The Complex Dielectric Permittivity

The interaction bet veen electromagnetic vaves and matter is quantified by the two complex physical quantities - the dielectric permittivity, s, and the magnetic susceptibility, fi. The electric components of electromagnetic waves can induce currents of free charges (electric conduction that can be of electronic or ionic origin). 

The magnitude of the dipole moment depends on the size and symmetry of the molecule. Molecules with a center of symmetry, for example methane, carbon tetrachloride, and benzene are apolar (zero dipole moment) whereas molecules with no center of symmetry are polar. Table 1.2 gives relative static dielectric permittivity 

From Maxwell s theory of electromagnetic waves it follows that the relative permittivity of a material is equal to the square of its refractive index measured at the same frequency. Refractive index given by Table 1.2 is measured at the frequency of the D line of sodium. Thus it gives the proportion of (electronic) polarizability still effective at very high frequencies (optical frequencies) compared with polarizability at very low frequencies given by the dielectric constant. It can be seen from Table 1.2 that the dielectric constant is equal to the square of the refractive index for apolar molecules whereas for polar molecules the difference is mainly because of the permanent dipole. In the following discussion the Clausius-Mossoti equation will be used to define supplementary terms justifying the difference between the dielectric constant and the square of the refractive index (Eq. (29) The Debye model). 

The temperature dependence of the dielectric constant of polar molecules also differs from that of nonpolar molecules. Change of temperature has a small effect only for nonpolar molecules (change of density). For polar molecules, the orientation polarization falls off rapidly with increasing temperature, because thermal motion reduces the alignment of the permanent dipoles by the electric field. In the following discussion we will see that it is possible to have increasing values of dielectric permittivity with increasing temperature. 

As discussed above, a molecule with a zero total charge may still have a dipole moment because molecules without a center of symmetry are polar. Similarly, a molecule may have a distribution of charge which can be regarded as two equal and opposite dipoles centered at different places. Such a distribution will have zero total charge and zero total moment but will have a quadrupole moment. Car- 

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Fig. 1.4 Dependence of the complex dielectric permittivity on frequency (s is the real part and e is the imaginary part, or the dielectric loss).

Inhomogeneity of the field-induced change in the characteristics of the medium, the complex dielectric permittivity esc = ei + >n particular (here e1>2 are real quantities), is a distinguishing feature of electrooptic effects in the space-charge region. The ranges of such inhomogeneities (10 4-10 5 cm)... 

As an important example, let us consider the effect of electroreflection due to inhomogeneity of the distribution of free carriers in the space-charge region of a semiconductor (plasma electroreflection). The contribution of the electrons to the complex dielectric permittivity (an n-type semiconductor is considered for illustration and the contribution of the holes is neglected) is given by the expression (see, for example, Ziman, 1972)... 

The temperature and frequency dependence of the complex dielectric permittivity a for both 2-chlorocydohexyi isobutyrate (CCHI) and poly(2-chlorocyclohexyl acrylate) (PCCHA) is reported. The analysis of the dielectric results in terms of the electric modulus suggests that whereas the conductive processes in CCHI are produced only by free charges, the conductivity observed in PCCHA involves both free charges and interfacial phenomena. The 4x4 RIS scheme presented which accounts for two rotational states about the CH-CO bonds of the side group reproduces the intramolecular correlation coefficient of the polymer. 

The analysis of the real and imaginary part of the complex dielectric permittivity allows one to distinguish between the two main relaxation processes (a and P). The a-process is correlated to the transition from the ferro to the paraelectric phase and the p-process is attributed to segmental motions in the amorphous phase. 

Most of the physical properties of the polymer (heat capacity, expansion coefficient, storage modulus, gas permeability, refractive index, etc.) undergo a discontinuous variation at the glass transition. The most frequently used methods to determine Tg are differential scanning calorimetry (DSC), thermomechanical analysis (TMA), and dynamic mechanical thermal analysis (DMTA). But several other techniques may be also employed, such as the measurement of the complex dielectric permittivity as a function of temperature. The shape of variation of corresponding properties is shown in Fig. 4.1. 

Figure 4.1 Variation of physical properties vs temperature, used to determine the glass transition (a) volume (V) or enthalpy (H) (b) expansion coefficient (a) or heat capacity (cp) (c) storage modulus (E ) (d) dissipation modulus (E") and dumping factor (tan 8) (e) real part of the complex dielectric permittivity (s ) (f) imaginary part of the complex dielectric permittivity (e").

This implies that the complex dielectric permittivity can be presented as follows ... 

As mentioned previously, the complex dielectric permittivity (g>) can be measured by DS in the extremely broad frequency range 10-6-1012 Hz (see Fig. 1). However, no single technique can characterize materials over all frequencies. Each frequency band and loss regime requires a different method. In addition to the intrinsic properties of dielectrics, their aggregate state, and dielectric permittivity and losses, the extrinsic quantities of the measurement tools must be taken into account. In this respect, most dielectric measurement methods and sample cells fall into three broad classes [3,4,91] ... 

The LF measurements (a) are provided by means of impedance/admittance analyzers or automatic bridges. Another possibility is to use a frequency response analyzer. In lumped-impedance measurements for a capacitor, filled with a sample, the complex dielectric permittivity is defined as [3]... 

