The Faradaic—reactive impedance for a sinusoidal electrode

Abstract

The faradaic—reactive impedance for a sinusoidal-shaped electrode is determined from a perturbation solution to the Laplace equation in which the ratio of the amplitude to period length ( A) is the perturbation parameter. Terms in the expansion up to and including fourth order in A are included, but A must be less than 1/2π for convergence. A formula for the impedance minus its infinite frequency limit is presented for the reference electrode far removed from the electrode surface. The Nyquist diagram of the imaginary versus the real component of the impedance for the sinusoidally rough surface is shown to be always perpendicular to real axis at both low and high frequency and is nearly semicircular with a diameter which, at a given amplitude, decreases asymptotically for low exchange current density but increases asymptotically for high exchange current density. At a given exchange current density, the diameter may decrease or increase with amplitude depending upon whether the exchange current density is low or high, respectively. If the Nyquist diagram for the faradaic—reactive impedance of the sinusoidal-shaped electrode is (incorrectly) interpreted as that from a planar electrode, the resulting apparent exchange current density and double-layer capacitance both differ from the actual values, and a means to correct the measurements to obtained the intrinsic values is given. The results show that for an irreversible reaction the apparent transfer coefficient determined from the slope of a plot of the inverse of the low frequency limit of the impedance versus the dc bias voltage may be nonconstant and vary with A. The impedance for an ideally blocking sinusoidal-shaped electrode is also presented. At low and high frequencies the phase lag is −π/2 and the modulus is inversely proportional to frequency, with a proportionality constant which is smaller at low frequency and is A dependent for all frequency; however, over a frequency range which is also A dependent, a deviation from the pseudo-planar behavior is found with the phase lag less than −π/2.