Langmuir 1999, 15, 5635-5648
5635
Four-Layer Complexation Model for Ion Adsorption at Energetically Heterogeneous Metal Oxide/Electrolyte Interfaces Robert Charmas* Department of Theoretical Chemistry, Faculty of Chemistry, Maria Curie Skłodowska University, Pl. Marii Curie Skłodowskiej 3, 20-031 Lublin, Poland Received July 14, 1998. In Final Form: December 8, 1998 Adsorption equations are developed for the four-layer model assuming that the anions and cations of the inert electrolyte (1:1) are adsorbed in two distinct layers, located at different distances from the surface; furthermore, the energetic heterogeneity of the oxide surface is taken into account. The theoretical development has been based on the idea of the triple-layer model along with the 2 pK charging mechanism. Because of some compensating effects, the titration curves 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 can even make discrimination among various models possible. The developed expressions for the four-layer model taking into account the energetic heterogeneity of the surface predict the difference between PZC and IEP values, similar to the homogeneous four-layer model. This new theoretical approach leads to adsorption equations which are easy to use like those developed for the popular triple layer model, but in both cases a simple computer program for performing calculations is needed.
Introduction Models of surface complexation are most commonly applied to represent the ion adsorption at the oxide/ electrolyte interfaces. A variety of such models have been employed, and all these models treat the reactions involving surface species in the same manner as those involving solution species. However, their construction can differ in modeling of the surface (charging mechanism of surface sites) and interfacial region (distribution of ions).1-4 Of all the models published so far, the most frequently used is the so-called triple-layer model (TLM)5-8 along with the 2-pK charging mechanism. The related schematic picture is shown in Figure 1A. All the assumptions accompanying the construction of the TLM were discussed in the paper by Robertson and Leckie.4 The most important of them are as follows: (1) the colloidal particle is represented by the planar surface, (2) the amphoteric products of surface hydrolysis are uniformly (homogeneously) distributed over the planar surface and they can be in one of three protonation states (0, 1, or 2 protons per sites2-pK charging mechanism), (3) electrostatic properties are calculated by assuming that the ion species are * Temporary address: INPL, ENSG, Laboratoire Environnemental et Mineralurgie, CNRS UMR 7569, BP 40, Rue du Doyen Marcel Roubault, F-54501 Vandoeuvre les Nancy, France. Fax: (33) 383 575404, E-mail:
[email protected]. (1) Davis, J. A.; Kent, D. B. In Mineral-Water Interface Geochemistry; Hochella, M. F., Jr., White, A. F., Eds.; Mineralogical Society of America: Washington, DC, 1990; p 117. (2) Sposito, G. The Surface Chemistry of Soils; Oxford University Press: New York, 1984. (3) Stumm, W. Aquatic Surface Chemistry: Chemical Processes at the Particle-Water Interface; Wiley: New York, 1987. (4) Robertson, A. P.; Leckie, J. O. J. Colloid Interface Sci. 1997, 118, 444. (5) Davis, J. A.; James, R. O.; Leckie, J. O. J. Colloid Interface Sci. 1978 , 63, 480. (6) Davis, J. A.; Leckie, J. O. J. Colloid Interface Sci. 1978, 67, 90. (7) Hayes, K. F.; Leckie, J. O. J. Colloid Interface Sci. 1987, 115, 564. (8) Hayes, K. F.; Papelis, C.; Leckie, J. O. J. Colloid Interface Sci. 1988, 125, 717.
point charges, (4) equilibria for the formation of the complexes between surface sites and solution species can be described in exactly the same way as for those composed solely of solution species. All of them were discussed in detail in the paper.4 Just recently, TLM has been subjected into a rigorous analysis by Sverjensky and Sahai,9-11 taking into account many experimental data. The modifications of TLM published in the literature went into three directions. One group of scientists has challenged the charging mechanism of the surface oxygens. They invoke Pauling’s principle12 of local neutralization of charge to show that the protonated surface oxygens will, in general, have fractional values of charge. That hypothesis is commonly called the 1-pK charging mechanism.13 Moreover, they argue that various surface oxygens may have a different 1-pK charging mechanism, depending on their local coordination to cations. That model is known in the literature as the “multisite complexation model” (MUSIC) of the oxide electrolyte interface.14-17 Another group of scientists has accepted the classical 2-pK charging mechanism, as shown in Figure 1A, but emphasized the importance of a small degree of crystallographic organization of the surface oxygens, compared to the situation in the interior of an oxide crystal. That (9) Sverjensky, D. A.; Sahai, N. Geochim. Cosmochim. Acta 1996, 60, 3773. (10) Sahai, N.; Sverjensky, D. A. Geochim. Cosmochim. Acta 1997, 61, 2801. (11) Sahai, N.; Sverjensky, D. A. Geochim. Cosmochim. Acta 1997, 61, 2827. (12) Pauling, L. In The Nature of the Electrostatic Bound; 3rd ed.; Cornell University Press: Ithaca, NY, 1967. (13) Bolt, G. H.; Van Riemsdijk, W. H. In Soil Chemistry, B. Physicochemical Models, 2nd ed.; Bolt, G. H., Ed.; Elsevier: Amsterdam, 1982; p 459. (14) Hiemstra, T.; Van Riemsdijk, W. H.; Bolt, G. H. J. Colloids Interface Sci. 1989, 133, 91. (15) Hiemstra, T.; De Wit, J. C. W.; Van Riemsdijk, W. H. J. Colloids Interface Sci. 1989, 133, 105. (16) Hiemstra, T.; Van Riemsdijk, W. H. Colloids Surf. 1991, 59, 7. (17) Hiemstra, T.; Van Riemsdijk, W. H. J. Colloid Interface Sci. 1996, 179, 488.
10.1021/la980880j CCC: $18.00 © 1999 American Chemical Society Published on Web 07/09/1999
5636 Langmuir, Vol. 15, No. 17, 1999
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Figure 1. (A) Diagramatic presentation of the triple layer model (TLM). ψ0, δ0, the surface potential and the charge density in the 0-plane; ψβ, δβ, the potential and charge coming from the specifically adsorbed ions (cations C+ and anions A-) of the inert electrolyte; ψd, δd, the diffuse layer potential and its charge; C1, C2, the electrical capacitances, constant in the regions between planes. (B) Diagramatic presentation of the four-layer model (FLM). ψo, δ0, ψd, and δd have the same meaning as in part A; ψC, δC, ψA, δA, the potentials and the charges coming from the specifically adsorbed ions (cations C+ and anions A-) of the inert electrolyte; C+, C-, Cx, and C2 are the electrical capacitances, constant in the regions between planes.
smaller degree of surface organization should lead to a different status of the surface oxygens, resulting in what is called “surface energetic heterogeneity”.18-25 The third refinement of the TLM is the four-layer model (FLM). Its idea was introduced to literature by Barrow et al.26 A new layer (the fourth one as the name indicates, but situated, as the second, next to the surface layer “0” where protons are adsorbed) was reserved first for the bivalent metal ions or anions of multiproton oxy-acids. Cations and anions of basic electrolyte were placed still (18) Kinniburgh, D. G.; Barkes, J. A.; Whitfield, M. J. J. Colloid Interface Sci. 1983, 95, 370. (19) Van Riemsdijk, W. H.; Bolt, G. H.; Koopal, L. K.; Blaakmeer, J. J. Colloid Interface Sci. 1986, 109, 219. (20) Van Riemsdijk, W. H.; De Witt, J. C. M.; Koopal, L. K.; Bold, G. H. J. Colloid Interface Sci. 1987, 116, 511. (21) Rudzinski, W.; Charmas, R.; Partyka, S. Langmuir 1991, 7, 354. (22) Rudzinski, W.; Charmas, R.; Partyka, S.; Thomas, F.; Bottero, J. Y. Langmuir 1992, 8, 1154. (23) Rudzinski, W.; Charmas, R.; Partyka, S.; Bottero, J. Y. Langmuir 1993, 9, 2641. (24) Rudzinski, W.; Charmas, R. Adsorption 1996, 2, 245. (25) Rudzinski, W.; Charmas, R.; Borowiecki, T. In Adsorption on New and Modified Inorganic Sorbents; Dabrowski, A., Tertykh, V. A., Eds.; Elsevier: Amsterdam, 1996; p 357. (26) Barrow, N. J.; Bowden, J. W.; Posner, A. M.; Quirk, J. P. Aust. J. Soil Res. 1981, 19, 309.
