Repulsion between Oppositely Charged Macromolecules or Particles

Repulsion between Oppositely Charged Macromolecules or Particles

M. Trulsson,* Bo Jo¨nsson, T. Åkesson, and J. Forsman

Theoretical Chemistry, Chemical Center POB 124, S-221 00 Lund, Sweden

C. Labbez

Institut Carnot de Bourgogne, UMR CNRS 5209, UniVersitede Bourgogne, F-21078 Dijon Cedex, France

ReceiVed April 26, 2007. In Final Form: August 20, 2007

The interaction of two oppositely charged surfaces has been investigated using Monte Carlo simulations andapproximate analytical methods. When immersed in an aqueous electrolyte containing only monovalent ions, two suchsurfaces will generally show an attraction at large and intermediate separations. However, if the electrolyte solutioncontains divalent or multivalent ions, then a repulsion can appear at intermediate separations. The repulsion increaseswith increasing concentration of the multivalent salt as well as with the valency of the multivalent ion. The additionof a second salt with only monovalent ions magnifies the effect. The repulsion between oppositely charged surfacesis an effect of ion-ion correlations, and it increases with increasing electrostatic coupling and, for example, a loweringof the dielectric permittivity enhances the effect. An apparent charge reversal of the surface neutralized by the multivalention is always observed together with a repulsion at large separation, whereas at intermediate separations a repulsioncan appear without charge reversal. The effect is hardly observable for a symmetric multivalent salt (e.g., 2:2 or 3:3).

Introduction

The interaction between charged surfaces in electrolytesolutions is important for essentially all colloidal and biologicalsystems. Consequently, these interactions have been studiedextensively during the last century, both experimentally andtheoretically. The theoretical approach has for many years beenbased on a combination of the Poisson-Boltzmann (PB) equationand van der Waals forces. This so-called DLVO theory1,2 treatsthe solvent as a continuous medium described through a dielectricpermittivity. The mean-field approximation in the PB equationmeans that ion-ion correlations are neglected, which becomesapparent when the electrostatic interactions increase.3,4 Thus,the DLVO theory can be expected to describe the behavior ofcolloidal systems at low electrostatic coupling. Perhaps the mostinteresting and well known behavior is the long-range screeningof the electrostatic interaction between two charged surfaces inan electrolyte solution.5,6 The DLVO theory is not onlyqualitatively but also quantitatively in agreement with experimentin this case.

For electrostatically highly coupled systems, the mean-fieldtheories fail to account even qualitatively for a number ofphenomena. Typical examples are charge reversal (or overcharg-ing),7,8 where ions accumulate close to a surface and create anapparent surface charge with the opposite sign of the bare surfacecharge, and the counter-intuivative phenomenon of attractionbetween surfaces of like charge and its converse, repulsion

between surfaces of opposite charge.9-11 More sophisticatedmethods, which include ion-ion correlations, such as the modifiedPB equation,12 the hypernetted chain theory,4 the correlation-corrected Poisson-Boltzmann,13 or Monte Carlo simulations(MC),3 are needed in order to describe these effects. The failureof the mean-field theory in the highly coupled regime is thus dueto the lack of ion-ion correlations (electrostatic and/or hardcore) and not to the use of a continuum model for the solvent.

In a previous study, inspired by the experiments of Bestemanet al.9and by the fact that oppositely charged particles are commonin cement paste, we investigated the origin of the repulsiveinteraction between two oppositely charged surfaces. It was foundthat if there is a charge reversal at one of the surfaces then therewill also be a repulsive interaction at large separations. Therepulsion will increase with decreasing separation and will alsoexist at intermediate separations, where theapparentsurfacecharge reversal has disappeared. For example, the maximum inrepulsion will never coincide with the charge reversal of one ofthe surfaces. The repulsion is not a direct consequence of thecharge reversal. Thus, the former should not be seen as an effectiveelectrostatic interaction between two apparently equally chargedsurfaces, except at large separations. At intermediate separationsthe repulsion is better described as an entropic effect due to thelarge amount of salt in the system. However, both phenomenahave the same origin, namely ion-ion correlations, which leadto a larger accumulation of counterions close to a charge surface.If the bulk salt concentration is high, then a large number of extracounterions, and their co-ions, can be sucked in between thecharged surfaces. If the salt consists of a multivalent counterionand monovalent co-ions, then the latter can make a significantcontribution to the internal pressure.

