Theoretical aspects of impedance spectroscopy at a disk electrode embedded in an insulating plane: Reciprocity relationships

An analytic solution for the impedance function of a disk electrode embedded in an insulating flat surface immersed in an electrolyte, in the absence of concentration variations, was obtained by applying an s -multiplied Laplace transform (with respect to the dimensionless time) to the solution for the potential step [J. Electrochem. Soc., 120, 1356 (1973)]. The results obtained were in excellent agreement with those reported earlier by Newman [J. Electrochem. Soc., 117, 198 (1970)], who used a much different approach. In addition, reciprocity relationships were invoked to show that solutions for either the impedance, admittance, or potential or current steps, can be found, using, as a starting point, the solution of any of the other three problems. Computational implementation of these relationships becomes particularly simple, as the time, in the step solutions reported by Niancioǧlu and Newman [J. Electrochem. Soc., 120, 1339 (1973); J. Electrochem. Soc., 120, 1356 (1973)], appears as a linear function in the argument of exponential functions. In particular, calculations based on this tactic yielded accurate eigenvalues and coefficients for the potential-step solution, directly from the corresponding eigenvalues and coefficients for the current-step solution [J. Electrochem. Soc., 120, 1339 (1973)].