Dielectric dispersion equation

It is known that the complex dielectric constant of the resonator materials follows the dielectric dispersion equation, which is expressed as the superposition of electronic and ionic polarization ... 

The dielectric dispersion for some solvents is poorly modeled by a multiple Debye form. Alternative, e(cu) distributions such as the Davidson-Cole equation or the Cole-Cole equation are often more appropriate. 

Thcse equations require that the dielectric constant decrease from the static to the optical dielectric constant with increasing frequency, while the dielectric loss changes from zero to a maximum value f" and back to zero. These changes are the phenomenon of anomalous dielectric dispersion. From the above equations, it follows that... 

Given that there had been questions about the validity of Equation (15.8), Tsurumi et al. [36] analyzed their pmn dielectric dispersion data in terms of a theory due to Ngai and White [37], The starting point for this theory is the universal dielectric function... 

The dielectric behaviour of pure water has been the subject of study in numerous laboratories over the past fifty years. As a result there is a good understanding of how the complex permittivity t = E — varies with frequency from DC up to a few tens of GHz and it is generally agreed that the dielectric dispersion in this range can be represented either by the Debye equation or by some function involving a small distribution of relaxation times. 

The pairs (4.8.30 and 31) and [4.8.32 and 33) illustrate that conductivity and dielectric dispersion essentially give the same information. The equations are... 

In cases where the individual relaxation mechanisms are closely related to each other the corresponding overall dielectric dispersion can be quantitatively fairly well represented by the Cole-Cole empirical equation ... 

This result has been confirmed by a correlation-function treatment/ For sufficiently fast rotational relaxation (i.e. Tch tj) we apparently obtmn Ta K T so that dielectric dispersion is controlled by Tr only. Equation (71) then becomes the dassical expression of (31) and (32). A measurable dielectric increment controlled by chemical relaxation may, however, be detected if Tch S tt- Its absolute magnitude wUl be proportional to... 

The dielectric relaxation processes of matter can be analyzed with an empirical model of dielectric dispersion, for example, the one described by Havriliak-Negami s equation. " We analyzed dielectric data obtained for our samples using a model of complex permittivity k with two dispersions (the main and the low-frequency dispersion of a space charge effect) and conductivity ao (caused by electrode discharge), as follows ... 

The last equation demonstrates that the starting point for the solution of the problem is the calculation of ci(double layer (this makes low-frequency dielectric dispersion [LFDD] measurements a most valuable electrokinetic technique). Probably, the first theoretical treatment is the one due to Schwarz [61], who considered only surface diffusion of counterions (it is the so-called surface diffusion model). In fact, the model is inconsistent with any explanation of dielectric dispersion based on double-layer polarization. The generalization of the theory of diffuse atmosphere polarization to the case of alternating external fields and its application to the explanation of LFDD were first achieved by Dukhin and Shilov [20]. A full numerical approach to the LFDD in suspensions is due to DeLacey and White [60], and comparison with this numerical model allowed to show that the thin double-layer approximations [20,62,63] worked reasonably well in a wider than expected range of values of both and ku [64]. Figure 3.12 is an example of the calculation of As. From this it will be clear that (i) at low frequencies As can be very high and (ii) the relaxation of the dielectric constant takes place in the few-kHz frequency range, in accordance with Equations (3.56) and (3.57). 

VAN KAMPEN et al. [5.49] proposed that the interaction forces for macroscopic media could be calculated by considering only the surface-mode solutions of Maxwell s equations at all interfaces. This method was extended [5.44,45] and is used in various applications. For the original case of two half-spaces separated by a gap d, the solutions are sought to the equations of electrostatics subject to the conditions V - D = 0 and vxE = 0 with no spacial dielectric dispersion, but with boundary conditions at the interfaces. By matching boundary conditions and requiring vanishing solutions at infinity, the dispersion relation... 

The most successful temperature dependence for the viscous flow [1,2], viscoelastic response [1], dielectric dispersion [3-5], nuclear magnetic resonance response [6-8] and dynamic light scattering [9-10] of polymers and supercooled liquids with various chemical stmctures is the Williams, Landel, and Perry (WLF) equation [11,12]... 

Dielectric dispersion of the solution was measured by using a transformer bridge, Ando Denki TR-IC model in the frequency range from 30 to 3M Hz. A platinum concentric cylinder cell was used in a thermostatted bath. The cell was calibrated with benzene. The concentration of the solution was 5 X 10 g ml to avoid the intermolecular interaction. The dipole moment of PBLG molecules was estimated by the Applequist-Mahr equation from the results of dielectric dispersion measurements. 

The results of these comparisons show the advantages of incorporating the viscoelastic properties of the environment in predicting dielectric behavior. In this approach, the broadening of dielectric dispersions is due to the shape of the viscoelastic relaxation process. Equations (25) and (79) have at most only 1 adjustable parameter that is related to the volume or shape of the moving segment. This, of course, is a significant achievement, but the question now is why are viscoelastic dispersions shaped the way they are. 

Alternating orientation polarization is feasible at frequencies ranging from v = 0 (steady field) to about ir = 10 Hz (lowest IR frequency). At higher frequencies, the dipoles can no longer follow the variations of an external alternating electric field, and here only electron distortion polarization is possible. In the frequency range from 0 to 10 Hz, s decreases monotonically from the highest value s = Ss at v = 0 (static dielectric constant) to the lowest value s = s (dielectric dispersion) at the upper limit of v. On the other hand, s" equals zero at v = 0 and at far IR frequencies. However, at intermediate frequencies. Equation 1.7 predicts a... 

Rigorous treatment of the self-action problem needs the transformation of Eq.(2.1), (2.5) into a system of integro-differential equations. However, if just some orders of group velocity dispersion and nonlinearity are taken into account, an approximate approach can be used based on differential equations solution. When dealing with the ID-i-T problem of optical pulse propagation in a dielectric waveguide, one comes to the wave equation with up to the third order GVD terms taken into account ... 

An alternative approach that was used in the past was to treat the photoelectrochemical cell as a single RC element and to interpret the frequency dispersion of the "capacitance" as indicative of a frequency dispersion of the dielectric constant. (5) In its simplest form the frequency dispersion obeys the Debye equation. (6) It can be shown that in this simple form the two approaches are formally equivalent (7) and the difference resides in the physical interpretation of modes of charge accumulation, their relaxation time, and the mechanism for dielectric relaxations. This ambiguity is not unique to liquid junction cells but extends to solid junctions where microscopic mechanisms for the dielectric relaxation such as the presence of deep traps were assumed. 

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