A numerical study of one-patch colloidal particles: from square-well to Janus
AA numerical study of one-patch colloidal particles: from square-well to Janus.
Francesco Sciortino, Achille Giacometti, and Giorgio Pastore Dipartimento di Fisica and CNR-ISC, Universit`a di Roma
La Sapienza , Piazzale A. Moro 2, 00185 Roma, Italy Dipartimento di Chimica Fisica, Universit`a Ca’ Foscari Venezia,Calle Larga S. Marta DD2137, I-30123 Venezia, Italy Dipartimento di Fisica dell’ Universit`a and CNR-IOM Democritos, Strada Costiera 11, 34151 Trieste, Italy
We perform numerical simulations of a simple model of one-patch colloidal particles to investigate:(i) the behavior of the gas-liquid phase diagram on moving from a spherical attractive potential toa Janus potential and (ii) the collective structure of a system of Janus particles. We show that, forthe case where one of the two hemispheres is attractive and one is repulsive, the system organizesinto a dispersion of orientational ordered micelles and vesicles and, at low T , the system can beapproximated as a fluid of such clusters, interacting essentially via excluded volume. The stabilityof this cluster phase generates a very peculiar shape of the gas and liquid coexisting densities, witha gas coexistence density which increases on cooling, approaching the liquid coexistence density atvery low T . I. INTRODUCTION
The synthesis of colloidal particles with controlledanisotropy is central in today research. One of the mainideas is the development of a set of colloidal molecules[1–7] which can be useful to generate, on the nano andmicron-scale, collective phenomena presently observedonly at atomic or molecular scale as well as additionalphenomena, induced by the possibility of controlling theparameters of the effective interaction potential betweencolloids. The self-assembly of a colloidal diamond crys-tal, to be used in photonic applications[8], is one of thesetechnological relevant goals. The anisotropy can be in-duced not only by building colloidal molecules (i.e. col-loids with peculiar non-spherical shapes), but also (with apromising alternative) via the process of patterning theparticle surface, generating in this way particles inter-acting in very different way according to their relativeorientation[1, 9–13].This vision of a material science, based on the designand self-assembly of materials with required propertieshas stimulated, beside the experimental work, a largeamount of theoretical and numerical studies based onsimple, primitive, anisotropic potentials[14–20]. Indeed,the possibilities of modifying the surface properties areendless, including the number of patches, their width,their location, their chemical specificity, and seem to poseno limits to the design of specific particle-particle inter-action potentials. In this large parameter space, primi-tive potentials — modeling repulsion as a hard-core andattraction as a square-well interaction — can providea useful reference system to deeply investigate the roleplayed by the number of patches, their width, their spa-tial location and the role of the attraction range. In addi-tion, there is some consensus that these studies can shedlight on the aggregation properties of proteins and onthe sensitivity of the aggregates on the protein surfaceproperties[21–24].Another useful aspect of the study of primitive mod-els is the possibility to accurately investigate theoreti- cally and numerically their phase diagram, casting theself-assembly process of these systems into a wider ther-modynamic perspective. Important questions about thenature of the self-assembly process and its reversibility,the relative stability of aggregates of different sizes, thecompetition between self-assembly and phase-separationcan in principle be addressed with an accurate study ofthese models. Finally, in some cases, numerical simu-lations can be compared with state of the art integralequation approaches for non-spherical potentials provid-ing a benchmark for the possibility of developing a fastand accurate prediction of the structural properties. Sev-eral efforts in this directions, capitalizing on studies ofmolecular associations[25–28], are taking place in thesedays[29–31].In this article we report a study of the phase behav-ior of a very simple primitive potential, proposed in 2003by Kern and Frenkel[14] with the aim of exploring thescaling of the critical parameters (including the reducedsecond virial coefficient) on the number