Langmuir 1995,11, 3199-3210
3199
Four Layer Complexation Model for Ion Adsorption at Electrolyte/Oxide Interface: Theoretical Foundations Robert Charmas, Wojciech Piasecki, and Wadyslaw Rudzinski” Department of Theoretical Chemistry, Faculty of Chemistry, Maria Curie Sktodowska University, PI. Marii Curie Sktodowskiej 3, 20-031 Lublin, Poland Received January 20, 1995. I n Final Form: May 16, 1995@ Adsorption equations are developed for the four layer model assuming that the anions and cations of the basic electrolyte are adsorbed in two distinct layers, located at different distances from the surface. The theoretical development has been based on the ideas first launched by Yates and Chan, and then elaborated in detail by Leckie and Morel, for the case of the triple layer model. Among various advantages of our new theoretical approach, the continuity of the electrical capacitances as pH functions is a very essential one. The developed expressions predict correctly what is found in the experiment, i.e., that the electrical capacitanciesdepend on the concentrationof the basic electrolyte. Our new theoretical approach predicts the differencebetween PZC and IEP values to be one of the fundamental features of these adsorption systems. This new theoretical approach leads to adsorption equations which are almost as simple and easy to use as those developed for the popular triple layer model.
Introduction The surface complexation model has appeared to be the most successful and commonly used approach to describe ion adsorption within the electric layer formed a t oxide/ electrolyte interface. This interface is probably the most important one for life on our planet, and for science and technology. Dozens of papers have been published reporting on the successful application of that approach to describe the behavior of a variety of oxidelelectrolyte adsorption systems, followed by several reviews and monographs. Of all the editions of the surface complexation approach, the so-called “triple layer model” is the most commonly used. The triple layer model was based on the ideas of Yatesl and C h a n 2 A detailed description and examples of its application were given first in the papers by Davis and L e ~ k i e and ~ - ~by Morel et a1.6 Despite the great success of the “triple layer model”, some suggestions are being made more and more frequently that this popular approach needs certain modifications. The assumption that anions and cations are located within the same electrical layer is often seen as much too crude. This is not only due to the usually different dimensions of anions and cations. A much different status of anions and cations arises also from the fact that cations create real chemical bonds to the outermost surface oxygens, whereas anions are kept by the doubly protonated oxygens through simple electrostatic interactions. There is still another problem accompanying applications of the triple layer model which does not affect strongly the fit by the related theoretical expressions of a most commonlystudied experimental data but may have some serious impact on fundamental studies.
* To whom correspondence should be addressed. Abstract published in Advance ACS Abstracts, July 15, 1995. (UYates, D. E.; Levine, S.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1 1974,70, 1807. (2) Chan, D.; Perram, J. W.;White,L. R.;Healy,T. W. J.Chem. Soc., Faraday Trans. 1 1976,71, 1046. (3)Davis, J. A,; James, R. 0.;Leckie, J. 0. J. Colloid Interface Sci. 1978,63,480. (4)Davis, J. A.; Leckie, J. 0. J.Colloid Interface Sci. 1978,67,90. (5)Davis, J . A.; Leckie, J. 0.In Chemical ModellingAqueous Systems; Jenne, E. A., Ed.; American Chemical Society: Washington, DC, 1979; Chapter 15. (6) Morel, F. M. M.; Yested, J. G.; Westall, J. C. In Adsorption of Inorganics a t Solid-Liquid Interfaces; Anderson, M. A., Rubin, A. J., Eds.; Ann Arbor Science: Ann Arbor, MI, 1981; Chapter 7. @
This is the assumed discontinuity of the electric capacity a t pH = PZC. This, along with some other reasons, has led to the modification of the triple layer model to what is now commonly called the “four layer model”. The idea of the four layer model was introduced to literature by Bowden et al.7,sas well as in the papers by BZUTOW.~J~ The primary interpretation covered the studies of adsorption of bivalent metal ions (Zn2+,Cu2+,Cd2+, Pb2+, etc.) or anions of multiproton oxy acids (e.g., the acid HsP04) on metal oxide surfaces. A new layer (the fourth as the name indicates but situated as the second, next to the surface layer “0”where protons are adsorbed) was reserved for such adsorbates. Cations and anions of basic electrolyte were placed in the same layer as in the triple layer model. Another approach was used by Bousse et al.ll who presented in their paper a diagram of the four layer model for the system where there are only basic electrolyte ions and potential determining ions H+. They believe that anions and cations of basic electrolyte are not located in the same layer, but in two separate ones. Although the principles of the four layer model seem to be well-established now, the literature still lacks its thermodynamic description. Such a description starting from the above physical model should provide theoretical expressions for all the experimentally measured physicochemical quantities (functions and parameters). Developingsuch a complete thermodynamic description based on the four layer model is the purpose ofthe present publications. The present theoretical description will be based on a picture of a geometrically and energetically homogeneous oxide surface. Accordingto this picture the outermost surface oxygens capable of forming surface complexes have a regular arrangement on the surface. Next, the adsorption properties of all the surface oxygens are the same. According to the modern, more realistic view, the surfaces of the really existing oxides are geometrically (7)Bowden,J.W.;Nagarajah,S.;Barrow,N. J.;Posner,A.M.;Quirk,
J. P. Aust. J. Soil Res. 1980,185,49.
(8) Bowden, J. W.; Posner, A. M.; Quirk, J. P. In Soil with Variable Charge; Theng, B. K. G., Ed.; New Zealand Society of Soil Science: Lower Hutt, 1980;p 147. (9)Barrow, N. J.Adu. Agron. 1986,38, 183. (10)Barrow, N. J.;Bowden, J. W. J.Colloid Interface Sci. 1987,119, 236. (11)Bousse, L.; de Rooij, N. F.; Bergveld, P.Surf. Sci. 1991,135,479.
