Influence of Confinement on the Electrostatic Interaction between

8164

J. Phys. Chem. B 2005, 109, 8164-8170

Influence of Confinement on the Electrostatic Interaction between Charged Colloids: a (N,V,T) Monte Carlo Study within Hyperspherical Geometry A. Delville† CRMD, CNRS, 1B rue de la Fe´ rollerie, 45071 Orle´ ans Cedex 02, France ReceiVed: NoVember 19, 2004; In Final Form: February 16, 2005

Monte Carlo simulations within closed hyperspherical geometry are used to analyze the ionic distribution around two confined charged colloids to determine the origin of the net attraction recently reported in the literature. A scaling procedure is used to compare our numerical results obtained with small ideal colloids with the conclusion of the measurements performed with large silica colloids. Although no electrostatic attraction is detected under confinement, our simulations exhibit a significant reduction of the electrostatic repulsion between charged colloids confined between two weakly charged walls. After rescaling to reproduce the electrostatic repulsion between large confined colloids, our numerical results are qualitatively consistent with the reported attraction because we reasonably expect a reduction of the electrostatic force between such confined colloids below the order of magnitude of their van der Waals attraction.

I. Introduction Charged colloids constitute a large class of natural (like clays, DNA or polysaccharides) or synthetic (like cement, latex, metallic oxides or polyelectrolytes) materials involved in numerous natural (DNA condensation, membrane fusion) or industrial (food, cosmetics, paint, drilling, civil engineering) processes because of their mechanical (swelling, setting) and dynamical (thixotropy, gelling) properties. The overlap of their diffuse layer1-4 of neutralizing counterions was early identified as the main mechanism responsible for their long range electrostatic repulsion whereas dispersion forces generally induce interparticle attraction. In the framework of the DLVO theory,3-5 ionic strength is the external parameter modulating the extend of this diffuse layer and thus the range of the electrostatic repulsions. The balance between these two antagonistic contributions was successfully used to interpret the stability of suspensions of charged colloids neutralized by monovalent counterions because interionic correlations6-10 then remain negligible. By contrast, under conditions of strong electrostatic coupling between charged colloids (i.e., in the presence of di- or trivalent counterions) correlations forces were shown to induce electrostatic attraction.11-14 Furthermore, more recent experiments5,15 exhibited attracto/repulsive mechanical behavior because of the layering of the solvent molecules confined between two lamellar interfaces. However, both phenomena occur only at reduced separations between the colloids, i.e., at only a few ionic or solvent diameters. In that context, recent experimental studies16-21 performed with confined charged colloids neutralized by monovalent counterions at low ionic strength were shown to exhibit a very surprising behavior: long range and apparently electrostatic attraction. Various theoretical22-29 and experimental16-21,30-32,63-70 studies were devoted to this observation but their validity is still subject to debate, probably because of the possible occurrence of various phenomena or artifacts including hydrodynamical effects,33 many body interactions between the colloids,34-39,62 dipolar coupling,41 overcharging42-44 or deple†

E-mail address: [email protected].

tion forces induced by the release of polyions from the latex particles. But obviously, the influence of the confining walls on the mechanical behavior of the charged colloids may be considered as a many body effect.34-39,62 A recent statistical analysis of observations performed on equilibrium configurations of confined silica colloids confirms21 the existence of such attraction whereas implication of attractive hydrodynamic artifacts is excluded.33 By contrast, simple theoretical arguments clearly exclude25 the existence of electrostatic attraction between charged colloids whatever their geometry if correlation forces are neglected. Because of the weak electrostatic coupling11-14 between these colloids and their large separation, this last hypothesis appears reasonably valid. To better understand the influence of confinement on the exact nature of the electrostatic interaction between charged colloids, we performed Monte Carlo simulations of the distribution of the neutralizing monovalent counterions around two charged spherical colloids under various confinement conditions. By carefully analyzing and quantifying how the confining charged interfaces disturb the ionic diffuse layers around the two colloids, we hope to elucidate the origin of this apparent electrostatic attraction. II. Method (A) Hyperspherical Geometry. A hypersphere is defined as the “surface” of a sphere within the Euclidian space of dimension 4 (noted E4). It is a closed isotropic space of dimension three (noted S3), allowing an exact formulation of the Coulomb potential,44-47 and thus well adapted for numerical simulations of the long range interactions between charged systems.13,24,48-49 Each point M on the hypersphere (Figure 1) is labeled by a set of generalized spherical coordinates: 4

