Monte Carlo cluster algorithm for fluid phase transitionsin highly size-asymmetrical binary mixtures

Douglas J. Ashton,1 Jiwen Liu,2 Erik Luijten,3 and Nigel B. Wilding1

1Department of Physics, University of Bath, Bath BA2 7AY, United Kingdom2Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign,Urbana, Illinois 61801, USA3Department of Materials Science and Engineering and Department of Engineering Sciences and AppliedMathematics, Northwestern University, 2220 Campus Drive, Evanston, Illinois 60208, USA

�Received 21 July 2010; accepted 10 September 2010; published online 16 November 2010�

Highly size-asymmetrical fluid mixtures arise in a variety of physical contexts, notably insuspensions of colloidal particles to which much smaller particles have been added in the form ofpolymers or nanoparticles. Conventional schemes for simulating models of such systems arehamstrung by the difficulty of relaxing the large species in the presence of the small one. Here wedescribe how the rejection-free geometrical cluster algorithm of Liu and Luijten �J. Liu and E.Luijten, Phys. Rev. Lett. 92, 035504 �2004�� can be embedded within a restricted Gibbs ensembleto facilitate efficient and accurate studies of fluid phase behavior of highly size-asymmetricalmixtures. After providing a detailed description of the algorithm, we summarize the bespokeanalysis techniques of �Ashton et al., J. Chem. Phys. 132, 074111 �2010�� that permit accurateestimates of coexisting densities and critical-point parameters. We apply our methods to study theliquid-vapor phase diagram of a particular mixture of Lennard-Jones particles having a 10:1 sizeratio. As the reservoir volume fraction of small particles is increased in the range of 0%–5%, thecritical temperature decreases by approximately 50%, while the critical density drops by some 30%.These trends imply that in our system, adding small particles decreases the net attraction betweenlarge particles, a situation that contrasts with hard-sphere mixtures where an attractive depletionforce occurs. © 2010 American Institute of Physics. �doi:10.1063/1.3495996�

I. INTRODUCTION AND BACKGROUND

Colloidal suspensions are a class of complex fluids thatcomprises systems as diverse as protein solutions, liquidcrystals, and blood. Technologically, colloidal suspensionsfeature in applications such as coatings, precursors to ad-vanced materials, and drug carriers.1 One of the key issues inall these systems is the phase behavior of the suspension ormore generally its stability. Attractive dispersion forces existbetween uncharged colloids that can engender phase separa-tion or irreversible aggregation resulting in a gel—undesirable features in many applications. Accordingly, oneseeks to control the phase behavior �as well as dynamicalproperties such as the rheology2� by modifying the form ofthe effective interactions between the colloidal particles.There are several routes to achieving this, including chargestabilization �via modification of the pH� and steric stabili-zation �via grafting of flexible polymers onto the colloidalsurface�.2,3 Alternatively, the effective interactions, andhence colloidal phase behavior, may be manipulated throughthe addition of nanoparticles, with nanoparticle size, concen-tration, and charge as control parameters.4–6 The simplestand most celebrated example concerns colloids which inter-act �to a good approximation� as hard spheres. Adding nano-particles in the form of small nonadsorbing polymers engen-ders an attractive “depletion” force between the colloidalparticles.7 This attraction can drive phase separation resultingin a colloid-rich �“liquid”� phase and a colloidal-poor �“gas”�

phase8—a phenomenon akin to the fluid–fluid transitions oc-curring in molecular liquids and their mixtures. Yet richerbehavior occurs when one transcends simple hard-sphere po-tentials between the nanoparticles and the colloids. For ex-ample, if the nanoparticles are weakly attracted to the col-loids but repel one another, they can form a diffuse�nonadsorbed� “halo” around each colloid particle.5,9–13 Thenet influence on the effective colloid-colloid interaction de-pends on the nanoparticle density in a nontrivial way.11,13,14

In view of the broad range of effects that can arise whennanoparticles are added to a colloidal suspension, their pro-totypical model representation, namely, a size-asymmetricfluid mixture, has attracted considerable theoretical and com-putational attention over the past years. Analytical ap-proaches typically either focus on drastically simplifiedmodels1,7 or attempt to render the size asymmetry tractableby integrating out the degrees of freedom associated with thesmall �nano�particles �see Ref. 15 for a review�. The latterstrategy yields a one-component system of colloids describedby an effective pair potential representing the net influenceof the small particles. One shortcoming of this approach isthat, for all but the simplest types of nanoparticles, the map-ping to a one-component system is approximate because itneglects many-body colloidal interactions that can consider-ably alter the nanoparticle distributions and hence the inter-actions induced by them. These effects may be significant inthe regimes of density at which phase separation occurs.16

