Clustering in Mixtures of SALR Particles and Hard Spheres with Cross Attraction.

Self-assembling complex fluids are often modeled as particles with effective competing isotropic interactions, combining a short-range attraction (SA) followed by a longer-range repulsion (LR). For moderately low temperatures and densities, SALR particles form clusters in equilibrium, at least provided that the potential parameters are appropriate. Here we inquire into the possibility that cluster formation in SALR fluids might be pushed by a foreign species even under thermodynamic conditions that would not allow for clusterization of the pure system. To this aim, we study by Monte Carlo simulations a mixture of hard-sphere two-Yukawa particles and hard spheres, with a cross interaction modeled by a square-well attraction, and we investigate the conditions of clustering in terms of strength of attraction and relative concentration of the two species. We find that clusters can occur in the mixture for the same temperature and density where the pure SALR fluid is almost structureless. In particular, we single out a cross attraction such that clusters are formed with a SALR concentration as low as 5%. We also find a situation where nearly pure droplets of hard spheres are held together by a shell of SALR particles. Conversely, we show that clustering can be undermined in the mixture under conditions for which this process takes place in the parent SALR fluid. Using a simple criterion, based on the second virial coefficients of the attractive part of interaction potentials (the so-called "reference attractive fluids"), we are able to predict accurately whether clustering is favored (or hindered) in the mixture, as compared to the pure SALR fluid.

Introduction

Nowadays, it is well recognized that the phase behavior of a complex fluid can be significantly rich and intriguing, owing to its ability to self-assemble into aggregates exhibiting modulated patterns at the mesoscale.[1−7] In this context, a large variety of colloidal dispersions, as well as block copolymer melts and solutions of amphiphilic molecules, can be described by effective pair interactions featuring a short-range attraction (SA) followed, at larger distances, by a long-range repulsion (LR) .[8−14] While the SA component, originating from van der Waals and/or depletion forces, favors particle aggregation, the LR component arises from electrostatic forces and imposes a penalty on the growth of a dense macroscopic phase. The distinguishing feature of SALR fluids is their attitude to self-assemble into equilibrium structures of various morphologies[15−23] characterized by spatial inhomogeneities whose typical length is larger than the particle size, but definitely not macroscopic. The size and shape of these patterns strongly depend on the detailed force parameters, as well as on the values of the thermodynamic variables (i.e., the temperature T and the density ρ). Generally speaking, the knowledge of this dependence can be exploited to steer the process of self-assembling toward the formation of the desired structure. For example, for moderately low values of T, one expects to see clusters in the dilute fluid, giving way to a percolated fluid at higher densities.[18,19,24,25] Using density-functional theory, it has been shown that, for sufficiently low temperatures, SALR particles give rise to periodic microphases exhibiting (in increasing order of density) clusters, cylinders, lamellae, inverted cylinders, and inverted clusters, just to name the simplest structures.[26−29] The equilibrium phase diagram of a cluster-forming fluid has been also reconstructed by a thermodynamic model and assessed by computer simulation.[30] The broad interest in the physics of SALR fluids is witnessed by four recent reviews on this subject.[31−34]

If we restrict to low-density conditions, such that the self-organized aggregates would essentially be clusters, the crucial problem is clearly how to establish the occurrence of clusters in the simulation sample. From a structural point of view, the characteristic condition for the development of microphases would be a maximum in the static structure factor S(k) at a wavenumber k0 ≠ 0, well below the location of the main diffraction peak;[34−36] by contrast, a diverging S(0) invariably implicates liquid–vapor separation. In fact, in a hard-sphere two-Yukawa (HSTY) fluid—a prototypical instance of an SALR fluid—the existence of a low-k peak in S(k) is not by itself sufficient to identify a cluster phase;[18,19,35,36] rather, this peak is indicative of the development of some kind of modulation (intermediate-range order), in the fluid. As a matter of fact, a low-k peak can occur under four different structural conditions, i.e., a monomer-dominated fluid, a cluster fluid, a random percolated fluid, and a cluster percolated fluid, as can be discriminated from the shape of the cluster-size distribution (CSD).[18,19] In particular, a monotonically decreasing CSD represents a fluid of nonbonded particles, whereas the presence of a maximum in the CSD unambiguously identifies a cluster fluid. When more than 50% of the particles are collected in the same network, the system undergoes percolation. If the CSD exhibits a single peak at a size comparable with the system size, then we speak of a random percolated fluid. On the other hand, if in addition to this peak the CSD also shows the maximum typical of a cluster fluid, then we have a cluster percolated fluid. Interestingly, when the height of the low-k peak exceeds a value ≈3, then the system is either a cluster fluid or a cluster percolated fluid.[18,19] This empirical rule-of-thumb, reminiscent of the Hansen–Verlet criterion for freezing of simple fluids,[37] is a useful means to locate the onset of a cluster phase. Other criteria, based on structural indicators, have been proposed to identify the onset of clustering in SALR fluids, either in terms of specific features of the low-k peak[38,39] or in terms of the properties of real-space correlations.[40−42] As for thermal properties, a maximum is generally expected in the constant-volume specific heat at the crossover between the monomer- and the cluster-dominated fluid, since here a randomly selected particle can equally be found as isolated or belonging to a cluster of nearly optimal size.[43] This leads to strong energy fluctuations and thus to a specific-heat peak, which then provides a practical, easily accessible tool to discriminate between different aggregation regimes.[43]

