3 spatial dimensions of pH, size (1/R), and K+(aq) chemical potential, and 1 time dimension of redox potential (E).

In this 4D phase diagram, we can:

• Two-phase coexistence is a 3D object (see a planar boundary moving in time).

• Three-phase coexistence is a 2D plane (a 1D line moving in time).

• 4-phase coexistence is a line (the 4-phase coexistence point moving in time)

• There is a 5-phase coexistence, which is at E = 0.53, where the 4-phase coexistence line terminates and meets another 4-phase coexistence line.

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

Jiadong Chen, Matthew J. Powell-Palm, Wenhao Sun

https://arxiv.org/abs/2404.05201