  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.