M B )R

ξib ei ∑ i)1

(1)

where R is the radius of the hypersphere, b ei is a local

10.1021/jp044711l CCC: $30.25 © 2005 American Chemical Society Published on Web 03/30/2005

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J. Phys. Chem. B, Vol. 109, No. 16, 2005 8165 of bipolar coordinates:

(ξi) )

( ) x1 - F2 sin φ x1 - F2 cos φ F sin ψ F cos ψ

(7)

where F ∈ [0, 1], φ ∈ [0, 2π] and ψ ∈ [0, 2π]. The new coordinates of the test point are uniformly generated within a cone of aperture δθ centered on the initial position of the ion. These bipolar coordinates are generated according to the scheme Figure 1. Schematic view of an hypersphere illustrating the colatitude and its local director.

(

orthogonal frame of E4 and

sin w sin V cos u sin w sin V sin u (ξi) ) sin w cos V cos w

)

(2)

with u ∈ [0, 2π], V ∈ [0, π] and w ∈ [0, π]. The local frame of E4 is defined by three orthogonal vectors of S3 (e bu, b eV, b ew) plus the fourth director (M B /R), which is perpendicular to the hypersphere. This local frame of S3 is defined by

() (

b eu b eV ) b ew

)

-sin u cos u 0 0 cos V cos u cos V sin u -sin V 0 cos w sin V cos u cos w sin V sin u cos w cos V -sin w (3)

F ) xR1sin2∂ϑ+cos2∂ϑ, φ ) (2R2 - 1)acos(cos(∂ϑ/F)) and ψ ) 2πR3, where R1, R2 and R3 are three chains of pseudorandom numbers uniformly distributed within [0, 1]. (B) Formulation of the Potentials. Interactions between ions and charged colloids of any shape are generally modeled in the framework of the primitive model,50 which includes long range electrostatic coupling and local finite size repulsion. The net force acting between a pair of equally charged colloids results then from a balance between colloid/colloid electrostatic repulsion, colloid/ion electrostatic attraction and colloid/ion contact repulsion. The accurate evaluation of this last contribution is generally difficult because it always requires extrapolating the ionic concentration profiles up to contact with the colloids.51 This difficulty is bypassed by replacing the ion/colloid excluded volume repulsion by a soft-core repulsion52-53

Erep )

|qionqcol| 12A ) 13 (aion + acol) 4π0r(aion + acol)2

dM B ) R[e bw dw + b e V sin w dV + b e u sin w sin V du] (4)

dΩ ) sin2 w sin V du dV dw

(5)

The volume of the hypersphere is obtained after integration of eq 5 over the whole range of variation of the set of coordinates (u, V, w) (see eq 2) leading to VT ) 2π2R3. Finally, the section of S3 by a cone with an angular aperture θ restricts the volume of the hypersphere to the residual volume V(θ)) πR3(2θ sin 2θ) and is limited by the interfacial area S(θ) ) 4πR2 sin2 θ. Within the curved hyperspherical space, the distance between two points (M and N) of S3 is measured along the geodesic by B and the product RθNM, where θNM is the angle between the M N B directors. For a potential depending only on the particle separation, the force is simply given by

B FMN ) -gradMVN(ϑMN) ) -

1 ∂V b e (M B) R ∂ϑ ϑ

(6)

where b eϑ(M B) is the director tangent to the geodesic joining points M and N. The uniform sampling of S3 during each Monte Carlo trial of displacement of an ion is ensured by using a set

rion-col12

(8)

where the coefficient A is determined by equating, at the distance equal to the sum of the ion (aion) and colloid (acol) radii, the magnitude of the electrostatic attractive force and soft-core repulsion:

The metric tensor of S3 is defined by the elementary displacement:

and elementary volumes of S3 are evaluated by the relationship dτ ) R3 dΩ, where the elementary solid angle satisfies

A

(9)

where qion and qcol are the ion and colloid electric charges, respectively. The simplest derivation of the electrostatic potential within hyperspherical geometry44-48 consider a pseudocharge resulting from the association of a charge and its neutralizing uniform background. Because we only deal with neutral systems, the contributions from the backgrounds cancel out. Assuming such a pseudocharge (qN) located at the North pole of the hypersphere (see Figure 1), its electric field generated at point M must satisfy Gauss’s relation:

