4526
J. Phys. Chem. 1988,92, 4526-4532
Electron Hopping between Locallzed Sltes. Effect of Ion Palring on Diffusion and Mlgration. General Rate Laws and Steady-State Responses Jean-Michel Sav6ant Laboratoire d'Electrochimie MolZculaire de 1'UniversitZ Paris 7, Unit; AssociZe au CNRS No. 438, 2 place Jussieu. 75251 Paris Cedex 05, France (Received: November 12, 1987; In Final Form: February 4, 1988)
Coupling between electron hopping and electroinactive counterion displacement by means of ion pairing between the immobile electroactiveions and the mobile electroinactivecounterions is investigated. The rate laws governing such systems are established for steady-state and transient conditions. They are then used for obtaining the steady-state current potential responses of a film sandwiched between two electrodes. The main effect of ion pairing is a rapid decrease of the charge-transport rate. The electroinactive mobile counterion thus controls the charge-transport rate through its ion-pairing association constant rather than by its own mobility. The effect of fixed electroinactive ions, possibly ion paired themselves by the mobile electroinactive counterions, is also discussed.
The kinetics of charge transport by electron hopping between fvted redox sites have received active recent attention in the context of the interest aroused by the development of redox polymer coatings on electrode surfaces.'s2 The propagation of electrons is accompanied by a movement of electroinactive counterions so as to maintain electroneutrality. The possibility of a kinetic control of charge transport by electroinactive counterion displacement has consequently been invoked frequently although no satisfactory model of its coupling with electron hopping has been developed so far. One approach to the problem is to view the coupling between these two components of the charge transport as solely based on the maintenance of electroneutrality. Electroinactive counterion displacement then plays a role in the migrational part of electron hopping, Le., in the effect of an electric field on the hopping rate. With regard to the propagation of electrons under a chemical potential gradient, it has been shown3 to be formally equivalent to the "diffusion" of the immobile redox centers with a diffusion coefficient
(k', standard rate constant of electron transfer between two adjacent sites; COE, concentration of redox sites; Ax, mean distance between two adjacent redox sites) which is the same for the ox and red forms. The effect of an electric field on the hopping rate is formally equivalent to "migration" of the immobile redox ions. It, however, does not obey the classical Nernst-Planck law but rather a second-order r e l a t i ~ n s h i p( x~,~distance; a, potential; C,, CB, concentrations of the ox and red forms, respectively; JE, (1) (a) For an overview, see b. (b) Murray, R. W. In Electroanalytical Chemistry; Bard, A. J., Ed.;Dekker: New York, 1984; pp 191-368. (2) (a) Based on a large body of previous experimental data by various workers, a general discussion of the factors controlling charge transport in redox polymer films is given in ref lb, pp 334-339; see also b-I. (b) Facci, J. S.; Schmehl, R. H.; Murray, R. W. J. Am. Chem. Soc. 1982,104,4959. (c) Buttry, D. A.; Anson, F. C. J. Am. Chem. Soc. 1983,105,685. (d) Majda, M.; Faulkner, L. R. J. Electroanal. Chem. Interfacial Electrochem. 1984, 169, 77. ( e ) J . Electroanal. Chem. Interfacial Electrochem. 1984, 169, 97. (f) Elliott, C.; Redepenning, J. G. J. Electroanal. Chem. Interfm'al Electrochem. 1984,181, 137. (9) Chen, X.;He, P.; Faulkner, L. P. J. Electroanal. Chem. Interfacial Electrochem. 1987, 222, 223. (h) Jernigan, J. C.; Murray, R. W. J. Phys. Chem. 1987, 91, 2030. (i) Jernigan, J. C.; Murray, R. W. J. Am. Chem. Sac. 1987, 109, 1138. Lange, R.; Doblhofer, K. J. Electroanal. Chem. Interfacial Electrochem. 1987, 216, 241. (k) Doblhofer, K.; Lange, R. J. Electroanal. Chem. Interfacial Electrochem. 1987, 229, 239. (1) Feldman, B. J.; Murray, R. W. Inorg. Chem. 1987, 26, 1702. (3) (a) Andrieux, C. P.; SavCnt, J. M. J. Electroanal. Chem. Interfacial Electrochem. 1980, I l l , 377. (b) Laviron, E. J. Electroanal. Chem. Interfacial Electrochem. 1980, 112, 1. (4) (a) Savtant, J. M. J. Electroanal. Chem. Interfacial Electrochem. 1986, 201, 21 1; J . Electroanal. Chem. Interfacial Electrochem. 1987, 227. 299. (b) SavZlant, J. M. J. Electroanal. Chem. Interfacial Electrochem. 1988, 262, 1. (c) Andrieux, C. P.; Savtant, J. M. J. Phys. Chem., in press.
