Kohlrausch, Williams, and Watts

Here tq is the relaxation time at equilibrium (Tf = T) at high temperatures, x is a structural parameter and measure of nonlinearity, with values 0 < x < 1, and AE is the activation energy for the relaxation processes and has an Arrhenius temperature dependence. The models also use the stretched exponential function of Kohlrausch, Williams, and Watts [1970] (KWW) to describe the distribution of relaxation times as... 

In practice dielectric relaxation curves for polymers (1-5) and glass-forming liquids (eg. Fig. 1) are far broader than that for an SRT process. Numerous empirical relaxation fiinctions have been proposed for this behavior (1,2,5). One function is the stretched exponential function of Kohlrausch, Williams, and Watts (KWW) (4-6), that is widely apphed to dielectric and other relaxation data for amorphous polymers ... 

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),... 

Much attention has been given to the relaxation of various properties toward their equilibrium values, especially in the glasses literature. Kohlrausch first proposed a stretched exponential form as a description of viscoelasticity, while Williams and Watts suggested the same form for dielectric relaxation exp[-(t/r) ], where 0 < 9 < 1 and 0 = 1 corresponds to the Debye limit. The master equation solution, Eq. (1.41), has a decaying multiexponential form that could lead to a wide variety of behavior depending upon the system. 

Studying relaxation phenomena, Kohlrausch [130] Williams and Watts [131] have used a decay function (j> with ... 

In practice, there exist many non-Debye relaxation processes, which can be described by a stretched exponential function, namely the Kohlrausch-Williams-Watts (KWW) equation (Kohlrausch 1854 Williams and Watts 1970), as given by... 

The relaxation function, can also be expressed in terms of a senuempirical function introduced originally by Kohlrausch (1897) and revived by Williams and Watts (1970), abbreviated as the KWW equation ... 

This function, originally introduced by Kohlrausch in 1854 to describe creep in silk and glass threads used as supports in magnetometers, is a very slow function of time and hence gives broad dispersion and loss curves when used in conjunction with equation (10). The integration cannot be expressed generally in closed form. For the special case jS=0.5 Williams and Watts showed that equations (10) and (25) give... 

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)... 

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. ... 

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).

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,... 

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... 

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