Depletion potentials in highly size-asymmetric binary hard-sphere mixtures:Comparison of simulation results with theory

Douglas J. Ashton,1 Nigel B. Wilding,1 Roland Roth,2 and Robert Evans3

1Department of Physics, University of Bath, Bath BA2 7AY, United Kingdom2Institut für Theoretische Physik, Universität Erlangen-Nürnberg, Staudtstrasse 7, 91058 Erlangen, Germany

3H. H. Wills Physics Laboratory, University of Bristol, Royal Fort, Bristol BS8 1TL, United Kingdom

We report a detailed study, using state-of-the-art simulation and theoretical methods, of the ef-fective (depletion) potential between a pair of big hard spheres immersed in a reservoir of muchsmaller hard spheres, the size disparity being measured by the ratio of diameters q ≡ σs/σb. Smallparticles are treated grand canonically, their influence being parameterized in terms of their pack-ing fraction in the reservoir, ηrs . Two Monte Carlo simulation schemes –the geometrical clusteralgorithm, and staged particle insertion– are deployed to obtain accurate depletion potentials fora number of combinations of q ≤ 0.1 and ηrs . After applying corrections for simulation finite-sizeeffects, the depletion potentials are compared with the prediction of new density functional theory(DFT) calculations based on the insertion trick using the Rosenfeld functional and several subsequentmodifications. While agreement between the DFT and simulation is generally good, significant dis-crepancies are evident at the largest reservoir packing fraction accessible to our simulation methods,namely ηrs = 0.35. These discrepancies are, however, small compared to those between simulationand the much poorer predictions of the Derjaguin approximation at this ηrs . The recently proposedmorphometric approximation performs better than Derjaguin but is somewhat poorer than DFTfor the size ratios and small sphere packing fractions that we consider. The effective potentialsfrom simulation, DFT and the morphometric approximation were used to compute the second virialcoefficient B2 as a function of η

rs . Comparison of the results enables an assessment of the extent to

which DFT can be expected to correctly predict the propensity towards fluid-fluid phase separationin additive binary hard sphere mixtures with q ≤ 0.1. In all, the new simulation results provide thefirst fully quantitative benchmark for assessing the relative accuracy of theoretical approaches forcalculating depletion potentials in highly size-asymmetric mixtures.

I. INTRODUCTION

Much of condensed matter physics and chemistry isconcerned with simplifying the description of a complexmany body system by integrating out certain subsetsof the degrees of freedom of the full system. Thus intreating atomic and molecular solids and liquids one of-ten resorts to integrating out, usually approximately, thehigher energy quantum mechanical degrees of freedom ofthe electrons in order to obtain an effective interatomicor intermolecular potential energy function that is thenemployed in a classical statistical mechanical treatmentto study the properties of condensed phases. Similarly inmetallic systems integrating out the degrees of freedomof the conduction electrons leads to an effective Hamil-tonian for the screened ions or pseudo-atoms. When oneturns to complex, multi-component fluids such as col-loidal suspensions or polymers in solution the basic ideais similar: one integrates out the degrees of freedom ofthe small species to obtain an effective Hamiltonian forthe biggest species; for an illuminating review see [1]. Inthis case all species can be treated classically and theformalism is essentially the famous one of McMillan andMayer [2] who developed a general theory for the equi-librium properties of solutions. These authors and manysubsequently recognized that integrating out is best per-formed when the smaller species (those constituting thesolvent) are treated grand canonically. Obtaining the fulleffective Hamiltonian is a tall order. A first step in any

theoretical treatment is to determine the effective poten-tial between a single pair of the biggest particles in a seaof the smaller species. There is a long history of work inthis field. Various statistical mechanical techniques havebeen developed to calculate these potentials for differenttypes of complex fluid. Well-known examples of effectivetwo-body potentials, forming cornerstones of colloid sci-ence, are the DLVO potential for charged colloids and theAsakura-Oosawa depletion potential for colloid-polymermixtures. Further examples, including polymer systems,are given in [1, 3].

The problem of determining the effective two-body in-teraction is particularly challenging to theory and com-puter simulation in the situation where – and we special-ize to a binary fluid – the bigger particles are much largerthan the smaller ones. Sophisticated theoretical and sim-ulation techniques, that will provide accurate results forthe effective potential, are only now becoming available.

Our present focus is on a simple model for a suspensionof a binary mixture of big and small colloidal particles,both species suitably sterically stabilized, i.e. we considera highly size asymmetric binary mixture of hard spheres.This can also serve as a crude model of a mixture of col-loids and non-adsorbing polymer and can be regarded asa reference system for a mixture of size asymmetric sim-ple fluids [4]. The hard sphere mixture is important inthat it provides a testing ground for theories of the effec-tive potential between two big hard spheres immersed inreservoirs of small ones, with different packing fractions

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ηrs , and indeed for theories of the full effective Hamilto-nian obtained when the degrees of freedom of the smallspecies are integrated out fully [5]. Generally speaking,the more asymmetric the mixture the more difficult it isto treat both species on equal footing and the more nec-essary it is to perform some integrating out to obtain an(approximate) effective Hamiltonian. Often this proce-dure is only feasible at the level of a Hamiltonian consist-ing of a sum of (effective) pair potentials, as obtained byconsidering a single pair of the big particles, along withzero and one body terms [1, 5]. One of the advantagesof the hard sphere system is that geometrical considera-tions indicate that three, four etc body terms become lessimportant as the size ratio q = σs/σb becomes small. σsand σb denote the diameters of the small and big species,respectively. Thus provided one can calculate an accu-rate effective pair potential, the pair description alonedetermines an effective Hamiltonian that should providean excellent description of the big-big correlation func-tions and the phase behaviour of the binary hard spheremixture when q is sufficiently small [5]. Note that in thispaper we consider additive hard sphere mixtures so thatthe big-small diameter σbs = (σb + σs)/2.

Since the studies of the phase behaviour of the hardsphere mixture by Dijkstra et al [5], whose simulations ofan effective one-component system used a rather crudeapproximate pair potential, there have been several newdevelopments in the theory of effective potentials. Mostof these are based on Density Functional Theory (DFT).It is important to assess whether i) the potentials derivedin recent studies are accurate and ii) use of these mightlead to different predictions for the properties of the mix-ture. In order to make such assessments it is necessaryto have accurate simulation results for the effective po-tential. Employing state of the art techniques we providewhat we believe are the most accurate results currentlyavailable for q ≤ 0.1 and packing fractions ηrs up to 0.35and make direct comparisons with the results of theoreti-cal approaches. The simulation techniques we employ donot allow us to work at very high values of ηrs but theydo allow us to compute accurate effective potentials, fora range of size ratios, in the regime of small sphere pack-ing fractions where the putative (metastable) fluid-fluidphase separation is predicted to occur [5]. By calculat-ing the second virial coefficient associated with the effec-tive potential we make new estimates of the value of ηrswhere the onset of this elusive phase transition occurs. Ofcourse real colloidal systems may not reach equilibriumon experimental timescales, particularly if the effectivepotential exhibits significant repulsive barriers [6]. Nev-ertheless, knowledge of the underlying phase behaviour isimportant for interpreting dynamical observations, suchas whether gelation or glassiness might set in [7, 8].

The comparison between DFT results and simulationaddresses recent suggestions [9, 10] that no existing the-oretical framework is reliable for calculating effective po-tentials for small values of q and physically relevant val-ues of ηrs . We examine and refute these suggestions in

the light of our present results.The effective potential between two big hard spheres

takes the form:

φeff (rbb) = φbb(rbb) +W (rbb) (1)

where φbb is the bare hard sphere potential between twobig spheres and W is the so-called depletion potential.This is attractive for small separations rbb of the bigspheres but decays in an exponentially damped oscilla-tory fashion at large separations. The physics of the at-traction is well understood: the exclusion or depletion ofthe small spheres as the big ones come close together re-sults in an increase in free volume available to the smallspecies leading to an increase of entropy. If this attrac-tion is sufficiently strong it can give rise to fluid-fluidphase separation. Such a phase transition is driven bypurely entropic effects: recall that all the bare interac-tions in the mixture are those of hard spheres. Of coursethe concept of an attractive depletion potential betweencolloids dispersed in a solution of non-absorbing polymer,or other depletants has a long history. The recent bookby Lekkerkerker and Tuinier [11] describes this and thegeneral importance of depletion interactions in colloidalsystems.

