Dielectric relaxation frequency dependence

The dramatic slowing down of molecular motions is seen explicitly in a vast area of different probes of liquid local structures. Slow motion is evident in viscosity, dielectric relaxation, frequency-dependent ionic conductance, and in the speed of crystallization itself. In all cases, the temperature dependence of the generic relaxation time obeys to a reasonable, but not perfect, approximation the empirical Vogel-Fulcher law ... 

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... 

The observation of complex dielectric constant frequency dependence (e = s — ie") shows that, at low frequencies, the different polarisations contribute to a high permittivity c value, beyond that, each kind of polarisation will create one resonance or one relaxation process e decrease and a maximum appears for e . 

In a manner similar to the dielectric constant, frequency-dependent Cp co) is defined as a dynamic susceptibility. Under equilibrium conditions, the heat that the system can adsorb from its surroundings during a AT change isq = H = CpAT, that is, the change in enthalpy per volume H. If the system contains (t) relaxing degrees of freedom after a T change, H = H t). For a time-dependent T variation, AT t) in a time interval 0 < < t is... 

At frequencies above the dielectric relaxation frequency, another diffraction with a threshold makes its appearance (see Fig. 4). This is the chevron regime, which is not hydrodynamic in origin but depends on interaction between the anisotropic dielectric properties of the fluid and the field. It represents a sinusoidal modulation of the... 

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) ... 

Relation [1] Is the frequency-dependent analogue of a formula proposed by Chasset and Thirion (2, 3) which has since been applied very frequently to relaxation measurements on cured rubbers. The next three equations are Inspired by similar relations In dielectrics (they are not derived from these) Equation [2] by the Cole-Cole and Equation [3] by the Davidson-Cole relation (15, 16). Both are special cases of the most general Equation [4] which contains five parameters (17). 

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. 

If the structure of water depends on distance from a surface, so must its physical properties, including its dielectric function. We noted in Section 9.5 that at microwave frequencies the dielectric function of water changes markedly when the molecules are immobilized upon freezing as a consequence, the relaxation frequency of ice is much less than that of liquid water. Water irrotationally bound to surfaces is therefore expected to have a relaxation frequency between that of water and ice. 

Dielectric relaxation study is a powerful technique for obtaining molecular dipolar relaxation as a function of temperature and frequency. By studying the relaxation spectra, the intermolecular cooperative motion and hindered dipolar rotation can be deduced. Due to the presence of an electric field, the composites undergo ionic, interfacial, and dipole polarization, and this polarization mechanism largely depends on the time scales and length scales. As a result, this technique allowed us to shed light on the dynamics of the macromolecular chains of the rubber matrix. The temperature as well as the frequency window can also be varied over a wide... 

This implies that the electronegativity difference between nitrogen and the metal decreases in the series leading to a decreased extent of dielectric polarization as actually observed. The frequency dependence of the tan 8 values in these complexes is marked. Evidently, the metal complexes possess a rigid structure where dipoles do not find sufficient time to reorient with the direction of applied frequency of alteration resulting thereby in a broad dielectric relaxation. 

While it is not clear how the constant frequency low field dielectric relaxation measurements mentioned above should be applied to reactions in liquids, save for a complete time-dependent theory of liquids, these effects are very significant. At short times (<10ps) the effective Onsager distance may be 20 nm, even in methanol or ethanol, but over the next two or three decades of time reduce to more nearly 2 nm. Such a change can reduce the rate of reaction much more rapidly than that which occurs by decay of the transient time dependence discussed in the previous sub-section. 

Hydrated Zeolites. Figure 3 gives a typical plot of the conductivity vs. the reciprocal temperature for hydrated NaF86.5. The other samples behave qualitatively in the same way. Conduction and dielectric absorption are superposed. The position of the maximum of dielectric absorption is frequency dependent it shifts to higher temperatures with increasing frequency. In some favorable cases a second conduction phenomenon is observed on the low temperature side of the relaxation phenomenon (Figure 3). Because of a lack of reproducibility we cannot interpret it. 

The dielectric tensor e in a viscoelastic medium is a function of the frequency at which it is measured. It can be represented in terms of a real and imaginary part e (co) = e (co) -ie"(a>). If the frequency dependence of e is determined by a single relaxation time, then the relationship between e and r is... 

