The TiO2/electrolyte solution interface II. Calculations by means of the site binding model

The TiO2/electrolyte solution interface II. Calculations by

means of the site binding model

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Janssen, M. J. G., & Stein, H. N. (1986). The TiO2/electrolyte solution interface II. Calculations by means of the site binding model. Journal of Colloid and Interface Science, 111(1), 112-118. https://doi.org/10.1016/0021-9797%2886%2990012-3, https://doi.org/10.1016/0021-9797(86)90012-3

DOI:

10.1016/0021-9797%2886%2990012-3 10.1016/0021-9797(86)90012-3

Document status and date: Published: 01/01/1986

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The TiO2/Electrolyte Solution Interface II. Calculations by Means of the Site Binding Model

M. J. G. JANSSEN AND H. N. S T E I N

Laboratory of Colloid Chemistry, Eindhoven University of Technology,

P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Received May 4, 1985; accepted August 30, 1985

Starting from experimental Cro and ~" potential values of TiO2 in KC1 and KNO3 solution, calculations were performed in order to check whether a self-consistent description of the data in terms of the site binding model can be obtained. Two adjustable parameters were used: the Stem layer capacitance and

the equilibrium constant for the association of surface hydroxyl groups with hydrogen ions (MOH + H + ,-~ MOH~). In the calculations, assumptions concerning activity coefficients of charged groups near the surface have to be introduced. If the quotients of surface activity coefficients of complexes and free surface hydroxyl groups (YMOn~Ca-/~/MOn and "~MO-H+/~MOtt) are thought to be independent of the charge in the adsorption plane, the association and dissociation constant of surface hydroxyl groups turn out to be pH-dependent. This is ascribed to the influence of local deviation from the average values of the potential at a certain distance from the surface. Neglect of this effect leads to internal inconsistencies of the model. © 1986 Academic Press. Inc.

I N T R O D U C T I O N

In a previous paper (1), data were presented on the colloid chemical properties of disper- sions of TiO2 in various aqueous solutions. The data comprised surface charge, ~" potential and stability toward coagulation. The m a i n result of this investigation was, that for pure TiO2 the colloid chemical properties men- tioned are not influenced by changes in solid- state properties (donor concentration).

The insensitivity towards changes in donor concentration (effected by pretreatment in 02 or in H2) entails independence of differences in surface conductivity. At the highest donor concentration obtained, the surface conduc- tivity apparently is still too low to influence adsorption equilibria; thus TiO2 can be re- garded, as far as surface charge and ion ad- sorption is concerned, as an insulator. The TiO2/aqueous solution interface, therefore is suitable to test models for such interfaces, in which the influence of changes in solid state parameters is disregarded.

0021-9797/86 $3.00

Copyright © 1986 by Academic Press, Inc. All ~ghf© nfr~nrrutuctinn in any form reserved.

112

Oxide/electrolyte solution interfaces are frequently described by the site binding model (2-8) based on the following hypotheses:

(1) The origin of surface charges can be de- scribed by association/dissociation equilibria of surface hydroxyl groups M O H :

MOH{2 +)~-0 M O H + H~ +) [ll

M O H *-~ M O (-) + H~ +). [2]

The suffix " s " denotes surface.

(2) Ions behind the electrokinetic slipping plane are all bound to charged sites of opposite sign; thus, in KC1 solutions:

M O H + K~ + ~ MO(-)K (+) + H~ +) [3]

M O H ~ C l - *-+ M O H + H~ +) + 0 2 . [41 (3) The "free ion concentrations near the surface" ([H~+)], [K~+)], [CI~-)]) are related to bulk concentrations by

yH+[H+]= yH~[H~]exp(--~T )

[5]

TiO2 ELECTROLYTE SOLUTION 113

7Kt[K~I = 3'Kg[Kglexp(-- k ~ ) [61

7o;[C1~] = 3'O~-[Clb]exp(+ e~,~ - ~ - j [7] in which ~0, ~a, and ~a, are the potentials of the solid phase and at distances fl and fl' from the phase boundary, fl and fl' are distances of closest approach of K + and CI-, respectively, to the solid phase. The suffix "b" denotes "bulk"; ~/s are activity coefficients.

