Self-assembly in a model colloidal mixture of dimers and spherical particles
Santi Prestipino, Gianmarco Munaò, Dino Costa, Carlo Caccamo
SSelf-assembly in a model colloidal mixture of dimersand spherical particles
Santi Prestipino ∗ , Gianmarco Muna`o, Dino Costa, and Carlo Caccamo Dipartimento di Scienze Matematiche ed Informatiche,Scienze Fisiche e Scienze della Terra, Universit`a degli Studi di Messina,Viale F. Stagno d’Alcontres 31, 98166 Messina, Italy (Dated: September 23, 2018)
Abstract
We investigate the structure of a dilute mixture of amphiphilic dimers and spherical particles, amodel relevant to the problem of encapsulating globular “guest” molecules in a dispersion. Dimersand spheres are taken to be hard particles, with an additional attraction between spheres and thesmaller monomers in a dimer. Using Monte Carlo simulation, we document the low-temperatureformation of aggregates of guests (clusters) held together by dimers, whose typical size and shapedepend on the guest concentration χ . For low χ (less than 10%), most guests are isolated andcoated with a layer of dimers. As χ progressively increases, clusters grow in size becoming moreand more elongated and polydisperse; after reaching a shallow maximum for χ ≈ χ further. In one case only ( χ = 50% and moderately lowtemperature) the mixture relaxed to a fluid of lamellae, suggesting that in this case clusters aremetastable with respect to crystal-vapor separation. On heating, clusters shrink until eventuallythe system becomes homogeneous on all scales. On the other hand, as the mixture is made denserand denser at low temperature, clusters get increasingly larger until a percolating network is formed. PACS numbers: 61.20.Ja, 64.75.Xc, 64.75.Yz ∗ Corresponding author. Email: [email protected] a r X i v : . [ c ond - m a t . s o f t ] F e b . INTRODUCTION Supra-molecular self-assembly, i.e. , the spontaneous formation of organized structuresfrom pre-existing simpler building blocks, is among the prominent features of soft mat-ter [1]. At present, the list of morphologies observed in experiments includes micelles ofvarious shapes, lamellae, gyroids, networks, and vesicles [2, 3]. The ability to control self-assembly is an important aspect in the fabrication of synthetic materials tailored for targetedapplications [4, 5].In simulations, self-assembly has often been studied in systems of “patchy” particles [6–15], i.e. , spherical particles with a handful of interaction spots on their surface, offeringthe advantage of a flexible angle-dependent interparticle potential that can be tuned toattain the desired equilibrium structures. In the last decade, these systems have gainedutmost importance owing to the possibility to synthesize colloidal particles mimicking awide assortment of patchy particles [5, 16, 17]. Interestingly, a wide variety of patternsis also observed in simpler systems of Janus dimers [18–27], i.e. , heterodimers where onemonomer is lyophilic and the other one is lyophobic. In essence, Janus dimers representthe “molecular” analogue of Janus spheres [28–31]; like their “atomic” counterparts, Janusdimers are able to organize into various super-structures, such as micelles and bilayers, aswell as to separate into vapor and liquid [32].Very recently, we have proposed heteronuclear dimers as encapsulating agents for spher-ical particles [33]. Aside from its fundamental interest, the reversible formation of cap-sules [34, 35] around a dissolved target species (“guest”) is an issue of growing importancein the pharmaceutical field, where it finds application in drug delivery [36–38], and for foodindustry as well, as a tool to improve the delivery of bioactive substances into foods [39, 40].In our model [33], a dimer consists of a pair of tangent hard spheres of different size, withan additional square-well attraction between small monomers and spheres. With a mini-mum number of parameters this model allows for the formation of non-trivial structuresat equilibrium, which depend on the relative size of dimers and spheres. For a mixture ofa small fixed density, we have demonstrated [33] the formation of coating layers of dimersonto guests (“capsules”), at least provided that the guest concentration is sufficiently low.This basic model is meant to represent, within an implicit-solvent description, a colloidaldispersion made of a guest species and a Janus dimer where the small monomer is at once2yophobic and shows a strong affinity for the guests. Such a system may be employed, forinstance, to describe at a basic level the coating of dispersed proteins by colloidal particles,where the proteins are represented in terms of either central potentials or site-distributedinteractions [41–43]. More important, colloids can be designed with the characteristics ofour model [5, 16, 17, 44, 45], implying that its self-assembly can be probed experimentally,at least in principle.In this paper we explore by Monte Carlo simulation the self-assembly of the same mixturestudied in Ref. [33], allowing the overall density and the sphere concentration to span overa wider range than previously considered. In the present analysis, the sphere size is fixedto that of the large monomers. By this choice, we aim to represent a generic colloid wherethe dimers are tailored to roughly match the size of the guest to be captured. We areinterested in working out the full self-assembling behavior of the system as a function ofthe sphere concentration χ at low density. In essence, our simulations show the