Analysis of dielectric or conductive system frequency response data using the Williams–Watts function

Abstract

The electrical, optical, and mechanical behavior of many materials, particularly polymers and glasses, have been analyzed using the Kohlrausch–Williams–Watts stretched exponential relaxation function in both the time and frequency domains. This function is currently of considerable experimental and theoretical interest. Unfortunately, no relatively simple and accurate approximation representing the small-signal frequency response of stretched exponential relaxation has been available. Thus it has been impractical to obtain accurate parameter estimates from fitting of frequency response data or to discriminate well between Williams–Watts response and that of other similar response models. Here we develop such an approximation for both dielectric systems and for intrinsically conducting ones (e.g., defect hopping materials). It is in complex form and allows fitting of both real and imaginary parts of all the data simultaneously (e.g., by complex nonlinear least squares) or of either part separately. For appropriate data, which need not be electrical, fitting with the new approximation can yield parameter estimates accurate to about 0.1%. Comparison of the results of the present fitting method to those of a more approximate one are presented.