Relaxation function, dielectric

The elastic contribution is also called elastic incoherent structure factor (EISF). It may be interpreted as the Fourier transformed of the asymptotic distribution of the hopping atom for infinite times. In an analogous way to the relaxation functions (Eq. 4.6 and Eq. 4.7), the complete scattering function is obtained by averaging Eq. 4.22 with the barrier distribution function g E) obtained, e.g. by dielectric spectroscopy (Eq. 4.5)... 

In all the above three polymers only a single process is apparently observed in the time window for PCS (10-6 to 100 s). The shape of the relaxation function is independent of temperature. The temperature dependence of (r) follows the characteristic parameters observed for mechanical or dielectric studies of the primary (a) glass-rubber relaxation. Relaxation data obtained by many techniques is collected together in the classic monograph of McCrum, Read and Williams41. The data is presented in the form of transition maps where the frequencies of maximum loss are plotted logarithmically... 

In contrast to polystyrene the observed intercepts for PMMA and PEMA in the glassy state remain high with values that are a substantial fraction of those observed in the equilibrium liquid state. Such a result should not be too surprising since it was shown above that a large part of the observed relaxation function above Tg was due to the secondary relaxation. The frequency of maximum dielectric or mechanical loss for the /9... 

Debye relaxation function was used for the solvent dielectric relaxation. From Ref. 32 with permission, from J. Chem. Phys. 90, 153 (1989). Copyright 1989, American Physical Society. 

Relation (14) gives equivalent information on dielectric relaxation properties of the sample being tested both in frequency and in time domain. Therefore the dielectric response might be measured experimentally as a function of either frequency or time, providing data in the form of a dielectric spectrum s (co) or the macroscopic relaxation function [Pg.8]

Several comprehensive reviews on the BDS measurement technique and its application have been published recently [3,4,95,98], and the details of experimental tools, sample holders for solids, powders, thin films, and liquids were described there. Note that in the frequency range 10 6-3 x 1010 Hz the complex dielectric permittivity e (co) can be also evaluated from time-domain measurements of the dielectric relaxation function (t) which is related to ( ) by (14). In the frequency range 10-6-105 Hz the experimental approach is simple and less time-consuming than measurement in the frequency domain [3,99-102], However, the evaluation of complex dielectric permittivity in the frequency domain requires the Fourier transform. The details of this technique and different approaches including electrical modulus M oo) = 1/ ( ) measurements in the low-frequency range were presented recently in a very detailed review [3]. Here we will concentrate more on the time-domain measurements in the high-frequency range 105—3 x 1010, usually called time-domain reflectometry (TDR) methods. These will still be called TDS methods. 

As mentioned above, the frequency dependence of the complex dielectric permittivity (e ) of the main relaxation process of glycerol [17,186] can be described by the Cole-Davidson (CD) empirical function [see (21) with a = 1, 0 < Pcd < 1], Now Tcd is the relaxation time which has non-Arrhenius type temperature dependence for glycerol (see Fig. 23). Another well-known possibility is to fit the BDS spectra of glycerol in time domain using the KWW relaxation function (23) < )(t) (see Fig. 24) ... 

Figure 41. Typical dielectric spectra of 20 mol% of glycerol—water mixtures at (a) 185 K (supercooled state) and (b) 218 K (frozen state), where solid and dashed curves show the real and imaginary parts of complex dielectric permittivity. Each relaxation process in the frozen state was fitted by (114) and by Cole-Cole and Debye relaxation functions, respectively, in order to separate the main process, the process due to interfacial water, and the process due to ice. (Reproduced with permission from Ref. 244. Copyright 2005, American Chemical Society.)...

The volumetric, elastic and dynamic properties of internally and externally plasticised PVC were studied and compared with those of unplasticised PVC. The glass transition temperature for the plasticised samples was markedly lowered and this decrease was more important for the externally plasticised ones. The positions of the loss peaks from dielectric alpha-relaxation measurements confirmed the higher efficiency of the external plasticisation. However, the shape of the dielectric alpha-relaxation function was altered only for the internally plasticised samples. The plasticisation effect was linked with a decrease in the intensity of the beta-relaxation process but no important changes in the activation energy of this process were observed. The results were discussed. 47 refs. 

Figure 4.3 Frequency-dependence of the imaginary (loss) part of the dielectric relaxation function for PDE at different temperatures. The lines are fits by the Cole-Davidson function, Eq. (4-2), with cu = 2nf and temperature-dependent exponent given in Fig. 4-4. (Reprinted from Physica, A201 318, Stickel et al. (1993), with kind permission from Elsevier Science - NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)...

It has to be mentioned that such equivalent circuits as circuits (Cl) or (C2) above, which can represent the kinetic behavior of electrode reactions in terms of the electrical response to a modulation or discontinuity of potential or current, do not necessarily uniquely represent this behavior that is other equivalent circuits with different arrangements and different values of the components can also represent the frequency-response behavior, especially for the cases of more complex multistep reactions, for example, as represented above in circuit (C2). In such cases, it is preferable to make a mathematical or numerical analysis of the frequency response, based on a supposed mechanism of the reaction and its kinetic equations. This was the basis of the important paper of Armstrong and Henderson (108) and later developments by Bai and Conway (113), and by McDonald (114) and MacDonald (115). In these cases, the real (Z ) and imaginary (Z") components of the overall impedance vector (Z) can be evaluated as a function of frequency and are often plotted against one another in a so-called complex-plane or Argand diagram (110). The procedures follow closely those developed earlier for the representation of dielectric relaxation and dielectric loss in dielectric materials and solutions [e.g., the Cole and Cole plots (116) ]. 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

Relaxation functions for fractal random walks are fundamental in the kinetics of complex systems such as liquid crystals, amorphous semiconductors and polymers, glass forming liquids, and so on [73]. Relaxation in these systems may deviate considerably from the exponential (Debye) pattern. An important task in dielectric relaxation of complex systems is to extend [74,75] the Debye theory of relaxation of polar molecules to fractional dynamics, so that empirical decay functions for example, the stretched exponential of Williams and Watts [76] may be justified in terms of continuous-time random walks. 

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. 

Similar heterogeneous model has been used to develop a relaxation function by Chamberlin and Kingsbury (1994), who consider the localized normal modes to be involved in the relaxation process. Localized (domains) regions are assumed to be present between Tg and T. They are described as dynamically correlated domains (DCD). A Gaussian distribution of the domain sizes has been assumed, with each domain characterized by a Debye relaxation time. Expressions for the dielectric susceptibility have been derived and used to fit the experimental susceptibilities of salol, glycerol and many other substances with remarkable agreement over 13 decades of frequency (even when only one adjustable parameter is employed). 

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