We have emphasized that the effective pair potentialis a key ingredient in an effective Hamiltonian descrip-tion of the mixture. However, this object is also impor-tant in its own right since it can be measured experi-mentally for colloidal systems using various techniques.More specifically, the effective potential between a singlecolloid, immersed in a suspension of small colloidal par-ticles or non-adsorbing polymer, and a flat substrate hasbeen measured; see for example [12] and the comparisonsmade between DFT results and experiment [13]. Crockeret al [14] measured the effective potential between two bigPMMA particles immersed in a sea of small polystyrenespheres and observed damped oscillations at high smallsphere packing fractions. Subsequently comparisons weremade with DFT results [15]. Ref. [16] provides a recentreview of direct experimental measurements of effectiveinteractions in colloid-polymer systems.

A. Previous Simulation Studies

In general, the task of accurately measuring effectivepotentials in highly size-asymmetrical fluid mixtures us-ing traditional simulation methods such as molecular dy-namics (MD) or Monte Carlo (MC) is an extremely chal-lenging one. The difficulty stems from the slow relaxationof the big particles caused by the presence of the smallones. Specifically, in order for a big particle to relax, itmust move a distance of order its own diameter σb. How-ever, for small size ratios q, and even at quite low valuesof ηrs , very many small particles are typically to be foundsurrounding a big particle and these hem it in, greatlyhindering its movement. In computational terms thismandates a very small MD timestep in order to control

3

integration errors, while in MC simulations that employlocal particle displacments, a very small trial step-sizemust be used in order to maintain a reasonable accep-tance rate. Consequently, the computational investmentrequired to simulate highly size asymmetric mixtures bytraditional means is (generally speaking) prohibitive atall but the lowest packing fractions of small particles.

Owing to these difficulties, most previous simulationstudies of hard sphere mixtures [9, 10, 17–19] haveadopted an indirect route to measuring effective poten-tials in the highly size asymmetrical regime q ≤ 0.1 basedon measurements of interparticle force. The strategyrests on the observation that the force between two bigparticles can be expressed in terms of the contact densityof small particles at the surface of the big ones [18, 20].By measuring this (angularly dependent) contact densityfor fixed separation rbb of the big particles and repeatingfor separations ranging from the minimum value rbb = σbto rbb =∞, one obtains the force profile F (rbb). This canin turn be integrated to yield an estimate of the depletionpotential. Generally speaking, however, the statisticalquality of the data obtained via this route seems typi-cally quite low, particularly at small q and high ηrs . Thispresumably reflects the difficulties of measuring contactdensities accurately and the errors inherent in numericalintegration.

Only a few studies have tried to measure the depletionpotential directly for q ≤ 0.1 (see ref. [21, 22] for hardsphere studies and refs. [23, 24] for more general poten-tials). In common with the present work, these studiesdeployed a cluster algorithm (to be described in sec. II A)to deal efficiently with the problem of slow relaxation out-lined above. However, they treated the small particlescanonically rather than grand canonically, which compli-cates comparison with theoretical predictions which aretypically formulated in terms of an infinite reservoir ofsmall particles [25]. Furthermore it seems that no pre-vious simulation studies have discussed (in any detail)finite-size effects in measurements of depletion potentials,the role of which we believe to be particularly significantat large size asymmetries. Consequently, while previouswork has evidenced good qualitative agreement betweensimulation and theory, there is to date a lack of data fromwhich one can make confident comparisons between thevarious theoretical approaches. This is remedied in thepresent work.

B. Previous Theoretical Studies

There are many of these and they are based on a vari-ety of techniques. Integral equation treatments aboundand some of these are summarized in the recent arti-cle by Boţan et al [26]. The studies of Amokrane andco-workers that implement sophisticated closure approx-imations within a bridge functional approach probablyconstitute the state of the art in integral equation treat-ments of asymmetric mixtures-see e.g. Refs. [27, 28]. A

related rational function approach was used recently byYuste et al [29]. Tackling highly asymmetric mixtures viaintegral equation methods, where one treats both specieson equal footing, is notoriously difficult and making sys-tematic assessments of the reliability of closure approx-imations is not straightforward and, of course, requiresaccurate simulation data.

Density functional (DFT) treatments are arguablymuch more powerful. There are several different waysof calculating the effective (depletion) potential betweentwo big hard spheres in a reservoir of small hard spheresor more generally of calculating the effective potential be-tween two big particles in a reservoir of small ones witharbitrary interactions between bb, bs and ss. The firstmethod is to fix the centres of the big (b) particles a dis-tance rbb apart and then compute the grand potentialof the small (s) particles in the external field of the twofixed b particles as a function of rbb for a given size ratioq and a reservoir packing fraction of ηrs . This method re-quires only a DFT for a single component fluid, the smallparticles. The big particles are fixed so they simply ex-ert an external potential on the small ones. DFT forone-component hard spheres is very well-developed; veryaccurate functionals exist and these are suitable for treat-ing the extreme inhomogeneities that arise for small sizeratios q. It follows that this brute force method should berather accurate. Its drawback is that the density profileof the small particles has cylindrical symmetry requir-ing numerically accurate minimization of the free energyfunctional on a two-dimensional grid.

Goulding [30, 31] performed pioneering brute force cal-culations using the Rosenfeld fundamental measure the-ory (FMT) [32] for a system with q = 0.2 and pack-ing fractions ηrs up to 0.314. This method has beenrefined recently by Boţan et al [26] who employed vari-ous hard-sphere functionals for more asymmetric systemsand higher values of ηrs . These authors, see also Oettelet al [33], also used DFT to calculate the depletion forcedirectly using the formula due to Attard [18, 20] that re-lates the force to the density profile of small spheres incontact with a big sphere. Once again the density profilehas cylindrical symmetry and careful numerical methodsare required.

A popular DFT method for hard-sphere systems isbased on what has become known as the insertion trickor insertion method [15, 19]. This is a general procedure,see Sec. III A, for calculating the depletion potential be-tween a big particle and a fixed object, e.g. a wall oranother big particle, immersed in a sea of small particles.The advantage of the method is that one requires onlythe equilibrium density profile of the small species in theexternal field of the isolated fixed object and this profileclearly has the symmetry of the single fixed object. Forthe case of two big spheres the profile ρs(r) has sphericalsymmetry. The disadvantage is that the theory requiresa DFT for an asymmetric mixture, albeit in the limitwhere the density ρb of the big particles approaches zero:ρb → 0. For hard-sphere mixtures the insertion method

4

is straightforward to implement and [15] provides a seriesof comparisons, using the Rosenfeld FMT [32], with thesimulation data that existed in 2000.

Further comparisons between results of the DFT in-sertion method and simulation studies were made inRefs. [9, 10, 26, 33]. In the present paper we seek to makemore quantitative comparisons, taking into considerationthe improved accuracy of our new simulation results andthe availability of improved DFTs for hard sphere fluids.

There is a further theoretical approach to calculatingdepletion potentials developed very recently in Ref. [33]and employed subsequently in Ref. [26]. This approachis based on morphological (or morphometric) thermody-namics [34]. The basic idea is that the depletion potentialis (essentially) the solvation free energy of the dumbbellformed by the two big spheres and that this quantity canbe separated into geometrical measures, namely the vol-ume, surface area and integrated mean and Gaussian cur-vatures. The coefficients of these measures are geometryindependent thermodynamic quantities, i.e. the pressure,the planar surface tension and two bending rigidities allof which can be obtained from simulations or from DFTcalculations of the single component fluid performed fora simple geometry.

The paper is arranged as follows: Section II describesthe grand canonical simulation methods that we have em-ployed for determining the depletion potential betweentwo big spheres for size ratios q from 0.1 to 0.01. In Sec-tion III we summarize briefly the DFT insertion methodand the three hard sphere functionals that we employ inour present calculations. We also discuss some of the lim-itations of the parameterization of the depletion potentialintroduced in [15]. Some details of the morphometric ap-proaches are also given here. Results are presented inSection IV. As a test of our simulation method we de-termine the depletion potential for two big hard spheresin a solvent of non-interacting point particles that havea hard interaction with the big spheres. For this case thedepletion potential is known analytically–it is the ven-erable Asakura-Oosawa potential [35, 36] of colloid sci-ence. For the additive binary hard sphere case we makecomparisons between results of simulation, DFT inser-tion method, the morphometric approach and the Der-jaguin approximation [37] for the depletion potential. Wealso compare simulation, DFT and morphometric resultsfor the associated second virial coefficient B2(η

rs). The

latter provides a valuable indicator of the propensity ofthe bulk binary mixture to phase separate into two fluidphases [38, 39]. We conclude in Section V with a discus-sion.

