The geometry of high-dimensional phase diagrams: II. The duality between closed and open chemical systems

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.