Real permittivity

Figure 1.5. Relative real permittivity (e ) and loss factor (e") depending on angular frequency of electromagnetic waves. Reprinted with the permission from [1].

The values of real permittivities (e ) and loss factors (e") for some common organic solvents are given in Table 1.2. 

The mechanisms that cause polarization depend on the frequency of the applied electric field and the composition of the material. Real permittivity increases as frequency decreases since polarizations accumulate (see Figures 1 and 2). Single-phase materials may experience three types of polarization electronic (r 10-16s), ionic (r 10-13s), and orientational (r =variable). Non-polar materials do not experience orientational polarization and have very... 

The electromagnetic fields of the right- and left-propagating polaritons, respectively, follow the wave equations with the speeds and damping rates of the different frequency components dispersed according to the frequency- and wavevector-dependent complex refractive index n = v/e(k, oj). A typical example of the dispersion of these modes is shown in Fig. 1 for the case of a real permittivity e. The term Ao(r,t) represents the envelope of the wavepacket on the phonon-polariton coordinate A. Note that this phonon-polariton coordinate is a linear combination of ionic and electromagnetic displacements, which both contribute to the polarization... 

Figure 2. BDS spectrum of the unfilled elastomer at 40°C. Solid squares and circles represent the experimental and resolved real permittivity components respectively. Hollow squares, circles and up-triangles represent the experimental complex imaginary permittivity, the resolved component of the imaginary permittivity and the resolved dc conductivity contribution.

Figure 3. BDS spectra of nanocomposite systems at 40°C. Solid symbols represent real permittivity and hollow symbols represent imaginary permittivity. Squares, circles, up-triangles, down-triangles and diamonds represent levels of 0, 2, 4 and 8% Cloisite respectively.

To obtain the best possible fit, a full regression analysis will be necessary, and results of this are not yet available. The outstanding question to be answered is whether the broad absorption is to be regarded as a relaxation process, or whether it is a resonance process, a molecular libration (partial rotation within a potential well) such as is responsible for submillimetre-wavelength absorption bands in many liquids it is possible that both processes contribute. A resonance process would produce an effect in the real permittivity similar to an anomalous dispersion a relaxation process merely produces a monotonic fall in the real permittivity. 

In passing it is noted that we met complex frequencies before in the Lifshits theory for dispersion forces. There, an integration of complex permittivities over real frequencies was replaced by an integration of real permittivities over imaginary frequencies. See [14.7.7]. This is just a mathematical device. 

Dielectric relaxation (DR) experiments measure the collective polarization response of all the polar molecules present in a given system. The DR time provides a measure of the time taken by a system to reach the final (equilibrium) polarization after an external field is suddenly switched on (or off). DR measures the complex dielectric fimction, s(w), that can be decomposed into real and imaginary parts as efca) = s (o) — is" (o) where s (co) and s fo ) are the real (permittivity factor) and imaginary (dielectric loss) parts, respectively. The total dipole moment of the system, at any given time t, M(t) = fift) where N is the total number of dipolar molecules and /Af is the dipole moment vector of the ith molecule. The complex dielectric function e((w) is given by the following relation. 

The microwave response both of polar solvents and electrolyte solutions is usually represented with the help of its frequency-dependent complex relative permittivity, s(co) = 8 ((o) -f je"(co), cf. Ref. The characteristic parameters of such investigations are the relaxation times or relaxation time distributions of molecular processes and the extrapolated real permittivities of zero (Eq) and inifinite (e ) frequencies of one or more relaxation regions. 

Figure 16 shows the s , a and AT traces for fructose when d.c. conductivity is contributing in the frequency range between 10 to 10 Hz, as can be seen by the slope in the log e plot being very close to unity, the invariance in s and the Debye-type peak in the modulus representation. At the lowest frequencies, the macroscopic effect of electrode polarization affects both s (lower slope) and s (large increase of the real permittivity component) vanishing in the modulus. 

