Fluid-fluid demixing and density anomaly in a ternary mixture of hard spheres

We report the grand-canonical solution of a ternary mixture of discrete hard spheres defined on a Husimi lattice built with cubes, which provides a mean-field approximation for this system on the cubic lattice. The mixture is composed by point-like particles (0NN) and particles which exclude up to their first (1NN) and second neighbors (2NN), with activities $z_0$, $z_1$ and $z_2$, respectively. Our solution reveals a very rich thermodynamic behavior, with two solid phases associated with the ordering of 1NN ($S1$) or 2NN particles ($S2$), and two fluid phases, being one regular ($RF$) and the other characterized by a dominance of 0NN particles ($F0$ phase). However, in most part of the phase diagram these fluid ($F$) phases are indistinguishable. Discontinuous transitions are observed between all the four phases, yielding several coexistence surfaces in the system, among which a fluid-fluid and a solid-solid demixing surface. The former one is limited by a line of critical points and a line of triple points (where the phases $RF$-$F0$-$S2$ coexist), both meeting at a special point, after which the fluid-fluid coexistence becomes metastable. Another line of triple points is found, connecting the $F$-$S1$, $F$-$S2$ and $S1$-$S2$ coexistence surfaces. A critical $F$-$S1$ surface is also observed meeting the $F$-$S1$ coexistence one at a line of tricritical points. Furthermore, a thermodynamic anomaly characterized by minima in isobaric curves of the total density of particles is found, yielding three surfaces of minimal density in the activity space, depending on which activity is kept fixed during its calculation.