0743-746319512411-3199$09.00/00 1995 American Chemical Society
3200 Langmuir, Vol. 11, No. 8, 1995 (A) s - H; S-
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Charmas et al.
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1. Triple Layer Model. The schematic picture of the triple layer model is shown in Figure 1A. While the pH of the solution in contact with the oxide surface is changed by bringing in acid or base, acid radical ions (most frequently oxygen-free or monovalent) or alkaline-metal ions are being introduced. As a result, besides potential determining ions Hf responsible for the pH of the solution, there are cations C+ and anions A- (e.g., K+, Na+, C1-, Br-, N03-, etc.) which according to this theory are found not only in the counterion (diffuse) layer but also in the compact one. To avoid complications caused by the increase of the concentration of anions A- or cations C+ while adding acid or base (when pH is changed), the electrolyte solution being in contact with the oxide surface already at the beginning of the experiment contains salt CA as a basic electrolyte so that C+ or A- addition with pH change will not change much the concentration of C+ and A-. Thus, it can be assumed that activities (concentrations) of cations ac and of anions UA are constant throughout the experiment and equal to ac = a A = a . According to the law of mass action, equations for intrinsic equilibrium constants of reactions taking place on the surface are as follows +it
2SOHo + H+
SOH:
s
so
c
Figure 1. (A) Diagrammatic presentation of the triple layer model of metal oxidelelectrolyteinterface. (B)Diagram of the charge distributionin the triple layer model. (C)Flat capacitors connected in a series as an equivalent of the triple layer model at the aqueous solutiodmetal oxide interface. Charge distribution on capacitorplates is obtained from the electroneutrality condition written in the form: 68 = (-60) + (-&I. distorted and energetically nonuniform (heterogeneous). In the recent series ofour publication^^^-'^ we have shown that a thermodynamic description based on the triple layer model and the picture of a homogeneous oxide surface is not able to represent the behavior of several important features of the really existing oxide/electrolyte interfaces. We ascribed that failure to the simplified picture of real pictures provided by the model of homogeneous oxide surface. Thus, there is one intriguing question which remains open. Specifically,this question deals with the extent to which that failure is due to some simplifications underlying the triple layer model and the extent to which it is due to the assumption that the oxide surface is homogeneous. Such a n initial fundamental study seems ultimately necessary before undertaking attempts to generalize the four layer approach for the case of an energetically heterogeneous surface. (12)Rudzidski, W.; Charmas, R.; Partyka, S.Langmuir 1991,7,354. (13)Rudzidski, W.; Charmas, R.; Partyka, S.; Foissy, A. New J.Chem. 1991,15, 327. (14)Rudzinski, W.; Charmas, R.; Partyka, S.; Thomas, F.; Bottero, J. Y . Langmuir 1992,8,1154. (15)Rudzinski, W.; Charmas, R.; Partyka, S.;Bottero, J. Y.Langmuir 1993,9,2641.
where [SOHo]and [SOHz+]are the concentrations of the surface complexes of singly and doubly protonated surface oxygens, respectively, [SO-I is the surface concentration of the free sites (unoccupied surface oxygens) and [H+adsl is the concentration of protons a t the surface. According to Boltzmann's statistics, the probability of finding a n ion at the surface is proportional to exp{- Izleq4zT);therefore, in the case of H+ ions which are located in the layer of the potential 1 ~ 0 ,[H'ads] = (aH)exp{ -eqdkT}, a H is the activity of protons in the equilibrium bulk solution. Besides the surface reactions l a and l b there are also reactions of formation of the surface complexes S O T + and SOHz+A-,which have the character of the ion pairing:
-so-c+
so- + c+
+
SOH2+ A-
K'A"t =
[SOH2+A-]
[A-adsl[SOHZ+l
K p
-
-
2nt A
SOH2+A-
[SOH2+A-1
exp{ (~A)[SOH~+]
%} (2b)
where [SO-C+l and [SOHz+A-I are the concentrations of the surface complexes of adsorbed cations or anions, and
Four Layer Complexation Model
Langmuir, Vol. 11, No. 8, 1995 3201
[C'adsl and [A-adsl are the concentrations of cations and anions close to the surface. It follows from Figure 1A that the adsorbed cations and anions are placed in the IHP layer called here the ,d layer in which the potential has the value ~ p By . use of the same procedure as in the determination of [H+,d,] concentration, the concentration of cations close to the surface will, accordingto Boltzmann's statistics, equal to [C'ads] = (ac) exp{-eWp/hT} and that of anions W a d s ] = ( a A ) exp{evp/kT) as shown by eqs 2. One can also consider two other equilibrium constants as fundamental ones in the description of the adsorption of the basic electrolyte cations and anions. These are the equilibrium constants of the following reactions SOH'
+ C'
*2nt C
SO-C'
*2nt A
SOH,'A-
SOH'
+ H'
eqs 4-6
(7) Both the surface charge and that of share layer can be determined from the experimental data. It should be stated that measurement methods used so far do not allow for a direct determination of the components of these charges, but only of some related to them. A total number of the sites capable of forming the surface complexeson the surfaceN, (here surface density in [sites/ m21)from the mass conservation law, is equal to NA
N, = -([SO-] Nx
+ [SOH'] + [SOH,'] +
In Figure 1B,Cthe quantities c1and cz are the electrical capacitances for capacitors connected in series, assuming that the capacitances are constant in the regions between planes. As can be deduced from that figure, the following relations between the charge and potential differences in the compact layer area exist:
+ H+ + A-
and obtained as a linear combination of reactions 1and 2 where = p'tp$t and *pit = Besides the above mentioned surface complexes there can also be formed SOHC+, SOHA- as well as SOHCA16 complexes, but their concentrations on the surface are so small that their presence is usually neglected. The triple layer model can also be presented by using the capacitors scheme shown in Figure lB,C. According to Figure lA, the surface charge density, 60, must be proportional to the sum of the concentrations of the following surface complexes:
pzt/pp.