and width of thepatches, and the possible implications for the phase be-havior of globular proteins. We limit ourselves to the casewhere the surface of the colloid particle is divided intotwo parts, respectively repulsive and attractive. On de-creasing the surface area corresponding to the attractivepart, the potential interpolates between the well-knownisotropic square well potential and the symmetric case ofan evenly divided surface, commonly indicated as Januspotential. We find that an unconventional phase dia-gram characterizes Janus particles, due to the onset ofa micelle formation process which takes place in the gasphase, providing additional stability to this phase as com-pared to the liquid one. We characterize the structuraland connectivity properties of the system in a wide rangeof temperatures T and number densities ρ , to clarify theorigin of the aggregation process and the mechanisms be-hind the stability of the cluster phase. Our study demon-strates how a small change in the attractive surface hasprofound consequences on the collective behavior of thesystem. a r X i v : . [ c ond - m a t . s o f t ] J u l II. MODEL AND SIMULATION TECHNIQUES
The Kern-Frenkel potential is a paradigmatic modelfor highly anisotropic interactions. In this model, a hard-sphere of diameter σ is complemented by a set of unit vec-tors { ˆ n i } , locating the position of the center of a patch onthe particle surface. Each patch can be reckoned as theintersection of the sphere with a cone of semi-amplitude θ and vertex at the center of the sphere. In the case stud-ied in this model, each particle has only one patch. Whenunit vectors the patch ˆn i of particle i and ˆn j of particle j form an angle smaller than θ , and in addition the dis-tance between the center of the two particles is between σ and σ + ∆, then an attractive interaction of intensity u is present. More precisely, the two body potential isdefined as: u ( r ij ) = u sw ( r ij ) f (Ω ij ) (1)where u sw ( r ij ) is an isotropic square well term of depth u and attractive range σ + ∆ and f (Ω ij ) is a functionthat depends on the orientation of the two interactingparticles Ω ij . The angular function f (Ω ij ) is defined as f (Ω ij ) = ˆ r ij · ˆ n i > cos θ patchon particle i andˆ r ji · ˆ n j > cos θ patchon particle j σ = 1 and u = 1.In practice, two particles interact attractively if, whenthey are within the attractive distance σ +∆, two patchesare properly facing each other. When this is the case,the two particles are considered bonded. The fraction ofsurface covered by the attractive patches χ is related to θ by the relation χ = − cos θ .Structural properties of the system have been evalu-ated by mean of standard MC simulations, for a systemof N = 5000 particles. Extremely long simulations, of theorder of 10 MC sweeps, have been performed to reach aproper equilibrium state of the system. Here a MC sweepis defined as an attempted random translation and rota-tion for each particle. To calculate the location of the gas-liquid critical point we perform grand canonical MonteCarlo (GCMC) simulations [32], complemented with his-togram reweighting techniques to match the distributionof the order parameter ρ − se with the known functionaldependence expected at the Ising universality class crit-ical point [33]. Here e is the potential energy density, ρ the number density and s is the mixing field parameter.We did not performed a finite size study, since we areonly interested in the trends with χ . We have studiedsystems of different sizes, up to L = 15. For each studied χ — using the methods described in [34] — we calculatedthe critical temperature T c and density ρ c for values ofcos θ between − . k B = 1.We also performed Gibbs ensemble simulations to eval-uate the coexistence curve. The GEMC method was de-signed [35] to study coexistence in the region where thegas-liquid free-energy barrier is sufficiently high to avoidcrossing between the two phases. Since nowadays thisis a standard method in computational physics, we donot discuss it here. We have studied a system of (total)350 particles which partition themselves into two boxeswhose total volume is 2868 σ , corresponding to an av-erage density of ρ = 0 . T this cor-responds to roughly 320 particles in the liquid box (ofside ≈ σ ) and about 30 particles in the gas box (of side ≈ σ ). Equilibration at the lowest reported T requiredabout one month of computer time.The model, in the case χ = 0 .
