Deformation of Fluid Interfaces Induced by Electrical Double-Layer

The problem of determining the electrical double-layer interaction between a rigid planar surface and a deformable liquid droplet is formulated as a p...
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Langmuir 1996, 12, 4197-4204

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Deformation of Fluid Interfaces Induced by Electrical Double-Layer Forces and Its Effect on Fluid-Solid Interactions D. J. Bachmann and S. J. Miklavcic* School of Applied Physics and Ian Wark Research Institute, University of South Australia, The Levels Campus, The Levels SA 5095, Australia Received January 29, 1996. In Final Form: May 28, 1996X The problem of determining the electrical double-layer interaction between a rigid planar surface and a deformable liquid droplet is formulated as a pair of coupled differential equations. The Young-Laplace equation, describing the shape of the droplet subject to double-layer pressures, is solved numerically, while the linearized Poisson-Boltzmann equation, which describes the double-layer interaction, is solved analytically. Results are provided for the three sets of boundary conditions of constant dissimilar surface potentials, constant dissimilar surface charges, and the mixed case of constant charge on one surface and constant potential on the other. Our principal object of interest is the net force between the surfaces evaluated as the integral of the normal stress tensor over the surfaces. We also provide information on the shape of the droplet interface and the distribution of the normal stress over that interface. Both of these items of information are vital for understanding the complex behavior of the net force. For constant charge surfaces of the same sign, as for the symmetric constant potential case, the results are qualitatively similar to those of our previously published work. For either constant dissimilar potential surfaces, for dissimilar constant charge surfaces, and/or the mixed case, however, we find greater diversity of qualitative features.

Introduction An interest in understanding the behavior of soft colloid systems is easily motivated by the myriad of problems encountered in mineral processing, in the preparation and processing of foods, or even in a vast range of biophysical phenomena. Soft colloids such as “dispersions” of bubbles or emulsion drops or vesicles are prolific in these circumstances, although it is not uncommon to also find mixed systems of fluid and solid particulate dispersions. In view of their abundance, we ask whether it is reasonable to continue to consider soft colloid systems as if the dispersed entities were solid? In this paper we concern ourselves with this question. Our recent work1 on fluid-solid colloidal interactions demonstrated unequivocally that fluid droplets are poorly represented by solid particles. In retrospect, this should have been expected, although what remains surprising is the magnitude of contrast between fluid and solid behavior. Under the assumption that the opposing surfaces are similarly charged, the repulsive double-layer force that results readily deforms the fluid surface. The deformation increases exponentially with decreasing distance under the exponential force until an effective barrier is reached which prevents any closer approach. Under asymptotic conditions it was shown that this limiting separation increases logarithmically with surface charge, decreases inversely with electrolyte concentration, and decreases logarithmically with the ambient pressure difference across the interface. With the combination of high surface charge, low electrolyte concentration, and microscopic to macroscopic droplets, the effective barrier appears at much larger distances than are normally associated with doublelayer stability. Given the nature of the colloidal interaction between fluid drops and solids, it is likely that the * Corresponding author. Present address: Department of Physics, Linko¨ping University, S-581 83 Linko¨ping, Sweden. E-mail: [email protected]. X Abstract published in Advance ACS Abstracts, July 15, 1996. (1) Miklavcic, S. J.; Horn, R. G.; Bachmann, D. J. J. Phys. Chem. 1995, 99, 16357.

S0743-7463(96)00088-1 CCC: $12.00

interpretation of experiments designed to estimate size will be affected; virial coefficients, for example, determined by light-scattering methods are measures of these types of interparticle interactions. The situation examined previously represented the special case of equal constant charge densities on the two opposing surfaces. Our interest now lies in more general situations, with the study of dissimilar interacting surfaces. We therefore consider the different possible limits of surface conditions: constant but dissimilar surface charge, constant but dissimilar surface potential, and constant charge on one and constant potential on the other surface. As was outlined previously,1 the problem is one of solving two differential equations simultaneously. The YoungLaplace equation relates the shape of the drop via its curvature to the difference in the normal stress across the interface. The Poisson-Boltzmann (PB) equation describes the electrical double layer between the two surfaces. If we focus our attention on describing general qualitative behavior, then we can simplify the task by invoking an assumption due to Derjaguin2,3 which advocates replacing the full three-dimensional Maxwell stress tensor generated by two curved objects with the tensor for infinite parallel plates. Further, by utilizing the Debye-Hu¨ckel approximation, whereby the surface charges and/or surface potentials are assumed to be low in magnitude, we avoid the complications associated with solving the nonlinear PB equation for the parallel plate geometry. We have performed calculations on a model of a liquid drop interacting with a plane solid surface through electrical double-layer forces. The equations and approximations described above are employed with arbitrary potential and/or charge boundary conditions. Inasmuch as we know that qualitatively different double-layer forces between solids can be found for different surface (2) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: Oxford, 1993. (3) Derjaguin, B. V. Kolloid Z. 1934, 69, 155.

