Inverse Wulff construction for surface energies of coexisting and missing surfaces of crystal particles

Abstract

Morphologies of nanoparticles are an important factor that defines various nanomaterials’ properties and applications. Wulff construction is an effective tool to understand and predict the morphologies of nanoparticles. Lack of surface energies, however, poses obstacles to applicability of this method. We use inverse Wulff construction, based on nonlinear optimization and crystal symmetries, to acquire surface energies from Wulff shapes. This method is capable of calculating not only the surface energies of those coexisting surfaces of a particle, but also lower bounds of surface energies of missing surfaces. We use the m3¯m crystal class to verify numerical surface energies by optimization against analytical results by thermodynamic equilibrium between coexisting surfaces. We further propose a general function for the surface-energy ratios. The method can be used to study evolution of surface energies from particles’ shapes for thermodynamic modeling that takes surface energies as input parameters.