in the same layer as in the TLM. Another kind of FLM was launched by Bousse et al.27 who presented in their paper a diagram of FLM, in which there are only ions of basic electrolyte and the potential-determining ions H+. They argued that anions and cations of the basic electrolyte are not located in the same layer (as in TLM) but in two separate ones. A schematic picture of such a FLM is shown in Figure 1B. A rigorous thermodynamic description based on that physical model, and providing theoretical expressions for all the experimentally measured physicochemical quantities, has been recently published by the author and his colleagues.28-30 These papers presented also a comparison of this model with the TLM. Recently, the related expressions for the heat effects accompanying ion adsorption have been developed.31 (27) Bousse, L.; De Rooij, N. F.; Bergveld, P. Surf. Sci. 1983, 135, 479. (28) Charmas, R.; Piasecki, W.; Rudzinski, W. Langmuir 1995, 11, 3199. (29) Charmas, R.; Piasecki, W. Langmuir 1996, 12, 5458. (30) Charmas, R. In Physical Adsorption: Experiment, Theory and Applications; Fraissard, J. P., Conner, C. W., Eds.; NATO-ASI-Series C, Vol. 491; Kluwer Academic Publ.: Dordrecht, 1997; p 609. (31) Charmas, R. Langmuir 1998, 14, 6179.
Model for Ion Adsorption
Langmuir, Vol. 15, No. 17, 1999 5637 Table 1. Surface Reactions and Equilibrium Equations
surface reaction
equilibrium equation
Ka1int
SOH2+ 798 SOH0 + H+
Ka1int )
Ka2int
SOH0 798 SO- + H+
Ka2int )
*KCint
SOH0 + C+ 798 SO-C+ + H+
*KCint )
*KCint ) *KAint
SOH2+A- 798 SOH0 + H+ + A-
*KAint )
*KAint )
In the present work, we are going to develop further FLM applying the theoretical methods used to describe adsorption on heterogeneous solid surfaces for the fourlayer model. The aim of this work is to complete thermodynamic description based on such modified FLM, along with the comparison with the four-layer model assuming that the oxide surface is geometrically and energetically homogeneous, as well as with the triple-layer model with the assumption that the oxide surface is energetically heterogeneous. Theory 1. The Definition of Charging Mechanism and Interfacial Region in the Triple- and Four-Layer Models. The potential determining ions H+, the cations C+, and anions A- of the basic electrolyte form, according to both TLM and FLM, the following surface complexes: SOH0, SOH2+, SO-C+, and SOH2+A-, where S is the surface metal atom. The concentrations of these complexes on the surface are denoted by [SOH0], [SOH2+], [SO-C+], and [SOH2+A-], respectively. [SO-] is the surface concentration of the free sites, “unoccupied” surface oxygens, which are likely to be engaged in adsorption of nondissociated water molecules. According to FLM, protons are located in the potential layer ψ0, the anions are situated within the layer of potential ψA, whereas the cations are within the layer of potential ψC. According to TLM both anions and cations are situated within the same layer of potential ψβ. For the reactions occurring at the oxide/electrolyte interface, the mass action equations and the related intrinsic equilibrium constants for both TLM and FLM are collected in Table 1. There, aC, aA, and aH are the activities of cations, anions, and protons, respectively. One can see in Table 1, that the equations describing the desorption of protons (reactions 1 and 2) are the same for both complexation models. The total number of the sites capable of forming the surface complexes on the surface, Ns, is given by
Ns ) [SO-] + [SOH0] + [SOH2+] + [SOH2+A-] + [SO-C+] (1) The surface coverages θi’s by the individual surface
(aH)[SOH0] [SOH2+] (aH)[SO-] 0
[SOH ]
{ } eψ0 kT
exp -
{ }
exp -
[SO-C+](aH) 0
[SOH ](aC) [SO-C+](aH) [SOH0](aC)
{
e(ψ0 - ψβ) (TLM) kT
{
e(ψ0 - ψC) (FLM) kT
exp -
exp -
[SOH0](aH)(aA) [SOH2+A-] [SO0](aH)(aA) [SOH2+A-]
eψ0 kT
}
{
}
exp -
{
exp -
}
e(ψ0 - ψβ) (TLM) kT
}
e(ψ0 - ψA) (FLM) kT
complexes (i)0,+,C,A) and free sites (i ) -) will be defined as follows
θ0 )
[SOH0] Ns
θ+ )
[SOH2+] Ns
θA )
θC )
[SOH2+A-] Ns
[SO-C+] Ns
θ- )
[SO-] (2) Ns
According to Figure 1, the surface charge density, δ0, must be proportional to the amount of the following surface complexes
δ0 ) Bs(θ- + θA - θ- - θC) B s ) N se
(3)
but the charges of the specifically adsorbed ions of the basic electrolyte in their planes are different in TLM and FLM. They are respectively given by
δβ ) Bs(θC - θA)
(4)
δC ) BsθC and δA ) -BsθA
(5)
For the whole compact layer, there must be fulfilled the electroneutrality condition (i.e., the sum of all charges must be equal to zero), so for both the models the diffuse layer charge, δd, is given by the same expression
δd ) Bs(θ- - θ+)
(6)
For both complexation models, the relationships between the capacitances, the potentials, and the charges within the individual electric layers were developed in our previous papers.28-30 To arrive at the adsorption isotherm equation, we have to consider an equivalent description of the reactions in Table 1 leading to the adsorption of ions onto so-called free sites [SO-] of the surface of oxide. This is presented in Table 2. The set of the nonlinear equations in Table 2 can be transformed into an equivalent one, having the form of multicomponent Langmuir-like adsorption isotherms of
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Table 2. An Alternative Description of the Surface Reactions Leading to the Adsorption of Ions onto Free Sites, and Their Equilibrium Constants equilibrium constant
surface reaction SO- + H+ T SOH0
K0 )
SO- + 2H+ T SOH2+
1
{ } { } { }
Ka2int 1 Ka1intKa2int
Ka1int Ka2int exp
2eψ0 (aH)2θ) kT θ+
Ka2int/*KCint exp
eψβ (aC)θ) (TLM) kT θC
K+ )
SO- + C+ T SO-C+
equilibrium equation
eψ0 (aH)θKa2int exp ) kT θ0
KC )
*KCint Ka2int
Ka2int/*KCint exp SO- + 2H+ + A- T SOH2+A-
KA )
1 Ka2int*KAint
Ka2int *KAint exp Ka2int *KAint exp
ions θi’s (i)0,+,A,C)
θi )
K if i 1+
∑i Kifi
i ) 0, +, A, C
,
(7)
where Ki values are the equilibrium constants defined in Table 2 and fi values, (i ) 0, +, A, C) are the functions of activity of protons and salt ions in the equilibrium bulk electrolyte. The functions f0 and f+ are the same for both complexation models and have the following form22,28
{ {
} }
f0 ) exp -
eψ0 - 2.3pH kT
f+ ) exp -
2eψ0 - 4.6pH kT
(8a) (8b)
The functions fC and fA are different. For the TLM they are given by22
{
fC ) aC exp -
{
fA ) aA exp -
}
eδ0 eψ0 + kT kTC1
eψ0 eδ0 - 4.6pH kT kTC1
(9a)
}
(9b)
whereas for the FLM they take the more complex form28
{
fC ) aC exp -
{
fA ) aA exp -
}
eδ0 eψ0 + kT kTC+
eδ0 eψ0 eδ* + kT kTC+ kTC+ eδ* - 4.6pH kTC-
(10a)
}
(10b)
where
δ* ) δA + δd ) Bs(θ- - θ+ - θA)
(10c)
{ } { {
eψC (aC)θ) (FLM) kT θC
} }
e(2ψ0 - ψβ) (aH)2(aA)θ) (TLM) kT θA e(2ψ0 - ψA) (aH)2(aA)θ) (FLM) kT θA
