The metal oxide - electrolyte solution interface revisited

Advances in Colloid and Interface Science, 28 (1988) 111-134

111

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

THE METAL OXIDE - ELECTROLYTESOLUTION INTERFACE REVISITED MIGUEL A. BLESA Departamento Quimica de Reactores, Comisi6n de Energ~a At6mica, Avenida del Libertador 8250, 1420 Buenos Aires, Argentina and NIKOLA KALLAY Laboratory of Physical Chemistry, Faculty of Science, University of Zagreb, Maruli~ev trg 19, 41001Zagreb, P.O. Box 163, Yugoslavia

CONTENTS I

ABSTRACT ........................................................

111

II

INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112

III

NERNSTIANAPPROACH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

IV

SURFACECOMPLEXATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

114

V

ASSOCIATIONOF COUNTER IONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

VI

CAPACITORS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

120

VII

INTERPRETATIONOF EXPERIMENTALDATA . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122

V I I I CONCLUDINGREMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

128

IX

APPENDIX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131

X

REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

132

I

ABSTRACT The Nernstian approach to the description of the m e t a l / e l e c t r o l y t e int er f ac e

is compared with descriptions based on the surface complexation models.

Dif-

ferences are from the assumptions made about the dependence of chemical potent i a l of the p o t e n t i a l determining ions ( p . d . i . ) at the s o l i d surface on the composition. The Nernstian approach is based on the constancy of the chemical p o t e n t i a l of p . d . i , a t / i n the s o l i d surface while the surface complexation model takes i n t o account the v a r i a b l e composition of the surface through the surface complexation of a l i m i t e d number of surface s i t e s .

The Nernst approach leads to

a l i n e a r r e l a t i o n s h i p between the surface p o t e n t i a l and the logarithm of p . d . i . a c t i v i t y with slope RT(In 10)/6 independent of i o nic strength.

On the other

hand, values of the surface p o t e n t i a l s calculated using the surface complexa-

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© 1988 Elsevier Science Publishers B.V.

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tion model are always lower in magnitude and decrease as the ionic strength increases.

The nature of the association of counter ions with the charged surface

sites and the problems connected with the introduction of planar capacitors characterised by the constant capacitance are discussed.

I t is concluded that

the present experimental data do not warrant the application of elaborate modelling and that f u r t h e r e f f o r t s to understand the nature of the e l e c t r o l y t e / h y drous oxide surface should focus on better and more diverse experimental studies. II.

INTRODUCTION The description of metal o x i d e / e l e c t r o l y t e solution interfaces has been the

subject of numerous experimental and theoretical studies (refs 1-20). Unfortunately, the majority of experimental evidence comes from one technique only ( r e f . 12); namely, the acid base t i t r a t i o n experiments.

These experi-

ments y i e l d the surface excess concentration of protons on various oxides as a function of solution variables (pH, ionic strength, nature of e l e c t r o l y t e ) and temperature.

Less popular e l e c t r o k i n e t i c measurements, which also give some

i n s i g h t into the d i s t r i b u t i o n of charged species in the e l e c t r i c a l double layer (ref. 21) have also been employed. Some f u r t h e r information has been gathered on the basis of spectroscopic measurements of adsorbed species (refs 22-26), and e l e c t r o l y t e adsorption measurements ( f o r both H+ and " i n e r t " ions) with radiotracers (refs 27-32). A general lack of d i r e c t experimental information is overcome by such techniques as Cylindrical Internal Reflection - FTIR Spectroscopy which yielded the states of ions in the i n t e r f a c i a l layer enabling a c r i t i c a l evaluation of assumptions, which are the basis of models describing the double layer e q u i l i b r i a ( r e f . 26). In this paper we examine and contrast some of these models paying p a r t i c u l a r attention to comparisons between "Nernstian" potentials and potentials calculated using surface complexation models. The thermodynamic description of adsorption should s t a r t from the Gibbs adsorption isotherm and i t s application to multicomponent systems. This subject w i l l be discussed in a separate a r t i c l e ( r e f . 33).

The purpose of the present

study is to analyze the i n t e r r e l a t i o n s h i p and contradictions (even incompatib i l i t i e s ) between the usual procedures describing the structure of the double layer in terms of potentials, charges and capacitances of various assumed layers. Two d i f f e r e n t thermodynamic approaches are often used in the l i t e r a t u r e to describe e q u i l i b r i a in the e l e c t r i c a l double layer.

The f i r s t

originates from

the treatment of the classical s i l v e r iodide c o l l o i d system and results in a Nernstian relationship between the a c t i v i t y of potential determining ion and the surface potential.

The second approach (surface complexation or s i t e binding)

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was developed s p e c i f i c a l l y f o r metal oxide interfaces.

Various models are used

which d i f f e r in some important aspects, but a l l of them can be traced back to the o r i g i n of surface charge (the same chemical phenomenon) namely, the amphoter i c behaviour of surface -OH groups.

In t h i s approach, the expressions derived

f o r the potential affecting charged surface sites are not l i n e a r with respect to the logarithm o f e c t i v i t y

of p . d . i .

(pH), but instead produce S-shaped curves.

The calculated surface potentials are always lower than those given by the Nernst equation but higher than the ~-potentials (refs 34-36). The d i s t i n c t i o n between H+ and OH- as potential determining ions from other ions i . e . " i n d i f f e r ent" ions and " s p e c i f i c a l l y adsorbed" ions is somewhat a r t i f i c i a l .