* Corresponding author. E-mail: [email protected](1) Derjaguin, B. V.; Landau, L.Acta Phys. Chim. URSS1941, 14, 633.(2) Verwey, E. J. W.; Overbeek, J. Th. G.Theory of the Stability of Lyophobic

Colloids; Elsevier: Amsterdam, 1948.(3) Guldbrand, L.; Jo¨nsson, B.; Wennerstro¨m, H.; Linse, P.J. Chem. Phys.

1984, 80, 2221.(4) Kjellander, R.; Marcˇelja, S.J. Chem. Phys.1985, 82, 2122.(5) Pashley, R. M.J. Colloid Interface Sci.1981, 83, 531.(6) Pashley, R. M.; Israelachvili, J. N.J. Colloid Interface Sci. 1984, 101, 511.(7) Pashley, R. M.J. Colloid Interface Sci.1984, 102, 23.(8) Sjostrom, L.; Åkesson, T.; Jo¨nsson, B.Ber. Bunsen-Ges. Phys. Chem.

1996, 100, 889.(9) Besteman, K.; Zevenbergen, M. A. G.; Heering, H. A.; Lemay, S. G.Phys.

ReV. Lett. 2004, 93, 170802.

(10) Besteman, K.; Zevenbergen, M. A. G.; Lemay, S. G.Phys. ReV. E 2005,72, 061501.

(11) Trulsson, M.; Jo¨nsson, B.; Åkesson, T.; Labbez, C.; Forsman, J.Phys.ReV. Lett. 2006, 97, 068302.

(12) Outhwaite, C. W.; Bhuiyan, L. B.J. Chem. Soc., Faraday Trans.1983,2 79, 707.

(13) Forsman, J.J. Phys. Chem. B2004, 108, 9236.

11562 Langmuir2007,23, 11562-11569

10.1021/la701222b CCC: $37.00 © 2007 American Chemical SocietyPublished on Web 10/05/2007

In this article, we extend our previous Monte Carlo simulationsof oppositely charged surfaces to investigate the effect of (i) asymmetric salt, (ii) adding a second salt, (iii) varying the surfacecharge density of the nonovercharged surface, and (iv) varyingthe dielectric permittivity. All is done in order to describe thegenerality of the repulsion between the two oppositely chargedsurfaces. The results are also rationalized with aneffectiVemean-field theory.

Model

The interaction between the oppositely charged surfaces ismodeled as two planar surfaces with uniform surface chargesdensities,σ-1 (negative) andσ1 (positive), separated a distanceh. The primitive model is used,14 with all ions explicitlyconsidered, whereas the solvent is treated as a dielectric continuumcharacterized by a dielectric permittivityεr. The temperatureTis set to 298 K, andεr is set to 78.7, corresponding to water atroom temperature. In the simulations, the ion-ion interactionsare calculated according to

u(rij) is the Coulomb energy between two ionsi andj with chargesqi andqj separated a distancerij. ε0 is the electric permittivityin vacuum, and the short-range repulsion is described by a hardcore interaction wheredi is the diameter of ioni. The ionic hardcore diameters were 4, 5, and 8 Å for mono-, di- and tri/tetravalentspecies, respectively. We will use the notation that the first speciesis a cation; that is, a 2:1 salt consists of divalent cations andmonovalent anions. It is sometimes numerically easier to calculatethe forces between soft core ions than between their hard corecounterparts, and for the simulations including symmetric 2:2and 3:3 salts, eq 3 was used. Then the pair potential takes theform

with σ ) 4 and 8 Å for 2:2 and 3:3 salts, respectively. The ionsalso interact with the walls through potentials given by3

k is the index for the different surfaces (-1 for the left surfaceand+1 for the right). The two surfaces are placed atz ) (h/2and have a size ofL2. A schematic picture of the system can beseen in Figure 1. The total energy of the simulation box can bedivided into several contributions

whereUii is the ion-ion, Uiw is the ion-wall, andUww is thewall-wall contribution, respectively, andUext is an external

correction termUext. The latter is calculated from the mean iondistribution in the slit. The external field is updated during thesimulation following the recipe of Torrie et al.,15 ensuring thatthe energy and ion distributions converge. The osmotic pressurecan be calculated at the midplane through the expression16

whereci(0) is the ion concentration of speciesi at the midplane.The direct interaction of two ions in either of the two halves giverise to a pressure,pcorr, and similarly the collision over themidplane of two ions gives rise to a hard core pressurephc. Wehave also introduced the apparent surface chargeσapp, definedas

Here,F(z′) is the total charge density atz′. The net pressure isgiven by the osmotic pressure in the slit minus the pressure inthe bulk:

