Generalized Distribution Function of Relaxation Times with the Davidson-Cole Model as a Kernel

Abstract

In this paper we propose a generalized distribution function of relaxation times (DFRT) considering the Davidson-Cole model as an elementary process instead of the standard Debye model. The distribution function is retrieved from the inverse of the generalized Stieltjes transform expressed in terms of iterated Laplace transforms. We derive computable analytical expressions of the generalized DFRT for some of the most known normalized impedance (or admittance) models including the constant phase element, the Davidson-Cole, Havriliak-Negami and the Kohlrausch-Williams-Watts models.

Figures (5)

• Figure 1: Plots of $g_c(\tau)$ in Eq. \ref{['eq:gctau']} for different combinations of the parameters $\alpha$ and $p$, and with $\tau_c=1$
• Figure 2: Plots of $g_{\beta}(\tau)$ in Eq. \ref{['eq:gbeta']} for different combinations of the parameters $\beta$ and $p$, and with $\tau_{\beta}=1$
• Figure 3: Schematic of a 1D semi-infinite two-impedance self-similar ladder network in the $x$-direction, coupled with a semi-infinite bifurcating binary tree network in the $y$-direction.
• Figure 4: In (a) Nyquist plot of real vs. imaginary parts of $Y_{eq} = 1/\sqrt{1+\sqrt{s\tau_{eq}}}$ for $\tau_{eq}=1$, frequency from 1 mHz to 10 kHz; In (b) plots of $g_{\nu}(\tau)$ in Eq. \ref{['eq:gHN']} for $\alpha=\beta=0.5$, $\tau_{\nu}=1$, and different values of $p$ (1.0, 0.5 and 0.3)
• Figure 5: In (a) Nyquist plot of real vs. imaginary parts of $Q_{\kappa}(s)$ (Eq. \ref{['eq:qofskappa']}) for $\tau_{\kappa}=1$ and different values of $\kappa$ (1.0, 0.7 and 0.5); the frequency range is 16 mHz to 100 Hz. In (b) plots of $g_{\kappa}(\tau)$ in Eq. \ref{['eq:gkappa']} for $\kappa=0.5$, $\tau_{\nu}=1$, and different values of $p$ (1.0, 0.8 and 0.51)