E(ϑM) S(ϑM) )

Q(ϑM) 0r

(10)

where S(θM) is the area resulting from the section of the hypersphere by a cone of angular aperture θM, Q(θM) is the fraction of the pseudocharge (ion plus background) limited by this cone, and V(θM) is the volume of S3 limited by the cone. Because the background is uniform, we obtain

Q(ϑM) ) qN(1 - V(ϑM)/VS3) ) qN

ϑ sin ϑ (π - θ + cos ) π (11)

leading to

8166 J. Phys. Chem. B, Vol. 109, No. 16, 2005

E(ϑM) )

(

π - ϑM

qN 2

2

4π 0rR sin2 ϑM

+ cot ϑM

Delville

)

(12)

The corresponding potential becomes

V(ϑM) )

qN 2

(-0.5 + (π - ϑM) cot ϑM)

4π 0rR

(13)

where the integration constant (-0.5) has been selected to satisfy the relationship

∫S V(ϑ) dτ ) R∫V(ϑ) S(ϑ) dϑ ) 4πR3∫V(ϑ) sin2(ϑ) dϑ ) 0

Figure 2. Schematic view of two equatorial colloidal particles confined between two parallel lamellae within hyperspherical geometry.

3

(14)

The electrostatic interaction between two charged colloids and their neutralizing counterions is derived from eq 13, whereas the force acting on the colloids is evaluated at equilibrium by using eq 12. To minimize curvature effects, the soft-core repulsion between the charged colloids and their condensed counterions is described in S3 by the relationship49

Vsoft(ϑM) )

A (R sin ϑM)12

(15)

where the parameter A is determined by eq 9. Within confined geometries, the hyperspherical volume available to the two colloid particles and their neutralizing counterions is limited by two symmetrical conical sections of the hypersphere with colatitude θN and π - θN (see Figure 2), generating two equivalent lamellar sections.13,24 Two charged colloidal particles are immobilized within the equatorial plane of the hypersphere (see Figure 2), and we evaluate only the component of the force exerted locally on the particles along the director tangent to the geodesic joining the two particles (i.e., inside the equatorial plane). These lamellae are either neutral or bear electric charges with the same sign as those of the charged colloids. Their neutralizing counterions are also confined between the lamellae and cannot be distinguished from those neutralizing the charged colloids. In the presence of charged lamellae, a supplementary potential must be introduced to describe the electrostatic coupling between the charged interfaces and the fixed colloids and labile counterions. If qlam is the net charge borne by a lamella, the electrostatic potential resulting from the pseudocharges generated by these two symmetrical lamellae and acting on a charge located at the colatitude θ is given by13

Vint(ϑ) )

qlam

((π - 2ϑ) cot ϑ - 1)

4π20rR

(16)

Because of its local orthogonality with the equatorial plane, the electrostatic force derived from eq 16 has no direct influence on the component of the force acting on the two colloidal particles in the direction of the tangent to their geodesic. Finally, the electrostatic energy must include the constant contribution resulting from the self-energy of the two charged lamellae and their mutual repulsion:13

Hlam,lam )

qlam2

(1 + (π - 4ϑN) cot ϑN)

4π20rR

(17)

TABLE 1

no.

colloid radius (µm)

colloid charge density (e/µm2)

lamellar separation (µm)

lamellar surface charge density (e/µm2)

1 2 3 4 5 6 7 8

0.01 0.01 0.01 0.01 0.01 0.02 0.01 0.04

7.96 × 105 7.96 × 105 7.96 × 105 7.96 × 105 2.39 × 106 7.96 × 105 7.96 × 105 7.96 × 105

