Contact topology and the structure and dynamics of cholesterics

Thomas Machon

doi:10.1088/1367-2630/aa958d

Abstract

Using tools and concepts from contact topology we show that non-vanishing twist implies conservation of the layer structure in cholesteric liquid crystals. This leads to a number of additional topological invariants for cholesteric configurations, such as layer numbers, that are not captured by traditional descriptions, characterises the nature and size of the chiral energy barriers between metastable configurations, and gives a geometric characterisation of cholesteric dynamics in any context, including active systems, those in confined geometries or under the influence of an external field.

Highlights

Cholesteric liquid crystals often display a high number of metastable configurations [1, 2], in confined geometries, such as droplets [3,4,5,6], shells [7,8,9], colloidal systems [10,11,12], or systems confined between parallel plates [13,14,15,16] where the interplay between chirality, elasticity and surface interactions can produce a large variety of structures
The theory we describe gives homotopy invariants of cholesteric textures in this regime originating in contact topology [28], allowing topological classification of cholesteric configurations in a way that cannot be accomplished in standard theories
In this paper we have only sketched the connection between contact topology and cholesterics, and there is a great deal of future work that may be done on this relationship

Summary

Contact Topology and the Structure and Dynamics of Cholesterics

This leads to a number of additional topological invariants for cholesteric textures, such as layer numbers, that are not captured by traditional descriptions, characterises the nature and size of the chiral energy barriers between metastable configurations, and gives a geometric characterisation of cholesteric dynamics in any context, including active systems, those in confined geometries or under the influence of an external field

INTRODUCTION

THE CONSERVATION OF LAYER STRUCTURE IN CHOLESTERICS

Layers flow along w Director evolution

LAYER NUMBER INVARIANTS

DOUBLE TWIST CYLINDERS AND OVERTWISTED DISKS

CONCLUSION