existenceof aggregates of spheres held together by dimers, reaching the largest size for a moderatevalue of χ . By performing a detailed microscopic analysis, we identify and characterize theclusters present in the system, from capsules to more open structures of tubular shape. Inone case only ( i.e. , at equimolar concentration and a moderately low temperature), particlesspontaneously self-assembled into a lamellar aggregate, which demonstrates the existence ofa competition between organized structures of different nature. Finally, upon increasing thesystem density at fixed concentration we observe the formation of a percolating network ofguests (“gel”).Mesoscale structures are usually the outcome of competing short-range attractive andlong-range repulsive interactions [46]; they are observed in such diverse fields as magneticalloys [47], Langmuir films [48], and protein solutions [49]. Characterization of clusters isalso relevant to nucleation (see, e.g., Refs. [50, 51] and references therein), even though inthis case they are ephemeral aggregates rather than permanent structures. Finally, spheri-cal, cylindrical, tetragonal, and other more exotic (but metastable) clusters [52] also occurin simple fluids, although exclusively within a two-phase coexistence region. In general, theformation of mesoscopic aggregates in fluids with competing interactions supersedes liquid-vapor coexistence, leading instead to microphase separation (also dubbed intermediate-rangeorder; see, for example, Refs. [46, 53–55]). We have computed an effective guest-guest poten-tial, whose profile changes with thermodynamic parameters in line with the aggregates found3n simulation: consistently, at low temperature this potential shows short-range attractiveand long-range repulsive components. The repulsive part provides the limiting factor forthe growth of clusters; in the present case, it results from the segregation of attractive sitesin the cluster interior.The outline of the paper is the following. After describing the model and the method inSection 2, we present and discuss our results in Section 3. Some concluding remarks andperspectives are finally given in Section 4. II. SYSTEM AND METHODA. Model
In our system, a dimer is modeled as a pair of tangent hard spheres with different diam-eters, σ and σ . We hereafter assume a fixed size ratio σ /σ of 3, which ensures that thedimers effectively isolate the guest from the solution in those cases where capsules form [33].Guest particles (species 3) are represented as hard spheres of diameter σ = σ . All particleinteractions are hard-sphere-like with additive diameters σ αβ = ( σ α + σ β ) /
2, except for theinteraction between the smaller monomer in a dimer (species 1) and a guest sphere, whichis given by a spherically-symmetric square-well potential: u ( r ) = ∞ for r < σ − ε for σ ≤ r ≤ σ + ∆0 otherwise , (2.1)∆ = σ being the square-well width. In the following, σ and ε are taken as units of lengthand energy respectively, which in turn defines a reduced distance r ∗ = r/σ and a reducedtemperature T ∗ = k B T /ε ( k B being Boltzmann’s constant). Finally, we denote by N and N the number of dimers and guests, respectively. Hence N = N + N is the total numberof particles. B. Simulation
Monte Carlo (MC) simulations have been performed in the canonical ensemble, using thestandard Metropolis algorithm. Canonical conditions seem more appropriate in our context,4ince they allow to represent a mixture of dimers and spheres in the realistic setting wherethe relative amount of the two species is fixed from the outset. In our runs, the numberof particles of each species is chosen according to the prescribed concentration of guests, χ = N /N with N = 400; only for the lowest analyzed concentration ( χ = 10%) we haveset N = 200. A small number of runs for larger samples have also been carried out, in orderto check the irrelevance of finite-size effects. Particles are initially distributed at randomin a cubic simulation box of volume V , with periodic conditions at the box boundary; MCevolution from such a configuration thus mimics relaxation of the system to equilibriumafter a quench from very high temperature (we checked in a few cases that gradual coolingof the system from high temperature gives essentially the same results). For most of ourruns we work with an overall system density ρ ∗ = N σ /V = 0 .
05, a relatively low value.Moreover, we have considered one case ( χ = 33%) where the system is progressively madedenser at fixed temperature.A single MC cycle consists of N trial single-particle moves. Depending on the type ofparticle, one trial move is a simple translation or a random choice between center-of-masstranslation and rotation about a coordinate axis. The acceptance rule as well as the scheduleof the moves are designed in such a way that detailed balance is satisfied. The maximumrandom shift and rotation are adjusted during equilibration so as to keep the ratio of acceptedto total number of moves close to 50% for T ∗ ≤ .
20, and to 60 ÷
70% otherwise. To speedup execution we have implemented linked lists in our computer code, which turns out tobe especially useful at low temperature ( T ∗ ≤ .
15) where equilibration runs are longer.For χ = 33% we have also evolved the system with the aggregation-volume-bias (AVB)MC algorithm [56]. While the AVB simulation converged quickly than the standard MCsimulation for T ∗ ≥ .
20, the AVB algorithm turns out to be far less efficient than theMetropolis algorithm for T ∗ = 0 .
