Crystal Shapes and Phase Equilibria: A Common Mathematical Basis

TL;DR

This paper reveals a unified mathematical framework underlying crystal shapes, phase equilibria, and interface energies, enabling new graphical methods and numerical tools for phase diagram calculations based on convex function analysis.

Contribution

It demonstrates a common geometric structure for diverse topics in thermodynamics and crystallography, leading to novel applications of graphical methods and phase diagram computations.

Findings

Unified geometric framework for solution thermodynamics and crystal growth.

New graphical methods for phase diagram calculations.

Analogies between thermodynamic equations and surface/interface theories.

Abstract

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.