Electrostatic Interaction Force between Planar Surfaces Due to 3-D

J. Phys. Chem. B 1998, 102, 2483-2493

2483

Electrostatic Interaction Force between Planar Surfaces Due to 3-D Distribution of Sources of Potential (Charge) Armik V. M. Khachatourian and Anders O. Wistrom* Department of Chemical and EnVironmental Engineering, UniVersity of California, RiVerside, California 92521 ReceiVed: April 30, 1997; In Final Form: September 26, 1997

A closed form solution for the electrostatic double-layer force between two planar surfaces having sources of potential (charge) distributed along the x-, y-, and z-axes has been derived. The double-layer force is evaluated using the expression for the Lorentz force by combining the tangential and normal components of the electric field on and between the two approaching surfaces. The general analytical solutions are developed for two cases: The first case is for when the distribution of charges on the two interfaces and intermediate layers are given. The second case is for mixed boundary conditions where the potential distribution of the interacting surfaces and charge distributions originating from intermediate layers (protrusions) are provided. Provided is a numerical example of the interaction force between two planar surfaces, each having a periodic potential distribution as well as discretely placed charges located on protrusions away from the surface, demonstrating that the electrostatic interaction force is always attractive at smaller plate separations, whereas at intermediate separations the magnitude (and sign) of the interaction force is dictated by the relative voltage, i.e., chargeto-potential ratio, electrolyte concentration, and relative lateral displacement.

I. Introduction Observed colloid aggregation and colloid deposition rates in the presence of repulsive interactions (usually double-layer repulsion) are several orders of magnitude larger than predicted by classical DLVO (Derjaguin, Landau, Vervey, Overbeek) theory.1-3 The opposite trend is observed for interacting silica surfaces and smooth mica surfaces which have been extensively studied using a surface force apparatus.4-6 For these latter cases it was found that the interaction force at short range was dominated by a repulsive force which prevented the surfaces from coming into a primary minimum, a finding which is contrary to the DLVO theory. Recent theoretical7 and experimental findings suggest that inconsistencies between the classical DLVO theory and experimental results may have an entropic origin, arising from the osmotic pressure between the surfaces.8,9 As a result, the origin of the observed short-range repulsive forces should be sought after at the surfaces themselves. Physical heterogeneity and surface potential (charge) heterogeneities have been suggested as plausible explanations for observed inconsistencies between theory and experiments.10-12 Kihira and Matijevic13,14 suggested that observed differences between experimental stability ratios of aggregating colloidal particles and theoretical predictions (based on the classical DLVO theory) can be reconciled if the surface charge is assumed to be discretely distributed. The importance of heterogeneous surface charge distribution on double-layer forces has been considered by Vreeker et al.15 They studied the coagulation stability ratio of nickel hydroxycarbonate particles at different electrolyte concentrations. By fitting a twodimensional sinusoidal variation of the surface potential, they attained a reasonable fit with observed results. However, in their analysis they used Kuins16 results, which were obtained using the superposition approximation restricting the validity * Corresponding author. E-mail: [email protected]. Fax: (909) 7873188.

of the analytical treatment to large separation distances. A more exact analysis of double-layer interaction forces is provided by Miklavic et al.,12 who considered the interaction force between two planar surfaces having a periodic distribution in two dimensions of charge or potential. We focus our attention on the double-layer contribution to the interaction force by relaxing the classical assumptions of the DLVO theory of molecularly smooth surfaces and smeared out sources of potential (charge). We derive the general analytical solutions for the double-layer interaction force between surfaces characterized by an arbitrary three-dimensional heterogeneity from the well-known laws of electricity (see section II). The model surfaces considered are heterogeneous along the three directional axes to model the effect of adsorbed surface active molecules, and, to a limited extent, physical roughness in combination with an arbitrary distribution of sources of potential (charge) at the interface. The derivations are carried out for two approaching planar surfaces. Other geometries relevant to colloid stability and colloid deposition, i.e., sphere-sphere and sphere-plate, will be considered in later contributions. Section I of this article determines the general solution for the double-layer force between two planar surfaces when sources of potential (charge) are discretely distributed along the x-, y-, and z-axes. Section II gives the boundary conditions governed by the distribution of potential and charge for two separate cases: (1) when the distribution of charges on the two interfaces and intermediate layers are given, and (2) when the potential distribution of the interacting surfaces and the charge distributions originating from intermediate layers are given (mixed boundary conditions). For both cases it is assumed that the potential (charge) remains constant during the interaction. Section III evaluates the double-layer interaction force. The evaluation consists of formalizing the use of the Lorentz force and combining the results with the tangential and normal components of the electric field on and between the two

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2484 J. Phys. Chem. B, Vol. 102, No. 14, 1998

Khachatourian and Wistrom

surfaces. Since the double-layer force is obtained directly, the results are valid for small separation distances. Section IV presents the analytical results for the double-layer force calculations for a case with mixed boundary conditions. Section V summarizes the major conclusions. II. Calculation of Surface Potential The Helmholtz equation for the electrostatic potential between two planar surfaces in a symmetric z:z electrolyte has the general form

∇2ψ ) λψ ∇2ψ ) 0

between the surfaces

(1a)

outside the surfaces

(1b)

where λ is a function of the Debye screening parameter. The functional form of λ has been thoroughly studied and has been well summarized by Hiemenz.17 Linearized Poisson-Boltzmann Equation. The PoissonBoltzmann equation

∇2ψ )

( )

∑

2ni0 zie zieψ sinh 0 kBT

2ni0zi2e2ψ ) λψ 0kBT

∑

2e2 0kBT

∑

zi2ni0

∇2ψ21 ) λψ21 ∇2ψ22 ) λψ22

∞ > z > z1 ) h z1 > z > z2 z2 > z > z3

l zN-1 > z > zN ∇2ψ2N-1 ) λψ2N-1 2 N N ∇ ψ2 ) λψ2 zN > z > zN+2 ∇2ψ3 ) 0

0 ) zN+1 > z > -∞

}

∞

∞

n)-∞ m)-∞ ∞

We note that λ1/2 ) κ is the so-called Debye-Hu¨ckel parameter in the linearized Poisson-Boltzmann equation. Distribution of Surface Potential or Charge. We have assumed the double-layer interaction between two infinite planar surfaces each to have an arbitrary distribution of surface potential or charge distribution. In other words, the distribution of surface potential or charge is considered nonuniform along the x-, y-, and the z-axes. A schematic representation of the system under study is shown in Figure 1. The surfaces have dielectric permittivities 1 and 3, respectively, and are separated by a dielectric continuum of permittivity 2, having a width h. We have also assumed that the medium is linear, isotropic, and homogeneous, i.e., 1, 2, and 3 has discrete values, and the displacement field, D B (E B), is proportional to the electric field, B E. The approach taken in this work explicitly takes into account the three-dimensional microstructure of the surfaces and is therefore an improvement over existing two-dimensional models (of the surface charges only along the surface itself). This approach allows the calculation of the double-layer force for a much wider selection of surfaces, including complex systems