The way of solving the set of eqs 7-9 for TLM and the set of eqs 7, 8, and 10 for FLM, to calculate the individual adsorption isotherms θi and the surface charge δ0 as a function of pH, was shown in our previous papers.21,22,28,29 2. Generalization of the Triple- and Four-Layer Models by Considering Adsorption on the Heterogeneous Solid Surface. It is now commonly realized that the crystallography and chemical composition of the actual solid surfaces do not represent an extrapolation of appropriate bulk crystal properties. The really existing solid surfaces are characterized by a more or less decreased crystallographic order, leading also to variations in a local chemical composition. This, in turn, causes the variation in adsorptive properties of adsorption sites, across the surface. That phenomenon is known as the energetic surface heterogeneity for adsorption.32-34 The surface complexation models can be generalized in terms of adsorption on a heterogeneous solid surface, by assuming that the adsorption of ions runs via formation of complexes with various surface oxygens. Because of a small degree of oxide surface organization, different surface oxygens may have a different status with respect to the electrostatic interactions and the strength of the chemical bonds formed with the adsorbed protons or other ions. The fact that the surfaces of the real oxides are geometrically distorted and energetically heterogeneous is crucial for almost all catalytic reactions running on the oxide surfaces. Various experimental techniques have been used to study that important phenomenon. These findings were reported by Somorjai35 and Hirschwald.36 The latter review includes the images of the surfaces of the oxides obtained by scanning tunneling microscopy, which shows the great geometric nonuniformity of the oxide surfaces. The quantitative measure of the degree of the surface (32) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surface; Academic Press: New York, 1992. (33) Jaronic, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (34) Rudzinski, W., Steele, W. A., Zgrablich, G., Eds. Equilibria and Dynamics of Gas Adsorption on Heterogeneous Surfaces, Studies in Surface and Catalysis; Elsevier: Amsterdam, 1997; Vol. 104. (35) Somorjai, G. A. Chemistry in Two Dimensions: Surfaces; Cornell University Press: Itaca, NY, 1981. (36) Horschwald, W. In Non-Stoichiometric Compounds, Surfaces, Grain Boundaries and Structural Defects; Nowotny, J., Wepper, W., Eds.; Kluwer: Dordrecht, 1989; p 203.
Model for Ion Adsorption
Langmuir, Vol. 15, No. 17, 1999 5639
energetic heterogeneity in the model of one-site-occupancy adsorption is the differential distribution χ() of the fraction of surface sites among corresponding values of adsorption energies , normalized to unity in their physical domain Ω. Very often χ() is approximated by a simple analytical function and Ω is assumed to be either (0,+∞) or (-∞,+∞) interval for the purpose of mathematical convenience. Such simplifications do not usually introduce a significant error in the further theoretical calculations, except for some extreme physical regimes.32 The exact function χ() for a real physical surface is expected to have a complicated shape in general. However, to a first crude approximation, it is often approximated by a simple, smooth Gaussianlike function. So, in our calculations we represent χ() by the following Gaussian-like function21-25,32
{ } { }]
1 - 0 exp c c χ() ) - 0 1 + exp c
[
2
(11)
centered at ) 0, the spread (variance) of which is described by the heterogeneity parameter c. (The variance σ is equal to πc/x3). Using such a simple function χ() in practical calculations is necessary to avoid introducing many unknown parameters. That procedure is common for the majority of theoretical works on adsorption on heterogeneous solid surfaces.32 The adsorption of H+ or OH- ions causes protonization or deprotonization of the surface, creating the surface structures SO-, SOH, and SOH2+. In addition, the adsorption of cations C+ and anions A- of inert electrolyte leads to formation of the surface complexes SO-C+ and SOH2+A-, which have the character of ion pairs. Each of these surface complexes has its own adsorption (binding) energy. As discussed previously,22,32 the intrinsic Ki constants can be written as follows
Ki ) Ki′ exp
{ }
i , kT
i ) 0, +, A, C
The experimentally measured adsorption isotherms have to be related to the following averages, θit,
θit({a},T) )
(37) Barrow, N. J.; Gerth, J.; Bru¨mmer, G. W. J. Soil Sci. 1989, 40, 437.
Ω
(13)
where {a} is the set of the bulk concentrations {aH, aC, aA}, {} is the set of the adsorption energies {0, +, A, C}, Ω is the physical domain of {}, and χ({}) is a multidimensional differential distribution of the number of adsorption sites among various sets {}, normalized to unity.The integration result in eq 13 depends on the degree of the correlations between adsorption energies of various complexes on various adsorption sites. In our hitherto studies21-23 we took into account two physical situations: (1) The case of high correlations between adsorption energies of different surface complexes, where the energies i and j*i change on passing from one center to another but their difference ∆ji ) j - i remains unchanged. Therefore the function χ({}) in eq 13 reduces to the onedimensional differential distribution χi(i), the shape of which is the same for all surface complexes. (2) The case of lack of correlations between adsorption energies on various centers. Then the function χ({}) in eq 13 becomes a product of one-dimensional distribution functions, having different shapes for different surface complexes. In both cases we arrived at multicomponent isotherm equations which have been known in gas adsorption theories for some time. In the first case we obtained the generalized Langmuir-Freundlich isotherm,19,20 whose form, derived22,23 for the ion adsorption on the oxide surfaces, is as follows
θjt )
[
Kj0fj
∑j
Kj0fj
∑j Kj0fj]kT/c ∑j
1+[
(14)
Kj0fj]kT/c
j ) 0, +, A, C
(12)
where i is the adsorption (binding) energy of the ith surface complex and Ki′ is related to its molecular partition function. The surface heterogeneity will cause the variation of i across the surface, from one oxygen to another. Then, for the reasons explained in our previous publication,22 we accept the random model of surface topography. According to this model, adsorption sites characterized by different adsorption energies are distributed on a solid surface at random. Of course, certain variations in the local Coulombic force fields ψ0 and ψβ37 are expected to occur, but we will neglect them because the Coulombic interactions are long-ranged. In the case of random topography, this will cause “smoothing” of these variations over the local structure of the outermost surface oxygens. Thus, as in the model of a homogeneous oxide surface, we will consider ψ0 and ψβ to be functions of the average composition of the adsorbed phase. Physisorption interactions (chemical binding forces) are short-ranged, so the local variations in i must be taken into account. The different values of adsorption energies i are accompanied by the changes of Ki′, but it is generally believed that these changes are of secondary importance compared to the changes of the chemical bond energy i.
