Pair Correlation Functions and a Free-Energy Functional for the Nematic Phase

TL;DR

This paper calculates pair correlation functions in nematic phases of spherical particles using MSA and PY theories, compares results with existing analytical solutions, and develops a free-energy functional to study phase transitions.

Contribution

It introduces a free energy functional incorporating symmetry parts of the correlation function and assesses its effectiveness in modeling the nematic phase.

Findings

MSA results agree with analytical solutions by Holovko and Sokolovska.

PY theory produces pair correlation functions with long-range tails.

The free energy functional with only the principal order parameter yields accurate results.

Abstract

In this paper we have presented the calculation of pair correlation functions in a nematic phase for a model of spherical particles with the long-range anisotropic interaction from the mean spherical approximation(MSA) and the Percus-Yevick (PY) integral equation theories. The results found from the MSA theory have been compared with those found analytically by Holovko and Sokolovska (J. Mol. Liq. $\bf 82$, 161(1999)). A free energy functional which involves both the symmetry conserving and symmetry broken parts of the direct pair correlation function has been used to study the properties of the nematic phase. We have also examined the possibility of constructing a free energy functional with the direct pair correlation function which includes only the principal order parameter of the ordered phase and found that the resulting functional gives results that are in good agreement with the original functional. The isotropic-nematic transition has been located using the grand thermodynamic potential. The PY theory has been found to give nematic phase with pair correlation function harmonic coefficients having all the desired features. In a nematic phase the harmonic coefficient of the total pair correlation function $h({\bf x_1},{\bf x_2})$ connected with the correlations of the director transverse fluctuations should develop a long-range tail. This feature has been found in both the MSA and PY theories.