Bond-valence methods for pKa prediction: critical reanalysis and a new approach

Pergamon

Geochimica et Cosmochimica Acta, Vol. 68, No. 9, pp. 2025–2042, 2004Copyright © 2004 Elsevier Ltd

Printed in the USA. All rights reserved0016-7037/04 $30.00 � .00

doi:10.1016/j.gca.2003.11.008

Bond-Valence methods for pKa prediction: critical reanalysis and a new approach

BARRY R. BICKMORE,1,* CHRISTOPHER J. TADANIER,2 KEVIN M. ROSSO,3 WILL D. MONN,1 and DENNIS L. EGGETT4

1Department of Geology, Brigham Young University, Provo, UT 84602-4606, USA2Department of Geological Sciences and Charles E. Via Department of Civil and Environmental Engineering, Virginia Tech,

Blacksburg, VA 24061, USA3Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, P.O. Box 999, MSIN K8-96, Richland, WA 99352, USA

4Center for Statistical Consultation and Collaborative Research, Department of Statistics, Brigham Young University, Provo, UT 84602, USA

(Received July 14, 2003; accepted in revised form November 11, 2003)

Abstract—Bond-valence methods for the prediction of (hydr)oxide solution monomer and surface functionalgroup acidity constants are examined in light of molecular structures calculated using ab initio methods. Anew method is presented that is based on these calculated structures, and it is shown that previously publishedmethods have neglected one or more of four essential features of a generalized model. First, if the apparentpKa values of solution monomers are to be used to predict intrinsic pKa values of surface functional groups,similar electrostatic corrections must be applied in both cases. In surface complexation models, electrostaticcorrections are applied by representing a charged surface as a uniform plane of charge density, and ananalogous correction can be made to solution monomers by treating them as charged spheres. Second, it mustbe remembered that real surfaces and real monomers are not homogeneous planes or spheres. Rather, chargedensity is distributed rather unevenly, and a further electrostatic correction (which is often quite large) mustbe made to account for the proximity of electron density to the point of proton attachment. Third, theunsaturated valence of oxygen atoms in oxyacids, hexaquo cations, and oxide surfaces is strongly correlatedwith acidity after electrostatic corrections are made. However, calculation of unsaturated valence for oxyacidsand oxide surfaces must be based on realistic Me–O bond lengths (taking into account bond relaxation), whichcan be obtained from ab initio structure optimizations. Finally, unsaturated valence must be divided betweenpossible bonds (four for oxygen atoms) to reflect the fact that O-H bonds are localized to particular regionsof the O atoms.

Empirical models that take all these factors into account are presented for oxyacids and hexaquo cations.These models are applied to the gibbsite (100), (010), (001), and cristobalite (100) surfaces, and it isdemonstrated that the model for oxyacids predicts reasonable intrinsic pKa values for oxide surfaces.However, the prediction of surface pKa values is complex, because the protonation state of one functionalgroup affects the pKa values of neighboring groups. Therefore, calculations of larger periodic systems,
progressively protonated and reoptimized, are needed. Copyright © 2004 Elsevier Ltd
1. INTRODUCTION

Acid-base reactions are among the most fundamental chem-ical interactions in natural systems, but the accessibility ofthese reactions to experimental investigation varies widely.Particularly troubling is the problem of isolating the contribu-tion of specific functional groups on complex solution speciesand solid surfaces to the total acidity. In the case of solidsurfaces, surface complexation models (SCMs) include elec-trostatic parameters that are usually experimentally inaccessi-ble, along with “intrinsic” acidity constant (pKa

int) values forsome set of functional groups. Potentiometric titration data aretypically explained with an SCM by numerically fitting theelectrostatic parameters. In addition, both the pKa

int values ofthe functional groups and even the number of sites are usuallyfitted, and this abundance of fitted parameters has the unfortu-nate consequence that any number of models can be made topredict titration data accurately, even when generic functionalgroups are hypothesized that have little basis in reality (Westalland Hohl, 1980; Davis and Kent, 1990; Dzombak and Morel,1990; Stumm, 1992; Kosmulski, 2001).

* Author to whom correspondence should be addressed
([email protected]).
2025

A convenient way to reduce the number of fitted parametersin SCMs is to estimate the distribution of sites and surface pKa

values theoretically for the different types of functional groups.The distribution of sites can be estimated by combining crys-tallographically derived models of the dominant surfaces on acrystalline sample with morphologic data regarding the propor-tion of total surface area accounted for by these surfaces. Theestimation of surface pKa values is more complicated, butattempts have been made to calibrate empirical models on thepKa values of solution monomers, which are more easily mea-sured. These empirical models have then been applied to sur-face functional groups with some success (Hiemstra et al.,1989, 1996; Rustad et al., 1996a, 1996b; Wasserman et al.,1999; Tossell and Sahai, 2000; Rustad, 2001; Bickmore et al.,2003).

The most successful and popular of these methods (Hiemstraet al., 1996) uses “bond valences” to predict pKa values, as doesthe method of Bickmore et al. (2003). The concept of bondvalence has eluded a strict physical interpretation, but has beenvery successful as a quantitative empirical description of the“bonding power” involved in cation-anion interactions. Thetotal valence of an ion is its formal charge, and the averagevalence of bonds extending from a cation or anion in a crystal
is equivalent to Pauling’s electrostatic bond strength (EBS):

2026 B. R. Bickmore et al.

EBS �zi

CNi(1)

where zi is the formal charge (total valence) of the ion and CNi

is the coordination number (Pauling, 1960). In real moleculesand crystals, however, coordination polyhedra are often dis-torted, and the strength of different bonds extending from anion may be substantially dissimilar. To account for this, Brownand Altermatt (1985) related bond valence to bond length byfitting an empirical expression to crystallographic data forthousands of mineral structures. The generalized expression is

s � exp� �R0–R�

B � (2)

where s is bond valence in valence units (v.u.), R is bond length(A), and R0 and B are fitting parameters specific to a givencation-anion pair. B is usually equal to 0.37 A. This formulationhas proven very helpful in determining crystal structures andpredicting the strength of various chemical interactions(Brown, 2002).

The appeal of bond-valence methods for pKa prediction isthat structural information, which is often fairly simple toobtain, can be translated directly into predictions of chemicalbehavior. However, Bickmore et al. (2003) pointed out thatthese methods have not always been used to their full potential.Whereas bond length/valence correlations are calibrated tobond lengths in crystalline structures, which can be determinedquite accurately, simplified structures (which do not account forbond relaxation) for solution monomers and solid surfaces havebeen employed to predict chemical behavior. Bond lengths ofrelaxed structures can be determined using ab initio structureoptimizations.

Ideally, such phenomenological models would not be neces-sary, and pKa values would be obtained directly from ab initioenergy calculations. But while this is theoretically possible,deficiencies in currently available treatments of solvation makeits accuracy questionable (Lim et al., 1991; Tossell and Sahai,2000; Rustad, 2001). Only very recently have ab initio calcu-lations been used to predict Ka values of small organic mole-cules in water to an accuracy of �0.5 log units (Liptak andShields, 2001; Liptak et al., 2002; Saracino et al., 2003).

In this paper we critically reexamine several published bond-valence methods for pKa prediction as applied to oxyacids andoxide surfaces, and then outline a new approach based on bondlengths from quantum mechanical calculations. The newmethod addresses some of the shortcomings of previous modelsby explicitly considering (1) realistic metal-oxygen (Me–O)bond lengths and (2) oversimplifications in the treatment ofelectrostatic considerations in surface complexation modelsand the interpretation of solution pKa values.

2. METHODS

Previously published bond-valence methods have first calibratedcorrelations with bond valence on the pKa values of solution mono-mers, which are more easily determined than those of individualsurface functional groups. In this study, these methods are criticallyevaluated using ab initio calculated structures of solution monomers. Anew approach to the problem is derived using the calculated structuresof the monomers, and then applied to calculated structures of solid
surfaces. This new approach requires Me–O bond lengths obtained
from ab initio calculations and associated bond valences, Me–H dis-tances for solution species, and estimates of Me–O bond ionicity. Inaddition, the model was calibrated against measured pKa values of thesolution monomers examined. The following subsections describe howthese parameters were obtained, whereas discussions of how they areapplied to the problem of pKa prediction are left to the “Results andDiscussion” section.

2.1. Me–O (and Me–OH) Bond Lengths and Valences inSolution Monomers

Me–O bond lengths for oxyacids and Me–OH bond lengths forhexaquo cations were needed to calculate bond valences using Eqn. 2,and to estimate Me–H distances in solution monomers. Me–O bondlengths for solution monomers were taken from the literature whereboth calculated structures and reliable experimental pKa values couldbe obtained. Calculated gas-phase structures for tetrahedral oxyacidmolecules, optimized using density functional theory (DFT) with theB3LYP hybrid Hartree-Fock/DFT functional, were taken from Rustadet al. (2000). Gas-phase structures were chosen because they are morereadily available, and because solvation should not affect Me–O bondlengths much, due to the inability of polar water molecules to hydrogenbond with the central metal cation (e.g., see the bond lengths forsolvated and gas-phase silicic acid molecules in Sefcik and Goddard,2001). However, in the future solvation will be explicitly included inour model as ab initio molecular dynamics simulations become morefeasible for large systems like solvated surfaces. Calculated structuresof hexaquo cations were taken from Rosso and Morgan (2002) andRosso et al. (2002). In Rosso and Morgan (2002), optimization wasperformed using the B3LYP functional; and solvation of the molecule(second shell water and beyond) was simulated using a polariziblecontinuum model. In Rosso et al. (2002), optimization was performedin the gas-phase using the local density approximation. See Table 1 forspecific values.

The bond lengths were translated into bond valence values usingEqn. 2. Calculated values of s (in v.u.) and the R0 values used to obtainthem (Brown and Altermatt, 1985) are reported in Table 1. Where morethan one deprotonated group existed on a conjugate base, the one withthe longest Me–O bond was included in the table, because this wouldbe the one with the most unsaturated valence (i.e., valence not used forbonding,) and hence the most attraction for a proton.

2.2. Me–H Distances in Solution Monomers

Me–H distances for the acids among the acid-base pairs investigatedwere needed to estimate the electrostatic work required to move aproton away from conjugate base molecules. The starting distancebetween the proton and the center of the charged base (assumed to bea homogeneous sphere) was assumed to be equal to the Me–H distancein the acid molecule before proton removal. Only Me–O bonds wereavailable in the literature, however, so they were estimated by assum-ing that oxygen and hydrogen atoms adopt the bond angles and dis-tances of a water molecule, and that a line cutting through the centersof the metal and oxygen atoms would bisect the water dipole (seeHiemstra et al., 1989). See Table 1 for Me–H distances calculated inthis manner.

2.3. Surface Structures

Optimized bulk and surface structures of gibbsite (001), (100), and(010) and cristobalite (100) were calculated using the pseudopotentialplane wave density functional theory method as implemented inCASTEP (Payne et al., 1992). The approach applied here is, for themost part, identical to that which was reported previously (Rosso et al.,2001; Bickmore et al., 2003). Thus, only new details pertinent to thecurrent calculations will be described here. The generalized gradientapproximation (GGA) was applied using the Perdew-Wang (Perdewand Wang, 1992) parameterization of the exchange-correlation func-tional, modified to work with plane wave calculations (White and Bird,1994). We used the CASTEP parameterization of ultrasoft pseudopo-tentials (Vanderbilt, 1990) without core corrections. Optimization wasperformed using a cutoff energy of 380 eV and a conjugate gradient
electronic minimizer using a density mixing scheme (Kresse and Furth-

2027Bond-valence methods for pKa prediction

muller, 1996). One k-point was used (� point). Cell parameters andatomic coordinates of the bulk structures were optimized withoutsymmetry restrictions (i.e., P1 symmetry).