Monte Carlo Simulations

The Monte Carlo simulations were carried out in the grandcanonical ensemble (µ, V, T), keeping the chemical potential,volume, and temperature constant. For this purpose, we used asimulation box with a side length of approximately 200 Å andtwo impenetrable surfaces. In a grand canonical simulation, saltpairs move in and out of the simulation box, allowing the saltcontent of the slit to vary with separation. The chemical potentialsof the salts were obtained from a separate bulk simulation withthe desired concentration and composition. The bulk simulationswere done in the canoncial ensemble (N, V, T) holding both thenumber of salt pairs and volume constant in a simulation boxof dimensions 2003 Å3. The chemical potentials were measuredusing a modified Widom method.17,18The simulations used theMetropolis Monte Carlo algorithm.19Equilibrium conditions were

(14) Hill, T. L. Statistical Mechanics; McGraw-Hill: New York, 1956.

(15) Torrie, G. M.; Valleau, J. P.J. Phys. Chem. 1982, 86, 3251.(16) Sjostrom, L.; Åkesson, T.J. Colloid Interface Sci.1996, 181, 645.(17) Widom, B.J. Chem. Phys.1963, 39, 2808.(18) Svensson, B. R.; Woodward, C. E.Mol. Phys. 1988, 64, 247.(19) Metropolis, N. A.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A.;

Teller, E.J. Chem. Phys.1953, 21, 1087.

u(rij) )qiqj

4πε0εrrijrij >

di + dj

2(1)

u(rij) ) ∞ rij <di + dj

2(2)

u(rij) )qiqj

4πε0εrrij+ kBT(σ

rij)6

(3)

Φk)-1,1(z) )σkL

πε0εr[ln xL2/2 + (z + (kh/2))2 + L/2

x(L/2)2 + (z + (kh/2))2-

2|z + (kh/2)| ×

(arcsin(L/2)4 - (z + (kh/2))4 - (L2/2)(z + (kh/2))2

((L/2)2 + (z + (kh2))2)2+ π

2)]

Utot ) Uii + Uiw + Uww + Uext (4)

Figure 1. Schematic picture of the model system with two oppositelycharged surfaces in equilibrium with a bulk solution containing amultivalent electrolyte.

posm) kBT∑i

ci(0) + pcorr + phc -σapp(0)2

2εrε0

(5)

σapp(z) ) σ-1 + ∫-h/2

zF(z′) dz′ (6)

p ) posm- posmbulk (7)

Repulsion between Oppositely Charged Species Langmuir, Vol. 23, No. 23, 200711563

obtained by sampling 105 and 107 configurations in the bulk andslit simulations, respectively (from a random starting configu-ration). Data were then sampled from the 106 and 108 followingconfigurations for the two systems. Periodic boundary conditionsand the minimum image approach were used for the explicitinteractions in two dimensions in the slit simulations and in threedimensions in the bulk simulations. The mean-field potential forthe slit was calculated by integrating the charge distribution fromthe left and right sides and adding the two terms

whereΦD(h) is the Donnan potential,20 which depends on theseparationhand ensures that the midplane potential goes to zerowhenh f ∞. The Donnan potential is defined as

whereµibulk is the chemical potential in the bulk andµi

slit is thechemical potential in the slit for speciesi. Note that bothµi

slit andΦD vary with separation but that their sum is equal to the constantµi

bulk. Chemical potentials for the slit were obtained in the samemanner as in the bulk simulations with the modified Widommethod. Uncertainties in chemical potentials varied with ionvalency, but for monovalent species, they were less than 0.01kBT (both for the slit and the bulk) and the accuracy in thepotentials was slightly better than 1 mV. The uncertainty in thepressure (p/RT) was estimated to be about 1 mM.

Effective Mean-Field Theory

Ahighlychargedsurfaceneutralizedbymultivalent counterionscan undergo an apparent charge reversal. That is, the originallynegatively charged surface and its counter- and co-ion cloudbecomes net positively charged. This phenomenon is driven byion-ion correlations, and it is outside the scope of simple mean-field theory. However, we can still try to use a linear Poisson-Boltzmann equation in order to describe the interaction of theapparently charged reversed surface(s) provided we can estimatetheir surface potentials. This is what we try to do in this section,and those who are not interested of this aspect can go directlyto the Results section.

