Kohlrausch-Williams -Watts

A particular characteristic feature of dynamic processes in the vicinity of the glass transition is the ubiquity of the Kohlrausch-Williams-Watts (KWW) stretched exponential relaxation 1,7-9... 

Figure 21 Coherent intermediate scattering functions at the position of the amorphous halo versus time scaled by the a time, which is the time it takes the scattering function to decay by 70%. The thick gray line shows that the a-process can be fitted with a Kohlrausch-Williams-Watts (KWW) law.

Where p defines the shape of the hole energy spectrum. The relaxation time x in Equation 3 is treated as a function of temperature, nonequilibrium glassy state (5), crosslink density and applied stresses instead of as an experimental constant in the Kohlrausch-Williams-Watts function. The macroscopic (global) relaxation time x is related to that of the local state (A) by x = x = i a which results in (11)... 

The considered model of a straight line of M nanoparticles illustrates only general features of dielectric losses caused by an M nanoparticle cluster in polymer matrix. Actually such cluster is a complex fractal system. Analysis of dielectric relaxation parameters of this process allowed the determination of fractal properties of the percolation cluster [104], The dielectric response for this process in the time domain can be described by the Kohlrausch-Williams-Watts (KWW) expression... 

The molecular relaxation process has been studied by the autocorrelation function of normal modes for a linear polymer chain [177]. The relaxation spectrum can be analyzed by the Kohlrausch-Williams-Watts function [177,178] ... 

It has the familiar form of the Kohlrausch-Williams-Watts (KWW) equation [17], except that p and x are not empirical constants here, and they will be discussed in the next two sections. 

The Kohlrausch Williams-Watts and Havriliak Negami formalisms are equally capable of representing real experimental data, and this is their main value, rather than an ability to explain the underlying relaxation processes. They are rooted in the time and frequency domains, respectively, and there is no analytical way of transforming from one to the other, but their effective equivalence has been convincingly demonstrated by numerical methods (Alvarez, Alegria and Colmenero, 1991). 

One of the features observed in many glass-forming liquids is the non-linear nature of any relaxation processes that occur around and below Tg. The relaxation rate is found to depend on the sign of initial departure of actual sample from the equilibrium state. The relaxation rate is described well by the Kohlrausch-Williams-Watts (KWl O empirical equation. ... 

It is an experimentally demonstrated fact that the a relaxation in the time domain fits the stretch exponential decay function (0 or the Kohlrausch-Williams-Watts (KWW) equation (7,8)... 

The Mittag-Leffler function has interesting properties in both the short-time and the long-time limits. In the short-time limit it yields the Kohlrausch-Williams-Watts Law from stress relaxation in rheology given by... 

Figure 12. The solid curve is the Mittag-Leffler function, the solution to the fractional relaxation equation. The dashed curve is the stretched exponential (Kohlrausch-Williams-Watts Law), and the dotted curve is the inverse power law (Nutting Law).

The model considerations outlined above permit one to clarify the results presented in Table XI. For example, from the explicit definition of the Kohlrausch-Williams-Watts stretched exponent on the barrier height... 

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,... 

Another important characteristic of viscous liquids close to Tg is nonexponential relaxation. Consider the response of a system to a perturbation, such as the polarization in response to an applied electric field, the strain (deformation) resulting from an applied stress, the stress in response to an imposed deformation, the volume response to applied pressure, or the temperature response to a heat flux. It is found experimentally that the temporal behavior of the response function 0(t), following an initial instantaneous response, can often be described by the stretched exponential, or Kohlrausch-Williams-Watts (KWW) function (Kohlrausch, 1854 Williams and Watts, 1970),... 

If the first scenario were real, the slower secondary relaxation should express its presence as an excess wing on high frequency side of the a- relaxation peak. To check this we superimposed dielectric loss spectra of octa-O-acetyl-lactose measured above and below Tg to that obtained at T=353 K. Next we fitted a master curve constructed in this way to the Kohlrausch-Williams-Watts function... 

This behavior was attributed to activated transport of the injected electron back to the oxidized sensitizer as the rate-determining step for charge recombination. Charge transport and recombination in sensitized Ti02 have both been shown to be second order. Nelson has modeled recombination data with the Kohlrausch-Williams-Watts model that is a paradigm for charge transport in disordered materials.138-140 The rates increased significantly when additional electrons were electrochemically introduced... 

KWW (Kohlrausch-Williams-Watts) plots for PhPPV films with various TNF concentrations. Reprinted from [66], copyright 2002, with permission from the American Institute of Physics. 

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