Langmuir 1992,8, 1154-1164
1154
On the Nature of the Energetic Surface Heterogeneity in Ion Adsorption at a Water/Oxide Interface: The Behavior of Potentiometric, Electrokinetic, and Radiometric Data W. Rudzinski,*>tR. Charmas,t S. Partyka,J F. Thomas,§ and J. Y. Botteroll Department of Theoretical Chemistry, Faculty of Chemistry, Maria Curie-Skbdowska University, PI. M . Curie S k b d o w s k i e j 3, Lublin 20-031, Poland, Laboratoire de Physico-Chimie des Systemes Polyphases, L A 330, U S T L , Place Eugene Bataillon, 34060 Montpellier Cedex, France, Centre de Pkdologie Biologique, L P 6831 d u CNRS associk 6 I'Uniuersitk de N a n c y I , B.P. 5, 54501 Vandoeuvre Cedex, France, and Centre de Recherche sur la Valorisation des Minerals, B.P. 40, 54501 Vandoeuvre Ceder, France Received September 27, 1991. I n Final Form: January 2, 1992 The fact that the surfaces of the actual solids are energetically heterogeneous has long been realized by scientists investigating the adsorption of gases and nonelectrolyte mixtures onto oxides. However, that important factor was almost totally ignored in the studies of ion adsorption at the waterioxide interface. The authors show the reason for that. These are special features of the most frequently investigated experimental data-the titration curves. Because of some compensation effects,these compositeadsorption isotherms are almost insensitive to surface energetic heterogeneity. It is also shown that electrokinetic data and radiometrically measured individual adsorption isotherms of ions are very sensitive, and make even discrimination among various heterogeneity models possible. The most realistic model seems to be that assuming small correlations between the adsorption energies of different ions, adsorbed on various surface sites distributed on an oxide surface at random. That heterogeneity model is able to explain the differences between the experimentally estimated point of zero charge and isoelectric point values.
Introduction The adsorption of ions and the formation of the electric double layer at the waterioxide interface are physical phenomena whose importance in life and technology can hardly be underestimated. So it is no surprise that hundreds of papers have already been published concerning these phenomena, and it would be difficult to list even the most important of them. Still more and more refined theoretical treatments were proposed, but one very important physical factor was forgotten in all these works, the energetic heterogeneity of the actual (really existing) oxide surfaces. Such a situation may only surprise us since the energetic heterogeneity of oxide surfaces has long been recognized by the scientists investigating adsorption of gases onto oxides. One might quote here dozens of works, but our attention in this work is focused on ion adsorption at waterioxide interfaces. Davis and Leckiel-sand Kinniburgh4were first to report about heterogeneity effects in these adsorption systems. Some few years ago, Koopal and Van Riemsdijk5-9
* Author t o whom correspondence should be addressed. + Maria Curie-Skiodowska University. t Laboratoire de Physico-Chimie des Systemes Polyphases. Centre de PBdologie Biologique. 11 Centre de Recherche sur la Valorisation des Minerals. (1) Davis, J. A.; Leckie, J. 0. J . Colloid Interface Sci. 1978,67,90. (2) Beniamin, M. M.; Leckie, J. 0. J . Colloid Interface Sci. 1981,79, 209. (3) Hayes, K. F.;Leckie, J. 0.A C S S y m p . S e r 1986,No.323 (Geochem. Process Miner. Surf.), 114. (4) Kinniburgh, M. M.;Barkes, J. A.; Whitfield, M. J . ColloidInterface Sci. 1983,95, 370. (5) Van Riemsdijk, W. H.; Bolt, G. H.; Koopal, L. K.; Blaakmeer, J. J . Colloid I n t e r f a c e Sci. 1986,109, 219. ( 6 ) Van Riemsdijk, W. H.; de Wit, J. C. M.; Koopal, L. K.; Bolt, G . H . J . Colloid I n t e r f a c e Sci. 1987, 116, 511. (7) Van Riemsdijk, W. H.;Koopal, L. K.;de Wit, J. C. M . J . Agric. Scr. 1987,35, 241. 0743-7463/92/2408-1154$03.00/0
published first quantitative analysis of heterogeneity effects in ion adsorption within the electric double layer. They considered a variety of problems, but their attention was mainly focused on the behavior of potentiometric titration and electrokinetic data. The energetic heterogeneity of oxide surfaces was taken into account by Rudzinski et al.1° in their analysis of calorimetric effects accompanying ion adsorption at the water/oxide interface. Their theory led to a reasonable simultaneous fit of both experimental titration curves and the experimentally measured heat of ion adsorption. Though the picture of an energetically heterogeneous waterioxide interface seems to be fairly well established, there are some questions which deserve further extensive theoretical studies. The most important of them is the topographical model of the surface heterogeneity of oxides. The first two papers by Van Riemsdijk et al.5~6 were based on the model of random surface topography. It is assumed in this model that the adsorption sites (the outermost surface oxygens) having different adsorption properties are distributed on the oxide surface at random. A quantitative fit of titration curves did not, however, lead these authors to well-established conclusions concerning the heterogeneity effects. As pointed out recently by Rudzinski et al.,1° there exists a certain compensation of heterogeneity effects in this case, as the potentiometric curves are not individual but composite adsorption isotherms. Thus, Koopal and Van Riemsdijk considered later on a model of a patchwise surface t o p ~ g r a p h y Apparently, .~~~ such a model might be useful in cases where rentgenographic analysis of the investigated oxide sample shows 1
( 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. (10) Rudzinski. W.; Charmas, R.; Partyka, S . L a n g m u i r 1991,7,354.
