The electrical double layer on oxides: Site-binding in the porous double layer model

Colloids and Surfaces, 51 (1990)

371

371-388

Elsevier Science Publishers B.V., Amsterdam

The electrical double layer on oxides: Site-binding porous double layer model

in the

J. Mieke Kleijn Department Dreijenplein (Received

of Physical and Colloid Chemistry, Wageningen 6,6703 HB Wageningen (The Netherlands) 10 May 1990; accepted

Agricultural

University,

19 July 1990)

Abstract The porous double layer model for the oxide-solution interface introduced by Lyklema is combined with the ionizable surface group model of Healy and White. The influences of the various parameters of the model on the surface charge and surface potential are evaluated and experimental surface charge density and zeta potential data as a function of pH are simulated. The extended porous double layer model describes adequately all properties typical for the electrical double layer on oxides.

INTRODUCTION

The classical Gouy-Chapman-Stern-Graham (GCSG) model for the electrical double layer [ 1 ] describes in an adequate manner the electrochemical properties of, e.g., mercury and silver iodide-solution interfaces. The behavior of oxide-solution systems is significantly different, however, and apparently the GCSG model is not directly applicable. Surface charge measurements on different oxide sols have revealed that oxides generally exhibit surface charge densities which are considerably higher than the maximum of ca 10 PC cm-’ found for mercury and silver iodide. (Several examples are given by Lyklema [ 2,3]. ) The surface charge versus pH curves for all oxides are convex, indicating a progressive increase in H+, respectively OH- adsorption with increasing 1pH -pH” I, pH” being the point of zero charge (P.z.c.). The high surface charges do not lead to particularly high zeta potentials or colloidal stability, as might be anticipated from the GCSG model. Only if for the capacitance of the Stern layer a high value of several hundreds of ,uF cm-’ is chosen, reasonable fits of experimental data can be obtained [ 431. However, this is not in accordance with the physical interpretation of the Stern layer as a charge-free layer with a thickness equal to the radius of a hydrated counterion and a dielectric constant lower than that of the bulk solution.

0166-6622/90/$03.50

0 1990 -

Elsevier Science Publishers

B.V.

372

The Nernst equation can be used to calculate the potential drop over mercury and silver iodide-solution interfaces. It is now generally accepted [ 61 that the surface potential I,U~ of oxides does not a priori obey the Nernst equation ~~=y/~=2.303

(kT/e) (pH’-pH)

(1)

in which k is the Boltzmann constant, T the absolute temperature and e the unit charge. From a thermodynamic point of view, this would be only true if the chemical potential of the potential-determining (p.d. ) ions in the interface would be constant, that is, independent of variations in the surface potential or surface charge. Several authors [ 6-101 have proposed alternative equations for the relation between v. and pH. Various models for the electrical double layer on oxides have been developed, one of the most important being the site-binding or ionizable group model of Healy and White [ 111. This model can, at least qualitatively, account for most of the observed interfacial properties of oxides. However, it does not explain the large charge build-up approaching or even beyond the density of surface -OH groups without a sign of saturation effects, which has been found for a number of oxides. Examples are silica [ 12,131, alumina oxide [ 131, hematite [ 141, tin oxide [ 151, rutile [ 151 and magnetite [ 161. In 1968 Lyklema introduced the concept of a porous double layer [ 21: it is assumed that surface groups as well as counterions are present not only in a two-dimensional array in the surface layer and adjacent liquid layer, but also in a volume phase of finite thickness extending from the surface into the interior of the solid. The counterions that penetrate the solid screen the surface charge effectively and keep the potential relatively low; high surface charge densities can be built up without increasing the colloidal stability. Lyklema elaborated the porous double layer (PDL ) model in a semiquantitative manner, which is summarized below. The dependence of the potential distribution and the penetrated counter charge on several parameters (the penetration depth, electrolyte concentration, adsorbability of the counterion, and surface charge density) was evaluated. The dependence of the surface charge and the surface potential on the concentration of p.d. ions, i.e., on the pH, was not considered however. In this paper a synthesis between the PDL model and the site-binding model is presented. This leads to the introduction of the concentration of p.d. ions and offers the opportunity to verify the validity of the model with experimental data. THEORY

