Debye behavior

The so-called Boson peak is visible as a hump in the reduced DOS, g(E)IE (Fig. 9.39b), and is a measure of structural disorder, i.e., any deviation from the symmetry of the perfectly ordered crystal will lead to an excess vibrational contribution with respect to Debye behavior. The reduced DOS appears to be temperature-independent at low temperatures, becomes less pronounced with increasing temperature, and disappears at the glass-liquid transition. Thus, the significant part of modes constituting the Boson peak is clearly nonlocalized on FC. Instead, they represent the delocalized collective motions of the glasses with a correlation length of more than 20 A. 

The role of specific interactions was not recognized for a long time. An important publication concerning this problem was the work by Liebe et al. [17], where a fine non-Debye behavior of the complex permittivity (v) was discovered in the submillimeter frequency range. The new phenomenon was described as the second Debye term with the relaxation time T2, which was shown to be very short compared with the usual Debye relaxation time td (note that td and 12 comprise, respectively, about 10 and 0.3 ps). A physical nature of the processes, which determines the second Debye term, was not recognized nor in Ref. [17], nor later in a number works—for example, in Refs. 54-56, where the double Debye approach by Liebe et al. was successfully confirmed. 

Careful investigations revealed that the VDOS varies as the square of the frequency below 10 cm for deoxyMb solutions, as predicted from a Debye model, but reveal significant deviations from Debye behavior at higher frequencies, reminiscent of the boson peak observed in amorphous solids. Interestingly, these deviations were not observed for deoxyMb crystals. 

The use of calculated from T] according to Eq. (37), instead of Tl, in the correlation of In kg with the time of relaxation led to a single linear dependence valid for Debye and non-Debye solvents. Such behavior is illustrated by the plot prepared [169, 170] for the CoCq /Coc system in 12 solvents, including methanol, ethanol, propanol, and propylene carbonate, which also exhibits a non-Debye behavior [202]. This plot is shown in Fig. 7. 

Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal fluctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker-Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. 

Since this Arrhenius temperature dependence implies that the molecular motions are unimpeded by intermolecular constraints, our finding reveals that the loss of intermolecular cooperativity is governed by the dynamics, or at least they have the same control parameter. Since the relaxation time determines the shape of the relaxation function ° , we can conclude from the results herein that at the onset of intermolecular cooperativity the breadth of the relaxation function is constant, independent of pressure. Of course, since in the absence of intermolecular constraints on the molecular motions we expect Debye behavior, the shape of the relaxation function should also be invariant for all T greater than Ta-... 

No Debye behavior characteristic of the Gaussian coil has been observed. Such a dependence can be studied by making a Zimm plot in the range qR < 1 with q the scattering vector and R the radius of gyration. 

The asymmetric shape of the normalized plot of tan d (tan is a non-Debye behavior and can be interpreted as the consequence of the distribution of relaxation time. The normalized plot of tan d (tan shown in Figure 29 dis-... 

Although the well-established measuranent of dielectric loss is not, in its narrowest sense, strictly impedance spectroscopy, a discussion of relaxation behavior is central to the family of techniques that use the interaction of a time-varying electromagnetic signal with a material to deduce microscopic detail. The generalization of the treatment of systems with a single relaxation time (Debye behavior) to those with multiple relaxations or distributions of relaxation times is discussed in Section... 

In this chapter, the binary mixture of GB particles of different aspect ratios has been studied by molecular dynamics simulation. The composition dependence of different static and dynamic properties has been studied. The radial distribution function has been found to show some interesting features. Simulated pressure and overall diffusion coefficient exhibit nonideal composition dependence. However, simulated viscosity does not show any clear nonideality. The mole fraction dependence of selfdiffusion coefficients qualitatively signals some kind of structural transition in the 50 50 mixture. The rotational correlation study shows the non-Debye behavior in its rank dependence. The product of translational diffusion coefficient and rotational correlation time (first rank) has been found to remain constant across the mixture composition and lie above the stick prediction. 

Let us now apply this Debye model to the orientational relaxation modes of a nematic with fixed director, n (f) = const, assuming that each corresponding to the orientational modes in Eq. (10-6) follows the Debye behavior (11). Using Eq. (10-5) and (10-6), we find... 

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