Biaxial order in liquid crystals and their mixtures: A Potts-Ising model

A lattice model for binary mixtures of prolate and oblate molecules is studied, by a mapping onto a 13-state Potts-Ising model, followed by an approximate renormalization-group analysis. Four different types of phase diagram are obtained, exhibiting prolate uniaxial, biaxial, and oblate uniaxial phases. Local disorder has the nonordering property of an effective vacancy and is thus included into our mapping. Such effective vacancies can cause first-order phase transitions. Thus the biaxial phase can disorder either directly through a ridge of first-order transitions, or via an intermediate uniaxial phase which vanishes at a multicritical point. In the biaxial region, the system is shown to be related to the six-state clock model, so that the latter point may be replaced by a segment of algebraic (Kosterlitz-Thouless) order in films. Similar considerations are applied to one-component systems of biaxially shaped molecules.