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

Once the binary mixtures 0NN-1NN ($z_{2}=0$), 0NN-2NN ($z_{1}=0$) and 1NN-2NN ($z_{0}=0$) are the boundary planes of the $(z_{0},z_{1},z_{2}$) space for the ternary case, it is interesting to start the presentation of results by discussing in detail the thermodynamic behavior of such planes NT0NN2NN ; NT0NN1NN-1NN2NN .

For the 0NN-1NN mixture — the plane $(z_{0},z_{1},0)$ — we have found two thermodynamic stable phases NT0NN1NN-1NN2NN : a disordered fluid ($F$) phase, where the recursion relations (RRs) for the ratios assume a homogeneous solution $A_{1,j}=B_{1,j}=\ldots=C_{2,j}=D_{2,j}$ with $j=0,1,2$, and the solid $S1$ phase, associated with the ordering of the 1NN particles, where the RRs take the form $A_{i,j}=B_{i,j}=C_{i,j}=D_{i,j}=r_{i,j}$, with $i=1,2$ and $j=0,1,2$, and $r_{1,j}>r_{2,j}$ for $j=0,1$, while $r_{1,2}<r_{2,2}$, when the sublattices with index $1$ are the ones more occupied. Namely, in this phase one has $\rho_{j}^{(A_{1})}=\cdots=\rho_{j}^{(D_{1})}>\rho_{j}^{(A_{2})}=\ldots=\rho_{j}^{(D_{2})}$, for $j=0,1$ and the opposite for $j=2$. [Note that these fixed points and densities are for these phases in the ternary case; and obviously $\rho_{2}^{(X)}=0$, $\forall$ sublattice $X$, in the 0NN-1NN mixture.] For small $z_{0}$ there is a continuous $F$-$S1$ transition, while for large $z_{0}$ such transition becomes discontinuous. The critical and the coexistence $F$-$S1$ lines meet at a tricritical point, located at $(z_{0},z_{1},z_{2})_{TC}=(0.5958,1.1277,0)$ [see Fig. 2]. Interestingly, this mixture presents a thermodynamic anomaly, characterized by minima in isobaric curves of the total density of particles as function of one activity ($z_{0}$ or $z_{1}$). A line of minimum density (LMD) exists within the fluid phase, starting at $z_{0}\approx 0.20$ for $z_{1}\rightarrow 0$ and ending close to the tricritical point in a region where the $F$ phase is metastable (see Fig. 2).

A similar anomaly has also been observed in the 0NN-2NN mixture — the plane $(z_{0},0,z_{2})$. In this case, the LMD starts at $z_{0}\approx 0.333$ for $z_{2}\rightarrow 0$ and, as in the 0NN-1NN case, it ends inside the region where the fluid phase is metastable NT0NN2NN (see Fig. 2). Actually, beyond the regular fluid ($RF$) phase (whose RRs have the symmetry just mentioned for the $F$ phase), the 0NN-2NN mixture displays another stable disordered fluid phase characterized by a dominance of 0NN particles, reason for which it was baptized as the $F0$ phase in Ref. NT0NN2NN . The RRs are also featured by $A_{1,j}=B_{1,j}=\ldots=C_{2,j}=D_{2,j}=r_{j}$ in the $F0$ case, but with $r_{1}\approx 1$, $r_{2}\approx 0$ and $r_{3}\approx 0$, while in $RF$ case the values of $r_{j}$ strongly depend on the activities. These very same symmetries apply for the particle densities: while in $RF$ phase $\rho_{0}$, $\rho_{1}$ and $\rho_{2}$ considerably vary with $z_{0}$, $z_{1}$ and $z_{2}$, in the stable region of the $F0$ phase one has $\rho_{0}\gg\rho_{1}$ and $\rho_{0}\gg\rho_{2}$. Beyond these two fluid phases, the solid $S2$ phase is also present in the phase diagram, which is associated with the ordering of the 2NN particles and characterized by a fixed point with $A_{1,j}=A_{2,j}>B_{1,j}=C_{1,j}=\ldots=C_{2,j}=D_{2,j}$, for $j=0,1,2$, when the sublattices $A_{1}$ and $A_{2}$ are the more populated ones. This yields densities $\rho_{j}^{(A_{1})}=\rho_{j}^{(A_{2})}>\rho_{j}^{(B_{1})}=\ldots=\rho_{j}^{(D_{2})}$, for $j=0,1,2$. A stable fluid-fluid demixing is observed in this system, whose discontinuous $RF$-$F0$ transition line ends at a critical point (CP), located at $(z_{0},z_{1},z_{2})_{CP}=(0.6297,0,5.4243)$, so that for $z_{2}<z_{2,CP}$ the $RF$ and $F0$ phases cannot be distinguished. Hereafter, whenever this happens we will refer to these phases simply as the fluid ($F$) phase. The $S2$ phase is the most stable one for large $z_{2}$ and it is separated from the two fluid phases by discontinuous transition lines. The three coexistence lines $RF$-$F0$, $RF$-$S2$ and $F0$-$S2$ meet at a triple point (TP) located at $(z_{0},z_{1},z_{2})_{TP}=(0.6774,0,6.5671)$ [See Fig. 2].