II. SIMULATION METHODS

A. Geometrical cluster algorithm

An efficient cluster algorithm capable of dealing withhard spheres mixtures was introduced by Dress and

Krauth in 1995 [40]. It was subsequently generalizedto arbitrary interaction potentials by Liu and Luijten[23, 41] who dubbed their method the Geometrical Clus-ter Algorithm (GCA). Here we describe the applicationof the GCA to a size asymmetrical binary mixture of hardspheres.

The particles comprising the system are assumed tobe contained in a periodically replicated cubic simula-tion box of volume V . The configuration space of theseparticles is explored via cluster updates, in which a sub-set of the particles known as the “cluster” is displaced viaa point reflection operation in a randomly chosen pivotpoint. The cluster generally comprises both big and smallparticles and by virtue of the symmetry of the point re-flection, members of the cluster retain their relative posi-tions under the cluster move. Importantly, cluster movesare rejection-free even for arbitrary interparticle interac-tions [23]. This is because the manner in which a clusteris built ensures that the new configuration is automati-cally Boltzmann distributed.

For hard spheres, the cluster is constructed as follows:one of the particles is chosen at random to be the seedparticle of the cluster. This particle is point-reflectedwith respect to the pivot from its original position toa new position. However, in its new position, the seedparticle may overlap with other particles. The identi-ties of all such overlapping particles are recorded in alist or “stack”. One then takes the top-most particle offthe stack, and reflects its position with respect to thepivot. Any particles which overlap with this particle atits destination site are then added to the bottom of thestack. This process is repeated iteratively until the stackis empty and there are no more overlaps.

In this work we shall be concerned with measurementsof the radial distribution function gbb(r) for a system con-taining a pair of big particles among many small ones.To effect this measurement we modify the GCA slightlyas follows: we choose one big particle to be the seedparticle, which we place uniformly at random within ashell σbb < r < L/2, centred on the second big parti-cle, with L the linear box dimension. The location ofthe pivot is then inferred from the old and new positionsof the seed particle. Thereafter clusters are built in thestandard way. This strategy satisfies detailed balanceand improves efficiency by ensuring that we only gener-ate separations of the big particles that lie in the ranger = [σbb, L/2] for which gbb(r) is defined for hard spheresin a cubic box. The correctness of this technique waschecked by comparing with the standard GCA approachdescribed above.

Small particles are treated grand canonically in oursimulations. In practical terms this means that in paral-lel with the cluster moves, we implement insertions anddeletions of small particles, subject to a Metropolis ac-ceptance criterion governed by an imposed chemical po-tential. The choice of chemical potential controls thereservoir packing fraction of small particles.

For the systems of interest in this work, we find that

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the GCA is efficient for reservoir packing fractions ηrs .0.2. Above this value one finds that practically all theparticles join the cluster, which merely results in a trivialpoint reflection of the entire system. For single compo-nent fluids this problem can be ameliorated by biasingthe choice of pivot position to be close to the positionof the seed particle [23]. Doing so has been reported toextend the operating limit to ηrs ' 0.34. However forthe case of highly asymmetrical mixtures we find thatthis strategy does not significantly decrease the numberof particles in the cluster because as soon as a second bigparticle joins the cluster and is point reflected it causesmany overlaps with small particles.

B. Staged insertion algorithm

Our second MC approach for obtaining effective po-tentials for size asymmetrical mixtures is based on thestaged insertion of a big particle [42–45]. The methodinvolves first fixing one big particle at the origin and thensampling the free energy change associated with insert-ing a second big particle at a prescribed distance rbb fromthe origin. Essentially this amounts to an estimation ofthe chemical potential of the second particle µex(rbb). Assuch our approach is close in spirit to one proposed veryrecently by Mladek and Frenkel [46], although their im-plementation did not employ staged insertion and wastherefore restricted to low density systems or those in-teracting via very soft potentials.

The effective potential for big particle separation rbb issimply

W (rbb) = µex(rbb) + C (2)

where the additive constant

C = − limrbb→∞

µex(rbb) (3)

can be determined as the excess chemical potential ofa single big particle in the reservoir of small particles.To estimate µex(rbb) we follow the strategy described inRef. [44]. In outline, one imagines that the second bigparticle can exist in one of M possible ‘ghost’ states inwhich it interacts with a small hard particle (a distancerbs away) via the potential

βφ(m)g (rbs) = −[1−Θ(2rbs − σb)] lnλ(m) .

Here m = 1 . . .M (an integer) indexes the ghost states,while the associated coupling parameter 0 ≤ λ(m) ≤ 1controls the strength of the repulsion between the bigparticle and the small one. Owing to the step functionΘ, the potential is uniformly repulsive over the volumeof the big particle, and zero elsewhere. Moreover, forλ(m) > 0 the repulsion is finite so that overlaps betweensmall particles and the big one can occur. If we denote by

No the number of such overlaps at any given time, thenthe configurational energy associated with the ghost bigparticle is

βΦ(m)g = −No lnλ(m) . (4)

Clearly for λ(m) = 0, the big particle acts like a normalhard sphere, while for λ(m) = 1 there is no interactionand the big particle is effectively absent from the system.To span this range we set the extremal states λ(1) = 0 andλ(M) = 1, and define some number of intermediate statesthat facilitate efficient MC sampling over the range m =1 . . .M , i.e. that permits the ghost particle to fluctuatebetween being a real hard sphere and being absent. Themeasured value of the relative probability of finding thesystem in these extremal states yields the excess chemicalpotential:

µex(rbb) = ln

[p(λ(M))

p(λ(1))

]. (5)

Now since W (rbb) is spherically symmetric, it can beestimated from Eqs. 5 and 2, by measuring µex(rbb) forvalues of rbb ≥ σb along a 1D grid. Moreover since eachsuch measurement is independent of the others, the ap-proach is trivially parallel and thus can be effectivelyfarmed out on multi-core processors.

Details of a suitable Metropolis scheme for samplingthe full range of m = 1 . . .M have been described in de-tail previously [44, 45]. The basic idea is to performgrand canonical simulation of the small particles, supple-mented by MC updates that allow transitions m→ m±1for the ghost big particle. These transitions are acceptedor rejected on the basis of the change in the configura-tional energy Eq. 4. However, for this strategy to realizethe aim of sampling the relative probability of the ex-tremal states, it is necessary to bias the transitions suchas to ensure approximately uniform sampling of the Mghost states. This is achieved by determining a suitableset of weights which appear in the MC acceptance prob-ability [47].

Additionally it is important to choose sufficient inter-mediate states and to place them at appropriate valuesof λ such that transitions m→ m± 1 are approximatelyequally likely in both directions and have a reasonablyhigh rate of acceptance. To achieve this we perform apreliminary run in which we consider a single big ghostparticle in the reservoir of small ones. We first defineM = 1000 values of λ in the range (0, 1), evenly spacedin lnλ, and (in short runs) measure the distribution ofoverlaps p(No|λ) for each. From this set we then pickout those values of λ for which successive p(No|λ) exhibitan overlap by area of approximately 20%. This criteriayields a suitable set of intermediate states.

Efficiency benefits result from noting that the rateof transitions in m depends on how quickly the num-ber of overlaps No relaxes after each successive transi-

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tion. To enhance this relaxation we preferentially per-form grand canonical insertions and deletions of smallparticles within a spherical sub-volume of diameter 1.2σbcentred on the second big particle. Updates within thesub-volume occur with a frequency 100-fold that outsidethe sub-volume. Our approach –which satisfies detailedbalance– greatly reduces the time spent updating smallparticles whose coordinates are relatively unimportantfor the quantity we wish to estimate.

The validity of the staged insertion technique was pre-viously verified via comparisons with GCA in the con-text of grand canonical ensemble studies of phase be-haviour in highly size asymmetrical binary mixtures ofLennard-Jones particles [45]. In the present context ofhard sphere depletion potentials, we have explicitly ver-ified for q = 0.1, ηrs = 0.2 that the staged insertion tech-nique yields results that agree to within statistical errorswith those determined using the GCA technique.

A further innovation, applicable to highly size-asymmetrical hard sphere mixtures, stems from the ob-servation that it is not actually necessary to insert a bighard sphere in order to calculate the effective potential.Instead it is sufficient and (generally much more efficient)to insert a hard shell of infinitesimal thickness. The ba-sic idea is that when fully inserted a hard shell parti-cle encloses a number of small particles. These remainin equilibrium with the reservoir (i.e. their number canstill fluctuate), but are fully screened from the rest ofthe system because their surfaces cannot penetrate theshell wall. Accordingly their contribution to the free en-ergy is independent of the position of the shell particleand hence their net effect is merely to shift the valueof the additive constant in the measurement of µex(rbb).Since the latter is anyway set by hand to ensure thatlimrbb→∞W (rbb) = 0, one does not need to know thecontribution to the free energy from the enclosed parti-cles.

