Stretched exponential deviations from

Several paths exist for improving the original renormalization group calculation. Merriam and Phillies(7) have since extended the author s original calculation(6) to determine the five-point chain-chain-chain-chain-chain hydrodynamic interaction tensor. The deviation of the observed stretched-exponential behavior from simple calculations yielding pure-exponential behavior was predicted to arise from the concentration dependence of the chain radius. Dielectric relaxation measures both a relaxation time and a chain radius. Analysis demonstrated that chain contraction accounts quantitatively for the form of the stretched-exponential concentration dependence of the dielectric relaxation time(8). 

Fig. 7 Left Arrhenius plot of the approximate folding and unfolding rates of 15. Right At low temperatures, the relaxation is better fitted by a stretched exponential or biexponential the stretching exponent is shown here as a convenient measure of the deviation from single exponentiality...

Rb and 1H SLR rate as a function of temperature is a very important parameter which shows the suppression of phase transition and reveals the frustration in the mixed system. Temperature dependence of Ti in any ordered system can be described by the well known Bloembergen-Purcell-Pound (BPP) type expression. However, disordered systems show deviations from BPP behaviour, showing a broad distribution of relaxation times. The magnetization recovery shows a stretched exponential recovery of magnetization following M(t)=Mo(1 — 2 exp (— r/Ti) ) where a is the stretched exponent. 

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. 

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. 

In the above, virtually the entirety of the published literature on polymer self-diffusion and on the diffusion of chain probes in polymer solutions has been reviewed. Without exception the concentration dependences of Dg and Dp are described by stretched exponentials in polymer concentration. The measured molecular weight dependences compare favorably with the elaborated stretched exponential, eq. 16, except that, when P M or M P, there is a deviation from eq. 16, that deviation referring only to the molecular weight dependences. The deviation uniformly has the same form The elaborated stretched exponential overestimates the concentration dependence of Dp, so that at elevated c the predicted Dp/Do is less than the measured Dp/Dp. Contrarywise, almost without exception the experimental data on solutions is inconsistent with models that... 

Richter et al. carried out neutron spin echo measurements at the minimum position of S(Q) on the same polymer (PB) as that described above [82]. The intermediate scattering functions were described by a stretched exponential function as well, but could not be scaled to a master curve using a shift factor a-j. The relaxation times extracted from the observed stretched exponential functions are plotted in the relaxation time map in Fig. 9, from which it is seen that they deviate from the relaxation time of the a-process and the temperature dependence of Tjg is well described by the Arrhenius formula. It was confirmed that the process observed in PB at the minimum position Q ,j in S(Q) is the JG process. The fact that the JG process is observed at Q jjj suggests that the process is not a cooperative motion but an isolated one. 

When all measurements needed for the analysis were reported, it was found that polymers that do not contract at elevated concentration have simple exponential dependences of r on c. Polymers that do contract with increasing c have stretched-exponential dependences of r on c. The degree of chain contraction accounts quantitatively for the deviation of r (c) from a simple exponential. We return to a consideration of dielectric spectroscopy after considering other static scattering methods for determining Rg. 

Transition concentrations ct[ /] are 60, 50, and 25 for ethylbenzene, CCI4, and ethylacetate, respectively. For c> q one sees upwards deviations from stretched-exponential behavior, perhaps consistent with a larger-concentration power law. However, with only a few points showing this behavior in any system it is diflBcult to be precise as to the deviation s functional form. 

Because these deviations are only seen in a few systems, the larger-c upward deviation, and the small-c downward deviation from the stretched exponential are therefore reasonably interpreted as involving specific chemical effects. 

In a series of systems in which (r ) and r were both determined, was found to follow a stretched exponential in c. The deviation of r (c) from a simple-exponential concentration dependence is quantitatively explained by the concentration dependence of (r ), namely r exp(ac Experimentally, the... 

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