Invited article Statistical mechanics of …Statistical mechanics of inhomogeneous model colloid–polymer mixtures JOSEPH M. BRADER1, ROBERT EVANS2,3*y and MATTHIAS SCHMIDT4z 1Institute

Invited article

Statistical mechanics of inhomogeneous

model colloid–polymer mixtures

JOSEPH M. BRADER1, ROBERT EVANS2,3*y and MATTHIAS SCHMIDT4z1Institute of Physiology, University of Bern, Buhlplatz 5, 3012 Bern, Switzerland

2H. H. Wills Physics Laboratory, University of Bristol, Royal Fort, Tyndall Avenue, Bristol BS8 1TL, UK3MPI fur Metallforschung, Heisenbergstraße 3, D-70569 Stuttgart, Germany

4Soft Condensed Matter, Debye Institut, Utrecht University, Princetonplein 5, 3584 CC Utrecht,The Netherlands

(Received 22 September 2003; revised version accepted 10 November 2003)

We describe two strategies for tackling the equilibrium statistical mechanics of inhomogeneouscolloid–polymer mixtures treated in terms of the simple Asakura–Oosawa–Vrij (AO) model, inwhich colloid–colloid and colloid–polymer interactions are hard-sphere like, whereas thepolymer–polymer interaction is zero (perfectly interpenetrating polymer spheres). The firststrategy is based upon integrating out the degrees of freedom of the polymer spheres to obtainan effective one-component Hamiltonian for the colloids. This is particularly effective forsmall size ratios q ¼ �p=�c<0:1547, where �p and �c are the diameters of colloid and polymerspheres, respectively, since in this regime three and higher body contributions to the effectiveHamiltonian vanish. In the second strategy we employ a geometry based density functionaltheory (DFT), specifically designed for the AO model but based on Rosenfeld’s fundamentalmeasure DFT for additive mixtures of hard-spheres, that treats colloid and polymer on anequal footing and which accounts for the fluid–fluid phase separation occurring for largervalues of q. Using the DFT we investigate the properties of the ‘free’ interface between colloid-rich (liquid) and colloid-poor (gas) fluid phases and adsorption phenomena at the interfacebetween the AO mixture and a hard-wall, for a wide range of size ratios. In particular, forq ¼ 0:6 to 1.0, we find rich interfacial phenomena, including oscillatory density profiles at thefree interface and novel wetting and layering phase transitions at the hard-wall–colloidgas interface. Where appropriate we compare our DFT results with those from computersimulations and experiment. We outline several very recent extensions of the basic AO modelfor which geometry based DFTs have also been developed. These include a model in which theeffective polymer sphere–polymer sphere interaction is treated as a repulsive step functionrather than ideal and one in which the depletant is a fluid of infinitely thin rods (needles) withorientational degrees of freedom rather than (non-interacting) polymer spheres. We commenton the differences between results obtained from these extensions and those of the basicAO model. Whilst the interfacial properties of the AO model share features in commonwith the those of simple (atomic) fluids, with the polymer reservoir density replacing theinverse temperature, we emphasize that there are important differences which are related tothe many-body character of the effective one-component Hamiltonian.

1. Introduction

It is well established that certain colloidal suspensionsbehave as hard-sphere systems. Pioneering studies by

Pusey, van Megen and co-workers in the 1980s estab-lished that polymethylmethacrylate (PMMA) particles,sterically stabilized by chemically grafted poly-12-hydroxystearic acid (PHSA), dispersed in a solventwhose refractive index matches that of the particles,exhibit phase behaviour which mimics closely that ofpure hard spheres [1]. In particular, coexistence ofcolloidal fluid and crystal phases was found for colloidpacking (volume) fractions, �c, in the range 0:494 <�c < 0:545, values consistent with computer simulations

*Author for correspondence. e-mail: [email protected] Physics Lecture, Liblice Conference, Czech

Republic, June 2002. Invited paper for journal.zOn leave from Institut fur Theoretische Physik II,

Heinrich-Heine-Universitat Dusseldorf, Universitatsstraße 1,D-40225 Dusseldorf, Germany.

MOLECULAR PHYSICS, 10–20 DECEMBER 2003, VOL. 101, NOS. 23–24, 3349–3384

Molecular Physics ISSN 0026–8976 print/ISSN 1362–3028 online # 2003 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals

DOI: 10.1080/0026897032000174263

of hard spheres. For �c < 0:494 there is a single fluidphase in which the diffusive dynamics of the colloidalparticles, as measured by dynamic light scattering, isconsistent with hard-sphere behaviour [1]. Other colloi-dal systems also behave as hard spheres. For example,the equation of state, obtained from X-ray measure-ments of the colloid density profile in a gravitationalfield, for charged polystyrene spheres suspended inwater (with HCl) is in excellent agreement with thehard-sphere equation of state, on both the fluid andcrystal branches [2]. Of course, the effective pairpotential between colloids cannot be perfectly hard;this cannot jump discontinuously to infinite repulsionat some precise diameter. However, as the particlesapproach each other the effective potential rises bymany kBT over a separation of one or two nanometresor so—a distance which is very small compared with theparticle diameter (typically 6–700 nm, in the experimentsmentioned above). Entropically driven freezing isthe only phase transition that can occur in a system of‘hard-sphere’ colloids.Suppose now that non-adsorbing polymer is added to

the suspension of colloids. The interactions are, ingeneral, complicated but one can consider the case offlexible polymer chains under athermal ‘good’-solventconditions, where the interactions between all thespecies are hard-core repulsive, i.e. polymer chains aremutually self-avoiding and are excluded from the hard-sphere colloids. From a statistical mechanics point ofview, only packing constraints are relevant and theproperties of the system, in particular any phasetransitions, are determined by purely entropic effects.It then comes as something of a surprise to learn that thephase behaviour of colloid–polymer mixtures can bevery rich. A series of experiments, performed mainlyby groups in Utrecht, Bristol and Edinburgh in the1980s and early 1990s, confirmed that adding sufficientnon-adsorbing polymers can cause phase separationinto two fluid phases, one rich in colloid and the otherpoor; references to the original papers can be foundin [1, 3–5]. In order for fluid–fluid phase separationto occur there must be a mechanism that generatesan effective attractive interaction between the colloidalparticles. Moreover, this mechanism must be entropicin origin since all ‘bare’ interactions between speciesare hard-core repulsive. An appropriate mechanismwas described as early as 1954 by Asakura and Oosawa[6] and this is now termed the depletion effect ordepletion attraction. Asakura and Oosawa consideredtwo big spherical colloidal particles in a ‘sea’ of rigidmacromolecules. The latter were treated as hardspheres as regards their interaction with the colloidswhereas macromolecule–macromolecule were set to zero,so that the solution of macromolecules was treated as an

ideal gas. Such an assumption should be appropriate fora dilute solution or for theta solvents where the secondvirial coefficient for macromolecule–macromoleculeinteractions vanishes. The centre of a macromoleculesphere is excluded from the surface of a colloid by adistance equal to �p=2, so there is a depletion zone (orexcluded volume region) in which there are no macro-molecule centres of mass. If two colloids approach eachother, so that the depletion zones overlap, then there isan increase in free volume for the macromoleculespheres, i.e. an increase in their translational entropy,leading to an effective attractive (depletion) interactionbetween the two colloids. One can also view theattraction as arising from an unbalanced osmoticpressure pushing the colloids together as the macro-molecules are expelled from the gap between the twocolloids. By calculating the volume of the overlapbetween the two depletion zones, Asakura andOosawa [6] obtained an explicit expression for thedepletion force between two colloids whose centres areseparated by a distance R, immersed in an ideal solutionof macromolecules of fugacity zp. The depletion poten-tial �AOðR; zpÞ vanishes when the depletion zones nolonger overlap; the attractive potential has a finiterange equal to �p. Increasing the concentration of themacromolecule sea increases zp and the depth of thepotential well, whereas increasing �p extends the rangeand, hence, the integrated strength of the attraction. Itis clear that depletion attraction is a possible mechanismfor driving phase separation. In a second paper, Asakuraand Oosawa [7] focused on the depletion potentialand discussed the possible aggregation of suspendedcolloidal particles. By assuming the polymers can betreated as spherical macromolecules, internal conforma-tional degrees of freedom are ignored. Independently,in 1976 Vrij [8] proposed the same depletion mechanism,deduced the depletion potential between a pair ofcolloidal spheres and the associated second virial coeffi-cient and went an important step further by writingdown an explicit model Hamiltonian for the binarymixture of hard-sphere colloids and ideal polymer(macromolecule) spheres; this is the Asakura–Oosawa–Vrij or AO model—see section 2. Vrij’s paper providesthe basis for a full statistical mechanical treatmentfor an (idealized) model of a colloid–polymer mixture.Vrij pointed out that his interparticle potentials definea ‘non-additive’ hard-sphere model in the the theory ofliquids. We return to this aspect in section 2. He alsosuggested that the colloid–polymer hard-sphere dia-meter should be ð�c þ �pÞ=2 with �p=2 � Rg, the radiusof gyration of the polymer. The next importantdevelopment in the theory was made by Gast et al. [9]who calculated the phase behaviour of the AO modelusing thermodynamic perturbation theories for an

3350 J. M. Brader et al.

approximation that assumes an effective colloid–colloidpair potential, �eff ðR; zpÞ, consisting of the bare hard-sphere potential plus the attractive depletion potential�AOðR; zpÞ. Gast et al. predicted that the effect of addingpolymer depends sensitively on the polymer to colloidsize ratio q ¼ �p=�c. For q9 0:3, adding polymer merelyaugments, in �c, the colloidal fluid–crystal coexistenceregion above the range found for pure hard spheres,whereas for larger values of q a stable colloidal liquidphase can exist. In the (�c, zp) plane there is a triplepoint where gas, liquid and crystalline colloidal phasescoexist [9] and for q00:6 the phase diagrams resemblethose of simple atomic fluids, with zp playing the samerole as inverse temperature, 1/T. Thus, entropicallydriven depletion attraction can become sufficientlystrong and sufficiently long-ranged to generate a vander Waals-like scenario for the gas-liquid transition.As noted by Gast et al., one significant drawback to

their approach is that for size ratios q > 0:1547, theeffective one-component Hamiltonian should containthree- and higher-body effective interactions betweenthe colloidal particles—see section 2.1; these interactionsare tedious to evaluate and cumbersome to incorporateinto perturbation theories or computer simulationsof the liquid and crystalline phases and are thereforeusually ignored. However, the effects of many-bodyeffective interactions are not necessarily small and weshall argue below that these can play a crucial rolefor both bulk and interfacial phase behaviour.An alternative approach to determining the phase

behaviour of the AO model, which does not rely uponmapping to an effective one-component Hamiltonianand which does capture some of the effects of many-body terms, is the so-called free-volume theory ofLekkerkerker et al. [4]. Some details of this approachwill be given in section 2.2 but the approximationconsists of replacing the exact (average) free volumeavailable to the ideal polymers by the correspondingquantity evaluated in the low-density limit, zp ! 0, forwhich an analytical approximation is available. Havingan explicit expression for the free volume fraction,�ð�cÞ, allows one to transform readily from the reser-voir representation, where zp or, equivalently, �rp, thepacking fraction of ideal polymer is specified, to thesystem representation where �p, the packing fraction inthe actual system is specified: �p ¼ ��rp. Experimentalphase diagrams are plotted in the (�c, �p) plane. Thefree-volume theory predicts a stable colloidal gas–liquidtransition for q00:32 and the phase diagrams exhibitsimilar trends to those from the approach of Gast et al.,although there are some significant differences—see [10]for detailed comparisons and [11] for a discussionof the connections and distinctions between the twoapproaches. Subsequently Meijer and Frenkel [3]

performed pioneering Monte Carlo simulations fortwo separate models of colloids dispersed in a dilutepolymer solution. The first was a lattice–polymer modelin which polymers are represented by ideal (non-self-avoiding) chains confined to a cubic lattice. However,lattice sites occupied by colloidal hard-spheres areinaccessible to polymer. The second was the AO model,but with the polymer spheres restricted to a cubic lattice.Phase diagrams, calculated in the (�c, zp) plane, for thetwo models were rather close to each other and for thesize ratios q ¼ 0:5 and 0.7 considered the simulationresults corresponding to the (lattice–polymer) AO modelwere in fair agreement with those from free-volumetheory.

To summarize, by 1994 various theoretical andsimulation studies of idealized models had shown thatfor small polymer–colloid size ratios the phase diagramis of the fluid–solid type, whereas for larger size ratios,where the depletion potential becomes longer rangedand many-body interactions become important, stablecolloidal gas–liquid phase separation occurs. This trendin the phase behaviour was consistent with earlierexperimental observations. Subsequent experiments[5, 12] provided convincing support for the scenariosuggested by theory. In particular, three-phase coex-istence was reported at certain size ratios for chargedcolloidal polystyrene particles mixed with hydroxyethylcellulose [12] and for sterically stabilized PMMAparticles mixed with random coil polystyrene (PS) in acis-decalin solvent [5]. An admirable account of thework of the Edinburgh group on the latter system,which the author describes as a ‘model colloid–polymermixture’, is given in the recent topical review by Poon[13]. We direct readers wishing to learn more aboutphase behaviour, equilibrium structure, phase transi-tion kinetics, gels and glasses in real mixtures to thisilluminating article. Another well-characterized system,much studied by the Utrecht group, consists of stericallystabilized silica particles and polydimethylsiloxane(PDMS) in cyclohexane. Silica has the advantage overPMMA that small particles are available (diametersas small as 20 nm) which is important in studies ofinterfaces—see section 3.

The thrust of the present article is quite different fromthat of Poon. Having established that the simple AOmodel captures the main features of the experimentalbulk phase diagrams, here we enquire what are theproperties of inhomogeneous colloid–polymer mixtures,as described by the same AO model. Inhomogeneoussituations, where the average density profiles of bothspecies are spatially varying, occur in the context ofadsorption at a solid substrate, at the planar interfacebetween two coexisting (colloid-rich and colloid-poor)phases, for mixtures confined in capillaries or porous

Statistical mechanics of inhomogeneous model colloid–polymer mixtures 3351

media or, indeed, in a colloidal crystal. Nucleationphenomena also require a description of the inhomoge-neous fluid. Our strategy is not to attempt any realisticmodelling of a colloid–polymer mixture which is, ofcourse, a highly complex system involving multiplelength and time scales, rather we seek to understand thebasic adsorption and interfacial properties that arise inthe context of the simple model fluids. Moreover, werestrict consideration to equilibrium aspects.Whether thepredictions from the model are relevant to real mixturesis a separate issue. In defence of our strategy we remark(as humble theorists!) that without detailed theoreticaland simulation studies of the Lennard-Jones and othermodel fluids, we would have little fundamental under-standing of the properties of simple, atomic liquids atinterfaces or under confinement—the experimentalistsare certainly catching up but experiments on fluidinterfaces are notoriously difficult. One might go furtherand argue that studies of the Ising or lattice gas model,with the substrate treated as an external field, led to awealth of predictions for surface phase transitions mostof which were found in adsorption experiments. Let usbe clear, however. We are not suggesting that the AOmodel has the same significance in statistical physics asthe Ising model! Nevertheless, the AO model has muchappeal to the theorist. There is only one parameter, thesize ratio q, and by varying q different types of bulkphase behaviour emerge. Moreover, we find that thesame model predicts striking interfacial phenomena,some of which are very different from those found forsimple fluids and which reflect the special character ofthe effective interactions in the AO model.Those readers who are interested in recent develop-

ments in the theory of more realistic models of colloid–polymer mixtures are referred to the topical review byFuchs and Schweizer [14] who describe liquid-stateintegral equation theories for tackling structural corre-lations starting at the polymer segment level. Thepolymers are treated as connected chains of segmentswhich experience excluded-volume forces among them-selves and with the hard-sphere colloids. Althoughthe integral equation approaches provide much insightinto the nature of correlations for a wide range of sizeratios (including q � 1) and for a variety of polymerconcentrations, they appear to be restricted to homo-geneous (bulk) fluids and it is difficult to envisageextensions to inhomogeneous systems or to crystallinephases, i.e. to the full phase diagram. Very recently,Bolhuis and co-workers [15] have performed MonteCarlo simulations of the bulk phase diagram of a modelin which the colloids are treated as hard spheres andthe polymers as self-avoiding walks that are mapped toan effective pair potential. We shall make referenceto their work in section 5.2, but for more details of

this powerful ‘polymers as soft colloids’ computationalapproach readers should consult the original paper andreferences therein.

The present article is organized as follows. Insection 2 we describe the AO model Hamiltonianand two different strategies for tackling its statisticalmechanics. The first is to integrate out the degrees offreedom of the polymer spheres, thereby obtaining aneffective one-component Hamiltonian for the colloidalhard spheres. As mentioned above, this approach isparticularly effective for small size ratios, q < 0:1547,where the one-component Hamiltonian consists ofzero, one and two-body contributions only; there areno higher-body effective interactions. The secondstrategy is to tackle the binary AO mixture directly;the two components are treated on equal footing bymeans of a geometry-based density functional theorythat is specifically designed for the inhomogeneousAO mixture [16, 17]. The procedure for constructingthe DFT is based on the fundamental measure theorydeveloped by Rosenfeld [18] for additive hard-spheremixtures. For uniform (bulk) fluids the free energyobtained from the DFT is identical to that given bythe free-volume theory of Lekkerkerker et al. [4]alluded to above. Section 3 describes an application ofthe DFT to the planar interface between demixed fluidphases, one rich in colloid, the other poor. We findthat for coexisting states well away from the criticalpoint, at high polymer fugacity, both the colloidand polymer density profiles exhibit oscillations onthe colloid-rich (liquid) side of the interface. We alsodiscuss the behaviour of the surface tension comparingwith experimental results. In section 4 we consideradsorption of the AO mixture at a planar hard wall.When a colloidal particle is sufficiently close to a hardwall, such that the two depletion zones overlap, thereis an effective attractive potential between the twowhich is similar to the potential, �AO, between twocolloidal particles, i.e. expulsion of polymer induces anattractive wall–colloid depletion potential �wall

AO whoserange is equal to �p. For small size ratios q, very largecontact densities are found for the colloid profile �cðzÞ,reflecting the form of �wall

AO ðz; zpÞ. For larger q fluid–fluid phase separation can occur and we find novelentropic wetting and layering transitions using theDFT—see section 4.2. Results of recent computersimulations of the AO mixture, with q ¼ 1, adsorbedat a hard wall are described in section 4.3; these alsopredict wetting and layering transitions [19]. Section 5describes a recent extension of the DFT approach thatincorporates, albeit in a simple way, polymer–polymerinteractions [20]. In the ‘soft colloid’ picture [21] ofpolymers, segment–segment repulsion, averaged overconformations of the chains, leads to an effective


interaction, �pp, between the polymer centres of massthat is repulsive, soft and penetrable. By making thecrude approximation that �ppðrÞ is a step-function pairpotential of height �, it is possible to construct ageometry-based DFT for a model that has as its limit-ing cases: (i) the AO model (� ¼ 0, non-interactingpolymer) and (ii) the binary hard-sphere mixture(� ! 1). We present results for fluid–fluid phaseseparation, comparing with those for the AO modeland with computer simulations. In section 6 we sum-marize briefly some other recent applications andextensions of the DFT for the AO model. Several ofthese are concerned with bulk properties only but wealso describe recent work on mixtures of hard-spherecolloids and hard needles, modelling experimentallyrealizable stiff colloidal rods, where interfacial proper-ties were considered [22–24]. This model system sharesmany features in common with the AO model—the needles act as the depletant. It has the additionalfeature of orientational degrees of freedom for theneedles which can introduce orientational ordering atan interface, even though the bulk fluid is isotropic(the infinitely thin rods cannot interact). We concludein section 7 with some remarks and an outlook onfuture work.What follows is not intended to be a review of what

is a large and rapidly developing literature on theexperimental, theoretical and simulation aspects ofcolloid–polymer mixtures; we make no attempt to givea comprehensive overview or an exhaustive list ofreferences. For example, we concentrate on theso-called ‘colloid limit’, q91, rather than the equallyimportant ‘protein or nanoparticle limit’, q � 1, wherethe physics is, arguably, less understood. The choiceof material reflects the personal viewpoints of theauthors on the subject area and the presentation isbased, in part, on the Molecular Physics Lecture givenby R. Evans at the Liblice Conference in June 2002.Although much of the theoretical background hasbeen published elsewhere, almost all the resultspresented in sections 3 and 4 appear for the firsttime: they are from the unpublished thesis of Brader[25]. Section 5.2 also contains new results. A briefsummary of some of the earlier work can be found ina conference article [26].

