CNLS fits and Kramers–Kronig validation of resonant EIS data

Abstract

Often, unusual electrochemical immittance spectroscopy (EIS) data are reported, characterized by scattered data points in complex impedance coordinates (−Im( Z) vs. Re( Z)) and corresponding sharp resonance-like peaks on Bode amplitude characteristics (log(∣ Z∣/ Ω) vs. log( f/Hz)) represent the state of the electrode close to Hopf bifurcation under current control (galvanostatic, gc) which is stable under voltage control (potentiostatic, pc). An example of such data reported here was obtained for anodic dissolution/passivation of a rotating copper disk electrode in a solution of copper sulphate and sulphuric acid. These data are especially cumbersome when complex nonlinear least squares (CNLS) fits of suitable models including electrical equivalent circuits (eqc) are attempted and this may in some cases even raise doubts as to data validity. It was therefore of importance to validate data with Kramers–Kronig (KK) transformation and this procedure was always successful for our data in their admittance representation while KK transformation in impedance representation failed for data presenting differential negative resistance (nr) i.e. non-minimum phase (nmp) characteristics. CNLS fits were done using 2nd, 3rd and 4th order rational functions of the frequency as the most suitable models, while the electrical equivalent (R, C) circuits and immittance in zero-pole representation were subsequently derived from these functions. It resulted that some forms of the (R, C) eqcs were unsuitable because their calculated parameters were complex numbers. In the case of the 4th order function it was possible to extrapolate data to a very low, experimentally inaccessible frequency, to reveal an additional low time constant.