1754
Langmuir 1993,9, 1754-1765
Heterogeneity of Proton Binding Sites at the Oxide/ Solution Interface Cristian Contescu,f Jacek JagieUo,*and James A. Schwarz* Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, New York 13244 Received January 15,1993. I n Final Form: April 26,1993 Proton adsorption isotherms were used to calculate the affhity distributions of proton binding sites for y alumina and F-modified alumina samples. The isothermswere obtained by potentiometric titration over as broad a pH interval as allowed by the samples being studied. The affinity distributions were calculated using the local solution of the adsorption integral equation. A smoothingspline procedure was employed, and acriterionfor optimalsmoothing was proposed. The method was testedby model calculations and verified on simple chemical compounds with known proton affiity characteristics. The apparent distribution spectra for oxides, with electrostaticeffeds neglected, showed well-defined peaks that were assigned to protonatioddeprotonationprocesses of distinct -O(H) or -OH(H) groups in various surface configurations. A correlation was obtained between the evaluated log K values for proton association equilibriaandvalues predided by structuralmodels. These resulta confirm that proton adsorptionisotherms are sensitiveto the heterogeneityexistingat oxide surfaces. We find that ionic strength effectaon apparent distributions were minor when compared with the effecta of surface structure and chemical composition. Introduction In general, an accurate description of proton transfer reactions at the oxidelsolution interface requires a combination of a particular site binding model and an electrostatic model which is needed to account for the electric field effects at the interface. Previous attempts to separate between chemical energy and electrostatic energyterms that characterize the interactions of charged ions at the solid/electrolyte solution interface were not completely One possible reason is the emphasis given in the earliest models to electrostatic interactions and neglect of the significance of specific chemical energy contributions. The classical Gouy-Chapman-Stern-Grahame theory was satisfactorilyusedto describe the chargingmechanism at mercury and silver halide surfaces, where the chemical energy term was assumed constant and the experimental data were solely used to compute the electrostatic energy term. In the case of the oxide/electrolyte interface, the situation is fundamentally different. Here the potential determining ions (H+, OH-) are no longer intrinsic constituents of the unperturbed solid phase, but reactants from the liquid phase. Customarily,the chemical energy parameters on the surface of oxides were found by extrapolation of experimental data to zero charge conditions, where the electrostatic energy term vanishes. However, as has been already shown’, this procedure always implied the use of a particular electrostatic model of the interface. This has, in turn, influence on the course of the extrapolation (through the corresponding model equations and the adjustable parameters involved) and consequently on values obtained for the chemical energy terms. Westall and Hohl showed in their 1980 review] that variouselectrostaticmodels for the oxide/solutioninterface
* Author to whom correspondence should be. addressed. + Permanent address: Institute of Physical Chemistry, Romanian Academy, Spl. Independentei 202, Bucharest 77208, Romania. t Permanent address: Institute of Energochemistry of Coal and Phyeicochemistry of Sorbents,University of Mining and Metallurgy, 30-069 Krakow, Poland. (1)Westall, J.; Hohl, H. Adu. Colloid Interface Sci. 1980, 12, 265. (2)Westall, J. In Aquatic Surface Chemistry;Stumm, W., Ed.; Wiley: New York, 1987;p 3.
0743-746319312409-I754$O4.W/0
can adequately represent experimental data. However no valuable physical meaning could be attached to the resulting ensemble of parameters which were modeldependent values. Based on new experimental data, Westal12concluded in 1987that the site binding model at the interface should be revisited. He showed that either a two-states (one pK) model or a three-states (two pK) model with pK values sufficiently close to each other can simultaneously explain both surface charge and surface potential data. On the other hand, the heterogeneity of the surface of oxideshas long been recognized. The surfaceheterogeneity results from differences in coordination of oxygen atoms which differ from each other by the number and type of surrounding metal cations. These differences in the configuration determine the occurrence of various types of surface hydroxyls on partially dehyrated surfaces, as identified in IR spectra and characterized by their acid/ base proper tie^.^ In addition, actual surfaces contain structural defects, coordinative imperfections, and impurities which also contribute to their overall heterogeneity. The heterogeneity effects in adsorption in oxide/ electrolyte solutions were f i s t considered by Leckie and A few years ago more detailed quantitative co-~orkers.~*~ models were published by Koopal and van Riemsdijk.SB The authors concluded that in the case of a random topography of sites titration curves were not sensitive to heterogeneity effects.- When a patchwise topography was assumeds for physical mixtures of oxides, the overall behavior of the mixture could be describedby the weighted average of the components. ~
(3) Tanabe, K.; Mieono, M.; Ono,Y.;Hattori, H.New Solid Acids and Bases; Studies in Surface Science and Catalysis;Elsevier: Amsterdam,
1989; Vol. 51. (4) Davis, J. A.; Leckie, J. 0. J. Colloid Interface Sci. 1978, 67, 90. ( 5 ) Benjamin, M. M.; and Leckie, J. 0. J. Colloid Interface Sci. 1981, 79,209. (6) Van Riemsdijk, W. H.; Bolt, G. H.; Koopal, L. K.; Blaakmeer, J. J. Colloid Interface Sci. 1986.109. -,- , 210. (7) Van RieAdijk, W. H.; De Witt, J. C. M.; Koopal, L. K.; Bolt, G. H. J. Colloid Interface Sei. 1987, 116, 511. (8)Koopal, L. K.; Van Riemsdijk, W. H. J.Colloid Interface Sci. 1989, 128.188. ~-~
(9)Gibb, A. W.M.; Koopal, L. K. J.Colloid Interface Sci. 1990,134, 122.
