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We report the grand-canonical solution of a ternary mixture of discrete hardspheres defined on a Husimi lattice built with cubes, which provides amean-field approximation for this system on the cubic lattice. The mixture iscomposed by point-like particles (0NN) and particles which exclude up to theirfirst (1NN) and second neighbors (2NN), with activities $z_0$, $z_1$ and $z_2$,respectively. Our solution reveals a very rich thermodynamic behavior, with twosolid phases associated with the ordering of 1NN ($S1$) or 2NN particles($S2$), and two fluid phases, being one regular ($RF$) and the othercharacterized by a dominance of 0NN particles ($F0$ phase). However, in mostpart of the phase diagram these fluid ($F$) phases are indistinguishable.Discontinuous transitions are observed between all the four phases, yieldingseveral coexistence surfaces in the system, among which a fluid-fluid and asolid-solid demixing surface. The former one is limited by a line of criticalpoints and a line of triple points (where the phases $RF$-$F0$-$S2$ coexist),both meeting at a special point, after which the fluid-fluid coexistencebecomes 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 oftricritical points. Furthermore, a thermodynamic anomaly characterized byminima in isobaric curves of the total density of particles is found, yieldingthree surfaces of minimal density in the activity space, depending on whichactivity is kept fixed during its calculation.

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