Simplified impedance/frequency-response results for intrinsically conducting solids and liquids

Abstract

The small‐signal ac response is considered of a system containing a single species of positive charge and a single species of negative charge. The charge carriers may be of many different types (ions, electrons, vacancies, etc.) and are assumed to have arbitrary mobilities and valences. Quite general boundary conditions are considered which encompass the range from complete blocking to zero blocking (infinite reaction rate at the electrodes) for positive and negative charges separately. The present paper deals primarily with approximations to an earlier exact solution of the problem which, in general, lead to an equivalent circuit made up of three parallel R C combinations in series. The elements of one of these parallel circuits, associated only with bulk effects, are frequency independent, and those of another, which are associated with nonzero blocking, may often be well approximated as independent. The third R C section arises from diffusion effects, involves frequency dependent elements, and exhibits approximate Warburg frequency response over a considerable frequency range. In general, Cole‐Cole or Nyquist impedance plane plots show three connected arcs, two of which are frequently good semicircles. These arcs are directly associated with the three R C sections. Under many conditions, only two of the three arcs may appear and melding of arcs into each other can also occur. Very simple, as well as less simple, approximations are developed for the impedances of the individual R C sections as well as the over‐all impedance of the system. The accuracy of these approximations is evaluated, and it is shown how they may be used in unambiguous cases to analyze frequency response data to yield estimates of mobilities, valences, electrode reaction rate parameters, and bulk charge concentrations. Surprisingly, it is found that for a certain mobility ratio range, the center arc, associated with electrode reactions, well approximates a depressed semicircle of the Cole‐Cole relaxation time distribution type, yet no distribution of relaxation times is present. Further, in the completely blocking case, where the center approximate semicircle is not depressed but has an infinite radius, the lower frequency portion does not necessarily begin with a vertical segment as frequency decreases but may be curved away from vertical over a considerable frequency range. General impedance results for the present unsupported conduction situation are found to be quite different in some ways from those following from supported electrolyte treatments. Such treatments should thus not be used to analyze unsupported situations. Finally, it is found that the parallel capacitance of the system, which may be far greater than ordinary double layer values, can exhibit appreciable regions of ω^−m frequency response with 0≤m ≤2, and with m =0.5, 1, 1.5, and 2 values especially prominent.