Havriliak-Negami relaxation function

In order to determine the structural relaxation times for the octa-O-acetyl-lactose we analyzed dielectric loss spectra of this carbohydrate with use of the Havriliak- Negami function. The temperature dependence of logioTa was fitted to the Vogel- Fulcher- Tammann (VFT) function... 

The temperature dependences of relaxation times of the p and y processes of lactose and the secondary mode of octa-O-acetyl-lactose are presented in Fig. 6. In order to determine relaxation times of P- and y- modes of lactose and octa-O-acetyl-lactose the Cole-Cole and Havriliak-Negami functions were used respectively. Activation energies of all secondary relaxations were estimated from the Arrhenius fits... 

The imaginary part of the dielectric a-relaxation can be analyzed by using the phenomenological Havriliak-Negami function ... 

Figure 8. Havriliak-Negami central relaxation time, Thn, as a function of the reciprocal temperature, 1/T, for p and 7 processes. Experimental data (A, A) 60/40 and ( , ) 40/60 (by wt) PBT-PC copoljuners the solid and open symbols represent the (3 and 7 processes, respectively. The sjunbols x and + represent (3 and 7 processes for PBT. The solid lines are fittings to the VFT equation...

Dielectric measurements were used to evaluate the degrees of inter- and intramolecular hydrogen bonding in novolac resins.39 The frequency dependence of complex permittivity (s ) within a relaxation region can be described with a Havriliak and Negami function (HN function) ... 

Alvarez F, Alegria A, Cohnenero J (1991) Relationship between the time-domain Kohlrausch-Williams-Watts and frequency-domain Havriliak-Negami relaxation functions. Phys Rev B Condens Matter 44 7306-7312... 

Equations (1.23a), (1.23b) and (1.23c) are, respectively, Cole-Cole (C-C) (0Davidson-Cole (D-C) (0Havriliak-Negami (0empirical laws. The calculations of permittivity on the base of Eq. (1.22) with relaxation function corresponding to KWW law (see Eq. 1.20) yield Eq. (1.23c) with y8 = a - [30]. Expression (1.23c) delivers pretty good description of experimental data obtained by dielectric spectroscopy, radiospectroscopy and quasielastic neutron scattering. It can be shown, that the physical mechanism, underlying the expressions (1.23) is the distribution of relaxation times in a system. Namely, Equation (1.23) can be derived by the averaging of simple Debye response (1.21) with properly tailored distribution function of relaxation times F(x) ... 

Figure 15 Imaginary part e" vs. frequency for amorphous poly(ethylene terephtalate) at temperatures as indicated in K (temperature steps 2.5 K). The solid line is the fit using two superimposed Havriliak-Negami relaxation functions. If not indicated, the error bars are not larger than the size of the symbols. Taken from Kremer, F. Schdnhals, A. In Broadband Dielectric Spectroscopy Kremer, F. Schonhals, A., Eds. Springer, 2003 Chapter 4 with permission.

The dielectric relaxation data obtained were deconvoluted using a sum of the model function introduced by Havriliak-Negami [106] ... 

It is well known that the a-process has a wide distribution of the relaxation time and hence there is a possibility that the E-process is a tail of the a-process. Due to the wide distribution of the relaxation time the intermediate scattering function due to the a-process is described by a stretched exponential function [exp(-(t/T) ) 0< < ]. In order to check this possibility, we fitted a dynamic scattering law, Sjjn(Q>(o)> derived from the Havriliak-Negami (HN) function (see Eqs. 32 and 33) to the observed S(Q,co). The HN function (co) and SjjN(Q.ro) are given by... 

This function is assumed to represent the superposition of many Debye functions [Eq. (6.5)] with various relaxation times (Bottcher and Bordewijk 1978). In terms of the Havriliak-Negami model, a complete description of a real (non-Debye) relaxation process in a polymer requires calculation of four parameters the dielectric strength (Ae), a parameter related to the relaxation time of the process at the temperature of the scan (xhn), and two shape parameters (0 < ttHN < 1 and 0 < Phn 1). The latter describe the width and the asymmetry of the loss peak, respectively, but lack a physical meaning. 

