Structure of twist-grain-boundary–$C$ phases

We study properties of the Renn-Lubensky twist-grain-boundary–$C$ $\left({\mathrm{TGB}}_C\right)$ phase, with layer normal rotating in a plane perpendicular to the pitch axis, and the Bordeaux ${\mathrm{TGB}}_C$ phase, with the layer normal rotating on a cone parallel to the pitch axis near the upper critical twist $k_{c2\mathrm{}}$ marking the transition to the cholesteric phase. We introduce a generalized model free energy for the smectic-$C$ phase that allows either ${\mathrm{TGB}}_C$ phase to be stable, and we calculate $k_{c2\mathrm{}}$ and the order-parameter profile, which shows only modest spatial variation, for both phases.