The porous double layer

In the elaboration of the PDL model given by Lyklema [ 21 only negatively charged surfaces and adsorption of positive counterions were considered, but

373

his analysis can equally well be applied to positive surface charges and negative counterions. Another point of departure was that the number of surface charges per unit of volume in the solid (n__ ) depends on the distance from the surface (cc) according to: n_(x) =n_ (0)ecax

(2)

The penetration depth a -’ is assumed to be much smaller than the radius of curvature of the smooth surface (flat double layer). The adsorption of counterions in the surface layer is assumed to obey a Langmuir-type equation. The maximum number of counterions that can be accommodated in a layer of thickness d_rat a distance x from the surface is determined by n_ (x) . Combination with Poisson’s law results in a second-order differential equation for the potential in the porous double layer, the Poisson-Langmuir equation: d2y/d_r2=o!e-U/(1+Be-y)

(3)

in which only dimensionless parameters occur: y is the reduced potential (y = ey/lhT) an d u is the normalized distance to the surface (u = ax). B = (c/ 55.5)exp( -$/lzT), with c the concentration of the counterion in the bulk solution and $ its specific adsorption potential; cy= ze 2n_ (0) / ( eskTa2), with es the dielectric constant of the solid. As boundary conditions to solve the Poisson-Langmuir equation y (co ) = 0 and (dy/du) u_o3= 0 were used. This implies that as a reference point for the potential the bulk of the solid phase is chosen, instead of the bulk of the solution phase, which is more common. Because no boundary condition at x = 0 was used, no connection was made to the diffuse double layer at the solution side of the interface. In his paper, Lyklema gives some numerical solutions of Eqn (3 ), as well as the penetrated counter charge a, as a function of the surface charge a,. With increasing surface charge the quantity o,+ o,, that is, the effective surface charge, becomes independent of 0,. In other words, the adsorption of p.d. ions is eventually compensated by an equal adsorption of counterions. Site-binding in the porous double layer In this paper, for all types of ions the site-binding concept is introduced in the porous double layer model. Now, the total number of binding sites per unit of volume, N(x), is taken to be an exponential function of the distance x to the surface: N(x) =NOeeax

(4)

Consider the amphoteric oxide surface, for which H+ and OH- are the p.d. ions: -OH,+ S-OH+H+

(5a)

374

-OH*-O-

+H+

(5b)

with intrinsic acidity constants K,, and Ka2, respectively. sociate with the binding sites, the p.z.c. is given by pH”= -0.5

If only p.d. ions as-

log(KalKa2)

(6)

The p.z.c. in the absence of specific adsorption is called the pristine point of zero charge (p.p.z.c.) [ 171, a property of the uncontaminated oxide surface itself. Generally, in the site-binding model ions other than H+ and OH- are considered to bind to -OH,+, -OH and -O- sites. For simplicity, it is assumed that such ions bind only to sites with opposite charge sign, which is energetically the most favourable option. Thus, if the supporting electrolyte is designated as MA, consisting of the monovalent ions M+ and A-, the binding sites are in one of the following states: -OH,+, -OH, -O-, -OH2A, or -0M. At distance x from the surface their total number equals N(x): N(x)=n+

(X) +n,(x)+n_

(X) +n*(x)

+&(X)

(7)

Considering the porous double layer as an open system for the various ions present and for energy exchange, one can write down the grand-canonical partition function. Using the “maximum term approximation” (i.e., the expectation value for each type of binding site ni(3t) is approximately equal to that value of ni ( X) corresponding to the maximum term in the partition function), the Bragg-Williams approximation (i.e., the distribution of the ions over the binding sites is not affected by lateral interactions), and the Stirling approximation (In N! = N In N-N), it can be derived that* no(x)/n_

(x) =Ale+‘(X)‘kT

A1 = ad&

(8a)

n, (x)/n_

(x) =A,e-2’W’“)‘kT

A2=a~IKlKa2)

