12083
J . Phys. Chem. 1993,97, 12083-12086
Ionic Size Effects on the Force between Planar Electrical Double Layers Scott E. Feller and Donald A. McQuarrie' Department of Chemistry and Institute of Theoretical Dynamics, University of California, Davis, Davis, California 95616 Received: July 6, 1993; In Final Form: September 10, 1993'
The effect of varying ionic size on the force between planar electrical double layers is studied for several systems with different surface potentials and ionic concentrations, through the solution of the three-point extension hypernetted chain/mean spherical approximation theory. We include conditions modeling clay particle swelling pressure experiments that have known a strong dependence on the choice of counterion. Our results show that the pressure between charged surfaces does depend on the ionic size, though they do not explain the clay swelling experiments. The increased repulsive force at small separations for the finite sized ions over the point ions is so great that the addition of an attractive van der Waals component, as is common in the colloid literature, does not result in a nonmonotonic force vs distance curve even at small separations.
Introduction
The structure of an electrolyte solution at its interface with a charged surface, the electrical double layer, and the interaction between two electrical double layers are classical problems of colloid science, having been studied theoretically for some 50 years, beginning with the work of Verwey and Overbeek.' This theory has enjoyed great success in explaining many properties of colloidal dispersions. More recently experiments have shown that under some conditions, such as small surface separation2or the presence of divalent ions? the Verwey-Overbeek(VO) theory cannot explain certain experimental observations. Additionally, Monte Carlo computer simulations of the primitive model electrical double layer have shown that the Poisson-Boltzmann equation, on which the VO theory is based, can be in error under these same conditions as well as for concentrated monovalent ele~trolytes.~~s Monte Carlo simulations have also shown that interacting double layers can be described much more accurately using one of the integral equation theories developed in the statistical mechanics of fluids. For the electrical double layer the most successful class of integral equation theories is based on the hypernetted chain (HNC) a p p r o ~ i m a t i o n . ~We ~ ~ have previously solved the three-point extension hypernetted chain/ mean spherical approximation (TPE HNC/MSA) of LozadaCassou et al. by a convenient variational method.* In the present work we will extend our earlier results to compare systems with different values for the ionic size. We hope to show that subtracting the calculated electrical double-layer force from experimentalmeasurement and ascribingthe differenceto solvent based effects may be in error if the point ion theory is used.
Theory The derivation of the TPE HNC/MSA theory has been presented in refs 7 and 9. Briefly, the TPE HNC/MSA theory treats the surface as a hard smooth planar surface of infinite extent with a uniformly distributed charge density and a fixed electrostatic potential. The solvent is taken to be a dielectric continuum described solely by its dielectric constant, e. The dielectric constant of the surface material is assumed to be the same as that of the solvent, thus image forces are neglected. The ions are treated as charged hard spheres of equal diameter, a, with point charges, z+ and z-,embedded in their centers. Though the TPE HNCf MSA theory could be extended to ions of unequal diameter, this is unnecessary for studying the problem of the electrical double layer, at least at moderate and high surface 0
Abstract published in Aduance ACS Abstracts, November 1, 1993.
