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Langmuir 1997, 13, 2608-2613
Origin of 1-pK and 2-pK Models for Ionizable Water-Solid Interfaces Michal Borkovec Swiss Federal Institute of Technology, ETH-ITO, Grabenstrasse 3, 8952 Schlieren, Switzerland Received December 27, 1996. In Final Form: March 13, 1997X
A discrete charge Ising model planar ionizable water-solid interface is developed. This model treats all individual ionizable sites explicitly and is solved with Monte Carlo simulation techniques. It is shown that a mean-field cluster expansion provides a very accurate scheme for the calculation of the titration curves in this system. From the first two iterations of this approximation scheme one recovers the classical 1-pK and 2-pK models. In most cases, the 1-pK model provides a sufficiently accurate description of the titration behavior of the interface, while in special situations the use of the 2-pK model may be required. The quick convergence of this approximation scheme is related to the long-range nature of the interaction potential between ionizable residues, a feature which is characteristic for water-solid interfaces.
metal center2,6,7
Introduction Ionization of water-oxide interfaces is often rationalized in terms of surface complexation models. Most commonly, such models assume two consecutive protonation steps1-3
XOH2+ h XOH + H+
(1)
XOH h XO- + H+ where X represents a surface group. In such a 2-pK model the surface group X is sometimes understood to represent a single metal center at the surface and both protons are assumed to bind sequentially to a single oxygen atom. However, this interpretation of eq 1 is incorrect.3 Whenever two protons bind to a single atom, the difference between the consecutive pK values is very large (∆pK > 10) and only one of the two protonation steps given in eq 1 can be operational in the entire experimentally accessible pH window. As a simpler interpretation, one may just consider a single protonation step and introduce the 1-pK model.3-5 For the water oxide interface such a model was derived by applying the valence bond principle to the metal oxide surface. For an oxygen coordinated to a single trivalent metal center the surface complexation model now only involves a single protonation step3,4
[-MeOH2]+1/2 h [-MeOH]-1/2 + H+
(2)
Basically, the same 1-pK model was introduced in the context of latex particles with one type of ionizable group.5 In this situation, however, the 1-pK model arises naturally, as only integer values of charges occur. The proper interpretation of the 2-pK model given in eq 1 considers a pair of neighboring ionizable groups, as for example two oxygen atoms coordinating a single X
[-Me(OH2)2]+ h -Me(OH2)OH + H+
(3)
-Me(OH2)OH h [-Me(OH)2]- + H+ This mechanism has the structure of eq 1, but now X refers to an entire -MeOH2 surface group. Such a picture immediately explains the small splitting of the consecutive pK values, as mostly derived from experimental oxide data (∆pK ∼ 2-4). As a basic unit of a 2-pK model, it would be equally reasonable to consider two neighboring oxygen atoms, each of which is coordinated to a different metal center. One might invoke the same idea to construct such a model for a latex surface with one type of ionizable groups: the basic unit would consist of two neighboring groups, and depending on its protonation state it would be either neutral, singly charged, or doubly charged. Adequate representations of experimental titration data of oxides (and other types of water-solid interfaces) are usually possible within the 1-pK model as well as the 2-pK model. But, clearly, both models give different results for certain parameters, and one wonders which model is actually the “better” model of a real ionizable watersolid interface. For water-oxide interfaces, the real situation is quite a bit more complicated. The surface of an oxide particle consists of various crystallographic planes. Even if these individual surface planes would be free of defects, they are heterogeneous, as they bear various nonequivalent singly, doubly, or triply coordinated oxygens atoms. The heterogeneity of the surface originates from the different intrinsic proton affinities of all these oxygen atoms.1,3,8,9 However, there is an additional complication, which not only arises for a water-oxide interface but already exists for a perfectly homogeneous planar interface composed of a single type of ionizable groups, such as a fatty-acid monolayer or the surface of a latex particle. In
Abstract published in Advance ACS Abstracts, April 15, 1997.
