Self-consistent effective interactions in charged colloidal suspensionsJuan A. Anta and Santiago Lago Citation: The Journal of Chemical Physics 116, 10514 (2002); doi: 10.1063/1.1479140 View online: http://dx.doi.org/10.1063/1.1479140 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/116/23?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Self-Consistent Ornstein-Zernike Approximation (SCOZA) and exact second virial coefficients and theirrelationship with critical temperature for colloidal or protein suspensions with short-ranged attractive interactions J. Chem. Phys. 139, 164501 (2013); 10.1063/1.4825174 Effective interaction between large colloidal particles immersed in a bidisperse suspension of short-rangedattractive colloids J. Chem. Phys. 131, 164111 (2009); 10.1063/1.3253694 Screening effects on structure and diffusion in confined charged colloids J. Chem. Phys. 126, 154902 (2007); 10.1063/1.2720386 Comparison of colloidal effective charges from different experiments J. Chem. Phys. 116, 10981 (2002); 10.1063/1.1480010 Effective interactions, structure, and isothermal compressibility of colloidal suspensions J. Chem. Phys. 113, 4799 (2000); 10.1063/1.1288921

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Self-consistent effective interactions in charged colloidal suspensionsJuan A. Antaa) and Santiago LagoDepartmento de Ciencias Ambientales, Universidad ‘‘Pablo de Olavide,’’ Ctra. de Utrera, Km. 1,41013 Sevilla, Spain

~Received 21 November 2001; accepted 26 March 2002!

We use an integral equation scheme to obtain self-consistently the effective interaction betweencolloids in salt-free charged colloidal suspensions. The colloid–counterion direct correlationfunction ~DCF! is obtained for the fixed colloid–colloid pair structure by solving the correspondinghypernetted-chain equation~HNC!. This DCF is then used to formulate an effective colloid–colloidpair potential for which the one-component reference hypernetted-chain equation is solved. Bothprocesses are iterated until self-consistency is achieved. Counterion–counterion correlations areconsidered linear and uncoupled from the rest of the correlations. The method is based on a similartreatment utilized in liquid metals@Phys. Rev. B61, 11400~2000!# and provides equivalent resultsto those obtained using the standard multicomponent HNC equation for mixtures of charged hardspheres. The theory proves rather accurate when compared with molecular dynamic simulations ofcharged hard and soft spheres for colloidal charges of up to 300. We study in detail the existence ofnet attractions between colloids in certain cases~especially in the presence of divalent and trivalentcounterions! and how this attraction may lead to phase instability. The problem of the lack ofsolution of the integral equation for more realistic cases~larger charges! is also discussed. ©2002American Institute of Physics.@DOI: 10.1063/1.1479140#

I. INTRODUCTION

Charged colloidal suspensions have been the focus ofextensive experimental and theoretical research over the past50 years. The reason for this interest has been mainly due tothe wide range of industrial applications of these systems.Additionally, a theoretical description of colloidal suspen-sions remains a challenge. Particularly difficult is the evalu-ation of the effective pair potential between colloidal par-ticles, which is the key ingredient to obtain the colloidalmicroscopic structure and the phase behavior of thesuspension.1 From the statistical mechanics point of view,colloidal suspensions are multicomponent systems character-ized by a large asymmetry in size and charge. For this reasonit is very awkward the use of atomistic approaches to obtainthe effective potential. Due to that, the standard and mostefficient way of dealing with the problem is thecoarse-graining method. By coarse graining we mean to eliminatethe degrees of freedom of the smaller and less charged par-ticles so the mixture is treated as an effective one-componentsystem~OCS! of large particles. The OCS can then be stud-ied by means of standard techniques like, for instance, mo-lecular simulation. The most prominent example of a coarsegraining procedure is the well-known Derjaguin–Landau–Verwey–Overbeek~DLVO! theory and effective potential,2

widely used nowadays in colloidal science.In spite of its success, the DLVO theory has been chal-

lenged in recent times by a number of experiments that sug-gest anomalous, non-DLVO behavior at rather low saltconcentrations.3 In addition, there are also experiments thatdepart from the DLVO predictions at very large salt

concentrations.4 It is in this context that various extensionsor alternative approaches to the DLVO theory have been putforward.5–9 Additionally, techniques of common use in thegeneral field of Statistical Mechanics10 such as MolecularSimulation11–14 and Integral Equation~IE! theories15,16 havebeen applied extensively to colloidal problems. In this paperwe will focus on the IE approach to describe charged colloi-dal suspensions. As opposed to simulation techniques, IEsare rapid and system-size independent. Moreover, some IEformulations such as the Reference Hypernetted-Chain~RHNC! theory17–19or the approximation of Barrat, Hansen,and Pastore16 ~BHP! are virtually exact, at least for low col-loidal charges. However, for highly charged colloids~whenthe Coulombic couplingbetween the colloidal particles andthe counterions is very large!, common IE theories do nothave a solution. In spite of this, IEs have been applied ex-tensively to charged20–22 and neutral23 colloidal suspensionsand used to obtain the effective potential betweencolloids24,25 at moderate size and charge asymmetries.

