Procedure for Obtaining the Analytical Distribution Function of Relaxation Times for the Analysis of Impedance Spectra using the Fox $H$-function

TL;DR

The paper tackles recovering the distribution of relaxation times (DFRT) from impedance spectra by casting the impedance model into Fox $H$-function form and performing two successive inverse Laplace transforms to obtain the step-response $A(t)$ and the DFRT $g(\tau)$. The method leverages the property that the Laplace transform of an $H$-function is itself an $H$-function, enabling closed-form expressions for $A(t)$, $D(\lambda)$, and $g(\tau)$ from $Z(s)$. It yields analytic DFRTs for Constant Phase Element, Davidson–Cole, Cole–Cole, and Havriliak–Negami models, with reductions to known limits when parameters take classical values. The work provides a unified framework that complements numerical inversions and enhances physical interpretation of electrochemical processes by linking impedance models directly to their time-domain relaxation spectra.

Abstract

The interpretation of electrochemical impedance spectroscopy data by fitting it to equivalent circuit models has been a standard method of analysis in electrochemistry. However, the inversion of the data from the frequency domain to a distribution function of relaxation times (DFRT) has gained considerable attention for impedance data analysis, as it can reveal more detailed information about the underlying electrochemical processes without requiring a priori knowledge. The focus of this paper is to provide a general procedure for obtaining analytically the DFRT from an impedance model, assuming an elemental Debye relaxation model as the kernel. The procedure consists of first representing the impedance function in terms of the Fox $H$-function, which possesses many useful properties particularly that its Laplace transform is again an $H$-function. From there the DFRT is obtained by two successive iterations of inverse Laplace transforms. In the passage, one can easily obtain an expression for the response function to a step excitation. The procedure is tested and verified on some known impedance models.

Figures (4)

• Figure 1: Basic relations between the functions $Q(s)$, $A(t)$, $D(\lambda)$ and $g(\tau)$ ($g(\tau)= \tau^{-1} D(\tau^{-1})$ with $\tau= \lambda^{-1}$)
• Figure 2: Plots of (a) $Q_c(s)$ (Eqs. \ref{['eq:QsCPE']} and \ref{['eq:QsCPE1']}) in Nyquist form of real vs. imaginary parts for $\tau_c=1$ and $\alpha=$ 0.98, 0.80 and 0.50. In dot-dashed we plot the discretized version of Eq. \ref{['eq:Zf02']} with $g(\tau)=g_c(\tau)$ for values of $\tau$ varying from 0.5 to 1000 s at a step of 0.1 s. In (b) we plot the functions $A_c(t)$ and $g_c(\tau)$ for the same parameters: $\tau_c=1$ and $\alpha=$ 0.98, 0.80 and 0.50
• Figure 3: Plots of (a) $Q_{\gamma}(s)$ (Eq. \ref{['eq:QgammaH']}) in Nyquist form of real vs. imaginary parts for $\tau_c=1$ and $\gamma=$ 0.98, 0.80 and 0.50. In (b) we plot the functions $A_{\gamma}(t)$ (Eq. \ref{['eq:Agammat20']}) and $g_{\gamma}(\tau)$ (Eq. \ref{['eq:gDC']}) for the same parameters: $\tau_{\gamma}=1$ and $\gamma=$ 0.98, 0.80 and 0.50
• Figure 4: Plots of (a) $Q_{\nu}(s)$ (Eq. \ref{['eq:QsalphaHN']}) in Nyquist form of real vs. imaginary parts for $\tau_{\nu}=1$, $\alpha=0.5$ and $\gamma=$ 1, 0.80 and 0.50. In (b) we plot the functions $A_{\nu}(t)$ (Eq. \ref{['eq41']}) and $g_{\nu}(\tau)$ (Eq. \ref{['eq:gHN']}) for the same parameters $\tau_{\nu}$, $\alpha$ and $\gamma$