(1) Schindler, P. W.; Stumm, W. In Aquatic Surface Chemistry; Stumm, W., Ed.; Academic Press: New York, 1987. (2) Westall, J. In Aquatic Surface Chemistry; Stumm, W., Ed.; Academic Press: New York, 1987. (3) Hiemstra, T.; van Riemsdijk, W. H.; Bolt, G. H. J. Colloid Interface Sci. 1989, 133, 91. (4) Bolt, G. H.; van Riemsdijk, W. H. In Soil Chemistry. B. Physiochemical Models; Bolt, G. H., Ed.; Elsevier: Amsterdam, 1982. (5) Healy, T. W.; White, L. R. Adv. Colloid Interface Sci. 1978 9, 303.
S0743-7463(96)02132-4 CCC: $14.00
(6) Hingston, F. J.; Posner, A. M.; Quirk, J. P. J. Soil Sci. 1972, 23, 177. (7) Pulver, K.; Schindler, P. W.; Westall, J. C.; Grauer, R. J. Colloid Interface Sci. 1984, 101, 554. Ludwig, C.; Schindler, P. W. J. Colloid Interface Sci. 1995, 169, 284. (8) Hiemstra, T.; van Riemsdijk, W. H. J. Colloid Interface Sci. 1996, 179, 488. (9) Rustad, J. A.; Felmy, A. R.; Hay, B. P. Geochim. Cosmochim. Acta 1996, 60, 1563.
© 1997 American Chemical Society
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Langmuir, Vol. 13, No. 10, 1997 2609
such a system, whether a given site will be protonated or not will be determined not only by the solution pH but also by the individual protonation states of all neighboring sites. (The microscopic pK of a given ionizable group in a polyprotic molecule also depends on the protonation states of all other groups.) In the limit of an infinite lattice, the result of this competition is nontrivial. The calculation of the titration curve of an ionizable interface is a typical many-body problem, which is equivalent to the so-called Ising model on a lattice. Such problems are notoriously difficult to solve, and exact solutions are available for special situations only.10 One must therefore recognize that any simplified approach, such as the 1-pK or 2-pK model, necessarily does invoke some kind of approximations. However, the quality of these approximations has never been properly tested. This study aims precisely at this issue. We shall model the ionization of water-solid interfaces by formulating this problem in terms of the Ising model. Such a model treats all ionizable sites explicitly and has already been widely used to discuss the ionization of various polyprotic systems, such as proteins, polyelectrolytes, and the corresponding oligomers.11-17 The Ising model can be accurately solved by brute force Monte Carlo simulation techniques. Approximate solutions are also possible by means of a so-called mean-field cluster expansion.18,19 As will be shown below, the two lowest order approximations of this expansion lead precisely to the 1-pK model and the 2-pK model, respectively. Comparison with simulation results reveals that both approximate models usually provide an excellent description of the ionization of watersolid interfaces. The rapid convergence of this approximation scheme for ionizable water-solid interfaces is related to the fact that the site-site interaction potential acts over very long distances. The same type of approximation scheme is much poorer in systems where sitesite interaction potentials are short ranged. Discrete Charge Ising Model of Planar Water-Solid Interfaces Ionization of polyprotic systems can be modeled with the Ising model.11-17 This model treats all ionizable sites explicitly; the protonation state of each ionizable site i (i ) 1, ..., N) is described by a two-valued state variable si. This variable is chosen such that si ) 0 if the site is deprotonated and si ) 1 if the site is protonated. The ionization properties can be derived from a free energy which consists of two terms
1
Eijsisj ∑i µisi + 2∑ i*j
F (s1, ..., sN) ) -
(4)
The first term involves a sum over all sites where µi denotes the chemical potentials for each site. For identical sites µi ) µ the chemical potential µ is given by βµ/ln 10 ) pK (10) Chandler, D. Introduction to Modern Statistical Mechanics; Oxford University Press: New York, 1987. (11) Tanford, C.; Kirkwood, J. C. J. Am. Chem. Soc. 1957, 79, 5333. (12) Bashford, D.; Karplus, M. Biochemistry 1990, 29, 10219. (13) Honig, B.; Nicholls, A. Science 1995, 268, 1144. (14) Ullner, M.; Woodward, C. E.; Jo¨nsson, B. J. Chem. Phys. 1996, 105, 2056. (15) Marcus, R. A. J. Phys. Chem. 1954, 58, 621. (16) Smits, R. G.; Koper, G. J. M.; Mandel, M. J. Phys. Chem. 1993, 97, 5745. (17) Borkovec, M.; Koper, G. J. M. J. Phys. Chem. 1994, 98, 6038. Borkovec, M.; Koper, G. J. M. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 764. (18) Burley, D. M. In Phase Transitions and Critical Phenomena; Domb, C., Green, A. B., Eds.; Academic Press: New York, 1985; Vol. 2. (19) Gilson, M. K. Proteins 1993, 15, 266.