5, can be relatedto the experimental system that is currently underinvestigation[11]. In these newly synthesized Janus parti-cles, the repulsive interaction has an electrostatic originand the attractive part is hydrophobic At the presenttime, experiments by Granick’s group[11] have focusedon the analysis of the structure of the aggregates sedi-mented on the bottom surface due to gravity. Micelleformation has been observed. While the interaction rangein the Granick’s experimental system is about 0.1 of theparticle size, nano-sized particles synthesized with thesame protocol would indeed give rise to potential rangessimilar to the one we have selected. It is also in prin-ciple possible to modify the range of the interaction bytuning the physical properties of the solvent (for exam-ple the ionic strength or the solvent dielectric constant).Another interesting possible experimental realization oflonger range interactions can be achieved by mean of therecently measured Casimir critical forces [for a highlight-ing review see for example Ref. [36] or Ref. [37]] dissolvingthe particles close to the critical point of the solvent. Bytuning the distance from the critical point, the range ofthe interaction can be controlled and tuned close to thevalue we have explored. Moreover, in the critical Casimireffect, the coating of the two hemispheres will controlsif the interaction is attractive or repulsive. Other pos-sibilities of probing different ranges will be offered bystudies of Janus-like proteins. Indeed, the hydrophobinproteins (extracted from fungi) are good candidates fornano-Janus particles, as discussed in Ref. [38]. Even inthis protein case, the bulk phase diagram has never beenexplored.
III. RESULTS: FROM SQUARE-WELL TOJANUS
The Kern-Frenkel model offers the possibility to con-tinuously change the coverage interpolating from the rs k B T / u c =1 SW c =0.9 c =0.8 c =0.7 c =0.6 c =0.5 FIG. 1: Phase diagram of the one-patch Kern-Frenkel po-tential with attractive range 0 . σ for different values of thecoverage χ , interpolating between the square-well potential( χ = 1) and the Janus potential ( χ = 0 . isotropic square-well potential to the symmetric Janus-like one, when the coverage moves from χ = 1 to χ = 0 . χ . Fig. 1 depictsthe gas-liquid phase coexistence for several χ values, ex-tending the original data by Kern and Frenkel[14]. Whilethe critical density presents a significant decrease withdecreasing χ , the density of the liquid branch does notshow any significant reduction, consistent with the pos-sibility of forming six or more bonds with neighboringparticles when χ > .
5. Recent studies on the role ofthe valence (defined as the maximum possible number ofbonded nearest neighbors)[15, 39], have suggested a pro-gressive reduction of the critical temperature T c and crit-ical density ρ c on decreasing the valence. Indeed, whenthe valence decreases below six, a significant reductionof the density of the liquid branch has been reported[16].Estimates of the critical parameters have been calculatedusing grand-canonical simulations, i.e. simulations inwhich the chemical potential µ , T and the volume V arekept constant. In GCMC simulations, the number of par-ticles fluctuates. When µ and T are close to their criticalvalues, the number of particles fluctuates widely (sincethe compressibility, which is a measure of the varianceof the density fluctuations diverges). The distributionof sampled densities, close to the critical point, followsan universal curve which depends only on the class ofuniversality of the critical phenomenon.The critical parameters, resulting from the grand-canonical simulations, are summarized in Table I. Thelast column indicates the size of the largest simulationbox, which has been progressively increased to compen-sate the decrease of the critical density and the associatedshift towards smaller number of particles of the densityfluctuations. Table I also reports the critical chemical c T c T c c r c s r c s FIG. 2: Coverage χ dependence of the critical density ρ c andtemperature T c . potential and the value of the second virial potential B c ,normalized to the hard-sphere value B HS B c /B HS = 1 − χ [(1 + ∆ σ ) − u k B T c ) − . (3)also evaluated at the critical point. Previous work[19] hasshown that B c /B HS becomes smaller and smaller withdecreasing valence. The actual value of B c /B HS canprovide an estimate of the effective valence of the system.A comparison with Table 1 of Ref. [19] suggests that when χ reaches the value 0.6, the effective valence becomes lessthan 4 and that when χ = 0 . T c and ρ c are also shown graphically in Fig. 2. In agreement withthe progressive reduction of the valence, T c and ρ c dodecrease with decreasing χ . Note that results presentedin Figure 1 have a counterpart in the case where theattractive part is spread over two patches distributed atthe opposite poles of the sphere (see Fig.2 in Ref.[31]). TABLE I: Critical parameters as a function of the coverage.The last column indicates the size of the largest studied sim-ulation box. χ ρ c T c β c µ c µ c B c /B HS L1 0.31 1.22 -2.955 -3.601 -2.020 70.9 0.30 1.01 -3.048 -3.079 -2.253 70.8 0.27 0.800 -3.270 -2.613 -2.790 110.7 0.24 0.610 -3.684 -2.248 -3.828 150.6 0.20 0.446 -4.482 -1.200 -6.187 150.5 0.15 0.302 -6.371 -1.924 -14.68 15
When χ = 0 . χ > . χ we have not presentlybeen able to evaluate the phase diagram for two differ-ent reasons: i) the temperature region where the criticalpoint is expected (around T = 0 .