© 1996 American Chemical Society

4198 Langmuir, Vol. 12, No. 17, 1996

Bachmann and Miklavcic

conditions,4-8 we would expect that an equal variety of characteristics exists for fluid interfaces. In fact, fluid interfaces exhibit a greater variety; we find cases where qualitatively similar forces between solids, even though of similar magnitude, can correspond to qualitatively different forces between fluid and solid. The reason lies in the ease with which fluid interfaces can deform.1 What has become clear to us in the course of this work is that the issue of thermomechanical stability of fluid disperions with respect to fluctuations of surface shape, charge, or tension is in need of being addressed. In a similar regard, we admit to leaving aside for the moment the question of the stability of the fluid drop involved in this study under the action of the supposed colloidal forces. Determination of the Droplet Profile The Laplace pressure in the liquid drop is dependent on the meniscus curvature and the surface tension.

∆PL ) γ(κ1 + κ2)

(1)

where κ1 and κ2 are the principal curvatures calculated in two orthogonal planes: constant r and constant θ. As suggested by Figure 1a, the curvature of the drop is not constant but is a function of r (the profile being symmetric about the z-axis (through the apex)). Expressions for κ1(r) and κ2(r) are

κ1(r) )

κ2(r) )

z′′(r) (1 + z′(r)2)3/2 z′(r) r(1 + z′(r)2)1/2

In the absence of gravity, or when the densities of the droplet and the bulk phases are equivalent, a droplet in a macroscopic fluid will be spherical with radius R so that the Laplace pressure is 2γ/R. In other cases, the Laplace pressure balances all other pressures acting on the droplet. A free droplet sessile under gravity on the end of a capillary tube in the presence of a continuous water phase is influenced only by the gravitational pressure head difference produced via the manometer arrangement illustrated in Figure 1b. Incorporated into this gravitational pressure is a corrective term which accounts for the variation in the gravitational pressure across the curved surface of the droplet meniscus. The correction considers the weight of macroscopic fluid extending from the height of the droplet apex down to the surface of the droplet and is therefore an implicit function of the droplet profile. At equilibrium:

γ(κ1 + κ2) ) R2z(r) - Papp

(2)

where R2 ) (Ffluid - Fwater)g is the correction factor with the macroscopic liquid phase denoted as water and the droplet as fluid. Substitution of the κ1(r) and κ2(r) expressions into eq 2 provides us with a differential equation which can only be solved numerically,

[

γ

z′(r)

z′′(r)

(1 + z′(r)2)3/2

+

]

r(1 + z′(r)2)1/2

) R2z(r) - Papp

(3)

In general, the most commonly found surface forces which act between two surfaces are the electrical double-layer, van der (4) Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1993, 155, 297. (5) Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1993, 161, 260. (6) Carnie, S. L.; Chan, D. Y. C.; Stankovich, J. J. Colloid Interface Sci. 1994, 165, 116. (7) Carnie, S. L.; Chan, D. Y. C.; Gunning, J. S. Langmuir 1994, 10, 2993. (8) McCormack, D.; Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1995, 169, 177.

