Influence of the Relative Orientation of Two Charged Anisotropic

Sep 29, 2001 - Influence of the Relative Orientation of Two Charged Anisotropic Colloidal Particles on Their Electrostatic Coupling: A (N,V,T) Monte C...
3 downloads 10 Views 143KB Size
Download PDF
10684

J. Phys. Chem. B 2001, 105, 10684-10690

Influence of the Relative Orientation of Two Charged Anisotropic Colloidal Particles on Their Electrostatic Coupling: A (N,V,T) Monte Carlo Study S. Meyer, P. Levitz, and A. Delville* CRMD, CNRS, 1b rue de la Fe´ rollerie, 45071 Orle´ ans Cedex 02, France ReceiVed: March 29, 2001; In Final Form: July 5, 2001

Monte Carlo simulations are performed in the Canonical ensemble to determine the equilibrium configurations of the counterions and co-ions in the vicinity of two charged hard disks at fixed separations. Derivation of the electrostatic energy and the ionic configurational entropy is used to determine the relative stability of two charged platelets as a function of their separation and relative orientation. An effective pair potential is derived from the free energy variation and may be used in the framework of the one component plasma to model suspensions of charged platelets. Finally, the variation of the free energy of two charged colloids in the presence of salt may be reproduced by Yukawa potential with a residual charge equal to 7% of the nominal electric charge of the colloids.

I. Introduction The long-ranged repulsion between charged colloidal particles neutralized by monovalent counterions is generally well understood on the basis of the overlap of their diffuse layers.1-4 However suspensions of charged anisotropic colloids were recently the subject of numerous theoretical5-10 and experimental11-24 studies since their mechanical and dynamical behavior is far from being fully understood. As an example, dilute suspensions of clay platelets were shown to display large heterogeneities of density at low ionic strength24 and a plateau, as occurring for a gas/liquidlike first-order transition, at high ionic strength.13-18 Both observations require some effective attraction between the clay particles although their swelling pressure is characteristic of a material composed of strongly repulsive particles. In the framework of the primitive model,25 the net force acting on charged colloids results from a balance between the attractive contributions (long-range van der Waals and electrostatic forces) and the repulsive contributions (contact forces). For colloids neutralized by monovalent counterions, correlation forces remain negligible4,26,27 and the balance between these attractive/ repulsive contributions is always a repulsion resulting from the overlap of the diffuse layers of condensed counterions surrounding each particle. For anisotropic colloids, one may expect to find relative orientations of the colloids minimizing this overlap of the diffuse layers, leading to a net stabilization of the colloidal suspensions. This mechanism is expected to occur for suspensions of Laponite clays in the presence of salt, since the thickness of their diffuse layer then reaches the same order of magnitude as their spatial extent. We already suggested that the pseudoplateau displayed by their equation of state results from the reorganization13 of the repulsive clay particles. A firstorder transition was predicted by Onsager,28 but it requires a parallel alignment of the neighboring clay platelets29,30 and thus increases the overlap of their diffuse layer. By contrast, previous Monte Carlo simulations have shown a net reduction of the electrostatic energy of two charged clay platelets oriented * Corresponding author. E-mail: [email protected].