10. Rejection-free moves [57] is another alternative to bareMetropolis, which could be considered for future studies.In order to decide whether the system has reached equilibrium for assigned values of χ and T we look at the evolution of the potential energy U with the number of MC cyclesperformed: a total energy fluctuating around a fixed value for long is the hallmark of stableequilibrium. For T ∗ = 0 .
20 or larger, 5 × MC cycles are well enough for attainingequilibrium, while we had to generate from 3 to 9 × cycles (depending on χ ) beforecomputing the structure quantities for T ∗ = 0 .
10. Indeed, only for this temperature (which5s the lowest temperature investigated) the value of U needed a conspicuous number of cyclesto level off.In the production runs, which are typically 10 cycles long, we compute the radial distri-bution functions (RDF) g αβ ( r ) (with α, β = 1 , , i.e. , every 1000 cycles) and classifying as a function of size connected struc-tures of guest spheres by the Hoshen-Kopelman algorithm [58] adapted to continuous space.Any such structure clearly represents the backbone of a cluster. The criterion used to qualifytwo spheres that are close to each other in space as bound is a relative distance smaller thanthe abscissa r min of the g first minimum for T ∗ = 0 .
10 ( r ∗ min varies in the interval 1 . . χ ranges from 10% to 80%). Then, for each cluster we compute its “size”, namely thenumber of spheres it contains, the binding energy E b (that is, the number of 1-3 contacts),and also count the number N dim of dimers hosted in the cluster ( i.e. , those dimers that arebound with at least one sphere of the cluster). The analysis is completed by determiningthe statistical distribution of the number n NN of guests that are nearest neighbors to thesame sphere and the distribution of the angle α formed by two 3-3 bonds having one spherein common. III. RESULTSA. Energy per guest particle
In our previous study [33], we found that guests with size comparable to that of dimers( σ = σ / σ = σ , with σ = σ /
3) can be encapsulated for all concentrations upto 20%, at least provided that the temperature is sufficiently low ( T ∗ = 0 .
15 or smaller,see Table 3 of Ref. [33]). As the guest size increases, progressively lower concentrationsare required to obtain encapsulation. For large guest sizes and not too low concentrations( σ = 3 σ and χ ≥ IG. 1: Left: MC evolution of the potential energy U per sphere, for various χ (see the legend)and for T ∗ = 0 . , . , .
20 from bottom to top (for χ = 50% and 80%, the case T ∗ = 0 .
15 isomitted). Right: average potential energy per sphere as a function of temperature. Error bars aresmaller than the symbols size. ( σ = σ ).We first comment on the simulation results for a system of density ρ ∗ = 0 .
05. In theleft panel of Fig. 1 we report some representative cases of U evolution in the course of thesimulation run. While equilibrium is attained relatively quickly for T ∗ ≥ .
20, thermalrelaxation is slower for T ∗ = 0 .
15, and even more so for T ∗ = 0 .
10, where we needed toproduce several hundred million cycles before U could level off. For example, for χ = 33%energy roughly stabilizes only after 7-9 × MC cycles, while relaxation is somewhat fasterfor the lowest and highest concentrations investigated in this work. After the U curve hasflattened to a sufficient degree we compute the average potential energy (cid:104) U (cid:105) over a trajectoryslice made of 10 cycles. This quantity is plotted in the right panel of Fig. 1 as a functionof temperature for the various concentrations. At every temperature, |(cid:104) U (cid:105)| is higher thelower the concentration of guests. Indeed, as it appears from the subsequent analysis, when7ew guests are present, dimers tend to coalesce around them, at least so long as spacepermits, this way maximizing the number of attractive contacts. Moreover, all (cid:104) U (cid:105) curvestend to merge near T ∗ = 0 .
25, which may thus be taken as an estimate of the maximumtemperature for which well-definite aggregates occur in the system. For low concentrations,the number of dimers that are bound to a single sphere grows substantially on cooling;apparently, this number has not yet saturated for T ∗ = 0 .
10. For χ = 50%, the value of (cid:104) U (cid:105) slightly rises as we turn from T ∗ = 0 .
15 to 0.10, indicating that longer equilibrationswould be necessary for T ∗ = 0 .
10, and the same conclusion applies for χ = 33% with regardto the transition from 0.10 to 0.05. We further point out that the energy plots for largersamples with N = 4000 , χ = 10% and N = 2400 , χ = 33% are hardly distinguishable fromthe corresponding plots for N = 2000 and N = 1200, respectively, hence finite-size effectsare negligible in the present case.Visual inspection indicates that finite-size aggregates of guests glued together with dimers(“clusters”) are formed at low T for all concentrations. These clusters have also been ob-served in our previous work on the same model [33], but no systematic study of their structurewas attempted there. The nature of such clusters is various: clusters have the form of cap-sules for χ = 10% or lower [33], while they are more elongated and polydisperse for higherconcentrations (a thorough analysis of the cluster shape and structure as a function of χ and T is provided in the next Section 3.2). Clearly, the value of ∆ in Eq. (2.1) is crucial todetermine what kind of aggregates are observed at low temperature (a smaller, yet non-zero∆ would result in a lower T ∗ threshold for cluster formation and, in addition, in a slowerrelaxation to equilibrium). In one thing all clusters are equal: once they have ceased to growand their shape has become relatively stable, almost all the small monomers contained areburied under the surface.Now turning to the case of T ∗ = 0 .