eNA
60 = N,([SOH~']
+ [SOH2+A-] - [SO-] - [SO-C']) (4)
and the charge coming from the specifically adsorbed ions of the basic electrolyte in the /? plane, 68,is given by
For our further purposes we shall consider the following equivalent description of the reactions leading to the formation of free sites [SO-I: SOH'
-
+ H' SO- + 2H+ SO-
so-c+- so- + c+ - + +
SOH2+
SOH2'A-
SO-
2H+
(1la) (lib)
(llc) A-
Introducing surface coverages ei's by the individual surface complexes (i = 0, +, C, A) and free sites (i = -)
R, = [SOH0]+ [SOH,']
+ [SOH,'A-I + [SO-C+l
where e is the elementary charge, N A is the Avogadro number, and Nx is the factor combining the surface with the solution volume in [m2of surface/dm3 of solution]. Equations 4 and 5 neglect shares of free ions H+ in the plane of charge 60as well as of ions C+ and A- in the plane p. Equations 1-3 point to the presence of such ions a t the surface in the compact area of electric double layer. Appropriate calculations show that the charge comingfrom the free ions is so small compared with that coming from the localized charges that it can be neglected. As within the compact layer, there must be fulfilled the electroneutrality condition:
6,
+ 6, + 6,
=0
(6)
The value of the diffuse layer charge can be obtained from (16)Johnson, R. E.,Jr. J. Colloid Interface Sci. 1984, 100,540.
(lid)
+ [so-]
we obtain the following set of equations corresponding to reactions 11
(13a)
Charmas et al.
3202 Langmuir, Vol. 11, No. 8, 1995
Consequently, from the nonlinear set of eqs 17 and eq 14, the following equation is obtained: where aH is the proton activity in the equilibrium bulk solution, U A and ac are the activities of anion and cation in the bulk phase, ?,bo is t h e surface potential, and vp is the potential in the plane of the specifically adsorbed counterions, given by eq 9. According to eq 4 the surface charge density, 60,can be expressed as follows
where
B = N,e and N , in eq 14 is the density of surface sites, [sites/m21. The activities of ions ai’s, (i = A, C),can be determined from the ion concentrations and the activity coefficients y l s . We assume further that yi is given by the equation proposed by Davies”
i=A,C For a certain ionic strength of solution Z [molldml, the values of the activity coefficient yi for z-valence ion “i” can be calculated by using A given by the equation
A = 1.825 x
lo6
(16)
(€,T)3’2
+,
Kifi 1
K+f++ KPfA 1
- KC&
-
+ CKfi
(20)
i
i = 0,
+, A, C
the solution of which allows one to calculate 60 for each value of pH. Equation 20 which is nonlinear with respect to 60 can be easily solved by a n iteration method (e.g., Mueller-S). Having calculated 60 values, one can calculate the individual isotherms of ions from eq 17. In this triple layer model two values of the parameter c1are assumed to exist, depending on the sign of the charge of the surface:18
for pH < PZC a n d c1 = c1(2) for p H > PZC (21)
c1
The parameter c1 is connected with the distance between the surface layer “0”and the layer ‘‘Pwhere cations and anions of basic electrolyte are located. For pH < PZC, i.e., in the acidic medium, there are many more ofSOHz+Acomplexes formed by anion adsorption, but when pH > PZC, the SO-C+ complex is strongly predominant. Two different values ofparameter cl suggest that the adsorbed anion and adsorbed cation are situated a t different distances from the surface. Expansion of this reasoning has led to a four layer model. 2. Four Layer Model
where T is the temperature and cr is the relative permittivity of solvent. A set of nonlinear eqs 13 can be numerically solved to calculate next the individual isotherms of adsorption of ions 0:s (i = 0, A, C). For that purpose the set of eqs 13 is transformed into the form:
6 .=
d0=B
(17)
+ CKfi
Figure 2A shows the physical model called the “four layer model”. The reactions taking place on the surface and leading to the formation of surface complexes are the same as in the triple layer model. The important difference is that anions are situated within the layer of potential whereas cations are within the layer of potential ~ I C . Thus only the agents of electrostatic reactions change in the equations describing the equilibrium constants ELt and
Ett:
-so-c+
i
so- + c+
i = 0, +, A, C
dnt C
where
KO= udst
K+ = upntdnt a 1 a2 K C -- *EctfE$t KA = l l h ? $ * ~ (18)
AS follows from eq 9, fi’s(i = 0, +, A, C) are the following
+
SOH2+ A-
Kint A
-
SOH2+A-
functions of activity of protons and salt ions:
fo = exp{ - W O - 2.3pH},
f+=
feo
(19ab) (19c)
According to the physical model underlying our present consideration, shown in Figure 2A, the equilibrium
(17)Davies, C. W. In Ion Association; Buttexworths: London, 1962.
(18)Blesa, M. A.; Kallay, N. Adv. Colloid Interface Sci. 1988,28, 111.
