Stretched exponential relation

Figure 7. The self intermediate scattering function for GB particles in the T range of 1050-1150K (defined in the text). The dashed curves are a fit of the stretched exponential relation, Fs q, t) 0 exp[—(t/r) ] to the long-time data, where the short-time decay arises from the inertial atomic dynamics. The inset shows a power fit of r to 7"— 7, where Tq and y are adjustable parameters as in previous measurements and simulations. Figure 7 was originally published in [16], National Academy of Sciences.

Huber D.L. Statistical model for stretched exponential relation in macroscopic systems. Phys. Rev. B 1985 31 6070-6071... 

Stretched exponential relation suggests that the relaxation process is governed by interactions with impurity ions in the present sample of mineral belite. 

The PL lifetime values r obtained by fitting a stretched exponential function decrease with increasing PL peak energy PPL. For micro PS dried in a vacuum, for which the PL energies range from 1.5 to 3.5 eV, this dependence can roughly be fitted to the empirical relation ... 

The relation between collective and self-motion in simple monoatomic liquids was theoretically deduced by de Gennes [233] applying the second sum rule to a simple diffusive process. Phenomenological approaches like those proposed by Vineyard [ 194] and Skbld [234] also relate pair and single particle motions and may be applied to non-exponential functions. The first clearly fails to describe the PIB results since it considers the same time dependence for both correlators. Taking into account the stretched exponential forms for Spair(Q.t) (Eq. 4.21) and Sseif(Q>0 (Eq 4.9), the Skold approximation ... 

In order to understand the stretched exponential behavior of DCF (23), let us discuss Gibb s phase exponent r G = —lnp(p,q-,t). This quantity plays a special role in statistical mechanics and relates to the entropy of the system. If Gibb s exponent obeys the fractional evolution equation... 

By looking at Eqs. (27) and (28), Eq. (32) confirms the customary way of relating P of the stretched exponential function, Eq. (19), to the relaxation time spectrum. The glassy state relaxation is dominated by the part of the spectrum having longer relaxation times. The fractal dynamics of holes are diffusive, and the diffusivity depends strongly on the tenuous structure in fractal lattices, v is the exponent in the power-law relationship between local diffusivity and diffusion length ... 

It is apparently a general characteristic of glassy disorder, although there has been considerable debate over the relation between the stretched exponential decay and the microscopic relaxation mechanisms. 

The aimealing kinetics of the light-induced defects are shown in Fig. 6.29. Several hours at 130 °C are needed to anneal the defects completely, but only a few minutes at 200 C. The relaxation is nonexponential, and in the initial measurements of the decay the results were analyzed in terms of a distribution of time constants, Eq. (6.78) (Stutzmann, Jackson and Tsai 1986). The distribution is centered close to 1 eV with a width of about 0.2 eV. Subsequently it was found that the decay fits a stretched exponential, as is shown in Fig. 6.29. The parameters of the decay-the dispersion, p, and the temperature dependence of the decay time, t - are similar to those found for the thermal relaxation data and so are consistent with the same mechanism of hydrogen diffusion. The data are included in Fig. 6.23 which describes the general relation between x and D,. The annealing is therefore the process of relaxation to the equilibrium state with a low defect density. 

Hassler et al. [40, 41] observed that the stretched exponential behavior was related to the active phase of the enzyme (busy phase) while the less active phase (lazy phase) exhibited a single exponential phase (Fig. 4.20). The behavior which was originally observed by Edman et al. 1999 [39] at a much lower sensitivity had been observed again, however, with a much better intensity and time resolution. As can be seen from the trace, the turnover frequency is more than 4-fold higher in the busy periods. It is tempting to relate this to the memory effect decribed in the next chapter (Fig. 4.21). 

CdO and smaller amounts of CdCU, YF3 and LaFs). The modulus relaxation spectrum has been fitted to stretched exponential function and it has been found that p has a roughly constant value in the glassy state and above Tg, it decreases rapidly. Since decoupling index also decreases rapidly above the glass transition temperature, it is suggestive of an implicit relation between and / . It is also noted that Rr Tg itself is inversely correlated to the corresponding p values and P decreases linearly with log at Tg (also see Hunt, 1994)... 

