On the macroion virial contribution to the osmotic pressure in charge-stabilized colloidal suspensions E. Trizac NSF Center for Theoretical Biological Physics, UCSD, La Jolla, CA 92093-0374 USA and LPTMS, Univ. Paris-Sud, UMR 8626, Orsay F-91405 and CNRS, Orsay F-91405 Luc Belloni 7 0 Laboratoire Interdisciplinaire sur l’Organisation Nanom´etrique et Supramol´eculaire CEA/SACLAY, 0 Service de Chimie Mol´eculaire, 91191 Gif sur Yvette Cedex, France 2 J. Dobnikar n a Institut fu¨r Chemie, Karl-Franzens-Universit¨at, Heinrichstrasse 28, 8010 Graz, Austria and J Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia 1 3 H.H. von Gru¨nberg Institut fu¨r Chemie, Karl-Franzens-Universit¨at, Heinrichstrasse 28, 8010 Graz, Austria ] t f R. Castan˜eda-Priego o Instituto de F´ısica, Universidad de Guanajuato, 37150 Leon, Mexico s . (Dated: February 6, 2008) t a Ourinterest goes tothedifferentvirialcontributionstotheequationofstateof chargedcolloidal m suspensions. Neglect of surface effects in the computation of the colloidal virial term leads to - spurious and paradoxical results. This pitfall is one of the several facets of the danger of a naive d implementation of the so called One Component Model, where the micro-ionic degrees of freedom n are integrated out to only keep in the description the mesoscopic (colloidal) degrees of freedom. o Ontheotherhand,dueincorporation of wall induced forces dissolves theparadox brought forth in c [ the naive approach, provides a consistent description, and confirms that for salt-free systems, the colloidal contribution to the pressure is dominated by the micro-ionic one. Much emphasis is put 1 on the nosalt case but thesituation with added electrolyte is also discussed. v 4 7 I. INTRODUCTION 7 1 0 In a complex mixture where several species with widely different characteristic time and space scales coexist, it 7 is common practice to resort to a coarse grained description integrating from the partition function all degrees of 0 freedom that do not belong to the main (larger) constituent [1, 2, 3, 4]. This leads to a state dependent effective / t Hamiltonianforthemainconstituent,therebyallowingaOneComponentModel(OCM)description. Themotivation a for such a procedure is not only to facilitate contact with experiments, where mostof the time the smallconstituents m cannotbe probeddirectly,butalsoto simplify the theoreticaltreatment. Indeed,onecanthenuse thewelldeveloped - statistical mechanics tools from the theory of simple liquids to study the OCM. This transposition from simple to d n complex fluids is however paved with practical difficulties, see e.g. [5, 6]. It is the purpose of the present paper to o discuss one such pitfall arising in the context of chargedcolloidal suspensions. c The system we will consider is made up of N charged spherical hard particles (colloids) immersed in a solvent c : v with dielectric constant ε, which fills a box with volume V limited by a neutral hard wall. The colloid’s interior is i assumedtohavethesamedielectricconstantasthesolvent. EachcolloidbearsachargeZ ewhereeistheelementary X c chargeandZ 1. Themedium outsidethe containerisa structurelessdielectric continuumwithdielectric constant c r ε′. To ensure e≫lectroneutrality,the solution contains N Z microscopic counterions,assumed monovalent. Additional a c c microions may also be present due to the dissociation of an added salt and the total number of microions is denoted N . TheparticlesinteractthroughCoulombforcesandhardsphereexclusion,althoughinthesubsequentanalysis, micro the hard-core interaction will turn out to be irrelevant. The paper is organized as follows. In the situation where ε = ε′, we consider in section II the different virial contributions to the equation of state. In the salt-free case, we argue that the colloidal contribution P has to be ocm negligible compared to the microionic one (P ). We then show that a naive implementation of the OCM picture micro leads to a violation of the constraint P P . Sections III for ε = ε′ and IV for ε′ < ε are devoted to the ocm micro ≪ resolution of this apparent paradox. It will be shown that in a closed cell, the surface contribution