The implementation of a coarse-graining procedure inthe context of IE theories implies to solve an integral equa-tion for a multicomponent system. This brings about a hugenumerical and ‘‘coding’’ complexity, as well as the afore-mentioned lack of solution at large colloidal charges. In ad-dition, pursuing a high degree of accuracy requires a largenumber of optimizable parameters. This makes the numericalsolution of the integral equation very cumbersome. Theproblem is even more serious in charged systems, where thenumber of constraints is smaller than the number of param-eters due to the electroneutrality condition.23,25 Somehow,these shortcomings have hindered the application of IE ap-proaches to the study of colloids.

In this work we present a different view in the applica-a!Electronic mail: jaantmon@dex.upo.es

JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 23 15 JUNE 2002

105140021-9606/2002/116(23)/10514/9/$19.00 © 2002 American Institute of Physics

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tion of IE theories to charged colloidal systems. We startfrom the idea that colloid–colloid, colloid–ion, and ion–ioncorrelations do not have to benecessarilytreated on the samefooting. In fact, the essence of the coarse-graining procedureis that there are species with dissimilar behavior in the sys-tem, and this is what makes it convenient to use atwo-stepstrategy to describe it. As recently pointed out by Warren,8 inthe regions of interest for experiments,3 colloid–ion andion–ion correlations can be treated in a linearized~Debye–Huckel! level, whereas higher degrees of accuracy are re-quired only to describe the colloid–colloid interaction. Thisis a very similar situation to that encountered in liquidmetals10,26and, in fact, this work is closely related to a recentapplication19 of the Quantum-Hypernetted-Chain~QHNC!theory for these systems.27 Also, it is worth it to note that asimilar treatment has been devised by Lo¨wen, Hansen, andMadden,28 but in the field of molecular simulation. Theseauthors use Density Functional Theory to describe the de-grees of freedom controlling the correlation between colloidsand counterions, whereas colloid–colloid interactions arehandled using ordinary Molecular Dynamics. Both processesare solved self-consistently and ‘‘on the fly.’’ Their techniqueis nothing else but the colloidal version of the well-knownCar–Parrinello method of electronic systems.29 In a some-how parallel manner, the method described in this paper isthe colloidal version of the QHNC theory of liquid metals.

In essence, the integral equation theory presented herecorresponds to a self-consistent coarse-graining strategy.Roughly speaking, it consists of the successive iteration oftwo steps:~1! solution of the colloid–counterion problem inthe HNC approximation; and~2! the solution of the colloid–colloid effective one-component problem in the RHNC ap-proximation. Step~1! is the alternative to the solution of thePoisson–Boltzmann equation for a charged spherical particlesurrounded by charged counterions, which is thefirst stepinthe derivation of the DLVO potential.30 Step~2! consists ofsolving a simple one-component integral equation for an ef-fective potential that has been obtained~self-consistently!from step~1!. Proceeding in this way, we follow a ‘‘divide-and-conquer’’ approach that permits us to separate conve-niently the different correlations involved in the problemwithout neglecting the fact that they all depend on eachother. Since our approach contains a coarse-graining proce-dure, we will refer to the integral equation theory presentedhere as CGHNC.

Apart from the methodological part of this work, wepresent and discuss results for colloid–counterion mixturesfor which there are simulation results available in the litera-ture. In this way we test the approximations made in thederivation of the CGHNC approach. We then use the theoryto obtain effective potentials for different charges of the col-loids and the counterions and look at the effect on the effec-tive interactions. Thus, we observe a clear non-DLVO behav-ior that is related to the existence of effective attractionsbetween colloids. This observation, already mentioned byother authors,14,21 is here connected to the thermodynamicstability of the mixture. This analysis should not be taken,however, as proof of the observed phase separation at verylow ionic strengths, as suggested by experiments.3 Even if a

colloid–counterion mixture can be considered as a simplemodel of a highly deionized colloidal suspension, to addressmore rigorously this problem we would need to extend thepresent theory to the case of added salt.8 We would then beable to look at the phase behavior for different concentra-tions of the electrolyte. On the other hand, to reproduce theexperimental situation, we need to solve the integral equationat much higher charges than those presented in this work.Both alternatives of an extension of the present work,~1! thecase of added salt and~2! numerical solution at largercharges, are discussed here in relation to the advantages ofthe CGHNC theory with respect to other IE formalisms.

