Measurement of the anisotropy of the energy of an isotropic-smectic interface in a smectic cylinder: Application to the L₃-L_α interface

In order to measure the relative anisotropy of a smectic-isotropic interfacial energy, we propose an alternative to the classic Wulff construction. We derive the Gibbs-Thomson equation that describes the equilibrium of a droplet of isotropic phase in a smectic sample confined within a cylindrical capillary. We then discuss the influence of the curvature energy term and show that the profile of the droplet gives the variation of the surface energy, either analytically or by means of a modified Wulff construction. From this approach, we measure the relative anisotropy of the sponge-lamellar interfacial energy in the dilute Cetylpyridinium Chloride/hexanol/brine surfactant system. We quantitatively confirm and complete previous observations of this interface, which displays strong crystalline features, such as cusps and forbidden orientations.