Lattice statistical models for the nematic transitions in liquid-crystalline systems

TL;DR

This paper explores lattice models for nematic liquid crystals, connecting them with a two-tensor formalism to accurately reproduce phase diagrams, including isotropic, uniaxial, biaxial, and tricritical points, beyond mean-field approximations.

Contribution

It introduces simple lattice models that replicate complex nematic phase behavior and extend analysis beyond mean-field theory using a two-tensor formalism.

Findings

Reproduces isotropic, uniaxial, and biaxial nematic phases.

Identifies tricritical points consistent with recent predictions.

Models are computationally simple and extendable beyond mean-field.

Abstract

We investigate the connections between some simple Maier-Saupe lattice models, with a discrete choice of orientations of the microscopic directors, and a recent proposal of a two-tensor formalism to describe the phase diagrams of nematic liquid-crystalline systems. This two-tensor proposal is used to formulate the statistical problem in terms of fully-connected lattice Hamiltonians, with the local nematic directors restricted to the Cartesian axes. Depending on the choice of interaction parameters, we regain all of the main features of the original mean-field two-tensor calculations. With a standard choice of parameters, we obtain the well-known sequence of isotropic, uniaxial, and biaxial nematic structures, with a Landau multicritical point. With another suitably chosen set of parameters, we obtain two tricritical points, according to some recent predictions of the two-tensor calculations. The simple statistical lattice models are quite easy to work with, for all values of parameters, and the present calculations can be carried out beyond the mean-field level.