Figure 1. Schematic representation of discrete charge Ising model of a planar ionizable water-solid interface. The solid circles represent the ionizable sites which are arranged on a regular lattice within a solid medium with a dielectric constant d. This solid is in contact with an aqueous electrolyte solution with a dielectric constant w and a Debye screening length κ-1. Top view of (a) individual sites on a square lattice and (b) pairs of closely spaced sites on a rectangular lattice. (c) Side view.
- pH, where pK and pH are the negative common logarithms of the microscopic dissociation constant of the ionizable group and of the proton activity, respectively. The second term involves a sum over all pair interaction energies, which are, to a first approximation, of electrostatic origin. The interaction energy of two sites i and j is given by Eij ) W(|ri - rj|), where W(r) > 0 represents the electrostatic pair interaction energy of two unit charges at a distance r ) |ri - rj| apart (ri denotes the position of site i within the planar lattice). For a planar interface (see Figure 1) it is possible to calculate the site-site interaction potential W(r) within the linearized Poisson-Boltzmann approximation explicitly.20 We approximate the ionizable sites by point charges, which are buried in a solid of dielectric constant d at a distance a away from a planar interface to the electrolyte solution of dielectric constant w. For this geometry, the resulting interaction potential between two elementary charges can be written as
W(r) )
[
e2 1 4π0d r
p- k
]
∫0∞wwp + ddk J0(kr) e-2ak dk
(5)
where p ) (k2 + κ2)1/2, κ is the inverse Debye length of the electrolyte, and J0(z) is a Bessel function of the first kind.21 The first term corresponds to a bare Coulomb interaction in the dielectric medium, while the second term represents a polarization contribution. The potential given by eq 5 can be evaluated numerically in a straightforward fashion. One observes the following behavior. At small separation distances, the polarization term can be neglected and the bare Coulomb term dominates the interaction potential. At large distances, on the other hand, the polarization term almost cancels the Coulomb term and the overall potential decays as r-3. This large distance behavior arises, since the point charge inside the dielectric generates its image charge of opposite sign in the electrolyte, and the electrostatic potential induced by these two charges decays in a dipole-like fashion at large distances. This slow decay of the interaction potential causes a given ionizable site in a water-solid interface to interact with a large number of neighbors; an aspect which will become important later. The titration curve, which is of direct experimental interest, represents a plot of the average degree of (20) Stillinger, F. H. J. Chem. Phys. 1961, 35, 1584. Richmond, P. J. Chem. Soc., Faraday Trans. 2 1974, 70, 1067. Medina-Noyola, M.; Ivlev, B. I. Phys. Rev. E 1995, 52, 6281. (21) Gradsteyn, I. S.; Ryzhik, I. M. Table of Integrals, Series, and Products; Academic Press: New York, 1980.
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Letters
a polyprotic molecule. However, we assume that all sites, which do not belong to this cluster, do influence the sites inside the cluster only through their average degree of protonation. At this point, we are making an approximation. Each site is influenced by the actual protonation states of the individual sites and not just through their average. But we shall see later that this (mean-field) approximation can be excellent indeed. Consider now the free energy (eq 7) for such a cluster. The state variables si of the sites which belong to the cluster are kept, but we replace all si for the sites which do not belong to the cluster by its average value 〈si〉 ) θi. Collecting terms, we find that the free energy of a cluster is again given by eq 7, but with two modifications. (i) Both sums over the state variables run over sites inside the cluster only; the total number of sites N must be reinterpreted as the number of sites within the cluster. (ii) The chemical potential µi, which enters eq 7, now also includes the mean-field contributions and must be replaced by an effective chemical potential19
µ˜ i ) µi -
∑j Eh ijθj
(8)
where coupling energies are given by Figure 2. Pictorial representation of the 1-pK, 2-pK, and 4-pK models on a square lattice. The outlined sites represent the cluster where the interactions between the sites are treated explicitly; the shaded regions represent the remaining sites, which are treated in the mean-field approximation.
