Relaxation Debye model

Figure 9.15 Dielectric function of water at room temperature calculated from the Debye relaxation model with r = 0.8 X 10 11 sec, eQcl = 77.5, and e0l, = 5.27. Data were obtained from three sources Grant et al. (1957), Cook (1952), and Lane and Saxton (1952).

The reflectivity of human skin is expected to decrease with increasing frequency. This is due to the reduction in the real part of the refractive index as well as an increase in absorption with frequency. In vivo studies have been conducted and used to derive parameters fitting a double Debye relaxation model for the impedance of human skin [58],... 

The fact that estimates of Xq are the correct order of magnitude suggests that there is an approximate relationship between xp and p, at least for liquids which obey the Debye relaxation model (see section 4.5). This point is illustrated in fig. 6.2 where Xj) is plotted against with logarithmic scales for the aprotic solvents listed in table 6.1. A very good correlation is found, confirming that equation (6.3.9) is approximately correct. On the basis of a least-squares fit for 14 solvents, the relationship is... 

The time dependent solvation funetion S(t) is a directly observed quantity as well as a convenient tool for numerical simulation studies. The corresponding linear response approximation C(t) is also easily eomputed from numerical simulations, and can also be studied using suitable theoretical models. Computer simulations are very valuable both in exploring the validity of such theoretical calculations, as well as the validity of linear response theory itself (by comparing S(t) to C(t)). Furthermore they can be used for direct visualization of the solute and solvent motions that dominate the solvation process. Many such simulations were published in the past decade, using different models for solvents such as water, alcohols and acetonitrile. Two remarkable outcomes of these studies are first, the close qualitative similarity between the time evolution of solvation in different simple solvents, and second, the marked deviation from the simple exponential relaxation predicted by the Debye relaxation model (cf Eq. [4.3.18]). At least two distinct relaxation modes are... 

The frequency effect of K mainly originates from Ac because the remaining parameters are all independent of frequency in the low frequency region. Based on Debye relaxation model, Ac has following form ... 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

Debye-Onsager model for C(t) and the longitudinal relaxation time Tj... 

Considerable progress has been made in going beyond the simple Debye continuum model. Non-Debye relaxation solvents have been considered. Solvents with nonuniform dielectric properties, and translational diffusion have been analyzed. This is discussed in Section II. Furthermore, models which mimic microscopic solute/solvent structure (such as the linearized mean spherical approximation), but still allow for analytical evaluation have been extensively explored [38, 41-43], Finally, detailed molecular dynamics calculations have been made on the solvation of water [57, 58, 71]. 

An Evaluation of the Debye-Onsager Model. The simplest treatment for solvation dynamics is the Debye-Onsager model which we reviewed in Section II.A. It assumes that the solvent (i) is well modeled as a uniform dielectric continuum and (ii) has a single relaxation time (i.e., the solvent is a Debye solvent ) td (Eq. (18)). The model predicts that C(t) should be a single... 

For a reasonable set of the parameters the calculated far-infrared absorption frequency dependence presents a two-humped curve. The absorption peaks due to the librators and the rotators are situated at higher and lower frequencies with respect to each other. The absorption dependences obtained rigorously and in the above-mentioned approximations agree reasonably. An important result concerns the low-frequency (Debye) relaxation spectrum. The hat-flat model gives, unlike the protomodel, a reasonable estimation of the Debye relaxation time td. The negative result for xD obtained in the protomodel is explained as follows. The subensemble of the rotators vanishes, if u —> oo. 

In the low-frequency range (with x spectral function L(z) depends weakly on frequency x. Then Eq. (32) comes to the Debye-relaxation spectrum given by Eq. (33). Its main characteristics, such as the dielectric-loss maximum Xd and its frequency xD, are given by Eq. (34). A connection between these quantities and the model parameters becomes clear in an example of a very small collision frequency y. In this case, relations (34) come to... 

We shall show below that the hybrid model gives a satisfactory description of the usual Debye relaxation at the microwave region and explains a quasiresonance absorption band in the FIR region. 

Models 1 and 2 were applied to simple nonassociated polar liquids. Wideband Debye relaxation + FIR spectral dependencies of the permittivity s (v) and absorption ot(v) were successfully described. Usually only one quasiresonance absorption band (at v between 10 cm-1 and 100-200 cm-1) and one non-resonance Debye loss e" peak (at microwaves) arise in these fluids. Although a spatial model gives, unlike a planar one, a correct value of the integrated absorption J ° ve"(v)dv, the calculated spectral dependencies resemble those found for motion in a plane. 

In our early work33 [50] the constant field model was applied to liquid water, where the harmonic law of particles motion, corresponding to a parabolic potential, was actually employed in the final calculations of the complex permittivity. In this work, qualitative description of only the libration band was obtained, while neither the R-band nor the low-frequency (Debye) relaxation band was described. Moreover, the fitted mean lifetime x of the dipoles, moving in the potential well, is unreasonably short ( ().02 ps)—that is, about an order of magnitude less than in more accurate calculations, which will be made here. 