For the sake

of uniformity, H+ and OH- ions w i l l be considered to be the p.d. ions in both approaches. III.

NERNSTIANAPPROACH

The equilibrium d i s t r i b u t i o n of potential determining ions (e.g. H+ and OHf o r oxides, Ag+ and I - f o r Agl) between the solution bulk (b) and the i n t e r f a cial layer ( I ) is described by equating the electrochemical potentials in the two phases Ub and Ul (refs 2,37). ~b(H+) = U°b(H+) + RT In aH+ = ~I(H +) = Ul(H +) + F~0

(i)

In Eq ( I ) , 4o is the e l e c t r o s t a t i c potential difference between the interface and the solution bulk.

This relationship leads to Nernstian behaviour provided

that the chemical potential of H+ in the i n t e r f a c i a l layer, Ul' is constant. This imposes certain special conditions such as constant composition of i n t e r face with respect to H+.

In terms of mole f r a c t i o n , ×, of i n t e r f a c i a l H+ ions

~I(H +) = ~°I(H+ ) + RT In xI(H +)

(2)

The a p p l i c a b i l i t y of the Nernst equation implies that ×I(H +) has to remain s u f f i c i e n t l y constant upon v a r i a t i o n of aH+ (and/or 4o) (refs 2,8). binding approach f o r oxides both FH+ and

In the s i t e

FOH- should be large with respect to

the excess rH+ - rOH- even in the i s o e l e c t r i c region where FH+ ~ FOH-. Such a behaviour is more reasonable f o r s i l v e r iodide than for metal oxides. Due to the large amount of p.d. ions in the Agl crystal l a t t i c e an excess of Ag+ with respect to I - would not s i g n i f i c a n t l y change the mole fraction of Ag+. However, f o r metal oxides H+ and OH- ions are not constituent ions. The charged sites are d i s t r i b u t e d along the surface among a l i m i t e d number of act i v e surface s i t e s . With these assumptions concerning the mole f r a c t i o n of H+ in the surface phase we can return to Eq ( I ) by rewriting i t in a more convenient form

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(Nernst type) : O

~b (H+) - ~I (H+) 40 =

F

+ RT In aH+ F

(3)

At ~o = 0; pH = pHo and in the simplest of cases pHo corresponds to the isoelectric point (i.e.p.) and point of zero charge (p.z.c.); pHo = pH(iep) = pH(pzc) so that ~o = RT In (aH+/aH+(iep)) = RT In 10 (PHo _ pH) F F

(4)

An analogous result is obtained when considering the distribution of OH- ions 4o .

~(0H-) . -. ~l(0H-) . F

RT . In K~ + RT In aH+ F F

(5)

where K~ is the standard equilibrium constant for the dissociation of water (K~ = aH+ , a0H-). By setting 4o = 0 at the i.e.p, one obtains Eq (4). Since there exists an interrelationship between H+ and OH- activities, i t is possible to write just one equation to describe the equilibrium raH+ (~°(H+) - ~°(OH-))b + RT In ~-~OH_Jb = (~(H +) - ~(OH-)) 1 + 2F ~o

(6)

which is equivalent to

¢o : A(~°(H+) - ~°(OH-)) + 2FRTIn (~0~_)b

(7)

The advantage of Eq (7) over Eq (3) is the symmetrical treatment of H+ and OH- ions that makes i t more suitable to describe experiments in which K~ varies, e.g. owing to temperature variation or in studies with mixed solvents (refs 2, 38-40). Equation (7) is the Nernst type equation and assumes constant mole f r a c t i o n s of p.d. ions in the i n t e r f a c i a l IV

layer so that ~I = ~!"

SURFACECOMPLEXATION Rather than simply accepting the accumulation of charged species at the interface, the surface complexation or site binding approach assumes that a specific

115

chemical reaction generates t h i s surface charge.

This proposed reaction i n -

volves amphoteric surface s i t e s , whose number is l i m i t e d and determined by the surface s t r u c t u r e .

Charging of the i n t e r f a c e is commonly described by the f o l -

lowing reactions (refs 1-20). MOH + H+ ~ MOH ~

MOH~

(8)

MO- + H+

(9)

where M denotes a metal ion in the s o l i d surface. In order to be more general we may w r i t e the equations in a d i f f e r e n t form: S + H+~

SH+

S + OH-~

(10)

SOH"

(11)

where S is an active surface s i t e (e.g. MOH). One common approach would be to consider the reactions (8,9 or 10,11) as two step processes, thus introducing an " i n t r i n s i c state" (refs 9-11).

For the re-

action (10) the two steps would be

H+(b)~

H+(I) (12)

H+(1) + S(1) ~"--"~=-SH+(1) The f i r s t

step is t r a n s f e r of H+ from s o l u t i o n bulk to the i n t e r f a c e ( " i n -

t r i n s i c s t a t e " ) ; the corresponding e q u i l i b r i u m is described by the Boltzmann distribution. The second step is the formation of a chemical bond between the active surface s i t e and the " i n t r i n s i c " H+ ion. called i n t r i n s i c e q u i l i b r i u m constant.

The l a t t e r step is characterised by a so The i n t r o d u c t i o n of the two step process

is unnecessary since relevant information is not a v a i l a b l e .