2. The Asakura–Oosawa–Vrij model

We consider a suspension of sterically stabilizedcolloidal particles immersed together with non-adsorb-ing polymers in an organic solvent. The interactionbetween two sterically stabilized colloidal particles in anorganic solvent is close to that between hard spheres,whereas dilute solutions of polymers in a theta-solventcan be represented by non-interacting or ideal polymers.

A simple idealized model for such a colloid–polymermixture is the so-called Asakura–Oosawa–Vrij (AO)model [6–8]. This is an extreme non-additive binaryhard-sphere model in which the colloids are treated ashard spheres with diameter �c and the interpenetrable,non-interacting polymer coils are treated as pointparticles but which are excluded from the colloids to acentre-of-mass distance of ð�c þ �pÞ=2. The diameter ofthe polymer sphere is �p ¼ 2Rg with Rg the radius ofgyration of the polymer. The pairwise potentials in thissimple model are given by

�ccðRijÞ ¼1, for Rij < �c,

0, otherwise ,

�

�cpðjRi � rjjÞ ¼1, for jRi � rjj <

12ð�c þ �pÞ,

0, otherwise,

(

�ppðrijÞ ¼ 0, ð1Þ

where R and r denote colloid and polymer centre-of-mass coordinates, respectively, with Rij ¼ jRi � Rjj andrij ¼ jri � rjj. Note that a general non-additive binaryhard-sphere mixture is described by the ‘diameters’ �11,�22 and �12, where subscripts 1 and 2 denote species 1and 2. The cross-term is given by �12 ¼ ð�11 þ �22Þ�ð1þ�Þ=2, where the non-additivity parameter � is zerofor additive mixtures. The AO model corresponds to�22 ¼ 0 and � ¼ q, the fixed size ratio �p=�c. TheHamiltonian of the AO model consists of (trivial)kinetic contributions and a sum of interaction terms:H ¼ Hccþ Hcp þHpp, where

Hcc ¼XNc

i<j

�ccðRijÞ,

Hcp ¼XNc

i

XNp

j

�cpðjRi � rjjÞ,

Hpp ¼XNp

i<j

�ppðRijÞ ¼ 0 ð2Þ

and we consider Nc colloids and Np ideal polymers in avolume V at temperature T. The solvent is regarded asan inert continuum. See figure 1 (a) for an illustrationof the model.

There are several ways of tackling the statisticalmechanics of the AOmodel. Brute force, direct computersimulations of this model mixture are not straightfor-ward since problems of slow equilibration and non-ergodicity can arise for large size asymmetries, wherehuge numbers of polymers are required per colloidparticle. Recently, however, simulation studies have beencarried out for several values of the size ratio q ¼ �p=�c


and we shall report on these in section 4.3. Here wedescribe two alternative approaches which have beensuccessfully employed. The first is most appropriatefor mixtures where the size ratio q � 1; it dependsupon integrating out the degrees of freedom of theideal polymer to obtain an effective one-componentHamiltonian for the colloids. The second approachemploys a density-functional theory designed specificallyfor the AO mixture; this treats both species on an equalfooting.

2.1. The effective one-component HamiltonianFor a general binary mixture it is possible to construct

an effective one-component Hamiltonian for one of thecomponents—usually the larger species—by integratingout the degrees of freedom of the other—usually thesmaller species [27]. As usual it is convenient to workin the semi-grand ensemble where (Nc, zp,V ,T) arefixed variables, i.e. the fugacity zp ¼ L� 3

p exp ð��pÞ ofa reservoir of polymer is fixed. �p is the chemicalpotential, � ¼ 1=kBT and Lp is the thermal de Brogliewavelength of the polymer. In order to treat inhomo-geneous fluids it is also convenient to allow the colloidsand polymer to couple independently to two externalfields. Thus one adds to the Hamiltonian H contribu-tions

Vextc ¼

XNc

i¼1

Vextc ðRiÞ, Vext

p ¼XNp

i¼1

Vextp ðriÞ, ð3Þ

which can produce inhomogeneous density profiles. It isstraightforward to show that the semi-grand free energyFðNc,V , zpÞ is given by

exp ð��FÞ ¼1

Nc!L3Ncc

ZdRNc exp ��ðHcc þ Oþ Vext

c Þ� �

,

ð4Þ

where Lc is the thermal de Broglie wavelength of thecolloid and

exp ð��OÞ ¼X1Np¼0

zNpp

Np!

ZdrNp exp ��ðHcp þ Vext

p Þ

� �:

ð5Þ

O is simply the grand potential of the (ideal) polymer inthe presence of the external field arising from (a) a fixedconfiguration fRNcg of Nc colloids and (b) any appliedfield Vext

p . Provided one can determine O explicitly, orat least a good approximation to O, the binarymixture problem is reduced to a simpler one-componentproblem: equation (4) describes a system of colloidsinteracting through an effective Hamiltonian: Heff ¼

Hcc þ Oþ Vextc .

In a general mixture O consists of zero, one, two, . . . ,many-body contributions [28] and the resulting Heff isunwieldy. However, for the particular case of the AOmodel the contributions simplify [10, 11] because thepolymer is ideal, i.e. �pp ¼ 0. Moreover, for a homo-geneous fluid, with no external potentials, the seriestruncates after the two-body term, provided thatq < 2=31=2 � 1 ¼ 0:1547 [9, 10]. For such high asymme-try there is no triple overlap of excluded volume regions,even when three colloids are in simultaneous contact.Thus, for q < 0:1547, geometrical considerations ensurethat there is an exact mapping from the bulk binarymixture to an effective Hamiltonian that contains onlytwo-body interactions plus structure (configuration)independent contributions. The effective pair potentialis given by [9, 10]

�eff ðR; zpÞ ¼ �ccðRÞ þ �AOðR; zpÞ, ð6Þ

where �AOðR; zpÞ is the well-known AO pair (depletion)potential between two hard-sphere colloids in a sea ofideal polymer [6, 7], see figure 1 (b) for an illustration.�AOðR; zpÞ is attractive and has a finite range equal to �p.Its strength is proportional to zp and it can be expressedas a polynomial in s ¼ R=�c:

��AOðR; zpÞ

¼

�p6�3pzpð1þ q�1Þ

3 1�3s

2ð1þ qÞþ

s3

2ð1þ qÞ3

� �,

1 < s < 1þ q,

0, s > 1þ q:

8>>><>>>:

ð7Þ

Figure 1. (a) Sketch of the Asakura–Oosawa–Vrij model ofhard spheres (grey circles) with diameter �c and idealpolymers (white circles) with diameter �p. (b) Depletionzones (dashed lines) that are inaccessible to polymercentres. Overlapping depletion zones (light grey shapes)are indicated in two cases, that between two colloids andthat between a colloid and a hard wall.


Perhaps more remarkably, Brader et al. [11] showedthat for the AO mixture in contact with a hard walldefined by the external potentials

Vextc ðzÞ ¼

1, z < �c=2,

0, z � �c=2,

(

Vextp ðzÞ ¼

1, z < �p=2,

0 z � �p=2,

( ð8Þ

where z is the centre of the particle measured normalto the wall, there is also an exact mapping of thebinary mixture to a simple effective Hamiltonian.Using geometrical considerations (see figure 2 of [11])they showed that for q < 0:25 all many-body terms On,with n� 2, are unaltered from their form in the bulk(homogeneous) fluid by the presence of the hard wall.For larger values of q the two-body potential becomesa complicated function of R1 and R2 and is not simplya function of the separation jR1 � R2j. Specifically, forq < 0:1547, the effective Hamiltonian for the inhomo-geneous fluid at the hard wall reduces to

Heff ¼ Hcc þXNc

i¼1

Vextc ðziÞ þ

XNc

i¼1

�wallAO ðzi; zpÞ

þ Obulk0 þ Obulk

1 þXi<j

�AOðRij ; zpÞ: ð9Þ

The additional one-body term �wallAO represents the

attractive AO depletion potential between the colloidand the planar hard wall. This potential has the samerange, �p, as �AO and it has similar form [11]. In theabsence of the wall equation (9) reduces to the effectiveHamiltonian of the homogeneous system. The structure-independent terms are given by

��Obulk0 ¼ zpV ,

��Obulk1 ¼ �zpNcp ð�c þ �pÞ

3=6 ¼ �zp�cð1þ qÞ3V , ð10Þ

where �c ¼ ðp=6Þ�3cNc=V is the colloid packing fraction.

Having such a simple effective Hamiltonian for thehomogeneous fluid means that it is straightforwardto perform computer simulations or to implementstandard one-component integral equation closures.Recall that integral equation approaches for highly asym-metric mixtures are especially problematical whereasPercus–Yevick (PY) and similar closures provide areliable description of the pair correlation functionfor a one-component fluid whose pair potential consistsof a hard core plus an attractive tail. Of course, the usualissues of thermodynamic inconsistency remain.Dijkstra et al. [10] carried out extensive investigations

of the properties of the bulk mixture with q ¼ 0:1 using

the effective Hamiltonian. The equilibrium structure,i.e. the colloid–colloid radial distribution function gccðRÞand the structure factor SccðkÞ, is determined solely by�eff ðR; zpÞ, equation (6), for this size ratio and MonteCarlo and PY results at various colloid packing fractionsare presented in [10]. Note that since the terms Obulk

0

and Obulk1 depend linearly on Nc or V they have no effect

on bulk phase equilibria and the latter is, once again,determined solely by the pair potential �eff ðR; zpÞ [10, 28].(These terms do contribute to the total pressure andtotal compressibility of the mixture, however [29].) Thecomplete equilibrium phase diagram of the mixture wasdetermined by simulation of the effective one-componentsystem for q ¼ 0:1. It is extremely rich, displaying a verybroad, in �c, fluid–solid coexistence curve which liesat lower polymer densities than a (metastable) fluid–fluid coexistence region. There is also an isostructural(fcc) solid–solid transition which is very slightly meta-stable with respect to the fluid–solid transition [10, 11].Note that as each term in the effective Hamiltonian isproportional to zp (� polymer density in the reservoir,�rp, for ideal polymer) this variable plays the same role asinverse temperature in simple atomic fluids. Thus, phasediagrams plotted in the (�c, zp) plane often displayfeatures similar to those plotted in the (density, 1=T)plane for simple fluids. The simulation results for thebulk system provide a valuable benchmark against whichapproximate theories, e.g. integral equation, densityfunctional or perturbation theory approaches, can betested [10].

Similar remarks apply to the Monte Carlo simula-tion results obtained using the effective Hamiltonian(9) for the mixture at a planar hard wall. Brader et al.[11] considered the AO mixture, with q ¼ 0:1, for twofixed values of the colloid packing fraction, �c ¼ 0:3and 0.4, and increasing amounts of polymer. Addingpolymer (increasing zp) leads to increased depletionattraction for the colloids at the hard wall whichleads, in turn, to pronounced effects on the densityprofile near the wall. We shall comment further onthis phenomenon in section 4.1 where we compareresults from density functional theory with those ofthe simulations.

We conclude this subsection by making some remarksabout the form of the effective Hamiltonian to be usedfor other types of inhomogeneity. If the wall–fluidpotentials are soft or exhibit attractive portions thenthe integrating out of the polymer can lead to morecomplex contributions, even for small size ratios q. Onthe other hand, for spontaneously generated inhomo-geneities, where the density profiles of colloid and ofpolymer are spatially varying in vanishing externalfields, the appropriate effective Hamiltonian is that ofthe bulk, homogeneous system [11]. Relevant examples


are the planar interface between demixed fluid phases,colloidal crystals where the densities vary periodically,and the fluid–crystal interface. Although the distributionof polymer in a colloidal crystal or in the region ofany interface is very different from that in a bulk fluidthis does not give rise to different effective interactions,in the sense that the effective Hamiltonian needs to bemodified. Since we work in the semi-grand ensemble,with a reservoir of polymer, it is solely the fugacity zpthat determines all the effective interactions in thesystem. As zp is constant throughout the inhomogeneoussystem then so too are the effective interactions betweencolloids; these do not depend on the local polymerdistribution. Also we note that although the polymerdegrees of freedom have been integrated out it is stillpossible, at least in principle, to recover informationabout the polymer distribution from properties thatare determined by the effective Hamiltonian. Forexample, the polymer density �pðrÞ can be obtained byfunctional differentiation of the free energy F withrespect to Vext

p ðrÞ and expressed in terms of n-bodycorrelation functions of the colloids, which can bedetermined using the effective Hamiltonian [11]. Whenq < 0:1547 this procedure yields an exact and tractableformula for the free-volume fraction �ð�c; zpÞ � �p=�

rp

(the ratio of polymer density in the mixture to that inthe reservoir) of a bulk fluid mixture [11].Finally, we should remark that the one-dimensional

version of the AO model, in which the colloids aremodelled by hard rods and the polymer by idealparticles that are excluded from the colloids by a certaindistance, can be solved exactly by mapping the binarymixture to an effective one-component Hamiltonian[30]. The construction of the effective Hamiltonianproceeds as in three dimensions and the effective pairpotential �eff ðX ; zpÞ ¼ �ccðXÞ þ �1D

AOðX ; zpÞ, where thedepletion potential �1D

AO is now the difference inaccessible length for the polymers when the colloidsare separated by a distance jX j and when their separa-tion is infinite. This potential is linear in jX j, pro-portional to zp and vanishes for separations beyondthe length of the polymer. Since the mapping reducesthe binary mixture problem to that of a one-componentfluid in one dimension, in which the particles interact viaa nearest-neighbour potential, the statistical mechanicscan be solved using the standard Laplace transformtechniques. Results for the (osmotic) equation of stateand free-volume fraction �ð�c; zpÞ are given in [30],where they are compared with the corresponding resultsfrom the approximate free-volume theory.

2.2. Density functional approachWe have recently developed a density functional

theory (DFT) for the binary AO model using the

techniques of fundamental measure theory (FMT). Ournew functional [16, 17] treats arbitrary size ratios q andis thus able to incorporate the effects of many bodyinteractions which arise in the effective one-componentdescription at larger values of q, and which represent animportant feature of the model. Unlike DFT treatmentsof interfacial phenomena in simple fluids, where theattractive portion of the fluid–fluid pair potential istreated separately from the repulsive part in a perturba-tive fashion that is equivalent to a mean-field treat-ment (correlations are ignored) [31], here the effectiveattractive interactions emerge naturally from the DFTand are treated non-perturbatively. Of course, thepresent DFT is still mean-field like in that bulk criticalfluctuations or, indeed, interfacial fluctuations are notincorporated. The excess, over ideal, Helmholtz freeenergy functional is given by a spatial integral over areduced free energy density F which is a function ofa set of species-dependent weighted densities

�FAOex ½�cðrÞ, �pðrÞ� ¼

ZdrFðfncg, fn

pgÞ, ð11Þ

where the function F ¼ F1 þ F2 þ F3 consists of threeterms given by

F1 ¼ nc0 � ln ð1� nc3Þ þnp3

1� nc3

� �� np0 ln ð1� nc3Þ,

F2 ¼ ðnc1nc2 � nc1 n

c1Þ

1

1� nc3þ

np3ð1� nc3Þ

2

" #

þnp1n

c2 � n

p1 n

c2 þ nc1n

p2 � nc1 n

p2

1� nc3,

F3 ¼ðnc2Þ

3� 3nc2ðn

c2 n

c2Þ

24p1

ð1� nc3Þ2þ

2np3ð1� nc3Þ

3

" #

þðnc2Þ

2np2 � np2ðnc2 n

c2Þ � 2nc2n

c2 n

p2

8pð1� nc3Þ2

: ð12Þ

The species-dependent weighted densities are given byconvolutions of the density profiles �iðrÞ, i ¼ c, p, withweight functions wi

ðrÞ

niðrÞ ¼

Zdr0�iðr

0Þwiðr� r0Þ: ð13Þ

The four scalar and two vector weight functions arefunctions characteristic of the geometry of the hardparticles

wi3ðrÞ ¼ �ðRi � rÞ,

wi2ðrÞ ¼ �ðr� RiÞ,

wi1ðrÞ ¼

�ðr� RiÞ

4pRi,


wi0ðrÞ ¼

�ðr� RiÞ

4pR2i

,

wiv2ðrÞ ¼

r

r�ðr� RiÞ,

wiv1ðrÞ ¼

r

r

�ðr� RiÞ

4pRi, ð14Þ

where Ri denotes the radius of species i so that Rc ¼ �c=2and Rp ¼ �p=2. �ðrÞ is the Heaviside step functionand �ðrÞ is the Dirac distribution.The procedure for constructing the DFT is based on

the successful FMT developed by Rosenfeld [18] foradditive hard-sphere mixtures. In order to obtain thereduced free energy density F appropriate to the AOmodel, Schmidt et al. [16, 17] considered the zero-dimensional limit which corresponds to a cavity that canhold at most one hard-sphere colloid but can hold anarbitrary number of ideal polymers if no colloid ispresent. Full details of the derivation of the DFT and itsapplications to the determination of bulk fluid thermo-dynamics and structure are given in [17]. A summary isprovided in section 5.1 of the present article where weconsider an extension of the AO model. Here it sufficesto make some remarks about the status of the theory.Note that in the original papers [16, 17] a tensor weightfunction was included; this contribution is omitted inthe above formulation and in the calculations to bedescribed later. Inclusion of the tensor (see equation (33)below), while essential for calculating crystalline pro-perties, makes negligible difference for inhomogeneousfluid states and so would only serve to complicatematters without altering the basic phenomena. It isimportant to recognize that the functional can also beregarded as a linearization, in the polymer density �pðrÞ,of the original Rosenfeld hard-sphere functional.For a homogeneous fluid mixture the functional yields

a reduced excess bulk free energy density

�FAOex =V ¼ �f AO

ex ð�c, �p; qÞ ¼ �fHSex ð�cÞ � �p ln �ð�cÞ,

ð15Þ

where

�ð�cÞ ¼ ð1� �cÞ exp ð�A � B2 � C3Þ, ð16Þ

with ¼ �c=ð1� �cÞ, A¼ 3qþ 3q2 þ q3, B¼ 9q2=2þ 3q3

and C ¼ 3q3. fHSex is the excess free energy density of pure

hard spheres in the scaled particle (PY compressibility)approximation; an explicit expression is given later inequation (36). This result can be shown to be identicalto the free-volume theory of Lekkerkerker et al. [4, 10],where �ð�cÞ is interpreted as the ratio of the free volumeaccessible to a single test polymer sphere and the systemvolume. It is not immediately obvious that the DFT

approach should be equivalent to free-volume theory.The starting points for the two theories are quite dif-ferent and it is usually the semi-grand free energy that isconsidered in free-volume theory; connections betweenthe two approaches are discussed in [17]. Free-volumetheory treats the semi-grand free energy as the sum of ahard-sphere (colloid) part plus a contribution from anideal gas of polymers in the free volume left by thecolloids, which is treated as an expansion in the fugacityzp truncated at the term linear in zp [4, 10]. This linearityin zp, or equivalently in �p, is built into the DFT.

The bulk pair direct correlation functions cð2Þij ,

obtained by taking two functional derivatives ofFAO

ex ½�c, �p�, are given analytically [17]. The Ornstein–Zernike relations then provide the partial structurefactors SijðkÞ. Linearization in the polymer density hasthe consequence that cð2Þpp ¼ 0, as in the PY approxi-mation for this AO model. However, the other directcorrelation functions cð2Þcc and cð2Þcp are not the same asthose from PY approximation, even though c

ð2Þij ðrÞ

vanishes for r > Ri þ Rj in both the DFT and the PYtreatments. In an exact treatment we would expectcontributions to c

ð2Þij ðrÞ beyond Ri þ Rj. An important

advantage of the DFT over integral equation theories isthat the partial structure factors and radial distributionfunctions gijðrÞ obtained from the Ornstein–Zernikeequations yield a spinodal consistent with that from thebulk free energy (15), i.e. the free-energy and structuralroutes to the fluid–fluid spinodal are consistent. (Notethat the spinodal can be obtained analytically fromthe canonical free energy (15)—see [17].) Such aproperty is especially advantageous when consideringinterfacial properties at or near two-phase coexistence.Schmidt et al. [17] also investigate the asymptotic decay,r ! 1, of gijðrÞ in different regions of the bulk phasediagram. Since the partial structure factors SijðkÞ aregiven analytically it is straightforward to determine thepoles of these functions in the complex plane and hencelocate the so-called Fisher–Widom line [32–35] wherethe ultimate decay of rgijðrÞ crosses over from oscillatoryto purely monotonic, exponential decay. Examples ofthe cross-over lines and of (mean-field) critical pointbehaviour of SijðkÞ for various size ratios q are givenin [17].