(8b)

nM(x)/n_

(x) =BMe-“v(“)‘kT

BM= (aM/55.5)eeeMlkT

(8c)

nA(x)/n_

(x) =AZBAe-ev(x)‘kT

II*= (aA/55.5)e-@AA/kT

(8d)

aH, aMand aAare the activities of the H+, M+, and A- ions in the bulk solution, respectively; in our calculations oM and aA are simply replaced by c, the electrolyte concentration of the solution. @Mand @Aare the adsorption potentials of the M+ and A- ions. From Eqn (8b) it follows immediately that v/(x) =hT/e){l

n an-O.5

ln(K,1K,2)

-0.5

ln[n+ (x)/n_

(x)]}

(9a)

*For more details see Healy and White [ 111: these authors use a similar derivation in their sitebinding model. Using Langmuir-type equations for the adsorption of cations to -O- and for the adsorption of anions to -OH,’ would give the same results

which can be rewritten, using Eqn (6)) as v(x) =2.303 (M’/e){

(pH’-pH)

-0.5 log[n+ (x)/n-x)]}

(9b)

For x=0 the resulting expression for the surface potential cy, is equivalent to the corrected Nernst equation given by Levine and Smith [ 81 and Wright and Hunter [ 9 1. The relation between the potential v(x) and the space charge density p(x) is given by Poisson’s law: d2~/d.r2= -_P(x)/E~=

- (e/es) [n+(x) -n_

(x)]

(10)

Equations (4), (7), (8) and (10) can be combined to give I_Aze-%““)lkT d2y eNoeVax -l+ (A, +BM +APBA)e-ew(x)‘kT+A2e-2eW(x)‘kT d.X2Q3

(11)

From this second-order differential equation the potential distribution in the porous double layer can be obtained, using the following boundary conditions: 1. The potential in the interior of the solid has a constant value, so (dy/ld-r),+,

(12a)

=0

2. At the interface, the potential distributions in the porous layer and in the diffuse double layer are connected, according to: es (G/h)

xio = CL(W/h

)rto

tL is the dielectric constant of the solution. In reality, es and eLwill depend on x and v(x) will be smooth at x = 0, but, for simplicity, here es and eLare taken to be constant. Using the Gouy-Chapman theory for the diffuse double layer, the second boundery condition becomes: es (dV/ld.r).,,= ( 8eLIZTNAc) “‘sinh (ey/,/BM’)

(12b)

with NA Avogadro’s number. The analytical surface charge a,, which is usually determined by potentiometric acid-base titration of the surface groups, is given by (13) while the penetrated counter charge a, is a,=e

7 [nM(x) -n*(x)

I&

(14)

.X=0

The effective surface charge oeff= a0+ 0,. After solving Eqn (11)) o. and o,

376

can be calculated. The maximum attainable surface charge density is determined by the total number of binding sites in the porous layer per unit area: CJ O,max =e

s

N(x)&=eN,/a

(15)

XC0

Equation ( 11) cannot be solved analytically. Numerical solutions have been obtained using a Digital PDP11/73 computer and a Digital VAX8700/VSM computer. The boundary condition (dy/ldx) = 0 for x-+ GOhas been replaced by (dv/dx) = 0 for x = lOa -I. For simulations of experimental surface charge versus pH curves, an optimization program of Birta [ 181 was modified. The sum )” for all experimental points in a set of titration curves of (~0,calculated ~O,exp was minimized using the above-mentioned VAX computer. RESULTS

AND DISCUSSION

Potential and charge distribution in the porous double layer Figure 1 gives the potential distribution w(x) in the porous double layer for dpH =pH - pH” = 1.5 and three different electrolyte concentrations. Both cations and anions have been assigned a rather high adsorbability (GM= @A= - 4 IzT). The values of the other model parameters are indicated. In the oxide phase, v(x) approaches the values of the Nernst potential V/N.This is in accordance with the experimental fact that the open circuit potential of oxide electrodes often shows (pseudo-) Nernstian behavior [ 19-211. In this case, the surface potential I,Y~ is much lower than VN and strongly depends on the electrolyte concentration. The corresponding distribution of binding sites and -100,

I

I

1

1

%4 _____________--__--_-_---_--,

g(x) (mV)

I

0

1

I 2

I

4

3 normalized

distance

(lx

Fig. 1. Potential distribution in a porous surface layer for three electrolyte concentrations; dpH = 1.5. PDL model parameters: a-‘= 5 8, ApK=pK.,-pK.,=3.6, N,=1.25-1028m-3, &=$*A= -4kT,