(4) The equilibrium constants describing the equilibria [3] and [4] can be determined by linear extrapolation of pQ*+ and PQ~l- as functions of aa, toward c~ = 0. Here the fol- lowing definitions are employed:

[MO-K +] pQ~+ = pH + log "¢K~(K~) -- log

[MOB]

[81

PQ~I- = pH - log 'ycg[Clg]

[MOH] - log [MOH~-C1-] [9] o~ = I([MO-K +] - [MOH~C1-])/Nsr [10] with Ns = the total number of sites per unit surface area.

Recently, Johnson (9) formulated the site- binding model with the surface site density and the standard free energies of adsorption of the ions concerned as parameters. Onoda and Casey (10) stressed the influence of transfer of oxygen ions from the solid to the liquid phase, involving nonstoichiometry in the solid. In the present paper, this possibility is not taken into account because at the TiO2/electrolyte so- lution colloid chemical phenomena are not influenced by the nonstoichiometry of the solid to a noticeable degree (1).

Among the assumptions of the site-binding model in particular those about the existence of single valued potentials ~bo, ~ba, and ff~,, ap- pear to us questionable, at least for interfaces with an insulating solid phase. It is true that the theory can be formulated in a way, in which these potentials are average potentials

at the phase boundary and at distances/3 or /3' from it; the local deviations from those av- erage values are then accounted for by the sur- face activity coefficients (11). However, it should be realized that these surface activity coefficients, by the close mutual proximity of ionic charges near the interface, will deviate substantially from unity; much more so than bulk activity coefficients in a solution which is in equilibrium with the interface concerned. Thus, all assumptions concerning the be- havior of activity coefficients, which are intro- duced during the calculations, must be re- garded with caution.

Although the site binding model has fre- quently been applied in recent years, it is not the only theory which aims at describing sur- face charge generation at oxide/electrolyte so- lution interfaces.

Among the alternatives, we mention the porous gel model by Lyklema (12-14) and the stimulated adsorption model (15-17). In the former, a hydrolyzed surface gel is considered to be in equilibrium with the electrolyte so- lution; this region is supposed to be permeable to all ions in the electrolyte solution. The stimulated adsorption model, on the other hand, stresses the importance of local devia- tions from the average values of the potential; through such a deviation, cations are adsorbed and promote in turn the adsorption of anions. In the present paper we describe the method of calculation by the site binding model in some detail, because the assumptions involved are not always given due consideration.

M E T H O D S

The experimental method employed has been described in the preceding paper (1). The data employed in the calculations refer to or0 and ~" values for TiOz DP 25 in both KC1 and KNO3 solutions; phenomena near the inter- face TiO2 M808/electrolyte solution are con- sidered to be too much influenced by chemi- sorbed impurities to be amenable for a sim- plified model of this interface.

From the ~" potential, the net charge behind the electrokinetic slipping plane (a~) was cal-

1 14 JANSSEN AND STEIN

culated by the relation between o r and ~" for flat interfaces (18) (it will be recalled that ~" potentials had been calculated from electro- phoretic mobilities by the formula for Ka > 1, on the basis of the argument that in electro- phoresis the movements of flocs rather than of primary particles is followed (1)). For com- parison's sake, we calculated o r also by the approximate relation (19) k T o f = 4"n'%e r - - K a 2 z e z e ¢ + 4 , ze¢'] × 2 sinh 2kT - - t a n n Ka

4~J

[ l l l with a = the primary particle radius as cal- culated from the BET surface area for spherical particles. When relation [ 11 ] was used, the ~" potential was employed as calculated by the Wiersema-Loeb-Overbeek method (19). The results obtained by these calculations are qualitatively similar to those obtained by using the o r vs ~relation for flat interfaces; although in quantitative respect differences exist. For more details, see Ref. (20).