protonation θ as a function of the pH (or the chemical potential µ). This quantity can be written as
θ)
1
N
(6)
where the average degree of protonation of an individual site is given by the thermal average of the corresponding state variable
∑s ,...,s sie-βF (s ,...,s ) θi ) 〈si〉 ) ∑s ,...,s e-βF (s ,...,s ) 1
N
N
1
1
∑R W(|rR + xi - xj|)
(9)
where rR is the lattice vector of the superlattice of the clusters and xi are the relative positions of each site within the cluster. The index R runs over all lattice vectors except in the case of i ) j, where the zero vector is omitted in the summation. 1-pK Model
∑θi N i)1
1
E h ij )
N
(7)
N
for i ) 1, ..., N.
Let us first assume that our imaginary cluster consists of a single site only (see Figure 2). The protonation of the single cluster site proceeds in a single step, as in the case of a monoprotic acid. As this site is part of an ionizable surface, the ionization of this site is modified by the presence of all neighboring sites. In the mean-field approximation, the equilibrium constant becomes a function of the average degree of protonation of the neighboring sites. For a single site within the cluster, we exactly recover the isotherm of the 1-pK model with a constant capacitance term. Inserting eqs 4 and 8 into eq 7, in our notation the isotherm becomes
Mean-Field Cluster Expansion Without introducing any approximations, the titration curve of a system involving a large number of ionizable sites can be calculated by means of the Monte Carlo simulation technique.10,22 Until recently, the application of this technique did require much computer time, and therefore, various approximate solution schemes have been proposed in the past. The most important among these is the mean-field approximation. In this approximation, one normally considers just a single site explicitly, and the presence of all neighboring sites is approximated by the mean field generated by these sites.10 But here we shall discuss a generalization of this approach where entire clusters of sites are considered.18,19 Applied to our problem of an ionizable interface, such a mean-field cluster expansion works as follows (see Figure 2). Let us split the lattice into imaginary clusters. All clusters are identical and may consist of one or more sites. Now, pick one of these clusters. All individual sites within the cluster are treated explicitly as if they were part of (22) Binder, K.; Landau, D. P. Adv. Chem. Phys. 1989, 76, 91. (23) Ja¨ger, I. Surf. Sci. 1991, 254, 300.
θ)
z 1+z
(10)
where z ) eβµ˜ and the effective chemical potential is given by
µ˜ ) µ - E hθ
(11)
where eq 9 reduces to the full lattice sum
E h )
1
∑W(|ri - rj|)
N i*j
(12)
which defines the strength of the mean field. Note that we have dropped the indices, as we are dealing with a single site. This expression corresponds to the sum of all pair interactions which are felt by a given site. This energy defines the overall strength of the mean-field potential which is generated by all other sites. 2-pK Model Now, let us assume that the cluster consists of a pair of neighboring sites (see Figure 2). The existing interaction between these two sites will cause the protonation of
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Langmuir, Vol. 13, No. 10, 1997 2611
the entire cluster to proceed in two steps, as in the case of a diprotic acid. The other sites, which do not belong to the cluster, influence these ionization equilibria. Within the mean-field approximation, both equilibrium constants become functions of the average degrees of protonation for the two different sites. For two neighboring sites within the cluster, the 2-pK model is now recovered. Inserting eqs 4 and 8 into eq 7, we obtain a coupled set of equations
θ1 )
z1 + uz1z2 1 + z1 + z2 + uz1z2
(13)
θ2 )
z2 + uz1z2 1 + z1 + z2 + uz1z2
(14)
where z1 ) eβµ˜ 1, z2 ) eβµ˜ 2, and u ) e-βE′ with E′ being the pair interaction energy between both sites within the cluster. The effective chemical potentials are now given by
h 11θ1 - E h 12θ2 µ˜ 1 ) µ1 - E
(15)
h 21θ1 - E h 22θ2 µ˜ 2 ) µ2 - E Even in the case of identical sites (µ1 ) µ2 ) µ) it is possible that eqs 13 and 14 have multiple solutions (i.e., θ1 ) θ2 and θ1 * θ2), but here we shall consider the case where θ ) θ1 ) θ2. The isotherm is now given by
θ)
z + uz2 1 + 2z + uz2
(16)
