Characterization of Geometric Structures of Biaxial Nematic Phases

TL;DR

This paper introduces a new geometric interpretation of the ordering matrix for biaxial nematic liquid crystals, providing tools for classification, comparison, and energy formulation of molecular systems.

Contribution

It proposes the geometric order parameter, a modified form of the ordering matrix, and develops the anisotropy diagram for analyzing tensorial quantities in biaxial nematics.

Findings

Defined the degree of order as the singular value of the geometric order parameter.

Established a diagrammatic relation between microscopic and macroscopic tensors.

Formulated a prescription for Landau-de Gennes free energy for molecules of arbitrary shape.

Abstract

The ordering matrix, which was originally introduced by de Gennes, is a well-known mathematical device for describing orientational order of biaxial nematic liquid crystal. In this paper we propose a new interpretation of the ordering matrix. We slightly modify the definition of the ordering matrix and call it the geometric order parameter. The geometric order parameter is a linear transformation which transforms a tensorial quantity of an individual molecule to a tensorial quantity observed at a macroscopic scale. The degree of order is defined as the singular value of the geometric order parameter. We introduce the anisotropy diagram, which is useful for classification and comparison of various tensorial quantities. As indices for evaluating anisotropies of tensorial quantities, we define the degree of anisotropy and the degree of biaxiality. We prove that a simple diagrammatic relation holds between a microscopic tensor and a macroscopic tensor. We provide a prescription to formulate the Landau-de Gennes free energy of a system whose constituent molecules have an arbitrary shape. We apply our prescription to a system which consists of D_{2h}-symmetric molecules.