potentials, since the number of counterions near the surface is so much greater than the number of co-ions. Thus the size of the ions is always taken to be that of the counterion. The TPE HNC/ MSA is a hybrid theory since the wall-ion interactions are approximated by the hypernetted chain approximation and the ion-ion interactions are taken to be those of the mean spherical approximationfor the bulk electrolyte. This combinationgreatly simplifies the solution to the TPE HNC/MSA theory since the MSA has an analytic solution. Additionally, this hybrid method has been used previously with excellent results for bulk electrolytes and isolated electrical double layers of spherical, cylindrical, and planar geometries. The TPE HNC/MSA theory allows the calculation of the ionic structure within a pore formed by the two parallel surfaces. In the limit of zero ionic diameter the TPE HNC/MSA theory reduces to the Poisson-Boltzmann equation, which forms the basis of the Verwey-Overbeek theory of interacting double layers. The pressure between the two plates is given in terms of the double layer structure by941 n
n j=I
where a0 is the surface charge density and gj(J2) is the surfaceion radial distribution function evaluated at the plane of closest approach of the ions. The quantities denoted by the superscript or subscript out correspond to the respective values at infinite plate separation. Equation 1 isexact within the restricted primitive model of electrolytes, and thus the only approximation involved in this method of pressure calculation is that required to calculate the electrolyte structure. Results All calculations were done using the method described in ref 8, with the exception of the varying ionic size, a. The dielectric constant, E, was taken to be 78.5 and the temperature was set to 25 "C. The model electrolyte is monovalent. In the version of the three point extension hypernetted chainfmean spherical approximation (TPE HNC/MSA) which we have used in this work, the potential at the surface, $0, is fixed as the surfaces are brought nearer each other and the pressure as a function of separation is calculated through eq 1. We have chosen to use an arbitrary set of ionic diameters in our calculations rather than attempt to model any specific counterions, as the precise size of an aqueous ion is not a well-known quantity. In Table I1 of ref
0022-365419312097-12083%04.00/0 0 1993 American Chemical Society
Feller and McQuarrie
12084 The Journal of Physical Chemistry, Vol. 97, No. 46, 1993
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separation/A Figure 2. Net pressureas a functionof surface separation,as calculated from the TPE HNC/MSA theory, for a range of electrolytediameters. For each ion size, the surface potential is fixed at a different potential asdescribed in the text such that each surfacecorrespondsto the indentical surfacechargedensityofa. 103 C m-2at infinite separation. The meaning of the curves is the same as in Figure 1. the same trend that we report. Their constant charge MGC theory results showed that the pressure increased with greater ionic diameters. In Figure 2 we have shown a set of pressure curves for the same electrolytecomposition as Figure 1, but this time the fixed surface potential has been chosen such that at infinite plate separation all the surfaces have the identical surface charge density of -0.103 C m-2, This surface charge density corresponds to the value obtained for the a = 4.25 A case in Figure 1; thus, the a = 4.25 A curve is identical in Figures 1 and 2. The trend with differing ionic diameter is the same as that seen in Figure 1. With the exception of the small separation region, the curves arevery close to each other and to the results of Figure 1. This is despite the fact that the potential on the surfaces range from 40 = -148 mV for the a = 0.0 case to 40 = -187 mV for the ion of diameter 6.375 A. The magnitude of the observed differences in the small separation region is seen to greatly increase over that seen in Figure 1 . This is an important observation as a large number of experiments on the force between interacting double layers are well explained by the point ion Gouy-Chapman theory at long range, but in these same experiments, the theory does not hold up in the same separation region. The effect of ionic size as seen through the short-range ion-ion correlations will increase with greater electrolyte concentration and will be more complicated than the effect through the varying location of the plane of closest approach. In this theory two types of ion-ion correlations exist, those which are electrostaticin origin and those that arise from the hard spherehard sphere interactions between the ions. The strength of these short range ion-ion correlations will increase with increasing size in the TPE HNC/ MSA theory. The electrostatic correlations would be expected to reduce the plate-plate repulsion and have been shown previously to lead to attraction in concentrated electrolytes.8 The effect of the hard sphere part of the ion-ion correlations is more complicated but has been shown to lead to oscillating regions of attraction and repulsion between uncharged surfaces.8 These ionic size effects will be in addition to the effect from the varying plane of closest approach. In Figure 3 we show results for a system with a fixed surface potential of -90.0 mV and a monovalent electrolyte with concentration 1.0 M. The range of surface charge densities a t infinite plate separation, uout,was from -0.141 C m-2 for a = 6.375 A to -0.327 C m-2 for a = 0.00 A. The effect of the short range ion-ion correlations is shown quite well here. For the smaller ionic diameters, a = 0.00 A and a = 2.125 A, the force is always repulsive. For the a = 4.25 A case, a single minimum in the pressure vs distance curve is observed where a weak attraction between the plates is found. As the ion size is increased to 6.375 A, a deeper attractive region is observed and the pressure is seen to be a complicated function of plate separation, with two minima observed.
