Langmuir 1991, 7, 354-362
354
Calorimetric Studies of Ion Adsorption at a Water/Oxide Interface. Effects of Energetic Heterogeneity of Real Oxide Surfaces W. RudziAski,*l+R. Charmas,+and S. Partykaz Department of Theoretical Chemistry, Faculty of Chemistry, Maria Curie-Sklodowska University, ul. Nowotki 12, Lublin 20-031, Poland, and Laboratoire de Physico-Chimie des Systemes Polyphases, L A 330, U.S.T.L., Place Eugene Bataillon, 34060 Montpellier Cedex, France Received February 27, 1990. I n Final Form: July 20, 1990 The dependence of the heat of immersion of an outgassed oxide sample on the pH of the immersional solution can be used as a source of information about the enthalpies of formation of various surface complexes. Developingequations for the quantitative analysis of these heats of immersion was the purpose of the present publication. Two sets of such equations are developed here: One is based on the popular theories of the electric double layer, assuming that the oxide surface is homogeneous. The other set is based on these theories modified here by assuming that the oxide surface is energetically heterogeneous. The model of a homogeneous oxide surface reproduced only qualitatively the behavior of the experimental heats of immersion, whereas the model of a heterogeneous oxide surface leads to a quantitative agreement between theory and experiment. On the contrary, the experimental titration curves can be fitted well by the model of both a homogeneous and heterogeneous solid surface. That means, the enthalpies of adsorption of ions are much more sensitive to the oxide surface heterogeneity than the adsorption isotherms of ions.
Introduction The mechanism of the formation of the electric double layer at waterloxide interfaces has been studied thoroughly in hundreds of papers and it would take far too long to review even the most fundamental of these studies. Its importance for various areas of surface chemistry can hardly be overestimated. So, it is no surprise that various techniques have been used to measure proton and accompanying ion adsorption on the outermost surface oxygens of oxides. The most popular of these techniques are potentiometric titration and [-potential measurements. In addition radiometric methods allow the adsorption of individual ions to be monitored. The interpretation of electrokinetic data is accompanied by some assumptions, which introduce a certain degree of uncertainty. On the contrary, the potentiometric titration and the radiometric methods yield directly the adsorption isotherms of ions. Thus, the most fundamental conclusions have been drawn from the suitable theoretical analysis of these adsorption isotherms. As the accuracy of the adsorption isotherm measurements increased, the necessity to fit the experimental data quantitatively led to more and more refined theories of the electric double layer. However, these more refined and complicated theories failed to correlate with the adsorption isotherms in some systems.’ Among various physical factors that were taken into account in these still being refined theories, one important factor has not been considered until very recently. This is the possible dispersion in the adsorption energy of ions adsorbed on various surface oxygens. This may only surprise some, because effects of surface energetic heterogeneity have long been known for their importance in
* Author to whom correspondence should be addressed. + Maria Curie-Sklodowska University. f
U.S.T.L.
(1) Blesa, M. A.;Figliolia, N. M.; Maroto, A. J. G.; Regazzoni, A. E. J. Colloid Interface Sci. 1984, 101,410.
0743-7463/91/2407-0354$02.50/0
adsorption of gases, gas mixtures, and solutions of nonelectrolytes on the surfaces of solids. To explain this situation, it is necessary to trace the historical development of electric double layer theories. At the early stage of the theories of adsorption onto oxide surfaces, the emphasis was given to electrostatic interactions. The fact that adsorption frequently involves chemical bonding as well was not so commonly recognized until more recently. This implies a dispersion of chemical bonding energies is possible similar to nonelectrolyte adsorption on solid surfaces. During the last decade, experimental evidence for the important role of surface energetic heterogeneity of oxides for ion adsorption has been obtained. In the late 1970s, Davis and Leckie2 and Benjamin and Leckie3 reported that the adsorption of Me2+ ions onto ferrihydride cannot be described in terms of the theories of ion adsorption onto homogeneous solid surfaces. An agreement between theory and experiment could be obtained only by assuming a large dispersion of site affinities. Two years later Kinniburgh et a1.4 demonstrated that Toth’s5p6 isotherm equations for adsorption on heterogeneous surfaces applies best to these adsorption systems. This problem was considered next by van Riemsdijk et al.738 They criticized Kinniburgh et al.4 for ignoring the electrical properties of the interface. They demonstrated that by taking the coulombic interactions between adsorbed ions into account, the agreement between theory (2) Davis, J. A.; Leckie, J. 0. J. Colloid Interface Sci. 1978, 67, 90. (3) Benjamin, M. M.; Leckie, J. 0. J. Colloid Interface Sci. 1981, 79, 209. (4) Kinniburgh,D. G.;Barkes, J. A.; Whitfield, M. J.Colloidlnterface Sri 95 Sci. 1983, 370. - .. -19RI - --, -95, - , 2717 - . -. (5) Toth, J. Acta Chim.Hung. 1974,82, 11. (5) ( 6 ) Toth, J.; Rudzidski, W.; Waksmundzki, A.; Jaroniec, M.; Sokolowski, S. Acta Chim.Hung. 1974, 82, 11. (7) Van Riemsdijk, W. H.; Bolt, G. H.; Koopal, L. K.: K.; Blaakmeer, J. JJ.. Colloid Interface Sci. 1986, 109,219. (8) Van Riemsdijk, W. H.; DeWitt, J. C. M.; Koopal, L. K.; Bolt, G. H. J. H. J. Colloid Interface Sci. 1987, 116, 511.
