Kinetics of electrochemical reactions mediated by redox polymer films

J. Electroanal. Chem., 169 (1984)9-21

Elsevier Sequoia S.A., Lausanne

9

Printed in The Netherlands

K I N E T I C S O F E L E C T R O C H E M I C A L R E A C T I O N S M E D I A T E D BY R E D O X POLYMER FILMS NEW FORMULATION AND STRATEGIES FOR ANALYSIS AND OPTIMIZATION

C.P. ANDRIEUX, J.M. DUMAS-BOUCHIAT and J.M. SAVI~ANT Laboratoire d'Electrochimie de l'Universitb de Paris VII, 2, place Jussieu, 75 251 Paris Cedex 05 (France)

(Received 4th August 1983; in revised form 16th December 1983)

ABSTRACT A new formulation of the kinetics of electrochemical reactions mediated by redox films is proposed for the case of an irreversible cross-exchange reaction in the context of rotating disc electrode voltammetry. It involves the normalization of the substrate concentration versus its value at the film boundary interface. The overall kinetics are then shown to depend on only two dimensionless parameters. Procedures for analysing the effects of the rotation speed and the other experimental parameters are proposed. The construction of a kinetic zone diagram summarizing the variation of the kinetic control with the various parameters is useful for defining the optimal performances of the film. INTRODUCTION It is n o w well recognized that the rate of a n electrochemical reaction mediated by a redox p o l y m e r film deposited o n the electrode surface results, besides the diffusion of the substrate into the solution, from the interplay of three p h e n o m e n a occurring in the film, n a m e l y the reaction between the substrate a n d the m e d i a t i n g centres, the diffusion of the substrate a n d the diffusion-like t r a n s p o r t of the charge (see ref. 1 a n d references cited therein). W i t h regard to the following reaction scheme: M e d i a t e d process (0) (1)

P+e-~Q k~ Q+A~B+P

Direct electrochemical process

(K)

A_+e-~B

where P / Q is the m e d i a t o r couple i n c o r p o r a t e d in the film, A is the substrate a n d B its r e d u c t i o n (or o x i d a t i o n ) product, the kinetics were first analysed for the case of a n irreversible cross-exchange reaction ( K = oe) [1,2], then for the case of a self-exc h a n g e reaction ( K = 1) [3], a n d ultimately for a reversible cross-exchange reaction 0022-0728/84/$03.00

© 1984 Elsevier Sequoia S.A.

10 with an arbitrary value of the equilibrium constant [4]. These analyses were given in the context of rotating disk electrode voltammetry and dealt with the plateau current of the mediated reduction (or oxidation) as well as that of the direct electrode reduction (or oxidation) of the substrate across the film. Although developed in less detail, an analysis of the dependence of the half-wave potentials on the various kinetic parameters was also given [4]. The interplay of the various phenomena governing the overall kinetics was first formulated by dimensionless parameters expressing the competition between the three rate-limiting factors in the film and the diffusion in the solution [1]. This led to a formulation in which the kinetics depend, for the irreversible case, on three dimensionless parameters [1]. An equivalent but more transparent formulation was proposed later on, using four characteristic current densities [2-4] which correspond to the four rate-limiting factors: (i) Diffusion of the substrate in the solution: (2)

i A = Fc~,D/8

(Equal to the plateau current at a bare electrode of the same surface area.) (ii) Diffusion of the substrate through the film: (3)

i s = Fc~xDs/q)

(iii) Diffusion-like transport of the charge through the film: i E = FDc~DE/e

P

= F F °DE/q~2

(4)

(iv) Cross-exchange reaction in the film: i k = Fc~Kklc~4)

= Fc~klF

°

(5)

c~, is the bulk concentration of the substrate in the solution, c~, is the total concentration of the mediator in the film, F o is the total surface concentration of the mediator in the film, D is the diffusion coefficient of the substrate in the solution, D s is the diffusion coefficient of the substrate in the film, D E is the "diffusion" coefficient of the charge propagation in the film, K is the partition coefficient of the substrate between the film and the solution, k~ is the second order rate constant of the cross-exchange reaction, 8 is the diffusion layer thickness, and q, is the film thickness. In the irreversible case, the kinetics then depend on the ratios of three of the current densities with the fourth. In the reversible case, the system depends on a fourth parameter, i.e. the equilibrium constant of the cross-exchange reaction. Although the kinetics can be analysed in the general case by finite difference resolution of the master differential equation, limiting kinetic behaviours were defined leading to closed-form expressions of the current [1,4[. The latter were symbolized by combinations of the three letters R, S and E, for cross-exchange reaction, substrate diffusion, charge "diffusion", respectively, emphasizing the nature of the rate-determining steps [1-4]. The cases where the cross-exchange reaction has a surface character occurring