17 for χ = 0 .
4, by aquadratic extrapolation of the data reported in Table I)requires significant computational resources. ii) prelimi-nary tests have detected the formation of lamellar phasesfor χ = 0 . T > .
17. Despite the disappear-ance of the liquid phase as an equilibrium phase in thepresence of anisotropic potentials is a potentially rele-vant issue[40], we can not at the present time addressthis point for the present model. A study of the stabil-ity of the liquid phase as compared to the (unknown)ordered phases will be the argument of a future study,which will require the use of algorithms to identify thepossible crystal structures[41, 42] as well as free-energyevaluations to establish the relative stability compared tothe liquid phase. A similar study for tetrahedral patchyparticles has been recently reported[40, 43]. Indeed, itcould well be, as suggested in a recent study of particleswith two patches[31], that the reduction of the bond-ing angle could play a role analog to the reduction ofthe range in spherically interacting potential[44, 45] andlimited valence potentials[40], where the liquid phase dis-appears when the range is smaller than ≈ . σ . IV. RESULTS: JANUS
The gas-liquid coexistence for the Janus ( χ = 0 .
5) par-ticles is enlarged in Fig. 4. As discussed in a preliminarypublication[46], the phase diagram has a very odd behav-ior. The gas coexisting density, which typically is a de-creasing function of the temperature, here increases pro-gressively on cooling, approaching the coexisting liquiddensity. In a simple liquid, coexistence between gas andliquid is established on the basis of a compensation be-tween the gas and liquid free energies. The lower energyof the liquid phase is compensated by a larger entropy ofthe gas phase, which is acquired by significantly increas-ing the volume per particle. As discussed in Ref. [46] (anddetailed more in the following), the uncommon behaviorobserved in the Janus case arises by a completely dif-ferent compensation mechanism between the liquid andthe gas. The gas becomes the energetically stable phase(due to the formation of orientationally ordered aggre-gates, micelles and vesicles) and the liquid phase insteadis stabilized by the larger orientational entropy of theparticles. This odd behavior give rises to a gas-liquid co-existence curve in the P − T plane which is negativelysloped and to an expansion of the system on crossingfrom the gas to the liquid phase on cooling along isobars(see Fig. 4 in Ref.[46]).This anomalous thermodynamic behavior arises fromthe progressive establishment in the gas phase of clustersof particles which — due to the surface pattern proper-ties of the particle — organize themselves in particularly stable structures. Typical cluster shapes for different val-ues of the cluster size s are shown in Fig. 3. For smallcluster sizes ( s < ∼
20) clusters are of micellar type, i.e.formed by aggregates in which the attractive part con-stitutes the core of the aggregate. For larger size, thecluster organization changes in favor of a double layerstructure, reminiscent of vesicles, in which the inner andouter surfaces are repulsive and the inner core is attrac-tive. Here, and in the rest of the manuscript, clustersare defined as set of particles connected by an uninter-rupted path of bonds, where we define bonded any pairof particles whose pair potential energy is − u .In this article we explore in details the properties of theclusters which develop in the gas phase, their structure,energy and abundance. We then investigate the gas andliquid phases with the aim of characterizing the collectivestructure of the system (both in real and wave-vectorspace) as well as its connectivity properties (percolation). A. Critical Micelle Concentration
The onset of micelles can be demonstrated and quan-tified by the study of the relation between the density ρ of particles in monomeric state (i.e. un-bonded par-ticles) and the system density ρ . Fig. 5 shows that for T < ∼ .
