Liquid-crystalline ordering in two-dimensional systems with discrete symmetry

Abstract

The mean-field theories of liquid-crystalline (nematic) ordering developed for three-dimensional systems are applied to describe two-dimensional systems of both geometrically anisotropic and anisotropically interacting particles. Systems with discrete symmetry (lattice models) for which long-range order is possible are considered on the base of the Landau free-energy expansion. It is shown that the Hamiltonian describing the energy of intermolecular interactions may be written in a common form for lyotropic and thermotropic systems. The mean-field theory gives a continuous phase transition (second-order phase transition) for a square lattice, whereas for a triangular lattice it gives a phase transition with latent heat (first-order phase transition) like for three-dimensional systems. These results are compared with results of the exact theories (two-dimensional Ising and Potts models). It is concluded that for realistic two-dimensional models the orientational in-plane ordering is not sharper than a second-order phase transition.