4260
J . Phys. Chem. 1990, 94, 4260-4268
Double-Layer Structure at an Ion-Adsorbing Surface Dongqing Wei, G . N. Patey,* Department of Chemistry, University of British Columbia, Vancouver, B.C., Canada V6T 1 Y6
and G . M. Torrie* Department of Mathematics and Computer Science, Royal Military College, Kingston, Ontario, Canada K7K 5LO (Received: September I , 1989)
The effect of adding a strong, short-range, ion-specificadsorption potential to a fully molecular model of the electrical double layer is investigated within the framework of the reference hypernetted-chain (RHNC) theory. This theory may be solved numerically for potentials strong enough to adsorb anions onto an intrinsically neutral surface so as to produce charge densities equivalent to as much as one electron per 180 A*. Detailed predictions are made for the ionic structure, solvent ordering, and electrostatic potential in the interface, and these are compared with previous RHNC results for smooth surfaces carrying uniformly distributed intrinsic surface charges of similar magnitude. Many of the novel features of these earlier results, such as a very rapid neutralization of the surface charge, dubbed “fast screening”, and a highly stable icelike orientational ordering of the surface layer of solvent are found to be much the same in the presence of adsorbed surface charge. On the other hand, this somewhat more realistic model of the surface does not exhibit the anomalous behavior of the mean electrostatic potential found for uniformly charged smooth surfaces. The sensitivity of the results to the details of the adsorption force and the effect of ion adsorption on intrinsically charged surfaces are also considered briefly.
Introduction Now that the theory of molecular fluids has progressed to the point that accurate statistical mechanical treatments of wholly molecular models of electrolyte solutions are similar theories of the electrical double layer can also be formulated. In both cases a model Hamiltonian is constructed in which the effective pair potentials between all molecular constituents are specified (such pair potentials usually have very few parameters, making the models highly simplified) and then the machinery of liquid-state physics is brought to bear in order to determine the microscopic structure and the macroscopic properties of the model. In the first work of this type for the double-layer problemk9 the solvent was modeled as a fluid of dipolar hard spheres, and the mean spherical approximation (MSA) was used to calculate the structure of the double layer in the limit of low concentration and surface charge. More recently, Kusalik and Patey4v5have shown how to solve the reference hypernetted-chain (RHNC) theory for more elaborate multipolar hard-sphere models of bulk electrolyte solutions. Although lacking the analytic tractability of the MSA, the R H N C theory is expected to yield more accurate results; moreover, it can be readily applied to more realistic solvent models and, in the case of the double layer, to systems far from the potential of zero charge (pzc). In fact, the recent extension of the R H N C theory of bulk electrolyte solutions to the double-layer problem has resulted in a wealth of detailed theoretical predictions for the solvent orientational structure and the ionic density profiles next to charged surfaces for a number of aqueous electrolytes over a wide range of surface charge densities and salt concentrat ions. ( I ) Adelman, S. A.; Chen, S. H. J. Chem. Phys. 1979, 70, 4291. (2) Pettit, B. M.; Rossky, P. J. J . Cfiem. Pfiys. 1986, 84, 5836. (3) Caillol, J . M.;Levesque, D.; Weis, J. J.; Kusalik, P. G.; Patey, G. N. Mol. Phys. 1987, 62, 461, (4) Kusalik. P. G.; Patey, G. N. J . Cfiem. Pfiys. 1988, 88, 7715. (5) Kusalik, P. G.;Patey, G. N. J . Chem. Phys. 1988, 89, 5843. (6) Carnie, S. L.; Chan, D. Y . C. J . Chem. Phys. 1980, 73, 2949. (7) Carnie, S. L.; Chan, D. Y . C. J. Chem. Soc., Faraday Trans. 2 1982, 78, 695. (8) Blum, L.; Henderson, D. J . Cfiem. Phys. 1981, 74, 1902. (9) Vericat, F.; Blum,L.; Henderson, D. J . Chem. Phys. 1982, 77, 5808. (IO) Torrie, G. M.; Kusalik, P. G.; Patey, G. N. J. Cfiem.Phys. 1988,88, 7826. ( I I ) Torrie, G.M.; Kusalik, P. G.; Patey, G. N. J . Chem. Phys. 1988, 89, 3285. (12) Torrie, G. M.; Kusalik, P. G.; Patey, G. N. J . Cfiem.Pfiys. 1989, 90, 4513. (13) Torrie. G. M.; Kusalik, P. G.; Patey. G. N . J . Chem. Phys. 1989, 9 / , 6367.