Finally, in the 1NN-2NN mixture — the plane $(0,z_{1},z_{2})$ —, we have found the fluid $F$ and the two solid $S1$ and $S2$ phases NT0NN1NN-1NN2NN . Similarly to the 0NN-1NN mixture, continuous and discontinuous $F$-$S1$ transition lines are observed, which meet at a tricritical point, located at $(z_{0},z_{1},z_{2})_{TC}=(0,1.5273,2.5016)$. The solid $S1$ and $S2$ phases are observed in the regions of large $z_{1}$ and $z_{2}$, respectively, and are separated by a discontinuous transition line. Another coexistence line exists between the $F$-$S2$ phases, which meet the $F$-$S1$ and $S1$-$S2$ ones in a triple point, located at $(z_{0},z_{1},z_{2})_{TP}=(0,2.3102,7.8746)$ [See Fig. 2]. In contrast with the previous binary mixtures, no density anomaly was found in the 1NN-2NN case.

The summary of the behavior for the binary mixtures, displayed in Fig. 2, indicates that the 3D phase diagram for the ternary case is quite complex and interesting. For instance, the presence of the two fluid-$S2$ discontinuous transition lines in the planes $(0,z_{1},z_{2})$ and $(z_{0},0,z_{2})$ strongly suggests the existence of a coexistence fluid-$S2$ surface in the 3D parameter space. Similarly, the $F$-$S1$ transitions in the planes $(z_{0},z_{1},0)$ and $(0,z_{1},z_{2})$ strongly indicates that a critical and a coexistence $F$-$S1$ surface exist in the full phase diagram, both meeting at a line of tricritical points (a TC line). Besides these features, everything else is less clear. For example, it is unclear from the binary mixtures what can be expected for the LMD lines found in the planes $(z_{0},z_{1},0)$ and $(z_{0},0,z_{2})$, since the values of $z_{0}$ where they start in each plane do not coincide. So, this could indicate either a complex scenario with two (or more) surfaces of minimum density, or that one or both of such surfaces do not exist. In the same token, it is not possible to known at this point whether the fluid-fluid demixing is a particular feature of the case $z_{0}=0$, or if it extends for $z_{0}>0$, giving rise to a $RF$-$F0$ coexistence surface, as well as to a line of triple points (where the phases $RF$-$F0$-$S2$ coexist) and a critical line in the 3D phase diagram. In what follows, these points will be addressed in great detail.

From all thermodynamic properties observed in the binary mixtures discussed above, the fluid-fluid demixing transition in the 0NN-2NN ($z_{1}=0$) system is the most surprising one, in face it is absence in other $k$NN mixtures studied so far. For this reason, we will start the building up of the 3D phase diagram for the ternary mixture by analyzing the $RF$-$F0$ transition.

