Dielectric complex permittivity

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).

The general equation for complex dielectric permittivity is then given by Eq. (11) ... 

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. 

It would be important to find analogous mechanism also for description of the main (librational) absorption band in water. After that it would be interesting to calculate for such molecular structures the spectral junction complex dielectric permittivity in terms of the ACF method. If this attempt will be successful, a new level of a nonheuristic molecular modeling of water and, generally, of aqueous media could be accomplished. We hope to convincingly demonstrate in the future that even a drastically simplified local-order structure of water could constitute a basis for a satisfactory description of the wideband spectra of water in terms of an analytical theory. 

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").

When applying an alternating electric field to a polymer placed between two electrodes, the response is generally attenuated and the output current is out of phase compared with the input voltage. This response stems from the polymer s capacitive component and its conductive or loss component, as represented by a complex dielectric permittivity measured frequencies f, and temperatures T ... 

It follows from Maxwell s equations that three material parameters are needed to describe interactions between electromagnetic waves and the medium complex dielectric permittivity e s + je", electrical conductivity... 

The dispersion of the dielectric response of each contribution leads to dielectric losses of the matter which can be mathematically expressed by a complex dielectric permittivity ... 

A frequency dependence of complex dielectric permittivity of polar polymer reveals two sets or two branches of relaxation processes (Adachi and Kotaka 1993), which correspond to the two branches of conformational relaxation, described in Section 4.2.4. The available empirical data on the molecular-weight dependencies are consistent with formulae (4.41) and (4.42). It was revealed for undiluted polyisoprene and poly(d, /-lactic acid) that the terminal (slow) dielectric relaxation time depends strongly on molecular weight of polymers (Adachi and Kotaka 1993 Ren et al. 2003). Two relaxation branches were discovered for i.s-polyisoprene melts in experiments by Imanishi et al. (1988) and Fodor and Hill (1994). The fast relaxation times do not depend on the length of the macromolecule, while the slow relaxation times do. For the latter, Imanishi et al. (1988) have found... 

Much of the unnecessary failure to use the easier, modern theory of van der Waals forces comes from its language, the uncommon form in which the dielectric permittivity is employed. For many people "complex dielectric permittivity" and "imaginary frequency" are terms in a strange language. Dielectric permittivity describes what a material does when exposed to an electric field. An imaginary-frequency field is one that varies exponentially versus time rather than as oscillatory sinusoidal waves. 

Figure 5. Real s (a) and imaginary part e (b) of complex dielectric permittivity against frequency / of the nanocomposites indicated on the plot at ambient temperature and relative humidity conditions.

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]. 

Advanced data visualization and preprocessing tools for displaying complex dielectric permittivity data and selecting appropriate frequency and temperature intervals for modeling... 

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 ... 

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