Relieving nematic geometric frustration in the plane

TL;DR

This paper investigates how nematic director fields in a plane can be geometrically relaxed, revealing that planar spirals are the unique global solution for quasi-uniform distortions, with most solutions being asymptotic spirals.

Contribution

It proves that the only way to fill the entire plane with a quasi-uniform distortion is via a planar spiral, providing a complete geometric classification of such nematic configurations.

Findings

Planar spirals uniquely fill the entire plane with quasi-uniform distortions.

Most relieving distortions are asymptotic spirals, limited to half-planes.

The geometric approach clarifies nematic frustration relief mechanisms.

Abstract

Frustration in nematic-ordered media (endowed with a director field) is treated in a purely geometric fashion in a flat, two-dimensional space. We recall the definition of quasi-uniform distortions and envision these as viable ways to relieve director fields prescribed on either a straight line or the unit circle. We prove that using a planar spiral is the only way to fill the whole plane with a quasi-uniform distortion. Apart from that, all relieving quasi-uniform distortions can at most be defined in a half-plane; however, in a generic sense, they are all asymptotically spirals.