0022-3654/90/2094-4260$02.50/0
In this work the simplest possible model has been used for the surface itself, viz., a perfectly smooth (and slightly curved) impenetrable wall carrying a uniformly distributed surface charge and bounding an electrically inert continuum. Although this is an appropriate choice when the primary objective is to gain an understanding of the electrolyte side of the electrified interface, such a complete lack of molecular structure in the surface is, of course, incompatible with the level of modeling of the solution phase. In particular, there are two aspects of the double-layer structure that emerged from this recent work in which the presence of a perfectly smooth surface might be suspected of playing a role. First, the R H N C results10are among a number of studies of molecular models of liquid ~ a t e r ’ that ~ * ’show ~ the formation of a very specific orientational order, loosely described as icelike, in the presence of a smooth, flat, neutral surface. This ordering also forms the basis for our understanding of the R H N C results for the short-range components of both the solvent and ionic responses to nearly flat charged surfaces in the double 1ayer.lb12 Yet the theory also predicts that this ordering extends only two or three molecular diameters from contact with the surface and cannot be fully established in any case unless the surface has very little or no curvature. This raises the possibility that a macroscopically flat but molecularly rough surface would not support the same degree of orientational order in the adjacent solvent layers. For example, there are simulations of water models at more irregular surfaces that seem to show quite different solvent ordering in the vicinity of an adsorbed ion.16 Second, in the RHNC theory of aqueous s y ~ t e m s ’ ~ Jand * J ~even in the MSA treatment of ion-dipole mixtures’ the spatial variation through the interface of *(r), the mean electrostatic potential, remains something of an enigma. Apart from oscillatory behaviour and a much larger magnitude than P ( r ) in continuum solvent models, both expected results, the electrostatic potential in the molecular solvent systems also exhibits a very large and very sharp maximum at the solvent-surface contact distance whose sign is opposite to that of the surface charge itself. This peak appears to originate in the contribution to the total potential from those solvent molecules directly in contact with the surfa’ce. For a perfectly smooth surface these contact molecules are completely unscreened from the bare field of the surface charge, and the reaction field arising from the resulting high degree of alignment (14) Lee, C. Y.; McCammon, A. J.; Rossky, P. J. J . Cfiem. Phys. 1984, 80, 4448. (15) Valleau, J . P.: Gardner, A. A. J . Chem. Phys. 1987, 86, 4162. (16) Kjellander, R.; MarEelja, S. Chem. Pfiys. Lett. 1985, 120, 393.
0 1990 American Chemical Society
Double-Layer Structure at an Ion-adsorbing Surface in this layer produces the unnatural looking result in *(r). Thus, although the potential at the surface always has a plausible value, * ( r ) varies in some cases by more than a volt over a distance of 1 A, putting into question the very notion of a well-defined surface potential. To a certain extent, this might be dismissed as a theoretical artifact arising from the attempt to extend down to molecular distance scales the usage of a function that is ordinarily employed in the domain of macroscopic continuum electrostatics. However, in the absence of either simulation or experimental data with which to test directly the detailed structural predictions of the theory, we must for now rely primarily on comparison of the theoretical predictions for macroscopic quantities with classical electrochemical data. The outstanding example of such data remains the behavior of the differential capacitance C, of aqueous electrolytes at a mercury electrode,” and so the surface potential is a crucial quantity. In an attempt to address these issues we have applied the R H N C theory to a model identical with that used in the earlier double-layer work for KCI solutionsI2 except that the smooth surface is not intrinsically charged but exerts instead an additional attractive force on only the CI- ions. This force is chosen to be very strong and very short ranged so that it has the effect of adsorbing anions essentially into contact with the underlying smooth surface. Obviously, one interpretation of such a model is as a highly idealized instance of specific adsorption; this approach was pioneered by Carnie and Chan,’ who studied the effect of introducing a &function adsorption potential into their MSA treatment of ion-dipole mixtures at flat walls. Our emphasis here will be on the ensuing competition for surface sites between solvent molecules and adsorbed ions and the consequences of this for the particular solvent orientational order and the peculiar behavior of q ( r ) that characterize the R H N C results for aqueous systems at smooth uniformly charged surfaces.
Model and Results Model. The bulk electrolyte solution is modeled by a mixture of decorated hard spheres in which the solvent molecules have diameter d, (2.8 A) and embedded point multipole moments (to quadrupole order) of a magnitude and symmetry appropriate to fully polarized bulk liquid water. The K+ and CI- ions are singly charged s heres of diameters d+ = 1.08d, (3.02 A) and d- = 1.16 d, (3.25 ), respectively, and the solvent and ionic densities are chosen to correspond to experimental values for 0.1 M KCI. As before, the surface is introduced into this model solution as a fourth s ecies with a very large diameter d,, here held fixed at 30d, (84 but present as a zero-concentration “impurity”. Consequently, the bulk properties of the 0.1 M solution are unaffected, while the correlation functions between this macroparticle and the other species in the system describe the structure of the solution near a slightly curved surface. In earlier work, this surface was assigned a uniformly distributed negative charge of various densities. By Gauss’ law the electric field of this surface charge is identical with that produced by an equal quantity of charge located at the center of the macroparticle which may therefore be treated in the theory simply as an additional ionic species, albeit with somewhat extreme values of size and valence. Most of the results we report in this paper are for systems in which the macroparticle is nominally neutral, but we have introduced an additional term into the Hamiltonian:
w
1)
This central potential is a function of the macroparticle-anion separation, rm-, and depends on the inverse nth power of the distance of the CI- ion from the macroparticle surface. We have used A > 0 and, except where noted below, n = 10; thus, uads is a short-range attractive potential that leads to a very localized adsorption of anions onto the smooth surface of the macroparticle. If we regard these adsorbed charges as now forming part of an (17) Grahame,
D.C.Chem. Reu. 1947, 41, 441.