For cristobalite (100), centrosymmetric vacuum-terminated surfaceslabs were created from optimized bulk structures. Protons were addedto the surface Si–O sites in various configurations to be included in theall-atom optimizations, as will be discussed in more detail below in the“Results and Discussion” section. Because these calculations alwayshave three-dimensional periodic boundary conditions, surfaces wereconstructed by building in a vacuum layer (Rosso, 2001). The cristo-balite (100) slab was �10 A thick. The vacuum layer separating themodels between adjacent cells was always � 10 A. The surface wasoptimized with fixed optimal bulk cell parameters using the inversioncenter symmetry (i.e., P1� symmetry). Various numbers of protons wereadded to the surface to investigate bond length dependencies on pro-tonation states. Some proton configurations yielded a charge neutralunit cell; others did not. For the latter, the uniform background chargemethod was used to neutralize the unit cell.

Optimization of gibbsite surfaces was performed in a similar fashionexcept, because of its highly anisotropic structure, only the basalsurface was optimized as a two-dimensional slab as described above.

Table 1. Several parameters used to predict pKaint values for solution

acid-base reaction, we include the Me–O bond length (Me–OH for hethe reaction (Rustad et al., 2000; Rosso et al., 2002; Rosso and Morga(R) in Eqn. 2 to calculate the valence of this bond. The Me–H bond ler0 in Eqn. 12 to calculate the electrostatic work involved. The Me–O bthe NIST database (National Institute of Standards and Technology, 1where noted.

Acid f Base � H�Me–O

(Å) R0

H3AsO4 f H2AsO4– � H� 1.654 1.798

H2AsO42– f HAsO4

2– � H� 1.709 1.798HAsO4

2– f AsO43– � H� 1.755 1.798

H2CrO4 f HCrO4– � H� 1.606 1.799

HCrO4– f CrO4

2– � H� 1.657 1.799H3PO4 f H2PO4

– � H� 1.507 1.653H2PO4

– f HPO42– � H� 1.561 1.653

HPO42– f PO4

3– � H� 1.606 1.653H4SiO4 f H3SiO4

– � H� 1.563 1.652H3SiO4

– f H2SiO42– � H� 1.607 1.652

H2SO4 f HSO4– � H� 1.486 1.676

HSO4– f SO4

2– � H� 1.526 1.676H3VO4 f H2VO4

– � H� 1.623 1.793H2VO4

– f HVO42– � H� 1.687 1.793

HVO42– f VO4

3– � H� 1.740 1.793

Acid f Base � H�Me–OH

(Å)

V3� · 6H2O f VOH2� · 5H2O � H� 1.819Cr3� · 6H2O f CrOH2� · 5H2O � H� 1.839CrOH2� · 5H2O f Cr(OH)2

� · 4H2O � H� 1.906Fe3� · 6H2O f FeOH2� · 5H2O � H� 1.781FeOH2� · 5H2O f Fe(OH)2

� · 4H2O � H� 1.923Mn2� · 6H2O f MnOH� · 5H2O � H� 1.992Fe2� · 6H2O f FeOH� · 5H2O � H� 1.881FeOH� · 5H2O f Fe(OH)2 · 4H2O � H� 2.041Co2� · 6H2O f CoOH� · 5H2O � H� 1.901CoOH� · 5H2O f Co(OH)2 · 4H2O � H� 2.004Y3� · 6H2O f YOH2� · 5H2O � H� 1.971Ga3� · 6H2O f GaOH2� · 5H2O � H� 1.778In3� · 6H2O f InOH2� · 5H2O � H� 2.006Rh3� · 6H2O f RhOH2� · 5H2O � H� 1.958Sc3� · 6H2O f ScOH2� · 5H2O � H� 1.810Al3� · 6H2O f AlOH2� · 5H2O � H� 1.732

a Values taken from Baes and Mesmer (1976).b Values taken from Perrin (1982).

The (001) slab was three octahedral sheets thick. The (100) and (010)

surfaces are edge surfaces and were treated as polymer-like one-dimensional repeat units one sheet thick, as was done for pyrophylliteedges in Bickmore et al. (2003). Two mutually perpendicular vacuumlayers were present for each edge model. All three gibbsite surfacemodels were designed with an inversion center and optimized in the P1�space group.

2.4. Me–O Bond Ionicity

The ionicity of Me–O bonds in solution monomers and solid surfaceswas calculated using electronegativity values obtained from thermo-chemical data (Allred, 1961) and the following equation taken fromPauling (1960):

I � 1 � exp� ��XA � XB�

2

4 � (3)

where I is fraction ionic character and (XA � XB) is the electronega-tivity difference between the metal and oxygen atoms. See Table 1 for

ers using calculated molecular structures are tabulated here. For eachations) for the oxygen atom in the base that has been deprotonated in). R0 values for the specific Me–O pairs are used with the bond lengthr the proton in the acid that will be removed in the reaction is used asicity is calculated using Eqn. 3. The pKa

app values are tabulated fromr oxyacids and Baes and Mesmer (1976) for hexaquo cations, except

Me–O(v.u.)

Me–H(Å)

Me–Oionicity pKa

app

1.36 2.42 0.328 2.241.17 2.48 0.328 6.961.03 2.58 0.328 11.51.66 2.41 0.547 0.2a

1.45 2.49 0.547 6.511.35 2.28 0.323 2.1481.16 2.33 0.323 7.1981.03 2.44 0.323 12.381.18 2.31 0.447 9.841.05 2.36 0.447 13.21.45 2.29 0.169 –1.98b

1.30 2.38 0.169 1.991.63 2.43 0.559 4a

1.37 2.51 0.559 8.551.19 2.62 0.559 14.3

Me–OH(v.u.)

Me–OH(Å)

Me–Oionicity pKa

app

3 0.81 2.656 0.559 2.264 0.73 2.615 0.547 4.14 0.61 2.629 0.547 5.559 0.94 2.656 0.422 2.199 0.64 2.670 0.422 3.480 0.58 2.799 0.591 10.594 0.67 2.754 0.477 9.54 0.44 2.773 0.477 11.12 0.57 2.721 0.456 9.652 0.43 2.732 0.456 9.159 1.14 2.925 0.708 7.70 0.88 2.614 0.485 2.62 0.75 2.803 0.498 43 0.64 2.665 0.286 3.49 1.11 2.755 0.661 4.31 0.80 2.556 0.567 4.97

monomxaquo cn, 2002ngth foond ion997) fo

R0

1.741.721.721.751.751.791.731.731.691.692.011.731.901.791.841.65

Me–O bond ionicity values calculated using Eqn. 3.


2.5. pKa Values of Solution Monomers

Measured solution pKa values for tetrahedral oxyacids were takenfrom the NIST Critically Selected Stability Constants database (Na-tional Institute of Standards and Technology, 1997) where possible, oralternatively from Baes and Mesmer (1976) and Perrin (1982). Valuesfor hexaquo cations were taken from Baes and Mesmer (1976). All pKa

values extrapolated to infinite dilution (ionic strength � � 0). SeeTable 1 for specific pKa values.

3. RESULTS AND DISCUSSION

In this section we critique various bond-valence methods forpKa prediction that have been proposed by referring to thereactions listed in Table 1 and the calculated structures ofsolution monomers associated with them. Next, we derive anew approach to pKa prediction based on these calculatedstructures. We then discuss the surface structures calculated forthis paper and suggest a possibility for how this new approachmay be applied to them. Finally, we point out areas of ambi-guity in the new approach and suggest avenues for furtherresearch.

3.1. The MUSIC Method

The MUSIC model is a three-plane or Basic Stern SCM thatincludes multiple surface functional groups whose identity,spatial distribution, and pKa values are predetermined. Thus,the fitted parameters in this model typically only include Helm-holz surface layer capacitances (Hiemstra et al., 1996). Theidentity and distribution of reactive sites are determined bycrystallographic methods combined with morphologic charac-terization of the sample; and the pKa values have typically beenestimated by a bond-valence method similar to that of Bleam(1993). For our purposes, we will distinguish between the“MUSIC model,” which is the full SCM, and the “MUSICmethod,” which is the bond-valence method of Hiemstra et al.(1996) for estimating surface pKa values.

Hiemstra et al. (1996) calibrated the MUSIC method on pKa

values of solution monomers, then applied it to surfaces. How-ever, to make this connection they had to deal with the conceptof “intrinsic” acidity constants (pKa

int). For charged surfaces,the apparent pKa (pKa

app) of an acid-base reaction is affected byboth the ionic strength and the concentration of protons near thesurface (which is dependent on the electrical potential at thesurface). These factors can be accounted for in an SCM bytreating the regions of inner- and outer-sphere adsorption asparallel plate capacitors, and by correcting apparent acidityconstants for the work involved in moving a proton away fromthe surface, which is treated as a plane of uniform chargedensity. After applying these corrections, one is left with apKa

int value that is supposedly independent of electrostaticfactors. In addition, Hiemstra et al. (1989) noted that when aproton is removed from a charged base, the electrostatic workinvolved also affects the apparent pKa. Therefore, if surfacepKa values are to be related to those of solution monomers interms of strictly “chemical” factors such as valence, electro-static corrections must be applied to solution monomers aswell.

For instance, H4SiO4 dissociates into H3SiO4� and H�, so

that the free energy of dissociation (�G°) is equal to the sum of
the “intrinsic” free energy of dissociation (�G°int) and the free
energy associated with the work of moving a proton away froman ion with �1 electron charge. (Here “intrinsic” is used tomean “everything other than electrostatic work.”) Thus

�G� � �G�int � �G�elec (4)

Hiemstra et al. (1989) calculated the molar free energyinvolved in the removal of a proton (�G°elec)from a conjugatebase with the following equation from Bockris and Reddy(1976):

�G�elec � �NAzi

2e�2

8��r�(5)

where NA is Avogadro’s number, zi is the number of electroncharges on the conjugate base (e.g., �1 for H3SiO4

� in theexample above), e0 is the charge on an electron, � is thedielectric constant of water, �0 is the permittivity of a vacuum,and r0 is the initial Me–H distance. The pKa

int for the dissoci-ation reaction can then be obtained by converting �G°elec to pKunits (using the equation �G�elec � � RTlnKelec) and sub-tracting from pKa

app.It was then assumed that differences in pKa

int values forsolution monomers are related solely to valence saturation.That is, the oxygen atoms on the conjugate bases of oxyacidsand hexaquo cations are “valence saturated” if the total valenceof bonds reaching these atoms is 2 v.u. If the total valence ofbonds reaching an oxygen atom is � 2 v.u., it is “oversatu-rated,” and “undersaturated” if the total valence of bonds is 2 v.u. The valence saturation state of the oxygen atoms wasthought to correlate with the acidity of the functional group,e.g., an undersaturated oxygen atom would still have valenceavailable for bonding with a proton, and so would be morebasic than an oversaturated or saturated oxygen. The method ofHiemstra et al. (1996) for calculating the valence saturationstate of an oxygen atom in a solution monomer can be sum-marized by the following examples. First, we will consider thefirst deprotonation of silicic acid.

