Fluid-fluid demixing and density anomaly in a ternary mixture of hard spheres
FFluid-fluid demixing and density anomaly in a ternary mixture of hard spheres
Nathann T. Rodrigues ∗ and Tiago J. Oliveira † Departamento de F´ısica, Universidade Federal de Vi¸cosa, 36570-900, Vi¸cosa, MG, Brazil (Dated: May 18, 2020)We report the grand-canonical solution of a ternary mixture of discrete hard spheres defined on aHusimi lattice built with cubes, which provides a mean-field approximation for this system on thecubic lattice. The mixture is composed by point-like particles (0NN) and particles which excludeup to their first (1NN) and second neighbors (2NN), with activities z , z and z , respectively. Oursolution reveals a very rich thermodynamic behavior, with two solid phases associated with theordering of 1NN ( S
1) or 2NN particles ( S RF ) and theother characterized by a dominance of 0NN particles ( F F ) phases are indistinguishable. Discontinuous transitions are observed betweenall the four phases, yielding several coexistence surfaces in the system, among which a fluid-fluidand a solid-solid demixing surface. The former one is limited by a line of critical points and a lineof triple points (where the phases RF - F S F - S F - S S S F - S F - S I. INTRODUCTION
In two recent works [1, 2], we have investigated threeathermal binary mixtures of hard spheres placed on thecubic lattice, where they are approximated by k NN parti-cles, i.e., particles which exclude up their k th neighbors.The mixtures considered were composed by pairs of thethree smallest “spheres” (0NN-1NN, 0NN-2NN and 1NN-2NN) and grand-canonical phase diagrams for these sys-tems were obtained through their solution on a Husimilattice built with cubes [1, 2] [see Fig. 1(a)]. Such mean-field solutions display very rich entropy-driven thermody-namic behaviors. In a brief, in the 0NN-1NN case, a fluid( F ) and a solid ( S
1) phase (associated with the orderingof 1NN particles) were found, separated by a continuousand a discontinuous transition line, both meeting at atricritical point [1]. The same phases and behavior wereobserved in the 1NN-2NN mixture, but now another solid( S
2) phase (associated with the ordering of 2NN parti-cles) is present in the system and it is separated from the F and S F - S RF ) and another characterized by adominance of 0NN particles ( F S RF - F RF - S F S ∗ [email protected] † [email protected] minima in isobaric curves of the total density of parti-cles. These different behaviors in the binary systems —specially the absence of the density anomaly in the 1NN-2NN case and the existence of the fluid-fluid demixingonly the 0NN-2NN mixture — lead us to inquiry howthese things are connected in the more general and in-teresting case of the 0NN-1NN-2NN ternary mixture. Inthis paper we address this by tackling the challengingproblem of building up the three-dimensional (3D) phasediagram for this ternary model, once again by solving iton a cubic Husimi lattice.We remark that ternary mixtures are central in avast number of systems. To name a few, we may citeternary quasicrystals [3], several chemical reactions in-volving three species (e.g., 2H + O (cid:11) O), mixturesof cholesterol and lipids which play a key role in cell mem-branes [4, 5], water-oil-surfactant mixtures [6, 7], and soon. Moreover, colloids and granular matter in general areusually modeled as hard spheres [8]. In fact, (thermal)models of ternary mixtures have recently been used toinvestigate, e.g., critical Casimir forces among colloidalparticles dispersed in a binary solvent [9, 10] and the per-colation of patchy colloids [11, 12]. Regarding the ather-mal case, ternary mixtures of hard spheres have beentheoretically investigated using different approximationmethods [13–16] and in several works such approacheswere compared with both simulations [17–23] and exper-iments [24–29].Despite these works considering the hard spheres inthe continuous space, as long as we know, ternary mix-tures of them on the lattice (i.e., of k NN particles) havenever been investigated so far. Actually, even binary mix-tures of k NN particles on the cubic (or any other three-dimensional) lattice have never be studied before our re-cent works discussed above, based on the cubic Husimilattice solution [1, 2]. On the other hand, the 0NN-1NN a r X i v : . [ c ond - m a t . s o f t ] M a y mixture has been considered by several authors on thetriangular [30–32] and square lattice [33–35], beyond amean-field solution on the Bethe lattice [36]. A centraltopic in these studies is the possibility of a fluid-fluiddemixing transition in k NN-binary mixtures. We remarkthat in nonadditive mixtures, as is the case in k NN ones,the demixed