(5b)

(5c)

ψ21 )

∞

∑ ∑ exp[2iπ(nx/a + my/b)] ×

n)-∞ m)-∞ [Bnm1 ∞ ∞

ψ22 )

exp(-k2z) + Cnm1 exp(k2z)]


n)-∞ m)-∞ [Bnm2 ∞

ψ2N-1 )

exp(-k2z) + Cnm2 exp(k2z)] l

∞


n)-∞ m)-∞ [BnmN-1 ∞ ∞

ψ2N

)

exp(-k2z) + CnmN-1 exp(k2z)]


n)-∞ m)-∞ [BnmN ∞

ψ3 )

}

∑ ∑ Anm exp[2iπ(nx/a + my/b)]exp(-k1z)

ψ1 )

(4)

(5a)

λ is the linearized dependence of the ionic distribution densities on the mean potential according to eq 4. Similarly to Miklavic et al.,12 we express an arbitrary surface potential distribution for all sources of potential by a Fourier expansion along x- and y-coordinates, eqs 6a and 6c. To account for surface protrusions and/or adsorbed materials we have added sources of potential at intermediate locations along the z-axis 0 < zi < h, eq 6b.

(3)

where the functional form of λ is given by

λ)

∇2ψ1 ) 0

(2)

relates the divergence of the gradient of the electric potential, ψ, at a given point to the charge density, F, at that point where ni0 and zi are the concentration in the bulk and the valency of electrolyte ions of species I, respectively, e is the elemental charge, 2 is the dielectric constant of the solvent, and kBT is the temperature in energy units. If the Debye-Hu¨ckel approximation is invoked, i.e., if it is assumed that zeψ is small compared to kBT, for eψF/kBT zi > h, represented by the symbol b, can be specified by delta, step, or Fourier series depending on the interface. Note all cells with dimensions of a × b are equivalent.

k1 ) 2π

n2 m 2 1/2 + a b

[( ) ( ) ]

(7a)

k2 ) [λ + k12]1/2

(7b)

k3 ) k1

(7c)

The Fourier coefficients Anm, Bxmn, Cxmn, and Dnm depend on the periodicity of the source function expressed in k1 and the magnitude of the Debye-Hu¨ckel parameter expressed in k2. Once the boundary conditions, i.e., distribution of sources of potential and charge on the surfaces and intermediate layers are given, the evaluation of the Fourier coefficients can proceed. Boundary Conditions. In addition to the requirement that the potentials, ψ, be either finite or periodic at infinity, the tangential component of the electric field, B E(-∇ B ψ), must be continuous (Faraday’s law), irrespective of whether the distribution of surface charges or surface potentials is given. In general, the solutions satisfying the continuity of the tangential component of B E, which is equivalent to the continuity of ψ, are

ψ1|z)z1 )

ψ21|z)z1

ψ21|z)z2 ) ψ22|z)z2 ψ21|z)z3 ) ψ23|z)z3 l ψ2N-2|z)zN-1 ) ψ2N-1|z)zN-1 ψ1N-1|z)zN ) ψ2N|z)zN ψ2N|z)zN+1 ) ψ3|z)zN+1

}

(8a)

∂ψ22 ∂ψ21 - 2 2 | ) σ2 ∂z ∂z z)z2 ∂ψ23 ∂ψ22 - 2 2 | ) σ3 ∂z ∂z z)z3 l N-1 ∂ψ2 ∂ψ2N-2 2 |z)zN-1 ) σN-1 - 2 ∂z ∂z N N-1 ∂ψ2 ∂ψ2 2 |z)zN ) σN - 2 ∂z ∂z

)

)

)

∂ψ3 ∂ψ2N 3 | ) σN+1 - 2 ∂z ∂z z)zN+1

(9b)

(9c)

Equations 8 and 9 constitute a set of 2(N + 1) linearly independent equations which can be easily solved for the unknown Fourier coefficients, Anm, Bxmn, Cxmn, and Dnm. When the distribution of sources of potential on the surface and sources of charge at intermediate layers are given, i.e., mixed boundary condition, the boundary conditions are

ψ1|z)z1 ) ψ21|z)z1 ) Φ1

( ( ( (

) )

∂ψ22 ∂ψ21 2 | ) σ2 - 2 ∂z ∂z z)z2 3 2 ∂ψ2 ∂ψ2 2 | ) σ3 - 2 ∂z ∂z z)z2 l ∂ψ2N-1 ∂ψ2N-2 - 2 2 |z)zN-1 ) σN-1 ∂z ∂z ∂ψ2N ∂ψ2N-1 - 2 2 |z)zN ) σN ∂z ∂z

)

)

ψ2N|z)zN+1 ) ψ3|z)zN+1 ) ΦN+1 (8b)

}

(9a)

}

(10a)

(10b)

(10c)

Again, the Fourier coefficients, Anm, Bxmn, Cxmn, and Dnm, are found by sequentially evaluating eqs 8 and 10 at each boundary. A listing of the coefficients and procedures for the solutions is provided in Appendixes A and B, respectively. III. Double-Layer Interaction Forces

(8c)

Significantly different interaction forces (often with sign reversal) can result when either charge or potential distribution is permitted to change as a function of plate separation distance. In this article we consider only the limiting cases where the distribution of sources of potential (charge) is independent of separation distance between the two plates. The two cases of the following discussion assume: (1) the distribution of sources of charge on the surface and at intermediate layers and (2) the distribution of sources of potential on the surface and sources of charge at intermediate layers. When the distribution of sources of charge along the boundaries is given, the discontinuity of the normal component of the displacement field, D B (E B), is proportional to the charge distribution, σ (Gauss’s law). The magnitude of the charge density at the boundaries is obtained by evaluating the derivatives of the potentials at all locations 0 e z1 e h so that

The force on the top plate is evaluated using the expression for the Lorentz force:

B F)

∫FEB dV

(11)

where F and B E are the charge per unit volume and the electric field, respectively, integrated over volume V. The Lorentz force is a uniquely defined experimental force, analogous to Newton’s law of gravity.18 Calculation of the Lorentz force yields the force vector for the interaction which is the quantity of interest in dynamic colloidal systems. The general analytic solutions for the double-layer force are developed for two sets of boundary conditions. The different boundary conditions evolve from the presence of sources of potential and charge on the interacting plates and intermediate layers. The two cases considered here are as follows: (1) sources of charge are present on the interacting surfaces as well as on “protrusions” at intermediate layers, and (2) sources of



potential are found on the surfaces, and sources of charge are present at intermediate layers. When sources of charge are assumed on both surfaces and intermediate locations, the electric field on the top plate and the charge distribution on the top plate (Figure 1) are given by