11 either at the film/solution interface or at the electrode/film interface were also treated [1,2,4]. They were obtained as limits of the volume reactions when the reaction layer relative to substrate diffusion and charge "diffusion" respectively became so thin as to reach molecular dimensions. This is met when the cross-exchange reaction is fast a n d / o r the substrate diffusion or the charge transport, respectively, are slow. More recently, another description of the kinetics of the mediated process in the case of an irreversible cross-exchange reaction has been given by Albery and Hillman [5]. Although the basis of their analysis and their main conclusions are the same as ours, the comparison between the two presentations proved fruitful in the sense that it allowed a simpler and more complete formulation of the problem and description of the results as made clear in the following. Let us first note that the question of the possible surface character of the cross-exchange reaction is treated in quite a different way. In addition to the volume reaction, the possible occurrence of a surface reaction at the film/solution interface is considered while we dealt with the problem of the volume reaction becoming a surface reaction either at the film/solution or the electrode/film interface as the corresponding reaction layers reach molecular dimensions. The comparison of the two presentations should therefore be done at the level of the volume reactions deleting the terms corresponding to the additional surface reaction in the main equation of ref. 5 (i.e. making k " = 0 in eqns. 9 and 10 in ref. 5). In this context, the interplay of the various rate-limiting factors is expressed by two distances which are the thicknesses of the two reaction layers relating the rate of the cross-exchange reaction to the diffusion of the substrate and to the charge propagation, respectively [5]. We do not find this representation appropriate for the following reasons. Two unknown quantities are contained in these two distances, namely the concentration of the mediator at the film/solution interface and the concentration of the substrate at the electrode/film interface. These two concentrations are functions of the parameter (rate constants, total concentrations, film thickness, rotation rates) governing the system and can only be obtained as a result of the resolution of the differential equation describing the overall kinetics. However, the most important point of our present concern is that the kinetics of the overall process could be dependent on only two parameters, as suggested by the analysis given in ref. 5. In the following, we will show that this is indeed the case also when defining these two parameters as ratios of characteristic current densities containing the actual parameters (rate constants, total concentrations, film thickness, diffusion layer thickness) that govern the overall kinetics. This has two important implications. The first is the possibility of describing rigorously and quantitatively how the variations of the actual parameters shift the system from one limiting kinetic behaviour to another and hence how the parameters should be adjusted for an optimal mediation process to be obtained. The second is that as soon as a relatively simple descripticxn of the kinetics involving only two competition parameters can be given for the above-mentioned reaction scheme, the treatment of more complicated reaction schemes, likely to be encountered in practice, appears reasonably tractable.

12

This will be illustrated in a forthcoming publication where a pre-activation mechanism involving Michaelis-type kinetics will be treated. In the present paper we will also propose new plotting procedures, besides those of Levich and Koutecky-Levich, giving a more general and more realistic estimation of the effect of the rotation rate on the kinetics of the overall process. NEW FORMULATION OF THE KINETIC PROBLEM

We are looking for the expression of the plateau current for the mediated process in the context of reactions (0) and (1), the latter being irreversible. As shown previously [2], the differential equation expressing the overall kinetics as a function of the various parameters is d2a dy 2

ik

1 + is a -- a o - y

is a

~E

=0

-d-'yY i

with, f o r y = 0, ( d a / d y ) o = 0 and for y = 1, 1 - a I = (is/iA)(da/dy)l, density being given by

the current

ii/i s = ( d a / d y ) l where iA, is, iE, ik are defined by eqns. (2)-(5), and y = x/go, the space being normalized vs. the film thickness, a = CA/KCT, meaning that the concentration of a is normalized towards the bulk concentration multiplied by the partition coefficient. The current density is thus apparently dependent on three dimensionless parameters, for example, ik/i s, is/i E and i s / i A. There is no symmetry between the concentration variations of A and Q and this is because of the depletion of A outside the film. In order to render the formulation symmetrical towards A and Q, let us normalize the concentration of A in a different way: a

cA

1 - i,/i A

c~(1 - i,/iA)

a*-

which amounts to normalizing cA towards the concentration of A at the film/solution interface on the film side. We simultaneously introduce two other expressions for the substrate diffusion and cross-exchange reaction current densities: i~ = ik(l