Assuming constant potentials at the surfaces, motivated byPerel and Shklovskii,21 one can derive an expression for thepressurestarting fromthe linearizedPoisson-Boltzmannequation

whereκ is the inverse Debye screening length andΦ-1 andΦ1

are the constant surface potentials. If the separation is largecompared to the Debye length, then eq 10 reduces to

Assuming constant potentials, the surfaces will adjust themselves

as they approach and change their surface charge density. Thesurface potential can be related to the surface charge density:

Equation 12 allows for an adjustment of the surface chargedensities as the surfaces approach, and one can derive anexpression describing at which separation charge reversal takesplace:

In eq 13, it is assumed thatΦ1 > Φ-1. The condition for zeropressure, that is, the change from repulsion to attraction, will,however, occur when

The ratio between the surface potentials will determine theoutcome. Apparently, zero pressure appears at shorter separationthan charge reversal, which means that there will be a regimewith oppositely charged surfaces and a repulsive interaction. Inthe PB approximation, this is an entropic repulsion due to theremaining counterions necessary for the neutralization of thesystem. The linear PB equation is surprisingly accurate for thesurface potentials in our case, and using the nonlinear PB willchange the pressure by at most 10-20% in the relevant regime.

To use the above expressions, one needs to know the (effective)surface potentialsΦ-1 andΦ1. One way to get these is throughsimulations. The effective potentials,Φ-1

eff and Φ1eff were

extrapolated to contact by fitting an exponential function to thesimulated mean-field potential profile. The function used was ofthe form

whereΦeff andR are two fitting parameters. From the fittingprocedure, we find thatR ≈ κ, whereκ is the Debye-Huckelinverse screening length calculated from the salt concentrationin the bulk solution. The effective potentials had to be determinedat some separation (because the effective potentials are assumedto be constant). The ideal choice would be when the two surfacesdo not interact, that is, one surface next to an infinite bulk solution.This situation was mimicked in a separate grand canonicalsimulation similar to the ones discussed above, the differencebeing that only one of the surfaces was charged (σ-1 or σ1)whereas the other was neutral and was placed at a separationlarge enough to ensure bulklike conditions.

The other alternative, obtaining the effective potentials froman approximate theory, has been attempted by Zhang andShklovskii.21,22 They proposed an expression for the effectivesurface potential based on a Wigner crystal approach neglectingthe contribution of co-ions; for more details see the abovereferences. We will discuss both routes in the following sections.

Results

Figure 2 shows simulated pressure curves between twooppositely charged surfaces, with charge densities of-2 and 1e/nm2, respectively. The slit is in equilibrium with 2:1 and 3:1bulk electrolyte solutions, respectively. At low salt concentrations,

(20) Donnan, F. G.Z. Electrochem.1911, 17, 572.(21) Perel, V. O.; Shklovskii, B. I.Physica A1999, 274, 446. (22) Shklovskii, B. I.Phys. ReV. E 1999, 60, 5802.

Φ(z, h) ) - 12ε0εr

[∫-h/2

h/2 |z - t|F(t, h) dt + σ-1(z + h2) +

σ1(h2 - z)] + ΦD(h) (8)

µibulk ) µi

slit(h) + qiΦD(h) (9)

p(h) )ε0εrκ

2

2 [2Φ-1Φ1( 1

4 cosh2(κh/2)+ 1

4 sinh2(κh/2)) -

(Φ-12 + Φ1

2)

sinh2(κh) ] (10)

p(h) ≈ 2ε0εrκ2[Φ-1Φ1 exp(-κh) -

(Φ-12 + Φ1

2) exp(-2κh)] (11)

σ ) -ε0εr(dΦdz ) (12)

cosh(κh2 ) ) x1

2( Φ1

Φ-1+ 1) (13)

cosh(κh2 ) ) x1

2(12 (Φ-12 + Φ1

2)

Φ-1Φ1+ 1) (14)

Φ(z) ) Φeff exp(-Rz) (15)

11564 Langmuir, Vol. 23, No. 23, 2007 Trulsson et al.

the surfaces attract each other as expected from the PB equation.At increasing concentration, however, a repulsion appears atintermediate separations. The repulsion is not seen in an aqueoussolution with a 1:1 salt, indicating that it is a result of ion-ioncorrelations. It is interesting that the systems showing a repulsiveinteraction always experience an apparent surface charge reversalat large separations (i.e., the surface charges are (over)-compensated for by multivalent counterions). However, therepulsive interaction and the charge reversal are only indirectlyrelated. For example, it is possible to have a net repulsiveinteraction in a system that does not show a charge reversal. Thisis illustrated in Figure 2, where we have indicated the region ofcharge reversal by a cross. (Charge reversal is found to the rightof the symbols.) It is clear that there is a repulsive interactionwithout charge reversal; however, both phenomena have thesame origin: ion-ion correlations. The repulsion at intermediateseparation is largely a consequence of the high salt concentrationin the slit. That is, with two oppositely charged surfaces it isalways energetically advantageous to increase the salt concentra-tion in the slit significantly above the bulk value and ion-ioncorrelations enhance this effect. Thus, the repulsion should ratherbe viewed as an entropic effect from an “ideal” osmotic pressurerather than an electrostatic interaction between two surfaces ofequal apparent charge; see also Figure 3.

