Stretched exponential decay

It is obvious that (21) is equivalent to a stretched exponential decay function of the general form... 

In real systems, a distribution in the characteristic time may lead to a stretched exponential decay. In the thermally activated regime where the relaxation of the magnetization is due to the Orbach mechanism, the temperature dependence of the relaxation time may be described by an Arrhenius law of the form ... 

In addition to the red PL band, an IR PL band between about 0.8 and 1.3 eV is usually present in most PS samples [Fa6, Pi3, Mal7, Mo7, Ku4, Pe5], as shown in Fig. 7.9. At RT the intensity of the IR band is weak. At cryogenic temperatures it becomes much stronger and can even be more intense than the red band. The PLE spectra for the red and the IR bands are identical despite their large difference in PL peak energy, as shown in Fig. 7.11a. Furthermore a correlation between the peak position of the two bands has been observed, as shown in Fig. 7.13 [Ku6]. The PL decay, however, is found to be different For the IR band a if2 dependence of PL intensity on time, with little dependence on temperature, is observed, while a stretched exponential decay with strong temperature dependence is observed for the red band, as shown in Fig. 7.11b. 

An approximation of the lifetime in PS at RT using an electron-hole pair density equal to one pair per crystallite and the radiative recombination parameter of bulk silicon give values in the order of 10 ms [Ho3]. The estimated radiative lifetime of excitons is strongly size dependent [Sa4, Hi4, Hi8] and increases from fractions of microseconds to milliseconds, corresponding to an increase in diameter from 1 to 3 nm [Hy2, Ta3], as shown in Fig. 7.18. For larger crystallites a recombination via non-radiative channels is expected to dominate. The experimentally observed stretched exponential decay characteristic of the PL is interpreted as a consequence of the randomness of the porous skeleton structure [Sa5]. 

Flomenbom. O, Velonia, K.. Loos. D.. Masuo. S.. Cotlet, M, Engelborghs, Y., Hofkens. J.. Rowan, AE., Nolte, R.J.M., de Schyver. F.C, and Klafter, J. (2005) Stretched exponential decay and correlations in the catalytic activity of fluctuating single lipase molecules. Proc. Natl. Acad. Sci. U.S.A., 102, 2368-2372. 

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. 

It is an experimentally demonstrated fact that the a relaxation in the time domain fits the stretch exponential decay function (0 or the Kohlrausch-Williams-Watts (KWW) equation (7,8)... 

Figure 16.19 shows the results from employing a stretched exponential decay model to fit the data (46). 

The intensity correlation function C t) is well fitted over more than three decades, from a delay time of 1 /s up to more than 1 ms with values of the parameters similar to those given above (Fig. 4). In particular, the deviations from the single exponential decay are well accounted for, at least in the time domain of interest. It is clear that a purely stretched exponential decay can be observed only at extremely long time where the amplitude of the correlation function is much less than the noise. 

Contrary to expectations, on the millisecond time scale the initial drop //< in the luminescence studies of Mays and Ilgenfritz [24] did not increase significantly with the temperature-induced cluster growth but remained constant even when an infinite percolation cluster was present. Furthermore, the observed decays were always exponential (Fig. 9). Evidently, the initial drop no longer reflects the cluster size. The process responsible for it is over within 50 //s and should perhaps rather be looked upon similarly to the transient active sphere part of normal diffusion-controlled decay. The diffusion in this case is a random walk performed by the quencher on a (percolation) cluster. A stretched exponential decay would be expected for a random walk deactivation on a static cluster, as was observed close to the percolation threshold in earlier studies [23,24]. Those measurements were performed over a time window of about 50 //s, which is close to the reported value of the cluster lifetime from electrical birefringence measurements [60]. It is very likely that... 

Thus, we again find for inhomogeneously cross-linked gels a stretched exponential decay of the relaxation modulus G(t) on scales larger than the average size of the structmal inhomogeneities (cross-link agglomerations). Remarkably, the stretched exponential index here, 1 /3, coincides with that foimd by Sommer for polydisperse polymer networks [142], see Sect. 7.4. This is an expected result, since a ladder model is equivalent to a linear Rouse chain, see Sect. 6.2. 

Fig. 15.19 Stretched exponential decay curves for several values of p. For small P values, the curve decays faster for t < tq, and is followed by a slower tail afterwards...

These results should be asymptotically correct in the limit N oo and t 0.05(6 t)Vts For finite N, the exponents predicted in Eq. 132 may change slightly due to the stretched exponential decay of Cp t) given in Eq. 131. The limiting laws for the observables of interest in the frame of this review, the mean squared displacement and the (intrasegment) spin-lattice relaxation rate, are summarized in Table 2 (compare [49]). 

In a study focused on structure of the composites, DEER measurements on 7-SL-UTMA, 9-SL-UTMA, and 11-SMJTMA in organoclay, PS midocomposites, and PCL nanocomposites were performed to reveal the distribution of spin-labeled surfactants. No well-defined distances were found. The data could be fitted by a stretched exponential decay 7(t) =exp(-at° ), where D is the fradal dimension of the homogeneous distribution of spin labels and a is a generalized density parameter. 

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