Minimizers of the Landau–de Gennes Energy Around a Spherical Colloid Particle

We consider energy minimizing configurations of a nematic liquid crystal around a spherical colloid particle, in the context of the Landau–de Gennes model. The nematic is assumed to occupy the exterior of a ball Br0, and satisfy homeotropic weak anchoring at the surface of the colloid and approach a uniform uniaxial state as |x|→∞$${|x|\to\infty}$$. We study the minimizers in two different limiting regimes: for balls which are small r0≪L12$${r_0\ll L^{\frac12}}$$ compared to the characteristic length scale L12$${L^{\frac 12}}$$, and for large balls, r0≫L12$${r_0\gg L^{\frac12}}$$. The relationship between the radius and the anchoring strength W is also relevant. For small balls we obtain a limiting quadrupolar configuration, with a “Saturn ring” defect for relatively strong anchoring, corresponding to an exchange of eigenvalues of the Q-tensor. In the limit of very large balls we obtain an axisymmetric minimizer of the Oseen–Frank energy, and a dipole configuration with exactly one point defect is obtained.