The Journal of Physical Chemistry, Vol. 94, No. 10, 1990 4261
1 0.
0
0.0
1.0
.
2.0
3.0
1
4.0
5.0
,
6.0
Figure 1. Relative density profile g,(r) for anions near surfaces with different values of the adsorption parameter A in eq 1: (-) A = 9.1 1; (---) A = 7.25; A = 0. (e**)
irregular “surface”, then by variance of the contact value A in eq 1 charge densities can be induced on this surface that match those used in previous work on uniformly charged surfaces,’* and a direct assessment can be made of the effects of the surface “granularity”. This adsorption potential is easily incorporated into the theoretical framework used in the previous work on smooth surfaces and fully described in those papersl0‘l2 and references therein. The R H N C closure is applied to the Ornstein-Zernike equations for the mixture of macroparticle and electrolyte solution and solved by transform methods for the full correlation functions between the macroparticle and the ionic and solvent species. As before, these correlation functions are represented as arrays of 2048 points on a real space grid of 0.024 However, we follow the procedure of Kusalik and Patey4 of using a finer grid of 0.005d, for each function within O.ld, of contact where the variation can be quite rapid, especially in the case of the macroparticle-anion g(r). All aspects of the model and the numerical solution of the theory not explicitly stated are unchanged from this earlier work.’*12 Ionic Structure in the Interface. The general nature of the surface region can be inferred from Figure 1 in which is shown the macroparticle-anion radial distribution function gm-(r),Le., the CI- relative density profile, for two values of the adsorption parameter, A = 7.25 and A = 9.11. The dotted line in this figure is the profile for CI- at a neutral nonadsorbing surface. The inset in this figure shows clearly the near &function character of the adsorption with well over 90% of the interfacial chloride ions lying virtually in contact with the macroparticle. Almost immediately, however, the two profiles for the adsorbing surfaces assume short-range structure similar to that found for CI- ions at a completely inert neutral surface. This latter structure, which for CI- and similar sized ions is characterized by the double peak evident in the dotted line in the figure, represents the effect on such ions of the corresponding short-range structure in the solvent. Its reemergence here is the first of several indications that the surface ordering of the solvent is perturbed very little by the adsorbed ions. As we have come to expect in these models, the solvent structural effects are once again very short-range, and already at four diameters from contact the CI- profiles exhibit the smooth decay to unity from below characteristic of the behavior of co-ion density profiles in a structureless solvent. To establish a basis for comparison of these results with the earlier ones for smooth charged surfaces, we must make a suitable choice of “surface“ charge in the present context. To this end, we consider the ionic charge profile Q(r), the net number of elementary charges within a distance r of the macroparticle. This function is shown in Figure 2 for three adsorption systems and also for a smooth surface with a uniformly distributed charge of 126 electrons. As a consequence of the very short range of the
The Journal of Physical Chemistry, Vol. 94, No. 10, 1990
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Q !I.> *O
1
\ O (Q= -126)
-1
-130
I
l
l
l
l
l
l
l
l
l
l
l
l
0.0
0.5
2.0
1.0
1.5
(1.