0 1993 American Chemical Society
Heterogeneity of Proton Binding Sites
Langmuir, Vol. 9, No. 7, 1993 1766
Similar conclusions about the insensitivity of titration curves to the heterogeneity effects were reached by Rudzinki et al.loJ1 However, electrokineticll and heats of immersionlodata confirmed the heterogeneous character of the investigated oxidelsolution interfaces. The objective of this paper is to study the heterogeneity of acid-base properties at oxidelsolution interfaces. We evaluate the proton affinity distributions from potentiometric titration curves consisting of a large number of data points collected over a broad pH range. At a fiist approximation we neglect electrostatic effects in the calculationof affiiity distributions. In this way we obtain so-called apparent distributions which do not exactly represent intrinsic properties of the oxide surface but are related to the conditions under which the interface was studied. Nevertheless, they are free from the ambiguity of any particular electrostatic model. We demonstrate that using this simple approach we are able to extract meaningful information about the acid-base properties of oxides. The features revealed by the apparent affinity distributionsobtained for pure and fluoride-dopedalumina samples were successfully correlated with available structural models.12
Background The customary representation of proton transfer at the oxidelsolutioninterface has been through reactions of the type -M-OH; -M-OH
Q
Q
-M-OH
-M-0-
+ H:
+ H:
Kay
(1)
Kdintr
(2)
Here the surface is viewed as comprised of a certain number (N,) of perfectly amphoteric -MOH groups which may either associate or dissociate protons, depending on the values of the solution pH and the intrinsic values of the two equilibrium constants, Kimb. In view of the existing heterogeneity of real oxide surfaces, this 2-pKlone site model is too simple and various attempts13were made to adopt more realistic reaction models. However, neither the 2-pE/two sites model of zwitterionic surfaces, advanced by James and Parks,14who proposed the existence of separate basic and acidic sites which may associate or dissociateprotons,nor the 2-pKlmultisitesgeneralization due to Kita and Tanabe,16who proposed that the above sites might be differently distributed according to their number and strength, gained sufficient interest to be applied to oxide surfaces in aqueous solutions. The representation of perfectly amphoteric sites was more appealingsince it offered a simple ionization scheme and allowed for an easier description of the point of zero charge: PZC = (pK,1+ pKd)/2. Based on this reaction scheme, more and more complicated electrostatic models were proposed, but these failed to give an accurate description of the electrical properties of the interface. Attempts to determine the total number of rapidly exchangeablesites gave almost alwaysvaluessmaller than the actual site density of oxide surfaces, so that this parameter was often consideredas an adjustable one. Also, it was shown that it is possible to fit the experimental data
by using a wide combination of acidity constants, so that the published intrinsic values of the equilibriumconstants had only a model-dependent meaning. Another characteristic of the oxides, the anomalously high surface capacitance for adsorption of protons and its dependence on sample preparation, was never explained satiafactorily.lJ4 A procedure to calculate independentlythe differenceof acidity constants (DPK = pKd - pKa1) and the surface density of ionizable size (N,) from linearization of the surfacecharge data around the PZC was proposed by Noh and Schwarz.lB Though still using the 2-pK model, the method no longer assumed the prevalence of one kind of surface group even far fromthe PZC. Their results showed that the values of DPK and N , were strongly dependent on the pH range used for analysis of experimental data, although a good correlation coefficient was always found in the regression analysis. Generally, the narrower the pH range, the smaller were the values for DPK and N,, so that within each pH interval, a defimite number of sites having pKai values very closeto eachother could be found.16 A more general picture2vBJof the proton transfer reactions (l-pK model) considers the surface as an ensemble of surfaceoxygen atoms which can exist in either protonated or deprotonated states -M-OH+1/2cs -M-O-1/2 + H:
K, = (Ka1Kd>'/2 (3)
where the fractionalchargesassociatedwith surface group should be considered as formal, used as a better bookkeeping of charges, and Karefersto the intrinsic reactivity of surface oxygen. We use the assumptions of the l-pK model and append the concept of the intrinsic heterogeneity of surface sites with respect to their proton binding affinities. Our l-pKI multisites model pictures the oxide surface as consisting of nonequivalent oxygen groups which are characterized by a continuous distribution of proton affiiities. In the previous site binding models, various assumptions of the distribution of surface sites have been made. Instead of making a priori assumptions about a possible distribution and then fittingthe experimentaldatato that distribution, our approach is to rely on the experiment and to fiid the distribution from deconvoluted proton adsorption isotherms. A similar approach was used recently for the analysis of proton affinity distributions and ion binding on humic ~ u b s t a n c e s . ~ ~ J ~ We assume that the proton affinity of various oxygen groups at the oxidelsolution interface is determined by the same factors (number and type of subadjacent coordinating metal ions) which are responsible for the differences in acidlbase properties of the corresponding -OH groups on a partially dehydrated oxide surface. For the purpose of the present paper we will neglect the binding of electrolyte counterions. When a single population of binding sites is considered, the degree of association or the obindingcurve", with the meaning of an adsorption isotherm, can be expressed as (4)
where Bi represents the fraction of the total population (10)Rudzinski, W.; Charmas, R.; Partyka, S. Langmuir 1991,7,354. which has bound the offered species (protons), [HI (11) Rudzinski, W.; Charmas, R.; Partyka, S.; Thomas,F.; Bottero, J. represents the molar concentration of unbound species at Y.Langmuir 1992,8,1154. (12)Knozinger, H.; Ratnasamy, R. Catal. Rev. Sci. Eng. 1978,17,31. (16)Noh, J. S.;Schwarz, J. A. J. Colloid Interface Sci. 1990,139,139. (13)HeLmy, A. K.;Ferreiro, E. A.; de Buaaetti, S. G. 2.Phys. Chem., (17)De Wit, J. C. M.; Van Riemedijk, W. H.; Nederlof, M. M.; Leipzig 1980;261, 1065. (14)James, R. 0.; Parks, G. A. Surf. Colloid Sci. 1982,12,119. Kinniburgh, D. G.; Kwpal, L. K.A n d . Chim. Acta 1990,232,189. (15)Kita,H.;Henmi,N.;Shimazu,K.;Hatt~ri,H.;Tauabe,K.J.Chem. (18)Nederlof, M.Analpie of Binding Heterogeneity; PhD. The&, SOC.,Faraday Trans. 1 1981, 77,2451. Wageningen Agricultural University, Wageningen, 1992.
