Transmission Lines for Conducting Polymers
Modeling ionic and electronic conduction through membranes, polymer-modified electrodes, and solid-state oxide electrodes has received much recent attention. In particular using transmission lines as circuit elements has been advocated by Rubinstein et al. (1) and Buck.(2) The classical transmission line used by these authors is illustrated in Fig. 4.1. The distributed capacitance connects a resistive line to a wire of zero resistance. Classically previous authors have simply driven the current through the circuit element with the voltage. We show that this is a serious error that undermines much previous work. Furthermore the distributed capacitance is dominated by Nernst and Donnan terms, which describe the layer charging. We develop the correct model and extend it to allow for differential mobility in two rails. We also develop the theory of a polymer where the Donnan exclusion is not complete, so we have ions of both charges present. For conducting polymers we develop an even more complicated model in which the rate of electron transfer in the polymer is limited by electron hops between segments of conducting polymer. We include effects of a charge transfer resistance and double-layer capacitance at either the electrode/polymer or the polymer/electrolyte interface. In our papers(3–6) we describe the detailed mathematics; in Chapter 4 we do not repeat the mathematical arguments but instead concentrate on the results and their implications. We then compare results from theoretical models with experimental results from electrodes coated with polyvinylferrocene and such conducting polymers as polythiophene and polypyrrole. For polyvinylferrocene excellent agreement is found between theoretical and experimental results for the dual-rail transmission line model. For a conducting polymer, such as polypyrrole, we argue that to secure agreement, an extra process, the kinetics of electron transfer at the polymer/electrode interface, must be included.
KeywordsTransmission Line Charge Transfer Resistance Conducting Polymer Circuit Element Impedance Plot
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