0 1991 American Chemical Society
Langmuir, Vol. 7, No. 2, 1991 355
Ion Adsorption at WaterlOxide Interface
and experiment was greatly improved. Finally, they concluded that their ion adsorption isotherms were rather insensitive to surface heterogeneity. No essential improvement was obtained in their numerical fit oftheoretical adsorption isotherms to experimental values when they assumed that the surface was heterogeneous. The apparent contradiction between the conclusions drawn by Davis and Leckie and Kinniburgh and those by Koopal et al. may be due to two reasons: First, the surfaces of ferrihydrides are probably more heterogeneous than those of titanium dioxides. Second, it has long been known that adsorption isotherms are not very sensitive to possible effects of surface energetic heterogeneity. Much more sensitive are enthalpies of adsorption, and this fact has been demonstrated in a series of papers by RudziAski et al.9-'6 concerning nonelectrolyte solution adsorption on oxides and other surfaces. However, not one paper to our knowledge has been published to date in which enthalpies of ion adsorption have been analyzed theoretically to elucidate information about the effects of surface energetic heterogeneity of oxides in the formation of the electric double layer. Meanwhile, some experimental works have recently been published that seem to make such an analysis possible. Among them were the results obtained in our laboratory in Montpellier, which will be subjected here to such an analysis. These are the heats of immersion of outgassed Ti02 (anatase) samples into solutions of varying pH, measured by Foissy and Partyka.17J8 Similar experiments were carried out earlier by Fuerstenau and c o - w o r k e r ~using ~~,~~ a-A1203samples. Very recently studies of that kind were published by Wierer et a1.,21Machesky and Anderson,22 and Fokkink et al.23 using somewhat different (displacement calorimeter) techniques. All these experimental results have been collected by us in Figures 1 and 2 for the readers' convenience. The general features of these experimental curves will be discussed in terms of our theory presented in the next section.
protons, aggregates composed of two protons, cations, and anions on the empty surface sites SO-. Let M denote the number of these sites a t a solid/solution interface. Now let N1, N2, Nc, and NAdenote the number of the adsorption sites occupied by one and two protons and cations and anions, respectively. While neglecting the repulsive interaction between the adsorbed ions and assuming that all the surface sites are the same, one can write the system canonical partition function Q N , N ~ =~ Q in the following form
M! X = N,! Nz! Nc! NA! ( M - N1- N2 - Nc - NA)! q>qzN2qCNcqANA
(l)
where q1, q2, qc, and q A are the surface molecular partition functions of one and two adsorbed protons, a cation, and an anion, respectively. We will write them in the following form
where €1 is the "intrinsic" part of the ion-site interactions, whereas q 1 O includes the coulombic ion-oxide attractive interactions. Using the Stirling formula and the method of maximum term, from eq 1, we obtain
i = 1,2, C, A where
N~ = M -
Theory 1. General Expressions. We will consider formally the formation of the surface complexes SOHo, SOH2+, SOT+, and SOHZ+A-as a competitive adsorption of single (9)Rudzidski, W.; Lajtar, L.; Zajgc, J.; Wolfram, E.; Paszli, J. J. Colloid Interface Sci. 1983,96, 339. (10)Rudzihski, W.; Zajec, J.; Hsu, C. C. J. Colloid Interface Sci. 1985, 103,528. (11)Rudzihski, W.;Zajgc, J.;Dekany, J.;Szanto,F.J. Colloidlnterface Sci. 1986,112, 473. (12)Rudzidski, W.; Zajgc, J.; Wolfram, E.; Paszli, J. Colloids Surf. 1987,22,317. (13)RudziAski, W.; Nerkiewicz, J.; Partyka, S. J. Chem. SOC.,Faraday Trans. 1 1981,77, 2577. (14)Rudzidski, W.; Narkiewicz, J.; Partyka, S. J. Chem. SOC.,Faraday Trans. 2 1982,78, 2361. (15)Rudzidski, W.; Zajqc,J. Acta Geodaet. Geophys. Montanist.Hung. 1985,20, 337. (16)Rudzihski, W.; Zajgc, J.; Michalek, J.; Partyka, S.In Proceedings of the International Conference on Fundamentals of Adsorption; Mayers, A. L., Belford, G., Eds.; Engineering Foundation: New York, 1983; p 513. (17)Foissy, A.; Lamarche, J. M.; Partyka, S. Cal. Anal. Therm. 1985, 16, 20. (18)Foissy, A. Ph.D. Thesis, Universith de Franche-Comth Bescan$on, 1985. (19)Roy, P.; Fuersteneu, D. W. J. Colloid Interface Sci. 1960,26,102. (20)Griffiths, D. A.; Fuerstenau, D. W. J. Colloid Interface Sci. 1981, 80,271. (21)Wierer, K.Ph.D. Thesis, Fakultat fur Chemie und Pharmazie der Universitiit Regensburg, 1987. (22)Machesky, M. L.;Anderson, M. A. Langmuir 1986,2,582. (23)Koopal, L.Private information.