This paper can be outlined as follows: In Sec. II wedescribe the derivation of the CGHNC theory and the nu-merical strategy used to implement it. In Sec. III we presentresults for the correlation functions of colloid–counterionmixtures at different sizes and charges and compare themwith simulation data. In Sec. IV we report the effective po-tentials derived from the CGHNC theory and how they arerelated to the stability of the mixture. In Sec. V we discussthe issue of the nonsolution boundaries of the integral equa-tion formalism. Finally, in Sec. VI the main conclusions ofthis work are summarized.

II. THE CGHNC THEORY

In the present work we consider a system of colloidalparticles with positive chargezc in the presence of negativelycharged counterions of charge2zi . The number density ofthe colloids isrc whereas the density of the counterions issubject to the restriction of charge electroneutrality, hencer i52(zc /zi)rc . These species interact via pair potentials ofthe type

umn~r !5umnSR~r !1

zmzne2

4per~m,n5c,i !, ~1!

whereuSR is a short-range interaction ande is the permittiv-ity of the solvent, which is taken as a continuum. In order toderive equations that describe conveniently this system weproceed as follows:

~i! Step 1: Many-body problem. In the IE formalism,many-body interactions are treated by means of theOrnstein–Zernike~OZ! equations.10 For a binary mixturethey can be written as31

Scc~k!5@12r iCii ~k!#/D~k!,

Sci~k!5Arcr iCci~k!/D~k!,

Sii ~k!5@12rcCcc~k!#/D~k!,

D~k!5@12r iCii ~k!#@12rcCcc~k!#2rcr i@Cci~k!#2,

~2!

where the Smn(k)’s ~c5colloid, i5counterion! are theAshcroft–Langreth partial structure factors.32 These are re-lated to the total correlation functions~TCF! hmn’s via

Smn~k!5dmn1~rmrn!1/2EVdr eik"rhmn~r !

5dmn1~rmrn!1/2hmn~k!, ~3!

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with the Cmn(k)’s being the direct correlation functions~DCF!. The OZ equations are exact relations betweenallTCF’s, which describe the pair structure of the system, andthe second-order functional derivatives of the excess free en-ergy represented by the DCFs.10,19,31 In a pure, one-component fluid, the OZ equations~2! reduce to the familiarform

S~k!512rh~k!51

12rC~k!. ~4!

~ii ! Step 2: Coarse graining.We reduce the problem toan effective one-component system~OCS! by taking21,19

S~k!5Scc~k!, ~5!

whereS(k) is the structure factor of the OCS. Thus, the OCSis defined as the fluid whose characteristic pair structure isidentical to the colloid–colloid pair structure of the originaltwo-component system. Note that this method of coarsegraining is substantially different to that utilized, for in-stance, in Ref. 6, where a pair interaction is obtained byintegrating the degrees of freedom corresponding to thecounterions~and coions! in the total partition function. As aconsequence of this, the effective pair interaction containscounterion-dependent terms that are nota priori pairwiseadditive.21 On the contrary, within the present approach, wemake sure that the effective pair potential describing theOCS contains, by definition, all many-body contributions as-sociated to the counterionic degrees of freedom via the OZequations. The effective potential so constructed is thenstatedependentbut pairwise additive by definition. Thus, by com-bining Eqs.~2!, ~4!, and ~5! we find the following relation-ship between the DCFs of the OCS and the mixture:

C~k!5Ccc~k!1r i@Cci~k!#2

12r iCii ~k!. ~6!

~iii ! Step 3: Effective potential. In order to find an ex-pression for the interaction characteristic of the OCS, i.e., theeffective potential, we utilize the same strategy as before butnow applied to the pair distribution functionsgmn(r )5hmn(r )11 and their corresponding potentials of meanforce wmn(r ),10

g~r !5exp@2bw~r !#5gcc~r !5exp@2bwcc~r !#, ~7!

where b51/kBT. The potential of the mean force can berelated in turn to the DCFs;10 hence

2bueff~r !1h~r !2C~r !2B~r !