1756 Langmuir, Vol. 9, No. 7, 1993
equilibrium, and Ki is the affiiity constant which relates the equilibrium between the bound and unbound species:
Si + H * H-Si In the case of binding of charged species, the electrostatic effects are usually included in order to derive an intrinsic affinity constant; otherwise, the affiiity constant is an apparent or microscopic constant whose value might be conditioned experimentally:
However, we will avoid the use of complicated electrostatic models. Our analysis will, therefore, provide apparent rather than intrinsic constants, Le., the effects of the ionic composition of the electrolyte are expected. We will show later that ionic strength has minor effects in the deconvoluted distributions. The lateral variations in the Coulombic field at the interface, as expressed by the specific potentials in the electrostaticmodels, may be neglected because Coulombic interactions are long ranged. The smeared out electrical potential at the interface depends on the solution and surface composition but may be viewed (geometrically)as a constant potential superimposed on the local field associated with various sites. The nonequivalence of surface hydroxyls caused by local differences in their configuration is reflected in the chemical binding forces which are short ranged forces. For a heterogeneous population of sites with a discrete affinity distribution, the overall degree of protonation, 8, is the weighted sum of the degrees of protonation of the different categories of sites; for a continuous affinity spectrum, the summation is replaced by an integral
where 6 (K,[H]) is the local adsorption isotherm corresponding to eq 5 andf(l0gK) is the normalized distribution function of intrinsic proton affinity constants, K, which characterizesthe population of proton acceptor sites. This problem is, indeed, common to many disciplines, where the analog of our solid/liquid interface takes many identities. Complex distributions for receptors with different binding affinities for a single chemical species are encountered in immunology, pharmacology, endocrin ~ l o g yanalysis ,~~ of aquatic systems,2O and humic materials.21*22Our first level formulation allows for dealing with the simplest case of affinity distributions where the local isotherm is described by a Langmuir type relationship. A second-levelanalysis can also be formulated and will be discussed elsewhere.23 Experimental Section Materials. Titration of pure compounds in homogeneous solutions was used as a preliminary check of the computational procedures. Monoprotic (sodiumacetate and pyridine),diprotic (glycine),and triprotic (citric acid) weak organic acids or bases were used without any additional purification. (19) Thakur, A. K.; Muneon, P. J.; Hunston, D. L.; Rodbard, D. Anal. Biochem. 1980,103, 240. (20) Perdue, E. M.In Organic Acids in Aquatic Ecosystems; Perdue, E. M.,Gjesaing, E. T., Me.; Wiley: New York, 1990;p 111. (21)Shuman, M.S.; Collins, B. J.; Fitzerald, P. J.; Oson, D. L. In A q w t i c and Terrestrial Humic Materials; Chrietman, R. F., Gjesaing, E. T., Us.; Ann Arbor Sci., Butterworthe, 1983, p 349. (22)De Wit, J. C. M.;Van Riemedijk, W. H.; Nederlof, M. M.; Kinniiburgh, D. G.; Koopal, L. K. Anal. Chim. Acta 1990,232, 189. (23) To be published.
Contescu et al.
A sample of high purity y alumina supplied by American Cyanamid (identificationNo. S.N.-7063) wascharacterized(BET surface area 140 mZ/g,pore volume 0.85 cm*/g, impurity lev& 0.01% Cu, 0.007% Fe, 0.004% Na, 0.001% S and Pd, O.OOOl% Mo and As) and ueed as a prototype solid oxide. The fluorine-dopedaluminas(0.2,1,and 2% F) were prepared by the incipient wetness method, using N W solutions. After drying (296K, 18 h) the modified aluminas were calcined (876 K, 3 h) and stored tightly bottled. All chemicalsusedwere certifiedanalyticreagentsfrom Aldrich and Fisher Scientific and were used without any additional purification. Procedures. Potentiometric Titrution. The experiments were carried out by using a 666 Dosimat (Metrohm) miemburet which allowed for an accurate dosing of the titrant solutions (fO.OO1 mL). A thermostated double-walled Pyrex vewl equippedwith magnetic stirrer and a lid with holes for electrodes and inert gae was used. The pH was measured with a digital Fisher Accumet pH-meter Model 806-Mp (aO.01 pH units). A combination glass electrode (Coming) was used that was standardized before and after each experiment with certified Fieher Scientific buffers (pH 4.00,7.00, and 10.00). In all experiments the inert electrolyte was prepared from certified A.C.S. d u m nitrate and used without further purification. The titrant solutions were certified volumetricstandards of 0.1 N nitric acid and 0.1N sodium hydroxideand were used as received. The acid and base solutions were standardized versus each other by potentiometric titration. In potentiometrictitration of oxides the pH has a rapid change occurring in the f i t few minutes after addition of acid (or base) and this is followed by a much slower variation. Only the f i t rapid change of pH is usually attributedw* to the reactions of H+/OH- ions at the oxide surface/electrolytainterface, and this will be what is reported in the following sections. Before titration, the solid samples were equilibrated with the electrolyte solution for 18-20 h with a continuous but gentle stirring and protected from contact with air by a continuous stream of purified nitrogen gas. During this time the surface was rehydrated slowly and an initial surface equilibrium was reached with respect to the electrolyte ions and H+/OH- ions. After equilibrationwas complete,a stable pH value was read and then a small amount of either acid or base was added in order to shift the pH toward one of the ends of the pH inverval selected for titration. We found that less than 20 min was long enough to reach a new quasi-stable equilibrium which was considered the starting point of titration. Once this starting point was reached, titration was conducted by a d d q small incrementa of either acid or base which only slightly shifted the surface equilibria. Thus the whole course of titration went,continuously, through a wries of (pseudo)equilibrium states which were not kinetically limited. As a standard procedure, the amount of oxide sample was adjusted in order to keep the concentration of surface sites (calculatedon a geometricalbasis) as close as possible within the recommended range%of (0.5-1) X 1 W mol/L. For an initial volume of electrolyte of 60 mL, the amoluit of alumina was 0.6 g. The incrementaladdition of either acid or base and the waiting time after each addition were then selected within the range of 1-6 mmoUmin. The totaltime needed to titrate an oxide sample was usually 3 h. A titration curve consisted usually of approximately 100-120points. The procedure of potentiometric titration assumes that the only reactions in the system are with H+ and OH- ions either adsorbing or desorbing from the oxide surface and that no other secondary reactions exist (such asdissolution and bulk speciation). The proton consumption function is obtained from subtraction of a reference curve from the sample curve. The main problem which must be addressed in interpreting the data are the choice of the reference curve and the method to minimize the errors. (24) Ahmed, S. M.J. Phys. Chem. 1969, 73,3646. (26) (a) Blok, L.; De B N ~ P. , L. J. Colloid Interface Sci. 1970,32, 518. (b) Blok, L. The Ionic Double Layer on Zinc Oxide in Aqueous ElectrolyteSolutione; Dodoral Thesis,Bronder Offset, Rottardam, 1968. (26) B a a , C. F.;Mesmer,R. E. The Hydrolysis of Cations; Wiley: New York, 1976.