So far, our discussion has been limited to one-component SALR fluids, though it is clear that the self-assembling scenario of a binary mixture is potentially far richer. In this regard, one way to induce a spatially modulated pattern in a mixture with attractive and repulsive short-range forces would be to employ a suitable molecular structure for one of the species; see, e.g., ref (44). Recently, we have followed this route in a number of papers[45−51] where we have reported on a variety of self-assembling structures in a mixture made up of amphiphilic dimers and hard spheres. A second possibility would be to adopt a fully coarse-grained view to the problem, by considering a mixture of monatomic species, but then we must leverage on the shape of interactions to produce stable microphases. Examples of such mixtures are found in refs (52 and 53). In these studies, the particle core of both species is harshly repulsive, self-interactions are of SALR type, and the cross interaction is repulsive in the core and attractive at short distances. The force parameters are so tuned that, for sufficiently low T and ρ, a one-component system of SALR particles forms clusters. In another study, the self-interaction of one species was purely attractive outside the core.[54] Then, if the cross interaction is weak, this species undergoes liquid–vapor separation at low T. For a strong cross attraction, the resulting clusters are composed of the two species in more or less equal proportions.

In the present paper, we study a mixture of two monatomic species: type-1 particles interact via a HSTY potential, whereas type-2 particles are simple hard spheres (HS). Outside the core, the cross interaction is a square-well (SW) potential of depth ε12. The proposed model would represent a generic colloidal mixture where species 1 is a cluster-former, whereas particles of species 2 are relatively blind to each other, even though—due to its affinity to species 1—they will actively participate in the formation of aggregates. Moreover, our system can describe the large class of soft materials—as for instance solutions of biomolecules—in which solutes interact through screened-Coulomb potentials with other species. Our mixture has similarities with (and also differences from) the model investigated in ref (54). For instance, the type-2 interaction in the latter model reduces to hard-core interaction in the limit of vanishing attraction. On the other hand, the cross attraction is different, being a SW here and a Yukawa attraction in ref (54). Our choice possibly allows one to more effectively discriminate the role played by the cross interaction in the overall organization of the mixture. Finally, in ref (54), the mechanism of clustering is mainly investigated through a thermodynamic model, whereas our approach is exclusively simulation-based.

Once established the above-mentioned setup, two questions will be addressed in the following: first, the role played by HS particles in the formation of clusters of HSTY particles, as a function of ε12 and relative concentration. Second, the conditions under which the HSTY particles are able to induce droplets of HS particles. In particular, it would be interesting to know whether the HSTY particles can form a coating shell around the HS particles, as it occurs for amphiphilic dimers in mixtures with a small fraction of hard spheres.[47,49−51]

The rest of the paper is organized as follows. In Section we describe the model and the simulation method to perform the investigation. Sections –V contain the core of our study: first, we focus on the pure HSTY fluid (Section ), establishing the temperature below which a cluster phase is stable for a specific low value of the density. This analysis is preliminary to the study of a HSTY–HS mixture, for which we examine the effect of strengthening the cross interaction (Section ) or changing the relative concentration (Section ), until we clarify the exact terms of the interplay between the two species with regard to clustering. Some concluding remarks follow in Section .

Model and Methods

Our mixture consists of hard-sphere two-Yukawa particles and hard spheres, henceforth labeled 1 and 2 respectively, with the same core diameter σ. Beyond the hard core, the HSTY interaction is given bywhere r* = r/σ; ε11 is the potential minimum at σ; za, and zr are the inverse range of attraction and repulsion, respectively; α is the ratio of strength of repulsion to attraction. The range of the attractive well extends up toi.e., the zero of u11(r). In particular, we choose za = 10, zr = 0.5, and α = 0.1, and therefore, r0 ≃ 1.242σ. With this parametrization, u11(r) corresponds to the HSDY1 case studied in ref (18). The cross interaction u12(r) is provided by a SW attraction of depth ε12 (with the ratio ε12/ε11 varying between 0.01 and 1) and range r0. With this prescription, the HSTY and SW interactions have the same attractive range. In Figure , we plot the shapes of u11(r) and u12(r) for all the values of ε12 investigated in this work. We take σ and ε11 as units of length and energy, respectively. Then, the density is given in units of σ–3 while the temperature is measured in units of ε11/kB, where kB is the Boltzmann constant.

Figure 1

Interatomic potentials investigated in this work: the HSTY potential u11(r) (eq , black line), and the SW cross interaction u12(r), plotted for all ε12 investigated in this work (in the legend). The HS interaction u22(r) is not shown.

