0 ≤ β ≤ 1 which controlled the width of the distribution and β = 1 for Debye relaxation. The smaller the value of β, the larger the distribution of relaxation times. The real and imaginary parts of the Cole-Davidson equation are given by (14) (15) (16) Both the Cole-Cole and Cole-Davidson equations were empirical and could be considered to be the consequence of the existence of a distribution of relaxation times rather than that of

the single relaxation time (Debye equation). After 15 years, in 1966, S. Havriliak and S. J. Negami reported the Havriliak-Negami (HN) equation which combined the Cole-Cole and Cole-Davidson equations for 21 polymers [82–84]. The HN equation is (17) The real MK5108 cost and imaginary selleck screening library parts of the HN equation are given by (18) (19)

(20) where α and β were the two adjustable fitting parameters. α was related to the width of the loss peak and β controlled the asymmetry of the loss peak. In this model, parameters α and β could both vary between 0 and 1. The Debye dielectric relaxation model with a single relaxation time from α = 0 and β = 1, the Cole-Cole model with symmetric distribution of relaxation times followed for β = 1 and 0 ≤ α ≤ 1, and the Cole-Davidson model with an asymmetric distribution of relaxation times follows for α = 0 and 0 ≤ β ≤ 1. The HN equation had two distribution parameters α and β but Cole-Cole and Cole-Davidson equations had only one. HN model in the frequency domain can accurately describe the dynamic mechanical behavior of polymers, including the height, width, position, and shape of the loss peak. 17-DMAG (Alvespimycin) HCl The evolution map for Debye, Cole-Cole, Cole-Davidson, and HN model is shown in Figure 3. Figure 3 Evolution map for Debye, Cole-Cole, Cole-Davidson, and HN model. A theoretical description of the slow relaxation in complex condensed systems is still a topic of active research despite the great effort made in recent years. There exist two alternative approaches to the interpretation of dielectric relaxation: the parallel and series models [54]. The parallel

model represents the classical relaxation of a large assembly of individual relaxing entities such as dipoles, each of which relaxes with an exponential probability in time but has a find more different relaxation time. The total relaxation process corresponds to a summation over the available modes, given a frequency domain response function, which can be approximated by the HN relationship. The alternative approach is the series model, which can be used to describe briefly the origins of the CS law. Consider a system divided into two interacting sub-systems. The first of these responds rapidly to a stimulus generating a change in the interaction which, in turn, causes a much slower response of the second sub-system.