Williams-Watts distribution

For describing this heterogeneous distribution of jump rates, a symmetric log-Gaussian distribution is not satisfactory, as mentioned. An asymmetric distribution with a more rapid decay at low rates, similar to the Williams-Watts distribution, yields an excellent agreement with the experimental data. 

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. 

On the other hand, some phenomenological distributions of relaxation times, such as the well known Williams-Watts distribution (see Table 1, WW) provided a rather good description of dielectric relaxation experiments in polymer melts, but they are not of considerable help in understanding molecular phenomena since they are not associated with a molecular model. In the same way, the glass transition theories account well for macroscopic properties such as viscosity, but they are based on general thermodynamic concepts as the free volume or the configurational entropy and they completely ignore the nature of molecular motions. 

On the other hand, some phenomenological distributions of relaxation times, such as the well known Williams-Watts distribution (28)... 

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

Traditionally, wide-ranging relaxation-time distributions P i) relative to population-average times r have been described by stretched exponentials of the Kohlrausch-Williams-Watts type given by (see Chapter 1)... 

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). 

The molecular domains in amorphous structure behave like an ensemble of autonomous substates, each following unique relaxation kinetics during annealing (Kawakami and Pikal 2005). This relaxation distribution is often expressed using an empirical Kohlrausch-Williams-Watts (KWW) equation (Eq. 14.3) ... 

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. 

Note, however, that the described classification of release mechanisms is valid for a given single microcapsule. A mixture of microcapsules usually includes a distribution of capsules varying in size and wall thickness. Since any spray-dried powder produced from an emulsion is essentially such a mixture of microcapsules with variations in their properties, the parameter n in Eq. 6.6 varies depending on the properties of the powder. Equation 6.6 is essentially analogous to the equation of Kohlraush-Williams-Watts (KWW). This relationship can be expressed as (Williams and Watts, 1970)... 

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