It is important to emphasize that, within the frame-work of DFT, the Ornstein–Zernike route is not theonly one to the pair correlation functions. The alter-native is the test particle route, whereby one fixes aparticle of species i at the origin and determines (bysolving the appropriate Euler–Lagrange equations,obtained by minimizing the functional) the inhomoge-neous one-body density profile �jðrÞ arising from theexternal potential exerted by the fixed particle i; thengijðrÞ ¼ �jðrÞ=�jð1Þ. All that is required to solve the


relevant equations are the one-body direct correlationfunctions c

ð1Þi ðrÞ, which involve only a first derivative

of FAOex . One can show that in the limit �c ! 0, the

test particle route for the DFT yields the exact resultgccðrÞ exp ð��eff ðr; zpÞÞ, where �eff is the effectivedepletion potential defined in equation (6) [16, 17]. Inother words, the geometrically based DFT describescorrectly the depletion effect between two hard-spherecolloids when implemented within the test-particleprocedure. There are other good reasons to believethat the test-particle route to gijðrÞ should be moreaccurate than the Ornstein–Zernike route [17].

3. The fluid–fluid interface

Phase separation in the AO model is a well-studiedproblem [4, 8–10]. As remarked earlier, for small sizeratios q, fluid–fluid phase separation is pre-empted bya fluid–solid transition so the former is, at best, meta-stable. For larger values of q, however, there is stable,entropically driven, fluid–fluid phase separation. Free-volume theory for the AO model predicts stable liquid(colloid-rich)–gas (colloid-poor) phase coexistence forq > 0:32. It follows that our present DFT predicts thesame behaviour. Figure 2 shows the bulk phase diagramobtained from the present theory for a size ratio q ¼ 0:6for which there is a stable fluid–fluid demixing transi-tion with a critical point at �rp, crit � 0:495. The polymerreservoir packing fraction is defined as �rp ¼ ðp=6Þ�3

p�rp,

with �rp ¼ zp for ideal polymer. Note that the tielines connecting coexisting states are horizontal in this(reservoir) representation. It should be emphasizedthat the fluid–fluid and fluid–solid phase boundariespresented here are those of the original free-volumetheory [4, 10]y. While the fluid–fluid phase boundaryshown is precisely that given by the functional, the truesolid–fluid boundary from DFT would require a fullminimization of the functional for a solid-like distribu-tion. This is outside the scope of the present study.Also shown in figure 2 is the Fisher–Widom (FW) linewhich divides the phase diagram into regions where theasymptotic decay of the three bulk pairwise correla-tion functions, rgijðrÞ, is either monotonic or exponen-tially damped oscillatory. The FW line has importantconsequences both for our study of the free fluid–fluidinterface and for adsorption at a wall as the asymptoticbehaviour predicted for the bulk pair correlationsapplies also to the one-body density profiles [33, 34].

We turn now to the free interface between demixedfluid phases. The density profiles for colloid andpolymer are obtained by minimizing the grand potentialfunctional

O½�cðrÞ, �pðrÞ� ¼ F id½�cðrÞ, �pðrÞ� þ FAOex ½�cðrÞ, �pðrÞ�

þXi¼c, p

ZdrðVext

i ðrÞ � �iÞ�iðrÞ , ð17Þ

where FAOex is the excess Helmholtz free energy func-

tional of the AO mixture given in equation (11), F id

denotes the functional for the ideal mixture, �i is thechemical potential (fixed by the reservoir) and Vext

i ðrÞ isthe external potential coupling to species i with i ¼ c, p.In the case of the free interface Vext

i � 0. The factthat the functional is linear in the polymer densitymakes solution of the AO Euler–Lagrange equationsconsiderably simpler than for the more familiar binaryhard-sphere Rosenfeld functional [18]. In the lattercase, the two minimization conditions �O=��1 ¼ 0 and�O=��2 ¼ 0 give rise to two coupled equations for �1and �2 which must be solved self-consistently. TheAO functional can be minimized explicitly with respectto the polymer density and the level of computationalcomplexity reduced to that of minimizing a functionalwith respect to a single density field [17].

The colloid density profiles and corresponding surfacetensions are shown in figure 3. The surface tension isplotted here in mNm�1 to facilitate comparison withthe experimental results of [36, 37] which we shall

yNote that in calculations, e.g. [10], based on free volumetheory the Carnahan–Starling approximation is often used forfHSex whereas in DFT the PY compressibility approximationmust be employed.

0 0.15 0.3 0.45 0.6ηc

0

0.4

0.8

1.2

1.6

ηrp

q=0.6

F+F F+S

S

a

b

cd

Oscillatory

Monotonic

I

II

Figure 2. The bulk phase diagram calculated from thefunctional for q ¼ 0:6. �c is the packing fraction of thecolloid and �rp that of polymer in the reservoir. F denotesfluid and S solid. The long dashed line shows thefluid–fluid spinodal and the short dashed line shows theFisher–Widom line. The latter intersects the binodalat �rp,FW ¼ 0:53. The horizontal tie lines (a), (b), (c) and(d) connect coexisting fluid states. Horizontal arrowsindicate the paths I and II by which the phase boundaryis approached for the adsorption studies in section 4.2.


return to later but which correspond to size ratio q � 1,i.e. we take the colloid diameter to be 26 nm in accor-dance with the experiments. The polymer profilesare given in figure 4 and demonstrate the polymerpartitioning between the two phases. We show fourprofiles for states between the critical and triple points.

The labels (a)–(d) in figure 3 correspond to the tie linesin figure 2; (d) corresponds to the free interfacial profilesbetween coexisting densities near to the critical pointand (a) to a near triple point state. For state (d)the colloid profile is smooth and reminiscent of theprofiles of [38] calculated using an effective one-com-ponent Hamiltonian and employing a square gradientfunctional. For states approaching the triple point theinterfacial width is found to be approximately �c, similarto values inferred from ellipsometric measurements ona colloid–polymer mixture [39] where a tan h functionwas fitted to the refractive index profile through theinterface.

While the profiles near the critical point are smoothand rather unsurprising, for states nearer to the triplepoint striking oscillations develop on the colloid-rich‘liquid’ side of the profiles. Such oscillations have beenfound previously in theoretical studies [33] of the freeliquid–vapour interface of the square-well fluid andin simulation studies of the liquid–liquid interface [40]in a simple model of a liquid mixture. The presenceof oscillations in one-body density profiles at interfacesis intimately connected to the asymptotic decay of bulkpairwise correlations [33, 34]. For the present mixtureoscillatory profiles arise at the free interface whenthe colloid density in the coexisting liquid is greaterthan the colloid density where the FW (Fisher–Widom)line intersects the binodal, i.e. for all states with �rp >�rp,FW ¼ 0:53 for q ¼ 0:6—see figure 2. The generaltheory of asymptotic decay of correlations in mixtureswith short-range forces predicts [33, 34] that the longest-range decay of the density profiles should be

�iðzÞ � �i � �iAi exp ð��0zÞ, z ! 1, ð18Þ

on the monotonic side of the FW line and

�iðzÞ � �i � �i ~AAi exp ð� ~��0zÞ cos ð�1z� �iÞ, z ! 1,

ð19Þ

on the oscillatory side; �i is the bulk density of species i.Equivalent definitions exist for z ! �1, with appro-priate bulk densities �i. On the FW line, �0 ¼ ~�0�0.The decay lengths ��1

0 and ~��10 and the wavelength of

oscillatory decay 2p=�1 are common to both species andare properties of the bulk fluid. These are determinedby the (common) poles of the structure factors SijðkÞ[34]. The amplitudes Ai, ~AAi and the phase �i are speciesdependent and although there is knowledge about theamplitude ratios in binary mixtures [34], there is notheory for the absolute amplitude of the oscillations.We have confirmed that for states where the colloidprofile oscillates, the corresponding polymer profilesalso exhibit oscillations on the same, colloid-rich side

−15 −10 −5 0 5 10z /σc

0

0.2

0.4

0.6

0.8

1ρ cσ

c3

0 0.1 0.2 0.3 0.4ηc

l– ηc

g

0

5

10

γ(µN

/m)

q=0.6

q=1.0a

b

c

d

Figure 3. Colloid density profiles at the free interfacebetween demixed fluid phases for a size ratio q ¼ 0:6. Thepolymer reservoir packing fractions correspond to the tielines in figure 2, i.e. �rp ¼ 1:0 (a), 0:8 (b), 0:6 (c) and 0:52(d) (near critical point). States (a), (b) and (c) lie on theoscillatory side of the FW line. The inset shows thesurface tension versus the difference in the colloidalpacking fraction in coexisting liquid (l) and gas (g) phasesfor q ¼ 0:6 and 1:0. The colloid diameter is taken to be26 nm to compare with experimental data of [36] (points)where q ¼ 1:0.

z/σc

0

0.5

1

1.5

2

ρpσp

3

(a)

(b)

(c)

(d)

0 5 10−5−10

Figure 4. Polymer density profiles for q ¼ 0:6 correspondingto the colloid profiles shown in figure 3. The polymerreservoir packing fractions correspond to the tie linesin figure 3, i.e. �rp ¼ 1:0 (a), 0:8 (b), 0:6 (c) and 0:52 (d)(near critical point). States (a), (b) and (c) lie on theoscillatory side of the FW line.


of the interface, with identical wavelength and decaylength to those of the colloid. As the bulk density ofthe polymer is low in the colloid-rich ‘liquid’ phase,the amplitude of the polymer oscillations is very small;figure 5 shows an enlargement of the oscillations inthe polymer profile for state (a). For states just abovethe FW intersection point, �rp,FW, the amplitude of thecolloid oscillations can also become extremely smalland this is why the profile (c) does not show oscillationson the scale of figure 3. For state (a), with �rp ¼ 1, whichis not especially close to the estimated triple point, theamplitude of the oscillations is substantial and appearsto be larger than the corresponding amplitude for asquare-well fluid very near its triple point where theoscillations have a wavelength of about one atomicdiameter [33]. Our present results resemble closelythose of a recent DFT treatment of binary mixtures ofrepulsive Gaussian core particles which exhibit fluid–fluid phase separation [41], although in that case thereare states for which oscillations occur on both sides ofthe interface.We note that all DFT treatments are mean-field-like

in that they ignore fluctuation effects, both of the bulkliquid (critical fluctuations) and of the interface.Thermally induced capillary wave fluctuations of theinterface will act to erode the oscillations in the ‘bare’(mean-field) density profiles but it is argued that atleast some of the oscillatory structure will remain inthe ‘dressed’ colloid profiles. The standard method ofincorporating capillary wave fluctuations is to assumethat DFT calculations yield a ‘bare’ or ‘intrinsic’ profileand that interfacial fluctuations can be ‘unfrozen’ by an

appropriate renormalization of the mean-field profile[31, 33, 34, 42]. Because of the extremely low surfacetensions which occur in colloidal systems one mightexpect the oscillations to be completely washed outby inclusion of these fluctuation effects but we will arguethat this is not necessarily the case. In the simplesttreatment of capillary wave fluctuations the intrinsicinterface is smeared by a Gaussian convolution over theinterfacial thermal roughness ?. For oscillatory profileswhich decay as equation (19) the wavelength and decaylength are unaltered by this convolution but the ampli-tude is reduced by a factor exp ½�ð�2

1 � ~��20Þ

2?=2� [33, 34,

42]. The roughness ? depends upon both the interfacialarea L2 and the external potential, i.e. the Earth’sgravitational field. For zero gravitational field, 2? ¼

ð2p�Þ�1 ln ðLmax=LminÞ, where Lmax and Lmin are upperand lower cut-off wavenumbers for the capillary wavefluctuations. If we choose Lmax ¼ 2p= and Lmin ¼

2p=L, where ¼ ~��10 is the correlation length of the bulk

coexisting ‘liquid’ phase, it follows that the amplitudeof the oscillations in the density profile will be reducedby a factor

� ¼ ðL= Þ�!½ð�1= ~��0Þ2�1�, ð20Þ

where we have introduced the dimensionless parameter

! ¼kBT

4p 2, ð21Þ

which measures the strength of the capillary wavefluctuations. Smaller surface tensions give rise to largervalues of ! and, as a result, the amplitude of theoscillations is damped more severely. Moleculardynamics simulations of a liquid–liquid interface havebeen performed by Toxvaerd and Stecki [40]. Theseauthors (see also Chacon and co-workers [43, 44] whoperformed Monte Carlo simulations of the liquid–gasinterface of a model of a metal) found a decrease in theamplitude of oscillation with increasing L, consistentwith a Gaussian renormalization of the oscillatoryintrinsic interface. We find for the present model that! takes values of a similar order of magnitude as thosefor simple fluids [38]. This is due to the fact that thebulk correlation length scales roughly as �c [16, 17] andthe tension as kBT=�2

c [38, 45]. From our explicitcalculations of and we find that at state point (a),for which the profile has pronounced oscillations, seefigure 3, ~��0�c ¼ 0:77, �1�c ¼ 6:74 and the reducedsurface tension � ¼ ��2

c ¼ 1:13 which implies that! ¼ 0:042 and ð�1= ~��0Þ

2� 1 ¼ 75:12. We thus find that

� ¼ ðL= Þ�3:23 which suggests that detecting oscillationsof the colloid profile should be no more difficult thanfor simple fluids where the exponent is expected to take

z/σc

0.016

0.017

0.018

ρcσc

3

0.5 1.5 2.5 3.5 4.5 5.5

Figure 5. An enlargement of the polymer profile for q ¼ 0:6and �rp ¼ 1:0, i.e. state (a) shown in figure 4. Theoscillations, although not visible on the scale of figure 4,are present in the polymer profiles and display the sameperiod and decay length as the corresponding colloidprofiles (figure 3).


a similar value. Indeed, from an experimental viewpointsuch oscillations in a colloidal system offer an interest-ing opportunity for experimental study as it should bemore favourable to investigate such structuring insystems where the period is of colloidal size than inatomic fluids, where the period is on the Angstrom scale.We conclude this section by returning to the results

for the surface tension. The inset in figure 3 shows thetension for size ratios q ¼ 0:6 and 1:0. We find that thetension calculated using the present DFT is consistentlylarger than that calculated [38] using the effective one-component Hamiltonian treated by square gradientDFT. The level of agreement with the experimentallymeasured tension for mixtures of a silica colloid, coatedwith 1-octadecanol, and polydimethylsiloxane (PDMS)in cyclohexane at T ¼ 293K [36] is somewhat better thanin [38]. The size ratio for this mixture is approximately1:0. As mentioned earlier, in order to compare our DFTresults with experiment we choose �c ¼ 26 nm, the meandiameter of the particles investigated in [36]; there areno adjustable parameters in the model [25, 26, 46].Note that the measured tensions are typically

3–4 mNm�1, values which are about 1000 times smallerthan the tensions measured for simple atomic fluids neartheir triple points. Such small values of the tension arenot so surprising when one recalls (i) that the tensionscales roughly as kBT=�2

c , provided the state is wellremoved from the critical point and (ii) that �c � 100atomic diameters [38, 45]. In figure 3 the tension isplotted against the difference between �c in the coexis-ting liquid (l) and gas (g) phases. Within the presentmean-field treatment vanishes as ð�lc � �gcÞ

3 on appro-aching the critical point. Incorporating critical fluctua-tions should lead to even faster decay; for Ising-likecriticality the exponent 3 should be replaced by2=� 3:9. The experiments of [36] were not performedsufficiently close to the critical point to examine scalingbehaviour. However, Chen et al. [47] do report resultsfor the density difference and surface tension, in asimilar mixture of silica particles and PDMS incyclohexane, taken near the critical point. They reportvalues for < 1�Nm�1 and a good fit to Ising-likescaling.

4. Adsorption at a hard wall

In this section we consider the AO mixture adsorbedat a planar hard wall described by the external potentials(8). Such a model constitutes the simplest frameworkin which one can address the statistical mechanics ofcolloidal adsorption or, more specifically, the effectsof entropic depletion forces on the distribution ofboth colloids and polymer near a (repulsive) substrate.Of course, other more complex wall–fluid potentialscan be considered which include soft repulsion and/or

attractive interactions. The advantages of the hard-wallmodel are (i) it encompasses the key feature of depletion-induced wall–colloid attraction and (ii) as emphasized insection 2.1, it can be mapped exactly to a very simpleeffective Hamiltonian (9), for q < 0:1547, that can beused efficiently in simulation studies.

We focus first on the case q ¼ 0:1, for which there areMonte Carlo results [11] for the colloid density profileagainst which we can test the reliability of our DFT.Afterwards we consider larger size ratios where bulkfluid–fluid phase separation occurs. This enables us toinvestigate entropic wetting phenomena at the hard wallusing the DFT approach.

4.1. Test case: q ¼ 0.1In order to test the performance of the DFT for

adsorption studies we first calculate density profilesagainst a hard wall for q ¼ 0:1. For this small size ratio,simulation results are available [11] for the colloidprofiles. These make use of the exact mapping tothe effective Hamiltonian (9); only a pairwise additivefluid–fluid potential (�AO) and an explicit one-bodywall–fluid depletion potential (�wall

AO ) are involved. Aspreviously, we minimize the functional (17) but nowfor a hard wall with external potentials (8). The resultsobtained from the present DFT (figure 6) should becompared and contrasted with those of [11] wherethe effective one-component Hamiltonian was treatedby means of a one-component mean-field DFT whichtreats the hard-sphere contribution by Rosenfeld’sFMT and the attractive contribution, arising from�AOðRÞ, in mean-field fashion. We find that the presentmixture functional provides an equally good descriptionof the colloid profiles, in particular the dramatic increasein the wall contact value, �cð�

þc =2Þ � �wc, as polymer is

added; the present functional clearly incorporates thedepletion attraction between the wall and the colloids. Itshould be noted that the binary mixture AO functionalgenerates internally all depletion effects between thecolloids and between the colloids and the hard wall,whereas in the previous one-component treatment theseare essentially put in by hand. We choose polymerreservoir packing fractions �rp ¼ 0:05 and �rp ¼ 0:1 and afixed bulk colloid packing of �c ¼ 0:3, as Monte Carlosimulation results exist for these state points. Weintentionally stay away from the vicinity of the solid–fluid phase boundary which is at about �rp ¼ 0:16 [10].

The present functional tends to give significantly morestructured colloid profiles than does the effectiveone-component DFT of [11]. In fact it appears thatthe binary mixture AO DFT consistently overestimatesthe degree of structuring (more pronounced maxima andminima than in simulation) whereas the one-componentDFT gives an underestimate. When there is no polymer


in the system, �rp ¼ 0, see figure 6 (a), both functionalsreduce to the Rosenfeld functional for pure hard spheresand, as is well known, this performs very well for thefull range of bulk packing fractions. The oscillationsarise from packing effects at the hard wall.

On adding a small amount of polymer, �rp ¼ 0:05,there is a pronounced increase in the contact value,figure 6 (b), as a result of the wall-induced depletionwhich is now implicitly incorporated into the mixtureDFT. We obtain a contact value of �wc�

3c � 7:56 which

is slightly higher than the value 6:41 obtained from theeffective one-component DFT [11]. For such small sizeratios the wall–colloid depletion potential is stronglyattractive and of short range (0:1�c) and it thus becomesdifficult to achieve good simulation statistics close to thewall [11]. The simulation results have been extrapolatedto contact and yield a value �wc�

3c � 4:28. For �rp ¼ 0:1,

see figure 6 (c), the wall–colloid depletion attractionbecomes even stronger and the present DFT gives acontact value �wc�

3c � 20:29; again this is larger than the

corresponding result from the one-component func-tional, which is 15:52. Note that these very high valuesof the local density decay to roughly unity over therange of the wall–colloid depletion potential, i.e. 0:1�c.The insets in figures 6 (b) and 6 (c) show that the DFTcaptures correctly the non-trivial ‘triangular’ structureof the second peak in the profile but for both �rp ¼ 0:05and 0:1 the first minimum is deeper than the simula-tion result and the height of the second maximum isoverestimated. As �rp is increased the oscillations becomedamped more rapidly and �cðzÞ is close to the bulkvalue after a distance of approximately 4�c. Numericalresults were checked using the hard wall sum rule,�P ¼ �cð�

þc =2Þ þ �pð�

þp =2Þ, where P is the total pressure

[11]. Recall that for each colloid profile we present herewe also have the corresponding polymer profile. Thesum rule was satisfied to better than 0:1% in all cases. Itshould be noted that comparing the DFT resultswith those of simulation for such a highly asymmetricmixture constitutes a severe test. The fact that theAO mixture DFT performs well under such difficultconditions is thus most encouraging. In addition, thefree-volume theory, which gives a bulk free energy(15) identical to that of the AO functional, becomesincreasingly accurate for larger q values. We mightreasonably expect the performance of the functionalto improve accordingly. Although the bulk free energyobtained from the functional improves with increasingq, this does not guarantee that one-body profiles will beany better. Nevertheless, since the profiles obey thehard wall sum rule, the contact values should becomecloser to those of simulation as q increases, inaccordance with the increasing accuracy of the free-volume bulk pressure. These considerations are relevant

0 1 2 3 4z/σc

0

4

8

12

16

20

ρcσc

3

0 1 2 3 40.3

0.5

0.7

0.9

(c)

Figure 6. The colloid density profiles calculated from thebinary mixture DFT compared with simulation results fora size ratio q ¼ 0:1 and bulk colloid packing fraction�c ¼ 0:3. The packing fractions of the polymer reservoirare (a) �rp ¼ 0 (hard spheres), (b) 0:05 and (c) 0:10,respectively. The circles are the Monte Carlo results [11]and the solid lines are the DFT results. The insets showthe results on an expanded scale. Note the rapid increasein the density near contact as �rp is increased.

