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

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, $\lambda$, and the parameter $h$, which is a measure of the cuboid height. Theoretically, we prove the existence and uniqueness of the global minimiser 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 $h$ and $\lambda$, 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 $\lambda$ and $h$.