From a computational standpoint, the task of insert-ing a hard shell is much less challenging than that of in-serting a hard sphere: essentially the chemical potentialgrows with the particle size ratio like (1/q)2 rather than(1/q)3. Consequently, far fewer intermediate stages Mare required to effect the insertion, which reduces sub-stantially the computational expenditure in measuringµex(r) accurately. We have explicitly verified that theshell insertion approach yields results for the effectivepotential that agree to within statistical error with thoseresulting from sphere insertion.

Whilst the staged insertion technique is not as straight-forward to implement as the GCA for highly asymmetri-cal mixtures, it does not suffer the very rapid decrease inefficiency that renders the GCA inoperative for ηrs & 0.2.The computational expenditure required to obtain effec-tive potentials via staged insertion does increase with ηrs ,but more gradually than for the GCA. Thus we were ableto attain (for q = 0.1), the considerably larger reservoirpacking fraction of ηrs = 0.35. This was achieved fora computational expenditure of circa 2 weeks on a 150-

core computing cluster, the bulk of which is associatedwith decorrelating the configurations of the small par-ticles at this volume fraction. At the somewhat lowervolume fraction of ηs = 0.32, only 2 days were requiredon the same machine. These figures sugest that while itmay be feasible to go to somewhat higher volume frac-tions than ηs = 0.35, the computational cost would behigh.

C. Correcting for finite size effects

The effective potential W (r) between two big parti-cles is defined in terms of the radial distribution functiong(r) ≡ gbb(r), with r = rbb, measured in the limit ofinfinite dilution

−βW (r) = limρb→0

ln[g(r)] , (6)

for r > σb. In our simulation studies this limit is approx-imated by placing a single pair of big hard spheres in thesimulation box. A finite-size estimate to g(r), which weshall denote gL(r), is then obtained by fixing the first ofthese particles at the origin and measuring (in the formof a histogram) the probability of finding the second bigparticle in a shell of radius r → r + dr, i.e.

gL(r) =P (r)

Pig(r), (7)

where, the normalization relates to the probability offinding an ideal gas particle at this radius:

Pig(r) =4πr2

V. (8)

Now the principal source of finite-size error in gL(r)arises from the normalization of Pig by the system vol-ume. Specifically, for a finite sized system, the volumeoccupied by the hard sphere at the origin is inaccessibleto the second particle. Accordingly, the accessible systemvolume is

Ṽ = V − v1 , (9)

where v1 = (1/6)πσ3b is the hard sphere volume. More

generally, one should define an effective excluded volumeṽ1 for use in Eq. 9, which allows for the fact that thesmall particles can mediate additional repulsions and/orattraction between the two big particles. In principle ṽ1is given by

ṽ1 = 4π

∫ ∞0

[1− g(r)]r2dr . (10)

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It follows from Eqs. 7-10 that the principal finite-sizecontribution to gL(r) is just an overall scale factor:

g(r) =Ṽ

VgL(r) . (11)

Accordingly gL(r) approaches V/Ṽ at large r instead ofunity, whilst the calculated effective potential, W (r) de-

cays to ln(Ṽ /V ) instead of zero.One can conceive of a number of possible approaches

for dealing with this finite-size error. One is simply tominimize it by choosing a very large system volume V sothat V/Ṽ is close to unity. The disadvantage of this ap-proach is that in a size asymmetrical mixture, in whichthe big particles are in equilibrium with a reservoir ofsmall ones, a huge number of small particles will neces-sarily fill the extra space available in a bigger box. Allthe interactions arising from these small particles thenneed to be computed – which can become prohibitivelyexpensive.

Another route, which we have adopted in the presentwork, is to attempt to correct gL(r) by estimating theoverall scale factor in Eq. 11, thus ensuring that g(r)→ 1at large r. An expedient approach to doing so, whichutilizes as much as possible of the information in gL(r),proceeds by determining the cumulative integral of gL(r):

G(R) =

∫ R0

gL(r)dr . (12)

In practice, this integral was observed to tend towardsa smooth linear form quite rapidly as the upper limit Rincreases, a fact illustrated for typical data in Fig. 1. Alittle thought then shows that if when left uncorrectedg(r) tends to V/Ṽ at large r, the limiting gradient m

of G(R) is m = V/Ṽ , which thus provides the requisitecorrection factor for use in Eq. 11. Thus one corrects themeasured histogram gL(r) by first fitting G(R) to obtainan estimate of the limiting gradient of the linear part,and then scaling gL(r) according to g(r) = m

−1gL(r).

III. THEORETICAL METHODS

As mentioned in the Introduction we choose to makecomparisons between our simulation results and thosefrom the DFT insertion method and from the morpho-metric and Derjaguin approximations.

A. DFT

The DFT insertion method is described in detail inRoth et al [15]. It is based on an exact result fromthe potential distribution theorem, for an arbitrary mix-ture, that expresses the effective potential between twobig particles in terms of the one-body direct correlation

1 1.05 1.1 1.15 1.2 1.25r/σ

b, R/σ

b

0

0.2

0.4

0.6

0.8

1

gL(r

), G

(R)

gL(r)

G(R)

Gradient 0.41004

FIG. 1. The measured form of gL(r) for q = 0.05, ηrs =

0.2 obtained for a system size L = 2.5σb. Also shown is

the cumulative integral G(R) =∫ R0gL(r)dr, together with an

asymptotic linear fit, the gradient of which yields the finite-size correction factor for g(r).

function of the big species in the limit where the den-sity ρb of that species vanishes. For DFT treatments ofhard sphere mixtures that employ the fundamental mea-sures theory (FMT) approach [32] the calculation of theeffective potential requires the computation of the den-sity profile ρs(r) of the small spheres in the neighbour-hood of a single fixed big sphere as well as knowledgeof the weight functions and the excess free energy den-sity of the (binary) mixture [15]. The FMT must besufficiently accurate to describe a very asymmetric bi-nary mixture, i.e. one with small values of q, in the limitρb → 0. In the original DFT studies [15, 19] the Rosen-feld (RF) FMT [32] was employed. In the present workwe employ RF, the White Bear (WB) version [48] and itsmodification the White Bear Mark 2 (WB2) version [49].These versions differ from RF in the choice of the coef-ficients φα entering the excess free energy function. RFyields thermodynamic quantities that are the same asthose from Percus-Yevick (compressibility) approxima-tion, whereas WB incorporates the accurate Mansoori-Carnahan-Starling-Leland (MCSL) empirical bulk equa-tion of state. In WB2 additional self-consistency require-ments are imposed on the pressure. The consistency ofthe WB2 version was demonstrated in calculations of thesurface tension and other interfacial thermodynamic co-efficients for a one-component hard-sphere fluid adsorbedat a hard spherical surface [49]. Ref. [50] provides anoverview of recent developments and describes compar-isons between different versions of FMT.

Boţan et al [26] carried out DFT insertion method cal-culations as well as explicit (brute force) free energy mini-mization for two fixed big spheres using different versionsof FMT. These authors provide a compendium of the in-gredients entering the FMT functionals as well as thethermodynamic coefficients required in the morphomet-ric approximation and we refer readers to Appendix B

8

of [26] for the explicit formulae used in the present cal-culations. Their paper is important in pointing to theregimes where the DFT insertion method is likely to fail.In particular for ηrs = 0.419 and q = 0.1 and 0.2 thereare substantial differences between the results for the de-pletion potential calculated by brute force and from theinsertion method. At higher reservoir packing fractionsthe differences can be even larger. Moreover differentFMTs can give rise to quite different potentials at highsmall sphere packings. The comparisons made by Oettelet al [33] for the depletion force using the RF functionalsuggest that for q = 0.05 the insertion method is not es-pecially accurate at ηrs = 0.314 and 0.367. Of course oneis assuming that the brute force minimization is the moreaccurate method as this requires only a reliable functionalfor a single component hard sphere fluid-not one for theasymmetric mixture.

However, our present study focuses on smaller values ofηrs than those considered in [26]. Previous studies [15, 19]showed generally good agreement between DFT insertionresults and those of simulation for q = 0.1 and 0.2 andηrs typically up to 0.3. Since we are concerned primar-ily with investigating the depletion potential for highlyasymmetrical mixtures in regimes, accessible to simula-tion, near the onset of fluid-fluid phase separation, we donot concern ourselves with very high values of ηrs wherethe DFT insertion method is likely to be inaccurate.