∫ ... ∫ θi({},{a},T)χ({}) d0 d+ dA dC
where
{
Kj0 ) Kj′ exp
}
i0 + ∆ij kT
(14a)
and where c is the heterogeneity parameter and i0 is the most probable adsorption energy of one of the surface complexes, chosen as a reference complex. The assumption of lack of correlations led us to an equation of the type which has been known in the theories of mixed-gas adsorption on solid surfaces for a long time, and was called the “multicomponent Langmuir-Freudlich isotherm”. Its form, derived21,23 for ion adsorption on the oxide surfaces, reads
[Kj0fj]kT/ci
θit ) 1+
∑i [Ki fi] 0
(15) kT/ci
i ) 0, +, A, C where
Ki0 ) Ki′ exp
{ } i0 kT
(15a)
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papers,22,23 and now we are going to derive such a method for het-FLM. The method of solving a set of eqs 15, 8, and 10 is similar to that described in the homogeneous FLM.28 At first, an attempt is made to reduce the number of the four nonlinear equations in the system. The starting point to calculate the function θit is to solve a set of two nonlinear equations obtained from the set of eqs 3 and 10c and taking into account eqs 15
F1(δ0,δ*) ) Bs
1 - [K+0f+]kT/c+ - [KA0fA]kT/cA 1+
Figure 2. The relief and countur plots of different energy distributions according to Rusch et al.40
and where ci values are the heterogeneity parameters, whose values can be different for different surface complexes, i ) 0, +, A, C. The multicomponent Langmuir-Freundlich equation was used as an empirical equation at the beginning, and its first derivation based on the assumption of a random mutual blocking of the surface sites by competing components was proposed by Rudzinski et al.38,39 This assumption led Rudzinski to the postulate that it reflects the situation when no correlations exist between adsorption energies of various components. Recently, however, the multicomponent Langmuir-Freundlich isotherm has been subjected to a rigorous analysis by Rush et al.40 who showed that the analytical form of eq 15 corresponds, in fact, to the situation when small correlations exist between adsorption energies of various components (complexes). So, now we will use this isotherm, assuming however existence of such small correlations. Figure 2 shows the schematic plots of different energy distributions taking into account the correlations between adsorption energies. One can compare easily the case of small correlations with the fully uncorrelated energy distribution.40 Our hitherto studies seem to suggest that the most realistic model is the one assuming small correlations to exist between the adsorption energies of different complexes. It was proved based on heterogeneous TLM (het-TLM), by analysis of the behavior of potentiometric, electrokinetic, and radiometric data,22 as well as of the bivalent ion adsorption at low ion concentrations.23 Very recently, impressive support for this view comes from our studies of the heats of proton adsorption (titrationcalorimetry).41 So, developing the extended version of FLM for the case of the heterogeneous surfaces (het-FLM) we will consider only the case of small correlations between adsorption energies of different surface complexes. The analytical form of the set of eqs 15 for the individual adsorption isotherms θit remains unchanged for both hetTLM and het-FLM (except for the form of the functions fi, eqs 8 and 9 for het-TLM and eqs 8 and 10 for het-FLM), but the mathematical way of solving these sets to calculate the surface charge δ0 as the function of pH is quite different. For het-TLM it was shown in our previous (38) Rudzin´ski, W. In Chromatographic Theory and Basis Principles; Jonsson, J. A., Ed.; Marcel Dekker: New York, 1988; p 245. (39) Rudzin´ski, W.; Nieszporek, K.; Moon, H.; Rhee, H.-K. Heterogeneous Chem. Rev. 1994, 1, 275. (40) Rusch, U.; Borkovec, M.; Daicic, J.; Van Riemsdijk, W. H. J. Colloid Interface Sci. 1997, 191, 247. (41) Rudzinski, W.; Charmas, R.; Piasecki, W.; Thomas, F.; Villieras, F.; Prelot, B.; Cases, J. M. Langmuir 1998, 14, 5210.
∑i [Ki fi] 0
- δ* ) 0
kT/ci
(16a)
i ) 0, +, A, C F2(δ0,δ*) ) Bs
[K+0f+]kT/c+ + [KA0fA]kT/cA - [KC0fC]kT/cC - 1 1+
∑i
-
[Ki0fi]kT/ci δ0 ) 0 (16b)
i ) 0, +, A, C where fC ) fC(δ0) and fA ) fA(δ0,δ*), to calculate δ* and δ0 for each value of pH. Further transformations (for details see Appendix A) lead to the following equivalent set of equations:
F1′(δ0,δ*) ) 2 + [K00f0]kT/c0 (1 +
∑i
[
[Ki0fi]kT/ci) 1 +
F2′(δ0,δ*) ) [KC0fC]kT/cC + (1 +
∑i
]
2δ* + δ0
[Ki0fi]kT/ci)
Bs
δ* + δ0 Bs
) 0 (17a)
) 0 (17b)
Evaluating (1 + ∑i[Ki0fi]kT/ci) from eqs 17, and next comparing these equations, we calculate δ* ) δ*(δ0)
δ* ) -
δ0(2 + [K00f0]kT/c0 + [KC0fC]kT/cC) + Bs [KC0fC]kT/cC 2 + [K00f0]kT/c0 + 2[KC0fC]kT/cC (18)
After substituting the above relation into eq 16b (variable δ* appears in eq 16b only inside the function fA), this equation becomes an implicit one only with respect to the variable δ0. Then, eq 16b can be numerically solved using one of the iteration methods to obtain δ0 values for each pH. Finally, having calculated the values of surface charge δ0, we calculate individual isotherms of ions θi from eqs 15. 3. Calculation of Surface Potential and ζ-Potential. The crucial points of our calculations are the definitions of the surface potential ψ0 and of ζ potentials. At the beginning, we repeat the IUPAC definitions42 introduced into the description of double electrical layer. (42) Somasundaran, P.; Goddard, E. D. Modern Aspects of Electrochemistry; Plenum Press: New York, 1979; p 207.
Model for Ion Adsorption
Langmuir, Vol. 15, No. 17, 1999 5641
One of them is the point of zero charge (PZC) which is the concentration of potential-determining ions (PDI) at which the surface charge is equal to zero (δ0 ) 0). Another one is the isoelectric point (IEP) defined as the concentration of PDI at which the electrokinetic potential is equal to zero (ζ ) 0). Bousse et al.27,43,44 elaborated a method of experimental determination for the surface potential ψ0 of metal oxide surfaces using in their experiments the so-called chemically sensitive electronic device (CSED) operating on the field-effect principle. The theoretical model for that determination of the surface potential ψ0 elaborated by Bousse et al. was found to be in a good agreement with experiment.43 It was stated that the PZC value, and the difference between pKa2int and pKa1int determined from ψ0(pH) measurements, were in a good agreement with the results of the surface charge measurements for the colloidal suspensions. Another important finding was that the surface potential ψ0 is much less sensitive to the surface complexation with the counterions than the surface charge δ0. The theoretical approach to this problem was first limited to the application of the Nernst equation (ψNernst) which predicts the potential change equal to 2.3 kT/e volts for a pH unit. This equation is satisfied by a glass electrode as the most popular device for pH measurements in electrochemistry. In turn, in the case of CSED device, the potential change value was much below the Nernst value. Therefore, the Nernst equation cannot be applied for the insulator such as a metal oxide, which is neither an electric nor ionic conductor.43 The surface potential defined in the equations for the equilibrium constants collected in Table 1 can be theoretically calculated as ψ0(pH), from the equation developed by Bousse et al.
( )
eψ0 eψ0 + sinh-1 2.303(PZC - pH) ) kT βkT
4. Relations between Equilibrium Constants of the Surface Reactions. Except for a few rare cases, the experimental titration curves corresponding to different concentrations of the basic (inert) electrolyte have a common intersection point (CIP) in PZC (PZC ) pHδ0)0,ψ0)0 ) -log H, where H is the activity of protons in the bulk solution at PZC). It means that the PZC value for a given system of oxide and electrolyte does not practically depend on salt concentration in the bulk solution. We were first to draw the reader’s attention to the fact, that such independence of PZC of salt concentration can be formally expressed as the set of two equations when we assume that a ) aC ) aA22
δ0(pH)PZC) ) 0
β)
( )
2e Ns Ka2int 1/2 CtkT K int a1
∂δ0(pH)PZC) ) 0 (21) ∂a
The above formal criterion applied for each investigated complexation model (TLM, FLM, het-TLM, and het-FLM) decreases by two the number of the equilibrium constants Ka1int, Ka2int, *KCint, *KAint. Such interrelations are very important because they reduce the number of the independent equilibrium constants (best-fit parameters) determined from fitting suitable experimental data. 4.1. Homogeneous Triple-Layer Model (TLM). The interrelations between the equilibrium constants inherent in TLM have been known in the literature for a long time.5,6 They are very easy to obtain without criterion 21
H2 ) Ka1int Ka2int
and
H2 ) *KCint *KAint (22ab)
Their logarithmic forms are the following:
(19)
where the dimensionless quantity β has the form43 2
and
(20)
and where Ct is the linear capacitance of the double electrical layer that can be calculated in the way proposed by Bousse et al.43,44 Please note, that the quantity β is not a best-fit parameter but the function of equilibrium constants of proton adsorption onto the surface. As pointed by Bousse et al.,43 the simplified, linear form of eq 19 (valid in the vicinity of PZC), i.e., ψ0 ) (β/(β + 1))ψNernst, shows, that for the possible relatively large values of β, β/(β + 1) = 1, the potential change corresponding to one pH unit becomes equal to that predicted by the Nernst equation. The next experimental source of information is the electrokinetic effects accompanying the formation of the electric double layer. The ζ-potential measurements are carried out very frequently and their theoretical interpretation is apparently simple but often leads to a poor agreement between theory and experiment. In our calculations we applied the equations outlined in our previous publications,28 obtained from the solution of the Poisson equation for the Stern model for 1:1 electrolyte.