H4SiO4fH3SiO4� � H� (6)

In H3SiO4� the central Si4� ion is 4-coordinated, so we assume

that the valence of each Si–O bond is 1 v.u. (�4 v.u. divided bythe coordination number). However, the average coordinationnumber of oxygen in known compounds is 4 (see e.g., Brown,2002), so we assume that three possible bonding sites or “or-bitals” remain on the oxygen from which the proton wasremoved. These orbitals are available for weak hydrogen bond-ing with surrounding water molecules or strong bonding with aproton. Weak hydrogen bonds are assumed to have a valence of0.20 v.u., and strong O-H bonds are assumed to have a valenceof 0.80 v.u. (Hiemstra et al., 1996; Brown, 2002; Bickmore etal., 2003). The total valence of bonds reaching an oxygen atom(sj) is calculated with the following equation (cf. Fig. 1a):

�sj � �sMe � msH � n�1 � sH�� (7)

Here sMe is the valence of the Me–O bond(s), m is the numberof strong (donating) hydrogen bonds of valence sH (0.80 v.u.)and n is the number of weak (accepting) hydrogen bonds ofvalence 1 � sH (0.2 v.u.). The values used for m and n in Eqn.
7 depend on the bonding environment of the oxygen consid-


ered: m � n � 3 for solution monomers, whereas m � n �2 for surface oxygens. It is assumed for steric reasons that m� n � 2 for singly coordinated surface oxygens. Becausetriply coordinated surface oxygens only have 1 remainingbonding orbital available, m � n � 1 for these functionalgroups. Depending on the valence of Me–O bonds in doublycoordinated surface groups, m � n may be 1 or 2. PlottingpKa

int vs. valence undersaturation on hydroxo- and oxo-solu-tion monomer complexes yields an empirical expression forpKa

int:

pKaint � � A��sj � V� (8)

where A � 19.8 , sj is calculated using Eqn. 7, and V is theformal valence of the surface oxygen (�2) (Hiemstra et al.,1996). Thus the predicted pKa

int for the first deprotonation of

Fig. 1. Bond graphs illustrating how Hiemstra et al. (1996) calculateunsaturated valence on oxygen atoms of solution monomers. (a) On anH3SiO4

– ion the unsaturated valence of the bare oxygen is calculated byassuming all Si–O bonds are 1.0 v.u. and weak H-bonds reaching theremaining three bonding sites are 0.2 v.u. each. This leaves 0.4 v.u. ofunsaturated valence. (b) The unsaturated valence of an OH group on anFe(OH)2 molecule is calculated by assuming all Fe–O bonds are0.33 v.u, all strong O-H bonds are 0.8 v.u., and all weak H-bonds are0.2 v.u. One bonding site on the oxygen is taken by the Fe–O bond, oneby the strong O-H bond, and two by weak H-bonds, leaving anunsaturated valence of 0.47 v.u.

silicic acid (sMe 1 v.u., m � 0 , and n � 3 ) is 7.9. Given

a pKaapp of 9.86 (Baes and Mesmer, 1976), and applying Eqn.

5 to determine the electrostatic contribution, Hiemstra et al.(1989) determined that the experimental pKa

int should be 9.24.Next, we will consider the deprotonation of FeOH� � 5H2O:

FeOH� · 5H2OfFe(OH)20 · 4H2O � H� (9)

Here the central Fe2� ion is 6-coordinated, so sMe 0.333 v.u.,and m � 1 since the oxygen from which the proton detachesis left with one strong O-H bond. Therefore, n � 2, since theadditional two bonding orbitals on the oxygen are available forweak hydrogen bonds. From Eqn. 7 we obtain sj � 1.533(see Fig. 1b). There is no electrostatic contribution to theremoval of a proton from a neutral species, so we calculatepKa

int � 9.25 , which can be compared directly to the mea-sured value of 11.1 (Baes and Mesmer, 1976).

To apply this model to oxide surfaces, Hiemstra et al. (1996)had to assume that the solvation of oxygen atoms on solutionmonomers and surfaces is (sometimes) different. For somesurface functional groups, it was postulated that one of the fourbonding orbitals on an oxygen atom is sterically hindered fromparticipating in strong or weak bonds with H. Thus, for surfaceoxygen atoms coordinated by one metal cation, m � n� 2 . For surface oxygen atoms bonded to two metal cations,m � n � 1 or m � n � 2 . Finally, m � n � 1 for surfaceoxygen atoms bonded to three metal cations.

For surfaces, values of sMe are obtained by terminating thebulk crystal structure along the proper plane, and then using theMe–O bond lengths at the surface to calculate bond valence.Thus, bond relaxation at the surface is explicitly assumed to benegligible.

A number of objections can be raised against the MUSICmethod, beginning with its treatment of �G°elec in Eqn. 5.Hiemstra et al. (1989) used Eqn. 5 to describe the contributionof electrostatic work to the dissociation energy of a proton fromthe conjugate base in a solution monomer, but they obtainedthis equation from Bockris and Reddy (1976), who derived it todescribe the discharging of a charged sphere by “leaking” outinfinitesimal amounts of charge. We suggest that it is morephysically realistic to represent this process as the removal of apoint charge (a proton) from the surface of a charged spherewith a radius equal to the starting Me–H distance (cf., Rossoand Rustad, 2000). We derive an equation for this as follows.

First, the work of removing a proton is equal to negative theelectrostatic interaction force (F) integrated over the distance ofremoval, which we can assume to be infinity.

�Gelec0 � � �

r�

�

Fdr (10)

The molar electrostatic interaction force between chargedspheres in a dielectric medium can be represented as follows:

F �NAZHzBe�2

4��r2 (11)

where zH is the number of electronic charges on the proton(�1), zB is the number of electronic charges on the conjugatebase, and r is distance. See discussion of Eqn. 5 for the other
parameters. Here the force is negative when attractive and


positive when repulsive. Substituting Eqn. 11 into Eqn. 10 andintegrating, we obtain

�G�elec � �NAZHzBe�2

4��r�(12)

Hiemstra et al. (1989, 1996) calculated �G°elec for only thefirst deprotonation of a number of oxyacids, and for the dep-rotonation of hexaquo cations involving a neutral conjugatebase. Therefore, we estimate that their calculations of the effectof �G°elec on pKa

int values of oxyacids were too small by �0.6log units, and their treatment of hexaquo cations would beunaffected, since no electrostatic correction was needed. Sincethe errors involved in their estimations of pKa

int values werefairly small, the predictions obtained by applying Eqn. 7 and 8should still be fairly accurate if these equations have accuratelycaptured the dependence of pKa

int on unsaturated valence.This proposition can be tested by applying the MUSIC

method to successive deprotonations of solution monomers. Itwill be recalled that Hiemstra et al. (1996) assumed the sym-metrical distribution of valence among all Me–O bonds in asolution monomer, and that the Me–O bond valences add up tocompletely neutralize the valence of the Me ion. It follows fromthis assumption that differences in pKa

app values of successivedeprotonations of a solution monomer ought to be relatedsolely to differences in �G°elec. Figure 2 illustrates how theunsaturated valence of a deprotonated oxygen on an H2SiO4

2�

ion would be calculated using the MUSIC method. Comparingto Figure 1a, it can readily be seen that pKa

int for the first andsecond deprotonations of silicic acid should be the same, i.e.,�7.9. Table 2 reports “measured” pKa

int values (i.e., electro-statically corrected pKa

app values) for the reactions in Table 1,calculated by using the original Me–H distance (Table 1) as r0

in Eqn. 12. Also reported are the pKaint values predicted by the

MUSIC method; predicted vs. actual pKaint values are plotted in

int

Fig. 2. Bond graph illustrating how the unsaturated valence ofdeprotonated oxygen atoms on an H2SiO4

2– ion would be calculatedusing the MUSIC method of Hiemstra et al. (1996). Since all Si–Obonds are assumed to be 1.0 v.u., the calculated unsaturated valence ofan oxygen atom on H2SiO4

2– would be exactly the same as that of anoxygen atom on H3SiO4

– (see Fig. 1a).

Figure 3. While the MUSIC method predicts pKa for the first

deprotonation of silicic acid (8.49) fairly well, its prediction ofpKa

int for the second deprotonation is off by �2.6 log units.The situation only gets worse for oxyacids, where three suc-cessive deprotonation reactions can be examined. For instance,the MUSIC method predicts a pKa

int value of 2.97 for the first,second, and third deprotonations of arsenic, phosphoric, andvanadic acids, but this prediction is off by up to 7.8 log units.

In fact, neither the assumption that all Me–O bond valencesin a solution monomer are the same, nor the assumption thatthey add up to completely neutralize the valence of the centralcation, is necessarily true for oxyacids. For instance, the com-bined valence of Si–O bonds in the H4SiO4 molecule is3.77 v.u. While it may have been possible for the DFT calcu-lations to have overpredicted the bond lengths by this amount,due to systematic errors, the same type of calculations returnedcombined Si–O bond valences of 3.58 v.u. for H3SiO4

� and3.38 v.u. for H2SiO4

2�. In our experience, DFT calculationstend to predict Si–O bond lengths that are slightly too short, soa more likely explanation for the larger bond lengths in thesolution monomers is that the Si atoms are not completelyvalence saturated. Table 2 reports the calculated sum of Me–Obond valences for the acid and base in each reaction listed.Similarly, Me–O bond lengths in these monomers are consid-erably shorter than Me–OH bond lengths. For example, theSi–O bonds in H2SiO4

2� are 1.05 v.u., but the Si–OH bondsare 0.65 v.u. Nor are these bond valences static among silicamonomers, since Si–O bonds in H3SiO4

� are 1.18 v.u. andSi–OH bonds are 0.79 to 0.82 v.u. (Rustad et al., 2000).

The Me–O bond valence of the base in each reaction con-sidered is listed in Table 1, and it is apparent that the unsatur-ated valence of oxygen is not the same for monomers withdifferent protonation states, at least with respect to tetrahedraloxyacids. Bickmore et al. (2003) used this fact to show thatdifferences in pKa

int values among chemically identical func-tional groups on monomers and surfaces may be related todifferences in bond valence, rather than differences in solva-tion, as the MUSIC method suggests. Tossell and Sahai (2000)had earlier suggested this possibility.

On the other hand, the assumption that differences in �G°elec

completely account for differences in pKaint seems to hold true

for at least some hexaquo cations. The calculated structures andpKa

app values needed to analyze successive deprotonation re-actions are available for only four hexaquo cations(Cr3� � 6H2O, Fe3� � 6H2O, Fe2� � 6H2O, Co2� � 6H2O). Ta-ble 2 shows that in all but the case of Co2� � 6H2O, the pKa

int

values for the first and second deprotonation reactions of thesehexaquo cations are within 0.5 log units of each other, despitethe fact that Me–OH bond valences in the bases for thesereactions are significantly different. This observation will befurther discussed later in this paper, but for now it is enough toput forward the suggestion that the processes governing theacidity of oxyacids and hexaquo cations may be, in some sense,physically distinct. If so, different models may be required topredict their acidity.

The application of the MUSIC method to solid surfaces willalso be discussed further in connection with our new approachto acidity constant prediction. However, it has been conclu-sively demonstrated here that the MUSIC method does notadequately predict the pKa

int values of solution monomers,
especially oxyacids. The major reason for this seems to be that


its treatment of bond valence is overly simplistic, and does nottake into account bond relaxation in response to differences inprotonation. Since the MUSIC method is calibrated to pKa

int

values of solution monomers, we have reason to suspect that itstreatment of surface functional groups will not always besuccessful.

3.2. The Method of Brown (1981)

Brown (1981, 2002) proposed a more generalized bond-valence model for the prediction of solution monomer pKa

values, including those for acids that do not contain oxygen.We briefly discuss it here because it has recently been adaptedfor use with oxide surfaces.

In this model, pKaapp values are related to “Lewis base

strength” (Sb) of a conjugate base, rather than total unsaturatedvalence.

We will illustrate how Sb is calculated with the followingtwo examples. First, consider the second deprotonation ofsulfuric acid:

� 2� �

Table 2. Tabulated here are several parameters used to evaluate vario1. The “measured” pKa

int values are simply the measured pKaapp value

12. Sb values are the Lewis base strengths of the oxygen atoms in the“Me–O acid” values represent the summed bond valence of all Me–Osame thing for the base. Also listed are the pKa

app values predicted ususing Eqn. 15 (oxyacids) and Eqn. 17 (hexaquo cations).