h ′θ with E h′ ) E h - E′. Equation where z ) eβµ˜ and µ˜ ) µ - E 16 has precisely the structure of the isotherm obtained from the classical 2-pK model with a constant capacitance term. However, the mean-field interaction energy E h ′ turns out to be smaller than that in the 1-pK model derived above. While, in the 1-pK model, all interactions are treated within the mean-field approximation, in the 2-pK model, one pair interaction is treated explicitly, and precisely this interaction must be omitted in the calculation of the mean field. Clearly this approximation scheme can be continued. We can consider clusters with increasing number of sites and thus devise a 3-pK, 4-pK model, etc. (see also Figure 2). These models will become increasingly complicated but more accurate approximations of the exact solution of the ionization problem with an infinite number of sites. However, one should realize that for a given type of interaction matrix Eij it is far from obvious whether this approximation scheme converges rapidly or slowly toward the true solution of the Ising model. Model Results Let us now discuss two examples to illustrate the effectiveness of this approximation scheme for watersolid interfaces. Recall that we are only considering interfaces with rather low charge densities such that the linearized Poisson-Boltzmann approximation remains valid. We assume the charges to be buried at a depth of a ) 0.25 nm in a solid of a dielectric constant d ) 3. The planar surface of this solid is in contact with a monovalent electrolyte of an ionic strength of 0.5 M (κ-1 = 0.43 nm) and a dielectric constant d ) 80. Furthermore, we assume that identical sites with a microscopic pK ) 10 are arranged in a regular lattice and that they are uncharged in the deprotonated state. (Note that, in the linearized model, the charge of the sites only affects the microscopic
Figure 3. Titration curves of ionizable sites arranged on the regular latices calculated by Monte Carlo and from 1-pK, 2-pK, and 4-pK models. (a) Identical sites on a square lattice (Figure 1a) and (b) pairs of sites on a rectangular lattice (Figure 1b). Note that the simulation data are described well with all models in part a, while for part b the 1-pK model does not represent the two-step ionization behavior.
pK value, which results only in a parallel shift of the titration curve along the pH axis. Thus the presented results are more general.) The chosen system parameters are meant to be illustrative; we do not intend to provide an accurate representation of any realistic system. Two situations will be discussed here: (i) sites are arranged in a square lattice with a lattice constant of 1.1 nm, and (ii) pairs of sites with a mutual distance 0.6 nm are arranged horizontally on a rectangular lattice with a unit cell of 2.0 nm times 1.6 nm (see Figure 1a and b). Case i represents the simplest plausible situation, where identical sites are arranged on a simple lattice. Case ii models the situation given in eq 3; pairs of strongly interacting sites are arranged in a simple lattice. In both cases titration curves were calculated by Monte Carlo simulation on lattices with periodic boundary conditions consisting of up to 400 sites. By varying the total number of sites, it has been verified that the systems considered were independent of the system size and thus representative for the N f ∞ limit. Figure 3 compares the simulation results with the results based on the 1-pK, 2-pK, and 4-pK models. Let us first consider results for the square lattice. As evident from Figure 3a the simulation results are described almost perfectly with all mean-field models. Already the simple 1-pK model does extremely well; the maximum deviation in the degree of protonation is about 0.02. The 2-pK model performs similarly, while the 4-pK model describes the Monte Carlo data to better than (0.01. In this case all models give basically the same results, a feature which was already pointed out earlier.4 The parameters of these models are summarized in Table 1. This table also illustrates the workings of the mean-field cluster expansion. If the cluster contains just a single site, all the site-site interactions are buried in the strength
2612 Langmuir, Vol. 13, No. 10, 1997
Letters
Table 1. Summary of Parameters of the Various Mean-Field Models for a Regular Lattice of Identical Sites with Microscopic pK ) 10 βEij/ln 10b model i
a
pKi
1
2
3
βE h ij/ln 10c 4
1
2
3
4
1-pKd 1 10.00
7.32
1 10.30 0.75 2 8.95 0.75
3.27 3.29 3.29 3.27
2-pK 4-pK 1 10.60 0.75 0.75 0.30 0.98 1.55 1.55 2 9.63 0.75 0.30 0.75 1.55 0.98 1.45 3 8.56 0.75 0.30 0.75 1.55 1.45 0.98 4 7.59 0.30 0.75 0.75 1.45 1.55 1.55
1.45 1.55 1.55 0.98
1-pKe 1 10.00
8.08
1 10.30 3.39 2 6.31 3.39
2.30 2.39 2.39 2.30
2-pK 4-pK 1 10.60 3.39 0.39 0.31 0.83 1.00 1.08 2 9.65 3.39 0.31 0.39 1.00 0.83 1.08 3 6.25 0.39 0.31 3.39 1.08 1.08 0.83 4 5.31 0.31 0.39 3.39 1.08 1.08 1.00
1.08 1.08 1.00 0.83
a Macroscopic pK values of the isolated cluster. b Pair interaction energies within the cluster. c Mean-field pair interaction energies generated by sites outside the cluster. d Sites on a square lattice (see Figures 1a and 3a). e Pairs of sites on a rectangular lattice (see Figures 1b and 3b).
