Multistability for Nematic Liquid Crystals in Cuboids with Degenerate Planar Boundary Conditions

SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 756-781, April 2024.
Abstract. We study nematic configurations within three-dimensional (3D) cuboids, with planar degenerate boundary conditions on the cuboid faces, in the Landau–de Gennes framework. There are two geometry-dependent variables: the edge length of the square cross-section, [math], and the parameter [math], which is a measure of the cuboid height. Theoretically, we prove the existence and uniqueness of the global minimizer with a small enough cuboid size. We develop a new numerical scheme for the high-index saddle dynamics to deal with the surface energies. We report on a plethora of (meta)stable states, and their dependence on [math] and [math], and in particular how the 3D states are connected with their two-dimensional counterparts on squares and rectangles. Notably, we find families of almost uniaxial stable states constructed from the topological classification of tangent unit-vector fields and study transition pathways between them. We also provide a phase diagram of competing (meta)stable states as a function of [math] and [math].