Langmuir, Vol. 9, No. 7,1993 1757
Heterogeneity of Proton Binding Sites
,
I
I
1.5
A
10.5 (R c
I
8
O
a
-0.5 -1
-2
a
9.5
-3
' 0
10.5 I
I
I
I
I
I
1
0.25
0.5
0.75
1
1.25
1.5
1.75
m (9)
- 4 1 2
"
'
~
"
4
"
'
'
"
"
"
6
"
"
8
10
12
PH
Figure 1. Check of the influence of aluminadissolutionon proton consumption isotherms, following ref 26.
Figure 2. Normalizedproton consumption functionsdetermined for compounds with known proton binding affiiities.
In a recent analysisSchulthessand Sparksn showedthat results obtained by using either a theoretical reference or various experimentalreferences were similar, but still not free of errors. To avoid subtraction of two experimental curves which would compound errore, we preferred to use a theoretical reference and the proton balance equation
a reference sample were titrated starting from either acidic or basic Solutions, and the starting pH was varied within pH 3-4 and pH 10-10.5,respectively. The titration curves were then compared with those measured in separate experiments when titration was started from neutral pH and conducted to both pH extremes. #en the contact time of aliquots of the reference sample with solution with aggressive pHs was kept as short as poseible2Bp80 the titration curves were quantitatively identical, within the limits of our estimated experimental error. Finally we measured by atomic absorptionthe total aluminum released during several titration and compared this value to the total H+ reacted during the titration. The average ratio was 0.076, which is small enough for us to conclude that under our experimental conditions alumina dissolution in either extreme pH is not contributing significantly to the proton consumption functionand surface protonreactionsrather thansolutionproton reactions are dominating. Errore in the calculated values of A Q may arise from errore in c. or c b but arise more likely either from errors in pH measurements or from underestimation of dissolution effecta. Though all necessary precautions were taken to avoid errore in pH measurements, even under the best conditions pH readings may be affected by errore in the range of 0.01-0.03 units (which may come from junction potentials or from fouling of the glass electrode by sample particles). The maximum effect of such pH errore on the proton consumption function as estimated by Perdueadis 0.2-0.7 mequiv/L (0.01to 0.03 mequiv/M)mL) at pH < 2 and pH > 12,which is very small compared to the level of the measured values in our experiments. We estimated errore associatedwith a toooptimisticapproach of the solubilityeffects. Thus, from the rate of dissolution of A l 2 0 3 (10-8 mol m-2 h-l at pH 3P1we dculated the amountcontributedby proton-catalyzed dissolution of alumina under our experimentalconditions. The result (0.004 mequid50 mL) is even lower than the usual error associated with pH readings. In this calculation it was assumed a contact time of 1 h with the 'aggressive* solution, which is a significant overestimation. The method of potentiometric titration gives accurate results within a pH "window" determined both by the properties of the investigated samples (such as dissolution behavior at extreme pH values) and by the errors in pH readings at very low and very high pHs. The use of a calculated reference curve is sufficiently accuratewithin the pH interval of 3-11. Outsidethierange elight deviationsoccurred between the calculated and the experimental blanks (Figure 21,and they became important at pH < 2 and pH > 12. This effect which was described as the 'buffering power of watefmobscuresthe titration of acids with pK < 2 or of bases with pK > 12 when present. Method of Calculation of the Affinity Distribution. The affinitydistribution functionis calculated usingthe approximate method proposed by Rudzinski and Jagiello (RJ)Bafor the
H+,,mp,n
vo(c,- c b ) + AvNt- (vo + AV)([Hlf- [OHIJ
(8) where C. and c b are analytical concentrations of acid (or base) added initially, [Hlf and [OHlf are actually measured concentrations, VOis the volume of solutioncorrespondingto the starting point of titration, AV is the cumulative volume of titrant added, and Nt is the normality of titrant (negative for titration with base). The concentrations [H]f and [OHIfwere calculated from the measured pH values and by using Davies' equationm for activity coefficients log y = -0.51152?(
-)
- 0.31
where I is the ionic strength for each point of the titration. The normalized proton consumption function, AQ(pH), was defiied with respect to the amount of titrated sample (moles for organic acids or bases, grams for oxides). The above procedure does not account for the supplementary proton consumption due to sample dissolution during titration. Alumina is soluble to some degree in both acidic (pH < 3) and basic (pH > 10.611)solutions. We intended to apply corrections accountingfor possiblesolubility duringthe fast adsorptionstep. As indicated by Blok and De Bruyn,= the amount of protons consumed or released in dissolution reactions does not depend on the amount of oxide present in suspension but only on the volume of solution and the solubility difference at the final and initial pHs. This is true when dissolution equilibriumis reached. At pH values of 2.5-3 the time to reach dissolution equilibrium is ca. 20 min.@* We checkedin a number of experimentscarried out with a constant volume of electrolyte that the amount of protons consumedat constant pH varied linearlywith the amount of oxide in suspension and passed through the origin, as shown in Figure 1. However, when the dissolution equilibrium is not reached, such plots would still pass through the origin because dissolution rates are first order in surface area. We estimated the impact of dissolution in the acidic range, and those estimates are presented below. In an attempt to find those experimentalconditionswhich allow us to use a broad pH rangeandminimizethedissolutioneffecta,the followingsequence of experiments were carefully conducted. Different aliquota of (27) Schultheae, C. P.; Sparks, D.L. Soil. Sei. SOC.Am. J. 1986, 50, 1406. (28) Stumm, W.; Morgan, J. J. Aqwtic Chemistry; Wiley: New York, 1981. (29)Mieth,J. A.;Huang, Y. J.; Schwarz, J. A. J. Colloid Interface Sci. 1988,123,366. (30)Mieth, J. A.;Schwan, J. A. Appl. Catal. 1989,55, 137.
