Excess Wings in Broadband Dielectric Spectroscopy

TL;DR

This paper introduces a fractional evolution equation approach to model excess wings in broadband dielectric spectroscopy data of glass-forming materials, achieving superior fits over traditional methods with fewer parameters.

Contribution

It presents a novel fractional semigroup model that accurately fits dielectric spectra with only three parameters, outperforming traditional Havriliak-Negami fits and providing new time-dependent relaxation functions.

Findings

Fractional evolution equations effectively model excess wings in dielectric spectra.

The model fits data over up to 10 decades in frequency with only three parameters.

Relaxation times follow the Vogel-Tammann-Fulcher law closely.

Abstract

Analysis of excess wings in broadband dielectric spectroscopy data of glass forming materials is found to provide evidence for anomalous time evolutions and fractional semigroups. Solutions of fractional evolution equations in frequency space are used to fit dielectric spectroscopy data of glass forming materials with a range between 4 and 10 decades in frequency. We show that with only three parameters (two relaxation times plus one exponent) excellent fits can be obtained for 5-methyl-2-hexanol and for methyl-m-toluate over up to 7 decades. The traditional Havriliak-Negami fit with three parameters (two exponents and one relaxation time) fits only 4-5 decades. Using a second exponent, as in Havriliak-Negami fits, the $\alpha$-peak and the excess wing can be modeled perfectly with our theory for up to 10 decades for all materials at all temperatures considered here. Traditionally this can only be accomplished by combining two Havriliak-Negami functions with 6 parameters. The temperature dependent relaxation times are fitted with the Vogel-Tammann-Fulcher relation which provides the corresponding Vogel-Fulcher temperatures. The relaxation times turn out to obey almost perfectly the Vogel-Tammann-Fulcher law. Finally we report new and computable expressions of time dependent relaxation functions corresponding to the frequency dependent dielectric susceptibilities.