Acid f Base � H�pKa

int

(measured)

OH3AsO4 f H2AsO4

– � H� 0.95H2AsO4

– f HAsO42– � H� 4.45

HAsO42– f AsO4

3– � H� 7.88H2CrO4 f HCrO4

– � H� –1.09HCrO4

– f CrO42– � H� 4.01

H3PO4 f H2PO4– � H� 0.78

H2PO4– f HPO4

2– � H� 4.53HPO4

2– f PO43– � H� 8.54

H4SiO4 f H3SiO4– � H� 8.49

H3SiO4– f H2SiO4

2– � H� 10.56H2SO4 f HSO4

– � H� –3.34HSO4

– f SO42– � H� –0.63

H3VO4 f H2VO4– � H� 2.72

H2VO4– f HVO4

2– � H� 6.07HVO4

2– f VO43– � H� 10.73

HexaV3� · 6H2O f VOH2� · 5H2O � H� 4.60Cr3� · 6H2O f CrOH2� · 5H2O � H� 6.48CrOH2� · 5H2O f Cr(OH)2

� · 4H2O � H� 6.73Fe3� · 6H2O f FeOH2� · 5H2O � H� 4.53FeOH2� · 5H2O f Fe(OH)2

� · 4H2O � H� 4.65Mn2� · 6H2O f MnOH� · 5H2O � H� 11.70Fe2� · 6H2O f FeOH� · 5H2O � H� 10.63FeOH� · 5H2O f Fe(OH)2 · 4H2O � H� 11.10Co2� · 6H2O f CoOH� · 5H2O � H� 10.79CoOH� · 5H2O f Co(OH)2 · 4H2O � H� 9.15Y3� · 6H2O f YOH2� · 5H2O � H� 9.83Ga3� · 6H2O f GaOH2� · 5H2O � H� 4.98In3� · 6H2O f InOH2� · 5H2O � H� 6.22Rh3� · 6H2O f RhOH2� · 5H2O � H� 5.74Sc3� · 6H2O f ScOH2� · 5H2O � H� 6.56Al3� · 6H2O f AlOH2� · 5H2O � H� 7.41

HSO4NSO4 � H (13)

The valence of sulfur (�6 v.u.) in SO42� is divided among

four S-O bonds (1.5 v.u. per bond). This leaves a total unsat-urated valence of 2 v.u. Oxygen has an average coordinationnumber in known compounds of 4, so we assume the existenceof four bonding “orbitals” per oxygen, three of which areavailable to form bonds with H. This leaves us with 12 possiblebonds over which we can divide the total unsaturated valence toobtain Sb � 0.17 v.u. (2 v.u./12 possible bonds).

Consider also the first deprotonation of silicic acid (see Eqn.6). Assuming Si–O bonds of 1.00 v.u. and O-H bonds of0.80 v.u. in H3SiO4, the oxygen atom not connected to ahydrogen would have 1.00 v.u. of unsaturated valence, and theOH groups would have 0.2 v.u. of unsaturated valence. In theprevious example, it was assumed that each oxygen atom couldform four bonds, but Brown (2002) explained that the OHgroups have enough unsaturated valence for only one weakbond, so we assume a coordination number of 3 for oxygenatoms in OH groups, while the deprotonated oxygen is 4-co-ordinated. The 1.6 v.u. of unsaturated valence is divided amongthe 6 possible bonds to obtain Sb � 0.27 v.u. (1.6 v.u./6

-valence methods with respect to the acid-base reactions listed in Tablee 1) corrected for the electrostatic work of proton removal using Eqn.

here a proton is lost, obtained using calculated molecular structures.in the acid for each reaction, and “Me–O base” values represent themethod of Brown (1981) and the pKa

int values predicted in this study

.)

Me–Oacid(v.u.)

Me–Obase(v.u.)

pKaapp

(Brown)pKa

int

(this study)

s4 4.51 4.91 4.1 1.407 4.91 4.24 7.0 5.242 4.24 4.13 8.8 8.053 5.97 5.89 –1.7 0.084 5.89 5.79 3.3 4.478 4.48 3.90 4.1 1.539 3.90 4.23 7.0 5.283 4.23 4.12 8.8 8.014 3.77 3.58 9.9 7.748 3.58 3.38 11.4 10.453 5.29 5.23 –1.7 –4.122 5.23 5.21 3.3 –1.074 5.10 4.95 4.1 1.081 4.95 4.77 7.0 6.381 4.77 4.74 8.8 10.13

tions3 2.88 2.87 N/A 6.533 3.07 3.01 N/A 6.424 3.01 2.98 N/A 6.428 3.01 3.05 N/A 5.279 3.05 2.97 N/A 5.270 2.18 2.15 N/A 11.584 2.13 2.14 N/A 10.552 2.14 2.00 N/A 10.556 2.09 2.08 N/A 10.355 2.08 1.97 N/A 10.351 3.23 2.83 N/A 7.890 3.17 3.14 N/A 5.863 2.99 2.92 N/A 5.979 3.13 3.22 N/A 4.034 3.20 2.90 N/A 7.468 3.13 2.98 N/A 6.60

us bonds (Tablbases wbonds

ing the

Sb

(v.u

xyacid0.210.270.320.110.180.210.270.320.270.310.180.230.120.210.27

quo Ca0.190.230.290.120.270.310.260.380.310.380.030.160.220.270.040.19

bonds).


Brown (1981) related Sb to pKaapp using the following equa-

tion:

pKaapp � 14.3ln� Sb

0.135� (14)

Substituting the Sb values calculated above, we obtain pKaapp

values of 3.0 and 9.9, which compare favorably to the values1.99 and 9.86 listed in Table 1. Table 2 lists Sb and pKa

app

values of the oxyacids considered here, calculated using thismethod (see Brown, 1981), and Figure 4 shows these valuesplotted against each other, along with a line representing Eqn.14. The standard error for Eqn. 14 with respect to this sampleis 2.6 log units.

One advantage of this approach is the division of unsaturatedvalence into “possible bonds,” ignoring the contribution ofweak H-bonding with surrounding water molecules. If weconsider the approach of a hydronium ion in solution toward anoxygen atom in a base, it is apparent that there can be nointervening water molecules between the hydronium and theoxygen just before bond formation. However, the MUSICmethod seems to imply the opposite by including the contri-bution of weak H-bonds at all possible bonding sites in thecalculation of unsaturated valence.

On the other hand, Brown’s (1981, 2002) method suffersfrom one of the same problems as the MUSIC method, in thatit assumes that the Me–O bond valences in a solution monomerare all equal and add up to completely neutralize the centralcation. We have already shown that this is not necessarily thecase if the calculated structures discussed here are correct, or

Fig. 3. Oxyacid pKaint values calculated using the MUSIC method

for first (crosses), second (circles), and third (squares) deprotonations.The method was calibrated only on first deprotonations, but shouldwork for subsequent deprotonations if the dependence of pKa

int onunsaturated valence has been properly described. While pKa

int values offirst deprotonations are predicted fairly well, predictions for subsequentdeprotonations become progressively worse.

even nearly correct.

It should be noted that the reason Brown (2002) assumed thatMe–O bond valences in solution monomers add up to com-pletely neutralize the central cations is that this is a naturalconclusion of the “valence sum rule,” which is a fundamentalaxiom of bond-valence theory. However, it must be remem-bered that this rule is derived solely from the analysis of thestructures of crystalline solids, and it does not apply to“strained” structures where some external stress has been ap-plied. It should be intuitively obvious to surface chemists thatthe valence sum rule is not followed by ions in solution orcharged surfaces, because their behavior can be approximatedso well by that of point charges and charged planes. This shouldonly be possible if some local undersaturation of valence inions is allowed. Still, larger calculations of explicitly solvatedsystems are needed to prove this point.

This discussion brings out the fact that, since it does notallow for local charge imbalances, classical bond-valence the-ory cannot be directly applied to existing surface complexationmodels without modification. Since SCMs treat bonding andpurely electrostatic effects separately, any bond-valence ap-proach to predicting SCM parameters must be modified toallow for local charge imbalances.

3.3. A Revision of the Brown (1981) Method

Bickmore et al. (2003) reasoned that Brown’s (1981) methodfor calculating Sb might be roughly equivalent to using ab initiocalculated Me–O bond lengths to calculate Sb for individualoxygen atoms. For instance, the calculated Si–O bond length inthe H3SiO4 ion is equivalent to 1.18 v.u. (Table 1). Therefore,the bare oxygen would have a total unsaturated valence of0.82 v.u. divided between three possible bonds, and Sb

� 0.27 , just as was obtained above from Brown’s (1981)

Fig. 4. Oxyacid pKaint values plotted vs. Sb values as they would be

calculated using the method of Brown (1981). The line representspKa

int values predicted by Eqn. 14.

method. Applying Sb values obtained in this way to Eqn. 14,


Bickmore et al. (2003) predicted reasonable pKaint values for

surface functional groups on calculated pyrophyllite and fer-ripyrophyllite surface structures.

This approach was somewhat naıve, in that electrostaticeffects were not separated from bond valence effects. However,in some cases this approach should work fairly well. Figure 5plots pKa

int for the oxyacid deprotonation reactions in Table 2vs. Sb values for the conjugate bases obtained from the calcu-lated molecular structures. A curve representing Eqn. 14 issuperimposed on the graph. The first detail to be noticed onFigure 5 is that the pKa

int values for successive deprotonationsof each oxyacid considered have a definite dependence on Sb

which is very similar from acid to acid, although the y-inter-cepts of these trends are shifted relative to each other. ClearlySb is strongly correlated with pKa

int, but that does not explainall the variation. Second, Eqn. 14 mimics the overall trend ofpKa

int values rather well, and the pKaint values of several

oxyacids match the curve fairly closely. Therefore, the reasonfor the success of Bickmore et al. (2003) in obtaining reason-able surface pKa

int values is apparent, but this approach cannotbe expected to be universally applicable.

Two important points came out of the study by Bickmore etal. (2003), however. First, it was shown that differences inacidity between solution monomers and analogous surfacefunctional groups may be related more to differences in Me–Obond valence than to differences in solvation. Essentially thesame point was previously made by Tossell and Sahai (2000).Second, if bond valence is related to acidity, calculated surfacestructures with different amounts of surface charge densityshow that the protonation state of one functional group can

Fig. 5. Oxyacid pKaint values plotted vs. Sb values as they are

calculated using Me–O bond lengths from calculated molecular struc-tures (Bickmore et al., 2003). The line represents pKa

int values pre-dicted by Eqn. 14. While about half the pKa

int values are predictedfairly well, the others are not. However, for each oxyacid, pKa

int valuesof successive deprotonations appear to have approximately the sameslope with respect to Sb.

affect the acidity of neighboring groups. These points highlight

both the necessity of using calculated atomic structures topredict acidity, and of creating a more universally applicablebond-valence method for that prediction.

3.4. A New Approach

A new approach to pKaint prediction has been formulated

based on the calculated molecular structures discussed above.Here we derive this method and do preliminary tests of itsapplicability to oxide surfaces.

3.4.1. Acidity of Solution Monomers

Figure 5 clearly shows that pKaint values of oxyacids are

strongly correlated with Sb, and the dependence is about thesame for each set of successive deprotonations. However, otherfactors are at work, and a generalized model based solely onthis correlation would not always be usefully accurate. If ad-ditional major predictors of pKa

int values could be identified,perhaps the situation would improve.

Me–O bond ionicities (see Table 1) appear to explain muchof the remaining variation in oxyacid pKa

int values. A multipleregression analysis of the oxyacid pKa

int values listed in Table2 with Sb and Me–O bond ionicity (I) used as predictors yieldsthe following equation:

pKaint� 0.83� � 61.5Sb � 22.5I � 19.2 (15)

where the parenthetical value represents the standard error ofthe regression. Predicted pKa

int values are listed in Table 2, andpredicted vs. experimental pKa

int values are plotted in Figure 6.Calculated p-values for each of the terms in Eqn. 15 indicate� 99% confidence that they should be included in the model.As the standard error in Eqn. 15 indicates, nearly all the pKa

int

values are predicted within 1 log unit of the experimental

Fig. 6. Oxyacid pKaint values predicted by Eqn. 15 plotted vs.

measured values. The line represents 1:1 correspondence.

values.


Correlation does not necessarily indicate causation, but it canstill be useful to attempt a physical justification for the phe-nomenological model represented by Eqn. 15, perhaps facili-tating further development on theoretical grounds. Unsaturatedvalence and related values, such as Sb, have been shown to begood measures of the “bonding power” of ions (Brown, 2002).However, we hesitate to relate unsaturated valence directly to“residual charge.” Bleam (1993) pointed out that bond valencecan be related to either local charge neutralization, or to theamount of electron density shifted from the cation to the anion,but that the charge neutralization interpretation cannot be ex-tended beyond very localized interactions. This would explainthe excellent success of bond-valence methods at predictinglocal coordination structures in crystals (Brown, 2002). SincepKa

int values are corrected for longer-range electrostatic inter-actions, we prefer to be intentionally vague about the physicalmeaning of Sb, using it merely as a predictor for the strength oflocalized cation-anion interactions (i.e., bond-making and-breaking).