Recent work has additionally raised concerns regarding the

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accuracy of effective potentials in this regime,17,18 and hasalso emphasized the significance of corrections to the en-tropic depletion picture in real colloids19,20 as well as theimportance of polydispersity.21,22 On the other hand, variouscomputational techniques, most notably Monte Carlo �MC�methods, are capable of explicitly incorporating fluctuationand correlation effects. Conventional MC techniques �whichattempt to displace or insert and delete particles� are re-stricted to fluids mixtures in which the size ratio is of orderunity, rendering them unsuitable for the simulation ofcolloid-nanoparticle suspensions, where typical size ratiosencountered can extend to one or two orders of magnitude.5

The computational bottleneck results from the effective“jamming” of the large species by even low volume fractionsof the small particles. However, this problem has been re-solved by means of the geometric cluster algorithm �GCA�of Liu and Luijten,23,24 in which configuration space issampled via rejection-free collective particle updates, each ofwhich facilitates the large-scale movement of a substantialsubgroup of particles �a “cluster”�. Although the original al-gorithm operates in the canonical ensemble, and hence can-not address phase separation phenomena directly, in previouswork25 we have developed a generalization that embeds theGCA in the restricted Gibbs ensemble �RGE�, such that clus-ters containing both large and small particles are exchangedbetween two simulation boxes of fixed equal volumes. Theresulting density fluctuations within one box can be analyzedto determine the phase behavior.

The purpose of the present paper is first to provide amore detailed description of the basic GCA-RGE algorithmthat was introduced in Ref. 25, and to integrate recent ad-vances that we have made in data analysis methods for de-termining coexistence and critical-point properties within theRGE.26 We then apply the improved methodology to studythe liquid-vapor coexistence properties of a mixture ofLennard-Jones �LJ� particles having a size ratio q=0.1. In sodoing we adopt one aspect of effective fluid approaches,namely, we focus on the liquid-gas phase coexistence prop-erties of the large species �colloids� which are assumed to beimmersed in a supercritical fluid of small particles of quasi-homogeneous density. This choice of perspective mirrors theexperimental reality, namely, that often only the colloidalparticles can be individually imaged. Accordingly, the phasediagrams that we present are single-component projections�i.e., referring to the large species� of the full phase diagram,obtained at a prescribed reservoir volume fraction of thesmall species, which we vary in the range of 0%–5%. Note,however, that the small particles are treated explicitly andexactly in our simulations, making this method superior toeffective-potential approaches which integrate out the de-grees of freedom associated with the small particles.

This paper is arranged as follows. Section II introducesour model system, a binary LJ fluid. The GCA-RGE MCalgorithm capable of simulating this system in the highlysize-asymmetrical limit is described in Sec. III together withan outline of techniques for determining phase coexistenceproperties and critical-point parameters within the RGE.Moving on to our results, Sec. IV presents measurements ofthe large-particle coexistence densities as a function of the

reservoir volume fraction of small particles. We also discussthe underlying reasons for the observed trends in the coex-istence properties in terms of measurements of the fluidstructure. Finally, Sec. V considers the implications of ourfindings, the efficiency of our simulation approach comparedto more traditional schemes, and an outlook for further work.

II. MODEL SYSTEM

The model with which we shall be concerned is a binarymixture of spherical particles, whose two species are denotedl �large� and s �small�. Pairs of particles labeled i and j �hav-ing respective species labels �i and � j� interact via a LJpotential

�ij�r� = 4��i�j����i�j

r�12

− ���i�j

r�6 , �1�

where ��i�jis the well depth of the interaction and ��i�j

setsits range based on the additive mixing rule ��i�j

= ���i+��j

� /2. ��iand ��j

represent the particle diameters. Inter-actions are truncated at rc=2.5��i�j

and we take �l as ourunit length scale.

In Sec. IV we study the case q�ss /�ll=0.1, i.e., a 10:1size ratio. We shall determine the phase coexistence proper-ties of the large particles as a function of temperature for aprescribed reservoir volume fraction �s

r of small particles, ascontrolled by the imposed chemical potential of small par-ticles �s. It should be noted, however, that since the smallparticles are not infinitely repulsive, their volume fraction isnotional in the sense that we use the value of �s as if it were

a hard-core radius, i.e., we take �sr=�N̄s�s

3 / �6V� where N̄s isthe average of the fluctuating number of small particles con-tained within the system volume V.

Since we adopt the viewpoint that the small particles actas a background to the large ones, we set �ss=�ls=�ll /10,which ensures that the small-particle reservoir fluid is super-critical in the temperature range of interest here, namely,down to well below the critical point of the large particles. Itis therefore natural to define the dimensionless temperature Tin terms of the well depth of the interaction between the

large particles, i.e., T=kBT̃ /�ll, where kB is Boltzmann’s con-

stant and T̃ the absolute temperature.

