Steric effects in a mean-field model for polar nematic liquid crystals

The existence of uniaxial liquid crystals comprising polar molecules, with all the dipoles aligned in a parallel pattern, is classically ruled out. Generally, there are two different avenues to a mean-field theory for liquid crystals: one is based on short-range, repulsive, steric forces, and the other is based on long-range, globally attractive, dispersion forces. Purely polar steric interactions have been shown to have the potential of inducing unexpected orientationally ordered states. In real molecules, anisotropies both in shape and in polarizability coexist; it has been shown that dispersion forces interaction can be combined with hard-core repulsion in a formal theory, based on a steric tensor. Starting from this, we build an interaction Hamiltonian featuring the average electric dipolar energy exchanged between molecules with the same excluded region. Under the assumption that the molecular shape is spheroidal, we propose a mean-field model for polar nematic liquid crystals which can exhibit both uniaxial and biaxial polar phases. By means of a numerical bifurcation analysis, we discuss the stability of the equilibrium against the choice of two model parameters, one describing the degree of molecular shape biaxiality and the other describing the relative orientation of the electric dipole within each molecule. We find only uniaxial stable phases, which are effectively characterized by a single scalar order parameter