Spontaneous Undulation of Equilibrium Interfaces with Positive Surface Stiffness

We report experimental evidence of spontaneous undulations at static phase boundaries stabilized by a thermal gradient $g$ and a positive surface stiffness. These periodic undulations are observed at the interface between a smectic-$C$ phase and its isotropic or nematic melt. We show that the mechanism driving this instability is the coupling between the surface tension anisotropy and the orientational degree of freedom of the liquid crystal. The period scales as $g^{\frac{-1\mathrm{}}{3}}$, as in the usual Herring instability.