perpendicularly6,13 to each other, but the resulting overlap of their diffuse layers was not yet quantified. The purpose of this study is thus to quantify both the energetic and the entropic contributions to the free energy of two lamellar charged colloids neutralized by monovalent counterions in order to determine their most stable configuration as a function of their separation. This analysis will help to determine the path, allowing the approach of two clay particles by minimizing their mutual repulsion.43 The resulting free energy will also be useful as an effective pair potential for modeling such suspensions of anisotropic charged colloids in the framework of the one component plasma,31 without requiring explicitly the knowledge of the distribution of their counterions whose contribution is already included in the effective potential. Finally, the same pair potential may be useful to analyze the energy barrier implied in the rotation of a single clay particle within the Wigner cell determined by its immobile first neighboring clay particles, explaining the origin of the long time scale implied in this local motion18 and the aging19 behavior reported recently by dynamic light scattering experiments on Laponite aqueous suspensions. II. Material and Methods (A) Modeling Laponite Clay Suspensions. Laponite is a synthetic clay resulting from the sandwiching of one octahedral layer of magnesium oxide between two layers of silicon oxides. Because of the substitution of some magnesium cations of the octahedral network by lithium cations, negative charges are localized within the equatorial plane of the clay network and are neutralized by exchangeable sodium counterions. In the aqueous dilute regime and at pH 10, Laponite clays behave as isolated disks13 (diameter 300 Å, thickness 10 Å) each bearing 1000 elementary charges. Thus, as long as electrostatic forces are considered, Laponite clay in alkaline pH11 is modeled as hard disks bearing 1000 electric charges uniformly distributed within each equatorial plane. No lateral charges are used, since the sites localized at the periphery of the Laponite particle are not ionized under alkaline pH.11,44 All ions are further described in the framework of the primitive model,25 i.e. by charged hard spheres with an ionic diameter of 4.5 Å to mimic solvated sodium counterions.32 The ionic diameter of anions is set equal

10.1021/jp011193v CCC: $20.00 © 2001 American Chemical Society Published on Web 09/29/2001

Electrostatic Coupling of Charged Colloidal Particles

J. Phys. Chem. B, Vol. 105, No. 43, 2001 10685 where 8, the dielectric constant of the continuum surrounding the replica of the central simulation cell, is set equal to 1. The summation over the reciprocal space is cut for |K B| g 5.2π/L, where L is the width of the simulation cell (L ) 1500 Å). The screening parameter κ is set to 3.3 × 10-3 Å-1, leading to an accuracy of the Ewald summation better than 0.005.34 Each of two hard disks (diameter 300 Å, thickness 10 Å) bear 1020 negative sites (noted Ns) uniformly distributed within a regular network in their equatorial plane. The disks are maintained at fixed relative geometry at the center of a cubic simulation cell and the distribution of the ions is thermalized by classical (N,V,T) Monte Carlo procedure. The clay/clay van der Waals attraction is evaluated as the sum of the interactions between two sets of elementary cubes, with a volume (noted Vcell) equal to 693 Å3 and centered on the 1020 charged sites of each disk. The van der Waals energy is given by

EVDW )

-HVcell2 π

Figure 1. Snapshot illustrating an equilibrium configuration of the counterions condensed in the vicinity of two charged disks with a face/ face parallel orientation.

to that of sodium cations. The solvent is described as a continuum characterized by its macroscopic dielectric constant. Figure 1 displays a snapshot of one equilibrium configuration of the sodium counterions condensed in the vicinity of the two parallel charged platelets. (B) Calculation of the Configurational Energy. The total energy is calculated for each configuration of the two clay particles and the ions as the sum of the clay/clay van der Waals attraction plus the ion/ion, ion/clay, and clay/clay electrostatic and contact interactions. In the framework of the primitive model, interactions between ions i and j is given by

uij(rij) )

qiqj 4π0rrij

uij(rij) ) ∞

if rij > aij if rij < aij

where aij is the sum of the radii of both ions. The dielectric constant of the solvent (r) is set equal to 78.5 to describe bulk water. Ewald summation33 is used to calculate this long-range electrostatic energy, in addition with the classical 3D minimum image convention and the periodic boundary conditions:

Eelect ) Edir + Erec + Emom + Eself Edir )

Erec )

1

Nj

∑qij)1,i*j ∑

qj erfc(κrij)

8π0r i)1

∑ 2V  KB,|KB|*0

exp(-|K B| /4κ )

0 r

[{

Ni

1

2

rij

|K B |2

2V(1 + 2∞)0r -κ 4π

3/2

0r

(

qib r i)2 ∑ i)1

∑i qi2

(2d)