phases fill the space more effectively thanthe mixed one, so that a demixing is expected; and it hasindeed been observed in several systems [37–47]. In the k NN systems, while van Duijneveldt and Lekkerkerker[31] claimed to have found a fluid-fluid demixing in the0NN-1NN mixture on the triangular lattice, strong evi-dence against this have been presented by other authors[30, 32], who observed only fluid-solid transitions in thismodel, similarly to what is found for this mixture on thesquare [33–35] and hierarchical lattices [1, 36].Therefore, it seems that among all k NN-binary mix-tures investigated so far the 0NN-2NN case on the cu-bic Husimi lattice is the single one presenting a sta-ble fluid-fluid demixing [2]. Hence, it is quite interest-ing to analyze what happens with this transition when1NN particles are introduced in this system. As will bedemonstrated below, the RF - F F - S II. MODEL DEFINITION AND SOLUTION ONTHE HUSIMI LATTICE
The model considered here consists of a ternary mix-ture of hard spheres, with diameter λ , defined on — andcentered at the vertices of — a cubic lattice. By assumingthe lattice spacing being a , the smallest (0NN) particlescorresponds to spheres of diameter λ = a , so that theyoccupy a single lattice site and effectively do not interactwith each other. The intermediate (1NN) and largest(2NN) particles are spheres of diameters λ = √ a and λ = √ a , respectively, which thus exclude up to theirnearest and next-nearest neighbors of being occupied byother particles. Let us remark that in our previous workson binary mixtures [1, 2], we have wrongly stated thatthe 1NN and 2NN particles would correspond to cubes oflateral sizes λ = √ a and λ = √ a , respectively, insteadof spheres . An activity z k is associated with each k NNparticle (with k = 0 , ,
2) in our grand-canonical study ofthe model. It is worthy recalling that while the pure 0NN model can be exactly solved and it does not present anyphase transition, the pure 1NN and 2NN models on thecubic lattice are known to undergo a continuous [48–51]and a discontinuous [51] phase transition, respectively,from disordered fluid to ordered solid phases.The translational symmetry breaking of these solidphases display sublattice ordering, as illustrated in Figs.1(b) and 1(c). In the ground state (the full occupancylimit) the solid phase for the pure 1NN system (the S S A i , B i , C i or D i ,with i = 1 ,
2, in Fig. 1(d)], so that its ground state isfour-fold degenerated. Thereby, to correctly capture thesymmetries of both solid phases in the 0NN-1NN-2NNmixture, we have to deal with eight sublattices, as de-fined in Fig.1(d).Following our recent studies on the corresponding bi-nary mixtures [1, 2], here we will investigate the ternary0NN-1NN-2NN case by defining the model on a Husimilattice built with cubes [see Fig. 1(a)]. It is importantto remark that the Husimi lattice consists of an infiniteCayley tree where each vertice is replaced by a polygonor a polyhedron. So, since the dimension of such lattice isinfinity, the critical transition behavior is lead by mean-field exponents. Despite this, solutions on the Husimilattice carry some degree of correlation of the system onthe relevant lattice (the cubic lattice here), usually giv-ing better results than other mean-field methods [52]. Infact, in some lattice gas systems, the Husimi solutioneven presents some quantitative agreement with simula-tion results on regular lattices [53, 54].To solve the ternary mixture on the Husimi lattice, wedefine a root site on an elementary cube [as shown in Fig.1(a)] and partial partition functions (ppf’s) according toits state. For each sublattice, the root site (and any otherlattice site as well) can be empty ( j = ∅ ), occupied bya 0NN ( j = 0), by an 1NN ( j = 1) or by a 2NN particle( j = 2), totaling 4 states [see Fig. 1(e)]. Hence, sincewe have 8 sublattices, there are 32 ppf’s for this system.One cube defines the 0-generation of the hierarchical lat-tice; so, by attaching the root sites of seven cubes to thevertices of such cube, with exception of its root site, oneobtains a subtree with 1-generation. Then, by attach-ing seven of such subtrees to the vertices of a new cubewith exception of its root site, a 2-generation subtree isbuilt. By repeating this process, we can create a ( M +1)-generation subtree from seven ones with M generations.Then, by summing over all possibilities of creating suchsubtree, by appropriately taking into account the parti-cle exclusions, with its root site kept fixed in the state s and sublattice g , one obtains a recursion relation for theppf g (cid:48) s as function of all ppf’s in the previous generation g j , with g = a , a , ..., d , d and s, j = ∅ , , ,
