The Geometry of the Cholesteric Phase

TL;DR

This paper introduces a mathematical framework for describing the cholesteric pitch axis in nematic liquid crystals, enabling comparison of defect structures across different liquid crystal phases.

Contribution

It presents a novel eigenvalue-based construction of the cholesteric pitch axis and a Frenet-Serret description for analyzing defect structures in liquid crystals.

Findings

Defined a cholesteric pitch axis as an eigenvalue problem.

Developed a Frenet-Serret framework for the orthonormal triad.

Compared defect structures across cholesterics, biaxial nematics, and smectics.

Abstract

We propose a construction of a cholesteric pitch axis for an arbitrary nematic director field as an eigenvalue problem. Our definition leads to a Frenet-Serret description of an orthonormal triad determined by this axis, the director, and the mutually perpendicular direction. With this tool we are able to compare defect structures in cholesterics, biaxial nematics, and smectics. Though they all have similar ground state manifolds, the defect structures are different and cannot be, in general, translated from one phase to the other.