Relaxation function, Debye

Debye relaxation function was used for the solvent dielectric relaxation. From Ref. 32 with permission, from J. Chem. Phys. 90, 153 (1989). Copyright 1989, American Physical Society. 

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-... 

Figure 41. Typical dielectric spectra of 20 mol% of glycerol—water mixtures at (a) 185 K (supercooled state) and (b) 218 K (frozen state), where solid and dashed curves show the real and imaginary parts of complex dielectric permittivity. Each relaxation process in the frozen state was fitted by (114) and by Cole-Cole and Debye relaxation functions, respectively, in order to separate the main process, the process due to interfacial water, and the process due to ice. (Reproduced with permission from Ref. 244. Copyright 2005, American Chemical Society.)...

The evolution of the many-molecule dynamics, with more and more units participating in the motion with increasing time, is mirrored directly in colloidal suspensions of particles using confocal microscopy [213]. The correlation function of the dynamically heterogeneous a-relaxation is stretched over more decades of time than the linear exponential Debye relaxation function as a consequence of the intermolecularly cooperative dynamics. Other multidimensional NMR experiments [226] have shown that molecular reorientation in the heterogeneous a-relaxation occurs by relatively small jump angles, conceptually simlar to the primitive relaxation or as found experimentally for the JG relaxation [227]. 

When a = 0 and = 1, this gives rise to the Debye relaxation function, which gives rise to a semicircle, when e" is plotted against e [ (ry) = (co) - ie co)]. When a alone is zero. 

For real dielectrics with complex and frequency dependent, inverse transformation of Equation 12 to obtain R(t) is not possible in any simple way, even for a Debye relaxation function, because of the awkward square root of in p and exp(-i2Z), but one expects a rounded and distorted version of the idealized progression of steps for c and p real, and Vq a step function. 

In this equation the quantity F(T/T )d ln(T/rQ) is the distribution of relaxation times, while the quantity in brackets is the Debye relaxation function, that is, see Eq. (5). A particularly useful form for representing the complex dielectric constant dependence on frequency is given by the following equation ... 

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. 

The correlation function C(t) is purely phenomenological. Interpretation of its time evolution is often based on theory in which the longitudinal relaxation time, tl, is introduced. This time is a fraction of the Debye relaxation time ... 

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 ... 

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).

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-... 

It follows from Eq. (32) that the spectral function L(z) actually determines the absorption coefficient. At high frequencies,13 such thatx y, this coefficient is proportional to xlm[x (x)]. In other limit, at low frequencies, one may neglect the frequency dependence L(z) by setting L(z) = L(iy). In this approximation, Eq. (32) yields the Debye-relaxation formula (VIG, p. 194) ... 

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... 

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. 

Different relaxation functions are derived assuming that the real (physical) ensemble of relaxation times is confined between the upper and lower limits of self-similarity. It is predicted that at times, shorter than the relaxation time at the lowest (primitive) self-similarity level the relaxation should be of classical, Debye-like type, whatever the pattern of nonclassical relaxation at longer times. The analysis of diffusion for a Brownian particle, where the assumption that the... 

P (z) are the Legendre polynomials [51] which now constitute the appropriate basis set), Eq. (132) may be solved to yield the corresponding results for rotation in space, namely, the aftereffect function [Eq. (123)] and the complex susceptibility [Eq. (11)], with x and Xo from Eqs. (81) and (84), respectively. Apparently as in normal diffusion, the results differ from the corresponding two-dimensional analogs only by a factor 2/3 in Xo and the appropriate definition of the Debye relaxation time. 

The picture in terms of the decoupled Eangevin equations (98) and (99) (omitting the inertial term in Eq. (98) is that the orientational correlation functions of the longitudinal and transverse components of the magnetization in the axially symmetric potential, Kvm sin" . are simply multiplied by the liquid state factor, exp(—t/T ), of the Brownian (Debye) relaxation of the ferrofluid stemming from Eq. (99). As far as the fenomagnetic resonance is concerned, we shall presently demonstrate that this factor is irrelevant. 

Relaxation functions for fractal random walks are fundamental in the kinetics of complex systems such as liquid crystals, amorphous semiconductors and polymers, glass forming liquids, and so on [73]. Relaxation in these systems may deviate considerably from the exponential (Debye) pattern. An important task in dielectric relaxation of complex systems is to extend [74,75] the Debye theory of relaxation of polar molecules to fractional dynamics, so that empirical decay functions for example, the stretched exponential of Williams and Watts [76] may be justified in terms of continuous-time random walks. 

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