Moreover, the " i n -

t r i n s i c state" is only h y p o t h e t i c a l ; i t assumes that the bond is formed without the change in the p o s i t i o n of H+; i . e . that there is no change in the e l e c t r o s t a t i c potential during the complexation process.

Thermodynamically, t h i s pro-

cedure is correct and i t is equivalent to s p l i t t i n g the Gibbs energy of reaction i n t o two parts: the e l e c t r o s t a t i c and the chemical c o n t r i b u t i o n s . The e l e c t r o s t a t i c c o n t r i b u t i o n can be interpreted also in terms of a c t i v i t y c o e f f i c i e n t s as discussed by Sposito ( r e f . 41).

Rather than introducing the " i n

t r i n s i c state" we w i l l consider e q u i l i b r i a which involve the reactants and f i n a l product only, i . e . Eqs (10,11).

The composition of the surface w i l l be express-

ed in terms of (mole) f r a c t i o n s , ×, with respect to the t o t a l number of surface

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sites while the standard state w i l l be defined by x° = I.

This does not i n f l u -

ence the values of the parameters (equilibrium constants).

The common surface

concentration r (in mol m-2) w i l l be also used. For reaction (10) at equilibrium: u°(H +) + RT In aH+ + u°(S) + RT In x S = ~°(SH +) + F~o + RT In XSH+

(13)

From Eq (13) the common expression for the surface equilibrium constant is obtained ,int KSH+ = ~SH+ • exp (-F~o/RT) .

xSH+ rSH+ . . . x s • aH+ r s • aH+

(14)

Since ~o = 0 is chosen to c h a r a c t e r i z e the standard state at the surface, Ki n t becomes the standard (thermodynamic) surface e q u i l i b r i u m constant and is r e l a t e d to the standard chemical p o t e n t i a l s by ,int -RT In ~SH+ : ~O(H+) + ~°(S) - ~°(SH +)

(15)

A s i m i l a r treatment f o r binding of OH ions Eq (11) leads to KSOH-

= vint "SON- " exp (F¢o/RT) -

XSOH-

-

x S • aOHFrom Eq (14) the surface p o t e n t i a l , tential

~o

rSOH-

(16)

r S • aOH-

or more p r e c i s e l y the e l e c t r o s t a t i c

po-

i n f l u e n c i n g the s t a t e of charged surface complexes is

RT In vint RT Xs KSH + + - - In F F xSH+

= --

+

RT In aH+

(17)

Equation (17) would express Nernstian behaviour only i f the ratio of the fractions of free to charged surface sites was constant. These two fractions could be constant i f the amounts of p.d.i, in the interfacial layer were much higher than the changes caused by the charging process (see ref. 42). However, for metal oxides 3-10 active sites per 100 X (I nm2) are usually found. In the experimental pH range up to 60% of the sites may be charged. In principle, the surface complexation approach can be applied to s i l v e r iodide with two different sets of active surface sites, one for Ag+ adsorption ( I - from the solid surface) and another one for I- adsorption (Ag+ from the solid surface). In this approach the j u s t i f i c a t i o n of the Nernst equation is not straightforward (refs 42,43).

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V

ASSOCIATIONOF COUNTER IONS The i n t e r p r e t a t i o n of adsorption data f o r various metal oxides leads to d i f -

f e r e n t values of the i n t r i n s i c e q u i l i b r i u m constants, defined by Eqs (14 and 16) f o r reactions (10,11) r e s p e c t i v e l y , when i o n i c strength is changed.

This d i s -

crepancy cannot be explained in terms of the change of a c t i v i t y c o e f f i c i e n t s ; i t suggests the adsorption of counter ions in the Stern layer (e.g. refs 9-12, 16, 44), i . e . t h e i r association with charged surface sites.

Other e f f e c t s , such as

counter ion s p e c i f i t y also provide evidence f o r such i n t e r a c t i o n s .

The l y o t r o p i c

e f f e c t is not very pronounced in the adsorption measurements (refs 44,45) but i t is s i g n i f i c a n t in coagulation k i n e t i c s which is also related to the double layer equilibrium.

The c r i t i c a l

coagulation concentration of e l e c t r o l y t e s was found

to be dependent on the size of counter ions f o r d i f f e r e n t c o l l o i d systems (e.g. Agl, AgBr, AgCl, Fe203, e t c . ) ( r e f s 46-50).

Since the counter ion size can hard-

l y be e f f e c t i v e in the d i f f u s e part of the double layer i t was necessary to assume that a s i g n i f i c a n t portion of counter ions was involved in close i n t e r a c t i o n s , such as association with charged species on the surface.

The association

of counter ions in the e l e c t r i c a l double layer is commonly described by the f o l lowing equations (refs 7,9-12). SH+ + A--.-T-~ SOH- + M÷ ~

SH'A

(18)

SOH.M

(19)

where A- and M+ denote anions and cations (other than OH- and H+), respectively. The schematic representation of the double layer in the a c i d i c region is shown in Figure I . The common approach takes into account the i n t r i n s i c e q u i l i b r i u m constants and the exponential term including the e l e c t r o s t a t i c potential in the B-plane (~B) where the associated counter ions are assumed to be located. the B-plane is f u r t h e r discussed below.

Accordingly:

.int KSH.A = ~SH.A " exp (~BF/RT) .