Figure 1. (a) Although in principle the calculations described in this paper are independent of the precise origin of the value of Papp, assuming only that it is constant, we present a realistic experimental example in which Papp can be obtained. This figure shows schematically how in our experimental apparatus a liquid drop is positioned sessile under gravity on the end of a capillary tube. The figure describes the pressures acting on the droplet in the presence of a continuous water phase. Computationally, the pressure Papp is the sum of the pressures shown above. Papp is approximately the gravitational pressure head difference in the liquid at the apex of the drop meniscus plus the weight of water of depth h1 on the droplet. Patm is the atmospheric pressure at the air-water interface, taken to be the reference level. It is evident that there is an additional pressure contribution corresponding to the weight of air of depth h2 on the liquid in the reservoir, namely Fairgh2, but this is negligibly small and can be omitted. Away from the apex of the drop, the gravitational pressure increases with the increasing depth of the drop surface, as described by the pressure R2z, where R2 ) (Ffluid - Fwater)g and z is the distance from the apex to any point on the profile. This pressure term can also be applied to the air-liquid interface in the reservoir, but the meniscus here is much flatter, so that the pressure contribution from the curved edges of this meniscus is negligible. (b). The droplet is modeled in cylindrical coordinates where the axis of symmetry is coincident with the z-axis. Note the separation between the drop and the adjacent solid wall is an implicit function of the radial distance from the apex. Waals, hydration, and hydrophobic forces.9 Of these, our understanding is currently limited to the electrical double-layer and van der Waals forces. Insofar as knowing their origin, we are able to predict the behavior of these forces, once the properties of the two interacting bodies are known. The remaining two types of forces are still not well characterized, so we are yet to ascertain their relevance to our system. We therefore focus our attention once more on the electrical double-layer force, extending our previous analysis1 to asymmetric double layers. van der Waals forces are significant at small separations, especially under high salt conditions. However, we leave aside this contribution until a better understanding of electrical double-layer forces is achieved. The forces which influence the droplet contribute to the total disjoining pressure in the thin film between the surfaces. Account of these surface pressures is made by including representations of the pressure across the surface of the droplet in the description of the profile, (9) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: San Diego, CA, 1992.

Deformation of Fluid Interfaces

[

γ

]

z′(r)

z′′(r)

(1 + z′(r)2)3/2

+

r(1 + z′(r)2)1/2

Langmuir, Vol. 12, No. 17, 1996 4199 Ψ(z1) ) Ψ1, Ψ(z2) ) Ψ2

) R2z(r) + Π(r) - Papp (4)

where we have denoted the disjoining pressure by Π(r). To be more precise, at any point of the droplet interface, the Maxwell stress tensor contributes to the difference between the normal components of stress across the interface, which must be balanced by the normal component of the surface tensile force. Π in eq 4 is, in principle, the normal component of this stress tensor contrived from the electrical double-layer interaction. In practice, it is quite an involved calculation to determine Π, because of the finite and changing curvature of the droplet. However, it has been shown2,9 that for radii large compared to the separations involved, it is reasonable to apply the Derjaguin assumption whereby the stress tensor between curved objects is replaced with the tensor for plane parallel surfaces. This is an assumption we invoke here; the disjoining pressure at a point (r,z) on the surface is taken to be the value of the disjoining pressure between plane surfaces at an equivalent separation

Substituting these boundary conditions into eq 6 gives expressions for the coefficients A and B

A)

Ψ2 - Ψ1e-κD eκD - e-κD

d2Ψ ) κ2Ψ dz2

(5)

Ψ(z) ) Aeκz + Be-κz

(6)

Ψ(z) )

(

eκD - e-κD

-dV dD

(7)

The potential energy due to the approach of the two surfaces is determined by the free energy of formation of a double layer,2,10 the surfaces being brought together from an infinite separation to a separation D.

V ) ∆F(D) - ∆F(∞)

(8)

The free energy terms require knowledge of both the charge and the potential on each surface. These can be deduced from eq 9 and the appropriate boundary conditions. The surface charge is related to the potential gradient in the usual manner:

σ ) -

∂Ψ ∂z

(9)

which is used to provide expressions for surface charge at the boundaries of the system.

∂Ψ σ1 ) - | ∂z z1

and

∂Ψ σ2 ) + ∂z z2

(10)

eκz +

)

Ψ1eκD - Ψ2 eκD - e-κD

e-κz

(11)

For constant (dissimilar) potential surfaces, the free energy of formation of an electrical double layer is10

∆F(D) )

-(σ1Ψ1 + σ2Ψ2) 2

(12)

Substituting the boundary conditions and eq 11 into eq 12 yields

∆F(D) )

κ [4Ψ1Ψ2 - (Ψ12 + Ψ22)(eκD + e-κD)] 2(eκD - e-κD)

Taking the limit of D approaching infinity, we get the free energy of the two isolated surfaces

is given by

Π(D) )

) (

Ψ2 - Ψ1e-κD

∆F(∞) )

where Ψ(z) is the potential at the distance z from a nominated surface, κ is the Debye-Hu¨ckel parameter, and A and B are constants determined by the boundary conditions. We are interested in three possible boundary conditions: (i) constant potential Ψ1, Ψ2 with Ψ1 * Ψ2; (ii) constant charge σ1, σ2 with σ1 * σ2; and (iii) constant potential and constant charge Ψ1, σ2 or σ1, Ψ2, where subscripts refer to surfaces 1 (at z1 ) z(r)) and 2 (at z2 ) z(r) + D(r)). The double-layer pressure is determined from the derivative of the free energy potential with respect to the separation between the surfaces, viz.:

eκD - e-κD

which for the double-layer potential give

D(r) ) D0 - z(r) We utilized this same assumption in our previous work.1 In line with the above comments, we once again consider the electrical double layer between two charged planar surfaces, separated by a dielectric medium of thickness D and permittivity . The solution to the linear PB equation