- 4,+ ) I 4
2.5
3.0
Figure 3. Relative density profile g,+(r) for cations near ion-adsorbing surfaces. Line types as in Figures 1 and 2. TABLE I: Parameters Used in the Adsorption Potential Eq I for Surfaces with Intrinsic Uniform Charges Q and the Resulting Values of the Adsorbed Charge Q..,, Q n A Pads 0
10
0 +63
6 10
7.25 8.41 8.92 9.1 1 8.41 5.71 7.48
-33.4 -64.5 -95.2 -122.2 -95.3 -62.1 -128.8
adsorption evident in the inset of Figure 1, the minimum in Q(r) is quite sharp and occurs very close to the surface, never more than O.lds from the CI- contact point. We take as a working definition of the surface charge Qads this minimum value of Q(r) in each case; these are tabulated in Table I. This obviously arbitrary choice reflects our interest in the adsorbed charge localized in the irregular surface and seems more appropriate than, say, the net excess of CI- in the interface which would include a small diffuse layer contribution of opposite sign. Qads defined in this way does, however, include a small positive contribution from those K+ counterions that lie within the distance at which Q(r) is a minimum; this component is never more than 5% of the total As the data in the table show, Qadsis an extremely sensitive function of the strength of the adsorption potential as measured by the contact value A. In fact, although the two values of A in Figure 1 differ by only 25% they nevertheless bound the range over which interesting results can be obtained. For A = 7.25, Qadsis only -33.4 elementary charges; this is about -2.4 p c cm-2 or 664 A2 of macroparticle surface per charge, just over half the smallest charge density considered in the smooth surface worki2 and a relatively modest surface charge by most standards. On the other hand, for A = 9.1 1, Qads= -122.2. This appears to be the largest value of A for which the RHNC equations have solutions for the present model, although the theory was readily solved for a smooth surface with a uniform charge of -243.1° (In the smooth surface work where a range of macroparticle diameters was considered, all surface charge densities were expressed in terms of the corresponding number of elementary charges for d, = 20ds and hence must be scaled by 9 / 4 to match the units used here.) From Figure 2 it is clear that one feature of the smooth surface results that is shared by the adsorption systems is the very rapid neutralization of the surface charge by counterions within one diameter of contact with the surface. The short-dashed line in the figure for Q = -1 26 is a typical smooth surface result in which there is a rapid initial rise in Q(r) to a point near Id, from contact representing a neutralization of fully 60% of the original surface charge. This is marked by an abrupt change in slope beyond which
Q(r) decays in a manner consistent with the more conventional neturalization characterized by the bulk solution screening length. The behavior of the comparable adsorption system, indicated by the solid line, is even more extreme: the change in slope is more abrupt [and, interestingly, seems to have shifted outward by a small amount roughly equal to the distance between the smooth surface and the “surface” defined by the minimum in Q ( r ) ] ;the neutralization at this point now corresponds to nearly 75% of The counterion profiles for these systems and for the neutral nonadsorbing surface are compared in Figure 3. These resemble the CI- profiles of Figure 1 in showing significant solvent structural effects only within two diameters of contact, notably the same double peak reminiscent of the structure at an inert surface. The cusp evident in the adsorption system curves occurs at the distance corresponding to contact of a K+ counterion with an adsorbed CIion touching the macroparticle surface. The counterion profile for the uniformly charged macroparticle is distinguished by an extremely high peak at the surface with a contact value of 69.2. This is “adsorption” of an entirely different nature from that induced by a potential such as eq 1 . It is solely a consequence of the Coulombic forces in the system, specifically the direct unscreened field of the uniformly charged surface acting on contact counterions. It is also very model dependent; it scarcely occurs at all, for example, with smaller counterions such as Na+, which interact more strongly with the ~ o l v e n t . ’We ~ can see here that it vanishes altogether in the adsorption systems where the “surface” formed by the adsorbed CI- ions in no longer perfectly smooth and impenetrable. What is less expected, perhaps, is the discovery that this Coulombic adsorption is not an essential factor in the “fast screening” phenomenon evident in Figure 2, which may instead be a more general feature of wholly molecular models of charged interfaces. Solvent Structure in the Interface. We remarked above that the ionic density profiles furnish some indirect evidence that much of the character of the solvent ordering at smooth intrinsically charged surfaces persists even in the presence of strong adsorption of ions. Figure 4 compares the solvent relative density profile g m S ( r in ) the adsorption system with Qads= -122.2 (solid line) to that at a uniformly charged surface with Q = -126 (broken line) and at an inert surface (dotted line). As for the counterion profiles, the most noticeable effect is the disappearance in the adsorption system of the strong Coulombic adsorption of contact solvent molecules by the unscreened charge of a smooth surface. There is a slight displacement of solvent by the high concentration of adsorbed ions at the surface and a very small feature at the distance corresponding to solvent in contact with the adsorbed layer of CI- ions. Otherwise, even in the system with the strongest adsorption the solvent packing is superimposable on that at the neutral nonadsorbing surface.
ea&.
The Journal of Physical Chemistry, Vol. 94, No. 10, 1990 4263
Double-Layer Structure at an Ion-adsorbing Surface
< cos 0, > 5.0
4.0
3.h
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The orientational order in the solvent should be much more sensitive to the presence of the adsorbed ions, but even here the results appear a little surprising at first. For example, solvent molecules in contact with the macroparticle are very nearly coplanar with the adsorbed ions and might therefore be expected to show a strong preference for dipole alignment parallel to the surface with little net polarization in the normal direction. An appropriate quantitative measure of such a preference is (P2(cos e,)), the average of the second Legendre polynominal of the angle formed between the surface normal and the dipoles of molecules at a particular distance r from the macroparticle. (P2(c0s OJ) is shown in Figure 5 as a function of distance from the macroparticle for a neutral nonadsorbing surface. (dotted line), a surface with a uniformly distributed charge Q = -126 (broken line), and the adsorption system with A = 9.1 1 (solid line, Qads= -122.2). Near the surface the results for the adsorption system are, in fact, perfectly coincident with those for the inert neutral surface, and in all three systems the identical maximum degree of parallel alignment with the surface occurs near 0.3ds from contact. On the other hand, the mean polarization per particle (cos e,) has altogether different behavior according to whether or not the surface charge resides in adsorbed ions. This is shown in Figure 6 where (cos e,) for the adsorption system A = 9.11 is compared to those for the nearly equivalent uniformly charged surface and the adsorption system with A = 7.25. Only for the uniformly charged case does (cos 0,) show a deep well at contact marking the high degree of alignment of the contact layer of solvent with
.