2. Somedetails on such recursion relations (RRs) are presentedin the appendix.
FIG. 1. (a) Illustration of part of a Husimi lattice built with cubes. The ground states of the solid S S j ) for the root sites. They can be empty ( j = ∅ ) or occupied by a0NN ( j = 0, open square), an 1NN ( j = 1, full square) or a 2NN ( j = 2, circle) particle. Since we are interested in infinite subtrees, and theRRs usually diverges in this thermodynamic limit, wewill work with ratios of them, defined as G j = g j g ∅ , (1)where G j = A ,j , A ,j , ..., D ,j , D ,j and j = 0 , , z , z , z )where Λ <
1, while Λ = 1 defines its spinodal.Similarly to the ppf’s, we can obtain the partition func-tion, Y , of the model on the tree by summing over allpossibilities of attaching the root sites of eight subtreesto a central cube. It can be written in a compact formin terms of the ppf’s, e.g., as Y = a , ∅ a (cid:48) , ∅ + z a , a (cid:48) , + z a , a (cid:48) , + z a , a (cid:48) , (2)= a , ∅ b , ∅ c , ∅ d , ∅ a , ∅ b , ∅ c , ∅ d , ∅ y, where y is a function of the ratios (Eq. 1) and the activ-ities z , z and z . Then, the density of k NN particles inthe sublattice A at the central cube is given by ρ ( A ) k = A ,k Y ∂Y∂A ,k , (3)with k = 0 , ,
2. A similar equation holds for the othersublattices, by replacing A with A , B , . . . , or D . Oncewe have the density of a k NN particle in all sublattices, its total density is ρ k = ρ ( A ) k + ρ ( A ) k + ... + ρ ( D ) k + ρ ( D ) k .From the partition function we can also calculate the bulkfree energy per site. Following the ansatz proposed byGujrati [52], and discussed in detail for a Husimi latticebuild with cubes in [2], for the ternary mixture one has φ b = −
18 ln (cid:20) A , ∅ B , ∅ C , ∅ D , ∅ A , ∅ B , ∅ C , ∅ D , ∅ y (cid:21) , (4)where A , ∅ ≡ a (cid:48) , ∅ b , ∅ c , ∅ d , ∅ a , ∅ b , ∅ c , ∅ d , ∅ , (5)and the other ratios can be obtained by the sublatticepermutation scheme described in the appendix. The bulkfree energy is handy to determine where a first-ordertransition takes place. In a region where two or morephases have Λ <
1, the equality of their bulk free ener-gies determines the point, line or surface of coexistenceamong these phases.
III. THERMODYNAMIC BEHAVIOR OF THEMODELA. Summary of results for the binary mixtures
Once the binary mixtures 0NN-1NN ( z = 0), 0NN-2NN ( z = 0) and 1NN-2NN ( z = 0) are the bound-ary planes of the ( z , z , z ) space for the ternary case,it is interesting to start the presentation of results bydiscussing in detail the thermodynamic behavior of suchplanes [1, 2].For the 0NN-1NN mixture — the plane ( z , z ,
0) —we have found two thermodynamic stable phases [1]:a disordered fluid ( F ) phase, where the recursion rela-tions (RRs) for the ratios assume a homogeneous solu-tion A ,j = B ,j = . . . = C ,j = D ,j with j = 0 , , S A i,j = B i,j = C i,j = D i,j = r i,j , with i = 1 , j = 0 , ,
2, and r ,j > r ,j for j = 0 ,
1, while r , < r , ,when the sublattices with index 1 are the ones more oc-cupied. Namely, in this phase one has ρ ( A ) j = · · · = ρ ( D ) j > ρ ( A ) j = . . . = ρ ( D ) j , for j = 0 , j = 2. [Note that these fixed points and densitiesare for these phases in the ternary case; and obviously ρ ( X )2 = 0, ∀ sublattice X , in the 0NN-1NN mixture.] Forsmall z there is a continuous F - S z such transition becomes discontinuous. The crit-ical and the coexistence F - S z , z , z ) T C = (0 . , . ,
0) [seeFig. 2]. Interestingly, this mixture presents a thermo-dynamic anomaly, characterized by minima in isobariccurves of the total density of particles as function of oneactivity ( z or z ). A line of minimum density (LMD)exists within the fluid phase, starting at z ≈ .
20 for z → F phase is metastable (see Fig. 2).A similar anomaly has also been observed in the 0NN-2NN mixture — the plane ( z , , z ). In this case, theLMD starts at z ≈ .
333 for z → RF ) phase (whose RRs have the symmetry justmentioned for the F phase), the 0NN-2NN mixture dis-plays another stable disordered fluid phase characterizedby a dominance of 0NN particles, reason for which it wasbaptized as the F A ,j = B ,j = . . . = C ,j = D ,j = r j in the F r ≈ r ≈ r ≈
0, while in RF case the values of r j strongly depend on the activi-ties. These very same symmetries apply for the particledensities: while in RF phase ρ , ρ and ρ considerablyvary with z , z and z , in the stable region of the F ρ (cid:29) ρ and ρ (cid:29) ρ . Beyond thesetwo fluid phases, the solid S A ,j = A ,j > B ,j = C ,j = . . . = C ,j = D ,j , for j = 0 , ,
2, when the sublattices A and A are the morepopulated ones. This yields densities ρ ( A ) j = ρ ( A ) j >ρ ( B ) j = . . . = ρ ( D ) j , for j = 0 , ,
2. A stable fluid-fluiddemixing is observed in this system, whose discontinu-ous RF - F z , z , z ) CP = (0 . , , . z < z ,CP the RF and F F ) phase. The S z and it is sep- FIG. 2. Phase diagrams for the binary mixtures in the( z , z , z ) space. The solid and dashed lines are continu-ous and discontinuous transition lines between the indicatedphases they are separating. The squares are triple points, thecircles are tricritical points and the star is a critical point.The dotted lines are the LMDs. arated from the two fluid phases by discontinuous tran-sition lines. The three coexistence lines RF - F RF - S F S z , z , z ) T P = (0 . , , . , z , z ) —, we have found the fluid F and the two solid S S F - S z , z , z ) T C = (0 , . , . S S z and z ,respectively, and are separated by a discontinuous tran-sition line. Another coexistence line exists between the F - S F - S S S z , z , z ) T P = (0 , . , . S , z , z ) and ( z , , z )strongly suggests the existence of a coexistence fluid- S F - S z , z ,
0) and (0 , z , z ) stronglyindicates that a critical and a coexistence F - S z , z ,
0) and ( z , , z ), sincethe values of z where they start in each plane do not co-incide. So, this could indicate either a complex scenariowith two (or more) surfaces of minimum density, or thatone or both of such surfaces do not exist. In the sametoken, it is not possible to known at this point whetherthe fluid-fluid demixing is a particular feature of the case z = 0, or if it extends for z >
0, giving rise to a RF - F RF - F S B. The Fluid-Fluid transition
From all thermodynamic properties observed in the bi-nary mixtures discussed above, the fluid-fluid demixingtransition in the 0NN-2NN ( z = 0) system is the mostsurprising one, in face it is absence in other k NN mixturesstudied so far. For this reason, we will start the buildingup of the 3D phase diagram for the ternary mixture byanalyzing the RF - F z . Some rel-evant examples of such slices are shown in Fig. 3. Forsmall z , specifically for z < . z = 0 . z = 0 is found, with the phases RF , F S RF - F RF - S F S RF - F S RF - F z = 0 case, being afeature of the ternary mixture.Quantitatively, we observe that while the ( z , z ) coor-dinates of the TP line mildly change with z , the z onefor the CP line present a strong variation with z , leadingthe stable region of the RF - F z increases. For instance, the RF - F z ≈ .
14 for z = 0, but only∆ z ≈ .
76 for z = 0 .
1. In fact, by increasing z one ob-serves that the CP line becomes closer to the TP line, andat the special point ( z , z , z ) ∗ = (0 . , . , . stable fluid-fluid demixingtransition disappears at this point. This gives rise to thephase diagram displayed in Fig. 3(b), for z = z ∗ , withthe (now completely metastable) RF - F RF - S z < z ∗ now becomes a F - S F S z > z ∗ , one still finds the RF - F z = 1 .
1, but now this line is also metastable,occurring inside the region where the S z gives place to a simple F - S z > z ∗ the fluid-fluiddemixing transition becomes preempted by the fluid-solidtransition. Hence, at the special point the CP line be- z z S2 (a) RF F0Fluid z z S2 (b) RF F0Fluid z z S2 (c) RF F0FluidS1
FIG. 3. Phase diagrams for fixed (a) z = 0 .
10, (b) z =0 . z = 1 .
10. In all panels, the dashed linesrepresent stable coexistence lines between the phases they areseparating, while the dashed-dotted lines are the metastable RF - F F - S RF - F S RF - F comes metastable and the RF - F S FIG. 4. Part of the 3D phase diagram for the ternary mixturein the activity space, highlighting the region around which thefluid-fluid demixing surface [the shaded (orange) one in theplot] takes place. All the surfaces presented here (separatingthe phases RF - or F - S F S RF - F
0) are coexistencesurfaces. The triangle is the special point, where the CP linemeets the RF - F S z , , z )plane. verify how further the RF - F z increases. First, we notice that, regardless the value of z , the metastable part of this surface is always limitedfrom above, i.e., it ends when it meets the spinodal of the RF phase, as shown in Figs. 5(a) and 5(b). Second, forlarge z the S z and z , as seen in Fig. 3(c),and the fluid phase let to be stable in such region. Thisis clearly seen by comparing the phase diagram in Fig.5(c) with the spinodals of the fluid phases in Fig. 5(b).Curiously, by increasing z the spinodal of the fluid phasedevelops a cusp, which approximates of the (metastable)CP line as z increases, see Fig. 5(a) for z = 2 .
