Langmuir 1992,8, 1659-1662
1659
Theory of Dissociative Electrical Double Layers: The Limit of Close Separations and “Hydration”Forces Jan J. Spitzert Department of Chemistry, The University of Lethbridge, Lethbridge, Alberta, Canada Received November 11, 1991. In Final Form: April 6, 1992 The premises of the theory of dissociative electrical double layers (DEDL), i.e., the requirement of electrostatic self-consistency and the assumption of partial double-layer dissociation, are reformulated and discussed in detail. The DEDL theory is shown to yield a new limitinglaw that predicts large repulsive forces in the limit of very close separationsof rigid charged planar surfaces. This limiting law is contrasted with Langmuir’s expression for the limit of infinite potentials, and it is also discussed in relation to “hydration”forces. These large electrostatic forces arise from compressing the diffuse counterion charge into a smaller volume, when the charged surfaces are pushed together, while the electroneutrality of the overlapping double layers is being maintained. These forces depend only weakly on bulk ionic strength. Their magnitude is determined by the ease of dissociation of the Stern layer, which in turn is determined by the hydration of the Stern and diffuse ions. In this sense these electrostatic forces can be termed hydration forces.
Introduction In previous papers the site-atmospheric ion-binding model of the electrical double layer was derived in an ad hoc manner for applications involving highly charged The model was found to account for the strong repulsive pressures in montmorillonite gel^,^^^ and for the Stern potentials (rpotentials) of swellingclays. Recently, the model was found to account for the equilibrium separation distance (“coagulation”) of swelling clays in the presence of divalent counterions using the DLVO approach to colloidal ~tability.~ These experimental data cannot be explained by the classical Gouy-Chapman-Stem (GCS) theories.“’ The theory also allowed the derivation of two new (possibly fundamental) limiting laws in the limits of infinite and very close plate separations.* The purpose of this paper is to reformulate and justify the previous ad hoc assumptions in and thereby derive a new electrostatic theory of dissociative electrical double layers (DEDL). The important theoretical feature of this theory is its complete electrostatic self-comistency in respect to the average charge and potential distributions. The utility of this theoretical approach is demonstrated by the discovery of a new double limiting law (the limit of close separations and the limit of zero bulk ionic strength) that appears to account for the existence of “hydration” forces at charged rigid surfaces. Theoretical Section Nature of the Model. The model is set up as a macroscopic electrostatic problem; Le., the ions (hard charged spheres) are assumed to be smeared out into surface and volume charge densities by their thermal + Present address:
BASF Canada Inc., Dispersions R&D, 453
Christina St. S., Sarnia-Clearwater, Ontario N7T 721,Canada. (1)Spitzer, J. J. Nature 1984, 310,396. (2)Spitzer, J. J. Colloids Surf. 1984,12, 189. (3)Spitzer, J. J. Langmuir 1989,5, 199. (4) Spitzer, J. J. In NATO Aduanced Workshop on Clay Swelling and Expansiue SoiLs; Bavey, P., McBride, M., Eds.; Kluwer Academic: Dordrecht, to be published. (5)Lubetkin, S. D.; Middleton, S. R.; Ottewill, R. H. Philos. Trans. R. SOC.London 1984, A311, 133. (6)Viani, B. E.; Low, P. F.; Roth, C. B. J. Colloid Interface Sci. 1983, 96,229. (7)Horikawa, Y.; Murray, R. S.; Quirk, J. P. Colloids Surf. 1988, 32, .e.
101.