2
v
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0
0.15
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I
I
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Figure 6. Profile of (cos e,) for solvent near ion-adsorbing surfaces. Line types as in Figures 1 and 2.
the bare field of the macroparticle; the feedback effect of this on the solvent density near contact was pointed out in the discussion of Figure 4. In the adsorption systems the analogous "contact" layer of solvent is located at 1.16ds from contact with the macroparticle next to the adsorbed CI- ions. The corresponding well here in (cos 8,) is much smaller; if reversed, it would resemble in size and shape the positive peak in this function at 1. O W s for the smooth surface case (broken line in Figure 6), which in that system represents the alignment of solvent next to the Coulombically adsorbed K+ counterions. The extreme behavior of both the solvent density and (cos e,) at contact for the uniformly charged macroparticle appears to be an artifact of the lack of molecular structure in this model of the surface. In both kinds of systems (cos e,) has short-range structure extending about four diameters from the macroparticle, after which it assumes continuum solvent behavior. The amplitude of this asymptote is noticeably smaller in the adsorption system, even for comparable values of Q and Qads,reflecting the faster neutralization in the adsorption system as already shown in Figure 2. This apparent paradox of systems with very different polarization densities sharing nearly identical number densities and ( P2(cos 0,)) profiles is not new; smooth surfaces with different uniform charge densities show very similar behavior.I0 In both cases it is necessary to make a more detailed examination of the orientational structure near the macroparticle. Following past practice, we do so by considering the function p(B), the probability density for the angle between the surface normal and either the dipole (e,) or the OH bond (eoH)of molecules at a fixed distance from the macroparticle. These functions were introduced by Lee et aI.l4 to elucidate the structure of water at a neutral wall as observed in a molecular dynamics simulation and have since been profitably employed in analyzing R H N C results for charged surfaces.l0,l2 The four panels of Figure 7 show p(BoH)for solvent particles at four distances near contact with the macroparticle for three different systems: an inert neutral surface, the adsorption system with A = 9.1 1 and the comparable uniformly charged surface, Q = -126. Lee et aI.l4 were the first to interpret the inversion that occurs between the first and third panels, i.e., between contact and 0.66ds from contact, as indicating a significant contribution to the fluid structure from an icelike arrangement in which the contact molecules are oriented with one O H bond pointing directly into the surface (eoH = 1 SO0). Clearly, both kinds of charged surfaces have this same feature and, by implication, similar OH-bond networks in the surface region. The corresponding data for the dipole angle distributions are shown in Figure 8. The bimodality in p(0,) at contact and at 0.66ds from contact for the inert surface and the splitting of this degeneracy by a uniform surface charge have been discussed previously.I0 They are consistent with the notion of an icelike orientational order that is largely undisturbed by the surface. charge but is nonetheless easily polarized through molecular dipole re-
4264
Wei et al.
The Journal of Physical Chemistry, Vol. 94, No. 10, 1990
i ! 9.11
@ ,,
@,,(degrees)
(degrees)
0l . . j
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0
30
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I
60
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90
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I
120
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\
150
,
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1
180
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Figure 7. Probability density p(BoH)for the angle between solvent OH bonds and the surface normal for solvent at various distances r from contact with ion-adsorbing surfaces. Line types as in Figures 1 and 2. (a) r = 0; (b) r = 0.34d,; (c) r = 0.66d,; (d) r = d,.