of the mean-field. As one moves to larger and larger clusters, more and more site-site interactions are treated explicitly. The explicit consideration of individual interactions between sites (as measured by the pair interaction energies Eij) leads to different macroscopic pK values of the individual cluster. At the same time, the overall strength of the mean field (as measured by the coupling parameters E h ij) becomes smaller. Note that for all models the sum of all interactions always remains the same and is given by eq 12. Only their partitioning between the inside and the outside of the cluster changes. The broad one-step ionization behavior shown in Figure 3a is very generic for identical sites arranged in a simple lattice on a planar ionizable water-solid interface. The classical mean-field approximation, or the 1-pK model, represents an excellent description this situation. Such behavior is expected for most types of simple lattices and realistic parameters (e.g., site density, ionic strength, dielectric constants, etc.). This behavior can be captured with two parameters only: a microscopic pK value and the strength of the mean field E h . In this generic situation, which certainly covers most cases of interest, the 1-pK model is entirely sufficient and there is no need to consider the 2-pK model or any more elaborate approaches. Let us now consider a different lattice structure, as shown in Figure 3b. Again, we have arranged identical sites on a lattice, but now these sites were placed such that pairs of two sites are always very close and interact strongly. This setting may mimic the situation anticipated in eq 3. Within the pairs, site-site interaction energies are substantially larger than interaction energies between any other pairs in the lattice. In this example, the Monte Carlo results are not well described with the simple 1-pK model, and it is necessary to resort to the 2-pK model in order to obtain an acceptable agreement with the simulation data. The 4-pK model describes the situation equally well. Within our description of the water-solid interface, it must be stressed that a situation as shown in Figure 3b can only arise if a given site interacts with one particular site more strongly than with all the remaining ones. Such
a situation may indeed arise, if two oxygens are coordinated to a single metal center, but the overall site density is rather low and thus all other interactions remain sufficiently weak. Note that for metal oxide surfaces, such an arrangement of sites is not very common but possible. One such example is the 001 face of rutile. Even in this setting, the interactions may not be sufficiently distinct, and one may be confronted with the simple situation shown in Figure 3a rather than with the situation shown in Figure 3b. It must be stressed again that all the results presented here are intended to be illustrative; they are not meant to describe any realistic systems quantitatively. Nevertheless, the present model is quite realistic for latex particle surfaces5,24 or for ionizable fatty acid monolayers25 with site densities less than 1 nm-2. For higher site densities, as typical for water-oxide interfaces,3,8 the present description will only be applicable in the immediate vicinity of the point of zero charge. Unfortunately, an extension of the present approach to higher charge densities is rather difficult. First of all, at shorter distances (say below 0.5 nm), the interaction potential W(r) will be modulated by the molecular surface architecture, and eq 5 will thus apply at larger distances only. (For the same reason, in the example shown in Figures 1b and 3b the smallest distance of 0.6 nm between the ionizable groups should be interpreted with caution; one should rather consider the strength of the site-site interaction Eij, which corresponds to ∆pK = 4, see Table 4). In addition to this problem, the linearization of the Poisson-Boltzmann equation will be no longer permissible for potentials above 25-50 mV; the Ising free energy (eq 4) must be extended with higher order terms, such as triplet interactions.16 As effects of surface charge heterogeneities do weaken in the nonlinear Poisson-Boltzmann regime,26 one may thus speculate that the mean-field approximation could remain sound at high surface potentials as well. The same generic behavior as shown in Figure 3a would thus be expected to persist to much higher site densities. High surface potentials are typical for water-oxide interfaces; the