{
fc = a c e x p -e% + e60} kT kTcl ~
~
~~~~~
Langmuir, Vol. 21, No. 8, 1995 3203
Four Layer Complexation Model
The meaning of the capacitances c+ and cx is illustrated in Figure 2A, and the capacitance c- is the one obtained by connecting the capacitors having capacitances c+ and cx in the following series:
Using the above relation, one can transform eqs 25 and 26 to the following form:
c+
wc = w o - 60
(29)
The surface charge density do and the diffuse layer charge density 6a are still calculated from eqs 4 and 7 as in the triple layer model. The former layer ,I3 is divided now into two layers, the charges of which are calculated as follows: SC
6,
B
6
6,
-6
-ls,s,l '&+&I
-s#
Sd
Figure 2. (A) Diagrammmaticpresentation of the four layer model of metal oxiddelectrolyteinterface. (B)Diagram of the charge distribution in the four layer model. (C) Flat capacitors connected in a series as an equivalent of the four layer model at the aqueous solutiodmetal oxide interface. Charge distribution on capacitorplates is obtainedfrom the electroneutrality condition written in the form: 6c = (-60) + [ - ( 6 ~+ &)I.
constants
(31) and
(32)
*R:tand *E? are now defined as follows: SOH'
+ C+
*dnt C
SO-Cf
*$nt
A
SOHZfA-
SOH'
The equations for the desorption reactions 11are the same as in the triple layer model. By use of the definition of surface coverage (12), the set of the equations corresponding to these reactions can be rewritten in the following form:
+ H+
+ H+ + A-
Equations 1remain unchanged for this model. Figure 2B,C presents the capacitors scheme of the four layer model. Treating a surface system as a set of flat capacitors connected in a series, and employing the electroneutrality condition
6,
+ 6, + 6, + 6, = 0
(24)
one arrives a t the following relations between the potentials and the charges within the individual electric layers: wo
- wc = c+
60
(25)
The above set of equations can be transformed to a similar form as in eq 17, where the equilibrium constants K,'s (i = 0 , +, C, A) are defined by eqs 18, and according to eqs 29 and 3O,f,'s(i = 0 , +,A, C) for this model are the following functions of proton and salt ion activities: fo = exp{ - ev0 k~ - 2.3pH},
f+ =f
(34ab)
3204 Langmuir, Vol. 11, No. 8, 1995
Charmas et al.
- 4.6pH} (34d) Provided that cations and anions are placed in the same plane as in the triple layer model, the following relations will hold: V A=VC
= Wp,
c- = c+ = c1,
c, - 00) 6 , + 6, = 6, (35)
and the above equations will reduce to those derived for the triple layer model. The method of solving equation set 17,18,34is similar to that described in the case of the triple layer model. First, a n attempt is made to reduce the number of the four nonlinear equations in the system to a smaller one. In order to attain it, the electroneutrality condition 24 is employed, and eq 34d is transformed to the followingform:
The starting point to calculate the function 6i is to solve a set of two nonlinear equations obtained from equation set 14 and 37 taking into account eqs 17, 18, and 34
Fl(d0,6*)= B
1 - K+f+ - KdA 1
+Z K h
- 6" = 0 (38a)
i
i = 0, +, A, C
1
+ZKf, i
i = 0, +, A, C where fc = fc(60) and f~= f~(60,6*),in order to calculate 6" and 60for each value of pH. Further transformations (for details see the Appendix) lead to the following equivalent set of equations: F1(d,,d*) =
+
Evaluating (1 ZiK$) from eqs 39 and then comparing these equations, we calculate 6" = 6*(60):
M e r the above relation is substituted into eq 38b (variable 6* appears in eq 38b only inside the function f A ) this
equation becomes a n implicit one only with respect to the variable 60. Then, eq 38b can be numerically solved using the Mueller-S method (or another one to find the root of the function) to obtain the 60 value for each pH. Comparing the four layer model with the triple layer model, one can see that the capacitances c+ and c- in this model have their counterparts in the triple layer model, i.e., cl(1)and ~ ( 2 ) . When anion adsorption is prevailing (pH < PZC), the parameter c- can be identified with the parameter c1(1),but when cation adsorption is prevailing, then (pH > PZC) c+ corresponds to c ~ ( z ) .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 triple layer model when c1 was equal to cl(l) and in another pH range to c ~ ( z ) This . continuity must change the values of the physicochemical quantities determined from the experimental data to some extent. As for the function values, it is to be expected to change them mostly around PZC. 3. Determination of the Surface Potential q,,
Bousse, de Rooij, and B e r g - ~ e l d l ' ~elaborated ~ ~ - ~ ~ on a method of an experimental determination of the surface potential ~ J Oof metal oxide surfaces, using in their experiments the so-called chemically sensitive electronic device CSED operating on the field-effect principle. The change of the semiconductorsurface potential is modulated by potential changes somewhere inside the structure, usually a t the insulator (e.g., metal oxide)/electrolyte interface. The measurements of pH which can be made by employing insulators made of commonly used semiconductors such as SiOz, Si3N4, or A 1 2 0 3 are of great interest in technology. The whole class of devices was called ion-sensitive (or unipolar) field-effect transistors ISFET's. They were described by B e r g - ~ e l and d ~ ~are the most frequently used devices in this field. The theoretical model for determining the surface potential vo elaborated on by Bousse, de Rooij and Bergveld was found to be in a good agreement with the results of a previous experiment.20,21It was stated that the PZC value and the difference between and pRZt determined from q~o(pH)measurements were in good agreement with the results of surface charge measurements for colloidal suspensions. Another important finding was that the surface potential ~ J Ois much less sensitive to surface complexation with counterions than the surface charge The theoretical approach to this problem was first limited to the application of the Nernst equation which predicts potential change equal to 2.3kTle V 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 ISFET's device, the potential change value was far below the Nernst value. Therefore, the Nernst equation cannot be applied to a n insulator such as a metal oxide which is neither an electric nor ionic conductor. (19) Bousse, L.J . Chem. Phys. 1982, 76, 5128. (20) Bousse, L.;Bergveld, P. J . Electroanal. Chem. 1983, 152, 25. (21) Bousse, L.;de Rooij, N. F.; Bergveld, P. IEEE Trans. Electron Devices 1983, 30, 1263. (22) Bousse, L.;Maindl, J. D. ACS Symposium Series (Geochem. Process Miner. Surf.); Davis, J.A., Hayes, K. F., Eds.; American Chemical Society: Washington, DC, 1986; No. 323, p 79. (23) Van den Vlekkert, H.; Bousse, L.; de Rooij, N. F. J . Colloid Interface Sci. 1988, 122, 336. (24) Bergveld, P.IEEE Trans. Biomed. Eng. 1970, 17, 7 0 .