ES=15C@

377

0

1

2

L

3 normalized

distance

ax

Fig. 2. Normalized binding site distribution in a porous surface layer; ApH= 1.5 and c=O.l M. The PDL model parameters used are the same as in Fig. 1. ( n _ - n + ) corresponds to the effective space charge density and

(n_ + nM- n, - nA) to the analytical (titratable)

space charge density.

the charge distribution for 0.1 A4 salt are shown in Fig. 2. At the surface, approximately 10% of the binding sites are negatively charged (n_ ), and only ca 0.2% are positively charged (n, ); for increasing distance X, n_ and n, tend to become equal. The fraction of binding sites at which a cation is adsorbed, nM, is rather constant throughout the surface layer and at this dpH value the penetration of anions is negligible (nA g 0). The analytical surface charge is higher than the effective surface charge; in this particular case a,= -7.4 PC! cm-’ and a,,= - 2.9 PC cm-‘. Influence of the model parameters on a0 and t,uo The effects of varying the model parameters a, dpK=p& -p&,N,, @Aand GMon the o,-pH and vo-pH curves are shown in Fig. 3. Increasing the penetration depth a -’ implies that the total number of binding sites and therefore the maximum attainable surface charge also increase: a,,,,, = eNo/a. The surface charge a0 increases less than linear with a - ’ (Fig. 3a). The surface potential is not very sensitive for variation-of the penetration depth (Fig. 3b); v. increases somewhat going from 1 to 5 A, but is rather constant in the range 520 A. Since a0 increases more strongly with increasing penetration depth than cy,, the differential capacitance of the double layer increases. Calculations using dpK= 3,5 and 7 were performed (Fig 3c and d). By increasing dpK, the surface charge and the surface potential decrease to the same extent, so that the differential capacitance of the double layer remains the same. The higher the value of dpK, the less “reactive” is the amphoteric oxide

APH

I

I

-2

,

,

,

,

,

-1

0

1

2

3

2

3

APH

-3

-2

-1

0

1 APH

-100

-3

-2

-1

0

1

2

3

3

APH

surface. For a large dpK, n_ and nM are negligible if 0, B 0, whereas n, and nA can be neglected for o. < 0. This leads to a non-Nernstian response of v/~,which is clear from the r//, (pH) relation given by Eqn (9b): the last term on the right-hand side is relatively large. For small values of dpK the number of charged surface groups n_ and n, is always large compared to their variations with pH. As a consequence the surface potential v. behaves pseudo-Nernstian (see also Ref. [22] ). By increasing the number of binding sites No, both a0 and ry, increase (Fig. 3e and f) and the differential capacitance is constant. The surface charge increases less than linear with No, because the increasing potential inhibits the progressive charge formation in the surface layer. The effect of the total number of binding sites is in accordance with Ref. [ 111, in which it has also been

379

APH

-2

-1

0

1

2

3

2

3

APH

-3

-2

-1

0

1 APH

APH

Fig. 3. Influence of PDL model parameters on $he analytical surface charge a, and the surface potential 1l/0Unless indicated otherwise: a-‘=5 A, &K=3.6, N,,= 1.25*1028m-3, &M=@*= -2kT, es= 15 co. For comparison, in the ~/o-pH plots the Nernst potential I++is also indicated.

shown that potentials at 10 and 20 A out of the surface (in the solution) remain remarkably constant over a wide range of binding site densities, i.e., N,, has only a minor effect on the colloidal stability of the system. Taking the same values for the adsorption potentials for cations and anions, @*=&, the a,,-pH curves and the ~/o-pH curves are symmetrical around the p.z.c. (Fig. 3g and h). If $A and &, are not equal, the curves become asymmetrical and the p.z.c. shifts to higher pH values if the anions are the stronger adsorbates (Fig. 3i). For pH < pH” ,GAis the important parameter, for pH > pH”, & has the greater