.

xSH.A . . XSH+'aA-

KSOH.M

= ~int "SOH.M " exp (-~BF/RT) =

The nature of

FSH'A

(20)

?SH+.aA-

XSOH-M XSOH-.aM+

=

FSOH.M

(21)

TSOH-'aM+

In the f o l l o w i n g paragraph we shall analyze these e q u i l i b r i a in the same way as i t was done f o r the binding of H+ and OH- ions at the 0 surface plane Eq (13). For the reaction (18) ~°(SH +) + F~o + RT In xSH+ + ~°(A-) + RT In aA- = ~°(SH.A) + ~eI(SH'A) + + RT In xSH.A

(22)

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~iiiiiiiiiiiii!iiii Q; ch

0 plane

Fig, I. Schematic representation of the metal oxide/electrolyte solution interface: O oxide ions, ~ M 2+ ions,

• H+ ions,

(~ counterions (anions).

This simplified scheme views a [100] face of a cubic metal oxide surface, and assumes complete hydroxilation of a l l exposed oxide ions. The potential ~o appears in Eq (22) because i t determines the state of the SH+ site which is located in the 0 plane. The quantity Pel(SH-A) is the contribution to the chemical potential caused by the effect of electric f i e l d on the ion pair; at ~o = O, Pel (SH'A) = O. The following analysis w i l l be based on the treatment introduced by Yates, Levine and Healy (ref. 7). The electric f i e l d contribution to the chemical potential of SH.A species Uel(SH.A) w i l l be calculated by treating the ion pair (at the surface) as a dipole exposed to an electric f i e l d . The energy is given by the product of the dipole moment and the electric f i e l d strength (-p.E). The dipole moment and electric f i e l d strength are related to the distance between separate charges +e and -e and to the local potentials ~+ and ~. acting on positive and negative sides of the dipole p =

e'b ;

E = (~+ - ¢_)/b

(23)

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Since the difference in local potentials is equal to the difference of the potentials in the 0 and B-planes, where the separated charges are located, we obtain ~eI(SH.A) = F(~ ° - ~B )

(24)

By introducing (24) into (22) one obtains ~°(SH+) + RT In XSH+ + p°(A-) + RT In aA- = ~°(SH'A) - F~B +

(25)

+ RT In xSH.A

which is consistent with (20) and gives the relationships between the i n t r i n s i c equilibrium constant and standard chemical potentials i n t = uo(SH.A) _ o(SH +) _ pO(A- ) -RT In .mSH'A

(26)

This analysis enables the correlation of the two apparently d i f f e r e n t approaches to the equilibrium in the Stern layer of ionic c r y s t a l s , i . e . the common model based on the i n t r i n s i c equilibrium constants (refs 9-12) is consistent with the model based on the assumption that the ion pair is acting as a d i pole at the solid l i q u i d interface ( r e f . 7). However, both approaches leave several questions unresolved. As defined by Eqs (20,26) .mSH'A i n t should have a r e l a t i v e l y high value in order to s a t i s f y the experimental data which implies that association should occur even when ~o = O.

Calculations for the magnetite/

KNO3 system indicate that at high ionic strengths ion pairs are more abundant than free charged sites (refs 34,44).

However, one would expect association

in the double layer only f o r those pairs of ions which associate in the bulk of solution.

The association of CI04 with SH+, thus, would not be in agreement i n t and with the strong a c i d i t y of perchloric acid. Instead, s i g n i f i c a n t .mSH.A Kint values were obtained even when experimental evidence f o r the solution SOH.M does not support this hypothesis. The assumption of some (additional) kind of chemical bond is ruled out by spectroscopic studies.

Tejedor-Tejedor and An-

derson ( r e f . 26) showed that the IR spectra of neutral counter ions in the double layer do not change with respect to those in the bulk. This problem may be analyzed by combining the chemical reactions shown by equations (10 and 18): S + H+ + A--~-~- SH.A

(27)

and analyzing the contributions of the ion pairing in the bulk H+ + A - ~

H'A

(28)

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and the binding of the ion pair with the surface site S + H.A ~

SH'A

(29)

The Gibbs energy of reaction (27) includes both the " i n t r i n s i c " and dipole contribution.

The l a t t e r should be associated with the second step (29) i . e .

location of a dipole in the electrical f i e l d at the surface.

Consequently, the

i n t r i n s i c contribution is given by (28), i . e . by the ion pairing in bulk.

If

ion pairing does not occur in the bulk (28), one cannot expect ion pairing in the double layer, especially at the isoelectric point where no dipole contribution exists.

Furthermore, contrary to the fact that the i n t r i n s i c equilibrium

constant for ion association in the double layer should not depend on ionic strength, the analysis of experimental ~SH.A'intand ~SOH'M'intstrongly depends on the ionic strength as w i l l be shown later.

Alternatively, one may apply s t a t i s t i -

cal mechanical treatments, which would take into account the influence of the overall double layer potential and the local f i e l d of the charge fixed on the surface.

Such a procedure w i l l be the subject of a separate publication (refs

51,52) and would imply negligible adsorption of monovalent counter ions at the isoelectric point.

This approach does not assume that the associated counter

ion is located at any particular point or distance from the surface.

A statis-

tical treatment leads to a certain distribution of counter ions around the central surface charge, and when expressed by the mass action law relationship, the "equilibrium constant" ceases to be constant, or a product of a constant and a simple exponential term such as in Eqs (20,21).

