Within the framework of continuum theory, we draw a parallel between ferromagnetic materials and nematic liquid crystals confined on curved surfaces, which are both characterized by local interaction and anchoring potentials. We show that the extrinsic curvature of the shell combined with the out-of-plane component of the director field gives rise to chirality effects. This interplay produces an effective energy term reminiscent of the chiral term in cholesteric liquid crystals, with the curvature tensor acting as a sort of anisotropic helicity. We discuss also how the different nature of the order parameter, a vector in ferromagnets and a tensor in nematics, yields different textures on surfaces with the same topology as the sphere. In particular, we show that the extrinsic curvature governs the ground state configuration on a nematic spherical shell, favouring two antipodal disclinations of charge on small particles and four disclinations of charge located at the vertices of a square inscribed in a great circle on larger particles.
Gaetano Napoli, Oleksandr V. Pylypovskyi, Denis D Sheka, Luigi Vergori. Nematic shells: new insights in topology- and curvature-induced effects, Soft Matter (2021) DOI: 10.1039/d1sm00719j, arXiv:2102.13497