Me–O bond ionicity is a measure of the fraction of theelectron density associated with a bond that is shifted from thecation to the anion, and so would be a predictor of the localelectron density associated with the oxygen atoms of the oxy-acids considered. It makes sense that greater Me–O bond io-nicity would result in a larger pKa, as Eqn. 15 predicts, butpresumably we have already removed electrostatic effects bycorrecting pKa

app for the work involved in removing a proton toobtain pKa

int. Why do we need to include this extra term thatalso appears to correct for electrostatic effects? Our electro-static pKa correction was based on the treatment of the protonas a point charge and the conjugate base as a charged sphere. Aspherically symmetrical distribution of charge can be treated asequivalent to a point charge of the same magnitude at the centerof the sphere, and in that case variations in bond ionicity wouldhave no effect. However, the electron density of the conjugatebases considered is not spherically symmetrical. Therefore, it ispossible that the ionicity term in Eqn. 15 represents a furtherelectrostatic correction arising from the difference in electrondensity distribution between a real molecule and a hypotheticalcharged sphere.

This interpretation is supported by the work of Sverjenskyand Sahai (1996) and Sahai (2002), who showed that the pointof zero net proton charge (PZNPC) of a number of oxides canbe predicted using the average bond valence and the averagedielectric constant of each crystal structure. Sahai (2002) re-lated the dielectric term to Me–O bond ionicity through polar-izability.

One apparent omission in Eqn. 15 is anything having to dowith solvation. For instance, Tossell and Sahai (2000) pointedout that ions associated with successive deprotonations of anoxyacid would have very different energies of hydration, due todifferent overall charge. Eqn. 15 seems to imply the assumptionthat any differences in solvation energy are related to the Sb

term. If we perform another multiple regression analysis likethe one discussed above, using Sb and I to predict pKa

app, weobtain the following relationship:

pKaapp� 1.08� � 72.3Sb � 23.91 � 20.0 (16)

The most significant difference between Eqn. 16 and Eqn. 15

is the magnitude of the Sb coefficient. This supports the ideathat differences in solvation energy between ions associatedwith successive deprotonations of an oxyacid are included inthe Sb term. In fact, Brown (2002) has shown a strong corre-lation between the enthalpy of solution of various cations andtheir cation bonding strength in v.u. (Cation bonding strength isthe valence of the cation divided by the number of watermolecules in its first coordination shell.) Certainly there couldbe an analogous correlation with Sb. In addition, the division ofthe unsaturated valence into “possible bonds” to derive Sb mayimplicitly recognize that water molecules surrounding a func-tional group will compete for some of the unsaturated valence.

It has already been mentioned that hexaquo cations appear tobehave somewhat differently than the oxyacids consideredhere. For instance, the pKa

int values for successive deprotona-tions of hexaquo cations are often very similar to each other,even though the Me–OH bonds differ in valence. The pKa

int

values for hexaquo cations are plotted against their Sb values inFigure 7. There does not appear to be a significant correlation.However, if we assume an average Me–O bond valence(valence/6) of 0.333 for divalent cations and 0.5 for trivalentcations, regardless of how many protons have been lost, thenperform a multiple regression relating Sb (calculated using theaverage bond valences) and Me–O bond ionicity to pKa

int, weobtain the following relationship:

pKaint� 1.10� � 57.1Sb � 9.1I � 18.6 (17)

Predicted vs. experimental pKaint values for the hexaquo cat-

ions are plotted in Figure 8. Once again, p-values for eachcoefficient in Eqn. 17 obtained in the regression analysis indi-cate � 99% confidence that they should be included in themodel.

Fig. 7. Hexaquo cation pKaint values plotted vs. Sb values obtained

from calculated molecular structures. There does not appear to be astrong correlation between the two.

The major difference between Eqn. 17 and Eqn. 15 is the


smaller dependence on I in Eqn. 17, and of course the Sb valuesare calculated differently for the hexaquo cations. It is difficultto assign physical meaning to such observations, but we mayspeculate that the –OH and –OH2 groups in hexaquo cations aremore easily interchangeable, leading to an averaging effect. Infact, the average coordination numbers of hydrated cations areoften something other than integral numbers, so obviously agood deal of water exchange occurs (Machesky et al., 2001).Furthermore, the smaller dependence on I in Eqn. 17 may bedue to geometric effects. That is, we have already hypothesizedthat the ionicity term is related to the difference in the electro-static work of removing a proton from a real ion and a chargedsphere. The distribution of charge in a hexaquo cation is morespherically symmetrical than that in a tetrahedral oxyanion.Therefore, we can expect the dependence on I in Eqn. 17 to besmaller than that in Eqn. 15.

Whatever the reason for these discrepancies, we have pre-sented evidence against the assumption (Hiemstra et al., 1996)that the acidity of oxyacids and hexaquo cations can be treatedwith the same model. The remaining question is whether –OHand –OH2 groups on oxide surfaces must be treated differently.

3.4.2. Acidity of Oxide Surface Functional Groups

It has long been observed that the pKaint values of surface

functional groups appear to roughly correlate with those ofanalogous groups on solution monomers (Sahai, 2002). Howare the two related? It has been suggested that differences insolvation account for differences in monomer and surface acid-ity (Rustad, 2001), but Tossell and Sahai (2000)) and Bickmoreet al. (2003) used calculated bond lengths on surfaces, mono-mers, and oligomers to show that acidity differences may wellbe related mostly to differences in unsaturated valence onfunctional groups. This is also consistent with our treatment of

Fig. 8. Hexaquo cation pKaint values predicted by Eqn. 17 plotted vs.

measured values. The line represents 1:1 correspondence.

solution monomer acidity. Our analysis suggests that the Sb

terms in Eqn. 15 to 17 implicitly include any differences insolvation energy, and this, in turn, suggests that differences insolvation energy are a purely local phenomenon. If so, it shouldbe possible to use the same treatment to predict acidity offunctional groups on both solution monomers and surfaces.

The same is true for our localized electrostatic correctionbased on Me–O bond ionicity. We calculated the long-rangeelectrostatic pKa correction for monomers by approximatingthe conjugate bases as uniformly charged spheres. Similarly,surface complexation models of oxides treat long-range elec-trostatic forces by approximating surfaces as flat planes ofuniform charge density. However, for both solution monomersand surface functional groups, excesses (or deficiencies) ofelectronic charge are concentrated around the oxygen atoms toan extent largely determined by the ionicity of the Me–Obonds. Therefore, it is reasonable to suppose that we could usesomething like Eqn. 15 and/or 17 to predict pKa

int values ofsurface functional groups.

We have tested this hypothesis on calculated surfaces of two(hydr)oxides: cristobalite (SiO2) and gibbsite (Al[OH]3). Atfirst it was supposed that Eqn. 15 could be used to predict theacidity of –OH groups, and Eqn. 17 could be used for –OH2

groups. However, we found that Eqn. 17 completely failed toproduce realistic pKa

int values, and instead Eqn. 15 could beused for both –OH and –OH2 groups. (When calculating the Sb

of the conjugate base of an –OH2 group, one merely subtractsthe Me–O bond valence and 0.8 v.u. for the O-H bond from thevalence of the oxygen and divides the unsaturated oxygenvalence between the two remaining possible bonds.)

It will be recalled that the main difference between Eqn. 15and 17 is the magnitude of the Me–O bond ionicity term. Themagnitude of this term may be smaller for hexaquo cations thantetrahedral oxyanions because the distribution of charge in thehexaquo cations more nearly approximates a sphere. The elec-tric field of a spherically symmetrical configuration wouldbehave exactly the same as that of a point charge, and thereforethe actual location of the electron density with respect to thecenter of the sphere would have no effect. We have hypothe-sized that the ionicity terms in Eqn. 15 and 17 represent thedifference between the electrostatic work of removing a protonfrom a charged sphere (which was the assumption made whencalculating pKa

int from pKaapp) and that of removing a proton

from a real ion. If so, we can rationalize the fact that only Eqn.15 appears to work with the oxide surface functional groupstreated below by postulating that the difference between thework of removing a proton from a charged sphere and atetrahedral ion is similar in magnitude to the difference be-tween the work of removing a proton from a homogeneousplane of charge and a real oxide surface. Whether Eqn. 15 isuniversally applicable to oxide surfaces remains to be seen.

In the following subsections, we report on the results of theab initio surface structure optimizations and associated bulkstructure optimizations. Eqn. 15 is then applied to predict theacidity of the surface functional groups on these surfaces, andthese results are rationalized with respect to experimental data.Although the calculations presented here represent preliminaryinvestigations of manageably small systems, the results aresuggestive and point the way to supercell treatments of these
surfaces.


3.4.3. Gibbsite

The calculated unit cell dimensions of the gibbsite bulkstructure (a � 8.806 A, b � 4.873 A, c � 9.743 A, � 92.463�) match experimentally determined values (a� 8.684 A, b � 5.078 A, c � 9.736 A, � 94.54�)reported by Saalfeld and Wedde (1974) fairly well. The com-putational procedure used has previously been validated forsimilar materials (Rosso et al., 2001).

Figure 9 shows the calculated structures of the (001), (100),and (010) faces of gibbsite. The basal (001) surfaces are pop-ulated entirely by �Al2OH groups, whereas the edge faceshave �Al2OH, �AlOH2

�1/2, and �AlOH�1/2 groups. Table 3lists some of the dominant reactions likely to take place on eachface, along with the density of each site type and the predictedpKa

int values obtained by applying Eqn. 15.We can compare these predictions with those of Hiemstra et

al. (1996, 1999), who used their MUSIC method to predict thefollowing reactions and associated pKa

int values for the func-tional groups on gibbsite surfaces:

�AlOH2�1⁄2f�AlOH–1⁄2 � H� pKa

int � 9.9 (18)

�Al2OHf�Al2O� � H� pKa

int � 11.9 (19)

�Al2OH2�f�Al2OH � H� pKa

int � 0.0 (20)

According to their analysis, the reaction represented by Eqn. 18should dominate the surface charging behavior of gibbsite inthe environmental pH range.

Hiemstra et al. (1999) synthesized gibbsite samples withdifferent proportions of edge and basal surface area and ana-lyzed potentiometric titration data for these samples in thecontext of their predictions. In general, the titration data con-formed very well to their predictions. For instance, the mea-sured PZNPC of all samples was consistently about pH 9.9,which is what one would expect if the reaction in Eqn. 18 isdominant. Also, the amount of surface charging observed wasstrongly positively correlated with the proportion of edge sur-face area.

This would appear to be a good confirmation of the MUSICmethod predictions, but one aspect of the titration results wassomewhat anomalous. That is, the magnitude of surface chargeobserved could not be explained by charging of the edgesurfaces alone, a point later confirmed by Rosenqvist et al.(2002). The basal surfaces had to participate, as well, andHiemstra et al. (1999) explained this anomaly by postulatingthe presence of defects on basal surfaces. They supported thisby showing that a longer aging time for gibbsite preparationsled to more acid production during aging, which may have beendue to the consumption of �AlOH2

�1/2 defects on (001) sur-faces to form �Al2OH groups. Also, titration curves of prep-arations with larger aging times exhibited lower overall charg-ing. Rosenqvist et al. (2002) used atomic force microscopy(AFM) imaging of synthesized gibbsite particles to show thataging does smooth the basal surfaces. On the other hand, theirdata also indicated that the unaged sample (which would havea larger proportion of singly coordinated aluminols) actuallyhad a lower overall charge than the aged sample. After account-ing for the proportion of edge and basal surface area, proton
adsorption on aged gibbsite samples was still over 3 times the
number of singly coordinated aluminol sites, and they con-cluded that some of the doubly coordinated aluminol groupsmust sorb and desorb protons. The surface charging measured

Fig. 9. Ball-and-stick representations of chain or slab calculatedstructures for neutral (a) (100), (b) (010), and (c) (001) surfaces ofgibbsite. White balls represent H, grey balls represent O, and blackballs represent Al. The chain structures in (a) and (b) are oriented sothat the reader can see the different kinds of functional groups presenton the edge. The slab structure in (c) is oriented so that the reader canview the �Al2OH groups present on the (001) surface.

by Rosenqvist et al. (2002) was larger than that measured by


Hiemstra et al. (1999), but it was shown that this was an effectof the equilibration time used in the titrations. It was assumedthat reactions involving doubly coordinated aluminols wereslower than those involving singly coordinated groups, and thetitrations of Hiemstra et al. (1999) were too rapid to allow fullprotonation of the doubly coordinated groups. This would alsoexplain why the unaged gibbsite sample of Rosenqvist et al.(2002) exhibited a lower overall charge than the aged sample.