(2e)

rij -6

(3)

B Frec )

[

qi∑qj ∑ i∈p j∉p -1

0rV

]

2κrij exp(-κ2rij2) + xπerfc(κrij) b r ij

qi ∑ ∑ i∈p K B,|K B|*0

rij3

4π3/20r

(4a)

K B exp(-|K B|2/4κ2)

∑j qj sin(KB × br ij)

|K B |2

(4b)

The van der Waals contribution to the force is given by

B FVDW )

-6HVcell2 π2

∑∑ i∈P j∈P 1

2

b r ij rij8

(5)

Finally, the contact force is evaluated from the local density of ions in the direct vicinity of the clay platelet [noted ci(0)]

B Fcont ) kT

(2c)

∑ ∑

i)1,i∈P1 j)1,j∈P2

(2b)

×

1

Eself )

B Fdir )

2

∑i qi cos(KB × br i)}2 + {∑j qj sin(KB × br j)}2] Emom )

(2a)

Ns

where the Hamaker constant (H ) 2.2 × 10-20 J) describes the van der Waals attraction between two mica2 particles immersed in water. (C) Calculation of the Net Interparticle Force. The net force acting on each charged platelet results from the balance between three contributions: sthe colloid/colloid and ion/colloid electrostatic force sthe colloid/colloid van der Waals attraction sthe ion/colloid contact force By using Ewald summation, the electrostatic contribution to the force is evaluated in the direct and reciprocal spaces33

(1a) (1b)

2

Ns

∫surfci(0)nb ds

(6)

where b n is the internal normal to the platelet surface. (D) Calculation of the Configurational Entropy. As already for previous numerical simulations of colloidal suspensions performed in the framework of the density functional theory,3,46 the entropy characterizing the distributions of the cations and the anions may be decomposed in an ideal and an excess contributions.47 The ideal mixing entropy of the microions is given by

Sid ) -k

∑s ∫dbr Fs(r) {ln(Fs(r)/Fs0) - 1}

(7)

10686 J. Phys. Chem. B, Vol. 105, No. 43, 2001

Meyer et al.

where Fs(r) is the local density of ions s and Fs° its average value. This equation is easily digitalized for each equilibrium configurations of the microions ({nj}).10 For that purpose, the total volume of the simulation cell (V) is divided into M nonoverlapping elementary volumes (noted Vj), each containing ncj cations and naj anions, respectively. The entropy is then evaluated by using10

Sid ) k[ln(PS({ncj})) + ln(PS({naj}))] M

ln(PS({nj})) )

(8a)

N°Vj

nj ln ∑ nV j)1

(8b)

j

where N° is the total number of cations or anions in the whole simulation cell. Because of the steep variation of the counterion density near the clay surfaces, we used a triple set of cubic boxes with lengths respectively equal to 15, 3, and 0.5 Å. At large separation from the platelets (i.e., outside the disks of 360 Å diameter and 60 Å thickness centered on each platelet), the ionic density is coarsesampled with the largest set of cubes, and the ionic sampling is gradually refined when this separation decreases. We have checked that using twice more cubes modifies only the evaluated entropy difference by a few percent. The cubic boxes from these different sets are selected in order to map the whole volume of the simulation cell without any overlap. The excess contribution to the entropy of the microions is derived from previous (N,V,T) Monte Carlo simulations48 of 1:1 electrolyte in aqueous solution at ionic strengths varying between 0.01 and 4.8 M. We have derived simple analytical formula to reproduce the excess electrostatic energy (Exs) and the excess Helmoltz free energy47 (Fxs) of these ionic solutions:

Exs ) -(0.88 ( 0.03)xI + (0.33 ( 0.03)I NkT (0.021 ( 0.004)I2 (9a) Fxs ) ln γxs ) -(0.73 ( 0.05)xI + (0.53 ( 0.05)I + NkT (0.058 ( 0.007)I2 (9b) From these two formulas, the excess contribution to the entropy of ionic solutions resulting from the electrostatic and the excluded volume interactions is given by

