Bivalent defect configurations in inhomogeneous nematic shells

We present a theoretical study of the director fields and energetics of nematic liquid crystal shells with two pairs of surface defects. The pairs of defects can undergo abrupt transitions from a configuration of maximum separation to a state in which the defects are confined to the thinner hemisphere. We construct a phase diagram that maps out the stability and coexistence of these two configurations as a function of shell thickness and thickness inhomogeneity. Our results compare favourably with the experimentally observed transitions in nematic double emulsion droplets and explain their hysteretic character.