Accurate fitting of immittance spectroscopy frequency-response data using the stretched exponential model

Abstract

A new method of accurately calculating two stretched-exponential (Kohlrausch-Williams-Watts (KWW)) models and fitting them by complex non-linear least squares (CNLS) to small-signal frequency-domain data is described and used for the detailed analysis of data for the disordered materials Li 2OAl 2O 32SiO 2 glass at 24°C and Na 2O · 3SiO 2 from 303 K to 398.5 K. Fitting was carried out with two different KWW models, KWW0 and KWW1, and with others, and included possible electrode polarization effects and ϵ D∞, the high-frequency-limiting dielectric constant, taken as a free parameter. For conductive-system dispersion, ϵ D∞ and ϵ ∞ are usually unequal. The present most-physically-appropriate KWW model, the KWW1, was much superior for the present data to all other models investigated. In particular, the power-law or ‘Jonscher’ model was found to be inferior for fitting the trisilicate data, contrary to earlier conclusions of Nowick and Lim, based on their comparison of the fitting utility of the power-law model and the Moynihan KWW modulus formalism. In addition, serious limitations of the modulus formalism were found and are illustrated; indicating that it should not be considered for future fitting. For the Na 2O · 3SiO 2 data, very-high-accuracy CNLS KWW1 fitting disclosed a small change in activation energy near 341 K and somewhat irregular, but well-determined, temperature dependence of the β exponent of the KWW1 model. Although the differences between fit predictions and the trisilicate data are too small to distinguish on ordinary M″( ω) or −ρ″(ω) plots, the very small relative residuals of the fit nevertheless show appreciable serial correlation, rather than random behavior, indicating that some systematic errors still remain.