Topological Phases and Curvature-Driven Pattern Formation in Cholesteric Shells

We study the phase behaviour of cholesteric liquid crystal shells with different geometries. We compare the cases of tangential and no anchoring at the surface, focussing on the former case, which leads to a competition between the intrinsic tendency of the cholesteric to twist and the anchoring free energy which suppresses it. We then characterise the topological phases arising close to the isotropic-cholesteric transition. These typically consist of quasi-crystalline or amorphous tessellations of the surface by half-skyrmions, which are stable at lower and larger shell size respectively. For ellipsoidal shells, defects in the tessellation couple to local curvature, and according to the shell size they either migrate to the poles or distribute uniformly on the surface. For toroidal shells, the variations in the local curvature of the surface stabilises heterogeneous phases where cholesteric or isotropic patterns coexist with hexagonal lattices of half-skyrmions.