Ψ1eκD - Ψ2

B)

-κ (Ψ12 + Ψ22) 2

The potential energy of interaction due to the approach of the two surfaces is (from eq 8)

VΨ ) ∆F(D) - ∆F(∞) )

κ [(Ψ12 + Ψ22)(1 - coth κD) + 2 2Ψ1Ψ2 csch κD] (13)

And, finally, applying eq 7 for the expression for the doublelayer pressure between two planar surfaces of constant, dissimilar, surface potential, we have

Π(D) )

-dVΨ -κ2 ) [(Ψ12 + Ψ22) csch κD dD 2 sinh κD 2Ψ1Ψ2 coth κD] (14)

(ii) Case of Constant Dissimilar Surface Charge σ1, σ2 with σ1 * σ2. The boundary conditions in this case are

σ(z1) ) σ1, σ(z2) ) σ2 Since the charge is related to the potential gradient as in eq 9, the derivative of eq 6 is used to determine the coefficients A and B. Proceeding in the same manner as before yields

Ψ(z) )

1 ((σ1e-κD + σ2)eκz + (σ1eκD + σ2)e-κz) κ(eκD - e-κD) (15)

For surfaces of constant (dissimilar) charge, the free energy of formation of an electrical double layer is10

∆F(D) )

(σ1Ψ1 + σ2Ψ2) 2

(16)

and the interaction potential is

(i) Case of Constant Dissimilar Surface Potential Ψ1 Ψ2 with Ψ1 * Ψ2. The boundary conditions in this case are

Vσ ) ∆F(D) - ∆F(∞) )

(10) Verwey, E. J. W.; Overbeek, J. Th.G. Theory of The Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (11) Handbook of Physics and Chemistry, 51st ed.; Chemical Rubber Company: Cleveland, OH, 1970.

The double-layer pressure between two planar surfaces of constant, dissimilar, surface charge becomes

1 [(σ2 + σ22)(coth κD - 1) + 2κ 1 2σ1σ2 csch κD] (17)

4200 Langmuir, Vol. 12, No. 17, 1996 Π(D) )

Bachmann and Miklavcic

1 -dVσ ) [(σ2 + σ22) csch κD + dD 2 sinh κD 1 2σ1σ2 coth κD] (18)

(iii) Case of Constant Surface Potential on “Plate” 1 and Constant Surface Charge on “Plate” 2: Ψ1, σ2. The result here will be applicable to the case where potential and charge are interchanged. The boundary conditions in this case are

Ψ(z1) ) Ψ1, σ(z2) ) σ2 For a constant charge surface interacting with a constant potential surface, the electrical double-layer free energy of formation is10

(-σ1Ψ1 + σ2Ψ2) 2

∆F(D) )

(19)

Again using similar reasoning we get

VΨσ ) ∆F(D) - ∆F(∞) )

(σ22 - κ22Ψ12) (tanh κD - 1) + 2κ σ2Ψ1 sech κD (20)

for the interaction free energy, while for the double-layer pressure between the two surfaces we have

Π(D) )

(

(κ22Ψ12 - σ22) -dVΨσ 1 ) sech κD + dD cosh κD 2

)

κσ2Ψ1 tanh κD

∫ Π(D(r))r dr ∞

0

R)

2γ Papp - Π(D0)

which suggests that, for repulsive double-layer pressures approaching the pressure head difference, the droplet profile about the apex will tend to zero curvature. This will be confirmed by the numerical examples in the next section. Unfortunately, there are difficulties associated with the experimental measurement of the deformation of a drop or bubble as a function of separation, for example in the atomic force microscope (AFM), where there is presently no means for obtaining any information about the bubble surface. With these constraints in mind, there is little value in us presenting quantitative self-consistent data on the normalized force taking correct account of deformation of the liquid interface at the apex. Instead, for our force normalization we adopt the undeformed drop radius. That is, the mean radius of curvature at the apex of the drop when at infinite separation from the solid wall (i.e. without double-layer interaction). The undeformed radius is simply

R∞ )

2γ Papp

Comparisons between the interaction free energy for planar surfaces and the force between the fluid drop and the plane solid normalized with R∞ are made in the following section.

