electrolyte solution interface

The TiO2/Electrolyte Solution Interface II. Calculations by Means of the Site Binding Model M. J. G. JANSSEN AND H. N. STEIN

Laboratoryof ColloidChemistry,EindhovenUniversityof Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlands Received May 4, 1985; accepted August 30, 1985 Starting from experimental Croand ~"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 coefficientsof complexesand free surface hydroxyl groups (YMOn~Ca-/~/MOnand "~MO-H+/~MOtt)are thought to be independent of the charge in the adsorption plane, the associationand dissociationconstant of surface hydroxylgroups turn out to be pH-dependent. This is ascribed to the influence of local deviation from the averagevalues of the potential at a certain distance from the surface. Neglect of this effectleads to internal inconsistencies of the model. © 1986AcademicPress.Inc. INTRODUCTION

In a previous paper (1), data were presented on the colloid chemical properties of dispersions of TiO2 in various aqueous solutions. The data comprised surface charge, ~"potential and stability toward coagulation. The main result of this investigation was, that for pure TiO2 the colloid chemical properties mentioned are not influenced by changes in solidstate 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 conductivity apparently is still too low to influence adsorption equilibria; thus TiO2 can be regarded, as far as surface charge and ion adsorption 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.

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 described by association/dissociation equilibria of surface hydroxyl groups MOH: MOH{2+)~-0 M O H + H~+) M O H *-~ M O (-) + H~+).

[ll [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~+) M O H ~ C l - *-+ M O H + H~+) + 0 2 .

[3] [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]

112 0021-9797/86 $3.00 Copyright© 1986by AcademicPress,Inc. All ~ghf©nfr~nrrutuctinnin any formreserved.

Journal of Colloid and Interface Science, Vol. l 11, No. 1, May 1986

TiO2 ELECTROLYTE SOLUTION

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 following definitions are employed: pQ~+ = pH + log "¢K~(K~) -- log

[MO-K +] [MOB]

[81 PQ~I- = pH - log 'ycg[Clg] [MOH] - log [MOH~-C1-] o~ = I([MO-K +] - [MOH~C1-])/Nsr

[9] [10]

with Ns = the total number of sites per unit surface area. Recently, Johnson (9) formulated the sitebinding 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 solution 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~,, appear 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

113

at the phase boundary and at distances/3 or /3' from it; the local deviations from those average values are then accounted for by the surface 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 behavior of activity coefficients, which are introduced during the calculations, must be regarded with caution. Although the site binding model has frequently been applied in recent years, it is not the only theory which aims at describing surface charge generation at oxide/electrolyte solution 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 solution; 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 deviations 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. METHODS

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 interface TiO2 M808/electrolyte solution are considered to be too much influenced by chemisorbed impurities to be amenable for a simplified model of this interface. From the ~"potential, the net charge behind the electrokinetic slipping plane (a~) was calJournal of Colloid and Interface Science, Vol. ! 11, No. l, May 1986

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 electrophoretic mobilities by the formula for Ka > 1, on the basis of the argument that in electrophoresis the movements of flocs rather than of primary particles is followed (1)). For comparison's sake, we calculated o r also by the approximate relation (19)

kT

o f = 4"n'%er - ze

Ka 2

ze¢+ 4 , ze¢'] -Ka -tann

4~J

2 sinh 2kT

×

[lll

with a = the primary particle radius as calculated 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-] -

[MO-]

-

[MO-K+]}

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

[131 [141

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

K~+ =

K~,

+ [MO-K+] + [MO-I

[151

-rMO-K+[MO-K+ITH~[H~+1 7MoH[MOH]TK:[KI~]

[16]

" Y M O H [ M O H ] y H + [ H +] =

YMOH~cr[MOH~CI-] [MOH]YHg[H$] Kal = [MOH~-]

K,~ =

[MO-]y~:[Hs+]

[MOH1

together with [5]-[7]. Journal of Colloid and Interface Science, Vol. 111, No. 1, May 1986

In Eqs. [13] and [14], e is the charge of a proton. [MOH~], [MOH~C1-], etc. are numbers 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 Kc,- =

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

[MOH~CI-]

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

"/MOH

e x p ( - ~T (¢~' - ~0))Ko-

YMOH~-C1-

[211 where YMOr~, '~MOI-I~CI-and ¢~, - ¢0 values should be substituted valid at a~ = 0. Thus, Kcl- will be equal to the thermodynamic 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 [MO-K +] = [MOH~C1-] which gives a~ = 0. This situation will arise whenever the tendencies of the

blo

alO,

p~

[17]

pa~. k

T

[18] [191

[20]

ct,,¢.102

Ct#X-lO2

L FIG. 1. pQ~- and pQ~+ vs

aa

in KCI solutions.