B E)B E1 + B E21

(12a)

F ) σ1(x,y,z1) δ(z-z1)

(12b)

where σ1(x,y,z) is the charge per unit area on the top plate and δ(z-z1) is the Dirac delta function representing the top surface. From eqs 11 and 12 it follows that the force between two infinite planes is given by

B F)

∫-∞∞∫-∞∞∫-∞∞ σ1(x,y,z1) δ(z-z1)[EB1(x,y,z1) +

sources of charge located on both plates which is represented by the terms in B1mn. The decay length, 1/k2, is a function of both the Debye-Hu¨ckel parameter, κ, and the relative displacement of the two plates along the xy-plane. The second term represents the contribution by sources of charge within the unit cell located at the top plate. Its value is affected by the magnitude of the surface charge(s) as well as the periodicity of the sources of charge. The general solution for the double-layer force for the case with mixed boundary conditions, i.e., when sources of surface potential are present on the plates instead of sources of charge, is developed in a similar manner. The charge distribution, σ1(x,y,z1), is now found by evaluating the discontinuity of the normal component of the displacement field, D B (E B), on the top plate given in eq 9a. From eqs A10-A18, 14, and 15a the double-layer force between the two plates becomes

B E21(x,y,z1)] dx dy dz )

∞ ∞

∫-∞∫-∞ σ1(x,y,z1)[EB1(x,y,z1) + ∞

∞

B E21(x,y,z1)]

dx dy

(13)

a/2 b/2 σ1(x,y,z)[E B1(x,y,z1) + B E21(x,y,z1)] dx dy ∫-a/2 ∫-b/2

(14) The double-layer force, B F, can be evaluated once the magnitude of the electric fields on the top plate has been determined. The electric fields are evaluated from the potentials by

E21(x,y,z1) ) -(∇ B ψ1 + ∇ B ψ21)|z)z1 (15a) B E1(x,y,z1) + B Substituting the values of the Fourier coefficients determined earlier and the derivatives of the potentials on the top plate (eq 15a) yields ∞ ∞

B E1(x,y,z1) +

B E21(x,y,z1)

)-

∑ m)-∞ n)-∞

{4iπβ1(22k2B1mn exp[-k2z1] ∑ m)-∞ n)-∞

This integral is infinite; however, the corresponding interaction force on a unit cell with dimensions a by b is finite:

B F)

B F ) ab

exp[2iπ(nx/a + my/b)] 1k1 + 2k2

×

(1k1 + 2k2)β1) (nxˆ /a + myˆ /b) + (22k2B1mn exp[-k2z1] (1k1 + 2k2)β1)(-2k2B1mn exp[-k2z1] + (k2 - k1)β1)zˆ} (16b) where β1 and B1mn are given in Appendix A. Both terms on the right-hand side include expressions for the charge density multiplied by the electric field. Summing over the unit cell yields the contribution to the double-layer force along the xyplane and z-direction, respectively. Contributions to the doublelayer force from both plates are embedded in B1mn, including the dielectric constant for both surfaces as well as the sources of potential (charge) located at intermediate layers. It is useful to examine the asymptotic behavior for eqs 16a and 16b. We note that the Lorentz force or double-layer force evaluated in eq 11 is the absolute force. The interaction force or gauge force, i.e., the force that is experimentally determined (if such an experiment were indeed possible), is found by subtracting the double-layer force at infinite separation from the absolute double-layer force, so that

{(42k2Bnm1 exp[-k2z1] + 2R1)(2iπ(nxˆ /a) + 2iπ(myˆ /b)) + (-2k1k2(1 + 2)Bnm1 exp[-k2z1] + (-k1 + k2)R1)zˆ} (15b) where xˆ , yˆ , and zˆ are Cartesian unit vectors. The interaction force in x-, y-, and z-directions follows from eqs 14 and 15 and becomes

Fxyz - B F∞ B Fi ) B At infinite separation eq 16a reduces to ∞ ∞

B F∞ ) -ab

B F) -ab

∑ m)-∞ k n)-∞

∞ ∞

∑

m)-∞ n)-∞

1

{4iπ(22k2B1mn exp[-k2z1] + R1) × 1k1 + 2k2

(nxˆ /a + myˆ /b) + (-2k1k2(1 + 2)B1mn exp[-k2z1] + b/2 a/2 σ1(x,y,z1) exp[2iπ(nx/a + ∫-b/2 ∫-a/2

where R1 and B1mn are provided in Appendix A. The details of the derivation of eq 16a are provided in Appendix B. The first term on the right-hand side of eq 16a is the electric field evaluated from the derivatives of the potential at the top plate according to the right-hand side of eq 15a. The first term also recognizes the contribution to the interaction force from

1 1

1

{4iπR1(nxˆ /a + myˆ /b) +

+ 2k2

b/2 a/2 σ1(x,y,z1) exp[2iπ(nx/a + ∫-b/2 ∫-a/2

(k2 - k1)R1zˆ} my/b)] dx dy

(18a)

and for mixed boundary conditions eq 16b reduces to

(-k2 - k1)R1)zˆ}

my/b)] dx dy (16a)

(17)

∞ ∞

B F∞ ) -ab

∑

{4iπ(nxˆ /a + myˆ /b) +

m)-∞ n)-∞

(k2 - k1)zˆ} (1k1 + 2k2)β12

(18b)

Consequently, to obtain the interaction force between two plates having only sources of charge, eq 18a is evaluated and then subtracted from eq 16a. If the interacting surfaces have mixed

Electrostatic Interaction Force between Planar Surfaces

J. Phys. Chem. B, Vol. 102, No. 14, 1998 2487 of sources of charges in eq 19b at c relative to the sources of point charges at z1 - c is represented by the term gs ) cos(2mπd/a) cos(2nπd/a), where a is the length of the unit cell and d/a is the relative displacement distance. The general solution for the double-layer force between two infinite planes having mixed boundary conditions, i.e., both sources of potential and charge, in the z-direction is given by ∞ ∞

(22k2B1mn exp[-k2z1] ∑ m)-∞

B Fz ) ab Figure 2. Schematic representation of two infinite planar surfaces used in numerical example. Cells, a × a, each have an equivalent distribution of sources of potential on the surface and sources of charge at intermediate locations, c, i.e., mixed boundary conditions.