0 1 2 3 4z/σc

0

0.5

1

1.5

2

2.5

ρcσc

3

(a)

0 1 2 3 4z/σc

0

2

4

6

8

ρcσc

3

0 1 2 3 40.3

0.5

0.7

0.9

(b)


when we focus on larger size ratios (q > 0:32) wherestable fluid–fluid demixing occurs and investigateinterfacial phase transitions at the hard wall–fluidinterface.The physical significance of the simulation results for

q ¼ 0:1 were discussed in [11] and we do not dwell uponthis here. Rather we simply emphasize that sincethe colloid density profiles decay so rapidly from theirvery high contact densities over the short range of thewall–fluid depletion potential, �wall

AO , the amount ofcolloid adsorbed in the contact ‘layer’ is rather small(fraction of a monolayer). Indeed the Gibbs adsorptionof colloid does not increase rapidly with increasing�rp [11]. From examination of the one-body profilesand simulation snapshots there was no evidence of wall-induced local crystallization at the polymer concentra-tions we examined. However, the state points weconsidered were still well removed from the bulkfluid–solid phase boundary. The issue of when andhow wall-induced crystallization occurs is an importantone, both conceptually and because there are severalexperimental papers reporting evidence for the phenom-enon (in colloid–colloid as well as in colloid–polymermixtures) occurring for state points well below the bulkphase boundary—see [11] for a summary.

4.2. Entropic wetting and layering: q� 0.6For these larger size ratios the bulk phase diagram

exhibits three stable phases, as in figure 2. In determin-ing the adsorption characteristics, we choose to fix �rpand approach the bulk fluid–fluid phase boundary fromthe colloid poor side. This is analogous to performinga gas adsorption isotherm measurement for a simpleatomic fluid, recalling that �rp plays a role equivalentto inverse temperature in the AO model. Dependingon the value of �rp chosen, we find the adsorptionbehaviour changes dramatically. For the present AOmodel one might expect to observe similar behaviour asthat pertaining to simple fluids adsorbed at attractivesubstrates; we have a system of colloids interactingvia an effective Hamiltonian which gives rise to fluid–fluid phase separation in bulk and a depletion inducedattraction acting between the colloids and the wall.However, in the present case we have effective many-body colloid–colloid and many-body wall–colloidpotentials for the size ratios of interest, i.e. values of qlarge enough to give rise to a stable bulk fluid–fluidtransition. For large size ratios the effective wall–colloidpotential ceases to be a simple one-body potentialacting on individual colloids and becomes a complicatedfunction of multiple colloid coordinates. We find thatthese complex wall potentials do indeed give rise to newphenomena which are quite distinct from those seen insimple fluids. As an example we consider size ratio

q ¼ 0:6 and describe some of the phenomena encoun-tered as we increase �c, for fixed �rp, towards the bulkphase boundary. We consider, in turn, paths I and IIshown in figure 2.

We first choose a path just above the critical point,�rp ¼ 0:55, i.e. follow path I in figure 2. The layer ofliquid-like colloid density grows continuously againstthe wall and becomes macroscopically thick as the phaseboundary is approached. The colloid density profiles infigure 7 show the onset of complete wetting by thecolloid-rich phase. We have confirmed that for stateswith smaller values of �rp (but above the critical value�rp, crit) the equilibrium film thickness, teq, or, equiva-lently, Gc, the adsorption of colloid, grows logarithmi-cally with the deviation from the bulk phase boundary.Although a detailed investigation of the amplitudewas not performed this should be the bulk correlationlength of the wetting (colloid-rich) phase, as is appro-priate to a system where all the interactions are of finiterange—see e.g. [48]. The corresponding polymer profilesare shown in the inset and indicate how polymerbecomes more depleted as the colloid-rich layer grows;the polymer can be effectively regarded as a ‘slave’species with profiles determined by the distributionof colloid [17]. Strictly speaking, macroscopicallythick wetting films can only occur when the densityof coexisting liquid lies on the monotonic side of the

0 1 2 3 4 5 6 7 8 9 10z/σc

0

0.5

1

1.5

2

ρ cσc3

0 1 2 3 4 5 6 7 8 9 10z/σc

0

0.5

1

ρ pσp3

Figure 7. Colloid density profiles for q ¼ 0:6 showing theonset of complete wetting of a hard wall by the colloid-rich phase at �rp ¼ 0:55 as the bulk phase boundary isapproached along path I in figure 2. Bulk colloid fractionsare �c ¼ 0:04, 0:06, 0:07, 0:076, 0:0775, 0:0778 and 0:0779(from bottom to top). The coexisting gas density is�c ¼ 0:078 12. The inset shows the polymer profiles for thesame values of �c (from top to bottom). Note that thepolymer distribution becomes progressively more depletedas the colloid-rich layer grows in thickness.


FW line in figure 2, i.e. for �rp < �rp,FW, the point ofintersection of the FW line and the binodal. Onecan envisage a coarse grained description of wettingphenomena whereby the surface excess free energy canbe regarded as an effective wall or binding potential,FwallðtÞ, acting on a single degree of freedom, the filmthickness t [49]. For �rp, crit < �rp < �rp,wet and �rp < �rp,FWthe potential FwallðtÞ exhibits a single minimum at afinite equilibrium film thickness teq provided the stateis off bulk coexistence. �rp,wet denotes the polymerreservoir packing fraction at the wetting transition.As the value of �c is increased towards coexistencethe position of this minimum, and hence teq, increasescontinuously to infinity. If we are in the region where�rp, crit < �rp < �rp,wet and �rp > �rp,FW the decay of thedensity profiles is oscillatory. It can be shown [50] thatin this region the effective binding potential FwallðtÞpossesses a corresponding oscillatory decay. As a resultthe minimum, teq, of the binding potential will alwayslie at a finite value. Such oscillatory binding potentialswill stabilize very thick but finite films, which wouldotherwise be infinite, even at bulk coexistence. This isbecause the global minimum lies at the trough of oneof the oscillations and not at infinity. Such situationscan easily lead to numerical difficulties when usingiterative methods because of the presence of a largenumber of metastable minima and care must be taken toensure that the global minimum of the excess (over bulk)free energy is reached. By choosing �rp ¼ 0:55 we avoidmany of these complications and can easily obtain filmsof thickness 20 or 30�c. We confirmed that in the flatportion of the profiles the densities of colloid and polymerare equal to their values in the coexisting colloid-richphase. In order to investigate such thick wetting filmsit is necessary to work very close to bulk coexistenceand solving the Euler–Lagrange equations resultingfrom minimizing the functional can be very slow.Very different scenarios arise for other paths

approaching bulk coexistence. As �c is increased alongpath II, see figure 2, at fixed �rp ¼ 0:7 the colloids behaveessentially as an ideal gas in the presence of the wall–colloid depletion potential but with some enhancementof the contact value due to packing effects. A selec-tion of profiles calculated along this path is shownin figure 8. The profiles remain largely unstructureduntil �c ¼ 0:0198 where a first-order phase transitionoccurs and a second liquid-like layer is adsorbedagainst the wall. The inset to figure 8 shows the disconti-nuous jump in the (reduced) Gibbs adsorption of thecolloids, Gc, at the transition. The Gibbs adsorption isdefined as

Gc ¼ �2c

Z 1

0

dz �cðzÞ � �cð1Þð Þ, ð22Þ

where �cð1Þ is the density of colloid in the bulk. Thislayering transition is most unusual and is quite differentfrom the transitions between layered liquid-like filmsfound in DFT studies of simple fluids at attractivesubstrates—see e.g. [51] which considers a Yukawa fluidagainst an attractive Yukawa wall. The transitions foundin [51] are for temperatures very close to the triple pointand the sequence always leads to complete wettingat coexistence. In the present case we are far away fromthe free volume triple point at �rp � 1:43 (although itslocation within a full DFT treatment of the solid isnot known). Moreover the transition is to only a singleextra layer; the adsorption remains finite all the way tocoexistence. Thus the wall is partially wet for this valueof �rp. Since for �

rp ¼ 0:7 the wall is partially wet, whereas

for �rp ¼ 0:55 the wall is completely wet, we can inferthat there exists an analogue of the wetting transitiontemperature, i.e. there is a wetting polymer reservoirpacking fraction �rp,wet, at which along coexistence thereis a transition from partial wetting to complete wetting.It should be noted that when the second layer is adsorbedits density is not that of the coexisting liquid, rather itis at some lower liquid-like density. For values of �caway from the layering transition, numerical solutionof the Euler–Lagrange equations is extremely rapiddue to the very low bulk density of colloid. As thetransition is approached convergence slows consider-ably and large numbers of iterations are required toovercome the free energy barrier separating layered

0 0.5 1 1.5 2 2.5 3z /σc

0

0.5

1

1.5

ρ cσc3

0 0.01 0.02ηc

0

0.2

0.4

0.6

0.8

1

Γc

Figure 8. Colloid density profiles for size ratio q ¼ 0:6showing the first layering transition at �rp ¼ 0:7 corre-sponding to path II in figure 2. Bulk colloid fractionsare �c ¼ 0:010, 0:015, 0:018, 0:019 and 0:020 (frombottom to top); the transition occurs between 0:019 and0:020. The inset shows the corresponding jump in theGibbs adsorption Gc. Gc remains finite at bulk coexistence�c ¼ 0:0203, i.e. the interface is partially wet by thecolloid-rich phase.


and unlayered profiles. In the sequence of layeringtransitions calculated in [51] the appearance of layer nis accompanied by a jump of density in the precedinglayers, primarily effecting the (n� 1)th layer. The jumpin adsorption which occurs at the transition thus receivesa contribution from all layers, not only layer n. A similareffect can be seen at the present layering transitionwhere the first contact peak in the colloid profile jumpsas a second layer is adsorbed.The layering transition which we find from the AO

functional, distinct from the wetting transition, appearsto result from effective many-body wall–fluid andfluid–fluid potentials acting on the colloids. It does notappear to have a direct counterpart in the adsorption ofsimple fluids—see section 7. In order to test this asser-tion we have calculated colloid density profiles using themean-field DFT employed in [11]. As described in theprevious subsection, we treat the AO pair potential as amean-field perturbation to a hard sphere referencesystem and apply an external potential consisting of ahard wall plus the one-body AO wall–colloid potential,see equation (9). We find that whilst a wetting transitiondoes exist, there is no sign of the layering which weobtain from the AO mixture DFT. The phase boundarywas approached for different fixed values of �rp above�rp,wet. In each case the adsorption was found to increasesmoothly to a finite value at bulk coexistence.In order to map out the full interfacial diagram and

locate further layering transitions we calculate densityprofiles along the coexistence curve starting at largevalues of �rp working down towards the critical point,�rp, crit. Using the transition points (jumps) on the phaseboundary as a guide we then take slices across the phasediagram, increasing �c for a fixed value of �rp, so we canlocate any lines of first-order transitions which mayextend into the single phase (dilute in colloid) region.We have determined interfacial phase diagrams for sizeratios q ¼ 0:6, 0:7 and q ¼ 1:0 in order to identify anyvariation of the topology with size ratio. Figure 9 showsthe interfacial phase diagram for q ¼ 0:6; it is very richand features not only the wetting and first layeringtransitions discussed above but also two further layeringtransitions. Figure 10 shows the colloid density profilescalculated along the bulk coexistence curve for a numberof values of �rp and the corresponding adsorption Gc

is shown in figure 11. Since these layering transitionsare rather unusual we give a brief description of howthey are located.We begin mapping out the phase diagram by calculat-

ing the profiles at coexistence for large �rp, region (a) infigure 9. In this region the profiles have little structureoutside the contact region and the numerical iterationscheme used to solve the Euler–Lagrange equationsconverges rapidly. Decreasing �rp we encounter the point

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07ηc

0.55

0.6

0.65

0.7

0.75

0.8

ηp

r

1st Layering

2nd Layering

3rd Layering

Wetting

q=0.6(a)

(b)

(c)

(d)(e)

Transition

Transition

Transition

Transition

Figure 9. The interfacial phase diagram for a colloid–polymer mixture adsorbed at a hard wall for size ratioq ¼ 0:6. The full curve is a portion of the bulk fluid–fluidcoexistence curve shown in figure 2. We find a wettingtransition at �rp,wet ¼ 0:596 and three separate layeringtransitions at higher values of �rp. The first layeringtransition line extends from coexistence to deep into thesingle phase region (dashed line) and, when crossed, givesrise to a jump in the adsorption as shown in figure 8for �rp ¼ 0:7; it ends in a surface critical point near�rp ¼ 0:62. The second and third layering transition linesand any prewetting line lie very close to the coexistencecurve so the transition lines are simply denoted by circles.

0 1 2 3 4 5z/σc

0

0.5

1

ρcσc

3

q=0.6

(a)

(b)

(c)

(d)

(e)

Figure 10. The colloid density profiles calculated at bulkcoexistence on decreasing �rp (bottom to top) for a sizeratio q ¼ 0:6. The labels (a)–(e) indicate groups of profilescalculated on the different sections of the coexistencecurve, see figure 9, and show clearly the four differenttransitions, e.g. the higher profile of (a) jumps to the lowerprofile of (b) at the first layering transition. The first threejumps are layering transitions and the final one, (d) to (e),is the wetting transition. As �rp is decreased the adsorptionjumps discontinuously as each layering transition isencountered. For �rp90:596 the wall is competely wetby the colloid-rich phase.


at which the first layering transition line intersectsthe phase boundary. Near this point the solutionof the Euler–Lagrange equation becomes unstable; thisinstability is reflected in the number of iterations requiredto obtain a converged solution. At �rp ¼ 0:722 theadsorption jumps discontinuously and a new layer isadsorbed, see figure 11.Moving further down the coexistence curve, region

(b), the adsorption increases smoothly until a secondlayering transition occurs at �rp ¼ 0:642, where there isagain a discontinuous jump in adsorption as a thirdlayer is adsorbed at the wall. Again the new layercorresponds to a density lower than the coexisting bulkliquid density but it is still liquid-like in character.As the third layer appears the local density in thesecond layer also jumps significantly, to a value whichappears to correspond more closely to that of thecoexisting liquid density, i.e. the jump in adsorptionat the second transition is only partially due to theappearance of the third layer. Similarly, for �rp ¼ 0:608,we find a third layering transition, (c) to (d), wherebya fourth adsorbed layer develops, with an accompany-ing increase in density of layer three, giving rise toanother jump in Gc. As �rp is reduced further Gc remainsfinite at bulk coexistence until �rp ¼ 0:596, where thetransition to complete wetting occurs. Having locatedthe transition points on the phase boundary, we thentake slices at fixed values of �rp to determine whetherthe layering transitions extend into the single phaseregion. As remarked earlier, the first layering transi-tion extends away from the phase boundary and ends

at a surface critical point near �rp ¼ 0:62. In determin-ing the transition line we simply mark the locus ofpoints where the adsorption jumps discontinuously.Such an approach will always be subject to somehysteresis effects, i.e. the actual transition line may becloser to the phase boundary as, for a given �rp, we mayhave to increase �c above its value at the equilibriumtransition in order to move to the next minimum inthe free energy. In order to assess the extent of thiseffect we reversed some of the paths across the phasediagram, starting with a converged solution at coexis-tence and decreasing �c until the transition is located.Although hysteresis effects do exist, these are small,and are not visible on the scale of the line in figure 9.Unlike the first transition, the second and third transi-tion lines are extremely short in �rp. As it is verydifficult to determine these accurately we have simplyrepresented the transitions as large circles in figure 9.The wetting transition appears to be of first order,i.e. Gc appears to diverge discontinuously, see figure 11.However, it is difficult to determine any prewett-ing line, which should emerge tangentially from thecoexistence curve at the wetting transition [49]. Wecan say with certainty that any prewetting line isextremely short. We repeated the calculations forq ¼ 0:7 and we find the same pattern of three layeringtransitions and a wetting transition as was found forq ¼ 0:6. The transitions occur at different values of �rp,see figure 12. Both the layering transitions and thewetting transition move to larger values of �rp on thephase boundary but the distance, in �rp, betweenthe first layering transition and the wetting transitionremains roughly the same. The first layering transitionline is shorter in �rp for q ¼ 0:7 than for q ¼ 0:6 andlies closer to the phase boundary. Figure 13 showsthe colloid profiles calculated at bulk coexistencein different regions of the phase diagram, and thecorresponding adsorption is shown in figure 14.

For q ¼ 1 the distance, in �rp, along the bulk phaseboundary between the first layering transition and thewetting transition increases and a fourth layeringtransition appears above the wetting transition—seefigure 15. The sequence of colloid density profiles isshown in figure 16. It is interesting to note from figure 17that at subsequent layering transitions the jump inadsorption is slightly less than that at the precedingone. Since the profiles shown in figure 16 would indicatethat the amount adsorbed in each new layer is roughlythe same, the difference in adsorption at each transitionis due chiefly to the contribution of jumps in the localdensity at the preceding layer. At the first layeringtransition the first peak in the colloid profile shows aclear jump, whereas at the fourth transition the localdensity in the third layer shows little change. We find

Figure 11. The Gibbs adsorption of the colloids Gc alongthe bulk coexistence curve corresponding to the profilesshown in figure 10, and the phase diagram of figure 9.At each of the layering transitions the adsorption changesdiscontinuously by a finite amount. At the wetting transi-tion the adsorption diverges discontinuously as the wall iswet completely by the colloid-rich phase.


that the first layering transition line is very shortfor q ¼ 1 and lies extremely close to the coexistencecurve so we simply represent this as a circle in figure 15.

4.3. A simulation study for q ¼ 1All the results we have described so far for wetting

and layering were based on the DFT for the binary AOmixture. It is important to enquire how much of therich behaviour predicted by DFT for this model canbe found in simulation studies. After the completionof our DFT calculations [25, 26, 46], Dijkstra andvan Roij [19] developed a novel Monte Carlo schemefor tackling the equilibrium statistical mechanics of

both homogeneous and inhomogeneous model colloid–polymer (AO) mixtures for arbitrary size ratios q,including those where effective many-body interactionsbetween the colloids play an important role. Theirapproach is based on the exact effective one-compo-nent Hamiltonian entering equation (4). Because of theideality of polymers, �pp ¼ 0, the grand potential ofpolymer in the external potential Vext

p and in the staticconfiguration fRNcg of the Nc colloidal hard spheresis given exactly by O ¼ ��1zpVf , where

Vf ¼

Zdr exp ð��Vext

p ðrÞ�Nc

i¼1½1þ fcpðjr� RijÞ� ð23Þ

0 0.01 0.02 0.03 0.04 0.05 0.06ηc

0.6

0.65

0.7

0.75

0.8

0.85

0.9

ηp

r

1st Layering

2nd Layering

3rd Layering

Wetting

q=0.7(a)

(b)

(c)

(d)

(e)

Transition

Transition

Transition

Transition

Figure 12. As in figure 9 but for size ratio q ¼ 0:7.As for q ¼ 0:6, we find three layering transitions butthe first layering transition line is slightly shorter in �rpand lies closer to the bulk coexistence curve. The wettingtransition occurs at �rp,wet ¼ 0:664.

0 0.005 0.01 0.015 0.02 0.025 0.03ηc

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

ηp

r

1st Layering

2nd Layering

3rd Layering

4th Layering

Wetting

q=1.0(a)

(b)

(c)

(d)

(e)

(f)

Transition

Transition

Transition

Transition

Transition

Figure 15. As in figure 9 but for size ratio q ¼ 1:0. Wefind four layering transitions but now all the transitionlines, including the first, lie extremely close to the bulkcoexistence curve (solid line). The wetting transitionoccurs near �rp,wet ¼ 0:845.

0 1 2 3 4 5z/σc

0

0.5

1

ρcσc

3

q=0.7

(a)

(b)

(c)

(d)

(e)

Figure 13. The colloid density profiles at bulk coexistenceon decreasing �rp (bottom to top) for q ¼ 0:7. The labels(a)–(e) indicate groups of profiles calculated on differentsections of the coexistence curve, see figure 12.