III. METHODOLOGY

A. The GCA-RGE algorithm

In the original GCA,23 a fixed number of particles islocated in a single, periodically replicated simulation box ofvolume V. These particles are then moved around via clustermoves, in which a subset of the particles �identified bymeans of a probabilistic criterion� is displaced via a geomet-ric symmetry operation. To realize density fluctuations, weemploy two simulation boxes and exchange particles be-tween both boxes, as in the Gibbs ensemble.27 However,rather than exchanging individual particles, we use the GCAto exchange entire clusters of particles, so that we retain theprimary advantage of the GCA, namely, the rapid decorrela-tion of size-asymmetric mixtures. As in the original GCA, a

194102-2 Ashton et al. J. Chem. Phys. 133, 194102 �2010�

variety of symmetry operations is possible; to connect to theoriginal description,23 we phrase the algorithm here in termsof point reflections with respect to a pivot. Since a pointreflection will generally displace some particles outside ofthe original simulation cell, we need to adopt periodicboundary conditions for both simulation cells. Moreover, aswill transpire below, all particles that belong to a cluster andthat are part of the same simulation cell will retain theirrelative positions during the cluster move. Thus, the twosimulation cells must have the same dimensions. This sym-metric choice, in which both cells have an identical, constantvolume V, is referred to as the RGE.

We first describe the GCA-RGE for the case of a singlespecies of particles that interact through an isotropic pairpotential V�r�. N0=N1+N2 particles are distributed over thetwo simulation cells, with N1 particles in simulation cell 1and N2 particles in simulation cell 2. N0 is chosen to match adesired average density 0=N0 / �2V�. A cluster move withinthe GCA-RGE proceeds as follows. A pivot is chosen at arandom position within simulation cell 1 and a second pivotis placed at the corresponding position within simulation cell2. One of the N0 particles is chosen as the seed particle of thecluster. This particle i, which thus can be located in eithersimulation cell, is point-reflected with respect to the pivot �inits own simulation cell� from its original position ri to thenew position ri�. However, rather than placing the particle atthe new position �modulo the periodic boundary conditions�in its original box, we place it at the corresponding positionr̄i� in the other box. Subsequently, in keeping with the meth-odology of the GCA, all particles in the first box that interactwith particle i in its original position �the “departure site”� ri

as well as all particles in the second box that interact withparticle i in its new position �the “destination site”� r̄i� areconsidered for point reflection with respect to the pivot pointin their respective box and subsequent transfer to the oppo-site box. These particles, which we refer to with the index j,are point-reflected and transferred with probability

pij = max�1 − exp�− �ij�,0� , �2�

where =1 / �kBT� and �ij =−V��ri−r j�� if i and j reside�prior to the transfer of particle i� in the same cell. If i and jinitially reside in different cells �and hence do not interactprior to the transfer of particle i�, �ij =V��r̄i�−r j��. This pro-cess is repeated iteratively, i.e., for each particle j that istransferred to the opposite box, all neighbors that interactwith j either near its departure site or near its destination site,and that have not yet been transferred in the present clusterstep, are considered for point reflection and transfer as well.This process proceeds until there are no more particles to beconsidered; all particles that are indeed point-reflected andtransferred are collectively referred to as the cluster. Observethat the pair energy of all particles that are part of the clusterremains unchanged: If two particles reside in the same simu-lation cell prior to the cluster construction and both becomepart of the cluster, their separation remains constant. Like-wise, if two particles reside in different cells prior to thecluster construction and both are transferred, then their inter-action energy is zero before and after the cluster move. Thesame holds true for the pair interactions between all particles

that are not part of the cluster. Thus, the total energy changeinduced by the cluster move originates from the change inpairwise interactions between members of the cluster andparticles that are not part of the cluster. In the terminology ofRef. 23 such “bonds” are either broken if a particle is in-cluded in the cluster whereas a neighbor near its departuresite is not, or formed if a particle interacts with a neighbornear its destination site, and this neighbor does not becomepart of the cluster.

Although the cluster formation process is probabilistic,we note that pij only depends on the pair potential betweenparticles i and j, rather than on the total energy change re-sulting from the displacement and transfer of particle j. As aresult, the cluster algorithm is self-tuning: Overlaps of repul-sive particles will be avoided and strongly bound particlestend to stay together. Indeed, owing to the choice of the bondprobability pij, Eq. �2�, no further acceptance criterion needsto be applied upon completion of the cluster, leading to arejection-free algorithm in which large numbers of particlesare moved nonlocally. The proof of detailed balance is iden-tical to that provided in Ref. 24 for the original GCA, whereit was demonstrated that the ratio of the probability of con-structing a cluster in a given configuration X leading to aconfiguration Y �the transition probability T�X→Y�� and thereverse transition probability T�Y →X� is the inverse of theratio of Boltzmann factors of the respective configurations.The presence of two simulation cells simplifies rather thancomplicates the proof, just like �ij in Eq. �2� is a special caseof the original expression,23 owing to the fact that two par-ticles do not interact if they reside in opposing boxes.