52bucc~r !1hcc~r !2Ccc~r !2Bcc~r !, ~8!

whereueff(r) is the effective potential and theBmn’s are theso-calledbridge functions. The bridge functions are relatedto the ‘‘higher-than-two’’ functional derivatives of the excessfree energy functional with respect to the density profiles19

and neglecting their contribution leads to the well-knownHNC approximation.10,31 On the contrary, by including themin Eq. ~8!, we start from a formalism that is completely exactso far. Nevertheless, it is at this point where we have tointroduce the first of the approximations of the presenttheory;

Approximation1: B~r !'Bcc~r !; ~9!

i.e., the bridge function of the OCS is essentially identical tothe colloid–colloid bridge function of the real system.33 As-suming this, we can eliminate the bridge functions from Eq.~8! along with the pair distributions~which are identical bydefinition!. We next make use of Eq.~6! to arrive to thefollowing expression for the effective pair potential betweencolloids:

ueff~r !5ucc~r !2E Cci~k!x i i ~2k!Cci~2k!dk, ~10!

with

x i i ~k!5r i

12r iCii ~k!, ~11!

which is thelinear-responsefunction of the fluid of counte-rions. Equation~10! shows that the effective interaction be-tween colloidal particles is made up of two contributions:~1!direct Coulombic repulsion and~2! a counterion-mediatedattraction between colloids and the ionic ‘‘atmospheres’’ ofneighboring colloids, this ‘‘atmosphere’’ represented by thecounterion density profile around a colloidal particlenci(k)5x i i (k)Cci(k). This is exactly analogous to the expressionfor the effective potential between positive metallic ions inliquid metals10 if we regard the colloid–counterion DCF asan ion–electronpseudopotential.19,27 The only difference isthat in the present case thebackgroundfluid ~electrons inliquid metals, counterions in colloidal suspensions! is not ofa quantum nature and therefore can be treated on the samegrounds as the rest of the components of the fluid.

~iv! Step 4: Counterionic background. The expressionintroduced above for the effective potential depends on thecounterion–counterion correlations via the response func-tion, which, in turn, depends on the DCF. In the presentapproach, we choose a simple expression for this:

Approximation2:

Cii ~r !5H 2buii ~s i !g~r i ,s i !, r ,s i ,

2buii ~r !, r .s i ,~12!

wheres i is the diameter associated to the counterions andgis a constant parameter that is chosen such thatCii is con-tinuous atr 5s i and consistent with the condition of ex-cluded volume for the colloidal cores. It can be shown6 thatan appropriate choice for this is

g~r i ,s i !5S kDs i

11kDs iD , ~13!

with kD being the inverse Debye length, i.e.,

kD2 5

4pr izi2

kBTe. ~14!

Note that this procedure is equivalent to treating thecounterion–counterion interactions in theMean SphericalApproximation~MSA!,10 i.e., the excluded volume conditionat short distances plus the equivalence between the interac-tion potential and the DCF at long distances. Nevertheless,leaving aside the actual approximation orclosureused, what

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is especially important in the present context is thatcounterion–counterion correlations are made independent oruncoupledfrom the rest. In other words, colloids move in asea of counterions and interact with them with a responsefunction that behaves as if the colloids were absent. Hence,we need only thepolarizationproperties of the counterionicfluid treated as aone-component plasma.8 Approximation2then plays the same role as thejellium approximation in thetheory of liquid metals,10,19 where the interaction of the ionswith the electrons is treated by means of the response func-tion of thehomogeneouselectron gas. The numerical impli-cations of this in the context of the IE method are immediate:we do not have to worry about counterion–counterion corre-lations, and the complexity of the solution is substantiallyreduced.

~v! Step 5: Colloid–counterion correlation. We treat thisinteraction in the HNC approximation, which means that thecolloid–counterion mean-force potential is obtained through

Approximation3: wci~r !5buci~r !2hci~r !1Cci~r !.~15!

This equation is solved in conjunction with the colloid–counterion part of the OZ equations~2!,

hci~k!5Cci~k!1rcCci~k!hcc~k!1r iCii ~k!hci~k!, ~16!

whereCii is given by Eq.~12! andhcc , which describes thecolloid–colloid pair structure, is aninput to this part of the

problem. In other words, the distribution of counterionsaround a colloidal particle is obtained by solving the multi-component HNC equation for a given,fixed, colloid–colloidpair distribution. As mentioned in the Introduction, this is theIE alternative to the solution of the Poisson–Boltzmannequation, whose numerical demands for a multiparticle con-figuration of colloids are well known.9

In order to check the accuracy of ‘‘step 5,’’ we plot inFig. 1 the HNC colloid–counterion mean-force potentialalong with its Debye–Hu¨ckel counterpart, which corre-sponds to the solution of the linearized Poisson–Boltzmannequation at infinite dilution.34 A comparison with simulationdata~upper panel! shows that the HNC prediction is the onlyone that catches conveniently the oscillations originated by afinite concentration of colloids~defined by a fixed pair dis-tribution!. Only at very low colloidal packing fractions~lower panel! the HNC mean-force potential approaches theDebye–Hu¨ckel limit. This result illustrates the main reasonwhy the IE approach is superior to Poisson–Boltzmanntheory, that is, it takes proper account of the influence of thecolloid–colloid correlations on the counterionic atmosphereof colloids.

