Cooling a spherical nematic shell

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