0 1992 American Chemical Society
Energetic Surface Heterogeneity 4
Figure 1. Model of adsorbent accepted by Bakaev in his Monte Carlo study of adsorption on oxide surfaces. The white randomly packed circles represent oxygen atoms (ions) of adsorbent, and the shadowed circle is an adatom which “rolls”over the solid surface (supplied kindly by Bakaev).
the existence of different crystal faces. However, in view of some experimental and theoretical studies, the rentgenographic information seems to be misleading. This is because it contains information not only about the outermost surface layer but also about a part of the underlying bulk phase. Bakaevll believes that for the majority of actual, i.e., really existing, solid surfaces, the picture of an amorphous surface phase should be more realistic than that of a crystalline surface with defects. He directs attention to an interesting piece of experimental evidence supporting his view, coming from the calorimetric experiments of Rouqueroll and co-workers.12J3 They investigated the enthalpy of adsorption of nitrogen and argon on a crystalline rutile and a rutile partially coated by amorphous silica. The enthalpies of adsorption of nitrogen were different, due to the particular electric field pattern near the rutile surface, “felt”by nitrogen molecule. Argon atoms, on the contrary, are insensitive to the difference in electric fields, so the close similarity of its enthalpy of adsorption on pure and coated rutile must testify to similarities in their surface structure. As the silica-coated rutile is, presumably, an amorphous surface, the similarity of the enthalpies of adsorption of argon indicates the amorphous structure of pure rutile too. The adsorption potential of argon is mainly determined by the large oxygen ions, since their polarizability is much larger than that of cations. So, the physical adsorption of argon on oxides is mainly determined by the amorphous structure of the oxygen atoms near the surface. Thus, for argon adsorption on oxides, a representative model of surface phase could be the Bernal model,14 Le., a dense random packing of hard balls. The surface of such a model amorphous solid may be visualized as the surface of a heap of ball bearings on a plate. The two-dimensional schematic visualization of such an atomic arrangement is presented in Figure 1. The white circles in Figure 1represent the atoms (ions) of adsorbent and the shadowed circle an adatom which “rolls” over the heterogeneous surface. The main objective of Bakaev’s earlier works15-18 was to calculate the threedimensional adsorption potential, U(x,y,z),of such an ada(11) Bakaev, V. A. Surf. Sci. 1988, 198, 571. (12) Furlong, D. N.; Rouqueroll, F.; Rouqueroll, J.; Sing, K. S.V. J . Colloid Interface Sci. 1980, 75, 68. (13)Furlong, D. N.; Rouqueroll, F.; Rouqueroll, J.; Sing, K. S.V. J. Chem. Soc., Faraday Trans. 1 1980, 76, 774. (14) Finney, J. L. Modellingof Atomicstructure (AmorphousMetallic Alloys); Luborsky, F. E., Ed.; Butterworthw. London, 1983; pp 42-57. (15) Bakaev, V.A. Proc. Int. Workshop;‘Adsorption on Microporous Adsorbents”; held in Berlin-Eberswalde 1987, 2 , 33.
Langmuir, Vol. 8, No. 4, 1992 1155
tom. It was done practically in the following way: A set of vectors (ri)was created in the computer memory representing a dense random packing of 2500 hard balls. The balls were contained in a square box with an area (20 X 20)D2,where D is the ball diameter. Periodic boundary contributions were imposed in the x and y directions, but no restrictions were imposed in the z direction, perpendicular to the surface. Each ball represented an oxygen anion with a diameter D = 0.28 nm. Argon adatoms were assumed to interact with oxygen balls via a Lennard-Jones potential. The next crucial mathematical operation in Bakaev’s computations was finding the minima of the potential function U(x,y,z).Thus, an average number of 226 local minima have been found by Bakaev in his unit cell. That means, that the area associated with one local minimum is about 0.14 nm2,compared to a value 0.15 nm2 accepted in the BET method utilizing argon adsorption data to determine the area of Ti02.19 The calculated distribution of the number of adsorption sites (local minima) among the values of these minima was compared next with Drain and Morrison’s adsorption energy distribution for argon adsorbed on rutile.20 The range of adsorption energies was similar, except that Drain and Morrison’s energy distribution suggested alarger contribution from less active sites. This may be the contribution from cations, neglected in Bakaev’s computer simulation. If we realize, however, that this is a purely ab initio result, we must be impressed by the success of Bakaev’s model. Bakaev’s computer simulation provides an impressive support for random topography of oxide surfaces. Of course, some degree of surface organization should exist, and it will increase when going more and more deeply into the solid bulk phase. The outermost layers of surface atoms (ions) may be amorphous, but the interior may have a well-defined structure. However, his simulations do not yet provide an answer for another very important question related to the topography model. This is the question of how much the adsorption energies of different ions are correlated on different adsorption sites. Koopal and Van Riemsdijk assumed a high degree of correlation, stating that no matter which was the adsorption site, the difference between the adsorption energies of various cations was still the same. Further, the adsorption energy distribution is the same for all adsorbing ions. In their recent work on the calorimetric effects of adsorption, Rudzinski et a1.10 followed this assumption. However, an analysis of some accurate experimental results obtained recently by Bottero and co-workers21,22 seems to put that high correlation into question, and also the assumption that different ions have the same adsorption energy distribution (the same curve, shifted only on the energy scale by some value). Bottero and co-workers have collected recently an accurate set of experimental data including the titration (composite) isotherms, the individual isotherms measured by radiometric methods, and electrokinetic data. Having such a complete and accurate set of data, one may try to discriminate between various theoretical models of adsorption on the basis of fitting the experimental data by (16) Bakaev, V.A,;Dubinin, M. M. Dokl. Acad. Nauk SSSR 1987,296, 369. (17) Bakaev, V. A. Izu. Acad. Nauk SSSR, Ser. Khim. 1988, 7 , 1478. (18) Bakaev, V. A.; Chelnokova, 0. V. Surf. Sci. 1989, 215, 521. (19) Gregg, S.J.;Sing, K. S.V. Adsorption, Surface Area and Porosity; Academic Press: New York, 1967; p 91. (20) Drain, L. E.; Morrison, J. A. Trans. Faraday SOC.1952, 48, 316. (21) Thomas, F.Ph.D. Thesis, Universite de Nancy, 1987. (22) Thomas, F.;Bottero, J. Y.; Cases, J. M. Colloid Surf. 1989, 37, 281.
