Spherical Particle in Nematic Liquid Crystal Under an External Field: The Saturn Ring Regime

We consider a nematic liquid crystal occupying the exterior region in R3$${\mathbb {R}}^3$$ outside of a spherical particle, with radial strong anchoring. Within the context of the Landau-de Gennes theory, we study minimizers subject to an external field, modeled by an additional term which favors nematic alignment parallel to the field. When the external field is high enough, we obtain a scaling law for the energy. The energy scale corresponds to minimizers concentrating their energy in a boundary layer around the particle, with quadrupolar symmetry. This suggests the presence of a Saturn ring defect around the particle, rather than a dipolar director field typical of a point defect.