u)
SCHEME I Dj-1
Dj
It Cj-1
It t
Cj
It t
Cj+l
+
+
Cj.1
Cj
Cj+l
+
+
+
Fj- 1
Fj
Fj+l
It
It
It
Gi
Gj+l
b
b.1
1 $(
Potential :
Djcl
b-1
+
electron flux; n, number of electrons exchanged between two adjacent sites):
The flux of the mobile electroinactive counterions (zC,charge number; Cc, concentration; De,diffusion coefficient) is given by the usual Nernst-Planck relationship:
The two fluxes are coupled through the maintenance of the electroneutrality condition. Application of this approach to films sandwiched between two electrodes containing a single mobile electroinactive counterion, or, additionally, an immobile electroinactive ion, has allowed the derivation of the steady-state current-potential responses under the assumption that activities do not depend upon the redox state of the film and that ion pairing is absent.4b It appears that the steady-state current is independent of the mobility of the electroinactive counterions. These are indeed macroscopically im-
0022-3654/88/2092-4526$01.50/00 1988 American Chemical Society
Electron Hopping between Localized Sites
The Journal of Physical Chemistry, Vol. 92, No. 15, I988 4521
+
mobile in the sense that their diffusion and migration exactly balance each other. This does not fit the concept of a partial kinetic control of charge transport by electroinactive counterion displacement as evoked earlier.1b*2d-f The potential step transient responses" do depend upon the electroinactive counterion mobility. However, the lower the latter, the larger the response which again does not fit the idea of electroinactive counterion displacement being a slow controlling step in the overall charge-transport process. The preceding therefore suggests the search of modes of coupling between electron hopping and electroinactive counterion displacement other than the simple maintenance of electroneutrality. In this connection, what follows aims at a description of the effect of ion pairing between the immobile redox ions and the mobile electroinactive counterions on the rate of charge transport. In view of their ionic concentrations and of the hydrophobicity of the polymer backbone, ions in most redox polymer coatings are indeed expected to undergo extensive ion pairing or even higher a g g r e g a t i ~ n . ~The mechanism that we investigate is shown in Scheme I. The ion-paired redox centers (D) are immobile and considered not to participate to electron hopping. The latter involves the free fixed ions (A), whereas the electroinactive counterions (C) can move. For the sake of simplicity we consider the case of where A bears a single positive charge, C bears a single negative charge, and B, the reduced form of the electroactive center, is neutral. We will also discuss the effect of electroinactive fixed unipositive ions (F) that may undergo ion pairing with the mobile electroinactive counterions (C) (lower part of Scheme I). The fixed electroinactive ion will be designated as the "supporting ion" since it plays a role similar to that of the supporting electrolyte in solution studies. Thus, strictly speaking, the following analyses apply to covalently attached electroactive centers that are unipositive in their oxidized form and neutral in their reduced form, the oxidized form being ion paired by uninegative mobile electroinactive counterions. They are also valid for attached ion-paired redox ions of higher charge number provided that it can be considered that the redox species that participate predominantly in electron hopping are the unipositive monodissociated ion aggregates