Rudzinski et al.
1156 Langmuir, Vol. 8, No. 4,1992 the related theoretical expressions. Performance by appropriate model calculations to see which model predicts the behavior of the actual systems better is the purpose of the present work. Theory 1. T h e Homogeneous Surface Model. Our consideration in this paper will be based on the most popular triple layer model proposed by Davis et al.1,23,24 from conceptualizations of the electrical double layer discussed by Yates et al.25 and Chan et a1.26 General reviews and representative applications of this model have been given by Davis and LeckieZ7and by Morel et a1.28 Thus, we write the surface proton reaction dissociations and the coadsorption of anions A- and cations C+reactions cause the formation of other surface complexes, SOHz+Aand SO-C+
+ -
+ H+ SOH' SO- + H+ SOH,+A- SOH' + H++ ASOH' C+ SOX!+ + H+ SOH,'
SOH'
where is the proton activity in the equilibrium bulk phase and a~ and ac are the bulk activities of anion and cation. Further, $0 is the surface potential and $8 is the mean potential at the plane of specifically adsorbed counterions, which is given by
(la) (1b)
= $0 -
+p
< 60
(5)
where the surface charge, 60
(IC)
6, = B[O+ + OA - 0- - O,] where B = N,e
(6)
(Id)
c1 in eq 5 is the first integral capacitance and N , in eq 6 is the surface density (sites/m2). The equilibrium constants Kalint,KaZint,*KA~"~, and *Kcint of these reactions were defined in l i t e r a t ~ r e . ~ s ~ ~ * ~ ~ We can evaluate the activity of ions ai (i = A, C) having the activity coefficient yi and the concentrations of ions. For our purposes, it will be useful to consider also the We assume that yi is given by the equation proposed by following equivalent reactions: DaviesZ9
- + - + so-c+- so- + c+ SOH'
SO-
H+
(2a)
SOH2+
SO-
2H+
(2b)
SOH,+A-
-
SO- + 2H+ + A-
= [SOH']
i=A,C
(2c) (2d)
Introducing the notation
c
(7)
where the values of the activity coefficient yi for z-valent ion "i" can be calculated from the ionic strength of the suspension, I (mol/dm3), and where A is given by
+ [SOH,'] + [SOH;A-] + [so-c+l+[so-]
A = 1.825 X lo6 (e,
13312
One can solve numerically the system of the nonlinear eqs 4 to obtain the individual adsorption isotherms of ions, Oi (i = 0, +,A, C). For this purpose we rewrite the equation system 4 to the form Kifi
0: = 1
i
we obtain the following set of equilibrium equations, respectively, to reactions 2 (23) Davis, J. A.; James, R. 0.;Leckie, J. 0. J . Colloid Interface Sci. 1978, 63, 480.
(24) Davis, J. A.; Leckie, J. 0. J. Colloid Interface Sci. 1980, 74, 32. (25) Yates, D. E.; Levine, S.; Hearly, T. W. J . Chem. SOC.,Faraday Trans. 1 1974, 70, 1807. (26) Chan, D.;Perram, J. W.; White, L. R.; Hearly, T. W. J . Chem. SOC.,Faraday Trans. 1 1975, 71, 1046. (27) Davis, J . A.; Leckie, J. 0. In Chemical Modelling in Aqueous Systems; Jenne, E. A., Ed.; American Chemical Society: Washington, DC, 1979; Chapter 15. (28) Morel, F. M. M.; Yested, J. C.; Westall, J. C. In Adsorption of Inorganics at Solid-Liquid Interfaces; Anderson, M. A., Rubin, A. J., Eds.; Ann Arbor Science: Ann Arbor, MI, 1981; Chapter 1.
+ZKfi
(9)
i = 0 , +, A, C
where KO= 1/Ka2int K+ = 1/(Ka,intK,2int) K, = * ~ ~ i n t / ~KA , = ~ 1/(Ka2 ~ ~ ~int*KA int )
(10)
and where f , ( i = 0 , +, A, C) are the following functions of proton and salt concentrations (29) Davies, C. W. Ion Association; Butterworths: London, 1962.
