Dispersion Debye

There also exist dispersion, or London-van der Waals forces that molecules exert towards each other. These forces are usually attractive in nature and result from the orientation of dipoles, and may be dipole-dipole (Keesom dispersion forces), dipole-induced dipole (Debye dispersion forces), or induced dipole-induced dipole... 

Phonons At least two phonon branches are involved in the observed absorption the acoustic phonons and the optical 46-cm "1 branch. Our model includes a single acoustic branch [with cutoff frequency f2max, and isotropic Debye dispersion hfiac q) = hQmaxq/qmax] and an optical dispersionless branch (Einstein s model, with frequency /20p). 

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

For an isolated dipole rotating under an oscillating field in a viscous medium, the Debye dispersion relation is derived in terms of classical mechanics. 

The dielectric constant is a natural choice of order parameter to study freezing of dipolar liquids, because of the large change in the orientational polarizability between the liquid and solid phases. The dielectric relaxation time was calculated by fitting the dispersion spectrum of the complex permittivity near resonance to the Debye model of orientational relaxation. In the Debye dispersion relation (equation (3)), ij is the frequency of the applied potential and t is the orientational (rotational) relaxation time of a dipolar molecule. The subscript s refers to static permittivity (low frequency limit, when the dipoles have sufficient time to be in phase with the applied field). The subscript oo refers to the optical permittivity (high frequency limit) and is a measure of the induced component of the permittivity. 

Figure 2. Spectrum plot for nitrobenzene in a 7.5 nm pore at T = —4 °C this plot yields two distinct dielectric absorption regions. The solid and the dashed curves are fits to the Debye dispersion relation.

The dielectric behavior described by equation (7-28) is known as the Debye dispersion.2 Note the key assumption of first-order relaxation in equation (7-21). 

First we consider in detail only the above-mentioned dispersion ranges (a)-(c), while the range (d) comprising the minimum-loss point and the Debye-dispersion... 

When referring to a Debye dispersion, it should be kept in mind that Debye s theory applies exclusively to rotational polarization of molecules with permanent dipoles, without net charge transfer (33, 90). The occurrence of an appreciable ionic transfer complicates the picture, as in ice doped with ionic impurities. A further complicating factor is aging. Steinemann observed that thin crystals (0.1 to 0.2 cm.) showed a decrease of both the low-frequency dielectric constant and the low-frequency conductivity, suggesting difiusion of impurity (HF) out of the sample into the electrodes. Thicker samples (of the order of 1 cm.) were not affected. 

At lower frequencies (Debye dispersion region), charge transport is limited by the minority carriers which form a bottleneck/ Applied field and current are no longer in phase this results in a polarization and rise of the dielectric constant from its high-frequency value. The conductances of the different defect types act as though in series. The minority carrier determines the over-all conductivity. 

Figure 18, Frequency dependence of the a-c conductivity and of the dielectric constant after Steinemann (140), (1) Pure ice, (2) Slightly impure ice, (a) Conductivity, (b) Dielectric constant. Curves for pure ice closely follow Equations 12a and 14, except for an incipient low-frequency dispersion that may result from very slight impurity content or from electroae polarization. Debye dispersion between 10 and 10 cps. As the impurity content increases (curves 2), the low-frequency dispersion (Steinemann s F dispersion) becomes more prominent and tends to coalesce with the Debye dispersion. Interpretation then becomes difficult. At still higher concentrations, the two dispersions separate again (see Ref. 140). A slight anisotropy of the dielectric constant, observed by Decroly et al. (34) for measurements parallel and perpendicular to the c axis of single crystals, has not been considered...

Mechanical Relaxation. Mechanical relaxation in pure and doped ice was investigated experimentally (136) and theoretically (4). Relaxation occurs in the frequency range of the dielectric Debye dispersion, but the variation in the measured quantity (logarithmic decrement) is only about 1/100 of that for dielectric measurements, The relaxation time, T, is given by... 

Fig. 9.3. The real part e and imaginary part e" of the complex dielectric constant near a region of Debye dispersion.

The expression for e given on the RHS of equation (9.27) is called the Debye dispersion relation. Writing the dielectric constant s = s — is", it follows immediately from equation (9.27) that... 