0.65 0.7 0.75 0.8ηp

r

0

0.5

1

1.5

2

2.5

Γc

q=0.7

1st Layering

2nd Layering

3rd Layering

TransitionWetting

Transition

Transition

Transition

(a) (b)

(b) (c)

(d)

(c) (d)

(e)

Figure 14. The Gibbs adsorption of the colloids Gc along thebulk coexistence curve corresponding to the profilesshown in figure 13 and the phase diagram in figure 12.


is the free volume. The integral in equation (23) is overthe total volume V of the system and fcpðrÞ is the colloid–polymer Mayer bond: fcp ¼ �1 for 0 < r < Rc þ Rp andzero otherwise. Of course, the shape of the free volumeis, generally, irregular and non-connected but compu-tationally efficient methods were developed to calculatethis [19]. Bulk simulations were performed for thecase q ¼ 1 where effective many-body interactions areexpected to be crucially important. The semi-grand freeenergy was obtained using thermodynamic integration,at fixed colloid packing fraction �c, with respect to thepolymer fugacity zp (see also [28]).Phase coexistence was then determined by standard

common tangent constructions at fixed zp. The inset

to figure 18 shows the resulting phase diagram in thereservoir, �rp versus �c, representation. Three phases arepresent; there is colloidal gas–liquid separation, with acritical point �rp, crit � 0:70, and a liquid–solid transitionwhich is almost independent of �rp. The triple point isat �rp, t � 6:0, i.e. there is a very large stable gas–liquidcoexistence region, much larger than the corresponding(inverse) temperature region for simple atomic fluids.Also plotted in this figure is the phase diagram obtainedfrom the free-volume theory of [4]. Note that the lattercan be viewed as a first-order perturbation theory thatapproximates Vf by its average in the pure hard-spherefluid [10, 11]; it sets Vf ¼ �ð�cÞV , where �ð�cÞ is thefree-volume fraction in equation (16). On the scale of theinset, the free-volume theory, and therefore our DFTapproach which yields the same free energy, providesan accurate description of the simulation data. Thereare significant deviations, as indicated on the expandedscale of the main figure where the gas branch of thecoexistence curve is plotted, but the overall agreementis quite remarkable. As emphasized by Dijkstra andvan Roij [19], it is important to compare the ‘exact’simulation phase diagram for q ¼ 1 with that obtainedin [10] from simulations in which solely the pairwiseeffective potential�eff ðR; zpÞ, equation (6), was employed.

0 0.01 0.02 0.03ηc

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

η pr

0 0.2 0.40

2

4

6

Figure 18. Simulation results (adapted from [19]) for bulkand surface phase diagram of the AO model (size ratioq ¼ 1) as a function of the colloid packing fraction �c andthe polymer reservoir packing fraction �rp. The main figureis a blow-up of the saturated bulk gas branch, separatedinto a regime of complete wetting (thick curve, opencircles), and partial wetting by colloidal liquid (thin curve)at a planar hard wall. The first (filled triangles), second(filled squares), and third (filled circles) layering transitionlines extend from bulk coexistence into the single phase(gas) region. The inset shows the gas–liquid and fluid–solid bulk coexistence (open squares) to full scale. Thedashed curves denote the bulk binodals obtained fromfree-volume theory.

0 1 2 3 4 5 6 7z/σc

0

0.5

1

ρcσc

3

q=1.0

(a)

(b)(c)

(d)(e)

(f)

Figure 16. The colloid density profiles at bulk coexistenceon decreasing �rp (bottom to top) for a size ratio q ¼ 1:0.The labels (a)–(f) indicate groups of profiles calculated indifferent sections of the coexistence curve, see figure 15.

0.82 0.87 0.92 0.97 1.02ηp

r

0

0.5

1

1.5

2

2.5

Γc

q=1.0

1st Layering

2nd Layering

3rd Layering

TransitionWetting

Transition

Transition

Transition

4th LayeringTransition

(a) (b)

(b) (c)

(c) (d)

(d) (e)

(e) (f)

Figure 17. The Gibbs adsorption of the colloids Gc along thecoexistence curve corresponding to the profiles shown infigure 16 and the phase diagram of figure 15.


For the latter, �rp, crit � 0:5 and �rp, t � 0:8. In otherwords, the effective many-body interactions extendgreatly the region in �rp over which stable gas–liquidcoexistence can occur. Moreover, these many-bodycontributions appear to be incorporated into the free-volume/DFT approximation. For completeness weshould also note that Bolhuis et al. [15] have investigatedthe bulk phase behaviour of the AO binary mixtureusing Gibbs ensemble Monte Carlo simulations for sizeratios q ¼ 0:34, 0.67 and 1.05. They find that forq ¼ 0:34 the fluid–fluid binodal is (weakly) metastable,consistent with the prediction of the free-volume theory.For q ¼ 0:67 and 1.05 their simulation results forthe gas–liquid coexistence curves are in good overallagreement with those of the free-volume theory. Theseauthors also performed simulations which incorporateexcluded-volume interactions between polymers but weshall return to this aspect later.The main part of figure 18 displays the surface phase

diagram obtained in [19] for the q ¼ 1 AO mixtureadsorbed at a planar hard wall. Once again Vf (nowgiven by equation (23) with Vext

p corresponding to thehard-wall potential) is calculated within the simulationso that all many-body interactions are incorporated,including the modification of pair and higher-bodyinteractions that occurs when two colloids are close tothe wall—see section 2.1 and [11].For small polymer reservoir packings �rp, crit < �rp <

1:05 there is strong evidence for complete wetting, i.e.formation of a thick film of colloidal liquid at the hardwall–colloidal gas interface with the adsorption Gc

increasing continuously (logarithmically) as bulk coex-istence is approached. In this regime the colloid densityprofiles (see figure 2 (a) of [19]) are reminiscent of theDFT results in figure 7. By contrast, for about �rp > 1:1,there is partial wetting, i.e. Gc remains finite at bulkcoexistence. The authors conclude that there is a wettingtransition at some value of �rp,wet in the range 1:05 <�rp,wet < 1:1. They find no evidence for an accompanyingprewetting transition out of bulk coexistence, i.e. thereappears to be no thin–thick transition in the same rangeof values of �rp. For about �

rp > 1:1, however, they find

jumps in the adsorption which they attribute to layeringtransitions of the type we found in the DFT calculationsdescribed in section 4.2. Indeed the colloid densityprofiles near the layering transitions (see figure 2 (b)of [19]) are very similar in form to those shown infigures 10, 13 and 16. Note that the latter refer totransitions encountered on reducing �rp at bulk coex-istence, whereas the simulation data is for a path at fixed�rp, increasing �c towards bulk coexistence. Figure 18describes three separate layering transition lines, eachof them rather short, lying close to bulk coexistenceand very far from the bulk triple point. A reasonable

person would conclude that the wetting and layeringphenomena found in the simulations for q ¼ 1 mimicthose found in the DFT calculations. There is notperfect agreement. The sequence of layering transitionsgleaned from the simulations for q ¼ 1, is arguablycloser to that observed in DFT for q ¼ 0:6 and 0.7 thanfor q ¼ 1; the layering transitions extend very littleout of bulk coexistence in the last case (see figure 15).But one should not expect an approximate DFT toprovide a completely accurate account of what arevery subtle interfacial phase transitions, resulting froma complicated competition between wall–fluid and fluid–fluid interactions. Given that the bulk gas–liquidcoexistence curves obtained from DFT and simulationdiffer significantly on the scale relevant to the surfacephase diagram it is not surprising that there are differ-ences in the surface phase behaviour. What is moreimportant is to establish that it is the many-body inter-actions entering the effective interfacial Hamiltonianfor the colloids which give rise to the pattern of inter-facial transitions that is observed. We shall return to thisissue later.

5. Treating polymer–polymer interactions within

a geometry-based DFT

So far in this paper we have treated the polymersas mutually non-interacting: �ppðrÞ ¼ 0. Real polymers,when dissolved in a good solvent, experience repulsivemonomer–monomer interactions. As described in theintroduction, when averaged over polymer conforma-tions this repulsion at the polymer segment level givesrise to a soft, penetrable and repulsive interactionbetween polymer centres of mass. The range of thiseffective pair potential is 0 the polymer radius ofgyration and its strength (at zero separation betweenthe two centres of mass) is of the order of the thermalenergy, kBT . Within this ‘soft colloid’ picture theeffective polymer–polymer potential can be well repre-sented by a Gaussian pair potential [52]. In the followingwe do not attempt to model this interaction realistically,rather we summarize the approach of [20] where aminimal model was considered that displays the essentialfeatures of polymer–polymer repulsion. With a simplegeometric picture in mind, the interactions betweenpolymers are represented by a repulsive step-functionpair potential

�ppðrÞ ¼�, for r < 2Rp,

0, otherwise,

�ð24Þ

whilst the colloid–colloid and colloid–polymer poten-tials remain hard-sphere-like. Note that in the limit�=kBT ! 0 we recover the AO model with non-interacting polymers, equation (1) with �p � 2Rp,


whereas for �=kBT ! 1 the model reduces to that of anadditive binary hard-sphere mixture. A similar treat-ment of polymer interactions was introduced by Warrenet al. [53], although in that work �ppðrÞ was assumed tohave a range of Rp, the polymer radius of gyration. Ourcurrent (longer-ranged) choice is more consistent withthe effective (Gaussian) potentials of Louis et al. [52],which extend even beyond 2Rp. Our aim is to develop aDFT which retains most of the features of the DFT forthe original AO model but which incorporates, albeitcrudely, polymer–polymer interactions. We brieflyreview how the DFT of section 2.2 can be extended tothe current case, following [20], and then describe thebulk fluid–fluid phase separation which arises in thistheory.

5.1. Construction of the functionalIn constructing functionals for hard-sphere fluids it

has been shown [54, 55] that requiring an approximatefunctional to recover the properties of the system inthe zero-dimensional (0d) limit, where the partitionsum can be calculated exactly, is a powerful constraintwhich has guided the development of DFTs for othermodels, such as the AO model. Thus we first considerthe current model in the 0d limit, in which particlecentres are confined to a volume v0d whose dimensionsare smaller than all relevant length scales in the system.The microstates accessible are then completely speci-fied by the occupation numbers of particles of bothspecies and each microstate is assigned a statisticalweight according to the grand ensemble. The grandpartition sum for an arbitrary binary mixture in thislimit is

X ¼X1Np¼0

zNpp

Np!

X1Nc¼0

zNcc

Nc!exp ð��totalÞ, ð25Þ

where the (reduced) fugacities are zi ¼ ðv0d=L3i Þ �

exp ð��iÞ,Li is the thermal wavelength, �i is the chemicalpotential of species i and �total is the total potentialenergy in the situation where all particles have vanishingseparation. Note that for hard-core interactions, theBoltzmann factor vanishes for forbidden configurations,which then limits the upper bounds in the summationsin equation (25). For the present case, where �cc and �cp

are hard-body interactions, we obtain

X ¼ zc þX1Np¼0

zNpp

Np!exp ��NpðNp � 1Þ=2

� �, ð26Þ

where the Np dependence in the Boltzmann factor stemsfrom the counting of pairs of polymers. There aretwo important limiting cases. For zc ¼ 0 (no colloids),

the limit of one-component penetrable spheres [56] isrecovered, whereas for �� ¼ 0, equation (26) reducesto the AO result [16, 17], X ¼ zc þ exp ðzpÞ. In order toobtain the Helmholtz free energy, a Legendre transformmust be performed, and the dependence on the fugaci-ties replaced with the dependence on the mean (occupa-tion) numbers of particles, �i ¼ zi@ ln X=@zi, i ¼ c,p.Taking the particle volume of species i as the referencevolume, �i is also the 0d packing fraction of species i.Subtracting the ideal contribution, one calculatesthe excess Helmholtz free energy, �F0d ¼ � ln XþP

i¼c, p �i ln ðziÞ �P

i¼c,p �i½ln ð�iÞ � 1�. In the presentcase (as for pure penetrable spheres [56]), this cannotbe expressed analytically. As we are interested in thecase of small � (small deviations from the AO model),we perform an expansion in powers of ��, and obtain

�F0dð1Þ ¼ ð1� �c � �pÞ ln ð1� �cÞ þ �c þ

��

2

�p2

1� �c,

ð27Þ

which is exact up to lowest (linear) order in ��. In thelimit �� ! 0, equation (27) reduces to the AO result[16, 17], which is �F0d,AO ¼ ð1� �c � �pÞ ln ð1� �cÞ þ �c.In the absence of colloids, �c ! 0, we obtain a mean-field-like expression, F0d,MF ¼ ��p

2=2.Since some terms of higher than first order can be

obtained analytically, we write the free energy asFð1Þ0d þ�F0d, and we find that up to cubic order in ��

��F0d ¼ ��p

2ð��Þ2

4ð1� �cÞþ

�p2

1� �cþ

2�p3

ð1� �cÞ2

� �ð��Þ3

12: ð28Þ

For large ��, the 0d free energy must be calculatednumerically. This is straightforward [20].

Returning to three dimensions, we write the excessHelmholtz free energy functional of an inhomogeneoussystem as

�F ex½�cðrÞ, �pðrÞ� ¼

ZdrF fncg, fn

pg

� �, ð29Þ

which is the same as in the ideal polymer case,equation (11). The weighted densities fnig are definedas convolutions with the bare density profiles throughequation (13). In addition to the weights (14) we alsointroduce Tarazona’s [57] tensor weight functionwwim2ðrÞ ¼ wi

2ðrÞ½rr=r2 � 11=3�, where 11 is the identity

matrix.The free energy density is composed of three parts

F ¼ F1 þ F2 þ F3 , ð30Þ


which are defined as

F1 ¼Xi¼c, p

n0i ’i nc3, n

p3

� �, ð31Þ

F2 ¼X

i, j¼c, p

ni1nj2 � niv1 n

jv2

� �’ij n

c3, n

p3

� �, ð32Þ

F3 ¼1

8p

Xi, j, k¼c, p

1

3ni2n

j2n

k2 � ni2n

jv2 n

kv2

�

þ3

2niv2n

jm2n

kv2 � tr nim2n

jm2n

km2

� � �’ijk nc3, n

p3

� �,

ð33Þ

where tr denotes the trace. The quantities ’i, ’ij etc. arederivatives of the 0d excess free energy

’i...kð�c, �pÞ �@m

@�i . . . @�k�F0dð�c, �pÞ: ð34Þ

In the absence of polymer, F1 and F2 are equivalent tothe free energy densities for hard spheres introducedin [18] and F3 is equivalent to the tensor treatmentfor pure hard spheres in [57]. Equations (31)–(33) aregeneralizations of these earlier treatments that includesummations over species. If we set �� ¼ 0, ideal poly-mer, and take the derivatives of �F0d,AO we recover theexplicit expressions given in equation (12) for the AOmodel.

5.2. Fluid–fluid phase separationIn bulk, the one-body densities of both species are

spatially uniform: �iðrÞ ¼ const: This leads to simpleanalytic expressions for the weighted densities. Theexcess free energy density, equations (30)–(33), is easilyevaluated provided the analytic approximations (27),(28) are employed for F0d. If we retain only the linearterm in ��, i.e. employ equation (27), then

�fexð�c, �pÞ ¼�F exð�c, �pÞ

V¼ �fHS

ex ð�cÞ � �p ln �1ð�cÞ

þ� ~��ppð0Þ

2�p

2½1� ln �2ð�cÞ� , ð35Þ

where the integrated potential is ~��ppð0Þ ¼4p

Rdr r2�ppðrÞ ¼ 4p��3

p=3. fHSex is the scaled-particle

(Percus-Yevick compressibility) approximation, and isgiven by

�fHSex ¼

3�c½3�cð2� �cÞ � 2ð1� �cÞ2 ln ð1� �cÞ�

8pR3cð1� �cÞ

2; ð36Þ

and is the same quantity that enters equation (15).The quantities �1 and �2, which depend solely on �c andthe size ratio q ¼ �p=�c, are given by

ln �1 ¼ ln ð1� �cÞ �X3m¼1

Cð1Þm m, ð37Þ

ln �2 ¼ �1

8

X4m¼1

Cð2Þm m, ð38Þ

where the dependence on colloid density is through ¼ �c=ð1� �cÞ, and the coefficients are polynomialsin the size ratio, given as C

ð1Þ1 ¼ 3qþ 3q2 þ q3,

Cð1Þ2 ¼ ð9q2=2Þ þ 3q3, C

ð1Þ3 ¼ 3q3 and C

ð2Þ1 ¼ 8þ 15qþ

6q2 þ q3, Cð2Þ2 ¼ 15qþ 24q2 þ 7q3, C

ð2Þ3 ¼ 18q2 þ 15q3

and Cð2Þ4 ¼ 9q3.

For �� ¼ 0, ideal polymer, our result is identical tothat of free-volume theory for the AO model—seeequations (15) and (16). The quantity �1 in equation (37)is identical to a, the free-volume fraction of a singlepolymer sphere in the AO model. �2 can be interpreted[20] as the free-volume ratio for pairs of overlappingpolymers. �1 and �2 both decrease monotonically withincreasing �c due to the increasing excluded volume.However, �2 > �1 over the whole density range, whichmay be due to correlations between polymer pairs [20].At fixed �c, both �1 and �2 decrease monotonically withincreasing size ratio.

The total canonical free energy is given by F=V ¼

fex þ kBTP

i¼c, p �i ½ln ð�iL3i Þ � 1�. It is convenient to

transform to the semi-grand ensemble, where thepolymer chemical potential �p is prescribed instead ofthe system density �p. The appropriate thermodynamicpotential is the semi-grand free energy Osemi, related toF via a Legendre transform: Osemi=V ¼ F=V � �p�p,where �p is given as

��p ¼ @ð�F=VÞ=@�p

¼ ln ð�pL3pÞ � ln �1ð�cÞ � � ~��ppð0Þ�p½1� ln �2ð�cÞ�,

ð39Þ

which is a transcendental equation to be solved for �ponce �p is prescribed. This result is a generalization ofthe standard free-volume expression ��p ¼ ln ðL3

p�p=�Þpertaining to the original AO model [4].

As usual, phase coexistence is determined by requiringequality of the total pressure, of the chemical potentials�i, and of the temperatures in the coexisting phases.This can be carried out in the system representation(�c, �p) [20] or, using common tangent constructions onOsemi, in the polymer reservoir representation (�c,�p).

Stable fluid–fluid phase separation (with respect tothe fluid–solid transition) is observed in experiments


on colloid–polymer mixtures only at sufficiently largepolymer-to-colloid size ratios. We consider the size ratioq ¼ 0:57, for which experimental data are available forPMMA colloid and polystyrene (PS) in cis-decalin [5].Figure 19 shows the calculated phase diagrams with andwithout polymer interactions. For non-interacting poly-mers (�� ¼ 0), our result is identical to that from freevolume theory for the AO model. In order to apply ourtheory to the experimental situation, we have prescribedthe potential energy barrier to be �� ¼ 0:5. This is basedon considerations of the second virial coefficient of apure polymer solution [20]. In order to achieve higheraccuracy than is provided by the linear expansion ofthe 0d free energy, equation (27), we use the cubic orderexpression, equation (28), to determine the excess freeenergy. Figure 19 shows a comparison of the calculatedtheoretical binodal with the experimental data of [5]in the system representation.Although the measured single-phase (fluid) state

point at high colloid packing fraction lies inside thetheoretical two-phase region, it is clear that our theorypredicts a shift in the correct direction comparedwith the non-interacting (ideal) binodal. The theoryalso predicts that the coexisting colloidal gas phase ismore strongly dilute in colloids, as compared with thenon-interacting case.In figure 20 the phase diagram is shown in polymer

reservoir representation: a reservoir of interacting purepolymer is considered that is in chemical (osmotic)equilibrium with the system with respect to exchangeof polymers. The tie lines are horizontal in thisrepresentation. Away from the critical point a gas,

dilute in colloids, coexists with a liquid that has veryhigh packing fraction of colloids, much higher than inthe ideal case. We consider next the case of equally-sizedspecies, q ¼ 1. Phase diagrams are shown in the systemrepresentation (figure 21) and in the reservoir represen-tation (figure 22). Again a marked shift of the criticalpoint towards higher �c is found and the single phaseregion is larger (in the system representation) comparedto the case of ideal polymer. Figure 22 shows that thepacking fraction of coexisting liquid increases quiterapidly with increasing �rp, which has repercussions forthe location of the triple point. All these results can beunderstood in terms of a free energy penalty arisingfrom polymer–polymer interactions. These manifestthemselves primarily in the colloidal gas phase as onlya small penalty arises in the colloidal liquid phase, wherepolymers are strongly diluted.

