Phase diagram of highly asymmetric binary hard-sphere mixtures

Marjolein�

Dijkstra, Rene´ van� Roij, and Robert EvansH.�

H. Wills Physics Laboratory, University of Bristol, Royal Fort, Bristol BS8 1TL, United Kingdom�Received 23 December 1998�

We�

study the phase behavior and structure of highly asymmetric binary hard-sphere mixtures. By firstintegrating�

out the degrees of freedom of the small spheres in the partition function we derive a formalexpression� for the effective Hamiltonian of the large spheres. Then using an explicit pairwise� depletion

potential� approximation to this effective Hamiltonian in computer simulations, we determine fluid-solid coex-istence�

for size ratiosq � 0.033, 0.05, 0.1, 0.2, and 1.0. The resulting two-phase region becomes very broad inpacking fractions of the large spheres asq becomes� very small. We find a stable, isostructural solid-solidtransition forq � 0.05� and a fluid-fluid transition forq � 0.10. However, the latter remains metastable withrespect to the fluid-solid transition for all size ratios we investigate. In the limitq � 0� the phase diagrammimics that of the sticky-sphere system. As expected, the radial distribution functiong(r) and the structurefactor S(k)

�of the effective one-component system show no sharp signature of the onset of the freezing

transition and we find that at most points on the fluid-solid boundary the value ofS(k) at its first peak is muchlower than the value given by the Hansen-Verlet freezing criterion. Direct simulations of the true binarymixture of hard spheres were performed forq � 0.05� in order to test the predictions from the effective Hamil-tonian. For those packing fractions of the small spheres where direct simulations are possible, we find remark-ably good agreement between the phase boundaries calculated from the two approaches—even up to thesymmetric limitq � 1 and for very high packings of the large spheres, where the solid-solid transition occurs.In both limits one might expect that an approximation which neglects higher-body terms should fail, but ourresults support the notion that the main features of the phase equilibria of asymmetric binary hard-spheremixtures are accounted for by the effective pairwise depletion potential description. We also compare ourresults with those of other theoretical treatments and experiments on colloidal hard-sphere mixtures.�S1063-651X� 99� 07805-8� �

PACS number� s� :� 82.70.Dd, 61.20.Gy, 64.70.� pI. INTRODUCTION

The�

theory of the structure and phase behavior of simple�atomic ! fluids relies heavily on our knowledge of the hard-

sphere" system, which serves as a standard reference systemfor#

determining the properties of more realistic models. Assuch" the hard-sphere system has been studied in great detailduring$

the past few decades and its phase behavior is nowwell% understood. In particular, it was shown by computersimulations" that a system of identical hard spheres has awell-defined% freezing transition& 1,2' driven$ by purely en-tropic(

effects. Pure hard spheres do not undergo a liquid-gastransition(

since this requires a source of attractive interac-tions.(

The state of affairs for the binary hard-sphere mixtureis less clear-cut and the phase behavior is still hotly debated.This system plays a similar) reference* role for binary mix-tures(

of simple fluids and also serves as a model for mixturesof+ colloids and polymers, or other colloidal systems. Themain issue is whether a binary fluid mixture of large andsmall" hard spheres is miscible for all size ratios and compo-sitions" or whether a fluid-fluid demixing transition takesplace, and, if it does, whether such a transition is stable ormetastable with respect to the fluid-solid transition. The dis-cussion- was instigated in 1991 by Biben and Hansen, whoshowed" within an integral equation theory that the binaryhard-sphere.

mixture exhibits a spinodal instability in a high-density$

fluid when the size ratio of the two species is moreextreme/ than 1:50 31 2 . As this result was in contradiction witha classic study by Lebowitz and Rowlinson, who had con-cluded- that the mixture is stable against demixing regardless

of+ the state point and the size ratio3 44 ,5 it initiated renewedinterest6

in this system. Despite all the work that has sincebeen7

devoted to this issue, it remains an unresolved questionas to whether a stable fluid-fluid demixing transition exists inhard-sphere mixtures. The physical mechanism behind8 pos-,sible" 9 demixing$ in hard-sphere mixtures is the depletion ef-fect. This is based on the idea that clustering of the largespheres" allows more free volume for the small ones whichmay: lead to an increase in the entropy, i.e., to a lowering ofthe(

free energy. The depletion effect is known to lead todemixing$

in colloid-polymer mixtures but it is not knownwhether% this is sufficiently strong to bring about demixing inadditive; mixtures: of hard spheres. In such mixtures the pair-wise% potential between species 1 and 2 is described by adiameter$ <

12 which% is the mean of those for like-like inter-

actions: = 12> (? @ 11ACB 22)D /2. Colloid-polymer mixtures areusuallyE treated by a model which assumes the polymer-polymer, interactions to be ideal so that the hard-sphere di-ameters are nonadditive. Here we focus on additive binaryhard-sphere mixtures, thereby ignoring recent work on bi-nary hard-core mixtures of nonspherical particlesF 5,6G H and polydispersity, I 7J K ,5 and we mention briefly results for nonad-ditive$

mixtures where these are relevant.OneL

might suppose that computer simulations shouldhave resolved the issues concerning the phase behavior.However,M

direct simulations of highly asymmetric binarymixtures: are prohibited by slow equilibration when the pack-ing fraction of the small spheres becomes substantial andthere(

have been no systematic attempts to calculate phasediagrams$

for the asymmetric cases which are of most inter-

PHYSICAL REVIEW E MAY 1999VOLUME 59, NUMBER 5

PRE 591063-651X/99/59N 5O P /5744Q R 28S T /$15.00Q 5744 ©1999 The American Physical Society

est./ On the theoretical side it is now well accepted that thoseapproaches which attempt to treat both species on an equalfooting,#

e.g., integral equation theories or virial expansionsof+ the mixture equation of state, yield results which are no-toriously(

sensitive to the details of the approximations. TheUnonV W existence/ and location of the spinodal instability is par-

ticularly(

susceptible but the location of the fluid-solid phaseboundary7

is also very sensitive to the choice of approxima-tion.(

In the present paper we adopt a different strategy whichtakes(

advantage of the large-size asymmetry: we integrateout+ the degrees of freedom of the small spheres and obtain aneffective/ Hamiltonian for the larger ones. The effectiveHamiltonian consists of zero-body, one-body, two-body, andhigher-body.

interactions which depend on the density of thesmall" spheres. We employ a simplified version, which con-sists" of only pairwise additive depletion potentials betweenthe(

larger spheres, in Monte Carlo simulations. We find afluid-fluidX

demixing transition for size ratios of 1:10 or moreextreme./ However, this transition is metastable with respectto(

the fluid-solid transition which occurs at strikingly smallvalues� of the packing fraction of the large spheres. Moresurprisingly," perhaps, we also find an isostructural solid-solidtransition(

at high packing fractions of the large spheres. Thistransition(

becomes stable for a size ratioY 0.05.Z The richnessof+ the predictions from the approximate effective Hamil-tonian(

leads us to attempt direct simulations of the true bi-nary hard-sphere mixture. Theseare; feasible for size ratios[

0.05,Z

provided the density of the small spheres is fairlylow. We find remarkably good agreement between the resultsof+ the two sets of simulations even in the solid phase\ s" ] and for#

relatively large size ratioŝ0.2Z _

where% one might expectthe(

approximation of pairwise additivity to fail. The successof+ our comparison implies that an effective Hamiltonian ap-proach, based on a pairwise depletion potential descriptionshould" provide an accurate, albeit approximate, account ofthe(

main features of the phase behavior of highly asymmetrichard-sphere mixtures.

The paper is organized as follows: In Sec. II we give anhistorical.

overview of research on additive binary hard-sphere" mixtures. This is not intended as a comprehensivereview; rather, it should provide a more detailed introductionto(

earlier work and motivation for our present approach. InSec.`

III the effective Hamiltonian is derived by integratingout+ the degrees of freedom of one of the species in a binarymixture. We emphasize that this derivation is not restrictedto(

a binary hard-sphere mixture; it applies to any two-component- mixture where the species interact via short-ranged pairwise potentials. The theory is applied to additivebinary7

hard-sphere mixtures in Sec. IV where explicit formu-lasa

are introduced for the one-body and two-bodyb depletion$potential, c contributions.- Results of computer simulationsbased7

on the approximate effective Hamiltonian are pre-sented" in Sec. V. The phase diagrams and the pairwise cor-relationd functions are calculated for a range of size ratios. InSec.`

VI we present the results of the direct simulations of thetrue(

binary mixture and compare the phase diagrams withthose(

obtained from the effective Hamiltonian. Section VII Amakes: comparisons between our present results and those ofexperiment/ and previous theories or simulations, while inSec.`

VII B we make a connection with the so-called freevolume� theory of binary hard-sphere mixtures, showing how

that(

approach relates to the present. Finally, in Sec. VII C wemake: some concluding remarks. Some of our results havebeen7

presented in short communicationse 8,9f g .

II. HISTORICAL OVERVIEW

Inh

the early days of liquid state physics it was not clearwhether% attraction was necessary to drive a freezing transi-tion(

in simple fluids. In 1957 Wood and Jacobsoni 1j and Alderk

and Wainwright l 2m n showed" by computer simulationsthat(

a system of purely repulsive hard spheres has a well-defined$

freezing transition. These results were disputed for along period, but nowadays it is generally accepted that asystem" of identical hard spheres does have a fluid-solid tran-sition" o 10p . The origin of this freezing transition is purelyentropic/ and occurs because the entropy of the crystallinephase, is higher than that of the fluid phase at sufficientlyhigh.

densities.Inh

a simple picture of a solid all molecules are confined tocells- centered at lattice sites. This confinement results in adecrease$

in entropy. At sufficiently high density, however,the(

molecules in a dense fluid will be more jammed than in asolid," where the molecules can move freely within cells. Thisincrease in free volume per molecule in a solid results in againq in entropy that can outweigh the loss in entropy in-curred- by confining the molecules to cells.

