Havriliak-Negami relaxation

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... 

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.

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. 

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],... 

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. 

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). 

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... 

The Kohlrausch Williams-Watts and Havriliak Negami formalisms are equally capable of representing real experimental data, and this is their main value, rather than an ability to explain the underlying relaxation processes. They are rooted in the time and frequency domains, respectively, and there is no analytical way of transforming from one to the other, but their effective equivalence has been convincingly demonstrated by numerical methods (Alvarez, Alegria and Colmenero, 1991). 

Like other retardation processes, the strength of the mechanical glass-rubber relaxation can, in principle, be determined by means of the empirical Havriliak-Negami equation (34)... 

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],... 

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