Debye dielectric dispersion

Raicu, V. 1999. Dielectric dispersion of biological matter Model combining Debye-type and universal responses. Phys. Rev. E 60 4667-80. 

Fig. 1.1 Dielectric dispersion spectra for a polar solvent with a single Debye relaxation process in the micro-wave region and two resonant transmissions in the IR and UV ranges [5 b].

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. 

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. 

For many of the systems being studied, the relationship above does not sufficiently describe the experimental results. The Debye conjecture is simple and elegant. It enables us to understand the nature of dielectric dispersion. However, for most of the systems being studied, the relationship above does not sufficiently describe the experimental results. The experimental data are better described by nonexponential relaxation laws. This necessitates empirical relationships, which formally take into account the distribution of relaxation times. 

Figure 1 For pure bulk ice samples, (a) Temperature dependence of the dielectric relaxation time r and (b) Cole-Cole plots of pure ice crystal (parallel to the c-axis) at -10 °C. The dielectric dispersion is of the Debye type (a=0.99, p=1.00).

Bulk ice samples show dielectric dispersion of the Debye type. On the other hand, samples consisting of packed ice particles tend to show dielectric dispersion of the Davidson-Cole type. 

In these equations the subscript 1 is used to identify the properties of the matrix, while 2 is used for the particles. These equations are for the special case of a highly insulating matrix of constant dielectric properties containing a small amount of well-dispersed spherical particles that are somewhat conductive. The important aspect of this result is that the particles produce a Debye-like dispersion centered at a frequency of roughly Oj/eq. With a conductivity of, say, 10 7 S/m (Siemens/meter) and 0 = 8.84 x 1CT12 F/m (Farads/meter) the frequency of the MWS dispersion will be around 10 kHz, where it can be easily confused with a dipolar relaxation process. As shown by equation (7-57), the magnitude of the MWS dispersion should increase linearly... 

In short-chain alcohols the dielectric dispersion is described by three relaxation times (three dispersion ranges), the predominant of which is the low-frequency Debye-type process. The experimental results may be interpreted in accordance with various assumptions. 

Figure 3.4 Dielectric (1R-2C) model circuit for a debye single dispersion. No DC conductance.

The polarizations discussed above have different relaxation times, as they are governed by the different physical origins. Figure 7 schematically shows the wide frequency spectrum of the dielectric properties of a heterogeneous system. All polarizations are depicted in Figure 7 on the basis of their relative relaxation times. The dielectric dispersion of the electronic polarization appears at the highest frequency, more than lO FIz. With the polarization entity size increase, the dielectric dispersion peak gradually appears at low frequencies in the sequence of the atomic, Debye, interfacial polarization, and the electrode polarization. The Debye polarization usually appears at 10 Hz, the interfacial polarizations appears around 1000 Hz, and the electrode polarization appears below 100 Hz. The Debye, interfacial, and electrode polarizations arc rather slow processes as compared with the electronic and the atomic polarizations. Usually, the former three... 

In Figure 17 (curve 1), the dielectric loss spectrum for PS at room temperature as taken from Bur [18] is presented. There are no pronounced relaxation loss peaks due to a- and / -processes in this polymer which is considered to be nonpolar , although in fact it possesses a smaU dipole moment due to the asymmetry at the phenyl side group. The loss tangent is seen to be constant and relatively small over a very broad frequency range from subaudio to 10 Hz. A loss peak occurs at vs2 x 10 Hz, a very high frequency for Debye relaxation dispersion. It appears to be the 5-peak which has been n asured by McCammon et al. at 46 K (1 kHz) with an activation energy of 12 kJ mole" ... 

FIGURE 5-3 Parallel (A) and series (B) equivalent circuit versions for a Debye single dispersion in lossy dielectrics. Circuit (C) represents realistic Debye dispersion with lossy relaxation... 

Ya.B. applied formal perturbation theory to the interaction of an atom with the electrons of a metal, where the latter are assumed to be free. Meanwhile, Casimir and Polder and Lifshitz neglected the spatial dispersion of the dielectric permittivity of the metal. Therefore, in the region of small distances, frequencies of order ui0 are important at small distances in the sense indicated above, as are arbitrarily small frequencies at large distances. In both limits the dielectric permittivity of the metal is not at all close to one. Meanwhile, the perturbation theory used by Ya.B. corresponds formally to an expansion in powers of e - 1. and is therefore not applicable in this case. Neglecting the spatial dispersion is valid, however, only at distances r > a (a is the Debye radius in the metal) of the atom from the surface. At the opposite extreme, r a, the wave vectors kj 1/r > a vF/u>0 Me of importance (vF is the electron speed at the Fermi boundary). In this region of strong spatial dispersion perturbation theory can be applied, and the (--dependence satisfies Zeldovich s law. 

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. 

The high frequency limit of for this second process is therefore n. The result of the fit is shown in Table III where the mean values of the various parameters and their associated 95% confidence intervals are given. Considering the small amplitude of the second dispersion both in absolute t rms and in relation to the main dispersion the parameters 6m, n and Y are quite well defined, and therefore it may be concluded that the double Debye representation is an acceptable description of the dielectric behaviour of water up to around 2THz. Other alternative interpretations are clearly possible but no attempt has been made here to follow these up at this stage. What is clear is that a small subsidiary dispersion region in the far infrared is necessary to account for all the presently available permittivity data, and that such a dispersion is centred around 650GHz and has an amplitude of about 2.4 in comparison with that of the principal dispersion which is approximately 75. 

B. Protein Solutions. The dielectric properties of proteins and nucleic acids have been extensively reviewed (10, 11). Protein solutions exhibit three major dispersion ranges. One occurs at RF s and is believed to arise from molecular rotation in the applied electric field. Typical characteristic frequencies range from about 1 to 10 MHz, depending on the protein size. Dipole moments are of the order of 200-500 Debyes and low-frequency increments of dielectric permittivity vary between 1 and 10 units/g protein/100 ml of solution. The high-frequency dielectric permittivity of this dispersion is lower than that of water because of the low dielectric permittivity of the protein leading to a high-frequency decrement of the order of 1 unit/g protein/... 

Equation (11.25) is called the Debye dispersion relation or the Debye equation. The complex dielectric constant is defined to be... 

Relaxations observed in polymers show broader dispersion curves and lower loss maxima than those predicted by the Debye model, and the (s" s ) curve falls inside the semicircle. This led Cole and Cole (1941) to suggest the following semi-empirical equation for dielectric relaxations in polymers ... 

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