in the oxidized form and the neutral undissociated ion aggregates in the reduced form. Higher degrees of dissociation may also be considered, but this would imply the knowledge of the relative participation of the resulting redox species in the electron-hopping process. Extension to the case of a fixed mononegative reduced form of the redox centers ion paired by a monopositive mobile electroinactive ion and a fixed neutral oxidized form of the redox center is, of course, immediate. Adaptation to electrostatically bound redox couples is also possible, provided the charge numbers of the redox couple are kl/O as above. The general rate laws featuring the above mechanism will be first established. The effect of ion pairing will then be explained, in terms of current-potential responses, for a redox polymer film sandwiched between two parallel electrodes at steady state.2f,2g,4b
monolayer (ion pairing) and the two adjacent (subscripts j 1 and j - 1) monolayers (electron hopping and electroinactive counterion translocation). It is assumed, as before? that the electron-hopping rate constant, kE,depends upon the potential difference between the two adjacent sites, A@, according to a Volmer-Butler law with a transfer coefficient of 0.5:
General Rate Laws Let us first consider the system without fixed electroinactive ions, as sketched in the upper part of Scheme I. For obtaining the rate laws we follow an approach similar to that already used for the same problem in the absence of ion pairing3a,4a. kE, k I , kD,and k A denote the rate constants for electron hopping, electroinactive counterion translocation, dissociation, and association of the ion pairs, respectively, and A , B, C, and D denote the concentrations of the various species. The variations of the latter with time in an equivalent monolayer (subscriptj) involve the same
- = koEIAjBj-l
(5) (a) Eisenberg, A. Macromolecules 1970, 3, 147. (b) Eisenberg, A.; King, M. Ion-Containing Polymers; Academic: New York, 1977. (c) Komoroski, R. A,; Mauritz, K. A. In Perfuorinated Ionomer Membranes; Eisenberg, A., Yeager, H. L., Eds.; ACS Symposium Series 180; American Chemical Society: Washington, D.C., 1982; pp 113-138 and references cited therein. (d) Mauritz, K. A,; Hopfinger, A. J. In Modern Aspects ofElectrochemistry; Bockris, J. 0.M., Conway, B. E., White, R. E., Eds.; Plenum: 1982; Vol. 16, pp 425-508.
Similarly, the electroinactive counterion translocation rate constant can be expressed as kI = k01 exp(
FAO
z)
It follows that
(
- = k o l Cj-l exp dCj dt
-(aj
[2iT
]
aj-,) +
exp
]
- O,-J - Cj exp
[--(aj+,
[--(aj 2:T
-
- Oj)] +
2iT
_ dDj - kAAjCj - k f i j dt
We then linearize the exponential terms and group the resulting potential-independent terms on one hand and the potential-dependent terms on the other: dAj _ - koE[-AjBj-l + Aj-iBj - AjBj+i + Aj+lBj] - kAAjCj + dt
kDDj
EF + ko 2RT -[-(@j
- @j-l)(AjBj-i + Aj-lBj) + (Oj+1 @ j ) (AjBj+l
dBj dt
+Aj+I B ~I )
koEF
- Aj-lBj + AjBj+l - Aj+lBj] + -[(Oj 2RT @,-1)(AjBj-I + Aj-lBj) - ( @ j + 1 - @j)(AjBj+l+ Aj+lBj)l
d Cj _ - koI(Cj_l - 2Cj + Cj+I) - kAAjcj + kDDj + dt
koIF ~
~
[
(
-@@j-I)(Cj-l j + Cj) - t @ j + 1 - @j)CCj+l+ Cj)l dDj _ - kAAjCj - k f i j
dt Since the mean distance, Ax, between the monolayers is small, t h e j - 1 a n d j + 1 concentrations and potentials can be expressed as functions of the corresponding j quantities and of their first and second space derivatives. It follows that
4528
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
SavEant
On the other hand, since C is electroinactive Jc = O, i.e.