to the colloidal virial P is comparable to the bulk term, while only the latter is considered in the naive picture. Hence its failure, ocm resulting from a gross overestimation of P . As a consequence, the knowledge of a good effective potential in the ocm 2 bulk is insufficient when it comes to directly computing the colloidal virial in a closed box. Concluding remarks are drawn in section V, where we discuss in particular how the effective potential can be used –indirectly but from a standard procedure– to compute the total pressure of the system. While most of the analysis holds without salt, the situation of an added electrolyte is also briefly addressed. II. EQUATION OF STATE, EFFECTIVE INTERACTIONS AND ONE COMPONENT MODEL VIRIAL A. The equation of state We startby the simplestsituationwhere ε=ε′ andconsiderall chargedspecies inthe solution. The virialtheorem allows to write the total osmotic pressure P (with respect to pure solvent) in the form β βPV = N +N + r Fint , (1) c micro 3 * i· i + i∈col+micro X where β = 1/(kT) is the inverse temperature and the summation runs over colloids and microscopic ions, therefore involving N + N terms. In Eq. (1) the angular brackets denote a statistical average (that coincides with c micro time average) and Fint is the (internal) force exerted on particle i at position r , due to hard core and Coulombic i i interactionsin the solution. Adding the force exertedby the wallto Fint wouldtherefore providethe total forceFtot i i felt by particlei. We note here that itis possible to expressthe pressurein Eq. (1) as a surfaceintegraloverthe wall of the total (colloid + micro-ions) concentration. Applying the virial theorem to the microions only, we have 1 1 kT N kT + r Ftot = 0 = N kT + r Fint ρ (r)r d2S , (2) micro 3*i∈micro i· i + micro 3*i∈micro i· i +− 3 (cid:28)Ibox micro · (cid:29) X X where the surface integralwith normal oriented outwardruns over the box confining the system. Inserting the latter equality into (1), we obtain 1 kT P = ρ kT +P +P with P = r Fint ; P = ρ (r)r d2S , (3) c ocm micro ocm 3V *i∈col i· i + micro 3V (cid:28)Ibox micro · (cid:29) X where ρ =N /V andρ (r) denotes the totalmicroiondensity at pointr. Within mean-fieldapproximation,this c c micro equation may be found in [1]. The first term on the right hand side of Eq. (3) is the colloid ideal gas term which can be safely neglected in practice for the parameter range of interest here (see below). The second term –of central interesthere–isthe colloid-colloidvirialcontributionandis indexedbythe subscriptOCMsinceitwouldbe the only termconsidered(apartfromthe idealgasone)inthe OCMapproach,restrictedto the mesoscopicdegreesoffreedom r . Indeed, the statistical average ... may be performed in two steps : { i}1≤i≤Nc h i 1 Nc 1 Nc 1 Nc P = r Fint = r Fint = r Feff , (4) ocm 3V * i· i + 3V * i· i micro+ 3V * i· i + Xi=1 col+micro Xi=1 (cid:10) (cid:11) col Xi=1 col wherewehaveintroducedthemicroionaveragedeffectiveforceFeffexertedoncolloidiforagivencolloidconfiguration. i The third term in (3), P , accounts for the direct coupling between colloids and microions. In principle, this micro third term is to be averaged over the colloidal degrees of freedom. However, even at simplified or mean-field level, a fullN -colloidsimulationiscomputationallydemanding[7,8,9],andfurthersimplificationsarehelpful. Ofparticular c interestaretwosuchsimplifications,bothbelongingtothePoisson-Boltzmannfamily,thatreducetheinitialN -body c problem onto a N = 1 body situation. The first one is the common cell model approach originating from a solid c statepointofviewwheretheWigner-Seitzcellaroundacolloidisconstructedandthen“sphericalized”forthesakeof simplicity. The Poisson-Boltzmannequationis solvedwithin this cell,andfromthe microionicdensityprofileone can then estimate P . The second model is the renormalized jellium model [10] where a liquid state point of view is micro adopted: thecolloid-colloidpairdistributionfunctiong (r)isconsideredstructurelesssothatothercolloidsarounda cc taggedmacroionbehaveasacontinuousbackground. Thechargeofthisbackgroundisaprioriunknown,andenforced to coincide with the effective charge. This