3. Suchapproximation occurs until z ≈ . RF and F z = 2 .
5, the stability region of RF phase becomesnow limited to a closed domain of the ( z , z ) phase dia-gram, and the (metastable) RF - F F RF phase. Therefore, the CP line endsat z ≈ . z . However, by increasing z , oneobserves that the domain of the RF stability shrinks,yielding a decreasing in the in RF - F z = 5 . C. The phase diagram for the ternary mixture
From the results in the previous subsection, one knowsthat for z < z ∗ the 3D phase diagram presents threecoexistence surfaces ( RF - F RF - S F S RF - F S z > z ∗ , but not so large, this z z FluidF0 (a) RF z z (b) RF F0 z z (c) S2 F0S1
FIG. 5. Spinodals of the disordered fluid phases (solid lines)and the metastable RF - F z = 2 . z = 2 .
5. The lineswith triangles, circles and crosses are the RF , F RF and F z = 2 . z = 2 . behavior gives place to a simple F - S F - S z fixed (andlarge enough), giving rise to a critical F - S z z F S1S2 (a) z z F S1S2 (b)
FIG. 6. Phase diagrams for fixed (a) z = 0 .
20 and (b) z = 1 .
20. The solid and dashed lines represent critical andcoexistence lines, respectively, between the indicated phasesthey are separating. The black squares and the circle denotethe TP and the TC point, respectively. the 3D space, as expected from the behavior of the bi-nary mixtures. Such surface starts at z = 0 . z ) of the F - S z , z , F - S z , asconfirmed in Fig. 5(c). These features are better seenin slices of fixed and small z , as the one in Fig. 6(a)for z = 0 .
2, where one finds a phase diagram similar tothe one for z = 0, with a continuous and a discontin-uous F - S F - S z or z , we have accurately determinedthe TC line, which connects the two tricritical pointsfound in the planes ( z , z ,
0) and (0 , z , z ). Interest-ingly, it presents a complex non-monotonic behavior, at-taining a maximum point, with respect to z and z , at( z , z , z ) T C,max = (0 . , . , . F - S FIG. 7. Part of the 3D phase diagram for the ternary mix-ture in the activity space, highlighting the region where thecritical F - S F - S F - S S depicted in Fig. 7, which shows this part of the 3D phasediagram in detail.The z -slices, as those in Figs. 6(a) and 6(b), alsoconfirm the presence of the discontinuous F - S S S
2) demixing in the ternary mixture. Moreover, onealways observes that the F - S F - S S S F - S S , z , z ) plane] and extends to z , z , z → ∞ ; its be-havior is also shown in Fig. 7. It is important to noticethat, since the TC line (and then the critical F - S z , for large values of thisactivity everything we find in this part of the 3D phasediagram are the F - S F - S S S F - S S z , z , z → ∞ can be justified by the existence of a F S
1, a F - S S S A i, = B i, = C i, = D i, = 1 and A i,j = B i,j = C i,j = D i,j = 0 for i = 1 , j = 1 , S A ,j = B ,j = C ,j = D ,j = 1for j = 0 ,
1, while all others RRs vanish, when the sub-lattices indexed by 1 are the more populated ones. In the S A and A are the ones morepopulated, one has A i,j = 1 and B i,j = C i,j = D i,j = 0,for i = 1 , j = 0 , ,
2. Using these solutions, it issimple to calculate the free energies in this limit, being φ ( F ) = (1 + z ) , φ ( S = (1 + z + z ) and φ ( S =(1 + z + z + z ) . By making φ ( F ) = φ ( S = φ ( S andsolving in terms of z , one finds z = z + z (6) FIG. 8. Phase diagram for the ternary mixture in the activityspace, indicating all phases and transition surfaces and lines,following the same scheme of symbols and colors from theprevious figures. and z = z + 4 z + 5 z + 2 z . (7)Therefore, as z → ∞ this TP line behaves has z ≈ z and z ≈ z . This result can be understood as follows:at full occupancy an 1NN particle effectively occupies thevolume of two 0NN ones, while a 2NN particle occupiesthe volume of four 0NN ones, in agreement with the ex-ponents above. The calculated free energies also allowus to determine the behavior of the F - S F - S S S z , z , z → ∞ ,where one finds z ≈ z and z ≈ z for the F - S F - S z , these surfacesshall appear as straight lines at constant z and z , re-spectively. The S S z ≈ ( z + z ) , being a quadratic func-tion of z ( z ) for fixed z ( z ). In fact, these results aresomewhat confirmed in Fig.6(b) for z = 1 .
