Debye-type expression

Before doing that, however, it is useful to understand qualitatively the anticipated behavior of the susceptibilities. Let us consider the Debye-type expression Eq. (4.107) for the linear susceptibility. The drastic change of the exponential factor reduces all the limiting behavior to just two cases. The first is oytjo 1, specifically, the system is rather cold and stiff. Thus the response is close to zero. The second case is oar [Pg.457]

As has been shown previously by us [3] and independently by Tarasov [4], an allowance for the regularity of elastic wave propagation in crystals gives a Debye-type expression in which the exponent is one (n — 1) for one-dimensional structures (filaments), two (n = 2) for laminar structures, and three (n = 3) for isotropic space structures. In all the intermediate cases, the exponent n will have an intermediate value. However, in contrast to Tarasov, we did not consider the behavior of filaments or layers and their interaction with each oiher, but the special features of occupation by figurative points corresponding to the excitation of vibrations in the phase k space of anisotropic substances. 

Here, Eq is the permittivity of free space. For a simple Debye-type relaxation process, Eq. (4), and owing to the incorrect representation of the high-frequency limit inherent in any expression for K (to) consistent with an exponential decay function for the electric moment, one obtains for the high-frequency limit of a(to) from Eqs. (4) and (6) (30) ... 

Since simple ions have mean ionic diameters of the order 0.3-0.6 nm, it is evident that deviations from the Debye-Htickel expression due to ion association are likely to occur for charge types of 2-1 or greater. These deviations are then handled by Bjerrum s ion-pair... 

However, for higher charge types matters become more difficult. When 2-2 electrolytes were considered, the predicted values and the Debye-Hiickel values for differed by up to 20%. Values for 1-3 and 3-1 electrolytes are expected to lie between these values and those for 1-2 electrolytes. The inevitable conclusion is that the Debye-Hiickel expression cannot be used as a base-line for these high charge types. 

Many materials display non-Debye dielectric behavior by a broader asymmetric loss peak. This non-Debye a.c response can be described by a combination of Cole-Cole [23] and Davidson-Cole [24] functions, and an empirical expression proposed by Havrilink-Negami [25]. The natural gum Arabica is found to obey a non-Debye type of response [25,26] and may be described [27] by the Havrilink-Negami function. 

For a non-Debye process, a good approximation is given by an HN-type expression where the frequency is changed to its reciprocal value [4,153] ... 

As an illustration of the observed behavior and types of functions to be transformed. Figure 2 shows the reflection R(t) from Equation 5 for a step-like voltage pulse and dielectric with "Debye" relaxation expressed by... 

Small proteins and biopolymers of 1 pm sizes also possess the Debye-type orientation relaxation in MHz (for small molecules of several angstroms this relaxation would be in the GHz range) that can overlap with the P-relaxation. Small dipoles and molecules exhibiting rotational orientation, the relaxation mechanism can be approximated as spherical particles of radius a in solvent of viscosity i), where their charge z can often be assumed to be unity [6]. The high-frequency relaxation times corresponding to this phenomenon can be described in a simplified expression ... 

A few remarks are due about this feature. The nonuniformity above is a formal expression of breakdown of the local electro-neutrality assumption in concentration polarization, described in the previous chapter. Essentially, this reflects the failure of a description based upon assuming the split of the physical region into a locally electro-neutral domain and an equilibrium double layer where all of the space charge is concentrated. The source of this failure, reflected in the nonuniformity of the corresponding matched asymptotic expansions, is that the local Debye length at the interface tends to infinity as the voltage increases. In parallel a whole new type of phenomena arises, which is not reflected in the simplistic picture above. The... 

Most work has been done by the same group of workers,360 363, 364, 367 433, 435,440 and practically all examples are from type B systems except for a study of the annular tautomerism (Section IV,A1) of 4-methyl-6-phenylpyrazolo[3,4-d]-u-triazole (90),367 a type A system. Values shown here are expressed in Debyes (D), using dioxane as solvent. 

In the early 1960s, Fowkes [88,89] introduced the concept of the surface free energy of a solid. The surface free energy is expressed by the sum two components a dispersive component, attributable to London attraction, and a specific (or polar) component, y p, owing to all other types of interactions (Debye, Keesom, hydrogen bonding, and other polar effects, as similarly described before in Sec. II. C... 

The equation 18.214 expresses the deformation energy of interaction of two dipoles and forces of this type do not depend to a first approximation on the temperature. For the calculation of the energy of orientation interaction it is possible to proceed in the same manner as in the deduction of the Debye equation (see above) but it is also possible to proceed directly from equation 18.214. 

Although both these types of analyses are well exploited, more information may be gleaned by comparing the experimental data with simulations made using a model, based either on a molecular structure expressed in the form of Debye spheres to facilitate calculation of model data, or a model constructed ab initio. This approach is especially favored when determining protein structure and has been used to great effect by Svergun and coworkers to obtain detailed structural information about proteins in solution. ... 

In this still relatively simple molecule all pair interactions between the nine H-atoms at the carbons, the H at the hydroxyl, the four carbons and the oxygen, have to be accounted for. Treating the OH-group interaction also via a Lennard-Jones interaction plus an added (ideal) dipole contribution is already an approximation because intermolecular distances are too short to treat dipoles as ideal. In mathematical terms, the expressions derived for Debye- and Keesom-type interactions (1,4,4c) are only first approximations, the more so because the rotation of the dipole is restricted. In practice there is often no alternative than to make clever guesses about the various Uy r) functions. It is always possible to group some types of interaction together, to obtain more detailed expressions for yA. Such an equation for dumb-bell types of molecules have been given by Alejandre et al. ) and by Harris 2). 

It is not easy to quantify precisely the supersaturation levels of given species generated in mixed salt systems or in solutions in which ion association occurs. Relationships such as equations 3.67 3.69 cannot be simply applied because of the difficulty of expressing the true reference condition of equilibrium saturation. It is first necessary to identify all the possible single species, ion pairs and solid liquid phase equilibria that can occur in the system. The relevant thermodynamic association/dissociation constants (K values) must be known. The activity coefficients for the various ionic species must be calculated, e.g. by means of Debye-Hiickel type equations (section 3.6.2). Equilibrium concentrations of all the possible species present are then evaluated by iterative procedures. 

The functions f are the electrostatic Debye-Htickel terms and the 5 and C coefficients are electrolyte-specific fitting parameters. Factors involving the charge numbers and stoichiometric coefficients have to be included for electrolyte types other than 1 1. Two universal constants, b =. 2 and a = 2.0 are employed in the full expressions for B and C as well as the solvent- and temperature-dependent A

Debye-Htickel theory. For details the series of papers by Pitzer and coworkers should be consulted, starting with (Pitzer and Mayoraga 1973). 

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