Surface,size and topological effects for some nematic equilibria on rectangular domains

Abstract

We study nematic equilibria on rectangular domains, in a reduced two-dimensional Landau–de Gennes framework. These reduced equilibria carry over to the three-dimensional framework at a special temperature. There is one essential model variable, $ϵ$ , which is a geometry-dependent and material-dependent variable. We compute the limiting profiles exactly in two distinguished limits: the $ϵ\to$ 0 limit relevant for macroscopic domains and the $ϵ\to \infty$ limit relevant for nanoscale domains. The limiting profile has line defects near the shorter edges in the $ϵ\to \infty$ limit, whereas we observe fractional point defects in the $ϵ\to$ 0 limit. The analytical studies are complemented by some bifurcation diagrams for these reduced equilibria as a function of $ϵ$ and the rectangular aspect ratio. We also introduce the concept of ‘non-trivial’ topologies and study the relaxation of non-trivial topologies to trivial topologies mediated via point and line defects, with potential consequences for non-equilibrium phenomena and switching dynamics.