ac o = -ax with DE = koECoEAx2 and DI = kIAx2
(eoE, total concentration of redox sites). At steady state, the fluxes (J) are given by
F -C RT
a@ ax
showing that the steady-state current response will not depend upon the diffusion coefficient, DI, of the mobile electroinactive counterions as in the case where ion pairing was not taken into c ~ n s i d e r a t i o n .In ~ ~addition A + B D = COE, A = C, D KAC
+
and the potential difference between the two electrodes, V, is expressed as
where the last term is the potential difference across the film. In addition, the film is conservative in C, B, and A + D (the three conditions are equivalent). Thus, for example
The foregoing two sets of equations can be used to obtained the transient and steady-state responses, respectively, taking into consideration the initial and boundary conditions appropriate to each particular investigated system. Electroneutrality and conservation of matter result in two additional equations: A=C
and A + B + D = C O E
Under steady-state conditions, ion pairing remains at equilibrium leading to the additional relationship ( K = k A / k D ,association equilibrium constant)
where p is the starting ratio of reduced over oxidized fixed redox site concentrations, thus defining the redox composition of the film. The resolution of the problem is simplified when formulated in dimensionless terms by introduction of the following variables and parameters: y = x / l , a = A/COE,b = B/CoE, c = C / C o E , d = D / C o c , K = KCOE, $ = I/Id with I, = FDECoE/l, q5 = ( F / R T ) @ ,E = (F/RT)V. The foregoing equations then become
D = KAC In the case where an attached "supporting ion" is present, the above equations are still valid with the exception of that concerning the mobile electroinactive counterion, C, which becomes
~ + b + d = l , U=C, d=Kac
ac _ --
where k < and k f Dare the association and dissociation rate constants of the ion-pairing reaction F C e G. In addition
+
(The subscripts 0 and 1 mean y = 0 and y = 1, respectively.) In addition
P I 1 b dy = - or, equivalently, l+P
+ Conservation of matter leads to an additional equation (COR total concentration of supporting ion including ion pairs):
F+G=Co~ Under steady-state conditions, the above set of equations giving the fluxes is still valid. The second ion-pairing reaction (F + C F'C G ) remains also at equilibrium. Thus (K'= k',/kb, ion-pairing equilibrium constant) G = KfFC Steady-State Responses We consider the case of a membrane, sandwiched between two parallel plane electrodes, conservative in the mobile electroinactive counterions, Le., the same system as discussed earlier without taking ion pairing into consideration."b At steady state, the density of the current flowing at the negatively polarized electrode (taken as the origin of space, x = 0, whereas the other electrode corresponds to x = I; I, thickness of the membrane) is given by
(3)
KU')
dy =
1 l+P
-( 5 )
Taking ( 2 ) and (3) into account, (1) becomes da $ = ( 2 - a) dY Integration then leads to $y = 2(a - ao)
a2 - a02
-2
and to
On the other hand $a
dy = ( 2 -~a') da
and $a'
Thus
dy = ( 2 ~ '- a 3 )d a
(7)
The Journal of Physical Chemistry, Vole92, No. 15, 1988 4529
Electron Hopping between Localized Sites
uI3- uo3 - ao2- -
1
a dy = ! ( a : 4
-[
1 x 1 a 2dy = 4
3
-I
2(a13- ao3) uI4- uo4 4
It follows from (5) that
+ 2K-3
uI2- uo2 -(ul3
1
K
- ao3)- -(a14 - ao4)= 4 l+P
Le., eliminating J, between (8) and (9):
2
--l +
P
-
-
1
+
1+-
2(1
+!
p) ](a1
+ uo) +
With regard to the potential drop across the film:
and thus
and +1
-
C#J~
= In
a1 -
a0
Equation 5 thus becomes
5 = In
U12(l
- Uo - KU): (13)
Uo2(1 - a1 - KO,')
For each [, ul and a. can be obtained from the simultaneous resolution of (9) and (10). J, is then obtained from (8) and thus the dimensionless current-potential response, J,([). Once J, is known, (7)gives the concentration profile of a and (3) those of 6,c, and d. The potential profile in the film ensues by using (1 1). Let us investigate the variations of some remarkable features of the response with the extent of ion pairing, Le., with the parameter K.
The current potential response is obviously symmetrical taward the [ = 0, J , = 0 point ( V = 0, Z = 0) showing a cathodic limiting current, J,L, that is obtained for a. = 0 and/or bl = 0, i.e., al + K U , ~= 0 (the anodic limiting current has the same absolute value). J,L depends upon the redox composition of the film, i.e., upon p. Its maximal value, J,LM, is reached when simultaneously a. = 0 and al K U = ~ 1. ~ Then
+
(1 a1 =
+ 4K)It2-
1 (14)
2K
and thus (1 J,LM
=
+ 4K)1/2 - 1 - [ ( I + K
4K)'l2
o-z 01
QO-4
Figure 1. Maximal limiting current, +L,, (a), and corresponding redox composition, p M (b), as functions of the extent of ion pairing ( K ) . Zero-current conductance, ($/& at p = p ~ as, a function of K (c).