In order to do this, we investigate the phase behavior of slices of the 3D space for fixed values of $z_{1}$. Some relevant examples of such slices are shown in Fig. 3. For small $z_{1}$, specifically for $z_{1}<0.3177$ [see Fig. 3(a) for the case $z_{1}=0.1$], the same thermodynamic behavior of the case $z_{1}=0$ is found, with the phases $RF$, $F0$ and $S2$ pairwise separated by discontinuous transition lines which meet at a triple point, while the fluid-fluid demixing transition ends in a critical point. This demonstrates that in the 3D phase diagram there exist coexistence surfaces separating the $RF$-$F0$, $RF$-$S2$ and $F0$-$S2$ phases, all of them meeting at a line of triple points (the $RF$-$F0$-$S2$ TP line). Moreover, there is a line of critical points (a CP line), where the $RF$-$F0$ coexistence surface ends. Substantially, this confirms that the fluid-fluid demixing transition is not restricted to the $z_{1}=0$ case, being a feature of the ternary mixture.

Quantitatively, we observe that while the $(z_{0},z_{2})$ coordinates of the TP line mildly change with $z_{1}$, the $z_{2}$ one for the CP line present a strong variation with $z_{1}$, leading the stable region of the $RF$-$F0$ coexistence to decrease as $z_{1}$ increases. For instance, the $RF$-$F0$ coexistence line occurs in an interval $\Delta z_{2}\approx 1.14$ for $z_{1}=0$, but only $\Delta z_{2}\approx 0.76$ for $z_{1}=0.1$. In fact, by increasing $z_{1}$ one observes that the CP line becomes closer to the TP line, and at the special point $(z_{0},z_{1},z_{2})^{*}=(0.6987,0.3177,7.0280)$ both lines meet, so that the stable fluid-fluid demixing transition disappears at this point. This gives rise to the phase diagram displayed in Fig. 3(b), for $z_{1}=z_{1}^{*}$, with the (now completely metastable) $RF$-$F0$ coexistence line ending at the special point, so that the $RF$-$S2$ coexistence surface for $z_{1}<z_{1}^{*}$ now becomes a $F$-$S2$ coexistence, which meets the $F0$-$S2$ coexistence line at the special point.

For not so large values of $z_{1}>z_{1}^{*}$, one still finds the $RF$-$F0$ coexistence ending at the CP line, as shows Fig. 3(c) for $z_{1}=1.1$, but now this line is also metastable, occurring inside the region where the $S2$ phase is the most stable one. Therefore, the more complex scenario observed for small $z_{1}$ gives place to a simple $F$-$S2$ discontinuous transition. Namely, for $z_{1}>z_{1}^{*}$ the fluid-fluid demixing transition becomes preempted by the fluid-solid transition. Hence, at the special point the CP line becomes metastable and the $RF$-$F0$-$S2$ TP line ends. A summary of these behaviors in the 3D space is presented in Fig. 4.

Although in general we will not be interested in discussing metastable transitions here, it is interesting to take a close look on this for the fluid-fluid demixing, to verify how further the $RF$-$F0$ coexistence surface goes as $z_{1}$ increases. First, we notice that, regardless the value of $z_{1}$, the metastable part of this surface is always limited from above, i.e., it ends when it meets the spinodal of the $RF$ phase, as shown in Figs. 5(a) and 5(b). Second, for large $z_{1}$ the $S1$ phase also appears in the phase diagram in the region of small $z_{0}$ and $z_{2}$, as seen in Fig. 3(c), and the fluid phase let to be stable in such region. This is 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 $z_{1}$ the spinodal of the fluid phase develops a cusp, which approximates of the (metastable) CP line as $z_{1}$ increases, see Fig. 5(a) for $z_{1}=2.3$. Such approximation occurs until $z_{1}\approx 2.3731$, after which the cusp meets the $RF$ and $F0$ spinodals (in the point where there was the CP line), changing completely the scenario of such spinodals. In fact, as demonstrated in Fig. 5(b), for $z_{1}=2.5$, the stability region of $RF$ phase becomes now limited to a closed domain of the ($z_{0},z_{2}$) phase diagram, and the (metastable) $RF$-$F0$ coexistence line ends at the point where the spinodal of the $F0$ phase crosses the one for the $RF$ phase. Therefore, the CP line ends at $z_{1}\approx 2.3731$, while the fluid-fluid demixing still exists for larger values of $z_{1}$. However, by increasing $z_{1}$, one observes that the domain of the $RF$ stability shrinks, yielding a decreasing in the in $RF$-$F0$ coexistence line, which finally disappears at $z_{1}=5.7157$.