-

-

il/iA)

i ~ = i S ( 1 -- il/iA)

while keeping i E the same, which again amounts to introducing the concentration of A at the film/solution interface instead of its bulk concentration. This has the apparent disadvantage of introducing a variable, il/iA, into the parameters. As will be discussed in more detail in the following, this drawback is only apparent since i l / i A can be obtained directly from the experimental data. The expression of the overall kinetics in terms of a* is thus given by d2a* dy 2

•

'~a* l + ~ - [ a * - a ~ - y i~'

tE [

(da-) ]) = 0 ~-Y

1

(6)

13

with, f o r y = O, ( d a * / d y ) o by

= 0 and f o r y = 1, a~' = 1, the current density being given

i,/i~ = (da*/dy)~ A n alternative expression can be obtained in terms of q, the normalized concentration of Q (q = CQ/C~): dY 2

(

iE q l + - - l ~

q--ql +(1--Y)

(dq)ll ~Y 0

=0

(7)

with, for y = 0, q0 = 1 and for y = 1, ( d q / d y ) l = 0, the current density now being given by i , / i E = -- ( d q / d y ) o The symmetry between a* and q is now complete: if q is replaced by a*, i E by i~ and 1 - y by y, the q differential equation and its b o u n d a r y conditions (7) yields the a* differential equation with its b o u n d a r y conditions (6). It is also seen that the system depends on only two dimensionless parameters that t ;' .S / ';k. J~1/2 and ( iE/i~ ) 1/2 can be chosen, for example, as ~ LIMITING BEHAVIOURS AND KINETIC ZONE DIAGRAM

The limiting expressions of the current density for the plateau of the first wave are readily deduced from those of Table 1 in ref. 2, using the newly introduced i~' and i~' characteristic current densities. They are summarized in Table 1 using the same c o m b i n a t i o n of the three letters R, S and E as before [1,2] to symbolize the nature of the rate-limiting phenomena. It is seen that the new formulation clearly shows that the roles of substrate diffusion (S) and charge propagation (E) are symmetrical provided the effect of mass transport in the solution is taken into account by the introduction of (1 - i l / i A ) into i s and i k. Table 1 also gives the expression of the current density at the plateau of the second wave. Mass transport in the solution is here taken into account by the introduction of 1 - (i t + i 2 ) / i A into i S and ik: i~'* = is[1 - - ( i t + i 2 ) / i A ] i~'* = ik[1 - - ( i 1 + i 2 ) / i a ] where (i I + i2) represents the total current density at the second wave. Returning to the plateau current of the first wave, a kinetic zone diagram [6] is shown in Fig. 1 which allows the passage from one limiting kinetic behaviour to another to be seen. Each point of the plane corresponds to a given kinetic state of the system, i.e. to two given values of the two dimensionless parameters \t ~; *S // ~; .k~ /!2 and ( i E / i ~)1/2. The boundaries between the zones have been derived on the basis of 5% accuracy of the determination of the plateau current, maximizing the simplest limiting behaviour. For example, the b o u n d a r y between R + E and R is obtained

14

I \

+

H

~s

Z"

I

~t

+

.=

I

+

II

15

.

rs

C~

... DE

°

kI

-c

• Cp

¢

-2,

-2.

Fig. 1. Kinetic zone diagram. The boundary lines are based on a 5% accuracy of the measurement of the plateau current of the first wave.

when (iE/i~) reaches a value such as the p l a t e a u c u r r e n t for (R + E) is 95% of the p l a t e a u current for (R). Similarly, the b o u n d a r y between the general case a n d S R + E is anf[(i~/i~) 1/2, (iE/i'~) t/2 ] = 0 curve such as when crossing the curve the p l a t e a u c u r r e n t in the general case is d i s t a n t b y 5% from the p l a t e a u c u r r e n t for ( S R + E). All the o t h e r b o u n d a r i e s were d e t e r m i n e d a c c o r d i n g to the same p r o c e dure. T h e d e t e r m i n a t i o n of the b o u n d a r y lines is s t r a i g h t f o r w a r d for the t r a n s i t i o n b e t w e e n the limiting b e h a v i o u r s as featured b y the c l o s e d - f o r m expression given in T a b l e 1. F o r passages from the general case to the various limiting zones, a finite difference resolution of eqn. (6) or (7) with the a c c o m p a n y i n g b o u n d a r y c o n d i t i o n s is required. F o r this purpose, we used the same " c u r v a t u r e r a d i u s " technique as b e f o r e [1]. T h e effect of the p a r a m e t e r s ~,, c~,, c~,, x, k 1, D E a n d D s is s u m m a r i z e d in the u p p e r l e f t - h a n d c o r n e r of Fig. 1 in terms of log units. It is then easy to p r e d i c t how the v a r i a t i o n of one of these p a r a m e t e r s will c h a n g e the kinetic c o n t r o l of the overall m e d i a t e d reaction, which can be used for o p t i m i z i n g the efficiency of the m e d i a t i o n . T h e effect o f the r o t a t i o n speed a p p e a r s through the q u a n t i t y (1 - iJiA). A c t u a l l y the effect of the v a r i a t i o n s of the a b o v e - m e n t i o n e d p a r a m e t e r s s h o u l d be a n a l y s e d while keeping ( 1 - il/iA) c o n s t a n t b y m e a n s of an a p p r o p r i a t e v a r i a t i o n of the r o t a t i o n speed. This indicates the a d v a n t a g e of using plots of the p l a t e a u currents vs.