10, it is difficult to say whether true equilibrium hasbeen reached in our simulations. As just said, after an initial stage of fast growth, the sizeand shape of clusters stop to change appreciably. By looking at the snapshot of the systemat regular intervals, we see that its late MC evolution mostly consists of tiny adjustmentsof the clusters already present; the reason is that a strong attraction between dimers andguests is an obstacle to achieving local equilibration, because it prevents substantial particlereshuffling among the clusters. However, rather than merely signaling a slow approach toequilibrium, the persistent decrease of (cid:104) U (cid:105) at low T might be the clue to a regime of very8low kinetic aggregation. We are perfectly aware that Markov-chain dynamics has littleto do with the true, Hamiltonian dynamics of the model; we nevertheless expect that theequilibration stage in a MC simulation still keeps some of the characteristics held by thetrue non-equilibrium dynamics, as could be observed in a molecular-dynamics simulation.As is known, two distinct regimes of irreversible aggregation are found in colloids (see, e.g. ,Ref. [59]): (i) diffusion-limited aggregation, which occurs when colloidal particles attract eachother strongly, so that the aggregation rate is solely limited by the time taken for particlesto encounter each other by diffusion; (ii) reaction-limited aggregation, which instead occurswhen there is still a substantial repulsion between the particles, so that the aggregation rateis limited by the time taken for two of them to overcome this barrier by thermal activation.These regimes respectively correspond to the cases of fast and slow aggregation. In oursystem, it is unlikely but definitely not impossible that two clusters whose attractive spotspreferentially lie under the surface can meet in the course of the run and join together toform a bigger aggregate. According to the above classification, this can be recognized as (arather extreme case of) reaction-limited aggregation. B. Clusters structure
In Fig. 2, the last system configuration generated in our MC runs is reported for T ∗ = 0 . χ investigated ( χ = 10%, case (a) in Fig. 2), we notethat nearly compact capsules occur in large number, in a medium of isolated dimers. Mostguests are associated in pairs and fully covered with dimers in such a way that no largepores are visible in the coating layer. Hence, in a regime of concentrations lower than χ = 10% the guest particles would not be in direct contact with solvent particles (however,here only implicitly present). The overall looking of the mixture is similar in a larger systemof N = 400 guests and N = 4000 dimers, as confirmed by a nearly identical U evolution(not shown). Moving to χ = 20% (case (b) in Fig. 2), we find larger clusters coagulatingin the system in the long run, with an appreciable fraction of guest spheres exposed totheir surface. The biggest aggregates now occurring are elongated curved objects, whichno longer resemble spheroidal capsules. For χ = 33% (cases (c) and (d) in Fig. 2) clustersare even larger than before, but still non-percolating (bottom-left panel). Upon doublingthe temperature (bottom-right panel), clusters reduce in size and the “mobility” of particles9 a) χ = 10% , T ∗ = 0 .
10 (b) χ = 20% , T ∗ = 0 .
10 (c) χ = 33% , T ∗ = 0 .
10 (d) χ = 33% , T ∗ = 0 . χ = 50% , T ∗ = 0 .
10 (f) χ = 50% , T ∗ = 0 .
20 (g) χ = 80% , T ∗ = 0 .
10 (h) χ = 80% , T ∗ = 0 . FIG. 2: Final system configurations for different χ and T ∗ (red: spheres; cyan: large monomers;blue: small monomers). increases substantially, as witnessed by the faster energy relaxation and the much larger U fluctuations in the course of the run.While clusters are still big at equimolar concentration (cases (e) and (f) in Fig. 2), verymuch as for χ = 33%, the average cluster size is definitely smaller for χ = 80% (cases (g)and (h) in Fig. 2), where the number of unbound spheres is high even for T ∗ = 0 .
10. Thereason is clear: the few dimers present are insufficient to bind all the guests and, as a result,very large clusters are not formed. Again, heating acts against aggregation ( i.e. , clustersshrink on increasing T ), making evaporation of particles from the cluster surface easier. Asa last comment, we note that ring-shaped portions occasionally appear in some cluster (seean example in Fig. 3, which refers to χ = 33% and T ∗ = 0 . IG. 3: Snapshot of a long-equilibrated system for χ = 33% and T ∗ = 0 .