versals within this tetrahedral arrangement. We have already seen in Figure 6 that polarization effects are generally smaller in the adsorption system, and this is reflected in the probability densities in both Figures 7 and 8. The greatest differences are for molecules in contact with the macroparticle where we would not have expected the adsorption system to show such large effects as the charged macroparticle in any case. But it is clear from Figure 8b-d that even near the plane of adsorbed surface charge in this system the structural effects of the solvent polarization are no more than mild perturbations upon the orientational order at a neutral uncharged surface. The evidence is quite strong, then, that the unique orientational ordering of the solvent observed previously in RHNC calculations for both neutral and uniformly charged surfaces is mostly unaffected by the presence of adsorbed ions even at quite high effective surface charge densities. This seems a little surprising since the excluded volume and high local fields of the adsorbed ions might have been expected to frustrate the establishment of a structure known to depend on highly directional forces and to be unable to accommodate much curvature in a smooth surface.I0 This argument is not as strong as it first appears, however. First, this icelike orientational order, once established, is also known to be extremely robust. Second, the theory as we have formulated it for the surface problem describes only the average behavior over the entire surface. This average could well conceal significant structural changes in the vicinity of individual adsorbed ions because even our largest value of Qads, -1 22.2, corresponds to a cross-sectional area of adsorbed C1- ions that is only 5% of the total macroparticle surface. Viewed in this way, the failure of
such adsorption to rupture the smooth surface solvent structure may be less surprising, but it remains a significant result since many real systems have surface charge densities no larger than this. Electrostatic Potential. In a molecular solvent model the ionic and solvent components of P(r) are computed separately from the individual correlation functions [the explicit recipe for the present model is given in eq 9 of ref 121. Far from the surface these opposing contributions will nearly cancel leaving a net po) the bare ionic term just as in a continuum tential that is O( l / ~ of solvent model. Our interest here, however, is primarily in the erratic behavior of P ( r ) close to the surface. The broken lines in Figure 9a show the solvent and ionic components of P ( r ) for the uniformly charged smooth surface with Q = -126. The resulting total potential, shown in Figure 9b, has a reasonable value on the surface (at - 0 . 5 4 in the figure), but the steep rise in the solvent contribution at contact near the top of Figure 9a results in the high positive spike in the total potential at the solvent contact distance in Figure 9b. (The dimensionless potential plotted in these figures and quoted in the text is P* P e l k T ; P = P*X 25.7 mV.) The solid lines in these figures show the corresponding potentials in the comparable adsorption system, A = 9.1 1, Qads = -122.2. The situation here is clearly quite different, largely because of the absence of any regions of extreme solvent polarization by unscreened surface charge. The peculiar looking spike in the potential of the smooth surface case has disappeared altogether, and the largest positive oscillation has a maximum of only 3.37 (87 mV). Carnie and Chan7 observed a qualitatively similar change in P ( r ) upon substitution of adsorbed for uniform
The Journal of Physical Chemistry, Vol. 94, No. 10, 1990 4265
Double-Layer Structure at an Ion-adsorbing Surface
P (@,I
P (@,I (a)
T=O
(c) 0
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(Q= -126)
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1.2
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60
120
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90
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1.2-
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1
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1
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1
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1
1
1
1
1
1
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1
Figure 8. Probability density p(B,) for the angle between solvent dipoles and the surface normal for solvent at various distances r from contact with ion-adsorbing surfaces. Line types and r values as in Figure 7.
charge in their MSA treatment of the double layer in ion-dipole mixtures so it appears that the extreme sensitivity of \k(r) to the details of the molecular structure of the surface is a fairly general result and that the sharp positive spike in the smooth surface \k(r) is an artifact of the oversimplified model of the surface. This still leaves open the question of what would constitute an appropriate choice of “surface potential” in a model such as we have used here. Of course, the macroscopic quantity is the potential on (and in the interior of) the macroparticle, -4.44 for the system with A = 9.1 1 shown in Figure 9. If we interpret the model as a representation of a neutral surface to which potential-determining ions have been adsorbed by the specific interaction, eq 1, then this value, about -1 14 mV, can be regarded as the potential of zero charge (pzc) resulting from such adsorption. This is a large effect since the pzc for a neutral surface in the same model of 0.1 M KCI in the absence of adsorption is very nearly zero.’* To relate this to experimental data on specific adsorption would require a battery of calculations with a fixed adsorption potential and different intrinsic surface charges from which the differential capacitance could be estimated; this is not a line that we have pursued here. Instead, we have been more concerned with using eq 1 as a device for creating an irregular charged surface in the model that could still be treated with our existing theoretical apparatus. From this point of view the “surface” is a charged one and the plane of this charge is well localized near the point of contact of a CI- ion with the macroparticle on account of the short range of the adsorption potential. Here, \k*(r) is much larger in magnitude than on the macroparticle surface and, interestingly,
has a minimum value of -13.95 (-358 mV) that is quite comparable to that (-13.44) on the surface of a macroparticle with a uniform surface charge of similar size. We lack the data to determine whether this is mere coincidence, but it is not the first time that the theory has predicted that very different microscopic surface structures with the same surface charge will have nearly identical surface potentials as well.’3 Variation in the Adsorption Potential. We mention briefly here some calculations in which the exponent n in eq 1 was changed from 10 to 6 in order to determine the extent to which our results depend on details of the adsorption potential. For the same value of A in the potential function, Le., for the same contact value, QadS is much larger for the inverse sixth power attraction as shown by the data in Table I. This is still a very short-range attraction, however, and for all practical purposes the adsorbed charge is just as highly localized near contact as when n = 10. Thus, if we adjust the contact value A for the two potentials to produce identical values of Qads,then all other properties of the two systems are virtually identical even very close to the macroparticle. This is illustrated in Figure 10, where we have plotted Q ( r ) for three systems: A = 8.41 for n = 10 (dotted line) and n = 6 (broken line), and A = 8.92 for n = 10 (solid line). Not only do the latter two curves have nearly identical values of Qads(-95.3 and -95.2), they are also coincident beyond a distance of d- from contact with the macroparticle and are very nearly so within that distance as well. The same may be said of the individual ion density profiles in these two systems, and there is no significant difference in any of their solvent properties. The differences between the two curves
The Journal of Physical Chemistry, Vol. 94, No. 10, 1990
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Wei et al.
