Debye dispersion equation

Figure 1 is a TEM photograph of the Cu (10wt%)/Al2O3 catalyst prepared by water-alcohol method, showing the dispersed state of copper and was confirmed the particle sizes from XRD data. Figure 2 is X-ray diffraction patterns of above-mention catalysts, was used to obtain information about phases and the particle size of prepared catalysts. Metal oxide is the active species in this reaction. Particle sizes were determined fix)m the width of the XRD peaks by the Debye-Scherrer equation. 

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

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. 

Using the Green function method and some decoupling approximations corresponding to the RPA and the Zeeeman reduced splitting smaller than Debye phonon quantum, it is possible to get the dispersion equation... 

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). 

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 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... 

XRD is a fast and nondestructive test which is frequently used to characterise thermoset materials and their composites. Crystalline materials are characterised by sharp peaks, whereas amorphous materials show broad humps. Thus, the degree of crystallinity can be estimated. When a crystalline material such as clay is dispersed in a thermoset matrix, one can study the intercalation and exfoliation behaviour (see subsequent chapters). If the crystallites of the power are very small, the peaks of the pattern will be broadened. From this broadening one can determine an average crystallite size using the Debye-Scherrer equation ... 

Except at absolute zero, every molecule must have one or more sources of electrostatic potential. Even if a molecule bears no net electrostatic charge, atomic dipoles which result from the motion of electrons in their orbits around a nucleus will give rise to dispersive van der Waals or London interactions. These atomic dipoles insure that a solute molecule has a small but finite electrostatic interaction energy with the surrounding solution molecules. For example, in aqueous ethanol solutions, the dipole—dipole interaction between ethanol and water molecules becomes the primary interaction energy. Ethanol molecules are not ionic, so use of the Debye—Huckel equation (57), based on Coulombic interaction, carmotbe used to determine the activity coefficient of an aqueous ethanol solution. At sensible... 

Diffusion controlled recombination of an ion pair is influenced by the random dispersive forces (also present for non-charged species) and the strong Coulombic electrostatic interactions. The diffusion equation [13, 14] governing the diffusive motion of charged species is known as the Debye-Smoluchowski equation [15], which can be expressed as... 

As we have already noted, all molecules display the dispersion component of attraction since all are polarizable and that is the only requirement for the London interaction. Not only is the dispersion component the most ubiquitous of the attractions, but it is also the most important in almost all cases. Only in the case of highly polar molecules such as water is the dipole-dipole interaction greater than the dispersion component. Likewise, the mixed interaction described by the Debye equation is generally the smallest of the three. 

The theoretical inconsistencies inherent in the Poisson-Boltzmann equation were shown in Section 11.4 to vanish in the limit of very small potentials. It may also be shown that errors arising from this inconsistency will not be too serious under the conditions that prevail in many colloidal dispersions, even though the potential itself may no longer be small. Accordingly, we return to the Poisson-Boltzmann equation as it applies to a planar interface, Equation (29), to develop the Gouy-Chapman result without the limitations of the Debye-Hiickel approximation. 

Bagchi et al. have derived analogous equations for a solvent with two Debye times associated with two overlapping dispersion regimes [53]. 

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. 

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 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 application of static light scattering to polymers is based on the theoretical equations of Debye (1944, 1947) and the methodology of Zimm (1948). The principles apply equally to polysaccharides (Sorochan et al., 1971). In total intensity light scattering, monochromatic light (436 and 546 nm) at constant T passes through the dispersion and becomes plane polarized the horizontal beam is scattered in accordance with the equation (Hiemenz, 1986)... 

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. 

Intramolecular interactions were introduced for the first time by van der Waals in 1873 he thus attempted to explain the deviation of the real gas from the ideal gas. In order to apply the ideal gas law equation to the behavior of real gases, allowance should be made for the attractive and repulsive forces occurring between molecules. From that time on, the dipole moment theory of Debye (1912) and the dispersion energy or induced dipole theory by London (1930) were the main driving forces of the research about intermolecular interactions. 

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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