The�

location of this hard-sphere freezing transition wasdetermined$

from simulations by Hoover and Ree, who foundthat(

the packing fractions of the coexisting fluid and face-centered- cubic solid phaser fccs are given by t fluidu v 0.494Zand w solidx 0.545,Z which corresponds to a pressureP y 3z /{ k| BT } 11.69 with ~ the( diameter of the hard spheres11 . We also note that it has been shown recently that at

coexistence- the fcc crystal is indeed more stable than thehexagonal.

close-packed hcp. crystal- 12 .In the last decade, it was found that binary hard-sphere

mixtures show extremely rich phase behavior. A densityfunctional#

treatment of Barrat, Baus, and Hansen showedthat(

starting from the pure limit, the freezing transition of themixture changes from a spindlelike transition via an azeotro-pic, to an eutecticlike one when the two species become moredissimilar$

in size 13 . In the case of spindlelike phase be-havior, a narrow coexistence is found between a fluid and asubstitutionally" disordered fcc crystal. Here ‘‘narrow’’ refersto(

the width of this coexistence region expressed in terms ofthe(

composition difference between the two coexistingphases., When the spheres become more dissimilar in size,the(

fluid-solid region broadens and an azeotropic point ap-pears., At higher packing fractions a coexistence region be-tween(

two substitutionally disordered fcc solids appears inthe(

phase diagram when the spheres become sufficiently dis-similar." When this miscibility gap in the solid phase inter-venes� with the fluid-solid coexistence the phase diagram be-comes- eutecticlike. Computer simulations, other densityfunctional approaches, and a scaled particle approach re-vealed� that these predictions are qualitatively correct 14–19 : the transition from a spindle to azeotropic type of phasediagram$

is predicted at a size ratioq C 2 /{ 1 0.94Z 13,14and the transition from azeotropic to eutectic atq 0.92Z 13or+ 0.875 14 . The diameters of the large and small spheresare, respectively, 1 and 2 . Note that only substitutionally

PRE 59 5745PHASE

DIAGRAM OF HIGHLY ASYMMETRIC BINAR Y . . .

disordered$

fcc solids are found for these size ratios.In recent experiments, however, complex crystalline order

is found forq 0.58Z and 0.62 20 . For sterically stabilizedpolymethylmetacrylate, PMMA spheres," AB  2 and AB  13 su-"perlattice, structures are found, whereA  denotes$ the largerspecies." The presence of these superlattice structures asstable" phases was subsequently confirmed by computersimulations" for 0.41¡ q ¢ 0.625Z £ 21¤ and by density func-tional(

approaches¥ 22¦ . In addition, it was shown that thefluidX

phase never undergoes fluid-fluid demixing for thesesize" ratios. Thus the phase behavior of binary hard-spheremixtures: is well understood forq § 0.4.Z

This�

state of affairs should be contrasted with the case ofmore asymmetric binary hard-sphere mixtures, the phase be-havior of which is still under debate. For instance, it is stillunclearE whether ä stable" © fluid-fluidX demixing transition canexist/ for any binary hard-sphere mixture. In 1964, Lebowitzand Rowlinson showed, using the Percus-Yevickª PY« clo--sure" of the Ornstein-Zernike¬ OZL equations,/ that the homo-geneousq fluid phase of a binary mixture of large and smallhard.

spheres is stable with respect to demixing, regardless ofthe(

diameter ratio, composition, or pressure® 4̄ . A similarprediction, emerged from the generalization of Mansooriet° al. of+ the Carnahan-Starling equation of state for mixtures;i.e.,6

no spinodal instability was found in the fluid phase± 23m ² .Computer³

simulations of binary hard-sphere mixtures forq´ 0.6Z µ 24¶ ,5 0.909 · 25̧ ,5 0.5 and 0.33¹ 26º ,5 0.33 » 27¼ ,5 0.909,0.6,Z

0.33, 0.2, and 0.05½ 28¾ ,5 performed for packings whereergodicity/ problems do not prevent equilibration, did notprovide, any evidence for a demixing transition, and it wastherefore(

generally believed that binary hard-sphere mixturesnever phase separate into two fluid phases. In 1990, how-ever,/ Biben and Hansen showed that the pair distributionfunction#

diverges at contact, within the Percus-Yevick ap-proximation,, in the extreme asymmetric limitq ¿ 0Z À 29m Á .OneL

year later, the same authors showed that the Rogers-Young  RYà closure,- which for the pure hard-sphere systemis known to be more accurate than the Percus-Yevick clo-sure," yields a fluid spinodal instability whenq Ä 0.2Z Å 31 Æ . Theyattributed the fluid spinodal to the stickiness of the largeparticles, arising from the so-calleddepletionÇ effect. This ef-fect,#

which has long been known to drive phase separation incolloid-polymer- mixtures È 30,311 É ,5 induces effective attrac-tions(

between largeÊ colloidal- Ë spheres" at small separationsdue$

to an unbalanced pressure of smallÌ polymeric, Í spheres."Ank

alternative description of the same effect is that a freevolume� of the small spheres is gained due to the overlap ofthe(

excluded volume of clustering large spheres; the result-ing6

gain of entropy of the small spheres drives this cluster-ing,6

and thus induces effective attractions between the largespheres." In 1993, this latter picture of the depletion effectwas% employed by Lekkerkerker and Stroobants in a phenom-enological/ approach based on scaled particle expressions forthe(

free volume Î 321 Ï . In qualitative agreement with the re-sults" of Ref. Ð 31 Ñ ,5 these authors found a fluid-fluid spinodalÒ321 Ó

. However, the spinodal instability in Ref.Ô 321 Õ shifts" tohigher.

packing fractions of the small spheres when the sizeratiod becomes more asymmetric, while the opposite trend isto(

be expected since the depletion effect becomes strongerfor more asymmetric hard-sphere mixtures. Moreover, it wasshown" by Amokrane and Regnaut that the location of the

spinodal" instability is very sensitive to the precise expressionusedE for the free volumeÖ 331 × . In 1995 Rosenfeld employedhis fundamental-measures density functional theory to asym-metric binary hard-sphere systems, and predicted a fluid-fluidX

demixing transition for a size ratioq Ø 0.25Z Ù 341 Ú .These early theories for asymmetric binary hard-sphere

mixtures: did not consider the crystalline solid phase explic-itly.6

However, experiments on colloidalÛ nearlyV Ü hard-sphere.particles, suggest that any demixing transition is stronglycoupled- to the freezing transition. Unfortunately, surfacecrystallization- and sedimentation effects preclude makingdefinite$

conclusions. For instance, Sanyalet° al. performed,experiments/ on mixtures of polystyrene spheres with a diam-eter/ ratio of 0.2 Ý 351 Þ ,5 and observed cluster formation in thesediment" at the bottom of their samples. In contrast, whenthey(

suspended the mixture in a density-matched solvent,neither sedimentation nor flocculation was seen. In the ex-periments, of van Duijneveldtet° al.,5 a phase instability wasfound#

in a fairly narrow concentration range of small andlarge silica particles withq ß 0.1667Z à 361 á . However, in theseexperiments/ conclusions could only be drawn during the firstfew#

hours, as sedimentation becomes important at longertimes.(

Thus the authors could not determine whether thephases, formed initially in the samples outside this narrowconcentration- range were stable fluid phases or metastablefluidsX

or glasses. They also concluded that the crystallization,observed+ in the sediment after several weeks, could becaused- either by slow concentration-dependent kinetics or bydensification$

to the freezing point arising from sedimenta-tion.(

Experiments on mixtures of polystyrene particles withsize" ratios 0.069â q ã 0.294Z by Kaplanet° al. showed" the ex-istence6

of either a single homogeneous disordered phase, acoexistence- between two disordered phases, a coexistencebetween7

a disordered phase and a crystal on the sample wall,or+ coexistence between two disordered bulk phases and asurface" crystallization ä 371 å . However, bulk crystallizationwas% never observed. A possible reason why no surface crys-tallization(

was found in the experiments of van Duijneveldtet° al. is the fast settling of silica spheres compared to poly-styrene" particles. Dinsmoreet° al. reportedd results on mix-tures(

of polystyrene particles with 0.083æ q ç 0.149Z è 381 é . Inaddition to surface crystallization, they observed fluid-solidphase, separation at higher packing fractions of small spheres.Whenê

they increased the packing fraction of the smallspheres" even further, they observed that the clusters in thesediment" rapidly form a ‘‘loose’’ gel instead of a crystal.Independently,h

Imhof and Dhont found a fluid-solid type ofphase, separation in experiments on silica spheres withqë 0.1075Z ì 391 í . Moreover, two types of glassy states, distin-guishedq by the different mobilities of the small spheres, werefound#

at high packing fractions of the large spheres. In bothcases- the large ones form a glasslike structure whereas thesmall" ones are mobileî fluidlikeï in one case and rather im-mobile in the other.

Inspiredh

by these experimental results, Poon and Warrenð40ñ ò

and Dinsmoreet° al. ó 41ñ ô extended/ the free volume ap-proach, of Lekkerkerker and Stroobantsõ 321 ö to( the solidphase, and concluded that the fluid-fluid demixing transitionshould" be metastable with respect to a broad fluid-solid co-existence/ region. In addition they found that the presence ofsmall" spheres causes crystallization of large spheres next to a

5746 PRE 59DIJKSTRA,÷

van ROIJ, AND EVANS

hard wall at concentrations well below those correspondingto(

bulk phase separation, in agreement with the experimentalobservations.+ A broad fluid-solid coexistence has also beenfound#

in density functional calculations based on the modi-fied weighted density approximation for a binary hard-spheremixture: with q ø 0.1Z ù 42ñ ú . Recently Caccamo and Pellicaneshowed," using a single-phase entropic freezing criterion andthe(

thermodynamically self-consistent Rogers-Young theory,that(

for a single state point the phase instability in binaryhard-sphere mixtures withq û 0.1Z is consistent with fluid-solid" phase separationü 43ñ ý . Another freezing criterion wasemployed/ by Saija and Giaquintaþ 44ñ ÿ . A different approach,based7

on results for the first five virial coefficients of thebinary7

hard-sphere mixture, was adopted by Coussaert andBaus� �

45,46ñ �

. In addition to a fluid-solid transition they finda thermodynamically stable fluid-fluid transition for size ra-tios(

as large asq � 0.15Z andq � 0.33Z � 45ñ � ,5 if the fourth andfifth�

virial coefficients are taken from results of Saijaet° al.�47 . However, in a subsequent erratum which uses the

fourth and fifth virial coefficients from Encisoet° al. 48� ,5they(

find that the fluid-fluid transition is at such high packingfractions#

that they argue it should be metastable with respectto(

a broad fluid-solid transition forq � 0.2Z and a narrow onefor q 0.1Z � 46� . Yet another theoretical treatment by Vegapredicts, a narrow fluid-solid coexistence forq � 0Z � 49ñ � . Insummary" the predicted phase behavior of asymmetric binaryhard-sphere mixtures is very sensitive to the details of thetheoretical(

approaches, and the character of the fluid-fluidand fluid-solid transitions and their interplay remains poorlyunderstood.E

In this paper we explore an alternative route to the phasebehavior7

of asymmetric binary hard-sphere mixtures. It isbased7

on our present knowledge of the depletion force be-tween(

two big spheres immersed in a fluid of small particles.Whenê

the separation of the big spheres is less than the diam-eter/ of the small ones, there is an unbalanced pressure of the‘‘sea’’ of small spheres which gives rise to the attractivedepletion$

force between the big spheres. As mentioned abovethis(

mechanism was first described by Asakura and Oosawafor#

colloid-polymer mixtures� 301 � . The depletion force wasinvestigated6

in detail by Attard and co-workers withinhypernetted-chain-based approximations� 50G � and in simula-tions( �

51G �

. Mao et° al. calculated- the depletion force up tothird(

order in the density of the smaller spheres� 52G � ,5 havingfirst�

made the Derjaguin approximation to relate the forcebetween7

the two big spheres to that which arises when thesmall" particles are confined between two planar walls. Theyfound#

good agreement with the simulation results forq� 0.1Z of Biben et° al. � 53G � . Götzelmann( et° al. � 54G � have. re-cently- assessed various theoretical treatments of the deple-tion(

force and the corresponding depletion potential dep! (? r).DDepletion potentials have also been ‘‘measured’’ experimen-tally.(

