Characterization of geometric structures of biaxial nematic phases

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 D2h-symmetric molecules. PACS codes: 61.30.Cz, 61.30.Gd, 83.80.Xz