Lattice Model for Biaxial Liquid Crystals

Abstract

The possibility that the repulsive interaction between platelike molecules might give rise to biaxial liquid crystal order is examined by considering a molecular-field lattice model in which molecules are represented by rectangular plates. We find that plates which are neither very square nor very rodlike in shape may exist in a phase characterized by biaxial long-range order in the orientation of the plates. This biaxial phase occurs at high pressure. At some lower pressure the biaxial phase undergoes a second order phase transition to a uniaxial phase. At a still lower pressure the uniaxial phase undergoes a transition to an isotropic phase. This last phase transition is usually first order in that there are discontinuities in the density and the orientational order in going through the transition. However the size of the discontinuities depend strongly on the shape of the plates. For example for plates of length 6 and width approximately 2.22 and thickness 1, the discontinuities are zero, and the transition is accidentally second order. The discontinuities at this uniaxial-isotropic transition are always very small for plates which have the potential to give rise to a biaxial phase. This last result is shown to have a direct physical basis which should transcend the limitations imposed by the approximate nature of our calculation. Thus we conclude that for real substances likely to have biaxial phases of the type we have studied, the value of the nematic order parameter at the nematic-isotropic transition should be much lower than for other nematic liquid crystals.