Sxs ) -(0.15 ( 0.08)xI - (0.19 ( 0.08)I Nk (0.08 ( 0.01)I2 (10) and the total excess contribution to the entropy of the co-ions distribution in the suspension is calculated by integration of eq 10 on the basis of the local ionic density49 within the suspension. As stated above, this formula is valid for ionic strength less than 4.8 M, which corresponds to a coupling parameter

Γ)

z2e2 8πI 1/3 e2 4π0rkT 3

( )

(11)

This approach extends the validity of the Debye-Hu¨ckel limiting formula (the first term of eqs 9a,b and 10) to ionic concentrations of the order of magnitude of 1 M, which occurs frequently in the vicinity of the polyions because of the ionic condensation (see Figures 1 and 2a-c).

Figure 2. Same as Figure 1, for two charged disks with edge/edge parallel (a), edge/face perpendicular (b), and edge/edge perpendicular (c) orientations.

Electrostatic Coupling of Charged Colloidal Particles

Figure 3. Variation of the total free energy (9) with its energetic (2) and entropic (1) contributions as a function of the center to center separation of two charged hard disks with a face/face parallel orientation.

(E) Monte Carlo Simulation Procedure. Equilibrium distribution of the ions is obtained in the (N,V,T) ensemble by using the classical Metropolis sampling procedure,35 with block averages to reduce statistical noise. The size of the blocks is at least equal to 10 times the number of ions. All simulations were performed for T ) 298 K. The ionic distributions are calculated at fixed geometry of the two clay platelets. In addition to the face/face parallel alignment of the two clay particles shown in Figure 1, three other relative geometries were considered: san edge/edge parallel alignment (the equatorial planes of the two particles coincide, see Figure 2a) san edge/face perpendicular orientation (the two clay particles forming a T), generally considered to occur in the house card model42 of clay suspensions (see Figure 2b) san edge/edge perpendicular orientation (Figure 2c), obtained by twisting along the common interparticule axis the configuration illustrated in Figure 2a To determine the influence of added salt, the simulations were performed either for two clay particles in the presence of their 2040 neutralizing sodium counterions only or for two clay particles immersed in a 3 × 10-3 M NaCl aqueous solution. III. Results and Discussion (A) Salt-Free Suspensions. The variation of the total free energy is drawn in Figure 3 as a function of the separation between two parallel disks (cf. Figure 1), in addition to its energetic (eqs 2 and 3) and entropic (eqs 8 and 10) contributions. Because of the strong condensation of sodium counterions on the basal surfaces of the charged disks (see Figure 1), the ion/ disk electrostatic attraction overcomes the ion/ion and disk/disk electrostatic repulsions, leading to a net attraction. Note that, in all cases, the energetic contribution from the van der Waals attraction remains negligible in the range of interparticle separations investigated here. Furthermore, the leading contribution to the entropy of the microions is the ideal component since the excess contribution to the entropy remains smaller than 6% of the total entropy. By contrast to the energetic contribution, the entropic contribution is purely repulsive. It results from the overlap of the clouds of counterions surrounding each particle: when the disk separation decreases, the fraction of counterions localized between the two parallel disks increases,7 reducing the configurational entropy of the counterions.10 Since the entropic repulsion is larger (in absolute value) than the energetic

J. Phys. Chem. B, Vol. 105, No. 43, 2001 10687

Figure 4. Variation of the longitudinal component of the net force exerted between two charged disks with a face/face parallel orientation. The straight line is derived from the fit of total free energy (eq 12, see text).

attraction, the net behavior of such charged disks neutralized by monovalent counterions is purely repulsive at all separations. As a consequence, one may expect that an increase of temperature will induce an increase of the swelling behavior of such colloidal suspensions, since the net repulsion between charged disks is driven by the entropic contribution. Finally, the variation of the free energy is fitted by an empirical power law (Figure 3)

