Compact and diffuse double layer interaction in unsupported system small-signal response

Abstract

Previous exact results for the small-signal impedance of an unsupported electrode/material/electrode system which include effects of the finite size of charge carriers are simplified and discussed. The material contains non-recombining charges of opposite sign with the positive one immobile and uniformly distributed. General boundary conditions which encompass the range from no electrode reaction to ohmic electrode behavior are employed. In the presence of an electrode reaction, the interaction of the compact and diffuse double layers leads to considerably more complexity in the equivalent circuit than might appear in simple treatments of the supported case, in which the diffuse double layer capacitance is neglected or the compact double layer and diffuse double layer capacitances are placed in series. Two different approximate equivalent circuits made up of frequency-independent elements are found which yield remarkable agreement with the exact results over the entire frequency range of interest. The first involves the ordinary approximate circuit (OAC) previously found in the absence of compact layer effects plus a series compact layer contribution involving a parallel resonant circuit with quality factor at resonance which may approach unity. Pseudo-inductance effects are found to be extremely significant in this representation. The second approximate equivalent circuit, simpler and almost as accurate as the first, has the same form as the original OAC but with its reaction element values altered by the presence of the compact layer. For non-Butler-Volmer electrode kinetics an upper limit is found for the experimentally determinable apparent reaction rate constant, a feature of practical importance for thin films or membranes. The response of thin films and membranes, including compact layer effects, can very readily be erroneously confused with pure bulk response, yielding entirely incorrect values for the geometrical capacitance and bulk resistance of the material.