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, [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.

Tailored nematic and magnetization profiles on two-dimensional polygons

“…Mathematically, this limit is much simpler than the → 0 limit, since we lose the nemato-magnetic coupling in this limit. Referring to [41], the leading order equations, in this limit, are: ). In the Q 1 plot, the corresponding vector n 1 in ( 13) is represented by white lines and the order parameter subject to the Dirichlet conditions (10).…”

“…This is precisely the solution along the Ring branch for large , in the bifurcation diagrams Figures 1 and 3, which is the unique energy minimizer in this limit. Following the methods in [41], the limiting solution, (Q ∞ , M ∞ ) of ( 26) is an excellent approximation to the solutions, Q , M of (6)-( 9), for fixed c, subject to the same boundary conditions, for large enough i.e.,…”

Multistability for a Reduced Nematic Liquid Crystal Model in the Exterior of 2D Polygons

“…We choose this temperature, partly for comparison with previous work in [16] and [9], and partly because for this special temperature, solutions in the rLdG model also exist as solutions for the full LdG model in 3D i.e. the minimizers of the rLdG model at A = −B 2 /3C will exist as critical points of the full LdG free energy in 3D settings, for suitably defined boundary conditions.…”

Tailored Nematic and Magnetization Profiles on 2D Polygons

“…Mathematically, this limit is much simpler than the → 0 limit, since we lose the nemato-magnetic coupling in this limit. Referring to [41], the leading order equations, in this limit, are:…”