boundary conditions, eq 18b is evaluated and then subtracted from eq 16b.

n)-∞

(1k1 + 2k2)β1)(-2k2B1mn exp[-k2z1] + (k2 - k1)β1)zˆ (20) Evaluating eq 20 with the input conditions defined in eqs 19a and 19b yields ∞ ∞

2 3 (-2k22){(f 1nm + f 2nm)(f 1* ∑ nm + f nm) + f nm} m)-∞

Fz ) a

2

IV. Numerical Example with Mixed Boundary Conditions

(21)

n)-∞

The model system studied here relaxes the assumption of smeared out potential (charge) distributions on molecularly smooth surfaces. However, we will consider the double-layer force between two planar surfaces when sources of potential are distributed on the two opposing surfaces in combination with sources of charge located on “protrusions” away from the surface itself. The proposed model system could represent the interaction of two complex surfaces, each having a sparse layer of adsorbed surface active molecules which give rise to charges located away from the surface itself. For example, a metal oxide surface having a polymer grafted to its surface could be represented by a potential distribution along both surfaces while the polymer itself is represented by a distribution of charges at intermediate locations (see schematic in Figure 2). Interaction Force in z-Direction. Both plates have an identical rectangular lattice structure composed of cells with dimensions a × b. To model the double-layer force between “complex” surfaces, we have selected to represent the heterogeneity of the source distribution of potential at the surface itself with a continuous, sinusoidal, function,

Φtop(x,y,z)|z1 ) Φ0 cos(2πx/a) cos(2πy/a) Φbottom(x,y,z)|z1 ) Φtop(x - d, y - d, z1)

(19a)

The periodicity of the potential distribution is represented in this particular example by a but could of course vary to represent both shorter and longer wavelengths. Centered within each rectangular unit cell is a source of charge protruding a distance c away from the surface,

σtop(x,y,z1 - c) )

σ0a2 4

∞ ∞

∑ enem{δ(x - d + na) +

m)0 n)0

δ(x - d - na)}{δ(y - d + ma) + δ(y - d - ma)} σbottom(x,y,c) ) σtop(x - d, y - d, c)

(19b)

where en and em are the Neumann numbers for which e0 ) 1 and en ) 2 for all n > 0. The magnitude of the double-layer force is influenced not only by the separation distance between the plates but also by the relative position of the two plates. The lateral displacement of sources of potential in eq 19a is represented by the term g′ ) cos 2πd/a and the displacement

where

f 1* nm )

f 1nm )

(

)

Φ0 (1 - g′2) k2z1 k1 - (δn,1δm,1 + + tanh 4 sinh k2z1 2 k2 δn,-1δm,-1 + δn,1δm,-1 + δn,-1δm,1)

(

(21a)

)

Φ0 (1 - g′2) k2z1 + tanh (δn,1δm,1 + δn,-1δm,-1 + 4 sinh k2z1 2 (21b) δn,1δm,-1 + δn,-1δm,1)

[

]

σ0 k2z1 cosh k2c - tanh sinh k2c × 2k2 2 k2z1 1 1 + (1 - gs) coth -1 2 2

f 2nm ) -

[

(

(

)

)]

Φ02 (1 - g′2)2 (δn,1δm,1 + δn,-1δm,-1 + 16 sinh2 k z 2 1 δn,1δm,-1 + δn,-1δm,1)

f 3nm ) -

(21c)

(21d)

Each term in eq 21 decays exponentially with separation, z1. Let us, on the one hand, consider the case of no sources of charge located between the plates; it follows from eq 21c that f 2nm ) 0, and the remaining terms are nonzero. It is worth noting that, when the periodicity of the sources of potential increases (see eq 19a), the double-layer force also increases at all separation distances. The reason for this increase is that for small wavelengths the lateral screening is relatively effective, since the lateral mobility of the ions parallel to the surface is reduced. If we simplify the boundary conditions by assuming a uniform potential distribution on both plates, we note that eq 21 reduces to the familiar expression for the double-layer force between two plates at constant potential:

Fz ) -2a2κ2Φ′02(tanh(κz1/2))2

(22)

On the other hand, if the potential on the top and bottom plate 1 is zero it follows from eqs 21a, 21b, and 21c that f 1* nm ) f nm ) 3 2 f nm ) 0 and f nm remains the sole, nonzero, contribution of Coulomb forces to the double-layer force between the two plates. Furthermore, if the charge distribution is uniform and offset by a distance c from the plates, eq 21 reduces to

2488 J. Phys. Chem. B, Vol. 102, No. 14, 1998 ∞ ∞

(


)

kz a2σ02 Fz ) cosh k2c - tanh 2 1 sinh k2c 2 m)-∞ 2

∑

(23)

n)-∞

We note that the magnitude of the double-layer force is effectively increased by the presence of protruding charges. Before we proceed with the numerical example, we remind ourselves that we have defined the double-layer force as being the absolute force and the interaction force as being the gauge force between the two surfaces. The interaction force is found by subtracting the double-layer force at infinite separation from the absolute force. Evaluating the Interaction Force. For the numerical study the lattice size a × a is 1000 Å × 1000 Å, and the area of the unit cell is thus 106 Å2. The sinusoidal varying potential is fixed to Φ ) -0.025 cos(2πx/a) cos(2πy/a) V. Sources of charge are attached to protrusions extending to a σa/2 ) -0.0025 V. To investigate the relative importance of sources of potential combined with sources of charges, we have concentrated on interactions taking place at low ionic strength. In the following evaluation we have assumed a monovalent electrolyte with concentrations of 9.29 × 10-6, 5.80 × 10-3, and 3.16 × 10-2 M. The corresponding values for the dimensionless group κa are 1, 25, and 58.3, respectively. To assess the relative importance of sources of potential and sources of charge on the double-layer force, we will study each contribution to the force one at a time, and in combination. First, we will consider the case of a sinusoidal varying potential distribution on the two plates, while the intermediate charges are set to zero. Second, we will examine the contribution to the double-layer force solely due to sources of charge discretely distributed away from the plate surfaces while the plates themselves are held at zero potential. Third, the double-layer forces will be evaluated when both sources of potential and sources of charge are present. a. Sinusoidal Varying Potential Distribution. The doublelayer force between two plates having a sinusoidal varying potential is obtained by evaluating eq 21 with eqs 21a, 21b, and 21d. Curves of the double-layer force, B Fz, are plotted as a function of displacement and ionic strength for different separation distances in Figure 3, a, b, and c. We find that the double-layer force varies significantly as a function of displacement and is most repulsive when the two plates are displaced a quarter wavelength, d/a ) 0.25. See eq 19a. When the two plates are in direct opposition, or displaced half a wavelength, d/a ) 0.50, the double-layer force is practically zero, a result to be expected when the positive and negative potential contributions from each plate cancel out. As a consequence, the double-layer force appears to be independent of the electrolyte concentration at this displacement. We note in Figure 3a-c that the double-layer force can attain both positive and negative values, depending on separation distance and electrolyte concentration. Generally, sign reversal satisfies the equation

g′ ) 1 + 2

2x2(x - k1/k2) (2x - k1/k2)(1 - x2)

