Relaxation, Debye distribution

In Refs. 80 and 81 it is shown that the Mittag-Leffier function is the exact relaxation function for an underlying fractal time random walk process, and that this function directly leads to the Cole-Cole behavior [82] for the complex susceptibility, which is broadly used to describe experimental results. Furthermore, the Mittag-Leffier function can be decomposed into single Debye processes, the relaxation time distribution of which is given by a mod-... 

The next issue to concern us will be anomalous relaxation in which the smearing out of a relaxation spectrum (i.e., the deviation of complex susceptibility from its Debye form) is associated with the concept of a relaxation time distribution. As is well known, this concept implies an assembly of dipoles with a continuous distribution of relaxation times of Eq. (379). 

Thus, the chain of transitions considered above is effectively reduced to exponential, Debye-like relaxation with the mean relaxation time (x). In other words, the concept of a relaxation time distribution implies Debye-like relaxation of a system. However, it is evident that the relaxation will become nonexponential, should a system be characterized by a complex susceptibility of, say, Cole-Cole type. 

Thus, it can be concluded that the relaxation time distribution concept applies to Debye-like relaxation (even though its frequency dependence may be smeared-out), whereas it becomes inapplicable for still slower relaxation patterns. In the latter situation, the distribution of relaxation times over a selfsimilar, fractal ensemble seems a physically more reasonable assumption. As is well known, the fractality of geometrical objects implies their non-integer dimension however, a more exact definition of the fractal concept with respect to the ensemble of relaxation times is in order. 

All the above descriptions use the Debye model, characterized by an arc of a circle in the plot e" vs. , and a unique relaxation time. In most cases (polymers, glasses, liquids ), however, the spectrum does not correspond to an arc of a circle and is frequently interpreted in terms of a relaxation time distribution. The latter broadens with increasing temperature. Such distributions can be either intrinsic (disordered compounds) or due to lack of accuracy in the measurements fixed frequency measurements with too-widely spaced intervals, or insensitive apparatus. As we shall see later, protonic conductors give rise to better defined but more complex spectra because of the existence of various protonic and polyatomic species corresponding to fixed or mobile charges strong dipoles lead frequently to ferroelectric phenomena. 

Fig. 1.9 Relaxation time distribution functions calculated on the basis of Debye law (D), Cole-Cole law (CC), Davidson-Cole law (DC) and...

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... 

Processes during a cell cycle are evidenced to be closely controlled cooperative events, including synchronisation within the ensemble. This caused us to describe relaxation within each cell by a Debye process, the relaxation time of which should increase with the size of the cell involved ( finite-size effect ). In that way ensemble structure and relaxation processes of cell ensembles are strictly interrelated. The universal energy density distribution and the universal relaxation mode distribution turn out to be copies of each other. Consequently, the spectrum depends only on the universal properties of the ensemble structure, i.e. on the value of p. Since all the cell populations studied here belong to the / = 3 class, the linear relaxation behaviour should show the same features. 

The type of relaxation time distribution can be easily determined from plots of e" versus e for a broad range of frequencies (so-called Cole-Cole plot or Argand plot, Fig. 8.29). The Cole-Cole plot for a single relaxation time is a semicircle between a = 8qo and e =eo centred on the e axis (Debye model) or below the e axis (Cole-Cole model). 

These relationships are known as the Debye formulae. The Debye process has a relaxation time distribution, which is symmetrical around /niax= niax/2n and has a full width at half-maximum of 1.14 decades in frequency for the dielectric loss. In most cases, the half width of measured loss peaks is much broader than the predicted by eqn [26] and in addition, their shapes are asymmetric and with a high-frequency tail. This is the non-Debye (or nonideal) relaxation behavior found in many glass formers. In the literature, several empirical model funaions, mostly generalization of the Debye function, have been developed and tested which are able to describe broadened and/or asymmetric loss peaks. Among these empirical model functions, the most important are the Kohlrausch-Williams-Watts (KWW), Cole-Cole (CC), Cole-Davidson (CD), and the Havriliak-Negami (HN) function. The HN function, with two shape parameters, is the most commonly used funaion in the frequency domain. 

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. 

In real systems, there will always be a distribution of the relaxation time meaning that Equation 5.3 should more adequately be written as given in Ref. [8] (generalized Debye model) ... 

Another arena for the application of stochastic frictional approaches is the influence of ionic atmosphere relaxation on the rates of reactions in electrolyte solutions [19], To gain perspective on this, we first recall the early and often quoted triumph of TST for the prediction of salt effects, in connection with Debye-Hiickel theory, for reaction rates In kTST varies linearly with the square root of the solution ionic strength I, with a sign depending on whether the charge distribution of the transition state is stabilized or destabilized by the ionic atmosphere compared to the reactants. 

It can not be described by means of a single Debye process, but more complicated relaxation functions involving distributions of relaxation times (like the Cole-Cole function [117]) or distributions of energy barriers (like log-normal functions [118]) have to be used for its description. Usually a narrowing of the relaxation function with increasing temperature is observed. The Arrhenius temperature dependence of the associated characteristic time is ... 

The dielectric behaviour of pure water has been the subject of study in numerous laboratories over the past fifty years. As a result there is a good understanding of how the complex permittivity t = E — varies with frequency from DC up to a few tens of GHz and it is generally agreed that the dielectric dispersion in this range can be represented either by the Debye equation or by some function involving a small distribution of relaxation times. 

Debye obtained his result by solving a forced diffusion equation Ci.e., with torque of the applied field included) for the distribution of dipole coordinate p - pcosS, with 6 the polar angle between the dipole axis and tSe field, and the same result for the model follows very simply from equation (3) using the time dependent distribution function in the absence of the field (5). The relaxation time is given by td = 1/2D, which for a molecular sphere of volume v rotating in fluid of viscosity n becomes... 

S nuclear quadrupole coupling constants have been determined from line width values in some 3- and 4-substituted sodium benzenesulphonates33 63 and in 2-substituted sodium ethanesulphonates.35 Reasonably, in sulphonates R — SO3, (i) t] is near zero due to the tetrahedral symmetry of the electronic distribution at the 33S nucleus, and (ii) qzz is the component of the electric field gradient along the C-S axis. In the benzenesulphonate anion, the correlation time has been obtained from 13C spin-lattice relaxation time and NOE measurements. In substituted benzenesulphonates, it has been obtained by the Debye-Stokes-Einstein relationship, corrected by an empirically determined microviscosity factor. In 2-substituted ethanesulphonates, the molecular correlation time of the sphere having a volume equal to the molecular volume has been considered. 

The proposed method is based upon the quantitative measurement of the contribution of differently charged nitroxide probes to the spin-lattice relaxation rate (1/T i) of protons in a particular molecule, followed by the calculation of local electrostatic potential using the classical Debye equation (Likhtenshtein et al., 1999 Glaser et al., 2000). In parallel, the theoretical calculation of potential distribution with the use of the MacSpartan Plus 1.0 program has been performed. 

For many of the systems being studied, the relationship above does not sufficiently describe the experimental results. The Debye conjecture is simple and elegant. It enables us to understand the nature of dielectric dispersion. However, for most of the systems being studied, the relationship above does not sufficiently describe the experimental results. The experimental data are better described by nonexponential relaxation laws. This necessitates empirical relationships, which formally take into account the distribution of relaxation times. 

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

In many cases—in particular in disordered media—static inhomogeneities (accompanied by a broad distribution of jump probabilities) have to be taken into account which obviously also result in a non-Debye relaxation and comparable frequency dependencies (see e.g., a recent review on this topic).220... 

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