For molecular relaxations like the primary (a-)relaxation in amorphous polymers, which exhibit an asymmetric broadening, i.e., an asymmetric underlying distribution in relaxation times, the most versatile Havriliak-Negami (HN) (Havriliak and Negami 1967) function is typically applied. A set of (additive) HN functions together with a term accounting for electrical conductivity (cf Eq. 7) is the typical choice to fit feature-rich dielectric loss spectra e"(co) like the example given in Fig. 7 ... 

Suitable theoretical models (e.g., the Havriliak-Negami (HN) model [24]) for analyzing the relaxation behavior of polymer blends (including amorphous/crystalline and thermosetting polymer mixtures) as a function of composition, temperature, crystallization and curing times can be used to evaluate the dielectric relaxation strength of a particular relaxation process and provide suitable information about the number of segments that contribute to the different relaxation processes. In... 

The memory effect is described by the integral term of Eq. (10-22), which is absent in the standard approach. Equation (10-22) holds for a liquid crystal with a Debye type of relaxation but it can be easily modified or generalized to other relaxation models (such as Cole-Davidson, Havriliak-Negami and other models with different functional forms of a(t — f) [5, 17]). 

Here e , is the high frequ y limit of s, So is the static dielectric constant (low frequency limit of s ). So - Soo = A is the dielectric increment, fR is the relaxation frequency, a is the Cole-Cole distribution parameter, and P is the asymmetry parameter. The relaxation frequency is related to the relaxation time by fa = (27It) A simple exponential decay of P (oc,P = 0) is characterised by a single relaxation time (Debye-process [1]), P = 0 and 1 < a < 0 describe a Cole-Cole-relaxation [2] with a symmetrical distribution function of t whereas the Havriliak-Negami equation (EQN (4)) is used for an asymmetric distribution of x [3]. The symmetry can be readily seen by plotting s versus s" as the so-called Cole-Cole plot [4-6]. 

Independently of the specific dielectric technique used, the results of dielectric measurements are usually analyzed in the form of complex dielectric permittivity e(o ) = e w) — ie (w) at constant temperature by fitting empirical relaxation functions to e(o ). In the examples to be given later in this chapter, the two-shape-parameter Havriliak-Negami (HN) expression [22]... 

FIGURE 5.5 Left logarithmic derivative of the relative permittivity of 1.9% L-336 (a) and 2% L-530 (b) suspensions as a function of the angular frequency (co = 2 tv) of the field, in 0.5 mM KCl and the SP concentrations indicated (key % volume fraction of LP-% volume fraction of SP). Right dielectric increment for the same suspensions. The lines are the best fits to the Havriliak-Negami relaxation function. 

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. 

Another way of describing experimentally observed dielectric relaxation in polymers is based on an empirical function formulated by Havriliak and Negami (1967) for the frequency domain ... 

These observations underlie many of the empirical functions commonly used to describe dielectric loss peaks, for instance, the ones proposed by Cole and Cole (100), Davidson and Cole (101), and Havriliak and Negami (102). In the time domain, the empirical KWW relaxation function 0(t) oc exp[—(t/t) ] often provides a reasonable description of experimental data (103). Since the response function is calculated as the negative derivative of 0(t), it behaves as a power law for short times. Moreover, the ubiquitous occurrence of power laws in (dielectric) spectra explains why log-log representations often are preferable power laws present themselves as straight lines when a log-log scale is used. 

Dielectric spectra are usually fitted by the empirical relaxation function suggested by Havriliak and Negami ... 

There were several attempts to generalize the Debye function like the Cole/Cole formula (Cole and Cole 1941) (symmetric broadened relaxation function), the Cole/Davidson equation (Davidson and Cole 1950, 1951), or the Fuoss/Kirkwood model (asymmetric broadened relaxation function) (Fuoss and Kirkwood 1941). The most general formula is the model function of Havriliak and Negami (HN function) (Havriliak and Negami 1966,1967 Havriliak 1997) which reads... 

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