The geometry of high-dimensional phase diagrams: III. Engineering relative stability in four dimensions

Designing thermodynamic conditions to improve (or reduce) the stability of a target material is a key task in materials engineering. For example, during materials synthesis one aims to enhance the stability of a target phase relative to its precursors or competing byproduct phases. If an undesired phase forms in experiment, one aims to destabilize the undesired phase by dissolution or corrosion. When multiple thermodynamic knobs are available to engineer relative stability, it can be difficult to navigate the corresponding high-dimensional phase diagram to identify optimal pathways to promote or destabilize a target phase. We propose that instead of mapping the absolute phase boundaries of a target material, we can invoke a generalized Clausius-Clapeyron relation, which provides a 'compass' to point out which directions on a high-dimensional phase diagram are best to stabilize or destabilize a target phase. Phase boundaries on high-dimensional phase diagrams are also high-dimensional objects, and can represent phase coexistence between numerous phases simultaneously. On a 2D temperature-pressure phase diagram, phase boundaries are 1D lines separating two phases, with a slope given by the Clausius-Clapeyron relation, dP/dT = {\Delta}S/{\Delta}V. Here, we derive a parametric form of the Clausius-Clapeyron relation that readily scales to high-dimensional phase boundaries. The gradient of a phase boundary guides us on how to increase or decrease the relative stability of a target compound, meaning this generalized Clausius-Clapeyron relation enables us to engineer relative stability with respect to multiple thermodynamic conditions simultaneously. Using this approach, we analyze the acid stability of manganese oxide catalysts on a 4-dimensional Pourbaix diagram with axes of pH, redox potential, nanoparticle size, and aqueous [K+] ion concentration.