Multi-component diffusion approach to transport across electroactive-polymer films with two mobile charge carriers

Ekrrochimica Acts, Vol. 41, Nos. 718. pp. 1375 1381. 1996 Copyright G 1996 Elwier ScienceLtd. Printed IIIGreat Britain. All rights reserved @X3-4686/96 115.00 + 0.00

Pergamon 0013-4686(95)00459-9

MULTI-COMPONENT DIFFUSION APPROACH TRANSPORT ACROSS ELECTROACTIVE-POLYMER WITH TWO MOBILE CHARGE CARRIERS M. A.

VOROTYNTSEV*~,

J.

P.

BADIALI~

and E.

TO FILMS

VIEIL

C.E.A., Centre d’Etudes Nucleaires de Grenoble, DRFMC-SESAM, Lab.d’Electrochemie Moleculaire, France, and $ Lab. Structure et Reactivite des Systemes Interfaciaux, Universite Paris-& France (Received 8 May 1995) Abstract-This paper aims to apply general relations between the thermodynamical forces (gradients of electrochemical potentials) and resulting fluxes of the species to transport phenomena in a uniform film of the electroactive polymer in contact with some other conducting media, metal(s) and/or solution(s), in the case of a low-amplitude perturbation imposed. Two kinds of mobile charged species are assumed to be present inside the film, the “electronic” and “ionic” ones. The coefficients in the above relations (friction coefftcients) are expressed through the experimentally measurable macroscopic transport parameters, the total high-frequency conductivity, migration transference numbers, binary diffusion coefficient and differential redox capacitance of the film. The non-stationary diffusion equation is found to be valid for several local characteristics of the film, in particular for electron or ion charge density, or for the low-frequency current density. This equation has been solved analytically for three usual geometries of the system, metal/film/metal, metal/film/solution and solution/film/solution, upon a sinusoidal variation of the electrode potential. The final expressions for complex impedance contain contributions of the bulk film, interfacial charge-transfer resistances and (in contact with solution(s)) bulk solution. The functional form of their frequency dependence as well as the shape of complex-impedance plots has turned out to be highly simple for all geometries, being in accordance with those derived earlier within the framework of the Nernst-Planck-Einstein equations. However, the parameters of those dependences have a form different with respect to the previous expectations, leading to a modification of the procedure to interpret experimental data. Key words: electroactive-polymer coupling of the fluxes.

films, electron-ion transport, impedance, irreversible thermodynamics,

NOTATION

CmII

c:,co

c, D

Da, Dr

perturbations of individual concentrations (e = electron, i = ion, fx= summation index), section 2 “maximum” concentration, eqn. 15 unperturbed concentrations, e = electron, i = ion (co, for symmetrical electrolyte, equation (2)) redox capacitance per unit volume, equation (24) “binary electron-ion diffusion coefficient”, equations (2)V(26), (27) individual diffusion coefticients (e = electron, i = ion)

* On leave from the A. N. Frumkin Institute of Electrochemistry, Moscow, Russia. t To whom correspondence should be addressed at Department of Applied Physics, Fukui University, 9-1 Bunkyo &home, Fukui-shi, 910 Japan.

F

i, , ii

L R

in the Nernst-PlanckEinstein model, equation (2) Faraday number instantaneous overall current density, equation (28) values of instantaneous partial current densities (e = electron, i = iqn), equation (18) “residual” current density, equation (28) instantaneous flux densities (e = electron, i = ion, p = polymer matrix), equation (6) combinations of friction coefficients (e = electron, i = ion), equation (21) friction coefficients (a, indices, B = summation e = electron, i = ion, p = polymer matrix), equation (6) film thickness gas constant

1376

M. A. VOROTYNTSEV et al.

Rf

t,

1

ti

T

X

zm/f/m) ZE,f,S -Tn/f/s ’

