Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2020 / Paper 2 / Section B / 22 / iii

homeotropic anchoring favors the nematic director normal to a surface. For a flat wall at $z=0$, uniform $\mathbf{n}=\hat{\mathbf{z}}$ is a simple compatible profile, shown as perpendicular rods. If a remote wall demands another orientation, elastic bending interpolates between the boundaries.

A common surface energy is $f_s=(W/2)[1-(\mathbf{n}\cdot \mathbf{\nu }{)}^2]$, with wall normal $\mathbf{\nu }$ and $W>0$. Its minima have $\mathbf{n}$ parallel or antiparallel to $\mathbf{\nu }$. The profile is controlled jointly by this preference and Frank elastic energy.

planar anchoring favors the nematic director tangent to a surface. A flat wall at $z=0$ permits uniform $\mathbf{n}=\hat{\mathbf{x}}$, as in the sketch. A rubbed surface can select one in-plane axis; degenerate planar anchoring selects only the tangent plane.

For the degenerate case, $f_s=(W/2)(\mathbf{n}\cdot \mathbf{\nu }{)}^2$ has minima at $\mathbf{n}\cdot \mathbf{\nu }=0$. A remote incompatible boundary produces elastic distortion. Tangency alone does not specify an azimuthal direction within the surface.

Strong homeotropic anchoring requires radial orientation just outside a sphere, while the remote director is uniform. A plausible dipolar texture contains a compensating hyperbolic hedgehog defect on one side. The particle's radial boundary has charge $+1$ under a consistent convention, balanced by an exterior point defect of charge $-1$ so the far field is uniform. The original cross-section depicts this configuration.

An alternative has an equatorial Saturn-ring defect. Its local disclination winding is half-integer; that local winding is not itself the net three-dimensional compensating charge. Particle size, elastic constants, confinement and anchoring strength select between configurations. The elastic theory explicitly compares dipolar, Saturn-ring and surface-ring states.^[ref]

A sphere with degenerate planar anchoring can have a bipolar texture with two surface boojums at opposite poles along the far-field axis. Surface director lines remain tangent, curve between the polar cores and relax toward uniform alignment far away. The original cross-section shows both cores.

A globally regular tangent orientation field cannot cover a sphere; the associated surface winding indices sum to its Euler characteristic, two. These are surface defects, not two arbitrarily placed bulk hedgehogs. Finite anchoring allows deviations near the cores, while an azimuth-selective coating could favor a different texture.