Mesoscopic continuum theory for liquid crystals

Abstract

A summary of the mesoscopic theory of liquid crystals is presented. Mesoscopic field quantities are defined on the enlarged space of position, time and orientation. An orientation distribution function is defined as the fraction of the mesoscopic mass density over the total mass density. For the mesoscopic field quantities of mass, momentum, angular momentum and energy balance equations on the enlarged domain are formulated. They allow the derivation of a differential equation for the orientation distribution function. Measurable quantities are macroscopic ones. They are obtained from the mesoscopic quantities by averaging with the orientation distribution function. In addition to the usual macroscopic fields, alignment tensors are obtained by the averaging procedure. These tensors of successive order are a measure of the orientational order, present on the mesoscopic level. For these alignment tensors equations of motion are derived here. They form a set of coupled equations for infinitely many alignment tensors, and a closure is needed. At the lowest level, only the alignment tensor of second order is an independent variable, and the fourth order tensor is substituted by an algebraic closure relation. At this level of approximation Landau theory of phase transitions can be recovered in case of vanishing flow field. In a refined approach, the second and the fourth order alignment tensor are considered as independent variables, and the sixth order tensor is eliminated by an algebraic closure relation. This refined alignment tensor dynamics is considered in detail in the case of a uniaxial orientation distribution function.