The Geometry of Phase Boundaries on High-Dimensional Phase Diagrams

There are four main varieties of phase diagrams today: 1) Temperature–Pressure; 2) Temperature–Composition; 3) Ellingham (<i>T</i>, <i>μ</i>_O2); and 4) Pourbaix diagrams (<i>p</i>H, Redox potential). However, these 2D phase diagrams do not account for many other forms of thermodynamic work that may be operative during nucleation and growth—such as surface, elastic, electromagnetic and electrochemical work. Here, I will describe a process to lift 2D phase diagrams into higher dimensions, exposing these additional forms of thermodynamic work on the axes. In particular, this involves deriving the geometry of high-dimensional phase boundaries and phase-coexistence regions, which leads to generalized forms of Gibbs' Phase Rule and Clausius-Clapeyron relations. From this geometric foundation, we can construct and deploy phase diagrams with any axes of mixed intensive or extensive variables—providing a powerful new theoretical framework for the design and synthesis of advanced functional materials.