Langmuir, Vol. 11, No. 8, 1995 3205
Four Layer Complexation Model
The paper by Van den Vlekkert, Bousse, and de Rooijz3 presented the measurements of temperature dependence of the surface potential V O at the y-Alz03/electrolyte interface. According to these authors, such investigations are of double significance. Firstly, because this device can serve as miniature sensors of pH measurements, e.g., in the clinical measurement of blood acidity. This application requires great accuracy and temperature dependence of the sensor must be known to use a suitable correction. Secondly, the function dVddT has some relations to the thermal effects accompanying adsorption of one proton or two protons on the oxide surface. Knowing these values, one can determine indirectly the values of these heats. The heat values obtained in this way can be compared with those from the analysis of calorimetric experimental results. The surface potential defined in the equations for the equilibrium constants of the surface complexation 1 in the triple or the four layer model can be theoretically calculated as qo(pH) from the equation developed by Bousse and co-workers21,2z
6;;)
+
2.303(PZC - pH) = eW0 - sinh-' kT
(41)
where the nondimensional quantity j3 has the formz3
P=-(-) 2e2Ns PAt ctkT
1/2
(42)
tg ;
In eq 42 ct is the linear capacitance of the double electrical layer. The value ct can be theoretically calculatedzO~zl (depending on salt concentration in the solution) as the capacitance of capacitors connected in series: the Stern compact double layer as well as the Gouy and Chapman diffuse one, because
1 --1-+ - 1 Ct
cs
cd
Taking into account the relation dd
eWd
= -(8~,c,,kTI)'~sinh -
2kT
(43)
obtained from the solution of the Poisson equation for the Stern model of 1:l electrolyte, and remembering that c d = -ddddqd and that the function cosh (eqd2kT) appearing in the differentiation q d tends to unity for small q d values, one arrives at the following relation
(44) where cr is the relative permittivity of solvent, EO is the permittivity of vacuum, and cs can be assumed, as the (Stern) compact layer capacity, equal to 0.2 Flm2. As it can be seen, the value of j3 depends on solution concentration and pH through ct. However, for high concentrations of base electrolyte ct G cs, B becomes dependent only on the kind of the oxide under investigation. In the vicinity of the point of zero charge (PZC), eq 41 reduces to the following linear one:
+
1) = 1 and the potential change values p, pI(j3 corresponding to one pH unit becomes equal to that predicted by the Nernst equation. According to van den V l e k k e ~ %for ,~~ the oxide y-Al203 having PZC = 8, p = 5. For such a n oxide, eq 45 is satisfied within the range of pH = 6 to pH = 10, with some more serious deviations outside this range. All model calculations presented in this paper were mae by takinginto account the rigorous form (41)ofthe function ~ o ( p H ) .The comparisons between the functions qo(pH) calculated by accepting its simplified form eq 45 and its rigorous form given by eq 41 are shown e1~ewhere.l~ 4. Equations Describing [-Potential Effects
The next experimental source of information which we are also going to explore is the electrokinetic effects accompanying the formation of the electric double layer. The 5-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. A common feature of electrokinetic phenomena is a relative motion of the charged surface and the volumetric phase of the solution. The charged surface is affected by the electric field forces and the movement of such surfaces toward each other which induce the electrical field. That is a question of shearing plane between the double layer and medium. The layer bounded by the plane a t the distanced from surface (OHP)can be treated as immobile in the perpendicular direction to the surface because the time of ion residence in the layer is relatively long. Mobility of ions in the parallel direction to the interfacial surface should also be taken into account. However, it seems that ions in the double layer and the medium surrounding it constitute a rigid whole, and that the layer from x = 0 to x = d is immobile also in the sense of resistance to the tangent force action. There is no reason why the boundary plane of the solution immobile layer should overlap accurately with the OHP plane. It can, as well, be placed deep in the solution. The potential in the boundary plane of the solution immobile layer is called potential 5. Strictly speaking it is not an interface potential because it is created in the liquid phase. It can be considered as the difference of potentials between a point far from the surface (in the bulk solution) and that in the shearing plane. However, many a ~ t h o r s ~assume ~ - ~ ' equality of potentials 5 = ?,bd at least for low values of surface potentials and low concentrations of electrolytes in the bulk phase. At higher values of potential and higher concentrations, viscosity close to the surface increases due to the increase of surface concentration. Subsequently the boundary plane of the mobile layer moves deep into the solution, and the anticipated value 151becomes lower than the value l q d l . Both potentials 5 and q d are diffise ones and therefore must be of the same sign and must behave in the same waywith the change of electrolyte concentration. Despite the difficulties in determining the accurate location of the plane with potential 5, one value of 5 potential can be determined from the double layer theory; Le., it is certain that 5 = 0 when q d = 0. Moreover, if the adsorption is not specific &e., 60 = -dd = 0 ) it can be also stated for sure that 5 = 0 when ~0 = 0, i.e., PZC = IEP. If one assumes that the relation 5 = q d is valid for low
(45)
This approximation is valid around PZC for IcqdkTI -= j3. Looking at eq 45 one can deduce that for relatively large
(25) Lyklema, J.; Overbeek, J. Th.G . J . Colloid Sci. 1961, 16, 501. (26) Hunter, R.J. J. Colloid Interface Sci. 1966,22, 231. (27) Smith, A.L. In Dispersions ofpowders and Liquids; Parfitt, G . D., Ed.; Elsevier: London, 1969; p 39.