Langmuir, Vol. 8, No. 4, 1992 1157
Energetic Surface Heterogeneity
{3
fo = exp - -- 2.3pH), f+ = f t
U =~ (11)
@
~ , i ~n t~
~
*Kci"t +i~ n~ t ~*KAint ''~1 = O i n t
(18) -
KaZint
where
H = 10-pzc (19) For our further purposes we write eq 18 in another form The nonlinear equation system 9 can be transformed into a one nonlinear equation, taking into account eq 6
6,u = B
K+f+ + KAfA - KCfC -
(12)
1+ C K f i i
i = 0, +, A, C This nonlinear equation for 60 can next be solved easily by means of the Mueller-S iteration method, to give the value of 60 for each pH. Having these values, we can evaluate the individual adsorption isotherms from eq 9. To express +o-pH dependence, which occurs in the equations for individual adsorption isotherms and for the surface charge 9-12, we accept here the relation used by Yates et and by Bousse et al.30931
where @ is given by
In eq 14 CDL is the linearized double-layer capacitance. The value of CDL can be calculated theoretically (depending on the salt concentration in the solution) in the way described in Bousse's work30 -1 = CDL
2kTle 1 (8t,tokT~)"~ CStern
(15)
where t, is the relative permittivity of solvent, to is the permittivity of free space, and c is the concentration of the electrolyte (iondm3). The value of the cstemis assumed as 0.2 F/m2. Around the point of zero charge (PZC) eq 13 can be linearized to yield
+ +
+ pKazint)- -21log 11 (*KCint/Ka2int)~C PZC = -(pKalint 1 2 (20) Experimental studies show that in the majority of the investigated systems the value of the point of zero charge (PZC) does not depend practically upon the salt concentration in the bulk ~ o l u t i o n . ~ ~ ~ Further, 3 - 3 ~ ~except ~ ~ for very low and very high PZC values and very low salt concentrations, we can assume that ac = Q A = a. Thus, the independence of PZC of the salt concentration can formally be expressed as follows:
Solving the set of eqs 18 and 21, we obtain
Correlations 22 between the parameters Kalint,Kapint and *Kcint,*K~intreduce the number of the parameters, intrinsic equilibrium constants, to be found by fitting experimental data from four to two. While accepting correlations 22 we obtain the following simplified expressions for the value of the point of zero charge (PZC) PZC = ,(pKal 1 int + pKa2int) PZC = L(p*Kcint+ p*KAint) 2 from which the following condition holds
(23a) (23b)
~ K aint 2 - p*Kcint = p*KA - pKalint
(24)
pKaiint= -log Kaiint, i = 1, 2
(25a)
p*Kiint= -log *Kiint, i = C, A
(25b)
where
We rewrite condition 24 as This approximation is valid in the region around the PZC, where (e+olkT) < 8. The observed behavior of experimental adsorption systems suggests that some correlations exist between the intrinsic equilibrium constants Kalint,Ka2int,*Kcint,and *K~int.The fact is that titration curves corresponding to different salt concentrations have a common crossing point, PZC. This point of zero charge is defined by the condition 6,(pH = PZC) = 0 (17) Now, let us transform eq 12 using eqs 10 and 11 for pH = PZC, where 60 = 0 a r d +O = 0, to the following form (30) Bousse, L.;de Rooij, N. F.; Bergveld, P. IEEE Trans.Electron Devices 1983, 30, 8. (31) Vander Vlekkert,H.;Bousse,L.;deRwij,N.F. J . Colloidlnterface
Sci. 1988, 122, 336.
log Kcationint = log Kmionint (26) where the intrinsic equilibrium constants Kcationintand Kanionint are the constants related to the followingreactions
so-c+F! so- + c+ SOH2+A-F! SOH2++ A-
(27a)
(27b) Various a ~ t h o r s ~report ~ ~ ~that ~ * ~their analysis of adsorption systems (based on the model of homogeneous oxide surfaces) suggests that condition 23 is usually fulfilled, but its theoretical interpretation is difficult. As a matter of fact it looks somewhat strange that these two (32) Sprycha, R. J. Colloid Interface Sci. 1984, 102, 173. (33) Foissy, A,; MPandou, A.; Lamarche, J. M.; Jaffrezic-Renault, N. Colloids Surf. 1982, 5, 363.
Rudzinski et al.
1158 Langmuir, Vol. 8, No. 4, 1992
So far, only one extreme physical situation of very high correlations was considered by Koopal and c o - ~ o r k e r s . ~ ~ They assumed that no matter which is the adsorption site, the difference between the adsorption energies t, and tj is still the same and equal to Ai,. So, the one-dimensional distributions of the number of adsorption sites among ti, x,(t,)look like those on Figure 2. Of course, the sequence of the most probable adsorption energies €10 can change from one oxide to another. Thus, the intrinsic equilibrium constant KJ can be written as
€1
Figure 2. Schematic visualization of the adsorption energy distributions x , ( c , ) (i = 0, +, A, C) for the model of surface
where
heterogeneity assuming the high correlations between the adsorption energies of various surface complexes.
different reactions are characterized by the same equilibrium constant. 2. Adsorption on Heterogeneous Surfaces. The Case of High Correlations between the Adsorption Energies of Various Surface Complexes. According to the consideration in our previous paperlo the "intrinsic" constant K, can be written as
where ti is the adsorption (binding) energy of ith surface complex and Ki' is related to its molecular partition function. On a heterogeneous solid surface with random topography ei will vary across the surface from one site to another. This is the consequence of the randomly varying surrounding of the surface sites, the outermost surface OXygens. Of course, there will be also variations in the local Coulombic force fields $0 and $@,but we will neglect them for the following reason: Coulombic interactions are longranged, what will cause a "smoothing" of these variations over the local structure of the outermost surface oxygens. Thus, similarly as in the model of a homogeneous oxide surface we will consider $0 and $@to be functions of the average composition of the adsorbed phase. Physisorption interactions (chemical binding forces) are shortranged, so, the local variations in ti must be taken into account. The experimentally measured adsorption isotherms O i , have to be related to the following averages, flit
eit((a),n= S,SO,((t),(a),nx((tl, deo de+ deAdec
(29)
where (a)is the set of the bulk concentrations (aH, ac, a d , (e) is the set of the adsorption energies (€0, e + , t ~ec), , R is the physical domain of (e), and ~ ( { t ) is ) the multidimentional differential distribution of the number of adsorption sited among various sets (€1, normalized to unity $ * . * $ x ( { t ) )deo de+ deAdec
=1 (30) R Now we have to consider the fundamental physical question, whether the variables ti are totally independent. In other words, where the values of t i for a given adsorption site do affect in some way t,+i. The degree of the correlations between ti and tj will affect the result of the integration in eq 29.
