inequality are discussed.