Solution Landscapes in the Landau-de Gennes Theory on Rectangles

We study nematic equilibria on rectangular domains, in a reduced two-dimensional Landau-de Gennes framework. We have one essential model variable---$\tilde{\epsilon}$ which is a geometry-dependent and material-dependent variable. We analytically study two distinguished limits---the $\tilde{\epsilon} \to 0$ limit which is relevant for macroscopic domains and the $\tilde{\epsilon} \to \infty$ limit which is relevant for nano-scale domains. We report a new stable nematic state featured by thin transition layers near the shorter rectangular edges, in the nano-scale limit. We numerically compute bifurcation diagrams for the solution landscapes as a function of $\tilde{\epsilon}$ and the rectangular aspect ratio. We also investigate relaxation mechanisms for non-trivial topologies and how such mechanisms are dominated by defect annihilation, near vertices or in the interior, or through transient almost isotropic states and such relaxation mechanisms have potential relevance for switching mechanisms in nematic-based devices.