Kaplanet° al. performed, experiments for a large spherenearV a wall in a suspension of small spheres. By tracking theBrownian trajectory of the large sphere with video micros-copy,- they were able to determine the potential depth, whichthey(

estimated to be of the order of a fewk|

BT for a size ratioof+ 0.028 " 55G # . Ye et° al. measured: the structure factor of thebig7

colloidal particles in a neutron scattering experiment oncolloid-polymer- mixtures. By fitting these structure factors tothose(

calculated from the Asakura-Oosawa depletion pair po-

tential( $

usingE the random phase approximation% ,5 they foundthat(

the depth of the depletion potential increases linearlywith% & lowa ' polymer, concentration, as predicted by theAsakura-Oosawa approximation( 56G ) . Laser radiation experi-ments were used to measure the minimum laser intensity thatis6

required to blow off a large polystyrene latex particle fromthe(

wall as a function of polymer concentration* 57G + . Theselaser intensities were translated into depletion forces andagain good agreement was found with the Asakura-Oosawadepletion$

force. Depletion forces of colloidal hard spherestrapped(

in a vesicle were measured by Dinsmoreet° al. , 58G - .They found that the large particles are more likely to betrapped(

close to regions of the surface which have a largecurvature- and they argue the importance of their results forbiological7

systems. 58G / . Very recently, Rudhardtet° al. 0 59G 1measured the depletion force on a large polystyrene sphereimmersed in a solution of small, noncharged polymers as afunction#

of thedistanceÇ

to(

a flat glass surface and ofpolymer2concentration3 usingE total internal reflection microscopy.They also find good agreement with theoretical predictions459G 5

. Finally, we mention that the depletion potential wasmeasured: recently using optical tweezers6 607 8 and goodagreement with the Asakura-Oosawa depletion potential wasclaimed.-

OfL

course, one should bear in mind that real colloidalmixtures: or colloid-polymer mixtures are not strictly binaryhard-sphere.

mixtures, and so there are possible influencesfrom screened Coulomb forces, polydispersity, etc., whichmake direct comparison between experiment and theorybased7

on idealized models a matter for some caution.Inh

view of all the theoretical and experimental effortwhich% has been expended on determining depletion poten-tials(

it is somewhat surprising, perhaps, that these potentialshave.

not been used to calculate the phase behavior of binaryhard-sphere.

mixtures. One might envisage performing simu-lations or carrying out theoretical studies for an effectiveone-component+ system in which the big spheres interact viaa pairwise potential9 eff: ;=< 11>=? dep! ,5 where @ 11 is the hard-sphere" potential between the two big spheres andA dep! is6 thedepletion$

potential obtained from the theoretical or simula-tion(

treatments described above. Recall thatB dep! depends$ onthe(

density of the small spheres. Such a strategy was em-ployed, by Gastet° al. C 617 D in a pioneering investigation of thephase, behavior of colloid-polymer mixtures. However, a sys-tematic(

derivation of an effective Hamiltonian, obtained byintegrating6

out the degrees of freedom of the small spheres,was% not carried out, and so the status of the effective pair-wise% depletion potential description has remained uncertain.Moreover,�

Ref. E 617 F specializes" to the case of ideal polymers,for#

which the Asakura-Oosawa potential is appropriate,rather than the case of additive binary hard-sphere mixtures.GWeê

make contact with the results of Ref.H 617 I in later sec-tions.( J

Inh

the present work we derive an effective Hamil-tonian(

by formally integrating out the degrees of freedom ofthe(

small spheres. We show that this Hamiltonian has zero-body,7

one-body, two-body, and higher-body terms. The two-body7 K

pairwise, L term( is precisely that which is given by thedepletion$

potential description and we show that the zero-body7

and one-body terms play no role in determining phaseequilibria./ Ignoring the higher-body terms we perform simu-lationsa

with the effective pairwise potential defined above

PRE 59 5747PHASE

DIAGRAM OF HIGHLY ASYMMETRIC BINAR Y . . .

and M dep! obtained+ from Ref. N 54G O . We thus determine thephase, behavior of binary hard-sphere mixtures for size ratiosq P 1.0, 0.2,Z 0.1,Z 0.05, and 0.033. As the results of these ef-fective#

one-component calculations predict several strikingfeatures at relatively low values of the density of smallspheres" we were motivated to perform direct simulations ofthe(

binary hard-sphere system in order to test these predic-tions.(

The results from the two sets of simulations are inremarkably good agreement for those values ofq and pack-ing fractions for which direct simulations are possible,thereby(

justifying the use of the effective pairwise depletiondescription.$

III. GENERAL THEORY OF BINARY MIXTURES

A. Mapping onto an effective one-component system

Here we formally map a system with interaction Hamil-tonian(

H that(

describes a binary mixture onto an effectiveone-component+ system with HamiltonianHQ effR by7 integratingout+ the degrees of freedom of one species. We are concernedwith% a classical fluid of two species, labeled 1 and 2, in amacroscopic volumeV. For particle numbersN

S1 and NS 2,5 the

total(

Hamiltonian T=U KV W HQ is6 a sum of kinetic energyKVand interaction energyH X H11Y H12Z H22,5 given by

KV []\

i ^ 1N_

1 P`

i2

2m

ma 1 bcjd e

1

N_

2f

pg jd22m

ma 2 ,5

HQ

11h]ii j jdN_

1 k11l Rm i j n ,5

H22 o]pi q jdN_

2 r22 s ri j t ,5

H12u]vi w 1N_

1 xjd y

1

N_

2 z12{ Ri | rjd } . ~ 1

Here ma 1 and ma 2 are the masses,Pi and pg jd the( linear mo-menta,: andRm i and r jd the( positions of the particles of species1 and 2, respectively. The spherically symmetric pair poten-tials(

are denoted 11,5 22 ,5 and 12,5 while Rm i j Rm i Rm jd and ri j ri rjd .

Atk

fixed inverse temperature 1/k| BT ,5 the relevant ther-modynamic: potential of the canonical (NS 1 ,5 NS 2 ,5 V,5 T)D en-semble" is the Helmholtz free energyFc (? NS 1 ,5 NS 2 ,5 V,5 T)D , givenby7

exp/ F c 1NS

1! 13z N_ 11

NS

2 ! 23z N_ 2f Tr

�1 Tr�

2 exp/ HQ ,5 2m

where% i h  /{ ¡ 2m ¢ ma ik| B£ T denotes$ the thermal wavelengthof+ speciesi¤ ¥ 1,2 as follows from the integration over themomenta.: The trace Tr1 is6 short for the volume integral¦

VdÇ

RN_

1 over+ the coordinates of the particles of species 1,and similarly for Tr2.

It proves more convenient to consider the system in the(?NS

1 ,5 § 2 ,5 V,5 T )D ensemble, in which the chemical potential¨2 © (? ª F c /{ « NS 2 )D N_ 1 ,V,T of+ species 2 is fixed instead of the

corresponding- number of particles,NS 2 . The associated ther-modynamic potential is denotedF(

?NS

1 ,5 ¬ 2 ,5 V,5 T )D , and is re-lateda

to F

c by7 the Legendre transform

F

NS

1 ,5 ® 2 ,5 V ¯±° F c ² NS 1 ,5 NS 2 ,5 V ³±´µ 2 NS 2 ,5 ¶ 31 ·where% we omitted the explicitT dependence.$ Equivalently,we% can write

exp/ ¸¹º F »¼¾½N_

2 ¿ 0ÀÁ

exp/ ÂÃÄÆÅ F c ÇÈ 2 NS 2 ÉËÊÌ 1

NS

1! Í 13z N_ 1Tr1 exp/ ÎÏÐÆÑ H11ÒÔÓÖÕË× ,5 Ø 4Ù

where% Ú is defined in terms of the fugacityzÛ 2ÜÔÝ2 Þ 3z exp(/ ßáà 2)D of species 2 as

exp/ âãäæåèçé¾êN_

2 ë 0Àì

zÛ 2N_ 2NS

2!Tr�

2 exp/ íîïÆð HQ 12ñ HQ 22òËó . ô 5G õ

Noteö

that ÷ depends$ not only onNS 1 ,5 zÛ 2 ,5 andV,5 but also onthe(

instantaneous coordinatesRi for i¤ ø

1,2, . . . ,NS

1 of+ the

canonically- treated component 1. In fact, the right hand sideof+ Eq. ù 5G ú can- be interpreted as the grand partition sum of afluid of species 2 in the external field of a fixed configurationof+ NS 1 particles, of species 1. Thus we writeû]üÔý]þ ÿ

R�;NS

1 ,5 zÛ 2 ,5 V � . � 67 �The reason why this (N

S1 ,5 � 2 ,5 V,5 T)D ensemble is conve-

nient can be seen from Eq.� 4� ,5 as the right hand side is thecanonical- partition sum of a one-component system of spe-cies- 1 with an effective interaction Hamiltonian

HQ effR � HQ 11� . � 7J

OnceL �

,5 and thusHeffR ,5 is known for all values ofzÛ 2 ,5 thethermodynamics(

and the phase behavior of the mixture canbe7

determined from standard techniques for one-componentsystems." We focus therefore on the calculation of� .Throughout�

we assume the volumeV to(

be macroscopicallylarge.a

B. Mayer expansion of �In order to calculate� explicitly,/ we first introduce the

Mayer functionsf�

and g� associated with the pair potentials�12 and �

22,5 respectively,f�

i j � f� � Rm i ,5 r jd ��� exp/ �����! 12" Rm i # r jd $&%(' 1,g� kl) * g� + rk) ,5 rl, -�. exp/ /�0�1!2 22 3 rk) 4 rl, 5&6(7 1. 8 8f 9

Inh

terms of these Mayer functions, we rewrite Eq.: 5G ; as exp/ @?BA(CED

NF

2 G 0ÀH

zÛ 2NF 2NI

2! VdÇ

rNF

2J K

i L 1NF

1 MjN O

1

NF

2J P

1 Q f� i j RTSk) U

l,

NF

2J V

1 W g� kl) XY 1 Z zÛ 2

VdÇ

r[ 1 \i ] 1NF

1 ^1 _ f� i1 `�a zÛ 222m VdÇ r[ 1dÇ r[ 2

bdci e 1NF

1 f1 g f� i1 hTi 1 j f� i2 kTl 1 m g� 12n

o zÛ 23z3!1

VdÇ

r[ 1dÇ r[ 2 dÇ r[ 3z pi q 1NF

1 r1 s f� i1 tTu 1 v f� i2 w

xzy1 { f� i3z |T} 1 ~ g� 12T 1 g� 13T 1 g� 23 � O zÛ 24 .

9

5748 PRE 59DIJKSTRA,÷

van ROIJ, AND EVANS

Weê

now introduce a diagrammatic technique 627 ,5 in which i6 each/ black circle represents a factorzÛ 2 and an integral ofr[ i over+the(

volumeV,5 ii6 each/ line between two black circles represents ag� bond,7 and iii6 each/ open big circle connected with a blackcircle- represents anf� bond7 and a summation over all different particles of species 1 at positionsRi for i¤ 1, . . . ,NI 1. Then theexpansion/ of Eq. 9 is6 equivalent to

exp/ ��@B( 1 ¡  the( sum of all distinct diagrams consisting of one or more black circles and some or nog� bonds7 and of one or more open big circles with one or moref� bonds7 ¢ £ 10¤

or,+ explicitly,

¥11¦

where% § (? ¨B© )D nª « (•)? m¬ denotes$ all diagrams involvingn or+more distinct particles of species 1 andm® black7 circles withor+ without g¯ bonds.7

Using°

the Goldstone theorem± 627 ² ,5 i.e., using ln(1³ x´ )Dµ·¶nª ¸ 1¹ (? º )D nª » 1x´ nª /{ n ,5 the only diagrams that survive after

taking(

the logarithm of the diagrammatic expansion are the

connected- ones ¼ i.e.,6 those which are extensive inV)D . Thuswe% can write

½12¾

Each of the diagrams above can be classified according to thenumbern of+ open big circles in it. This amounts to a decom-position, of ¿ that( can be written as

À�Á@ÂdÃÅÄÇÆnª È 0ÀNF

1 É@Ênª ,5 Ë 13Ì

where% n labels the number of particles of species 1 that si-multaneously interact with the ‘‘sea’’ of particles of species2 at fugacity zÛ 2. Below we show thatÍ nª corresponds- ton -body interactions between the large spheres, and we giveexplicit,/ albeit formal, expressions forÎ nª for# n Ï 0,1,2,3.Z Forconvenience- later we introduce the notation

HÐ

12

Ñnª ÒÔÓÖÕ

i × 1nª Ø

jN Ù

1

NF

2J Ú

12Û RÜ i Ý r[ jN Þ ,5 ß 14àwhich% describes the interaction betweenNI 2 particles, of spe-cies- 2 andn á 1 of species 1.

