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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>M</a:mi><a:mo>+</a:mo><a:mn>1</a:mn></a:mrow></a:math> phases, as would normally be allowed by Gibbs' phase rule at generic temperature in a mixture of <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>M</b:mi></b:math> molecular species, can be made to coexist in equilibrium. While such “super-Gibbs” phase coexistence is possible in the grand-canonical ensemble by tuning interactions among the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>M</c:mi></c:math> species, there is no straightforward equivalence in the canonical ensemble: Only a subset of the grand-canonical phases will generically be realized. Here, we show that, upon further design of interface tensions, it is possible to stabilize a super-Gibbs number of phases also in the experimentally relevant canonical ensemble, thus effectively restoring equivalence to the grand-canonical one. Using a graph-theoretical approach, we determine a sufficient set of inequalities for the interfacial tensions for which all grand-canonical phases are realized. 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.