self-consistency requirement leads to a unique and well defined effective charge [10]. It has been shown that for salt-free suspensions, these two models – cell and jellium – both lead to a pressure P that is in excellent agreement with existing experimental data [11] and primitive model simulations micro 3 for P [12, 13], see e.g. [10, 14, 15]. We note that P may be coined a “volume” term [2, 17, 18], since – at least, micro within the cellmodel andjellium approaches– itdoes notdepend onthe colloidaldegreesoffreedombutonly onthe mean colloidal density. The good agreement one obtains with the exact pressure P for both models implies that for salt-free systems P P . This is corroborated by a recent study of finite stiff-chain polyelectrolytes [19]. From micro ≃ Eq. (3) where the ideal gas contribution (ρ kT) is neglected, this may be transposed into the following requirement: c P P . (5) ocm micro ≪ A similar conclusion was reached in Ref. [20]. B. Effective interactions BothPoisson-Boltzmanncellandjelliumapproachesarenotonlyusefultoestimatethepressure,butalsotoderive effective parameters for solvent + microions averaged colloid/colloid interactions. By construction, the effective potential is that which leads to the correct colloid-colloid pair structure encoded in the potential of mean force g , cc assuming pair-wise colloid-colloid interactions within the OCM model (see e.g. [1]). Although the effective potential has a clear-cut definition, there is no rigorous operational route to construct this object. In general, when microionic correlations do not invalidate the mean-field picture [13], a good approximation is to write the effective potential as a sum of pair-wise Yukawa terms of the form 2 exp(κ a) exp( κ r) βv (r) = Z2 λ eff − eff (6) eff eff B 1+κ a r (cid:18) eff (cid:19) with a the colloidradius,λ =βe2/ε the Bjerrumlength, andZ andκ the effective chargeandinversescreening B eff eff length computed within the cell or jellium model [10, 21, 22, 23]. Such a “DLVO”-like expression [1, 2, 4] would accurately reproduce the large distance interaction of two colloids in a salt sea [1, 2, 4]. Its relevance in the no-salt casewill notbe discussed. As will become clearbelow,we areinterestedherein ordersofmagnitude, thatshouldnot depend on the precise form of (6). C. An apparent paradox Within the jellium model, the salt-free equation of state takes a particularly simple form βP =Z ρ . (7) micro eff c Withinthecellmodel,thisexpressionisnotexactbutapproximatelycorrect. Forahighlychargedmacroion,onehas Z 1 which allows to neglect the ideal gas term in (3). In spite of its simplicity, the expression βP = ρ Z eff micro c eff ≫ hides a complex density dependence through Z and is in excellent agreement with the exact pressure P found eff experimentally or in primitive model simulations, as emphasized above. In addition, the effective screening length reads [10] κ2 =4πλ ρ Z (8) eff B c eff The constraint embodied in Eq. (5) may therefore be rewritten βP Z ρ . (9) ocm eff c ≪ Alternatively,inthe lowelectrostaticcoupling regime(whereZ coincideswith Z ), oneshouldrecoverthe idealgas eff c pressure βP ρ (1+Z ). Given that in this limit, βP ρ Z , we recover the requirement (9), that will be c eff micro c eff ≃ ≃ an important benchmark for the following analysis. We now turn to the formulation of the apparent paradox. In the bulk of the suspension, the effective potential (6) provides the effective force acting on a colloid i Nc Nc Fieff = Fiejff = − ∇rveff(r) . (10) j=1 j=1 (cid:12)r=ri−rj X X (cid:12) (cid:12) Consideringnaively thatPocm appearingin (3)and (4) is dominated in(cid:12)a very largesystembyits bulk behaviour,we insert (10) into (4) to approximate P by P∗ with ocm ocm 1 Nc 1 Nc P∗ = r Feff = r Feff , (11) ocm 3V * i· ij + 6V * ij· ij + iX,j=1 col iX,j=1 col 4 where r =r r . We will subsequently omit the subscript “col” indicating the degrees of freedom involved in the ij i j − average. Introducing the colloid-colloid pair correlationfunction g (r), we can write cc ρ2 ∞ dβv (r) βP∗ = c g (r) eff rd3r (12) ocm − 6 cc dr Zr=2a 2πρ2Z2 λ (κ a)2 ρ2 ∞ = c eff B 1+ eff + c [g (r) 1](1+κ r)βv (r)d3r. (13) κ2 