  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.