We carry out canonical-ensemble Monte Carlo (MC) simulations of a mixture composed of N1 HSTY particles and N2 HS particles, with N = N1 + N2 = 2048, enclosed in a cubic box with periodic boundary conditions to reduce surface effects. The concentration N1/N of HSTY particles is denoted χ. Before moving to the mixture, we have studied a one-component system of 500 HSTY particles, to fix reference thermodynamic conditions for the clustering of the pure fluid. Thereby, we have set ρ = 0.05, T in the range 0.20–0.25, whereas χ varies over the full range of relative concentrations. In our simulations, u11(r) is truncated at half the box length (10.77σ for the pure HSTY fluid and 17.23σ for the mixture). The chosen density is high enough to allow for cluster formation at low temperatures, but still sufficiently low to prevent packing effects from playing any role. In the selected temperature range, the pure HSTY fluid undergoes the transformation from a weakly inhomogeneous state to a deeply clustered state.

The particles are initially distributed at random (T = ∞); then, the system is quenched at a given temperature T and equilibrated. From our experience on self-assembling mixtures we know that a huge number (higher than 108) of MC cycles is typically required to properly equilibrate the system at low temperatures. Therefore, we have optimized our MC code in the attempt to make the numerical algorithm converge faster. In particular, we have implemented two distinct values for the maximum random displacement (mrd) of a particle, so as to account for the possible existence of two different (intracluster and intercluster) length scales in the system. The two mrd values are adjusted during the equilibration run until the acceptance of translational MC moves is around 30% for the longer moves and 80% for the shorter ones. This gives mrd values of ≈0.1σ and ≈0.01σ, respectively. Furthermore, we have introduced swapping moves, whereby the positions of two randomly chosen HSTY and HS particles are exchanged. For both moves, acceptance is decided based on the standard Metropolis test. The schedule of all moves, the canonical moves and the new ones, is such that detailed balance holds exactly at each step.

The structure of the mixture is investigated by computing the radial distribution function g(r) and the structure factor S(k), with i and j labeling the two different species. Structure factors are calculated as Fourier transforms of g(r), according to the formula[37]

To gain knowledge on the self- and cross-correlations of global variables, we also compute the Bathia–Thornton number–number, concentration–concentration, and number-concentration structure factors of the mixture, defined as[55,56]respectively.

Aggregates are characterized through a cluster analysis performed by the Hoshen–Kopelman algorithm.[57] According to our convention, two particles (of whatever species) are bonded if they are separated by less than r0, i.e., if the particles fall within the range of the SA/SW potential. Following refs[18 and 58], the CSD is taken to bewhere n(s) is the average number of clusters with size s in a single configuration. The distribution N(s) is normalized in such a way that ∑N(s) = 1. In our study, the distribution of cluster sizes is computed by averaging over 1000 equispaced configurations extracted from the last part of the production run.

Pure HSTY Fluid

Information about the structure attained by the HSTY fluid long after the initial quench can be gained from the profile of the total structure factor SNN(k) ≡ S11(k), reported in Figure . For T = 0.25, the height of its low-k peak, located at k0σ ≈ 1, attains a value ≈1.8, which would suggest, according to the heuristic criterion of refs (18 and 19), that the fluid exhibits intermediate-range order. This condition preludes to the subsequent development of more structured aggregates for T = 0.22, where indeed S11(k0) rises to ≈4.2, i.e., above the clustering threshold of ≈3. Based on the structural evidence, for T = 0.20 [where S11(k0) exceeds 10] clusters appear to be well structured. As for the real-space structure, we report in the inset of Figure the radial distribution function g11(r) for the same three temperatures. While for T = 0.25 the system is little structured, the multiple solid-like peaks of g11(r) for T = 0.22, and even more so for T = 0.20, indicate a high degree of order in the arrangement of particles around a reference particle, i.e., within the linear extent of a cluster. The subsequent long-range oscillations in the spatial correlations for r ≳ 4σ are another well-known indication that a modulated phase is taking over in the fluid.

Figure 2

Structure factor and radial distribution function (inset) of the pure HSTY fluid for ρ = 0.05 and three different temperatures (in the legend).

The local arrangement of the HSTY fluid is better elucidated by a visual inspection of microscopic configurations, see Figure . For T = 0.25 (A), in the wake of development of intermediate-range order, the fluid appears to be overall homogeneous, even though local inhomogeneities are occasionally seen. Instead, for T = 0.22 (B), the tendency of particles to aggregate is more evident, with clusters starting to form in the system. For T = 0.20 (C), very few particles are isolated and the cluster shape is approximately spherical. Information on the cluster structure can be gained from the CSD, plotted in Figure D: this decays monotonically for T = 0.25, certifying that the system almost behaves as a dispersed fluid. For T = 0.22 a shallow peak develops around s = 15, witnessing the presence of clusters with this typical size. The sign of clustering becomes definitely sharp at T = 0.20, where a distinct peak is visible in the CSD at s ≈ 20.

Figure 3

Microscopic configurations of the pure HSTY fluid. (A) T = 0.25; (B) T = 0.22; (C) T = 0.20; (D) CSD at the same temperatures.

