Equation Havriliak-Negami

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. 

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.

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... 

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... 

The Havriliak-Negami equation is given by the following formula [86] ... 

Table 7. Parameters of the Havriliak-Negami equation for DGEBA oligomers [10]...

Results of dielectric studies of a series of nanocomposites based on a semiaromatic PA-1 ITIO with HAp helped to explain the structure-property relationships [Sender, 2008], For dry, neat PA-llTlO, a symmetric depolarization peak, detected by TSC near Tg measured by DSC, was attributed to dielectric dynamic glass relaxation. By DDS, two distinct high-temperature processes were distinguished when plotting the experimental data points versus HT. The dependence is displayed in Figure 13.2, where the two dashed lines show the complex dielectric permittivity fitted by the Havriliak-Negami equation ... 

FIGURE 21.5. Data from Fig. 21.4 as fitted (solid line) using the Havriliak-Negami equation [Eq. (21.1)]. The fit parameters are shown in Table 21.1. Taken from [2] with permission. 

To study the effects of interaction of starch with silica, the broadband DRS method was applied to the starch/modified silica system at different hydration degrees. Several relaxations are observed for this system, and their temperature and frequency (i.e., relaxation time) depend on hydration of starch/silica (Figures 5.6 and 5.7). The relaxation at very low frequencies (/< 1 Hz) can be assigned to the Maxwell-Wagner-Sillars (MWS) mechanism associated with interfacial polarization and space charge polarization (which leads to diminution of 1 in Havriliak-Negami equation) or the 5 relaxation, which can be faster because of the water effect (Figures 5.8 and 5.9). 

The normalized loss curve which is obtained from the frequency dependence of e" is described in terms of Havriliak-Negami equation[30],... 

Table I. Havriliak-Negami equation and VFTH equation fit parameters for 95 days degraded vs. untreated samples at 60 C.

The combination of the Cole-Cole equation (eq. (10.20)) and the Cole—Davidson equation (eq. (10.24)) is, after the inventors, referred to as the Havriliak— Negami equation ... 

Haviviliyaku-Negami equation, 367 Havriliak-Negami equation, 367 Heat capacity, 331 Heat flux, 350... 

Neither the a-process nor the normal mode equal a single-time relaxation process. A good representation of data is often achieved by the use of the empirical Havriliak—Negami equation, which has the form... 

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]. 

Liquid crystals in the nematic phase are dielectrically anisotropic, and the real (s ) and imaginary (s ) parts of the dielectric permittivity have two independent components, corresponding to the parallel and perpendicular orientations, respectively, of the applied electric field with respect to the nematic director. The imaginary part of the permittivity [12] is given by the phenomenological Havriliak-Negami equation... 

Figure 6. Isothermal plots of the dielectric loss component, e , vs. frequency, P, in the merging region for neat PBT and two PBT/PC copolymers (weight ratios indicated in the figure). The solid lines represent the results of fitting experimental data to the sum of two Havriliak-Negamy equations (Eq. 1) -ith a conductivity term. The dashed lines at the highest temperature show the separate contributions of (3 and 7 processes...

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