C. Expressions for â nãIt can be shown explicitly that the sum of diagrams with-

out+ any open big circle, i.e., representingä�å@æ 0À ,5 can bereexponentiated to give

exp/ ç�è�é@ê 0À ë(ìEíNF

2J î 0Àï

zÛ 2NF 2JNI

2!Tr2 exp/ ð�ñ�ò H22 ó(ôöõ 0À ÷ zÛ 2ø ,ù V ú ,ùû

15üwhereý the grand partition sumþ 0À (ÿ zÛ 2ø ,ù V)� is that of a puresystem� of species 2 at fixed fugacityzÛ 2 in a thermodynamicvolume� V. Extensivity requires that

PRE 59 5749PHASE�

DIAGRAM OF HIGHLY ASYMMETRIC BINAR Y . . .�

�0À � zÛ 2ø ,ù V ��� Vp � zÛ 2ø ,ù � 16�

whereý p� (ÿ zÛ 2ø )� is the negative of the grand-potential density, orthe�

pressure, of this one-component system.The�

diagrams involving only one open big circle, repre-senting� ����� 1,ù are recovered exactly by the identification

exp� ������� 1� � ! 1 " zÛ 2 ,ù V #$0À % zÛ 2 ,ù V &

NF

1

,ù ' 17(whereý ) 0À is* defined in Eq.+ 15, and- where.

1 / zÛ 2 ,ù V 0132NF

2 4 0À5

zÛ 2øNF 2NI

2! Vd6

rNF

2J

exp� 7�8�9 H12: 1 ;=< exp� >�?�@ H22AB18C

is*

the grand partition sum of a system of volumeV consistingDofE a single particle of species 1 and a ‘‘sea’’ of particles ofspecies� 2 at fugacityzÛ 2ø . It follows from Eq. F 17G that� H 1 canDbeI

written as J1 K NI 1 ,ù zÛ 2ø LM NI 1 N 1 O zÛ 2ø P ,ù Q 19R

whereý S 1 is defined byexp� T�U�VXW 1 Y zÛ 2 Z\[

]N̂F

2J _ 0À`

zÛ 2NF 2JNI

2ø ! Vd6 r[ NF 2 exp� acb�d HÐ 12e 1f=g exp� h�i�j HÐ 22ø k

l mNF

2 n 0Ào

zÛ 2NF 2NI

2ø ! Vd6 r[ NF 2J exp� pcqsr HÐ 22ø t

u 1vxw exp� y�z�{ HÐ 12| 1 }=~ z 2. 20

The�

brackets z 2 denote a statistical average in the res-ervoir� of particles of species 2. It follows directly from Eqs.17 and- 19 that� 1(ÿ zÛ 2)� is actually the grand-potential dif-

ference

between a sea of small spheres at fugacityzÛ 2ø withýand- without a single particle of species 1. Another interpre-

tation�

of 1 stems� from the analogy between Eq. 20 and-the�

Widom-insertion theorem 62 ; i.e., 1(ÿ zÛ 2)� is seen to bea- contribution to the chemical potential of species 1 due tothe�

presence of a sea of species 2 at fugacityzÛ 2. This latterinterpretation is, of course, consistent with the linear depen-dence

on NI

1 in*

Eq. 19 .It can also be shown explicitly that the exponential of the

sum� of connected diagrams involving two open big circles isgiven by

exp� c�� 2 �i jNNF

1  2ø ¡ Ri j ;zÛ 2ø ,ù V ¢ /£ ¤ 0À ¥ zÛ 2ø ,ù V ¦§

12 ¨ zÛ 2 ,ù V © /£ ª 0À2 « zÛ 2 ,ù V ¬ ,ù 21®

whereý ¯ 1 is given in Eq. ° 18± and-²

2ø ³ Ri j ;zÛ 2ø ,ù V ´µ·¶

NF

2J ¸ 0À¹

zÛ 2øNF 2JNI

2ø ! Vd6 rNF 2 exp� ºc»�¼ H12½ 2ø ¾=¿À

exp� Á�Â�à H22Ä . Å 22ÆOneÇ

recognizes thatÈ 2(ÿ RÜ i j ;zÛ 2 ,ù V)� is the grand partition sumofE a system in a volumeV containingD twoÉ particlesÊ of species1 Ë at- positionsRÜ i and- RÜ jN )� and a ‘‘sea’’ of particles of species2 at fugacityzÛ 2ø . It follows directly from Eq. Ì 21Í that�

Î2 ÏÑÐ RÒ ;NI 1 ,ù zÛ 2 ÓÔÖÕ

i × jNNF

1 Ø2 Ù Ri j ;zÛ 2 Ú Û 23Ü

is a pairwise sum of the pair potentialÝ 2 defined byexp� Þ�ß�àXá 2â ã Ri j ;zÛ 2â ä\å æèç 2â é RÜ i j ;zÛ 2â ,ù V ê /£ ë 0À ì zÛ 2â ,ù V íî

12â ï

zÛ 2â ,ù V ð /£ ñ 0À2â ò zÛ 2ó ,ù V ôõ÷ö exp� ø�ù�ú H12û 2üþý Ri j ÿ���� z 2�

exp� ���� HР12 1 ���� z 22 . � 24 �

Arguments�

along the same lines yield, for the exponentialofE the three-body contribution� 3� ,�

exp� ������ 3� �����i � jN k!

NF

1 "3# $ RÜ i, jN ,k! ;zÛ 2% ,& V '(

0À ) zÛ 2 ,& V *

+13# ,

zÛ 2 ,& V -.2 / Ri j ;zÛ 2 ,& V 0�1 2 2 Rik ;zÛ 2 ,& V 3�4 2 5 RjkN ;zÛ 2 ,& V 6 ,& 7 258

where9 the : nª for n; < 0,1= , and 2 are defined in Eqs.> 15? ,@A18B ,@ and C 22D E ,@ and where F 3G (H RÜ i, jN ,k! ;zÛ 2I ,@ V)J is the grand-

canonicalK partition sum of the sea of species 2 at fugacityzÛ 2in the external field of three particles of species 1 at positionsRi, jN ,k! . It follows directly from Eq. L 25M thatN

O3G PRQ

i S jN T k!NF

1 U3G V RÜ i, jN ,k! ;zÛ 2I W ,@ X 26Y Z

where[ the three-body potential\ 3G canK be rewritten in termsof] the corresponding interaction HamiltoniansHÐ 12(^ nª )_ for` n;a 1,2,3 as

expb c�degf 3G h Ri, jN ,k! i�jkml expb nporq H12s 3G tvu Ri, jN ,k! w�x�y zz 2J{ |}

mn¬ ~ i j , ik , jkN

expb � HÐ 12 1 � zz 2expb pr HÐ 12 2I v RÜ mn¬ �� zz 2J . 27

Y In principle this process can be continued for any integer

n; ,@ with the result

nª R  ;N¡ 1 ,¢ zÛ 2 £�¤ ¥i1 ¦ i2 §©¨v¨¨vª in«

NF

1 ¬nª Ri1 , . . . ,in« ;zÛ 2 ® ,¢ ¯ 28°

5750 PRE 59DIJKSTRA,±

van ROIJ, AND EVANS

where[ the interaction² nª between³ n´ particlesµ of species 1 canbe³

expressed in terms of grand-canonical averages of theBoltzmann¶

factors associated with the HamiltoniansHÐ

12(^m¬ )_

with[ 1 · m® ¸ n´ ,¢ as we showed explicitly forn´ ¹ 1,2,3. Ofcourse,º we are still left with the problem of calculating» nªfor n´ ¼ 1, but the present analysis shows that the effectiveHamiltonian ½ 7¾ ¿ for species 1 is of the formHeffÀ ÁÃÂ

pÄ Å zÛ 2 Æ V Ç NÈ 1 É 1 Ê zÛ 2 Ë�ÌÎÍi Ï jNNF

1 Ð�Ñ11Ò Ri j Ó�ÔÖÕ 2 × Ri j ;zÛ 2 Ø�Ù

ÚÜÛi Ý jN Þ kß

NF

1 à3á â RÜ i jk ;zÛ 2 ã�ä ,¢ å 29Y æ

where[ the ellipsis represents the termsç nª forè n´ é 4ê , andwhere[ we used Eqs.ë 16ì ,¢ í 19î ,¢ ï 23ð ,¢ and ñ 26ò .

Weó

make four remarks on the results obtained so far.ô i õWeó

wish to emphasize that the derivation ofHeffÀ

holds foranyö two-component mixture with integrable pair interactions,andö is not restricted to binary hard-sphere mixtures. The rateof÷ convergence of the expansion of the effective Hamiltoniandependsø

on the particular form of the pair potentialsù 12 andöú22; one could expect a relatively fast convergence for

short-rangedû potentials, although correlations will generallycauseº the effective interactions to be longer ranged than thebare³

pair potentials ü 12 andö ý 22. þ ii ÿ As a check on theresults obtained it is instructive to consider the following:exp� �������� is the grand-partition sum of a fluid of species2Y

in the external field of a fixed configuration ofNÈ

1 part-µ

icles of species 1 and the decomposition of� given� inEq.

�

13� is equivalent to the factorization exp�����������nª � 0ÀNF 1 exp� ������� nª � . For NÈ 1 0! , i.e., no particles of species

1 in the sea of species 2, we recover from Eq." 15# that$exp� %�&�'�()+* exp� ,�-�.�/ 0À 0+132 0À (4 zÛ 2,¢ V).5 For NÈ 1 6 1 wefind7

exp8�9�:�;= exp� ?�@�A�B 0À C exp� D�E�F�G 1H+I3J 0À (4 K 1 /L M 0À )5 N3O 1,¢where[ we used Eqs.P 15Q andö R 17S . Similarly, we find, forNÈ

1 T 2,Y expU�V�W�XY+Z3[ 0À (4 \ 1 /L ] 0À )5 2(4 ^ 2_ ` 0À )/(5 a 1)5 2 b3c 2_ ,¢ andfor N

È1 d 3e that expf�g�h�ij+k3l 3á . For arbitraryNÈ 1,¢ we indeed

find7

that expm�n�o�pq canº always be factorized into zero-,one-,÷ two-, three-, . . . , and NÈ 1-body terms so that expr�s�t�uv+w3x

Ny

1,¢ as required. This scheme to factorize the par-

tition$

function is actually similar in spirit to that of Ref.z 63{ | ,¢where[ it is applied to a one-component imperfect gas.} iii ~The terms 0À pÄ (4 zÛ 2_ )5 V andö 1 NÈ 1 1(4 zÛ 2_ ,¢ V)5 that repre-sentû the zero- and one-body terms in the effective Hamil-tonian$

29Y

doø

not depend on the instantaneous coordinatesR of÷ the particles of species 1, and therefore do not affect

the$

structure of÷ the uniform system.û Moreover, these twoterms$

do not affect thephaseÄ behavior of÷ the two-componentsystemû because of the trivialNÈ 1 dependenceø or, equivalently,the$

trivial dependence on density 1 NÈ 1 /L V: 0À /L V is in-dependentø

of 1,¢ and 1 /L V dependsø only linearly on 1.Since

two coexisting phases must have the same chemicalpotentialµ of species 2, and hence the samezÛ 2_ ,¢ the two termsunder consideration do not affect the common tangent con-structions,û as explained further in the Appendix. This in-nocuous character of 0À andö 1 is to be contrasted withanalogousö terms for systems with long-ranged Coulomb in-teractions,$

for which the Mayer expansion diverges so thatthe$

present derivation does not apply directly. Recently, it

was[ shown 64,65{ that$ a system of charged colloids sus-pendedµ in a solvent with coions and counterions can bemapped onto an effective one-component colloidal systemby³

integrating out the degrees of freedom of the coions andcounterions.º The usual Derjaguin-Landau-Verwey-Overbeekscreened-Coulombû potentialµ was recovered for the two-

body³

term, as expected. However, the charge neutrality con-straintû gives a nontrivial dependence of the one-body term onthe$

colloid density which has dramatic consequences for thephaseµ behavior at low salt concentrations 64,65{ . iv   Weóseeû from Eq. ¡ 24¢ that$ £ 2(4 Ri j ;zÛ 2)5 is the grand-potential dif-ferenceè

between the sea, of fugacityzÛ 2_ ,¢ containing two par-ticles$

of species 1 at a finite separationRi j andö at infinite

separationû RÜ i j ¤3¥ . In other words¦ 2 is the work done inbringing³

a particle of species 1 from infinity, but still in thereservoir§ of fixed zÛ 2_ ,¢ to a finite distance from another particleof÷ species 1 located at the origin.