To summarize this section, we have a coherent structural and microscopic picture showing that in the HSTY fluid with density ρ = 0.05 clusters are absent for T = 0.25; they start appearing for T = 0.22 and are fully developed for T = 0.20. In another study of the same model,[18] at about twice the density analyzed here, the fluid for T = 0.25 was found in a clustered state, suggesting that the characteristic temperatures marking the onset of the various aggregation stages—intermediate-range order first and clustered state afterward, possibly evolving into an arrested state along the so-called λ-line—show a shallow parabolic trend as a function of density, which is reminiscent of the liquid–vapor binodal in fluids with short-range attraction.

Equimolar HSTY–HS Mixture

Having clarified the behavior of the HSTY fluid, we move to the binary mixture obtained from the former by replacing half of the particles with hard spheres (i.e., χ = 0.5). We choose thermodynamic conditions—ρ = 0.05 and T = 0.25—such that, as shown in the previous section, no stable clusters are present in the HSTY fluid. Our purpose is to establish whether the presence of another species, able to provide a bridge between two nearby HSTY particles, is sufficient to promote cluster formation. In principle, this is one out of several scenarios that could hypothetically arise once the balance between attraction and repulsion in the mixture moves in favor of the former; other two plausible outcomes, related to the strength of cross attraction ε12, are that either the mixture stays almost homogeneous as the pure HSTY fluid or, at the opposite end, that the mutual attraction is now strong enough to overcome the competition with repulsion, thus giving rise to liquid–vapor separation. In this regard, it seems unlikely that demixing can occur, because ruled out by the 1–2 attraction.

To ascertain this issue we have simulated equimolar mixtures for values of ε12 in the range 0.01–1 (in units of ε11). Results for the potential energy are collected in Figure : as a preliminary observation, the equilibration time strongly depends on ε12, ranging from 106 MC cycles for ε12 = 0.01 to 108 or more MC cycles for ε12 = 1. To check whether equilibrium has been reached—especially for the deepest attractions, ε12 > 0.50—we have performed additional simulations for ε12 = 0.75 and ε12 = 1, using a big spherical cluster encompassing all particles as initial configuration, see Figure A. As for ε12 = 0.75, we have verified that the final state of the system along this route (in terms of potential energy, structural, and microscopic appearance, see Figure B–D) matches with the previous one, indicating that equilibrium has eventually been attained. On the other hand, for ε12 = 1, the time needed for the decay of the big-cluster configuration is much longer than we could achieve in our analysis. Our conclusion, also supported by the slow energy drift in Figure , is that for ε12 = 1 the system is still slowly approaching equilibrium, even though this fact would have no serious impact on the considerations presented below.

Figure 4

Equimolar HSTY–HS mixture for T = 0.25. Potential energy per particle vs MC cycles for different ε12, in the legend; results for the pure HSTY fluid are also shown, for comparison. Inset: asymptotic values (symbols) vs ε12 are fitted by two different straight lines.

Figure 5

Equimolar HSTY–HS mixture for T = 0.25 and ε12 = 0.75. (A) Big-cluster initial configuration, containing 2048 randomly dispersed HSTY and HS spheres. (B) After 2.5 × 107 MC moves, the initial cluster is reduced in fragments, now resembling the cluster fluid obtained from a fully homogeneous initial configuration; see Figure B below. After equilibration, the potential energy fluctuates around a fixed value, which is roughly independent of the initial configuration (C). Interestingly enough, the initial energy of the system in the big cluster is lower than in the final cluster-fluid configuration; hence, it is the relevant entropy gain achieved by forming many smaller clusters that ensures that the cluster fluid is more stable than the big cluster. Also the final structure factors are similar in the two runs (D), but for a small mismatch in the position and height of the low-k peak.

Figure 9

Microscopic configurations for the equimolar mixture at T = 0.25 (HSTY, green; HS, orange). (A) ε12 = 0.5; (B) ε12 = 0.75; (C) ε12 = 1; (D) CSD for the same values of ε12.

Turning back to the behavior of the potential energy per particle in Figure , it appears that for ε12 ≤ 0.50 the mixture behaves similarly to the HSTY fluid of same T; i.e., the system quickly attains equilibrium and the final energy is between −0.05 and −0.25. However, when ε12 is increased from 0.50 to 0.75, the scenario changes drastically, with the energy now exhibiting a sharp drop until reaching a final value of ≈−2 after about 107 MC steps. Upon further increasing the attraction strength to ε12 = 1, the energy approaches ≈−3 after about 5 × 107 MC steps. In particular, we see that the rate of (absolute) energy increase grows abruptly as ε12 overcomes 0.50. This can be appreciated in the inset of Figure , where the asymptotic energies are reported as a function of ε12: the energy values can be grouped in two sets (i.e., up to ε12 = 0.50 and from ε12 = 0.75 onward) which are fitted by different straight lines, in this way corroborating the idea that the crossover between the two regimes follows a threshold-like behavior.