(24a)

where g′ ) cos 2πd/a and x ) tanh-1 k1/k2. For example, when the electrolyte concentration is 5.80 × 10-3 M (κa ) 25), sign reversal takes place at a separation distance between 10.5 and 15.5 Å (see Figure 3b). It follows from eq 24a that sign reversal occurs in the region satisfying the equation

Figure 3. Electrostatic double layer (absolute) force per unit area [N/m2] versus displacement, d/a, for surfaces having a potential distribution Φtop(x,y,z)|z1 ) Φ0 cos(2πx/a) cos(2πy/a) [V] on the top surface and Φbottom(x,y,z)|z1 ) Φtop(x - d,y - d,z) [V] on the bottom surface. There are no discrete charges located at intermediate layers. The unit cell has dimensions a × a (1000 Å × 1000 Å). The double layer force is plotted for separation distances of 10.5 Å (+), 15.5 Å (-), 20.5 Å (*), and 30.5 Å (solid line). (a, top) κa ) 1 which corresponds to an 1:1 electrolyte concentration of 9.29 × 10-6 M, (b, middle) κa ) 25 which corresponds to an 1:1 electrolyte concentration of 5.80 × 10-3 M, and (c, bottom) κa ) 58.3 which corresponds to an 1:1 electrolyte concentration of 3.16 × 10-2 M. d/a range: 0.025 < d/a < 0.975.

(

)

1 - x1 - (k1/k2)2 2 2 tanh-1 e z1 e tanh-1 k1/k2 k2 k1/k2 k2

(24b)

The lower limit corresponds to g′ ) 0, and the higher limit to g′ ) 1; i.e., sign reversal occurs at smaller plate separations as g′ approaches zero and d/a ) 0.25. Sign reversal is triggered by a change in the electrolyte concentration, expressed in λ and embedded in k2 (eq 7b). The presence of discontinuity or sign reversal in double-layer force calculations stems from the definition of gauge and absolute force. Once the interaction


J. Phys. Chem. B, Vol. 102, No. 14, 1998 2489

Figure 4. Electrostatic interaction (gauge) force per unit area [N/m2] versus separation distance, Å, at displacement, d/a ) 0.25. Surfaces have the same potential distribution as in Figure 3 and no discrete charges at intermediate layers. The unit cell has dimensions a × a (1000 Å × 1000 Å). Comparison is made for different values of κa, 58.3 (+), 25 (-), and 1 (solid line). The interaction force is repulsive for all separation distances. Plotted for 10.0 Å < h < 38.0 Å.

force is calculated by subtracting the asymptotic value, the discontinuity cancels out. If the potential is both uniform and zero, the interaction force will also be zero, according to eq 22. Therefore, the near-field repulsive force observed in Figure 4 is due to the heterogeneity of the potential distribution. In our calculations, the heterogeneity of the potential distribution is embedded in coefficients k1 and k2 of eq 7. We note that at close proximity the interaction force is large, but decreases as a function of separation distance. Furthermore, we find that the overall interaction force increases when the periodicity of the potential distribution decreases. This behavior is a direct consequence of the fact that the interaction force is always positive for all displacements. Our results are comparable to the observations made by Kuin,19 who arrived at similar conclusions using a simplified approach. Our results are also in general agreement with DLVO theory, in that the interaction force becomes less repulsive with increasing separation length and decreasing electrolyte concentration. The interaction force decreases monotonically with separation distance with a first-order magnitude proportional to

(1 - g′2)/z

(24c)

which is shown in Figure 4 for the case when the two plates are displaced a quarter wavelength. The interaction force decreases further when modified by displacement and electrolyte concentration according to the following equation:

( )

- 1-

k1 2 k k2 2

(24d)

Short-range repulsive forces have been measured in surface force apparatus experiments and sometimes been attributed to a hydration effect, which is apparently due to strongly bonding surface groups modifying the hydrogen-bonded network of liquid water adjacent to them.19 This view has been challenged and the suggestion made that adsorbed organic impurities are responsible for reversing an attractive interaction force, predicted by DLVO theory, to a repulsive force.20 The results from the calculations presented here suggest a third possibility to explain the presence of short-range repulsive forces. Our calculations indicate that it is the periodic surface potential variations that result in a repulsive interaction force at close proximity,

Figure 5. Electrostatic double layer (absolute) force per unit area [N/m2] versus displacement, d/a, for surfaces having discrete charges at σtop(x,y,z1 - c)/2 ) -0.0025δ(x/a - d/a)δ(y/a - d/a) [V] and σbottom(x,y,c)/2 ) -0.0025δ(x/a - d/a)δ(y/a - d/a) [V] and c ) 10 Å. The potential on both surfaces is set to zero. The unit cell has dimensions a × a (1000 Å × 1000 Å). The double layer force is plotted for separation distances of 15.5 Å (-), 20.5 Å (*), and 30.5 Å (solid line) for (a, top) κa ) 1 (9.29 × 10-6 M), (b, middle) κa ) 25 (5.80 × 10-3 M), and (c, bottom) κa ) 58.3 (3.16 × 10-2 M). d/a range 0.025 < d/a < 0.975.

prompting us to speculate that the origin of the repulsive force may be found on the surface itself. b. Periodic Charge Distribution. The double-layer force between two plates having zero potential with periodic charge distribution at intermediate layers 10 Å from the center of each lattice cell is now evaluated using eqs 21 and 21c. At low electrolyte concentrations the effect of surface irregularities on the double-layer force is maximal. The results of the interaction force calculations for an electrolyte concentration of 9.29 × 10-6 M (κa ) 1) is plotted in Figure 5a. The attractive doublelayer force increases by approximately a factor of 2 when the plates are displaced completely out of register to d/a ) 0.50. Although the results reported here are for an otherwise smooth surface having discrete sources of charge located on protrusions,



Figure 6. Electrostatic interation (gauge) force per unit area [N/m2] versus separation distance, Å, at displacement, d/a ) 0.25. Surfaces have the same charge distribution as in Figure 5 and surface potentials are set to zero. The unit cell has dimensions a × a (1000 Å × 1000 Å). Comparison is made for different values of κa, 58.3 (+), 25 (-), and 1 (solid line). The interaction force is attractive for all separation distances and displacements (except d/a ) 0). Plotted for 11.75 Å < h < 38.0 Å.

