Generalized Gibbs' Phase Rule*

Gibbs’ Phase Rule describes the nature of phase boundaries on phase diagrams and is a foundational principle in materials thermodynamics. In Gibbs’ original derivation, he stipulates that the Phase Rule applies only to “simple systems”—defined to be homogeneous, isotropic, uncharged, and large enough that surface effects can be neglected; and not acted upon by electric, magnetic or gravitational fields. Modern functional materials—spanning nanomaterials, multiferrorics, materials for energy storage and conversion, colloidal crystals, etc.—are decidedly non-simple, often leveraging additional forms of thermodynamic work to achieve their functionality. Adding thermodynamic variables into a free-energy expression increases the dimensionality of its corresponding thermodynamic space. Here we revisit Gibbs’ original arguments on phase coexistence and show that phase boundaries in high-dimensional Internal Energy space, U(S,X_i,…), are simplicial convex polytopes—which are N-dimensional analogues of triangles and tetrahedra. From this geometric description we derive a generalized form of Gibbs’ Phase Rule; which can be combined with high-throughput DFT calculations of solid-state entropies, strain tensors, surface energies, magnetic structures, and polarization displacements to build entirely new classes of phase diagrams. These generalized phase diagrams can exist in multiple (≥3) thermodynamic dimensions, and exhibit elastic, surface, electromagnetic or electrochemical work on their axes. New phase diagrams are poised to expand the thermodynamic toolkit beyond the common T-P and T-x phase diagrams, enabling materials scientists to fully interrogate the complex thermodynamic environments of modern materials.