1 PZC ) (pKa1int + pKa2int) 2
(23a)
1 PZC ) (p*KCint + p*KAint) 2
(23b)
where pKaiint ) -log Kaiint, i ) 1, 2 and p*Kiint ) -log *Kiint, i ) C, A. The same interrelations 23 are obtained by applying criterion 21. However, for more complicated models such as FLM, het-TLM, and het-FLM, the interrelations can be obtained using only criterion 21 because of the complex form of δ0 (pH)PZC) ) 0 equation. 4.2. Homogeneous Four-Layer Model (FLM). So, using that criterion we arrived in our previous papers28,29 at the following relations for FLM
H2 ) Ka1int Ka2int Y*
and
1 X* (24ab)
H2 ) *KCint *KAint
or in the logarithmic form
1 PZC ) (pKa1int + pKa2int - log Y*) 2
(25a)
1 PZC ) (p*KCint + p*KAint + log X*) 2
(25b)
where (43) Bousse, L.; De Rooij, N. F.; Bergveld, P. IEEE Trans. Electron Devices 1983, 30, 1263. (44) Van den Vlekkert, H.; Bousse, L.; De Rooij, N. F. J. Colloid Interface Sci. 1988, 122, 336.
X* ) P* + a
∂P* ∂a
(26a)
5642 Langmuir, Vol. 15, No. 17, 1999
∂P* ∂a K
Y*) a2
P* ) exp δ* ) -Bs
H2 int int a2 *KA
Charmas
+1
(26b)
)}
(26c)
{ (
1 eδ* 1 kT C- C+
KCa )2 + K0H + 2KCa Bs
(
*KCinta 2Ka2int + H + 2*KCinta
2 + K0H 1 ∂P* ) P* ln P* + ∂a a 2 + K0H + 2KCa
(
) P* ln P*
1 a 2K
(
d ln
(26d)
)
1 1 C- C+ da
)
2Ka2int + H int a2
+ + H + 2*KCinta 1 1 d ln C- C+ da
(
)
eqs 27 for the het-TLM model:
)
(26e)
Determination of two equilibrium constants from eqs 25ab for given values of two others is not as simple as in the case of the TLM. For example, determination of Ka1int and *KAint for given Ka2int and *KCint is as follows: First the value *KAint is determined from eq 25b keeping in mind that X* depends on Ka2int. Having evaluated the value of *KAint, eq 25a is used next to calculate Ka1int employing the fact that Y* depends on *KAint. Of course, we can determine Ka1int and *KCint from the given (starting) values of Ka2int and *KAint, but it is impossible to use these equations for the starting pairs Ka1int and *KCint or Ka1int and *KAint. When we compare eqs 25ab for FLM and eqs 23ab for TLM we can see that when X* f 1 and Y* f 1, the reduction of FLM into TLM takes place. It occurs when C- ) C+, then P* f 1 and ∂P*/∂a f 0. 4.3. Heterogeneous Triple-Layer Model (het-TLM). Applying criterion 21 for this model, we arrived22 at the following set of equations:
(
) ( kT/c+
H2
Ka1intKa2int
(
+
kT/cC -1 kT/cA
)
H2a kT int cA K *KAint a2
)( ) ( ) int
*KC a
int
kT/cA
-
kT/cC
Ka2int
kT *KC a cC K int a2
Figure 3. Schematic picture of the reduction conditions for the investigated models.
-1)0 (27a)
kT/cC
)0 (27b)
For given values of the heterogeneity parameters kT/ci (i ) 0, +, A, C) and some given values of Ka2int and *KCint, one can easily calculate first the value of Ka1int from eq 27a and then *KAint from eq 27b. As for other starting pairs, the values of equilibrium constants can be calculated by using: (Ka1int; *KCint) and (Ka2int; *KAint). However, calculations become impossible for the starting values (Ka1int; *KAint). When we assume that the heterogeneity parameters kT/ci ) 1 (lack of energetic heterogeneity), we obtain the equation set equivalent to 22 for TLM. 4.4. Heterogeneous Four-Layer Model (het-FLM). We have never used criterion 21 to the het-FLM. Using the method described indetail in refs 21, 22, and 28, we obtain two equations which are very similar to the set of
(
)
H2 int Ka1 Ka2int
(
kT/cC kT/cA
(
kT/c+
+
1 -1 a ∂P* 1+ P* ∂a
) (
H2aP* kT cA K int *K int a2 A
)( )
kT/cA
1+
*KCinta
kT/cC
- 1 ) 0 (28a)
Ka2int
)
a ∂P* P* ∂a
( )
int kT *KC a cC K int a2
kT/cC
) 0 (28b)
where the general form of P* is the same as that in FLM (eq 26c), but now
δ* ) -Bs
(
(KC0a)kT/cC
(29a)
2 + (K00H)kT/c0 + 2(KC0a)kT/cC
2 + (K00H)kT/c0 kT/cC ∂P* ) P* ln P* + ∂a a 2 + (K 0H)kT/c0 + 2(K 0a)kT/cC 0 C 1 1 d ln C- C+ (29b) da
(
)
)
To verify the methods of calculating the equilibrium constants values for all models, for a given set of heterogeneity parameters kT/ci (i ) 0, +, A, C) we assume Ka2int and *KCint to be the starting values, calculating next Ka1int from eq 28a, and then *KAint from eq 28b. Because the het-FLM is the most complex model of those (it has more general assumptions), it should reduce for certain values of parameters to all the other models. These reduction conditions are presented in Figure 3. Contrary to TLM, other investigated models show that relations between the parameters are not independent of the concentrations of the electrolyte ions.This, at a first view, might be considered as a nonphysical result. However,our numerical exercises showed that the concentrations of anions and cations affect only slightly the calculated values of two parameters, when the two others are fixed. The explanation for that is as follows: strictly speaking, CIP does not exist. The experimentally observed
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Langmuir, Vol. 15, No. 17, 1999 5643 Table 3. Mathematical Relations between PZC and IEP
complexation model
interrelation between IEP and PZC IEP ) PZC IEP ) PZC + 1/2(β + 1) log Y*
TLM FLM het-TLM
het-FLM
1 - K+f+ 1+
∑i Kifi
IEP ) PZC +
kT/cC 1 c+ (β + 1) log 1 2 kT kT/cA
)0
(30a)
and for het-TLM and het-FLM
δd ) Bs(θ- - θ+) )
1 - (K+0f+)kT/c+ 1+
∑i (Ki fi) 0
) 0 (30b)
kT/ci
from which the conditions K+f+ ) 1 and (K+0f+)kT/c+ ) 1 follow, respectively. But both these conditions give the same fi/(Ka1intKa2int) ) 1, because in the second case kT/c+ * 0. While assuming that pH ) IEP in eq 8b we have
IEP ) -
1 eψ0 1 ln Ka1intKa2int 4.6 2.3 kT
(31)
To express the dependence of the surface potential ψ0 on pH, we can use22 the simplified linear form of eq 19, because the surface potential is calculated at the point pH ) IEP, which is usually close to PZC. After a certain rearrangement in eq 31, we obtain the relationship valid for all investigated models
IEP ) - (β + 1)
)
kT/cC 1 c+ (β + 1) log 1 - 1 (KCa)kT/cC 2 kT kT/cA
CIP is, as a matter of fact, a set of closely lying points which, at the present standard of accuracy of titration experiments, are commonly treated as one CIP. Thus, relations 25, 27, and 28 should be treated as a way of estimating the interrelations of intrinsic equilibrium constants to a degree of accuracy which seems to be satisfactory at the present accuracy of titration experiments. Using these relations is very essential to have more confidence in the elucidated values of these parameters from fitting numerically the theoretical expressions to the experimental data. Scientists involved into such computer exercises know how quickly their reliability decreases with the growing number of best-fit parameters. 4.5. Relationship between PZC and IEP in the Investigated Models. Theoretical interpretation of ζ-potential measurements often leads to a poor agreement between theory and experiment. The only quantity resulting from the electrokinetic measurements, the physical meaning of which is not model dependent, is the isoelectric point (IEP). The existence of the isoelectric point IEP is predicted only for the situation when the charge density in the d-plane is equal to zero. Remembering that θ- ) 1 - ∑iθi (i ) 0, +, A, C), from eq 6 we obtain for TLM and FLM
δd ) Bs(θ- - θ+) )
[ (
IEP ) PZC +
1 ln Ka1intKa2int - β PZC (32) 4.6
Taking into account that the product Ka1intKa2int is different
[ (
]
)
]
1 - 1 (KCa)kT/cC a ∂P* 1+ P* ∂a
for the investigated models and that we can express it using relations 22a, 24a, 27a, and 28a for TLM, FLM, het-TLM, and het-FLM, respectively, one can obtain the interrelations between PZC and IEP collected in Table 3. We believe, that this equation developed by us is of fundamental importance, because of the widely spread recent view that the difference between PZC and IEP is a common feature of these adsorption systems. Discovering that difference as a common feature is the result of the recent growing accuracy of experiments. In TLM, the simpler form of interrelations 25ab led to the conclusion that no difference between IEP and PZC should be observed in the systems exhibiting CIP. That conclusion has turned out to be in contradiction with the still growing body of more and more accurate data, on PZC and IEP reported by various authors.45-48 In TLM, different values of PZC and IEP could be predicted by certain sets of values of the four intrinsic constants treated as free best-fit parameters, but these sets will not satisfy relations 22ab. Developing these relations made it possible to show rigorously that in the TLM, having CIP, also PZC and IEP must be equal. Then, it was one of the fundamental discoveries made on the basis of FLMs, that this is the dependence of the electrical capacitances C- and C+ on the bulk electrolyte activity, which is the main source of the differences between PZC and IEP. In TLM the