IV. APPLICATION TO A BINARY MIXTUREOF¨

HARD SPHERES

In the previous section we showed that we can describe aclassicalº binary fluid by an effective one-component Hamil-tonian$ ©

Eq.

ª

29Y «�¬

. In this section we apply this approach tothe$

particular case of a mixture of large and small hardspheresû with diameters 1 andö ® 2_ ,¢ respectively. The sizeratio is denotedq¯ °²± 2 /L ³ 1 ´ 1. The pair potentialsµ i j(4 r)5 arethe$

usual additive hard-sphere potentials, i.e., infinite for 0¶r· ¸ (4 ¹ i º²» j¼ )5 /2 and zero otherwise. The zeroth-order term½0À is equal to the grand potential of a pure system of small

hard¾

spheres at fugacityzÛ 2_ :¿

0À À zÛ 2_ ,¢ V ÁÃÂÄ VphsÅ Æ zÛ 2_ Ç ,¢ È 30e É

where[ pÄ hs(4 zÛ 2)5 is the pressure of the small hard-sphere sys-tem.$

This pressure is accurately described by the Carnahan-StarlingÊ

equation of state for the fluid state values ofzÛ 2 of÷interest.

An explicit scaled particle expression forË 1(4 zÛ 2_ ) i5 sgiven� by Henderson, and consists of a volume, a surface, andaö Ì 1-independent termKÍ (4 zÛ 2_ )5 Î 66{ Ï :

NÈ

1 Ð 1 Ñ zÛ 2_ ÒÃÓ pÄ hsÅ Ô zÛ 2_ Õ V Ö 1 × 1 Ø q¯ Ù 3á ÚÜÛ hsÅ Ý zÛ 2_ ,¢ RÞ 1 ßáàãâ 12NÈ 1äKÍ å

zæ 2 ç NÈ 1 ,¢ è 31e éwhere[ ê hsÅ (4 zæ 2_ ,¢ RÞ 1)5 is the surface tension of a hard-spherefluid at a hard-spherical wall with a radiusR1 ë²ì 1/2,í andî

1 ïñðóò 13áNÈ

1/6í

V is the large-sphere packing fractionô 67{ õ .The two-body termö 2 canº be written as a sum of pair po-tentials$ ÷

2; see Eq.ø 23Y ù . In the case of hard spheres this pairpotentialµ can be identified with the depletion potential:

ú2 û RÜ i j ;zæ 2 üÃýÜþ depÿ � RÞ i j ;zæ 2 � . � 32e �

Attard � 50� � has derived an exact expression for the depletionforce�

between³

two large spheres and this has been employedin

simulation studies of depletion� 51� . Recently, Maoet al.calculatedº the depletion potential within the Derjaguin ap-proximationµ up to third order in � 2_r� ,¢ and found excellentagreementö with simulations forq¯ 0.1! and for � 2r� asö large as0.34! �

52,53� �

. Here � 2r is the packing fraction of a reservoir ofsmallû hard spheres at fugacityzæ 2_ . In this work, we use the

PRE 59 5751PHASE�

DIAGRAM OF HIGHLY ASYMMETRIC BINAR Y . . .�

simplerû expression given by Go¨tzelmann$ et al. which,[ up tothird$

order in � 2_r ,¢ is equally accurate and reads� 54� ����

depÿ � RÞ i j ����� 1 q

!2Y

q! " 3e

x# 2_ $ 2r %'& 9( x# ) 12x# 2_ *,+.- 2r / 2_0'1

36e

x# 2 30e x# 2_ 35476 2_r 8 3á 9 for : 1 ; x# < 0,! =33e >

where[ x# ? RÞ i j /í @ 2_ A 1/q! B 1. Contact corresponds toRÞ i j CED 1or÷ x# F�G 1. The total effective pair potential is from Eq.H 29I :

JeffÀ KML 11NMO depÿ ,¢ P 34e Q

where[ R 11 is the bare hard-sphere potential between twolargeS

spheres. Examples ofT effÀ areö shown in Fig. 1 for sizeratios q! U 0.2! and 0.1 at several values of reservoir packingfraction V 2_r . This pair potential contains a deep and narrowpotentialµ well close to the surface of the large sphere, fol-lowedS

by a small repulsive barrier. The range of the potentialis equal toq! times$ the large sphere diameter. For simplicityWseeû X 54� Y[Z we[ set \ depÿ ] 0! for RÞ i j ^E_ 1 `Ea 2,¢ and thus we ne-

glect� longer-ranged and weaker oscillations; we expect theseto$

be unimportant for the phase behavior of the mixture. It isworthwhile[ noting that exact expressions for the depletionpotentialµ were given in Ref.b 54� c within[ the context of theDerjaguin approximation. However, these expressions give apoorµ account of the simulation results of Ref.d 53� e forè q!f 0.1! and g 2r h 0.3,! thereby casting doubts on the validity ofthe$

Derjaguin approximation for these values ofq! andö i 2r .Inj

all our effective one-component calculations we setknl m 0! for n´ n 3e . This approximation was tested forq! o 0.1!

in computer simulations by Bibenet al.,¢ who found that thethree-body$

term, denotedp 3á above,ö contributes less than0.5%!

at q 2_r r 0.3! s 53� t . The neglect ofu 3á canº be made plau-sibleû by geometric arguments forq! v 0.154,! since then threeor÷ more nonoverlapping large spheres cannot simultaneouslyoverlap÷ with a small one; i.e., the first and dominant diagramin the three-body termw 3á vanishes.x However, it is important

to$

realize that other diagrams iny 3á areö not necessarily zero,even� for q! z 0.154,! so that the neglect of all three-body andhigher-body terms is an approximation. At this stage it is notevident� that q! playsµ the { formal| role of a small parameter.

Thus}

we arrive at the effective one-component Hamil-tonian$

HeffÀ ~

H0 Mi j¼N

1 effÀ R i j ,¢ 35e

where,[ as mentioned in Sec. III,H0 � pÄ hs zæ 2 , 1 1 1 q! 3á V hs zæ 2 ,¢ R1 12_ NÈ 1

K zæ 2_   NÈ 1 ¡ 36e ¢is

irrelevant for the phase behavior of the fluid, and where£effÀ is defined in Eq.¤ 34e ¥ . We have now mapped the binary

hard-sphere¾

mixture onto an ¦ approximateö § effective� one-componentº system of large spheres, which can be treatedwith[ standard techniques.

V.¨

RESULTS OF SIMULATIONS OF THE EFFECTIVEHAMILTONIAN

A. Phase diagram

At©

first sight, one might think that the phase behavior ofthe$

effective one-component system characterized by Eqs.ª34e «

andö ¬ 35e canº be determined by standard perturbationtheory$

using the pure hard-sphere system at the same pack-ing

fraction as a reference system. Using first-order pertur-bation³

theory forq! ® 0.1,! we did not find any indication of afluid-fluid spinodal for packing fractions¯ 1 ° 0.5.! This resultwas[ also found in Ref.± 54� ² . However, our simulations of thesystemû described by³ effÀ given� in Eq. ´ 34e µ yield¶ radial dis-tribution$

functionsg· (¸ r¹ )º that differ enormously from those ofthe$

reference hard-sphere fluid at the same» 1. This is illus-trated$

in Fig. 2, where we plotg· (¸ r)º for ¼ 1 ½ 0.35,! ¾ 2_r¿ 0.25,! and q! À 0.1.! We find thatg· (¸ Á 1)º Â 42, which shouldbe³

compared with the much lower contact valueà 3e for thehard-sphere¾

reference system. Similar large contact values,

FIG. 1. The effective large-sphere pair potential, i.e., the sum ofthe depletion potentialÄ 33Å and the large hard-sphere potentialÆ 11 ,Çof a binary hard-sphere mixture with size ratioÈ aÉ Ê qË Ì 0.2Í and Î bÏ Ðq Ñ 0.1, for several small-sphere reservoir packing fractionsÒ 2Ór .Ô

FIG. 2. The radial distribution functiong(ÕrÖ /× Ø 1) for the effective

one-component system with packing fractionsÙ 1 Ú 0.35,Í Û 2Ór Ü 0.25,Íand qË Ý 0.1 calculated using the depletion potentialÞ 33ß à with andwithout the small repulsive barrier. Note that this state point fallswell inside the fluid-solid coexistence regioná see Fig. 4â bÏ ãåä .

