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

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→ 0 limit relevant for macroscopic domains and the ϵ→∞ limit relevant for nanoscale domains. The limiting profile has line defects near the shorter edges in the ϵ→∞ limit, whereas we observe fractional point defects in the ϵ→ 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.