Results and Discussion (21)

It can be shown that this expression holds for the alternate case of constant charge on plate 1 and constant potential on plate 2, simply by interchanging the subscripts. The total surface force acting on the droplet, or equivalently the force on the plane solid surface, can be determined once the self-consistent pressure profile which shapes the droplet is known: the simplest course is to numerically integrate the doublelayer pressure (stress tensor) over the surface of the solid plane, which we have done using the extended trapezoidal rule. This procedure follows the form

F ) 2π

radius1 is defined by

(22)

where r is the radius measured from the apex of the droplet. Although the analytic expressions for the double-layer pressure derived earlier depend on the separation between the planes, the planar separation is related to this radius via

D(r) ) D0 - z(r) which must be determined numerically. The number of points used in the range of the integral in eq 22 depends on the step size determining the precision needed to generate an accurate droplet profile. In practice, the force is calculated over a range limited by the apex (lower limit at r ) 0) and a finite maximum radial distance from the apex corresponding to an interfacial separation, D, of 15 Debye lengths. At this outer extreme, the double-layer pressure is negligible. In its traditional form, the Derjaguin approximation3 states that for a sphere of radius R near a flat surface one can relate the total surface force to the interaction free energy between two planar surfaces by normalizing with respect to the sphere radius,

F )V 2πR In fact, for any curved solid surface, not necessarily spherical, this expression remains valid provided that one employs the mean surface curvature.2 For the system under study, the appropriate variable to use is the radius of curvature at the apex. Furthermore, if account is to be taken of the deformation of the liquid drop under surface forces, then naturally a self-consistently varying curvature radius would seem correct. The deformed

In this section, we shall first devote attention to interpreting the behavior of the net force between the surfaces as a function of their minimum separation. According to earlier statements, understanding the force depends on knowing the behavior of the double-layer pressure between parallel charged plates and subsequently the pressure distributed over the self-consistent profile of the fluid-water interface. Thus, although generally well-known, the double-layer pressure dependence on the separation between planes (eqs 14, 18, and 21) is included here as Figures 2a, 3a, and 4a and b for convenient reference. Our discussion is also aided by graphic particulars of the self-consistent profile shape under corresponding conditions (parts b or c of the figures). The range of conditions to be explored are the three cases outlined above, namely, constant dissimilar surface potential, constant dissimilar surface charge, and the mixed case of constant potential on one surface and constant surface charge on the other. However, our emphasis is on dissimilar surface charge conditions, so that we may complement our initial investigation,1 which covered the case of symmetric systems. We therefore only show one symmetric case for each of constant surface charge and potential. In the systems of dissimilar surfaces, only the set of constant charge cases explicitly includes values of opposite sign. In the case of constant dissimilar potential, opposite surface charge conditions occur naturally. Figure 2 describes the behavior of a system with constant dissimilar surface potential. When the surface potentials are identical in sign and magnitude (in Figure 2a this is the case of -50 mV), the double-layer pressure is always repulsive. However, a sufficient difference in magnitude in the surface potentials can lead to a negative pressure, i.e. an attractive force, between the surfaces for sufficiently small separations. This is despite the fact that the surface potentials are of the same sign. For example, in Figure 2a the case of surface potentials of -10 and -50 mV gives rise to an attractive force at separations below 45 nm. Below such transition separations there occurs a reversal in the sign of the surface charge on the body having the lower potential. This charge

Deformation of Fluid Interfaces

Figure 2. Graphs describing the interaction between a liquid droplet and a planar, rigid solid with constant dissimilar surface potentials, separated by a thin aqueous film. The concentration of the aqueous solution is 0.125 mM, and the temperature is 25 °C, which gives a Debye length of 27.48 nm. Where relevant, the applied pressure is 500 Pa. The surface tension of the liquid is 375 mN/m, corresponding to mercury.11 These constants are applicable to all subsequent figures: (a) double-layer pressure as a function of planar separation in decreasing order of repulsion (-50 mV, -50 mV; -50 mV, -30 mV; -50 mV, -20 mV; -50 mV, -10 mV); (b) droplet profiles with varying minimum separations for the constant dissimilar potential case of (-10 mV, -50 mV); (c) distribution of the double-layer pressure over the surface of the droplet for various minimum separations for the constant dissimilar potential case of (-10 mV, -50 mV); (d) total surface force as a function of minimum separation for the set of constant surface potentials used in (a) (inset: logarithmic scale).