(31)Stumm, W.; Furrer, G. In Aquatic Surface Chemietry; Stumm, W., Ed.; Wdey New York, 1987;p 197. (32) Rudzinski, W.; Jagiello, J.; Grillet, Y. J. Colloid Interface Sci. 1982,87,478.
Conteecu et al.
1758 Langmuir, Vol. 9, No. 7, 1993 -
2.5
1 -
2
1.2
,
I
0.8 Q
0.6
-
0.2 -
-0.5
-0.2 , 4
-1 5
6
7
8
9
10
J 0
0.005
0.01
0.015
0.02
Goodness of flt, s
log K
Figure 3. Distribution calculated wing CA and RJ approximations from the Langmuir (L) isotherm with log K 7 and
Figure 4. Effect of random errors on the dependence between the roughness of the spline function and the goodness of fit for the Langmuir isotherms generated with the imposed random errors of different standard deviations, s*.
calculation of adsorption energy distributions from gas-solid adsorptionisotherms. This method canbe considered as a special case of the exact local solution of the integral equation with a kernel of a Langmuir isotherm derived by Jagiello et al.= The exact local solution is given by the following infinite series
To approximate experimental adsorption isotherma and to calculate appropriate derivatives, we apply here a procedure of smoothing splinea described by Rsinech.88 This procedure was already employed by others in similar applications.18a In this approach, the function under consideration is approximated by a cubic spline function g(r) which minimizes the following function
from the isotherm generated for the Gaussian (G)distribution centered at log K = 7 with standard deviation UG = 1.
#%(log[Hl) f(l0gK) = $b”(
Wg[Hl)”+l
1
(10)
lq[w.-lqK
where b b coefficients are given by the formula
When expansion 10 is truncated after the f i s t term, the 80called condensationapproximation(CA)”is obtained;retainment of two terms in this expansion gives the RJ approximation. The advantages of local solutions were recognized for a long time%due to their applicabilityto isothermswhich are measured over a limited experimental range of concentrations. Approximations that belong to the same category and contain the f i b and third-order derivatives of experimentalisotherms were also developed by Nederlof et al.” on the basis of the local isotherm approximation. From the detailed discussions7of solution 10, it follows that with increasing number of terms retained in the series, the approximationis improved. However,the improvementobtained between two subsequentordersof approximationhas a decreasing trend and, therefore,the greatestimprovement is obtained when the CA and RJ approximationsare compared. Depending upon the broadness of the original distribution, the R J approximation may give very accurate results; this fact was acknowledged independently by Nederlof et a1.S The application of higher order approximations may improve the results provided the quality of experimental data allows one to calculate correctly higher order derivatives of an adsorption isotherm. As an example, we present in Figure 3 the results obtained, usingour method, for the Langmuir isotherm and for the isotherm generated for the Gaussian distribution of log K values with standard deviation UG = 1. It is seen that in the latter case the result of the RJ approximation is in excellent agreement with the assumed distribution. On the other hand, for the Langmuir isotherm, which corresponds to a delta Dirac distribution of log K values, the result is only a crudeapproximation. It was showng8 that the exact result is obtained in this case when the summation in eq 10is performedto infiity. Nevertheless, our approximation correctly locates the position of thispeak and shows a significant difference in width between the two distributions considered. (33) Jagiek, J.; Ligner, G.; Papirer, E. J. Coloid Interface Sci. 1990,
-1.77. - ., 12X.
(34)Harris,L. B. Surf. Sci. 1968,10, 128. (36)Cerofohi, G. F. Chem. Phye. 1978,33,423. (36) Nederlof, M. M.; Van Riemedijk, W. H.; Koopal, L. K. J. Colloid
Interface Sci. 1990, 135,410. (37) Jagiello, J.; Schwarz,J. A. J. Colloid Interface Sci. 1991,146,416.
where N is the number of experimentalpoints, xj and yj are their coordinates, and X is a Lagrangian parameter. The f i t term in this equation represents the average square difference between data points and the corresponding function values, while the integral is a measure of the “roughness*of the functiong(r). For the purpose of simple notation we denote these quantities by s2 and G,respectively, and rewrite eq 12a with the aid of the new quantities 8’ + XG = minimum (1%) The smoothing parameter X controls the balance between the “roughness”of the function and the goodness of fit. Putting X = 0 leads to the so-called interpolation spline function which passee exactly through the points (s = 0). By increasing the value of A, a smoother function is obtained, but at the price of a decrease in the goodness of fit. In the case of the data points containing certain experimental error, the optimal value of the X parameter would be that which smoothes out fluctuations due to the experimentalerror but retains maximum informationabout the true shape of the original function. The problem is to choose this optimal X value for a given set of data. R e i h s Bsuggeststhat if the standard deviation u of y values was known, then X should be chosen so that the goodness of fit Corresponds to the u value, s r u. For the case where the experimentalerror is not known, the generalized cross validation criterion was pr0posed.a It was found,l8 however, that this criterion does not always give the correct results, and therefore it was necewary to consider in addition to thiscriterion physical constraints to find the optimal smoothing parameter. Ourapproachisbaeedonthefactthattheadeorptionisotherme considered are smooth functions since they are weighted s u m of smooth Langmuir isotherms. In order to f i d the smoothing parameter, we analyze the dependence between the roughness of the spline function measured by log(@ and the goodness of fit, s. To compute the cubic spline function for a given s value, We s t a r t with the we w e the algorithm described by model calculations. In Figure 4 we show the results obtained for Langmuir isothermswhich were generated at N = 100equallyspaced points
(38) Reinsch, C. H.Numer. Math. 1967,10, 177. (39) Brauer, P.; Fader, M.; Jaroniec, M. Thin Solid Elma 1966,123, 246. (40) Craven, P.; Wahba, G. Numer. Math. 1979,31,377.
