Determination of the symmetry classes of orientational ordering tensors

TL;DR

This paper introduces a new algorithm to identify the symmetry group of nematic liquid crystal phases from the second-rank ordering tensor, addressing the challenge of symmetry determination from experimental or simulation data.

Contribution

The authors provide a novel method for classifying the symmetry of orientational order tensors and prove the existence of only five symmetry classes for these tensors.

Findings

Successfully identified phase symmetries in simulated systems

Proved the existence of five symmetry classes for the second-rank ordering tensor

Provided a canonical form for each symmetry class

Abstract

The orientational order of nematic liquid crystals is traditionally studied by means of the second-rank ordering tensor $\mathbb{S}$. When this is calculated through experiments or simulations, the symmetry group of the phase is not known \emph{a-priori}, but needs to be deduced from the numerical realisation of $\mathbb{S}$, which is affected by numerical errors. There is no generally accepted procedure to perform this analysis. Here, we provide a new algorithm suited to identifying the symmetry group of the phase. As a by product, we prove that there are only five phase-symmetry classes of the second-rank ordering tensor and give a canonical representation of $\mathbb{S}$ for each class. The nearest tensor of the assigned symmetry is determined by group-projection. In order to test our procedure, we generate uniaxial and biaxial phases in a system of interacting particles, endowed with $D_{\infty h}$ or $D_{2h}$, which mimic the outcome of Monte-Carlo simulations. The actual symmetry of the phases is correctly identified, along with the optimal choice of laboratory frame.