Langmuir, Vol. 12, No. 17, 1996 4201

reversal is the result of that surface accumulating, in this case, positive charge in order to maintain its constant surface potential condition. The qualitative difference between symmetric and asymmetric systems leads naturally to different features in the droplet shape at different minimum separations. Instead of a progressive flattening, which occurs for the symmetric case, we see in Figure 2b the distinctive shape found for the particular set of constant dissimilar potentials of -10 and -50 mV. At 100 nm, the profile bears a normal convex shape. However, as the separation decreases and the surfaces approach contact, the drop profile elongates, or extrudes, at the apex of the droplet. Referring to Figure 2a, this extrusion of the droplet is the result of the attractive force (negative pressure) at the apex, which decays away from the center, eventually becoming weakly repulsive. The elongation commences when the apex is less than 45 nm from the adjacent surface and becomes more pronounced as the attractive force increases with decreasing minimum separation. It should be mentioned that the elongation and curvature in general have been exaggerated in the figures by the choice of the scaling. It is questionable whether these double-layer interactions would have any directly visible effect on the macroscopic droplet shape. Naturally, over the surface of the drop the double-layer pressure varies according to the results shown in the previous two graphs. At 100 nm minimum separation, there is repulsion over the entire interface, with a maximum at the apex. When the surfaces are separated by a minimum distance of 50 nm, the surfaces lie between the charge reversal point at 45 nm and the distance where the pressure attains its maximum (∼60 nm). Thus, the pressure at the apex is less (but still repulsive) than the peak value found some distance away from the apex on the droplet’s convex shape. For the minimum separations of D0 ) 8 nm and D0 ) 2 nm, the pressure around the apex is attractive while on the remainder of the drop the pressure is repulsive: charge reversal has taken place over that part of the droplet within 45 nm from the adjacent surface. Although not visible on the scale of Figure 2c, there is a very strong attractive force at the apex (a large negative pressure of the order of -100 kPa). Integrated pressure curves (eq 19) corresponding to the four constant potential profiles depicted in Figure 2a are shown in Figure 2d. For the symmetric case of -50 mV, the double-layer pressure is repulsive at all separations over the entire drop surface. We have found before with constant identical charge surfaces (see also below) that the net surface force increases exponentially at first, until the appearance of a barrier which effectively prevents any further approach of the interacting bodies.1 This is also the case here when Papp is comparable with the maximum in the disjoining pressure. With significant asymmetry in surface potential, however, this limit in minimum separation does not occur. Although all the repulsive forces exhibit a maximum, these maxima decrease with increasing asymmetry. In the case of greatest asymmetry, but still with the same sign (-10 and -50 mV), there is a net attractive surface force present (Figure 2d). However, this occurs at a minimum separation of almost 1 Debye length, as repulsive contributions to the integral come virtually from over the entire droplet excluding the region near the apex where negative pressures are experienced. At very small separations these negative pressures become large enough to dominate the smaller positive contributions at other points on the droplet. For the case of constant surface charges (Figure 3) of the same sign but of different magnitude, the drop profile, the pressure profile, and the net force behavior are

4202 Langmuir, Vol. 12, No. 17, 1996

Bachmann and Miklavcic

Figure 3. Graphs describing the interaction between a liquid droplet and a planar, rigid solid with constant dissimilar surface charges, separated by a thin aqueous film: (a) double-layer pressure as a function of planar separation in order of decreasing repulsion (-1.0 mC/m2, -1.0 mC/m2; -1.0 mC/m2, +0.1 mC/m2; -1.0 mC/m2, +0.5 mC/m2; -1.0 mC/m2, +1.0 mC/m2); (b) droplet profiles with varying minimum separations for the constant symmetric charge case of (-1 mC/m2, -1 mC/m2); (c) distribution of the double-layer pressure over the surface of the droplet for various minimum separations for the constant symmetric charge case of (-1 mC/m2, -1 mC/m2); (d) droplet profiles with varying minimum separations for the case of (-1 mC/m2, +0.5 mC/m2); (e) double-layer pressure at varying minimum separations for the case of (-1 mC/m2, +0.5 mC/m2); (f) total surface force as a function of minimum separation for the set of constant surface charges used in (a).