the general principle would also apply to the interaction between two molecularly smooth surfaces separated by a thin layer of water containing specifically adsorbed counterions. The importance of surface alignment on adhesive forces between mica sheets has been investigated experimentally. In surface force apparatus experiments the attractive force decreased dramatically when the two mica surfaces were twisted out of alignment.21 Increasing the electrolyte concentration to 5.80 × 10-3 and 3.16 × 10-2 M (κa ) 25 and 58.3) shown in Figure 5, b and c, respectively, reduces this variation in force for all displacement values by masking the presence of protruding sources of charge. Even though increased electrolyte strength also lessens the interaction force, the two surfaces will always attract for all displacement values d/a * 0, shown in Figure 6. Hence, if a strong bond between interacting surfaces is desired, both surface alignment and electrolyte concentration are important considerations. These observation are contrary to DLVO theory which yields a repulsive electrostatic force for a uniformly distributed surface charge. Nevertheless, measured collision efficiency factors for colloid aggregation are orders of magnitude larger than predicted by DLVO theory and also support the notion that attractive electrostatic forces can indeed arise from surface heterogeneities.3,22,23 c. Sources of Potentials and Sources of Charge. The interaction force between two plates having both a sinusoidal varying potential and discretely distributed sources of charge is obtained by evaluating eq 21 with eqs 21a, 21b, 21c, and 21d. The double-layer force for mixed boundary conditions is plotted in Figure 7a-c as a function of separation distance, displacement, and electrolyte concentration. The interplay between sources of potential and sources of charge on the double-layer force is complex. In our numerical example, the overall magnitude of the interaction is controlled by sources of charge, while the variations as a function of relative displacement are ruled by the periodicity of the potential distribution. Specifically, the near-field double-layer force is dominated by the sources of charge which makes the interaction attractive. Even very small scale modifications due to the sources of potential are qualitatively identifiable, especially at low electrolyte concentrations. See Figure 7a. In Figure 7b with 7c we note that the sign reversal manifests itself also for the case

Figure 7. Double layer force per unit area [N/m2] versus displacement, d/a, for surfaces having mixed boundary conditions (potential distribution Φtop(x,y,z)|z1 ) Φ0 cos(2πx/a) cos(2πy/a) [V] on the top surface and Φbottom(x,y,z)|z1 ) Φtop(x - d,y - d,z) [V] on the bottom surface, and discrete charges at σtop(x,y,z1 - c)/2 ) -0.0025δ(x/a - d/a)δ(y/a - d/a) [V] and σbottom(x,y,c)/2 ) -0.0025δ(x/a - d/a)δ(y/a - d/a) [V] and c ) 10 Å). The unit cell has dimensions a × a (1000 Å × 1000 Å). The interaction force is plotted for separation distances of 10.5 Å (+), 15.5 Å (-), 20.5 Å (*), and 30.5 Å (solid line) at different electrolyte concentrations. (a, top) κa ) 1 (9.29 × 10-6 M), (b, middle) κa ) 25 (5.80 × 10-3 M), and (c, bottom) κa ) 58.3 (3.16 × 10-2 M). d/a range 0.025 < d/a < 0.975.

with mixed boundary conditions, but at a somewhat higher electrolyte concentration. However, once the interaction force is calculated by subtracting the asymptotic value of the doublelayer force value, the singularity is hidden from view. In Figure 8 the interaction force is plotted as a function of separation distance, when the two plates are displaced a quarter wavelength, d/a ) 0.25. For this numerical example we find a local interaction force maximum, when the separation distance is between 10 and 20 Å. We also note the absence of such a maximum, when there is no displacement between the two plates (not plotted). At intermediate separation distances, the attractive or repulsive nature of the interaction force is dictated by the



B1mn )

{[

]

1k1 + 2k2 [(3k3 + 2k2)R3 + (3k3 - 2k2)R4] + 22k2

}/

[(1k1 + 2k2)R2 + (3k3 - 2k2)R1 exp[-k2z1]] {(1k1 + 2k2)(3k3 + 2k2) - (1k1 - 2k2)(3k3 - 2k2) × (A2) exp[-2k2z1]} exp[k z ] b/2 a/2 2 j+1 Bj+1mn ) Bjmn σ × 22k2‚ab -b/2 -a/2 j+1 exp[-2iπ(nx/a + my/b)] dxdy for j e N -1 (A3)

∫ ∫

C1mn ) Figure 8. Interaction (gauge) force per unit area [N/m2] versus separation distance, Å, for surfaces having mixed boundary conditions, e.g. the same potential distribution and discrete charge distribution as in Figure 7. The unit cell has dimensions a × a (1000 Å × 1000 Å). Comparison is made for different values of κa ) 25 (-) and κa ) 1 (solid line). At small separations the interaction force is attractive, and at intermediate and large separations the magnitude and sign of the interaction force are dictated by the relative voltage, i.e., charge to potential ratio, electrolyte concentration, and relative displacement. Plotted for 11.0 Å < h < 35.5 Å.

relative voltage, i.e., charge-to-potential ratio, electrolyte concentration, and relative displacement. The numerical example presented here illustrates how the heterogeneity of the surface and the surrounding medium profoundly influences surface interactions. V. Conclusion A closed form solution for the interaction force between two plates having sources of potential (charge) distributed along all three axis has been derived. While we have not obtained any simple equation for the interaction force, we have derived a generalized model that can be used to explore the effects of surface heterogeneities on the interaction force between surfaces at close proximity. The results presented here should have a bearing on a wide range of physical phenomena in both natural and engineered systems, where interfacial properties are important. The derived equations have been used to study the combined effect of sources of surface potential and sources of charge located away from each surface on the interaction force. We have found that surface heterogeneity may produce both attractive and repulsive interaction forces and that the overall interaction is profoundly affected by the charge-to-potential ratio, electrolyte concentration, and relative displacement. Specifically, the periodicity of sources of potential in combination with the location of sources of charge dictates whether a system is overall repulsive or attractive, or both, at varying separation. For a better understanding of interfacial phenomena we need to specifically account for the sources of potential and charge that contribute to the interactions between complex surfaces across thin films. Both surface structure and the intervening media must be characterized at the atomic level in order to predict the interaction force. Appendix A Simple Boundary Conditions. Charge distribution on the two plates and at intermediate layers are provided. The coefficients are

Amn ) B1mn exp[-(k2 - k1)z1] + C1mn exp[(k2 - k1)z1] and Dmn ) BNmn + CNmn

(A1)