I t depends in a compli-

cated way on the surface potential, ionic strength, ionic size, etc. Saturation of the surface potential may also take place, giving rise to "catastrophic" counter ion co-adsorption in a fashion reminiscent of Manning's condensation model in polyelectrolyte systems, when a certain c r i t i c a l value of 4o is reached (ref. 53). Vl

CAPACITORS The interpretation of experimental data on ion adsorption is usually based on

the consideration of p.d.i, adsorption reactions Eqs (8,9, or 10,11) and the counter ion association reactions Eqs (18,19). The equilibrium constants of these reactions are defined by Eqs (14,16,20,21). In addition one applies mass balance and charge/potential relationships. The commonly accepted approaches are based on the Gouy-Chapman theory of the diffuse part of the interfacial layer and introduce several capacitors with constant capacitance in series to each other. The f i r s t capacitor (constant capacitance CI) is defined by the surface and B-plane. The adsorbed p.d. ions are located at the surface and the associated counter ions in the B plane. The most developed t r i p l e layer model (ref.

121

12) assumes an additional potential drop, i . e . the drop between the B and d planes.

The l a t t e r defines the onset of the d i f f u s e layer (see Fig. 2).

In the

t r i p l e layer model, the potential drop is taken to be proportional (with proport i o n a l i t y constant C2 I) to the surface charge (~s = ° o + ~ ) " Consequently, a second capacitor of constant capacity is introduced, not because of the e x i s t ence of a new charged plane but rather to improve the f i t .

The assumption is

that the hydrated ions from the d i f f u s e layer cannot approach the B-plane owing to t h e i r size.

Note that t h i s holds only i f the associated ions determining the

distance of the B-plane are dehydrated (as o r i g i n a l l y assumed by Stern f o r metal surfaces).

However, i f the associated counter ions were dehydrated t h e i r IR spec-

tra should change, and such changes have not been detected ( r e f . 26).

In addi-

t i o n , the potential drop in the double layer is caused by the i o n i c d i s t r i b u t i o n . I f counter ions only approach the d-plane but not B-plane, then a potential drop between the B and d-planes cannot occur.

The constant C2 should not be confused

with the capacitance of the d i f f u s e layer.

The d i f f u s e layer capacitor is char-

acterized by a f i x e d surface plane and the d i s t r i b u t i o n of charges in the plane on the bulk side of the capacitor.

The average distance and thus d i f f e r e n t i a l

capacitance of t h i s capacitor is a f u n c t i o n of the p o t e n t i a l , i o n i c strength, temperature and p e r m i t t i v i t y ( d i e l e c t r i c constant).

The d i f f e r e n t i a l capacitance

can be calculated from the Gouy-Chapman theory. Several other problems regarding the modelling based on the p a r a l l e l plane capacitors should be noted: (i)

In metal electrodes, i t might be reasonable to d i s t i n g u i s h between planes containing ions of d i f f e r e n t r a d i i

(although even then the r e a l i t y of t h i s

model is questionable); f o r metal oxides, any r e a l i s t i c approach should recognize the high i n t r i n s i c roughness of the surface, which has a much greater e f f e c t in determining the location of charges than the hypothetical separation between the surface, B and d-planes. (ii)

The distance between charged sites along any of these planes is large compared to the assumed separation of the capacitor planes. geneous charge d i s t r i b u t i o n cannot be assumed.

Therefore, homo-

A good approximation re-

quires the distance between planes to be much larger than the distance between charged surface s i t e s . (iii)

Also, i t would be more appropriate to consider the various capacitors to be connected in p a r a l l e l , rather than in series (refs 54,55).

The main

purpose of introducing capacitors into the i n t e r p r e t a t i o n of the e l e c t r o chemical experiments on m e t a l / s o l u t i o n interfaces has been to enable a b e t t e r f i t of the data. However, there are other p o s s i b i l i t i e s to improve the f i t ,

e.g. one can as-

sume that two kinds of amphotheric surface sites are present (refs 56,57), one being more active than the other.

This new model of the surface would be char-

122 acterized by two different i n t r i n s i c surface acid-base equilibrium constants.

C2

m I I

o

d

Z

Fig. 2. Schematic representation of potential profile according to the t r i p l e layer model. Vll

INTERPRETATIONOF EXPERIMENTAL DATA The adsorption of H+ and OH- ions on the metal oxides is mostly interpreted

on the basis of surface complexation. However, there are many variations around this basic approach to the problem. The simplest method is to extrapolate the total equilibrium constants Eqs (14,16) for the surface acid-base equilibria to ~o = 0 ( i . e . p . ) and to obtain the i n t r i n s i c values, which are different at each ionic strength (refs 15-17). The most elaborate method is a t r i p l e layer model (refs 9-12) taking into account surface acid-base @quilibria Eqs (8,9 or 10,11) together with the counter ion association reactions in the Stern layer described by Eqs (18,19). With respect to the location of charges three planes can be identified (see Fig. 2) ( i ) Surface or zero plane, where adsorbed p.d. ions are distributed (H+and OH-); ( i i ) B-plane, where associated counter ions are located; ( i i i ) d-plane, which is the onset of the diffuse layer. I f the total charge of the interfacial layer is assumed to be zero, then the

123

f o l l o w i n g r e l a t i o n s h i p s describe each plane: oo = F(FSH+ + FSH.A - FSOH_ - FSOH.M)

(30)

oB = F(FsoH. M - FSOH.A)

(31)

The charge in the d i f f u s e l a y e r (Od) is equal in magnitude to the charge bound to the surface (Os): os = -o d = F(FSH+ - FSOH-)

