Stretched exponential model

For statistically stationary isotropic turbulence, the stretched-exponential model has the form... 

FIGURE 7.16 (a) Example of fluorescence decays of Pro4[M + H] + ions at 303 and 438 K and the instrument response function (light solid curve) (b) fits to the measured decays by a stretched exponential model (solid black curves) showing the fit decay constant Linear and logarithmic intensity scales are used in (a) and (b), respectively. (From lavarone, A.T. Duft, D. Parks, J.H. J. Phys. Chem. 2006,110, 12714-12727. With permission.)... 

In derivations [34-40] of the stretched-exponential models, functional forms and numerical values for exponents and pre-factors were calculated, using various approximations (in particular that chain motion is adequately approximated by whole-body translation and rotation) that are not obviously appropriate if P and M differ greatly. Some exponential models [41,42] also include a transition between a lower-concentration regime, in which a stretched-exponential concentration dependence is found, and a higher-concentration regime, in which transport coefficients show power-law concentration dependences. This transition sometimes appears in viscosity data [41-43], typically at some c[ 7] > 35, but is very rarely found with Dg or Dp. 

The stretched exponential model gives the worst fitting to our C(t) data which is shown in Figure 8.7. 

The best experiments to properly measure thixotropy are those where the sample to be tested is sheared at a given shear rate until equilibrium is obtained, then as quickly as possible the shear rate is changed to another value. The typical response to such a step-wise change from one steady-state condition to another is, in terms of the viscosity, often characterised by the so-called stretched exponential model ... 

In derivations leading to stretched-exponential models, functional forms and numerical values for exponents and prefactors are obtained, subject to various approximations (2-5).Some derivations assume that chain motion is adequately approximated by whole-body translation and rotation, which may be appropriate if p M, but which is not obviously appropriate if P and M are substantially unequal. 

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. 

According to the Rouse model the mode correlators (Eq. 3.14) should decay in a single exponential fashion. A direct evaluation from the atomic trajectories shows that the three major contributing Rouse modes decay with stretched exponentials displaying stretching exponents jSof (1 13=0.96 and 2,3 jS=0.86). We note, however, that there is no evidence for the extreme stretching of in-... 

While the Rouse model predicts a linear time evolution of the mean-square centre of mass coordinate (Eq. 3.14), within the time window of the simulation t<9 ns) a sublinear diffusion in form of a stretched exponential with a stretching exponent of (3=0,83 is found. A detailed inspection of the time-dependent mean-squared amplitudes reveals that the sublinear diffusion mainly originates from motions at short times t[Pg.39]

The time dependence of the dynamic correlation function q t) was investigated numerically on the Ising EA model by Ogielski [131], An empirical formula for the decay of q t) was proposed as a combination of a power law at short times and a stretched exponential at long times... 

The Mittag-Leffler function, or combinations thereof, has been obtained from fractional rheological models, and it convincingly describes the behavior of a number of rubbery and nonrubbery polymeric substances [79, 85]. The numerical behavior of the Mittag-Leffler function is equivalent to asymptotic power-law patterns that are often used to fit experimental data, see the comparative discussion of data from early events in peptide folding in Ref. 86, where the asymptotic power-law was confronted with the stretched exponential fit function. 

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). 

An appropriate choice of the distribution function can account for any reasonable shape of the relaxation data. The unexplained feature of this model is that there is no apparent reason why the resulting decay should be a stretched exponential, as this does not result from any obvious distribution of lifetimes, such as a gaussian. The integral in Eq. 

The model exhibits the expected Rouse relaxation modes at high temperature. At lower temperature, the chain motions can still be decomposed into Rouse normal modes, but the normal modes no longer relax via single exponential decays. Instead, the decay of each mode is described by a stretched exponential, and the stretching increases decreases) as the mode number increases. In addition, the temperature-dependence of the relaxation rates is described by the VFTH expression. The emergence of these features of real glasses in such a simple model suggests that such features are insensitive to molecular details. 

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