Ensemble inequivalence in the design of mixtures with super-Gibbs phase coexistence

Designing the phase behavior of multicomponent mixtures is a rich area with many potential applications. One key question is how more than $M+1$ phases, as would normally be allowed by Gibbs’ phase rule at generic temperature in a mixture of $M$ molecular species, can be made to coexist in equilibrium. In the grandcanonical ensemble, such super-Gibbs phase equilibria can be realized by tuning the interactions among the $M$ species. This introduces $\sim M^2$ additional degrees of freedom and hence a superlinear number of phases that can coexist. We show that, surprisingly, there is no straightforward equivalence to the situation in the experimentally relevant canonical ensemble: here only a subset of the grandcanonical phases will generically be realized. This subset is determined by interfacial tensions in addition to bulk free energies. Using a graph-theoretical approach, we determine a sufficient set of inequalities for the interfacial tensions for which all grandcanonical phases are realized so that equivalence of ensembles is effectively restored. We illustrate the design method for a two-component mixture with four coexisting phases and point out the route for generalizing this to a higher number of components.

💡 Research Summary

The paper addresses the longstanding question of how to achieve coexistence of more than the Gibbs‑rule limit of M + 1 phases in a mixture of M components. In the grand‑canonical ensemble, where the chemical potentials of each component are fixed, the authors show that by treating the pairwise interaction strengths as design parameters one gains roughly M² additional degrees of freedom. Consequently, the maximum number of coexisting bulk phases can be pushed to n ≈ M², a result that follows directly from a generalized Gibbs‑phase rule: with D design parameters the total number of independent variables is M + D, and the pressure‑equality constraints reduce the freedom by n − 1, giving n = M + D + 1.

The novelty of the work lies in demonstrating that this “super‑Gibbs” coexistence does not automatically survive the transition to the canonical ensemble, where the total particle numbers of each species are fixed. In the canonical setting the parent composition ρ⁽⁰⁾ must be reproduced as a convex combination of the coexisting phase compositions ρ^(α) (the lever rule). Because all grand‑canonically designed phases share the same bulk free‑energy density and pressure by construction, the bulk contribution to the total free energy is identical for any admissible convex combination. Therefore, the selection of which set of phases actually appears at equilibrium is governed entirely by interfacial free‑energy contributions.

To incorporate interfacial effects the authors introduce a spatially varying composition field ρ(r) and write the total free energy as
F = ∫