The repulsion and charge reversal are enhanced if the bulk saltconcentration is increased. This simply reflects the fact that theentropic cost of increasing the slit concentration is reduced. There

is also a dependency on the valency of the multivalent salt particlethat cannot be explained by different ionic strengths. Withincreasing valency, the correlations become more important, andcharge reversal and repulsive interactions appear more readily.With a divalent salt the repulsive interaction appears at a saltconcentration of approximately 50 mM, whereas for a tri- ortetravalent salt repulsion sets in at 5 and 0.5 mM, respectively;see Figures 2 and 4.

Ion-ion correlation increases the accumulation of counterionsclose to the charged walls and, with multivalent cations, it actuallyovercompensates for the bare surface charge at large separation;see Figure 5a. That is, normally the potential a few ångstro¨msfrom a negatively charged surface would be negative, but forsufficiently high electrostatic coupling it changes sign and be-comes positive! Table 1 shows the effective potentials extractedfrom simulations. Note that the effective potentials for thenegatively charged surface (column 3) are all positive except forthe lowest concentration of the 2:1 salt. The weak couplingbetweenσ1 and the monovalent anions is not sufficient to createa charge reversal, which means that the effective potentials forthe positively charged surface have the same sign as the surfacecharge density.

Figure 6 shows how the net osmotic pressure changes withvarying σ1 in a 3:1 salt solutions. Ifσ1 ) 0, the pressure isrepulsive for all separations with some short-range oscillationsdue to packing effects of the counterions. Once the non-overcharged surface has a small positive charge, the pressurecurve is essentially independent ofσ1 at intermediate and largeseparations. This can be understood from PB theory, whichpredicts that the surface potential is a slowly varying function

Figure 2. Net osmotic pressure between two oppositely chargedsurfaces in equilibrium with a bulk salt solution. The surface chargedensities areσ-1 ) -2 andσ1 ) 1 e/nm2. The dashed curves areobtained from the linearized PB equation for two surfaces approachingat constant surface potentials; see eq 10. The potentials have thesame sign but different magnitudes; see the text. X symbols delimitthe region of charge reversal, which is found to the right of thesymbols: (a) 2:1 salt solution in the bulk with black) 100, red withcircles ) 50, and blue with squares) 10 mM and (b) 3:1 saltsolution in the bulk with black) 15, red with circles) 5, and bluewith squares) 1 mM.

Figure 3. Salt content between two oppositely charged surfaces inequilibrium with 1 and 5 mM 3:1 salt solutions. The surface chargedensities are-2 and 1 e/nm2, respectively.

Figure 4. Net osmotic pressure between two oppositely chargedsurfaces in equilibrium with a 4:1 salt solution. The surface chargedensities are-2 and 1 e/nm2, respectively. The dashed curves areobtained from the linearized PB equation for two surfaces approachingat constant surface potentials; see eq 10. The bulk concentrationsare black) 0.5 and red with symbols) 0.1 mM.


of the surface charge density. A similar behavior is seen with2:1 and 4:1 salt solutions.

The addition of a second salt affects the charge reversal, andit can both increase and decrease depending on concentration.23-26

Figure 7 shows that the addition of a 1:1 or 2:1 salt to a solutioncontaining a 3:1 salt leads to significantly increased repulsionat short separations. In disagreement with Martin-Molina et al.in their recent study,26 we see increased charge inversion whenadding small amounts of a 1:1 or 2:1 salt. The discrepancy is dueto the difference in the definition of charge reversal. Martin-Molina et al. calculate the charge reversal by integrating onlythe multivalent ion profile, which means that an additional 1:1salt will only indirectly affect the charge reversal through themultivalent ions. We believe, however, that our definition ofcharge inversion is more correct because it takes all ions intoaccount. The increased repulsion can be understood from the

increased chemical potential of anions in the bulk, which naturallygives a higher concentration in the slit. At the same time, theelectrostatic screening increases in the system, and at largerseparations, the repulsion is reduced upon salt addition. Furthersalt addition will eventually screen out all of the electrostaticinteractions, and the repulsion will of course disappear.