15 (here N = 1600 and N = 800; same notation as in Fig. 2). A clear ring is observed in a cluster located near the bottomof the simulation box. The foregoing discussion, based on the few sketches of the system structure provided inFig. 2, will now be substantiated by an explicit cluster analysis, already described in Sect. 2.2.For T ∗ = 0 .
10 and 0.15, averages have been computed from the last 10 cycles only, whilefor larger temperatures 5 × cycles are well enough for an accurate cluster statistics. Wefirst show a comparison between simulation data relative to various concentrations at fixedtemperature, T ∗ = 0 .
10. On the left in Fig. 4, we show the average number N cl of clustersof equal size in a single configuration, as a function of the size s (a cluster of size one is justan isolated sphere). On the right panels, we report the statistics of the number N dim ( s ) ofdimers hosted in a cluster, and of the number E b ( s ) of 1-3 contacts.We see that statistical distributions are noisy for all values of χ , which again provesthat system evolution for T ∗ = 0 .
10 is extremely slow. Gaps in the distributions reflectinsufficient statistics, a problem which is more severe for the largest sizes. As far as thecluster-size distribution N cl ( s ) is concerned, we find that the average number of guest spheresper cluster is about 2.5 for χ = 10%, 9.3 for χ = 20%, 49.6 for χ = 33%, 53.6 for χ = 50%,and 19.2 for χ = 80% (excluding isolated spheres from the counting). Hence we confirm thatthe biggest clusters occur at moderate concentration. For the same χ values the cluster-sizedistribution is very dispersed around the mean, while the sharp peak at 1 for χ = 80% simplyreflects the existence of a large number of isolated guests at this concentration. The averagenumber N dim of dimers hosted in a cluster is almost linearly dependent on size, suggesting ahomogeneous distribution of dimers within a cluster. The slope of N dim vs s slightly grows11 IG. 4: Left: cluster-size distribution for T ∗ = 0 .
10 and different χ (in the legend). Right: averagenumber of dimers (top) and attractive contacts (bottom) in a cluster. with decreasing χ , indicating a highest density of dimers in the clusters (mostly capsules)for χ = 10%. Also the cluster binding energy E b grows linearly with size, showing a strongcorrelation with N dim . In Fig. 5, the cluster statistics is reported at fixed concentration( χ = 33%) as a function of temperature. Note, in particular, how the statistical accuracyof N cl ( s ) substantially improves with increasing temperature, while its leading maximummoves towards lower sizes, until most guests become isolated for T ∗ = 0 . T clusters is their endurance,or resistance to add/lose particles, which makes the measured N cl a very irregular functionof size. Although the features of N cl ( s ) for T ∗ = 0 .
10 are hidden by noise, we expect tofind a two-peak structure at large χ where, besides the maximum at one, another shallowmaximum corresponds to the most probable s value in a broad distribution of sizes. As χ is reduced, the maximum at one gets strongly depressed, and has already washed away for12 IG. 5: Left: cluster-size distribution for χ = 33% and different T ∗ (in the legend). Right: averagenumber of dimers (top) and attractive contacts (bottom) in a cluster. χ = 50% (this maximum is eventually recovered upon heating, see Fig. 5 left panel). Theother maximum is roughly located near the average cluster size; hence, its abscissa wouldshow a non-monotonous χ dependence: it first grows on approaching χ = 50% from aboveand then drops, until finally reaching the value of two for χ = 10%. On the contrary, theaverage values of N dim ( s ) and E b ( s ) are much less uncertain (despite the many gaps present),since the relative errors are not so large as to obscure their linear behavior with s .The spatial distribution of guests within the clusters can be investigated by collecting inthe course of the run the statistics of a few indicators sensitive to the local sphere environ-ment. We first present results relative to (i) the distribution P ( α ) of the angle α formedby two nearest guest-guest bonds, i.e. , two bonds sharing one sphere (at the angle vertex),and (ii) the statistics of the number of guest spheres that are nearest neighbors to a givensphere ( i.e. , the “coordination number” of a sphere; clearly, all the neighbor spheres belongto the same cluster of the central sphere).The α distribution is reported in Fig. 6 for χ = 33% and T ∗ = 0 .
15 (its overall shape is13
IG. 6: Distribution P ( α ) for χ = 33% and T ∗ = 0 .
15 (line with dots). We also report the separatecontributions from vertex spheres characterized by a different coordination number n NN (in thelegend). similar for other concentrations, being poorly defined only for small χ ). In the same figure, P ( α ) has been resolved into separate contributions, according to the number of neighborsowned by the vertex sphere. We see three maxima occurring in P ( α ), at 60 ◦ , ≈ ◦ , and ≈ ◦ , respectively. The maximum at 60 ◦ is very sharp: it corresponds to three guestspheres that are in reciprocal contact. The other two maxima are much broader. Thecentral peak is actually originated from the merging of two distinct peaks: one centerednear 90 ◦ , which marks a distinct preference of the highest-coordinated spheres for a localsquare ordering (see more in Section 3.4 below); the other peak, occurring close to 109 . ◦ , isthe signature of a diffuse tetrahedral arrangement of fourfold-coordinated spheres within theclusters, indeed seen in the snapshots. The third maximum just results from the interpositionof a guest sphere, in close contact with the vertex sphere, between the other two spheresengaged in the angle. Finally, we show in Fig. 7 the distribution function of n NN . This14 IG. 7: Distribution P ( n NN ) of the sphere coordination number n NN , for T ∗ = 0 .