discovered experiment dependence of the capacitance C1 on the activity was a troubling problem which could not be explained in terms of that model. It means that contrary to the FLM, the derivative dC1/da does not appear in the theoretical considerations based on TLM. Thus, using different values C1 for various concentrations of the electrolyte is not justified in TLM but nevertheless there are assumed various values of C1 depending on the concentration of the inert electrolyte.49 This results from the fact that C1 is proportional to the interfacial dielectric constant of water which depends on the ionic strength. Proceeding in this way we assume in an artificial way that the same oxide/electrolyte type systems differing only in the concentration of the inert electrolyte are completely individual systems. Only in FLM does using different capacitance values arise in a natural way from the fundamental mathematical treatment. In the final part of this section we will discuss other best-fit parameters appearing in our calculations. Our TLMs (TLM and het-TLM) assume two values of the capacitance C1 as the best-fit parameters, depending on the sign of the charge of the surface49 (45) Smit, W.; Holten, C. L. M. J. Colloid Interface Sci. 1980, 78, 1. (46) Kallay, N.; Babicˇ, D.; Matijevicˇ, E. Colloids Surf. 1986, 19, 375. (47) Wood, R.; Fornasiero, D.; Ralston, J. Colloids Surf. 1990, 51, 389. (48) Thomas, F.; Bottero, J. Y.; Cases, J. M. Colloids Surf. 1989, 37, 281. (49) Blesa, M. A.; Kallay, N. Adv. Colloid Interface Sci. 1988, 28, 111.
5644 Langmuir, Vol. 15, No. 17, 1999
Charmas
C1 ) C1L for pH < PZC C1 ) C1R for pH > PZC
(33)
The capacitance C1 is connected with the distance between the surfacelayer “0 ” and the layer “β” where cations and anions of the inert electrolyte are located. For pH < PZC, i.e., in the acidic medium, there is much more of SOH2+Acomplexes formed by the anion adsorption, but when pH > PZC, SO-C+ complex is strongly predominant. The discovered two different values of parameter C1 suggest that the adsorbed anion and adsorbed cation are situated at different distances from the surface. So, comparing the TLMs with the FLMs, one can see that these capacitances C1L and C1R have their counterparts in the FLMs. When anion adsorption prevails (pH < PZC), the parameter Ccan be identified with the parameter C1L, but when cation adsorption prevails (pH > PZC), C+ corresponds to C1R. Besides setting in order and making real effects, using the four-layer model provides continuity of the functions describing physicochemical quantities in the whole pH range. This did not occur in the TLMs when C1 was equal to C1L and in another pH range to C1R. Results and Discussion One of the interesting results of using the models which are the refinement of the “classic” TLM is the possibility of theoretical prediction of the inequality of PZC and IEP values for a given oxide. Such a result can be obtained for the electrolyte 1:1 ions in the system (without invoking specific adsorption), using the relations collected in Table 3. The modeling exercises of calculating the IEP value for the FLM (equation in the second row of Table 3) are presented in our previous paper.29 Now, we start our analysis with the relation for the het-TLM, which is specified in the third row of Table 3. This relation, though using the model in paper,22 was developed for the first time here. Performing the analysis of this equation, strictly from the mathematical point of view, we can see that equality of IEP and PZC can be obtained only in the case when the heterogeneity parameters kT/cC and kT/cA, devoted to the cation and anion adsorption, are not only equal to unity (reduction of het-TLM into TLM) but also equal in general. Moreover, it can be clearly seen in this instance that kT/cC > kT/cA implies IEP < PZC, otherwise kT/cC < kT/cA implies IEP > PZC. The analysis of the relation linking IEP and PZC for the het-FLM is not as simple as the above despite the structurally similar form of the third and fourth equations in Table 3. Equation 4 is very complex in the respect of such an analysis. We can notice, making the modeling computer exercises, that a shift of the PZC and IEP values in the opposite directions can be caused by not only different kT/cC and kT/cA values (the same directions of the shifts as in the case of het-TLM) but the values of parameters (C+, C-, and d ln((1/C-) - (1/C+))/da strictly related to the four-layer modeling, as well. The situation is rather complicated as for different sets of the het-FLM model parameters the same results of calculations of the PZC and IEP relations can be obtained. Due to the fact that the parameters in the equation describing the difference between IEP and PZC are correlated to a great extent, we are not able to determine the values of those parameters in an objective way only knowing the experimental values of PZC and IEP for a given system. For a complete analysis there is needed a set of experiments differing in methodology of measure-
ments and providing independently some information about the electolyte/metal oxide system under the investigation. The set which may be composed of potentiometric titration curves, individual adsorption isotherms of cation and anion of the inert electrolyte, electrokinetic curves, and possibly results of calorimetric experiments made with the maximum precision under stable conditions should be subjected to the theoretical analysis. Also additional information obtained using spectroscopic methods should be taken into account. Such precision results also from the fact, as pointed by Kosmulski50 making the comparison of PZC values selected by different authors, that there is the difference of about 0.1 pH unit between PZC values at 20 and25 °C. Such an effect has not received proper attention in the experimental results. Some attempts are made to verify the existing literature data to eliminate the deviations caused by impurities or an inadequate experimental procedure.50 There is a quite common belief that if the simpler model (with a smaller number of parameters) described a given type of the experiment, it is sufficient evidence for its superiority over the model possessing more realistic and precise assumptions (but more parameters), which is often a direct development of the simpler model. But we have to realize that if in both such models the definitions of parameters are in agreement and we obtain their different values as the best-fit results, then undoubtedly the results obtained for the more complex model are more realistic. Of course, it should be taken into consideration if our results are good enough to justify application of a more complex model. Taking into account the assumptions of the models compared in this paper, it is obvious that the het-FLM is the most accurate. Moreover, all models include the same definitions of parameters: surface reaction equilibrium constants which make their comparison easy. The set of experimental results described at the beginning of the Discussion and required in the theoretical analysis is not available in the literature. Therefore, the paper presents the behavior of the elaborated het-FLM model with its prospective application in the actual adsorption system. The system TiO2 (anatase)/NaCl reported by Sprycha51 was chosen as an adsorption one. Sprycha analyzed the experimental titration curves for this system theoretically using the graphical extrapolation method for solving TLM equations. The parameters pKa1int, p*KCint, C1L, and C1R obtained by him by fitting these equations to the experimental titration curves are collected in the first column of Table 4. The theoretical equations of TLM with these values of the parameters generate the theoretical titration curve which matches accurately the experimental titration data for this system. So, these parameters were taken as a starting point for the analysis to point out the difference between the investigated models in the model exercises. The results of calculations are shown in Figures 4 and 5. Comparing the behavior of the TLM and FLM models in Figure 4B, there can be seen differences between the values of IEP predicted by corresponding sets of parameters in Table 4 (for TLM: IEP ) PZC and for FLM: the calculated IEP ) 5.62). In turn (what is not shown in the figures), for both heterogeneous models, if the heterogeneity parameters kT/ci (i ) 0, +, A, C) are equal to unity, then according to the diagram presented in Figure 3, the calculations give the same results for TLM and het-TLM as well as for FLM and het-FLM. (50) Kosmulski, M. Langmuir 1997, 13, 6315. (51) Sprycha, R. Habilitation Thesis, Maria Curie Skłodowska University, Lublin, 1986.