The evolution of potential energy with ε12 suggests that the system exhibits some form of aggregation for ε12 > 0.50, even though we cannot yet say whether it is liquid–vapor separation or clusterization that occurs. To clarify this point, we analyze in Figure A the partial structure factor S11(k) of HSTY particles: we see that, upon increasing ε12, the k → 0 limit of S11(k) does not change significantly; on the other hand, for ε12 > 0.50, S11(k0) jumps over 10, indicating that clusterization, rather than liquid–vapor separation, takes place in the system for sufficiently strong cross attraction. Correlations involving species 2 are shown in panels B and C of Figure , but they hardly add anything new to what we have already learned from S11(k), since S12(k) and S22(k) also show a remarkable increase of the low-k peak for ε12 > 0.50. Global variables, expressed according to eqs –6, are reported in panels D–F for the same values of ε12: the total structure factor SNN(k) (D) is similar to S11(k), both in shape and height of the peaks, confirming that the indications provided by the structure factor of the HSTY species alone hold for the mixture too. A similar trend is also observed in S (E), whereas S (F), behaves differently: in this case, besides the development of the low-k peak, we observe an increasing structuring with ε12, as indicated by the multiple peaks found for ε12 > 0.50. Therefore, significant concentration fluctuations are found over a wide range of k values, as can be expected in a cluster-forming system, due to the spatial rearrangement of particles inside the aggregates.

Figure 6

Equimolar HSTY–HS mixture for T = 0.25. (A) S11(k); (B) S12(k); (C) S22(k); (D) SNN(k); (E) SNc(k); (F) Scc(k). Values of ε12 are shown in panel A, with the same colors applying for all panels. The pure HSTY case is also reported in panels A and D for comparison. Note the semilogarithmic scales in panels A, C, D, and E.

To complete our thermodynamic/structural survey, we mentioned in the Introduction that a maximum in the specific heat at constant volume, cV, provides another reasonable indication of the onset of clustering.[43] In particular, the specific heat is related to energy fluctuations according to the expression[37](where H is the Hamiltonian of the system) and indeed such fluctuations become large when clusters form and disrupt with similar probability. In Figure , cV is reported as a function of ε12 at constant T = 0.25. It clearly emerges that the observed behavior corroborates our previous finding: cV shows a maximum for ε12 = 0.75, where, according to internal energy (see Figure ) and structure factors (see Figure ), clusters show up for the first time. For ε12 = 1 clusters are already formed and quite stable, therefore energy fluctuations are less marked and in turn cV attains smaller values.

Figure 7

Equimolar HSTY–HS mixture for T = 0.25. cV/k vs ε12. The error bars represent maximum dispersions from the mean as computed from 10 independent runs.

Once having ascertained that a second species is able to promote the cluster formation under proper conditions, we have computed a number of properties to characterize the structure of aggregates, see Figure A: the number of isolated particles monotonically decreases with increasing ε12 (left), signaling that aggregates encompass more and more particles; concurrently, the largest cluster size increases (middle) and the total number of clusters decreases (right), indicating that aggregates appear and grow in size with increasing ε12. This picture is reflected in the probability distribution of the number of bonds, plotted in Figure B. Here the particles are classified according to the number of bonded “neighbors”: the monotonic decay for ε12 = 0.5 is replaced by a trimodal distribution for ε12 = 0.75, with peaks at Nb = 0, 6, and 12, respectively associated with isolated particles, outer cluster particles, and inner cluster particles. In particular, the peak at Nb = 12 (more enhanced for ε12 = 1) strongly points to a liquid-like (if not even fcc-like) ordering inside clusters. The comparatively higher number of bonds formed for larger values of ε12 can be better appreciated in the inset of Figure B: here the probability to form more than two bonds is remarkably low for ε12 = 0.5, whereas the trend is reversed for ε12 = 0.75 and, even more so, for ε12 = 1, where 8 and 10 bonds per particle occur with a rather high frequency.

Figure 8

Equimolar mixture for T = 0.25. (A) Number of isolated spheres (left), largest cluster size (middle), and total number of clusters (right) for various ε12 values. Standard deviations are also shown. (B) Probability distribution of the number of bonds for a number of ε12 values, in the legend. In the inset, the same data are plotted vs ε12 for various Nb, in the legend.

The above picture is completed by the microscopic configurations reported in Figure : as ε12 grows from 0.50 (A) to 0.75 (B), up to 1 (C), clusters become progressively more structured. In analogy with the behavior of the pure HSTY fluid upon cooling (Figure ), for ε12 = 0.75 and 1, the shape of the clusters is approximately spherical, although in the latter case nonspherical aggregates also exist. Under these conditions, very few isolated particles appear in the box, due to a strong HSTY–HS interaction. A closer look at Figure C shows that cluster particles are instead tightly packed for ε12 = 1, similar to what occurs in the solid-like nuclei developing in a supercooled vapor or liquid which is about to crystallize.[59] Indeed, a high degree of order in the arrangement of particles within a cluster is implied by the behavior of g11(r) (not shown here), which displays multiple solid-like peaks akin to those observed in the inset of Figure for T ≤ 0.22. More refined investigations of the liquid–solid transition inside clusters, like those described in ref (60), are deferred to future studies. The cluster-size statistics (computed by making no difference between the species) is reported in Figure D: for ε12 = 0.50 the CSD is a monotonically decreasing function of the cluster size s, a profile compatible with a dispersed fluid. On the contrary, for ε12 = 0.75 the CSD has a well-definite maximum around s = 60, indicating this preferred cluster size. In addition, we observe a non-negligible fraction of larger clusters comprising up to 100 particles. For ε12 = 1, we see multiple peaks in the CSD, suggesting the existence of as many preferential cluster sizes, although the distribution is centered around the same size as for ε12 = 0.75. In addition, we see no sign of a percolated structure, in spite of the existence of large aggregates with up to 200 particles. Overall, the distributions reported in Figure closely resemble those in Figure , suggesting that the strength of cross attraction plays a role similar to inverse temperature in a HSTY fluid.