From the results in the previous subsection, one knows that for $z_{1}<z_{1}^{*}$ the 3D phase diagram presents three coexistence surfaces ($RF$-$F0$, $RF$-$S2$ and $F0$-$S2$), a $RF$-$F0$-$S2$ TP line and a CP line, which meet (and ends or becomes metastable) at the special point, as depicted in Fig. 4. Moreover, for $z_{1}>z_{1}^{*}$, but not so large, this behavior gives place to a simple $F$-$S2$ coexistence surface.

As shows Fig. 3(c), a critical $F$-$S1$ transition line is found in slices of the phase diagram for $z_{1}$ fixed (and large enough), giving rise to a critical $F$-$S1$ surface in the 3D space, as expected from the behavior of the binary mixtures. Such surface starts at $z_{1}=0.7284$ — which turns out to be the point of minimum (in relation to $z_{1}$) of the $F$-$S1$ critical line in the 0NN-1NN mixture [i.e., in the plane ($z_{0},z_{1},0$)] NT0NN1NN-1NN2NN — and exists in a limited region of the parameter space. Namely, it gives place to a discontinuous $F$-$S1$ transition surface for large $z_{1}$, as confirmed in Fig. 5(c). These features are better seen in slices of fixed and small $z_{0}$, as the one in Fig. 6(a) for $z_{0}=0.2$, where one finds a phase diagram similar to the one for $z_{0}=0$, with a continuous and a discontinuous $F$-$S1$ transition line meeting at a TC point. This confirms not only the existence of a critical and a coexistence $F$-$S1$ surface, but also the presence of a TC line where they meet in the 3D space. By analyzing several slices for fixed $z_{0}$ or $z_{2}$, we have accurately determined the TC line, which connects the two tricritical points found in the planes $(z_{0},z_{1},0)$ and $(0,z_{1},z_{2})$. Interestingly, it presents a complex non-monotonic behavior, attaining a maximum point, with respect to $z_{1}$ and $z_{2}$, at $(z_{0},z_{1},z_{2})_{TC,max}=(0.6719,1.4216,4.6678)$. This line, as well as the critical and coexistence $F$-$S1$ surfaces are depicted in Fig. 7, which shows this part of the 3D phase diagram in detail.

The $z_{0}$-slices, as those in Figs. 6(a) and 6(b), also confirm the presence of the discontinuous $F$-$S2$ surface, as well as unveil the existence of a surface of solid-solid ($S1$-$S2$) demixing in the ternary mixture. Moreover, one always observes that the $F$-$S1$, $F$-$S2$ and $S1$-$S2$ coexistence surfaces meet in a line of triple points (the $F$-$S1$-$S2$ TP line), as seen in Figs. 5(c) and 6. This line starts at the triple point of the 1NN-2NN mixture [i.e., in the ($0,z_{1},z_{2}$) plane] and extends to $z_{0},z_{1},z_{2}\rightarrow\infty$; its behavior is also shown in Fig. 7. It is important to notice that, since the TC line (and then the critical $F$-$S1$ surface) ends at a finite value of $z_{0}$, for large values of this activity everything we find in this part of the 3D phase diagram are the $F$-$S1$, $F$-$S2$ and $S1$-$S2$ coexistence surfaces meeting at the TP line [see Fig. 6(b)].

