Dielectric relaxation complex systems

Frequency dependent complex impedance measurements made over many decades of frequency provide a sensitive and convenient means for monitoring the cure process in thermosets and thermoplastics [1-4]. They are of particular importance for quality control monitoring of cure in complex resin systems because the measurement of dielectric relaxation is one of only a few instrumental techniques available for studying molecular properties in both the liquid and solid states. Furthermore, It is one of the few experimental techniques available for studying the poljfmerization process of going from a monomeric liquid of varying viscosity to a crosslinked. Insoluble, high temperature solid. 

The considered model of a straight line of M nanoparticles illustrates only general features of dielectric losses caused by an M nanoparticle cluster in polymer matrix. Actually such cluster is a complex fractal system. Analysis of dielectric relaxation parameters of this process allowed the determination of fractal properties of the percolation cluster [104], The dielectric response for this process in the time domain can be described by the Kohlrausch-Williams-Watts (KWW) expression... 

In the writing of this chapter we have not sought to cover every aspect of the dielectric relaxation of complex materials. Rather, our aim has been to demonstrate the usefulness of dielectric spectroscopy for such systems, using its application to selected examples as illustrations. 

The dielectric relaxation of many complex systems deviates from the classical exponential Debye pattern (16) and can be described by the... 

The dielectric relaxation properties in a sodium bis(2-ethylhexyl) sulfosuc-cinate (AOT)-water-decane microemulsion near the percolation temperature threshold have been investigated in a broad temperature region [47,143,147]. The dielectric measurements of ionic microemulsions were carried out using the TDS in a time window with a total time interval of 1 ps. It was found that the system exhibits a complex nonexponential relaxation behavior that is strongly temperature-dependent (Figure 8). 

Non-Debye dielectric relaxation in porous systems is another example of the dynamic behavior of complex systems on the mesoscale. The dielectric properties of various complex multiphase systems (borosilicate porous glasses [153-156], sol-gel glasses [157,158], zeolites [159], and porous silicon [160,161]) were studied and analyzed recently in terms of cooperative dynamics. The dielectric response in porous systems will be considered here in detail using two quite different types of materials, namely, porous glasses and porous silicon. 

As mentioned in Section II.B, the dielectric response in the frequency domain for most complex systems cannot be described by a simple Debye expression (17) with a single dielectric relaxation time. In a most general way this dielectric behavior can be described by the phenomenological Havriliak-Negami (HN) formula (21). 

Nandi N, Bhattacharyya K, Bagchi B. Dielectric relaxation and solvation dynamics of water in complex chemical and biological systems. Chem. Rev. 2000 100 2013-2045. 

Figure 1 shows the dielectric relaxation properties of the Tween microemulsions plotted on the complex permittivity plane (from Foster et al ( 1). The mean relaxation frequency (corresponding to the peak of each semicircle) decreases gradually from 20 GHz for pure water at 25°C to ca. 2 GHz for a concentrated microemulsion containing 20% water. Since the permittivity of the suspended oil/ emulsifier is 6 or less at frequencies above 1 GHz, this relaxation principally arises from the dipolar relaxation of the water in the system. Therefore, the data shown in Figure 1 clearly show that the dielectric relaxation times of the water in the microemulsions are slower on the average than those of the pure liquid. The depressed semicircles indicate a distribution of relaxation times (9), and were analyzed assuming the presence of two water components (free and hydration) in our previous studies. 

The principal result of our calculation is that the Debye theory (based on the Smoluchowski equation), when extended to fractional dynamics via a onedimensional noninertial fractional Fourier-Planck equation in configuration space, can explain the Cole-Cole anomalous dielectric relaxation that appears in some complex systems and disordered materials. A further result of our calculation is that the aftereffect solution [Eq. (66)] is, with slight modifications, the moment generating function of the configuration space distribution function. Hence the mean-square angular displacement of a dipole, and so on, may be easily calculated by differentiation. We must remark, however, that the fractional Debye theory can be used only at low frequencies (got < 1) just as... 

Thus we have demonstrated how the empirical Havriliak-Negami equation [Eq. (11)] can be obtained from a microscopic model, namely, the fractional Fokker-Planck equation [Eq. (101)] applied to noninteracting rotators. This model can explain the anomalous relaxation of complex dipolar systems, where the anomalous exponents ct and v differ from unity (corresponding to the classical Debye theory of dielectric relaxation) that is, the relaxation process is characterized by a broad distribution of relaxation times. Hence, the empirical Havriliak-Negami equation of anomalous dielectric relaxation which has been... 

Relaxation functions for fractal random walks are fundamental in the kinetics of complex systems such as liquid crystals, amorphous semiconductors and polymers, glass forming liquids, and so on [73]. Relaxation in these systems may deviate considerably from the exponential (Debye) pattern. An important task in dielectric relaxation of complex systems is to extend [74,75] the Debye theory of relaxation of polar molecules to fractional dynamics, so that empirical decay functions for example, the stretched exponential of Williams and Watts [76] may be justified in terms of continuous-time random walks. 

Polystyrene and polybutadiene homopolymers as well as random and block copolymers of these mers have been studied via dielectric relaxation spectroscopy and tensile stress-strain measurements. The results suggest that some block copolymer systems studied have styrene rich surfaces which appear to partially crosslink upon initial exposure to ozone even though surface oxygen concentrations are not significantly affected. After continued exposure these samples appear to then undergo chain scission. Complex plane analysis implies that after degradation... 

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. 

Obviously, the dieleetrie behavior of systems with ion pairing is much more complex than those without it. The presence of the ion pair gives the solution a higher permittivity at lower frequencies than it otherwise would have. This feature is important in understanding the equilibrium properties of these solutions. The permittivity data for the low-frequency process may be used to determine the ion pairing equilibrium constant and the rate constants for formation and breakup of this species. Thus, dielectric relaxation experiments in electrolytes provide valuable information about ion association equilibria. 

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