Havriliak-Negami model

Table 2. Fitting parameters of Havriliak-Negami model...

The Havriliak-Negami model was first established for the complex dielectric constant (equivalent to the complex compliance). Its mechanical translation can be written as... 

A comparison of results obtained for several networks based on the same epoxide-amine pair, but with variable amine/epoxide molar ratios, and thus variable crosslink densities, is shown in Table 11.3 (Tcharkhtchi et al., 1998). Havriliak-Negami and Perez models cannot be distinguished from one another by the quality of the fit of experimental curves, within experimental uncertainty. From a mathematical point of view, the Havriliak-Negami model is better than the Perez model because it has less parameters to fit (four parameters against five). In contrast, physical arguments could favor the Perez model, for which the parameters have a physical interpretation. 

However, this subject remains controversial. The KWW model is probably a relatively good approximation of both models, but only in a restricted range of a and y values for the Havriliak-Negami model (Alvarez et al., 1991), or a restricted range of % and % values for the Perez model. 

The a relaxation analysis can be performed using the Havriliak-Negami model [70], The parameters corresponding to this analysis are summarized in Tables 2.9 and 2.10. 

Fig. 2.51 Cole-Cole plots for the polymers studied (+) experimental data ( ) Havriliak-Negami model ( ) biparabolic model. References temperatures (a) 2,6-PDMPM, 167°C (b) 2,4-PDMPM, 109°C (c) 3,5-PDMPM, 106°C (d) 2,5-PDMPM, 111°C. (From ref. [42])...

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. 

The complex modulus components i and E" are frequently modeled in the frequency domain by means of the so-called Cole-Cole [43] plot, that is, E" =j ), and application of the Havriliak-Negami model [44], provided that there is no overlapping between a and relaxations ... 

Table 2. Parameters obtained from fit to the experimental data by the empirical Havriliak-Negami model function given by means of equation (8) for KN5, KN6, KN7, KN8, KN9 and KNIO, respectively.

Complex permittivity of a more realistic "lossy" dielectric where non-zero parallel DC conductivity ct(o)) exists can be represented on the basis of a more complex Havriliak-Negami model. This model also accounts for non-idealities of both capacitive and resistive components accounting for the asymmetry and broadness of the dielectric dispersion curve and resulting frequency-dependent conductivity a(to) and permittivity efoo) contributions ... 

As was discussed in Sectionl-2, the presence of conduction in parallel with capacitance and non-idealities in the capacitive and resistive components of the relaxation process described by Rg j, complicated Havriliak-Negami model (Eq. 3-8). 

The above dielectric (or complex capacitance) notation and Debye dispersion (Eq. 1-15) have often been used to describe a single bulk-media dielectric relaxation process in organic and polymeric (lossy) systems where at least two components with resistive and capacitive features exist [9, p. 33]. The permittivity of a lossy dielectric with negligible parallel DC conductance can be expressed on the basis of the Havriliak-Negami model (Eq. 1-16). Equivalent circuits representing a Debye model for lossy dielectric, where C, g = C g e, Cg - C j, R 1 /G [1, p. 65, p. 216], are shown in Figure 5-3. 

With regards to the translation of dynamic data in terms of static loading data through appropriate mathematical transformations, both the Havriliak-Negami and Perez models have been claimed to be counterparts of the KWW model ... 

In this system the a relaxation can be analyzed by the symmetric equation of Fuoss-Kikwood and a new model which is similar to Havriliak- Negami equation used in the analysis of dielectric spectroscopy. According to the Tg values calculated for these systems, the free volume can be appropriately described by the free volume theory. The analysis of these families of poly(methacrylate)s allow to understand in a good way the effect of the structure and nature of the side chain on the viscoleastic behavior of polymers [33],... 

As in the systems described above the a relaxation can be modeled by using the classical Havriliak Negami [70,87] procedure. 

The complex dielectric permittivity data of a sample, obtained from DS measurements in a frequency and temperature interval can be organized into the matrix data massive = [s, ] of size M x N, where Eq = ( >, , 7 ), M is the number of measured frequency points, and N is the number of measured temperature points. Let us denote by / =/(co x) the fitting function of n parameters x = x, X2,..., xn. This function is assumed to be a linear superposition of the model descriptions (such as the Havriliak-Negami function or the Jonscher function, considered in Section II.B.l). The dependence of/on temperature T can be considered to be via parameters only / =/(co x(T )). Let us denote by X = [jc,-(7 )] the n x N matrix of n model parameters xt, computed at N different temperature points 7. ... 

Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems, provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole-Cole, Cole-Davidson, and Havriliak-Negami equations of anomalous dielectric relaxation from a microscopic model based on a... 

Thus we have demonstrated how the empirical Havriliak-Negami equation [Eq. (11)] can be obtained from a microscopic model, namely, the fractional Fokker-Planck equation [Eq. (101)] applied to noninteracting rotators. This model can explain the anomalous relaxation of complex dipolar systems, where the anomalous exponents ct and v differ from unity (corresponding to the classical Debye theory of dielectric relaxation) that is, the relaxation process is characterized by a broad distribution of relaxation times. Hence, the empirical Havriliak-Negami equation of anomalous dielectric relaxation which has been... 

In the extended CM, the JG relaxation is just part of the continuous evolution of the dynamics. The JG relaxation should not be represented by a Cole-Cole or Havriliak-Negami distribution, as customarily assumed in the literature, and considered as an additive contribution to the distribution obtained from the Kohlrausch a-relaxation. Nevertheless, the JG relaxation may be broadly defined to include all the relaxation processes that have transpired with time up until the onset of the Kohlrausch a-relaxation. Within this definition of the JG relaxation, experiments performed to probe it will find that essentially all molecules contribute to the JG relaxation and the motions are dynamically and spatially heterogeneous as found by dielectric hole burning [180,283] and deuteron NMR [284] experiments. This coupling model description of the JG relaxation may help to resolve the different points of view of its nature between Johari [285] and others [180,226,227,280,281,283,284],... 

The dielectric relaxation processes of matter can be analyzed with an empirical model of dielectric dispersion, for example, the one described by Havriliak-Negami s equation. " We analyzed dielectric data obtained for our samples using a model of complex permittivity k with two dispersions (the main and the low-frequency dispersion of a space charge effect) and conductivity ao (caused by electrode discharge), as follows ... 

In general, plant tissue immittance data are not in accordance with Cole—Cole system models. Nonsymmetrical models of the Davidson Cole and Havriliak—Negami types have... 

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