The existence of the $F$-$S1$-$S2$ TP line in the limit of $z_{0},z_{1},z_{2}\rightarrow\infty$ can be justified by the existence of a $F0$-$S1$, a $F$-$S2$ and a $S1$-$S2$ coexistence line in the binary mixtures, all of them extending to infinity NT0NN2NN ; NT0NN1NN-1NN2NN . Since the RRs assume simple forms in such limit, it is straightforward to determine the behavior of the TP line there. For instance, the fixed point for the RRs in the fluid phase has the form $A_{i,0}=B_{i,0}=C_{i,0}=D_{i,0}=1$ and $A_{i,j}=B_{i,j}=C_{i,j}=D_{i,j}=0$ for $i=1,2$ and $j=1,2$. For the $S1$ phase one finds $A_{1,j}=B_{1,j}=C_{1,j}=D_{1,j}=1$ for $j=0,1$, while all others RRs vanish, when the sublattices indexed by $1$ are the more populated ones. In the $S2$ phase, if the sublattices $A_{1}$ and $A_{2}$ are the ones more populated, one has $A_{i,j}=1$ and $B_{i,j}=C_{i,j}=D_{i,j}=0$, for $i=1,2$ and $j=0,1,2$. Using these solutions, it is simple to calculate the free energies in this limit, being $\phi^{(F)}=(1+z_{0})^{8}$, $\phi^{(S1)}=(1+z_{0}+z_{1})^{4}$ and $\phi^{(S2)}=(1+z_{0}+z_{1}+z_{2})^{2}$. By making $\phi^{(F)}=\phi^{(S1)}=\phi^{(S2)}$ and solving in terms of $z_{0}$, one finds

and

Therefore, as $z_{0}\rightarrow\infty$ this TP line behaves has $z_{1}\approx z_{0}^{2}$ and $z_{2}\approx z_{0}^{4}$. This result can be understood as follows: at full occupancy an 1NN particle effectively occupies the volume of two 0NN ones, while a 2NN particle occupies the volume of four 0NN ones, in agreement with the exponents above. The calculated free energies also allow us to determine the behavior of the $F$-$S1$, $F$-$S2$ and $S1$-$S2$ coexistence surfaces in the limit of $z_{0},z_{1},z_{2}\rightarrow\infty$, where one finds $z_{1}\approx z_{0}^{2}$ and $z_{2}\approx z_{0}^{4}$ for the $F$-$S1$ and $F$-$S2$ cases, respectively. Therefore, in slices of the 3D phase diagram for very large and fixed $z_{0}$, these surfaces shall appear as straight lines at constant $z_{1}$ and $z_{2}$, respectively. The $S1$-$S2$ coexistence surface, on the other hand, behaves as $z_{2}\approx(z_{0}+z_{1})^{2}$, being a quadratic function of $z_{1}$ ($z_{0}$) for fixed $z_{0}$ ($z_{1}$). In fact, these results are somewhat confirmed in Fig.6(b) for $z_{0}=1.2$, where we see that the $F$-$S1$ and $F$-$S2$ lines occur at almost constant values (specially the former one), while the $S1$-$S2$ line is not so straight, indicating the existence of a nonlinear dependence between $z_{2}$ and $z_{1}$. Results [not shown] for slices for much larger values of $z_{0}$ indeed confirm the correctness of the predicted surfaces.

Figure 8 presents the complete 3D phase diagram in the space of activities, summarizing all the features discussed so far. In face of its complexity — it has a fluid-fluid, a solid-solid and several fluid-solid coexistence surfaces, with three of them ($F$-$S1$, $F$-$S2$, and $S1$-$S2$) extending to $z_{0},z_{1},z_{2}\rightarrow\infty$, beyond a critical fluid-solid surface, a CP line, a TC line and two TP lines — it is a bit hard drawing this phase diagram in an intelligible way. Anyhow, it is important to let clear how the partial diagrams depicted in Figs. 4 and 7 connects, what are the sizes of the transition surfaces (at least the ones which exist in limited domains) and so on.