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. 

Dielectric relaxation of complex materials over wide frequency and temperature ranges in general may be described in terms of several non-Debye relaxation processes. A quantitative analysis of the dielectric spectra begins with the construction of a fitting function in selected frequency and temperature intervals, which corresponds to the relaxation processes in the spectra. This fitting function is a linear superposition of the model functions (such as HN, Jonscher, dc-conductivity terms see Section II.B.l) that describes the frequency dependence of the isothermal data of the complex dielectric permittivity. The temperature behavior of the fitting parameters reflects the structural and dynamic properties of the material. 

The dielectric spectroscopy study of conductive samples is very complicated because of the need to take into account the effect of dc-conductivity. The dc-conductivity c>o contributes, in the frequency domain, to the imaginary part of the complex dielectric permittivity in the form of additional function a0/(so ). The presence of dc-conductivity makes it difficult to analyze relaxation processes especially when the contribution of the conductivity is much greater than the amplitude of the process. The correct calculation of the dc-conductivity is important in terms of the subsequent analysis of the dielectric data. Its evaluation... 

Luckily, the real and imaginary parts of the complex dielectric permittivity are not independent of each other and are connected by means of the Kramers-Kronig relations [11]. This is one of the most commonly encountered cases of dispersion relations in linear physical systems. The mathematical technique entering into the Kramers-Kronig relations is the Hilbert transform. Since dc-conductivity enters only the imaginary component of the complex dielectric permittivity the static conductivity can be calculated directly from the data by means of the Hilbert transform. 

The complex dielectric permittivity data of a sample, obtained from DS measurements in a frequency and temperature interval can be organized into the matrix data massive = [s, ] of size M x N, where Eq = ( >, , 7 ), M is the number of measured frequency points, and N is the number of measured temperature points. Let us denote by / =/(co x) the fitting function of n parameters x = x, X2,..., xn. This function is assumed to be a linear superposition of the model descriptions (such as the Havriliak-Negami function or the Jonscher function, considered in Section II.B.l). The dependence of/on temperature T can be considered to be via parameters only / =/(co x(T )). Let us denote by X = [jc,-(7 )] the n x N matrix of n model parameters xt, computed at N different temperature points 7. ... 

The classical approach to the fit parameter estimation problem in dielectric spectroscopy is generally formulated in terms of a minimization problem finding values of X which minimize some discrepancy measure S(, s) between the measured values, collected in the matrix s and the fitted values = [/(co,-, x(7 ))] of the complex dielectric permittivity. The choice of S(e,e) depends on noise statistics [132]. 

Simultaneous fit of both real and imaginary components of the complex dielectric permittivity data... 

Figure 14. The typical three imensional plot of the complex dielectric permittivity real...

Figure 15. Typical temperature dependence (for sample E [156]) of the complex dielectric permittivity of the real part of different frequencies ( 8.65 kHz 32.4 kHz A 71.4 kHz). (Reproduced with permission from Ref. 2. Copyright 2002, Elsevier Science B.V.)...

Three-dimensional plots of both the measured real part s and the imaginary part s" of the complex dielectric permittivity versus frequency and temperature for 20-pm-thickness PS sample are shown in Fig. 17a,b. From the figure, one can identify three distinct processes, marked by I, II, and III, defined as follows ... 

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) ... 

The dielectric relaxation at percolation was analyzed in the time domain since the theoretical relaxation model described above is formulated for the dipole correlation function T(f). For this purpose the complex dielectric permittivity data were expressed in terms of the DCF using (14) and (25). Figure 28 shows typical examples of the DCF, obtained from the frequency dependence of the complex permittivity at the percolation temperature, corresponding to several porous glasses studied recently [153-156]. 

The conducting properties of a liquid in a porous medium can provide information on the pore geometry and the pore surface area [17]. Indeed, both the motion of free carriers and the polarization of the pore interfaces contribute to the total conductivity. Polymer foams are three-dimensional solids with an ultramacropore network, through which ionic species can migrate depending on the network structure. Based on previous works on water-saturated rocks and glasses, we have extracted information about the three-dimensional structure of the freeze-dried foams from the dielectric response. Let be d and the dielectric constant and the conductivity, respectively. Dielectric properties are usually expressed by the frequency-dependent real and imaginary components of the complex dielectric permittivity ... 

In equation (2), C is the capacitance, is the capacitance without the dielectric and 5 is the angle by which current leads the voltage. Nitrobenzene was confined in porous silica (CPG and VYCOR), of pore widths H = 50 nm to 4 nm at 1 atm. pressure. The freezing temperature in the bulk is 5.6 °C (the liquid freezes to a monoclinic crystal). n = n —iK", the complex dielectric permittivity is measured as a function of temperature and frequency. 

In this section we apply to ice Ih, the two-fractional (mixed) molecular model that was briefly described in Section II and applied to water H20 in Sections III and IV. For convenience we give in Appendix IV the list of formulas used in this Section V. We calculate analytically the complex dielectric permittivity s(v) and absorption a(v) of ice in the far-infrared and submillimeter wavelength regions. 

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