Another interesting aspect of the data of Rosenqvist et al.(2002) is that they consistently observed a “bump” in thesurface charging curves centered around pH 5.2. They reasonedthat some reorganization of the surface must occur in this pHrange, and supported this hypothesis with FTIR data that indi-cated the appearance of new –OH groups in this pH range.

Rosenqvist et al. (2002) created a MUSIC model to accountfor their titration data. But although they concluded that somedoubly coordinated aluminols must participate in the observedcharging behavior of gibbsite, they accepted the pKa

int esti-mates made by Hiemstra et al. (1996, 1999) and instead pos-tulated reactions between the surface groups and electrolyteions (Na� and Cl�), where protons coadsorb with electrolyteanions, and electrolyte cations displace protons.

Our predictions in Table 3 can, at least qualitatively, accountfor the titration data of Hiemstra et al. (1999) and Rosenqvist etal. (2002) without appealing to the presence of a large numberof defects, ion pairing of H� and Cl� at the surface, ordisplacement of surface protons by Na�. For example, thepredicted pKa

int values for singly coordinated aluminol groupson (100) and (010) faces range from 7.4 to 11.4, with aweighted average of 10.1, assuming equal representation of(100) and (010). However, the range of pKa

int values predictedfor the deprotonation of �Al2OH2

� groups is stunning. On theedge faces, these groups are predicted to have pKa

int values of�0.5 and �2.2. However, on basal surfaces 4 of 6 groups havepredicted pKa

int values in the range �0.4 to �5.1, 1 group hasa pKa

int value of 5.2, and another has a pKaint value of 10.8.

This would likely result in a PZNPC of somewhere around pH10, but a larger surface charge than could be accommodated onthe edge surfaces, as long as some pKa

int values for the depro-

Table 3. Neutral, calculated gibbsite surface structures werusing Eqn. 15. The density of sites where these reactions coubasic form of the functional groups examined. Me–O ionicitsame reaction is listed, it indicates crystallographically distin

Acid f Base � H�Site density(sites/nm2)

Gib�AlOH2

�1/2 f �AlOH–1/2 � H� 8.1�Al2OH2

� f �Al2OH � H� 4.0Gib

�AlOH2�1/2 f �AlOH–1/2 � H� 4.7

�AlOH2�1/2 f �AlOH–1/2 � H� 4.7

�Al2OH2� f �Al2OH � H� 2.4

Gib�Al2OH2

� f �Al2OH � H� 2.3�Al2OH2

� f �Al2OH � H� 2.3�Al2OH2

� f �Al2OH � H� 2.3�Al2OH2

� f �Al2OH � H� 2.3�Al2OH2

� f �Al2OH � H� 2.3�Al2OH2

� f �Al2OH � H� 2.3

tonation of �Al2OH groups on basal surfaces are around 10 to

counterbalance the protonation of one of the �Al2OH groupsin this range. This seems quite likely, considering the range ofpredicted values for the protonation reaction, and it fits with thehigher-than-expected adsorbed proton densities above the PZCobserved by Rosenqvist et al. (2002). However, we have not yetperformed the relevant calculations to determine pKa

int valuesfor the deprotonation of �Al2OH groups.

Since edge surfaces account for a relatively small fraction ofgibbsite surface area, we would predict that basal surfacecharging would have a larger effect than Hiemstra et al. (1999)demonstrated, but Rosenqvist et al. (2002) showed that thecharging of the basal surfaces is enhanced with a longer equil-ibration time. Rosenqvist et al. (2002) observed about a 4.5�mol/L H�/m2 charge, relative to the PZC at pH � 3, with 9%estimated edge surface area. If we assume that half the singly-coordinated sites on the edge surfaces and 1 of 6 doubly-coordinated sites on the basal surfaces (i.e., the ones withpKa

int � 5.2 ) gain protons below the PZC, we obtain a totalpossible adsorbed proton density of 4.4 �mol/L H�/m2. Sinceit is possible, in our scheme, that some of the doubly coordi-nated groups with pKa

int � 10.8 gain protons below the PZC,this is a good confirmation that the new approach leads torealistic total charge densities.

In addition, it is interesting to note that the “bump” in thesurface charging curves of Rosenqvist et al. (2002) occurred inthe pH range (pH � 5.2) where we predict that 1 of 6 doublycoordinated aluminol groups on the basal surfaces should react.Therefore, if some sort of surface reconstruction does occur inthis pH range, as Rosenqvist et al. (2002) postulated, it is likelyinstigated by the protonation of these basal sites. For instance,protonation at these sites might instigate etch pit formation,which would add new –OH groups to the surface, as the FTIRdata indicates.

The idea that some doubly coordinated aluminol groups ongibbsite basal surfaces contribute to surface charging is sup-ported by analysis of potentiometric titration data for kaolinite,which has a gibbsite-like basal surface on one side of eachcrystal. Ma and Eggleton (1999) have pointed out that it isdifficult to explain the magnitude of kaolinite surface charging

ined to predict pKaint values for possible acid-base reactions

place is listed along with Sb values for oxygen atoms in thes were calculated using Eqn. 3. Where more than one of thewith different pKa

int values.

Sb

(v.u.)Me–O

ionicitypKa

int

(Eqn. 15)

00)0.292 0.567 11.60.068 0.567 –2.2

10)0.290 0.567 11.50.225 0.567 7.40.095 0.567 –0.5

01)0.188 0.567 5.20.280 0.567 10.80.078 0.567 –1.60.097 0.567 –0.40.066 0.567 –2.30.021 0.567 –5.1

e examld take

y valuect sites

bsite (1

bsite (0

bsite (0

without assuming some participation of the basal surfaces.


Similarly, Wieland and Stumm (1992) fit a triple-layer model tokaolinite titration data, and had to assume a pKa

int value of 5.5for doubly coordinated aluminol groups on the basal surfaces.Preliminary AFM force titration data on the gibbsite basalsurface indicates a major surface charging transition centeredaround pH 5 to 6 (B. R. Bickmore, unpublished data).

A simple termination of the corundum/sapphire (001) sur-face also should expose only doubly coordinated aluminolgroups, although surface reconstruction is possible (Noguera,2000). If only doubly coordinated groups are present, force andstreaming potential measurements on this surface have consis-tently supported the idea that doubly coordinated aluminolgroups contribute to surface charge under pH conditions notpredicted by the MUSIC method. Stack et al. (2001) performedAFM force titrations and second harmonic generation experi-ments to determine the point of zero charge for the corundumbasal surface, although the surface was miscut by �5°. Theyobserved a PZC of pH 5 to 6 for this surface, in contrast to thegenerally accepted figure of pH � 9 for alumina samples.Kosmulski (2003) objected that this low PZC likely resultedfrom surface contamination, but Stack et al. (2003) respondedthat Auger Electron Spectroscopy characterization of the sur-face in question did not reveal the presence of any contami-nants other than adventitious C. In addition, it was pointed outthat these results only applied to a single surface on corundum,rather than the entire surface, for which the accepted PZC of pH� 9 might still apply. Franks and Meagher (2003) used stream-ing potential measurements and AFM force titrations to obtainisoelectric points (IEP) of pH 5 to 6 for the (001) surface ofsapphire. On the other hand, an IEP of pH 9.4 was obtained foralumina powder. The higher IEP on the powder was attributedto a greater proportion of singly coordinated aluminol groups,and the lower IEP of the single faces was attributed to thepresence of more multiply coordinated aluminol groups. Forcetitrations with the AFM and surface forces apparatus have beenused on the same surface to determine PZC values between pH3.0 and 6.7 by several other workers (Horn et al., 1988; Duckeret al., 1994; Larson et al., 1997; Meagher et al., 2002). Al-though we have not yet examined alumina surfaces using ournew approach, our predictions for the acidity of doubly coor-dinated aluminol groups on gibbsite surfaces certainly supportthe hypothesis that there could be doubly coordinated aluminolgroups on alumina crystal faces with pKa values in the rangeneeded to produce a PZC of pH 3 to 6.7.

Although our predictions seem capable of qualitatively ac-counting for gibbsite titration data, they are not yet sufficient toinclude in a quantitative surface complexation model, for tworeasons. First, we have not examined all possible surface reac-tions. For instance, to predict pKa

int values for the deprotona-tion of �Al2OH groups, we would have to determine Sb valuesfor the conjugate bases (�Al2O� groups), and this wouldrequire optimization of charged surface structures. Second,Bickmore et al. (2003) showed that the protonation state of onesurface functional group could drastically affect the pKa

int

values of neighboring groups. For example, we obtained pKaint

values for the transition from �AlOH2 to �AlOH groups onthe neutral gibbsite (100) and (010) surfaces, but half of thesingly coordinated groups were doubly protonated, and there-fore we could not calculate pKa

int values for them. To account
for such effects, we would have to optimize a completely
deprotonated surface, analyze the predicted pKaint values of all

functional groups, protonate the group with the highest pKaint

value, reoptimize the structure, reanalyze the pKaint values of

the remaining functional groups, and so on.Ideally, the procedure outlined above would be performed on

systems with fairly large repeat units and, given the computa-tional cost, will have to wait for future studies. However, wehave performed such an analysis on a smaller scale for thecristobalite (100) surface, which we report below.

3.4.4. Cristobalite (100)

The calculated unit cell dimensions of the cristobalite bulkstructure (a � b � 4.957 A, c � 6.890 A, � � � �� 90� ) also closely match experimentally determined values(a � b � 4.97 A, c � 6.93 A, � � � � � 90� )(Griffen, 1992).

The cristobalite (100) surface has been suggested as a pos-sible analog for silica surfaces in general (Hiemstra et al.,1996). An illustration of the neutral (100) surface is shown inFigure 10. Each repeat unit has two unique silanol groups oneither side, those on one side related to those on the other bysymmetry. We have optimized the structure of this surface withboth silanol groups deprotonated (–4 unit cell charge), onesilanol group deprotonated (–2 unit cell charge), both silanolgroups protonated (neutral unit cell), and one silanol groupdoubly protonated (�2 unit cell charge).

Hiemstra et al. (1996) used the MUSIC method to predict thefollowing reactions and associated pKa

int values for silica sur-faces:

�SiOHf�SiO� � H� pKaint � 7.9 (21)

�SiOH2�f�SiOH � H� pKa

int � � 4.0 (22)

However, these pKaint values were not obtained exactly as

we have described above. That is, instead of assuming that onepossible bond to the oxygen atom is sterically hindered fromforming bonds with H, as with other singly coordinated surfacegroups, Hiemstra et al. (1996) assumed that all the possiblebonds were available. If they had treated silanol groups in thesame way they treated other singly coordinated groups, they

int

Fig. 10. Neutral (100) surface of cristobalite. White balls representH, grey balls represent O, and black balls represent Si.

would have predicted pKa values of 11.9 and 0.0 for the


reactions represented by Eqn. 21 and 22, respectively. Theyjustified this deviation by asserting that the surface layer ofsilica is penetrable by water molecules, and indeed, one wouldpredict a PZNPC of 1.9 using the pKa

int values of 7.9 and �4.0.This is in the range of reported values for silica (Kosmulski,2001), but pKa

int values of 11.9 and 0.0 would imply a PZNPCvalue of 6.0, which is not in agreement with experiment.

