Complex permittivity-frequency

Dielectric measurements were used to evaluate the degrees of inter- and intramolecular hydrogen bonding in novolac resins.39 The frequency dependence of complex permittivity (s ) within a relaxation region can be described with a Havriliak and Negami function (HN function) ... 

Knowledge of complex permittivities of appropriate electrolyte solutions is useful in assessing interactions of microwave radiation with biological tissues. A full study and analysis of complex permittivities of sodium chloride solutions as a function of concentration, temperature, and microwave frequency (207) has laid the foundations for a similar investigation of calcium salt solutions. 

The Impedance Analyzer was controlled by a 9836 Hewlett-Packard computer which also controlled the time-tempe ture of the press. Measurements at frequencies from 5 to 5 x 10 Hz were taken at regular Intervals during the cure cycle and converted to the complex permittivity. Further details of the experimental procedure has been given elsewhere [10]. 

The dielectric measurements were carried out in a plate capacitor and frequency dependences of complex permittivity e = e — is (e and e" being its real and imaginary part, respectively) were determined [33] in the range /= 20 Hz- 200 kHz. 

In particular, VF2/F3E copolymers have also been the subject of extensive research [6,17,96]. As an example to illustrate the dielectric behavior of these copolymers, the temperature dependence of the real and the imaginary part of the complex permittivity at two different frequencies (1 and 100 kHz) are shown in Figs. 23a and 23b respectively. The measurements correspond to the 60/40 copolymer. The data have been collected by using a sandwich geometry with gold evaporated electrodes [95]. Frequencies of 103 and 106 Hz have been used by employing a 4192 A HP Impedance Analyzer. From inspection of Fig. 23b... 

ASTM D2520, 2001. Standard test methods for complex permittivity (dielectric constant) of solid insulating materials at microwave frequencies to 1650°C. 

ASTM 1986. Standard Methods Qf Test for Complex Permittivity (Dielectric Constant) of Solid Electrical Insulating Materials at Microwave Frequencies and Temperatures 1q 1650°C. Document D 2520-86 (Reapproved 1990). Philadelphia, PA. American Society for Testing and Materials (ASTM). 

It should, however, be noted that there exist rather complex and nontransparent descriptions made [15] in terms of the absorption vibration spectroscopy of water. This approach takes into account a multitude of the vibration lines calculated for a few water molecules. However, within the frames of this method for the wavenumber1 v < 1000 cm-1, it is difficult to get information about the time/spatial scales of molecular motions and to calculate the spectra of complex-permittivity or of the complex refraction index—in particular, the low-frequency dielectric spectra of liquid water. 

We have introduced the effective complex susceptibility x ( ) = X,( )+ X ) stipulated by reorienting dipoles. This scalar quantity plays a fundamental role in subsequent description, since it connects the properties and parameters of our molecular models with the frequency dependences of the complex permittivity s (v) and the absorption coefficient ot (v) calculated for these models. 

In this section we calculate the complex permittivity (v) and the absorption coefficient ot(v) of ordinary (H20) water and of fluoromethane CH3F over a wide range of frequencies. We shall first write down the list of the formulas useful for further calculations. 

In this section we have to calculate the complex permittivity s (v) and the absorption coefficient a(v) of ordinary (H2O) water over a wide range of frequencies. It is rather difficult to apply rigorous formulas because the fluctuations of the calculated characteristics occur at a small reduced collision frequency y typical for water (in this work we employ for calculations the standard MathCAD program). Such fluctuations are seen in Fig. 13b (solid curve). Therefore the calculations will be undertaken for two simplified variants of the hat model. Namely, we shall employ the planar libration-regular precession (PL-RP) approximation and the hybrid model.26... 

In our early work33 [50] the constant field model was applied to liquid water, where the harmonic law of particles motion, corresponding to a parabolic potential, was actually employed in the final calculations of the complex permittivity. In this work, qualitative description of only the libration band was obtained, while neither the R-band nor the low-frequency (Debye) relaxation band was described. Moreover, the fitted mean lifetime x of the dipoles, moving in the potential well, is unreasonably short ( ().02 ps)—that is, about an order of magnitude less than in more accurate calculations, which will be made here. 

Starting with the important example of ordinary water, we choose temperatures 22.2°C and 27°C. We compare our theory with the recorded FIR spectra [42, 56] of the complex permittivity/absorption. At low frequencies we use for this purpose an empirical formula [17] by Liebe et al. these formulas were given also in Section IV.G.2.a. The values of the employed molecular constants are presented in Table VI and the fitted parameters in Table VII. The Reader may find more information about experimental data of liquid H2O and D2O in Appendix 3. 

Figure 28. Experimental frequency dependences of dielectric parameters recorded for liquid water (a) Real (curve 1) and imaginary (curve 2) parts of the complex permittivity at 27°C. The data are from Refs. 42 (solid lines) and 17 (circles), (b) Absorption coefficient. Solid line and crosses 1 refer to 1°C filled circles 2 refer to 27°C dashed line and squares 3 refer to 50°C. For lines the data from Ref. 17 were employed, for circles the data are from Ref. 42, for crosses and squares the data are from Ref. 53.

Figs. 32a-c illustrate the absorption spectra, calculated, respectively, for water H20 at 27°C, water H20 at 22.2°C, and water D20 at 22.2°C dotted lines show the contribution to the absorption coefficient due to vibrations of nonrigid dipoles. The latter contribution is found from the expression which follows from Eqs. (242) and (255). The experimental data [42, 51] are shown by squares. The dash-and-dotted line in Fig. 32b represents the result of calculations from the empirical formula by Liebe et al. [17] (given also in Section IV.G.2) for the complex permittivity of H20 at 27°C comprising double Debye-double Lorentz frequency dependences. 

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 frequency dependence of the complex permittivity pertinent to the vibration process is described by Eqs. (290a) and (293) ... 

Figure 35. Frequency dependence in the submillimeter wavelength region of the real (a, b) and imaginary (c, d) parts of the complex permittivity. Solid lines Calculation for the composite HC-HO model. Dashed lines Experimental data [51]. Dashed-and-dotted lines show the contributions to the calculated quantities due to stretching vibrations of an effective non-rigid dipole. The vertical lines are pertinent to the estimated frequency v b of the second stochastic process. Parts (a) and (c) refer to ordinary water, and parts (b) and (d) refer to heavy water. Temperature 22.2°C.

The dielectric behavior of PMCHI was studied by Diaz Calleja et al. [210] at variable frequency in the audio zone and second, by thermal stimulated depolarization. Because of the high conductivity of the samples, there is a hidden dielectric relaxation that can be detected by using the macroscopic dynamic polarizability a defined in terms of the dielectric complex permittivity e by means of the equation ... 

As a result of these very general considerations, one expects the dielectric response function, as expressed by the complex permittivity, k (oj), or the attenuation function, a(oi), of ordinary molecular fluids to be characterized, from zero frequency to the extreme far-infrared region, by a relaxation spectrum. To first order, k (co) may be represented by a sum of terms for individual relaxation processes k, each given by a term of the form ... 

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