Transport across an electroactive polymer film in contact with media allowing both ionic and electronic interfacial exchange

doi:10.1016/S0013-4686(98)00318-1

Electrochimica Acta

Volume 44, Issue 12, 1 January 1999, Pages 2105-2115

https://doi.org/10.1016/S0013-4686(98)00318-1 Get rights and content

Abstract

In this paper, the general theory of coupled electron-ion transport inside a film having a mixed electron-ion conductivity is applied, and the a.c. impedance expressions have been calculated for the situations where the film can exchange electrons and/or ions with the surrounding media. Important example of such system is given by a conducting polymer film between two solutions containing a redox active couple (“redox active solutions”) which is subjected to a faradaic process at the film surface with the participation of electronic species inside the film. As particular cases, the general formula describes also the systems where one or both interfacial boundaries can only be crossed by one type of charged species, electrons or ions. Besides the previously analysed situations, metal/film/metal, metal/film/background solution, or a film between two background solutions, we have obtained expressions for impedance of the film between a redox active solution and a background (without redox species) one, or between a metal and a redox active solution. Theoretical predictions for the shapes of complex impedance plots in the case of a redox active solution have been demonstrated.

Introduction

Complex impedance is one of the major techniques for experimental study of electroactive polymer films. The possibility to vary the perturbing frequency within a very wide interval enables it to provide characteristics of numerous bulk film and interfacial processes in those systems. However, the overlap of the frequency domains where each of these processes plays a significant role in the overall signal leads to the necessity of an adequate theoretical interpretation to unravel their respective contributions.

During an extended period, this method was applied to the simplest examples of “symmetrical” and “asymmetrical” systems (according to Buck1, 2) of this kind, a film between two metals[3] or two identical solutions[4], or metal/film/solution5, 6. In those cases the solution only contains ions which do not possess a redox activity (“background electrolyte”). The counter-ion is able to cross the film/solution boundary to retain the bulk film electroneutrality. Thus, in all the above systems there is only one charged species, electron or anion (the counter-ion involved for unsubstituted polymer films which become conductive upon oxidation), which can pass the current through each particular interface. Recent results in this direction, in particular with account of the Poisson equation instead of the local electroneutrality condition, may be found in references7, 8, 9, see also references in6, 10.

These three simpler systems have been recently analysed on the basis of the general theory of coupled electron-ion transport which generalises the Nernst–Planck equations[11]. The relevant impedance expressions were obtained as follows (denotations below correspond to those in reference[11]):

metal/film/metal (m/f/m)[3]: $Z_{m/f/m} (ω)=R^{m/f} +R_{f} + R^{f/m} +4t_{i}^{2} ΔR_{f} z_{th} z_{th} (ν)=ν^{−1} tanh ν$ $R_{f} = Lκ^{−1}, ΔR_{f} =L 4DC_{ρ}^{−1}, ν= jωL^{2} /4D^{1/2}$ redox inactive solution/film/redox inactive solution (s/f/s)[4]: $Z_{s/f/s} (ω)=R_{s1} +R_{i}^{s/f} +R_{f} +R_{i}^{f/s} +R_{s2} +4t_{e}^{2} ΔR_{f} z_{th}$ metal/film/redox inactive solution (m/f/s)5, 6: $Z_{m/f/s} (ω)=R^{m/f} +R_{f} +R_{i}^{f/s} +R_{s} +ΔR_{f} z_{cth} +(t_{e} −t_{i})^{2} z_{cth}, z_{cth} (ν)=ν^{−1} coth ν$ Those formulae contain: frequency, ω, film thickness, L, redox capacitance (quasi-equilibrium) per a unit volume of the film, Eq. (10), C_ρ, three macroscopic transport parameters of the polymer phase: D, binary diffusion coefficient, t_e=1-t_i, (migration) transference numbers of electrons and ions, κ, specific electron-ion conductivity, as well as resistances of the interfacial exchange with the corresponding charged species, electron for the metal/film boundary (R^m/f, R^f/m), or ion for the inactive solution/film one (R^s/f_i, R^f/s_i) and the ohmic resistances of the solutions in contact, depending on the systems (R_s1, R_s2, R_s).

One should keep in mind that capacitance terms due to the interfacial charging (“double-layer effects”) are disregarded in this derivation. As a consequence, all interfacial and bulk-solution contributions in , , , are simply added in those systems to the bulk-film ones. Expressions of those macroscopic transport parameters via microscopic “friction coefficients” are given in[11].