(8) Spitzer, J. J. Colloids Surf. 1991, 60, 71.
motions, and the solvent is represented by a linear dielectric continuum. In other words the macroscopic (“primitive”)model of the Debye-Hiickel theory of strong electrolytes is a d ~ p t e d The . ~ discrete surface charges are approximated by the surface charge density. The finite size of the counterions defines the thickness of the “Stern layer” of surface charge density (which can be assigned a low dielectric constant, if desired). The model is sketched in Figure 1. Premises of the Theory. There are two premises from which the theory is developed. The first premise is that the requirements of macroscopic electrostatics (additivity of potentials, proportionality of charges and potentials, and the equivalence of different charging processes) must be satisfied. The second premise is the explicit recognition that electrical double layers are not generally completely dissociated. These two premises can be elaborated as follows. (1) The First Premise: The Debye-Hiickel Approximation. It is known that the nonlinear PoissonBoltzmann (NLPB) equation is inconsistent10-14from the point of view of macroscopic electrostatics. In particular the NLPB equation violates the principle of additivity of electrostatic potentials, and the requirement that charges and potentials are proportional to each other; therefore, calculations of electrostatic interaction energies by different (though electrostatically equivalent) charging processes give inconsistent results. Nevertheless, the use of the NLPB equation has remained customary.” In the area of electrolyte solutions the NLPB equation has been rejected because it is also inconsistent from a purely statistical-mechanical ~iewp0int.l~ It is also known1@-14 that the linearized Poisson-Boltzmann equation (LPB) is consistent with macroscopic electrostatics (and also with statistical mechanics). Less known is the observation, however, that the LPB equation is not to be regarded as (9) Debye, P. J. W.; Hiickel, E. Phys. Z. 1923, 24, 85. (10)Robinson,R. A.;Stokes,R. H.ElectrolyteSolutio~;Butterworths: London, 1959;Chapter 4. (11)Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry; Plenum Publishing: New York, 1973;Vol. 1, Chapter 3;Vol. 2,Chapter 0
I.
(12)Bennetto, H. P.; Spitzer, J. J. J. Chem. SOC., Faraday Trans. 1 1976, 72,2108.
(13)McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1976;Chapter 15. (14)Spitzer, J. J. J. Chem. Soc., Faraday Tram. 1 1978, 74,2418.
0143-1463/92/2408-1659$03.00/00 1992 American Chemical Society
Spitzer
1660 Langmuir, Vol. 8, No. 6,1992
$ (b)=kT/z+e
........... ..
Figure 1. Electrostatic model of dissociative electrical double layers. ao(+) is the primary surface charge density, ua(-) is the undissociated counterlayer of charge (Stern layer) of thickness a, q ( 6 ) is the thermal electrostatic potential of counterions at the co-ionexclusionboundary 6, and d is the midpoint separation distance between two double layers.
an approximation to the NLPB equation valid for low electrostatic energies only. This point is made clearly in the classic book Electrolyte Solutions,lo from which it is worth quoting directly “At distances up to a few ionic diameters, however, the approximation exp(-zieQ/kT) = 1- zie\k/kT made in the Debye-Hiickel treatment cannot be justified on the grounds that zieq is small compared to kT as is usually claimed: instead it must be justified on grounds of mathematical expediency in order to obtain a distribution functionconsistent with the principle of the linear superposition of fields.” In other words the linearization is in fact a derivation of a new ionic distribution law that is consistent with macroscopic electrostatics regardless of the magnitude of electrostatic energies. The paradox that the linearization ‘approximation” gets worse with decreasing ionic strength (where the Debye-Huckel limiting laws become fundamentally valid) is notablelo but can be underst00d.l~ The linear ionic distribution laws have been used improperly, even within the conventional Debye-Huckel theory; particularly troubling is the prediction of negative concentrations of co-ions when (paradoxically) a good agreement with experiment is obtained.14 The proper physical interpretation of these negative co-ion concentrations is that the co-ions are absent from the regions where their concentration would otherwise become negative. (To put it colloquially, the ions “know their electrostatics”, and hence they will on average distribute themselves to yield electrostatically consistent charge and potential distributions.) The correct use of the linear distribution laws then requires the existence of co-ion exclusion boundaries which allow the construction of a completely self-consistent macroscopic theory. A new boundary is then defined in such a way that co-ions are repelled (excluded) from within this boundary, b - a, into the conventional Debye-Huckel region, d - b, as shown in Figure 1. This co-ion exclusion boundary b is defined as W b ) = kT/(z+e)= 9 :