At this point, it is important to stress that we have carefully looked for new phases in the general case, but only the four phases already present in the binary mixtures were found.

As discussed in subsection IIIA, two lines of minimum density (LMDs) were found in the binary mixtures inside the fluid phase, one in the 0NN-1NN ($z_{2}=0$) and other in the 0NN-2NN ($z_{1}=0$) case. In the former system, the LMD starts at $z_{0}\approx 0.333$ when $z_{1}\rightarrow 0$, while in the last one it starts at $z_{0}\approx 0.20$ when $z_{2}\rightarrow 0$, and in both cases they end at the spinodal of the fluid phase, inside regions where the corresponding solid phases are more stable than the fluid NT0NN1NN-1NN2NN ; NT0NN2NN . The difference between these starting points suggests a complex scenario for the ternary mixture with at least two surfaces of minimum density (SMDs). Before discussing them, however, it is important to remark that such starting points were calculated through a limiting process. For example, for the 0NN-1NN mixture, we located the ($z_{0},z_{1}$) coordinate where the total density — defined as $\rho_{T}=\rho_{0}+2\rho_{1}+4\rho_{2}$ — presents a minimum for a given pressure, $P$ ¹¹1In our grand-canonical formalism $P=-\phi/a^{3}$., i.e., along an isobaric curve of $\rho_{T}\times z_{0}$ (or $z_{1}$) (see, e.g., Fig. 5 in Ref. NT0NN2NN ). Then, by varying $P$ we can build up the LMD curve and determine it as close as we want of the $z_{0}$ axis by making $z_{1}\rightarrow 0$. Exactly on the $z_{0}$-axis, however, one has $\rho_{T}=z_{0}/(1+z_{0})$, so that no anomaly exists there.

Then, if we take slices of fixed $z_{2}>0$, one still finds LMDs in the way just described, forming a SMD in the 3D space. Such $z_{2}$-SMD (since it is obtained for fixed values of $z_{2}$) starts at the LMD in the plane $(z_{0},z_{1},0)$ and varies smoothly in the space, ending at the spinodal of the fluid phase, deep inside the region where the fluid is metastable. Although this surface does not exist exactly in the plane $(z_{0},0,z_{2})$, using the limiting process we can obtain it as close as we want of this plane, so that there exists a LMD for $z_{1}\rightarrow 0$ limiting this surface (see Fig. 9). Proceeding in the same way, but now considering slices of fixed $z_{1}$, a second SMD (the $z_{1}$-SMD) is obtained, which starts in the LMD on the plane $(z_{0},0,z_{2})$ and also ends at the spinodal of the fluid phase in the metastable region. In this case, the SMD does not exist exactly in the plane $(z_{0},z_{1},0)$, but it can be determined in the limit of $z_{2}\rightarrow 0$. This $z_{1}$-SMD is also shown in Fig. 9. The existence of these two SMDs indicates, for sake of completeness, the existence of a third one for fixed $z_{0}$. As shown in Fig. 9, such $z_{0}$-SMD indeed exists in the 3D space and, albeit it cannot be observed exactly in the planes $(z_{0},z_{1},0)$ and $(z_{0},0,z_{2})$, it can be obtained in the limits $z_{2}\rightarrow 0$ and $z_{1}\rightarrow 0$, respectively. In such limits, the starting point of the $z_{0}$-SMD is at $z_{0}\approx 0.50$ and, similarly to the other SMDs, it also ends in the spinodal of the fluid phase. As observed in Fig. 9, the three SMDs become quite close for large $z_{2}$ and they seem to be limited from above by a single line. However, it is quite hard to assure this numerically and it may be the case that they end at different (but very close) lines, or become a single surface before this.