ti
=
ti
+ Aji
The equation system 9 can then be written as
i, j = 0, + , A , C
where f , is the same as in eq 11, and the average quality 0jt reads
The problem formulated above is identical with that considered in dozens of papers on gas adsorption on heterogeneous solid surfaces. So, it is no surprise that Koopal and co-workers followed some solutions found in gas adsorption. Thus, they wrote the integrated form of eq 33 as follows:
Rjfj
o;, = -
[CRjf,1k*/' (34)
j = 0, +, A, C
Putting e,, from eq 34 to the left-hand side of eq 33, one can evaluate xi(ti) corresponding to the assumed form of 0jt. This is done by rewriting the integral on the righthand side of eq 6a into a Stieltjes transform and next evaluating the reverse Stieltjes t r a n ~ f o r m . 3 ~The J ~ obtained xi(c,) then takes the form (34) Koopal, L. K.; van Riemsdijk, W.;Roffey,M. G. J . Colloid Interface Sci. 1987, 118, 117. (35) Sips, R. J . Chem. Phys. 1948, 16, 490. (36) Sips, R. J . Chem. Phys. 1950, 18, 1024.
Energetic Surface Heterogeneity
sin
x(t) =
(y) exp{-)
(y)expi
(t
1+ 2 cos
Langmuir, Vol. 8, No. 4, 1992 1159
- to) C
(35) where c is the heterogeneity parameter. The function x(c) in eq 8a is bell-shaped and centered a t t = to. However, as it is temperature dependence, the function is not correct from a physical point of view. It is argued often that it degenerates "correctly" to Dirac delta distribution &(e - €0) as kT/c 1. Then, eq 7a reduces apparently to that one for a homogeneous solid surface. However, it is to be emphasized that even in the limit 1, function x(c) in eq 35 is still temperature dekT/c pendent. The fact that the exponent kT/c in eq 34 has the numerical value of unity does not change the fundamental fact that it is still temperature dependent. The consequences can easily be seen when one evaluated the enthalpy of adsorption from eq 34. The obtained expression does not reduce to the expression for a homo1. geneous solid surface in the limit kT/c For the purpose of the present work we summarize the above discussion as follows: Equation 34 is not a fully rigorous expression. However, as the function x(t) in eq 35 is not strongly temperature dependent, one can use it as an appropriate expression for Bit. Later we are going to show how that appropriate solution can be improved substantially in a simple way. To evaluate the surface charge, one must transform the nonlinear equation system 34 taking into account eq 6
-
(38) To solve numerically equation set 37 and 38, we transform this set to the following form
-
-
6, =
B
where Xcor(l) and Xcor(') are correction functions, arising from surface heterogeneity
-
X
-
When kT/c 1 and Xcor(l),Xcor(') 0, we arrive at eqs 18 and 21 for the homogeneous surface model. Solving the set of eqs 39 with respect to * K Aand ~ ~KaPt ~ we obtain
This equation can be solved in the same way as eq 12. Now, we are going to establish the correlations between the intrinsic equilibrium constants in a similar way as in the case of the homogeneous surface model, considered in the previous section. To that purpose we transform eq 36, using eqs 10 and 11 for pH = PZC, to the following form
IP ~ , ~ i ~n , 't i n t
+ ~ , ' i n t *KAint
+
= 0 (37)
where H is given by eq 19. This equation is nonlinear with respect to the proton concentration a t the point of zero charge and it cannot be transformed as in the case of the homogeneous surface model. The experimentally observed independence of PZC of salt concentration leads now to
*KAint
IP *Kcint+ Xco~')PIK,lint(Xco,(l) + 1 - aXcor('))]-l
(41b) Equations 40 and 41 form the system of four equations from which one can determine the four constants Kalint, We have applied an iteration K,zint,* K A ~and ~ ~*Kcint. , method to solve this equation system: In a first step kT/c was taken equal to unity, and the ~~ startingrelations,K,2ht = W/Kalintand * K A=~W/*Kcint, were accepted to evaluate Xcor(l) and XcoJ2)from eqs 41a,b, in which kT/c was decreased by a small value, A(kT/c). The evaluated corrections were accepted next to find K,zht and * K Afrom ~ ~ eqs ~ 41 corresponding to [kT/c - A(kT/c)]. ~ ~ ~were used Calculated in that way, K,2int and * K Avalues to calculate corrections Xcor(l)and Xcor(') for still lower values of the heterogeneity parameter, [kT/c - 2A(kT/c)l. These corrections were then used in eq 41 to calculate Kazintand *KA'"~ corresponding to this new value of the heterogeneity parameter [kT/c - 2A(kT/c)l. The iterations were continued until the heterogeneity parameter did not achieve the a priori assumed value, for which the v a l ~ e s K , 'and ~ ~ ~*KAintareto be calculated. The accuracy of the calculation increases obviously as A(kT/c)decreases.
Rudzinski et al.