3(1+κ a)2 6 cc − eff eff eff (cid:26) eff (cid:27) Zr=2a To estimate the above quantity, it is sufficient to keep the dominant term only, which is the first one on the rhs, arising from the long-range behavior of the pair correlation function (g 1 at large distances). In this term, the cc → curly brackets may be safely approximated by 1 since at low densities, κ a 1. Remembering Eq. (8), we obtain eff ≪ 2πρ2Z2 λ P∗ c eff B (14) ocm ≃ κ2 eff 1 Z ρ , (15) ≃ 2 eff c Thefactor1/2whichappearsisclassical(seee.g. [1]). Theimportantpointhereisthatestimation(15)byfarviolates theconstraint(9). AsimilarconclusionwouldbereachedincludingthefirstcorrectioninZ2 exp( κ r)/rtothelong distance behaviour g =1 when computing the integral on the rhs of (13): this yields P∗ eff Z −ρ /ef2f[1+ (κ λ )] ocm ≃ eff c O eff B with κ λ 1 in the dilute limit. The paradox here is that the very same approach that provides a contribution eff B ≪ P very close to the total pressure, gives an effective potential that apparently spoils the previous agreement, by micro grossly overestimating the colloidal virial contribution to the pressure. We will see that this feature is not ascribable to a failure of the functional form of Eq. (6), which provides a decent approximationfor the quantity P∗ . ocm D. How can the paradox be resolved ? The root of the paradox reported above is that approximating P by P∗ is incorrect: while P∗ provides ocm ocm ocm a reasonable estimate for the bulk contribution to P , surface effects make that in the vicinity of the wall, the ocm effective force felt by a colloid differs from (10). These surface induced terms play a key role here and contribute a large amount to the colloidal virial P , no matter how large the system is. It turns out that bulk and surface ocm induced contributions almost cancel each other, so that the resulting expression for P is much smaller than P∗ ocm ocm and therefore fulfills the requirement (9). Our goal in the remainder is to illustrate this cancellation explicitly, from a correct description of confinement effects. To this aim, it is judicious to simplify the problem by considering the limit of point colloids (a = 0), and by identifying the effective charge with the bare one Z . Considering charge c renormalization effects is here immaterial and focussing on dilute systems where κa is small, finite a effects do not affect our main conclusions. In the bulk of the suspension, the effective potential therefore takes a simple Yukawa form exp( κr) βv (r) = Z2λ − , (16) eff c B r with κ2 =4πλ Z ρ . B c c At this point, a comparison with simple electrolytes seems appropriate, for the aforementioned cancellation is already present. For our discussion, we may consider that the role of the colloids is played by the cations, and that the anions constitute the remaining “microions”. The pressure has to be close to [1, 4] κ3 βP ρ +ρ , (17) electrolyte anion cation ≃ − 24π with equal mean densities ρ =ρ . From the contact theorem, we deduce the densities at the wall anion cation κ3 ρ (wall)=ρ (wall) ρ . (18) anion cation anion ≃ − 48π Rewriting (3) in the form β βP = ρ + r Fint +ρ (wall) (19) electrolyte cations 3V * i· i + anion i∈cation X 5 we obtain from (17) and (18) β κ3 r Fint . (20) 3V * i· i + ≃ −48π i∈cation X Given that κ2 = 8πλ Z2ρ , we have κ3/(βP∗ ) κλ which is a small quantity for a dilute system. We explicitly see here thatBthe “cactoiollnoidal” virial [lhs ofo(c2m0)∝up toBa factor β] is by far smaller than the estimation P∗ . ocm III. WALL MEDIATED FORCES WITHOUT DIELECTRIC DISCONTINUITY In the vicinity of the wall, the colloids do not see a spherically symmetric environment. As a consequence, 1. the usual exp( κr)/r pair interaction is modified. − 2. the mean force acting on a colloid does not vanish. This is a one body, wall induced effect, mediated by the microions. It is therefore an internal force, that should be taken into account in (4). It should not be confused with the external (and short range) direct colloid-wall interaction. Evaluating the rhs of (4) therefore requires a carefulcomputation of both types of microionaveragedcolloidalforces. Tothisend,weneedthesolutionφ (ρ,z′)ofDebye-Hu¨ckelequation 2φ =κ2θ(z′)φ inthecasewhereatestcharge z z z ∇ is located in the solution a distance z from an infinite neutral wall. We have introduced the Heaviside function θ and cylindrical coordinates (ρ,z′) such that the test particle is located at (0,z) with z > 0. The planar geometry approximationfor the wall is sufficient provided the cell size or radius of curvature is much larger than Debye length 1/κ. We startby the situation of equaldielectric constants inside andoutside the solution (ε=ε′). The electrostatic potentialmaybewrittenintheformofaHankel(twodimensionalFourier)transform[26]whereqandρareconjugate quantities [24, 25] φ (ρ,z′) = Z λ ∞ k−q e−k(z+z′) + e−k|z−z′| 1 J (qρ)qdq ; k κ2+q2 (21) z c B 0 k+q k ≡ Z0 (cid:18) (cid:19) p The second term in the integrand (e−κ|z−z′|) gives exactly Z λ exp( κr)/r where r = [ρ2 +(z z′)2]1/2 is the c B − − distance to the source. This is the standard Debye-Hu¨ckel potential which dominates in the bulk. The remaining term, which vanishes at large distances (κz or κz′ 1) is due to the presence of the interface. ≫ A. One colloid ion average force The force felt by a colloid located a distance z from the planar interface follows from (21), considering the electro- static potential φ =φ Z2λ e−κr/r where the self term has been subtracted: z z − c B ∂ ∞ k q e βF = nZ βφ (0,z′) = Z2λ − e−2kzqdq( n). (22) c−wall c∂z′ z c B k+q − (cid:12)z,z′=z Z0 (cid:12) In this equation, n denotes the unibt vector peerpendi(cid:12)(cid:12)cular to the interface pointing outside tbhe solution. We coin the force (22) “colloid-wall” and for notational convenience, we henceforth omit the superscripts “int” and “eff”. This force repels the colloid from the wall [k = (κ2+q2)1/2 > q], as a result of microions imbalance between the half of b the colloid exposed to the wall, and the other hemisphere. Inserting (22) into (4) we have 1 Nc 1 ∞ r F = d2S ρ (z)r F (z)dz. (23) i i−wall c i−wall 3V *i=1 · + 3V Zwall Z0 · X To leading order, the above integral may be computed assuming a uniform density of colloids ρ (z) = ρ . In (23), c c r denotes the absolute position with r = s zn (s is therefore the orthogonal projection of r onto the wall). We − b 6 neglect the term in zn (that would contribute proportionally to the surface of the system), so that − 1 Nc r βFb βρc d2S ∞s F (z)dz (24) i i−wall i−wall 3V *i=1 · + ≃ 3V Zwall Z0 · X ρc Z2λ s nd2S ∞dz ∞ κ2+q2−q e−2z√κ2+q2qdq (25) ≃ −3V c B(cid:18)Zwall · (cid:19) Z0 Z0 pκ2+q2+q 1 ρ Z2κλ b p (26) ≃ −6 c c B κ3 . (27) ≃ −24π Incidentally, this is exactly the Debye-Hu¨ckel form for the excess pressure of an electrolyte [see Eq. (17)]. For dilute systems, this quantity is small compared to ρ Z , as emphasized earlier. The constraint (9) is therefore fulfilled. c c B. Colloid-colloid interactions Within the simple Debye-Hu¨ckel treatment, the potential of interaction between two colloids near the wall (one at z, the other at z′, with a lateral distance ρ between them) is Zcφz(ρ,z′)=Zcφz′(ρ,z). To calculate the force felt by the colloid at z due to all neighbors, we assume again a uniform distribution of neighbors: ∞ ∞ ∂Z βφ (ρ,z′) βF (z) = nρ dz′ 2πρdρ c z (28) col−col c Zz′=0 Z0 ∂z (cid:12)z′ (cid:12) = bnρ Z2λ ∞dq ∞2πρdρ κ2+(cid:12)(cid:12) q2−q 1 e−z√κ2+q2 1 J (qρ)qdq (29) − c c BZ0 Z0 pκ2+q2+q − ! κ2+q2 0 b p p The component of the force parallel to the wall vanishes upon averaging. Inserting this force into (4) and proceeding along similar lines as in Eqs. (24) sq, we have β Nc ∞ ∞ κ2+q2 q 1 r F ρ2Z2λ 2πρdρ − +1 J (qρ)qdq. (30) 3V *i=1 i· i−coll+ ≃ c c BZ0 Z0 −pκ2+q2+q ! κ2+q2 0 X p Both expressions (29) and (30) are of the form of a Hankel transform at the origin q = 0 of the inverse Hankel transform of a function A(q), with A = (... 1)e−kz/k in (29) and A = ( ...