F/kT ) (49760 ( 50) +

(16000 ( 1000) D0.78

(12)

where the center to center disk separation (D) was expressed in Å. Note that the variation of the total free energy cannot be described by an effective Yukawa potential, but only by a power law, as already shown for lamellar,45 spherical,50 and disklike7,17 polyions. As a consequence, in contrast with some recent statements,36 the counterions neutralizing charged colloids do not screen the repulsion between the particles. Equation 12 is valid in salt-free suspensions for center/center separations of the platelets smaller than 350 Å. Within homogeneous dispersions, this maximum separation corresponds to a minimum Laponite concentration of 4.4% (w/w). The repulsion between the two parallel charged disks may also be evaluated from the direct derivation of the net force exerted on the disks (Figure 4). Figure 4 also exhibits a perfect agreement between the net interparticle forces calculated from the equilibrium ionic configuration using Eqs 4-6 and the analytical derivation of eq 12 divided by 2, since the free energy is an additive thermodynamic property evaluated here for two charged disks. This agreement validates our numerical procedure used to determine the entropy of the small ions since the net force between the platelets is also evaluated independently from the derivation of the entropy. The same calculations are also performed for three other relative configurations of the charged disks, checking for the occurrence of a pathway allowing the approach of the two disks by minimizing their mutual repulsion. As shown in Figure 5a, the edge/edge parallel configuration (cf. Figure 2a) greatly decreases the electrostatic energy of the disks. Figure 5a also confirms the net reduction of the electrostatic energy of the edge/ face perpendicular configuration (cf. Figure 2b) of the two disks with reference to the face/face parallel configuration (cf. Figure 1). The entropic contributions (Figure 5b) display exactly the

10688 J. Phys. Chem. B, Vol. 105, No. 43, 2001

Meyer et al. separation. Below 200 Å, it is less stable than the face/face parallel configuration and more stable above 250 Å. Here also the relative stability of the two charged disks is mainly driven by the entropic contribution, greatly limiting the validity of predictions based on energetic arguments only. From the difference of free energy between the two extrema, one can estimate as about 130 kT the reorientation activation energy of two charged disks at separations larger than 300 Å. This order of magnitude may explain the long-time relaxation behavior18 and aging19 properties reported for Laponite suspensions in dilute regimes and low salt concentration. This comparison of the relative stability of charged anisotropic colloids was possible because we have simultaneously derived the energetic and entropic contributions to the free energy. The same information is usually derived from the integration of the variation of the net force acting on the particles as a function of their separation.37,38 But in addition to the numerical uncertainty resulting from this integration procedure, the total force calculated for the edge/face and edge/edge perpendicular configurations as well as the edge/edge parallel configurations are not accurate enough (cf. Figure 4) to allow such a comparison, probably because of the reduced extent of the cross section in such geometries. It is also possible to fit simultaneously the configurational free energies displayed in Figure 5c for the four different relative orientations of the charged disks by a simple analytical law, by using an effective site/site pair potential

F(rpq,φpq)/kT ) (-49840 ( 50) + (210 ( 50) cos(2φpq) (2400 ( 200) + (13) ( 0.1) (2.0( 0.2) is∈p js∈q r(0.6 r pq is,js

∑∑

Figure 5. Variation of the total free energy (c) together with its energetic (a) and entropic (b) contributions for different relative orientations: face/face parallel (1), edge/face perpendicular (2), edge/ edge parallel (3), and edge/edge perpendicular (4). Error bars are either drawn or comparable to the size of the points.