Another way of viewing this DFT insertion method isthat it is equivalent [15, 19] to calculating the big-bigradial distribution function gbb(r) for a binary mixtureusing the test particle route, i.e. one fixes a big sphereat the origin and computes the inhomogeneous densityprofile of the big spheres ρb(r) by minimizing the mixturefree energy functional for this spherical geometry. Thengbb(r; ρb) = ρb(r)/ρb and the depletion potential is givenby

−βW (r) = limρb→0

ln gbb(r; ρb) , for r > σb. (13)

In Refs. [15, 19], for all cases considered, it was demon-strated that a bulk packing fraction ηb = 10

−4 of the bigspheres was sufficiently small to ensure that the depletionpotential calculated from gbb(r) had converged to the lim-iting form. In a very recent paper Feng and Chapman[51] used the mixture WB theory to calculate gbb(r) viathe test particle route. For size ratios q = 0.1 they reportgood agreement with existing simulation results [52] forconcentrations of the big hard spheres as small as 0.002and total packing fractions as large as 0.4. However, thepacking fraction ηb is still too high to be appropriate fordetermining the depletion potential.

Roth et al [15] also introduced a parameterized formfor the depletion potential obtained from their DFT in-sertion method calculations. Their motivation was toprovide an explicit form W = 1/2(1/q + 1) W̃ (x, ηrs) ,with x = h/σs and h = r − σb the separation betweenthe surfaces of the big spheres, that would be valid for a

range of size ratios q and reservoir packing fractions ηrsand therefore efficacious in studies of the phase behaviourand correlation functions of binary hard sphere mixtures.The authors were influenced by the comprehensive sim-ulation studies of Dijkstra et al [5] which had employeda very simplified (third order virial expansion ) formulafor the depletion potential derived in ref. [37]. Roth etal aimed to provide a formula, convenient for simula-tions of an effective one-component fluid, that capturedboth the short-ranged depletion attraction and the long-ranged oscillatory behaviour of W (r). Such a formulais of course also useful in making comparisons betweentheory and experimental measurements of the depletionpotential. Their formula for W̃ consists of a third orderpolynomial at small x and an exponentially damped os-cillatory function at large x accounting for the correctasymptotic decay [15]. Comparisons made for ηrs be-tween 0.1 and 0.3 and different values of q showed thatthe parametrized form gave a good fit to the results ofthe numerical calculations. Largo and Wilding [39] em-ployed this parametrized form in simulation studies ofthe (metastable) fluid-fluid critical point of the effectiveone-component fluid, comparing their results with thosefrom the much simpler parametrized form used in [5].

In the present study we noticed that the parametriza-tion in [15] did not recover the correct Asakura-Oosawalimiting behaviour as ηrs → 0 and this restricts its regimeof application. Since we are concerned with makingdirect quantitative comparisons between DFT and thepresent simulation we performed a set of new numericalDFT insertion calculations, avoiding parametrization.

B. Derjaguin and Morphometric Approximations

A much used theoretical tool of colloid science is theDerjaguin approximation [53] that relates the force be-tween two large convex bodies immersed in a fluid con-sisting of much smaller particles or molecules to the inte-gral of the excess pressure of the same fluid containedbetween two parallel walls. In recent years there hasbeen considerable discussion about the regime of valid-ity of the Derjaguin approximation for our present caseof a fluid of small hard spheres confined between twofixed big hard spheres or between a planar hard walland a single big hard sphere. The reader is referredto Refs. [10, 15, 37, 54, 55] should they wish to savourthe arguments. Herring and Henderson [9, 10] performedsimulations for the wall-sphere case for q = 0.05 andηrs = 0.3 and 0.4, comparing their results for the deple-tion force with those from the Derjaguin approximationand from the DFT insertion method [15]. In the presentwork we perform equivalent comparisons, for the sphere-sphere case, using what we believe is much more accuratesimulation data for the depletion potential.

As shown in [37] the depletion potential difference forhard spheres obtained from the Derjaguin approximationcan be expressed succinctly as:

9

WDer(h)−WDer(σs) = −�π

2(σs + σb)(σs − h)

[1

2p(ηrs)(σs − h) + 2γ(ηrs)

]; 0 < h < σs (14)

where h is the separation between the surfaces, p(ηrs) isthe pressure of the small sphere reservoir and γ(ηrs) is thesurface tension between a single planar hard wall and thesmall sphere fluid. Within the Derjaguin approximationthe potential between a wall and a single big sphere isprecisely twice that between two big spheres: � is 1 forsphere-sphere and 2 for wall-sphere. Expressions for thepressure and surface tension are listed in Appendix A of[26]. Another expression for the surface tension due to D.Henderson and Plischke [56] as obtained by fitting sim-ulation data was used in [10]. For h > σs the depletionpotential depends on the excess grand potential of thesmall sphere fluid confined in the planar hard wall slitwhich must be obtained from simulation or DFT [26].

Morphometric thermodynamics [34] was developed tocalculate the solvation free energy (excess grand poten-tial) of large convex bodies immersed in a solvent. Its ap-plication to determining depletion potentials is describedin [26, 33] where it is shown that

WMorp(h) = −p∆V (h)−γ∆A(h)−κ∆C(h)−4πκ̄, (15)

for 0 < h < σs. Here ∆V (h) and ∆A(h) are the volumeand surface area of the overlap of exclusion (depletion)zones around the big spheres (or a wall and a big sphere)and ∆C(h) is the integrated mean curvature of the over-lap volume. The thermodynamic coefficients are the pres-sure p, surface tension γ and the two bending rigiditiesκ and κ̄ ; these four quantities are functions of ηrs .Thefourth term is the difference in integrated Gaussian cur-vatures between a dumbbell (4π) and two disconnectedspheres (8π). For h > σs the dumbbell separates into twodisconnected spheres. Thus WMorp(h > σs) = 0. Ex-plicit formulae are given in Ref. [26] for the geometricalquantities and for the four thermodynamic coefficients.Note that WMorp(σ

−s ) = −4πκ̄(ηrs), independent of size

ratio. This term is small in comparison with the others.

In order to connect with the Derjaguin approximationwe invoke the colloidal limit , i.e. q → 0. Then thedifference

WMorp(h)−WMorp(σs) = −�π

2(σs + σb)(σs − h)

[1

2p(ηrs)(σs − h) + 2γ(ηrs)

]− κ(ηrs)π2

√�(σs − h)(σs + σb)/2, (16)

for 0 < h < σs. Since κ is positive the morphometricapproach contributes an additional attractive term, aug-menting the attraction from the pressure (∆V (h)) term.γ is negative so the surface tension (∆A(h)) term givesa repulsive contribution to the depletion potential. Thephysical interpretation of the third term in Eq. 15 or 16is of a line contribution to the effective interaction asso-ciated with the circumference of the edge of the annularwedge formed between the two exclusion spheres wherethe line tension is −κπ/2 [33]. As this term is propor-tional to

√� (not to � ) Derjaguin scaling is violated [26].

The morphometric analysis must break down in thelimit h → σs where Eq. 15 or 16 predicts that the de-pletion force is singular, diverging as (σs − h)−1/2. Thereasons for this unphysical limiting behaviour are asso-ciated with problems of self-overlapping surfaces as ex-plained in Refs. [26, 33]. However, away from this limitone might expect the elegant geometrical arguments un-derlying the morphometric analysis to capture the essen-tial physics. Indeed the comparisons with brute forceDFT results for the depletion force in Ref. [33] indicated

rather good agreement for a range of q and ηrs = 0.314.In Sec. IV we compare the results of Eqs. 14 and 15

with our simulation data and with results from the DFTinsertion method.

IV. RESULTS

A. Test Case

We have tested the ability of the GCA to accuratelydetermine effective potentials by applying it to the caseof the Asakura-Oosawa (AO) Model [35, 36]. Thismodel describes colloidal hard-spheres in a solvent ofnon-interacting point particles modelling ideal polymerthat have a hard-particle interaction with the colloids.Although not the case of additive hard-spheres whichis our primary focus in this paper, the extremely non-additive AO model does provide a very useful testbed forour simulation methodology because the exact form ofthe depletion potential is known, taking the form: [36]

10

βWAO(r) =

−ηrs

(1+q)3

q3

[1− 3r2σb(1+q) +

r3

2σ3b (1+q)3

], σb < r < σb + σs

0, r ≥ σb + σs ,(17)

where σs is the ‘polymer’ diameter, i.e. the colloid-polymer pair potential is infinite for r < (σb + σs)/2.