(32)

The r e l a t i o n s h i p s between p o t e n t i a l s and charges f o r d i f f e r e n t layers are commonly introduced through the f o l l o w i n g equations: 00

cI

-

_

(~)

_

~o - ~B

~d = ~B - C2/°s

(34)

The Gouy-Chapman theory is used to describe the e q u i l i b r i u m in the d i f f u s e layer:

*d :

2RT

(

In \

~_.+

F

+ I

(35)

V8~RTI

where ~d is the potential in the d plane which defines the onset of the d i f f u s e layer. The potential at a distance z from the plane d is equal to -KZ

I + yo e

~d(Z ) = 2RT In ( ) F I - yo e-Kz

(36)

where - I + exp(FCd/2RT ) Yo =

I + exp(F~d/2RT )

(37)

and K is Debye-H~ckel's reciprocal thickness of the double l a y e r < = (~RT/2F21) - I / 2

(38)

(Note that a l l equations are given r a t i o n a l i z e d form i . e . in SI units) The i n t e r p r e t a t i o n s imply the use of Eqs (14,16,20,21,30-35) mass balance equations, and the f o l l o w i n g equation defining the t o t a l number of surface sites

124

(39)

rtot = rs + rSH+ + ?SOH- + FSH.A + rSOH.M The measured values are oo and pH = -Ig aH+ Regression analysis may be used to adjust the seven parameters:

rtot'

,,int KSH+'

.int .int Kint CI and C2 ~SOH-' ~SOH'M' SH'A'

Since the use of seven adjustable parameters to f i t the potentiometric t i t r a tion data is unreasonable, independent methods are sought to provide estimates for some of them. Several procedures and assumptions are used: ( i ) rto t may be estimated from the surface structure or from the results of tritium exchange measurements (ref. 58).

Either procedure might yield

high values because a l l surface sites (all exchangable hydrogen ions) do not necessarilly act as amphoteric sites. ( i i ) One of the surface acid-base equilibrium constants may be related to the another one through the point of zero charge (which is identical to i.e.p. i f no specific adsorption occur) by ivint wo vint pH(iep) = 0.5 Ig ~,,SH+ / .~ ,,SOH_~

(40)

Note that this procedure does not always improve the accuracy of the computation procedure. The agreement of i.e.p, from electrokinetic measurements with the intersection point of the ro(PH) curves at different ionic strength supports this procedure. ( i i i ) A certain value of C2 may be assumed and fixed, e.g. C2 = 20~F/cm2 is sometimes used (e.g. refs 7,9); this assumption is based on extrapolations of the behaviour of the mercury electrode.

(iv) A value of CI is often taken to be 140-170~F/cm2 (e.g. ref. 7). (v) Electrokinetic measurements can be used to obtain the {-potential; i t is sometimes assumed that Cd = ~ (refs 32,59,60). Another possible approach is to assume the ~-potential to be equal to the diffuse layer potential at the distance of the shear plane (z = 15 to 25 ~) i . e . ~ = = Cd(Z) which is then calculated via Eqs (36-38) (refs 52,61). The l a t t e r procedure does not reduce the number of adjustable parameters, but a capacitance is replaced by the distance of shear plane.

However,

the advantage is that in this procedure one imposes more restrictions on the number of adjustable constants. The distance of the shear plane may be assumed to be similar for a l l metal oxide/water systems. Such an assumption allows the reduction of the number of parameters, but one should then assume that Cd = CB' or even ~d = ¢o together with CB = Cd"

125

( v i ) The additional measurements, as f o r instance the excess of counter ions in the double layer might provide the data improving the faithfulness of the i n t e r p r e t a t i o n (refs 31,32). I t is not necessary to resort to such an elaborate model, which introduces the numerous assumptions j u s t described, in order to analyze the experimental data.

The s i m p l i f i e d potential p r o f i l e may be assumed i . e . no intermediate po-

t e n t i a l drop between the 0 and d-planes occurs (see Figure 3).

This approach

implies the use of Eqs (14,16) f o r the equilibrium constants, the Gouy-Chapman theory (35) f o r the diffuse part of the double layer together with charge balances (30-32), and mass balances for the surface sites (39).

The association

of counterions was described by apparent equilibrium constants KSH.A and KSOH.M i . e . no i n t r i n s i c equilibrium constants were used (20,21). Such an analysis was performed by using the data f o r magnetite, which had already been interpreted by means of the t r i p l e layer model ( r e f . 44). In this analysis, Fto t was assumed to be 1 • 10-5 mol/m 2 (as e a r l i e r ) , .~SOH i n t was related through Eq (40) with the experimental value f o r pH(iep

= pH(pzc) = 6.8.

,f

>

0

p

Z

Fig. 3. Schematic representation of potential p r o f i l e according to the present s i m p l i f i e d description of the double layer.

126

200

"b

MAGNETITE

"...

o'. d•I

100

E

0

9-

"t3 • e

-I00

"9

".,o

NERNST -200 3

J 5

r

l ?

l

"x.....