From o0 and or, the charge between the phase boundary and the electrokinetic slipping plane (~r) was calculated by

o~= o r - o0. [121

The set of equations to be solved consists of ~0 = e{[MOH~] + [MOH~CI-]

- [ M O - ] - [ M O - K + ] } [131

ae = e {[MO-K +] - [MOH~CI-1} [141

N, = [MOH] + [MOH~] + [MOHi~C1 -]

+ [MO-K+] + [MO-I [151 -rMO-K+[MO-K+ITH~[H~ +1 K~+ = [16] 7MoH[MOH]TK:[KI ~] " Y M O H [ M O H ] y H + [ H +]

K~,

=

[17]

YMOH~cr[MOH~CI-] [MOH]YHg[H$] Kal = [MOH~-] [18] [MO-]y~:[Hs +] K,~ = [191

[MOH1

together with [5]-[7].

In Eqs. [13] and [14], e is the charge of a proton. [MOH~], [MOH~C1-], etc. are num- bers of sites per unit surface area. Thus, o0 and o~ are surface charge densities.

The consecutive steps in the solution of these equations were

(1) For pH ~ pHpzc, la~J --- e[MOH~-CI-] and [MOH~] > [MO-], We can solve then [131-[15] for the three unknowns [MOH], [MOH~-CI-], and [MOH~], and calculate pQ~i- according to Eq. [9]. This was performed for various values of a~ (see Eq. [I0]), and pQ~> was plotted as a function of a0 (Fig. 1). Activity coefficients in the bulk solutions were taken from Harned and Owen (21). By extrapolation to ao = 0 we find

[MOH]yng[H~]yog[Cl~]

Kc,- = [20]

[MOH~CI-]

which differs from K&- as defined by Eq. [ 17] because

K ~ l - - "/MOH

YMOH~-C1-

e x p ( - ~ T (¢~' - ~0))Ko-

[211

where YMOr~, '~MOI-I~CI- and ¢~, - ¢0 values should be substituted valid at a~ = 0.

Thus, Kcl- will be equal to the thermody- namic constant K~- only on the assumption that ~{MOH/'YMOH~O- --" 1 and fie, --, if0 for a~ --* 0. The latter assumption should be kept in mind because it is possible that there is a surface charge on the solid while [ M O - K +] = [MOH~C1-] which gives a~ = 0. This situ- ation will arise whenever the tendencies of the

alO,

p~

k Ct#X-lO 2 L blo pa~.

T

ct,,¢.10 2

FIG. 1. pQ~- and pQ~+ vs a a in KCI solutions.

TiO2 ELECTROLYTE SOLUTION 1 15 K + and C1- ions for chemisorption are not

exactly equal.

The slope of the straight line in Fig. l a is given by

OpQ*v

O (

'~MOH~C1- / e Oo-0

0c~ - 0o~a log YMOH /

2.3kTC_ 0o~

[22] with C_ = ao(ffo - ff~'), considered to be in- dependent of a~. If in addition YMOH~C>/

"YMOH is thought to be independent of aa as well, we can calculate C_ from

OpQ~>

(2) Similarly, for pH >> pHpzc, }rr~] -~ e × [MO-K+], and [MO-] >> [MOH2~]. We solve Eqs. [ 13]-[ 15] now for the unknown [MOH], [MO-] and [MO-K+]; on calculating P*QR according to Eq. [8] for various values of c~e, we obtain Fig. 1 (right side). On extrapolating to a~ = 0 we obtain

[MO-K+][H~]yH¢

KK+ = [23]

[MOH][Kg]3,Kg differing from K~+ because

(e

)

K~+ - 7MO K+ exp + ~-~ ( ~ - ~0) KK+. TMOH

[24]

Again we can calculate from the slope of the straight line in Fig. Ib the parameter C~ = ~o/ (~0 - ~ ) on the assumption that C+ and ~MoH/YMo-K+ are independent of a¢. Similar data for KNO3 solutions are plotted in Fig. 2.