5752 PRE�

59DIJKSTRA,æ

van ROIJ, AND EVANS

which[ may signal a strong tendency for clustering, have beenobserved÷ in previous simulation and integral equation studiesç53,68� è

,¢ and also in experiments on colloidal hard-spheremixtures é 69{ ê . The enormous difference betweeng· (¸ r)º of thereference§ hard-sphere system and that of the effective systemsignalsû the breakdown of perturbation theory, and thus weresort to full numerical simulations for the free energyF of÷the$

effective system.Before describing these simulations we compareg· (¸ r)º for

aö depletion potential with and without the repulsive barrier.Figureë

2 shows that the contact value and most other struc-tural$

features are not sensitive to the repulsive barrier, apartfrom a small well nearr ì 1.07í 1,¢ which does reflect thepresenceµ of the barrier. We conclude that small differences inthe$

choice of depletion potential—such as our omission ofthe$

very weak oscillations forR i j î (ï ð 1 ñEò 2_ )ó —should nothave¾

a drastic effect on the resulting phase equilibria. Itprobablyµ suffices to have a good account of the potentialwell[ close to contact, while the first repulsive barrier, as wellasö the longer-ranged oscillations, should play only a minorrole. Note that the peaks ingô (ï r)ó nearr/õ ö 1 ÷ 1.74 and 2.0 aresimilarû to those found at state point ‘‘C’’ of Ref.ø 53� ù where[it

is argued that these arise from a particular local organiza-tion$

of particles equivalent to that in the sticky-sphere model.In order to determine the phase diagram of the effective

one-component÷ system, we first calculate the thermodynamicpotentialµ Fú ,¢ defined in Eq.û 4ê ü ,¢ as a function ofNÈ 1 ,¢ V,¢ andzý 2_ . We actually determine the dimensionless free energydensityø

f� þ

(ï ÿ

/6)õ �

13áF/õV asö a function of � 1 andö zý 2_ . For con-

veniencex we often replace the dependence onzý 2_ by³ that onthe$

reservoir packing fraction� 2_r� . As the free energy cannotbe³

measured directly in a Monte Carlo� MC� � simulation,û weuse thermodynamic integration to relate the free energy ofthe$

effective system to that of a reference hard-sphere systematö the same large-sphere packing fraction� 1. To this end weintroduce the auxiliary effective Hamiltonian

H �effÀ �

i � jN

1 ���11� R i j ������� dep� � R i j ��� ,¢ � 37e �

where[ 0 �!" 1 is a dimensionless coupling parameter: at#"$0!

the auxiliary Hamiltonian is that of the pure system ofNÈ

1 large hard spheres, while at%"& 1 it is the effectiveHamiltonian'

of interest ( forè fixed zý 2 andö V)ó . It is a standardresult ) 7¾ 0–72* that$

F + NÈ 1 ,¢ V,¢ zý 2 ,�- F . NÈ 1 ,¢ V,¢ zý 2 / 0! 01

01

d2 3 4

i 5 jN

1 6dep� 7 R8 i j 9

N

1 ,V,zz

2 , :,¢ ; 38e <

where[ Fú (ï NÈ 1 ,¢ V,¢ zý 2_ = 0! ) is the free energy of the pure refer-ence� system of large hard spheres (>"? 0! ), for which we usethe$

Carnahan-Starling expressions@ 73¾ A forè the fluid, and theanalyticö form for the equation of state proposed by HallB 74¾ Cfor the solid phase. In the latter case an integration constantis determined such that the known fluid-solid coexistence ofthe$

pure hard-sphere system is recoveredD 11E . The angularbrackets³ F G

N

1 ,V,zz

2H , I denoteø a canonical average over the

systemû of NÈ 1 particlesµ interacting via the auxiliary Hamil-

tonian$

H JeffÀ . The integrand in Eq.K 38e L can,º for a fixed M , b¢ emeasured in a standard MC calculation; for the numericalNintegration, we use a ten-point Gauss-Legendre quadratureO75¾ P

.In order to map out the phase diagramf

Q(ï R

1 ,¢ zý 2)ó must bedeterminedø

from S integrations for many state points(ï T

1 ,¢ zý 2_ )ó . We chose therefore to simulate relatively small sys-tems,$

with NÈ

1 U 108. As an illustration we plot, in Fig. 3,fQ asöaö function of V 1 atö several W 2r forè q! X 0.1.! For Y 2r Z 0.06! wefind7

that the solid branch offQ

becomes³

nonconvex, indicativeof÷ a spinodal instability. For[ 2r� \ 0.29! another spinodal in-stabilityû is found, but now in the fluid branch. This instabilitycanº be seen clearly in the inset of Fig. 3, wherefQ is shown at]

2_r� ^ 0.31.! For clarity, we have subtracted a linear fit in_ 1,¢

which[ does not affect the common tangent construction.Note`

that each point in the (a 1 ,¢ b 2r )ó plane is obtained by anindependent

c

integration.

In order to construct the fullphaseµ diagram we employ common tangent constructions atfixed7

zý 2_ to$ obtain the coexisting phases. We fitted polynomi-alsö to fQ andö computed the pressure and chemical potential ateach� d 1. The densities of the coexisting phases can then bedeterminedø

by equating the pressures and chemical potentialsin both phases. For more details we refer the reader to theAppendix.

The}

above procedure has been carried out to determinethe$

phase diagram for size ratiosq! e 0.2,! 0.1, 0.05, and0.033.!

In Fig. 4, we show the resulting phase diagrams in the(ï f

1 ,¢ g 2r� )ó plane. This representation, which is the natural onegiven� our approach, implies that the tie lines connecting co-existing� state points are horizontal. The shaded areas repre-sentû the h metastable i fluid-fluidj and solid-solid two-phase re-gions.� At k 2r� l 0! and for allq! we[ recover the known freezingtransition$

of the pure hard-sphere system.

FIG. 3. Reduced free energym fn * oqpsr F t ( u 0v wyx 1)z {�| 13} /~ V ver-sus 1 for q 0.1 at several 2r . The curves for 1 0.54 are thesolid branches, while the curves for lower 1 are the fluid branches.Note the difference in scale for 2r 0.31. For clarity, we subtracteda linear fit in 1 to the data for 2r 0.31 see inset . For a given 2rcommon tangent constructions can be made to determine the valuesof 1 at coexistence.

PRE 59 5753PHASE

DIAGRAM OF HIGHLY ASYMMETRIC BINAR Y . . .

For q! 0.2,! an enormous widening of the fluid-solid tran-sitionû is observed when 2_r� increases sufficiently. This im-pliesµ that the coexisting fluid and solid phases becomeprogressivelyµ more dilute and dense, respectively, upon in-creasingº 2_r� . This widening is consistent with findings byGast

et al. 61{ in perturbation theory studies of the Asakura-Oosawa

depletion potential model 30,31e ,¢ and has also beenobserved÷ in experiments on colloid mixtures, where smallamountsö of small spheres induce a rapid decrease in the lat-tice$

constant of the crystal 39e   . The shape of the coexistencecurveº for ¡ 2r ¢ 0.1! implies that the fluid phase only persists toveryx low values of £ 1.

The}

calculations also reveal the existence of a fluid-fluidtransition$

for q! ¤ 0.1.! However, this fluid-fluid coexistence ismetastable with respect to the broad fluid-solid transition forallö q! andö ¥ 2r . It is of interest to note that for allq! ¦ 0.1! thecriticalº point of the metastable fluid-fluid coexistence curveoccurs÷ at a value of § 2_r� that$ is about twice that of the fluid-solidû curve at the samë 1. Thus the fluid-fluid curve doesnot´ move deeper into the fluid-solid coexistence region upondecreasingø

q! ,¢ i.e., upon decreasing the range of the potential.This situation is in contrast to the findings of Ref.© 71¾ ª where[for a Yukawa pair potential decreasing the range of the at-traction$

lowers, in temperature, the fluid-fluid coexistence

into the fluid-solid region. Here there is no trend that sug-gests� a stable fluid-fluid curve atq! « 0.033.! For q! ¬ 0.2! we donot find a spinodal instability in the fluid branch for 2r®

0.46,!

while we do find a spinodal instability in the solidbranch³

for ¯ 2r� ° 0.12.! However, in Fig. 5 we show that assoonû as the spinodal instability appears in the solid phase,this$

instability is very broad and disappears in the fluidphase.µ We were not able therefore to find a metastable solid-solidû coexistence using the common-tangent construction.Rather, we plot in Fig. 4± aö ² the$ spinodal ³ dashedø curvéwhich[ is given by (µ 2_ fQ /õ ¶¸· 12_ )ó ¹ 0! . The presence of this spin-odal÷ instability on the solid branch may be important for thekineticsº

of the phase separation of the mixture» 36,76e ¼ .More surprisingly perhaps, the phase diagrams also show

the$

existence of an isostructural solid-solid transition forq!½0.1.!

For q! ¾ 0.1,! the solid-solid coexistence region is foundto$

be metastable with respect to the freezing transition, al-though$

the critical point of the solid-solid binodal is verycloseº to the stable fluid-solid phase boundary. For smallerq!the$

most striking feature is the downward shift of the solid-solidû with respect to the fluid-solid binodal, so that there is aregime with a stable solid-solid coexistence forq! ¿ 0.05.! Si-multaneously, the solid-solid critical point shifts closer tocloseº packing upon decreasingq! . These results are consistent

FIG. 4. Phase diagram of binary hard-sphere mixtures with size ratiosÀ aÁ q à 0.20,Ä Å bÆ Ç q È 0.10,Ä É cÊ Ë q Ì 0.05, and Í dÎ Ï q Ð 0.033Ä as afunction of the large-sphere packing fractionÑ 1 and the small sphere reservoir packing fractionÒ 2r as obtained from simulations of theeffective one-component Hamiltonian.F andÓ SÔ denote the stable fluid and solidÕ fccÖ phase.× F Ø S,Ù F Ú F, andSÔ Û SÔ denote,Î respectively, thestable fluid-solid, the metastable fluid-fluid, and the solid-solid coexistence regions. The dashed line inÜ aÝ denotesÎ the spinodal instability onthe solid branch. Note that the solid-solid coexistence forq Þ 0.05 becomes stable and that all the tie lines are horizontalß notà drawná .â

5754 PRE 59DIJKSTRA,ã

van ROIJ, AND EVANS

withä those of Refs.å 72,77¾ æ for square-well and Yukawa flu-ids,

where the authors find a solid-solid transition when therange§ of the potential is sufficiently short. The coexistingdensitiesø

for q! ç 0.2,! 0.1, 0.05, and 0.033 are tabulated inTables}

I, II, III, and IV.Withinó

the context of the Percus-Yevick approximation, ithas been shown that the binary hard-sphere mixture in thelimitS

of q! è 0! is equivalent to the one-component Baxtersticky-sphereû model é 78¾ ê . The latter is obtained from thesquare-wellû model by considering the limitë /õ ì 1 í 0,! îðïñóò withä ôöõ (ï 1/12)(÷ 1 /õ ø )ó expùûúýüÿþ�� finite, where � andö �areö the well depth and the well width, respectively. Note that� playsµ the role of a dimensionless nonlinear temperature. Inthe$

( � 1 ,¢ � )ó plane the sticky� or÷ adhesive� hard-sphere¾ modelexhibits� a vertical solid-solid binodal at close packing forinfinite,

while for any finite it exhibits a coexistence be-tween$

a close-packed solid and an infinitely dilute gas, withallö other phases at best metastable and probably unstable�72¾ �

. This pathological behavior of the adhesive hard-sphere

model is consistent with Stell’s analysis of the 12th virialcoefficient,º which was shown to be divergent 79¾ � . It followsfrom Eq. � 33� � that$ the depletion potential forq! � 0! and � 2_r�

0!

gives rise to a well depth������� 2_r� /õ q! ,¢ to a well width�/õ �

1 � q! ,¢ and hence to�� (1ï /q! )ó exp!#"%$ 2r/õ q! & whereä we ne-glect� irrelevant terms of order unity. Under the condition that'

2_r� ( q! ln(1/q! (1) *,+ )- )ó and .0/ 0! , this gives rise to1�2 0! for q!3 0! , i.e., to the ‘‘ground state’’ of the adhesive hard-sphere

model. If, however, 4 2_r is taken so small that 5 2_r6q! ln(1/q! (1) 7,8 )- )ó for q! 9 0! , then the high-temperature limit:�;=< is obtained, corresponding to the one-component hard-

sphereû system with its ‘‘normal’’ hard-sphere freezing tran-sition.û One can thus envisage two differentq! > 0! limitingprocedures:µ ? i @ the$ Baxter-like limit in which A is nonzero by

FIG.B

5. Reduced free energyC fn * DFE,G F H ( I 0v JLK 1)z MONQP 13} /6~ VversusR 1 for q S 0.2 at severalT 2r .â The curves forU 1 V 0.54 are thesolid branches, while the curves for lowerW 1 are the fluid branches.