-%I,zs2

7

0

PC7 Pi * PetPi

z,

0

*

v = (~oL.?/~D)"~

P=PetPi

0

Pe 3 Pi

PP

w

0

high-frequency bulk-film resistance, equations (2), (32) interfacial charge-transfer resistances (m = metal, f = film, s = solution), equations (3), (4) (33)-(36) time migration (high-frequency) transference numbers (e = electron, i = ion), equations (2), (29, (27) absolute temperature coordinate normal to both interfaces charge numbers of species in the film (a = summation index, e = electron, i = ion), equations (5), (8) complex impedance of the film in contact with two surrounding media (m = metal, f = film, s = solution), equations (1) (3) (4) (4% (41), (44) complex impedance of the semi-infinite solution having an interfacial boundary permeable only for the film counter-ions, equations (39, (36) perturbation of the local electrical potential electron-ion combined potential, equation (22) high-frequency bulk conductivity of the film, equations (25), (27) partial conductivities, equations (1 l), (12) perturbations of the electrochemical potentials (a = summation index, e = electron, i = ion), section 2. unperturbed electrochemical potentials (e = electron, i = ion), section 2 potentials chemical i = ion), (e = electron, equation (10) dimensionless function of frequency, equations (I), (38) perturbations of the electron and ion charge densities (e = electron, i = ion), equation (8) unperturbed electron and ion charge densities (e = electron, i = ion), equation (5) polymer-matrix charge density (“fixed charges”), equation (5) frequency

1. INTRODUCTION The charging process in electroactive polymer films at the electrode surface is realized via the interfacial electron and ion exchange followed by the transport of these species inside the polymer phase. The latter step may be a rate-determining one, especially for relatively thick films and/or high-frequency perturbations. Their description represents a basic problem at interpretation of experimental data obtained with the use of the potential-step, or impedance techniques. In the simplest case one can neglect effects due to co-ion redistribution so that one should consider electrodiffusion transport of two mobile charge carriers, electrons (eg “polarons”, “bipolarons”) and counter-ions. A straightforward idea is to apply the same Nernst-Planck-Einstein approach well substantiated for electrolyte solutions. The results for concentration and potential distributions depend crucially on the type of the permeability conditions at the interfaces, “symmetrical” and “asymmetrical” systems, according to Buck[l, 21. In the former case (eg the film between two metals) the concentration and potential perturbation profiles are odd functions with respect to the middle plane, and complex impedance contains the hyperbolic tangent function of the frequency divided by a combination of the film thickness L and the “binary electron-ion diffusion coefficient” D[3] : Z m/r/m= Rr + Wi/t,)z,&b Z,,,(V)= v-i tanh v, v = O’WL?/~D)“~,

(1)

R, being the high-frequency bulk-film resistance, D, and Di,the diffusion coefficients of electrons and ions, t, and ti, their transference numbers: R,= RTL[z2F2co(D, + Di)]-', D = 20, Di(D, + Di)-', t,= 1 - ti= D,(D,+ Di)-l

(2)

R, the gas constant, T,the absolute temperature, F, the Faraday number, co, the unperturbed concentration, the electronic and ionic charge numbers being assumed identical, z and -z. A similar expression has been derived for another symmetrical system, a film between two solutions of the same composition, with taking into account the interfacial chargetransfer (R,,,) contributions[4] : Z s/r/s= 2&/s + Rr + Rf(telti)ZJV)~

(3)

A somewhat more complicated formula takes place for the film between the metal and solution[5,6]: Z m/t/s= Rm/r + Rt/, + Rr + RX4te ti)- ’ ’ Czcdv) + Cte- ti)2zth(v)lt zo,,(v) = v-l coth v,

(4)

R m/fand Rf,,being the corresponding interfacial charge-transfer resistances. Much attention has been paid to the problem of “hopping” the electronic species at the array of redox