Y*(Y) 2501
4---\, I
solvent ionic
,, 0 (Q = -126)
i
.//'
-80, 0.0
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2.5
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4.5
Figure 9. Dimensionless mean electrostatic potential e*(r) for (-) an ion-adsorbing surface ( A = 9.1 I , Qsh= -122.2) and (---)a surface with a comparable uniformly distributed instrinsic charge Q = -126: (a) ionic (lower curves) and solvent components of P*(r); (b) resulting total P*(r).
e (1.1 1
ii
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-70
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I
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I
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I
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,
I
,
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Figure 11. Q ( r ) for surfaces with instrinsic charge Q with and without adsorption: ( - - - ) Q = -63, A = 0; Q = +63, A = 5.71 = -62.1); Q = +63, A = 7.48 (Qads = -128.8); (---) Q = +63, A = 0.
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for A = 8.41 in the figure also show that the rapid neutralization in these adsorption systems, like that for uniformly charged surfaces, is relatively more efficient the higher the surface charge. Thus, although for A = 8.41 the value of Qadsfor the inverse sixth
potential is nearly half again as large as for the inverse tenth power attraction, -95.3 versus -64.5, the difference in Q(r) for these same two systems at the abrupt change in slope near 1.16ds in the diagram is less than 20%, -33.0 versus -27.9. Adsorption by Intrinsically Charged Surfaces. To this point we have considered only systems in which the ion-adsorbing surface of the macroparticle is itself neutral. We turn our attention here to the case where this smooth surface carries a uniformly distributed positive charge, Q = +63, so that now there is a competition between intrinsic and adsorbed surface charge in determining the properties of the interface. In Figure 11 the Q ( r ) profiles for two adsorption potentials with n = 10 are compared to Q(r) for nonadsorbing surfaces with uniform charges of -63 and +63. For A = 5.71 the adsorbed charge at the minimum in Q(r) is -62.1, or nearly enough to just neutralize the uniform surface charge. For A = 7.48, Pads= -128.8 so that the net "surface" charge is negative and comparable in magnitude to the negatively charged nonadsorbing surface result included in the diagram. [ Q ( r ) in this figure includes the contribution of the intrinsic charge, but this constant term has been excluded from the numerical value of the minimum giving Qads.] In Figure 2 we saw that the neutralization of an adsorbed charge of -122.2 by K+ counterions was even more rapid than for a uniform surface charge of comparable magnitude; here, the neutralization in the system with A = 7.48 is not only faster than for the smooth surface with Q = -63, it also turns out on examination to be more efficient = -64.5 in Figure 2. This than for the neutral surface with (lads fast screening is evidently a very general feature of these models, but its precise magnitude is once again sensitive to the details of the local structure very near the surface. Another recurring motif of the RHNC theory of these charged surfaces has been a simple near additivity of highly localized short-range structural effects on one hand and long-range asymptotic behavior representative of a continuum solvent on the other. In systems with an adsorption potential there seems to be a similar additivity of even the different short-range effects. In particular, the local polarization of the solvent by the adsorbed charge appears to be largely independent of that due to the uniform surface charge. For example, the (cos 0,) profile of the system with A = 7.48, the dash-dot line in Figure 12, shows the same sharp rise at contact as for the identically charged nonadsorbing surface despite the adsorption in the former system of 129 negative charges. Yet, next to this adsorbed C1- at 1.16ds in the figure the negative peak in (cos e,,) is nearly the same as that in Figure 2 for a similar amount of adsorbed charge at a neutral surface rather than resembling, say, (cos 0,) for a neutral surface with a value of Pads corresponding to the same net charge of -66. This independence of the action of the adsorbed and intrinsic charges
Double-Layer Structure at an Ion-adsorbing Surface
< cos 0, >
The Journal of Physical Chemistry, Vol, 94, No. 10, 1990 4261
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is evident as well in the behavior of the solvent angular probability densities of these systems. Some of these are shown in Figures 13 [p(&)] and 14 [p(0,)] together with the corresponding results
Figure 14. Angular probability density ~ ( 0 , )for solvent at two distances r from contact with surfaces with intrinsic and adsorbed charges. Line types as for Figure 13. (a) r = 0; (b) r = 0.34d,.