10 and variousguest concentrations χ (in the legend). number counts how many guest spheres are nearest neighbors to a given reference sphere.This number typically grows with χ , moving from 1 at low concentration to 4 ÷
5. The sharpmaximum at zero observed for χ = 80% merely results from the large number of isolatedguests. C. Effective guest-guest interaction
The propensity of guest spheres to bind together at not-too-small χ provides evidence thatthe dimer-mediated interaction between guests has an attractive component. An effectivepair potential between guests can be constructed as follows. We first use the Ornstein-Zernike relation [61] to compute the direct correlation function c ( r ) from the numericalprofile of g ( r ). Then, both functions are plugged in the hypernetted-chain (HNC) clo-sure [61] to finally obtain the guest-guest potential φ ( r ). We expect this HNC scheme tobe valid up to at least moderate concentrations. Typical results are shown in Fig. 8, which15 IG. 8: Left: g ( r ) (with corresponding S ( q ) in the inset) for χ = 33% and various T ∗ (see thelegend). Right: guest-guest potential φ ( r ) obtained from the HNC inversion of g ( r ). refers to χ = 33% (the shape of φ ( r ) is only weakly dependent on χ ). We see that a highlystructured g ( r ) ( T ∗ = 0 .
15 and 0.20) goes along with a non-trivial profile of φ ( r ), showinga hard-core, a short-range well, and a repulsive hump for larger distances. This profile isconsistent with the existence of clusters characterized by an average cross radius of ≈ σ .At sufficiently high temperature, the spatial distribution of guests is nearly homogeneousand the effective interaction becomes more hard-sphere-like.We have not tried to use closures more sophisticated than the HNC; on the other hand,the simpler low-density approximation, i.e. , g ( r ) = exp {− βφ ( r ) } is insufficient to obtainnon-trivial structure in φ ( r ) at low temperature, while it reproduces the HNC outcome athigh temperature. In the inset of Fig. 8 we show the computed guest-guest structure factor S ( q ). Intermediate-range order should be evidenced in the presence of a peak at a small q value, but no such peak is present in S ( q ). The reason of this is merely numerical: a muchlarger system size and much longer simulations would be needed in order to see this featureeventually appear at low temperature. 16 IG. 9: g ( r ) (left) and g ( r ) (right) for T ∗ = 0 .
10 and various χ (in the legend). Insets show amagnification of the short-distance region. A more complete set of the relevant RDFs of the system is plotted in Fig. 9 for T ∗ = 0 . i.e. , the radial structure within distances smallerthan the cross radius of a typical cluster) is well accounted for by the conventional RDF. Asa first comment, we note that the development of an enormously large g value at contact isthe most distinct signature of the formation of sphere aggregates at low temperature. Thisis evident for all concentrations, even the higher ones. These aggregates are more extendedfor intermediate χ values, where the short-distance structure in g is richer. As to the g ( r ) function, its short-distance profile is sharper for the lowest concentrations, where thenumber of dimers that are in close contact with the same sphere is higher. Instead, thesomewhat high third-neighbor peak at intermediate concentrations is another signature ofthe existence of more extended aggregates of dimers and guests for such χ values.Finally, we have computed the second virial coefficient B of the mixture as a function17f the temperature, seeking for its vanishing at fixed χ . The value of T where B = 0 (theBoyle temperature T B ) gives an indication of the threshold below which attractive forcesstart to be effective in the system. The expression for B is: B = (1 − χ ) B DD2 + 2 χ (1 − χ ) B DS2 + χ B SS2 , (3.1)where in the rhs the partial contributions to B are from a dimer pair (DD), a dimer anda sphere (DS), and a sphere pair (SS; for hard spheres, B SS2 = (2 / πσ ). For instance, thedimer-dimer contribution reads: B DD2 = − π V (cid:90) d R d Ω d R d Ω (cid:0) e − βU − (cid:1) , (3.2)where, e.g. , R and Ω are the coordinates defining the position and orientation of dimer 1,and U denotes the interaction energy between dimers 1 and 2. We have computed B DD and B DS by Monte Carlo integration (see Ref. [62] for details). In Fig. 10 we report our resultsfor T B as a function of χ ( B has been computed in steps of ∆ T ∗ = 0 . T ∗ ≈ . T B ( χ ) at χ (cid:39)
50% will conform withthe expectation that the larger the clusters, the wider is their range of stability.The B data in Fig. 10 reveal that attractive forces between spheres and dimers becomeincreasingly strong upon cooling. A large negative B would suggest a tendency of themixture to phase separate at low T ; on the other hand, we may look at the sign of thesecond virial coefficient relative to spheres only, as computed from the effective interactionbetween spheres, φ ( r ) in Fig. 8. From the right panel, which reports φ ( r ) for χ = 33% atvarious temperatures, we see that the effective potential is largely positive for T ∗ = 0 . B is positive as well. Hence, we surmise that the clusterfluid is stable and phase separation, if any, would only occur for temperatures below 0.15. D. Evidence of lamellar structure at low temperature
We found an unexpected behavior when relaxing the χ = 50% system for T ∗ = 0 . × cycles, the energy dropped below the T ∗ = 0 .
