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

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.