10•

blo

8 t

Pi q

8

°'l"

2~ 2 2 0 2 pQ~. , o ~ "~ 2 t, 6 8 1'0

FIG. 2. pQ~, and pQ~- versus ~a in KNO3 solutions.

(3) With the values of C+ and 6"_ thus ob- tained, and substituting KK* for K~+ and

Ko-

for K~>, we solved the equations men-

tioned analytically following the method out- lined by Dousma (22) and Bousse (23). As ad- justable parameters we used the differential Stern capacitance

CSte~n ---- & o / d ( ~ o - ~d) [25]

with ~bd = ~', and/fat as given by Eq. [18]. Note that we have only two adjustable pa- rameters since

(pKa~ + PK~2) = 2*pHpzc [26]

from Eqs. [18] and [19].

The adjustable parameters were chosen such as to obtain a fit to the experimental Cro and ~" values.

RESULTS

Table I contains the values for the adjustable parameters giving the best fit to our data, to- gether with other parameters (Ns, C+, C_, KK+,

Kc>).

Figures 3-5 show the final fit of the a0 and ~ data for KNOa solutions; similar results were obtained for KC1 solutions.

From this table it can be seen that the model necessary for fitting the experiments comprises the following features:

(1) The planes where anions and cations are adsorbed do not coincide.

(2) Different pK, I values are necessary for pH < pHpzo and for pH > pHpzc. In other words: the Kal values necessary to obtain a reasonable fit to both surface charge and ~'po- tential data, are a function of the sign of the surface charge and of the concentration of supporting electrolyte.

(3) The Stern capacitance is a function of the concentration of supporting electrolyte. The value of the Stern capacitance, however, is not critical; a rather broad range of values could be used.

(4) Agreement between model calculations and experimental data is particularly good at high electrolyte concentration.

116 JANSSEN AND STEIN TABLE I

Parameters Used in the Model Calculations

TiO2 in KNO3 Solutions TiO2 in KCI Solutions Parameter Unit Value Remarks Value Remarks

N~ C+ C_ K~o; or Ko- pKal Cstern rn -2 1.2 X 10 '8 Ref. (6) 1.2 X 1018 Refi (6) F. rn -2 1.4 + 0.3 1.5 _ 0.3 F. m -2 0.9 + 0.4 0.9 +__ 0.4 - - (5 + 0 . 2 ) X 10 -8 (4 + 0 . 2 ) X 10 -8 M -2 (2.5 _ 0.1) × 10 -6 (2 + 0.1) × 10 -6 9.1 10 -3 M; pH < pHp~ 10.1 10 -3 M; pH > pH~c 7.4 10 -2 M; 6.8 10 -2 M; pH < pHp~ pH < PHpzc 9.5 10 -2 M; 9.6 10 -2 M; pH > pHp= pH > pH w 6.8 10 -1 M; 6.5 10 -1 M; pH < pH w pH < p H ~ 8.5 10 -1 M; 8.9 10 -1 M; pH > pHp~ pH > pHp~ F. m -2 0.06 10 -~ M 5 ~ < p H ~ < 10 0.16 10 -2 M 0.16 10 -2 M 5 ~ < p H ~ < 10 5 ~ < p H ~ < 10 0.6 10 -1 M 0.6 10 -I M 6~<pH~< 10 5 ~ < p H ~ < 10 -1 -2( -3C

¢

(mVl /

T \.

o/°

/ \° / '~ o 1 - ; ) o" '° ~ ; o /

£

51c~

-'20 -% -12 - 08 "0~- '08

FIG. 3. Surface charge density (©) and zeta potential (e) of TiO2 in 0.1 M KNO3 solutions.

