Compatible director fields in $\mathbb{R}^3$

TL;DR

This paper revisits the compatibility conditions for director fields in three-dimensional liquid crystals, providing a clearer derivation and characterizing compatible states using vector calculus.

Contribution

It offers a more transparent derivation of compatibility equations and characterizes a broad class of compatible director fields in D.

Findings

Derived compatibility equations using vector calculus.

Characterized a wide class of compatible director fields.

Clarified the geometric conditions for liquid crystal phases.

Abstract

The geometry and interactions between the constituents of a liquid crystal, which are responsible for inducing the partial order in the fluid, may locally favor an attempted phase that could not be realized in $\mathbb{R}^3$. While states that are incompatible with the geometry of $\mathbb{R}^3$ were identified more than 50 years ago, the collection of compatible states remained poorly understood and not well characterized. Recently, the compatibility conditions for three-dimensional director fields were derived using the method of moving frames. These compatibility conditions take the form of six differential relations in five scalar fields locally characterizing the director field. In this work, we rederive these equations using a more transparent approach employing vector calculus. We then use these equations to characterize a wide collection of compatible phases.