The Journal of Physical Chemistry, Vol. 97, No. 46, 1993 12085
Force between Electrical Double Layers $,=-90.0 mV 1:l electrolyte 1.0 mol L"
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separationIA Figure 3. Net pressure as a function of surface separation, as calculated from the TPE HNC/MSA theory, for a range of electrolyte diameters. Thesurfacepotentialis fixed at -90.0mV for each curve as the separation is decreased. The meaning of the curves is the same as in Figure 1. 7.0
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separationlA Figure 4. Net pressure as a function of surface separation,as calculated from the TPE HNC/MSA theory, for a range of electrolyte diameters. The surfacepotential is fixed at -299 mV for each curve as the separation is decreased. The meaning of the curves is the same as in Figure 1. Lubetkin et al. have performed experiments on clay-water dispersionswhere the pressure has been measured as a function of plate separation and have observed a dependenceon the choice of counterion in the surrounding electr01yte.l~ If the charged clay particles are well dispersed, then their interactions should be well described by the electrical double layer theory. The experiments of Lubetkin et ai. were carried out with Wyoming bentonite, uo = -0.116 C m-2, and a series of aqueous group 1A chloride salts at a reservoir electrolyte concentration of 10-4 M. They found that the pressure curves followed the trend Li+(aq) > Na+(aq) > K+(aq) > Cs+(aq). They noted in their conclusions that it was difficult to assess if this trend was the result of electrical double-layer forces or if it was caused by formation of associated units of clay platelets. The cesium sample especially seemed to form associatedunits which would cause an error in their pressure measurements. To assess the role that electrical double-layer forces might have played in their clay swelling pressure experiments, we have made calculations for the various ion sizes at the same electrolyte concentration and at a surface potential approximating that of the clay particles. The surface potential, 40 = -299 mV, we have chosen is that which leads to a surface charge density value of 4.116 C m-2 at infinite plate separation in the a = 4.25 A case. For plate separations of less than 100 A, the curves lie nearly on top of each other, thus we have plotted only the two extremes of a = 0.00Aand a = 6.375A. Our results are presented in Figure 4. The same trend is observed under these conditions as was observed in Figure 1, at most separations the larger ions result in larger values of the pressure. Assuming that the Li+ ion is the smallest in the 1A metal series and Cs+ is the largest, our results are in complete disagreement with the experimental observations. Additionally, the magnitude of the differencesin our calculated pressures is much smaller than that observed in the experiment.
0
10
20
300
40
50
separationIA Figure 5. Net pressure as a function of surface separation after the addition of the van der Waals pressure component to the results of Figure 1. The meaning of the curves is the same as in Figure 1. It has been common in the literature to add an attractive van der Waals component to the electrical double-layer force when comparing theory with experiment in the study of the interaction of charged surfaces.14 This approach is the basis for the DLVO theory of colloid stability. Since the development of the direct force measurement apparatus of Israelachvili and co-workers,15 much experimental work has been done on the interaction of charged surfaces with various intervening electrolytes. At large surface separations they find excellent agreement with the DLVO theory, but at small separations they find that an additional repulsive forceis present which has an oscillatory fine structure.2J6 This additional force has been investigated in great detail by Pashley and co-workers and has been termed the hydration force.16 We have taken the electrical double-layer force results of Figure 2 for the systems with ionic diameters of 0.00 and 6.375 A and added the van der Waals pressure component
