Estimation and correction of instrument artefacts in dynamic impedance spectra

Dynamic impedance spectroscopy is one of the most powerful techniques in the qualitative and quantitative mechanistic studies of electrochemical systems, as it allows for time-resolved investigation and dissection of various physicochemical processes occurring at different time scales. However, due to high-frequency artefacts connected to the non-ideal behaviour of the instrumental setup, dynamic impedance spectra can lead to wrong interpretation and/or extraction of wrong kinetic parameters. These artefacts arise from the non-ideal behaviour of the voltage and current amplifier (I/E converters) and stray capacitance. In this paper, a method for the estimation and correction of high-frequency artefacts arising from non-ideal behaviour of instrumental setup will be discussed. Using resistors, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\hbox {Fe(CN)}_6]^{3-/4-}$$\end{document}[Fe(CN)6]3-/4- redox couple and nickel hexacyanoferrate nanoparticles, the effect of high-frequency artefacts will be investigated and the extraction of the impedance of the system from the measured dynamic impedance is proposed. It is shown that the correction allows acquiring proper dynamic impedance spectra at frequencies higher than the bandwidth of the potentiostat, and simultaneously acquire high precision cyclic voltammetry.


S1 Effect of the Intensity of the Multisine
The role of the nonlinear response on the noise level of the impedance spectra was investigated.
The current response contains in addition to the fundamental frequencies, the corresponding harmonics and intermodulations. The dc component is influenced by the first harmonics while the fundamental harmonic is influenced by the second order harmonics [1]. The signal-to-noise ratio relative to the nonlinear component (R k ) of the response of the fundamental frequency can be described as [1]: where ∆U ac denotes the amplitude of the multisine signal, A is the second order harmonic response and A j is the intensity of the fundamental harmonics at frequency f j . ω k and ω j are the k-th and j-th frequency of the multisine signal [1]. Equation 1 can be rewritten as: Equation 3 allows for R k to be estimated from the slope of admittance (1/Z(ω k )) acquired at different intensity versus ∆U 2 ac . The error due to the nonlinear component at the intensity used in the acquisition of the impedance can then be described as: where Y 0 is the intercept of the plot 1/Z(ω k ) versus ∆U 2 ac . The result obtained for dynamic impedance of the redox couple using a multisine intensity of 50 mVpp and for NiHCF nanoparticles using a multisine intensity of 50 mVpp is shown in Fig. 6 of the main paper. The result indicates that the error introduced by the nonlinear components at the used intensity of the multisine in both cases were less than 1%. We considered a multi-sine intensity of 50 mVpp as a good trade off between signal intensity and error arising from the nonlinear components.

S2 Fitting
The transimpedance of the potentiostat (Z tr ) and the stray capacitance (C st ) between WE -CE were extracted using the mathematical description of the electrical circuit of the potentiostat: where Z s is the impedance of the system which was obtained from the low frequency data points, while Z m is the measured impedance.

S3 Limit of the Correction Method at Different Current Ranges
The correction method cannot be used under all conditions, because it tends to amplify all the errors. For this reason, we looked at the limits in the application of the correction. The permissible limit set in this work is an error of 1% for magnitude of the impedance and 1 • for phase. The error in magnitude (err |Z |) and error in phase ( err (ϕ)) are given by err(|Z |) = (|Z corr | − |Z s |)/|Z s | and err(ϕ) = ϕ corr − ϕ s , where |Z | and φ indicate modulus and phase of the impedance, and the subscript corr and s indicate the corrected and real values respectively.
The results suggest that the error in magnitude and phase were below the limit for all frequencies (up to 1 MHz) for the current ranges from 100 mA to 1 mA. For 100 µA current range (Fig. S3a and Fig. S3b), the result indicates that the error of the measured impedance is below the 1% limit