10 level (see Fig. 11top panel). Looking at the system snapshot beyond the crossing point, a new type of self-assembly emerges (Fig. 12) where most of the particles are arranged into an ordered lamellar18
IG. 10: B vs. T ∗ for various χ (see the legend). In the inset, T ∗ B is plotted as a function of χ . structure, that is a planar arrangement where the guest spheres are bound to a double layerof dimers. In Fig. 12 the observed lamellae are actually three, fused together along an edge,and we also see a short straight tube floating in the simulation box. A clue to how particlesare arranged in a lamella can be obtained from the distribution of the angle α between twonearest guest-guest bonds.Now, the leading maximum by far occurs for α = π/ n NN distribution reaches its maximum for a number of neighbors equal to four (Fig. 12,right-bottom panel), which is consistent with the proposed picture. The structure of thetube is different: four rows of spheres are alternated to four rows of dimers, all beingwrapped around another chain of spheres (an arrangement somehow reminiscent of that19 IG. 11: MC evolution of the potential energy U per sphere under different concentration, tem-perature, and density conditions. of metal atoms in nanowires [63]). The significance of this outcome is that a lamellar solidcoexisting with vapor competes for stability at low T with a fluid of clusters, at least for guestconcentrations close to 50%. The present scenario bears a strong similarity to what occursin a one-component system of one-patch spheres [60], where the cluster fluid is metastableat low temperature.To summarize, for T ∗ = 0 .
15 the simulated sample eventually succeeded to reach equi-librium, thanks to a non-negligible particle diffusivity, and apparently separates into vaporand lamellar solid, as it does the system studied in Ref. [60] in a range of thermodynamicparameters (see Fig. 10 therein). For the lower temperature of T ∗ = 0 .
10, clusters are stillformed very early in the system but they did not subsequently evolve, probably for lack oftime.Even though we are not in a position to draw an accurate phase diagram, we attempta sketch in Fig. 13 for ρ ∗ = 0 .
05. Similarly as argued for χ = 50%, for T ∗ = 0 .
10 we20
IG. 12: Properties of the mixture for χ = 50% and T ∗ = 0 .
15. The top panels report the sameconfiguration as seen from different perspectives. The total number of guests is 400: 363 spheresbelong to the lamellar aggregate, while 33 stay in the tube; finally, four spheres are isolated.Bottom: distribution of the angle α formed by two nearest guest-guest bonds (left, same notationas in Fig. 6) and the distribution of the sphere coordination number n NN (right). conjecture the metastability of clusters throughout the whole range of guest concentrations(red symbols in Fig. 13), especially considering that the average potential energy seems tostill evolve in such conditions (see Fig. 1).The present results can be compared with those obtained in Ref. [71], where a mixture ofspheres and spherical patchy particles with two or more patches is studied in two dimensionsby MC simulation. Leaving aside the question of the different dimensionality, the strongestsimilarity with our mixture is reached for patchy particles with two equal wide patches. In21 IG. 13: Putative phase diagram of the mixture for ρ = 0 .
05. Cyan and red symbols denote stableand possibly metastable clusters, respectively. that case, at relatively high density various types of vapor-solid separation are reported atlow temperature, similarly as found in our equimolar mixture.
E. Gelation
A final question to address concerns the existence of a sol-gel transition (gelation) in amixture of heteronuclear dimers and spherical guests. Gelation is a structural transition(the mean cluster size diverges within finite time) [64–66]. Colloidal particles with a mutualattraction much stronger than the thermal energy exhibit a fluid-to-solid gelation transfor-mation at a critical, temperature-dependent volume fraction [67]. The sol-gel transition ismanifested in the onset of a spanning cluster , which gives rise to the divergence of viscosityas the transition point is approached from the sol phase, and to a vanishing elastic modulusas the transition is approached from the gel phase (see, e.g. , Ref. [68]). A percolating, orspanning cluster has at least one particle on each of the six faces of the simulation box.22
IG. 14: Final system configuration for χ = 33% , T ∗ = 0 .
15, and ρ ∗ = 0 .