for nonadsorbing surfaces with uniform charges of 0, -63, and +63. The inversion in p(0OH) between parts a and b of Figure 13 as well as the remarkable invariance of p(0,) at 0.34ds from contact in Figure 14b are the familiar signatures of the characteristic orientational order at an inert surface, as expected. The distributions of both angles for molecules in contact with the macroparticle, however, show polarization effects that are dominated by a response to the adjacent uniform surface charge and show very little influence from the adsorbed charge. In the case of 0, it is the sign of the intrinsic surface charge that determines which of the nearly equivalent peaks in the neutral surface distribution is amplified, with all three of the systems with Q = +63 giving similar results. Likewise, the depression of the peak in p(OoH) at 180' for such systems (and the accompanying enhancement of the peak near 70') indicates a substitution of some oxygen-lone pair bonds for OH bonds directed into the surface in response to the positive intrinsic charge, also regardless of adsorbed charge. Since a nonzero uniform surface charge always induces a sharp contact peak in (cos 0,) regardless of any adsorbed charge (Figures 7 and 12), the electrostatic potential for intrinsically charged systems with adsorption has the same large spike of opposite sign to the surface charge as it has for nonadsorbing charged surfaces. \ k * ( r ) for the three systems with Q = +63 are compared in Figure 15. The difference of 5.19 between the broken and dash-dot lines at -0.5d, represents the lowering of the nonadsorbing surface potential \k*, by the adsorbed charge of -128.8 for A = 7.48. This reduction is comparable in magnitude to the effect of a similar
4268
The Journal of Physical Chemistry, Vol. 94, No. 10, 1990
ii i
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1
1
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Figure 15. Total 3 * ( r ) for surfaces with intrinsic charge Q = +63 and various adsorbed charges. Line types as for Figure 11.
quantity of adsorbed charge on the potential at a neutral surface since q*sof -4.44 for Qads= -122.2 in Figure 9b is 4.56 below the small potential given by the theory for 0.1 M KCI at an inert surface. That is, these limited data suggest that the effect on qS of a fixed amount of adsorbed charge is not strongly dependent on the intrinsic surface charge density. However, these similar quantities of adsorbed charge were induced by quite different adsorption strengths, A = 7.48 for Q = +63 versus A = 9.1 1 for the neutral surface. This certainly leaves open the possibility that the charge dependence of *s, Le., the differential capacitance, for a fixed adsorption potential could depend strongly on that potential. A proper investigation of this question would require far more extensive data than we have generated in this study, however. Conclusion
The ability to construct accurate theories of wholly molecular models of bulk electrolyte solutions is making possible much improved treatments of the solution side of the electrified interface as well. Once the double layer can be modeled at such a level, however, the conventional treatment of the surface as smooth and structureless is no longer appropriate and will eventually have to be superseded by a fully integrated statistical mechanical treatment of both phases of the interface. Some initial work of this type for ion-dipole mixtures at a metallic surface has already appeared.'8,19 Our objective here was a more modest one, to probe
Wei et al. for possible artifacts in the behavior of a molecular solvent model of the solution phase arising from this naive model of the surface. Thus, our emphasis has been on replacing the uniformly charged smooth surface with a less regular one in which granularity of both charge and excluded volume of the surface are represented to some degree, rather than on the construction of a realistic model of ion adsorption in a particular real system. Happily, the results seem to confirm the generality of the most interesting features of the earlier RHNC work on smooth surfaces while invalidating just those aspects of the behavior at smooth surfaces that seemed the most artificial. In the former category are the rapid neutralization or fast screening of the surface charge and the remarkable stability and unique polarization properties of the ice-like orientational order of the solvent at a flat neutral surface. In the latter category are the quite high, narrow peaks that appear in the counterion and solvent density profiles and especially in the polarization profile at contact with a smooth charged surface. These arose from a highly artificial situation in which the perfectly smooth surface admits no screening whatsoever of its field on such contact particles. Although the bare field of any individual adsorbed ion in the present model is higher still than that of the smooth surface, the local reaction field of the adjacent coplanar solvent can act to reduce the Maxwell field even for a counterion in contact with the adsorbed ion in a way that is geometrically forbidden at a smooth charged surface. In this respect, the highly simplified model used here seems more faithful than a smooth surface to the topology of most real surfaces where such intermingling of solvent and charge centers must be the norm. The bizarre-looking spike in the mean electrostatic potential that arose from this contact effect at smooth surfaces is likewise not found in systems having only adsorbed surface charges. Even in these systems, however, q ( r ) still varies in some cases by as much as half a volt over a distance of one molecular diameter. The reconciliation of this behavior with the notion of a well-defined surface potential seems likely to be one of the principal challenges in the construction of a complete molecular theory of the charged interface.
Acknowledgment. This work has been supported in part by the Defence Research Board of Canada under ARP Award 3610-645 and in part by the Natural Sciences and Engineering Research Council of Canada. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. (18) Badiali, J . P.; Rosinberg, M. L.; Vericat, F.; Blum, L. J . Elecfroanul. Chem. 1983, 158, 253. (19) Schmickler, W.; Henderson, D. J . Chem. Phys. 1983, 80,3381.