Summing up, a sufficiently large ε12 can sustain clusters in an equimolar HSTY–HS system, notwithstanding their absence in the pure HSTY fluid of same density and temperature. The same conclusion was already drawn in ref (54) but using mutually attractive spheres as foreign species. Our results demonstrate that even particles that do not attract each other are able to provide the extra attraction needed by HSTY particles to undergo clusterization.

To rationalize our evidence, we turn to the interpretation of clustering given in refs (18 and 19). Therein it is argued that the clustering threshold in HSTY systems falls upon (roughly) the same thermodynamic conditions for which a modified HSTY fluid with the long-range repulsion switched off would phase-separate into liquid and vapor. We hereafter refer to this modified HSTY fluid [i.e., with u11(r) truncated at r0] as the “reference attractive” HSTY fluid. In other words, adding an appropriate long-range repulsion on top of a purely attractive tail breaks the liquid–vapor mixture into separate, noninteracting fragments (the clusters). A similar scenario has been described in regard to spinodal decomposition in ref (23), where the behavior of a fluid interacting via a generalized Lennard-Jones plus a repulsive-Yukawa tail was investigated in the regime where arrested states hold the stage. If the same argument applies for the present mixture, then for T = 0.25, χ = 0.5, and ε12 = 0.75 the liquid–vapor binodal line of the reference attractive HSTY–HS mixture will lie above that of the reference attractive HSTY fluid. Accordingly, a measure of the overall amount of attraction (i.e., the “glue” keeping particles together) will be larger for the mixture than for the pure HSTY fluid alone.

To check this conjecture, we have employed the second virial coefficient B2, which provides an integrated measure of the strength of interactions in the system. The virial coefficient of the reference attractive mixture is[61]withfor ij = 11 or 12, and B222 = 2πσ3/3. Only the first term on the r.h.s. of eq (with χ = 1) needs to be considered for the reference attractive HSTY fluid. In Figure we plot B2 and B211, for χ = 0.5 and different temperatures, as a function of ε12: for a given T, the intersection between B2 and B211 falls atwithand

Figure 10

Second virial coefficient of the reference attractive fluids for the equimolar mixture (full lines) and the pure HSTY (dashed lines) vs ε12 for various temperatures, in the legend. Intersections are marked by crosses.

For the temperatures in Figure , we have the pairs [T, ε12*] = [0.20, 0.5621], [0.22, 0.5465], [0.25, 0.5272], and [0.30, 0.5032], each marked by a cross in the figure. In particular, for T = 0.25, the reference attractive mixture will be more attractive than the reference attractive HSTY fluid for ε12 > ε12* = 0.5272 (see red lines and cross). Accordingly, the liquid–vapor binodal of the former falls above the binodal line of the latter for ε12 > ε12*, while lying below otherwise. This prediction is in gratifying agreement with results shown in Figures –9, confirming the validity of our approach: for T = 0.25 the equimolar mixture with ε12 = 0.50 is unable to sustain cluster formation, similar to the HSTY fluid; conversely, for ε12 = 0.75, the clustering tendency of the mixture will be enhanced with respect to the HSTY fluid.

We may wonder whether the same comparison also applies in reverse, so that clusterization is preempted in the mixture for the same density and temperature where clusters occur in the pure HSTY fluid. To verify this hypothesis, we take T = 0.22, where clusters are present in the pure HSTY fluid (see Figure ). At this temperature, the crossing of B2 lines occurs for ε12* = 0.5465 (see Figure , blue lines and cross). Structural data are presented in Figure for ε12 = 0.30 and 0.75, i.e., two values on opposite sides of 0.5465. While the height of the low-k peak is ≈4 in the pure HSTY fluid, the same peak falls below 2 in the mixture with ε12 = 0.30, meaning that with a proper choice of ε12 (to be discriminated on the basis of the B2 criterion) clustering can be suppressed. On the other side of the threshold, i.e., for ε12 = 0.75, the attraction is so intense that clusterization is strongly enhanced [S(k0) ≈ 30].

Figure 11

Equimolar mixture for T = 0.22. S11(k) for ε12 = 0.30 and 0.75. Results for the pure HSTY fluid of the same density and temperature are also reported for comparison.