1160 Langmuir, Vol. 8, No. 4, 1992
the physical backgound of which was discussed in an earlier paper by Rudzinski et al.39 Then, the RJ approach is used to calculate single isotherms @it
+
Oit = -Xi(tic) corr
(44)
X i ( € )= sxi(ci)dei
(45a)
where
---
€1
and
tic
is found from the condition
Figure 3. Schematic visualization of the adsorption energy distributions x,(cl) (i = 0, +, A, C), for the model of surface heterogeneity assuming a lack of correlations between the adsorption energies of various surface complexes.
Our computer exercises showed that the correction was very small and practically equal to coefficient Xcor(2) zero. They also showed that the members of the equations of set 41, in which cation or anion activities occur, can be neglected because the values of the calculated parameters Kazintand * K Aare ~ ~the~ same for the various substituted values of activity. The latter result was to be expected for the basic reasons leading to eq 38. The above findings make the following simplification of eqs 41 possible:
(45b) The correction term reads
where B is Bernouli's number. When kT/ci 50.9, the correction term corr on the righthand side of eq 44 can be neglected, and the averaged single isotherm @it takes the form (46) i = 0, +, A, C where
The above two equations are independent. The first one (42a) is a nonlinear equation with regard to Kazintand can be solved by means of the Mueller-S iteration method. The other, eq 42b, is linear with regard to * K A ~ ~ ~ . Thus, in a similar way as in the case of the homogeneous surface model, we managed to reduce the number of the intrinsic equilibrium constants from four to two. From the comparison of eqs 42 and eqs 22 it follows that equality 24 is not satisfied for the case of a heterogeneous oxide surface. Thus the equilibrium constants of reactions 27a and 27b are not longer equal to each other. Such a physical situation seems to be more probable. 3. T h e Model of Surface Heterogeneity Assuming a Lack of Correlations between t h e Adsorption Energies of Various Surface Complexes. Now, we are going to consider another extreme model of surface heterogeneity, when Ai,'s are not correlated at all. That model has already been elaborated by Rudzinski and coworker~~ for~ the J ~ case of adsorption of liquid mixtures of nonelectrolytes on heterogeneous solid surfaces. The relations between the one-dimensional adsorption energy distributions xi(€{)are, for instance, as shown in Figure 3. The starting point is the hypothetical one-component adsorption which, in the case of gaseous adsorption, is a fully physical case. All the functions x L ( t Zare ) represented by the bell-shaped function
(47) and where OV = 1 - @it is the function of the *vacancies", i.e., free surface sites-oxygen ions. Now let us consider our case of multicomponent adsorption. When the adsorption energies ci, tj#i are not correlated, the presence of other components in the adsorbed phase will affect the adsorption of "in only by a random blocking of adsorption sites. It means, that the fraction of vacancies 8, has to be expressed as follows:
4 = 1-
EOit
(48)
1
The solution of the equation system (46 and 48) yields [KiOfi]kTIC' Oit =
(49) 1+ x[KiOfilkT'Ci
i = 0, +, A, C
Soc., Faraday Trans. 2 1985, 81, 553.
Now let us remark that while accepting the function 43 to represent x i ( 4 in eq 33 and applying the RJ approach outlined in eqs 44 and 45, one arrives again at eq 34. The present derivation, however, shows clearly that considering the limit kTlc 1 in both eqs 34 and 49 does not make sense. The essential condition of the RJ approach to be applied is that the variance ci be larger than kT.
(38) Rudzinski, W. Retention in Liquid Chromatography. In Chromotographic Theory and Basic Principles; Jonsson, J. A., Ed.; Marcel Dekker: New York, 1988; p 245.
1987, 406, 295.
(37) Rudzinski, W.; Michalek, J.; Suprynowicz, Z.; Pilorz, K. J. Chem.
-
(39) Rudzinski, W.; Michalek, J.;Brun, B.; Partyka, S.J. Chromatogr.
Energetic Surface Heterogeneity
Langmuir, Vol. 8, No. 4, 1992 1161
To evaluate the surface charge in this model, one must transform the nonlinear equation system 49 taking into account eq 6
[K+"f+IkTlc+ [KAofA]kTlcA - [KCofC]kT/CC -1 6, = B 1
(50)
+ ~[Kiofi]kTjci i
The above equation can be solved in the same way as eq 12. Now, let us consider the correlations between the intrinsic equilibrium constants predicted by the present model of surface heterogeneity. To this purpose we transform eq 50, using eqs 10 and 11, for pH = PZC (the conditions: 60 = 0 and $0 = 0)
(
I
~ , ~ ff i ~n , ,t i n t ) k T / c +
+
(
~ , ~ i *KAint n t
-
Applying the condition that PZC is independent of salt concentration, we obtain
source was the individual adsorption isotherms for surface complexes SOH2+A- and SO-C+, measured separately by means of radiometric methods. The third source of information which we are also going to explore is the electrokinetic effects accompanying the formation of the electric double layer. The {-potential measurements are carried out quite frequently and their theoretical interpretation is apparently simple, but often leads to a poor agreement between theory and experiment. The Gouy-Chapman theory is used to describe the equilibrium in the diffuse layer for 1:l electrolytes e
+ (L+ (54)
(8e,~~kTZ)'/~ 8e1e&TZ
1)1J2]
where $d is the potential in the d-plane which defines the onset of the diffuse layer, e, is the relative permittivity of solvent for water a t T = 25 "C, el. = 78.25, EO = 8.854 X 10-12 F/m is the permittivity of free space, and I is the ionic strength of the suspension (ions/m3). The charge density in the d-plane, ad, reads 6, = B(8- - 8,)
(55)