J 9

pH Fig. 4. Surface potential for magnetite/NaNO3 aqueous solution system calculated . i n t+ = 105 by using Eqs (14,16,20,21,30-32,35,39,40) assuming ~SH rto t = I • 10-5 mol dm-3 from titration data (ref. 40) [NaNO3]/mol dm-3: C)i0-I

~ 1 0 -3 [-]~-potential as obtained from electrophoretic mobility measurement, I = 10-2 mol dm-3 Dotted line corresponds to the Nernst equation. Consequently this treatment reduces the number of adjustable parameters to vint only one i.e. ~SH+ . Instead of f i t t i n g this intrinsic equilibrium constant, all other quantities (KsH-A" KSOH.M" ¢o etc.) were calculated from experimental ~o and pH data by as,int suming certain reasonable values of KSH+. Figure 4 illustrates the calculated

127

values of surface potential as a function of pH f o r two d i f f e r e n t ionic strengths i n t = 105 . This value was found to be appropriate, (10 -3 and 10-I mol dm-3 ) for KSB+ because, when lower values of K ~

were used, the ( t o t a l ) association equilibrium

constants KSH.A and KSOH.M became independent of ~o or even worse, they decrease i n+t would lead to as (~o) rose; e i t h e r case is unreasonable. A higher value of vKSH negative amounts of ~ssociated counter ions which is also impossible. Figure 4 shows that the surface potential decreases in magnitude with increasing ionic strength.

This is in disagreement with the Nernstian behaviour.

Also note, that

calculated surface potentials were always lower than those obtained from the Nernst equation, even at low ionic strength (10 -3 mol dm-3). 3 units from the i . e . p , th~ difference is 25 mV.

I f pH deviates by

The e l e c t r o k i n e t i c ~-potential

measured at I = 10-3 mol dm-3 is much lower than the surface potential.

This

enables one to calculate the separation distance of the slipping plane using Eqs (36-38). In the described procedure, no assumption was made about ion association in the double layer; instead, the procedure enabled the evaluation of relevant (tot a l ) association constants KSH.A and KSOH.M. This result allows us to consider the a p p l i c a b i l i t y of the commonly used method of dividing the equilibrium constants into an " i n t r i n s i c " and an exponential term Eqs (20,21).

Combining

equations (20) and (21) with the relationship (33) between ~o and ~B gives the general expression RT In Kass - 14oi = R_~TIn Kintass - l~°iF F CI

(41)

which is applicable for both cation and anion association in the Stern layer. Figure 5 describes Eq (41) for two ionic strengths and f o r association of NO3 ions with H+ on the surface as well as f o r Na+-surface OH- ion pairing. The model requires mas s'int to be independent of ionic strength, but extrapolation to ~o = 0 showed a s i g n i f i c a n t difference in the i n t r i n s i c equilibrium constants. .int A This discrepancy cannot be improved by adjusting the chosen value of mSH+. f u r t h e r test of the model can be performed by examining the slopes of the l i n e s . According to (41) they are equal to the reciprocal value of the capacitance, CI (Table I ) .

A comparison of data f o r positive (pH < pH(pzc)) and negative (pH >

pH(pzc)) charge regions shows that the apparent capacitance is somehow lower f o r the association of Na+ with respect to NO3 ions. plained by the larger size of hydrated Na+ ions.

This finding can be ex-

However, no reasonable expla-

nation can be offered for the influence of the ionic strength; the higher capacitances at higher ionic strengths are contradictory to tions on which the model is based; capacitor with planes separated determined by the sizes of ions. I t has been noted ( r e f . 40) that

significantly the assumpby a distance the e f f e c t of

temperature on CI suggests that this magnitude should be considered only as a

128

fitting

parameter. Table I

The capacitance CI of the Stern layer as obtained from slopes in Figure 5 according to Eq (41) for magnetite ( r e f . 40). -2 CI I ~F cm

SURFACE INTERACTION

1:I0 -3 mol dm-3

I=10 - I mol dm-3

SH+ + NO3 80

130

100

290

(positive region) SOH- + Na+ (negative region) VIII.

CONCLUDINGREMARKS

The Nernst approach used f o r s i l v e r iodide ( r e f . 37) was developed by equating the electrochemical potentials of potential determining Ag÷ or I - ions in the bulk solid and l i q u i d phases. s t i t u e n t ions

of the solid phase.

In this case, the p.d. ions are also the conFor metal oxides, p.d. ions are not s o l i d

phase constituents; therefore the surface complexation model is used, and the s o l i d surface and solution bulk are considered to be in equilibrium.

However,

the extent to which the i n t e r i o r of the oxide p a r t i c l e s may exchange p.d. ions can change appreciably from one oxide to another.

Since the d i f f u s i o n of p.d.

ions through the s o l i d phase is r e l a t i v e l y slow, the results of the measurements would depend on the time scale of the experiment. The existence of f a s t and slow e q u i l i b r a t i o n of metal oxides with e l e c t r o l y t e solutions has been noted by many authors, and most of the experimental acid-base t i t r a t i o n data refer to the so-called " f a s t - t i t r a t i o n " procedures (e.g. r e f . 5). The i s o e l e c t r i c point of an oxide is determined to a large extent by the acid-base reactions taking place at the surface, but i t is also l i k e l y that the conditions in the bulk of the s o l i d phase have an e f f e c t as examined by Onoda and Casey ( r e f . 62).

I f the bulk solid is completely equilibrated with the solution

phase the Nernstian potential would characterize the i n t e r i o r of the p a r t i c l e s . This process would influence the surface acid base e q u i l i b r i a through the potential. The net r e s u l t would depend on the equilibrium constants and on the degree of e q u i l i b r a t i o n of the solid bulk phase. This hypothesis might explain as to why the i . e . p , changes upon aging.

