Debye’s model

To appreciate the predictive properties of Kieffer s model, it is sufficient to compare calculated and experimental entropy values for several phases of geochemical interest in table 3.1, which also lists entropy values obtained through apphcation of Debye s and Einstein s models. One advantage of Kieffer s model with respect to the two preceding formulations is its wider T range of applicability (Debye s model is appropriate to low frequencies and hence to low T, whereas Einstein s model is appropriate to high frequencies and hence to high T). 

Figure 18 Saturation correlation factor versus dipole interaction, for the models of Debye and Piekara. With Debye s model, Rs is always positive, decreasing from to 0 as dipole coupling increases. With Piekartfs model, in the case of nearly antiparallel pairs, Rg falls rapidly from 1 to 0, is negative for dipole coupling Y > 1.33, and then vanishes for association into a rigid pair of antiparallel dipoles...

Moreover, an equation for /as a function of absorptive crystal temperature, was obtained from the Debye s model for solids ... 

We shall now examine briefly the calculation of the vapour pressure on the basis of statistical models of the solid and liquid states. Debye s model of the solid state, which we have discussed briefly in chapter XII, 5, allows us to evaluate the vapour pressure of a solid, but the agreement with experiment is very poor. We obtain only a rough order of magnitude. ... 

Calculation of the dielectric permittivity of an isotropic polar material involves the problem of the permanent dipole contribution to polarizability and the problem of calculation of the local field acting at the molecular level in terms of the macroscopic field applied. Debye s model for static permittivity considers the local field equal to the external field. This assumption is valid only for gases at low density or dilute solutions of polar molecules in nonpolar solvents. Several workers... 

Debye s model The Debye model could be built with these assumptions, and polarization and permittivity become complex as described by Eq. (18) where n is the refractive index and t the relaxation time ... 

The real and imaginary parts of the dielectric permittivity of Debye s model are given by Eqs. (20) and (21) ... 

Debye s model assumptions are not valid for supercritical water, because the dilute limit is doubted. Microscopically there are many degrees of freedom and all these motions are not totally decoupled from the others, because the eigenstate of motions are not well known the structurally disordered matter. Dielectric measurements can only probe slow dynamics which can be described by stochastic processes and classical Debye s model could be rationalized [65, 66). 

This plot of experimental values is a convenient graphical test of the applicability of Debye s model. The effect of the last term on the shape of the diagram can be seen in Fig. 1.13. The larger the conductivity, the further the actual diagram departs from Debye s semicircle. 

When a is dose to unity this again reduces to Debye s model and for a smaller than unity an asymmetric diagram is obtained. The Cole-Cole diagram arise from symmetrical distribution of relaxation times whereas the Cole-Davidson diagram is obtained from a series of relaxation mechanisms of decreasing importance extending to the high-frequency side of the main dispersion. 

In contrast with condensed phases, intermolecular interactions in gases are negligibly small. The dipole moment found in the gas phase at low pressure is usually accepted as the correct value for a particular isolated molecule. The molecular dipole moment calculated for pure liquid using Debye s model gives values which are usually very different from those obtained from gas measurements. Intermolecular interactions in liquids produce deviations from Debye s assumptions. 

In Debye s model of dielectric relaxation, the polarisation process has a single relaxation time. The model has both electrical circuit and viscoelastic model analogues (Fig. 12.14). The electrical circuit is the dual of the mechanical model, because the voltages across the capacitor and resistor... 

The dielectric constant at angular frequent w is e (tu) = e (.dielectric constants are Su (sometimes written ) and (sometimes written f,). In Debye s model the relaxing element (which represents a polar molecule) is a sphere of radius a containing an electric dipole of dipole moment p-qd (see Figure 4.36). The sphere is immersed in a liquid of viscosity i). Under an electric field E the torque on the dipole is pE sin d. The rotation of the dipole under this torque is resisted by the Stokes firictional torque The dipole will follow the field for... 

Consider, for instance, the sign inversion frequency of the dielectric anisotropy As [16, 17]. According to Debye s model the frequency dependence of the dielectric constant s is [2]... 

An important result of Debye s model is that the relationship between the macroscopic, X, and molecular relaxation times, t, is given by Eq. (9). For example, Eq. (9) predicts that materials with Eg 100 and e = 2 the ratio of times is 25.5, with the molecular time being the faster one. 