/ 0.06 0.06 0.06 0.06 0.12 0.04 0.24

/ / 2 × 104 6 × 104 6 × 104 2 × 104 2 × 104 2 × 104

(C) Systems Investigated. Because of the very large number of counterions necessary to neutralize charged silica or latex particles with a radius in the micrometric range, it is impossible to perform Monte Carlo simulations of the ion distribution around such charged colloids to obtain the resulting electrostatic force. The radius of the colloids is thus reduced by 2 orders of magnitude (acoll ∼ 0.01 µm) whereas the surface charge density of the colloidal particles is set equal to that of silica54 particles (σcoll ∼ 8 × 105 e/µm2). As a consequence, the total number of counterions treated in these simulations remains limited and varies between 2000 and 13000. To mimic solvated sodium counterions, the ionic radius is set equal to 2.5 Å.55 In the framework of the primitive model,50 the dielectric constant of the solvent is set equal to that of bulk water and no salt is added into the solution. The radius of the hypersphere is 0.074 µm. Average forces and standard deviations are evaluated by using block averages with block sizes at least equal to 10 times the number of ions. The simulations were performed for a temperature of 298 K. We performed various numerical simulations to probe the influence on the net colloidal repulsion of the different parameters characterizing the geometry of the confined colloids and the intensity of their electrostatic coupling. Table 1 summarizes the systems studied: •First, the behavior of two charged colloidal particles within isotropic conditions is used as a reference to probe the influence of confinement on the net electrostatic repulsion between the particles. •Second, and for the same reason, we analyze the repulsion between these colloidal particless confined between two neutral lamellae with a separation equal to six particle radii21 (0.06 µm). •Third, the surface charge density of both lamellae (σlam) is set equal to 2 × 104 e/µm2, i.e., the same order of magnitude as that reported in the literature. •Fourth, to investigate the influence of the lamella electric charge, its value is increased by a factor of 3. •Fifth, the surface charge density of the colloids is increased by a factor of 3 while the ratio σcoll/σlam is kept constant

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J. Phys. Chem. B, Vol. 109, No. 16, 2005 8167

•Sixth, the particle radius is increased by a factor of 2 while the surface charge densities and the reduced interlamellar separation are kept constant. •Seventh, the reduced interlamellar separation is set equal to four particle radii. •Finally, one simulation is performed for a particle radius increased by a factor of 4 while the surface charges densities and the reduced interlamellar separation are kept constant. To compare the net electrostatic repulsion calculated between these different sets of charged colloids, the force is normalized according to F/z ) Fz/F0 where F0 ) qcoll2/4π0racoll2, leading to dimensionless results. III. Results and Discussion We performed Monte Carlo simulations to avoid any approximation concerning the exact behavior of the effective electrostatic interaction between confined charged colloidal particles. The simulations are thus performed by using the exact nominal electric charge of the particles evaluated from realistic surface charge density of silica surfaces54 and not from some apparent effective charge by using screened Coulomb potential like Yukawa or Debye-Hu¨ckel potentials.21,23 As a consequence, it is necessary to reduce the size of the colloidal particles to maintain the number of neutralizing counterions within manageable range (i.e., a few thousand ions, see above). Because of the reduced surface charge density (σcoll ∼ 8 × 105 e/µm2) and of the presence of monovalent counterions, attractive correlation forces are not expected to occur.11-14 Second, no salt is added to the solution to reduce numerical uncertainty on the derivation of the force between the colloidal particles by propagating their electrostatic repulsion to separations as large as possible.5,56-57 Finally, we selected the use of hyperspherical geometry because it allows an exact reproduction of the long range of the Coulomb potential without using Ewald summation. This procedure was already used successfully to describe the behavior of electrolytes,45-46 charged interfaces13 and confined colloids24 under strong coupling conditions. To check for the influence of the curvature of the hypersphere on the mutual repulsion between the colloids, some simulations were performed within classical 3D Euclidian space by using minimum image convention and Ewald summation to derive the force between two charged colloids immersed in a large simulation cell (width 2000 Å). No significant deviation was detected by comparison with the results obtained from the simulations performed within S3, validating our procedure. Figure 3 displays the net repulsion calculated for two charged colloidal particles either immersed within an isotropic S3 space (no. 1) or confined between two neutral and symmetrical lamellae (no. 2) with a separation equal to three colloid diameters21 (0.06 µm) (see Figure 3 and Table 1). In the present case, we do not detect any influence of this confinement on the net repulsion between the particles. This result, obtained by Monte Carlo simulations, is in contradiction with the results previously obtained by numerical simulations using the Density Functional Theory,22 whose validity for treating electrostatic coupling is not fully established. By contrast, we detect a noticeable reduction of the electrostatic repulsion between two particles (Figure 3) confined between weakly charged lamellae (no. 3), even for a lamellar surface charge density nearly 2 orders of magnitude smaller than that of the colloidal particles. One can note that according to experimental5,56-57 and numerical5,53,56 evidences, the long range behavior of the electrostatic repulsion between two charged colloids immersed within isotropic 3D space without salt varies according to a power law (no. 1 in

Figure 3. Variation of the normalized electrostatic force (see text) between isotropic charged colloids (no. 1) and colloids confined between neutral (no. 2) or charged (nos. 3 and 4) lamellae.