C

j+1 mn

R1 )

exp[-k2z1] [R - (1k1 - 2k2)B1mn exp[-k2z1]] 1k1 + 2k2 1 exp[-k2zj+1] b/2 a/2 ) C mn + σ × 22k2‚ab -b/2 -a/2 j+1 exp[-2iπ(nx/a + my/b)] dx dy

∫ ∫

j

(A5)

b/2 a/2 σ1 exp[-2iπ(nx/a + my/b)] dx dy ∫-b/2 ∫-a/2

1 ab

for j e N - 1 R2 ) R3 )

(A4)

(A6)

b/2 a/2 σN+1 exp[-2iπ(nx/a + my/b)] dx dy ∫-b/2 ∫-a/2

1 ab

(A7)

N

1

∑exp[k2zj]∫-b/2 ∫-a/2σj exp[-2iπ × b/2

a/2

ab j)2

(nx/a + my/b)] dx dy 1

N

(A8)

∑exp[-k2zj]∫-b/2 ∫-a/2σj exp[-2iπ × ab j)2

R4 ) -

b/2

a/2

(nx/a + my/b)] dx dy

(A9)

Mixed Boundary Conditions. Potential distribution on the two plates and charge distribution at intermediate layers are provided. The coefficients are

Amn ) exp[k1z1]β1

B1mn )

[ ]

(A10)

1 [R + R4] + [β2 - β1 exp[-k2z1]] 22k2 3 1 - exp[-2k2z1]

(A11)

exp[k2zj+1] b/2 a/2 σ × 22k2‚ab -b/2 -a/2 j+1 exp[-2iπ(nx/a + my/b)] dx dy for j e N - 1 (A12)

Bj+1mn ) Bjmn -

∫ ∫

C1mn ) exp[-k2z1](β1 - B1mn exp[-k2z1]) exp[-k2zj+1] b/2 22k2‚ab -b/2 exp[-2iπ(nx/a + my/b)] dx dy

Cj+1mn ) Cjmn +

(A13)

a/2 σj+1 × ∫ ∫-a/2

for j e N - 1 (A14)

Dmn ) β2

(A15)

R3 ) eq A8

(A16)

R4 ) eq A9

(A17)

b/2 a/2 Φ1 exp[-2iπ(nx/a + my/b)] dx dy ∫-b/2 ∫-a/2

β1 )

1 ab

β2 )

1 ab

(A18)

b/2 a/2 ΦN+1 exp[-2iπ(nx/a + my/b)] dx dy ∫-b/2 ∫-a/2



Appendix B The expression for the force, B F, on a unit cell due to randomly distributed sources of charge, σz, is outlined below. We first consider the boundary condition given in eq 8a, which is

ψ1|z)z1)h ) ψ21|z)z1)h

(B1)

Substituting from eq 6a and 6b for ψ1 and ψ21 in eq B1 yields ∞ ∞

∞ ∞

Anm exp[2iπ(nx/a + my/b)] exp[-klz1] ) ∑ exp[2iπ(nx/a + my/b)]{Bnm1 exp[-k2z1] + Cnm1 exp[k2z1]} ∑ n)-∞ m)-∞

m)-∞

(B2)

n)-∞

The two terms in eq B2 are linearly independent, and it follows that

Anm exp[-k1z1] ) Bnm1 exp[-k2z1] + Cnm1 exp[k2z1]

(B3)

Next, we consider the distribution of charges along the boundary, eq 9a, which is

(

)

∂ψ21 ∂ψ1 - 1 | ) σ1(x,y,z)|z)z1 ∂z ∂z z)z1

2

(B4)

Again substituting from eqs 6a and 6b for ψ1 and ψ21 and taking the derivatives as indicated ∞ ∞

∑

m)-∞ n)-∞

exp[2iπ(nx/a + my/b)]{2k2(-Bnm1 exp[-k2z1] + Cnm1 exp[k2z1]) + 1k1Anm exp[-k1z1]} ) σ1(x,y,z)|z1

(B5)

Multiplying both sides of eq B5 by exp[-2iπ(n′x/a + m′y/b)] and integrating over x and y yields ∞ ∞

{2k2(-Bnm1 exp[-k2z1] + Cnm1 exp[k2z1]) + 1k1Anm exp[-k1z1]}∫-a/2 ∫-b/2exp[2iπ((n - n′)x/a + ∑ m)-∞ a/2

b/2

n)-∞

(m - m′)y/b)] dx dy )

a/2 b/2 σ1(x,y,z)|z ∫-a/2 ∫-b/2

1

×exp[-2iπ(n′x/a + m′y/b)] dx dy

(B6)

Due to orthogonality a/2 b/2 exp[2iπ((n - n′)x/a + (m - m′)y/b)] dx dy ) δn,n′δm,m′ab ∫-a/2 ∫-b/2

which allows eq B6 to then be rewritten as

ab{2k2(-Bnm1 exp[-k2z1] + Cnm1 exp[k2z1]) + 1k1Anm exp[-k1z1]} ) a/2 b/2 σ1(x,y,z)|z ∫-a/2 ∫-b/2

1

exp[-2iπ(n′x/a + m′y/b)] dx dy

(B7)

By repeating the steps above for the remaining boundary conditions in eqs 8 and 9, we then obtain eqs B8 and B9, respectively. The

Bnm1 exp[-k2z1] + Cnm1 exp[k2z1] ) Anm exp[-k1z1] Bnm1 exp[-k2z2] + Cnm1 exp[k2z2] ) Bnm2 exp[-k2z2] + Cnm2 exp[k2z2] Bnm2 exp[-k2z3] + Cnm2 exp[k2z3] ) Bnm3 exp[-k2z3] + Cnm3 exp[k2z3] l BnmN-2 exp[-k2zN-1] + CnmN-2 exp[k2zN-1] ) BnmN-1 exp[-k2zN-1] + CnmN-1 exp[k2zN] BnmN-1 exp[-k2zN] + CnmN-1 exp[k2zN] ) BnmN exp[-k2zN] + CnmN exp[k2zN] Anm ) BnmN + CnmN

}

(B8a)

(B8b)

(B8c)

Electrostatic Interaction Force between Planar Surfaces 2k2[-Bnm1 exp[-k2z1] + Cnm1 exp[k2z1]] + 1k1Anm1 exp[-k1z1] )


∫ ∫

1 ab

a/2

b/2

-a/2 1

-b/2

σ1(x,y,z1) exp[-2iπ(nx/a + my/b)] dx dy

1 2k2[-Bnm exp[-k2z2] + Cnm exp[k2z2]] + 2k2[Bnm exp[-k2z2] - Cnm exp[k2z2]] ) ab 1 2k2[-Bnm3 exp[-k2z3] + Cnm3 exp[k2z3]] + 2k2[Bnm2 exp[-k2z2] - Cnm2 exp[k2z2]] ) ab l 2