,
EN,
(4)
i
i = 1,2, C, A The partial surface potentials p1, pz, pc, and p~ are given by
i, 1 = 1,2, C, A At equilibrium f
kT In aH
+ pc = pco + k T l n ac pA = 2kHo + pAo + 2kT In a H + kT In a A ~2
= 2 p ~ O 2kT In UH
(6) (7)
(8) (9)
where pH0, pc0, and pA0 and aH, ac, and aA are the standard chemical potentials and activities of protons, cations, and anions, respectively, in the equilibrium bulk solution. In the case of a homogeneous solid surface, eqs 5-9 yield
Rudziizski et al.
356 Langmuir, Vol. 7, No. 2, 1991 PHO
Fo = k~ + In q1 - 2.3pH - In 1- 0,
00
- 0,
- 0, - oc
=O
(loa)
where
+ [SOH;A-]
6, = M([SOH;]
- [SO-] - [SO-C+I)
(17)
is the first integral capacitance, and M denotes the number of adsorption sites in appropriate units of charge. For our further purposes, it is more convenient to consider the following equivalent reactions: c1
FA = 2pHo + kT kT
+ 2H+ + Aso-c++ co- + c+
+ In qA- 4.6pH + 2.3 log aA In
8,
1-
- e+ - oA - oC
SOH2+A-* SO= 0 (10d)
where 80 = N1/M, O+ = N2/M, Oc = Nc/M, and OA = NA/ M. 2. Homogeneous Surfaces. 2.1. Relation to Previous Works. The notation used in this paper follows the notation commonly used in physisorption and chemisorption processes, where coulombic interactions are not involved. However, it should be helpful for the readers to see the links between our present notation and the traditional notation used to describe ion adsorption a t the waterloxide interface. For that reason, we write the surface proton dissociation reactions SOH2++ SOH'
(18) (19)
We will denote further [SOH;]
= O+, [SOH'] = Bo, [SO-C+] = Oc
[SOH,+A-] = OA, [SO-] = 1- 80 - 8, - Oc - OA = 0-
(20)
We may also write eqs 1 2 and 13 and 18 and 19 in the following way:
+ H+ In 0, -= 0 (21~) 0-
where UH is the proton activity in the equilibrium bulk phase. Instead of eqs 11 and 12, we may consider also the equivalent set of reactions 12 and 13
SOH2++ SO- + 2H+
4.6pH - In 0, = 0 (21d) 0From the comparison of eqs 10 and 21, we have
(134
The coadsorption of anions A- and cations C+ causes the formation of the other surface complexes, SOHz+Aand SO%+
+ H+ + ASOHo + C+ s SO-C+ + H+
SOH2+A-F= SOH'
(144
(14b) and the related equations for thermodynamic equilibria read
And yet we define
i = 1,2
(234
p*K? = -log *Kiint, i = C, A
(2%)
pK,?
= -log K,?,
To express $'-pH dependence, we will accept here the relation used by Bousse et al.24and van der Vlekkert et al.25 where aA, and ac are the bulk activities of anion and cation, is the surface potential, and $B is the mean potential at the plane of specifically adsorbed counterions, which is given by
~
~~
(24) Bousse, L.; de Rooij, N. F.; Bergveld, P. IEEE Trans. Electron Deuices 1983, 30, 8. (25)vanderVlekkert,H.;Bousse,L.;deRooij,N. F.J. ColloidInterface Sci. 1988, 122, 336.