Although the work of Glarum and Cole was criticized, we will not continue along these lines because for the present purposes they are relatively minor albeit elegant criticisms see, for example, the work of Fatuzzo and Mason. There are two other studies that need to be reviewed. The first one is by Gamant who observed that the viscosity in Debye s model should be time dependent (see Fig. 3). By means of a series of heuristic arguments, he replaced tj with a time-dependent one that is, 17(f). We examined these equations in some detail and could not obtain his results for the time dependence of e (w). 

The second work to be reviewed is that of DiMarzio and Bishop (D-B) who introduced a time-dependent viscosity into the hydrodynamic equations of Debye s model and then solved the hydrodynamic equations... 

This expression shows that the low-temperature heat capacity varies with the cube of the absolute temperature. This is what is seen experimentally (remember that a major failing of the Einstein treatment was that it didn t predict the proper low-temperature behavior of Cy), so the Debye treatment of the heat capacity of crystals is considered more successful. Once again, because absolute temperature and dy, always appear together as a ratio, Debye s model of crystals implies a law of corresponding states. A plot of the heat capacity versus TIdo should (and does) look virtually identical for all materials. 

In the case of Debye s model, the 3N oscillators no longer have the same fundamental frequency. By applying the equation ... 

Debye s model, which allows for a frequency distribution given by relation [1.9], has only yielded correct values of the specific heat capacity at constant volume (see section 1.8) for fairly low temperatures. Other authors have improved the model by modifying that frequency distribution. For example, Bom and Karman took a new approach to the establishment of the frequency distribution, this time supposing that the solid was no longer a continuum, but instead was represented by a periodic lattice of particles, which led them to the distribution function as shown in Figure 1.4(a). The distribution function reaches its peak very near to the limit frequency. 

In fact in this types of models, with a frequency distribution modified compared to Debye s, we can keep the developments obtained using Debye s model, but as if the Debye temperature varied with temperature. 

For rigid molecules the frequency dependence of the orientational polarization in isotropic liquids can be calculated using Debye s model for rotational diffusion. This may be modified to describe rotational diffusion in a liquid crystal potential of appropriate symmetry, but the resulting equation is no longer soluble in closed form. Martin, Meier and Saupe [34] obtained numerical solutions for a nematic pseudopotential of the form ... 

In the study of dielectric relaxation, temperature is an important variable, and it is observed that relaxation times decrease as the temperature increases. In Debye s model for the rotational diffusion of dipoles, the temperature dependence of the relaxation is determined by the diffusion constant or microscopic viscosity. For liquid crystals the nematic ordering potential contributes to rotational relaxation, and the temperature dependence of the order parameter influences the retardation factors. If rotational diffusion is an activated process, then it is appropriate to use an Arrhenius equation for the relaxation times ... 

Debye s model gives only an approximate deseription of the vibrational properties of real solids, espeeially for solids eontaining different atoms or having certain lattice structures. However, it is very convenient in situations where an analytical expression for the distribution functions is necessary, because more rigorous models give analytical solutions for one- or two-dimensional systems only. Since the spectra of surface vibrations are much more complicated, this model is often used. There are also some empirical combinations of Debye and Einstein distributions (in the classical limit) ... 

The application of this method requires knowledge of the explicit form of at least one of the two funetions (F(/)F(0)> or y t), in order to find a solution of the equation. Variants of the approaeh, developed up to the present time are based on different ways of modeling of the dissipation term y(t), conneeted with the secondary zone of atoms [ 18-20]. Adelman and Garrison use Debye s model for phonons of the solid and obtain an equation for the dissipation term whieh, ean be solved numerically. Doll and Dion propose y t) as a linear combination of conveniently chosen functions, where the coefficients are determined by numerieal self-consistency. Another possibility is to model the microscopic interactions in the lattice of the solid in order to derive a dissipation term. Tully presents the friction as a white noise or positionally autocorrelated function of a Brownian oscillator, including both oscillation and dissipation terms. 

We assume that the contribution of mode 4 satisfies the Debye s model with the exponential decay of aft — f) andLorenzian behavior for a ([Pg.234]

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