Application of our new method to the cristobalite (100)surface also explains the charging behavior of silica, but doesso by treating silanol groups the same as any other. Table 4 liststhe predicted pKa

int values (using Eqn. 15) of the silanol groupson this surface in the different calculated protonation states.The pKa

int values for the deprotonation of �SiOH groups rangefrom 3.8 to 7.6, while the pKa

int values for the deprotonation of�SiOH2

� groups range from �3.5 to �6.6. All of these valuesseem somewhat low, and would lead to a PZNPC below therange of reported values for silica. However, it must be remem-bered that although Eqn. 15 appears to be reasonably accuratein the environmental pH range, it does not explain all thevariation in pKa

int values even for the oxyacids. Indeed, Table2 shows that the pKa

int values predicted for silicic acid depro-tonation are too low, and the discrepancy gets larger with lowerSb values. On the other hand, if we simply plot the silicic acidpKa

int values vs. Sb and fit a line through the points, we obtainthe following regression equation:

pKaint � 46.9 Sb � 4.34 (23)

New pKaint values for the cristobalite (100) silanol groups

obtained by applying Eqn. 23 are also listed in Table 4. Nowthe predicted pKa

int values for the deprotonation of �SiOHgroups range from 5.5 to 8.4, while the predicted pKa

int valuesfor the deprotonation of �SiOH2

� groups range from �0.1 to�2.5. This would likely lead to a PZNPC of � pH 3; andaccording to Kosmulski (2001), the mean reported PZC valuefor all types of silica is 2.92.

This analysis demonstrates that the approximate nature of thepredictions of Eqn. 15 should always be kept in mind. Whenpossible, a relationship such as that in Eqn. 23 should bedeveloped to predict surface acidity unless (as with the alumi-nol groups considered above) there are no suitable solution

int

Table 4. Calculated cristobalite (100) surface structures in differentvalues as pH is varied. Both Eqn. 15 (derived from the pKa

int values ospecies only) were used to predict pKa

int values for reactions on silanolone of the same reaction is listed, it indicates crystallographically dist

Acid f Base � H�Site density(sites/nm2) (

Cristobalite (100)—�SiOH f �SiO–1 � H� 2.8 0�SiOH f �SiO–1 � H� 2.8 0

Cristobalite (100)�SiOH f �SiO–1 � H� 2.8 0�SiOH2 f �SiOH � H� 2.8 0

Cristobalit�SiOH2 f �SiOH � H� 2.8 0�SiOH2 f �SiOH � H� 2.8 0

Cristobalite (100)—�SiOH2 f �SiOH � H� 2.8 0

monomers with which to relate pKa and Sb. Also, it should be

remembered that extremely high or low predicted pKaint values

are prone to larger errors.The predicted pKa

int values in Table 4 provide insight intohow this surface might change as it is progressively protonated.Figure 11 illustrates the order in which the silanol groups wereprotonated and the calculated pKa

int values for the silanolgroups at each step. We begin with a completely deprotonatedsurface (Fig. 11a), and treat the silanol groups as the conjugatebases in reactions where both silanol groups go from �SiOH to�SiO�. The predicted pKa

int values for such reactions are 5.5and 8.4. We next protonate the silanol with the higher pKa

int

value, since it is more basic, leaving a surface with one �SiOHand one �SiO� (Fig. 11b). Again, silanols on this surface aretreated as conjugate bases in reactions where both groups havelost a proton. Now the pKa

int value for the deprotonation of thesecond group changes from 5.5 to 6.1. Therefore, the initialvalue of 5.5 would never effectively come into play. The pKa

int

value of �0.1 obtained for the transition from �SiOH2� to

�SiOH on the first group would also never come into playbecause we would next protonate the second group to obtain aneutral surface (Fig. 11c). Calculated pKa

int values for the lossof protons to obtain a neutral surface are �1.2 and �2.5.However, if we next protonate the group with the higher pKa

int

value (Fig. 11d), the second value changes from �2.5 to �2.1.Therefore, by this analysis, half the silanol groups would havepKa

int values of �1.2, and the other half �2.1, for the transitionfrom �SiOH2

� to �SiOH. Also, half the silanol groups wouldhave pKa

int values of 8.4, and the other half 6.1, for thetransition from �SiOH to �SiO�. Taking the average valuefor each type of reaction, we obtain a PZNPC of 2.6, which isreasonable for a silica surface.

However, once again we have not quantitatively applied theabove predictions to titration data for silica surfaces. Onereason for this is that while we have examined the effect of theprotonation state of neighboring functional groups on pKa

int,the surface unit cell used only included two unique silanolgroups. We now know that changing the protonation state ofone surface group can change the proton affinity of adjacentgroups. This effect is analogous to the proximity effect ob-served for semiconducting mineral surfaces (Becker et al.,

f protonation were examined to predict the evolution of surface pKaint

l oxyacids) and Eqn. 23 (derived from the pKaint values of silicic acid

. Eqn. 23 is thought to be more reliable, in this case. Where more thans with different pKa

int values.

Me–Oionicity

pKaint

(Eqn. 15)pKa

int

(Eqn. 23)

ites deprotonated0.447 3.8 5.50.447 7.6 8.4

site deprotonated0.447 4.5 6.10.447 –3.5 –0.1

—neutral0.447 –5.0 –1.20.447 –6.6 –2.5

doubly protonated0.447 –6.2 –2.1

states of severagroupsinct site

Sb

v.u.)

both s.209.271—one.222.090e (100).066.040one site.047

2001; Rosso and Becker, 2003), but it arises from different


causes. Over what length scale does this “proximity effect”extend? Similarly, Schlegel et al. (2002) modeled X-ray reflec-tivity data on quartz surfaces to show that surface relaxationextends inward �14 A. Our calculated structures show thatsurface relaxation is more pronounced for highly charged sur-faces, but the slabs in our cristobalite calculations were only 12to 13 A thick, and included an inversion center. Therefore,although we have shown that the more limited treatment givenabove is capable of predicting reasonable pKa

int values, thegeometry of significantly larger systems of atoms must be

Fig. 11. Diagrams representing the sequence by which the cristob-alite (100) surface was protonated, and the structure reoptimized ateach step. (a) Both sites deprotonated. (b) The most basic site proton-ated, and the other deprotonated. (c) Both sites protonated (neutralsurface). (d) The most basic site doubly protonated, and the other singlyprotonated.

optimized to fully treat surface relaxation and the effect of the

protonation state of neighboring groups. And if larger surfaceunit cells are used, a greater number of calculations will beneeded to progressively protonate the surface.

3.4.5. Implications for Multisite Complexation Models

We have purposely examined two types of surfaces (gibbsiteand silica) that are rather well described by the MUSIC method(Hiemstra et al., 1996), but they exhibit some behaviors that areanomalous with respect to that model. Specifically, Hiemstra etal. (1996, 1999) had to assume the existence of a large numberof defects on the gibbsite basal surfaces, and had to predict theacidity of silanol groups differently than other singly coordi-nated functional groups. Our new bond-valence approach qual-itatively explains the major features of gibbsite and silicasurface charging curves, including the “anomalous” behaviors,by taking into account the effect of bond relaxation at surfaces.

Our results seem to indicate that quantitative application ofthese methods to surface charging curves via multisite compl-exation models is possible, but a greater number of surfacestructure optimizations, performed on larger periodic systems,will be necessary. This is considerably more work than it wouldtake to predict surface pKa

int values using the MUSIC method,but we argue that the results may be well worth the effort.Multisite complexation models are still dependent upon somefitted parameters, such as Helmholz layer capacitances. If ourultimate goal is to predict surface-charging behavior withoutthe aid of titration data, it will be necessary to predict theparameters we now fit to such data. To reach that point, we willneed to fit these parameters in a number of systems where othervariables, such as pKa

int values, are accurately described. Forinstance, a surface complexation model using “average” pKa

int

values for �SiOH protonation and deprotonation could cer-tainly be made to fit titration data, but at the expense ofinaccurate determination of layer capacitance. However, as-signing different effective pKa

int values for silanol groups (seeTable 4) to various fractions of the surface sites would likelyresult in the fitting of more realistic capacitances.

Beyond this, our analysis leads to the conclusion that over-simplified methods of pKa prediction can seriously undercut thebasic purpose of multisite modeling. For example, Hiemstra etal. (1999) and Rosenqvist et al. (2002) may have postulatedreactions on gibbsite basal surfaces that do not, in fact, occur.If the purpose of multisite modeling is to understand surfacereactions on a molecular level, more accurate methods of pKa

int

prediction must be developed.

4. CONCLUSIONS

Bond-valence methods for pKa estimation have in the pastneglected one or more crucial factors. First, if the pKa values ofsolution monomers are to be used to predict pKa

int values ofsurface functional groups, similar electrostatic corrections mustbe applied in both cases. In surface complexation models,electrostatic corrections are applied by representing a chargedsurface as a uniform plane of charge density, and an analogouscorrection can be made to solution monomers by treating themas charged spheres. Second, it must be remembered that realsurfaces and real monomers are not homogeneous planes or
spheres, respectively. Rather, charge density is distributed


rather unevenly, and a further electrostatic correction (which isoften fairly large) must be made to account for the proximity ofelectron density to the point of proton attachment. Third, theunsaturated valence of oxygen atoms in oxyacids, hexaquocations, and oxide surfaces is strongly correlated with acidityafter electrostatic corrections are made. However, calculationof unsaturated valence for oxyacids and oxide surfaces must bebased on realistic Me–O bond lengths, which can be obtainedfrom ab initio structure optimizations. Finally, unsaturated va-lence must be divided between possible bonds (4 for oxygenatoms) to reflect the fact that O-H bonds are localized toparticular regions of the O atoms.

We have incorporated all of these factors into a new empir-ical bond-valence model, which appears to predict pKa

int valuesto an accuracy of �1 log unit in the environmental pH range.While this degree of accuracy is perhaps not ideal, it is asignificant step forward. Furthermore, if our attempt to explainphysically the success of this new model is correct, it may bepossible to improve on this approach in the future.

Possible future improvements might include the following.First, it may be possible to characterize localized variations inelectrostatic forces not accounted for by double layer treat-ments more precisely than by including crude Me–O bondionicity values (derived from electronegativity differences) in amultiple regression analysis. For instance, we might determineelectrical potential at the point of proton attachment directlyfrom calculated structures. Second, R0 values for the determi-nation of bond valence (Eqn. 2) are fitted to bond lengths in realcrystal structures, but here we have applied these values tocalculated structures, which are subject to systematic errors thatmight differ among various Me–O pairs. It may be possible toaccount for these systematic errors by fitting new R0 values tobonds in calculated crystal structures, where the calculationmethod is the same as will be used for structure optimizationsof solution monomers and surfaces. Third, we have assumedthat all strong O-H bond valences are the same (0.8 v.u.), butthis may not be the case. Inclusion of this factor in a generalmodel of site acidity might greatly improve its predictivepower. Finally, larger surface structure calculations must be runto better account for surface relaxation and the effect of theprotonation state of neighboring functional groups.

Acknowledgments—Portions of this work were made possible by grantACS PRF #38775-GB5S from the Petroleum Research Fund of theAmerican Chemical Society to BRB. Work by CJT was made possibleby grants from the U.S. Department of Energy (DE-FG02-02ER15326)and the National Science Foundation (EAR-0103053). A portion of theresearch described in this manuscript was performed at the W. R. WileyEnvironmental Molecular Sciences Laboratory, a national scientificuser facility sponsored by the U.S. Department of Energy’s Office ofBiologic and Environmental Research and located at Pacific NorthwestNational Laboratory. PNNL is operated for the Department of Energyby Battelle. WM thanks the BYU undergraduate mentorship programfor supporting his contribution to the paper. Three anonymous review-ers contributed comments that significantly improved the manuscript.We also thank Professors G. V. Gibbs and I. D. Brown for helpfuldiscussions.