Langmuir, Vol. 9, No. 7, 1993 1769
Heterogeneity of Proton Binding Sites -3
0.4
$s,
0.3
P 8
=
0.2
0.1
0
-1
I
4 0
0 005
0 01 Goodnoas of flt, s
0 015
0 02
3
2
J 1
f
c
m o
P -1
-2
0
0 005
0 01 Goodness of fit, s
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3
log K
Figure 6. Effect of the number of points, N, on the roughness curves for the Langmuir isotherms generated with the same random error, s* = 0.01.
-a
1
0 015
0 02
Figure 6. Comparisonof the roughness curves obtained for the Laugmuir and bi-Langmuirianisotherma. Differentrealizations of the same isotherm are obtained by independent error generations. in the ra& of 2 to 12 on the pH scale with the imposed,normally distributed, random errors of different aeaumed standard deviations,~*.Allcurveslog(G)vasehowthesametypicalcharacter. They start with a convex shape for lows values; they pass through an inflection point for a certain s below s* value; for s > s* they decline slowly and almost linearly. This shape reveals the mechanism of smoothing, namely the f i t part of the curves corresponds to the process of smoothing out the fluctuations cnused by the imposed error while the second part, where the G values do not change much with ihcreasing s, corresponds to oversmoothing. In Figure 5 we show the effect of the number of experimental points. The Langmuir isotherms were generated at different numbers of equally spaced points in the same range as before with assumed error of the same value of s* = 0.01. We observe essentially similar curves to those in Figure 4. We see, however, that as the number of points, N, grow so does the steepness of the initialconvex part of these curveswhich makes the transition between the two regions more pronounced. On the other hand if the number of points is too small, no transition is observed; thishappens here for N = 20. It follow that certainoversampling in terms of the number of points is necessary in order to detect the actual order of smoothing. In this figure, as well as in the previousone, all curvesfall onto one for higher values of s because they are related to the same isotherm. In Figure 6 we compare the results obtained for a Langmuir isotherm and for a bi-Langmuirian isotherm which is a simple nosum of two Langmuir isotherms. For each case we present log(G) va s curvesobtained for three different realizations of these isotherms. Different realizations of the same isotherm are obtained by independent error generations. This simulates different sets of experimentaldata measured for the same system with the same equipment. We see that different realizations of the isotherm may affect the firstpart of the curve since the error may be differentlydistributed alongthe isotherm. In the second part of the m e , corresponding to the functions for which the error is smoothed out, all realizations of a given isotherm fall
Figure 7. Effect of smoothing the isotherm on the calculated CA distribution. The results are obtained from the bi-Lagmuirian isotherm generated with s* = 0.005 whose roughness curve is presented in Figure 6. onto the same line. On the other hand, we see that the roughness of the bi-Langmuirian isotherm is substantially lower than that of a simple Langmuir isotherm. To illustrate the effect of smoothing the isotherm on the distribution calculated, in Figure 7 we present condensation approximationdistributions,which are simplythe f i t derivativm of the isotherm, calculated from the bi-Langmuirian botherm using different smoothing parameters. The isotherm was generated with the imposed random error of s* = 0.005. From Figure 7 we see that smoothing, which corresponds to the goodness of fit better than s = 0.006, leaves sharp fluctuations in the distribution. On the other hand, for s > 0.01 oversmoothingis observed which leads to neglecting the actual details of the distribution. &om the above discussion, supported by illustrations,we draw the following practical conclusion. It is possible to detect the order of smoothing for the data with unknown random error by plotting log(G) va s. The distinct bend of thie plot indica- the transition between under- and oversmoothing. We found that it is d e to take the-value of s a little larger than the position of the bend. Obviously, the degree to which the details of the original function can be recovered depends on the quality of the experimental data in the sense of number and accuracy of experimental points.
Rssults Homogeneous (OnePhase)Systems. We have used the titration of weak organic acids or bases to check the reliabilityof the experimentaland computational methods. Figure 2 shows the normalized proton consumption function derived from titration of sodiumacetate, pyridine, glycine, and citric acid. The ionic strength of the electrolyte was 0.001 N. The AQ(pH) function is a measure of the net exceas of bound protons with respect to a reference level. For homogeneous titrations this reference corresponds to the species formed in solution by dissolving the titratable compound at the pH of the inert electrolyte before any addition of strong acid or strong base. For example,sodium acetate exists in the reference state (pH 7.4) as the CH3COO- ion, pyridine exieta as neutral molecules (pH 7.6), glycine forms the zwitterion +HsN-CHdOO- (pH 6.41, and citric acid is mainly in the triply protonated form (pH 2.9). Positive variations of AQ(pH) in Figure 2 indicate binding of protons in excess of thisreference level, while negative variations show dissociation of protons. It can be seen from Figure 2 that the first acidic group in citric acid is partially ionized in the reference etate. The calculated proton affinity spectra are shown in Figures 8 and 9. Table I collects the values of apparent acidity constants (pKL), obtained from the positions of maxima in calculated distributions, and of corrected
Contescu et ul.
1760 Langmuir, Vol. 9, No. 7, 1993 45 4
35 3 sz. 8 2.5
P
8
i! '
0.8
Alumina
-0-
1:
0.8 0.4
E 05
0.2
0
0
-0 5
2
3
4
5
7
6
8
9
10
0
11
0.002
0.006
0.004
0.008
0.01
Goodnesa of 111, s
log K
Figure 8. Proton affiity distributions calculated for standard compounds using the RJ approximation.
Figure 10. Roughneescurvea obtained for selected experimental systems. 0.9
1 A
0.8
P 8 0.7 E. 0.6
0.5
2
3
4
5
6
7
8
9
2
3
5
4
6
7
log K
log K
Figure9. Proton affiiity distributiona calculatedfor citric acid using different numbers of terms in eq 10.
Figure 11. Effect of smoothing the experimental isotherm for citric acid on the calculated CA distribution.