Langmuir, Vol. 7,No.2, 1991 357
Ion Adsorption at WaterlOxide Interface
where
and /3 is given by
In eq 26, N , is the surface density and CDL is the linearized double-layer capacitance. Around the pzc, eq 24 can be linearlized to yield $o=--
p+l
where the nonconfigurational values are
2*3kT(PZC- pH) e
This approximation is valid in the region around the pzc, where (e+o/kT) < /3. Therefore, eq 27 is expected to be valid roughly from pH = 6 to 10, with some deviation outside that range. The parameter ,f3 for Ti02 (anatase) was estimated to 6.9 using Bousse'sZ4method and the data presented in Sprycha's work.26 While calculating $0 (pH), we will use the more accurate eq 24 instead of eq 27 to get more precise results. 2.2. Heat of Adsorption of Proton on Homogeneous Oxide Surfaces. Now, let us consider the calorimetric effects accompanying the formation of the surface complexes SOHo, SOHz+, SOHz+A-, and SO-C+. In the immersion experiments reported by Griffiths and Fuerstenau20 and by Foissy and Partykal7J8 the (cumulative) heat of immersion into the solution of a certain pH, Qim (pH), is given by the obvious expression
where M is the number of adsorption sites in equivalent units and 80, Q+, Qc, and QA are the molar differential heats accompanying the formation of the SOHo, SOHz+, SOHz+A-, and SO%+ complexes
The differentiation with respect to 1/T runs a t constant where (ei) is used to the note the set of values Bi. Neglecting (dc1/d(l/T)), as Bousse et al.24did, we will operate with the molar differential heats of adsorption, Qc, QA,given by eqs 30c,d, in which the term (e/?")[% (So/cl)/d(l/T)] is put equal to zero. On the other hand from the work by Blesa et a1.l it follows that the temperature dependence of c1 exists and the function cl(T) is almost linear. The last term in eqs 30c,d of molar differential heats of adsorption, Qcand QA, will be of the following form: (e60T/c12)(dcl/dT). The difference QZ = Q+ - QO is the molar heat of adsorption accompanying the attachment of the second proton to the already existing complex SOHo
(Oil,
The nonconfigurational values, Qal and Qaz, are very interesting, since they describe that part of the molar heats of adsorption, which is connected with the change of the molecular status of an isolated proton, when adsorbed on SO- (i = 2) and SOHo (i = 1) sites. We will accept further that the derivative (d+o/d(l/ T))pH is of the form developed by van der Vlekkert et al.25 The only difference is in the expression 2.3-
apzc --
Q,1
+ Q,z
(33) 2kp Let us note that Qal and Qaz have the same meaning as A& and AH,z in the paper published by van der Vlekkert et al.25 Thus dT
2kT (294
where the meanings of PSO-c+,PSOH2+A-,PSOH, and P S O H ~ + are obvious. Van der Vlekkert et al.25have shown that (dpH/aT) is small for two kinds of applied buffers. Neglecting the temperature dependence of pH, QA, and ac, we arrive a t the following expression for Qo, Q+, Qc, and QA (26) Sprycha, R.J. Colloid Interface Sci. 1984, 102, 173.
- l)](pH
-
k 0 Qa1+ Qa2 (34) PZC) - e0 + l 2kT This means the experimentally measured Qim(pH) is the function of pH directly, and indirectly through (dei/dpH) (i = 0, +), and contains two new parameters, Qal and Qa2. The last two parameters can be measured also in an independent experiment as described by van der Vlekkert et aL25 (measurement of the temperature dependence of $0).
358 Langmuir, Vol. 7, No. 2, 1991
Rudzihski et al.
The derivatives (d&/apH) (i = 0, +, A, C) from eq 28 are to be evaluated from the equation system (21). For instance
from one to another site. Let tio denote the most probable value of ci. Then for a j t h kind of adsorption site, we have
K , = Kio exp tTl rl where
(35)
While considering the correlation between Aii's, we follow here the assumption made in the work by Koopal and ~ o - w o r k e r sconcerning ~~ ion adsorption on heterogeneous solid surfaces, i.e.