+1)/k2 in (30). This is nothing but − − A(0) [26],which vanishesin bothcases. Therefore,withthe approximationsproposed,the forcein (29) andthe virial term in (30) vanish. To be more specific, we compute explicitly the integrals in (30): β Nc 1 r F ρ Z ( 1+1). (31) i i−coll c c 3V * · + ≃ 2 − i=1 X The term in +1 in the parenthesis arises from the term in +1 in Eq. (30), which gives the usual “bulk” e−κr/r pair interaction, as already mentioned. The associated virial is βP∗ = Z ρ /2, as obtained in (15). The present ocm c c calculation shows that this term is canceled by an opposite wall induced contribution. If the simplifying assumption g = 1 is relaxed, the resulting expression for (31) no longer vanishes but remains negligible with respect to ρ Z . cc c c On the other hand, relaxing the assumption of a uniform profile ρ (z) leaves the result unaffected, as will be seen in c section IV. We conclude here that summing the two contributions from Eqs. (27) and (31) provides a value for P that is ocm compatible with the constraint (9). IV. ANALYSIS IN PRESENCE OF A DIELECTRIC DISCONTINUITY In this section, we extend the previous analysis to the situation where the dielectric constants are not matched: η = ε′/ε = 1. The relevant parameter range corresponds to η < 1 e.g for water droplets in air in a spray-drying 6 experiments. The first important difference with the η = 1 case is that the equation of state (3) takes a different 7 form. The pressure is indeed not solely given by the contact densities of charged species at the wall, but contains additional electric contributions (polarizationor image effects). On the other hand, Eqs (1) and (2) are still formally correct providedone also includes in the “internal” forces the electric forces from the wall. The resulting equation of state reads 1 kT 1 P = ρ kT + r Fint + ρ (r)r d2S + r TeldS . (32) c 3V *i∈col i· i + 3V (cid:28)Ibox micro · (cid:29) 3V (cid:28)Ibox · (cid:29) X Here ε ε Tel = E2I E E (33) 8π − 4π ⊗ is the Maxwell tensor, with E the local electric field and I the isotropic tensor. The counterpart of (21) now reads: φ (ρ,z′) = Z λ ∞ k−ηq e−k(z+z′) + e−k|z−z′| 1 J (qρ)qdq ; k κ2+q2. (34) z c B 0 k+ηq k ≡ Z0 (cid:18) (cid:19) p As in the case η =1 and as long as η =0, the corresponding interaction between two colloids decays as ρ−3 at large 6 distancesparalleltothewall(see[27]foradiscussionofthisdipolar-liketerm). Whenη =0,the wallcanbeformally removed considering the electric image located symmetric to the z =0 plane. The colloid-colloid and colloid-wall interactions readily follow from (34). At short distances z 0, the latter → diverges like z−1(1 η)/(1 + η) [25], which corresponds to the unscreened interaction of a particle with its own − image. This divergence means that the uniform colloid density cannot be invoked when it comes to computing (23). To obtain the leading order behaviour, we can assume that the colloids are distributed with the Boltzmann weight ρ (z)= ρ exp[ βφ (z)], where F = ∇φ and the potential φ deriving from (34) vanishes for c c c−wall c−wall c−wall c−wall − − z . The precise knowledge of this potential is however not required since →∞ 1 Nc ρ ∞ r F = c d2S r F (z) exp[ βφ (z)]dz. (35) i i−wall c−wall c−wall 3V *i=1 · + 3V Zwall Z0 · − X ρ kT [exp( βφ (z))]∞ (36) ≃ c − c−wall 0 ρ kT. (37) c ≃ − This term therefore cancels the ideal gas one on the rhs of (32). Thewallinducedcolloid-colloidcontributiontothecolloidalvirialmaybecomputedalongsimilarlinesasinsection IIIB. An expression involving again a Hankel transform composed with its inverse is again obtained, with now a function A(q)= κ2+q2−ηq 1 1 ∞dzρ (z)e−z√κ2+q2 (38) c pκ2+q2+ηq − ! κ2+q2 Z0 p p Since A(0)=0, we conclude here that Nc r F 0, (39) i i−coll * · + ≃ i=1 X so that the total colloidal virial [including colloid-colloid and colloid-wall interactions] is close to ρ kT, which is a c − small quantity compared to the microionic contribution Z ρ kT. Equation (32) can finally be rewritten c c kT 1 P ρ (r)r d2S + r TeldS . (40) micro ≃ 3V · 3V · (cid:28)Ibox (cid:29) (cid:28)Ibox (cid:29) V. CONCLUDING REMARKS Before briefly discussing the situation where a salt is added, two comments are in order. 