Table I. Acidity Constants Determined from Proton Adsorption Isotherms and Selected Values from Ref 41 acidity constante PK.' PK. eelectad (measured) (corrected) ref valueaa 4.61 4.64 4.48-4.76 Na acetate 5.35 5.22-5.54 pyridine 5.32 2.44 2.23-2.78 glycine PKl 2.42 PK2 9.68 9.70 9.30-10.10 citric acid pK1 3.12 3.14 2.63-3.44 PK2 4.54 4.66 4.11-5.02 PKS 5.96 6.27 5.34-6.55 0 The valuea selected have a reported uncertainty of ApK between kO.04 and +0.005.
glycine concentration of 5.3 X 10-9 mol/L) because the H+ concentration at the inflection point of the isotherm exceeded that of the titrated sample. The distributions presented in Figure 8 were calculated using the RJ approximation, while Figure 9 show three orders of approximations calculated for citric acid, using eq 10. Citric acid is a good example of the resolution of our method. As seen from Figure 2, the isotherm in this case does not show pronounced features corresponding to the three dissociation steps. On the other hand, the deconvolutionof this isothermreveals distinct peaks which become sharper for higher orders of approximation. For the above calculations the correct smoothing of isothermswas chosen accordingto the proceduredescribed in the previous section. We show in Figure 10 selected examples of curves based on experimental data, they behave in an analogous way to those obtained for model calculations. The well-pronounced bend enables us to select the proper smoothing parameter in each case. The effect of smoothing on the CA distribution is shown for citric acid in Figure 11. Strong "noise" on the distribution is obtained when the smoothingcorresponds to an s value lower than the bend point in Figure 10. On the other hand, the details of the curve are lost when the smoothing is too large. Taking an s value slightly larger than that at the bend point resulta in a smooth distribution which still contains its significant features. Heterogeneous (Two Phases) Systems. Our analysis of surface heterogeneity from titration data makes no reference to any electrostatic model and thus the knowledge of an absolute value of surface charge is not needed. Consequently, a reference charge condition could be arbitr@y chosen at any point of the titration curve. We used as an operational reference that charge condition
thermodynamic constants (pKd. To obtain the latter values the following relationship28was used 0.5 ("HA2 - Z A2)I112 pK, = pK,' (13) 1 Ill2 where ZHA and Z A are the charge of the protonated and deprotonated species in equilibrium. The salt effect correction is especiallyimportant in the case of citric acid, where multiply charged species are formed in solution during titration. In the same table we show the range of reference values from l i t e r a t ~ r e . ~ ~ The proton adsorption isothermswere perturbed at both ends of the pH scale by the increased concentration of H+ and OH- ions. An example is the glycine curve shown in Figures 2 and 8, where the f i i t dissociation equilibrium (pK,1 = 2.35) could not be accurately measured (for a
+
(41) Sergeant,E. P.; Dempsey, B. Ionization Conetanta of Organic Acidu in Aqueous Solutione; LUPAC Chemical Data Series No. 23; Pergamon Prese: Oxford, 1979.
Langmuir, Vot. 9, No. 7,1993 1761
Heterogeneity of Proton Binding Sites 1.2
I
1
I
’
I
0.15,
-P E
0.8
-
0.4
-
0
-
-0.4
-
1
I
n
0
Y
0
a
I
2
pH
4
*
.
I
.
10
12
Figure 12. Proton adsorption isotherms for alumina, measured at different ionic strengths. 1
I
m -. n
0.6
E
v
0
-0.6 I
2
. 4
I
I
1
6
8
10
.
J
12
PH
Figure 13. Example of proton adsorption isotherms measured at differentionic strengthsfor a fluoride-doped alumina sample (0.20% F).
acquired when the oxide had been equilibrated with the neutral electrolyte. If AQ(pH) data were converted to “absolute”a(pH) curves, by shiftingthe u(pH) curves along the surface charge axis and setting the u = 0 level at the pH corresponding to a common intersection point (CIP), if any, similar results would have been obtained for the distribution functions. This is a consequence of the fact that, as derivatives,the distribution functions reflect the shape of the relative proton consumption curves and are insensitive to the absolute values of the surface charge. Figures 12and 13show the proton consumption function per unit mass, AQ(pH), for alumina and a fluoride-doped alumina sample at different ionic strengths. As shown in the insert of Figure 12,a CIP was obtainedfor pure alumina at pH 7.05. For the fluoride-modified sample, two common points might be suspected, but a closer examination of data around pH 7 and pH 4 showed that no definite CIP exists. This possibly indicatesan increasedheterogeneity of the modified alumina sample. Discussion Interpretation of the data presented in this paper is based on the following model of the oxide surface. The oxide/solution interface exhibits variable proportions of low index crystal planes of oxide. Each crystalface consists of a close packed regular array of -0 or -OH groups that differ from each other by (i) the number of surrounding metal cations and (ii) the coordination number of these cations. We assume that the configurationof oxygen and hydroxyl groups on the surface does not differ from that predicted by simple crystallographic models of ideally
perfect surfaces. Hence the same configurations of -0 and -OH groups will be considered at the oxide/solution interface as those used in assignment of IR bands for partially dehydroxylated surfaces.12 The number of possible configurationsfor oxygen or hydroxyl groups on a low index plane oxide surface is limited mainly to terminal and bridging positionswith respectto neighboring metal ions. The diversity of possible configurations for a polycrystallinealumina sample is, however, increased by the presence of metal cations with differentcoordination numbers (4 and 6) and also by the possibility of exposure of more than one low index plane. We adopt the picture of a vicinal monolayer of strongly bound water that completescoordinationof surface metal atoms in places were anion vacancies existed on the dry surface. The region of the strongly adsorbed layer of hydrogen-bonded water molecules is probably confined to a monolayer42 and adjacent water is more or less b~lklike.4~ The layer of strongly bound water precedes any counterions in the electrical double layer.42 The protons of the bound water become distributed over oxygen atoms on the surface. If all surface groups were actually equivalent, the surface would be homogeneous and the proton population would depend only on solution pH and the electrostatics of the double layer. The pH of solution where the activity of surfacesites with associated protons equals the activity of the remaining empty sites is often called the pristine point of zero charge (PPZC) of the homogeneous surface. Its value reflects the relative preference for adsorption as compared to desorption of protons. Quantitatively, [H+Ippu:= K-l where [H+],, represents the concentration of protons at PPZC and K is the equilibrium constant for proton adsorption. Thus the PPZC point of a homogeneous surface is similar to the pK, point of an acid, i.e. the pH point where the acid and its conjugate base have equal activities in a dilute solution. The surface at the PPZC can be viewed as a dilute ideal solution of acidic (protonated) and basic (dissociated)sites of equal concentration and existing without lateral interactions. This idealized description is not valid for real surfaces. On various crystal planesthere exist structural differences between oxygen atoms which determine differences in the magnitude and sign of the uncompensated electriccharge. The later may be estimated using Pauling’s concept of effective strength of electrostatic bonds. According to Pauling’s electrostatic valence rule, the net electric charge in a stable ionic structure should be zero or nearly equal to zero. To satisfy this requirement water molecules distribute their protons differently between various (nonequivalent)categories of surface sites. For a particular category of sites (a particular surface configuration)a specific pH exists (which has the significance of the PPZC) where half of the available positions are filled with adsorbed protons and half are empty, correspondingto the log Ki = pHipp condition. At the same pH other categories of sites (with different configurations) may be empty (log Kj > pH) or filled (log Kj < pH) with adsorbed protons. We still may assume the lack of lateral interactions between surface sites of the same sign because the charge densities attainable in potentiometric titrations usually never exceed 10% of a monolayer. A t this density, the sites will statistically be distributed within a medium of high dielectric constant (close to that of bulk water).42 This is a t least in (42) Griffithe, D. A.; Fuerstenau, D. W. J. Colloid Interface Sci. 1981, 80,271. (43) Lyklema, J. Chem. I d . 1987,741.
