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 $B_{r_0}$, and satisfy homeotropic weak anchoring at the surface of the colloid and approach a uniform uniaxial state as $|x|\to\infty$. We study the minimizers in two different limiting regimes: for balls which are small $r_0\ll L^{\frac12}$ compared to the characteristic length scale $L^{\frac 12}$, and for large balls, $r_0\gg L^{\frac12}$. 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.