The generalization to multiple species is straightforwardand does not lead to any conceptual changes in the algo-rithm. Indeed, the GCA shows its primary advantages in thesimulation of size-asymmetric mixtures, as it realizes nonlo-cal moves without the usual decrease in acceptance ratio.23

However, whereas there is no limitation on the number ofspecies, the overall volume fraction must be kept below athreshold value. Above this threshold, which is related to thepercolation threshold and depends on system compositionand interaction strengths between the particles,24 the clusterfrequently contains the majority of all particles. This is det-rimental to the performance of the algorithm, as it is compu-tationally expensive to construct such clusters, whereas theconfigurational change in the system is very small. The ex-istence of this threshold also necessitates the use of an im-plicit solvent, as is common in the simulation of colloidalsuspensions. Moreover, for reasons explained in detail inSec. III C, in our simulations of binary mixtures we combinethe geometric cluster moves with grand-canonical moves forthe small species. It is important to emphasize that the dif-ferent types of MC moves are independent. Thus, the smallspecies fully participate in the cluster construction processand the advantage of nonlocal rejection-free moves is re-tained, yet the density of small particles in both simulationcells is controlled by a chemical potential �s.

In Ref. 24, a number of technical improvements to theGCA are described. These can all be applied to the GCA-RGE algorithm. Most notably, it is possible to decrease theaverage cluster size, and hence increase the packing fractions

194102-3 Algorithm for highly size-asymmetric mixtures J. Chem. Phys. 133, 194102 �2010�

that can be simulated efficiently, through biased placement ofthe pivot. Furthermore, for mixtures of particles with largesize disparities, the cluster construction process can be facili-tated by employing multiple subcell structures and corre-sponding neighbor lists.24,28

Lastly, we note that, during the preparation of our origi-nal work,25 Buhot29 proposed an approach that has signifi-cant similarities to the GCA-RGE method. His method alsoemploys two boxes of identical size and exchanges clustersof particles. However, rather than the GCA �Ref. 23� he usesthe original geometric algorithm of Dress and Krauth,30

which is only applicable to hard spheres. Moreover, for eachpoint reflection it is decided at random whether a particle istransferred to the opposing box or not. If this decision weremade only once per cluster �i.e., upon selecting the seedparticle of the cluster�, this would amount to an alternation ofthe GCA-RGE with regular GCA moves. On the other hand,if it is decided independently for each particle that is addedto the cluster, the average cluster size will be larger than inthe GCA-RGE, generally an undesirable situation. The mostimportant difference, however, between, our approach andRef. 29 is that the latter can only be used for the idealizedcase of symmetric binary mixtures, where the critical com-position is known a priori. By contrast, in our method weemploy the relationship between the RGE and the grand-canonical ensemble to derive a prescription for locating thecritical point and coexistence curve for general binary mix-tures.

B. Locating phase coexistence and criticality in theRGE

The absence of volume exchanges between both simula-tion boxes in the symmetrical restricted Gibbs ensemble im-plies that, unlike for the full Gibbs ensemble,27 there is noautomatic pressure equality and hence no guarantee that themeasured particles densities are representative of coexist-ence. In this section we outline how one can neverthelessextract coexistence properties from RGE simulations withoutresorting to direct measurements of pressure. A fuller ac-count of the theoretical basis of the methods we describe canbe found in Ref. 26.

Within the RGE framework for our mixture, the totaldensity 0 of large particles across the two boxes is fixed.However, the one-box density of large particles, N1 /V,fluctuates. For any given choice of 0, the form of the prob-

ability distribution of , P̂L��, depends both on the tempera-ture T and on the choice of the chemical potential �s of thesmall particles. As shown in Ref. 26, measurement of the

form of P̂L�� for a range of values of 0 provides a route tothe coexistence and critical-point parameters. The basic strat-egy is as follows. Within the RGE, one explores the coexist-ence region by varying 0 at fixed T and �s. For 0 suffi-ciently far inside the coexistence region, the distribution

P̂L�� exhibits a double-peaked form, with peaks located atdensities − and +. In general, however, these peak densitiesdo not coincide with the gas and liquid coexistence densitiesgas and liq—a situation which contrasts with the full Gibbsensemble. An important exception is when 0 equals the co-

existence diameter density d�gas+liq� /2, for which onefinds26

�− = gas

+ = liq when 0 = d. �3�

Another important case is when 0= �gas+d� /2 for whichone finds

�− = gas

+ = d when 0 = �gas + d�/2. �4�

Enforcing consistency between Eqs. �3� and �4� suffices topermit determination of d and hence �via Eq. �3�� the coex-istence densities. It is convenient to achieve this graphically�see Fig. 1� by plotting the low density peak − both against0 and against 20−−: The value of 0 at which the twocurves intersect provides an estimate for the coexistence di-ameter d and one can simply read off the coexistence den-sities from the corresponding values of − and +. In Ref. 26this “intersection method” was shown to be very accurate fordetermining coexistence properties and to exhibit finite-sizeeffects comparable to those found in grand-canonical simu-lations. Indeed it turns out to be much more accurate than thetechnique we proposed previously for determining the coex-istence diameter in the RGE,25 wherein one determines d as

the value of 0 at which the variance of P̂L�� is maximized.We have found this latter procedure to be considerably moresensitive to finite-size effects than the intersection method,and it was therefore not used here.

