Computing equilibrium states of cholesteric liquid crystals in elliptical channels with deflation algorithms

Abstract

We study the problem of a cholesteric liquid crystal confined to an elliptical channel. The system is geometrically frustrated because the cholesteric prefers to adopt a uniform rate of twist deformation, but the elliptical domain precludes this. The frustration is resolved by deformation of the layers or introduction of defects, leading to a particularly rich family of equilibrium configurations. To identify the solution set, we adapt and apply a new family of algorithms, known as deflation methods, that iteratively modify the free energy extremisation problem by removing previously known solutions. A second algorithm, deflated continuation, is used to track solution branches as a function of the aspect ratio of the ellipse and preferred pitch of the cholesteric.

Structural Landscapes in Geometrically Frustrated Smectics

“…Close examination of the energy landscape, together with the corresponding solution set, shows many small discontinuous jumps that result from delicate commensurability effects, whereby certain domain sizes are compatible with a given periodicity of the layers, as well as from variations in the number of defects and their detailed placement. Similar effects have been observed when other periodic liquid crystals such as cholesterics are confined in domains that promote geometric frustration [74].…”

Control of Bifurcation Structures using Shape Optimization

“…The bifurcation diagrams displayed in this section are computed with deflated continuation [23,24], complemented with arclength continuation. These techniques have been successfully applied to a wide range of physical problems such as the deformation of a hyperelastic beam [25], liquid crystals [20,78], Bose--Einstein condensates [10,13,14], and fluid dynamics [11]. However, our optimization strategy is not tied to a specific algorithm for bifurcation analysis, and other options may be used.…”

Control of bifurcation structures using shape optimization

“…The bifurcation diagrams displayed in this section are computed with deflated continuation [19,20], complemented with arclength continuation. These techniques have been successfully applied to a wide range of physical problems such as the deformation of a hyperelastic beam [21], liquid crystals [17,65], Bose-Einstein condensates [8,10,11], and fluid dynamics [9]. However, our optimization strategy is not tied to a specific algorithm for bifurcation analysis, and other options may be used.…”