2, where wesee that the F - S F - S S S z and z . Results [not shown]for slices for much larger values of z indeed confirm thecorrectness of the predicted surfaces.Figure 8 presents the complete 3D phase diagram inthe space of activities, summarizing all the features dis-cussed so far. In face of its complexity — it has a fluid-fluid, a solid-solid and several fluid-solid coexistence sur-faces, with three of them ( F - S F - S
2, and S S
2) ex-tending to z , z , z → ∞ , beyond a critical fluid-solidsurface, a CP line, a TC line and two TP lines — it isa bit hard drawing this phase diagram in an intelligibleway. Anyhow, it is important to let clear how the par-tial diagrams depicted in Figs. 4 and 7 connects, whatare the sizes of the transition surfaces (at least the oneswhich exist in limited domains) and so on.At this point, it is important to stress that we havecarefully looked for new phases in the general case, butonly the four phases already present in the binary mix-tures were found. FIG. 9. Surfaces of minimum density in the activity space.The dash-dotted (green), continuous (black) and dashed (red)lines are the z -, z - and z -SMD, respectively. The doted(black) line is the LMD where all three SMDs seems to end.The shaded (blue) surface is part of the F - S D. The surfaces of minimum density
As discussed in subsection IIIA, two lines of minimumdensity (LMDs) were found in the binary mixtures insidethe fluid phase , one in the 0NN-1NN ( z = 0) and otherin the 0NN-2NN ( z = 0) case. In the former system,the LMD starts at z ≈ .
333 when z →
0, while inthe last one it starts at z ≈ .
20 when z →
0, and inboth cases they end at the spinodal of the fluid phase,inside regions where the corresponding solid phases aremore stable than the fluid [1, 2]. The difference betweenthese starting points suggests a complex scenario for theternary mixture with at least two surfaces of minimumdensity (SMDs). Before discussing them, however, itis important to remark that such starting points werecalculated through a limiting process. For example, forthe 0NN-1NN mixture, we located the ( z , z ) coordinatewhere the total density — defined as ρ T = ρ + 2 ρ + 4 ρ — presents a minimum for a given pressure, P [ ? ], i.e.,along an isobaric curve of ρ T × z (or z ) (see, e.g., Fig.5 in Ref. [2]). Then, by varying P we can build up theLMD curve and determine it as close as we want of the z axis by making z →
0. Exactly on the z -axis, how-ever, one has ρ T = z / (1 + z ), so that no anomaly existsthere.Then, if we take slices of fixed z >
0, one still findsLMDs in the way just described, forming a SMD in the3D space. Such z -SMD (since it is obtained for fixedvalues of z ) starts at the LMD in the plane ( z , z , z , , z ), using the limiting process we canobtain it as close as we want of this plane, so that thereexists a LMD for z → z , a second SMD (the z -SMD) is obtained,which starts in the LMD on the plane ( z , , z ) and alsoends at the spinodal of the fluid phase in the metastableregion. In this case, the SMD does not exist exactlyin the plane ( z , z , z →
0. This z -SMD is also shown in Fig. 9.The existence of these two SMDs indicates, for sake ofcompleteness, the existence of a third one for fixed z .As shown in Fig. 9, such z -SMD indeed exists in the3D space and, albeit it cannot be observed exactly in theplanes ( z , z ,
0) and ( z , , z ), it can be obtained in thelimits z → z →
0, respectively. In such limits,the starting point of the z -SMD is at z ≈ .