1 1.02 1.04 1.06 1.08 1.1 1.12 1.14r/σb

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

βWA

O(r

)

ηsr=0.05

ηsr=0.1

ηsr=0.15

ηsr=0.2

FIG. 2. (Color online) A comparison of GCA simulation mea-surements of the depletion potential of the AO model with theexact analytical form. The size ratio is q = 0.1 and data areshown for four values of the reservoir packing fraction. Sym-bols are results from GCA measurements of g(r) for a pair ofbig particles, corrected for finite-size effects and transformedvia βW (r) = − ln[g(r)]. Statistical errors are smaller thanthe symbol size. Lines are the exact AO effective potential,Eq. 17.

Simulation measurements of g(r) ≡ gbb(r) were per-formed for the AO model using the GCA for a systemcomprising a pair of hard spheres in a cubic box of lineardimension L = 3σ in equilibrium with a reservoir of smallparticles having size ratio q = 0.1. Since the small parti-cles are mutually non-interacting, the chemical potentialof the reservoir is just that of an ideal gas. The depletionpotential was calculated as βW (r) = − ln[g(r)] and theresults were corrected for finite-size effects according tothe procedure described in Sec. II C. In Fig. 2 we com-pare the results of simulations of the effective potentialwith the exact result. Data is shown for various values ofthe reservoir packing fraction. In each instance, the sim-ulation results (symbols) are indistinguishable from theanalytical form (lines) within the very small statisticalerrors, a finding that supports the validity and accuracyof the simulations and the procedure for correcting finite-size effects.

B. Effective potentials for additive hard spheres

We turn now to our measurements of the effective po-tential for highly size asymmetrical additive hard spheresand the comparison with DFT calculations. Similarlyto the case of the AO model, our simulations treat thesmall particles grand canonically, i.e. their number fluc-tuates under the control of a chemical potential µrs. Thevalue of µrs is chosen to yield some nominated value ofthe packing fraction of small particles, ηrs , in the notionalreservoir. Thus the simulations require prior knowledgeof µrs(η

rs). In principle, one could employ the Carnahan-

Starling (CS) approximation [57] to estimate the requi-site chemical potential. However in tests we found thisapproximation to be insufficiently accurate for our pur-poses. For instance, taking ηrs = 0.32 as an example, if weemploy the CS value for the chemical potential we actu-ally measure η̄rs = 0.3195, which while close to the target,lies outside the range of fluctuations in ηrs that occur ina large simulation box. In order to determine µrs moreaccurately we therefore performed a series of accurategrand canonical simulations for the pure fluid of smallhard spheres in a large box of L = 50σs. We then em-ployed histogram reweighting to extrapolate to the pre-cise values of the chemical potential that corresponds tothe various values of ηrs that we wished to study. Theseresulting estimates are listed in table I.

ηrs βµrs

0.05 -1.9079(1)

0.10 -0.6770(3)

0.15 0.3923(2)

0.20 1.5105(1)

0.32 5.0472(1)

0.35 6.2659(2)

TABLE I. Measured values of the reduced chemical potentialβµrs corresponding to each of the packing fractions η

rs listed.

The data were obtained by histogram reweighting the resultsof grand canonical simulations of hard spheres obtained atthe nearby value of βµrs predicted by the CS approximation.The simulation cell size was L = 50σs. The definition of µ

rs is

subject to the convention of choosing the thermal wavelengthto equal the hard sphere diameter.

Measurements of the radial distribution function g(r)were made for a pair of big hard spheres in equilibriumwith a reservoir of small hard spheres, for the combina-tions of values of ηrs and size ratio q shown in table II.The system size was L = 3σb for q = 0.1, while forq = 0.05, 0.02, 0.01, where the range of the depletion po-tential is shorter, L = 2.5σb was used. In all cases the

11

depletion potential was obtained as βW (r) = − ln[g(r)]with corrections for finite size effects applied as describedin Sec. II C.

q ηrs

0.10 0.05 0.1 0.15 0.20 0.32 0.35

0.05 0.05 0.1 0.15 0.20 - -

0.02 0.05 0.1 0.15 - - -

0.01 0.05 - - - - -

TABLE II. Combinations of particle size ratio q and reservoirvolume fraction ηrs for which we compare simulation estimatesof depletion potentials with DFT predictions. Values shown innormal typeface were studied by simulation using the GCAdescribed in Sec. II A, while those in boldface were studiedusing staged insertion MC as described in Sec. II B.

1. Comparison of simulation and density functional theoryresults

We now examine a selection of the measured effectivepotentials. Data for q = 0.1, ηrs = 0.2 is shown in Fig. 3.Despite our use of a rather small histogram bin size ofjust δr = 0.001 to accumulate estimates of βW (r), thestatistical fluctuation is sufficiently small that one cansimply connect the data points by lines. This allows us tobetter discern differences between the simulation resultsand those of the DFT calculations using the insertionmethod, which are also included on the plot. Data forthree versions of DFT are shown, namely the Rosenfeld(RF), White Bear (WB) and White Bear 2 (WB2) func-tionals. Clearly for these parameters the overall agree-ment is very good. To quantify the extent of the accord,the two insets to Fig. 3 show a comparison in the range ofseparations close to hard sphere contact (left inset) andaround the first maximum (right inset). These show thatnear contact, WB2, fares slightly better than WB, whichis in turn better than RF. Near the first maximum in thepotential however, the trend is reversed and RF has thegreatest accord with the simulation data, while WB isbetter than WB2.

A similar picture emerges for q = 0.05, ηrs = 0.2 asshown in Fig. 4. Although here the simulation data isnot as smooth as for q = 0.1, the form and magnitude ofthe deviations from the DFT are similar. We commentlater on the results of the morphometric approximationEq. 15.

Generally speaking, the smaller the size ratio, q, thelower the maximum packing fraction ηrs for which we canobtain good statistics with the GCA. Data for q = 0.02,with ηrs = 0.1 and ηs = 0.15 is shown in Fig. 5(a) and5(b), respectively. For this size ratio and these (low)small sphere packings the various versions of FMT per-form very well. Data for q = 0.01 with ηrs = 0.05 isshown in Fig. 6. In this extreme case the insertion DFTresults are almost indistinguishable from each other and

1 1.1 1.2 1.3 1.4 1.5r/σb

-4

-3

-2

-1

0

1

βW(r)

GCA simulationRosenfeldWhite BearWhite Bear 2

1 1.0025 1.005 1.0075 1.01-3.5

-3

-2.5

-2

1.06 1.08 1.10

0.2

0.4

0.6

0.8

FIG. 3. (Color online) Simulation and DFT results for thehard sphere depletion potential βW (r) for q = 0.1, ηrs = 0.2.The abscissa is the separation of hard sphere centres expressedin units of the big particle diameter σb. The two insets expandthe region close to hard sphere contact (left panel) and aroundthe first maximum (right panel).

1 1.05 1.1 1.15 1.2 1.25r/σb

-7

-6

-5

-4

-3

-2

-1

0

1

2

βW(r

)

GCA SimulationDFT (RF)DFT (WB)DFT (WB2)Morphometric (RF)Morphometric (WB2)

1.02 1.03 1.04 1.05 1.060

0.25

0.5

0.75

1

1.25

1.5

1 1.0025 1.005-7

-6.5

-6

-5.5

-5

FIG. 4. (Color online) As for Fig. 3 but with q = 0.05, ηrs =0.2. Also shown are the results of the morphometric approx-imation Eq. 15.

from the results of simulation. However one should notethat there is still a maximum in W (r) – one is not yet inthe AO limit although the contact value is close to theAO value given by Eq. 17.

For our system, the GCA is operable for ηrs . 0.2. Togo beyond this limit we have employed the staged inser-tion algorithm outlined in Sec. II B. Simulation resultsfor q = 0.1, ηrs = 0.35 are compared with those fromDFT calculations in Fig. 7. While the simulation dataare somewhat noisier, they show that in this regime, quitesignificant discrepancies with the DFT insertion methodhave emerged. The principal form of the discrepancy, i.e.DFT underestimates the height of the first maximum, is

12

1 1.02 1.04 1.06 1.08 1.1r/σb

-6

-5

-4

-3

-2

-1

0

1βW

(r)

GCA SimulationRosenfeldWhite BearWhite Bear 2

1 1.002 1.004 1.006 1.008 1.01-8

-7

-6

-5

-4

-3

-2

1.01 1.02 1.03-1

-0.5

0

0.5

1

(a)

1 1.02 1.04 1.06 1.08 1.1r/σb

-12

-10

-8

-6

-4

-2

0

2

βW(r)

GCA SimulationRosenfeldWhite BearWhite Bear 2

(b)

FIG. 5. (Color online) As for Fig. 3 but with (a) q =0.02, ηrs = 0.1 and (b) q = 0.02, η

rs = 0.15.

similar in form but greater in magnitude to that seen us-ing the GCA at smaller values of ηrs (cf. Fig. 3). Onceagain RF fares better than the two WB functionals butunderestimates the first maximum by about 0.5kBT . Re-sults from the three functionals agree quite well with oneanother and with simulation near contact.

