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

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.