To sum up, the crossing of the B2 vs ε12 lines falls in the narrow interval ε12 ≈ 0.50–0.56 in the T range 0.20–0.30, where the pure HSTY fluid experiences the crossover from a rather homogeneous fluid to a strongly structured cluster fluid. Our method provides a viable means to tune the onset of clustering in the equimolar HSTY–HS mixture; this tuning acts in both directions, namely enhancing or depressing the tendency to clusterization by just increasing or decreasing the strength of cross attraction at a given temperature. We have thus provided another assessment of the clustering picture proposed in refs (18, 19, and 23), according to which clusters develop for the same thermodynamic conditions where the reference attractive fluid undergoes liquid–vapor separation.

Our analysis also confirms that S(k0) ≈ 3 is a sensitive indicator of the onset of clustering. As we move around this value (in the present case, in the temperature range 0.22–0.25, see Figure ), clusters can be reversibly created or destroyed by adjusting ε12 on the basis of the relative balance in B2 between the (attractive) mixture and HSTY fluid. However, this tuning no longer works when we penetrate too much in the clustering region. Indeed, for T = 0.20 [where S(k0) ≈ 10 in the pure HSTY fluid, see Figure ] we find S(k0) ≳ 12 for ε12 = 0.75 and S(k0) ≳ 6 for ε12 = 0.30. Therefore, a change in ε12 will certainly affect the quality of the clustered state, but with no possibility to recover homogeneity.

Changing the Relative Concentration

Finally, we elucidate the effect of varying the relative concentration of the species on the formation of clusters. We set T = 0.25 and ε12 = 0.75 (allowing for the presence of clusters in the equimolar mixture, as shown in the previous section) and span the whole range of HSTY concentrations. The behavior of S11(k) is shown in Figure A, where, for the sake of comparison, we have also included S(k) of the pure HSTY fluid. We see that S11(k0) > 3 in almost all cases; in particular, S11(k0) falls between 4 and 5 for both χ = 0.05 and χ = 0.9, while rising above 10 between these extremes. As a result, at the given conditions of temperature and density, changing the concentration will only moderately affect the clustered nature of the mixture, with the only exception being χ = 0.01, where clusters are apparently absent. In particular, we can induce clustering in a fluid of hard spheres by replacing a fraction as low as 5% of HS particles with HSTY particles. Moreover, compared to the pure fluid, the main peak of S11(k) is more pronounced and better resolved; at the same time, k0 moves closer and closer to zero as hard spheres become progressively dominant in the mixture, pointing to the presence of increasingly large clusters. Therefore, the inclusion of HS particles invariably gives rise to stabilization of the clustered state, with the strongest effect near equimolarity, as demonstrated by the largest value of S(k0) attained for χ = 0.3 and 0.5. The tendency of HSTY particles to form clusters acts as a dragging mechanism for the structural properties of the whole mixture, as can be appreciated from the similarity between S11(k) and the total structure factor SNN(k) in Figure B. Some discrepancy only emerges for low χ, where indeed a large number of HS particles stay isolated, giving rise to a homogeneous background for clusters.

Figure 12

Mixture for T = 0.25 and ε12 = 0.75. (A) S11(k); (B) SNN(k). The whole range of HSTY concentrations is shown; see the legend.

Our conclusions, based so far on the sole structural evidence, are confirmed by the microscopic analysis of MC configurations. In Figure we show the CSD for various concentrations: for all concentrations examined, the CSD shows the typical features of a clustered state—specifically, a nonmonotonic decay accompanied by a well-definite local peak at the most probable size. All distributions vanish beyond s ≈ 100–200, therefore suggesting the absence of any tendency to percolation. Moreover, we see that comparatively larger clusters are stabilized in the range χ = 0.1–0.5, a feature already noted when analyzing structural correlations. Under the same conditions, the fraction of isolated (mainly HS) particles keeps relatively high, as shown in the inset.

Figure 13

CSD for T = 0.25, ε12 = 0.75, and a few selected values of χ (in the legend). Inset: fraction of isolated particles vs χ.

The simultaneous occurrence of large clusters and a high number of isolated HS for χ = 0.1 and 0.3 can be better appreciated in the snapshots reported in Figure ; in particular, a glance at panels A and B—where the same χ = 0.1 configuration is shown with and without isolated particles—indicates that HS (in orange) are homogeneously distributed across the sample. A few large clusters (formed by particles of both species in equal proportions) are immersed in the bath of isolated HS, thus corroborating the picture emerging from the structure. As the HSTY concentration increases (χ = 0.3, panel C) the coupling between HSTY and HS becomes more effective, with an ensuing smaller number of isolated HS. As said before, the optimal condition for mutual aggregation is at equimolarity where practically all HS particles are involved in the formation of aggregates (see Figure B and Figure , inset). Starting from χ = 0.7 onward (panels D and E), the opposite trend sets in, with a progressively higher number of isolated HSTY particles and the presence of many smaller clusters. The ultimate fate of this process is already known (see Figure A): as the HS are completely removed, clusters break apart, and the now pure HSTY fluid is nearly homogeneous.

Figure 14

Microscopic configurations for T = 0.25 and ε12 = 0.75. (A and B) χ = 0.1, with and without isolated particles; (C) χ = 0.3; (D) χ = 0.7; (E) χ = 0.9; (F) average eccentricity with standard deviation of the largest cluster as a function of χ.

