Phase diagram of a model for a binary mixture of nematic molecules on a Bethe lattice

TL;DR

This paper explores the phase diagram of a simplified model for binary nematic molecules on a Bethe lattice, revealing the stability of isotropic and uniaxial nematic phases but not biaxial phases, consistent with mean-field predictions.

Contribution

It introduces a discrete Maier-Saupe model with additional degrees of freedom analyzed on a Bethe lattice, providing insights into phase stability and ruling out biaxial phases.

Findings

No stable biaxial nematic phase found.

Thermodynamic and stability analyses agree with mean-field results.

Method developed for free energy calculation deep inside Cayley trees.

Abstract

We investigate the phase diagram of a discrete version of the Maier-Saupe model with the inclusion of additional degrees of freedom to mimic a distribution of rodlike and disklike molecules. Solutions of this problem on a Bethe lattice come from the analysis of the fixed points of a set of nonlinear recursion relations. Besides the fixed points associated with isotropic and uniaxial nematic structures, there is also a fixed point associated with a biaxial nematic structure. Due to the existence of large overlaps of the stability regions, we resorted to a scheme to calculate the free energy of these structures deep in the interior of a large Cayley tree. Both thermodynamic and dynamic-stability analyses rule out the presence of a biaxial phase, in qualitative agreement with previous mean-field results.