FIG. 1. �a� Illustration of the operation of the intersection method describedin the text for the determination of the coexistence diameter density. Dataare shown for the state point �s

r=0.01, T=0.8�=0.764Tc�. Plotted are mea-

sured estimates of the average value − of the low-density peak of P̂L� �0�,for a series of values of 0. The same data are also shown plotted against20−−. The value of 0 at which the two data sets intersect �0

=0.351�3�� serves as an estimate of the coexistence diameter density d. �b�and �c� The measured peaks of P̂L�� for 0=d, whose individual integratedaverages yield estimates of the coexistence densities.

194102-4 Ashton et al. J. Chem. Phys. 133, 194102 �2010�

Turning now to the matter of estimating critical param-eters within the RGE, an accurate technique for achievingthis has been described in detail in Ref. 26. The basic idea isto measure the “iso-Q� curve” introduced in Ref. 25, whichis simply the locus of points in 0−T space for which the

fourth-order cumulant ratio of P̂L��

Q �� − 0�2�2

�� − 0�4�, �5�

matches the independently known25 fixed-point value Q�

=0.711 901 appropriate to the Ising universality class and theRGE ensemble.31

Now, it transpires25,26 that the iso-Q� curve is essentiallya parabola in 0−T space, the position of whose maximumrepresents a finite-size estimator of the critical-point densityc and temperature Tc. This maximum can be accurately lo-cated via a simple quadratic fit to measured points on theiso-Q� curve. In practice we determine this curve as follows.A simulation is performed at some 0 and T to determine

P̂��. This distribution is then extrapolated in temperaturevia histogram reweighting32 to determine that temperaturefor which Q=Q�. The procedure is then repeated for a rangeof values of 0 allowing us to trace out the whole iso-Q�

curve. An example of the resulting form of this curve isshown in Fig. 2. Note that, in general, for reasons of com-putational economy, the majority of the points that we deter-mine on an iso-Q� curve are for densities lower than thecritical density, since the efficiency of the cluster algorithm isgreater at lower overall volume fractions of particles.

Estimates of the critical parameters obtained from theiso-Q� maxima for a range of system sizes can, in principle,be extrapolated to the thermodynamic limit using finite-sizescaling relations derived in Ref. 26, which fully account forboth field-mixing effects and corrections to scaling. Unfortu-nately, in the present work, the computational cost of simu-lating more than one system size was found to be prohibitive.However, the variations that we find in critical-point param-eters as a function of �s

r dwarf those that one might expect onthe basis of finite-size effects alone. Thus we are neverthe-less able to report reliable trends from our measurements.

C. Treatment of the small particles

As was argued in Sec. I, it is the relaxation of the largeparticles that constitutes the sampling bottleneck for highlysize-asymmetrical mixtures. Local MC updates of small par-ticles are computationally relatively unproblematic, irrespec-tive of whether one performs particle displacements or inser-tions and deletions. Consequently, one has the choice oftreating the small particles canonically so that their density isglobally conserved, or grand canonically, in which case thedensity fluctuates under the control of a prescribed chemicalpotential. The choice one makes in this regard greatly affectsthe manner in which the bulk phase behavior is probed.Moreover it transpires that only the grand-canonical treat-ment of the small particles is compatible with our intersec-tion method �Sec. III B� for determining coexistence param-eters.

To clarify these points, we show in Fig. 3 sketches of theisothermal bulk phase diagram of an exemplary size-asymmetrical binary mixture with large-particle density l,small-particle density s, and conjugate chemical potentials�l and �s, respectively. Figure 3 ��ai�� shows the phase be-havior in the l−s plane, while the corresponding phasediagram in the �l−�s plane is shown in Fig. 3 ��aii��. Inconstructing these sketches we have anticipated the behaviorof the model of Sec. II, namely, that the larger species hasstronger attractive interactions and thus phase separates onits own �vertical axis of Fig. 3 �ai�� at the chosen tempera-ture, while the small-particle fluid �horizontal axis� does not.With interaction strengths chosen in this way, the larger par-ticles will typically accumulate in the liquid phase, with itsshorter interparticle distances, as shown by the representativetie lines in the density representation.33 Note that inchemical-potential space, coexistence occurs on a line ofpoints, as shown in Fig. 3 ��aii��.

FIG. 2. Estimates of points on the iso-Q� curve for �sr=0.01, obtained for a

system of size L=10, as described in the text. A parabolic fit to the data�solid line� identifies the coordinates of the maximum of the curve whichserves as a finite-size estimate of the critical-point parameters.

FIG. 3. Isothermal cuts through the exemplary phase diagram of a binaryfluid mixture described in the text. Liquid-vapor coexistence is representedin terms of �i� densities �l−s� and �ii� chemical potentials ��l−�s�. In �a�the coexistence region is crossed along a path of constant s, whereas in �b�it is crossed along a path of constant �s.

