Quasi-stable configurations of liquid crystals in polyhedral geometries

Bistable director configurations are of great interest in liquid crystal display technologies, offering the possibility of higher resolution combined with reduced power consumption. One way to achieve such bistability is to use the cell geometry. As part of an ongoing programme to analyze quasi-stable configurations of liquid crystals in polyhedral geometries, we construct a topological classification scheme of unit-vector fields in convex polyhedra subject to tangential boundary conditions and obtain a general lower bound on the energy of these configurations. We also study the specific case of a unit cube, where we obtain lower and upper bounds for the energies of a family of topological types.