Figure 3). The same behavior is reported for colloidal particles confined between neutral walls (no. 2 in Figure 3) whereas an exponential long range decrease is detected in the presence of charged walls (no. 3 in Figure 3). The confining charged lamellae behave as a source of salt, reducing the range of the net repulsion between the colloids. By further increasing the electric charge of the confining lamellae (no. 4 in Figure 3) one notes a stronger decrease of the electrostatic repulsion between the colloidal particles. We analyze the ionic distribution around the two colloidal particles to understand how the charged lamellae reduce the electrostatic repulsion between the particles. First we consider the local ionic concentration as a function of the distance from the confining walls. In the presence of neutral walls (Figure 4a), the weak ionic density at contact with the lamellae originates from the diffuse layers surrounding the charged particles. For weakly charged lamellae (Figure 4b), one notes a significant overlap between the diffuse layers of counterions neutralizing the lamellae and those localized around the colloids. At a larger surface charge density of the confining lamellae (Figure 4c), the ionic clouds around the particles appear totally immersed in the diffuse layers originating from the lamellae. As a consequence one may expect a net influence of the confinement by charged lamellae on the ionic distribution around the particles and thus their electrostatic repulsion. But before deciding between the possible increase or decrease of their electrostatic repulsion, one needs to analyze the ionic distribution within the equatorial plane containing the colloids. One indeed expects an increase of the net repulsion between the colloidal particles if the confinement enhances the ionic density in the interparticle region, because it is similar to an increase of the overlap between the diffuse layers surrounding both charged particles. For the same reason, the opposite trend is expected if the confinement induces an increase of the counterion density in the region behind the charged particles. Figure 5 displays a radial plot of the ionic density within a layer of thickness 3 nm (i.e., onethird of the colloid radius) centered on the equatorial plane. Within statistical uncertainties, the ionic density in contact with the colloids appears isotropic in the equatorial plane both for charged particles confined between neutral and for charged lamellae (Figure 5), but deviations are detected at larger separations (Figure 6). A quantification of the influence of confinement of the net repulsion between the colloidal particle is obtained by dividing the local ionic density at half separation

8168 J. Phys. Chem. B, Vol. 109, No. 16, 2005

Delville

Figure 5. Local concentration [c(x,y;z)0), see text] of counterions condensed within the equatorial plane around the two colloids (see text) for neutral lamellae (no. 2).

Figure 6. Lateral view of the local concentration of condensed counterions [c(x;z)0), see text] for neutral lamellae (no. 2). The two horizontal lines help to visualize the local ionic densities in the inner and outer regions around the colloids quantifying the overlap of their diffuse layers (see text and Table 2).

TABLE 2: Average and Local Ionic Densities (mM)

Figure 4. Variation of the local average ionic concentration (c°(z)) as a function of the longitudinal separation from the confining lamellae: (a) for neutral lamellae (no. 2); (b) for weakly charged lamellae (no. 3); (c) for highly charged lamellae (no. 4). See Table 2 for more details.

between the particles by the ionic density at the same distance from each particle but in the external region. This ratio quantifies the distortion of these diffuse layers resulting from their overlap: any departure from unity is the fingerprint of a

no.

bulk average

wall contact

average equatorial

colloid contact

inner/outer equatorial density (see text)

2 3 4 5 6 7

0.85 1.84 3.81 5.51 2.32 2.82

0.085 3.2 22.8 22.8 2.2 3.9

2.0 2.5 2.7 7.2 4.4 3.0

680 ( 40 650 ( 40 680 ( 40 2900 ( 200 400 ( 30 720 ( 60

3.8 2.6 2.2 34 3.5 10

significant overlap between the diffuse layers of counterions centered on the colloidal particles, generating a net electrostatic repulsion between these charged particles. As shown on Figure 6, this ratio is large for colloidal particles confined by neutral lamellae and gradually reduces as a function of the charge density of the lamellae (see Table 2). These data are collected in Table 2: the confinement within charged lamellae reduces the electrostatic repulsion between colloidal particles with the same radius and surface charge density by reducing the overlap

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J. Phys. Chem. B, Vol. 109, No. 16, 2005 8169

Figure 7. Variation of the normalized electrostatic force (see text) between two confined charged colloids (nos. 3, 5, 6 and 7) illustrating the influence of the colloidal charge (no. 5), colloidal size (no. 6) and lamellae separation (no. 7) on the mutual repulsion between the colloids.