aFo aF, aF, aF, -
Aoj = A+j = AN = Acj = 4
aF, aF, aF, aF, -
The equation system (37) can then be written as follows:
ae, ao,
ad,
ao,
(43)
ae, ae, ae, ao,
aF, aF, aF, aF, -
ae, ae, ao, ae,
aFA aFA aFA aFA --
ae,
ad,
ao, ao,
t
While evaluating the partial derivatives (aFi/apH), one has to consider $0 as a function of pH defined in eq 27. In our present treatment we will neglect the effect of pH on ac and a,, i.e. put (yA,Cb/apH)equal to zero. Then, while evaluating the derivatives (aFi/aOj), (i, j = 0, +, A, C), one has to remember that according to eq 17
+
6, = ~ [ 2 e + , o, 20, - 11 (36) 2.3. Equations for Adsorption Isotherms of Proton in the Presence of Other Surface Complexes. The equation system (21) can be rewritten in the following form Kifi
Bi = 1
, i =O,+,A,C
+C K f i
i = 0, +, A, C For the whole heterogeneous solid surface, the experimentally determined adsorption isotherm Bit has to be related to the following average:
(37)
i = 0, +, A, C
i
where
The problem formulated above is identical with that, considered in dozens of papers on gas adsorption on heterogeneous solid surfaces. To represent the dispersion of A, we have chosen the following symmetrical function (38)
XU)
and f i (i = 0, +, A, C) are the following functions of proton and salt concentration:
kT fc=acexp
{--+2
I:;
(39)
e*, e60 4.6pH kT kTc, 3. Adsorption on Heterogeneous Surfaces. According to our statistical derivation in the first part of this publication, the "intrinsic" constant Ki can be written as follows: f,=a,exp
On a heterogeneous solid surface, the values of q will vary
where c is the heterogeneity parameter. In the case of gas adsorption, that kind of adsorption energy dispersion leads to the "generalized Freundlich" isotherm, which is probably the most popular equation to correlate experimental adsorption isotherms. The recent computer simulations by Bakaev2* of adsorption by solid surfaces provide a further support for choosing such symmetrical functions. Except for very extreme regimes, where proton or salt concentrations are either very small or very large, the finite integration limits (Amin,Am-) can be replaced by the interval (-a,+a), and the result of integration in eq 45 reads (27) Koopal,L. K.;van Riemsdijk,W.; Roffey, M. G. J.Colloidlnterface Sci. 1987, 118, 117. (28) Bakaev, V. A. Surf. Sci. 1988, 198, 571.
Ion Adsorption at WaterlOxide Interface
Langmuir, Vol. 7, No. 2, 1991 359
i = 0, +, A, C Let pxib denote the sum of all the bulk chemical potentials of the components, which, after being adsorbed on the same site, form the surface complex "i", (i = 0, +, A, C). In order to evaluate Qi, we have to first rewrite the equation system (47) in the following form:
&'- pi(iei),n= 0
(48)
Then we obtain
J
-U H
15 PH
Figure 1. (A) Experimental heats of immersion of Ti02 samples measured by Foissy and Partyka17J8 (0) and by Machesky and Anderson22( 0 ) .The lines are drawn to help the eye. The symbol AQimdenotes the value of Qimfrom which its minimum value was abstracted. (B) Experimental heats of immersion of A1203 samples measured by Griffiths and Fuerstenau20 (0) ,and by Wierer21 ( 0 ) . The lines are drawn to help the eye.
~
PH
-
PH
Figure 2. (A) Experimental heats of immersion of Fez03 samples measured by Wiererzl (0) and of a-FeOOH by Machesky and Anderson22 ( 0 ) . The lines are drawn to help the eye. (B) Experimental heats of immersion of SnOz samples (0) and Si02 ( 0 )measured by WiererS2lThe lines are drawn to help the eye.
where
The molar differential heat of adsorption Qi(c) will be given by
i = 0, +, A, C Thus, the energetic surface heterogeneity is a source of a configurational contribution to the molar heats of adsorption, expressed by the last (second) term on the righthand side of eq 52. After performing the differentiation of @i with respect to 1/T, we arrive a t the following explicit expression: (53)
i = 0, +, A, C The derivatives (d8it/8pH) are evaluated now from the equation system (50), using the way similar to eq 35. Results and Discussion Figures 1and 2 show the results of all the experiments on the heat of immersion Qim as the function of pH, reported to date in the literature. This type of experiment
is relatively recent, and consequently, reported results are not numerous. The most extensive research has been carried out in Professor Dobias' laboratory,21 but their measurements of the heats of immersion were not accompanied by the measurements of the titration curves. We face a similar situation in the interesting work published by Machesky and Anderson.22 Our strategy lies in a simultaneous fit of both experimental heats of adsorption (immersion) and of adsorption (titration) isotherms. The necessity of fitting the data obtained in two kinds of independent experiments imposes more rigour on the correctness of theory and brings one to a new level of understanding in the obtained results. Soon we will show that our titration curves can be fitted by accepting various adsorption models and sets of parameters. However, the necessity of fitting simultaneously the experimental heats of immersion decreases dramatically the field of maneuver. Among the published results there are only two that can be subjected to our analysis along the lines of our investigation strategy. These are the data for a-Al2O~ published by Griffiths and Fuerstenau20 and our results for TiO2. Of these two experiments our data are more suitable for the theoretical analysis, because PZC is close to 7, and consequently both the acidic and alkaline branches of the heat of immersion curve are represented equally well. In the case of the data by Griffiths and Fuerstenau, the alkaline branch is much shorter than the acidic one. The detailed description of our calorimetric investigation has been published elsewhere.17J8 Here we repeat only some of the most essential information. Ti02 was a commercially available compound, Degussa P.25 (anatase 95 72, rutile 5 % ), with a specific area of 52
RudziAski et al.