28, a sharp kink separates the ideal gas behavior(where all the particles are in a very dilute monomericstate and ρ = ρ ) from a rather insensitive density de-pendence of the number of monomers in solutions, a clearindication that, at low T , the addition of particles to aconstant volume system promotes the formation of ad-ditional aggregates. This behavior is indeed typical ofmicelle forming systems[47], and the location of the kinkprovides an estimate of the critical micelle concentration(c.m.c.), which in the present case varies from ρ = 10 − down to ρ = 10 − when T changes from T = 0 .
28 to T = 0 . T , the gas phase does not show significant clustering.Indeed, on increasing the density, the system transformsinto a liquid phase, by establishing an infinite size perco-lating clusters, in the attempt to minimize the potentialenergy, being the entropic loss in the free energy (asso-ciated to the restricted sampling of the system availablevolume) made less relevant by the small T . In the presentcase, the Janus potential allows for the establishment ofsignificantly bonded aggregates, i.e. with a small poten-tial energy, without the need of forming an infinite sizecluster. By exposing the hard-core part to the exteriors,these clusters do not feel any driving force toward furtherclustering. (a)s=11 (b)s=22 (c)s=33 (d)s=50 (e)s=63 (f)s=79 (g)s=117 (h)s=130 (i)s=191 FIG. 3: Typical cluster shapes of different size, extracted from simulations at T = 0 . r s k B T / u Gas-Liquid Critical Point r g r l non-percolatingpercolatinglamellae liquid phasegas phase lamellar phase FIG. 4: Phase diagram of the Janus particles. Filled (red) cir-cles indicate the gas-liquid coexistence lines with the (blue)triangle denoting the critical point. The filled and opensquares indicate the percolating and non-percolating statepoints, respectively, whereas (green) diamonds indicate thesimulations that show a lamellar phase. Dashed lines connectcoexisting state points whereas the two dotted lines refer tothe two paths followed in the calculation of the structure fac-tor of Fig. 8.
B. Cluster Size Distributions, percolation
To quantify the effect of clustering and its ρ and T dependence we show in Fig. 6 the cluster size distribution, N ( s ).Below the critical temperature, the gas phase becomespopulated by rather stable clusters. Particularly signif-icant are clusters of size between 10-15 (micelles) andclusters between size 40 and 50 (vesicles). Other sizesare also found, but their statistical relevance decreases ondecreasing T . This feature appears clearly in the clustersize distribution, N ( s ), shown in Fig. 6. On decreas-ing T , the monotonic decaying N ( s ) develops a shoulderaround T c , which evolves into a clear two-peaked functionon further cooling, signaling the appearance of micellesand vesicles (Fig. 6-(a)). Part (b) of Fig. 6 shows theevolution of N ( s ) on increasing ρ . The micelle peak pro-gressively empties in favor of the vesicles peak, which atlow T and large ρ are the most stable structures. Similar -4 -3 -2 -1 rs -5 -4 -3 -2 -1 r s T=0.25T=0.27T=0.28T=0.29T=0.3T=0.4T=0.5Ideal Gas
FIG. 5: Relation between the monomer number density ρ and the system number density ρ at different T . The flat re-gion for T < ∼ .