Figure 8. Comparison between the normalized electrostatic forces evaluated from the numerical results (nos. 1, 3 and 6) and that extracted from experimental data for isotropic and confined silica particles21 (see text).

of their diffuse layers. The plasma of counterions neutralizing the lamellae acts as a salt reducing the net repulsion between the charged particles. Figure 7 compares the different results obtained from numerical simulations performed by increasing the electric charge of the colloidal particles (no. 5), their radius (no. 6) and the geometrical confinement (no. 7). The normalization introduced in section II.C rescales perfectly these different data, allowing for a possible extrapolation of these predictions for real systems with much larger size (acoll ) 0.8 µm)21 or different electrostatic coupling. Of course, it is always possible to describe the long range electrostatic repulsion between charged colloids in the presence of salt by using a Yukawa potential with fitted effective electric charge21 but such an approach precludes any determination of the influence of confinement on their electrostatic repulsion because the functional describing this repulsion is then selected a priori. Figure 7 exhibits a perfect merging of the net forces between each pair of particles in contact because no counterion then screens the electrostatic repulsion between the charged particles. As shown in Figure 7, these different parameters have a large influence on the range of the electrostatic repulsions between the confined colloidal particles but in all cases the electrostatic forces between the particles remains positive. Figure 8 is plotted by exploiting this normalization to compare our numerical data obtained with small charged colloids (acoll ) 0.01 or 0.02 µm) with the order of magnitude of the electrostatic repulsion under the experimental conditions,21 i.e., with two large colloidal particles (acoll ) 0.8 µm, qcoll ) 10000 e)21 confined between two charged lamellae (σlam ) 2 × 104 e/µm2) with a separation equal to three colloid diameters.21 The experimental data plotted in Figure 8 are extracted from the slope displayed by Figure 1 in ref 21, after removing the van der Waals force between two spheres:58

repulsion between the charged colloidal particles induced by the confinement. In the absence of confinement, the reported electrostatic repulsion decreases exponentially,21 as expected for colloids in the presence of salt.53,56-57 The corresponding screening length (κ-1 ∼ 150 nm) is compatible with the ionic strength reported by the authors.21 As shown in Figure 8, we obtain a qualitative understanding of the influence of geometrical confinement on the reduction of the electrostatic repulsion between large silica colloids. The rescaling of the repulsion calculated for the smallest silica particles (no. 3 with acoll ) 0.01 µm) leads to predicted forces larger by 2 orders of magnitude in the range of interest for the reported attraction (r/(2acoll) ∼ 1.5),21 but simply increasing the size of the colloids by a factor of 2 and keeping the confining conditions constant (no. 6) already decreases the net repulsion by 40%. We do not expect a linear variation of the electrostatic force within the whole range of colloid size between 0.01 and 0.8 µm, but this result is encouraging because it allows us to expect a significant reduction of the electrostatic repulsion between large silica particles under confinement, in agreement with experimental data.21 Furthermore our predictions are performed for salt-free suspensions; in the presence of salt we expect a further decrease of the electrostatic repulsion between the confined colloids. Figure 8 may also help to understand the origin of the controversy30-32 concerning the experimental evidence of an effective attraction between confined colloids because such a decrease of the electrostatic repulsion between confined colloids below the order of magnitude of their van der Waals attraction depends on various experimental parameters (colloid size and surface charge density, lamellae separation and electric charge, ionic strength, ...). Another possible application of this treatment is the interpretation of the puzzling attraction59-61 reported for latex colloids under heterogeneous conditions (during solid/gas coexistence) because the outer surfaces of the macroscopic colloidal crystals may behave at large distances as confining solid/liquid charged interfaces.