2. Comparison with Derjaguin and morphometricapproximations

In this subsection we make comparisons between oursimulation and DFT results with those from the approxi-mations described in Sec. III B. Recall that the Derjaguinapproximation is specifically designed to tackle small sizeratios. In Fig. 4 we compare the results of the morpho-metric approximation Eq. 15 with those from simulationand DFT. Two sets of thermodynamic coefficient wereused: RF and WB2 [26]. Both versions underestimatethe maximum of the depletion potential and overestimatethe magnitude of the potential at contact for q = 0.05 andηrs = 0.2. By contrast for q = 0.1 and η

rs = 0.35, Fig. 7

1 1.01 1.02 1.03 1.04 1.05r/σb

-8

-7

-6

-5

-4

-3

-2

-1

0

1

βW(r)

GCA simulationRosenfeldWhite BearWhite Bear 2

FIG. 6. (Color online) As for Fig. 3 but with q = 0.01, ηrs =0.05.

1 1.1 1.2 1.3 1.4 1.5r/σb

-6

-4

-2

0

2

4

βW(r)

Staged insertion MCDFT (RF)DFT (WB)DFT (WB2)Morphometric (RF)Morphometric (WB2)

FIG. 7. (Color online) The depletion potential for q =0.1, ηrs = 0.35 obtained using the staged insertion simulationmethod and DFT. The simulation data points represent theresults of 150 independent measurements of βW (r) made atvarious fixed values of the big particle separation, though con-centrated in the range r < 1.12σb. Also shown are the resultsof the morphometric approximation Eq. 15.

shows that both versions of the morphometric approx-imation overestimate the maximum and underestimatethe magnitude of the potential at contact. Fig. 7 alsoshows a pronounced minimum for h close to σs. Thisfeature is absent in both simulation and DFT. It is asso-ciated with the unphysical divergence of the line tensioncontribution to the depletion force arising in the morpho-metric treatment. Recall that WMorph is zero for sepa-rations h > σs.

Fig. 8(a) compares the depletion potential differenceobtained from simulation and insertion method DFT forq = 0.1 and ηrs = 0.35 with results from the Derjaguin ap-proximation (where plotting the difference is the naturalchoice [37]) and morphometric approximations, Eqs. 14

13

and 15 respectively. For this value of q the packing frac-tion of the small spheres is sufficiently large to enter theregime where fluid-fluid phase separation might occur-asdiscussed below in Sec. IV B 3. Thus it is interesting toobserve how well these explicit approximations perform.Similar remarks apply for q = 0.05 and ηrs = 0.2 for whichcomparisons are presented in Fig. 8(b)

One sees in Fig. 8(a) that the Derjaguin approximationis very poor. Overall the morphometric approximationsfare considerably better than Derjaguin with RF betterthan WB2 near the maximum. However, both morpho-metric versions overestimate the magnitude of the con-tact value by about 0.5kBT . Note once again the mini-mum close to h/σs = 1 for this packing fraction. The sit-uation is clearly different in Fig. 8(b) where the Derjaguinand morphometric approximations are reasonably good;they bracket the simulation and DFT results. The twomorphometric versions yield results that are very closeand even in this difference plot one sees that these fallbelow the simulation results both at maximum and atcontact. At this smaller value of ηrs there is no mini-mum visible in the depletion potential. Although plot-ting the difference appears to improve the level of agree-ment between morphometric and simulation, one shouldrecall that it is the actual depletion potential displayed inFigs. 4 and 7 which matters, eg. in determining B2(η

rs),

to which we now turn.

3. Second virial coefficients

While the various simulation and DFT estimates of ef-fective potentials show generally good agreement at lowηrs , the differences grow with increasing η

rs and it is nat-

ural to enquire as to the likely implications for the prop-erties of the bulk mixture and in particular phase be-haviour. A useful indicator in this regard is the value ofthe second virial coefficient B2:

B2 = −2π∫ ∞0

(e−βφeff (r) − 1

)r2dr . (18)

where the effective pair potential is defined in Eq. 1.Previous work by Vliegenthart and Lekkerkerker [58]

and Noro and Frenkel [59] has shown that an extendedcorresponding states behaviour applies to fluids thatshare the same value of B2. Specifically, the measuredvalues of B2 at fluid-fluid criticality were found to besimilar across a wide range of model potentials. Sub-sequent work by Largo and Wilding [39], examined thiscriterion explicitly for the case of two DFT-based hardsphere effective potentials which had been fitted to an-alytical forms and parameterized in terms of the reser-voir packing fraction ηrs [5, 15]. Using simulation of asingle component fluid interacting via a pair potential(Eq. 1) with W (r) given by these parameterized deple-tion potentials, the value of ηrs at which the metastablefluid-fluid critical point occurs was determined using an

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1h/σs

-6

-4

-2

0

2

4

6

8

β[W

(h)-W

(σs)]

Staged insertion MCDFT (RF)DFT (WB2)Derjaguinmorph (RF)morph (WB2)

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1h/σs

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

β[W

(h)-W

(σs)]

GCA SimulationDFT (RF)DFT (WB2)Derjaguinmorph (RF)morph (WB2)

(b)

FIG. 8. (Color online) Comparison of βW (h) − βW (σs) ob-tained from simulation and DFT for (a) q = 0.1, ηrs = 0.35(cf Fig. 7) and (b) q = 0.05, ηrs = 0.2 (cf Fig. 4) with resultsof the Derjaguin (Eq. 14) and morphometric approximations(Eq. 15).

accurate approach based on finite-size scaling [39, 60].Interestingly for both q = 0.1 and 0.05 and both choicesof parameterized potentials, the value of B2 for the deple-tion potential at criticality was in quantitative agreementwith that of the adhesive hard sphere model (AHS) at itsfluid-fluid critical point as determined separately in simu-lations by Miller and Frenkel [61]. These authors reporta critical value BAHS2 = −1.207BHS2 , where the hardspheres second virial coefficient BHS2 = 2πσ

3b/3. The

level of agreement was much greater than that seen formore general model potentials (such as square well orLennard-Jones model studied by Noro and Frenkel), sug-gesting that the quasi-universality of the critical pointB2 value holds particularly closely for effective potentialswhose attractive piece is very short ranged in nature, aspertains to highly size asymmetrical hard sphere mix-tures. Further confirmation of this has been found veryrecently in simulations of the AO potential where, forq = 0.1, Ashton [62] has found that the metastable crit-ical point occurs at ηrs = 0.249(1), to be compared withthe prediction ηrs = 0.2482 based on matching to B

AHS2 .

14

In practical terms the universality of B2 at the fluid-fluid critical point implies that one can predict the criticalpoint value of ηrs for effective one-component treatmentsof hard sphere mixtures at small q simply by matchingthe corresponding B2 to the universal value. Conversely,it follows that comparison of B2 values as a function of η

rs

for different potentials provides a sensitive measure of theextent to which their phase behaviour differs. We havemade such a comparison for effective potentials obtainedfrom DFT, the morphometric approximation and simu-lation for q = 0.1, 0.05 and 0.02. The results are shownin Fig. 9(a-c) and demonstrate that at the two largervalues of q even the relatively small differences that weobserve between the DFT and simulation estimates of ef-fective potentials could lead to significant differences inthe small particle packing fractions at which fluid-fluidphase separation is predicted to occur. Specifically, forq = 0.1 based on this B2 criteria, it seems that the DFTwith the Rosenfeld functional underestimates the puta-tive critical point value of ηrs by some 13%, while theWB2 functional underestimates it by some 9% [63] Forq = 0.05 (Fig. 9(b)) the discrepancy between DFT andsimulation has fallen to about 4%, while for q = 0.02(Fig. 9(c)), the values of B2 for the hard spheres mixturesarising from the various DFT flavors agree very well withone another and with simulation, at least for the rangeof ηrs at which bulk phase separation is expected to oc-cur. They also agree well with the AO model, suggest-ing that the additive and extreme non additive modelswill have very similar phase behaviour at this value of q.Recall that for q < 0.154 the mapping of the binary AOmixture to an effective one-component Hamiltonian, withthe AO depletion potential Eq. 17, is exact and we mightalso expect that for very small q the phase behaviour ofthe full binary hard sphere mixture, at physically rele-vant (small) values of ηrs , is described accurately by thedepletion pair potential we calculate here. Many bodycontributions should be negligible.