1377

Multi-component diffusion approach centers[7-91 leading to another expression for the migration flux. This concept has been applied by Haas et al.[lO, 111 to treat the impedance problem for the “asymmetrical” system. Surprisingly, the frequency dependence of Z(o) turned out to be identical to that in equation (4)[6] although expressions for all impedance parameters were much more complex. It is demonstrated below that such coincidence is due to a much more genera1 reason. Experimental data for numerous polymer systems in both geometries have some resemblance to predictions of the above theories, a Warburg linear behaviour at high frequencies with a further increase of the slope in the complex impedance plane for the “modified electrode” geometry (see equation (4)), or with a loop for the symmetrical systems, equations (1) or (3) (see[4]). However, the low-frequency shape is not well reproduced by the theoretical equations. Moreover, some remarkable anomalies have been revealed for systems with comparable values of the electronic and ionic bulk-film resistancesC121. One of the reasons for this failure might be a simplified description of the bulk film transport. Ion subsystem in all above theories was considered within the framework of the “dilute solution” approximation, with no account for its short-range, or saturation effects[13]. Another important factor to be incorporated into the model is a direct coupling between the electronic and ionic fluxes[ 121. This paper is aimed to propose such a general theory of the coupled electron-ion transport inside the film, with further appfication to impedance phenomena for all three geometries of the film. The final expressions for complex impedance turned out to be quite astonishing. The functional form of Z(w) for all geometries (and therefore, the shapes of the complexplane plots) is extremely simple being similar to equations (l), (3) and (4), correspondingly, although the expressions for the impedance parameters through the microscopic characteristics (and consequently, their interpretation) are significantly modified.

tions: c: +

c, ,

co

+

Ci,

d+Pe~

!t+Pi*

as well as to non-zero fluxes of the species: J, + 0, Ji # 0. The amplitude of all perturbations is assumed to be small enough to apply a linear response theory, i.e. linear relations between the gradient of electrochemical potential of a species (e, or i) and the precisely, flux fluxes (more instantaneous densities)[ 141:

c,oV~,= c &dJ&

(6)

- J,/d.

Here, index c( means e or i, /3 runs all interacting components, electrons, ions and the polymer matrix (p), K,, are the “friction coefficients” characterizing the interaction between the corresponding fluxes which are generally some functions of unperturbed the irreversibleconcentrations c,“. Within thermodynamics treatment these functions are considered as known. An important advantage of such a treatment is an opportunity to describe the transport phenomena (eg, to derive expressions for complex impedance as a function of frequency, see below, and use those for treating experimental data for each charging level) without specifying the functional form of &(c:). On the other hand, one must address after it to a model theory to specify these functions for interpretation of thus found dependences of the parameters on the charging degree. It is convenient to take the polymer matrix as a reference point which means J, = 0. The latter relation is obviously valid if electron-ion transport is not accompanied by the local polymer matrix displacement. In the presence of a reversible swelling and the contraction of the film one should redefine the concentrations and charge densities relating them to a unit polymer-matrix volume. Then, the resulting set of relations (6} contains three microscopic transport characteristics, K,, , K~, and K,, This set should be supplemented with the continuity equation for each concentration, e or i : dc,/dt = - VJ, ,

2. BACKGROUND

TRANSPORT

RELATIONS

In a general case the film contains mobile electron and ion species as well as some “fixed” charges, eg attached ionogenic groups, or ions tightly bound to the polymer matrix. This “fixed” charge, pp, will be assumed as space and time independent. In the (quasi)equilibrium state the electron (~8) and ion (co) concentrations inside the film (beyond the space-charge regions near the interfaces) satisfy the electroneutrality condition (z,, i, the corresponding charges of the species): it) + PO + pp = 0,

~2, i = ze,i Fez,i

(5)

and are determined by the equilibrium conditions at the interfaces. Their electrochemical potentials in the equilibrium state, ~(0and pp, are also constant in this region. An external perturbation (eg, variation of the electrode potential) leads to a change of these distribu-

and the local electroneutrality tion (5): P,+pi=O,

condition,

(7) see equa-

Pc.i=Ze,iFc,.i

(8)