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Langmuir, Vol. 15, No. 17, 1999 5645
Table 4. Parameters for the Anatase/NaCl System Established in Such a Way That the Related Model Calculations Should Show the Difference between the Physical Functions Calculated on the Basis of Investigated Models, Respectivelya TLM pKa2int ) 8.00 p*KCint ) 5.80 C1L ) 1.10 F/m2 C1R ) 1.20 F/m2
het-TLM pKa2int ) 8.00 p*KCint ) 5.80 C1L ) 1.10 F/m2 C1R ) 1.20 F/m2
FLM
het-FLM
Parameters pKa2int ) 8.00 p*KCint ) 5.80 C- ) 1.10 F/m2 C+ ) 1.20 F/m2
kT kT ) ) 0.8 c0 cC
d ln
(
)
1 1 /da ) 0 C- C+
kT kT ) ) 0.8 c+ cA
pKa2int ) 8.00 p*KCint ) 5.80 C- ) 1.10 F/m2 C+ ) 1.20 F/m2
kT kT ) ) 0.8 c0 cC kT kT ) ) 0.8 c+ cA
(
d ln
from eqs 23 pKa1int ) 4.00 p*KAint ) 6.20
from eqs 27 pKa1int ) 4.00 p*KAint ) 6.20
Values Calculated from eqs 25 pKa1int ) 3.95 p*KAint ) 6.27
)
1 1 /da ) 0 C- C+
from eqs 28 pKa1int ) 3.88 p*KAint ) 6.35
a The parameters pK int and p*K int have the same values as those found by Sprycha51 for TLM, whereas the parameters pK int and a2 C a1 p*KAint were calculated by applying our criterion 21 to all models, respectively. The calculation corresponded to the physical situation when 2 the concentration of the basic electrolyte NaCl is equal to 0.01 M. The value of Ns ) 12 sites/nm was assumed according to Sprycha.51
Figure 4. Comparison of the investigated models: het-FLM (darker dash); FLM (lighter dash); TLM (- - -); het-TLM (‚‚‚) for the parameters collected in Table 4: (A) theoretical titration curves, δ0 vs pH; (B) theoretical ζ-potential curves.
As an example of kT/ci * 1 for both heterogeneous models, the theoretical curves generated while assuming kT/ci ) 0.8 are presented in Figures 4 and 5. On the basis of the earlier analysis of the equations in Table 3, one can see in Figure 4B that equality of IEP values for the TLM and het-TLM is directly connected with the equality kT/ cC ) kT/cA. Despite the equality of IEP for both these models, the course of ζ-potential curves is different.
Figure 5. Comparison of the investigated models: het-FLM (darker dash); FLM (lighter dash); TLM (- - -); het-TLM (‚‚‚). (A) Theoretical individual adsorption isotherms of θC and θA. (B) Theoretical individual adsorption isotherms of θ+ and θ-. All the theoretical curves were calculated for the parameters collected in Table 4.
Contrary to this, the assumption of the equality kT/cC ) kT/cA is not a sufficient factor to obtain IEP ) PZC in the het-FLM model. The calculated value IEP for a set of parameters in Table 4 is 5.19. As shown in our previous publication,28 a new parameter, d ln((1/C-) - (1/C+))/da, occurring only in the four-layer models has a significant
5646 Langmuir, Vol. 15, No. 17, 1999
Charmas
Figure 6. Theoretical ζ-potential curves calculated for hetFLM for different values of the parameter Z: (1) 0 Å; (2) 10 Å; (3) 20 Å. The other parameters are those collected in Table 4.
effect on the position of IEP in relation to PZC. Thus, for example, for the set of parameters in Table 4 (for the hetFLM), when d ln((1/C-) - (1/C+))/da ) -80, then IEP ) PZC, and IEP ) 5.60 for its value equal -40. Figure 4A is more evidence that the potentiometric titration curve is insensitive to the selection of the model and that many different sets of parameters for a given model make it possible to obtain the identical results of numerical calculations as was pointed out in our previous publications.22,25,28,29 This fact is of significant importance, as can be seen in Figure 5, the individual isotherms of ion adsorption, θi (i ) +, -, A, C), have a quite different course depending on the model under investigation. The “compensation effects” clearly seen in eq 3 affect such behavior of potentiometric titration curves (composite adsorption isotherms). The analysis of the experimental values θC and θA is essential in theoretical studies. Taking into consideration the analysis of ζ-potential curves presented in Figure 4B again, we can see that the slope of the curves obtained for both heterogeneous models has a much greater value. One should mention here a theoretical approach while introducing a new quantity, Z, as a distance of the shear plane from the diffuse layer used in the literature52-54 and in our calculations.22 Figure 6 shows the effect of this parameter on the curve obtained for the het-FLM model while making calculations for the parameters in Table 4. It should be added that the effect of Z on ζ-potential curves for other models is identical. Introducing Z parameter to the theoretical calculations, one should take into account the statement made by O’Brien55 that the model that chemists use to convert experimentally determined mobility to ζ-potential should be examined critically. The ζ-potential is expected to have higher absolute values56 than those obtained using the well-known Smoluchowski formula or related formulas for recalculating mobility.57 According to the currently considered het-TLM and hetFLM models, assuming small correlations between adsorption energies of various surface complexes, we may treat kT/c0, kT/c+, kT/cC, and kT/cA as independent free (52) Harding, I. H.; Healy, T. W. J. Colloid Interface Sci. 1985, 107, 382. (53) Kallay, N.; Tomic, M. Langmuir 1988, 4, 559. (54) Tomic, M.; Kallay, N. Langmuir 1988, 4, 565. (55) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (56) O’Brien, R. W.; Rowlands, W. N. J. Colloid Interface Sci. 1993, 159, 471. (57) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Marcel Dekker: New York, 1997; Chapter 12.
