Stretched exponential function,

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

The stretched exponential function, A = Ao exp(—tfxf, has been applied to the fluorescence of unstained tissue [82-84], In particular, researchers at Paul French s group at Imperial college [82], show that the use of the stretched exponential, the parameters of mean, and the heterogeneity parameter (the inverse of the degree of stretch, ft) gives better tissue contrast and better fit than the mono- or multiexponential models. 

Po2]. For microporous silicon a more complex time behavior extending well into the ms range is observed, as shown in Fig. 7.11b. The experimentally observed dependence of intensity P on time t has been fitted to an exponential [Ho3, Fi4, MalO] or a stretched-exponential function according to Eq. (7.4) [Ka6, Pal2, Ool]. 

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 data were fitted to a stretched exponential function (Eq. 4.9) setting the stretching parameter to its dielectric value. The solid lines included in Fig. 6.3 display the resulting curves. These fits lead to the Q-dependent characteristic relaxation times TKww(Q)> hich are converted to average relaxation times by Eq. 5.25 (see Fig. 6.4). 

Both forms have a distribution of relaxation times about t0 which contribute to s(Laplace transform can be obtained numerically, and in this case C(t) can be well fit to a stretched exponential function [46] ... 

A fluorescence decay of the ensemble of many CV molecules on a PMMA film is shown in Fig. 12. The decay was not fitted to a single exponential function and a stretched exponential function, but was well fitted to a biexponential function I(t) = Af exp(-t/xf) + Asexp(—t/ts), where tf and ts are time constants and Af and As are pre-exponential factors. We obtained tf = 0.43 ns and ts = 1.76 ns, and the ratio As/Af = 1.14. Compared with the excited state lifetime of CV (2-3 ps) in methanol and ethanol [5-8,58-68], the fluorescence lifetime of CV on a PMMA film increased more than two orders of magnitude [9-12] thus, so did the fluorescence quantum efficiency. The enhancement of the fluorescence efficiency of CV on a PMMA film made it possible to observe single CV molecules. Figure 13 shows fluorescent spots on a PMMA film on which a drop of 1-nM CV in methanol was spin-coated. The number of fluorescent spots in an image linearly increased with increasing concentration of a CV methanol... 

Measurements of the time decay of the remanent magnetization in powdered TDAE-C60 by SQUID showed below Tc the presence of a long tail, which could be described by a stretched exponential function [103]. Similar results were also obtained by ESR time decay measurements [104]. 

The above reasoning shows that the stretched exponential function (4.14), or Weibull function as it is known, may be considered as an approximate solution of the diffusion equation with a variable diffusion coefficient due to the presence of particle interactions. Of course, it can be used to model release results even when no interaction is present (since this is just a limiting case of particles that are weakly interacting). 

As mentioned earlier, typical three-dimensional plots of s and s" versus frequency and temperature (see Fig. 14) suggest superimposing two processes (percolation and saddle-like) in the vicinity of the percolation. Therefore, in order to separate the long-time percolation process, the DCF was fitted as a sum of two functions. The KWW function (64) was used for fitting the percolation process and the product (25) of the power law and the stretched exponential function (as a more common representation of relaxation in time domain) was applied for the fitting of the additional short-time process. The values obtained for Dp of different porous glasses are presented in the Table I. The glasses studied differed in their preparation method, which affects the size of the pores, porosity and availability of second silica and ultra-porosity [153-156]. 

This is none other than the time-temperature superposition principle. However, the exact shape of the function F(t) is not a generic feature of the theory. A good approximation is provided by the Kohlrausch stretched exponential function. In the frequency domain, Eqs. (23) define the shape of the susceptibility minimum usually observed in the gigahertz range, and an interpolation formula follows ... 

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 known that the density autocorrelation function or the intermediate scattering function F(q, ai) in the supercooled fluid phase can well be described by a stretched exponential function of the form F q,u>) A exp[—(t/computer simulations. This particular relaxation is called a relaxation, and such a characteristic decay manifests itself in a slower decay of the dynamical structure factor S q,u>) and of the a peak of the general... 