TEP
(1)
where W b ) is the potential at the boundary b, which is termed the thermal electrostatic potential of the co-ions (TEP),denoted by at+,and the other symbols have their usual meanings. The co-ions can still be present in the
co-ion exclusion region but only as electroneutral combinations with the counterions (no net contribution to the volume charge density). The important feature of eq 1is that the co-ion exclusion boundary b is a free boundary, the location of which is determined by the electrostatic solution of the model. (2) The Second Premise: The Incomplete Dissociation of the Double Layer. A number of experiments (e.g., electrophoretic mobilities, electrocapillarity, thermodynamics, and conductance) on charged systems have indicated that counterions are often not free to completely dissociate from the charged surface. In order to deal with this phenomenon, a classical (van’t Hoff-Ostwald-Arrhenius) empirical approach is adopted by assuming that a fraction of the counterions, CY,remains associated with the surface charge ad+), giving rise to the Stern layer of charge ua(-) (Figure 1). The behavior of this charged layer is determined by the appropriate solutions of the electrostatic model, in contrast to the usual Stern theory which is based on the ad hoc assumption of Langmuirian adsorption of bulk counterions. The degree of counterion association CY is defined in planar symmetry as = -ua(-)/uJ+) (2) where uo(+) is the primary charge density, and aa(-) is the associated surface charge density (Stern layer of charge). Equations 1and 2 are sufficient to solve the electrostatic problem as shown in Figure 1,using standard electrostatic formulas. Using these two premises, a variety of electrostatic problems can be formulated to suit particular experimental conditions and geometries. Here we discuss a new limiting law for repulsive electrostatic forces a t close separations of charged planar surfaces in relation to the so-called hydration forces. In the followingpaper in this issue the consequences of the explicit treatment of the co-ion exclusion are discussedin relation to attractive electrostatic forces in ordered colloidal dispersions. Electrostatic Repulsions in the Limit of Close Separations. The theoretical equations for the case of close separations of highly charged planar surfaces were derived before,ll3v4and the close-range limiting law can be obtained from these equations by the following procedure: by working out eqs 38 and 36 in ref 3 and then neglecting the small terms and taking the limit of zero separation, the theoretical limiting law for electrostatic repulsions is derived as CY
lim P ( d ) = 0.5[(1- h)uO(+)l2/[c,h2(d - al21 d-o
(3)
where UO(+) is the surface charge density, cr is the relative permittivity of the solvent, d is the midpoint separation distance, and a is the thickness of the Stern layer. (The detailed derivation of this law is availablefrom the author.) The quantity h is defined by analogyto the Debye constant K B s
h2 = [ ( ~ - e ) ~ n , l / ( c , k r ) (4) where z-e is the charge of the counterions, n+ is their bulk concentration, and kT is their thermal energy. The degree of double-layer association CY is related to the original ad hoc constant K A bylB CY = 1/(1+ KA) = - u , ( - ) / u ~ ( + ) (5) for these limiting conditions of close separations. The application of the theory3 to the extensive experimental determinations of the ionic strength dependence of repulsions in clay gels5 gave the appi oximate empirical
Langmuir, Vol. 8, No. 6,1992 1661
Dissociative Electrical Double Layers law, termed the LMO law, as
1 - a! = pK/(1 + PK) (6) where the parameter p determines the ionic strength dependence of the degree of dissociation, l - CY, of the Stern layer. A similar expression (in fact the limiting form of eq 6 for low ionic strengths) has been derived recently from experimental data on montmorillonites, making an ad hoc assumption that the Stern potential is independent of ionic strength.15 On substituting eq 6 into the limiting law given by eq 3, and taking the limit for low ionic strengths, a new double limiting law (infinite bulk electrolyte dilution and very close surface separation) is obtained as lim
d+a;A
P ( d ) = @’/t,)[uo(+)/(d - a)]’
(7)
The significant feature of this law is that the repulsive pressures decrease in inverse proportion to the square of separation of the surfaces, and that the ionic strength dependence of the pressures disappears entirely from the theoretical limiting law, eq 3. The magnitude of these forces can be very large, especially when the double layer is highly dissociated (large LMO parameter p ) . It may be noted that the semiempirical LMO law predicts no dissociation of the planar double layer in the limit of infinite dilution of bulk electrolyte. This prediction is in agreement with recent computer simulations,16which also showed no diffuse charge under these conditions.