same trends, but obviously with the opposite consequence on the net stability of the two charged disks. The final balance between these antagonistic effects (energetic stabilizations and entropic repulsion) is displayed in Figure 5c: with reference to the face/face parallel configuration (cf. Figure 1), the edge/edge parallel configuration (cf. Figure 2a) is less stable while the edge/edge perpendicular configuration (cf. Figure 2c) is more stable. The relative stability of the edge/face perpendicular configuration (cf. Figure 2b) varies as a function of the disk

where φpq is the angle between the two normals of the particles with the center to center separation rpq. The parameters given in eq 10 are evaluated by minimizing the least-squares deviation using a Simplex fitting procedure for a regular network of 38 sites uniformly distributed in the equatorial plane of the charged disks. This effective potential includes both the electrostatic and entropic contributions to the configurational free energy. This simple analytical law may be used for modeling such suspensions of charged disks in the framework of a one-component description of the platelets, after tracing out the microscopic degrees of freedom of the small ions. This treatment extends to salt-free suspensions the use of empirical potential, like the Yukawa potential,8,31 whose validity is restricted to colloidal suspensions in the presence of salt. (B) Suspensions with Added Salt. The same Monte Carlo simulations are performed for two charged disks under different relative geometries in the presence of a large amount of added salt (I ) 3 × 10-3 M). Under such conditions, the number of counterions pertaining to the electrolyte is 3 times larger than the number of counterions neutralizing the disks. Because of the Donnan exclusion phenomenon,45 the resulting Debye screening length (κ-1) must be evaluated on the basis of the ionic strength in a salt reservoir in equilibrium with the colloidal suspension. The salt concentration in such a reservoir is estimated from the salt concentration at the boundary of the simulation cell, at large separation from the two charged colloids. From the analysis of the ionic distributions, we obtain Ireservoir ) 2.2 × 10-3 M, leading to

κ-1 )

(

)

4π0rkT 2

2e I

1/2

(14)

Electrostatic Coupling of Charged Colloidal Particles

J. Phys. Chem. B, Vol. 105, No. 43, 2001 10689

Figure 7. Comparison between the variations of the free energy of two charged disks in the face/face parallel orientation without salt (2) and in the presence of 3 × 10-3 M NaCl (1).

to the previous statement, the range of the free energy variation is strongly reduced. Unfortunately, the statistical noise induced by the entropy calculation increases markedly, limiting the sensitivity of this numerical procedure. Because of these large uncertainties, it is not possible, from the free energy variations reported in Figure 6c, to discriminate between different effective pair potentials. Nevertheless, the variation of the free energy derived for the face/face parallel configuration may be fitted to the classical Yukawa potential (cf. Figure 7), by taking the free energy at infinite separation as the reference:

Zeff2e2 ∆F(rpq)/kT ) exp(-κrpq) 4π0rrpqkT

Figure 6. Same as Figure 5, for two charged disks immersed in a 3 × 10-3 M NaCl aqueous solution.

equal to 62 Å, i.e., 1 order of magnitude smaller than the disk diameter. The thickness of the diffuse layers surrounding each particle is thus strongly reduced and reveals their anisotropy better than in the case of salt-free suspensions.24 Figure 6c displays the total free energy together with its energetic (Figure 6a) and entropic (Figure 6b) contributions. The general trends are the same as in the case of salt-free suspensions: the edge/edge parallel configuration has again the lowest electrostatic energy but also the lowest entropy. The final balance between these antagonistic contributions corresponds to a large excess [(1200 ( 400) kT] of free energy for the edge/ edge parallel orientation as compared with the three other relative orientations of the charged disks. Note that, according

(15)

with Zeff ) 75 ( 10, where rpq is the center to center separation between the disks and κ-1 is the Debye length defined in eq 14. As a consequence, the electric charge of the disks is strongly reduced by the ionic condensation and only 7% of the total charge contributes to the long-range repulsion between the disks. (C) Limitations and Perspective. The stability of charged disks as a function of their relative orientation was deduced from a direct calculation of the total electrostatic energy and the ionic configurational entropy by using (N,V,T) Monte Carlo simulations of the ion distribution in the framework of the primitive model. Of course, the validity of this study is restricted by the validity of the primitive model, which neglects the molecular structure of the solvent2,39 and the atomic structure39,40 of the charged disks. This study further assumes an additivity of the free energy of a collection of such charged disks within a colloidal suspension. This approach thus neglects the possible occurrence of collective effects that were suggested to explain the anomalous behavior of suspensions of charged colloids at low ionic strength.41 Further simulations are now under progress in order to quantify such phenomena. IV. Conclusion (N,V,T) Monte Carlo simulations were used to determine the equilibrium distribution of monovalent counterions and co-ions in the presence of two charged disks at fixed separation and relative orientation. The relative stability of these different

