= Solution
<Homeotropic anchoring> means that the <nematic director> is locally normal to the particle surface, hence radial on a sphere. Far away, the uniform <nematic liquid crystal> has a fixed director $\mathbf n_\infty$, with $\mathbf n$ and $-\mathbf n$ physically equivalent.
One plausible configuration is a quadrupolar <Saturn-ring defect>: the director is radial near the particle, bends into $\mathbf n_\infty$ far away, and has a disclination loop just outside the particle's equator, in the plane perpendicular to $\mathbf n_\infty$. A meridional section cuts the loop at two points.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2023/paper-2-nematic-director.png]
{title=Homeotropic surface anchoring matched to a uniform far-field director}
{description=Original schematic Saturn-ring texture. Short rods represent a headless director; the two marked defect cores are intersections of a three-dimensional equatorial ring with the section. This is not a numerical free-energy minimizer.}
{source=paper-2-nematic-director.py}
The defect compensates the topological distortion introduced by the radial surface boundary condition. A dipolar configuration with a nearby hyperbolic point defect is another possible texture. Which is stable depends on particle size, anchoring strength, elastic constants and external confinement; the boundary conditions alone do not uniquely select one.<Director field configurations around a spherical particle in a nematic liquid crystal by Holger Stark>[Competing dipolar, Saturn-ring and surface-ring configurations]{ref}
Back to article page