380

effect on the analytical surface charge. Only near the p.z.c. the adsorption potentials of both anions and cations are important. If these parameters have a positive value, their effect on o. and y. is small. The ions hardly penetrate into the oxide. However, if they have a negative value, a, can attain high values, whereas I,V~ stays low. At low electrolyte concentrations, the effect of the adsorption potentials is very limited, as can be seen in Fig. 3k and 1. For c-0, cy, approaches the Nernst potential f#N.The a,-pH curves for high electrolyte concentration and negative adsorption potentials are strongly convex. The effect of the parameter tS is not shown, because it is not another independent variable. If eSa and No/a are kept constant, all three of these parameters can be varied without changing the calculated curves. Compensation of the surface charge by counterion penetration The extent of penetration of ions other than H+ and OH- into the porous layer is determined by the penetration depth a-‘, the adsorbability of the ions gA and GM,and their concentration c. The penetration of counterions results in a partial compensation of the surface charge ao. The total number of binding sites increases with increasing penetration depth. The compensation of the surface charge also increases with a - ‘, as can be seen in Fig. 4a. Apparently, the number density of adsorbed counterions, nM for pH> pH” or nA for pH
381 100

I

. & 80-

I

I

@

51 x

ApH=2.5

,=,O 60-

1.5

a

d

1.5

40

20

li_l--:ll 0

10

0.5 40-

05

5

zo-

15

0

20

4

a-’ (A)

10‘)

lo-*

I

I

2

0

-2

-4

-6

-8

OA=eM (kT)

10-l

1

c CM)

Fig. 4. Compensation of the analytical surface charge q by counterion penetration: effect of the penetration depth (a), the specific adsorption potentials (b)_and the electrolyte concentration (c). Default values of the model parameters: a -I= 5 A, &X=3.6, N0=1.25*1028m-3, &,=@*A= -2kT, es=15 co, c=O.l M.

the p.z.c. both counter- and coions penetrate into the porous layer, resulting in low values for a,. Simulation of experimental surface charge data Various sets of titration curves of the oxides hematite (cx-Fe,O,), rutile (Ti02), ruthenium dioxide (RuO,) and silica (SiO,) were obtained from the literature [4,12,14,20] and were fitted with the PDL model. The choice of starting values of the model parameters in the optimization procedure did hardly affect the final simulations. For different sets of initial parameter values, essentially the same optimized values were found. The accuracy of the obtained parameters is determined by the experimental error in the titration data, which

382 0~

x ©

e~

2g

~x

0 ~

.

.

.

~ X . .

.

e~

© Z~

=

©

0

0

~

~MdMd~

.~

~.~

×

oN

.~0~

383

is generally in the order of 0.5 ,d cmV2 [ 221 and, of course, by approximations inherent to the model. The adjustment of the parameters was stopped when the accuracy of the fit was equal to the estimated accuracy in the experimental data. In Table 1 the optimized parameters are given; for all oxides the dielectric constant in the surface layer is taken as 156,, to being the permittivity of a vacuum. (The dielectric constant of solid oxides varies between 5 and 15 co [ 231. ) On the basis of the specific densities and molar masses of the various oxides [ 231 it is possible to estimate a maximum value for No which is physically realistic. This value is 4-7. 1028m-3. According to literature data [ 24,251, the density of -OH groups at the surface of oxides is about 1 -OH group per 20 A”. This corresponds to a maximum value of ca 1028m-3. As can be seen from Table 1, the optimized values of No for all oxides under consideration are in this order of magnitude or lower, the only exception being silica. For each type of oxide, one of the experimental sets of a,-pH curves together with the corresponding calculated curves are displayed in Fig. 5. Breeuwsma and Lyklema [ 141 determined a,-pH curves of hematite in different types of electrolytes. The curves obtained in various concentrations of KC1 showed a common intersection point (c.i.p.) at pH 8.5, which was identified as the p.p.z.c. For both K+ and Cl- we find high positive adsorption

a-

-‘O- @

Fe203/

LlCl “E ” u Y.

-a -5 -

B

/ 3

4

I

I

I

I

I

5

6

7

8

9

10 PH

PH

3

f+

5

6

7

8 PH

9

2345678

9

10

PH

Fig. 5. Experimental surface charge-pH data (points) and PDL model simulations (curves). The optimized PDL model parameters are given in Table 1.