- -
When K PM
i.e., for extensive ion pairing, $JLM
-
(Z/v0,
-
a
8 K2
i.e.
Then The variations of the maximal limiting current, J,LM, and the corresponding value of the redox composition factor, pM, with the extent of ion pairing as represented by K are shown in Figure 1. When K 0, i.e., when ion pairing becomes negligible, J,LM = 1.5 and pM = 1.25 as expected from earlier studies.4b
-
2 / ~ ' and /~
The limiting current is a function of both the starting redox composition of the film (as defined by p) and the extent of ion pairing (as measured by 6). The variations of $JL with these two parameters are shown in Figure 2a. Another characteristic of interest is the apparent conductance of the membrane at zero current (and therefore at zero potential), i.e., in dimensionless terms, (J,/[)o = (RT/FZd)(I/V)o.It can be calculated as follows. When $J and 0, al - a. = Au 0, the common value of al and a. being the solution of the equation
- 112
The redox composition of the film corresponding to the maximal value of the limiting current pM is then derived from (9), replacing J, by J,LM. a. by 0 and al by its value from (14):
-
m,
2.
+
KU2
1. =1+P
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
4530
SavCant
Thus
(
t
ao(2 - ao)P
):o
= 2p
+ ( 1 + p)ao(l + 2 K U o )
wL
-li
The variations of ($loo at p = pM with the extent of ion pairing are shown in Figure IC. On the other hand, the variations of ($/E)o with p and K are shown in Figure 2b. (K)
Influence of a Fixed “Supporting” Electroinactive Ion The film now contains, in addition to the previously considered species, a fixed electroinactive monopositive ion, F, which can be ion paired to a variable extent by the mobile electroinactive counterion, C, giving rise to a fixed ion pair, G. The master flux equations characterizing the system are the same as in the preceding case ( ( 1 ) and ( 2 ) ) as well as (4). What changes are the electroneutrality and matter conservation laws. Introducing-?’ = C°F/CoE,i.e., the excess of attached electroinactive over attached electroactive species, we now have, in dimensionless terms a+f=c, a+b+d=l, f+g=f” (15) and the ion-pairing equilibrium conditions
d = iiac and g =
K ’ ~ C (K’
(16)
= K’CoE)
Let us first consider the case where the attached electroinactive ions, F, are not ion paired (K’ = 0 ) . Then f = f”. Elimination of b and c between ( 1 ) and (2) and ( 1 5 ) leads to
The current is thus given by $=
p + 2)(al - an) -
a12- ao2 ~
2
+ a1 -f”v+ 1) In f”f“ + an
On the other hand, the conservation of the total amount of oxidized species in the film implies that
Multiplication of (17) by a and integration leads to
$L
a dy =
1
-f”w + l ) ( a l - ao) + y2 +f” ( a
1 2- ao2)-
+ a1 +f”’Cf“ + 1 ) In f” -
a i 3- aO3 3
f” + a0
Figure 2. Variations of the limiting current, #L (a), and of the zerocurrent conductance, (#/& (b), with the redox composition, p , and the extent of ion pairing ( K ) . The numbers on the curves are the values of
log
K.
The current-potential curves, $(E) in dimensionless form, can thus be obtained for any set of values of p, K , and$ by simultaneous numerical resolution of ( 1 8 ) , (201,and (21), as well as their main characteristics, $LM, pM, and ($/E),,. Qualitatively it is expected that an increase of the amount of electroinactive fiied ions over the electroactive fixed ions, i.e., an increase of $, should reinforce ion pairing of the electroactive species A and therefore decrease the conductivity of the film. Let us illustrate this in quantitative terms by representing the variations of the maximal limiting current with the ion-pairing equilibrium constant (i.e., K in dimensionless terms) and the excess of “supporting” electroinactive ions, f”. The maximal current is obtained from (18), making a. = 0 and bl = 0; i.e.