Discussion Hydration Forces at Charged Rigid Surfaces. A number of experimental results at charged surfaces have been interpreted in terms of hydration or ‘structural” f o r ~ e s . 6 J ~The - ~ ~most clear-cut experiments that identify these forces are those dealing with the swelling of homoionic montmorillonite c1ays,5p6 and the measurements of forces between crossed mica cylinders in electrolyte solutions.18,21*22 (Because of the ion exchange phenomena on the potassium specific mica surface, particularly a t low ionic strengths, these experiments are somewhat less suitable for testing theories than the experiments on repulsive pressures in homoionic swelling clays.) The usual procedure has been to define these forces as deviations from the DLVO theory, and in fact a term “non-DLVO forces” has been coined in a recent review of such forces.23 In spite of these extensive experimental efforts, current theoretical understanding of these hydration forces remains incomplete and rather Nevertheless, it is assumed that there are at least two kinds of hydration forces: those that originate from the modified water structure (also termed structural or solvation forces), and those that originate from the Stern layer of counterions. ~
(15)Miller, S.E.;Low, P. F. Langmuir 1989,6, 572. (16)Skipper, N.T.;Refson, K.; McConnell, J. D. C. J. Chem. Phys. 1991,94,7434;Proceedingsof Internationul Symposiumon the Structure and Dynamics of Liquids; Institute of Physics and Royal Society of Chemistry: Oxford, 1991. (17)Low, P.F.Langmuir 1987,3, 18. (18)Horn, R. G.;Iaraelachvili, J. N. J. Chem. Phys. 1981,75, 1400. (19)Lis, L. J.; McAlister, N.; Rand, R. P.; Parsegian, V. A. Biophys. J. 1982,37,657. (20)Peschel, G.;Belouschek, M.; Muller, M. M.; Muller, M. R.; Konig, R. Colloid Polym. Sci. 1982,260,444. (21)Israelachvili, J. N.;Adams, G. E. J. Chem. SOC.,Faraday Trans. 1 1978,74,975. (22)Pashley, R. M. J. Colloid Interface Sci. 1981,80,153. (23)Christenson, H.G.J. Dispers. Sci. Technol. 1988,9,171. (24)Schiby, D.;Ruckenstein, E. Chem. Phys. Lett. 1983,100, 277. (25)Radif, N.; Marbelja, S. Chem. Phys. Lett. 1978,55, 377. (26)Ninham, B. W. Pure Appl. Chem. 1981,53,2135.
