Crystal shapes and phase equilibria: A common mathematical basis

Geometrical constructions, such as the tangent construction on the molar free energy for determining whether a particular composition of a solution is stable, are related to similar tangent constructions on the orientation-dependent interfacial energy for determining stable interface orientations and on the orientation dependence of the crystal growth rate which tests whether a particular orientation appears on a growing crystal. Subtle differences in the geometric constructions for the three fields arise from the choice of a metric (unit of measure). Using results from studies of extensive and convex functions, we demonstrate that there is a common mathematical structure for these three disparate topics and use this to find new uses for well-known graphical methods for all three topics. Thus, the use of chemical potentials for solution thermodynamics is very similar to known vector formulations for surface thermodynamics and the method of characteristics which tracks the interfaces of growing crystals; the Gibbs-Duhem equation is analogous to the Cahn-Hoffman equation. The Wulff construction for equilibrium crystal shapes can be modified to construct a “phase shape” from solution free energies that is a potentially useful method of numerical calculations of phase diagrams from known thermodynamical data.