360 Langmuir, Vol. 7, No. 2, 1991
I
l, 10
UH
3
6
9
is
12
10
DH
3
PH
8
9
12
-
15
PH
Figure 4. (A) Comparison between experimental titration data Figure 3. (A) Comparison between experimental titration data for the Ti02sample (0) reported by Foissy18and our theoretical for the salt concenfor a Ti02 sample reported by Foissyls (0) titration curve (-) calculated by fitting simultaneouslytitratration 0.5 mol/dm3 and our theoretical fit (-) based on the tion data and the experimental heats of immersion by our model of a homogeneous oxide surface. The theoretical curve equations developed by assuming the model of homogeneous was calculated by accepting that M = 192 &/cm2 (12 sites/ nmz), and that &lint = 3.6, K,zint = 8.8, *K,cint = 6.8, and * K A ~ ~ oxide ~ surface. The parameters used in our calculationsare those collected in the first row of Table 11. (B) Agreement between the = 5.5, as estimated by Smit31analysing Foissy's titration curves experimental heats of immersion Qimof Ti02 sample measured for other salt concentrations. For the salt concentration 0.5 mol/ by Foissy and Partyka17J8(a) and our theoretical heats of dm3investigated here, our computer found c1(1)= 0.85F/m2,c1(2) immersion (-) calculated by fitting simultaneously the titra= 1.00 F/m2as acidic and alkaline capacitance values leading to tion data and the experimental heats of immersion by our the best agreement between the theory and experiment. The equations developed by assuming that the Ti02 surface is above numerical estimation was done by introducingbulk activity homogeneous. The parameters used in our calculation are those coefficients y ~ = 0.614 ~ +and~yc1-b = 0.650. (B) Comparison of collected in the first row of Table 11. the experimental heats of immersion of Ti02measured by Foissy (a) at 30 O C with the typical behavior of our and Part~ka'~J8 Table 1. List of the Surface Reactions, the Related theoretical heat of immersion calculated by assuming the oxide Equilibrium Constants, and Heats of Adsorption surface is homogeneous and neglecting the temperature depenK,zint, dence of ql)and c1(2). The adsorption parameters KOlint, equilibrium heat of *Kcint,and * K Aare ~ ~the~ same as those in Figure 3A, and the constant reaction reaction type parameter /3 was estimated as equal to 6.9. The solid line SOHz+ SOH' + H+ Keiint 8 1 1 1 corresponds to the assumption that Qal = 20 kJ/mol, Quz = 26 SOHO SO- + H+ Kazint -Be2 kJ/mol, Q,A = 40 kJ/mol, and Q,c = 12 kJ/mol, whereas the * K A ~ ~ ~ SOHz+A- SOHo+ H+ + A4Qu\ - &2) broken line is calculated by assuming that Q,c has a slightly *Kcint SOHO + C+ S 0 - U + H+ -(Qa2 - B a d smaller value of 11 kJ/mol. SOHo SO- + H+ Kezint -Qe2 mz-g-l, and the major impurity was C1 ions (1000 ppm). SOH2+ SO- +2H+ KalintKezint -(Qal + Qe2) The powder was washed in boiling water until no chloride SOH*+A- SO- +2H+ + A*KqintKa2 -Qu\ could be detected in the supernatant. so-c+ so- + c+ Kazlnt/*Kclnt -8.c The surface charge was measured potentiometrically. All aqueous solutions were carefully deoxygenated and by Smit31for analyzing the titration curves of our anatase prepared with deionized water. Such an experimental sample. He analyzed two titration curves corresponding procedure made it possible to believe that the surface to NaCl concentrations of 0.001 and 0.01 mol/dm3. The charge was solely due to the amphoteric properties of the heats of immersion curve were measured at the salt Ti02 surface hydroxyl groups. concentration 0.5 mol/dm3; thus, we had to find new values Heat of immersion was measured by using the experof a double layer capacitance c1corresponding to the much imental set described in detail in other works by Partyka higher salt concentration. Similar to Smit, we assumed et al.2g Before immersion, the investigated samples were two values of c1: one for the acidic (cl(1))and the other for outgassed at temperature of 130 "C under vacuum of about alkaline ( ~ ~ ( 2branch )) of the titration curve. We felt, that 10-4 Torr, during a time period of 8 h. All immersion at such high salt concentration it was necessary to operate experiments were carried out a t 30 "C, and pH was with true activities, i.e., to introduce the bulk activity monitored by means of a microelectrode. coefficient y ~ and~ y c +~ - ~The ~. computer choice of c1(1) Before presenting our results, let us remark that = 0.85 F/m2 and c1(2) = 1.00 F/m2 led us to an excellent neglecting the formation of certain surface complexes fit of the experimental titration curve, shown in Figure below PZC and above PZC, in our theoretical treatment 3A. in sections 1-3, yields the popular expression PQai (at),30 Then, we accepted first the simplest possible assumption used so widely in the graphical estimation of the paramthat c1(1) and c1(2) are temperature independent. With eters pK,lint, pKrrzint,p*Kcint,and p*KAint, In other words, this assumption, we could not fit the heat of immersion our expressions for adsorption isotherms corresponding curve. Figure 3B shows typical shapes of Qim obtained by to the model of a homogeneous oxide surface are those choosing various sets of the parameters Qal, Qa2, Qat, and used in the most popular model of the electric double layer. Qu*.The alkaline branch is always almost vertical, and Thus, it was natural to see as a first step the consequences the minimum of theoretical curve is much below the of accepting that popular theory to predict the behavior minimum of the experimental values, which are located of the related heats of immersion. Figures 3 and 4 show close to the PZC. the results of that exercise. For the reader's convenience Then, it was found, that assuming even a slight we have prepared Table I, listing the parameters described temperature dependence of c1(1) and c1(2) changes the in a condensed manner. situation dramatically. Our computer exercises showed First we accepted the values of the adsorption paramthat assuming dcl(l)/dT = -0.007 and dcl(z)/dT = -0.002 eters pK,,lint, pKJnt, p*Kcint, and p * K ~ i "as~ determined leads already to a reasonable fit of the experimental heat of immersion curve. This is shown in Figure 4B. Fitting (29) Partyka, S.; Lindheimer, M.; Zaini, S.; Keh, E.; Brun, B. Langsimultaneously the titration and the heat of immersion muir 1986, 2, 101.
--.-. ---
(30)Davis, J. A.; James, R. 0.;Leckie, J. 0. J. Colloid Interface Sci.
1978, 63,480.
(31) Smit, W.
J. Colloid Interface Sci.
1986, 113, 288.
Langmuir, Vol. 7, No. 2, 1991 361
Ion Adsorption a t WaterlOxide Interface
Table 11. Parameters Obtained by Fitting Best Our Theoretical Equations for the Adsorption Isotherms of Ions and the Heats of Immersion in the Case of the Ti02 Sample Investigated by Foissy and PartykalTJB. PK.1
PK.2
PKC
PKA
3.6
8.8
7.1
5.3
kT/c
ciw Qai Qaz Homogeneous Surface Model
ci(i)
0.90
1.08
Qac
Q~A
5.50
39.8
-0.007
-0.002
7.15
40.1
-0.0051
-0.001
HFoiz 20.0
26.0
dcici)/dT
dcicz)/dT
Heterogeneous Surface Model 3.6
8.8
7.1
5.3
0.9
0.90
1.08
20.0
26.0
The values of Qim (pHin= 15) for homogeneous and heterogeneous surface models are, respectively, 21.91 and 23.36 kJ/mol. Units for parameters q,) (i = 1, 2) are F/m2, and for parameters Qai (i = 1, 2, C, A), kJ/mol. -: H 10
-;I
1
o
O -
-7---
-/.-e.-
-10
-20
-w
-300
-
3
9
6
PH
Figure 5. (A) Agreement between FoissyW titration data for and our theoretical titration curve (-) a T i 0 2 sample (0) calculated by fitting simultaneously titration data and heats of immersion of Ti02, by our equations developed by assuming Koopal's model of the surface heterogeneity of oxides. The parameters used in our calculations are those collected in the second row of Table 11. (B)Agreement between the experimental heats of immersion Qimof Ti02sample measured by Foissy and Partyka17Js (a) and our theoretical heats of immersion (-) calculated by fitting simultaneously the titration data and the experimental heats of immersion by our equations developed for the model of a heterogeneous oxide surface. The parameters used in our calculation are those collected in the second row of Table 11.
3
15
6
9
1
2
1
5
__ PH
Figure 7. (A) Contributions to Q h due to the formation of the and of SOH2+,Qim(+)(- -), surface complex SOHo, Qim(0)(-), calculated for the model of the heterogeneous solid surface, using the parameters collected in the second row of Table 11. (B) Contributions to Qimdue to the formation of the surface complex and of SOH2+A-, Qim(A)(- -), calculated for SO%+, Qim(C) (-), the model of the heterogeneous solid surface, using the parameters collected in the second row of Table 11.