Figure 8 shows the interaction between two oppositely chargedsurfaces in equilibrium with symmetrical divalent and trivalentsalt solutions. Depending on the salt concentration and the surfacecharge densities, one can observe overcharging at one or bothsurfaces, where we focus only on the former case. The repulsivemaxima seen in asymmetric electrolytes are strongly reduced,and the pressure is essentially attractive down to very smallseparations, where the trivial entropic repulsion sets in. (Notethat the limit of accuracy in the simulations is approximately 1

(23) Zhang, R.; Shklovskii, B. I.Phys. ReV. E 2005, 72, 021405.(24) Martin-Molina, A.; Quesada-Perez, M.; Galisteo-Gonzalez, F.; Hidalgo-

Alvarez, R.J. Phys.: Condens. Matter2003, 15, S3475.(25) Jonsson, B.; Nonat, A.; Labbez, C.; Cabane, B.; Wennerstro¨m, H.Langmuir

2005, 21, 9211.(26) Pianegonda, S.; Barbosa, M. C.; Levin, Y.Europhys. Lett.2005, 71, 831.

Figure 5. Electrostatic potential near the surfaces for a 3:1 salt atvarying concentration as indicated. Charge reversal appears only atthe negatively charged surface. (a)σ-1 ) -2 and (b)σ1 ) 1 e/nm2.Solid lines, simulated mean potential; dashed lines, extrapolation tocontact with the potential assuming the formΦ(z) ) Φeff exp(-Rz).

Table 1. Effective Surface Potentials from the SimulatedCharge Profiles for Various Salts and Concentrationsa

salt cs (mM) Φ-1 (mV) Φ1 (mV)

2:1 10 -7 1412:1 50 11 1002:1 100 20 863:1 1 28 1833:1 5 40 1453:1 15 48 1204:1 0.1 47 1994:1 0.5 67 188

a The numbers have been obtained from extrapolation as indicatedin Figure 5. The surface charge densities areσ-1 ) -2 andσ1 ) 1 e/nm2.

Figure 6. Net osmotic pressure as a function of separation forvarying surface charge density (e/nm2) of the positively chargedsurface as indicated.σ-1 is kept constant at-2 e/nm2, and the bulkconcentrations of a 3:1 salt are (a) 1 and (b) 5 mM.

Figure 7. Net osmotic pressure as a function of separation for asystem in equilibrium with 5 mM 3:1 salt (black line). A second saltof 50 mM 1:1 (red with circles) and 25 mM 2:1 (blue with squares)has been added. The surface charge densities areσ-1 ) -2 andσ1) 1 e/nm2.


mM.) For a 3:3 salt, the repulsion at short separations is to a largeextent due to the soft core repulsion, which explains the outwardshift of the pressure minimum. The pressure curves are almostindependent of salt concentration. The absence of a significantrepulsive regime in the case of a symmetric salt is probably dueto both a reduced number density and stronger correlations amongthe anions because they now are di- or trivalent. This also leadsto an enhanced accumulation of anions at the positively chargedwall with only a very small apparent surface charge density.That is, even if both surfaces have the same sign on their apparentsurface charge densities/potentials, the interaction between thesesurfaces should be rather small.

The previous results have shown that the repulsion betweenoppositely charged surfaces is a correlation effect, and it cantherefore be expected in systems with strong electrostaticinteractions. The easiest way, at least experimentally, to increasethe coupling is by using a solvent with a dielectric permittivitylower than that of water.10 For example, a 1:1 mixture of ethanoland water has a dielectric permittivity of 54. Figure 9 shows theeffect on lowering the dielectric constant in the 3:1 salt case withsurface charge densities equal to-2 and 1 e/nm2. As expected,the repulsion increases, and charge reversal becomes easier,favored by an enhancement of salt in the slit.

Discussion

The results presented above and supported by independentexperimental studies7,9,10,27,28clearly demonstrate the limitationsof the traditional mean-field approximation. That is, in a strongly

coupled system we can expect ion-ion correlations to play animportant role. This has earlier been shown to be the case forequally charged surfaces where the repulsion predicted by mean-field theory is incorrect and instead an attractive interactiondominates. This mechanism plays an important role in a numberof technical (cement29,30) and biological (DNA31,32) systems. Inthe present study, we have focused on the interaction betweentwo oppositely charged surfaces, where it turns out that ion-ioncorrelations are equally important. As for equally chargedsurfaces, ion-ion correlations play a key role in highly coupledsystem. Experimentally, this often turns out to be the case whenmultivalent counterions are present. However, we emphasizethat other parameters such as the surface charge density and thedielectric permittivity of the solvent are also important and canbe used to modify the importance of the correlations. It issometimes, at least within the primitive model, fruitful to lookupon the resulting forces as a result of the competition betweenentropic and energetic terms. That is, in an ideal system everythingis entropic whereas when interaction comes into play the ener-getic term becomes increasingly important. For example, theexchange of monovalent to divalent ions not only increases theinteraction but also reduces the entropy, halving the number ofparticles.