Figure 17. Evidence of the Maxwell-Wagner-Sillars effect in the real permittivity of the composite system nematic E7 dispersed over hydroxypropylcellulose-type matrix. The interfacial polarization can be described by a double-layer arrangement. At lower frequencies and higher temperatures, the real permittivity increases further due to electrode polarization.

Figure 20. Left real permittivity (full symbols) and loss factor (open symbols) obtained in a P-PVDF film at different fiequencies (see labels inside the graphics) (data adapted from [170]). Right Dynamic mechanical spectra of a p-PVDF film obtained at a frequency of 1 Hz, in tensile mode, performed along ftie longitudinal (solid lines) and transverse (dashed lines) directions, with respect to the stretch direction used to process the film (data adapted from [171]).

The actual losses can be computed by normalization of these losses with storage terms [i.e., by ratio of dielectric losses/imaginary permittivity is") with dielectric constant/real permittivity ( )] to quantity loss tangent (tan 8). In case of in-situ formed PANI/MWCNT nanocomposites, improvement of dielectric properties leads to high value of loss tangent (Figure 9.23b) which further increases with increase in MWCNT loading. 

Figure 9.29 Variation of (a) SE and SE, (b) real permittivity (e ), (c) imaginary permittivity (e ), and (d) loss tangent (tan 6) with frequency for PANl and its TBT-based NCs. Reprinted from Ref [6] with permission from RSC.

Where eo and Soo are the limiting low and high frequency real permittivities, and indicates a one-sided Fourier transformation. A simple form for (.t) is the single exponential decay function... 

Relaxation profiles obtained as loss peak in imaginary dielectric curve and as step jump of real permittivity line Partial miscibility might exhibit multiple relaxation peaks overlaying or towards individual components, or display diverse relaxation rate even >T, Non-isothermal crystallization by isochronal temperature sweep DSC (cold crystallization and melting discernible as sudden drop and steep rise of static dielectric permittivity, respectively) Crystallization onset predicted using DRS relaxation time in coupUng model faster than the experimental time... 

The real permittivity, at any temperature above of each polymer, as well as the intensity of the dielectric loss peaks observed for the main a-relaxation has been reported to be always greater for poly(AN-co-ATRIF) copolymer than for... 

Figure 20.16 displays isochronal scans temperature at fixed frequencies (103 Hz, 1.09 kHz, 11.6 kHz, and 123 kHz) of the real permittivity e (Figure 20.16a) and the dissipation factor tan5 (Figure 20.16b) for P4FST homopolymer. 

FIGURE 20.21 Temperature dependency of real permittivity e (T) and tanS collected for poly(MATRIF) homopolymer (a) and (b) poly(VCN-a/f-MATRIF) copolymer (c) and (d) [121],... 

For the sake of brevity, henceforth, e., e. and tan 5 will, in general, be referred to as the real permittivity, the imaginary permittivity and the loss, respectively. In the following discussion, the real permittivity will first be considered, before moving on to consider the imaginary permittivity/dielectric loss. 

In the case of a multi-component system such as a nanodielectric, an effective medium approach can be used to define the real permittivity of the whole based upon the composition and the properties of the individual components Myroshnychenko and Brosseau (2005) provide a good overview of the topic, as applied to binary systems. The Lichtenecker-Rother equation is an example of such a relationship ... 

Kochetov et al. (2012) described the dielectric response of a range of nanodielectrics based upon particulate nanofillers dispersed within an epoxy matrix. In all cases, with the exception of nano-silica, the inclusion of a low volume fraction (<5 %) of nanofiller resulted in a reduction in the measured real permittivity, below that of the host matrix, despite S In another epoxy-based... 

Applying such concepts to the real permittivity data described above, it is evident that this inequahty does not hold. This suggests that the situation in nanodielectrics is more complex, which implies that the presence of the nanoparticles is affecting the overall performance of the system in a much more subtle and complex way. 

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