The geometry of high-dimensional phase diagrams: I. Generalized Gibbs Phase Rule

Phase diagrams are essential tools of the materials scientist, showing which phases are at equilibrium under a set of applied thermodynamic conditions. Essentially all phase diagrams today are two dimensional, typically constructed with axes of temperature-pressure or temperature-composition. For many modern materials, it would be valuable to construct phase diagrams that include additional forms of thermodynamic work--such as elastic, surface, electromagnetic or electrochemical work, etc.--which grows the free energy of materials into higher (>3) dimensions. Here, we extend Gibbs' original arguments on phase coexistence to derive a generalized Phase Rule, based in the combinatorial geometry of high-dimensional convex polytopes. The generalized Phase Rule offers a conceptual and geometric foundation to describe phase boundaries on high-dimensional phase diagrams, which are relevant for understanding the stability of modern materials in complex chemical environments. We revisit Gibbs arguments on the equilibrium of heterogeneous substances and show that phase coexistence regions in high-dimensional Internal Energy space, U(S,Xi,...,Xj), are simplicial convex polytopes--which are N-dimensional analogues of triangles and tetrahedra. In the first of this three-part series, we examine how the combinatorial relationships between the vertices and facets of simplicial polytopes leads to a generalized high-dimensional description of Gibbs' Phase Rule. Because Gibbs' Phase Rule describes the nature of phase boundaries on phase diagrams, this isomorphism between the physical principles of equilibrium thermodynamics and the geometry of simplicial polytopes provides the foundation to construct generalized phase diagrams, which can exist in any dimension, with any intensive or extensive thermodynamic variable on the axes.