What is given above is a qualitative picture. In the simulations,the counterions are treated as charged hard spheres, and whenthe charge density of a surface neutralized by monovalentcounterions goes above approximately 1 e/nm2, then the collisionterm will contribute to the pressure and a more complicatedpicture emerges. We have in the present study tried to avoid thisrange of parameters, although a few cases reported above areclearly in this range.

There have been different attempts to rationalize the repulsionand overcharging phenomena. One approximate way is toincorporate the charge distribution near the surface into a new“renormalized” surface charge density to be used in a PBcalculation at constant surface charge density. This gives areasonably good description of the force behavior at long range

(27) Martin-Molina, A.; Maroto-Centeno, J. A.; Hidalgo-AÄ lvarez, R.; Quesada-Perez, M. J. Chem. Phys. 2006, 125, 144906.

(28) Lesko, S.; Lesniewska, E.; Nonat, A.; Mutin, J.-C.; Goudonnet, J.-P.Ultramicroscopy2001, 86, 11.

(29) Plassard, C.; Lesniewska, E.; Pochard, I.; Nonat, A.Langmuir2005, 21,7263.

(30) Delville, A.; Pellenq, R.; Caillot, J.J. Chem. Phys.1997, 106, 7275.(31) Jonsson, B.; Wennerstro¨m, H.; Nonat, A.; Cabane, B.Langmuir2004,

20, 6702.(32) Rouzina, I.; Bloomfield, V. A.J. Phys. Chem. 1996, 100, 9977.

Figure 8. Net osmotic pressure between two oppositely chargedsurfaces from simulations (solid lines) and from the linearized PBequation with surface potentials from simulations (dashed lines). (a)The surface charge densities areσ-1 ) -3.5 andσ1 ) 1 e/nm2, andthe bulk concentrations of a 2:2 salt are 20 mM (black) and 40 mM(red with symbols). (b)σ-1 ) -1 andσ1 ) 0.5, and the system isin equilibrium with 5 mM (black) and 1 mM (red with symbols) 3:3salt solutions, respectively. Only the negatively charged surface isovercharged.

Figure 9. Net osmotic pressure between two oppositely chargedsurfaces-σ-1 ) -2 andσ1 ) 1 e/nm2. The bulk solution containsa 3:1 salt of 5 mM (lines) and 1 mM (lines with symbols). Thedielectric permittivity has been varied:εr ) 78 (black) andεr ) 54(red).


for a system in equilibrium with an X:1 or 1:X salt, but it doesnot capture the transition to attractive pressure at short separation.However, if we estimate an effective potential at large separationand then perform a (linearized) PB calculation at constant surfacepotentials, then a non-monotonic curve will result. By keepingthe surface potentials constant, we allow the charge density torespond to the change in separation mimicking the change inovercharging seen in the simulations; see Figure 2. This procedureseems to capture the coupling between the overcharged surfaceand the multivalent counterions quite well, and what remains isthe contribution of monovalent ions to the overall pressure, whichis well described by the PB equation. It is, however, only a partof the ion-ion correlations that is taken into account, and theweakness is apparent for a system with a 2:2 electrolyte (Figure8), where the simulated pressure is virtually always attractiveand the renormalized PB calculations predict a significantrepulsion. In the same manner, the latter will never predict theattraction between two equally charged surfaces in equilibriumwith a multivalent salt. It should be emphasized that this typeof calculation cannot replace the simulations because it dependson the renormalized description of the surface. Unfortunately,there does not seem to be any experimental source from whichone could obtain an effective potential. Thus, the linearized PBcalculations can be used only to rationalize the simulated pressurecurves.

Shklovskii and Zhang21,22 have proposed an approximatemethod to calculate the effective potentials in the case ofovercharging surfaces. The idea is based on a division of thesystem into two parts. One is a strongly correlated liquid at thecharged surface and a more gaslike phase further away from thesurface. The latter is described by the PB equation, whereas thefree energy of the first part is approximated using a Wignercrystal (WC) approach. The partitioning of the system mightseem appealing, but the numerical results are definitely notquantitative and in some cases not even qualitatively correct(even if the interaction parameter is high). As an example, inFigure 10 we have plottedΦ-1 calculated from simulations andthe WC approach. The latter overestimates the effective surfacepotential and the overcharging of the surface for a large rangeof concentrations. The use of such potential values leads to anoverestimate of the repulsion. For example, consider the pressureat 10 mM 2:1 salt concentration, where simulations predict anegative surface potential but the WC theory leads to overchargingand strongly repulsive pressure; see Figure 11.