Figure 8 Structural relaxation times for quench-cooled glassy disaccharides as determined from enthalpy relaxation data. Structural relaxation times were obtained by a fit of the data to the stretched exponential function (see [37,50]). ( ) Data for sucrose obtained by differential scanning calorimetry on annealed samples [37], (O) Data for sucrose obtained by isothermal microcalorimetry [50]. (A) Data for trehalose obtained by isothermal microcalorimetry [50].

Now let us consider the correlation function for N = 1000. We adopt feq = 220 106, which is long enough to reach equilibrium imaging from Fig. 2. The correlation function Cp(t teq) is reported in Fig. 9a, and it is well approximated by the stretched exponential function [38]... 

We remark that a stretched exponential function exp[—xf3] with a small exponent (3 1 is indistinguishable from a power-type function in the region... 

However, the fitting function (16) well agree with numerical result even around 0.32 ln(f/410) = 1, whose two solutions are f 18,9330. We therefore adopt a stretched exponential function as an approximation of Cp(t feq). 

It seems natural that we regard Cp(t x) as a series of stretched exponential functions of t rather than power-type functions, since this function fits Cp(t x) in more than two decades of time (power-law fits of the correlation functions hold in one decade). Moreover, at equilibrium, Cp(t teq) is also a stretched exponential rather than a pure exponential, as shown in Fig. 9. 

Figure 11. Correlation function of momenta at r = 0, that is, Cp(t 0). The inset is magnification of the horizontal axis around t = 0 for N = 100. These numerical results are approximated by solid curves that are stretched exponential functions in Eq. (19). [Reproduced with permission from Y. Y. Yamaguchi, Phys. Rev. E 68, 066210 (2003). Copyright 2004 by the American Physical Society.]...

Let us proceed to investigate the origin of anomaly in diffusion. We focus on the behavior for N = 1000. The mean-square displacement ag(f) is perfectly determined by the correlation function Cp(t x) using Eq. (9), once we assume that Cp(t ij is a series of stretched exponential functions. We introduce three parameters, Cp(0 x), tcolr(x), and (3(x), to describe the stretched exponential function as... 

Diffusion is obtained by integrating the correlation function of momenta Cp(t x), and the correlation function is approximated by a series of stretched exponential functions Cp(t x) = Cp 0 x) exp[— (f/fCorr(x)) ]- Among the three parameters Cp(0 x), fcorr(x), and (3(x), the stretching exponent p(x) plays a crucial role to yield anomaly in diffusion. If we assume that (3(x) is a constant, we never observe anomaly in diffusion. This result is consistent with the fact that anomaly in diffusion does not appear in a (quasi-)stationary state, because the correlation functions Cp(t xj and, accordingly, (3(x) are almost invariant with respect to x. 

Both in quasi-stationary and in equilibrium stages, fcorT is proportional to N, and (3 is almost constant. These simple scaling laws imply that fitting by stretched exponential functions is valid irrespective of degrees of freedom. 

A model having predictions that are consistent with the aforementioned experimental facts is the Coupling Model (CM) [21-26]. Complex many-body relaxation is necessitated by intermolecular interactions and constraints. The effects of the latter on structural relaxation are the main thrust of the model. The dispersion of structural relaxation times is a consequence of this cooperative dynamics, a conclusion that follows from the presence of fast and slow molecules (or chain segments) interchanging their roles at times on the order of the structural relaxation time Ta [27-29]. The dispersion of the structural relaxation can usually be described by the Kohlrausch-William-Watts (KWW) [30,31] stretched exponential function,... 

Fig. 3. Normalized E-O coefficients of RT-9800/poly quinoline at 80 °C.The data points arethe experimental values. The dotted line is the KWW stretched exponential function fitting and the solid line is a biexponential function fitting...

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