3. RELATIONS TO THE PREVIOUS THEORIES The above quoted theoretical approaches disregard a direct coupling between the electron and ion fluxes due to the term with Kei, as well as a dependence of the electrochemical potential, eg, p’e on the concentration of another component, ci. Within these simplifying assumptions equation (6) is reduced to independent relations for each component:

J, = -(c:)~W,,)- ‘VP,, ~a = A(+’ 3cp>cm). (9) After separating p@into the electrical and chemical parts: & = Z,.i Fq + c~X3 Pt,*= PZfC,O 3CA

w

1378

M. A. VOROTYNTSEVet al.

one can represent the fluxes in diffusion-migration form :

the

standard

J, = - (K#/Z,F)Vcp- D, vc,

(11)

the partial conductivities and the diffusion coeflicients being expressed through the friction coeflicients and the chemical potentials, K, = (Z, Fc:)2(&,) - ‘7 D, = (c%K,)K#,

(dp:/dc,o)-

=

’ d&de:, ‘zf F2.

(12)

Low concentration behaviour (“dilute solution’J

Then, the friction coellicients are proportional to the concentrations, and the chemical potentials are a logarithmic function of concentrations, K ilp*c:T

o~const+RTlnc~, P’a

(13)

so that the diffusion coefficient approaches some low-concentration limit, and the ratio, KJD,, increases linearly with the concentration: DL1x D;“,

KID,

zz cx z;

F’(RT)-‘.

(14)

This case is identical to the Nernst-Planck-Einstein relations. Saturation efects

The concentration of species a may have an upper limitation, c8 G cr. It may be due to localization of the species at an array (eg, electrons), or due to geometrical factors for ions (number of cavities available). Then, one should consider the difference, Cmar_ cO as the concentration of vacancies (similar ti hole; in semiconductors) which leads to expressions : pz x const - RT In (c,““”- cz), D, Z ghigh = 3 z

(C,““”

K,, - (cy’ - cz),

Kw’D,

-

c;)zzf

F’(RT)- ‘.

(1%

Intermediate behaviout

One can expect[lS] the diffusion coefficient to vary between these limits whereas the partial conductivity should have at least one extremum since KID,

x c.“( 1 -

C: / cm= a )~(c:~:F*(W-~.

(16)

Using an expression for the chemical potential,

=

i,+i = z,.i FJ, ,i .

(18)

Then, relation (6) takes the form: V&/z, F) = -k, i, + kii

(19)

V(pJziF) = -kiii + ki,

(20)

the three new parameters previous ones :

are combinations

of the

k, = (K,p + K,J(P,0)-2t ki = (Kip + K&O)-2~ k = K,,(p: p;)- ‘.

(21)

Instead of two electrochemical potentials, p’eand pi, we shall use their combination (an analog of the chemical potential of the salt in the theory of solutions): ~ei = ~,/z,F - ~ilZiF.

(22)

It should be noted that this quantity is measured in units of the potential (rather than energy) which makes subsequent relations much more compact. The electric parts of electrochemical potentials are cancelled in equation (22) so that Qei (being a combination of the chemical potentials of electrons and ions) depends only on the local electronic charge, pc, which is equal (with minus) to the ionic one so that we’ll denote it below without an index: @ei= @e&Z, PO, PX P G P, = -Pi’

P = c&t

pz = const + RT In c~/(c~~ - cp)

K,

In the following treatment we shall start from general relations (6), in particular, taking into account the non-diagonal term with Kei. This direct coupling of the fluxes well known for concentrated electrolyte solutions is due to electrophoretic relaxation and similar effects. The set of relations (6)-(g) includes three microscopic kinetic parameters, K,, , Ki, and Kei, as well as quasi-equilibrium characteristics, chemical potentials of both species. Those all depend on concentrations cp and co, an explicit form of these dependences calls for some model theory. However, it is shown below that one can obtain extensive information for a low-amplitude response of the system without addressing such models. The subsequent relations have a much simpler form if the concentrations are replaced by the charge densities (equations (5), (8)), and the fluxes by the partial currents (more precisely, the partial current densities):