Figure 7. Theoretical δ0(pH) (part A) and ζ(pH) (part B) curves calculated for het-FLM for different values of the parameters kT/ci: kT/c0 ) kT/cC ) 0.8 and kT/c+ ) kT/cA ) 0.8 (s); kT/c0 ) kT/cC ) 0.7 and kT/c+ ) kT/cA ) 0.8 (- - -); kT/c0 ) kT/cC ) 0.8 and kT/c+ ) kT/cA ) 0.7 (‚‚‚). The other parameters are those collected in the fourth column of Table 4.
parameters. However, we also realize, that decreasing the number of such freely chosen parameters would result in growing confidence of our exercises. So, we decreased the number of the free parameters from four to two by accepting the following rationalization. The energetic heterogeneity of surface oxygens should affect in a similar way the surface complexes containing two protons. So, we put kT/c+ ) kT/cA in our calculations. Then, we assumed also, that the energetic heterogeneity will affect, in a pretty similar way, the adsorption of the first proton, and the adsorption of cations carrying the same charge. So, we put kT/c0 ) kT/cC. In that way we reduced the number of freely chosen heterogeneity parameters from four to two. Figures 7 and 8 show the results of our computer exercises, carried out by accepting that kT/c0 ) kT/cC and kT/c+ ) kT/cA for het-FLM. The theoretical curves correspond to three situations: (1) kT/c0 ) kT/cC ) kT/c+ ) kT/cA; (2) kT/c0 ) kT/cC < kT/c+ ) kT/cA; (3) kT/c0 ) kT/cC > kT/c+ ) kT/cA. As for Figure 7A, it shows that the titration curves are less sensitive for the changes of the heterogeneity parameters compared to the other results presented in Figures 7B and 8A,B. There is only a small difference which can be leveled by very slight changes in the values of capacitances C- and C+.The influence of the kT/ci values on the behavior of het-TLM is not shown but generally is of the same direction but not of the same extent as for the difference between the situations 1, 2, and 3 under investigation. Table 5 presents the influence of independently chosen kT/ci parameters on the calculated value of IEP for hetFLM, compared to the situation in which all kT/ci values are equal to 0.8. The calculations were performed changing
Model for Ion Adsorption
Langmuir, Vol. 15, No. 17, 1999 5647
values of ∆pKint ) pKa2int - pKa1int with the experimental points of zero charge. The developed expressions for the four-layer model taking into account the energetic heterogeneity of the surface predict the difference between PZC and IEP values, similar to the homogeneous four-layer model or the triplelayer model including energetical heterogeneity effects. This new theoretical approach leads to the adsorption equations which are relatively easy to use like those developed for the popular triple-layer model, i.e., in both cases the simple computer program for performing calculations is needed. The new theoretical model of FLM taking into account the heterogeneity of the real oxide surfaces will be presented in the forthcoming publication where the present theory will be extended further to handle the temperature dependence of the adsorption data and the calorimetric effects accompanying ion adsorption.Then it would be possible to analyze not only the behavior of adsorption isotherms in one temperature and the related electrokinetic behavior but also temperature dependence of adsorption and independently monitored calorimetric effects. Only such independent experiments make it possible to verify theoretical expressions with a sufficient degree of confidence.
Figure 8. Theoretical individual adsorption isotherm curves: (A) θC and θA and (B) θ+ and θ-, calculated for het-FLM for different values of the parameters kT/ci: kT/c0 ) kT/cC ) 0.8 and kT/c+ ) kT/cA ) 0.8 (s); kT/c0 ) kT/cC ) 0.7 and kT/c+ ) kT/cA ) 0.8 (- - -); kT/c0 ) kT/cC ) 0.8 and kT/c+ ) kT/cA ) 0.7 (‚‚‚). The other parameters are those collected in the fourth column of Table 4. Table 5. The Influence of the Independent Values of Heterogeneity Parameters, kT/ci (i ) 0, C, +, A), on the Position of IEP Value Calculated for the het-FLMa no.
kT/c0
kT/cC
kT/c+
kT/cA
IEP
∆IEPrelativeb
1c 2 3 4 5 6c 7
0.8 0.7 0.8 0.8 0.8 0.7 0.8
0.8 0.8 0.7 0.8 0.8 0.7 0.8
0.8 0.8 0.8 0.7 0.8 0.8 0.7
0.8 0.8 0.8 0.8 0.7 0.8 0.7
5.19 4.73 6.08 5.09 4.37 5.74 4.22
+0.46 -0.89 +0.10 +0.82 -0.55 +0.97
a
The other related parameters are collected in Table 4. b ∆IEP relative ) IEPfor the 1st row of the table - IEPfor a given row of a table. c The influence of this set of kT/c values is shown in Figures 7 and i 8.
any time each kT/ci value by 0.1. The calculated ∆IEPrelative values show that such influences are very complex. Within the pairs assumed before as equal, kT/c0 ) kT/cC or kT/c+ ) kT/cA, the “force” of changes of the IEP value is different and even in the case of kT/c0 and kT/cC behaves in opposite directions on pH axis.
Acknowledgment. The author expresses his gratitude to Professor W. Rudzin´ski for helpful discussions. This research was supported by KBN Research Grant No. 3 T09A 078 11. Also, the author wishes to express his thanks and gratitude to the Foundation for Polish Science for the grant making his one-year stay in the Laboratoire Environnement et Mineralurgie, E.N.S. Geologie, CNRSUMR 7569, Vandoeuvre les Nancy, France, possible. Appendix The two equations of equation system 16 are multiplied on both sides by (1 + ∑i[Ki0fi]kT/ci)/Bs, to obtain the following equivalent set of equations for i ) 0, +, A, C:
F1(δ0,δ*) ) 1 - [K+0f+]kT/c+ δ* [KA0fA]kT/cA - (1 + Bs
∑i [Ki0fi]kT/c ) ) 0 i
F2(δ0,δ*) ) [K+0f+]kT/c+ + [KA0fA]kT/cA δ0 [KC0fC]kT/cC - 1 - (1 + [Ki0fi]kT/ci) ) 0 (A.1b) Bs i
∑
Then eqs A.1a and A.1b are added to each other on both sides, and the obtained one is multiplied by (-1) on both sides, to yield the following:
[KC0fC]kT/cC + (1 +
δ* + δ0
∑i [Ki0fi]kT/c ) i
)0
Bs
(A.2)
Next, [K00f0]kT/c0 is added to both sides of eq A.2:
Conclusions As follows from the model studies, the het-FLM model describing the real oxide surface most accurately requires for its theoretical analysis a set of many independent experiments carried out with maximum precision. Furthermore, it should be essential to calculate the surface protonation constants pKa1int and pKa2int using the Sverjensky and Sahai method9,10 combining the predicted
(A.1a)
[K00f0]kT/c0 + [KC0fC]kT/cC(1 +
δ* + δ0
∑i [Ki0fi]kT/c ) i
)
Bs
[K00f0]kT/c0 (A.3) Subsequently, eq A.1a is subtracted from the above A.3, which leads to the following result:
5648 Langmuir, Vol. 15, No. 17, 1999
2δ* + δ0
∑i [Kifi]kT/c - 1 + (1 + ∑i [Kifi]kT/c ) i
Charmas
i
)
After certain rearrangement of eq A.5, one obtains:
Bs [K00f0]kT/c0 (A.4)
2 + [K00f0]kT/c0 - (1 +
To both sides of eq A.4 number 2 is added:
1+
2δ* + δ0
∑i [Kifi]kT/c + (1 + ∑i [Kifi]kT/c ) i
i
)
Bs 0
[K0 f0]kT/c0 + 2 (A.5)
∑i
[
[Kifi]kT/ci) 1 +
]
2δ* + δ0 Bs
)0 (A.6)
These transformations lead to a new set of equations equivalent to set 16, including eqs A.2 and A.6, which are rewritten as equation system 17 in this paper. LA980880J