194102-5 Algorithm for highly size-asymmetric mixtures J. Chem. Phys. 133, 194102 �2010�

Let us consider first the canonical scenario in which onetraverses the coexistence region at constant bulk density ofsmall particles, as expressed by the dashed trajectory in-cluded in Fig. 3 ��ai� and �aii��. Clearly the tie lines in Fig. 3��ai�� cross any line of constant s moving from smaller val-ues of l /s at the gas end to larger ones for the coexistingliquid. Hence this path generates a sequence of pairs of co-existence states, one for each tie line crossed. The same tra-jectory in terms of the chemical potentials �l−�s is shown inFig. 3 ��aii��. Here the path followed first meets the coexist-ence line, tracks along it for some distance and then sepa-rates from it.

The alternative �grand-canonical� scenario, in which thesmall-particle density is permitted to fluctuate at constantchemical potential �s, is illustrated in Fig. 3 ��bi� and �bii��.Here, as �l is varied at constant �s, the system crosses thecoexistence line at a single point. In density space the corre-sponding trajectory thus follows a particular tie line thoughthe coexistence region, as shown in Fig. 3 ��bi��.

In seeking to apply the RGE ensemble to study a binarymixture, it is therefore imperative that one adopts a grand-canonical treatment of the small particles. Doing so ensuresthat only a single pair of coexistence states is encounteredinside the bulk coexistence region, i.e., that one tracks a tieline of the bulk phase diagram. This is a prerequisite for thecorrect operation of our intersection method, which is de-signed to determine first the diameter density for a singlepair of coexistence states as a prelude to determining thecoexistence densities of the large particles themselves. Wefurther note that a grand-canonical treatment of the smallparticles corresponds more closely to common experimentalarrangements where one typically measures properties of themixture with respect to variations of a reservoir volume frac-tion of small particles.

IV. RESULTS

Equipped with the methods described above, we can setabout the task of determining the coexistence properties ofthe large particles in the presence of a sea of small ones. Tothis end we apply the GCA-RGE method to study liquid-vapor phase coexistence in a q=0.1 LJ mixture �see Sec. II�.In addition to the cluster updates which swap whole groupsof particles �including both large and small species� betweenboxes, small particles are sampled across both boxes using astandard local grand-canonical algorithm at constant chemi-cal potential �see Sec. III C�. As discussed above, we choose�s to yield �for each temperature of interest� a prescribedvolume fraction �s

r of small particles in the reservoir. Thisrequires prior knowledge of the reservoir equation of state�s

r��s ,T�, which we obtained via explicit simulation of thepure fluid of small particles. Note that the computational costof obtaining �s

r��s ,T� is low, particularly if one employshistogram extrapolation32 to scan a region of � and T sur-rounding each simulation state point.

The GCA-RGE simulations are performed using two cu-bic periodic simulation boxes of linear size L=10. We con-sider seven values of the reservoir volume fraction of thesmall particles, �s

r=0.005, 0.01, 0.015, 0.02, 0.03, 0.04, and

0.05. In the limit of low densities of large particles, thesevalues of �s

r correspond to average numbers of small par-ticles in the range of 104–105. The computational expendi-ture incurred in simulating such great numbers of small par-ticles places an upper bound on the value of �s

r for which itis feasible to perform a full determination of the coexistencebinodal. Further difficulties arise from the fact that the typi-cal cluster size at coexistence was found to grow steadily aswe increased �s

r. This is demonstrated in Fig. 4 which plotsthe distribution of the fraction of large particles in the clusterfor �s

r=0.01, 0.03, and 0.05 measured at the respective criti-cal point parameters. These distributions are bimodal, withsome fairly small clusters comprising just a few particles,and many clusters that comprise the vast proportion of largeparticles. Updating such clusters results in only relativelyminor alterations to a configuration and consequently, we areable to determine the coexistence binodal only for �s

r�3%,whereas for �s

r=0.04 and 0.05 we restrict ourselves to deter-mining critical-point parameters. It should be stressed how-ever, that the cluster sizes observed in the present study maynot provide a general guide to the maximum �s

r at which theGCA-RGE scheme will operate. This is because, as we shallshow, our choice of interspecies interactions engenders alarge depression in Tc with increasing �s

r which in turn pro-motes the formation of large clusters due to the temperaturedependence of the GCA bond-formation probability Eq. �2�.However, other choices of interactions can be expected tolead to a different temperature dependence of the criticalpoint parameters, hence allowing larger values of �s

r to beattained.

The critical-point parameters are determined, for each �sr

studied, from measurements of the iso-Q� curve. For �sr

�3%, the intersection method described in Sec. III B is de-ployed to determine the large-particle coexistence densitiesin the subcritical regime. Figure 5 presents our results for the−T binodal. The principal feature is, as previously men-tioned, a strong depression of the binodal to lower tempera-tures and lower densities as �s

r is increased. The scale of theassociated shifts in the critical parameters is made apparentin Fig. 6, which plots our estimates of the critical tempera-ture and density as a function of �s

r. One sees that as the

FIG. 4. Distribution of the fraction of large particles in the cluster, fc, atcriticality for the values of �s

r shown in the legend.