8 A. Closed cells versus periodic boundary conditions Fromthe previousdiscussion,itappearsthatthe equationofstate (3)holds whenthe systemis confinedby a hard wall,andwouldfailifperiodicboundaryconditions(pbc)wouldbeenforced. TheinadequacyofP∗ toapproximate ocm P may then be phrased in the following way ocm 1 Nc 3VPo∗cm ≡ 2*iX,j=1 Xn rij ·Fiejff(rij −Rn)+pbc (41) Nc = r Feff (42) 6 * i· i + Xi hard walls where in (41), the suminvolvesallperiodic imagesof the cellconsidered: n is a vector with components in Z3, which indexes the center Rn of a givenimage of the “central”cell. The centralcell has R0 =0 and since we dealhere with a short range effective potential, the sum over n may be truncated to retain only the 7 terms with n 1. However, for any simple fluid where the forces F are given, (42) would be an equality. Indeed we| h|a≤ve i simple fluid simple fluid 1 r F r F (43) i i ij ij * · + ≡ 2* · + Xi hard walls Xi,j hard walls where the rhs shows negligible dependence on the boundary conditions provided the system is large enough, and can then be computed with pbc provided the correct forces are considered [Fi = j nFij(rij −Rn)]. Hence simple fluid P Psimple fluid 1 ri Fi = rij Fij(rij Rn) (44) *Xi · +hard walls 2*Xi,j Xn · − +pbc Thedifferencebetweenequations(42)and(44)illustratesthe importantroleofmicroions. Wemayalsoconsiderthat the = sign in (42) arises from the density dependence of the effective pair potential. 6 A natural questionat this point is : does the knowledge of the “bulk” effective potential (6) between colloids allow to compute their virialP as it appearsin (3) ? The answeris positive in a closedcell, at the OCMlevel, provided ocm that due account is taken for the dielectric images of the colloids. In the following section, we address a related question, and discuss how the full pressure of the colloidal system may be recovered, assuming again that the only information at hand is that of the bulk effective colloid-colloid interaction. B. Back to the DLVO potential Weconsiderhereasimpleliquidthatinteractswithapair-wisepotentialgivenbyEq. (6),witheffectiveparameters Z∗ 1 and κ∗2 = 4πλ Z∗ ρ∗ (salt-free case, for simplicity). These parameters are fixed a priori, and chosen to coeiffnc≫ide with thefofse relevBanteffforca colloidal suspension at ρ = ρ∗. The potential of interaction is therefore density c c independent and the system, later referred to as “auxiliary”,can be studied for ρ =ρ∗. c 6 c We consider the parameter range (essentially low density) where the excess pressure of such a system is well approximated by P∗ in Eq. (14): ocm 2πρ2Z∗2λ 1 ρ2 βP∗ c eff B = c Z∗ . (45) ocm ≃ κ∗2 2 ρ∗ eff eff c Incidentally, the contact theorem indicates that the contact density in the case where the system is confined by a closed box, reads ρ (wall) ρ2Z∗ /(2ρ∗). This quantity is much larger than the mean density ρ (except when ρ is c ≃ c eff c c c extremely small, a limit of little interest here). This excess with respect to the mean density is to be contrasted with the depletion from the wall that is present in the original colloidal system containing microions: Eq. (22) for ε = ε′ shows a repulsive colloid-wall behaviour, and the depletion is even stronger when ε′ < ε due to like-sign images, see the discussion after Eq. (34). ThepressureofthesimpleliquidwithDLVOinteractions,closetoP∗ ,hasapriorinothingtodowiththepressure ocm Poriginal of the real colloidal system. It has also nothing to do with the colloid virial contribution entering Eq. (3). However, for ρ = ρ∗, the colloid-colloid structural information is the same for both original and auxiliary systems. c c 9 OnemaytheninvokeKirkwood-Buffidentity[28]whichstatesthattheinversecompressibilityoftheoriginalcolloidal suspension coincides with the long wave-length limit of the colloid-colloidstructure factor S (k) : cc ∂βPoriginal −1 χ = = S (0). (46) cc ∂ρ (cid:18) c (cid:12)T(cid:19) (cid:12) The compressibility in our auxiliary simple liquid with fixed(cid:12) potential of interaction is therefore the same at ρ =ρ∗ (cid:12) c c (and only at this density) ∂Poriginal ρc==ρ∗c ∂Po∗cm (47) ∂ρ ∂ρ c (cid:12)T c (cid:12)T,κ∗eff,Ze∗ff (cid:12) (cid:12) This offers a means to compute the equation of(cid:12)state of the origin(cid:12)al colloidal system from integrating the inverse (cid:12) (cid:12) compressibility of the auxiliary one. In this integration, due account must be taken of the density dependence of both Z∗ and κ∗ . The previous integration procedure therefore