Exponentials, stretched

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. 

Phillies [296] observed that several authors [87,270] using different underlying argnments have obtained stretched exponential form for diffusion in presence of polymer matrix ... 

Tokita et al. [394] measured diffusion of different molecules with molecular weights varying from 18 to 342 in polyacrylamide gels with constant percentage cross-linker and varying total acrylamide concentration. They found the data to be in good agreement with the stretched exponential of the form... 

Johnson et al. [186] measured diffusion of fluorescein-labeled macromolecules in agarose gels. Their data agreed well with Eq. (85), which combined the hydrodynamic effects with the steric hindrance factors. Gibbs and Johnson [131] measured diffusion of proteins and smaller molecules in polyacrylamide gels using pulsed-field gradient NMR methods and found their data to fit the stretched exponential form... 

Chrambach [307] also used the stretched exponential to describe electrophoresis of spherical latex particles in polymer solutions. 

Figure 6. Shown is the correlation between the liquid s fragility and the exponent p of the stretched exponential relaxations, as predicted by the RFOT theory, superimposed on the measured values in many liquids taken from the compilation of Bohmer et al. [50]. The dashed line assumed a simple gaussian distribution with the width mentioned in the text. The solid line takes into account the existence of the highest barrier by replacing the barrier distribution to the right of the most probable value by a narrow peak of the same area the peak is located at that most probable value. Taken from Ref. [45] with permission.

As mentioned in the introductory part of this section, quantum dots exhibit quite complex non-radiative relaxation dynamics. The non-radiative decay is not reproduced by a single exponential function, in contrast to triplet states of fluorescent organic molecules that exhibit monophasic exponential decay. In order to quantitatively analyze fluorescence correlation signals of quantum dots including such complex non-radiative decay, we adopted a fluorescence autocorrelation function including the decay component of a stretched exponential as represented by Eq. (8.11). 

It has been shown that the rate constants obtained from the slopes of In [intensity] versus plots approximate the rates of the highest-probability matrix sites. Hence, workers have utilized the temperature dependence of these values, or other empirically derived stretched exponential time dependencies, to estimate low temperature Arrhenius plots. The validity of such methods, however, depends critically on obtaining accurate time-dependence data on the fastest matrix sites, which is increasingly difficult as temperatures are raised. 

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

Fig. 5. The characteristic frequencies QR and time exponents (3 in the stretched exponential relaxation function obtained for the randomly labelled PDMS melt at 100 °C. (Reprinted with permission from [44]. Copyright 1989 Steinkopff Verlag, Darmstadt)...

Lifetime heterogeneity can be analyzed by fitting the fluorescence decays with appropriate model function (e.g., multiexponential, stretched exponential, and power-like models) [39], This, however, always requires the use of additional fitting parameters and a significantly higher number of photons should be collected to obtain meaningful results. For instance, two lifetime decays with time constants of 2 ns, 4 ns and a fractional contribution of the fast component of 10%, requires about 400,000 photons to be resolved at 5% confidence [33],... 

Lifetime heterogeneity itself can be the target of the measurement. In this case, high photon counts and alternative model functions like stretched exponentials and power-distribution-based models can be used [39, 43], These provide information on the degree of heterogeneity of the sample with the addition of only one fit parameter compared with single exponential fits. 

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. 

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 ... 

Fig. 8. Time decay of normalized light induced spin density for various temperatures. The solid circles are the data points while the solid lines are fits to the data using a stretched exponential time dependence (Jackson and Kakalios, 1988b).

Fig. 10. Arhennius plot of relaxation time obtained by stretched exponential fits of decay of nBx for n-type and p-type a-Si H (Street et al., 1988).

The slow relaxation of the occupied band tail density and of the conductivity a are accurately described by the stretched exponential time dependence... 

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