The values of 8- and 8+ are calculated in the way described in sections 1-3. Some authors assumed that $d = but this assumption often led to a poor agreement between theory and experiment. A more general approach is to assume f = $d(Z)42-44(2 is the distance of the shear plane), calculated from the equations f,32,40941
To solve the equation system (51 and 521, we evaluate the term where from eq 52 and put it into eq 51. Then we arrive a t the following nonlinear equations U1(Ka2'"')=
(
ff
~ , ~ i ~n , ,t i n t
)
kTlc+
kT/cc
+ (k~/c, - 1)X
Now we solve first the nonlinear equation Ul(K,#") usingfor example the Mueller-S iteration method toobtain the value of Kapintfor assumed values of the parameters Kalintand *Kcintas well as the heterogeneity parameters kT/c, (i = +, A, C). Next, we solve the second nonlinear equation U,(*KA'"~) for the same parameters Kalint,*Kcint,kT/ci (i = A, C) and for the new calculated value of K,@t. In the same way as before, we reduce the number of the intrinsic equilibrium constants from four to two. 4. Equations Describing Electrokinetic Effects. In sections 1-3 we described the two experimental sources of information about the formation of the electric double layer a t the water/oxide (anatase) interface: The first source was the adsorption isotherms, measured as a combined effect called the titration curves. The second
Yo =
exP(e$d(o)/2kn - 1 exp(e$J0)/2kn + 1
(56b)
and where K is DebyeHuckel's reciprocal thickness of the double layer
Now let us consider yet the problem which, to our personal feeling, did not receive enough attention in the past. This is the relation between PZC and IEP values. The IEP (isoelectric point) is the pH value a t which 6d = 0, i.e. no electrokinetic mobility is observed. Some a ~ t h o r have s ~ ~reported ~ ~ ~ about ~ ~ differences between PZC and IEP values in their experimental adsorption systems. It was also the case for all the four alumina samples investigated in the CNRS Laboratory in Nan~y.2~ So, let us remark at this moment that these differences cannot be explained in terms of the model of a homogeneous oxide surface. The condition 6d = 0 is fullfilled when 8+/8- = 1. Thus, from eqs 54 and 22 we arrive at the condition PZC = IEP. (40) Hunter, R. J.; Wright, H.J. L. J. Colloid Interface Sci. 1971,37, 564. (41) Sprycha, R.; Szczypa, J. J. Colloid Interface Sci. 1984,102,288. (42) Harding, I. H.; Healy, T. W. J. Colloid Interface Sci. 1985, 107, 382. (43) Kallay, N.; Tomic, M. Langmuir 1988, 4, 559. (44) Tomic, M.;Kallay, N. Langmuir 1988, 4, 565. (45) Smit, W.; Holten, C. L. M.J. Colloid Interface Sci. 1980, 78, 1. (46) Kallay, N.; Babic, D. Colloids Surf. 1986, 19, 375.
Rudzinski et al.
1162 Langmuir, Vol. 8, No. 4, 1992
Table I. Values of the Parameters Obtained by Fitting Best Experimental Data by the Equations Developed for the Three Models of Oxide Surfaces parameters calculated values pKslint 6.0
*pKeint 8.7
N, 12.
kTic kTlco kTlc+ Homogeneous Surface Model
Z 0.80
1.20
kTlcc
kTicA
10.
*PKA'"'
P
11.00
8.30
4.4
11.38
8.40
3.6
11.05
8.34
3.7
pKsnint
Heterogeneous Surface Model (High Correlation between Adsorption Energies) 6.2
12.
8.6
0.85
1.50
20.
0.80
Heterogeneous Surface Model (Lack of Correlations between Adsorption Energies) 5.9
8.7
12.
0.70
1.20
25.
0.83
0.93
0.90
0.86
The last three columns are the values calculated from the best fit parameters. c1.1 and c1.2 are the two values of c1: the former one for the acidic and the latter for the alkaline branch of the titration curve. The units for both c1 are Fim2 and the unit for Z is A.
I
-003
-006
1 l
\ \i i o\
5
6
7
a
9
10
11
PH
I
PH
Figure 4. Comparison between the experimentally determined of the Alumina Ma and the values of the surface charge bo (0) for the salt concentration theoretically calculated ones (-) 10-2 mol/dm3 for the three models of oxide surface and the three sets of parameters collected in Table I. The differences between three corresponding lines cannot be practically visible, so they are represented by the same solid line.
On the contrary, the differences between PZC and IEP are predicted by both models of heterogeneous oxide surfaces. However, the first heterogeneity model is not able to explain larger differences which are sometimes observed. This is because for physically reasonable values of k T / c varying between 0.7 and 0.9, the term in the square bracket on the right-hand side of eq 42a is not far from unity, i.e., eq 42 leads to interrelations between the intrinsic constants Kalintand Kazintclose to those predicted by the model of a homogeneous oxide surface. Thus, KC1 There is much larger flexibility in the case of the second model of surface heterogeneity, neglecting the correlations between the adsorption energies of various surface complexes. Our numerical exercises have shown, that even for k T / c , ( i = 0, +, A, C) close to 0.9, one may arrive at quite large differences between PZC and IEP values. Results and Discussion Four samples of amorphous alumina (gels) were studied at the CNRS Laboratory in Nancy: Alumina Ma, Alumina Mb, Alumina C, and Alumina D. The most detailed information concerning these experiments can be found in the Ph.D. Thesis by Fabien Thomasz1(in French), but
Figure 5. Comparison between the radiometrically measured, individual adsorption isotherms of SO-C+ ( 0 )and SOH*+A-(0) and the theoretical ones, corresponding to the model of a the first model of surface homogeneous solid surface (-), heterogeneity assuming high correlation between adsorption energies (- - -), and the second model neglecting the correlations between adsorption energies (- - -). The calculations were performed by using the parameters collected in Table I.