15. A percolating structureformed by all particles in the system is clearly visible on the left. On the right, the large monomershave been removed in order to highlight the cluster backbone. Note that the disconnected partsare an artifact of periodic boundary conditions.
Like percolation, gelation has many of the hallmarks of a critical phase transition. As thegelation point is approached, the sol viscosity, the shear modulus, and the mean clustersize all exhibit power-law behavior with characteristic exponents. Gelation is accompaniedby kinetic arrest due to crowding of clusters; like the glass transition, gelation is akin to ajamming transition.According to the evidence presented so far, no gelation occurs in our system for ρ ∗ = 0 . i.e. , no sign is ever found of a percolating network of guests. But it might as well be that thesimulated system is simply too dilute (and T ∗ = 0 .
10 is too low a temperature) for gelationto occur within affordable time. To check this intuition we have gradually increased thedensity of the χ = 33% system, up to ρ ∗ = 0 .
30, while keeping the temperature fixed at T ∗ = 0 .
15. As expected, the asymptotic value of the potential energy slightly decreases uponcompression (see Fig. 11 bottom panel). As ρ increases, the clusters occurring in simulationbecome larger and larger until, for ρ ∗ ≈ .
15, a single cluster encompassing almost all systemparticles appears for the first time and maintains practically unaltered in the subsequentpart of the MC evolution (see Fig. 14 for a visual representation of the gelified system; in thecase shown, only one sphere out of a total of 400 is not part of the spanning cluster). It goeswithout saying that we do not have the necessary resolution in density and temperatureto see whether the characteristics of gelation for the present model do actually conformwith the generic expectation of a “critical” transition. Finally, both n NN and bond-angle23istributions of the gelified system are found to be relatively independent of the particledensity (be it ρ ∗ = 0 .
15 or larger), being quite similar to the corresponding distributions for ρ ∗ = 0 . ρ - T diagram. Should the glass line meet the phase-coexistence locus on its high- ρ branch, the gel formed would be an arrested two-phase coexistence state. However, since wehave neither a precise idea of the overall mixture phase diagram, nor any knowledge of thelocation of the glass-transition line, the question about the nature of the percolating clusterremains presently unsettled. IV. CONCLUSIONS
We have investigated the self-assembly of a mixture of heteronuclear dimers and sphericalparticles. All interactions are of hard-sphere type; in addition, a square-well attraction ispresent between a guest sphere and the smaller particle in a dimer. We have fixed the spherediameter to that of the larger monomer, with the purpose to describe a colloidal mixturewhere all solute particles are similar in size. Using Monte Carlo simulation, we have mappedthe full emergent behavior of the system as a function of the sphere concentration χ , evenfar beyond the low- χ encapsulation regime.Our findings can be summarized as follows. At low temperature, the tendency of parti-cles to reduce internal energy produces quite distinctive organized structures in an overalldilute mixture. These structures have the character of spheroidal clusters for low or veryhigh concentrations, while looking more like curved tubes for intermediate χ values. Theseaggregates are rather different from the precipitates that form when small molecules attracteach other in a poor solvent, as well as from the liquid clusters spontaneously arising in asupercooled vapor. The difference mainly originates from the fact that in an initially homo-geneous mixture of dimers and spheres the formation of a large number of bound pairs soonresults in the saturation of the attraction, just for steric reasons. Indeed, as the simulationgoes on an increasing number of binding sites get entrapped in the interior of aggregates,24his way becoming unavailable for other bonds. As a result, the initial stage of fast clus-ter growth comes to a stop and clusters become relatively stable (at low temperature, thebinding energy is so strong that cluster particles only hardly get unstick). This is reflectedin the shape of the effective guest-guest potential, featuring a short-range attraction and along-range repulsion. At intermediate concentrations, the number of spheres and bindingsites exposed to the surface of the clusters are both sufficiently high during the growth stagethat big clusters eventually arise. However, only when the overall system density is highenough (at least 0.15 in reduced units) a spanning structure (gel) is formed in relativelyshort time.In one case only, that is χ = 50% and a temperature not so low that particle mobilityis significantly suppressed, Monte Carlo relaxation to equilibrium had an unexpected out-come: the formation of an ordered lamellar structure with a square arrangement of guestspheres and dimers. The very same existence of this structure indicates a low-temperaturecompetition between clusters and crystal-vapor separation. It is reasonable to expect that,while clusters are kinetically favored at low T , the separation between lamellar crystal andvapor is the solution adopted by the system in stable equilibrium.In a forthcoming paper, we plan to analyze in detail the case of guest spheres muchlarger than dimers, focusing our attention on the competition between clusterization and(macro-)separation into a guest-rich and a guest-poor phase [33]. An extension of the dimerichard-sphere model to include non-additive hard sphere effects [69] is also in program. [1] C. N. Likos, Phys. Rep. , 267 (2001).[2] J. N. Israelachvili,
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