Surface energies of non-centrosymmetric nanocrystals by the inverse Wulff construction method.

The Wulff construction is a well-known method for studying the morphologies of nanocrystal particles. The underlying Gibbs-Wulff theorem relies on Wulff points or particle centers, which are invalid for non-centrosymmetric crystals. In this study, we extend the method of inverse Wulff construction to study surface free energies of non-centrosymmetric crystals. A nonpolar (4[combining macron]3m) and a polar crystal system (6mm) are selected to show the difficulties and constraints caused by the lack of inversion centers. In addition to analytical and numerical results, we also present a general four-parameter function to simplify calculations of surface free energies from observed micro- or nanoparticle morphologies. The results reveal that non-centrosymmetric crystal morphologies entail the surface thermodynamics of coupling polar surfaces, with certain limitations originating from the combination of geometrical shapes and crystal symmetries.