  Modern materials are often synthesized or operated in complex chemical
environments, where there can be numerous elemental species, competing phases,
and reaction pathways. When analyzing reactions using the Gibbs free energy,
which has a natural variable of composition, it is often cumbersome to solve
for the equilibrium states of a complicated heterogeneous mixture of phases.
However, if one is interested only in the stability of a single target
material, it may be easier to reframe the boundary conditions around only the
target material-of-interest, with boundary conditions open to chemical exchange
with an external reservoir. The corresponding phase diagram would thus have a
chemical potential axis for the open volatile species, rather than a
composition axis. Here we discuss how to derive, compute, and interpret phase
diagrams with chemical potential axes, which are dual to the more common
composition phase diagram.
  In our ambition to construct high-dimensional phase diagrams featuring any
thermodynamic variable on its axes, here we examine the duality between
extensive and intensive conjugate variables in equilibrium thermodynamics. This
duality manifests from the distinction between closed and open boundary
conditions of a thermodynamic system, to the relationship between the Internal
Energy and its Legendre transformations, to the point-line duality in
calculating convex hulls versus half-space intersections. Here we focus on the
duality relationships of chemical work, with extensive composition variables,
N, and intensive chemical potentials, {\mu}. In particular, we explore mixed
composition-chemical potential diagrams for oxynitride synthesis, lithium-ion
cathode stability, and oxidation of high-entropy alloys. We further illustrate
how chemical potential diagrams reveal the driving forces for non-equilibrium
growth and dissolution kinetics.