2

1

∫ ∫ ∫ ∫ a/2

b/2

-a/2 a/2

-b/2 b/2

-a/2

-b/2

σ2(x,y,z2) exp[-2iπ(nx/a + my/b)] dx dy σ3(x,y,z3) exp[-2iπ(nx/a + my/b)] dx dy

∫ ∫ ∫ ∫

b/2 1 a/2 exp[-k2zN-1] + Cnm exp[k2zN-1]] + 2k2[Bnm exp[-k2zN-2] - Cnm exp[k2zN-2]] ) σ (x,y,zN-1) exp[-2iπ(nx/a + my/b)] dx dy 2k2[-Bnm ab -a/2 -b/2 N-1 b/2 1 a/2 N N N-1 N-1 exp[-k2zN-1] - Cnm exp[k2zN-1]] ) σ (x,y,zN) exp[-2iπ(nx/a + my/b)] dx dy 2k2[-Bnm exp[-k2zN] + Cnm exp[k2zN]] + 2k2[Bnm ab -a/2 -b/2 N N-1

N-1

2k3Anm + 2k2[BnmN - CnmN] )

N-2

∫ ∫

1 ab

a/2

b/2

-a/2

-b/2

N-2

σN+1(x,y,zN+1) exp[-2iπ(nx/a + my/b)] dx dy

}

(B9a)

(B9b)

(B9c)

solutions to the coefficients given in eqs B8 and B9 above are found in Appendix A. The force, B F, can now be evaluated once the magnitude of the electric fields and the charge distribution on the top plate has been determined. From eq 15 we have the relation

(E B1 + B E21)|z)z1 ) -(∇ B ψ1 + ∇ B ψ21)|z)z1 and it follows that ∞ ∞

+∞

+∞

∑ Anm exp[2iπ(nx/a + my/b)]exp[- k1z1][2iπ(nxˆ /a) + 2iπ(myˆ /b) - k1zˆ] + n)-∞ ∑ m)-∞ ∑ exp[2iπ(nx/a + m)-∞

(E B1 + B E21)|z)z1 ) -

n)-∞

my/b)]{(2iπ(nxˆ /a) + 2iπ(myˆ /b))(Bnm1 exp[-k2z1] + Cnm1 exp[k2z1]) + (-k2Bnm1 exp[-k2z1] + k2Cnm1 exp[k2z1])zˆ}} (B10) By using the relations provided in eqs A1 through A9 found in Appendix A, eq B10 can now be expressed in terms of Bnm1 only: ∞ ∞

(E B1 +

B E21)|z)z1

∑ m)-∞

)-

n)-∞

exp[2iπ(nx/a + my/b)] {(42k2Bnm1 exp[-k2z1] + 2R1)(2iπ(nxˆ /a) + 2iπ(myˆ /b)) + 1k1 + 2k2

(-2k1k2(1 + 2)Bnm1 exp[-k2z1] + (-k1 + k2)R1)zˆ}

(B11)

Substituting eq B11 into eq 14 then yields eq 16a ∞ ∞

∑ m)-∞ k

B F ) -ab

n)-∞

1

1

{4iπ(22k2B1nm exp[-k2z1] + R1)(nxˆ /a + myˆ /b) + (-2k1k2(1 + 2)B1nm exp[-k2z1] +

1 + 2k2

b/2 a/2 σ1(x,y,z1) exp[2iπ(nx/a + my/b)] dx dy ∫-b/2 ∫-a/2

(k2 - k1)R1)zˆ}

(16a)

The derivation of eq 16b follows along similar lines to the procedure outlined above. References and Notes (1) Gregory, J.; Wishart, A. J. Colloids Surf. 1980, 1, 313-334. (2) Kallay, N.; Nelligan, J. D.; Matijevic, E. J. Chem. Soc., Faraday Trans. 1983, 1, 65-74. (3) Vaidyanathan, R.; Tien, C. Chem. Eng. Sci. 1991, 46, 967-983. (4) Elimelech, M.; O’Melia, C. R. Langmuir 1990, 6, 1153-1163. (5) Grabbe, A.; Horn, R. G. J. Colloid Surf. Sci. 1993, 157, 375-383. (6) Chapel, J.-P. Langmuir 1994, 10, 4237-4243. (7) Iler, R. K. The Chemistry of Silica; Wiley: New York, 1979. (8) Spitzer, J. J. Colloid Polym. Sci. 1992, 270, 1147-1158. (9) Israelachvili, J.; Wennerstro¨m, H. Nature 1996, 379, 219-225. (10) Kuin, A. J. Faraday Discuss. Chem. Soc. 1990, 90, 235-244. (11) Vreeker, R.; Kuin, A. J.; Den Boer, D. C.; Hoekstra, L. L.; Agterof, W. G. M. J. Colloid Interface Sci. 1992, 154, 138-145. (12) Miklavic, S. J.; Chan, D. Y. C.; White, L. R.; Healy, T. W. J. Phys. Chem. 1994, 98, 9022-9032.

(13) Kihira, H.; Matijevic, E. Langmuir 1992, 8, 2855-2865. (14) Kihira, H.; Ryde, N.; Matijevic, E. J. Chem. Soc., Faraday Trans. 1992, 88, 2379-2386. (15) Vreeker, R.; Kuin, A. J.; Den Boer, D. C.; Hoekstra, L. L.; Agterof, W. G. M. J. Colloid Interface Sci. 1992, 154, 138-145. (16) Kuin, A. J. Faraday Discuss. Chem. Soc. 1990, 90, 235-244. (17) Hiemenz, P. C. Principles of Colloid and Surface Chemistry; Marcel Dekker Inc.: New York, 1986. (18) Lorrain, P.; Corson, D. R. Electromagnetic Fields and WaVes; W. H. Freeman and Co.: San Francisco, 1970. (19) Leiken, S.; Parsegian, V. A.; Rau, D. C. Annu. ReV. Phys. Chem. 1993, 44, 369-395. (20) Dunstan, D. E. Langmuir 1992, 8, 740-743. (21) McGuiggan, P. M.; Israelachvili, J. N. J. Mater. Res. 1990, 5, 22322243. (22) Gregory, J.; Wishart, A. J. Colloids Surf. 1980, 1, 313-334. (23) Kallay, N.; Nelligan, J. D.; Matijevic, E. J. Chem. Soc., Faraday Trans. 1983, 1, 65-74.

Electrostatic Interaction Force between Planar Surfaces Due to 3-D

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