It is instructive to make some comparisons betweenthe present results and the recent extensive Monte Carlosimulation results of Bolhuis et al. [15] mentioned in

0

0.2

0.4

0.6

0 0.1 0.2 0.3 0.4 0.5

η P

ηC

βε=0.5ideal polymer

liquid-gasfluid

Figure 19. Fluid demixing binodals in the system represen-tation, i.e. as a function of the colloid (�c) and polymer(�p) packing fractions for size ratio q ¼ 0:57. Results areshown for the case of non-interacting polymer (dashedline) and for interacting polymer (solid line) with strength�� ¼ 0:5. The crosses denote the critical points. Thesymbols refer to the experimental state points takenfrom [5]. The open circles denote single-phase (fluid) statesand the filled rhombuses two-phase (liquid–gas) states.

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

η Pr

ηC


Figure 20. Same as figure 19 (q ¼ 0:57), but in the reservoirrepresentation, i.e. as a function of �c and the packingfraction of polymer (�rp) in a reservoir of pure polymerin osmotic equilibrium with the system.

0

0.2

0.4

0.6

0 0.1 0.2 0.3 0.4 0.5

η P

ηC


Figure 21. Same as figure 19, but for size ratio q ¼ 1.


section 4.3. These authors consider hard-sphere colloidsbut obtain effective potentials for the polymer–polymerinteraction from simulations of a bulk system of self-avoiding walks (SAW) at various concentrations—aprocedure known to be reliable for polymers in a goodsolvent. The effective pair potentials �pp correspondto tracing out the monomer degrees of freedom andtreating the polymer as ‘soft colloid’; �pp dependson the volume fraction of the polymer and is calculatedby inversion of the centre-of-mass radial distributionfunction gppðrÞ. The colloid–polymer effective potentialalso depends on the volume fraction of polymer. It isobtained from simulations of a single hard sphere ina solution of SAW polymers by inverting the centre-of-mass concentration profile using the HNC equation—details are given in [15]. Phase diagrams are determinedby Gibbs ensemble simulations of the binary mixturedescribed by these effective pair potentials and resultsare presented for q ¼ 0:34, 0.67 and 1:05. We focushere on size ratios q ¼ 0:67 and 1.05 where gas–liquid(fluid–fluid) phase separation is stable with respect tofluid–solid.In the (�c, �

rp) representation including polymer–

polymer interactions increases the critical point valueof the packing fraction of colloids very slightly butincreases the critical polymer reservoir fraction �rp, crit byabout 0.25 for q ¼ 0:67. More significantly, the gas–liquid two-phase region is broadened significantly in �cand the triple point �rp, t is lowered considerably fromwhat is found in the ideal polymer (AO) case—seefigure 1 of [15]. The upshot is that there is a very narrow,in �rp, region of stable fluid–fluid phase coexistence wheninteractions are included. This is not totally differentfrom what we find in figure 20, for q ¼ 0:57, from theDFT; as freezing is expected to occur when �c � 0:5, thispermits only a small separation between �rp, t and �rp, critwhen interactions are incorporated. However, our DFTdoes not predict the substantial increase in �rp, crit which

is found in simulations when interactions are included.This is reflected clearly in the (�c, �p) representationwhere the binodal calculated with interactions appearsto be more separated from the non-interacting case (seefigure 2 of [15]) giving a larger single-phase region thanin the corresponding DFT results, although the latterare for a smaller value of q. The situation is exaggeratedfor q ¼ 1:05. Now the critical point is shifted from�c, crit � 0:11, �rp, crit � 0:75 to �c, crit � 0:18, �rp, crit � 1:12and the triple point is lowered from what is a very highnumber �rp, t � 6 to �rp, t � 1:65 when interactions areincluded. Again the range in �rp over which stable fluid–fluid phase coexistence occurs is narrowed considerably.A similar trend is found in the DFT results of figure 22for q ¼ 1 but the latter predict a small decrease in �rp, critrather than the substantial increase found in simulation.Moreover, the DFT does not predict the very strongshift of the binodal in the (�c, �p) representationfound in simulations (figure 3 of [15]) which results ina much larger single-phase region when interactionsare included.

In summary, the DFT introduced in [20] and descri-bed above captures several, but not all, of the effects ofpolymer–polymer excluded volume interactions that arefound in the computer simulations of bulk phase behav-iour. (Note, however, that the freezing transition wasnot considered explicitly in the DFT calculations.) Thisshould encourage applications of the theory to inhomo-geneous situations—the primary purpose of DFT.Such problems are not easily tackled by simulations.

Of course, this model of polymer–polymer interac-tions is highly idealized and a detailed critique is givenin [20]. The main limitations are: (i) �ppðrÞ is assumed tobe a step function rather than a Gaussian-type function,(ii) the range of the step function is set equal to 2Rp,twice the polymer radius of gyration, in order to derive ageometry-based DFT meeting ‘additive’ restrictions and(iii) the strength � and range of �pp are assumed to beindependent of the concentrations of polymer andcolloids, whereas simulation studies of SAW [15, 52]indicate that the effective polymer–polymer potentialshould depend on the volume fraction of polymer.

We conclude this section by emphasizing [20] that thepresent DFT approach is not a perturbative treatment ofpolymer–polymer interactions. One might attempt toconstruct a DFT which starts with the AO functionalfor the AO reference model and then simply adds a(perturbative) contribution to account for polymer–polymer interactions. The latter could be the type ofmean-field functional used recently to describe a purepolymer system [21], namely

FMF½�pðrÞ� ¼1

2

Zdr

Zdr0�pðrÞ�pðr

0Þ�ppðjr� r0jÞ: ð40Þ

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

η Pr

ηC


Figure 22. Same as figure 20, but for size ratio q ¼ 1.


This functional generates the pair direct correlationfunction which corresponds to the random phaseapproximation: cð2Þppðr, r

0Þ ¼ cð2Þppðjr� r0jÞ ¼ ��ppðjr� r0jÞ,known to be a good approximation at high densitiesfor penetrable potentials [21]. It has been used to goodeffect in investigations of repulsive Gaussian coreparticles adsorbed at a hard wall [58]. The generalizationof equation (40) to binary Gaussian core mixtures wasused in studies of fluid–fluid interfaces [41] and ofwetting phenomena [48]. At first sight the functionalFAO

ex þFMF would appear to provide a reasonabledescription of the excess free energy of the AO mixture;note that the hard-body interactions between colloidsand polymers are already included in FAO

ex . However, asshown in [20], this is not the case for the model describedabove. The perturbative contribution FMF dependssolely on the polymer density �pðrÞ and thereforeneglects exclusion of polymer from the volume occupiedby the colloids. By contrast the geometry-based DFTbuilds in excluded volume effects. One can understandthis by returning to the 0d free energy, equation (27).The final (polymer) contribution is enhanced over thecorresponding mean-field expression ��2p=2 by a factorð1� �cÞ

�1. The perturbative DFT exhibits some severefailings when applied to the calculation of phasebehaviour and similar failings are to be expected [20]if the same approach, i.e. simply adding FMF to FAO

ex ,were adopted for other models, e.g. mixtures of hard-core and Gaussian particles. Note that the approach ofWarren et al. [53] for the bulk free energy can be viewedas a perturbative treatment in which �ppðrÞ is regarded asa perturbation about the AO reference system. Schmidtet al. [20] rederive the theory of [53] in a frameworkthat allows them to understand relationships betweenthe various approaches. They argue that the geometry-based DFT provides a more realistic account of thebinodal than does the theory of [53].

6. Further developments

During the last two years several applications andextensions of the DFT for the AO model have beenreported. We summarize some of them in this section.An overview of geometry-based DFT, which lists awider range of developments, is given by Schmidt [59].We mention first two direct applications of the DFT

for the AO mixture described in section 2.2. As a planarhard wall prefers the colloid-rich (liquid) phase to thecolloid-poor (gas) phase, the hard wall–liquid surfacetension is lower than the hard wall–gas tension. Generalarguments [60] then imply that for the colloid–polymermixture confined between two, parallel, planar hardwalls, capillary condensation of the liquid phase shouldoccur when the reservoir fluid is in the gas phase, i.e.for a given �rp the value of �c is lower than the bulk

coexistence value. Brader observed this phenomenonin his PhD studies [25] using the DFT. Systematicinvestigations were carried out recently [61] using bothDFT and computer simulation. Lines of capillary con-densation were determined for size ratio q ¼ 1. Theseare most easily represented in the (�c, �

rp) plane, where

�c is the colloid chemical potential. Upon decreasing the(scaled) wall separation distance H=�c from ten to two apronounced shift of the capillary binodal towardssmaller values of �c occurs. The critical point shifts tolarger �rp, corresponding to lower temperature in thecase of a simple atomic fluid, and to larger �c uponreducing H. This latter trend seems to be much morepronounced in the simulation results than in those fromDFT. Preliminary investigations demonstrated that theshift of the binodal could be described reasonably wellusing the generalized (to binary mixtures) Kelvinequation for capillary condensation [62]. The practicalimportance of this investigation lies in possible experi-mental realizations by strongly confining (between glasssubstrates) real colloid–polymer mixtures [63]. From atheoretical viewpoint one sees that since all the bareinteractions in this system are either hard or ideal, thisis an important example of shifting a bulk phasetransition, via confinement, by means of purely entropic(depletion) forces.

The second application refers to the AO modelcolloid–polymer mixture exposed to a standing laserfield that is modelled as an external potential acting onthe colloids. This has a sinusoidal variation as a functionof the space coordinate in the direction of the beam[64]. DFT results indicate that the external potentialmay stabilize a ‘stacked’ fluid phase which is a periodicsuccession of liquid and gas slabs. The regions of largelaser intensity (where the external potential is small)are filled with colloidal liquid whereas the regions withsmall laser intensity are gas-like.

Several extensions of the AO model have been madeand the bulk phase behaviour compared with theoriginal. In order to take into account the effect ofpoor solvent quality on colloid–polymer mixtures, thesolvent was modelled as a distinct component. Speci-fically it was taken to be a binary mixture of a primarysolvent that is treated (as usual) as a homogeneous,inert background and a secondary cosolvent that istreated as an ideal gas of point particles [65]. Thesecosolvent particles are assumed to penetrate neitherthe polymers nor the colloids. In the absence of colloids,the polymer–cosolvent subsystem is the Widom–Rowlinson model [66] of a binary mixture in whichparticles of like species are non-interacting while unlikespecies interact with hard cores. Thus the cosolventinduces an effective many-body attraction betweenthe polymers, reminiscent of that caused by poor solvent.


It was found that worsening the solvent quality, byincreasing the cosolvent concentration, increases greatlythe tendency to demix, i.e. this shifts the correspondingbinodal to smaller colloid and polymer packing fractions[65].The shape and size of real polymers will be affected by

confinement effects, generated by the presence of thecolloidal particles in a colloid–polymer mixture. In orderto model the effect on the polymer size distribution,an AO model with polydisperse polymer spheres wasconsidered [67]. The polymer spheres are mutuallynon-interacting but are excluded from the colloids;their radii are free to adjust to allow for colloid-inducedcompression. The size (radius of gyration) probabilitydistribution in the polymer reservoir was taken to bethat of ideal chains. It was found that the presence of thecolloids reduces considerably the average polymer size.As a consequence, the bulk demixing binodal is shiftedtowards higher polymer densities, stabilizing the single,mixed phase as compared with the incompressibleAO case.For very large polymer-to-colloid size ratios the

assumption that colloids cannot penetrate polymersis no longer valid. A small colloidal sphere may wellpenetrate the open coil structure of a big polymer. Inorder to incorporate this effect on the level of an effectivesphere model a penetrable AO model was introduced[68]. The colloid–polymer interaction is assumed to be astep-function of finite height, in contrast to the hard-core repulsion of the conventional AO case. The rangeof this interaction is still taken to be Rc þ Rp, and thestrength is taken from the known immersion free energyof a single sphere in a dilute solution of polymer coilsor in a theta solvent. The colloid–colloid interactionremains hard-sphere-like and �pp ¼ 0. For large sizeratios, q03, there is a significant increase in the extentof the mixed region in the phase diagram compared tothe free-volume result for impenetrable polymer, i.e. forthe AO model [68]. These findings are in keeping withresults from more microscopic approaches which treatexcluded volume at the segment level [14].A system that is closely related to the AO model is a

mixture of spherical hard-sphere colloids and hard,needle-like particles. The latter may represent eitherstretched polymers or stiff colloidal rods. In the simplestmodel [69] the thickness of the rods is set to zero suchthat rod–rod interactions can be ignored but thereremains an excluded volume interaction between a rodand a hard sphere. For this model there is no liquidcrystalline order, because of the absence of rod–rodinteractions. However, simulation studies found iso-tropic gas and liquid phases as well as a solid phase[69]. The rods act as a depletant, in a similar fashionto the non-interacting polymer spheres in the AO

model, and give rise to an effective attraction betweenthe hard-sphere colloids. Bolhuis and Frenkel [69] alsodetermined the phase behaviour using a free-volumeapproximation for the free energy, similar to that ofLekkerkerker et al. [4] for the AO model, i.e. a first-order perturbation theory that sets the free-volumefraction for the needles equal to its average value in thepure hard-sphere fluid. A DFT was constructed [22]for this model and applied subsequently [23] to theplanar (free) interface between demixed fluid phases, oneof which, the liquid, is rich in spheres (and poor inneedles) and the other, the gas, is dilute in spheres (andrich in needles). Note that in constructing the DFT itwas necessary to introduce a weight function wSN

2 ðr,:Þ,where : refers to the orientation of the needles, whichcontains information about both species, the spheresand the needles, in order to generate the sphere-needleMayer bond. The corresponding sphere-needle weighteddensity, nSN2 ðr,:Þ, is a convolution of the sphere density�sðrÞ and wSN

2 ðr,:Þ, whereas all the remaining weighteddensities involve only variables of an individual species.For uniform fluids the DFT yields the followingexpression for the excess free energy density:

�fexð�S, �NÞ ¼ �fHSex ð�SÞ � �N ln �ð�SÞ, ð41Þ

where �S and �N refer to the number densities of spheresand needles, respectively. fHS

ex is, as usual, the excessfree energy density of pure hard spheres (36) and �ð�SÞis the free-volume fraction for a single, test needle oflength L in the hard-sphere fluid:

�ð�SÞ ¼ ð1� �Þ exp ½�ð3=2ÞðL=�Þ�=ð1� �Þ�, ð42Þ

with � ¼ p�S�3=6, the packing fraction of spheres ofdiameter �. Equation (41) is identical to the free-volumeresult [69] and can also be obtained by applying scaled-particle theory for non-spherical bodies [70] to thecurrent model. It has the same form as the excess freeenergy for the AO model—see equation (15). The fluid–fluid binodal resulting from equation (41) was found tobe rather close to that obtained by simulations for thecase L=� ¼ 1 [69]. Brader et al. [23] chose the same sizeratio for their DFT studies of the planar fluid–fluidinterface. The bulk phase diagram, plotted in thereservoir representation, i.e. �r� ¼ �rNL

2� versus �, hasa similar form to figure 2 with � ¼ �c and �r�, thereduced needle density in the reservoir, replacing thepolymer reservoir packing fraction �rp. The critical pointis located at �crit ¼ 0:158 and �r�, crit ¼ 14:64 whilst thetriple point, estimated from first-order perturbationtheory [69], is at �r�, t ¼ 24. As in figure 2, there is aFisher–Widom (FW) line intersecting the liquid branchof the binodal near �r�,FW ¼ 17, see figure 2 of [23].For coexisting states with higher values of �r�, on the


oscillatory side of the FW line, Brader et al. find spheredensity profiles �SðzÞ that are oscillatory on the sphere-rich (liquid) side of the planar interface. However, theamplitude of the oscillations is considerably smallerthan what was described in section 3 for the AO model.As expected, on the needle-rich (gas) side the densityprofile of the spheres decays monotonically into bulk forall states. The important new feature which arises in thisstudy concerns the nature of the needle density profiles.Within DFT one can determine the one-body density�Nðz, �Þ, where z is the perpendicular distance fromthe interface and � is the angle between the needleorientation and the interface normal, and hence both theorientation averaged needle density profile, ��NðzÞ, andthe orientational order parameter profile hP2ðcos �Þi.For states above the FW line, ��NðzÞ displays very weakoscillations on the liquid side of the interface arisingfrom the influence of the packing of the spheres on thedistribution of the needles. From the result forhP2ðcos �Þi, Brader et al. conclude that needles orderparallel to the interface on the needle-rich side. Thisappears to be similar to the biaxial ordering of needlesthat occurs at a planar hard wall; in the present casethe densely packed hard-sphere fluid acts as a ‘wall’ forthe needles. On the sphere-rich side the needles preferto orient themselves perpendicular to the interface andthis is interpreted as the needles protruding through thevoids in the first layers of the hard-sphere fluid. Thesegeneral features of the orientational order are indepen-dent of the size ratio L=�. As an optimal compromisebetween manageable needle particle numbers andsimulation box size, Bolhuis et al. [71] chose toperform Monte Carlo simulations with ratio L=� ¼ 3.The different preferential alignment of needles on eitherside of the interface is confirmed by the simulationresults. Moreover, there is remarkable quantitativeagreement between simulation and DFT results for thedensity profiles �SðzÞ, ��NðzÞ and the orientational orderparameter profile [71]. Brader et al. [23] also describeresults for the liquid–gas surface tension and discusswhat might be appropriate scaling factors to bringabout data collapse for different size ratios L=�.In a related study Roth et al. [24] considered the same

sphere–needle mixture, with L=� ¼ 1, adsorbed at aplanar hard wall—the same type of situation as thatdescribed in section 4. Density profiles calculated fromthe DFT for the binary mixture of spheres and needlesshow that complete wetting of the hard wall–gasinterface by the sphere-rich, liquid phase occurs uponapproaching the binodal on a path at fixed �r� ¼ 16; theprofiles are reminiscent of those in figure 7. Thereappears to be complete wetting for all the values of �r�considered. An effective one-component description ofthe same system was also investigated in [24]. This

follows the approach of section 2.1 in that a sphere–sphere and a wall–sphere depletion potential can beobtained by integrating out the degrees of freedom ofthe needles (rods) [72]. Since the needles are mutuallynon-interacting the depletion potentials depend linearlyon �rN. Geometrical arguments, similar to those for theAO model, show that the effective one-componentHamiltonian corresponding to equation (9) should beexact for L=� < 1� ð2ð31=2Þ � 3Þ1=2 ¼ 0:31875, whichshould be compared with the corresponding size ratioq ¼ �p=�c ¼ 0:1547 for the AO model. Thus, for thesystem under consideration, with L=� ¼ 1, there aremany-body effective interactions which are not includedin the analogue of equation (9). Ignoring these, Rothet al. [24] employed a one-component DFT, equivalentto that used in [11] for adsorption studies of the AOmodel with q ¼ 0:1; this treats the hard-sphere con-tribution by the Rosenfeld FMT and the depletionattraction between pairs of spheres by means of a mean-field approximation. The resulting fluid–fluid binodal isin fair agreement with that from the free-volume theory,equivalent to the result of the binary mixture DFT.The effective one-component treatment also predictscomplete wetting. However, there is also a partialwetting regime, at high �r�, followed by a sequence offive layering transitions and then a transition to thecompletely wet state as �r� is reduced following the gasbranch of the binodal. The pattern of surface phasetransitions (see figure 5 of [24]) is very similar to thatseen in figures 9 and 12 for the AO model, i.e. the firstlayering transition is quite extended in �r� and thelayering critical point, near �r� ¼ 20, is well removedfrom the binodal. Subsequent transition lines arereduced in extent and lie close to the binodal. Thus, itis tempting to argue that this effective one-componenttreatment of the adsorbed sphere–needle mixture mimicswhat we found for the AO model. But there we arguedthat it was the incorporation of effective many-bodyinteractions between the colloids that was responsiblefor the rich layering behaviour; our effective one-component treatment did not yield layering transitionsin the case of the AO model. A full understanding ofthe results for the sphere–needle model is yet to emerge.The precise location of the triple point is important.Indeed a preliminary simulation study of the bulksystem [73], using the same effective sphere–spherepotential as in [24], finds that �r�, crit 17:5 and�r�, t 22:7, i.e. gas–liquid coexistence is stable over arather narrow region of �r�. More significantly, the triplepoint lies below the onset of the first layering transitionfound in [24], suggesting that the layering could beoccurring in a region where the gas–liquid transition ismetastable with respect to gas–solid. This is not the casein the AO model where in both the DFT calculations of


section 4.2 and in the simulation studies of section 4.3,the onset of layering occurs well below the triple point.A further study [74] was devoted to developing a

geometry-based DFT for a ternary mixture of hard-sphere colloids, ideal polymer spheres and vanishinglythin hard needles. The model combines those of AO andof Bolhuis–Frenkel. Both the mutually non-interactingpolymers and mutually non-interacting needles act asdepletion agents so that rich fluid phase separationcan arise. As usual, colloid–colloid, colloid–polymerand colloid–needle interactions are hard. Schmidt andDenton [74] consider two cases: (i) the polymer–needleinteraction �PNðrÞ vanishes for all distances r and (ii) it ishard, i.e. �PN ¼ 1 if polymer and needle overlap, andzero otherwise. Constructing the DFT for the secondcase requires a generalization to needles of an earlierDFT treatment [75] of the Widom–Rowlinson model[66] since in the absence of colloids the polymer–needlemixture is of the Widom–Rowlinson type: interactionsbetween like species vanish while unlike particles interactvia hard-core repulsion. For case (i), two-phase coex-istence between colloid-rich and colloid-poor fluidphases is found. For case (ii), there is the possibilityof de-mixing between the polymers and the needles.Moreover, there can be a competition between thedepleting effects of (interacting) polymers and needles;the two species compete to generate the attractionbetween the colloid spheres. Striking demixing fluidphase behaviour is found, with various critical pointsand a three-phase coexistence region [74]. It would beof considerable interest to investigate structural correla-tions in the various bulk fluid phases and to considerinhomogeneous situations, such as fluid–fluid or wall–fluid interfaces, in this model system.A very different kind of confinement from that

considered so far is present for fluids adsorbed inporous media. Rather than the well-defined geometrieswhich occur at a planar wall or at fluid–fluid interfaces,irregular and random confinement acts on the adsorbedfluid. In order to model porous media one introduces theso-called quenched–annealed fluid, where the quenchedcomponents represent the porous medium in terms ofimmobilized fluid configurations. Since the quenchedcomponents act as an external potential on the annealed(equilibrated) components constituting the adsorbate,the one-body density distribution of the latter can beobtained using standard DFT methods. However, suchan approach is computationally demanding since thespatial distribution of the adsorbate is extremely com-plicated. An alternative approach, termed quenched–annealed DFT, was proposed recently. This treats thequenched components on the level of their one-bodydensity distributions [76]. The minimization conditiondiffers from that of equilibrium (fully annealed)

mixtures, as the quenched components are treated asfixed input quantities in the grand potential functional.For the AO model adsorbed in different types ofmatrices the results of this DFT approach were com-pared to those from liquid integral equation theory(replica OZ using the optimized random-phase approx-imation) for the corresponding effective one-componentmodel, where the polymers are integrated out and onlythe pairwise contributions to the effective Hamiltonianof section 2.1 are taken into account [77]. Thus, theeffective interaction between colloids is given by �eff