50 and,similarly to the other SMDs, it also ends in the spinodalof the fluid phase. As observed in Fig. 9, the threeSMDs become quite close for large z and they seem tobe limited from above by a single line. However, it isquite hard to assure this numerically and it may be thecase that they end at different (but very close) lines, orbecome a single surface before this. IV. CONCLUSIONS
We have determined the thermodynamic behavior of aternary mixture of hard spheres, defined on the cubic lat-tice, composed by point-like particles (0NN) and particleswhich exclude up to their first (1NN) and second (2NN)neighbors. We treat this model by solving it on a Husimilattice built with cubes, where four phases were found inthe phase diagram, being two fluid and two solid ones,separated by several coexistence surfaces and a criticalsurface, which end or meet in lines of critical, tricriticaland triple points. One of the disordered phases is a reg-ular fluid ( RF ), in the sense that the particle densities( ρ , ρ and ρ ) strongly depends on the activities, whilein the second disordered fluid ( F
0) phase one always has ρ (cid:29) ρ and ρ (cid:29) ρ . The solid S S
2) phase is fea-tured by a sublattice ordering of 1NN (2NN) particles.We notice that columnar and smectic phases, observedfor hard cubes on the cubic lattice [55, 56], are absent inour solution. Although this can be due to the hierarchi-cal structure of the Husimi lattice, we remark that thesephases have never been found in previous studies of thepure 1NN and 2NN models on the cubic lattice [48–51],strongly suggesting that they should indeed not appearin their mixture.A fluid-fluid ( RF - F
0) demixing surface is observed inthe system, which is limited from below [in the ( z , z , z )space] by a line of critical points (CP), whose coordi-nate z ,c increases fast with z ,c , while the CP line andthe whole RF - F z . This shows that, by increasing ρ , a large ρ is need to yield the demixing, confirming that the fluid-fluid transition is not a feature restricted to the 0NN-2NNmixture, but the presence of the 1NN particles turns itsappearance more difficult. In fact, although the demixingsurface extends for a large portion of the phase diagram, it is stable only in a small area, when compared withthe other transition surfaces. Such stable part is limitedfrom above by the RF - F S z , z , z ) ∗ = (0 . , . , . F - S z > z ∗ ,corresponding to a very small density of 1NN particles: ρ > ρ ∗ = 0 . k NN mixtures with larger k ’s are necessary toconfirm this.We have also observed an anomaly in isobaric curves ofthe total density of particles (versus z i , with i = 0 , , z , , z )and ( z , z ,
0) by making z → z →
0, respectively,and ends at the spinodal surface of the fluid phase. Forvery large z , deep inside the metastable fluid region,these three surfaces either become quite close or collapseinto a single surface. Although it is hard to decide thisnumerically, it is simple to figure out that this occursbecause in this region one has ρ (cid:29) ρ and ρ (cid:29) ρ , sothat ρ T ≈ ρ independently of the activity being fixed.We remark that density anomalies are typically found incomplex polymorphic fluids, whose modeling is usuallyvery elaborated (as is the case, e.g., in lattice gases withdirectional interactions) [57]. Therefore, the existenceof this kind of anomaly in the simple athermal systemanalyzed here indicates that mixtures of hard spheres(and hard disks as well [35]) might be useful as a startingpoint to understand anomalies in more complex fluids. ACKNOWLEDGMENTS
We acknowledge support from CNPq, CAPES andFAPEMIG (Brazilian agencies).
Appendix A: Recursion relations for the ternarymixture