~vi! Step 6: Solution of the effective colloid–colloidproblem. Once we have solved the colloid–counterion corre-lation, the effective potential between colloids is completelydetermined via Eqs.~10!, ~11!, and ~12!. In the OCS, thispotential induces a colloid–colloid pair structure that is ob-tained using the RHNC approximation:

Approximation4:

w~r !5bueff~r !2h~r !1C~r !1B0~r ;D !, ~17!

FIG. 1. Colloid–counterion potential of mean force as obtained from thesolution of the CGHNC equations~solid lines! and Monte Carlo simulations~Ref. 12! ~circles!. The mean-force potential in the Debye–Hu¨ckel ~DH!approximation~Ref. 34! is represented by the dashed lines. The upper panelcorresponds to a case where the correlation or coupling between colloidaldegrees of freedom is large~low charge, high packing fraction! and theCGHNC solution lies far from DH but close to the Monte Carlo results. Thiscalculation corresponds to a system composed of hard-core particles of sizes4 and 0.4 nm, respectively. The effect of neglecting the size of the counte-rions is represented by the dotted line. When the coupling between counte-rions is small~high charge, low packing fraction!, the CGHNC predictionapproaches the DH limit~low panel!.

FIG. 2. Colloid–colloid total correlation functions~TCF! for a mixture ofhard-core colloids and counterions of sizes 3 and 0.4 nm, respectively. Thesolid lines correspond to the estimates of the CGHNC equations when theOCS is solved in the HNC approximation, whereas the dashed line representthe TCFs when a hard-sphere bridge function with an optimized diameterDis introduced~see the text!. The circles are simulation results taken fromRef. 21. The DLVO prediction for the 20/22 case is also included as adotted line.

10517J. Chem. Phys., Vol. 116, No. 23, 15 June 2002 Self-consistent effective interactions

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whereB0 is the bridge function of areferencehard-spheresystem.17 This depends parametrically on the hard-sphere di-ameterD. As the mean-force potential is directly related tothe h(r ) via h(r )5exp@2w(r)#21, the colloid–colloid pairstructure is completely determined, for a givenD and effec-tive potential, by Eq.~17! coupled with the one-componentOZ equation~4!. Alternatively, the bridge function can beignored in Eq.~17!, such that the HNC approximation is alsoapplied in the colloid–colloid problem. As we will see be-low, this provides a good first approximation of both thecolloid–colloid and colloid–counterion correlations.

A. General strategy

Both steps 5 and 6 consist of solving an integral equationfor only one correlation, keeping the rest fixed. In view ofthis, we iterate over steps 5 and 6 until self-consistency isachieved. The result should be equivalent to the solution ofthe full multicomponent OZ equations with MSA, HNC, andRHNC closures for the counterion–counterion, colloid–counterion, and colloid–colloid correlations, respectively.

Still, the whole formalism depends parametrically on asingle parameter, the hard-sphere diameterD. In the presentapproach we optimizeD by requiring thermodynamicconsistency35 between thetotal compressibility and thetotalvirial pressure of the mixture,25

xT215S ]b Pn

]r DT

, ~18!

with

xT215124p(

m,n~rmrn!1/2E

Vdr cmn~r !, ~19!

bPn5r22

3pr2(

m,n~rmrn!1/2

3EVdr r

db umn~r !

drgmn~r !, ~20!

where thec’s, the direct correlation functions, and theg’s,the radial distribution functions, are given by the full self-consistent solution of the CGHNC equations@Eqs. ~6!, ~7!,~12!, and~16!#.

B. Numerical details

The HNC and RHNC integral equations are solved nu-merically on a grid of 4096 points with a grid size of 0.01–1nm in real space. The method of Ng36 combined withBroyles’ strategy37 to mix conveniently successive estimatesof the correlation functions is employed to enhance the con-vergence in the numerical solution of each integral equation.As regards the full CGHNC self-consistent procedure, westart from the HNC solution of the OCS described by thestandard DLVO effective potential. Full self-consistency isthen obtained normally in a few successive iterations of steps5 and 6. The whole process is repeated for different values ofD until Eq. ~18! is fulfilled.

FIG. 3. Colloid–colloid and colloid–counterion totalcorrelation functions for the soft-core counterparts~Ref.38! of the systems considered in Fig. 2.