Interestingly, the DFT data for B2 exhibit a broadminimum with increasing ηrs , as can be seen for q = 0.1in Fig. 9(a). The same feature has previously been re-ported in Ref. [38]. A similar minimum occurs within theDFT for q = 0.05 at ηrs ≈ 0.38 (not shown in Fig. 9(b)).The origin of the upturn in the value of B2 beyond theminimum appears to be due to the fact that the mag-nitude of the first repulsive maximum of W (r) increasesfaster with ηrs than the depth of the potential well atcontact. Unfortunately we could not corroborate the au-thenticity of this feature via simulation because it occursat larger values of ηrs than are currently accessible to us.Should it prove real (rather than being an artifact of theDFT), it raises the intriguing possibility that fluid-fluidphase separation may occur only within a certain rangeof ηrs .

Also plotted in Fig. 9 are the results of the morphome-tric approximation Eq. 15 for B2. Like the DFT resultsthese show minima at all q studied (though only that forq = 0.1 is visible in the plotted ranges). For q = 0.1,

0.1 0.2 0.3 0.4 0.5ηs

r

-8

-6

-4

-2

0

2

σb-3 B

2(η

sr )

SimulationDFT (RF)DFT (WB)DFT (WB2)AOAHSmorph (RF)morph (WB2) (a)

0 0.05 0.1 0.15 0.2 0.25ηs

r

-8

-6

-4

-2

0

2

σb-3 B

2(η

sr )

SimulationDFT (RF)DFT (WB)DFT (WB2)AOAHSmorph (RF)morph (WB2) (b)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14ηs

r

-20

-16

-12

-8

-4

0

σb-3 B

2(η

sr )

SimulationDFT (RF)DFT (WB)DFT (WB2)AOAHSmorph (RF)morph (WB2) (c)(c)(c)

FIG. 9. (Color online) (a) Comparison of the second virialcoefficient B2 (Eq. 18) derived from DFT, morphometric andsimulation measurements of the effective potential for q =0.1 and various ηrs . The horizontal dashed line indicates thevalue of BAHS2 = −2.527σ3b for which fluid-fluid criticalitywas found in the adhesive hard sphere model [39, 61]–see text.For values of B2 below this line, fluid-fluid phase separationis expected. (b) and (c) show corresponding plots for q =0.05 and q = 0.02 respectively. Unless error bars are shown,statistical errors in simulation data points do not exceed thesymbol size.

B2(ηrs) does not cross the AHS line, implying that the

theory fails to predict fluid-fluid phase separation at thissize ratio. At smaller q, the agreement with simulationis better, but still poorer than DFT. It is disappointingthat both versions of the morphometric approximationperform poorly for q = 0.02, where we find the results

15

are substantially different from those of the AO model.The discrepancy with simulation for B2 appears to ariseprimarily from a failure of the morphometric approxi-mation to correctly predict the additive constant in thepotential, as shown from the comparison of the poten-tials of Figs. 4 and 7 with the shifted representation ofFig. 8. Whilst morphometric results for the depletionforce [26, 33] might be in reasonable agreement with DFTand simulation, any additive shift is important for B2.

V. DISCUSSION

In summary, we have employed bespoke MC simula-tion techniques to obtain direct and accurate simulationmeasurements of depletion potentials in highly size asym-metrical binary mixtures of hard spheres having q ≤ 0.1.Small particles were treated grand canonically, the valueof the chemical potential being chosen to target pre-scribed values of the reservoir packing fraction ηrs . Thesimulation results were compared with new DFT calcu-lations (performed using the insertion method) based onthe Rosenfeld, White Bear and White Bear Mark 2 func-tionals. For ηrs ≤ 0.2 generally good agreement with sim-ulation was found at all size ratios studied, though on in-creasing the packing fraction to ηrs = 0.35 at q = 0.1 sig-nificant discrepancies between the various flavors of DFTand the simulation estimates were evident. In this latterregime, Rosenfeld (RF) was found to be somewhat betterthan the other functionals in reproducing the height ofthe first maximum of the effective potential, while WhiteBear 2 was marginally the best of the three with regard toits prediction for the contact value and for second virialcoefficients.

Overall our results show that the DFT insertionmethod provides a reasonably accurate description ofeffective potentials for highly size asymmetrical hardsphere mixtures at least in the range of small particlepacking fractions at which fluid-fluid phase separationis likely to occur. Indeed at ηrs = 0.35, and q = 0.1DFT was found to be more accurate than the morphome-tric and Derjaguin approximations, the latter providinga particularly poor prediction. This conclusion is partlyat odds with that of Herring and Henderson [9, 10] whoassert that both DFT and the Derjaguin approximationprovide descriptions that are almost equally poor com-pared to simulation data (see Fig. 7 of Ref. [10] whichrefers to q = 0.05 and ηrs = 0.3 and 0.4) and advocatein particular that neither approach should be used in theregime of ‘moderate’ ηrs to answer important questionssuch as the existence of fluid-fluid coexistence. While weconcur with this assessment in the case of the Derjaguinapproximation, Herring and Henderson’s conclusions re-garding the accuracy of DFT calculations were reachedon the basis of simulation estimates for the effective po-tential which were not obtained directly, but by integrat-ing measurements of the interparticle force as outlined inSec. I A. Perhaps as a consequence, their estimates are

much noisier (see Fig 6 of [10]) than those presented inthe present work and consequently –we feel– do not serveas a sufficiently reliably indicator of the accuracy of DFTespecially in the key regime where fluid-fluid phase sep-aration might occur.

Indeed, we have investigated the likely extent of theconsequences for predictions of phase behaviour arisingfrom discrepancies between theory and simulation esti-mates of depletion potentials via calculations of the de-pendence of the second virial coefficient on ηrs . Previoussimulation studies of phase behaviour in single compo-nent fluid interacting via effective potentials [39] haveshown that when the potential is very short ranged, theonset of fluid-fluid phase separation occurs at a near-universal value of B2 = −2.527σ3b . Based on this crite-rion, we found that compared to the effective potentialsobtained via simulation in the present work, the mor-phometric theory provides the poorest predictions of thecritical packing fraction of small particles (and fails topredict phase separation at all at q = 0.1). Those fromDFT underestimate the critical packing fraction of smallparticles by about 10% for q = 0.1 and about 4% forq = 0.05. While these are significant discrepancies, we donot feel that they constitute a “qualitative breakdown”of the DFT insertion method approach as suggested byHerring and Henderson [9, 10] on the basis of their sim-ulation data, at least not in the regime where phase sep-aration is expected. Herring and Henderson speak of ananocolloidal regime. We interpret this as size ratios q ofsay 0.1 to 0.01. Our present study shows that this regimeis amenable to accurate simulation studies up to values ofthe small sphere packing fraction that are relevant for in-vestigations of fluid-fluid phase separation and that DFTworks well in this regime -the focus of the present paper.For larger values of ηrs there are serious issues concerningthe accuracy of the existing DFT approaches and we shedno new light on this interesting but somewhat extremeregime.

Turning finally to the outlook for further work onhighly size asymmetrical mixtures, it would, of course,be of great interest to verify the existence (or otherwise)of the putative fluid-fluid critical point in the full twocomponent size-asymmetric hard sphere mixture. Thistopic remains ebullient. For example, a recent paper[64], based on a version of thermodynamic perturbationtheory, conjectures that additive hard spheres will ex-hibit fluid-fluid separation, albeit metastable with re-spect to the fluid-solid transition, for size ratios in therange 0.01 ≤ q ≤ 0.1. Our measurements of B2 for thedepletion potentials obtained in our simulations provide(potentially) accurate predictions for the small particlepacking fraction at which the critical point occurs. Weare currently employing a grand canonical version of thestaged insertion MC method [45] to investigate this mat-ter.

The simulation methods we develop here can be ap-plied to any short ranged potential and it would also beof interest to examine the influence on the effective poten-

16

tial of adding small amounts of finite attraction or repul-sion to the bs and ss interactions. This would allow us tobetter model real colloidal systems, where one can havea variety of interaction potentials, and where, even insystems (such as sterically stabilized PMMA) which ap-proximate hard spheres rather well, one expects residualnon-hard sphere interactions [65]. Previous work [38, 66]has suggested that the effects of such residual interac-tions may be represented in terms of a non-additive hardsphere mixture. It would be of interest to examine thisproposal explicitly using accurate simulation data andDFT calculations.

ACKNOWLEDGMENTS

This work was supported by EPSRC grantEP/F047800 (to NBW). Simulation results werepartly produced on a machine funded by HEFCE’sStrategic Research Infrastructure fund. The authors aregrateful to Martin Oettel for comments on the originalmanuscript and in particular on the morphometricapproximation.

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