10690 J. Phys. Chem. B, Vol. 105, No. 43, 2001 geometries was derived, at equilibrium, from the direct calculation of the total electrostatic energy and the ionic configurational entropy. In the absence of salt, the repulsion between the charged disks is driven by the entropic contribution to the total free energy, while the electrostatic energy is always attractive. The free energy variation is also fitted by an empirical power law, allowing us to model a collection of charged disks. The large difference reported for the free energy as a function of the disks orientation indicates a high activation barrier for the reorientation of a charged disk in the dilute regime, i.e., even without direct excluded volume interactions with its first neighbors. This result may explain the long-time relaxation and aging measured for suspensions of such anisotropic charged colloids. Despite the marked increase of the statistical noise in the presence of salt, a Yukawa effective potential may be used to describe the free energy variation of two charged disks, with an effective charge reduced to 7% of the nominal electric charge of the disks. Acknowledgment. We cordially thank Pr. P. G. de Gennes and J. P. Hansen for suggestions and stimulating comments on the colloidal stability and Dr. R. Setton for helpful discussions. The Monte Carlo simulations were performed either locally on workstations purchased thanks to grants from Re´gion Centre (France), or on workstations of the Gage computing facilities (Palaiseau, France) or on Nec supercomputers (IDRIS, CNRS). References and Notes (1) Overbeek, J. Th. G. J. Chem. Phys. 1987, 87, 4406. (2) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1985. (3) Lo¨wen, H.; Madden, P. A.; Hansen, J. P. Phys. ReV. Lett. 1982, 68, 1081. (4) Pellenq, R. J. M.; Caillol, J. M.; Delville, A. J. Phys. Chem. B 1997, 101, 8584. (5) Chang, F. R. Ch.; Sposito, G. J. Colloid Interface Sci. 1996, 178, 555. (6) Dijsktra, M.; Hansen, J. P.; Madden, P. A. Phys. ReV. E 1997, 55, 3044. (7) Delville, A. J. Phys. Chem. B 1999, 103, 8296. (8) Kutter, S.; Hansen, J. P.; Sprik, M.; Boek, E. J. Chem. Phys. 2000, 112, 311. (9) Rowan, D. G.; Hansen, J. P.; Trizac, E. Mol. Phys. 2000, 98, 1369. (10) Delville, A.; Levitz, P. J. Phys. Chem. B 2001, 105, 663. (11) Thompson, D. W.; Butterworth, J. T. J. Colloid Interface Sci. 1992, 151, 236. (12) Ramsay, J. D. F.; Lindner, P. J. Chem. Soc. Faraday Trans. 1993, 89, 4207. (13) Mourchid, A.; Delville, A.; Lambard, J.; Le´colier, E.; Levitz, P. Langmuir 1995, 11, 1942.