384

potentials, indicating that specific adsorption of these ions on hematite does not occur. With LiCl as the electrolyte, no c.i.p. was found (see Fig. 5a). The p.z.c. shifts to lower pH values with increasing LiCl concentration, from which it was derived that Li+ ions adsorb specifically on this oxide. It is interesting to see that indeed the optimized parameters for the sets of curves obtained in KC1 and LiCl solutions only differ significantly with respect to the adsorption potential of the cation. The adsorption potential of the Cl- ion is in both cases ca 5kT, and all parameters characteristic for the oxide are approximately the same. The maximum attainable surface charge o,,,,, is, according to our calculations, ca 30$ cm-‘, whereas Breeuwsma and Lyklema have found surface charges as high as 70 ,uC cm-’ with Mg(NOB)2 as the supporting electrolyte (like Li+, Mg2+ also adsorbs specifically on this oxide). Penners et al. [20] also measured titration curves of hematite. They were able to reproduce the results of Breeuwsma and Lyklema and showed that the a,-pH curves changed after the hematite sol was purified at pH values below 8. Its p.z.c. shifted by this procedure from pH 8.5 to pH 9.5. The a,-pH curves for this purified sol were also simulated with the PDL model (see Table 1). Because above the p.z.c. the surface charge was only determined over a very limited pH range, the value found for &+ is in this case not very accurate. Experimental a,-pH curves for Ru02 were determined by the present author (Ref. [4] and unpublished results). The curves obtained for different KNOB concentrations showed a c.i.p. at a pH of ca 5.7, which was taken to be the p.p.z.c. From the simulations of different sets of titration curves of RuO, in k+ and gNO,- are found (Table 1)) but in any case KN03, different values for qb both are positive or have a value near 0 kT, which means that there is almost no specific adsorption. Because there was no c.i.p. found for the a,-pH curves of Ru02 in KC1 and the p.z.c. shifts to higher pH values with increasing KC1 concentration, the Cl- ion adsorbs specifically on the oxide surface. The PDL model indeed gives a high negative value for @cl- . The experimental data for Ti02 were taken from Yates and Healy [ 261. The optimized parameters for the a,-pH curves of TiO, in KNOB give an excellent fit (Fig. 5c), but again the value for o,,,,, that can be calculated from these parameters is too small to account for the high surface charge densities measured in other electrolytes [ LiN03, Mg ( NO3 ) 2] [ 261. The set of titration curves for silica in KC1 solutions were obtained by Tadros and Lyklema [ 121. Again, it was possible to obtain a good fit, but the value found for No is too high compared to the density of the silanol groups at the oxide surface. The shape of the titration curves of silica differs from that of the other oxide sols under consideration. This is reflected in the high value of N 7 obtained for dpK, which is comparable to the value found by Healy and White [ 111 using their ionizable group model. For a-Fe203, RuO, and TiOz, dpK is less than 1. The high dpK value for Si02 masks the effect of specific

385

adsorption of the Cl- ions: a shift in the p.z.c. with increasing KC1 concentrations is hardly perceptible. Another eye-catching difference between silica and the other oxides is the value obtained for the penetration depth. For cu-Fe,O,, RuOz and Ti02, CI-~is only l-2 A. The surface roughness on the atomic scale of the oxide particles is in fact sufficient to account for such “penetration” depths. For silica the extent of the porous layer is larger, a - ’ z 7 A. The material used in the study of Tadros and Lyklema was found to be porous with respect to nitrogen gas. This does not a priori imply that the surface would be porous for ions. The latter type of porosity may solely exist in (aqueous) solutions where the surface structure has been affected by processes such as swelling or processes connected with the finite solubility of the solid, whereas the “nitrogen porosity” is measured on dried material. On the other hand, the origin of both types of porosity may be related. Lyklema [3] has shown that indeed for silica the surface charge does increase with increasing degree of “nitrogen porosity”. As an additional check of the computer simulations of the surface charge on -

a-

Fe,O,

I

KC1

-I_

-4o1

\ A

-60

5

’ 3

I 6

I 5

’ 4

I1 7

A I. 9

8

-6O-60

4

I 5

I 6

I 7

8

9

10

II

3

4

PH

5

6

7

8

9

IO

PH

Fig. 6. Comparison between experimental zeta potential points and calculated w( 20 A) -pH curves, using the optimized PDL model parameters from Table 1. The experimental data were obtained from Penners et al. [20] (a-Fe,O,), Kleijn and Lyklema [29] (RuO,), Wiese and Healy [30] (TiOz) and Wiese et al. [31] and Lynskey

[32]

(Si02).