FVDW ) -

[

r* A 1 1 + 12acoll (r*2 - 1)2 r*3 r*(r*2 - 1)

]

(18)

where r* ) r/(2acoll). This procedure neglects any influence of confinement on dispersion forces. The nonretarded Hamacker constant A is selected to describe the dispersion force between silica colloids immersed in water (A ∼ 10 kJ/mol).5,58 These experimental data confirm the net reduction of the electrostatic

IV. Conclusions We performed a set of Monte Carlo simulations of the counterion distribution around two charged colloidal particles to determine the influence of geometrical confinement on the

8170 J. Phys. Chem. B, Vol. 109, No. 16, 2005 net electrostatic repulsion between the particles. The electrostatic contribution to the net force acting on these confined colloids remains always repulsive, but our results illustrate how confinement by two weakly charged walls reduces the overlap between the diffuse layers of condensed counterions and hence the mutual electrostatic repulsion between the confined charged colloidal particles. By rescaling the electrostatic force, we are able to extrapolate our results to real systems and qualitatively explain the origin of the net attraction recently reported for large confined silica colloids under equilibrium conditions without invalidating simple theoretical arguments. Acknowledgment. We cordially thank Drs. G. Na¨gele, J. Puibasset and R. Setton for helpful discussions. The numerical simulations were either performed locally on workstations purchased thanks to grants from Re´gion Centre (France) or at the Gage Computing Facilities (Palaiseau, France). References and Notes (1) Langmuir, I. J. Chem. Phys. 1938, 6, 873. (2) Schofield, R. K. Trans. Faraday Soc. B 1946, 42, 219. (3) Derjaguin, B.; Landau, L. D. Acta Physicochim. URSS 1941, 14, 635. (4) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of Lyotropic Colloids; Elsevier: New York, 1948. (5) Israelachvili, J. N. Intermolecular and Surfaces Forces; Academic Press: London, 1985. (6) Kjellander, R.; Marcelja, S.; Pashley, R. M.; Quirk, J. P. J. Chem. Phys. 1990, 92, 4399. (7) Greberg, H.; Kjellander, R.; Akesson, T. Mol. Phys. 1997, 92, 35. (8) Hakem, F.; Jokner, A.; Joanny, J. F. Macromolecules 1998, 45, 726. (9) Allahyarov, E.; D’Amico, I.; Lo¨wen, H. Phys. ReV. Lett. 1998, 81, 1334. (10) Netz, R. R.; Orland, H. Europhys. Lett. 1999, 45, 726. (11) Valleau, J. P.; Ivkov, R.; Torrie, G. M. J. Chem. Phys. 1991, 95, 520. (12) Pellenq, R. J. M.; Caillol, J. M.; Delville, A. J. Phys. Chem. B 1997, 101, 8584. (13) Delville, A.; Pellenq, R. J. M.; Caillol, J. M. J. Chem. Phys. 1997, 106, 7275. (14) Linse, P.; Lobaskin, V. J. Chem. Phys. 2000, 112, 3917. (15) Israelachvili, J. N.; Pashley, R. M. Nature 1983, 306, 249. (16) Hug, J. E.; van Swol, F.; Zukoski, C. F. Langmuir 1995, 11, 111. (17) Crocker, J. C.; Grier, D. G. Phys. ReV. Lett. 1996, 77, 1897. (18) Larsen, A. E.; Grier, D. G. Nature 1997, 385, 230. (19) Bowen, W. R.; Sharif, A. O. Nature 1998, 393, 663. (20) Ramirez-Saito, A.; Chavez-Paez, M.; Santana-Solano, J.; ArauzLara, J. L. Phys. ReV. E 2003, 67, 050403(R). (21) Han, Y.; Grier, D. G. Phys. ReV. Lett. 2004, 92, 148301. (22) Goulding, D.; Hansen, J. P. Mol. Phys. 1998, 95, 649. (23) Nuesser, W.; Versmold, H. Mol. Phys. 1998, 94, 759. (24) Allahyarov, E.; D’Amico, I.; Lo¨wen, H. Phys. ReV. E 1999, 60, 3199. (25) Trizac, E.; Raimbault, J. L. Phys. ReV. E 1999, 60, 6530. (26) Trizac, E. Phys. ReV. E 2000, 62, R1465. (27) Sader, J. E.; Chan, D. Y. C. Langmuir 2000, 16, 324. (28) Mateescu, E. M. Phys. ReV. E 2001, 64, 013401. (29) Yuet, P. K. Langmuir 2004, 20, 7960.

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