194102-6 Ashton et al. J. Chem. Phys. 133, 194102 �2010�

reservoir volume fraction of small particles is increased inthe range of 0–0.05, the critical temperature decreases byapproximately 50%, while the critical density drops by some30%. The error bars shown on the estimates of the criticalparameters derive from a bootstrap analysis of the variousquadratic fits that are consistent with the uncertainties in thelocus of each iso-Q� curve �Fig. 7�.

To demonstrate the correctness of our method, we com-pare the binodal for �s

r=0.01 with that obtained using a quite

different approach, recently proposed by two of us.34 This isa fully grand-canonical MC scheme in which large particlesare gradually transferred to and from the system by means ofstaged insertions and deletions. To negate ensemble differ-ences that occur when comparing results in finite-size sys-tems, we transform the grand-canonical distribution of thelarge-particle density, P��, to the RGE using the exact trans-

formation P̂��= P��P�20−�.25,26 We then proceed to lo-cate coexistence as if the data had been generated in the RGEby treating 0 as a parameter of the transformation. The re-sulting coexistence densities are compared with those ob-tained via the GCA-RGE simulations in Fig. 8. The agree-ment is good, particularly at low temperature. The deviationsnear criticality arise from the difference in the system sizeused in each case �L=7.5�ll for the grand-canonical systemand L=10�ll for the RGE system�, and thus reflect that thecorrelation length exceeds the system size in the grand-canonical simulation.

It is instructive to attempt to relate the shift in the bin-odal occurring with increasing small-particle density to alter-ations in the underlying local fluid structure. An indication asto the factors at work here follows from a study of the effectof small particles on the effective potential between a pair oflarge particles

FIG. 5. Phase diagrams showing the liquid �diamonds� and gas �squares�coexistence densities of large particles for q=0.1. Data are shown for reser-voir volume fractions �top to bottom� �s

r=0, 0.005, 0.01, 0.02, and 0.03.Also shown in each case are the coexistence diameter �circles� and criticalpoint �asterisks�.

FIG. 6. �a� Critical temperature and �b� critical density vs �sr as determined

from the iso-Q� curves. Error bars derive from a bootstrap analysis with 100resamples.

FIG. 7. The measured iso-Q� curves for �top to bottom� �sr=0, 0.005, 0.01,

0.015, 0.02, 0.03, 0.04, and 0.05. Also shown are the estimated error barsfrom which we calculated the overall uncertainty in critical parameters via abootstrap analysis.

FIG. 8. Comparison for �rs=0.01 of the binodal obtained using the GCA-

RGE technique �crosses� and the grand-canonical approach �circles� de-scribed in Ref. 34. Statistical uncertainties are comparable to the symbolsizes. Differences in the results near criticality arise from the different sys-tem sizes used in the two cases.

194102-7 Algorithm for highly size-asymmetric mixtures J. Chem. Phys. 133, 194102 �2010�

W�r� lim→0

− ln�gll�r�� . �6�

An example is shown in Fig. 9�a� which compares W�r� forthe cases �s

r=0 �which simply corresponds to the bare LJpotential� with that for �s

r=0.05, at T=1.3. One sees that fortypical separations of large particles, the effective potential isless attractive than the bare interaction. Thus the net effect ofthe small particles is repulsive as shown by the differenceplot in Fig. 9�a�, a feature that accords with the reduction inthe critical temperature. A likely reason for this is to befound in the associated form of gls�r� describing the correla-tions between a large particle and a small particle, as shownin Fig. 9�b� at �s

r=0.05. This shows that small particles forma diffuse, nonabsorbing cloud around each large particle be-cause of their weak mutual attraction. Presumably, however,the free-energy cost arising when the clouds associated withtwo or more large particles overlap acts to reduce the intrin-sic attractions between large particles. Interestingly, the dif-ference plot of Fig. 9�a� shows that at very small separationsof large particles �corresponding to high overlap energy� theeffect of the small particles changes from being repulsive tobeing attractive.

Finally, we show in Fig. 10 a configurational snapshot ofour simulation boxes at coexistence �i.e., 0=d� for the case�s

r=3%, T=0.88Tc. This provides a visual impression of thecharacter of the coexisting phases and the extent to which thelarge particles are severely “jammed” by the small ones.

V. DISCUSSION AND CONCLUSIONS

In summary, we have described a variant of the GCA�Ref. 23� for the accurate determination of phase behavior inhighly size-asymmetrical fluid mixtures. The method �anearly version of which was previously described in Ref. 25�operates by swapping clusters containing large and smallparticles between two boxes of equal volume, the global den-sity of large particles being fixed. The resulting spectrum ofsingle-box fluctuations of the large particles can be analyzedwith respect to changes in their global density using the in-tersection method of Ashton et al.26 to yield accurate esti-mates of coexistence densities. Critical points can similarlybe located to high precision by using an appropriate finite-size estimator for criticality, namely, the maximum of theiso-Q� curve.