16

(1 - i J i A) rather than Levich or K o u t e c k y - L e v i c h plots in which the abscissa is the square root of the rotation speed or its inverse. It is seen, in this context, that the effect of varying (1 - i l / i A ) is exactly the same as that of varying ¢~, as expected from the observation that the pertitent quantity is the concentration of the substrate at the f i l m / s o l u t i o n b o u n d a r y which is equal to c~ (1 - il/iA). EFFECT OF THE ROTATION SPEED. HOW TO ANALYSE T H E EXPERIMENTAL DATA

The experimental data generally consist of a series of values of the plateau current density measured as a function of the rotation speed and of some other experimental parameters such as the film thickness, and the concentrations of substrate and mediator. Let us take two examples illustrating the procedures proposed for analysing the experimental data. In the first example, we consider a system passing from an " R " to an " R + E" situation u p o n varying the film thickness and the rotation speed, while keeping all the other parameters constant. Let us assume that the characteristics of the system are as follows. The plateau current at a bare electrode follows the Levich criterion: iA =

A*~I/2

where A* is a constant expressed in A (rpm) 1/2 cm 2. We vary the film thickness from an initial value q~o to 2 q~o, 4 #~o, 16 q5° and 32 q~o. The cross-exchange reaction current density for q5 = q~o is i~' = 12 A*, the charge propagation current density for q~ = #,° is i~ = 400 A* and the substrate current density for q) = #~o is i° s = 36840 A*, all the current densities being expressed in A c m - 2 . The Levich plots obtained for a

il//A cm -2

6o~ 5o.~ 4o.~ 3o.~ 2o~ 10/(

o

o

40

50

o

d2/rpm V2 10

20

30

60

Fig. 2. Levich plots for a system under R + E control. The various curves are obtained for film thicknesses values of (from the bottom upwards): #~o (O), 2 q~o ( + ) , 4 q)o (~), 8 q~o (zx), 16 #~o (E]), 32 ,~o (e). The straight line corresponds to the bare electrode.

17

i1 1- i.j iA

i I

iA!

2oo,~ 6OA*

~OA*

lso~ (a)

~OA* 100

30A*

2OA* 50

•

,

O

,

~

,

O

Z

f

,

CCC

I

0

I

0.5

0

05

~od

(-.A) 1

i

Fig. 3. Analysis of the data of Fig. 2 according to the rate law corresponding to R (a) and to ER (b), respectively.

series of rotation speeds: 100, 400, 900, 1600, 2500, 3600 rpm, are as shown in Fig. 2. Figure 3 shows how the experimental data of Fig. 2 can be analysed according to the rate law corresponding to an R situation (Fig. 3a) and to an ER situation (Fig. 3b), respectively. It is seen that the R rate law tends to be followed for small values of the film thickness and small values of the rotation speed and vice-versa for the ER rate law. From these limiting behaviours it is found that i~ = 12.37 A* and 32 (i~ i~) a/2 = 2205 A* and thus that i~ = 348 A*. We can now test the validity of the R + E rate law (Table 1). The result is shown in Fig. 4. The slight deviation from 1 is due to the fact that we used the limiting rate laws for determining i~' and i~ under conditions where they are followed only approximately. Figure 5 summarizes the shifts of the point representing the system in the zone diagram as the rotation speed and the film thickness are varied. The second example illustrates the transition between an SR and an E rate law upon varying the substrate concentration and the rotation speed while keeping all

WI [(1 -

,

0

~f$lvd)]

I

,

I

a.5

1

1.5

of the data of Fig. 2 according to the rate law corresponding to R + E.

Fig. 4. Analysis

ER

.. . .