These two kinds of forces can be further discussed in the light of available evidence as follows. A number of different experimental techniques and computer simulations have shown that the presence of a surface can change the properties of the adjacent solvent.6J6-18p27-31 However, it is difficult to unequivocally show that the modified structure of water (or solvent) can give rise to large repulsive forces. For example, Low has empirically related the hydration repulsive pressures to the thickness of the modified water layer between montmorillonite platelets, and concluded that the swelling of clays is due to hydration of their surfaces.17 This is perhaps a too sweeping conclusion, as other experimental evidence shows that the swelling of clays depends on the ionic strength and on the kind of counter ion^.^ The modified solvent structure has also been discounted as the cause of hydration forces at flexible uncharged surfaces.32 The question whether purely structural solvent effects can be translated into large repulsive forces has been partly answered experimentally by the force measurements with crossed mica cylinders: these structural forces appear as relatively weak force oscillations with maxima of about 1-3 mN/m in systems with no diffuse charges (silicone liquid);18 however, in the presence of free ions these oscillatory structural forces are shifted into a “high” range Thus, of 5-500 mN/m (and may remain o~cillatory).~~ the modified solvent structure per seis unlikely to account for the large magnitude of these close-range repulsions. Nevertheless, the contribution from the structural effects could be larger in the case of water, as the silicone liquid (octamethylcyclotetrasiloxane) is somewhat flexible. Hence, a contribution from purely structural forces cannot be ruled out.l7 In the measurements with crossed mica cylinders Pashley showed that the hydration forcesbecome operative only when there is a substantial Stern layer developed on the mica surface33at high bulk ionic concentrations. It seems unlikely, however, that the hydration forces are due to the dehydration of the Stern layer of counterions, as deviations from the theory become apparent at larger distances, as was shown in the case of montmorillonite gels.5@ Nature of ‘Hydration” Forces from the DEDL Theory. The first-order phenomenological insight into the origin of these large hydration forces is provided by the new theory of dissociative double layers (DEDL),1-4 as reformulated in this paper. To begin with, it is instructive to calculate the repulsive pressures in the limit of very close separations, as shown in Table I, using the example of lithium montm~rillonite.~~~ It can be seen that the ionic strength dependence of the forces is small (for other alkali-metal ions the ionic strength dependence is even smaller), and that these forces reach magnitudes of hundreds of megapascals. Clearly,these theoretical forces have the character of the hydration forces: they are very large and not very dependent on ionic strength. The results obtained previou~ly’~~ show that the amount of diffuse charge is constant regardless ofplate separation. (27)Derjaguin, B. V. B o g . Colloid Polym. Sci. 1987,74, 17. (28)Sun, Y.;Lin, H.; Low, P. F. J. Colloid Interface Sci. 1986,112, 556. (29)Oliphant, J. L.; Low, P. F. J. Colloid Interface Sci. 1983,95,45. (30)Kjellander, R.; Marfelja, S. S. Chem. Scr. 1985,25,73. Lozada-Cassou, M. J.Colloid Interface Sci. 1986, (31)Henderson, D.; 114,180. (32)Israelachvili, J. N.;WennerstrBm, H. Langmuir 1990,6, 873. (33)Pashley, R. M.; Israelachvili, J. N. J. Colloid Interface Sci. 1984, 101,511. (34)Robinson, R.A.; Stokes, R. H. Electrolyte Solutions;Butterworths: London, 1959;Chapter 8.
Spitzer
1662 Langmuir, Vol. 8, No. 6,1992 Table I. Repulsive Electrostatic Forces P.1 and Attractive van der Waals Pressures Psal vs Midpoint Separation Distance d at Different Ionic Strengths of 1:l Electrolyte
d (A) 4.25 4.5 5.0 6.0 7.0 8.0 9.0 10.0 12.5 15.0 20.0
Pvdw
(MPa) 1.55 1.28 0.89 0.47 0.27 0.17 0.11 0.073 0.031 0.015 0.005
P.1 (MPa)at various 1:l electrolyte concentrations (mol dm-3) 10" 10-4 10-2 10-1 3870 2470 4830 4940 1210 968 618 1240 242 88.1 309 302 60.4 38.1 77.2 75.4 26.8 16.6 34.3 33.5 15.1 9.1 19.3 18.9 12.1 9.6 5.7 12.4 6.7 3.8 8.6 8.4 4.3 4.2 3.3 1.6 1.9 0.8 2.6 2.5 0.9 0.2 1.2 1.2
The parameters are uo(+) = 11.6 pC cm-2,a = 4.0 A (surface contact), and p = 4.0 A.