28 indicates that the addition of monomerspreferentially results in the formation of aggregates. Theintercept between the flat curve and the ideal gas behaviorprovides an accurate estimate of the critical micelle concen-tration. trends are seen at other T or ρ .An indication of the relative stability of the clusters asa function of their size is provided by the cluster potentialenergy. Fig. 7 shows the average potential energy (cid:104) E ( s ) (cid:105) ,averaged over all clusters of the same size, vs. size s fordifferent T at fixed ρ . One notices that for each size thereare several distinct arrangements with different energiesand entropies. On cooling, lower energy clusters becomepreferentially selected and (cid:104) E ( s ) (cid:105) decreases. One alsonotices a plateau between size 10 and 30 followed by anadditional plateau, at low T , starting from s ≈
40. Theinitial location of the plateaus coincides with the size ofthe mostly represented clusters. This can be understoodby considering that the free energy of the system hasboth energetic and entropic contributions. When the po-tential energy per particle does not change with the sizeof the cluster, then it becomes convenient to the systemto favor the formation of the smallest possible clusterwith the same energy, since it will be possible in this wayto maximize the total number of clusters and hence thetranslational component of the entropy.Fig. 7 also shows the lowest energy configuration E min ( s ) found for each cluster size, independently from cluster size s10 -7 -6 -5 -4 -3 -2 -1 N ( s ) T=0.25T=0.27T=0.30T=0.35T=0.4T=0.5(a)10 cluster size s10 -8 -7 -6 -5 -4 -3 -2 -1 N ( s ) r =0.001 r =0.005 r=0.01r=0.05 (b) FIG. 6: Cluster size distributions at ρ = 0 .
001 for several T values (a) and at T = 0 .
25 for several densities (b). Thedistributions have been normalized so that (cid:80) s sN ( s ) = 1. temperature and density of the simulation. This curveprovides an estimate for the cluster ground state andclearly show that the lowest-energy structures are thevesicles, i.e. clusters composed between 40 and 60 parti-cles.We have examined the connectivity properties of thedifferent state points by evaluating the presence of clus-ters spanning the entire simulation box. When more than50 per cent of the configurations are characterized byspanning clusters, the state point is classified as perco-lating. Fig. 4 shows the location of the percolating statepoints in the phase diagram. The cluster gas phase isnever percolating, while the liquid states always are, sothat percolation properties can also be used to distin-guish between the two phases. We also confirm thatthe critical point is located inside the percolation re-gion, confirming once more that a pre-requisite for criti-cal phenomena is the existence of a percolating networkof interactions[48]. cluster size s -5-4-3-2-10 < E ( s ) > / ( N u ) T=0.25T=0.27T=0.29T=0.3T=0.32T=0.35T=0.4T=0.5E min (s)
FIG. 7: Average potential energy (per particle) of clustersof different size s at several T at ρ = 0 .
01. For each s , it isalso shown the lowest energy ever observed E min ( s ), indepen-dently from T or ρ . C. Structure Factor
It is particularly relevant to look at the structure fac-tor in this system both in the gas phase at low T , whenmicelles and vesicles become prominent, as well as inthe liquid phase. Fig. 8-(a) shows the evolution of thestructure factor on increasing density at T = 0 .