Similarly, multiplication of (17) by a2 and integration leads to
2
+a1 +f” (a,3 - ao3) - a14- aO4+F3(f”+ 1 ) In f” 3 4 f” + a0 ~
Linear combination of the two above equations and replacement into (19) finally leads to
-f”r + l ) ( a l - ao) + [ 1 + f”(12+ * ) ] ( a 1 2- ao2) +
The results are shown in Figure 3. As J” increases, an interesting limiting case is reached where the maximal limiting current is given by 1 ~LLM=
-
-
In this context, when ion pairing is small, Le., for ~ f ” 0, $LM 1 , and we obtain a migration-free. system with the usual current potential curve $=- 1 1 e-[
+
Equation 4 now becomes
(note that is then equal to 1 ) corresponding to an excess of supporting electrolyte. Another way of displaying the results, more fitted to experiments aiming at a systematic investigation of the concentration dependency of the electron hopping diffusion coefficient,’ is as follows. In such experiments the sum of the concentration of attached electroactive (A, B) and electroinactive (F) species is
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4531
Electron Hopping between Localized Sites
XE
*E
Figure 3. Maximal limiting current in the presence of a fixed ‘supporting” electroinactive ion as a function of the extent of ion pairing (as measured by K ) and the excess of “supporting”over redox fixed ions, $. The number on the curves is the value of log$.
+
maintained constant (CoE C°F= Co)while the ratio of electroactive over electroinactive species is systematically varied. The variable of interest is therefore the molar fraction of attached electroactive species, X E :
Figure 4. Influence of a non-ion-paired attached electroinactive ion. Variation of the apparent electron-hopping“diffusion”coefficient with the molar fraction of electroactive species, XE, and the extent of ion pairing as measured by K C E . The numbers on the curves are the values of log KCOR.
The algebra is simpler, although still cumbersome, if we take c as master variable instead of a , as done before. Using the above equations, ( 1 ) becomes K‘f”
(1
+ K’C)2
f ” -
-
+
C(1
K’C)
f”2
The purpose of this kind of experiment is then to investigate the variations of the apparent diffusion coefficient, PP,defined as iLM
FsCoE P ~ FSCOEDE = = J/LM 1
(23)
C(1
+
K’C)’
] dc dr
(24)
Integrating between y = 0 and y = 1 thus leads to the expression of )I as a function of c1 and co:
J/=
upon XE. Then r
The maximal limiting current, +LM, is then obtained by replacing, in the above equation, c1 and co by their values corresponding to uo = 0 and bl = 0. (1 co =
+
4K’P)’/2
-1
2K’ whereas c1 is one solution of the third-degree equation: KK’Ci3
The results are shown in Figure 4, which represents the variations of the apparent diffusion coefficient with X Eand the extent of ion pairing as measured by KCO (itself independent of X E ) . It P with XE is nonlinear. In the is seen that the variation of P absence of ion pairing the slight nonlinearity is a result of “migration”. The nonlinearity dramatically increases as the ion-pair association constant increases. What happens when the electroinactivefixed “supporting” ions, F, me significantly ion paired with the mobile electroinactive ions, C ( K ’ >> 0) is the question we discuss now. The master equations ( l ) , ( 2 ) , and ( 4 ) are still valid. The changes are in the matter conservation equations. We now have a+~ac+b=l a+f=c g + f = f ” and g’K’fC Thus f=
f” 1
+
K‘C
+ + K’)Ci2 + ( 1 (K
Kf”
- K’)Cl - ( 1
+ f”)= 0
(26)
(27)
and is comprised between co and
Figure 5 represents an example which illustrates the effect on the maximal limiting current of the presence of an electroinactive attached ion and of its ion pairing with the mobile electroinactive counterion. We start with a strong excess, J” = 100, of the attached electroinactive ion, F, over the electroactive species, C. If F is not ion paired by C (K’ 0), the effect of F is to render the decrease of the limiting current with the extent of ion pairing of the electroactive ion, A, by C, much more dramatic than in the absence of F (compare the extreme right-hand and left-hand curves in Figure 5). At this stage, when ion pairing of the fixed electroinactive ion is increased, Le., when K’increases, it is seen that the variation of the limiting current originally observed in the absence of (“supporting”) electroinactive fixed ion is progressively restored. It must be borne in mind that this description applies for a constant value of the total concentration of electroactive species, COE. If the variations of F concentration do not
-
4532
SavCant
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
Figure 5. Variations of the maximal liiting current, GLM,with the extent of ion pairing of electroactiveions (as measured by K ) in the presence of an excess of electroinactivefixed ion (f“ = 100) when the extent of ion pairing of electroinactive fixed ions by mobile electroinactive counterion (as measured by K’) increases. The number on each curve is the value O f log K’.