10518 J. Chem. Phys., Vol. 116, No. 23, 15 June 2002 J. A. Anta and S. Lago

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III. COLLOID–COLLOID AND COLLOID–COUNTERIONPAIR STRUCTURES

In order to check the performance of the CGHNC equa-tions, we have solved them in a variety of cases. Hence, wehave considered colloidal charges ranging from 20 to 300electrons, mono-, di- and trivalent counterions, and hard- andsoft-core38 short-ranged potentialsumn

SR(r ). Results areshown in Figs. 1–6.

In Fig. 2 the pair structure of hard-core colloidal mix-tures with charge asymmetries 20/21 and 20/22 is plotted.This case was already considered by Belloni,21 who foundvery good agreement between simulation and the BHP inte-gral equation.16 Nonetheless, the present calculations indi-cate that the CGHNC equations provide an equivalentresult.39 This fact validates, at least for this case, the approxi-mations on which the CGHNC theory is based. As regardsthe level of agreement achieved, we see that if we use theHNC approximation to describe the OCS we get a good firstapproximation of the colloid–colloid distribution. Alterna-tively, if we introduce in the OCS a bridge function with ahard-sphere diameter optimized as explained above, we thenobtain a perfect match with the simulation.

A similar outcome is found when the hard-core interac-tion is replaced by a soft-core one, as shown in Fig. 3. TheCGHNC theory yields excellent estimates for both thecolloid–colloid and the colloid–counterion structure. Theformer behaves in a like manner to the case explained beforewhen the HNC approximation is replaced by the optimizedRHNC. Nevertheless, as it could be expected, the effect ofthe OCS colloid–colloid bridge function is not visible in the

colloid–counterion distribution. This confirms the fact men-tioned before~see Fig. 1! that the HNC approximation aloneis able to describe very accurately the colloid–counterioncorrelation.

We have also explored systems with larger colloidalcharges, closer to regions of experimental interest and notaccessible, in principle, to standard simulation techniques. Inspite of this lack of ‘‘classical’’ simulation data, in Fig. 4 wehave compared our CGHNC calculations with the results ob-tained by Lowen, Hansen, and Madden using a hybrid simu-lation procedure.28 Similar conclusions as those mentionedbefore can be extracted from this comparison. Nevertheless,for the case of largest colloidal packing fraction, the HNCprovides surprisingly a better estimate of the colloidal distri-bution than the RHNC approximation.

In summary, the results presented in this section showthat CGHNC formalism is capable of producing a good de-scription of the colloid–colloid and colloid–counterion dis-tributions with simple approximations and a single optimiz-able parameter. Thus, to treat the counterionic correlations asan unperturbed background and the interaction of the col-loids with their ionic atmospheres in the HNC approximationproves to be both accurate and numerically convenient. Thisallows us to study with confidence effective intercolloidalinteractions and stability conditions, as we will see in thenext section.

IV. EFFECTIVE POTENTIALS AND STABILITY

Effective potentials cannot be compared with simula-tions, but the agreement found in the previous section indi-cates that the effective interactions between colloidal par-

FIG. 4. Colloid–colloid total correlation functions for hard-core colloidalparticles with pointlike monovalent counterions. Circles stand for the simu-lation results of Lo¨wen, Hansen, and Madden~Ref. 28!.

FIG. 5. Colloid–colloid effective potentials, as obtained from the solutionof the CGHNC equations, for hard-core colloidal mixtures with size asym-metry 3/0.4 nm and with charge asymmetries as shown.

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ticles extracted from the CGHNC calculations should beclose enough to the real ones. In Fig. 5 we plot the effectivepotentials for the hard-core mixtures considered in Fig. 2, butfor different charges of the colloids and the counterions, andwith the OCS solved in the HNC approximation. An essen-tial feature of these potentials is the occurrence ofnegativeminima in certain cases. This is especially visible in the pres-ence of divalent and trivalent counterions but can also hap-pen with monovalent counterions if the colloidal charge islarge enough. This means that colloids can effectivelyattracteach other if the Coulombic coupling of the mixture isstrong. This point was already mentioned by Belloni21 anddetected in simulations.11,14 From the form of the effectivepotential expressed in Eq.~10!, we see that the attraction ofthe colloids by the counterionic atmospheres of opposite signof other colloids can overcome the Coulombic repulsion be-tween them at certain conditions. This effect, which wasfirstly noticed by Langmuir in the 1940s,40 is not included inthe derivation of the DLVO effective potential, which is al-ways repulsive~in the absence of dispersion forces!. On thecontrary the present IE description describes adequately thismany-body effect, as shown in the calculations presentedhere.