(23)

Within the linear response theory the local perturbations of both quantities are simply proportional:

Lattice gas model for electrons

one arrives at the “hopping” law[8]

4. MACROSCOPIC TRANSPORT PARAMETERS

:

D, c.“(1 - c,O/cy)z,Z F2(R7’)- ’

(17)

which is practically a simple particular case of equation (16) at f(cz) = 1, supplemented by a constancy of the individual diffusion coefficient, Da = const[7].

9 POPei

(24)

the coefficient C,, the redox capacitance per unit volume[16], being a function of the background charging level (via pt, ~0) but not space- or timedependent explicitly. Let us relate the microscopic transport parameters (Kp, 9or k,) with the traditional transport characteristlcs, (high frequency) conductivity K, (migration) transference numbers, t, = 1 - t,, and binary diffu-

1379

Multi-component diffusion approach sion coefficient D. To introduce the latter one should consider two limiting situations: (1) The conductivity and transference numbers are defined in conditions of no concentration gradients (eg, at sufficiently high frequencies) where V(PJ’Z,F) = V(Pci/ziF) = VP, i = i, + ii = - KV~,

i,, i = t,, ii.

(25)

(2) Binary diffusion coefficient D is introduced upon an alternative condition of no instantaneous current passage, i = 0, according to the relation : J, = -DVc,,

Ji = -DVc,

(with taking into account the equality, i, = -ii). (26) Combining equation (25), or equation (26) with relations (19), (20) one can express the macroscopic parameters through the friction coefficients, equation (2 1) and vice uersa : DC, = (k, + k + ki)- ‘, K-

’ =

t,,i = DC&k,,, + k),

DC,(k, ki - k’).

i = i, + i.,,

i’ = tii, - teii.

i’ =

(overall current depends only on time),

-Dvp,

ap/at = -Vi’.

ap/at = Dv2p, aqi/at = Dv2c+,, ai’@= DV’i’. (30) At the formulation of the boundary conditions for these equations one can use relations for the partial electrochemical potentials of the species: ‘i & ti, cV@,i

(31) which can be easily integrated in view of a constancy of the overall current across the film: I&. i(O,t) - PLY. i(L, t)l(ze,i F)- ’ = -R, i f ti, e[@,#A t) - @ei(b 01. (32) Here, L is the film thickness, R, = L/K, the highfrequency bulk film resistance (per a unit film crosssection area). 5. BOUNDARY

ie, i’(0, t) = tii.

CONDITIONS In most cases each interface of the film with the surrounding medium is penetrable for only one kind

(33)

This electron-exchange condition at another interface, film/metal (which potential is taken as a reference point), would have the form: P&

0/z, F = iR,,, , i,(L, t) = 0,

i’(L, t) = ti i.

(34) For the case of a contact with the solution possessing an interfacial ionic exchange one should interchange indices e and i as well as to account for the bulk-solution contribution to impedance, Z,, or Z,,: E - ~~(0,t)/zi F = i(Z,, + RSir), i’(0. t) = -t, i

(29)

Equations (29), (24) show that any of three quantities satisfies the non-stationary diffusion equation:

V(u, i/z,. i F) = -K-

ii(O, t) = 0,

(28)

Comparison of equations (25) and (28) shows that the ratio, ii/i, vanishes at high frequencies. Then, one arrives at a very simple set of equations : Vi = 0

i = CE- ~(0, W, W4,,,X ‘,

(27)