25, acrossthe gas-liquid transition. At this low T , as discussed inSec. IV B, vesicles are the dominant clusters. The struc-ture factor indeed evolves from the one characteristic ofan ideal gas of spherical vesicles (and indeed the insetshows that S ( q ) is well represented by the form factor P ( q ) of a sphere of radius R ) P ( q ) = (cid:20) qR ) − qR cos( qR ))( qR ) (cid:21) , (4)to the one of interacting spheres, in the micelle-rich gasphase. Indeed, on increasing ρ , oscillations at qσ ≈ . ρ = 0 . qσ ≈ ρ = 0 . q , which progressively increases on approach-ing the phase separation. These are the standard criticalfluctuations which are expected to diverge on approach-ing a spinodal line. While in simple liquids, below thespinodal temperature the system phase separate in a gascoexisting with a liquid phase, here, the peculiar shapeof the gas-liquid coexistence line (Fig. 1) opens up newstable states, composed by interacting vesicles and S ( q )becomes peaked at the vesicle-vesicle distance. q s -1 S ( q ) r =0.001 r =0.01 r =0.1 r =0.4 r =0.60 r =0.675 r =0.75 r =0.85 -1 r =0.01Form Factor R=2.26 s (a) q s S ( q ) T=0.25T=0.27T=0.28T=0.29T=0.32T=0.35T=0.4T=0.5 (b)
FIG. 8: Structure factor at (a) T=0.25 for several densitiesand at (b) ρ = 0 . T . The inset in (a) shows the fitwith the form factor of a sphere (Eq. 4). The best-fit value forthe radius is R = 2 . σ . The above isothermal and isochorepaths are indicated with dotted lines in Fig.4 D. Angular Correlations
To provide evidence of the different ordering of theparticles in the gas and in the liquid phase, we have cal-culated the distribution of relative orientations betweenall pairs of bonded particles. More precisely, we haveevaluated the distribution of the scalar product ˆn · ˆn where ˆn and ˆn are the two unit vectors indicating thelocation of the patch center in each particle frame, for allbonded pairs. The distribution P ( ˆn · ˆn ) is expected toshow well defined peaks for an ordered state and to be flatin a completely disordered state. Fig. 9 shows such dis-tribution for several ρ values at low T . The orientationaldifferences between the ordered gas phase and the disor-dered liquid phase appear very clearly. On increasing ρ , P ( ˆn · ˆn ) evolves from a highly structured function withpeaks close to | n · ˆn | > . ˆn · ˆn close to − ˆn · ˆn close to 0 . q l ( i ) ≡ (cid:34) π l + 1 l (cid:88) m = − l | ¯ q lm ( i ) | (cid:35) / (5)where ¯ q lm ( i ) is defined as,¯ q lm ( i ) ≡ N b i N bi (cid:88) j =1 Y lm ( ˆr ij ) (6)Here N b i is the set of bonded neighbors of a particle i . The unit vector ˆr ij specifies the orientation of thebond between particles i and j . In a given coordinateframe, the orientation of the unit vector ˆr ij uniquely de-termines the polar and azimuthal angles θ ij and φ ij . The Y lm ( θ ij , φ ij ) ≡ Y lm ( ˆr ij ) are the corresponding sphericalharmonics. We have calculated the distribution of q l forthe present model. As shown in Fig. 9-(b), the result-ing distributions show systematic differences between gasand liquid phase, but it is hard to provide a clear con-nection with the degree of order in the system. E. Pathways for vesicles formation
Despite Monte Carlo simulations do not allow for a pre-cise definition of time, it is interesting to analyze how par-ticles rearrange themselves into large aggregates. Morespecifically, the analysis of the sequence of MC configura-tions may help understanding how vesicles are generated.We have examined graphically several processes of forma-tion of a vesicle starting from a monodisperse solutionof particles and one of these processes is documented inFig. 10. In all cases, the system starts by forming smallclusters that grow by incorporating isolated monomersor colliding to similar clusters, to form micelles (Fig. 10-(a-c)). The transition to vesicle requires the interactionbetween three micelles (d-e), with the formation of a longcylindrical micelle (f) which then self-restructures itselfinto the more energetic vesicles configuration (g).