It is interesting to investigate how these effective inter-colloidal attractions can drive phase separation. FollowingBelloni,22 we have monitored the behavior of the total iso-thermal compressibility of the mixture as a function of thedensity. Results for the systems considered in Fig. 5 are plot-ted in Fig. 6. We observe that it is only for the case in whichthere is no attractive minimum in the effective potential

~asymmetry 20/21!, where the compressibility does not ap-pear to diverge as we approach the low-density region. Thus,the existence of a ‘‘spinodal’’41 line is associated to the oc-currence of effective attractions in the OCS and not to theinfluence of ‘‘volume’’ terms in the total free-energy func-tional, as it seems to be the case in other situations.6,8 Nev-ertheless, we must say that the actual system studied herestill corresponds to a situation very different from those in-vestigated in the experiments.3 In fact, a similar analysis ap-plied to the case considered in Fig. 4, with a larger size andcharge, does not lead to the same conclusions. Although notshown in the figures, the effective potentials extracted fromthe CGHNC calculations in this case do not exhibit anynegative regions and, correspondingly, no divergences in thetotal compressibility are detected.

An extension of the present calculations to more realisticsituations~charges of the order of a thousand electrons! ishindered by the lack of solution of the CGHNC integralequation. This problem is addressed in more detail in thefollowing section.

V. NONSOLUTION REGIONS OF THE CGHNCINTEGRAL EQUATION

With regard to the reasons why the CGHNC equationsand related approaches do not exhibit solution in chargedcolloidal suspensions we should distinguish between fourdifferent scenarios.

~1! Lack of solution at very low colloidal densities. Dueto the electroneutrality condition in the colloid–counterionmixture, if we decrease the number of colloids in the system,we have to decrease accordingly the number of counterions.This causes the ionic strength of the system to becomesmaller and smaller and so the inverse Debye lengthkD . Asthe size of the counterionic atmosphere that surrounds thecolloids is directly related tokD

21, this makes the colloid–counterion correlation to become very rapidly long ranged aswe approach the low-density limit. As a consequence, thenumerical solution of the CGHNC integral equation becomesimpracticable. This is the reason why the 20/21 curve in Fig.6 does not terminate in the low-density region.

~2! ‘‘Gas–liquid’’ nonsolution boundary. At intermediatedensities and always below a certain critical temperature, theCGHNC bumps into a nonsolution region. This feature,which appears in this case associated to an increase of thetotal compressibility41,22and the low-wavelength limit of thecolloid–colloid structure factor, should not be considered asa shortcoming of the IE approach but as a manifestation ofthe gas–liquid transition in the theoretical description. Thiskind of ‘‘nonsolution’’ is the cause of termination for thecurves plotted in Fig. 6, except the 20/21 one.

~3! ‘‘Liquid –solid’’ nonsolution boundary. In the samemanner as the CGHNC ceases to have a solution when weapproach a gas–liquid transition, the numerical solution ofthe integral equation becomes unfeasible if the packing frac-tion is very large. This is identified by a sudden increase inthe height of the main peak of the colloid–colloid structurefactor and it is a clear indication of freezing. As in the pre-vious case, this feature should not be considered, in prin-ciple, as a shortcoming of the IE description.

FIG. 6. Isothermal compressibility versus colloidal density for the systemsconsidered in Fig. 5.

10520 J. Chem. Phys., Vol. 116, No. 23, 15 June 2002 J. A. Anta and S. Lago

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~4! Lack of solution at large colloidal charges. This isthe main problem that is faced when we try to match theoryand experiments.3,4 Traditionally, this drawback of the IEapproach has been mentioned in relation to the attempt ofsolving the IE for highly charged colloidal particles. Never-theless, the origin of this anomaly is somehow more compli-cated. In the present work it is observed that what reallydetermines whether the CGHNC equations have a solution ornot is the ratio between the colloidal charge and colloidalradius. This feature is illustrated in Fig. 7, where we showthe regions of solution and nonsolution at a given tempera-ture and packing fraction. As the charge-to-radius ratio isrelated to the electrostatic interaction between colloids andcounterionsnear the colloidal surface, we conclude that theintegral equation cannot be solved when this energy is verylarge in comparison with the thermal energy characteristic ofthe system. This is a very interesting finding because this isprecisely the condition for the occurrence ofionic condensa-tion in colloidal suspensions.42 When ions, or in this case,counterions, are said to becondensedon the surface of thecolloids, it is meant that a certain fraction of them becomeeffectively attached to the colloids instead of behaving asgenuine free particles. This is a highly nonlinear effect thatcannot be apprehended either by the simple Debye–Hu¨ckelapproximation8 or by the HNC approximation. In fact, thisproblem has been commonly addressed by means of chargerenormalization.43 In the present context, the existence ofcharges that lead to a very high charge-to-size ratio makes itimpossible to solve the CGHNC equations in its presentform.