With the use of equation (27) one can replace coefficients k in equations (19), (20) by the above macroscopic parameters. The resulting equations have a much simpler form if written for the overall instantaneous current, i, and a residual term, i’, introduced instead of the partial currents: i,,i = t, i + i’,

of species, electrons or ions. Generally, such a heterogeneous charge-transfer process possesses some resistance (interfaces with an equilibrium exchange, or completely blocked, represent particular cases corresponding to very low, or very high values of such resistance). In this paper we will disregard effects of the interfacial (double layer) charging. Consider the metal/film interface with an electron exchange as an example. At equilibrium (without a perturbation) the electrochemical potentials of the electron in the metal and film are equal. After perturbation the electrochemical potential in the metal of the species with charge z, is shifted by z,FE, E = E(t) being a value of the perturbed electrode potential (with respect to the reference electrode) at the moment t. If the instantaneous value of the electrochemical potential of this species in the film near the interface, ~~(0, t), is different from the metal one, the electronic current will pass across the interface which must be equal to the value of the overall current i at any point of the film (see equation (29)) while the ionic current should vanish at the interface :

(35) pi(L, t)/zi F = i(R,,, + Z,,),

i’(L, t) = -te i. (36)

Thus, for any geometry of the system, m/f/m, m/f/s or s/f/s, there are two boundary conditions for “residual” current i’ (equation (28)) which enable one to determine a certain solution from its diffusion equation (30). Then, equations (29) and (24) give a solution for the charge density, p, and Qei. After it, one should apply corresponding boundary conditions (33)-(36), coupling the values of electrochemical potentials at two different interfaces with the use of equations (32) and (for an “asymmetrical” system, m/f/s) equation (22). Realization of this program with respect to all geometries in the case of a low-amplitude sinusoidal perturbation is illustrated below. Equation (30) for the complex amplitude of the residual current takes the form: d2i’/dx2 = joi’.

(37)

6. METAL/FILM/METAL In view of the symmetry of equation (37) and boundary conditions (33), (34) with respect to the

M. A. VOROTYNTSEV et al.

1380

middle plane of the film, x = L/2, the solution should be the hyperbolic cosine of x - L/2, with the coefficient found immediately: i’(x) = tii cash v(2x/L - l)[cosh v]- ‘, v = (jwL2/4D)“*,

see equation (1).

(38)

Relations (24), (29) give a solution for p and p at any point: Qti = p/C, = -(jwC,)-

’ di’jdx.

Combining boundary conditions (33), (34) for the potentials with relation (32) for /.I~one obtains: E = f(R,/r + Rr + Rt/m) + riC@,i(O,r) - @ei(L t)l. (39) It leads to an expression for impedance: G,,r,m(~) = R,,r + R, + Rr,, + 4t; AR,r,,(v), ZJV) = v-i tanh v, AR, = L(4DCJ’.

(40)

7. SOLUTION/FILM/SOLUTION One can use the same formulae, with replacing the transference number and adding the solution contributions originating from equations (35), (36): Zs,r,s(w) = Z,, + R,,r + R, + &is + +

Zs2

4rf AR, z,,,(v).

(41)

8. METAL/FILM/SOLUTION Due to a non-symmetry of boundary conditions (33), (36) for i’, the solution contains both hyperbolic functions [6] : i’(x) = Bi[b cash v(2x/L - 1) - sinh v(2x/L - l)], b = (ti - t,) tanh v,

B = (2 sinh v))‘.

(42)

Results for p and p are again obtained by differentiation. Equations (33), (36), (32), (22) for the potentials given after removing the particular electrochemical potentials at the interfaces: E = i(R,,, + R, + Rr,, + Z,) + [email protected](O, t) + t, @JL9 t).