_ /

/ iR+S

//

/

/

--I

. __--0-

/

I

,’

R+S

I General

case

I I I SIX

-a5

0

I 6

log( &/id I as

L

10

Fig. 5. Location of the data shown in Fig. 2 in the kinetic zone diagram

19

I ,/~/A

6off

M-'~

sod nod 3od 2od

0

lb

2'0

~o

go

-t~l,)l/2/rpnlV2 I I I

so

60

Fig. 6. Levich plots for a system under S R + E control. The various curves are obtained for substrate concentrations of: ( ¢ ~ ) o ( © ) , 3.16 (c~)o ( + ), l 0 (¢~)o (z~), 31.6 ( c ~ ) o (E3), I00 ( c ~ ) 0 (O). The straight line corresponds to the bare electrode.

the other parameters constant. The plateau current is assumed to have the following characteristics. At the bare electrode: i A = C~I/2B

*

(B* in A cm mo1-1 (rpm) 1/2). The substrate concentration is varied from (c~,)0 to 3.16 (c~)0, 10 (c~) 0, 31.6 (c~,)0 and 100 (c~.)0. ik/c~ = 9.86 × 104 B* A cm mo1-1, iv. = 785 (c~,)0 B* A cm -2. The Levich plots obtained for this series of concentrations are shown in Fig. 6 under the form of il/c7, vs. ~1/2 plots. Figure 7 shows the analysis of the data according to the SR rate low (Fig. 7a) and the E rate law (Fig. 7b). The SR rate law tends to be followed for small c~, 's and small rotation speeds while the E rate law tends to be followed for opposite conditions. From these limiting behaviours it is found that (ik iS)I/2/C~, = 245 and that i E = 780 (c~,)0 B*. Using these values we can test the validity of the SR + E rate law corresponding to the formula given in Table 1. The result is shown in Fig. 8, while Fig. 9 summarizes the shifts of the point representing the system in the zone diagram as the substrate concentration and the rotation speed are varied. These two examples as well as the kinetic zone diagram in Fig. 1 show that kinetic situations such as R + E, S + E, ER + S and SR + E, in which the system depends on one dimensionless parameter, cannot be neglected from a practical viewpoint, considering only the situations such as R, ER and S + E in which the system does not depend formally on any competition parameter. Variations of the adjustable kinetic parameters within the practically accessible ranges may indeed not be sufficient to cross such one-parameter zones completely. The same is true for the general case which should be dealt with if iE/i ~ is not definitely larger or smaller

i l/C/{( 1 -il/iA)

i1

25od

2001

600(C~) 13

150[

(b)

450(c~) B*-..-

10(3

300(C,~) B*-

0.5

0

Fig. 7. Analysis of the data of Fig. 6 according to the SR rate law (a) and to the E

i,/(ik~is*/2 iE )E(1 "*4 i E2/i ~' iS*)VL'- 1]

I

i

i

-0.5

0

0.5

r

A ~LA;~

Fig. 8. Analysis of the data of Fig. 6 according to the SR + E rate law.

1

21 ER.S

case

-1.5~ S+E

-2.0 SR+E SR

-2

-1.5

-1.0

-0.5

Fig. 9. Location of the data of Fig. 6 in the kinetic zone diagram.

t h a n unity. T h e s e effects w o u l d e v e n be a m p l i f i e d if the e x p e r i m e n t a l a c c u r a c y is b e t t e r t h a n 5%, w h i c h l e a d s to a n e n l a r g e m e n t of the i n t e r m e d i a r y zones. ACKNOWLEDGEMENTS T h i s w o r k was s u p p o r t e d in p a r t b y the C . N . R . S . ( E q u i p e de R e c h e r c h e A s s o c i 6 e N o . 309 " E l e c t r o c h i m i e M o l 6 c u l a i r e " ) . W e t h a n k W.J. A l b e r y a n d A . R . H i l l m a n for m a k i n g a v a i l a b l e to us a p r e p r i n t of ref. 5 a n d for s t i m u l a t i n g discussions. REFERENCES 1 2 3 4 5

C.P. Andrieux, J.M. Dumas-Bouchiat and J.M. Sav6ant, J. Electroanal. Chem., 131 (1982) 1. C.P. Andrieux and J.M. Sav6ant, J. Electroanal. Chem., 134 (1982) 163. F.C. Anson, J.M. Sav6ant and K. Shigehara, J. Phys. Chem., 87 (1983) 214. C.P. Andrieux and J.M. Sav6ant, J. Electroanal. Chem., 142 (1982) 1. W.J. Albery and A.R. Hillman, Annual Reports C. (1981), The Royal Chemistry Society, London, 1983, pp. 317-437. 6 J.M. Sav6ant and E. Vianello, Electrochim. Acta, 8 (1963) 405.