VI. DISCUSSION AND CONCLUSIONS

In this paper we have tried to demonstrate the usefulnessof the integral equation formalism to study charge-stabilized

colloidal suspensions if adequate approximations are takeninto account. These approximations are inspired by similarconsiderations utilized in the field of liquid metals and areimplemented in a way that it resembles a self-consistentcoarse-graining procedure. For the systems studied here, thepresent approach, that we call CGHNC, provides almost ex-act results for the colloid–colloid and colloid–counterioncorrelation functions with just oneoptimizableparameter.This parameter, the diameter of the hard-sphere referencesystem, is involved in the description of the effectivecolloid–colloid one-component problem only.

The effective potentials obtained in the context of theCGHNC approach confirm the findings of previous authorsusing molecular simulation and more sophisticated integralequation formalisms. These findings prove that like-chargedcolloidal particles may attract each other as a consequence ofthe action of the counterions and that this may lead to phaseinstability. Nevertheless, this mechanism must not be re-garded as an explanation for the experimental evidence,which suggests that highly deionized colloidal suspensionscan phase separate. This is most probably due to ‘‘volume’’terms in the total free energy of the system6,8 rather than toattractions in the effective potential. This kind of instabilitycan, in principle, be predicted by the present IE formalismbecause it considers all correlations involved in the problemin a self-consistent manner. Nevertheless, the region of inter-est for the experimentalists is still beyond the reach of theCGHNC theory and other IE approaches. This is basically, asdiscussed above, a consequence of the lack of solution of theHNC colloid–counterion integral equation when the electro-static interaction between both species at the colloidal sur-face is very large. Any attempt to surmount this problemshould go either in the line of charge renormalization or theutilization of a more accurate closure for colloid–counterioncorrelations. The adequate separation of the colloid–counterion problem from the rest of the correlations accom-plished by the CGHNC approach makes easier any improve-ment upon this drawback of the IE technique. In addition, thegood performance of both the ‘‘jellium’’ approximation andthe HNC equation to describe the counterionic backgroundand the colloid–counterion structure, respectively, proves tobe very promising in any extension of the present formalismto the case of added salt. An investigation on both theseissues is currently in progress.

ACKNOWLEDGMENTS

This work has been supported by the SpanishDireccionGeneral de Investigacio´n Cientıfica y Tecnica under GrantNo. PB1998-0326 and theInstituto de Salud Carlos IIIunderGrant No. 01/1664. We also thank Dr. A. A. Louis and Dr.Mejıas-Romero for interesting comments and a critical readof the manuscript.

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2B. V. Derjaguin and L. D. Landau, Acta Physicoquim. URSS,14, 633~1941!; E. J. W. Verwey and J. Th. G. Overbeek,Theory of the Stability ofLyophobic Colloids~Elsevier, Amsterdam, 1948!.

FIG. 7. Solution~shaded! and nonsolution~white! regions of the CGHNCequations for a colloid–counterion mixture at 300 K and a packing fractionof 0.08. The black point shown in the figure corresponds to the thermody-namic state considered in Fig. 4.

10521J. Chem. Phys., Vol. 116, No. 23, 15 June 2002 Self-consistent effective interactions

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equivalence between the bridge functionals of very different systems.Nevertheless, bearing in mind that the bridge function normally representsa small correction in the mean-force potential, and that bridge functions ofvery different systems have the same functional form~tuned by an opti-mizable parameter!, we can consider this approximation as reasonable inthe present context.

34The solution of the linearized Poisson–Boltzmann equation for a sphericalparticle of radiusR5sc /2 corresponds to the electrostatic potentialC(r )5(zc /e)@1/(11kDR)#exp@2kD(r 2R)#/r . The colloid–counterionmean-force potential is thenwci(r )52bziC(r ).

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36K. Ng, J. Chem. Phys.61, 2680~1974!.37A. A. Broyles, J. Chem. Phys.33, 2680~1960!.38The soft-core potential employed here is the same considered in Ref. 13,

that is, umnSR(r )5Bmn /r , with Bmn chosen such that the minimum in

the total soft-core potential coincides with that in the hard-core one,Bmn52zmzne2smn

8 /(9e).39A plot of the BHP results of Belloni~Ref. 21! in Fig. 2 would produce a

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equation. However, the clear correlation between no solution and highvalues of the compressibility indicate that the true spinodal line would notlie far away from the state points for which the CGHNC equation cease tohave a solution. The real nature of this nonsolution boundary has beendiscussed by Belloni@J. Chem. Phys.98, 8080~1993! and Ref. 22#.

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10522 J. Chem. Phys., Vol. 116, No. 23, 15 June 2002 J. A. Anta and S. Lago

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