We have applied the method to a LJ mixture with sizeratio 10:1 to determine the coexistence properties of largeparticles for small-particle reservoir volume fractions in therange 0��s

r�3%. Additionally, critical-point parameterswere determined for �s

r=0.04 and 0.05. Our results show thatwhen the small particles are weakly attracted to the largeones, their net effect is to lower the degree of attractionbetween large particles. As a consequence, the coexistencebinodal shifts to lower temperatures, confirming the prelimi-nary findings of Ref. 25. Such a situation contrasts markedlywith the depletion effect applicable to small particles thatinteract with the large ones such as hard spheres,7 for whichthere is a net increase in the degree of attraction betweenlarge particles. Our measurements of local structure suggestthat in the case we have considered, the small particles forma diffuse �nonadsorbing� cloud surrounding each large par-ticle. The overlap of clouds necessary for two large particles

FIG. 9. �a� The measured form of the effective potential W�r� defined inthe text, at temperature T=1.3. Data are shown for the bare LJ potential��s

r=0, dashed line� and �sr=0.05 �solid line� and their difference �dashed-

dotted line�. �b� Form of gls�r� for ll→0 at �sr=0.05, T=1.3.

FIG. 10. Configurational snapshot of the two boxes in the restricted Gibbsensemble at coexistence for �s

r=3% and T=0.88Tc.

194102-8 Ashton et al. J. Chem. Phys. 133, 194102 �2010�

to approach one another appears unfavorable in free-energyterms, leading to a net decrease in the degree of attractionbetween large particles. This is reminiscent of the “nanopar-ticle haloing” effect.5,9

It is gratifying to note that the GCA-RGE method per-mits the study of phase behavior in regimes that are inacces-sible to traditional simulation approaches. Specifically, thephase diagrams we have presented could not have been ob-tained using even the most efficient traditional approach tofluid phase equilibria, namely, standard grand-canonicalsimulation.35 For instance, for �s

r 0.005 our tests show thatthe grand-canonical relaxation time is too large to be reliablyestimated. Nevertheless, a lower bound on the grand-canonical relaxation time, relative to that of the pure LJ fluid,can be estimated via a comparison of the large-particle trans-fer �insertion/deletion� acceptance probability pacc. For liq-uidlike densities of the large particles ��0.6�, pacc is of theorder of 10−4 at �s

r=0.005.25 Upon increasing �sr to a volume

fraction of merely 1%, this probability falls to pacc�10−6.These values are to be compared with pacc�10−1 for the pureLJ fluid. One can therefore expect the grand-canonical relax-ation time of the mixtures studied here to be several orders ofmagnitude greater than for the pure LJ fluid.

Notwithstanding the efficiency gains provided by theGCA-RGE approach, it should be stressed that the results wehave reported nevertheless entailed a significant computa-tional outlay. Specifically, runs to determine each coexist-ence point typically varied in length between 100 and 3000 hof central processing unit time on a 3 GHz processor. Theupper value in this range was that required at the highestvolume fraction of small particles studied �for which thereare very many small particles� and the lowest temperature�where most of the large particles are involved in each clus-ter update�. Thus while studies of phase behavior in highlyasymmetrical mixtures cannot yet be regarded as routine,they are now at least feasible.

With regard to future studies of highly asymmetricalmixtures, one barrier to attaining higher values of �s

r andsmaller values of q is simply the computational overheadassociated with large numbers of small particles, althoughwe note that the GCA has been successfully applied to sys-tems with millions of nanoparticles.9 Additionally, in thepresent model, the suppression of the critical temperaturewith increasing �s

r leads to a rapid growth in the cluster sizewhich renders the GCA-RGE approach increasingly less ef-ficient. More generally, however, in situations where thesmall particles induce an effective �depletion� attraction be-tween the large ones, we expect that the cluster size willremain manageable to rather larger �s

r than studied here.Finally, we mention an alternative approach, proposed

by two of us, for determining coexistence properties inhighly size-asymmetrical mixtures.34 This method utilizes anexpanded grand-canonical ensemble in which the insertionand deletion of large particles is accomplished gradually bytraversing a series of states in which a large particle interactsonly partially with the environment of small particles. Imple-menting this approach requires prior determination of a mul-ticanonical weight function to bias insertions of the particles,

and thus renders it less straightforward to use than the GCA-RGE. However, being fully grand canonical does have theadvantage of providing information on the chemical poten-tials of large particles, thereby permitting histogram reweigh-ing in terms of density as well as temperature. In future workwe hope to provide a systematic comparison of the relativecomputational cost of both approaches in various parameterregimes.

ACKNOWLEDGMENTS

This work was supported by EPSRC �Grant No. EP/F047800� �N.B.W.� and NSF �Grant No. DMR-0346914��E.L.�. Computational results were partly produced on a ma-chine funded by HEFCE’s Strategic Research Infrastructurefund.

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