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 andT-x phase diagrams, enabling materials scientists to fully interrogate the complex thermodynamic environments of modern materials.

This work was supported as part of GENESIS: A Next Generation Synthesis Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award No. DESC0019212.