In practice, the observed repulsion can partially explain thedegradations of hydrated cementitious materials, in particular,

concrete constructions. Concrete subjected to an internal (gypsum)or external (seawater, lake, etc.) source of sulfate is known topotentially undergo a progressive and profound reorganizationof its internal microstructure.34,35 In civil engineering jargon,this is called a sulfate attack. For example, concrete undergoingsulfate attack is often found to suffer from swelling, spalling,and cracking.33,34There is evidence indicating that the degradationalso causes a significant reduction of the mechanical propertiesof concrete,34,35which in the most severe cases requires partialreconstruction. The observed electrostatic repulsion betweenoppositely charged particles could explain the observed expansionof concrete for two reasons. First, hydrated cement is composedof a mix of different mineral hydrate phases bearing oppositecharges. The hydrated cement is formed through the dissolution/precipitation process of an initial anydrous powder dispersed inwater. The composition of the latter varies according to the typeof cement, but it mainly contains tricalcium silicate and smallamounts of tricalcium aluminate and gypsum. The precipitatedhydrates are calcium silicate hydrate (C-S-H) and a large panelof aluminate hydrates (monosulfoaluminate, hydroxyaluminate,carboaluminate, and ettringite).36C-S-H is negatively chargedbecause of the titration of the surface silanol groups, and aluminatehydrates possess a structural positive charge due to the substitutionof calcium ions by aluminum ions in the structure.36 Second, theexpansion is known to be more pronounced for cement rich inaluminate and at high sulfate concentration. Further experimentaland theoretical studies are performed to verify this hypothesis.

Conclusions

From the simulations, we find that the addition of asymmetricsalts to a system composed of oppositely charged particles maylead to a repulsion between the particles. This repulsion is strongfor asymmetric salts and is virtually zero when the salt issymmetric. The repulsion is a result of ion-ion correlations, thatallow a huge accumulation of salt in between the oppositelycharged particles/surfaces, thereby generating a large osmoticpressure. The requirement for significant repulsion is that onesurface has a high surface charge density and the multivalent ionof the asymmetric salt as the counterion. The surface chargedensity of the second surface is not crucial and can be close toneutral. Concomitant with the appearance of a repulsive pressure,one can also observe a charge reversal of the surface with

(33) Khan, M. O.; Jo¨nsson, B.Biopolymers1999, 49, 121.(34) St-John, D. A.; Poole, A. W.; Sims, I.Concrete Petrography: A Handbook

of InVestigatiVe Techniques; Arnold: London, 1998.(35) Skalny, J.; Marchand, J.; Odler, I.Sulfate Attack on Concrete; EFN

SPON: London, 2001.(36) Thorvaldson, T.; Vigfusson, V. A.; Larmour, R. K.Trans. R. Soc. Can.

1927, 21, 295.

Figure 10. Effective surface potentials from eq 15 (thick solidlines) and the corresponding WC potentials (thin lines with symbols)for a charged surface (σ ) -2 e/nm2) in equilibrium with a 2:1(black curves), a 3:1 (red curves), or a 4:1 bulk solution (blue curves).The curves show how the potential varies as a function of saltconcentration in the bulk.

Figure 11. Net osmotic pressure between two oppositely chargedsurfaces calculated from simulation (black) and the WC approach(red with circles). The charges areσ-1 ) -2 andσ1 ) 1 e/nm2,respectively. The bulk solution contains a 10 mM 2:1 salt.


multivalent counterions. Note, however, that there is a region ofrepulsive interaction before the charge reversal takes place. Thatis, both phenomena are due to ion-ion correlations, but therepulsive pressure is not a consequence of the charge reversalper se. The repulsion can be enhanced by adding small amountsof monovalent salt. Correlations become more important in highlycoupled systems, and, for example, a decrease in the dielectricpermittivity will lead to a stronger repulsion.

In practice, the observed repulsion can explain the earlyswelling of hydrated cement rich in aluminate (positively charged)and calcium silicate (negatively charged) phases. The swellingis known to be particularly pronounced at high sulfate concen-trations.

Acknowledgment. Stimulating discussions on the setting ofvarious cement types with Andre Nonat, Universite de Bourgogne,are gratefully acknowledged. This work was supported by theFoundation for Strategic Research, Sweden.

LA701222B(37) Taylor, H. F. W.Cement Chemistry; Thomas Telford: London, 1997.


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Repulsion between Oppositely Charged Macromolecules or Particles