TiO2 ELECTROLYTE SOLUTION 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 0c~

O(

'~MOH~C1- /

Oo-0

e

2.3kTC_0o~

- 0o~a log YMOH /

[22]

1 15

(3) With the values of C+ and 6"_ thus obtained, and substituting KK* for K~+ and Ko- for K~>, we solved the equations mentioned analytically following the method outlined by Dousma (22) and Bousse (23). As adjustable parameters we used the differential Stern capacitance CSte~n ---- & o / d ( ~ o - ~d)

[25]

with C_ = ao(ffo - ff~'), considered to be inwith ~bd = ~', and/fat as given by Eq. [18]. dependent of a~. If in addition YMOH~C>/ Note that we have only two adjustable pa"YMOH is thought to be independent of aa as rameters since well, we can calculate C_ from (pKa~ + PK~2) = 2*pHpzc [26]

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 KK+ =

[MO-K+][H~]yH¢ [MOH][Kg]3,Kg

differing from K~+ because

(e

[23]

)

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.

blo

10• 8t

Piq

°'l" 2~

pQ~. 8

, o ~ "~

2 2 2

0

2

t,

6

8

1'0

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

from Eqs. [18] and [19]. The adjustable parameters were chosen such as to obtain a fit to the experimental Croand ~" values. RESULTS Table I contains the values for the adjustable parameters giving the best fit to our data, together 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 ~'potential 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. Journal of ColloM and Interface Science, Vol. 111, No.

1, May 1986

JANSSEN AND STEIN

116

TABLE I Parameters Used in the Model Calculations TiO2 in KCI Solutions

TiO2 in KNO3Solutions Parameter

Unit

N~ C+ C_

rn -2 F. rn -2 F. m -2

K~o; or Ko-

M -2

Value

Remarks

1.2 X 10 '8 1.4 + 0.3 0.9 + 0.4

Ref. (6)

(2.5 _ 0.1) × 10-6 9.1

pKal

10-3 M; pH < pHp~ 10-3 M; pH > pH~c 10-2 M; pH < pHp~ 10-2 M; pH > pHp= 10-1 M; pH < pH w 10-1 M; pH > pHp~

10.1 7.4 9.5 6.8 8.5 F. m -2

Cstern

0.06

10-~ M 5 ~ < p H ~< 10 10-2 M 5 ~ < p H ~< 10 10-1 M 6~
0.16 0.6

¢ /

(mVl

/ o

'~ 1-; -1 -2(

)o"

'°

o/°

-'20

51c~ -%

-

08 "0~-

~

Refi (6)

6.8 9.6 6.5 8.9

0.16 0.6

10-2 M; pH < PHpzc 10-2 M; pH > pH w 10-1 M; pH < p H ~ 10-1 M; pH > pHp~

10-2 M 5 ~ < p H ~< 10 10-I M 5 ~ < p H ~< 10

DISCUSSION

-12

/

T \. \°

£

1.2 X 1018 1.5 _ 0.3 0.9 +__0.4

Remarks

(4 + 0.2) X 10-8 (2 + 0.1) × 10-6

(5 + 0.2) X 10 -8

--

Value

;

o/ '08

-3C

FIG. 3. Surface charge density (©) and zeta potential (e) of TiO2 in 0.1 M KNO3 solutions. Journal of Cofloid and Interface Science, Vol. 11I, No. 1, May 1986

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 in t h e p r e c e d i n g s e c t i o n , especially 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 are 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 are dep 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 the true thermodynamic constants K~ by 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 coefficients are 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 exp 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 for ~/MO-/3'MOn. I n itself this 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 coefficients o f c h a r g e d g r o u p s are 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 units, w h i l e near a charged interface local concentrations o f t h e l a t t e r are high.