qualitatively similar to our earlier findings1 as well as to the case of equal constant potentials, above. The doublelayer pressure is monotonically repulsive and decreases exponentially with distance (Figure 3a). The charge reversals suggested in Figure 2a are not possible, and the barrier eventually appears as the disjoining pressure diverges with vanishing separation. On the other hand with charges of opposite sign, the double-layer pressure is nonmonotonic. In fact, these are the reverse of the dissimilar potential cases: these are attractive at large separations and repulsive at short separations. Only when the charges are exactly equal and opposite is there a monotonic attraction. The behavior of the droplet profile and the surface pressure distribution are by now predictable, though nonetheless interesting (Figure 3b-e). Surfaces of equal sign induce progressive flattening; oppositely charged fluid surfaces first elongate and then flatten, except for the case of equal and opposite charge, for which only elongation results. The overall attractive contribution is, however, not terribly significant unless the magnitudes of the two opposite surface charges are

similar. And, understandably, the net force between the drop and the solid reflects the trend in the surface pressure. Again, only in the case of exact equal and opposite constant surface charge is there no barrier (Figure 3f). Finally, we consider in Figure 4 the mixed case of constant potential on one surface and constant charge on the other. Fixing the charge on one surface and varying the potential on the other (Figure 4a) results in similar qualitative responses to those of the alternative of fixing the surface potential and varying the surface charge (Figure 4b): an exponential repulsion at large separations which attains a maximum value at some separation and then decreases to a finite value at surface contact. Increasing the magnitude of the asymmetry shifts the maximum to larger separations and increases the attraction, which, incidentally, corresponds to reversing the sign of the surface charge on the constant potential surface. With the values taken for these mixed cases, the deformation is minimal and qualitative trends in the net forces, shown in Figure 4c, appear quite consistent with the features of Figure 4a and b.

Deformation of Fluid Interfaces

Langmuir, Vol. 12, No. 17, 1996 4203

Figure 5. Graphs comparing the normalized net surface force between the droplet and a plane wall with the interaction free energy between two planar surfaces. The arcsinh scale is used to allow representation of both attractive and repulsive forces and energies. The force curves are adapted from Figure 2. In this instance, constant dissimilar surface potentials are assumed. The open squares and solid line represent the symmetric case of -50 mV on each surface. The solid triangles and long dashed line represent the case of -50 mV on one surface and -20 mV on the other. The open circles and dot-dashed line represent the case of -50 mV on one surface and -10 mV on the other. The salt concentration is 0.125 mM, Papp ) 500 Pa, and the surface tension is 375 mN/m.

Figure 4. Graphs describing the interaction between a liquid droplet and a planar, rigid solid with constant surface charge on one surface and constant surface potential on the other, separated by a thin aqueous film: (a) double-layer pressure as a function of planar separation for a fixed surface charge and various constant surface potential; (b) double-layer pressure as a function of planar separation for a fixed surface potential and various constant surface charges; (c) total surface force with varying minimum separations for the asymmetric cases -10 mV, -0.1 mC/m2; -10 mV, -0.2 mC/m2; -10 mV, -1 mC/ m2; -10 mV, -10 mC/m2).

In the remainder of the figures we collate the results for the net forces in the different cases, normalized with respect to the radius of curvature of the undeformed drop. These are now in a form to compare, assuming the Derjaguin approximation, with the double-layer interaction free energy between plane solid interfaces. Derjaguin’s argument states3 that, for small enough separations between two convex bodies, the behavior of F/2πR is independent of curvature and in fact equals the V(D0) of eqs 13, 17, and 20. The smaller the separation, the better the approximation. One should keep in mind, throughout the comparisons made in Figures 5-7, that the double-layer interaction free energy reflects the qualitative behavior of the disjoining pressure between planes. The choice of normalizing the forces with the undeformed drop radius, rather than with the self-consistent mean radius of the deforming drop, is motivated above all by practical considerations. In experimental methods

Figure 6. Graphs comparing the normalized net surface force between the droplet and a plane wall with the interaction free energy between two planar surfaces. The arcsinh scale is used to allow representation of both attractive and repulsive forces and energies. The force curves are adapted from Figure 3. In this instance, constant dissimilar surface charges are assumed. The open squares and solid line represent the symmetric case of -1.0 mC/m2 on each surface. The solid triangles and long dashed line represent the case of -1.0 mC/m2 on one surface and 0.5 mC/m2 on the other. The open circles and dot-dashed line represent the case of -1.0 mC/m2 on one surface and 1.0 mC/m2 on the other. The salt concentration is 0.125 mM, Papp ) 500 Pa, and the surface tension is 375 mN/m.