TABLE I. The coexisting densitiesX expressedY in terms of thepacking fractionZ 1 of the large spheres[ at the fluid-solid transitionfor a binary mixture of hard spheres with size ratioq \ 0.2Ä andvarying packing fractions] 2r of^ small spheres in the reservoir asobtained from simulations with the effective Hamiltonian.

Fluid-solid_2r `

1 a fluidb c d 1 e solidf0.00Ä

0.494 0.5450.03Ä

0.487 0.5680.06Ä

0.492 0.5760.09Ä

0.484 0.6050.12Ä

0.478 0.6530.15Ä

0.466 0.6730.18Ä

0.440 0.6890.21Ä

0.381 0.6990.27Ä

0.131 0.7110.30Ä

0.0652 0.7150.33Ä

0.0252 0.7200.36Ä

0.00639 0.724

TABLE II. The coexisting densities at the fluid-solid, solid-solid, and fluid-fluid transitions for a binary mixture of hard sphereswith size ratioq g 0.1Ä and varying packing fractionsh 2r of^ smallspheres in the reservoir as obtained from simulations with the ef-fective Hamiltonian.

Fluid-solidi2r j

1 k fluidl m 1 n solido0.00 0.494 0.5450.02 0.491 0.5550.04 0.487 0.5670.05 0.487 0.5740.055 0.487 0.6050.06 0.483 0.6830.08 0.480 0.6920.09 0.472 0.7000.10 0.459 0.7060.12 0.426 0.7150.13 0.370 0.7180.15 0.264 0.7200.17 0.110 0.7250.20 0.00863 0.7270.31 0.00424 0.7280.34 0.00274 0.730

Solid-solidp2r q

1 r solid 1s t 1 u solid 2v0.06 0.613 0.6690.07 0.556 0.6870.08 0.553 0.6910.09 0.555 0.6980.10 0.555 0.7050.12 0.546 0.7140.13 0.541 0.715

Fluid-fluidw2r x

1 y fluid 1z { 1 | fluid 2}0.29 0.140 0.4080.30 0.0461 0.4350.31 0.0207 0.4440.32 0.00874 0.4520.34 0.00672 0.458

PRE 59 5755PHASE~

DIAGRAM OF HIGHLY ASYMMETRIC BINAR Y . . .

restricting§ attention to sufficiently small 2r or÷ ii the$ Stell-likeS

so-called 0 limit where 2_r is fixed resulting in � 0!unless 2_r� 0! 79 . It is the latter procedure that is relevantfor a comparison with the limitingq! 0! behavior of binaryhard-sphere¾

mixtures. The resulting phase diagram, in the(ï

1 , 2r�)ó

plane, of the effective one-component system withanö infinitesimal q! is shown in Fig. 6. There is a verticalsolid-solidû binodal at close packing in the limit 2_r 0! . How-ever,� for all 2_r� 0! the phase diagram shows coexistencebetween

a close-packed solid and an infinitely dilute gas,whileä all other phases are metastable 72 . Clearly, all trendsasö a function ofq! featured in Fig. 4 are consistent with thefactè

that the phase diagram approaches that of the adhesivehard-sphere¾

model in Stell’s double limitq! 0! and 2r 0.!In order to test the range of validity of the depletion po-

tential$

picture, we also studied the extreme limit ofq! 1,whereä the system of ‘‘large’’ spheres speciesû 1 is in equi-librium with a reservoir of ‘‘small’’ spheres speciesû 2whichä have exactly the same diameter and a packing fraction

2r� . The depletion potential  33� ¡ is based on the Derjaguin

approximationö which becomes exact in the limitq! ¢ 0! , but

cannotº be expected to be accurate forq! £ 1. In the limit q!¤ 1, the actual depletion potential between two hard spheressuspended¥ in a fluid of hard spheres with exactly the samediameter¦

is given by § k¨ B© T ln ª gô (ï r)ó « , i.e., the potential ofmean force¬ 54� . We therefore computedgô (ï r)ó in a simula-tion®

for a system of pure hard spheres at a packing fraction¯2°r� . In Fig. 7, we compare the depletion potential± 33� ² withä³ k¨ B© T´ lnµ ¶ gô (ï r· )ó ¸ for¹ º 2r� » 0.1,! 0.2, and 0.3. There is reasonably

good¼ agreement except in the repulsive barrier for½ 2r ¾ 0.3.!We¿

then performed simulations of the effective one-componentÀ system interacting with the depletion potentialÁ33� Â

for q! Ã 1. Figure 8 shows that the fluid-solid coexistencehardly varies with Ä 2°r

�andÅ remains narrow. This phase dia-

gram¼ agrees rather well with the theoretical prediction rep-resented by the dashed curve. The latter is obtained by equat-ing the pressure and the chemical potential of species 1 in thesolid¥ and fluid phases and by equating the chemical potentialofÆ species 2 to that of the reservoir. This amounts to solvingthe®

four equations

ÇPHallÈ ÉËÊ solidÌÎÍÐÏ PCSÑ ÒÔÓ fluidÕ ,

TABLE III. The coexisting densities at the fluid-solid, solid-solid, and fluid-fluid transitions for a binary mixture of hard sphereswith size ratioqÂ Ö 0.05Ä and varying packing fractions× 2r of^ smallspheres in the reservoir as obtained from simulations with the ef-fective Hamiltonian.

Fluid-solidØ2r Ù

1 Ú fluidÛ Ü 1 Ý solidÞ0.00Ä

0.494 0.5450.02Ä

0.490 0.5660.04Ä

0.490 0.5730.045Ä

0.487 0.5800.05Ä

0.487 0.7170.06Ä

0.473 0.7220.08Ä

0.320 0.7260.10Ä

0.093 0.7270.12Ä

0.0061 0.7280.15Ä

1.75ß 10à 4 0.7300.17Ä

4.94á 10â 6ã 0.7310.19Ä

3.30ä 10å 7æ 0.7350.21Ä

2.30ç 10è 7æ 0.740Solid-solidé

2r ê

1 ë solid 1ì í 1 î solid 2ï0.04Ä

0.658 0.7070.045Ä

0.619 0.7150.05Ä

0.587 0.7170.06Ä

0.558 0.722

Fluid-fluidð2r ñ

1 ò fluid 1ó ô 1 õ fluidb 2ö0.165Ä

0.0622 0.4370.17Ä

0.0540 0.4470.18Ä

0.0170 0.4750.19Ä

0.0058 0.483

TABLE IV. The coexisting densities at the fluid-solid, solid-solid, and fluid-fluid transitions for a binary mixture of hard sphereswith size ratioq ÷ 0.033Ä and varying packing fractionsø 2r of smallspheres in the reservoir as obtained from simulations with the ef-fective Hamiltonian.

Fluid-solidù2r ú

1 û fluidü ý 1 þ solidÿ0.00 0.494 0.5450.01 0.489 0.5570.02 0.488 0.5660.03 0.489 0.5760.04 0.469 0.7250.05 0.442 0.7260.06 0.349 0.7270.07 0.205 0.7280.08 0.06695 0.7290.10 8.816� 10� 5� 0.7310.15 0.00 0.733

Solid-solid�2r �

1�solid 1� � 1 � solid 2

0.025 0.676 0.71590.028 0.666 0.71640.03 0.664 0.71750.04 0.562 0.7253

Fluid-fluid

2r �

1 � fluid 1 � 1 � fluid 2�0.12 0.0943 0.4340.13 0.0161 0.4490.14 0.0108 0.4550.15 0.004 0.457

5756 PRE�

59DIJKSTRA,�

van ROIJ, AND EVANS

lnµ �

1solid�����

Hallex� ��� solid��� lnµ 1fluid !�"�# CSÑex

� $�% fluid& ,'(�)

2* + ln , 2*solid-�.�/ HallÈex

� 0�1 solid2 ,'3�4

2 5 lnµ 6 2fluid 7�8�9 CSÑex� :�; fluid< = ,'

for the four unknowns> 1,2fluid<

,' and ? 1,2solid withä @ fluid ACB 1fluid

precludeÇ using the form of the simulatedgô (ï r)ó as a criteriontoV

determine freezing in binary mixtures. In the thermody-namic limit gô (ï r)ó in the two-phase region would be theconcentration-weightedU sum of the radial distribution func-tionsV

of the coexisting fluid and solid phases. However, thenature of gô (ï r )ó obtained from finite simulations in a two-phaseÇ region depends on several factors making its interpre-tationV

rather complicated. Note that the peak nearrÈ 1.73É 1 found in Fig. 2 Ê atP Ë 1 Ì 0.35,[ Í 2*r� Î 0.25)[ is notpresentÇ at the state points investigated in Fig. 9. It appearsthatV

this particular feature only develops deep inside the two-phaseÇ region.

The structure factorSg

(ïkh)ó

is shown in Fig. 10 for the samestateN points. This was calculated directly, usingSg (ï kh )óÏ Ni Ð 1ÑÓÒ (ï kÔ )ó Õ (ï Ö kÔ )ó × ,' where Ø (ï kÔ )ó ÙÛÚ i Ü 1N

Ý1 exp(M iÞ kÔ Rß i)ó . For

eachM à 1 consideredU we observe an increase of theá extrapo-Mlated â

valueã of Sg (ï kh ä 0[ ) with increasing å 2r . An increase ofSg

(ïkh æ

0[

) was also found in experiments on hard-sphere col-loid

mixtures ç 69è é andP in other simulations of binary hard-sphereN mixtures based on effective pair potentialsê 53 ë . Asì

2*r is increased the depletion potential becomes more attrac-

tive;V

i.e., the well deepens, and simple random phase ap-proximationÇ í RPAî argumentsP indicate thatSg (ï kh ï 0[ ) shouldthenV

increase—provided the repulsive barrier does not be-

comeU too large ð 54 ñ . For the most dilute systemò 1ó 0.05,[ Sg (ï kh )ó is monotonically increasing withkh atP small khfor ô 2*r õ 0.15[ but for ö 2*r ÷ 0.20—a[ point in the fluid-solidcoexistenceU region—there is pronounced small-angle scatter-ing ø anP increase ofSg (ï kh ) aó s kh ù 0[ ú whichä reflects the growthofc dense clusters of the large spheres. For higher values ofû

1 againP we do not observe this increase ofSg (ï kh )ó at smallkh

untilü we enter the two-phase region. However, when thisoccursc the structure factor becomes quite noisy.

Anotherý

significant feature which is observed within thesingleN þ fluidÿ � phaseÇ is thatkh m� ,' the position of the first peakin Sg

(ïkh)ó, shifts to values that are higher than the value for

pureÇ hard spheres as� 2*r is increased at fixed� 1. This seemstoV

be a feature of potentials with short-ranged attraction giv-ing rise to a sharply peakedgô (ï r)ó and is accounted for quali-tativelyV

by the RPA. The height of the first peak does notvaryã rapidly with � 2r� andP we find that freezing occurs whenSg

(ïkh

m� )ó � 1.04 for � 1 � 0.05[ and � 1.52 for 1 0.30.[ Thesevaluesã of the peak height aremuch� smallerN than the peakheight S

g(ïkh

m� )ó � 2.85 which, according to the Hansen-Verletone-phasec � criterionU � 70� � ,' is supposed to signal the onset of

freezing in simple fluids. Note that upon increasing� 1 theVvalueã of Sg (� kh � 0[ ) is decreased andSg (� kh m� )ó , and the heights oftheV

subsequent maxima, is increased. These features simply

FIG.B

9. The radial distribution functiong� (� r� /~ � 1) for the effective one-component system, based on the depletion potential� 33� � with� sizeratio q � 0.1, small-sphere reservoir packing fractions� 2r � 0.00, 0.10, 0.15, and 0.20, and with large-sphere packing fractions a!#" 1$ 0.05, % bÆ &(' 1 ) 0.10, * cÊ +(, 1 - 0.20,Ä and . d/(0 1 1 0.30.Ä In each case the states with2 2r 3 0.20Ä lie in the solid-fluid coexistence region. In4 dÎ 5the state with6 2r 7 0.15Ä is just inside this coexistence region.

