Distribution function Williams-Watts

The mean value of the energy barrier for the 38 resulting phenyl ring flips was 45 d= 28 kj mol-1. The peak energy barriers showed a broad distribution, which could be fitted to a Williams-Watts distribution function with a between 0.1 and 0.2. By applying the transition state theory, the distribution of ring flip frequencies could be derived, as shown in Fig. 59. 

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 method of cumulants performs rather weakly for very broad distributions of the decay rate. In this case, the autocorrelation functions are better fitted by stretched exponentials (Williams and Watts 1970). The Williams-Watts analysis is mainly employed for phase transition in colloidal suspensions (Ruzicka et al. 2004 Katzel et al. 2007) and for polymer suspensions. 

Such distribution has been shown to span several orders of magnitude for instance in the case of local motions in the glassy state. Instead of the distribution function, other fitting functions for correlation fimction C (t) are considered. One of these functions is the stretched exponential Kohlraush-Williams-Watts function exp —(f/tkww) that has been found to fit the data quite universally (2). 

At each temperature and pressure study in the high viscosity region, the analytical expression C(t) = a ( 1 + b0 (t)) has been adjusted to the digital experimental correlation function Cgjjp (t). Here, 0(t) has been taken to be equal to the empirical Williams Watt (15) relaxation function exp(-(t/s y ), in which 0<9< is related to the width of the distribution function and "2 is nearly the time of its maximum amplitude which decays asymmetrically on both sides. This relaxation function has generally proved to adequately represent the experimental data obtained from viscous liquids. 

Sun and Wang report a series of studies of polystyrene polymethylmethacrylate mixtures (in benzene, dioxane, and toluene, respectively) using light scattering spectroscopy as the major experimental technique(78-80). Both polymers were in general nondilute. Neither polymer is isorefiractive with any of the solvents. The objective was to study the bimodal spectra that arise under these conditions and to show that the two relaxation times and the mode ampUtude ratio can be used to infer diffusion and cross-diffusion coefiBcients of the two components. Experimental series varied both the total polymer concentration and the concentration ratio of the two components. The theoretical model predicts a biexponential spectrum. The experimental data were fitted by a bimodal distribution of relaxation rates or by a sum of two Williams-Watts functions. The inferred self-diffusion coefiBcients of both species fall with increasing polymer concentration. 

Schematic of Williams-Watts distribution functions for three values of 6 according to equation (10). Reference (10).

The molecular dynamics associated with the glass transition of polymers are cooperative segmental dynamics. The relaxation process of the cooperative segmental motions is known as the a-relaxation process. At the glass transition, the length scale of a cooperative segmental motion is believed to be 1-4 nm, and the average a-relaxation time is 100 s [56]. The a-relaxation process is represented by a distribution of relaxation times. In time-domain measurements, the a-relaxation is non-exponential and can be described by a stretched-exponential function. The most common function used to describe the a-process is that of the Kohlrausch-Williams-Watts (KWW) [57, 58] equation ... 

The weakness of this approach as first introduced by Narayanaswamy [3] is the assumption that a single relaxation time can describe the physical aging process accurately. Some improvement can be achieved by introducing the Kohlrausch-Williams-Watts (KWW) stretched exponential function [8] to describe the distribution of relaxation times eq>ressed as ... 

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... 

---