AOðRÞwhile the colloid and matrix particles interact via eithera hard-sphere potential or �eff

AOðRÞ. Both approachespredict, consistently, capillary condensation or evapora-tion, depending on the nature of the matrix–polymerinteraction. If a matrix particle excludes both colloidand polymer, condensation occurs, whereas if the matrixexcludes only colloid (there is vanishing matrix–polymerinteraction) capillary evaporation occurs. The latterrefers to the situation where phase separation occurs inthe fluid adsorbed in the matrix at chemical potentialsfor which the reservoir would remain in a single, colloid-rich, liquid phase. The bulk pair correlation functionsfrom DFT are in good agreement with computersimulation results [77]. A very recent study is devotedto demixing and the planar fluid–fluid interface of theAO mixture adsorbed in a matrix of homogeneouslydistributed hard-sphere particles [78]. Two cases areconsidered: (i) colloid-sized matrix particles at lowpacking fractions and (ii) large matrix particles at highpacking fractions. The two cases exhibit very differentbehaviour; for details see [78].

7. Discussion

We have outlined two strategies for tackling thestatistical mechanics of inhomogeneous colloid–polymermixtures described by the simplest model that capturesthe effects of depletion attraction, namely the Asakura–Oosawa–Vrij (AO) model. The first of these, based onthe well-trodden ‘integrating out’ route of McMillan–Mayer theory, is most useful for highly asymmetricmixtures with small size ratio q < 0:1547, where thereare no three- or higher-body contributions to theeffective one-component Hamiltonian for the colloids.The second employs a geometry-based DFT for themixture that treats the two species on equal footing.Since there is no explicit integrating out of the polymerdegrees of freedom, depletion effects are generatedinternally by the geometrical structure of the functional.This DFT approach has the advantage that it applies toarbitrary size ratios, including those where many-bodycontributions are known to be important. The mainresults of our investigations of interfacial propertiesare presented in sections 3 and 4 where we describe the


density profiles and surface tensions for the free fluid–fluid interface and the properties of the AO mixtureadsorbed at a hard wall calculated using the functionalderived in [16, 17]. For the free interface we findoscillatory structure in the density profiles on thecolloid-rich (liquid) side of the interface for state pointswhere the coexisting liquid density lies on the oscillatoryside of the Fisher–Widom line and surface tensionswhich are in reasonable agreement with experiment.We find that for a highly asymmetric mixture, q ¼ 0:1,at state points away from any phase boundary, the AOfunctional gives a good account of the simulation datafor density profiles at a hard wall. For larger size ratios,where fluid–fluid coexistence occurs, the DFT predictsvery rich interfacial phase diagrams which display botha wetting transition and several novel layering transi-tions. The interfacial phase diagram was determined forsize ratios q ¼ 0:6, 0:7 and 1:0 and we find the topologychanges significantly with q. We make a comparisonwith recent Monte Carlo simulation results for theinterfacial phase diagram for size ratio q ¼ 1.Here we discuss further some features of our results.

The oscillations we observe in the free interface colloiddensity profiles are dramatic. They appear to have anamplitude larger than oscillations which have beencalculated for simple liquids at state points near thetriple point. We argue in section 3 that despite the verylow interfacial tension of the colloid–polymer system,the capillary wave fluctuation erosion of the oscilla-tions should not be greater than for a simple liquid. Thissuggests that colloid–polymer mixtures might providean excellent opportunity to investigate oscillatorystructure at fluid–fluid interfaces, as the large size ofcolloidal particles makes them well suited to ellipso-metric measurements [39, 79]. Oscillations at a freefluid–fluid interface arise when that interface is ‘stiff’, i.e.for high values of the appropriate reduced surfacetension �, and when the density difference between thetwo coexisting bulk phases is large. Under theseconditions the free interface (treated at mean-fieldlevel) behaves as a ‘wall’ and layered structure candevelop due to packing effects. These effects are mostpronounced near the triple point where both the reducedsurface tension and the density difference are largest.Considering the phase diagram for q ¼ 0:6, shown infigure 2, and identifying �rp with T�1, it is clear that theseparation between the critical and triple points is muchlarger in the present case than for a simple fluid such asargon. It is primarily for this reason that we can obtainsuch pronounced oscillations in the colloidal densityprofiles for states near the triple point. By contrast, asthe temperature is reduced in simple liquids theoscillations can only grow slightly before the liquid–vapour coexistence region runs out and the triple point,

i.e. the crystalline phase, intervenes. One could arguethat oscillatory liquid–gas interfaces would be acommon feature in simple fluids at sufficiently lowtemperatures were it not for the onset of crystallization.The AO model can generate oscillations of even largeramplitude than those shown in figure 3. As the size ratiois increased the separation in �rp between the critical andtriple points grows, e.g. for q ¼ 1, �rp, t=�

rp, crit � 8, see

figure 18, and � can become rather large—see figure 3.A similar scenario arises for binary mixtures of repulsiveGaussian core particles. There liquid–liquid coexistencecan occur over a very large range of total density, sincethere is no crystalline phase present, and the resultinginterfaces can be very ‘stiff’ leading to predictions ofvery weak erosion of the oscillations: the exponentin equation (20) is calculated to be about �0:1 [41].Pronounced oscillatory structure is also well known inthe study of liquid metals where there is a largeseparation between the triple and critical points andthe reduced liquid–gas surface tensions are very large.Simulations of the liquid–gas interface of alkali metals[80] and of Ga [81] and X-ray reflectivity experiments onboth Ga [82] and Hg [83, 84] all indicate stratificationof the ion-density profile, with a spacing of about oneatomic diameter, on the liquid side of the interface.

The recent Monte Carlo studies of Chacon andco-workers [43, 44] are significant in this context. Theseauthors constructed classical pair potential (no explicittreatment of the electrons) models of a metal so thatthe melting temperature Tm was deliberately suppres-sed relative to the critical temperature Tc, typicallyTm=Tc90:2. At low temperatures they find high surfacetensions and thus a ‘stiff’ interface which seems to be theorigin of the strongly stratified density profiles whichthey observe. Chacon and co-workers [44] also makethe important point that for models of the type theyconsider, strong stratification can occur over a fewatomic layers in the interface even when the bulk liquidis on the monotonic side of the FW line. However, weknow the ultimate asymptotic decay into bulk of thedensity profile must be governed by the behaviour of thebulk pair correlation functions. At very low tempera-tures T � Tm, they estimate the erosion of the oscilla-tions to be weak, with the exponent in equation (20)about �0:4. The calculated exponent increases with Tto a value of about �1, near the FW line. Monte Carloresults for the density profiles calculated for increasinginterfacial area are consistent with the power-lawprediction of equation (20) [44]. There are, of course,important differences between the liquid metal models,where the large ratio Tc=Tm arises from the inclusion,albeit empirically, of electronic effects into the pairpotential, and the AO model, where the large value of�rp, t=�

rp, crit arises from the presence of effective many-


body interactions between the colloids—recall that thelatter are responsible for separating the triple andcritical points at large size ratios—see the discussion offigure 18.We turn now to the layering transitions we find in

section 4.2. These are a very specific prediction of theAO functional and since they are found in simulation,appear to be a genuine feature of the AO model, ratherthan an artefact of the DFT.Although the pattern of the layering transitions

observed for q ¼ 1 in the simulations of [19] is notidentical to that from DFT, the shapes of the densityprofiles near the transitions are very close to those foundin the theory so the layering phenomena are certainlythe same. We argued in section 4.3 that the occurrenceof such transitions, far from �rp, t, requires the effectivemany-body interactions that are incorporated intothe DFT and into the simulations. Without these contri-butions the triple and critical points lie much closertogether and there is less scope for the layering(and wetting) transitions to manifest themselves at the(stable) gas–hard wall interface. We emphasize, onceagain, that simulations employing only the effective pairpotential �eff ðR; zpÞ yield a very narrow range, in �rp, ofstable liquid–gas coexistence. The form of the wall-induced effective interactions must also be importantfor the occurrence of layering transitions. The effectivepair interaction between colloids �eff ðRi;Rj ; zpÞ is acomplicated function of the coordinates Ri and Rj whenthe colloids are close to the wall [11]; the strength of theattraction is lowered compared with that of �eff

AOðRij ; zpÞ,which pertains to the homogeneous fluid. It is possiblethat this reduction in pairwise attraction, caused bythe wall reducing the overlap volume of the depletionzones around each colloid, competes with the attractiveone-body potential �wall

AO ðz; zpÞ and that this situationfavours layering [19, 25]. However, it is fair to say thatwe do not, as yet, have a full understanding of whatgives rise to these curious transitionsy. The issues arecompounded by the fact that layering transitions werefound in an effective one-component DFT calculationfor the analogous case of the sphere–needle mixture[24]. Although in this case, see section 6, there remain

questions as to how well the one-component theoryaccounts for the extent of stable liquid–gas coexistence.

Whether the layering transitions remain in modelswhich treat the wall and/or the polymer more realisti-cally remains to be seen. It would be of considerableinterest to investigate adsorption at a hard wall for thesimple model described in section 5. Incorporatingpolymer–polymer interactions is expected to reducesubstantially the separation between �rp, t and �rp, crit(this is also what is found in the simulation studies forSAW models of the polymer [15]) which might make itdifficult to observe any layering transitions and indeedthe pronounced oscillations found in the colloid densityprofiles at the free interface for the AO model. Directsimulation studies for colloids and interacting polymersnear a wall would be valuable, but extremely demand-ing! It seems unlikely that such layering transitionscould be observed in adsorption experiments as thetransition lines lie so close to the bulk coexistence curvethat remarkable experimental accuracy would berequired to resolve these. Moreover, polydispersity inboth colloid and polymer sizes, roughness of thesubstrate etc. would tend to eliminate what are probablyrather subtle and specific features of the AO mixtureadsorbed at a hard wall.

We conclude this discussion of layering transitions byreturning to the difference between the type of transi-tions found in the AO model and those found in DFT[51] and in simulation [87] studies of the adsorptionof simple gases at strongly attractive substrates. Inthe latter case the transitions always occur close to thetriple point where the density profiles are more highlystructured than those shown in section 4.2, i.e. theindividual layers are much sharper. When a transitionoccurs (at fixed T, increasing �) the jump in the Gibbsadsorption corresponds to the addition of (roughly)one dense ‘liquid’ layer. As the sequence of transitionsprogresses the magnitude of the jump in adsorptionincreases, reflecting the broadening of the outer peaksin the profile and the fact that the local density inone or two of the previous layers also increases atthe transition. There can be a very large number oftransitions—nine or ten could be discerned in DFTcalculations [51]—but their critical points all lie quiteclose to the triple point Tt. Indeed, within a mean-fieldtreatment the number of transitions can be infinite,i.e. complete wetting of the gas–substrate interfaceby liquid, which is expected to occur for all T > Tt forstrongly attractive substrates, can occur by an infinitesequence of layering transitions out of coexistence.When capillary wave fluctuations in the wetting film areincluded the number of transitions must be finite, sincethe liquid–gas interface is always rough (in the statisticalmechanics sense). By contrast, for the AO mixture

yIt is well known that wetting transitions can dependsensitively on the details of the wall-fluid interactions. Forexample, in a Landau treatment which includes a surface termof third order in the surface magnetization m1, which is ofopposite sign to the usual linear �h1m1 term, Indekeu [85]found a single thin–thick transition, followed by a continuouswetting transition on increasing T along the bulk coexistencecurve. An equivalent scenario was found by Piasecki andHauge [86] in a simple DFT treatment of a Yukawa fluidadsorbed at a wall exerting both an attractive exponential anda square-well wall-fluid potential.


adsorbed at a hard wall the onset of layering transitionsoccurs far below �rp, t where the density profiles of thecolloid are not as highly structured (see figure 8) and thejump in the adsorption, Gc, does not necessarily increaseas the sequence increases (but note that the results insection 4.2 refer to a path following the bulk coexistencecurve rather than following paths at fixed �rp, equivalentto fixed T.) It is feasible, nevertheless, that within ourDFT treatment of the AO mixture there could bean infinite sequence of layering transitions leadingto complete wetting of the ‘gas’–hard wall interface.Our numerical results do not rule out this scenario.Given that the layering transition lines get shorterand shorter as this sequence progresses it is difficultto ascertain the precise sequence. If this is the correctinterpretation, then, as observed, we would not expectto find a substantial pre-wetting line emerging from thewetting point. The simulation results [19] cannot shedmuch light on this issue; there one expects a finite butlarge number of transitions before the onset of completewetting if the attractive one-body wall–fluid potential�wallAO ðz; zpÞ is driving the wetting transition. Note that

layering transitions at temperatures above the bulktriple point have been found recently in Gibbs ensembleMonte Carlo simulations of TIP4P water in cylindricalpores [88].We complete this discussion of layering transitions

between distinct adsorbed fluid phases by pointing outthat there is much experimental evidence for suchinterfacial behaviour in molecular fluids adsorbed on agraphite substrate under near triple point conditions—see references in [51]. Ellipsometric studies of variousliquids adsorbed on Pyrex glass [89] and on mica [90] atlow temperatures also provide evidence for layering, i.e.steps in the adsorption isotherms. Steps in the relativeadsorption of 2.5-dimethylpyridine in the liquid mixturewith water, in contact with solid silica have also beenidentified as layering transitions [91]. Bonn et al. [92, 93]have argued on the basis of ellipsometric measurements,that a series of first-order layering transitions occurs at afluid substrate, i.e. at the liquid–gas interface in a binarymethanol–cyclohexane mixture at high temperatures.The wetting transition in colloid–polymer mixtures

is a much more promising candidate for experimentalinvestigation and the basic phenomenon of wettingof a ‘hard’ substrate by the colloid-rich phase should,in principle, be experimentally observable. Althoughexperiments with colloidal particles have an advantageover those using rare gases or fluids composed of smallmolecules in that they can be performed at roomtemperature, issues such as polydispersity obviouslycomplicate matters. Nevertheless, systematic measure-ments of the contact angle for the liquid–gas meniscusof phase-separated colloid–polymer mixtures, at various

compositions and size ratios, in contact with glass wallswould be most interesting. Indeed, there are alreadyindications, from contact angle measurements [94, 95],of a transition from partial to complete wetting for amixture of silica particles and PDMS, with size ratio� 0:93, in a solvent of cyclohexane at a glass substratecoated with the same organophilic material as the silicaparticles (on which PDMS does not adsorb). Thetransition appears to take place further from the criticalpoint, measured in terms of �lc � �gc , the difference incolloid packing fractions of the liquid and gas, than isfound in simulations or in DFT for the AO model—seesection 4. Measuring the contact angle for these systemsis not straightforward, however. Although the staticinterfacial profile (shape of the meniscus) can bemeasured using an optical microscope and the capillarylength extracted, thereby yielding the liquid–gas surfacetension , the contact angle depends very sensitively onthe precise location of the wall and studies by theUtrecht group [96] were unable to ascertain whether ornot a wetting transition occurred for a silica–PDMSmixture. The values of which they obtained were closeto those measured by the independent spinning droptechnique [36, 79], given in figure 3. More recent resultsfrom confocal scanning laser microscopy demonstratethe presence of a thick colloidal liquid layer at the glasswall, consistent with complete wetting [63].

As mentioned in section 6, it should also be possibleto design experiments which investigate, quantitatively,capillary condensation in simple confining geometries.Such experiments are difficult to perform for atomicfluids but for colloidal systems, where the particle sizes,and hence the confining length scales, are much larger,this is feasible and there are already some observationsof the phenomenon in a colloid–polymer mixture [63].One might also envisage studying some of the moresubtle capillary phenomena, such as those associatedwith interfacial transitions in wedge geometry [97, 98].

Turning now to small size ratios, where the onlyequilibrium phase transition is the fluid–solid transition,we would encourage further theoretical and simulationstudies of adsorption at a hard wall. As mentioned insection 4.1, we did not [11] make any effort to approachclosely the fluid–solid phase boundary so we could notaddress the important issue of how crystalline layersdevelop prior to bulk crystallization. Various scenariosare possible [11], but given that the wall–colloid deple-tion potential �wall

AO ðz; zpÞ is strongly attractive, for sayq ¼ 0:1, this might favour an infinite sequence of(crystalline) layering transitions, culminating in com-plete wetting of the hard wall–fluid interface by a nearlyclose-packed crystal. However, it is not known howclose to the phase boundary the bulk (reservoir) fluidmust be before the first adsorbed layer becomes


crystalline. Of course, understanding wall-induced crys-tallization is not an easy problem. For pure hard spheresat a planar hard wall it is not fully established whetheror not there is complete wetting by hard-spherecrystal—although recent simulation studies [73, 99]suggest that this is the case. The nature of the wettingbehaviour has important repercussions for heteroge-neous nucleation of the crystal [99].In sections 5 and 6 we considered several extensions

of the basic AO model that were designed to incorpo-rate additional physical features pertaining to a realcolloid–polymer mixture or to introduce entirely newingredients, such as orientational degrees of freedom inthe case of the sphere–needle mixture. All of theseextensions make use of geometry-based DFT; indeedthey were constructed with this purpose. Just howaccurate the DFT proves to be for the various modelsremains to be seen for the most part. Where the DFThas been tested in detail for an inhomogeneous situa-tion, it performs very well [71]. We envisage manyapplications of these extensions to various types ofinhomogeneity.We began this article by praising the virtues of simple

models so it is appropriate to ask what other physicalproblems might be tackled within the context of the basicAO model. In particular, are there situations wheredepletion attraction is expected to play an importantrole and where our DFT might prove effective? Recallonce more that entropic depletion effects are generatedinternally—unlike the case of DFT for simple fluidswhere van der Waals attraction is treated explicitly,usually in a perturbative or mean-field fashion [31]—inour geometry-based approach the attraction betweencolloids, or between a colloid and a hard wall, arisessolely from the geometrical structure of the functional.The solvation of a single (big) hard sphere in a binary AOmixture is of particular interest. Since the planar hardwall can be wet completely by the colloid-rich liquidphase it follows that a liquid film will develop around abig hard sphere immersed in a colloid-poor gas that isvery close to bulk coexistence. This circumstance leadsto pronounced effects in the solvation free energy orexcess chemical potential, measured as a function of theradius of the big sphere, Rb, and of the packing fractionof colloid, say. Indeed the situation mimics that of asimple liquid (solvent) adsorbed at a big solvophobicparticle (hard sphere), except there drying alwaysoccurs, so that a film of gas develops on the spherewhen the solvent is close to coexistence [100]. A newfeature arising in the AO case is the presence of thelayering and the wetting transitions. For sufficientlylarge Rb, these transitions are still present at the sphere,albeit rounded by finite-size effects, and lead to verystriking features in the adsorption, density profiles at

contact with the sphere and in the solvation free energy ofthe sphere [101]. A single hard sphere in the AO mixtureserves as an excellent model for investigating subtleeffects of curvature on interfacial properties, especiallywhen complete wetting occurs.

