Phase stacking diagram of colloidal mixtures under gravity

The observation of stacks of distinct layers in a colloidal or liquid mixture in sedimentation-diffusion equilibrium is a striking consequence of bulk phase separation. Drawing quantitative conclusions about the phase diagram is, however, very delicate. Here we introduce the Legendre transform of the chemical potential representation of the bulk phase diagram to obtain a unique stacking diagram of all possible stacks under gravity. Simple bulk phase diagrams generically lead to complex stacking diagrams. We apply the theory to a binary hard core platelet mixture with only two-phase bulk coexistence, and find that the stacking diagram contains six types of stacks with up to four distinct layers. These results can be tested experimentally in colloidal platelet mixtures. In general, an extended Gibbs phase rule determines the maximum number of sedimented layers to be $3+2(n_b-1)+n_i$, where $n_b$ is the number of binodals and $n_i$ is the number of their inflection points.