Cooling a spherical nematic shell

Within the framework of Landau–de Gennes theory for nematic liquid crystals, we study the temperature-induced isotropic-nematic phase transition on a spherical shell under the assumption of degenerate tangential anchoring. Below a critical temperature, a thin layer of nematic coating a microscopic spherical particle exhibits nonuniform textures due to the geometrical frustration. We find the exact value of the critical threshold for the temperature and determine exactly the nematic textures at the transition by means of a weakly nonlinear analysis. The critical temperature is affected by the extrinsic curvature of the sphere, and the nematic alignment is consistent with the Poincaré-Hopf index theorem and experimental observations. The stability analysis of the bifurcate textures at the isotropic-nematic transition highlights that only the tetrahedral configuration is stable.

Figure 1

Energies of the bifurcate configurations. For small temperature departures from the critical value $T_{\mathrm{cr}}$, the tetrahedral configuration (${\mathcal{C}}_5$) is a local minimizer of the energy functional. All the other configurations are unstable.