The subscripts refer to frequency, a sine wave parameter. Doo is the surface charge density at t = 0+, which is after the step but so early that only apparently instantaneous polarization mechanisms have come to effect (high frequency e.g., electronic polarization). The capacitor charging current value at t = 0 is infinite, so the model has some physical flaws. Do is the charge density after so long time that the new equilibrium has been obtained and the charging current has become zero. With a single Debye dispersion, this low-frequency value is called the static value (see Section 6.2.1). t is the exponential time constant of the relaxation process. 

Both values converge, at high frequencies with values smaller than at low frequencies. Thus with the parallel model of the two slabs in series, we have a classical Debye dispersion, with a capacitive decrement AC or Ae. This is without postulating anything about dipole relaxation in the dielectric. Dehye dispersion appears and is modeled by two capacitors and two resistors, or even with two capacitors and one resistor (one layer without conductivity) as shown in Section 12.2. If the components are ideal (frequency independent), the dispersion will be characterized by one single relaxation time constant. 

Hydrogen ions in the form of protons or oxonium ions contribute to the DC conductivity of aqueous solutions by migration and hopping. Pure water exhibits a single Debye dispersion with a characteristic frequency of approximately 17 GHz (Figure 4.2). 

Whole blood exhibits P-, y-, and 6-dispersion, but curiously enough it exhibits no a-dispersion (Foster and Schwan, 1989). The 3-dispersion has a dielectric increment of approximately 2000 centered at approximately 3 MHz (hematocrit 40%). Erythrocytes in suspension have a frequency-independent membrane capacitance with very low losses (Schwan, 1957). The impedance of lysed erjrthrocytes in suspension shows two clearly separated single relaxation frequencies (Debye dispersions). The a-dispersion is in the lower kilohertz range, and the p-dispersion is in the lower MHz range (Schwan, 1957 Pauly and Schwan, 1966). 

The permittivity locus of a Debye dispersion in the Wessel diagram is a complete half circle with the center on the real axis. Figure 9.8(a). An ideal resistor in parallel destroys the circle at low frequencies, upper right (see Figure 9.8(b)). The conductivity locus is equally sensible for an ideal capacitor in parallel at high frequencies. Figure 9.8(d) lower right. 

The imaginary part describes dissipation of energy due to molecular friction. It is called dielectric losses and equivalent to appearance of non-Ohmic electric conductivity. The frequency dependence of s can be written in the form of the Debye dispersion law [10]... 

NEWTON - Even if one adopts the concept of Tj based on a simple Debye dispersion model, one must recognize the ambiguity in evaluating associated with the choice of a value for (infrared or optical ). Furthermore, Wolynes has recently argued for the likely occurrence of an additional relaxation time of magnitude larger than T,, arizing from the molecularity of the solvent in the immediate vicinity of the ion. 

The Maxwell-Wagner dispersion effect due to conductance in parallel with capacitance for two ideal dielectric materials in series Rj Cj - Rj Cj can also be represented by Debye dispersion without postulating anything about dipole relaxation in dielectric. In the ideal case of zero conductivity for both dielectrics (R, — , R —> ), there is no charging of the interfaces from free charge carriers, and the relaxation can be modeled by a single capacitive relaxation-time constant. 

The above dielectric (or complex capacitance) notation and Debye dispersion (Eq. 1-15) have often been used to describe a single bulk-media dielectric relaxation process in organic and polymeric (lossy) systems where at least two components with resistive and capacitive features exist [9, p. 33]. The permittivity of a lossy dielectric with negligible parallel DC conductance can be expressed on the basis of the Havriliak-Negami model (Eq. 1-16). Equivalent circuits representing a Debye model for lossy dielectric, where C, g = C g e, Cg - C j, R 1 /G [1, p. 65, p. 216], are shown in Figure 5-3. 

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... 

The systems represented by the classical Debye dispersion model at constant temperature exist as multicomponent systems. So-called "electrorheolog-ical fluids" (Section 12-3) represent examples of such a system. In many other "real-life" systems the Debye capacitive transition is often not observed due to the presence of measurable ionic conduction (= 1/Rjx-) parallel with high-frequency bulk capacitance and resulting "lossy relaxation" with time constant T = Even in highly resistive "unsupported" systems there is... 

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