Limiting electrical response of conductive and dielectric systems, stretched-exponential behavior, and discrimination between fitting models

Abstract

Given a fitting model, such as the Kohlrausch–Williams–Watts (KWW)/stretched-exponential response, three plausible approaches to fitting small-signal frequency or time-response data are described and compared. Fitting can be carried out with either of two conductive-system formalisms or with a dielectric-system one. Methods are discussed and illustrated for deciding which of the three approaches is most pertinent for a given data set. Limiting low- and high-frequency log–log slopes for each of the four immittance levels are presented for several common models; cutoff effects are considered; and an anomaly in the approach to a single-relaxation-time Debye response for one of the conductive-system approaches is identified and explained. It is found that the temporal response function for the most appropriate conductive-system dispersion (CSD) approach, designated the CSD1, one long used in approximate form for frequency-response data analysis, does not lead to stretched-exponential transient behavior when a KWW response model is considered. Frequency-domain fitting methods and approaches are illustrated and discriminated using 321 and 380 K Na2O–3SiO2 data sets. The CSD1 approach using a KWW model is found to be most appropriate for fitting these data exceedingly closely with a complex nonlinear least-squares procedure available in the free computer program LEVM. Detailed examination and simulation of the approximate, long-used CSD1 modulus fitting formalism shows the unfortunate results of its failure to include separately the effects of the always present high-frequency-limiting dielectric constant, εD∞. The stretched-exponential exponent, β, associated with this fitting approach has always been misidentified in the past, and even after its reinterpretation, the result is likely to be sufficiently approximate that most physical conclusions derived from such fitting will need reevaluation.