V. CONCLUSIONS
This article reports a numerical study of a simple po-tential for Janus-like particles, i.e. colloidal sphericalparticles whose surface is divided evenly into two areas -1 -0.5 0 0.5 1 n · n P ( n · n ) ( a . u . ) r =0.001 r =0.01 r =0.1 r =0.4 r =0.60 r =0.675 r =0.75 r =0.85 r =0.9 (a) q l P ( q l ) ( a . u . ) q q q lines=gas phasesymbols=liquid phase(b) FIG. 9: (a) Distribution of the scalar product between theunit vectors of all bonded pairs of particles. The gas-likestate points are characterized by peaked distributions, reveal-ing the presence of orientational order. The vesicle configura-tions significantly contribute to the region n · n ≈
1, whilethe micelle configurations contribute to n · n ≈ −
1. (b)Distribution of the rotational invariant q l in the gas and inthe liquid phase. of different chemical composition. This study is moti-vated by the ongoing effort in the direction of synthesiz-ing optimal Janus colloids[9, 11–13, 52]. The possibilityof creating particles whose surface has different behavioron the two hemispheres significantly enlarges the rich-ness of the resulting collective behaviors. The phase di-agram, the ordered and disordered stable structures, theself-assembly properties of optimal clusters are indeedexpected to finely depend on the chemistry and physicalproperties of the particle surface. For this reason, theassembly behavior of Janus particles is receiving a con-siderable attention even from a theoretical and numericalpoint of view[10, 20, 53–55]. With proper choices of thechemico-physical surface properties, Janus particles canprovide the most elementary and geometrically simpleexample of a surfactant particle[56], in which solvophilicand solvophobic areas reside on different part of the sur-face of the same particle.Modeling of the phase behavior of these particles can be performed at different levels of realism. In this articlewe have chosen to implement the highest coarse-grainingprocedure, by considering the two sides of the particle asrepulsive and attractive, and modeling each of them withthe simplest corresponding potential, a hard-core plus asquare-well. Despite this strong simplification, the phasediagram of the system displays a very rich behavior, witha colloidal-poor (gas) colloidal-rich (liquid) de-mixing re-gion, which is progressively suppressed by the insurgenceof micelles, thus providing a model where micelle forma-tion and phase-separation are simultaneously observed.The study of this model shows that the suppression of thephase separation is driven by the possibility of buildinglow energy clusters which are shielded by the presence ofan external hard-sphere surface, diminishing the drivingforce for forming large-size aggregates. As a result, at low T , the system organizes into a dispersion of orientationalordered micelles and vesicles, essentially interacting viaexcluded volume only.Another advantage of studying the one-patch Kern-Frenkel potential is that the parameters of the potentialcan be tuned continuously from the square-well to theJanus potential, simply by changing the bonding angle,that is the coverage χ . This has made possible to followthe evolution of the gas-liquid coexistence curve with χ ,thus including the Janus case which displays the inher-intly interesting simultaneous presence of a critical pointand of a micelle formation. It would be interesting inthe future to explore more in details the transition fromthe standard gas-liquid coexistence to the more exoticJanus case exploring a more refined grid of χ values. Itis of course possible that the stabilization of micelle andvesicles can take place already at χ = 0 . T region where numerical simulations can not be performedat the present time.Several questions are still open including the role of theinteraction range. Experimentally realized Janus parti-cles are characterized by interaction ranges of the orderof 0 . σ or smaller. The range width is an important vari-able, since it controls both the internal flexibility of theaggregates (and hence the cluster entropy) and the ge-ometries of the lowest-energy aggregates (cluster energy).In this respect, it can not be a priori foreseen if micelleand vesicles will remain the most stable clusters on fur-ther decreasing the range. We are currently repeatingour study for smaller ranges, but this requires a majorcomputational effort, since clustering shifts to lower tem-perature, making studies of the phase diagram (in par-ticular Gibbs ensemble calculations) extremely hard withpresent time numerical resources. The sensitivity of thecluster shape to the cluster range would be an impor-tant observation and could teach us a lot about how tocontrol the shape of the assembly experimentally, and,perhaps, help us understanding geometric arrangementswhich are found in protein aggregates. In this respect itis worth noticing that micelles have been detected in themost recent experimental work of Granick[11] togetherwith Bernal-spiral clusters, which have not been observed (a)25.000 (b)125.000 (c)500.000 (d)750.000 (e)825.000 (f)950.000 (g)1.000.000 FIG. 10: Snapshot from a simulation at ρ = 0 .
005 and T=0.25 for several MC steps, indicated in the labels. The initialconfiguration, composed by isolated monomers, quickly evolves to form small micelles (a-c). The final vesicle (g) arises froma collision between three distinct micelles (d-e) which form an elongated transient tubular cluster (f). Each picture has a sidelength of 9 σ . with the present model, perhaps due to differences in theinteraction ranges. VI. ACKNOWLEDGEMENTS
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