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