Charmas et al.
3206 Langmuir, Vol. 11, No. 8, 1995
concentration^,^^-^^ 5 can be theoretically calculated from eq 43. Keeping in mind that the inverse function to the hyperbolic function sinh x is the function sinh-l x = In [x (x2 1)u2],for 1 : l electrolytes we obtain the following equation
+ +
where ?)d and d d are the potentials and the charge density, respectively, in the d plane. This means that the onset ofthe diffise layer, G,is the relative permittivity ofsolvent (for water a t T = 25 "C, = 78.251,c0 = 8.854 x 10-l2 F/m is the permittivity offree space, and I is the ionic strength of the suspension (ions/m3). The charge density in the d-plane, d d , therefore reads
6, = B[8- - e+]
where exp{s} -1 exp{
s}+
6&pH = PZC) = 0
(49)
Taking eqs 17-19 for the triple layer model into account, for pH = PZC (60= 0 and ?)o = 01, eq 20 can be transformed as follows:
(47)
The values of 8- and €I are +calculated in the way described in sections 1 or 2, depending on the model (triple layer model or four layer model) for which the value of potential 5 is calculated. If it is assumed that the slip plane of potential [ lies a t the distance Z from the plane of potential ?)d,18'32-35 then employing the accurate solution of the Poisson equation3e leads to the equation
gd =
electrolyte have a common intersection point (CIP)in PZC. That intriguing feature led us to study the consequences of treating it in a fully rigorous way. For the triple layer model, that way has been outlined in our previous theoretical works.l 3 7 l 4 Here we are going to repeat it briefly for readers' convenience, before considering the four layer model. A point of zero charge is determined by the condition
(48b) 1
K is the Debye-Huckel reciprocal thickness of the double layer
(484
where
(51) Then, eq 50 can be rewritten in another form
PZC =
Knowledge of the PZC experimental value makes it possible to eliminate one of the four parameters of surface reaction equilibrium constants, even for those few reported cases where CIP was not observed in the system.35 However, the reported experimental studies show that in most systems the PZC value does not practically depend on salt concentration in the bulk solution (Le., a common intersection point (CIP) occurs a t pH = PZC).4129137 Except for the region of very low salt concentrations, one can assume that ac = aA = a. Thus, the independence of PZC of salt concentration can be formally expressed as
-apzc _- aa
and ?)d(z) in eq 48a is the potential a t the distance Z from the diffise layer. Accordingly, [ = ?)d(z), and the value Z is usually a parameter in theoretical calculations. The value of the parameter Z affects the positions of the acidic and basic branches of the [-potential curve, but it does not affect the position of IEP. Therefore all model calculations presented in this paper were made by assuming that Z = 0. The influence of parameter Z on the electrokinetic curves was shown e1~ewhere.l~ 5. The Common Intersection Point and Its Consequences in Both Complexation Models Except for a few rare cases, the experimental titration curves corresponding to different concentrations of basic (281 Smit, W.; Holten, C. L. M. J.Colloid Interface Sci. 1980, 78, 1. (29) Sprycha, R.J. Colloid Znterface Sei. 1984, 102, 173. (30) Sprycha, R.;Szczypa, J.J.Colloid Interface Sci. 1984,102,288. (31) Sprycha, R.;Szczypa, J.J.Colloid Interface Sci. 1087,115,590. (32) Harding, T. H.; Healy, T. W. J.Colloid Interface Sci. 1985,107, 382. (33) Kallay, N.; TomiC, M. Langmuir 1988, 4, 559. Kallay, N. Langmuir 1088, 4, 565. (34) TomiC, M.; (35) Kallay, N.; Sprycha, R.; ToniC, M.; galec, S.; TorbiC, 2. Croatica Chem. Acta 1090, 63, 467. (36) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; John Wiley & Sons,Inc.: New York, 1990.
auaa au/aPzc = O
(53)
So, knowing that aU/aPZC f 0, we arrive a t
After solving the set of eqs 50 and 54, the following relations are obtained:
Equations 55 relating the parameters If,;" and PAt and *Pgt and *E? reduce the number of the unknown equilibrium constants determined from fitting suitable experimental data from four to two. From relations 55, the following expression for PZC value is obtained (56a) (37) Kallay, N.; BabiC, D.;MatijeviE, E. Colloids Surf. 1086,19,375.