Meyer et al. (14) Gabriel, J. Ch. P.; Sanchez, C.; Davidson, P. J. Phys. Chem. 1996, 100, 11139. (15) Pignon, F.; Magnin, A.; Piau, J. M.; Cabane, B.; Lindner, P.; Diat, O. Phys. ReV. E 1997, 56, 3281. (16) Kroon, M.; Vos, W. L.; Wegdam, G. H. Phys. ReV. E 1998, 57, 1962. (17) Philipse, A. P.; Wierenga, A. M. Langmuir 1998, 14, 49. (18) Mourchid, A.; Le´colier, E.; Van Damme, H.; Levitz, P. Langmuir 1998, 14, 4718. (19) Bonn, D.; Tanaka, H.; Wegdam, G.; Kellay, H.; Meunier, J. Europhys. Lett. 1998, 45, 52. (20) Pelletier, O.; Sotta, P.; Davidson, P. J. Phys. Chem. B 1999, 103, 5427. (21) Saunders, J. M.; Goodwin, J. W.; Richardson, R. M.; Vincent, B. J. Phys. Chem. B 1999, 103, 9211. (22) Pelletier, O.; Davidson, P.; Bourgaux, C.; Coulon, C.; Regnault, S.; Livage, J. Langmuir 2000, 16, 5295. (23) Nicolai, T.; Cocard, S. Langmuir 2000, 16, 8189. (24) Levitz, P.; Le´colier, E.; Mourchid, A.; Delville, A.; Lyonnard, S. Europhys. Lett. 2000, 49, 672. (25) Carley, D. D. J. Phys. Chem. 1967, 46, 3783. (26) Delville, A.; Pellenq, R. J. M.; Caillol, J. M. J. Chem. Phys. 1997, 106, 7275. (27) Allahyarov, E.; Lo¨wen, H.; Trigger, S. Phys. ReV. E 1998, 57, 5818. (28) Onsager, L. Ann N. Y. Acad. Sci. 1949, 51, 627. (29) Eppenga, R.; Frenkel, D. Mol. Phys. 1984, 52, 1303. (30) Forsyth, P. A.; Marcelja, S.; Mitchell, D. J.; Ninham, B. AdV. Colloid Interface Sci. 1978, 9, 37. (31) Fushiki, M. J. Chem. Phys. 1992, 97, 6700. (32) Cooker, H. J. Phys. Chem. 1976, 80, 1084. (33) Heyes, D. M. Phys. ReV B: Condens. Matter 1994, 49, 755. (34) Hummer, G. Chem. Phys. Lett. 1995, 235, 297. (35) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford Science Publishers: Oxford, U.K., 1994. (36) Gru¨nberg, H. H.; Van Roij, R.; Klein, G. Europhys. Lett. 2001, 55, 580. (37) Guldbrand, L.; Jo¨nsson, B.; Wennerstro¨m, H.; Linse, P. J. Chem. Phys. 1984, 80, 2221. (38) Delville, A.; Gasmi, N.; Pellenq, R. J. M.; Caillol, J. M.; Van Damme, H. Langmuir 1998, 14, 5077. (39) Delville, A. J. Phys. Chem. 1993, 97, 9703. (40) Bleam, W. F. ReV. Geophys. 1993, 31, 51. (41) Crocker, J. C.; Grier, D. G. Phys. ReV. Lett. 1996, 77, 1897. (42) Van Olphen, H. An Introduction to Clay Colloid Chemistry; Interscience Publishers: New York, 1963. (43) Levitz, P.; Delville, A.; Le´colier, E.; Mourchid, A. Prog. Colloid Polym. Sci., in press. (44) Mourchid, A.; Levitz, P. Phys. ReV. E 1998, 57, R4887. (45) Dubois, M.; Zemb, Th.; Belloni, E.; Delville, A.; Levitz, P.; Setton, R. J. Chem. Phys. 1992, 96, 2278. (46) Lo¨wen, H.; Hansen, J. P.; Madden, P. A. J. Chem. Phys. 1993, 98, 3275. (47) Hill, T. L. An Introduction to Statistical Thermodynamics; Dover Publications Inc.: New York, 1986; Chapter 18. (48) Valleau, J. P.; Cohen, L. K. J. Chem. Phys. 1980, 72, 5935. (49) Delville, A.; Gilboa, H.; Laszlo, P. J. Chem. Phys. 1982, 77, 2045. (50) Delville, A. Langmuir 1994, 10, 395.