Havriliak-Negami

As an example of the Curie transition for other VF2/F3E compositions, measurements of e and e" as a function of temperature and f = 1 kHz are shown in Figs. 25a and 25b. The complex permittivity function of polymeric materials has been shown to follow the Havriliak-Negami phenomenological equation [99] ... 

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. 

Table 11.3 Havriliak-Negami and Perez (see text) parameters for the glass transition region of DGEBA-ETHA networks differing in the amine/epoxide molar ratio. (After Tcharkhtchi et a/., 1998.)... 

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 ... 

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. 

We show in Figure 13.8 that in the case of a well-behaved piezoelectric relaxation (counterclockwise hysteresis) presented in Figure 13.7, the Kramers-Kronig relations are indeed fulfilled. Closer inspection of the data show that the relaxation curves can be best described by a distribution of relaxation times and empirical Havriliak-Negami equations [19]. It is worth mentioning that over a wide range of driving field amplitudes the piezoelectric properties of modified lead titanate are linear. Details of this study will be presented elsewhere. 

The a relaxation without the conductive contribution can be analyzed using the Havriliak-Negami (HN) equation [70] ... 

Table 2.4 Parameters of Havriliak-Negami equation (2.8) for a relaxation at indicated temperature. (From ref. [33])...

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],... 

Figures 2.21 and 2.22 are examples of the obtention of a clean a peak after subtracting the conductivity. Afterwards it is possible to fit an empirical Havriliak-Negami equation (87) to the experimental data following the usual procedure.

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. 2.39 Dielectric permittivity (A) and loss ( ) for P3M2NBM in the frequency domain at 105°C. The discontinuous straightline represents the conductive effects. Circular black points ( ) represent the resulting loss curve after subtraction showing the dipolar a - relaxation. The continuous line correspond to the Havriliak-Negami curve fit [35,70], (From ref. [35])...

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

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])...

The peak of the dielectric loss of this process reflects its viscoelastic nature by obeying the time-temperature superposition principle, wherein the peak is shifted to higher temperatures for shorter times (higher frequencies) and vice versa. This process has been described by the Havriliak-Negami empirical formula [106, 108]... 

The majority of cases of non-Debye dielectric spectrum have been described by the so-called Havriliak-Negami (HN) relationship [8,11,15] ... 

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. ... 

As mentioned in Section II.B, the dielectric response in the frequency domain for most complex systems cannot be described by a simple Debye expression (17) with a single dielectric relaxation time. In a most general way this dielectric behavior can be described by the phenomenological Havriliak-Negami (HN) formula (21). 

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