Langmuir, Vol. 11, No. 8, 1995 3207
Four Layer Complexation Model
Table 1. Examples of the Difference between PZC and IEP Values Taken from Literature lit. system value of PZC value of IEP Smit et aLZ8 a - A l z O a a B r PZC = 4.5 IEP = 3.1-3.5O
where pZ:t
i = 1 , 2 and p * q = -log
= -logK"t,
*e,i = C, A (57)
Now we are going to apply our criterion 53 to the four layer complexation model. While taking eqs 17, 18, and 34 into account, for pH = PZC, eq 38b can be rewritten in the following form
(58) where
Kallay et al.37 anataseNaN03 Wood et al.38 boehmiteKN03 Thomas et al.39 Ma aluminaKC1 a
PZC = 6.5 PZC = 8.5 PZC = 8.5
IEP = 4.5 IEP = 9.1 IEP = 8.4
Depending on the salt concentration.
The determination of the equilibrium constant pP2t from eq 63a and of the constant p*Ppfrom eq 63b for given pEAt and p*ICcnt values is not so simple as in the case of the triple layer model. Now the procedure is the following: first the value p * P p is determined from eq 63b keeping in mind that X* depends on PAt. Consequently, having evaluated the value p*F$ (also *E?), eq 63a is used to calculate P E ~ employing ~ , the fact that Y* depends on *E?.
(62b)
Discussion Despite the great success ofthe TLM (triple layer model), a large body of experimental data has been accumulated which can hardly be explained in terms of the TLM, or fitted satisfactorily into the related theoretical expressions. Perhaps the most striking of them is the difference between PZC and IEPvalues observed more or less clearly in many adsorption systems. Some of the examples taken from literature are collected in Table 1. Attempts to explain the difference between PZC and IEP values in terms of TLM were undertaken by Wood et al.38 They fitted simultaneously the titration isotherms and electrokinetic (-functions for the boehmiteKN03 system by adjusting numerically all the four equilibrium constants: PP;~,pPAt, P*E;~, and p*Eit. Figure 3 shows the agreement between their experimental data and their best-fit curves. In addition to the moderate agreement between their experimental and calculated data, their calculations must be put into question for the same reason as many other calculations of that kind. This is because their titration isotherms have CIP, as can be easily seen in Figure 3. In such a case application of our rigorous criterion 53 leads to the conclusion that only two of the four equilibrium constants can be chosen as best-fit parameters. The other two are to be calculated from eqs 55 and then the calculated IEP value will always be equal to PZC. Treating all the four equilibrium constants as independent best-fit parameters is unjustified, as it violates the rigorous criterion 53 for CIP to exist. Application of TLM to the adsorption systems exhibiting CIP will always lead to the conclusion that these systems must include PZC and IEP equally. This conclusion is an obvious contradiction with the reported experimental data. Nowadays, the general feeling is growing that the inequality of PZC and IEP is a fundamental feature of these adsorption systems more or less clearly observed, depending on the particular system under investigation. The difference between PZC and IEP is also related to
From relations 61 the followingexpressions for PZC value are obtained:
TLM equations to fit experimental titration curves in a wide range of salt (basic electrolyte) concentrations. This was first emphasized by Blesa et al.,40 but the most
* p t a
-B
c c
&At + H + 2*Pttac
(59b)
The independence of PZC of salt concentration can now be formally expressed as follows:
H2 p * + aH2 aP* -au_ aa pa\t*pi k;t*p;t aa
* g ; t
p t ;
-0
(60)
Solving the set of eqs 58 and 60, one arrives at the following relations
where
P = P * + a - ap*
aa
(62a)
and
another intriguing question raised by the application of
1
PZC = 5(pk,"1'
+ pZgt - log P)
(63a)
1 2
+ p * Z i t + logx*)
(63b)
PZC = -(P*P;~
(38) Wood, R.; Fornasiero, D.; Ralston, J. Colloids Surf. 1990,51, 389. (39) Thomas, F.; Bottero, J. Y.;Cases, J. M. Colloids Surf. 1989,37,
281. (40) Blesa, M. A.; Fignolia, N. M.; Maroto, A. J. G.; Regazzoni, A. E. J . Colloid Interface Sci. 1984, 101, 410.
3208 Langmuir, Vol. 11, No. 8, 1995
Charmas et al. d ln(
c-
5
6
7
8
9
1
0
PH
Figure 3. Comparison between experimental results and theory for the bohemite/KN03system reported by Wood et aL3* using the equations correspondingto TLM: (A) comparison of the experimental (+,0,O) and the theoretical values (-) of the surface charge 60 vs pH, for various concentrations of KNO3, (+) 0.1 M, (0)0.01 M, (0)0.001 M; (B)comparison of the experimental (0, 0 ) and the theoretical values (-1 of the f-potentialvs pH, for various concentrationsof KNO3, (0)0.01 M, (0)0.001M. exhaustive study of that intriguing problem has been reported by S p r y ~ h a Specifically, .~~ to fit titration curves corresponding to quite different salt concentrations, one has to accept different values of the capacities c1(1) and c1(2) for different salt concentrations. Meanwhile, the adsorption equations developed on the basis of TLM do not predict the dependence of either cl(1) or c1(2) on salt concentration. On the contrary, our equations developed on the basis of the FLM (four layer model) predict correctly the existence of such a dependence. The application of our criterion 53 leads to the interrelations 59-63 of the equilibrium constants which contain the term
,$-
d ln(
predicting the dependence of the capacitances c- and c+ on the salt concentration. Let us remark here that the application of our criterion 53 to TLM led to eqs 56 which do not contain derivatives of such a kind. Our numerical illustrative calculations based on the FLM show that the coincidence of PZC and IEP will be observed only for a certain nonzero (always negative)value of the derivative
k- dl
dln-f-
du
The greater is the difference between c+ and c-, as the more negative of (41)Sprycha, R. Habilitation Thesis; Maria Curie-Sklodowska University, Lublin, 1986.