One can easily construct the DFT for a ternarymixture consisting of (big) hard spheres, colloidal hardspheres plus mutually non-interacting polymer spheres,i.e. a solution of big hard spheres in the AO solvent.For such a mixture one can determine the solvent-mediated potential between two big spheres or betweena big sphere and a wall using a general DFT particleinsertion approach [102]. Since the solvent exhibitsfluid–fluid phase separation one can investigate theeffects of layering, wetting and solvent criticality on thesolvent-mediated potentials [101]. In order to implementthe DFT particle insertion procedure one requires adensity functional that can describe reliably a mixture ofthe solvent and the big particles in the limit of vanishingdensity of the big particles [102]. Such functionals arehard to come by (perturbative approaches are generallynot appropriate) but the geometry-based DFT for thisparticular ternary mixture fits the billy.

Finally we should mention briefly some of the topicsthat we have not covered in this article. There is arapidly expanding body of work on glass transitions incolloid–polymer mixtures. Strong evidence is emergingfor two types of glass; one is repulsion dominated (as inpure hard spheres) and the other is attraction dominated(as occurs when the addition of polymers gives riseto very deep depletion potentials). For an ‘Edinburghmodel mixture’, i.e. PMMA and PS with q ¼ 0:08,adding a little polymer (PS) to the hard-sphere colloidalglass, at fixed �c � 0:6, disrupts the caging structureand brings about crystallization. Adding more polymerresults in non-crystallization, there appears to be a re-entrant glass transition [13]. Simulations and modecoupling theory, based on an effective one-componentdepletion potential, have been used to investigate thisbehaviour [104, 105]. We have not considered in anydetail the situation where q � 1, i.e. the so-called‘protein limit’, where small particles such as proteinsor micelles replace the colloids. There is much interest-ing physics in this limit but it is quite different fromthe depletion-driven phenomena we have describedin this article. Interested readers should consult therecent articles [14, 106, 107] to obtain some overview ofthis regime. During the course of completing our paper

ySolvent-mediated potentials in the presence of wettinghave also been investigated for big Gaussian core particlesimmersed in a binary mixture of smaller Gaussian particlesusing the mixture generalization of the mean-field functional(40). See [103].


we were alerted to the work of Forsman et al. [108]who extended a DFT developed for polymer solutionsto the situation where solvent and monomer particleshave different diameters. These authors consider capil-lary-induced phase transitions when the confiningplanar surfaces are hard.The Urbana group [109] have investigated phase

diagrams and osmotic compressibilities for silica parti-cles and polystyrene in decalin comparing their resultswith the PRISM integral equation theory developed byFuchs and Schweizer [14] and with the free-volumetheory of [4]. These authors argue that PRISM accountsfor all experimental trends whereas the free-volumeapproach appears to miss certain aspects of the experi-mental behaviour, even for theta solvent conditions.Chen et al. [110] also present a critique of the AO modelfor treating real colloid–polymer mixtures. We werealso made aware of the work of Paricaud et al. [111]who consider bulk fluid–fluid phase separation in amodel in which the polymer–polymer and polymer–colloid excluded volume interactions are treated at thesame level of the monomeric segments making up thepolymer chain, using the Wertheim thermodynamicperturbation theory. Once again we direct interestedreaders to these papers.

Much of the work described in this article wouldnot have come to fruition without the substantial inputof our co-workers, A. R. Denton, A. Esztermann,A. Fortini, M. Fuchs, I. O. Gotze, G. Kahl, J. Kofinger,R. Roth, E. Scholl-Paschinger and P. P. F. Wessels.H. Lowen was instrumental in bringing the Bristol andDusseldorf groups together and provided much stimulusfor the research. We are grateful to our experimentalcolleagues in Utrecht, D. G. A. L. Aarts, E. H. A.de Hoog and, especially, H. N. W. Lekkerkerker, whoshared with us their love and knowledge of real colloid–polymer mixtures (and free-volume theory!y). We thankR. Roth for many inspiring discussions about depletionforces, barmaids and nurses and for valuable commentson the manuscript. P. G. Bolhuis, J.-P. Hansen and A. A.Louis kept us busy with a steady supply of pre-prints; weadmire their industry and regret that we cannot do morejustice to their work in the present article. R. van Roijprovided much illuminating insight into matters ofintegrating out and constructing effective Hamiltonians.We would not have embarked upon the researchdescribed here were it not for Marjolein Dijkstra.She introduced us to the subject, explained why it wasinteresting, carried out the key simulation studiesand motivated many of the DFT studies. Marjolein

also provided healthy criticism of most of the results.She must share some of the responsibility for the lengthof this article! The early stages of this research weresupported by the British–German ARC Programme(Project 104b) and by EPSRC. RE is grateful toS. Dietrich and his colleagues for their kind hospitalityand to the Alexander von Humboldt Foundation fortheir support under GRO/1072637 during his stay at theMPI in Stuttgart. He also wishes to thank the editorsof Molecular Physics for their patience in waiting for thearticle to be delivered.

References[1] PUSEY, P. N., 1991, Liquids, Freezing and Glass

Transition, edited by J. P. Hansen, D. Levesque andJ. Zinn-Justin (Amsterdam: Elsevier Science), Chap. 10.

[2] RUTGERS, M. A., DUNSMUIR, J. H., XUE, J. H.,RUSSEL, W. B., and CHAIKIN, P. M., 1996, Phys. Rev.B, 53, 5043.

[3] MEIJER, E. J., and FRENKEL, D., 1994, J. chem. Phys.,100, 6873.

[4] LEKKERKERKER, H. N. W., POON, W. C. K., PUSEY,P. N., STROOBANTS, A., and WARREN, P. B., 1992,Europhys. Lett., 20, 559.

[5] ILETT, S. M., ORROCK, A., POON, W. C. K., and PUSEY,P. N., 1995, Phys. Rev. E, 51, 1344.

[6] ASAKURA, S., and OOSAWA, F., 1954, J. chem. Phys., 22,1255.

[7] ASAKURA, S., and OOSAWA, F., 1958, J. Polym. Sci.,33, 183.

[8] VRIJ, A., 1976, Pure appl. Chem., 48, 471.[9] GAST, A. P., HALL, C. K., and RUSSELL, W. B., 1983,

J. Coll. Int. Sci., 96, 251.[10] DIJKSTRA, M., BRADER, J. M., and EVANS, R., 1999,

J. Phys.: condens. Matter, 11, 10079.[11] BRADER, J. M., DIJKSTRA, M., and EVANS, R., 2001,

Phys. Rev. E, 63, 041405.[12] LEAL CALDERON, F., BIBETTE, J., and BIASIS, J., 1993,

Europhys. Lett., 23, 653.[13] POON, W. C. K., 2002, J. Phys.: condens. Matter, 14,

R859.[14] FUCHS, M., and SCHWEIZER, K. S., 2002, J. Phys.:

condens. Matter, 14, R239.[15] BOLHUIS, P. G., LOUIS, A. A., and HANSEN, J. P., 2002,

Phys. Rev. Lett., 89, 128302.[16] SCHMIDT, M., LOWEN, H., BRADER, J. M., and EVANS, R.,

2000, Phys. Rev. Lett., 85, 1934.[17] SCHMIDT, M., LOWEN, H., BRADER, J. M., and EVANS, R.,

2002, J. Phys.: condens. Matter, 14, 9353.[18] ROSENFELD, Y., 1989, Phys. Rev. Lett., 63, 980.[19] DIJKSTRA, M., and VAN ROIJ, R., 2002, Phys. Rev. Lett.,

89, 208303.[20] SCHMIDT, M., DENTON, A. R., and BRADER, J. M., 2003,

J. chem. Phys., 118, 1541.[21] See the review of LIKOS, C. N., 2001, Phys. Rep., 348, 267.[22] SCHMIDT, M., 2001, Phys. Rev. E, 63, 050201(R).[23] BRADER, J. M., ESZTERMANN, A., and SCHMIDT, M.,

2002, Phys. Rev. E, 66, 031401.[24] ROTH, R., BRADER, J. M., and SCHMIDT, M., 2003,

Europhys. Lett., 63, 549.[25] BRADER, J. M., 2001, PhD Thesis, University of Bristol,

Bristol, UK.yThere is a special t-shirt describing the virtues of this theory

available at a small cost from the Van’t Hoff laboratory.


[26] EVANS, R., BRADER, J. M., ROTH, R., DIJKSTRA, M.,SCHMIDT, M., and LOWEN, H., 2001, Phil. Trans. Roy.Soc. (London) A, 359, 961.

[27] MCMILLAN, W. G., and MAYER, J. E., 1945, J. chem.Phys., 13, 276.

[28] DIJKSTRA, M., VAN ROIJ, R., and EVANS, R., 1999, Phys.Rev. E, 59, 5744.

[29] DIJKSTRA, M., VAN ROIJ, R., and EVANS, R., 2000,J. chem. Phys., 113, 4799.

[30] BRADER, J. M., and EVANS, R., 2002, Physica A, 306, 287.[31] See for example EVANS, R., 1992, Fundamentals of

Inhomogeneous Fluids, edited by D. Henderson(New York: Dekker), Chap. 3, p. 85.

[32] FISHER, M. E., and WIDOM, B., 1969, J. chem. Phys., 50,3756.

[33] EVANS, R., HENDERSON, J. R., HOYLE, D. C., PARRY,A. O., and SABEUR, Z. A., 1993, Molec. Phys., 80, 755.

[34] EVANS, R., LEOTE DE CARVALHO, R. J. F., HENDERSON,J. R., and HOYLE, D. C., 1994, J. chem. Phys., 100, 591.

[35] DIJKSTRA, M., and EVANS, R., 2000, J. chem. Phys., 112,1449.

[36] DE HOOG, E. H. A., and LEKKERKERKER, H. N. W., 1999,J. phys. Chem. B, 103, 5274.

[37] VLIEGENTHART, G. A., and LEKKERKERKER, H. N. W.,1997, Prog. colloid polym. Sci., 105, 27.

[38] BRADER, J. M., and EVANS, R., 2000, Europhys. Lett.,49, 678.

[39] DE HOOG, E. H. A., LEKKERKERKER, H. N. W., SCHULZ,J., and FINDENEGG, G. H., 1999, J. phys. Chem. B, 103,10657.

[40] TOXVAERD, S., and STECKI, J., 1995, J. chem. Phys., 102,7163.

[41] ARCHER, A. J., and EVANS, R., 2001, Phys. Rev. E, 64,041501.

[42] MIKHEEV, L. V., and CHERNOV, A. A., 1987, Sov. Phys.JETP, 65, 971.

[43] CHACON, E., REINALDO-FALAGAN, M., VELASCO, E., andTARAZONA, P., 2001, Phys. Rev. Lett., 87, 166101.

[44] TARAZONA, P., CHACON, E., REINALDO-FALAGAN, M.,and VELASCO, E., 2002, J. chem. Phys., 117, 3941.

[45] VRIJ, A., 1997, Physica A, 235, 120.[46] BRADER, J. M., EVANS, R., SCHMIDT, M., and LOWEN, H.,

2002, J. Phys.: condens. Matter, 14, L1.[47] CHEN, B., PAYANDEH, B., and ROBERT, M., 2000, Phys.

Rev. E, 62, 2369.[48] ARCHER, A. J., and EVANS, R., 2002, J. Phys.: condens.

Matter, 14, 1131.[49] See for example SCHICK, M., 1990, Liquids at Interfaces,

edited by J. Charvolin, J. F. Joanny and J. Zinn-Justin(Amsterdam: North-Holland), chap. 9, p. 415.

[50] HENDERSON, J. R., 1994, Phys. Rev. E, 50, 4836.[51] BALL, P. C., and EVANS, R., 1988, J. chem. Phys., 89, 4412.[52] LOUIS, A. A., BOLHUIS, P. G., HANSEN, J. P., and MEIJER,

E. J., 2000, Phys. Rev. Lett., 85, 2522.[53] WARREN, P. B., ILETT, S. M., and POON, W. C. K., 1995,

Phys. Rev. E, 52, 5205.[54] ROSENFELD, Y., SCHMIDT, M., LOWEN, H., and

TARAZONA, P., 1997, Phys. Rev. E, 55, 4245.[55] TARAZONA, P., and ROSENFELD, Y., 1997, Phys. Rev. E,

55, R4873.[56] SCHMIDT, M., 1999, J. Phys.: condens. Matter, 11, 10163.[57] TARAZONA, P., 2000, Phys. Rev. Lett., 84, 694.[58] LOUIS, A. A., BOLHUIS, P. G., and HANSEN, J. P., 2000,

Phys. Rev. E, 62, 7961.

[59] SCHMIDT, M., 2003, J. Phys.: condens. Matter, 15, S101.[60] EVANS, R., 1990, J. Phys.: condens. Matter, 2, 8989.[61] SCHMIDT, M., FORTINI, A., and DIJKSTRA, M., 2003,

J. Phys.: condens. Matter, 15, S3411.[62] EVANS, R., and MARINI BETTOLO MARCONI, U., 1987,

J. chem. Phys., 86, 7138.[63] AARTS, D. G. A. L., and LEKKERKERKER, H. N. W.,

private communication (unpublished), Utrecht University.[64] GOTZE, I. O., BRADER, J. M., SCHMIDT, M., and

LOWEN, H., 2003, Molec. Phys., 101, 1651.[65] SCHMIDT, M., and DENTON, A. R., 2002, Phys. Rev. E, 65,

061410.[66] WIDOM, B., and ROWLINSON, J. S., 1970, J. chem. Phys.,

52, 1670.[67] DENTON, A. R., and SCHMIDT, M., 2002, J. Phys.:

condens. Matter, 14, 12051.[68] SCHMIDT, M., and FUCHS, M., 2002, J. chem. Phys., 117,

6308.[69] BOLHUIS, P., and FRENKEL, D., 1994, J. chem. Phys., 101,

9869.[70] BARKER, J. A., and HENDERSON, D., 1976, Rev. mod.

Phys., 48, 587.[71] BOLHUIS, P. G., BRADER, J. M., and SCHMIDT, M., 2003,

J. Phys.: condens. Matter, 15, S3421.[72] ROTH, R., 2003, J. Phys.: condens. Matter, 15, S277.[73] DIJKSTRA, M., private communication (unpublished),

Utrecht University.[74] SCHMIDT, M., and DENTON, A. R., 2002, Phys. Rev. E, 65,

021508.[75] SCHMIDT, M., 2001, Phys. Rev. E, 63, 010101(R).[76] SCHMIDT, M., 2002, Phys. Rev. E, 66, 041108.[77] SCHMIDT, M., SCHOLL-PASCHINGER, E., KOFINGER, J.,

and KAHL, G., 2002, J. Phys.: condens. Matter, 14, 12099.[78] WESSELS, P. P. F., SCHMIDT, M., and LOWEN, H.,

2003, Phys. Rev. E., 68, 061404.[79] DE HOOG, E. H. A., 2001, Dissertation, Utrecht

University, The Netherlands.[80] HARRIS, J. G., GRYKO, J., and RICE, S. A., 1987,

J. chem. Phys., 87, 3069.[81] ZHAO, M., CHEKMAREV, D. S., CAI, Z. H., and RICE,

S. A., 1999, Phys. Rev. E, 56, 7033.[82] REGAN, M. J., KAWAMOTO, E. H., LEE, S., PERSHAN, P. S.,

MASKIL, N., DEUTSCH, M., MAGNUSSEN, O. M., OCKO,B. M., and BERMAN, L. E., 1995, Phys. Rev. Lett., 75,2498.

[83] MAGNUSSEN, O. M., OCKO, B. M., REGAN, M. J.,PENANEN, K., PERSHAN, P. S., and DEUTSCH, M., 1995,Phys. Rev. Lett., 74, 4444.

[84] DIMASI, E., TOSTMANN, H., OCKO, B. M., PERSHAN, P. S.,and DEUTSCH, M., 1998, Phys. Rev. B, 58, R13419.

[85] INDEKEU, J. O., 1989, Europhys. Lett., 10, 165.[86] PIASECKI, J., and HAUGE, E. H., 1987, Physica A, 143,

87.[87] FAN, Y., and MONSON, P. A., 1993, J. chem. Phys., 99,

6897.[88] BROVCHENKO, I., GEIGER, A., and OLEINIKOVA, A., 2001,

Phys. Chem. chem. Phys., 3, 1567.[89] LAW, B. M., 1990, J. Colloid Interface Sci., 134, 1.[90] BEAGLEHOLE, D., and CHRISTENSON, H. K., 1992, J. phys.

Chem., 96, 3395.[91] SELLAMI, H., HAMRAOUI, A., PRIVAT, M., and OLIER, R.,

1998, Langmuir, 14, 2402.[92] BONN, D., WEGDAM, G. H., KELLAY, H., and

NIEUWENHUIZEN, T. M., 1992, Europhys. Lett., 20, 235.


[93] BONN, D., KELLAY, H., and WEGDAM, G. H., 1993,J. chem. Phys., 99, 7115.

[94] WIJTING, W. K., BESSELING, N. A. M., and COHEN

STUART, M. A., 2003, Phys. Rev. Lett., 90, 196101.[95] WIJTING, W. K., BESSELING, N. A. M., and COHEN

STUART, M. A., 2003, J. phys. Chem. B, 107, 10565.[96] AARTS, D. G. A. L., VAN DER WIEL, J. H., and

LEKKERKERKER, H. N. W., 2003, J. Phys.: condens.Matter, 15, S245.

[97] RASCON, C., and PARRY, A. O., 2000, Nature (London),407, 986.

[98] BRUSCHI, L., CARLIN, A., and MISTURA, G., 2002, Phys.Rev. Lett., 89, 166101.

[99] AUER, S., and FRENKEL, D., 2003, Phys. Rev. Lett., 91,015703.

[100] EVANS, R., ROTH, R., and BRYK, P., 2003, Europhys.Lett., 62, 5360.

[101] ROTH, R., and EVANS, R., unpublished.[102] ROTH, R., EVANS, R., and DIETRICH, S., 2000, Phys. Rev.

E, 62, 5360.[103] ARCHER, A. J., and EVANS, R., 2003, J. chem. Phys.,

118, 9726.

[104] PHAM, K. N., PUERTAS, A. M., BERGENHOLTZ, J.,EGELHAAF, S. U., MOUSSAID, A., PUSEY, P. N.,SCHOFIELD, A. B., CATES, M. E., FUCHS, M., andPOON, W. C. K., 2002, Science, 296, 104, and referencestherein.

[105] DAWSON, K., FOFFI, G., FUCHS, M., GOTZE, W.,SCIORTINO, F., SPERL, M., TARTAGLIA, P.,VOIGTMANN, T., and ZACCARELLI, E., 2001, Phys. Rev.E, 63, 011401.

[106] SEAR, R., 2001, Phys. Rev. Lett., 86, 4696.[107] BOLHUIS, P. G., MEIJER, E. J., and LOUIS, A. A., 2003,

Phys. Rev. Lett., 90, 068304.[108] FORSMAN, J., WOODWARD, C. E., and FREASIER, B. C.,

2002, J. chem. Phys., 117, 1915.[109] SHAH, S. A., CHEN, Y. L., SCHWEIZER, K. S., and

ZUKOSKI, C. F., 2003, J. chem. Phys., 118, 3350.[110] CHEN, Y. L., SCHWEIZER, K. S., and FUCHS, M., 2003,

J. chem. Phys., 118, 3880.[111] PARICAUD, P., VARGA, S., and JACKSON, G., 2003,

J. chem. Phys., 118, 8525.


Documents

Invited article Statistical mechanics of …Statistical mechanics of inhomogeneous model colloid–polymer mixtures JOSEPH M. BRADER1, ROBERT EVANS2,3*y and MATTHIAS SCHMIDT4z 1Institute