1762 Langmuir, Vol. 9,No. 7,1993 0.8
,
,
. . , , . . , . , , , . . ,. ,
,
.
Contescu et al.
,
OH
OH
OH
OH
0.8 Y
8
"m
O.'
IR bards
37303735
3740-3745
10 5
10 25
0
charge
op~""'"'"''""'~ 2 4 e 8 -2
3700.3730
Net
0
22
io 98
12
log K
125
Figure 14. Apparent proton affinity distributions (I= 0.01 N) for alumina (upperpart) and the propoared speciation of oxo and hydroxogroup with varioussurfaceconfigurations (bottom part) which correspond with the peake observed in the distribution.
semiquantitative agreement with the high capacitance for proton adsorption on oxide surfaces (correspondingto the adsorbed water monolayer). This description of the surface of oxide particles in aqueous solution contains three essential features. First, there is no reconstruction of the surface;second,the surface metal atoms are only exposed to the solution through a surface oxygen and all proton transfer reactions involve the first layer of strongly adsorbed water; third, local differences in proton affinity constants between various surface groups dominate the charging behavior, so that the degreeof protonation of variousgroups dependsmainly on the solution pH. The first feature simplifies the problem by allowing information about the bulk structure of oxides to be extrapolated to the surface. The second implies that the reaction sites for adsorption or desorption of protons are the surface oxygen groups. They differ with respect to their uncompensated charge, which is a structural-dependent parameter, and thus exhibit different acidiclbasic properties. All proton transfer reactions involve the first layer of strongly adsorbed water. The third feature is perhaps the most critical. We assume that surface groups in different configurations behave as isolated from each other and the electrostatics of the double layer has only a small influence on their proton population. A simple calculation shows2*that the electrochemical work for adsorption of protons against a potential difference of, say, 120 mV, is 12 kJ/mol, which can be easily smaller than the variations in proton affinity between sibs of different configurations. We deliberately avoid any reference to double layer models, which would require the use of one or more adjustable parameters. Indeed, we report "apparent" or "microscopic"affinity constants derived directly from the experimental data. We show below how these constants characterize the identity of surface groups that may be expected on actual oxide surfaces. Alumina. The proton affinity distributions for our alumina sample were calculated from the normalized AQ(pH) curves. The correct smoothing of the isotherm was chosen as explained previously. An example of analysis of experimental data for I = 0.01 N is shown in Figure 10. The proton affiiity distribution determined at the same ionic strength is shown in the upper part of Figure 14. It is seen that four categories of surface sites participate in proton reactions between pH 3 and 11. Knozinger and F&tnasamy12showed that -OH groups maybe present in five specificconfigurationson the surface
I1b
IIP
IH
0.2
la
3760.3700
Ib
3705-3800
-0 25
cm"
45
Figure 15. Five types of -OHgroups on a partidly dehydroxylated alumina surface, their nomenclature and the correepondence with IR bands, according to ref 12. The increasing order of positive charge correspondswith the increasing order of acidity and the decreasing order of wavenumbere in IR spectra. of spinel-type alumina. Their occurrence depends on the relative contributions of various crystal faces. Based on crystallographicmodels and using Pauling's bond valence rule, these authors proposed a reasonable assignment of the five stretching modes of surface hydroxyls detected in IR spectra of partially dehydroxylated alumina, in the range of 3700-3800 cm-'. The decreasing order of the wavenumbers corresponds to the increasing order of positive charges and thus of the related acidities (seeFigure 15). An -OH group of type 111, coordinated to three octahedralaluminumcationsis the most acidic (net charge +0.5), while a terminal -OH coordinated to Al in an octahedral position (type Ib) is the most basic (net uncompensated charge -0.5). An -OH group of type IIb which links two Al cations both in octahedral sites hae a zero net charge and should therefore exhibit equal tendencies for protonation or deprotonation. The other two configurations,namely those of types IIa and Ia, are slightly acidic and slightly basic, respectively. Since the configuration of surface groups determines their net charge and the affinity for adsorption or desorption of protons, we come to the conclusionthat the maxima observed in the proton affinity distribution for alumina (Figure 14)have correspondence to the one-step protonation of different types of surface groups. We assign the sites which react around pH 9.5-9.8 to the most basic singly coordinated type Ib groups [(&h)*Hl4'.'
+:H * [(&h)-OH2IH" log Kb
9.5-9.8
The peak which is partially seen below pH 3 is assigned to the triply coordinated groups of type I11which are the most acidic: [(&h)3-Ol-''5
H:
* [(&h)3-OHlM*6 log K m
