Transmission line model for conducting polymers including cations and Donnan exclusion

Abstract

In conducting polymers, such as polypyrrole, polythiophene or polyaniline the electronic conduction in the polymer backbone is significantly greater than the ionic conduction in the ionic pores. Although it is likely that Donnan exclusion of mobile cations will be complete for the highly oxidised polymer this will not be true for the reduced polymer. A new transmission line is therefore proposed in which the electrons are assumed to be mobile and the cation and anion each have a resistive rail. The response of the line to a small potential step is derived and shown in general to be biphasic. A case diagram is constructed showing five different cases. These are each discussed in detail as to which ions move in which phase and as to the phase in which the oxidation of the polymer occurs. Expressions for the effective diffusion coefficient for the ions for each phase and each case are derived and discussed. General expressions for the a.c. impedance are derived and are again biphasic. Typical Argand diagrams for the a.c. impedance are calculated as a function of the percentage oxidation in the first phase and of the ratio of the effective diffusion coefficients for the two phases. In two of the five cases when the cation is more mobile than the anion and when there is Donnan exclusion the oxidation changes from being in the first phase to being in the second phase or vice versa. Under these conditions there must be a range of oxidation where both terms are significant and biphasic kinetics should be apparent.