As a second example, we consider liquid fluoromethane CH3F, which is a typical strongly absorbing nonassociated liquid. For our study we choose the temperature T 133 K near the triple point, which is equal to 131 K. The relevant experimental data [43] were summarized in Table IV. As we see in Table VIII, which presents the fitted parameters of the model, the angle p is rather small. At this temperature the density p of the liquid, the maximum dielectric loss and the Debye relaxation time rD are substantially larger than they would be, for example, near the critical temperature (at 293 K). At such small (5 the theory given here for the hat-curved model holds. For calculation of the complex permittivity s (v) and absorption a(v), we use the same formulas as for water. 

The calculated spectra are illustrated by Fig. 25. In Fig. 25a we see a quasiresonance FIR absorption band, which, unlike water, exhibits only one maximum. Figure 25b demonstrates the calculated and experimental Debye-relaxation loss band situated at microwaves. Our theory satisfactorily agrees with the recorded a(v) and e"(v) frequency dependencies. Although the fitted form factor/is very close to 1 (/ 0.96), the hat-curved model gives better agreement with the experiment than does a model based on the rectangular potential well, where / = 1 (see Section IV.G.3). 

Thus, evolution of semiphenomenological molecular models mentioned in Section V.A (items 1-6) have led to the hat-curved model as a model with a rounded potential well. This model combines useful properties of the rectangular potential well and those peculiar to the field models based on application of the parabolic, cosine, or cosine-squared potentials. Namely, the hat-curved model retains the main advantage of the rectangular-well model—its possibility to describe both the librational and the Debye-relaxation bands. 

Here multiplier 2 approximately accounts for doubling of integrated absorption due to spatial motion of a dipole, which is more realistic than motion in a plane to which LCs(Z) corresponds. For representation (235), only one (Debye) relaxation region with the relaxation time rD is characteristic. At this stage of molecular modeling it was not clear (a) why the CS potential, which affects motion of a dipole in a separate potential well, is the right model of specific interactions and (b) what is physical picture corresponding to a solid-body-like dipole moment pcs. 

A principal drawback of the hat-curved model revealed here and also in Section V is that we cannot exactly describe the submillimeter (v) spectrum of water (cf. solid and dashed lines in Figs. 32d-f). It appears that a plausible reason for such a difference is rather fundamental, since in Sections V and VI a dipole is assumed to move in one (hat-curved) potential well, to which only one Debye relaxation process corresponds. We remark that the decaying oscillations of a nonrigid dipole are considered in this section in such a way that the law of these oscillations is taken a priori—that is, without consideration of any dynamical process. 

In Section V the reorientation mechanism (A) was investigated in terms of the only (hat curved) potential well. Correspondingly, the only stochastic process characterized by the Debye relaxation time rD was discussed there. This restriction has led to a poor description of the submillimeter (10-100 cm-1) spectrum of water, since it is the second stochastic process which determines the frequency dependence (v) in this frequency range. The specific vibration mechanism (B) is applied for investigation of the submillimetre and the far-infrared spectrum in water. Here we shall demonstrate that if the harmonic oscillator model is applied, the small isotope shift of the R-band could be interpreted as a result of a small difference of the masses of the water isotopes. 

We conclude The submillimeter spectra calculated in terms of the harmonic oscillator model substantially differ from the spectra typical for the low-frequency Debye relaxation region. Such a fundamental difference of the spectra, calculated for water in microwave and submillimeter wavelength ranges, evidently reveals itself in the case of the composite hat-curved-harmonic oscillator model. 

Figure 5.3 shows the measured temperature dependence of the loss tangent of LaA103 single crystals and a fit employing the skm model plus a Debye relaxation term. The best fit was achieved for an activation energy of 31 meV, the estimated defect concentration is only 1016/cm3. It was suggested that aluminium atoms on interstitial lattice positions are responsible for the observed relaxation phenomena. This indicates, that the dielectric losses are extremely sensitive to very small concentrations of point defects [21],... 

Clear evidence of L-L transitions has been found only in /-Si modeled by the SW potential [269]. Sastry and Angell [288] performed MD simulations of supercooled /-Si using the SW potential. After cooling at ambient pressure, the liquid (HDL) was transformed to LDL at 1060 K. The Nc in LDL is almost 4, and the diffusivity is low compared with that in HDL. The structural properties of LDL, such as g(r) and Nc, are very close to those of LDA, which indicates that this HDL-LDL transition is a manifestation of the multiple amorphous forms (LDA and HDA) of Si. McMillan et al. [264] and Morishita [289] have also found structural fluctuations between LDL-like and HDL-like forms in their MD calculations for /-Si at 1100 K. Morishita has demonstrated that such a structural fluctuation induces spatial and temporal dynamical heterogeneity, and this heterogeneity accounts for the non-Debye relaxation process that becomes noticeable in the supercooled state [289]. 

In summary, we must say that unfortunately there is as yet no generally acknowledged opinion about the origin of the non-Debye dielectric response. However, there exist a significant number of different models which have been elaborated to describe non-Debye relaxation in some particular cases. In general these models can be separated into three main classes ... 

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