Debye single dispersion

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

Figure 3.5 Debye single dispersion relaxation, relative permittivity. Values are found in the text.

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 area per molecule as which appears in the preceding equation is evaluated at a distance d from the surface in contact with water and is curvature dependent. Expressions for ag are given by eq 7.15. The distance <5 is estimated as the distance from the surface in contact with water to the surface where the center of the counterion is located. k is the reciprocal Debye length, na is the number of counterions in solution per cubic centimeter, Cuon is the molar concentration of the singly dispersed ionic surfactant molecules in water, Cadd is the molar concentration of the salt added to the surfactant solution, andA Av is Avogadro s number. The last term in the right-hand side of eq 7.17 provides a curvature correction to the ionic interaction energy. For normal droplets, Cg = 2/(7 w + d) for reverse droplets, Cg = —2/(7 w — <5) and for flat layers, Cg = 0. 

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].

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. 

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). 

For certain types of gas-surface interactions, it may be useful to view the interaction as between the gas atom and a single surface atom. Weak attractive interaction between a pair of atoms can be due to dispersion forces (London [14, 15]) that represent the interaction of induced fluctuating charge distributions. In addition, molecules that possess permanent dipoles can further polarize each other (Debye [16, 17]) and can have dipole-dipole interactions (Keesom [18, 19]). All these pairwise interaction potentials fall off inversely as the sixth power of the distance. 

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...

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. 

Schwan emphasized the concept of dispersion in the field of dielectric spectroscopic analysis of biomaterials. Dispersion has already been introduced in Section 3.4.1 dispersion means frequency dependence according to relaxation theory. Biological materials rarely show a single time constant Debye response as described in Section 3.4.2. Knowing how complex and heterogeneous living tissue is, the concept of a distribution of... 

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 value of the relaxation time is based on dielectric constant studies of Oncley (140) at 25 , who showed that the protein underwent anomalous dispersion and conformed nicely to the simple Debye curve, exhibiting a single critical frequency ve — 1.9 X 10 cycles sec"S a low frequency dielectric increment of -f 0.33 g. liter and a high frequency increment of —0.11 g." liter. The data just presented have been discussed by Oncley (141) and by Wyman and Ingalls (241) with the aid of their nomograms. It appears from their analyses that the facts might reasonably well be reconciled with the assumption either of oblate ellipsoids with p = 3 and A = 0.3 — 0.4 or of prolate ellipsoids with p = H and = 0.3 — 0.4. On the assumption of prolate ellipsoids, however, it would be necessary to assume that there was no component of the electric moment parallel to the long axis (axis of revolution). In either case the two dielectric increments correspond to an electric moment of about 500 Debye units (140). 

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