Interconnection between frequency-domain Havriliak-Negami and time-domain Kohlrausch-Williams-Watts relaxation functions

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Abstract
The Kohlrausch-Williams-Watts (KWW) and the Havriliak-Negami (HN) relaxation functions have been widely used to describe the relaxation behavior of glass-forming liquids and complex systems over the last several years. The HN relaxation function is a frequency-domain function while the KWW function applies for the time domain. In a previous paper we discussed the interconnections between these two functions by presenting a method where we found that the best HN description in the frequency domain corresponds to a given KWW function in the time domain. From that work we proposed several empirical relationships that allow us to determine the HN parameters corresponding to a given set of KWW ones. It was also outlined how to proceed in the opposite way, i.e., to obtain the best KWW time-domain description corresponding to a given HN relaxation function in the frequency domain. This is what we develop in this work by varying the HN parameters in search of the values of the best KWW fits. Likewise we can put a limit to the region where the HN and the KWW functions are compatible in the sense that they can be equally used by just choosing the right parameter change. A confident confirmation of this procedure is that when the HN parameters reported in the literature for the α relaxation of glass-forming liquids are considered, their values fall directly upon this region, where we find that the HN and KWW functions deviate less. However, for other dynamical processes like secondary relaxations of polymers or the α relaxation of polymer blends the HN parameters reported indicate that a single KWW relaxation function is not able to describe the time-decay behavior.