Associate editor: C. M. Eggleston

REFERENCES

Allred A. L. (1961) Electronegativity values from thermochemical
data. J. Inorg. Nucl. Chem. 17, 215–221.
Baes C. F. and Mesmer R. E. (1976) The Hydrolysis of Cations. JohnWiley, New York.

Becker U., Rosso K. M., and Hochella M. F., Jr. (2001) The proximityeffect on semiconducting mineral surfaces: A new aspect of mineralsurface reactivity and surface complexation theory? Geochim. Cos-mochim. Acta 65, 2641–2649.

Bickmore B. R., Rosso K. M., Nagy K. L., Cygan R. T. and TadanierC. J. (2003) Ab initio determination of edge surface structures fordioctahedral 2:1 phyllosilicates: Implications for acid-base reactiv-ity. Clays Clay Miner 51, 359–371.

Bleam W. F. (1993) On modeling proton affinity at the oxide/waterinterface. J. Colloid Interface Sci. 159, 312–318.

Bockris J. O. and Reddy A. K. N. (1976) Modern Electrochemistry,Vol. 1. Plenum, New York.

Brown I. D. (1981) The bond-valence method: An empirical approachto chemical structure and bonding. In Structure and Bonding inCrystals 2 (eds. M. O’Keefe and A. Navrotsky), pp. 1–30. AcademicPress, New York.

Brown I. D. (2002) The Chemical Bond in Inorganic Chemistry: TheBond Valence Model. Oxford University Press, New York.

Brown I. D. and Altermatt D. (1985) Bond-valence parameters ob-tained from a systematic analysis of the inorganic crystal structuredatabase. Acta Crystallogr. B41, 244–247.

Davis J. A. and Kent D. B. (1990) Surface complexation modeling inaqueous geochemistry. Rev. Mineral. 23, 177–260.

Ducker W. A., Xu Z., Clarke D. R., and Israelachvili J. N. (1994)Forces between alumina surfaces in salt solutions: Non-DLVOforces and the implications for colloidal processing. J. Am. Ceram.Soc. 77, 437–443.

Dzombak D. A. and Morel F. M. M. (1990) Surface ComplexationModeling: Hydrous Ferric Oxide. John Wiley, New York.

Franks G. V. and Meagher L. (2003) The isoelectric points of sapphirecrystals and alpha-alumina powder. Colloids Surf., A 214, 99–110.

Griffen D. T. (1992) Silicate Crystal Chemistry. Oxford UniversityPress, New York.

Hiemstra T., Van Riemsdijk W. H., and Bolt G. H. (1989) Multisiteproton adsorption modeling at the solid/solution interface of(hydr)oxides: A new approach. I. Model description and evaluationof intrinsic reaction constants. J. Colloid Interface Sci. 133, 91–104.

Hiemstra T., Venema P., and Van Riemsdijk W. H.. (1996) Intrinsicproton affinity of reactive surface groups of metal (hydr)oxides: Thebond valence principle. J. Colloid Interface Sci. 184, 680–692.

Hiemstra T., Yong H., and Van Riemsdijk W. H.. (1999) Interfacialcharging phenomena of aluminum (hydr)oxides. Langmuir 15,5942–5955.

Horn R. G., Clarke D. R., and Clarkson M. T. (1988) Direct measure-ment of surface forces between sapphire crystals in aqueous-solu-tions. J. Mater. Res. 3, 413–416.

Kosmulski M. (2001) Chemical Properties of Material Surfaces, Vol.102. Marcel Dekker, Basel, Switzerland.

Kosmulski M. (2003) Comment on “Point of zero charge of a corun-dum-water interface probed with optical second harmonic generation(SHG) and atomic force microscopy (AFM): New approaches tooxide surface charge” by A. G. Stack, S. R. Higgins, and C. M.Eggleston. Geochim. Cosmochim. Acta 67, 319–320.

Kresse G. and Furthmuller J. (1996) Efficient iterative schemes for abinitio total-energy calculations using a plane-wave basis set. Phys.Rev. B. Condens Matter 54, 11169–11186.

Larson I., Drummond C. J., Chan D. Y. C., and Grieser F. (1997) Directforce measurements between silica and alumina. Langmuir 13,2109–2112.

Lim C., Bashford D., and Karplus M. (1991) Absolute pKa calculationswith continuum dielectric methods. J. Phys. Chem. 95, 5610–5620.

Liptak M. D. and Shields G. C. (2001) Accurate pKa calculations forcarboxylic acids using complete basis set and Gaussian-n modelscombined with CPCM continuum solvation methods. J. Am. Chem.Soc. 123, 7314–7319.

Liptak M. D., Gross K. C., Seybold P. G., Feldgus S., and Shields G. C.(2002) Absolute pKa determinations for substituted phenols. J. Am.Chem. Soc. 124, 6421–6427.

Ma C. and Eggleton R. (1999) Cation exchange capacity of kaolinite.
Clays Clay Miner. 47, 174–180.


Machesky M., Wesolowski D., Palmer D., and Ridley M. (2001) On thetemperature dependence of intrinsic surface protonation equilibriumconstants: An extension of the revised MUSIC Model. J. ColloidInterface Sci. 239, 314–327.

Meagher L., Maurdev G., and Gee M. L. (2002) Interaction forcesbetween a bare silica surface and an alpha-alumina surface bearingadsorbed polyelectrolyte and surfactant. Langmuir 18, 2649–2657.

National Institute of Standards and Technology (1997) NIST CriticallySelected Stability Constants of Metal Complexes Database, Ver. 3.0,NIST Standard Reference Database 46. U.S. Department of Com-merce.

Noguera C. (2000) Polar oxide surfaces. J. Phys.: Condens. Matter 12,R367–R410.

Pauling L. (1960) The Nature of the Chemical Bond. Cornell Univer-sity Press, Ithaca, NY.

Payne M. C., Teter M. P., Allan D. C., Arias T. A., and JoannopoulosJ. D. (1992) Iterative minimization techniques for ab initio total-energy calculations: Molecular dynamics and conjugate gradients.Rev. Mod. Phys. 64, 1045–1097.

Perdew J. P. and Wang Y. (1992) Accurate and simple analytic repre-sentation of the electron-gas correlation-energy. Phys. Rev. B: Con-dens. Matter 45, 13244–13249.

Perrin D. D. (1982) Ionization Constants of Inorganic Acids and Basesin Aqueous Solution. Pergamon, Oxford, UK.

Rosenqvist J., Persson P., and Sjoberg S. (2002) Protonation andcharging of nanosized gibbsite (alpha-Al(OH)3) particles in aqueoussuspension. Langmuir 18, 4598–4604.

Rosso K. M. (2001) Structure and reactivity of semiconducting mineralsurfaces: Convergence of molecular modeling and experiment. Rev.Mineral. Geochem. 42, 199–271.

Rosso K. M. and Rustad J. R. (2000) Ab initio calculation of homo-geneous outer sphere electron transfer rates: Application toM(OH2)6

3�/2� redox couples. J. Phys. Chem. A 104, 6718–6725.Rosso K. M. and Morgan J. J. (2002) Outer-sphere electron transfer

kinetics of metal ion oxidation by molecular oxygen. Geochim.Cosmochim. Acta 66, 4223–4233.

Rosso K. M. and Becker U. (2003) Proximity effects on semiconduct-ing mineral surfaces II: Distance dependence of indirect interactions.Geochim. Cosmochim. Acta 67, 941–953.

Rosso K. M., Rustad J. R., and Bylaska E. J. (2001) The Cs/K exchangein muscovite interlayers. An ab initio treatment. Clays Clay Miner.49, 500–513.

Rosso K. M., Rustad J. R., and Gibbs G. V. (2002) Trivalent ionhydrolysis reactions II: Analysis of electron density distributions inmetal-oxygen bonds. J. Phys. Chem. A 106, 8133–8138.

Rustad J. (2001) Molecular models of surface relaxation, hydroxyla-tion, and surface charging at oxide-water interfaces. Rev. Mineral.Geochem. 42, 169–197.

Rustad J. R., Felmy A. R., and Hay B. P. (1996a) Molecular staticscalculations for iron oxide and oxyhydroxide minerals: Toward aflexible model for the reactive mineral-water interface. Geochim.Cosmochim. Acta 60, 1553–1562.

Rustad J. R., Felmy A. R., and Hay B. P. (1996b) Molecular staticscalculations of proton binding to goethite surfaces: A new approach

to estimation of stability constants for multisite surface complexationmodels. Geochim. Cosmochim. Acta 60, 1563–1576.

Rustad J. R., Dixon D. A., Kubicki J. D., and Felmy A. R. (2000)Gas-phase acidities of tetrahedral oxyacids from ab initio electronicstructure theory. J. Phys. Chem. A 104, 4051–4057.

Saalfeld H. and Wedde M. (1974) Refinement of the crystal structure ofgibbsite, Al(OH)3. Z. Kristallogr. 139, 129–135.

Sahai N. (2002) Is silica really an anomalous oxide? Surface acidityand aqueous hydrolysis revisited. Environ. Sci. Technol. 36, 445–452.

Saracino G. A. A., Improta R., and Barone V. (2003) Absolute pKadetermination for carboxylic acids using density functional theoryand the polarizable continuum model. Chem. Phys. Lett. 373, 411–415.

Schlegel M. L. (2002) Structures of quartz (10-10)- and (10-11)-waterinterfaces determined by X-ray reflectivity and atomic force micros-copy of natural growth surfaces. Geochim. Cosmochim. Acta 66,3037–3054.

Sefcik J. and Goddard W. A. I. (2001) Thermochemistry of silicic aciddeprotonation: Comparison of gas-phase and solvated DFT calcula-tions to experiment. Geochim. Cosmochim. Acta 65, 4435–4443.

Stack A. G., Higgins S. R., and Eggleston C. M. (2001) Point of zerocharge of a corundum-water interface probed with optical SecondHarmonic Generation (SHG) and Atomic Force Microscopy (AFM):New approaches to oxide surface charge. Geochim. Cosmochim.Acta 65, 3055–3063.

Stack A. G., Higgins S. R., and Eggleston C. M. (2003) Response tocomment on “Point of zero charge of a corundum-water interfaceprobed with optical second harmonic generation (SHG) and atomicforce microscopy (AFM): New approaches to oxide surface charge.”Geochim. Cosmochim. Acta 67, 321–322.

Stumm W. (1992) The Chemistry of the Solid-Water Interface. JohnWiley, New York.

Sverjensky D. A. and Sahai N. (1996) Theoretical prediction of single-site surface-protonation equilibrium constants for oxides and sili-cates in water. Geochim. Cosmochim. Acta 60, 3773–3797.

Tossell J. A. and Sahai N. (2000) Calculating the acidity of silanols andrelated oxyacids in aqueous solution. Geochim. Cosmochim. Acta 64,4097–4113.

Vanderbilt D. (1990) Soft self-consistent pseudopotentials in a gener-alized eigenvalue formalism. Phys. Rev. B. Condens Matter 41,7892–7895.

Wasserman E., Rustad J. R. and R. F. A. (1999) Molecular modeling ofthe surface charging of hematite—I. The calculation of proton af-finities and acidities on a surface. Surf. Sci. 424, 17–19.

Westall J. C. and Hohl H. (1980) A comparison of electrostatic modelsfor the oxide/solution interface. Adv. Colloid Interface Sci. 12,265–294.

White J. A. and Bird D. M. (1994) Implementation of gradient-cor-rected exchange-correlation potentials in Car-Parrinello total energycalculations. Phys. Rev. B: Condens. Matter 50, 4954–4957.

Wieland E. and Stumm W. (1992) Dissolution kinetics of kaolinite inacidic aqueous solutions at 25 °C. Geochim. Cosmochim. Acta 56,3339–3355.

Bond-valence methods for pKa prediction: critical reanalysis and a new approach

Documents

Transcript of Bond-valence methods for pKa prediction: critical reanalysis and a new approach