Debye behavior relaxation

Abstract For three liquids, salol, propylene carbonate, and o-terphenyl, we show that the relaxation time or the viscosity at the onset of Arrhenius behavior is a material constant. Thus, while the temperature of this transition can be altered by the application of pressure, the time scale of the dynamics retains a characteristic, pressure-independent value. Since the onset of an Arrhenius temperature-dependence and the related Debye relaxation behavior signify the loss of intermolecular constraints on the dynamics, our result indicates that intermolecular cooperativity effects are governed by the time scale for structural relaxation. 

In summary, the NFS investigation of FC/DBP reveals three temperature ranges in which the detector molecule FC exhibits different relaxation behavior. Up to 150 K, it follows harmonic Debye relaxation ( exp(—t/x) ). Such a distribution of relaxation times is characteristic of the glassy state. The broader the distribution of relaxation times x, the smaller will be. In the present case, takes values close to 0.5 [31] which is typical of polymers and many molecular glasses. Above the glass-to-liquid transition at = 202 K, the msd of iron becomes so large that the/factor drops practically to zero. 

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. 

A comparison of either the dispersion c or the complex plane loci of e with the Debye relaxation function in eq 7 shows that the experimental curves extend over much broader ranges of frequencies, a behavior typical also of other polyelectrolyte systems and described within their accuracy by the empirical cir-... 

Fig. 14. Cole-Cole diagrams illustrating dipole relaxation behavior. a) Debye single relaxation time model, b) Williams-Watts expression with p = 0.5. c) Cole-Cote expression with...

A comparison between experimental and simulated main Debye relaxation time is presented in Figure 16-7. Simulation and experimental results show excellent agreement for not so dilute systems (p > 0.4g/cm3). However, below this density the experimental Debye time increases with decreasing density, whereas simulation results for this quantity keep decreasing and approaching the limiting behavior of a collection of free rotors. The extent of the loss of dynamic correlation between... 

In the most general sense, non-Debye dielectric behavior can be described in terms of a continuous distribution of relaxation times, G(x) [11]. 

Dielectric relaxation of complex materials over wide frequency and temperature ranges in general may be described in terms of several non-Debye relaxation processes. A quantitative analysis of the dielectric spectra begins with the construction of a fitting function in selected frequency and temperature intervals, which corresponds to the relaxation processes in the spectra. This fitting function is a linear superposition of the model functions (such as HN, Jonscher, dc-conductivity terms see Section II.B.l) that describes the frequency dependence of the isothermal data of the complex dielectric permittivity. The temperature behavior of the fitting parameters reflects the structural and dynamic properties of the material. 

Thus, the non-Debye dielectric behavior in silica glasses and PS is similar. These systems exhibit an intermediate temperature percolation process associated with the transfer of the electric excitations through the random structures of fractal paths. It was shown that at the mesoscale range the fractal dimension of the complex material morphology (Dr for porous glasses and porous silicon) coincides with the fractal dimension Dp of the path structure. This value can be obtained by fitting the experimental DCF to the stretched-exponential relaxation law (64). 

The situation is more complex in the case of the so-called non-Debye liquids -the protic solvents. Due to their internal structure, these liquids exhibit a complicated dielectric relaxation behavior. This group of solvents comprises alcohols, formamide, propylene carbonate, and some other liquids. One should remember that in the In vs. In Tl analysis (Sec. 3.1.3), the rate constants measured in these solvents deviated from the values measured in aprotic solvents. 

This short discussion shows that in the case of the non-Debye solvents further work is necessary on both the electrode kinetics and the dielectric relaxation behavior of the solvents in the presence of various electrolytes. There are also significant discrepancies between the results on the relaxation dynamics of these solvents reported by various authors (see [169]). 

The plot shows a distribution closely around a slope of unity indicated by the solid line in Figure 2 except for the alcohols and nitrobenzene. Such anomaly in alcohols is also reported for other chemical processes and time-dependent fluorescence stokes shifts and is attributed to their non-Debye multiple relaxation behavior " the shorter relaxation components, which are assigned to local motions such as the OH group reorientation, contribute the friction for the barrier crossing rather than the slower main relaxation component, which corresponds to the longitudinal dielectric relaxation time, tl, when one regards the solvent as a Debye dielectric medium. If one takes account of the multiple relaxation of the alcohols, the theoretical ket (or v,i) values inaease and approach to the trend of the other solvents. (See open circles in Figure 2.)... 

In order to demonstrate how the anomalous relaxation behavior described by the hitherto empirical Eqs. (9)—(11) may be obtained from our fractional generalizations of the Fokker-Planck equation in configuration space (in effect, fractional Smoluchowski equations), Eq. (101), we first consider the fractional rotational motion of a fixed axis rotator [1], which for the normal diffusion is the first Debye model (see Section II.C). The orientation of the dipole is specified by the angular coordinate 4> (the azimuth) constituting a system of one rotational degree of freedom. Electrical interactions between the dipoles are ignored. 

Acetonitrile (AcN) can be described by the Debye model using data up to 40 GHz [9]. When the frequency range is extended to 90 GHz, the data can only be fitted if a distribution of relaxation times is assumed. Dimethylsulfoxide (DMSO) and propylene carbonate (PC) have been studied up to 89 GHz [9]. These data can be fitted to the Debye model with two relaxation processes. DMSO is known to form dimers in the bulk [28]. This association is undoubtedly the reason for its complex relaxation behavior. The same is true for PC, a molecule with a very high dipole moment. 

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... 

Functions such as P(t) in Figure 2 with zero long time limit and finite area often have a long time behavior that is at least roughly an exponential decay P(t) = P(tf))exp[-(t-tm)Tgpp] for t > tw. (A case in point is Debye relaxation with Tgpp = tq + e - )d/c as... 

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