TiO2 ELECTROLYTE SOLUTION

1 17

During the calculations the assumptions 4.0 had been introduced that '~MO-K+/~MOH and (mV) 30 -~2 3'MOH~CV/3'MOHare independent of the ionic 20 08 environment near the surface. This was introduced both in the calculation of C_ and C÷ • Ot~ 10 from the slope of the lines in Figs. 1a and b, o , ° ~ , -~---~pH 0 respectively, and by the use of KK+ and K a in calculations for conditions at which aa = 0. -I0 \ Introduction of the independence of \. 2C 7MO-K+/~MOH and 7MOH~CI-/'~MOHof the ionic \. "30 environment leads to a conclusion, which is at variance with the same assumption for FIG. 5. As Fig. 3, in 0.001 MKNO3 solutions. the analogous case of '~MOH/'YMOH~ and "YMOH/'YMo-• There is a difference, to be sure, between Thus, it appears that the method to correct the cases of'~MOH~C1-/~YMOHand '~MO-K+/'~MOH for local activity coefficients near the surface, on the one hand, and '~MO-/'~MOH and 3~MOU~/ by extrapolating pQ* to o~a -- 0, is insufficient "YMOHon the other: the former refer to groups to take into account the influence of local dewhich, as a whole are uncharged while the latviations from the average potentials. ter refer to groups which do carry a charge. It This conclusion is supported by the work should be realized, however, that MOH~-CIof Johnson (9). and M 0 - K + are dipoles rather than uncharged groups: the charges of the CI- and the K + ions CONCLUSIONS are in the model considered to be at some disCalculations with the site binding model, tance from the phase boundary, where other based on the assumptions that local activity average potentials (¢~,, and ¢8) prevail than at the surface itself (~0), and where local de- coefficients near the surface are independent viations from these average potentials certainly of the charge at the interface, lead to concluwill differ from similar deviations at the sur- sions which are at variance with these asface. MOH, on the other hand, should be re- sumptions. garded as an uncharged group. ACKNOWLEDGMENTS

\.

~

t'20 -t6

o:

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

30

REFERENCES

20

1. Janssen,M. J. G., and Stein, H. N., J. Colloidlnterface Sci. 109, 508 (1986). 2. Rendall, H. M., and Smith, A. L., J. Chem. Soc. Faraday Trans. L 74, 1179 (1977). 3. Healy, T. W., Yates, D. E., White, L. R., and Chan, D., J. Electroanal. Chem. 80, 57 (1977). 4. Levine, S., and Smith, A. L., Disc. Faraday Soc. 52, 290 (1971). 5. Healy,T. W., and White, L. R., Adv. Colloidlnterface Sci. 9, 303 (1978). 6. Davis,J. A., James, R. O., and Leckie,J. O., J. Colloid Interface Sci. 63, 480 (1978).

10 0 -10 -20 -30 FIG. 4. As Fig. 3, in 0.01 MKNO3 solutions.

Journal of Colloid and ]nterface Science,

Vol.111,No. 1. May1986

1 18

JANSSEN AND STEIN

7. Drzymala, J., Lekki, J. O., and Laskowski, J., Colloid Polym. Sci. 257, 768 (1979). 8. Bowden, J, W., Bolland, M. D. A., Posner, A. M., and Quirk, J. P., Nature Phys. Sci. 245, 81 (1973). 9. Johnson, R. E., Jr., J. Colloid Interface Sci. 100, 540 (1984). 10. Onoda, G. Y., and Casey, J. A., J. Colloid Interface Sci. 103, 590 (1985). 11. Smit, W., and Holten, C. L. M., J. Colloid Interface Sci. 78, 1 (1980). t2. Lyklema, J., Croat. Chem. Acta 43, 249 (1971). 13. Lyklema, J., J. Electroanal. Chem. 18, 341 (1968). 14. Perram, J. W., Hunter, R. J., and Wright, H. J. L., Aust. J. Chem. 27, 461 (1974). 15. Siskens, C. A. M., Stein, H. N., and Stevels, J. M., J. Colloid Interface Sci. 52, 251 (1975). 16. Van Diemen, A. J. G., and Stein, H. N., J. Colloid Interface Sci. 67, 213 (1978).

Journal of Colloid and Interface Science, Vol. 111. No. 1, May 1986

17. Stein, H. N., Adv. ColloidlnterfaceSci. 11, 67 (1979). 18. Overbeek, J. Th. G., in "Colloid Science I" (H. R. Kruyt, ed.), p. 180, relation [48]. Elsevier, Amsterdam, 1952. 19. Loeb, A. L., Wiersema, P. H., and Overbeek, J. Th. G., "The Electrical Double Layer around a Spherical Particle." MIT Press, Cambridge, 1961. 20. Janssen, M. J. G., "The Titanium Dioxide/Electrolyte Solution Interface." Ph.D. thesis, Eindhoven, 1984. 21. Harned, H. S., and Owen, B. B., "The Physical Chemistry of Electrolyte Solutions." Reinhold, New York, 1967. 22. Dousma, J., "A Colloidal Chemical Study of the Formation of Iron Oxyhydroxide." Ph.D. thesis, Utrecht, 1979. 23. Bousse, L., "The Chemical Sensitivity of Electrolyte/ Insulator/Silicon Structures." Ph.D. thesis, Enschede, 1982.