Stretched exponential function may characterize the relaxations of the imaginary parts of any of the quantities listed earlier. They all exhibit similar loss peaks as functions of frequency. The corresponding full width at half maximum (FWHM) values are generally significantly greater than the Debye value and they are related to p. If FWHM is expressed as decades of width. A, then... 

Since fV(T,t) is equal to fVg(T) for ( <1, the relaxation function will be a simple exponential function for this time regime. For the region co t >1, it is more complex. It has been shown by Ngai and co-workers (Nagai, 1979 Nagai et al., 1984) that KWW stretched exponential function, exp[-(t/r) ], with j3= (1 - n), is a satisfactory solution to the above equation. When coJWg 1, the effective relaxation time r and the primitive relaxation time Vg = 1/Wg are related as... 

Vlad, M. O. Ross, J. Huber, D. L. Linear free energy relations and reversible stretched exponential kinetics in systems with static and dynamic disorder. J. Phys. Chem. B 1999, 103, 1563-1580. 

It is noteworthy that when Schweizer worked out his equation for entangled polymer melts, he did not obtain a stretch exponential that has exponent P = 0.57 or 0.59 that would have let him obtain the experimental or the dependence of the terminal relaxation time in order to be consistent with the experiment. On the other hand, as hinted by Schweizer (1989), Property (ii) of the CM when applied to terminal chain relaxation time xr of PI in blends with PtBS leads to the relation,... 

Whether this is more than a convenient parameterization is debatable. The stretched exponential in eq. (35) has no direct relation to stretched exponential relaxation of bulk magnetization (see Campbell et al. 1994). As will be discussed in sect. 8 the only clear-cut case is the highly dilute spin glass. It was shown by Uemura et al. (1984) that above the glass transition temperature root-exponential relaxation occurs, that is p = 0.5. That... 

Tinland and Borsali [123] used FRAP and QELS to measure Dp of 433 kDa dextran in aqueous solutions of 310 kDa polyvinylpyrrolidone (PVP) for 0 < c < 120g/L. M ,/Mn was 1.5 for the matrix polymer but ca. 1.9-1.95 for the probes. Dp from the two techniques do not agree. We analyzed Dp from FRAP measurements, because FRAP does not require the detailed model assumptions needed to relate the QELSS spectrum to diffusion coefficients. Tinland and Borsali s data agree well with stretched exponentials in c. 

For us, this clearly means, that the scaling dependence k [Mq]" is not determined by choice of the kinetic equation of postpolymerization (exponential law with two characteristic relaxation times on the basis of schemes (7.24), (7.25) and (7.26) or the stretched exponential law on the basis of scheme (7.1)) but by fundamental causes. However, at this the stretched exponential law requires the spectrum of the characteristic times of the relaxation characterizing by fractal properties to be known [6]. This is also proven by the scaling Ibrm of the dependence ki [Mq] 1 Evidently, between this dependence and the stretched exponential law there should be a relation. 

The stretched-exponential temporal response of Eq. (63), Section 2.1, a versatile and theoretically plausible correlation function, is one whose corresponding frequency behavior is now called Kohlrausch-Williams-Watts or just Kohlrausch [1854] model response, denoted here by Kk. It is also now customary to replace the a of the stretched-exponential equation by P or P, with A =D or 0. The k=D choice may be related to KD-model dispersive frequency response involving a distribution of dielectric relaxation (properly retardation ) times, and the A = 0 and 1 choices to two different distributions of resistivity relaxation times and thus to KO and K1-model responses, respectively. Note that the P parameter of the important K1 model is not directly related to stretched exponential temporal response, as are the other Kohlrausch models, but the DRTs of the KO and K1 models are closely related (Macdonald [1997a]). Further, although the KD and KO models are identical in form, they apply at different immittance levels and so represent distinct response behaviors. 

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