Therefore the origin of the repulsive forces lies in the compression of the diffuse charge into a smaller volume, which is a readily understandable electrostatic effect. The magnitude of the electrostatic forces is governed by the dissociation of the Stern layer: the higher the degree of dissociation (the value of the parameter p ) ,the larger the electrostatic forces are going to be at the same ionic strength. The dissociation of the Stern layer will depend in general on the surface charge density, on the kind of counterions, on the hydration phenomena within the double layer, and on the bulk ionic strength. The dependence on bulk ionic strength has already been established in terms of the approximate LMO law (eq 4). It may be concluded that the origin of these large shortrange forces is electrostatic but their magnitude is determined by the ease of dissociation of the Stern layer of charge. Such dissociation depends on the hydration of the Stern ions and of the diffuse ions, and hence in this sense these forces could be called the hydration forces. One important and general consequence of the theoretical calculations in Table I is that the electrostaticmodel can predict spontaneous colloidal redispersibility of "dry bulk powder", e.g., of swelling clays. (The van der Waals attractions are much weaker in the whole range of separations, c.f. Table I.) This theoretical prediction of redispersibility arises from the large magnitude and slow decay of the short-range electrostatic forces given by the new inverse square limiting law, eq 3. DEDL Theory and Other Theoretical Results. The premise of the incomplete double-layer dissociation in the DEDL theory does not allow useful comparisons with the solutions of the NLPB equation or those theories or computer simulations that assume a priori that the counterions of the primitive model are 100% dissociated. However, recent computer simulations16 based on the "civilized" model26(specific interaction potentials for the atoms of the surface,solvent, and ions) predict the absence of the diffuse double layer (no double-layer dissociation)
at zero ionic strength.16 This result is consistent with the semiempiricalLMO law, eq 6, which is deduced from the DEDL theory and from experimental data: and which also predicts the absence of the diffuse double layer at zero ionic strength. Therefore, the ad hoc assumption of complete double-layerdissociation inherent in the NLPB equation is questionable. To a first approximation, it appears that the double-layer dissociation may be viewed as the 'increased (two-dimensional)solubility"of the Stem layer at higher ionic strengths; the analogous increase of the (three-dimensional) solubility of ionic salts is wellkn0~n.34
In the literature there appear no theoretical limiting results except one expression derived by Langmuir a long time ago for the repulsive pressures at infinite surface potential^.^^ This Langmuir law is also an inverse square limiting law, written as lim p = (.rr/2)e,(k~/2ed)'
-4o"',
(8)
in centimeter-gram-secondunits; however, it is derived in the limit of infinitesurface potential from the nonlinear Poisson-Boltzmann equation, and it should not be mistaken for a different form of the new limiting laws given by eqs 3 and 7. In contrast, these new inverse equare limiting laws are derived in the limit of close surface separations (the range of hydration forces) from the linear Poisson-Boltzmann equation for any finite potential or charge density. It is then clear that the nature of the new inverse square laws given by eqs 3 and 7 is quite different from the Langmuir inverse square law (eq 8). There are two other major differences between these two laws: (i) the new inverse square laws are specific to the dissociation of the Stem layer, whereasthe Langmuir law is nonspecific, and (ii) the new inverse square law predicts decreased repulsions with increasing dielectric constant (in accordance with Coulomb's law for linear dielectrics), whereas the Langmuir law, rather surprisingly, predicts increased repulsions with increasing dielectric constant. Thus, it is questionablewhether Langmuir's inverse square law offers any useful insights into the double-layer behavior. Further comparisons can be made with other results given in Langmuir's paper.% Langmuir shows that at finite surface potentials the NLPB equation gives a zero power law in the limit of close separations, which leads to a constant repulsive force, i.e., to an upper limit on repulsions. In contrast the DEDL theory predicts forces that increase according to the new inverse square power laws (eqs 3 and 7). Hence, the NLPB equation, or any other theoretical approaches that give results similar to the NLPB equation, cannot account for the large magnitude of the hydration forces. Indeed, the incorrect prediction of the dependence of electrostatic repulsions on dielectric constant, as given by Langmuir's law (eq 8), may be indicative of another electrostatic inconsistency of the NLPB equation. (36) Langmuir I. J. Chem. Phys. 1938, 6, 873.