- $)/du f
has to be accepted to make the IEP value equal to PZC. While carrying out our illustrative calculations, we accepted the following strategy. Specifically, we took as a starting point the parameter values for the system Ti02 (anataseYNaC1, reported by S p r y ~ h awho , ~ ~analyzed theoretically the experimental data for this system using the TLM equations. The parameters obtained by fitting these equations to experimental titration curves are collected in Table 2. It should be emphasized here that our criterion 53 was published after Sprycha had carried out his theoretical analysis; therefore, all four equilibrium constants were treated by him as independent best-fit parameters. Meanwhile, the experimental titration curves measured by Sprycha have CIP, so to proceed correctly with the theoretical illustrative calculations based either on TLM or FLM model, we had to choose only two ofthe equilibrium constants reported by Sprycha as independent parameters. The other two were calculated using our criterion 53 for the systems exhibiting CIP. As the two independent equilibrium constants we chose pXAt and p*Xtt and they were assigned the values 8.0 and 5.8, respectively, as reported in Table 2. The corresponding, calculated values of pkAt and p*KT are displayed collectively in Table 3. In the case of FLM, the interrelations 61-62 of the equilibrium constants contain the term
which is another best-fit parameter. Thus, searching for a logical criterion for canying out our comparison between TLM and FLM, we accepted the following assumption. As the physical meaning of the capacitancies c1(1)and c- should be similar to some extent, we assumed cl(l) = c- in our illustrative numerical calculations. For the same reason we assumed c1(2) = c+ in these calculations. The FLM parameters established in this way are displayed collectively in Table 3. Using the parameters collected in Table 3 we calculated next the titration (60)and electrokinetic (5) curves predicted by TLM and FLM models, respectively. Their comparison is shown in Figure 4. One can deduce from Figure 4 that TLM and FLM models lead to similar values of the titration curves. As the titration curves were the most extensively studied experimental data, there was no dramatic need to improve TLM. On the contrary, TLM and FLM lead to quite different values of the calculated electrokinetic curves; but again, the necessity ofimproving TLM was not urgent, until all the four equilibrium constants could be chosen as best-fit parameters. After publishing our criterion 53 for CIP to exist, the possibility of choosing all the four parameters as best-fit ones does not exist any longer. However, even then TLM yields a pretty good fit of the experimental electrokinetic curves, as far as the data studied correspond to one salt concentration. The shortcomings of TLM become evident only when one is trying to fit simultaneously the data measured for much different salt concentrations. The TLM approach inheriting the assumption that cl(1) and CI(Z) are independent of salt concentration leads then to poorer agreement with the experimental findings. Figure 5 shows the individual adsorption isotherms of the surface complexes and the fraction of the free surface oxygens SO-, for the parameters used to carry out the
Four Layer Complexation Model
Langmuir, Vol. 11, No. 8, 1995 3209
Table 2. The Values of the Parameters Obtained by Sprycha" Using the Graphical Extrapolation Method for the System Anatase/NaCl, When the Concentration c of the Basic Electrolyte Was Equal to 0.01 M system: anatase/NaCl PZC = 6.0 N . = 12 sites/nm2 p e : = 4.0 c1(1)=
p e i = 8.0 p*p;t = 5.8 p * p i t = 6.2
c1(2)=
1.10 F/m2 1.20 F/m2 2
Table 3. Parameters for the AnataseiNaCl System Established in Such a Way That the Related Model Calculations Should Show the Difference between the Physical Functions Calculated on the Basis of TLM and FLM, Respectivelp triple layer model four layer model
parameters N B,= 12 sites/nm2 p e l = 8.00 = 5.80 ~ ( 1=) 1.10
c1(2) =
F/m2
1.20 F/m2
4
8
0
2
1
6
8
PH
PH
Figure 5. Comparison ofTLM and FLM models: (A) theoretical individual adsorption isotherms of 8c and 8 A for TLM (-) and
FLM (- - -1; (B)theoretical individual adsorption isotherms of 8+ and 8- for TLM (-) and FLM (- - -). All the theoretical curves were calculated for the parameters collected in Table 3.
parameters N , ,= 12 sites/nm2 pPAt = 8.0 p*Ptt = 5.80 c- = 1.10 F/m2 c+ =
1.20 F/m2
d ln((l/c-) - (l/c+))/du = 0 calcd from eqs 61-62 pPAt = 3.95 p * e = 6.27
calcd from eqs 55 p e t = 4.00 p*@ = 6.20
The parametersp p d and p*@ have the same values a? those found by Sprycha41for TLM, whereas the parameters p e : and p * c t were calculated by applying our criterion 53 to TLM and FLM, respectively. The calculation corresponded to the physical situation when the concentration of the basic electrolyte NaCl, c, is equal to 0.01 M.
-
PH
Figure 6. Theoretical [-potential curves calculated for FLM for different values of the parameter d ln((l/c-) - (l/c+))/da: (1)50; (2) -50; (3) -100; (4) -200. Curve 3 is practically the
same as the theoretical curvefor TLM for the same parameters. The other parameters are those collected in Table 3.
-
PH
-
PH
Figure4. Comparisonof TLM and FLM models: (A)theoretical
titration curves,60vs pH, for TLM (-) and FLM (- - -) models, calculated by using the parameters collected in Table 3. (B) Theoretical