Important additional information concerning the properties of the film may be obtained by measurements in presence of a redox active couple inside the solution(s) in contact. Such experimental studies have already been carried out for two geometries of the system, metal/film/redox active solution (m/f/es)12, 13, 14 and a film between two redox active solutions (es/f/es)[15]. The interpretation of those data requires the corresponding generalisation of the theory. It has been carried out in this paper within the framework of the assumption that the solution contains an excess of the “background” electrolyte (as earlier, it means no redox activity while its anion is able to cross the interfacial boundary). Redox active species are supposed to be only present in the solution phase but not inside the film, and they participate in the interfacial electron exchange with the polymer at the film/solution boundary. One can expect such a behaviour if the redox species have a sufficiently large size and/or they are repelled by the space-charge region at the film/solution interface as for a permselective membrane.

The analysis developed in this paper is based on the same concepts as those used in[11]. However, the existence of both electron and ion interfacial fluxes at the film/solution boundaries (such interfaces are referred below as “es”) leads to an essential modification of the boundary conditions for those transport equations. Their solution has enabled us to obtain a general analytical expression for the complex impedance of the system, which can correspond to different types of interfaces and contains, in a complicated manner, bulk parameters of all three phases as well as characteristics of the interfacial electron and ion transfer.

No symmetry of the system has been assumed in the course of its derivation so that the solutions in contact may have quite different compositions. The only restriction is an assumption of no concentration gradients in the unperturbed state, i.e. no direct current at zero amplitude of the perturbation.

The latter requirement specifies possible types of the system:

— es/f/es: symmetrical case (identical redox active solutions) with a zero bias voltage between the bulk solutions,

— s/f/es: background solution/film/redox active solution. It corresponds to an infinite value of the interfacial impedance of the electronic exchange. The bias voltage must be adjusted to realise a zero background anion flux across the system,

— m/f/es: metal/film/redox active solution. Surprisingly, the result for this case can be immediately obtained from the general formula if the interfacial resistance of ionic transfer is infinitely large while the bulk ohmic resistance of this medium is very small. In this last situation, the bias voltage must deliver a zero value of the electronic flux.

Section snippets

Transport relations and boundary conditions

Adequate description of transport phenomena in electroactive polymer films means to take into account numerous factors like a disordered type of the polymer phase, high concentrations of electronic and ionic species, their short-range interactions, inhomogeneity effects, “hopping” type of the electronic motion in certain systems (e.g. interchain transport in conducting polymers), coupling between the electronic and ionic fluxes, etc. Thus, application of the traditional set of the

Boundary conditions: mixed electron-ion exchange

In most cases, each interface of the film with the surrounding medium can be crossed by only one kind of charged 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 a resistance). In this case one can immediately write down a boundary condition for one of the partial currents, i.e. for i′, for

Current distribution and impedance

General solution of Eq. (11)may be written as: $i'(x)= i'(L) sinh 2νx/L+i'(0) sinh 2ν(1−x/L) / sinh 2ν$ so that $Φ_{ei} (x)=− jωC_{ρ}^{−1} d i'/ d x=2W_{f} −i'(L) cosh 2νx/L+i'(0) cosh 2ν(1−x/L) / sinh 2ν$ where, W_f is the Warburg impedance for the electron-anion transport inside the polymer film, $W_{f} =W_{f} (ν)=ν(jωLC_{ρ})^{−1} = ΔR_{f} /ν, ν and ΔR_{f}, see Eq. (2).$ Then, one should insert expression (31) into conditions (21) and (25) to get a set of two linear equations for i′(0) and i′(L) . Their solution may be written as: $i'(0)= iZ_{0}^{−1} t_{i} R_{i}^{s/f} −t_{e} Z_{e}^{s/f} Z_{e}^{f/s} +R$

Discussion

In view of the complexity of the equations describing the impedance of the system in the different possible geometries, their analysis does not allow to appreciate all relevant features. Therefore, a set of theoretical impedance diagrams was calculated, in order to establish the effect of various physical parameters. The diagrams presented hereafter correspond to several combinations of the rates of the exchange/transport processes, i.e. redox exchange at the f/es interface(s), the mass

Conclusions

This paper was aimed at providing a comprehensive description of the electrochemical behaviour of a film having mixed electron-ion conduction, and liable to exchange electrons and/or ions with surrounding media. The bulk transport processes were analyzed in the framework of the theory of coupled electron-ion transport, which accounts for the specific case of concentrated media. Interfacial exchanges were considered on a thermodynamic approach using electrochemical potentials for ions and

Acknowledgements

This work was supported by a NATO Programme HTECH.LG 941162.

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Transport across an electroactive polymer film in contact with media allowing both ionic and electronic interfacial exchange

Abstract

Introduction

Section snippets

Transport relations and boundary conditions

Boundary conditions: mixed electron-ion exchange

Current distribution and impedance

Discussion

Conclusions

Acknowledgements

J. Electroanal. Chem.

J. Electroanal. Chem.

J. Electroanal. Chem.

J. Electroanal. Chem.

Electrochimica Acta

Synth. Met.

J. Electroanal. Chem.

J. Electrochem. Soc.

J. Electrochem. Soc.