requires to consider the auxiliary system for several eff eff values of ρ for a given ρ∗ [to compute the derivative in the rhs of (47)], before scanning the range of interest for ρ∗. c c c Of course, the general procedure outlined here does not depend on the specific form of the effective potential, and is equallyvalidwhensaltis added. Itturnshoweverthatthe DLVOpotentialtogetherwiththe salt-freeapproximation (14) –which leads to (45)– provide a clear illustration of the procedure. From (45), we obtain the rhs of Eq. (47): ∂βP∗ ocm Z∗ at ρ =ρ∗ (48) ∂ρc (cid:12)T,κ∗eff,Ze∗ff ≃ eff c c (cid:12) Tocomputethelhsof(47),wemaycomebackt(cid:12)(cid:12)othejelliummodelwhichgivesβPoriginal Zeffρc. Inthisexpression, ≃ the effective charge may depend on the density, but for salt-free cases, this dependence is at most logarithmic for ρ 0 [10] and provides only a subdominant term to the compressibility, so that c → ∂βPoriginal Z . (49) ∂ρ ≃ eff c (cid:12)T (cid:12) Evaluating this expression at ρ = ρ∗ where Z = Z∗ , w(cid:12)e recover Eq. (48). This not only illustrates the identity c c eff eff (cid:12) (47) but also the consistency of the underlying DLVO potential. C. Situation with added salt Whenthesuspensionisdialyzedagainstasaltreservoir,mostofthetechnicalanalysiscarriedoutearlierisstillvalid. We consider a similar auxiliary system as in section VB, with effective screening length such that κ∗2 >4πλ Z∗ ρ eff B eff c due to the screening by salt ions [29]. The effective charge and screening lengths are again chosen to coincide with thoseofacolloidalsystemataparticulardensityρ∗,butareotherwisedensityindependent. Equation(47)stillholds c while P∗ is givenby (13). Neglecting againthe integralonthe rhs of (13), andinserting the resulting P∗ in (47), ocm ocm we obtain: ∂Poriginal 4πλ ρ Z2 = B c eff , (50) ∂ρ A κ2 c (cid:12)T eff (cid:12) where we have replaced Z∗ by Z and κ∗ by κ a(cid:12)fter computing the rhs of (47). Here, the prefactor reads eff eff eff eff (cid:12) A (κ a)2 = 1+ eff . (51) A 3(1+κ a)2 eff Is relation (50) compatible with P P = Poriginal ? Neglecting P (together with ρ kT) in (3), we have ocm ocm c P P which in the jellium model is g≪iven by κ2 /(4πλ ). With the help of [29], we arrive at ≃ micro eff B ∂P 4πλ ρ Z2 micro = B c eff . (52) ∂ρ κ2 c (cid:12)T eff (cid:12) Equations (50) and (52) give the same result provided(cid:12) is close to unity, which means κ a<1. We conclude here (cid:12) A eff that omitting the colloidal contribution to the pressure, P , is inconsistent when κ a > 1. It turns out however ocm eff that increases very mildly with κ a (e.g it is close to 1.2 for κ a =4). A more precise discussion would require eff eff A to consider the full rhs in (13), which is beyond the scope of this paper. Finally, we note that in the salt-free case where κ a = 3η Z λ /A with η = 4πρ a3/3 the colloidal volume fraction and Z λ /a on the order of 10 for eff c eff B c c eff B highly charged colloids, we have κ a<3 and therefore close to 1 even for packing fractions as high as 10%. eff A 10 D. Summary We have seen that for a salt-free colloidal suspension, the colloidal contribution P to the equation of state [as ocm written in Eq. (3)] is a negligible quantity. This feature may easily be overlooked in a naive implementation of the One Component Model, where only P∗ , the bulk contribution to P , is computed. The fact that P∗ is ocm ocm ocm of the same order of magnitude as the total pressure P of the suspension, is not compatible with the requirement P P P, that has emerged as a central constraint in our analysis. We have shown that no matter how ocm micro ≪ ≃ large the system is, surface effects that require the resolution of Poisson’sequationin the vicinity of a confining wall, contribute a largeamount to P . To zerothapproximation,these surface terms cancel the bulk value P∗ , so that ocm ocm one finally recoversP P. ocm ≪ Acknowledgments It is a pleasure to thank Y. Levin for fruitful discussions. R.C.P thanks PROMEP-Mexico and CONACyT (grant 46373/A-1) for financial support. J.D. acknowledges the Marie-Curie fellowship MEIF-CT-2003-501789. 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