titration curves broaden beyond the accessible pH window, and it will be impossible to titrate an ionizable site fully. However, the detailed shape of the titration curve in the nonlinear regime will differ from that of the linearized model discussed here. This difference will be analogous to the difference between the basic Stern model and the constant capacitance model.2 Discussion By considering a discrete charge Ising model, we have shown that simple mean-field models, which are commonly referred to as the 1-pK or the 2-pK model, usually provide an excellent description of the ionization process of watersolid interfaces. The success of such a simple mean-field description may seem surprising; for Ising models on various two-dimensional lattices it is well-known that mean-field models give poor results; particularly, if small clusters such as the ones we have employed here are considered.22,23 There is, however, one fundamental difference between the Ising models discussed often in the literature and the present situation. The site-site interaction potentials of the Ising models studied in the (24) Schubin, V. E.; Isakova, I. V.; Sidorova, P. M.; Men’shikova, P. M.; Evseeva, T. G. Kolloidn. Zh. 1990, 52, 935. (25) Zhao, X.; Ong, S.; Wang, H.; Eisenthal, K. B. Chem. Phys. Lett. 1993, 214, 203. Lovelock, B.; Grieser, F.; Healy, T. W. Langmuir 1986, 2, 443. Drummond, C. J.; Grieser, F.; Healy, T. W. J. Phys. Chem. 1988, 92, 2604. (26) Miclavic, S. J. Colloid Interface Sci. 1996, 171, 446.
Letters
Figure 4. Cumulative sum of the pair energies for the square lattice as a function of total number of nearest neighbors. Note the slow convergence to the plateau value which is also given in the first row of Table 1.
literature are usually short range, mostly just of the nearest neighbor type.10,22,23 This means that, only between sites that are nearest neighbors, the energy of interaction is nonzero. In the context of ionization problems, such short-range potentials are typical for polyelectrolytes. For a planar ionizable interface, on the other hand, the interaction potential is long ranged and decays slowly with distancesa given site interacts appreciably with a large number of neighboring sites. Figure 4 illustrates this point for the present example of a square lattice, where the cumulative contribution of nearest neighbors to the lattice sum given in eq 12 is shown. In order to account for the total interaction energy within kT, one must consider more than 100 nearest neighboring sites, which corresponds to a maximum distance of about 10 nm. But these are precisely the conditions under which a mean-field model should be applicable: a given site must
Langmuir, Vol. 13, No. 10, 1997 2613
interact with a large number of neighboring sites. In the case of nearest neighbor pair interactions (short-range interactions), for a square lattice a mere four nearest neighboring sites account for all interactions. In such a situation, a mean-field model performs poorly.22,23 In conclusion, on the basis of statistical mechanical arguments it has been shown that, for planar ionizable interfaces, mean-field approximations inherent to all surface complexation models are sound. The basic reason for the validity of the mean-field approximation is the long-range character of the site-site interaction potential. For most water-solid interfaces a simple 1-pK model, where only a single ionizable site is treated explicitly, will represent an excellent approximation of the ionization behavior. Essentially the same results can also be obtained within a 2-pK model, where two neighboring ionizable sites are treated explicitly, but this model is more complicated and invokes a larger number of parameters. Under certain conditions, however, it is conceivable that a 1-pK model might fail and the consideration of a 2-pK model (or possibly a 3-pK or 4-pK model) becomes essential for a proper description of the titration behavior. Such a description will become necessary if the surface is composed of small groups of sites (as in eq 3, for example) which do interact much more strongly among each other than with all other neighboring sites within the lattice. However, for an ionizable water-solid interface, such a situation is somewhat exceptional and the rule is certainly the simple behavior shown in Figure 3a. Acknowledgment. Pleasing and fruitful discussions with John Daicic, Ger Koper, Paul Schindler, Willem van Riemsdijk, and John Westall are acknowledged. LA9621325