386

the various oxides, experimental zeta potential data have been used. This cannot be more than a global check, because, in general, care should be taken with the interpretation of zeta potential measurements. In all electrokinetic studies, the location of the shear plane is uncertain, and moreover, there are also problems in calculating zeta potentials from measured electrokinetic data. If relaxation and retardation effects are not taken into account properly and if surface conductance effects play a role, zeta potential values that are too low will be obtained [ 27,281. With irregularly shaped particles or heterodisperse samples the calculation of zeta potentials will always be somewhat arbitrary, because both the shape and the size of the particles are important for a correct calculation of zeta potentials. Anyway, considering these uncertainties, we found that the optimized sets of PDL model parameters for all oxides considered in this paper, give a reasonable agreement between the calculated potential of the diffuse double layer at 20 A from the oxide surface and elektrokinetic mobility or zeta potential data available from the literature [ 20,29-321. In Fig. 6 this is shown. CONCLUDING REMARKS

The extended PDL model can account for all properties typical for the electrical double layer on oxides. The model parameters obtained by fitting of different sets of experimental a,,-pH curves also describe experimental electrokinetic data rather well. For the oxides hematite, ruthenium dioxide and rutile, the optimized values for the penetration depth, dpK and the number density of binding sites at the oxide surface are in the same order. This is in agreement with the uniformity of the electrical double layer on these oxides as pointed out by Fokkink et al. [ 331. The low values found for dpK imply that the surface potential v/Obehaves almost Nernstian, which has also been suggested in Ref. [ 321. Surface roughness effects would be sufficient to account for the low penetration depths found for these oxides, i.e., for the small degree of counterion penetration. In fact, also considering that from the PDL fits rather low values were found, the site-binding model [ 111 can describe the principal for HO,,,, features of these oxide-solution interfaces just as well. Silica behaves quite differently from the other oxides under consideration. Contrary to the PDL model, the site-binding model cannot describe its double layer properties in a satisfactory manner, i.e., with one set of model parameters: simulations of surface charge and zeta potential data require different dpKvalues [ 111. Lyklema [3] has shown that, at least for SiOZ and glass, the magnitude of the surface charge parallels the porosity of the surface. Only extensive counterion penetration can account for the low colloidal stability and zeta potentials. Of course, various refinements can be introduced in the PDL model. For example, Lyklema [ 21 assumed that counterion size effects are implicitly taken

381

into account via the parameter a, because a large ion radius virtually corresponds to a low penetration depth. To implement this in the extended model described in this paper, would mean that for every type of ion a different penetration depth should be defined. Although this would allow a,,,,, to vary for different types of counterions, it seems not very useful to introduce more parameters; here, specific properties of the different ions are all taken into account via the adsorption potentials. The usefulness of the addition of a Sterntype inner region in the solution side of the double layer has been examined in more detail. From simulations of experimental surface charge data, it was found that for the capacitance of such a layer very high values are obtained, implying that the effect of such a refinement is negligible. If the binding site distribution in the porous layer is changed from an exponential one into a step function (N(X) =N,, for 0 < x < L and N(n = 0 for x > L), the PDL model becomes in principle the same as the gel layer model of Perram et al. [ 341. In this way also good fits of surface charge measurements can be obtained. The optimized values for the thickness L of the surface layer do not deviate significantly from the penetration depths a -’ found using the exponential binding site distribution; the other model parameters do not change very much either. Introduction of a Gaussian distribution of the binding sites, which requires an extra parameter, does not lead to significantly different results. Apparently, the plain assumption that the surface charge and the counter charge reside partly in the interior of the solid is sufficient to describe the double layer properties of oxides like silica; the way in which the charges are distributed throughout the surface layer seems to be of minor importance. ACKNOWLEDGEMENTS

Although Professor Lyklema was not aware of the preparation of this paper, he has contributed to it by means of stimulating discussions. Mr B.C. Santing is gratefully acknowledged for developing part of the computer programs and examination of the effects of introduction of a Stern layer and alternative binding site distributions.

