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, , which is a geometry-dependent and material-dependent variable. We compute the limiting profiles exactly in two distinguished limits: the 0 limit relevant for macroscopic domains and the limit relevant for nanoscale domains. The limiting profile has line defects near the shorter edges in the limit, whereas we observe fractional point defects in the 0 limit. The analytical studies are complemented by some bifurcation diagrams for these reduced equilibria as a function of 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.