the most essential part of this information was also published by us.22 Here we repeat only that part of information which is related closely to further discussion of our theoretical results. Thus KC1 was used as the basis electrolyte, and the experiments were carried out at 25 OC. The titration and electrokinetic curves were monitored for three electrolyte concentrations, 10-1, 10-2, and 10-3 molidm3, but the radiometric measurements were made only for the salt concentration mol/dm3. A successful theory should, for a certain set of parameters, fit simultaneously experimental titration curves, electrokinetic curves, and the individual adsorption isotherms of K+ and C1- measured radiometrically. Thus, to follow this investigation strategy, we took into consideration only the experimental data, measured at the salt concentration mol/dm3. For this electrolyte concentration the calculated activity coefficients YA = yc are equal to 0.902. As we have already mentioned, four various alumina samples were investigated, but we selected Alumina Ma for our theoretical numerical analysis. The reason for that was as follows: while the quality of our experimental titration and electrokinetic curves was similar for all of them, the experimental radiometric individual isotherms of K+ and C1- adsorption covered the widest range of pH values, and the experimental points seemed to exhibit the smallest experimental scatter. The BET area of the Alumina Ma sample was 117 mz/g, whereas that obtained by the Harkins-Jura method was
Langmuir, Vol. 8, No. 4, 1992 1163
Energetic Surface Heterogeneity
PH
Figure 8. Comparison between the theoretical isotherms OA, BC, and Bo for the model of homogeneous solid surface (-) and the first (- - -) and the second (- - - -) models of a heterogeneous oxide surface, by using the parameters collected in Table I.
-401 6
5
8
7
9
1
0
1
1
PH
Figure 6. Comparison between experimentally determined values of {-potential ( 0 )and those theoretically calculated for the first model of surface a homogeneous surface model (-),for heterogeneity (- - -), and the second model of surface heterogeneity (- - - -). The calculations were performed for the parameters collected in Table I.
pendently of these three experimental sources (titration curves, electrokinetic curves and radiometric isotherms). The MINUIT4' subroutine was applied in our numerical calculations. Concerning the details of our calculations, we would like to mention that the surface charge 60 was recalculated from the published data by means of the equation
QPA x ~
8
0.03
W
V
9
I
\
\
where Qs is the amount adsorbed expressed in [pmol/gl, N A is Avogadro's number, e is the elementary electron is the BET surface [m2/gl. Thus, charge [Cl, and SBET 60 is expressed in [C/m21. The fractional coverage by SO-C+ and SOH2+A- complexes was calculated from the experimental radiometric curves by means of the equation
\\
\
\\
001
2
L
6, = SBET
6
8
1
0
1 2 PH
Figure 7. Theoretical values of 6'+ and 6'- calculated for the homogeneous surface model (-) and the first (- - -) and the second (- - - -) models of a heterogeneous oxide surface, by using the parameters collected in Table I.
84 m2/g. We used the BET value for our numerical calculations. The experimental titration curves have a CIP (common intersection point), at 60 = 0, so they can be analyzed in termsof our theory. The CIP is located a t 8.5 on pH scale, so, we accepted PZC = 8.5 in our theoretical analysis. The experimental IEP point was located a t a different pH value equal to about 8.2. For all the alumina samples the experimentally determined PZC and IEP values were different. This would, undoubtly, suggest that our theoretical analysis should be carried out in terms of the second heterogeneity model, assuming a lack of correlations between the adsorption energies of ions. We believe, however, that comparing the results obtained by applying all the three models should be very instructive for future theoretical consideration. Thus, Table I collects the values of the parameters found by fitting best simultaneously the data obtained inde-
where i = C, A, Qi is the measured adsorbed amount [pmoll g], and N , is the surface density of adsorption sites [sites/ nm21. Figure 4 shows the comparison between the experimentally measured and theoretically calculated surface charge 60. The solid line is for all three sets of the parameters collected in Table I. It means titration curves are practically insensitive to the heterogeneity effects. Some authors argue that they are practically insensitive to accepted physical model^.^*-^^ On the contrary, the radiometrically measured individual adsorption isotherms of K+ and C1- appear to be sensitive to surface energetic heterogeneity of oxides. This is shown in Figure 5. One can see there that the second model of surface heterogeneity leads to the best fit of the experimental data. Neglecting the surface heterogeneity leads to a serious underestimation of the concentration of SO-C+ complexes a t low pH values and of SOH2+Acomplexes a t high pH values. (47) James, F.; Roos, M. Comput. Phys. Commun. 1975, 10, 343. (48) Sposito, G. J. Colloid Interface Sci. 1980, 74, 32. (49) Sposito, G. J . Colloid Interface Sci. 1983, 91, 329. (50)Johnson, R. E., Jr. J. Colloid Interface Sci. 1984, 100, 540.
1164 Langmuir, Vol. 8, No. 4, 1992
Accepting the second model of surface heterogeneity is very essential for a proper fit of our experimental {-pH curves. This is because only that model can show the difference between the PZC and IEP values. This is shown in Figure 6. Figure 7 explains the reason for that. This is caused by the fact that the crossing point of the curves 8+ and 8- is not located a t PZC. Although they accept very small
Rudzinski et al. values, they determine { potential, as it can be deduced from eqs 54-56. Figure 8 shows the theoretical curves 80)OA, and flc for comparison. As we have already emphasized, titration curves are not sensitive to the surface energetic heterogeneity. On the contrary the electrokinetic curves appear to be quite sensitive.