REFERENCES 1 2 3 4 5 6 7

D.C. Grahame, Chem. Rev., 41 (1947) 441. J. Lyklema, J. Electroanal. Chem., 18 (1968) 341. J. Lyklema, Croat. Chem. Acta, 43 (1971) 249. J.M. Kleijn and J. Lyklema, J. Colloid Interface Sci., 120 (1987) 511. L.G.J. Fokkink, A. de Keizer and J. Lyklema, J. Colloid Interface Sci., 127 (1989) 116. D.N. Furlong, D.E. Yates and T.W. Healy, in S. Trasatti (Ed.), Electrodes of Conductive Metallic Oxides, Part B, Elsevier, Amsterdam, 1981, pp. 367-432. Y.G. BQrubkand P.L. de Bruyn, J. Colloid Interface Sci., 28 (1968) 92.

388 8 S. Levine and A.L. Smith, Discuss Faraday Sot., 52 (1971) 290. 9 H.J.L. Wright and R.J. Hunter, Aust. J. Chem., 26 (1973) 1183 and 1191. 10 T.W. Healy, D.E. Yates, L.R. White and D. Chan, J. Electroanal. Chem., 80 (1977) 57. 11 T.W. Healy and L.R. White, Adv. Colloid Interface Sci., 9 (1978) 303. 12 Th. F. Tadros and J. Lyklema, J. Electroanal. Chem., 17 (1968) 267. 13 J.A. Schwarz, C.T. Driscoll and A.K. Bhanot, J. Colloid Interface Sci., 97 (1984) 55. 14 A. Breeuwsma and J. Lyklema, Discuss Faraday Sot., 52 (1971) 324. 15 S.M. Ahmed and D. Maksimov, J. Colloid Interface Sci., 29 (1969) 97. 16 SM. Ahmed and D. Maksimov, Can. J. Chem., 46 (1968) 3841. 17 J. Lyklema, J. Colloid Interface Sci., 99 (1984) 109. 18 L.G. Birta, Simulation, 28 (1977) 113. 19 G. Lodi, G. Zucchini, A. De Battisti, E. Sivieri and S. Trasatti, Mater. Chem., 3 (1978) 79. 20 N.H.G. Penners, L.K. Koopal and J. Lyklema, Colloids Surfaces, 21 (1986) 457. 21 A. Daghetti, G. Lodi and S. Trasatti, Mater. Chem. Phys., 8 (1983) 1. 22 L.K. Koopal, W.H. van Riemsdijk and M.G. Roffey, J. Colloid Interface Sci., 118 (1987) 117. 23 R.C. Weast (Ed.), CRC Handbook of Chemistry and Physics, 58th edn, CRC Press, West Palm Beach, 1977. 24 J.J. Fripiat, A. Gelli, G. Poncelet and J. Andre, J. Phys. Chem., 69 (1965) 2185. 25 J.J. Jurinak, Soil Sci. Sot. Am. Proc., 30 (1966) 559. 26 D.E. Yates and T.W. Healy, J. Chem. Sot. Faraday Trans. 1,76 (1980) 9. 27 R.W. O’Brien and L.R. White, J. Chem. Sot. Faraday Trans. 1,74 (1978) 1607. 28 R.J. Hunter, Zeta Potential in Colloid Science, Academic Press, London, 1981, p. 386. 29 J.M. Kleijn and J. Lyklema, Colloid Polym. Sci., 265 (1987) 1105. 30 G.R. Wiese and T.W. Healy, J. Colloid Interface Sci., 51 (1975) 427. 31 G.R. Wiese, R.O. James and T.W. Healy, Discuss Faraday Sot., 52 (1971) 302. 32 B. Lynskey, Thesis, Liverpool Polytechnic, 1972 (from Ref. [6] ). 33 L.G.J. Fokkink, A. de Keizer, J.M. Kleijn and J. Lyklema, J. Electroanal. Chem., 208 (1986) 401. 34 J.W. Perram, R.J. Hunter and H.J.L. Wright, Aust. J. Chem., 27 (1974) 461.