'E
1
curves suggested slightly different values of p*Kcintand ~ * K than A ~those ~ ~found by Smitq31 They are collected in the first row of Table 11. However, a closer inspection of Figure 4B must bring one to the conclusion that the shape of the acidic branch of the heat of immersion curve is still reproduced poorly. Further, the minimum of the theoretical heat of immersion curve is still much below the experimental minimum. So, in the next step of our investigation (Figure 5) we introduced the concept of an energetically heterogeneous surface to see whether it can improve the agreement between the theoretical and experimental heats of immersion. The results were simply striking. Our best fit MINUIT32subroutine starting from the previous parameters has changed them only slightly. However, the
2
UH
-
Figure 6. (A) Individual adsorption isotherms of surface SO-C+, (--), and SOH2+A- (- - -) complexes SOHo (-), calculated by assuming the model of a heterogeneous oxide surface and by using the parameters collected in the second row of Table 11. (B) Individual adsorption isotherm of the surface complex SOH2+(-) calculated for the model of a heterogeneous solid surface, compared with the fraction of free surface oxygens SO(- -). The parameters used in this calculation are those collected in the second row of Table 11.
1
L
-
L
-a
-
----/,e ,
-10/
/
Figure 8. Comparisons between the contributions to Qim due to cation adsorption Qim(C) ( - - - ) and those coming from the formation of the surface complexes having a proton in their structure, Qim(0)+ Qim(+)+ Qim(A)(--). The solid line is our theoretical heat of immersion, being the sum of these two contributions. The calculation is done for the model of the heterogeneous surface, using the parameters collected in the second row of Table 11. Therefore, the experimental heats of immersion of Ti02 (a) have also been marked.
assumption that there is some dispersion in the adsorption energy of ions represented by the heterogeneous parameter k T / c = 0.9 led to an impressive agreement between the experimental and theoretical heats of adsorption. In other words, introducing the concept of an energetically heterogeneous water 1oxide interface is absolutely necessary to quantitatively reproduce the behavior of the heats of adsorption of ions. We would like to emphasize that introducing the concept of a heterogeneous solid surface was not necessary to reproduce well the behavior of the adsorption isotherms of ions (titration curve). This is why Koopal et al.,27 analyzing only the experimental adsorption isotherms, could not make a decisive judgement about the role of the ~~
~
~
(32) James, F.;Roos, M. Comput. Phys. Commun. 1975,10, 343.
RudziAski et al.
362 Langmuk, Vol. 7, No. 2, 1991 surface energetic heterogeneity of the TiOz/water interface. We would like also to draw attention to the fact that the concept of surface energetic heterogeneity is necessary to reproduce the "knee" on the acidic branch of the heat of immersion curve, which is much more clearly observed in the case of the a-AlzO3sample (Figure 1B-the first "knee" on the left side of PZC). Finally, we would like to report about a large uncertainty in our numerical estimation of Qal value. That parameter relates to the heat of attachment of the second proton to the already existing complex SOHo. The next Figure 6 suggests an explanation. There the individual adsorption isotherms of all the surface complexes are displayed. The formation of the surface complex SOHz+ is almost negligible and the derivative (dO+/dpH) is always small. Because of these small derivative values, Qim is not much sensitive to a particular choice of Qa1. Figure 7 separated the contributions to Qim due to the formation of various surface complexes, Qim(i) (54)
i = 0, +, C, A It can be seen that the minima in the experimental heat of immersion curves Q i m (pH) are due to the formation of the surface complexes with protons. These minima could
be even deeper, but they are counterbalanced to some extent by a positive contribution coming from cation adsorption. This is even more clearly seen in Figure 8. This figure suggests that the popular estimation of the enthalpies of proton adsorption based on the temperature dependence of PZC should be employed with caution since that estimation is accompanied by a underlying assumption that at the PZC cation adsorption and the accompanying heat effects are negligible. While investigating the temperature dependence of PZC for their Ti02 sample, Fokkink et al.33found Qaz = 17.6 kJ/mol, whereas their direct calorimetric measurements suggested Qaz = 22 kJ/mol. That value is closer to Qa2 = 26 kJ/mol, deduced here from our calorimetric experiments. The difference between the two last values is probably due to the somewhat different nature of the Ti02 samples investigated in our and in Kokkink's experiments.
Acknowledgment. W. Rudziiiski wishes to express his thanks to Professors H. Lyklema and L. Koopal for access to the experimental data obtained in Professor Lyklema's laboratory prior to publication and also to HeinrichHertz-Stiftung for the support making his extended visit to Professor Findenegg's laboratory at Ruhr-Universitat in Bochum possible, where a part of that work was done. Partial support from RP-1-08 is also acknowledged. (33) Fokkink, L. G. J.;Keizer, A.; Lyklema, J. J.Colloid Interface Sci. 1989, 127, 116.