129

150 ~o

MAGNETITE

100 0

~_o

0

mot d rn-3:

7"

0 10-1 -5£

l J 5 10 Imol/,uC cm-2

0

-15

Fig. 5. Plot according to Eq (41) for magnetite/NaNO3 aqueous solution system described in Fig. 4. [],

pH > pH(i.e.p.); negative region

C)OpH < pH(i.e.p.); positive region 0, I = 10-I mol dm-3 C ) E ] I = 10-3 mol dm-3 The Gouy-Chapman theory for the diffuse double layer assumes a homogeneous d i s t r i b u t i o n of charge along the solid surface.

This condition is usually

reached at the metal/solution interface unless specific adsorption of ions occurs. The origin of the charge on solid crystals is due to the presence of "fixed" charged groups. (ref. 63).

Therefore. the isopotential surfaces are not planar

Only at rather long distances from the O-plane (as compared to the

separation between surface charges) are the planar isopotential profiles from the Gouy-Chapman equation applicable.

130

The local electrostatic f i e l d is responsible for the association of counter ions.

The introduction of ion pairing reactions Eqs (18,19) is an obvious im-

provement in the interpretation of the behaviour of fixed charges. However, the electrostatic forces suffice to explain the formation of ion pairs.

The "equi-

librium constant" for their formation is controlled by both the local and the general potential and depends on ionic strength and ionic size parameters (refs 51,52).

Thus, there is no need to assume any kind of chemical contribution to

the free energy of counter ion association.

Spectroscopic data support this ar-

gument (ref. 26). The idea of surface ion pairs was applied by Te2ak (refs 64,65) who assum- ied that associated counter ions are located at Bjerrum's c r i t i c a l distance from the adsorbed p.d.i, and by Mirnik (refs 66,67) who assumed a distribution of counter ions around the central fixed charge as described by the Debye-HUckel theory.

The usual interpretation of the ion pairing equilibria is based on the

" i n t r i n s i c approach" Eqs (20,21) which is consistent with the description of Yates, Levine and Healy (ref. 7) who considered the ion pairs as oriented dipoles in the electric f i e l d of the double layer.

However, the i n t r i n s i c equi-

librium introduced by these treatments is not a true (mass action law) standard equilibrium constant, but i t is more a f i t t i n g parameter that changes with ionic strength and does not provide information on the nature of the association process. One important point to stress is the interdependence of the various f i t t i n g parameters of any procedure used to describe the double layer.

That is to say,

equilibrium constants reported in the literature should not be used outside the context of the model employed to derive them. The above description refers to the simplest possible cases, in which no ions adsorb specifically (except p.d. ions as H+ and OH-).

In order to take into ac-

count specific ionic adsorption, the involved modelling is much more elaborate and speculative.

Various aspects of specific ionic adsorption can be found in

refs (35,68-81).

Further complications also arise when semiconductor oxides are

considered (refs 54,55,82-86). The future progress in the understanding of metal oxides immersed in aqueous solutions w i l l depend not so much on the development of ever-increasingly sophisticated models, but rather on the a v a i l a b i l i t y of more precise and varied experimental information. Particularly important is the use of modern spectroscopic tools to probe the interface in 8i~-u, as i l l u s t r a t e d by the work of Tejedor-Tejedor and Anderson (ref. 26) for the case of ATR-FTIR techniques. Standard electro chemical measurements such as direct potential (ref. 87) and impedance measure=. ments (refs 88-90) (although not easy to implement in these systems) should also be of value in providing sound basic information on which to elaborate models. Coagulation studies should also serve as a test for basic ideas, e.g. the fact

131

that the rate constant of Agl coagulation does not depend on the I - a c t i v i t y in a broad pl region suggests a close c o r r e l a t i o n between the behaviour of ~d and ~-potentials ( r e f . 91).

Even some simple, but c a r e f u l l y designed experiments

regarding e l e c t r o p h o r e t i c m o b i l i t i e s and i . e . points could be h e l p f u l .

For i n -

stance, the measurement of the i . e . p , of two i d e n t i c a l samples, one prolonguedl y aged in acidic and another in basic s o l u t i o n could provide information on the importance of the s o l i d bulk e q u i l i b r a t i o n . IX.

APPENDIX

Symbols a

- activity separation distance between 0 and B planes

b

concentration

c

C

- capacitance proton charge

e

F I

the Faraday constant - i o n i c strength (concn. basis)

K t o t a l (apparent) e q u i l i b r i u m constant Ki n t _ i n t r i n s i c e q u i l i b r i u m constant p - dipole moment R

(molar) gas constant

T

thermodynamic temperature

X

mole f r a c t i o n

z

distance from d plane p e r m i t t i v i t y ( d i e l e c t r i c constant)

0

surface charge density

F

surface concentration Debye-HUckel reciprocal thickness of i o n i c atmosphere - e l e c t r o s t a t i c potential

K

e l e c t r o k i n e t i c potential, ~-potential chemical potential %

electrochemical potential

Notations l i q u i d bulk - s o l i d part of i n t e r f a c i a l layer anion

-

-

-

metal ion (cation) surface s i t e

132

O,B,d - planes in the interfacial layer p . d . i . - potential determining ion i.e.p.- isoelectric point (~ = O) p.z.c,- point zero charge (rSH+ + rSH.A = rso H + FSOH.M) pHo pH for which 4o = 0

X. I 2 3 4 5 6

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