The equilibrium shapes of crystals and of cavities in crystals

Surface free energies are assumed to be the sum of the excess free energies of bonding of molecules in or near the surface, and the stable form of a crystal or cavity is assumed to be the form that makes the sum of these excess free energies a minimum. When only plane surfaces are allowed, this model predicts the same shapes for crystals as an equation of Wulff (2. Kristallogr. 34, 449 (1901)), which is based on the macroscopic thermodynamic relation of Gibbs (“The Collected Works, Vol. 1.: Thermodynamics,” Longmans, Green, New York (1931)). The model predicts rounding of edges and corners of kinds which are not allowed by the Wulff relation and predicts that spherical forms of particles and cavities can be stable despite anisotropic surface free energies. The model provides a useful framework for analysis of whether unstable crystal or cavity shapes will evolve into stable or metastable forms. Some crystals and cavities that have been assumed to have equilibrium shapes instead have metastable shapes.