5758 PRE 59DIJKSTRA,8

van ROIJ, AND EVANS

reflect the effect of increased packing of the large spheres inthisV

effective one-component fluid.In9

Fig. 11 we plotgô (� r ): for the less extreme size ratioqo; 0.2[ at several< 2r for= > aP ?A@ 1 B 0.10,[ C b DFE 1 G 0.20,[ and H cU IJ1 K 0.30.[ Upon increasingL 2*r� atP fixed M 1 weä observe fea-

turesV

that are similar to those forqo N 0.10;[ namely, O i P theVcontactU value of gô (� r ): increases,Q ii_ R theV minimum nearrS 1.15T 1 becomes more pronouncedU note that the minimumis_

broader than forqo V 0.1,[ reflecting the difference in thedepletionR

potentials—see Fig. 1W ,' and X iii_ Y theV peak at rZ 2 [ 1 becomes more pronounced. Once again there is noclearU signature ingô (� r ): that phase separation has occurred.The structure factors for the same state points are shown inFig.

12. The main features are similar to those forqo\ 0.1:[ Sg (� kh ] 0[ ) increases with increasing^ 2r� for= fixed _ 1 andPfor states within the fluid-solid coexistence region the databecome

noisier. Upon increasing̀2*r� ,' the position of the firstmaximumk

hm� againP shifts to larger values. Finally we note

that,V

as for qo a 0.1,[ the values ofSg (� kh m� ): at the fluid-solidtransitionV

are much lower, for the three values ofb 1,' than thevalueã given by the Hansen-Verlet criterionc 70� d . Only whenfreezing occurs at high values of the packing fraction,e 1f

0.45,[

should we expect this criterion to be valid. For thepresentÇ system this would restrict its validity to the regime

g2r h 0.15[ for qo i 0.2[ and to j 2r k 0.1[ for qo l 0.1,[ where the

freezing is m essentiallyM n thatV of pure hard spheres. However,thisV

is not known in advance.

VI.o

RESULTS OF DIRECT SIMULATIONSOFp

THE MIXTURE

In9

the previous section, we determined the phase behaviorofc a binary mixture of hard spheres using an effective one-componentU Hamiltonian based on pairwise additive depletionpotentials.Ç It is important to keep in mind that this Hamil-tonianV

does not constitute an exact treatment of the binarysystem,N since three- and higher-body interactions were ne-glected.X One might expect the pairwise approximation tobreak

down at sufficiently high densitiesq e.g.,M in the solidphaseÇ r orc for less extreme size ratioss e.g.,M qo t 0.154,[ wherethreeV

nonoverlapping large spheres can overlap with a smallonec u 61è vxw ,' thereby casting doubt on the specific predictionsyin particular those for the solid-solid transitionz . Moreover,

theV

potential { dep� usedü in the simulations is approximated byanP empirical form that does not take into account the pres-enceM of longer-ranged oscillations| 8e } . To the best of ourknowledge these approximations—and therefore the deple-tionV

potential picture as a whole—have never been tested

FIG. 10. The structure factorS(k) for the effective one-component system, based on the depletion potential~ 33 with size ratioq 0.1, small-sphere reservoir packing fractions 2r 0.00,Ä 0.10, 0.15, and 0.20, and large-sphere packing fractions 1: a 0.05,Ä bÆ 0.10,Ä c0.20, and d 0.30.Ä These are the same state points as in Fig. 9.

PRE 59 5759PHASE

DIAGRAM OF HIGHLY ASYMMETRIC BINAR Y . . .

directlyR

by making a comparison with results of a full treat-ment of the true binary mixture. Given the richness of thepredictedÇ phase diagrams and the experimental and compu-tationalV

effort that is being put into the determination of thedepletionR

potential, it is important to perform such test. It hasbeen

argued by many authors that direct simulations are not

feasible=

for highly asymmetric binary hard-sphere mixturesbecause

of the ergodicity problems mentioned in the Intro-duction.R

However, the results in Fig. 4 show such interestingphaseÇ behavior at surprisinglyN low 2*r thatV we were moti-vatedã to perform direct simulations in this regime, using ascheme,N to be discussed below, designed to deal with thisregime, but not with the whole phase diagram. Our scheme

FIG. 11. The radial distribution functiong(r/~

1)z

for the effec-tive one-component system, based on the depletion potential 33with size ratioq 0.2, small-sphere reservoir packing fractions 2r 0.00, 0.10, 0.20, 0.25 and 0.30, and large-sphere packing frac-tions 1: a 0.10, bÆ 0.20,Ä and c 0.30.Ä Note that fluid-solid co-existence occurs when  2r ¡ 0.285 for ¢ a£ , ¤ 2r ¥ 0.255Ä for ¦ bÆ § , and¨

2r © 0.23 for ª cÊ « .â

FIG. 12. The structure factorS(k¬)z

for the effective one-component system, based on the depletion potential 33� ® with� sizeratio q ¯ 0.2,Ä small-sphere reservoir packing fractions° 2r ± 0.00,Ä0.10, 0.20, 0.25, and 0.30, and large-sphere packing fractions² 1:³á 0.10, µ b¶ 0.20, and· ç 0.30. These are the same state points as in

Fig. 11.

5760 PRE¹

59DIJKSTRA,º

van ROIJ, AND EVANS

complementsU the direct simulations performed recently byBuhot and Krauth» 68è ¼ . Their new algorithm is designed todealR

with small-size ratios but it does not seem feasible½ 68è ¾toV

implement the algorithm for most of the state points ofinterest_

here (¿ 1 ÀÂÁ 2* Ã 0.25)[ .TheÄ

scheme we use to calculate the ‘‘exact’’ phase dia-gramsX of binary hard-sphere mixtures by direct simulationemploysM the identity

ÅF Æ Ni 1 ,' V,' zÇ 2 ÈÊÉÌË F Í Ni 1 ,' V,' zÇ 2 Î 0[ Ï

Ð0ÑzÒ 2

dzÓ

2ÔÕ×Ö

FØ Ù

Ni

1 ,' V,' zÇ 2ÚxÛÜzÇ 2Ý . Þ 39

� ß

TheÄ

system atzÇ 2 à 0[ is the pure system of large hard spheres.Hence the first term of the right hand side of Eq.á 39� â isgivenX accurately by the Carnahan-Starlingã 73� ä free energy intheV

fluid phase and by that of Hallå 74� æ in_ the solid phase, asdiscussedR

already. The integrand in the second term can berewritten using Eqs.ç 4è andP é 5 ê asP

ëíìF î Ni 1 ,' V,' zÇ 2 ïð

zÇ 2ñóò

ôNi

2 õ zÒ 2zÇ 2 ,' ö 40

÷ ø

whereä ù Ni 2* ú zÒ 2 denotesR the average number of small particlesin_

the (Ni

1 ,' V,' zÇ 2* ): ensemble. This quantity can be measureddirectlyR

in a grand-canonical simulation of the ‘‘adsorption’’ofc small spheres from a reservoir at fugacityzÇ 2* ontoc a systemofc Ni 1 large spheres in a volumeV. Note that this scheme isalmostP identical to the widely knownû -integration schemepresentedÇ in Eq. ü 38� ý andP in Ref. þ 81e ÿ .

Before discussing the results of the direct simulations, wewishä to make two remarks. First, the scheme proposed inEqs.� �

39� �

andP � 40÷ � is_ merely a bulk analog of using the GibbsadsorptionP equation to determine the surface tension, where�Ni

2* �

zÒ 2� playsÇ the role of the adsorption,FØ thatV of the surfacetension,V

andNi

1 ,' V,' andqo characterizeU the ‘‘substrate.’’ Sec-ond,c it is important to realize that� Ni 2 zÒ 2 is_ not identical_ totheV

unweighted average adsorption from the reservoir onto asystemN of static� large hard spheres, since not all configura-tionsV

of large spheres carry the same statistical weight. Infact,=

this weight is proportional to exp���� H� eff� � ,' a quantitythatV

is not known exactly as it involves empirical pair poten-tialsV

and unknown higher-order interactions, as we have seenabove.P Consequently, the grand-canonical simulations thatmeasure� � Ni 2 � zÒ 2 must� be combined with asimultaneous� ca-Unonical average over the large-sphere configurations. Thisrequirement still leads to ergodicity problems at high� 2*r ,'althoughP the upper bound, which depends onNi 1 ,' V,' andqo ,'is sufficiently high even forqo asP small as 0.05 to permit us tostudyN interesting regimes.

We�

now return to the calculation ofFØ

from=

Eqs. � 39� � andP�40÷ �

. In Fig. 13, we plot� 2 � �"! 23# $ Ni 2 % zÒ 2/6& V,' as a function of'2*r ,' as measured in a simulation withNi 1 ( 32� for several) 1.

Results are shown forqo * 0.1[ and qo + 0.05.[ Here we con-vertedã zÇ 2 into , 2*r usingü the Carnahan-Starling expressionzÇ 2 - 2*3# . (6� / 2*r /& 0 ): exp1 (8� 2 2*r 3 94 5 2*r 2* 6 3� 7 2*r 3# )(1: 8:9 2*r ): ; 3# < ,' which isessentiallyM exact in the regime of interest. AlthoughNi 1= 32� may seem too small a number of large spheres to per-

form reliable simulations, one should recognize that> i ? theVpureÇ large-sphere free energy@ atP zÇ 2 A 0[ ) is taken from accu-rate independent sourcesB 73,74� C andP D ii E theV maximum valueofc F Ni 2 G zÒ 2 is_ about 5H 104

IdueR

to the small-size ratios. We

alsoP plot J 2* asP predicted by the scaled particle expressionsfor=

the zero- and one-body terms and the predictions of thefree volume approach of Ref.K 32� L ; these comparisons will bediscussedR

in more detail in Sec. VII.UsingM

the simulation data forN 2* asP a function of O 2*r orc zÇ 2* ,'weP calculateF(� Ni 1 ,' V,' zÇ 2): from Eq. Q 39� R byS numerical inte-gration.X OnceFØ is_ known we employ common tangent con-structionsN at fixed zÇ 2 toV obtain the phase boundaries shownbyS

the symbols in Fig. 14. The main observation is the strik-ingly_

good overall agreement with the effective one-componentU results for the three values of the size ratio,qoT 0.2,[ 0.1, and 0.05, that we consider. Such good agreementthroughoutV

the fluid-solid coexistence curve forqo U 0.2[ andatP high V 1 for qo W 0.1[ and 0.05 is rather unexpected, as onemight expect the depletion picture to break down in theseregimes. The only significant difference is that the isostruc-turalV

solid-solid transition forqo X 0.1[ at Y 2r Z 0.06[ turns out to

FIG. 13. The small-sphere packing fraction[ 2 of\ a hard-spheremixture with size ratio] â q _ 0.10 and ` ba b q c 0.05 versus that ofthe reservoird 2re for several large-sphere packing fractionsf 1.g Theasterisks denote simulation data while the solid lines denote theresults obtained from the expressi