(43)

Impedance expression is produced after insertion of p: -%,/r/r(~) = R,,, + R, + R,,, + Z, + ARrCs,lh + (r, - tYztJt z,,,,(v)= v-’ coth v, AR,, equation (40). (44)

9. DISCUSSION This paper aims to overcome a principal shortcoming of the preceding treatments of transport phenomena in such complex media as electroactive

polymer films containing high concentrations (several mol I - ’ in a charged state) of electronic and ionic species, which makes doubtful an application of very simplified approaches based on the NernstPlanck equations, or the dilute-gas model for electron hopping. Usual justifications for such simplification of the real picture are: (1) difficulties to derive a more fundamental theory and solve than its which equations calls often for numerical calculations[l 1, 171 making the final results not easily available for comparison with experimental data; (2) the final expressions of such general theories are expected to contain so many parameters that it will be hardly realistic to determine those reliably from experimental data. The surprising conclusion of this study is an opportunity to apply the most general transport theory for uniform systems with two mobile charge carriers and to obtain in an analytical form all final results (concentration and potential profiles, complex impedance) for the low-amplitude perturbation. Even more astonishing is the form of these final expressions. The frequency dependences for all geometries of the system are extremely simple even in this most general case, equations (40), (41) and (44) for three different geometries, being identical to the results derived earlier within the framework of the Nernst-Planck equations, equations (l), (3) and (4) (except for additional bulk-solution contributions). It means also a coincidence of the plots in the compleximpedance plane, at the corresponding choice of the parameters, This insensitivity of the impedance results to the model may be a reason for the success of the previous simplified treatment in interpretation of some basic features of experimental data. On the other hand, these results imply that some other experimental results (like a low-frequency behaviour) cannot be understood even within the most general treatment so that one should introduce additional physical factors, eg the film inhomogeneity, diffusion of neutral species, ion pairing and association, etc.[18, 191. Although the functional form being unchanged, the meaning of impedance parameters in the frequency-dependent terms has to be modified. One must keep in mind that formulae (I), (3), (4) rely upon expressions (2) for the bulk-film electron and ion resistances and migration transference numbers, as well as equation (45) for the redox capacitance, C, = z2FZc0(2RT)- ‘.

(45)

Although a linear dependence of the ion conductivity on the ionic charge has been found for some systems[20] a more complicated behaviour is usually observed, eg curves with a maximum at medium charging levels for redox capacitance, or electron conductivity[21]. One might believe that the same formulae (l), (3) and (4) (presented in the form containing no model parameters, like the unperturbed concentration co, or the individual diffusion coefficients D, and Q) can be applied in a more general case, beyond the dilute-solution limiting region. However, the results of this paper have demonstrated that such a generalization is not so straightforward. Therefore, the new formulae for impedance,

Multi-component diffusion approach (40), (41) or (44), should he applied in this case. The bulk-film contributions to those expressions for Z(o) contain 4 parameters, R,, AR,, t, = 1 - ti

equations

and a characteristic frequency, 4D/c, equation (I), which may be used to determine the overall conduc-

tivity K, binary diffusion coefficient D and the redox capacitance, C,, assuming thickness L to be known. A thorough analysis of the low- and highfrequency resistances of conducting polymers[ 121 has revealed profound anomalies in the potential region of comparable electron and ion conductivities. One can see that the theoretical relations for those limiting resistances hold only in the case of no direct coupling between the electronic and ionic fluxes, ie, at Kei = 0, see Section 3. At nonzero values of this friction coefficient the ratio of the limiting resistances is not expressed immediately through the product of transference numbers, thus giving hope for a more satisfactory interpretation of these data. Acknowledgements-We are grateful to the participants of the International Workshop on Electrochemistry of Electroactive Polymer Films (WEEPF’95) and the 3rd Electrochemical Impedance Spectroscopy Meeting (EIS-95) for stimulating discussions, especially to F. Beck, R. P. Buck, C. Deslouis, K. M. Jiittner, M. M. Musiani, P. G. Pickup and B. Tribollet. MAV expresses his thanks to the Lab. Structure et Reactivite des Systemes Interfaciaux and Lab. d’Electrochimie Moleculaire for the support of his stay in Paris and Grenoble, as well as to the Russian Foundation of Fundamental Investigations (Grant RFFI No. 93-034448).

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