let COE remain constant, this has to be taken into account through the definition of
+
I , being proportional to COP The above described behavior results from the fact that one effect of increasing F is to increase C and thus to enhance the extent of ion pairing of the electroactive ion A. Upon increasing K’, C decreases because it is trapped by ion pairing with F. As a consequence, the extent of ion pairing of A decreases. The presence of F creates another effect, namely, the decrease of “migration”. This explains why the increase of K‘ in the presence of a substantial concentration of F is not strictly equivalent to a decrease of K although this does represent the observed trend qualitatively. Another way of representing the effect of ion pairing of both the electroactive and electroinactive ions is again to consider experiments in which their relative proportions change while the sum of their concentrations is kept constant. The apparent electron-hopping diffusion coefficient, as defined by (23), is then a function of the molar fraction of attached electroactive species (22), of KCO and K’CO. It is obtained from (25-28) replacingj” by 1 - XE/XE, K by KCOX,, and K’ by K‘C’XE. The results are shown in Figure 6. It is seen that the increase of ion pairing of the attached electroinactive ions smooths the variations of the apparent electron-hopping diffusion coefficient with the molar fraction of 2 both electroactive ions. The variation tends toward X E 1 / where K and K’ are large. Conclusions
The foregoing analyses allow the calculation of all characteristics of steady-state current-potential curves as a function of the redox composition of the film, the extent of ion pairing of the electroactive ions by the mobile electroinactive counterions, the addition of an attached electroinactive ion (“supporting ion”), and the extent of its own ion pairing with the mobile electroinactive counterions. Qualitatively, the main following trends are observed for the variation of the charge-transport rate with the foregoing factors. As seen in Figure 1, the charge-transport rate falls quite rapidly as the extent of ion pairing between the attached electroactive ions and the mobile electroinactive counterions increases. The parameter measuring the extent of ion pairing, K = KCoE,is
XE Figure 6. Variations of the apparent electron-hoppingdiffusion coefficient with the ion-pairing association constants of the attached electroactive ( K ) and electroinactive(K? ions and with the molar fraction of attached electroactive ions (&). KCO: 0 (a); lo2 (b); 104 (c); lo6 (d). The numbers on the curves are the values of log K ‘ P . The values of D a g X e = 1 are 1.5 (a), 0.185 (b), 2 X (c), and 2 X (d).
proportional to both the ion-pairing equilibrium constant, K , and the total concentration of electroactive species, CoE. Within this framework, it is thus understood how the nature of the mobile electroinactive counterion can affect the chargetransport rate, even at steady state where its diffusion coefficient does not interfere directly. The control of the mobile electroinactive ion on the charge-transport rate arises from the strength of the ion pairs it may create with the electroactive ion: the stronger the ion pairs the slower the charge transport. Another important aspect of the problem concerns the variations of the apparent diffusion coefficient derived from the maximal plateau currents with the amount of electroactive material present in the polymer films. Experiments aiming at investigating these variations are usually carried out with systems containing electroactive and electroinactive attached ions. The apparent electron-hopping diffusion coefficient is then measured as a function of the relative proportion of the two sorts of ions while the sum of their concentration remains constant. As seen in Figures 4 and 6, considerable deviation from the usual proportionality law can be obtained depending on the relative magnitude of ion pairing of the mobile counterions with the electroactive and electroinactive attached ions, respectively. Another consequence of the rapid fall of the charge-transport rate upon ion pairing of the electroactive ions is the following. As ion pairing becomes stronger and stronger, charge transport tends to be vanishingly slow in the framework of the mechanism considered so far, in which the ion-paired electroactive ions do not participate directly in electron hopping. Under such circumstances, this mechanism is likely to be replaced by another one involving the direct participation of the ion pairs in the electron-transfer process. As discussed elsewhere,6 the coupling between electron hopping and electroinactive ion displacement is then of a different type, the two processes being possibly concerted. Acknowledgment. Discussions with C. P. Andrieux (UniversitC Paris 7) on the matter of the present paper were, as always, very helpful. (6) Saveant, J.
M.J . Phys. Chem. 1988, 92,
1011