DISCUSSION I n t h e f e a t u r e s o f t h e m o d e l d e s c r i b e d i n t h e p r e c e d i n g s e c t i o n , e s p e c i a l l y t h e s e c o n d o n e is i m p o r t a n t . It m e a n s t h a t Kai a n d K~2 a r e n o t t r u e t h e r m o d y n a m i c c o n s t a n t s , b u t a r e d e - p e n d e n t o n c o n c e n t r a t i o n s . Kal differs f r o m t h e t r u e t h e r m o d y n a m i c c o n s t a n t s K ~ b y a f a c t o r 'YMOH/'YMOH~. T h u s w e m u s t c o n c l u d e t h a t t h e a c t i v i t y c o e f f i c i e n t s a r e d e p e n d e n t o n c o n c e n t r a t i o n s i n a d i f f e r e n t w a y t h a n e x - p r e s s e d t h r o u g h t h e a v e r a g e p o t e n t i a l if0; t h e s a m e h o l d s f o r ~/MO-/3'MOn. I n i t s e l f t h i s is n o t s u r p r i s i n g b e c a u s e a c t i v i t y c o e f f i c i e n t s o f c h a r g e d g r o u p s a r e k n o w n t o s t r o n g l y d e p e n d o n t h e p r e s e n c e o f o t h e r c h a r g e d u n i t s , w h i l e n e a r a c h a r g e d i n t e r f a c e l o c a l c o n c e n t r a t i o n s o f t h e l a t t e r a r e high.

TiO2 ELECTROLYTE SOLUTION 1 17 During the calculations the assumptions

had been introduced that '~MO-K+/~MOH and 3'MOH~CV/3'MOH are independent of the ionic e n v i r o n m e n t near the surface. This was intro- duced both in the calculation of C_ and C÷ from the slope o f the lines in Figs. 1 a and b, respectively, and by the use of KK+ and K a - in calculations for conditions at which aa = 0.

Introduction of the independence of

7MO-K+/~MOH and 7MOH~CI-/'~MOH of the ionic e n v i r o n m e n t leads to a conclusion, which is at variance with the same assumption for the analogous case o f '~MOH/'YMOH~ and "YMOH/'YMo- •

There is a difference, to be sure, between the cases of'~MOH~C1-/~YMOH and '~MO-K +/'~MOH on the one hand, and '~MO-/'~MOH and 3~MOU~/ "YMOH on the other: the former refer to groups which, as a whole are uncharged while the lat- ter refer to groups which do carry a charge. It should be realized, however, that MOH~-CI- and M 0 - K + are dipoles rather than uncharged groups: the charges o f the CI- and the K + ions are in the model considered to be at some dis- tance from the phase boundary, where other average potentials (¢~,, and ¢8) prevail than at the surface itself (~0), and where local de- viations from these average potentials certainly will differ from similar deviations at the sur- face. M O H , on the other hand, should be re- garded as an uncharged group.

4.0 30 20 10 0 -I0 2C "30 (mV)

\.

o:

~

_{o , ° ~ , -~---~pH}

\

\.

\.

-~2 08 • Ot~

FIG. 5. As Fig. 3, in 0.001 MKNO3 solutions.

Thus, it appears that the m e t h o d to correct for local activity coefficients near the surface, by extrapolating pQ* to o~a -- 0, is insufficient to take into account the influence o f local de- viations from the average potentials.

This conclusion is supported by the work of Johnson (9).

CONCLUSIONS

Calculations with the site binding model, based on the assumptions that local activity coefficients near the surface are independent of the charge at the interface, lead to conclu- sions which are at variance with these as- sumptions. ACKNOWLEDGMENTS 30 20 10 0 -10 -20 -30 t'20 -t6

FIG. 4. As Fig. 3, in 0.01 MKNO3 solutions.

The authors thank F. N. Hooge, J. Th. M. G. Klein- penning, and (especially) W. Smit for the very valuable discussions on the subject of this paper.

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