To gain more information on the cluster structure, we have determined the shape distribution of the largest cluster across configurations, as a function of χ. To this aim, we have followed the method devised in ref (62), in which the cluster eccentricity ec is defined aswhere R and R are, respectively, the minimum and maximum values of the gyration radius Rg of the largest cluster and the average is taken over sufficiently many configurations. By definition, the value of ec ranges from 0 (perfect sphere) to 1 (ellipsoid). The method slightly differs from a similar protocol proposed in ref (63). The behavior of ec as a function of χ is reported in Figure F, along with standard deviations: clearly, no sensible dependence of ec on χ is found, with all values of eccentricity ranging between 0.23 and 0.32. This suggests that the largest clusters have a nearly spherical shape regardless of the relative concentration of the species.

One further remark is worth making in this context. We see in Figure D, corresponding to χ = 0.7, that HS are gathered in droplets essentially held together by an outer shell of SALR particles. This surprising outcome indicates the possibility of encapsulating inert particles by a second species with affinity to them, such that they are held apart from the solvent (here only implicitly defined).

We end this section by an analysis of B2 along the same lines followed before, now in terms of HSTY concentration for ε12 = 0.75 and fixed temperature. Results are shown in Figure : as seen, the two B2 lines cross for χ = 1 (which is trivial) and forFor the temperatures considered in the figure, we have the pairs [T, χ*] = [0.20, 0.1434], [0.22, 0.1432], [0.25, 0.1447], and [0.30, 0.1490], each marked by a cross. Hence, over the rather extended concentration range χ ≈ 0.14–1, the reference attractive mixture is more attractive than the reference attractive HSTY fluid. Accordingly, clusters would appear more structured in the mixture than in the HSTY fluid. At the same time, we must note a discrepancy between the B2 threshold (χ = 0.14) and the simulation threshold (χ ≈ 0.05). Despite this mismatch, our expectation that clusters are strengthened by replacing even a large fraction of SALR particles with HS is in accordance with the simulation.

Figure 15

Second virial coefficient of the reference attractive fluids for the mixture with ε12 = 0.75 (full lines) and the pure HSTY (dashed lines) vs χ for different temperatures, in the legends. Intersections are marked by crosses.

Compared to the ε12* dependence on T, now a weaker dependence of χ* on T is found. This is due to a minimum attained by χ* for T = 0.22; as a result, χ* slowly changes in the range from T = 0.20 to T = 0.30, which is a close neighborhood of the minimum. We finally observe that B2 of the reference attractive mixture reaches a minimum at χ ≈ 0.57. Coherently with our previous conclusion, the optimal conditions for maximizing the number of bonds (hence, for minimizing the energy) are met near equimolarity.

Conclusions

We have carried out Monte Carlo simulations of a model mixture of hard-sphere two-Yukawa particles and hard spheres, being coupled through an attractive square-well potential.

We have ascertained that the inclusion of HS particles can induce the formation of clusters under conditions for which the pure HSTY fluid would not allow for their appearance. As already recognized in ref (54), the tendency of HSTY particles to form clusters can indeed be enhanced by a second species having sufficient affinity to them. We have also verified that the same mechanism can be employed the other way around, so that clustering is hampered in the mixture for the same temperatures where they are present in the pure HSTY fluid.

By simulating the equimolar HSTY–HS mixture at sufficiently low temperatures, it emerges that clusters contain HSTY and HS particles mixed together in equal proportions. Upon moving away from equimolarity, we observe a net preference of the majority species at the cluster surface. In particular, hard spheres can be loosely encapsulated by HSTY particles and, in this way, separated from the (implicit) solvent.

For fixed strength of cross interaction, clustering turns out to be only weakly dependent on the HSTY concentration. In this regard, one of the most interesting results is that clusters can even be induced in a HS fluid by replacing a small fraction of spheres (as low as 5%) with HSTY particles, provided that a sufficiently large attraction (ε12 ≥ 0.75) exists between the species. Hence, given a sample composed by inert colloidal particles, clustering can be induced by even marginal chemical substitution with a species with competing interactions and a suitable degree of affinity to the host particles. The optimal condition for aggregation is realized near 50% concentration, where almost all particles participate in the formation of aggregates.

We have placed our results in the more general framework provided in refs (18 and 19), according to which cluster formation takes place under the same conditions where the reference system (without long-range repulsion) would give rise to liquid–vapor phase separation. To substantiate this idea, we have computed the second virial coefficient B2 of the reference attractive mixture and compared it with the B2 of the similarly modified HSTY fluid. It emerges that clustering in the mixture—for fixed temperature and density—is invariably obtained as it becomes more attractive than the pure HSTY fluid. Conversely, when the B2 of the reference attractive mixture falls below that of the pure fluid, clustering is undermined in the HSTY–HS mixture. This indication is in perfect agreement with the evidence got from the structural and the microscopic analysis.

In the near future, we plan to explore the space of model parameters further, in particular by considering other values of the square-well width and also varying the relative size of the two species. To this aim, aside from simulations, we plan to resort on refined, thermodynamically self-consistent integral equation theories of the fluid phase.