The Dielectric Relaxation Parameters

According to the Debye model there are three parameters associated with dielectric relaxation in a simple solvent, namely, the static permittivity s, the Debye relaxation time td, and the high-frequency permittivity Eoq. The static permittivity has already been discussed in detail in sections 4.3 and 4.4. In this section attention is especially focused on the Debye relaxation time td and the related quantity, the longitudinal relaxation time Tl. The significance of these parameters for solvents with multiple relaxation processes is considered. The high-frequency permittivity and its relationship to the optical permittivity Eop is also discussed. 

As defined, the Debye relaxation time is the reciprocal of a first-order rate constant. Thus, it is expected to depend on temperature according to the usual Arrhenius relationship 

The temperature dependence of the longitudinal relaxation time tl is also an important quantity. For a Debye solvent, tl is given by the relationship 

The enthalpy associated with the temperature dependence is defined as 

The temperature derivatives of the high- and low-frequency contributions to the permittivity are easily estimated from experimental data (see table 4.2). In some cases the temperature derivative of s, is not available but one may assume that it is approximately equal to the temperature derivative of Sop. 

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First comes the determination of the dielectric relaxation parameters entering the Debye equations ... 

The considered model of a straight line of M nanoparticles illustrates only general features of dielectric losses caused by an M nanoparticle cluster in polymer matrix. Actually such cluster is a complex fractal system. Analysis of dielectric relaxation parameters of this process allowed the determination of fractal properties of the percolation cluster [104], The dielectric response for this process in the time domain can be described by the Kohlrausch-Williams-Watts (KWW) expression... 

As expected, the capacitance of the cell increases when the frequency is decreased (Figure 1.25a) below the knee frequency, the capacitance tends to be less dependent on the frequency and should be constant at lower frequencies. This knee frequency is an important parameter of the EDLC it depends on the type of the porous carbon, the electrolyte as well as the technology used (electrode thickness, stack, etc.) [20], The imaginary part of the capacitance (Figure 1.25b) goes through a maximum at a given frequency noted as/0 that defines a time constant x0 = 1 lf0. This time constant was described earlier by Cole and Cole [33] as the dielectric relaxation time of the system, whereas... 

To connect these results to the dielectric relaxation in curing systems, it is useful to examine the post-cure data of Shito et al. 49 50) and of Blyakhman et al. 5I,52). Figure 25 shows the fit of Shito s data for xd (the reciprocal of fmax) to the WLF Equation for a series of DGEBA resins cured with a variety of anhydrides50). In this case, the data were fit to the universal WLF Equation, using the reference temperature as an adjustable parameter, which fell in the range 55-61 °C above Tg for the various samples. 

H NMR parameters used depend mostly on amplitude and fioquency of reorientations of the CH3 protons due to the motion of the siloxane chain. A restricted reorientation of one monomer unit provides already an effective source for T and T2 relaxation. On the other hand, the dielectric relaxation is caused by the reorientations of the Si-O dipoles of the main chain. A few adjacent adsorbed units might be involved in the relaxation caused by adsorption-desorption. 

The dielectric constant is a natural choice of order parameter to study freezing of dipolar liquids, because of the large change in the orientational polarizability between the liquid and solid phases. The dielectric relaxation time was calculated by fitting the dispersion spectrum of the complex permittivity near resonance to the Debye model of orientational relaxation. In the Debye dispersion relation (equation (3)), ij is the frequency of the applied potential and t is the orientational (rotational) relaxation time of a dipolar molecule. The subscript s refers to static permittivity (low frequency limit, when the dipoles have sufficient time to be in phase with the applied field). The subscript oo refers to the optical permittivity (high frequency limit) and is a measure of the induced component of the permittivity. 

The behavior of the dielectric spectra for the two-rotational-degree-of-ffeedom (needle) model is similar but not identical to that for fixed-axis rotators (one-rotational-degree-of-fireedom model). Here, the two- and one-rotational-degree-of-freedom models (fractional or normal) can predict dielectric parameters, which may considerably differ from each other. The differences in the results predicted by these two models are summarized in Table I. It is apparent that the model of rotational Brownian motion of a fixed-axis rotator treated in Section IV.B only qualitatively reproduces the principal features (return to optical transparency, etc.) of dielectric relaxation of dipolar molecules in space for example, the dielectric relaxation time obtained in the context of these models differs by a factor 2. 

Qualitative predictions of the classical Marcus theory of ET have been confirmed for a wide variety of systems. Given this success, theory is now challenged to calculate absolute ET rates on the basis of separately obtainable parameters. Regarding the solvent, the input is the dielectric relaxation spectrum. It can be measured or even calculated independently to a high degree of accuracy. Intramolecular modes participating in the ET... 

Analysis of dielectric relaxation parameters of this process allowed us to determine the fractal properties of the percolation cluster [70]. The dielectric response for this process in the time domain can be described by the... 

The results of the D-B and H-H models, as shown in a comparison of Figures 4-7 with Figures 32-35 are very similar. In both cases a = 0.6 and approximately independent of either the K or the shape parameter The )3 parameter changes in a sigmoidal pattern from about 0.35 to 1.0 for the D-B model and from 0.35 to about the 0.7-0.8 range for the H-H model. In other words, both the D-B and the H-N models predict the shape of the dielectric relaxation process to have nearly the same dependence on sphere diameter or segment asymmetry and both models have very different starting points. 

The relaxation time constant 0 would also influence the parameter A, finally p substantially. From either Eq.(73) and (75), one will find that p increases as 0 increases, that is, the ER effect will be stronger if the dielectric relaxation is slower. However, too slow relaxation time (tlien the slow response time) would make FR fluids useless. Generally, the FR response time around 1 millisecond is favorable, thus requiring the relaxation time be of the same time scale, i.e., the dielectric relaxation frequency around lO llz. Block presumably thought the polarization rate would be important in the ER response process, and too fast or too slow polarization is unfavorable to the ER effect [7J. Ikazaki and Kawai experimentally found that the FR fluids of the relaxation frequencies within the range 100-10 Hz would exhibit a large ER effect [21,31], supporting the derivation from Eq. (69). 

The parameter, a, is related to the angle, 6, between the radius of the arc at the point ( = e e" = 0) and the real axis by the expression 6 = anil. This is shown in Fig. 5.27 Malik and Prud homme [133] found that the temperature shift factors for the dielectric relaxation of several blends were very close to those for the shift of mechanical linear viscoelastic data, if a slightly different reference temperature was used for the two types of data. 

Dielectric constants and refractive indices, as well as electrical conductivities of liquid crystals, are physical parameters that characterize the electronic responses of liquid crystals to externally applied fields (electric, magnetic, or optical). Because of the molecular and energy level stractures of nematic molecules, these responses are highly dependent on the direction and the frequencies of the field. Accordingly, we shall classify om studies of dielectric permittivity and other electro-optical parameters into two distinctive frequency regimes (1) dc and low frequency, and (2) optical frequency. Where the transition from regime (1) to (2) occurs, of course, is governed by the dielectric relaxation processes and the dynamical time constant typically the Debye relaxation frequencies in nematics is on the order of 10 ° Hz. 

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