Stretched exponential small

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. 

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... 

Only a small number of studies have addressed the problem of direct measurements of the dynamics of the surfactant layer in a bicontinuous microemulsion [16, 73, 75, 76]. The Zilman-Granek (ZG) model, which assumes membrane Zimm dynamics on an ensemble of free membrane patches [75] is expected to be applicable to the problem. In the framework of this model the intermediate scattering functions should then be describable by a stretched exponential function of the type... 

Two data sets that did not include small-c measurements were fit to stretched exponentials, after including dilute solution measurements of Hadgraft, et al. [87] as adjusted for polymer molecular weight and viscosity and temperature of the particular solvent. For brevity, Figures for these more restricted data sets are omitted, von Meerwall, et al. [97] used PFGNMR to measure Dg of 10, 37.4, 179, 498, and 1050 kDa polystyrene in tetrahydrofuran at concentrations 6-700 g/L, and also Dg of tetrahydrofuran and hex-afluorobenzene in the same polymer solutions. Wesson, et al. [98] used FRS to measure Dg of polystyrenes in tetrahydrofuran and benzene. Polystyrenes had M of 32, 46, 105, 130, and 360 kDa, and were observed for 40 < c < 500g/L. Dg covered four orders of magnitude, but only for Dg/Do substantially less than 1. 

A joint stretched exponential in c cmd polymer molecular weight thus fits most data sets well. However, Xuexin, et al. [99] cover an extremely broad range of M, so that 1/ varied substantially with molecular weight. A fit of all of Xuexin, et al. s data to eq. 16, with V held constant, therefore worked poorly for large or small P. A fit to intermediate P worked well and is displayed (solid curves). 

Tokita et al. [160], using small-molecule probes having molecular weights of 18-342 Da in crosslinked polyacrylamide, report that Dp of the probes ftdls as a stretched exponential in probe molecular weight and matrix concentration. They interpret the dependence on P as a dependence on the probe radius R, namely exp(—Matsukawa and Ando [159], studying polyethylene glycol probe polymers in crosslinked poly-(N,N-dimethylacrylamide) with P up to 20 kDa, find the very similar Dp/Dpp = exp(—... 

The rate dependencies of the ferroelectric material properties are also reflected in the dynamics after fatigue. Initially, most of the domain system will be switched almost instantaneously [235], and only a small amount of polarization will creep for longer time periods [194]. A highly retarded stretched exponential relaxation was observed after bipolar fatigue treatment [235], and these observations correlated well with the thermally activated domain dynamics. If the overall materials response was represented in a rate-dependent constitutive material law 236], however, then a growing defect cluster size would retard the domain dynamics considerably. Hard and soft material behaviors were also representable as different barrier heights to a thermally activated domain wall motion, as demonstrated by the theoretical studies of Belov and Kreher [236]. 

If the exchange is governed by diffusion away from the surface, then a stretched-exponential type of behavior F(t) = r(0) exp[-(t/r) ] is observed for the decay in the surface excess of the initially adsorbed pol5mier. The value of the exponent p can vary between 0 and 1, depending on the rate of readsorption and the bulk diffusion rates (35,36). If diffusion is slow and the chain exhibits a strong tendency to readsorb, p will have a small value p < 1/2). The time constant for exchange in this case scales as where D is the diffusion coefficient of the polymer chain in the matrix of adsorbed chains. 

SuiQ, E) = SiiQ, E) 0 Sr Q, E), that is, a convolution of the translational dynamic structure factor, Si(Q,E), and the rotational one, 5r(<2, ) In addition, for small Q spectra, Q < 1 A the 5r(<2, E) can be made negligibly small, hence 5 h((2, E) Si(Q, E) and its Fourier transform will give the self-intermediate scattering function F Q, t) that have a stretched exponential FniQj) = exp [ - r (g) r] long-time decay. When the T is above the room temperature, P 1. A situation for which the exponential form Eh(Q, t) exp(—r(g)/) can be approximately used, or equivalently, in frequency domain theSnCg, E) of water is approximated as a Lorentzian shape function [67],... 

The short-time response was dominated by an overshoot and was followed by a stretched exponential or damped oscillations depending on the applied shear rate. The subsequent behavior was a slow variation towards steady state that could be related to the small undershoot observed in a(t) [144,191]. 

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...

One recalls that the spectral modes here are stretched exponentials with very different stretching exponents. Unlike pairs of pure-exponential modes, one cannot simply identify a mode as "fast" or "slow" because it decays sooner or later than another mode stretched-exponential modes decay to some extent on a multiplicity of time scales. However, the observations that (i) the slow modes have a common, diffusive, q-dependence, the same for large and small probes, and (ii) the fast modes have a different, non-diffusive (b bj 0) but also common q-dependence, nearly the same for large and small probes, supports our grouping of modes as "fast" or "slow". It would have been equally possible to refer to the modes as sharp, with p 1, and broad, with Pf < p. 

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