currently geared to measuring surface forces between solids and fluids, such as the atomic force microscope,12-14 it is difficult to normalize with anything but the undeformed radius. At this stage we are as yet unconvinced that to do otherwise would prove pedagogically beneficial. Normalizing with R∞ at least retains only the dependence of F on separation. Departures from V are due then to the behavior of F as opposed to the behavior of the convolution of two functional dependencies. In essence, only when the disjoining pressure is weak or, alternately, when the deformation is small are we assured agreement with V. As one expects, these conditions are always met at sufficiently large separations, and Figures 5-7 confirm this. The system of -10 mV and (12) Ducker, W. A.; Xu, Zh.-H.; Israelachvili, J. N. Langmuir 1994, 10, 3279. (13) Butt, H.-J. J. Colloid Interface Sci. 1994, 166, 109. (14) Mulvaney, P.; Perera, J.; Biggs, S.; Grieser, F.; Stevens, G. J. Colloid Interface Sci., in press.

4204 Langmuir, Vol. 12, No. 17, 1996

Figure 7. Graphs comparing the normalized net surface force between the droplet (symbols) and a plane wall with the interaction free energy between two planar surfaces (lines). The arcsinh scale is used to allow representation of both attractive and repulsive forces and energies. The force curves are adapted from Figure 4c: (open squares and solid line) -0.1 mC/m2, -10 mV; (solid triangles and long dashed line) -1.0 mC/m2, -10 mV; (open circles and dot-dashed line) -10.0 mC/ m2, -10 mV. The salt concentration is 0.125 mM, Papp ) 500 Pa, and the surface tension is 375 mN/m.

-0.1 mC/m2 surfaces found in Figure 7 is one case in which the criteria appear to hold necessarily over all separations: the deviation of F/2πR∞ from V being negligible. We can conclude that Derjaguin’s approximation can be trusted under low deformation conditions, for fluid and solid bodies alike. Carnie, Chan, and Gunning7 have reported that the Derjaguin approximation quite accurately describes the electrical double-layer interaction between rigid spheres or between a rigid sphere and a plane, at separations which are small compared to the mean radius of curvature of the two bodies. On the basis of Figures 5-7, this can no longer be asserted to hold with confidence for fluid interfaces. What is more, the stronger the double-layer interaction, whether attractive or repulsive or nonmonotonic, the greater the deviation from the theoretically equivalent planar interaction. Certain circumstances, for example the constant potential case shown of one surface at -50 mV and the other at -20 mV, can even result in an interaction which differs qualitatively from the planar force behavior. It is important to note that in such circumstances normalizing with respect to the selfconsistent radius does not correct for the qualitative divergence. Thus, the above conclusions are not altered by the fact that we use the undeformed radius. We have also shown previously1 that, even under the straightforward case of flattening under monotonic repulsion,

Bachmann and Miklavcic

normalizing the total force with the separation-dependent radius is not always sufficient to compensate for the departure from Derjaguin’s approximation. On a much more general level, one should appreciate that, for a given droplet size, reasonably small differences in surface conditions (especially in the case of constant surface potential) can be quite decisive in demarcating fluid-colloid stability and instability. As the presence or absence of the barrier is determined by the relative magnitudes of Papp and Π, there is reason to wonder what distinctive macroscopic behavior might eventuate for marginally different surface conditions. Consider, for example, the recent observations of Craig et al.15 on bubble coalescence in different electrolytes. It would be of interest to establish (possibly by means of the atomic force microscope) exactly what surface potentials are found on bubble interfaces in those different electrolytes and whether different size bubbles have different potentials in similar environments. We might then be in a better position to remark on whether the observations of bubble coalescence or alternatively stability can be reconciled with what can be deduced from our findings. Conclusions By calculating the electrical double-layer interaction between a deformable fluid interface and a planar, rigid solid, assuming dissimilar surface conditions, we find that the fluid interface shape and the net force between the two bodies exhibit a variety of qualitative behavior distinctive for each of the representative cases. In comparison with the interaction free energy between planes, the normalized forces demonstrate the failure of the Derjaguin approximation, which normally justifies such a comparison. As quite different qualitative force behavior is possible under marginally different electrical double-layer conditions, we are at a point where we must consider how very slight variations in potential or pressure, or even the presence of surface capillary wave-type disturbances, could lead to changes in the overall stability of fluid dispersions. Acknowledgment. We are grateful for the financial support of the Australian Research Council. We extend our sincere gratitude also to Professor R. G. Horn, who continues to be involved with our research efforts on fluidsolid interactions and has been a great source of inspiration and enthusiasm. LA960088N (15) Craig, V. S. J.; Ninham, B. W.; Pashley, R. M. J. Phys. Chem. 1993, 97, 10192.