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

Many physical phenomena, such as the polarization of dielectrics and viscoelastic materials, or porous electrode/electrolyte interfaces cannot be modeled by a single relaxation time of the Debye type. To account for this fact, data analysis based on distribution functions of the Debye relaxation times (DFRT) is conventionally used. In this study, a generalized DFRT is proposed considering the Davidson‐Cole model as an elementary process instead of the standard Debye model. The distribution function can then be retrieved from the inverse of the generalized Stieltjes transform, expressed in terms of iterated Laplace transforms, applied on a given frequency function. Computable analytical expressions of the generalized DFRT are derived for some of the most known impedance (or admittance) models including the constant phase element, the Davidson‐Cole, Havriliak–Negami and the Kohlrausch–Williams–Watts models. The obtained distributions will be valuable to interpret and compare with inverted frequency‐domain data using numerical methods, in addition to the usual way of equivalent circuit modeling. The proposed theory and results can be easily translated to other physical systems that are known to exhibit local frequency dispersion features.