The recursion relations (RRs) for the partial partitionfunctions (ppf’s) of the 0NN-1NN-2NN mixture were ob-tained using the method described in Sec. II. For theroot site in the sublattice A , they are given by0 a (cid:48) , ∅ = b , ∅ c , ∅ d , ∅ a , ∅ b , ∅ c , ∅ d , ∅ { (1 + z B , )(1 + z C , )(1 + z D , )(1 + z A , )(1 + z B , )(1 + z C , ) (A1a)(1 + z D , ) + z ( B , + C , + D , + A , + B , + C , + D , )+ z [ B , ( D , + A , + B , + C , ) + C , ( B , + C , + D , ) + D , ( A , + C , + D , ) + A , C , + B , D , ]+ z [ B , D , ( A , + C , ) + B , A , C , + C , B , D , + D , A , C , ]+ z B , D , A , C , + z z [ B , ( D , + A , + B , + C , ) + C , ( B , + C , + D , )+ D , ( B , + A , + C , + D , ) + A , ( B , + D , + C , ) + B , ( B , + C , + D , )+ C , ( B , + C , + D , + A , ) + D , ( C , + D , + B , )]+ z z [ B , D , ( A , + B , + C , ) + B , A , ( B , + C , ) + B , B , C , + C , B , ( C , + D , )+ C , C , D , + D , B , ( A , + C , + D , ) + D , A , ( C , + D , ) + D , C , D , + A , B , ( D , + C , ) + A , D , C , + B , B , ( C , + D , ) + B , C , D , + C , B , ( C , + D , + A , ) + C , C , ( D , + A , ) + C , D , A , + D , C , ( D , + B , ) + D , D , B , ]+ z z [ B , D , A , ( B , + C , ) + B , D , B , C , + B , A , B , C , + C , B , C , D , + D , B , A , ( C , + D , ) + D , B , C , D , + D , A , C , D , + A , B , D , C , + B , B , C , D , + C , B , C , ( D , + A , ) + C , B , D , A , + C , C , D , A , + D , C , D , B , ]+ z z ( B , D , A , B , C , + D , B , A , C , D , + C , B , C , D , A , )+ z z [ B , D , ( A , + C , ) + B , A , ( D , + C , ) + B , C , ( D , + A , ) + C , B , D , + C , D , B , + D , A , ( B , + C , ) + D , C , ( B , + A , ) + A , C , ( B , + D , ) + B , D , C , ]+ z z ( B , D , A , C , + B , A , D , C , + B , C , D , A , + D , A , B , C , + D , C , B , A , + A , C , B , D , ) + z z ( B , D , A , C , + B , D , C , A , + D , A , C , B , + B , A , C , D , ) + z ( B , + C , + D , + A , + B , + C , + D , )+ z ( B , B , + C , C , + D , D , ) + z z ( B , B , + C , C , + D , D , + B , B , + C , C , + D , D , )+ z z ( B , B , + C , C , + D , D , + B , B , + C , C , + D , D , ) } a (cid:48) , = b , ∅ c , ∅ d , ∅ a , ∅ b , ∅ c , ∅ d , ∅ { (1 + z B , )(1 + z C , )(1 + z D , )(1 + z A , )(1 + z B , )(1 + z C , ) (A1b)(1 + z D , ) + z ( C , + A , + B , + D , ) + z ( C , B , + C , D , + B , D , ) + z C , B , D , + z z [ C , ( B , + C , + D , ) + A , ( B , + D , + C , ) + B , ( B , + C , + D , ) + D , ( C , + D , + B , )]+ z z [ C , B , ( C , + D , ) + C , C , D , + A , B , ( D , + C , ) + A , D , C , + B , B , ( C , + D , ) + B , C , D , + D , C , ( D , + B , ) + D , D , B , ]+ z z ( C , B , C , D , + A , B , D , C , + B , B , C , D , + D , C , D , B , )+ z z ( C , B , D , + C , D , B , + B , D , C , ) + z A , } a (cid:48) , = b , ∅ c , ∅ d , ∅ a , ∅ b , ∅ c , ∅ d , ∅ { z ( C , + A , + B , + D , ) + z [ C , ( A , + B , + D , ) (A1c)+ A , ( B , + D , ) + B , D , ] + z [ C , A , ( B , + D , ) + C , B , D , + A , B , D , ]+ z C , A , B , D , + z ( C , + A , + B , + D , ) + z ( C , ( B , + D , ) + B , D , )+ z C , B , D , + z z [ C , ( B , + D , ) + B , ( C , + D , ) + D , ( C , + B , )]+ z z ( C , B , D , + B , C , D , + D , C , B , )+ z z ( C , B , D , + C , D , B , + B , D , C , ) + z A , } a (cid:48) , = b , ∅ c , ∅ d , ∅ a , ∅ b , ∅ c , ∅ d , ∅ (1 + z A , + z A , + z A , ) (A1d)where A i,j , B i,j , C i,j and D i,j , with i = 1 , j =0 , ,
2, are the ratios defined in Eq. 1. The RRs forthe other sublattices can be obtained from cyclic permu-tations of the sublattice labels: A → B , B → C , C → D , D → A , together with C → D , D → A , A → B , B → C . Following this order, we can findthe RRs for the sublattices D , C and B . Those forsublattices A , B , C and D can be obtained from the1ones for A , B , C and D , respectively, by exchang-ing all the sublattice indexes i (1 ↔ [1] N. T. Rodrigues and T. J. Oliveira, Phys. Rev. E ,032112 (2019).[2] N. T. Rodrigues and T. J. Oliveira, J. Chem. Phys. ,024504 (2019).[3] Quasicrystals: Structure and Physical Properties , editedby H.-R. Trebin (Willey, 2006).[4] R. Elliott, I. Szleifer, and M. Schick, Phys. Rev. Lett. ,098101 (2006).[5] D. W¨ustner and K. Solanko, Biochim. Biophys. Acta , 1908 (2015).[6] S. Abbott, Surfactant Science: Principles & Practice (Destech Publications, 2017).[7] W. F. C. Sager. Microemulsion Templating. In:
Nanos-tructured Soft Matter: Experiment, Theory, Simulationand Perspectives , edited by A. V. Zvelindovsky, Springer,2007, p. 3-44.[8]