P,, = - A / 6 m 3 (2) where A is the Hamaker constant, 2.2 X J m-1, and T is the separation between the surfaces. In Figure 5 we show our results for the DLVO theory pressure, where the van der Waals component has been added in. Here one can see the dramatic differences between the point ion Gouy-Chapman theory and the more rigorous TPE HNC/MSA theory. Due to the logarithmic scale that we have used, the differences between the various ionic sizes in Figure 2 may have appeared small; however, in Figure 5 it is seen that the pressure at small plate separations is so much greater for the finite-sized ions that no attractive region is observed and in fact the pressure remains nearly monotonic down to the smallest possible surface separation of one ionic diameter. In the TPE HNC/MSA theory, the pressure can only be calculated down to a separation of one ionic diameter because after this point the force diverges due to the hard platehard sphere interaction. The differences between the finite sized and point ions are also great when the van der Waals component is added to the results of Figure 1, where all curves correspond to an identical surface potential. Conclusions We have used the three-point extension hypernetted chain/ mean spherical approximation to study the effect of electrolyte size on the force between charged planar surfaces immersed in a restricted primitive model electrolyte. For a dilute electrolyte where the TPE HNC/MSA theory is in relatively good agreement with the classical theory of interacting double layers of Gouy and Chapman, the pressure is seen to increase with increasing ionic diameter. The differences between the two approaches becomes large at small separations, however, and it is possible that the common practice in the literature of ascribingdifferencesbetween the Gouy-Chapman theory (with added van der Waals component) and experimental observations entirely to solvent effects
Feller and McQuarrie
12086 The Journal of Physical Chemistry, Vol. 97, No. 46, 1993
may lead to some error in the estimation of the magnitude of the solvent effects. For a concentrated electrolyte the effect of the ion diameter is more pronounced, leading to attraction between the two like charged plates, a result which cannot be predicted with the classical theory. Though not as relevant for explaining experimental observations,the concentratedsystems are of interest since they show attraction between the charged surfaces without the addition of a van der Waals component. Our results show that small changes in the ionic diameter of only a few angstroms cause the systems to behave in completely different manners, ranging from monotonic repulsion between the surfaces to complicated behavior including regions of both attraction and repulsion with multiple local minima in the force curve. In summary, the use of a point ion theory as is common in the colloid literature may lead to erroneous results at small surface separationseven in dilute monovalent electrolyte. The variational method for the solution of the three-point extension hypernetted chain/mean spherical approximation theory described in ref 8 was especially useful for investigating the interaction between the surfaces at small separations since it avoided some of the convergence difficulties encountered with other numerical methods under conditions of small separation. Acknowledgment. This work has been supported by the National Science Foundation under Grant NSF EAR 8910530.
References and Notes (1) Venvey, E. J.; Overbeck, J. TI. G. Theory of Stability ofLyophobic Colloids; Elsevier: Amsterdam, 1948. (2) Pashley, R. M.; Israelachvili, J. N. J. Colloid Interface Sci. 1984, 101, 511. (3) Kjellander, R.; Marcelja, S.;Pashley, R. M.; Quirk, J. P. J. Chem. Phys. 1990, 92, 4399. (4) Valleau, J. P.; Ivkov, R.;Torrie, G. M. J . Chem. Phys. 1991,95,520. (5) Kjellander, R.;Akesson, T.; Jonsson, R.; Marcelja, S . J . Chem. Phys. 1992, 97 (2). 1424. (6) Kjellander, R.; Marcelja, S.J. Chem. Phys. 1988, 88, 7138. (7) Lozada-Cassou, M.; Diaz-Hcrrera, E. J. Chem.Phys. 1990,92,1194. (8) Feller, S.E.; McQuarrie, D. A. J . Colloid Interface Sci., in press. (9) Lozada-Cassou, M. J. Chem. Phys. 1984,80, 3344. (10) Olivares, W.; McQuarrie, D. A. J . Phys. Chem. 1980,84, 863. (11) Wcnnerstrom, H.; Jonsson, E.;Linse, P. J. Chem. Phys. 1982, 76, 4665. (12) Huerta, M. M.;Curry, J. E.; McQuarrie, D. A. Clays Clay Miner. 1992, 40, 491. (13) Lubetkin, S.D.; Middleton, S.R.; Ottewill, R. H. Philos. Tram. R. SOC.London, A 1984, 311, 353. (14) Derjaguin, B. V.; Landau, L. Acta Phys. Chim. U.R.S.S.1941, 14, 633. (15) Israelachvili, J. N.; Adams, G. E. Nature 1976, 262, 774. (16) Ducker, W. A.; Pashley, R. N. J . Colloid Interface Sci. 1986,131, 433.