In order to compare frequencies of maximum loss to l/( r), the frequency dependent susceptibility corresponding to the WW function was calculated as a function of / . The values of ra>max and r/( z ) are listed as a function of P in Table 1. The frequency of maximum loss correlates much better with the parameter r and depends only weakly on p. The exact relationship presented in the Table should allow data obtained from PCS to be compared properly to dielectric or mechanical relaxation data. 

In addition to knowing the temperature shift factors, it is also necessary to know the actual value of ( t ) at some temperature. Dielectric relaxation studies often have the advantage that a frequency of maximum loss can be determined for both the primary and secondary process at the same temperature because e" can be measured over at least 10 decades. For PEMA there is not enough dielectric relaxation strength associated with the a process and the fi process has a maximum too near in frequency to accurately resolve both processes. Only a very broad peak is observed near Tg. Studies of the frequency dependence of the shear modulus in the rubbery state could be carried out, but there... 

The concentration dependence of ionic mobility at high ion concentrations and also in the melt is still an unsolved problem. A mode coupling theory of ionic mobility has recently been derived which is applicable only to low concentrations [18]. In this latter theory, the solvent was replaced by a dielectric continuum and only the ions were explicitly considered. It was shown that one can describe ion atmosphere relaxation in terms of charge density relaxation and the elctrophoretic effect in terms of charge current density relaxation. This theory could explain not only the concentration dependence of ionic conductivity but also the frequency dependence of conductivity, such as the well-known Debye-Falkenhagen effect [18]. However, because the theory does not treat the solvent molecules explicitly, the detailed coupling between the ion and solvent molecules have not been taken into account. The limitation of this approach is most evident in the calculation of the viscosity. The MCT theory is found to be valid only to very low values of the concentration. 

In order to assess the orientational stability of the poled state, the temperature dependence of the dipole mobility of the side groups was examined through dielectric relaxation measurements. (13) No low temperature relaxation below Tg was observed in the frequency range studied (100 Hz-100 kHz). In addition, the dielectric constant was approximately equal to the square of the refractive index, indicating that below T only electronic and no significant orientational contributions to the dielectric displacement are present. Thus, it was expected that a given orientational state of the ensemble would be stable at temperatures significantly below Tg. 

The shape of the frequency dependence of e" has been compared in Fig. 109 in terms of reduced units s / max an(i ///max> at various temperatures. The peak is asymmetric, being broader on the high-frequency side, especially at 10 °C. A gradual narrowing occurs on both the high- and low-frequency sides with increasing temperature. These results show that the motional processes involved in the dielectric j3 relaxation have a distribution of correlation times and that this distribution becomes narrower as temperature increases. 

The absorption of ultrasonic energy is also influenced by relaxation effects. At the frequencies of near 100 MHz that are employed, the relaxation times are of the order of ns, rather than the ps for dielectric relaxation. The relevant quantity is the absorption coefficient, a, divided by the square of the frequency, f2. Values of a//2 in 10-15 s2 nr1 have been measured for many solvents near 25 °C at the frequency of 104 to 107 MHz (Heasall and Lamb 1956 Krebs and Lamb 1958) and are shown in Table 3.10, being considered accurate within 2%. For a few solvents the ratio a//2 depends strongly on the frequency as it decreases somewhat for all solvents, e g., for carbon disulfide a//2/ (10-15 s2 nr1) = 2068 at 104 MHz and 776 at 189 MHz and for dichloromethane it decreases from 779 at 107 MHz to 550 at 193 MHz... 

Obviously in the limit of low and high frequencies the parameters of the two models determine the frequency dependence of the dielectric loss in the a relaxation [162,163],... 

In the temperature interval of —70 to 0°C and in the low-frequency range, an unexpected dielectric relaxation process for polymers is detected. This process is observed clearly in the sample PPX with metal Cu nanoparticles. In sample PPX + Zn only traces of this process can be observed, and in the PPX + PbS as well as in pure PPX matrix the process completely vanishes. The amplitude of this process essentially decreases, when the frequency increases, and the maximum of dielectric losses have almost no temperature dependence [104]. This is a typical dielectric response for percolation behavior [105]. This process may relate to electron transfer between the metal nanoparticles through the polymer matrix. Data on electrical conductivity of metal containing PPX films (see above) show that at metal concentrations higher than 5 vol.% there is an essential probability for electron transfer from one particle to another and thus such particles become involved in the percolation process. The minor appearance of this peak in PPX + Zn can be explained by oxidation of Zn nanoparticles. 

The full lines indicate the approximate temperature dependence of the a, / , and y relaxation frequencies determined by dynamic mechanical and dielectric measurements... 

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