Kohlrausch-Williams-Watts model

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... 

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 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 way to extract information on ([)(/) is to assume a stretched exponential function, as described by the Kohlrausch-Williams-Watts (KWW) model [9,10] ... 

Moynihan s formulation [5] of the Tool-Narayanaswamy [7] model is used in tins woilc. In Moynihan s equations, the Active temperature, Tf, originally d ned by Tod [78], is used as a measure of the structure of the glass. The evdution of Active temperature is represented by the generalized stretched exponential Kohlrausch-William-Watts(KWW) function [76,77] ... 

The stretched-exponential temporal response of Eq. (63), Section 2.1, a versatile and theoretically plausible correlation function, is one whose corresponding frequency behavior is now called Kohlrausch-Williams-Watts or just Kohlrausch [1854] model response, denoted here by Kk. It is also now customary to replace the a of the stretched-exponential equation by P or P, with A =D or 0. The k=D choice may be related to KD-model dispersive frequency response involving a distribution of dielectric relaxation (properly retardation ) times, and the A = 0 and 1 choices to two different distributions of resistivity relaxation times and thus to KO and K1-model responses, respectively. Note that the P parameter of the important K1 model is not directly related to stretched exponential temporal response, as are the other Kohlrausch models, but the DRTs of the KO and K1 models are closely related (Macdonald [1997a]). Further, although the KD and KO models are identical in form, they apply at different immittance levels and so represent distinct response behaviors. 

The data can he obtained from dynamic mechanical analysis (DMA) tests used to construct the stress relaxation master cnrves at different levels of conversion, within the frame of time-temperature superposition principle. Curves of stress relaxation modulus can he modeled using the stretched Kohlrausch-William-Watts (KWW) exponential function at each level of conversion, as follows ... 

Maintaining polar order in a poled pol5uner is of great importance for second-order applications (88,89). The dielectric relaxation process leading to decay in the orientation of ordered polymers has been studied extensively and is the subject of another article (see Dielectric Relaxation). Several models that describe the chromophore reorientation for NLO materials have been proposed, including the Kohlrausch-Williams-Watts (KWW) model (90,91), biexponential and triexponential decay models (92), time-dependent Debye relaxation time models (93), and the Liu-Ramkrishna-Lackritz (LRL) model (94). For further information on... 

Relaxation functions, describing the time dependence of the modulus, are either derived from a model or simply an empirically-adopted fitting function. Only the former are amenable to interpretation. However, an empirical function with some theoretical basis is the Kohlrausch-Williams-Watts equation [6], which describes a variety of relaxations observed in many different materials [7]... 

The a- and ajS-processes are characterized by a broad asymmetric dielectric relaxation spectrum, which can be well represented by the Kohlrausch Williams-Watts (KWW) decay function (cf. eqn. (4.17)). The major factor leading to the broad DR spectra for a- and ajS-relaxations is that chain segments relax in cooperation with their environment. In order to explain the mechanism of this relaxation, the concepts of defect diffusion and free-volume fluctuation are used. For example, Bendler has proposed a model in which the KWW function is interpreted as the survival probability of a frozen segment in a swarm of hopping defects with a stable waiting-time distribution At for defect motion. 

These relationships are known as the Debye formulae. The Debye process has a relaxation time distribution, which is symmetrical around /niax= niax/2n and has a full width at half-maximum of 1.14 decades in frequency for the dielectric loss. In most cases, the half width of measured loss peaks is much broader than the predicted by eqn [26] and in addition, their shapes are asymmetric and with a high-frequency tail. This is the non-Debye (or nonideal) relaxation behavior found in many glass formers. In the literature, several empirical model funaions, mostly generalization of the Debye function, have been developed and tested which are able to describe broadened and/or asymmetric loss peaks. Among these empirical model functions, the most important are the Kohlrausch-Williams-Watts (KWW), Cole-Cole (CC), Cole-Davidson (CD), and the Havriliak-Negami (HN) function. The HN function, with two shape parameters, is the most commonly used funaion in the frequency domain. 

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... 

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