Relaxation time Relaxed” permittivity

Fig. 11a and b. Debye single relaxation time model for dipole orientation showing a) permittivity and b) loss factor as a function of the product of the angular frequency w and the dipole realxation time rd. The relaxed permittivity is e, and the unrelaxed permittivity is e ... 

Figure 21 shows the permittivity and loss factor for an isothermal cure (137 °C) of DGEBA (EPON 825 n = 0) with diaminodiphenyl sulfone (DDS)47. To a first approximation, the data are the mirror image of Fig. 20, supporting the idea that the effect on electrical properties of the increase in Tg during isothermal cure might be similar to the effect of a decrease in temperature at fixed Tg. Examination of Fig. 21 shows one important difference between the temperature and cure dependences, namely that the relaxed permittivity decreases with cure time under isothermal conditions. This is a direct result of the changing chemistry, as discussed further in Section 4.3. The detailed behavior of the dipolar mobility is examined in Section 4.4, and of the ionic mobility in Section 4.5. 

Fig. 22. Plot of relaxed permittivity versus time for a low molecular weight DGEBA resin (EPON 825) cured isothermally with DDS at temperatures between410 K and460 K. Crosses represent experimental data the solid curve represents the model described in the text. (Reprinted from Ref. 71 > with permission of the Society of Plastics Engineers)...

In Section 4, we have examined, from a fundamental point of view, how temperature and cure affect the dielectric properties of thermosetting resins. The principal conclusions of that study were (1) that conductivity (or its reciprocal, resistivity) is perhaps the most useful overall probe of cure state, (2) that dipolar relaxations are associated with the glass transition (i.e., with vitrification), (3) that correlations between viscosity and both resistivity and dipole relaxation time are expected early in cure, but will disappear as gelation is approached, and (4) that the relaxed permittivity follows chemical changes during cure but is cumbersome to use quantitatively. 

Relaxed permittivity of PDS against time for a series of selected isothermal crystallization temperatures at a frequency of 25 kHz. 

Interfacial or Maxwell-Wagner polarization is a special mechanism of dielectric polarization caused by charge build-up at the interfaces of different phases, characterized by different permittivities and conductivities. The simplest model is the bilayer dielectric [1,2], (see Fig. 1.) where this mechanism can be described by a simple Debye response (exponential current decay). The effective dielectric parameters (unrelaxed and relaxed permittivities, relaxation time and static conductivity) of the bilayer dielectric are functions of the dielectric parameters and of the relative amount of the constituent phases ... 

The accessibility of chitin, mono-O-acetylchitin, and di-O-acetylchitin to lysozyme, as determined by the weight loss as a function of time, has been found to increase in the order chitin < mono-O-acetylchitin < di-O-acetylchitin [120]. The molecular motion and dielectric relaxation behavior of chitin and 0-acetyl-, 0-butyryl-, 0-hexanoyl and 0-decanoylchitin have been studied [121,122]. Chitin and 0-acetylchitin showed only one peak in the plot of the temperature dependence of the loss permittivity, whereas those derivatives having longer 0-acyl groups showed two peaks. 

Relaxation processes are probably the most important of the interactions between electric fields and matter. Debye [6] extended the Langevin theory of dipole orientation in a constant field to the case of a varying field. He showed that the Boltzmann factor of the Langevin theory becomes a time-dependent weighting factor. When a steady electric field is applied to a dielectric the distortion polarization, PDisior, will be established very quickly - we can say instantaneously compared with time intervals of interest. But the remaining dipolar part of the polarization (orientation polarization, Porient) takes time to reach its equilibrium value. When the polarization becomes complex, the permittivity must also become complex, as shown by Eq. (5) ... 

In contrast with Eq. (5), Eq. (11) gives the frequency behavior in relation to the microscopic properties of the studied medium (polarizability, dipole moment, temperature, frequency of the field, etc). Thus for a given change of relaxation time with temperature we can determine the change with frequency and temperature of the dielectric properties - the real and imaginary parts of the dielectric permittivity. 

In order to quantify diffiisional effects on curing reactions, kinetic models are proposed in the literature [7,54,88,95,99,127-133]. Special techniques, such as dielectric permittivity, dielectric loss factor, ionic conductivity, and dipole relaxation time, are employed because spectroscopic techniques (e.g., FT i.r. or n.m.r.) are ineffective because of the insolubility of the reaction mixture at high conversions. A simple model, Equation 2.23, is presented by Chem and Poehlein [3], where a diffiisional factor,//, is introduced in the phenomenological equation, Equation 2.1. 

Static solution permittivity, e(c), and static solvent permittivity, es(c), for solutions of various electrolytes at various concentrations (c) have been obtained by dielectric relaxation spectroscopy [44]. Ion-pairs contribute to permittivity if their lifetime is longer than their relaxation time. However free ions do not contribute to permittivity. Thus,... 

Figure 1.3 Dielectric spectra for range of alcohols in the frequency range of 107-10n Hz. The absolute permittivities at low frequencies fall as the size of the alcohol increases and they began to respond to the microwave fields at lower frequencies because their relaxation times become longer. The loss factors that control the efficiency of conversion of microwave into thermal energies also reach their maxima at lower frequencies. The loss tangent is the ratio of the loss factor and permittivity at that frequency. (Idealised from the raw data illustrated in Ref. 10.)...

As a second example, we consider liquid fluoromethane CH3F, which is a typical strongly absorbing nonassociated liquid. For our study we choose the temperature T 133 K near the triple point, which is equal to 131 K. The relevant experimental data [43] were summarized in Table IV. As we see in Table VIII, which presents the fitted parameters of the model, the angle p is rather small. At this temperature the density p of the liquid, the maximum dielectric loss and the Debye relaxation time rD are substantially larger than they would be, for example, near the critical temperature (at 293 K). At such small (5 the theory given here for the hat-curved model holds. For calculation of the complex permittivity s (v) and absorption a(v), we use the same formulas as for water. 

The role of specific interactions was not recognized for a long time. An important publication concerning this problem was the work by Liebe et al. [17], where a fine non-Debye behavior of the complex permittivity (v) was discovered in the submillimeter frequency range. The new phenomenon was described as the second Debye term with the relaxation time T2, which was shown to be very short compared with the usual Debye relaxation time td (note that td and 12 comprise, respectively, about 10 and 0.3 ps). A physical nature of the processes, which determines the second Debye term, was not recognized nor in Ref. [17], nor later in a number works—for example, in Refs. 54-56, where the double Debye approach by Liebe et al. was successfully confirmed. 

Equations (281b) and (282) determine the frequency dependence of the reorienting complex permittivity e r(v). One can estimate the principal (Debye) relaxation time by using the relation... 

The hat-curved-harmonic oscillator model, unlike other descriptions of the complex permittivity available now for us [17, 55, 56, 64], gives some insight into the mechanisms governing the experimental spectra. Namely, the estimated relaxation time of a nonrigid dipole (xovib 0.2 ps) is close to that determined in the course of very accurate experimental investigations and of their statistical treatment [17, 54-56]. The reduced parameters presented in Tables XIVA and XIVB and the form of the hat-curved potential well (determined by the parameters u, (3, f) do not show marked dependence on the temperature, while the spectra themselves vary with T in greater extent. We shall continue discussion of these results in Section X.A. 

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