Debye behavior relaxation function

In Refs. 80 and 81 it is shown that the Mittag-Leffier function is the exact relaxation function for an underlying fractal time random walk process, and that this function directly leads to the Cole-Cole behavior [82] for the complex susceptibility, which is broadly used to describe experimental results. Furthermore, the Mittag-Leffier function can be decomposed into single Debye processes, the relaxation time distribution of which is given by a mod-... 

A comparison of either the dispersion c or the complex plane loci of e with the Debye relaxation function in eq 7 shows that the experimental curves extend over much broader ranges of frequencies, a behavior typical also of other polyelectrolyte systems and described within their accuracy by the empirical cir-... 

Since this Arrhenius temperature dependence implies that the molecular motions are unimpeded by intermolecular constraints, our finding reveals that the loss of intermolecular cooperativity is governed by the dynamics, or at least they have the same control parameter. Since the relaxation time determines the shape of the relaxation function ° , we can conclude from the results herein that at the onset of intermolecular cooperativity the breadth of the relaxation function is constant, independent of pressure. Of course, since in the absence of intermolecular constraints on the molecular motions we expect Debye behavior, the shape of the relaxation function should also be invariant for all T greater than Ta-... 

Dielectric relaxation of complex materials over wide frequency and temperature ranges in general may be described in terms of several non-Debye relaxation processes. A quantitative analysis of the dielectric spectra begins with the construction of a fitting function in selected frequency and temperature intervals, which corresponds to the relaxation processes in the spectra. This fitting function is a linear superposition of the model functions (such as HN, Jonscher, dc-conductivity terms see Section II.B.l) that describes the frequency dependence of the isothermal data of the complex dielectric permittivity. The temperature behavior of the fitting parameters reflects the structural and dynamic properties of the material. 

In theoretical studies, one usually deals with two simple models for the solvent relaxation, namely, the Debye model with the Lorentzian form of the frequency dependence, and the Ohmic model with an exponential cut-off [71, 85, 188, 203]. The Debye model can work well at low frequencies (long times) but it predicts nonanalytic behavior of the time correlation function at time zero. Exponential cut-off function takes care of this problem. Generalized sub- and super-Ohmic models are sometimes considered, characterized by a power dependence on CO (the dependence is linear for the usual Ohmic model) and the same exponential cut-off [203]. All these models admit analytical solutions for the ET rate in the Golden Rule limit [46,48]. One sometimes includes discrete modes or shifted Debye modes to mimic certain properties of the real spectrum [188]. In going beyond the Golden Rule limit, simplified models are considered, such as a frequency-independent (strict Ohmic) bath [71, 85, 203], or a sluggish (adiabatic)... 

As an illustration of the observed behavior and types of functions to be transformed. Figure 2 shows the reflection R(t) from Equation 5 for a step-like voltage pulse and dielectric with "Debye" relaxation expressed by... 

Functions such as P(t) in Figure 2 with zero long time limit and finite area often have a long time behavior that is at least roughly an exponential decay P(t) = P(tf))exp[-(t-tm)Tgpp] for t > tw. (A case in point is Debye relaxation with Tgpp = tq + e - )d/c as... 

For polymeric systems in the most cases, the measured dielectric loss is much broader and in addition the loss peak is asymmetric. This is called non-Debye or nonideal relaxation behavior. Formally such a non-Debye-like behavior can be described by a supposition of Debye functions... 

After the introduction of the various interrelated response functions and basic concepts like the Debye-process and the derived spectral representations we come now in the second part of this chapter to the description and discussion of actual polymer behavior. In fact, relaxation processes play a dominant role and result in a complex pattern of temperature and frequency dependent properties. 

Note that at low frequencies (<10 kHz), Cp is the function of the frequency of the electric field. This is attributed to the relaxation of the double layer surrounding the particle, which depends on the radius of the particle with respect to the Debye length. This situation is complex, and no satisfactory model exists to describe this phenomenon. The counterions of the double layer do not have enough time to move in case of high frequencies. Thus, at high frequencies, the particle is nondispersive compared to that at low frequencies. Hence, Cp and high frequencies. Here, the polarization is primarily due to the particle with respect to the surrounding medium. The behavior of the particles switches... 

These relationships are known as the Debye formulae. The Debye process has a relaxation time distribution, which is symmetrical around /niax= niax/2n and has a full width at half-maximum of 1.14 decades in frequency for the dielectric loss. In most cases, the half width of measured loss peaks is much broader than the predicted by eqn [26] and in addition, their shapes are asymmetric and with a high-frequency tail. This is the non-Debye (or nonideal) relaxation behavior found in many glass formers. In the literature, several empirical model funaions, mostly generalization of the Debye function, have been developed and tested which are able to describe broadened and/or asymmetric loss peaks. Among these empirical model functions, the most important are the Kohlrausch-Williams-Watts (KWW), Cole-Cole (CC), Cole-Davidson (CD), and the Havriliak-Negami (HN) function. The HN function, with two shape parameters, is the most commonly used funaion in the frequency domain. 

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