Williams-Watts expression

Broadness of the relaxation spectrum, as attested, for example, by the low value of the exponent p in the Kolrausch-Williams-Watt expression. 

Figure 14 compares the Cole-Cole diagrams for a single relaxation time (P = 1), the Cole-Cole expression with p = 0.5, and the Williams-Watts expression with... 

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... 

As mentioned in Chapter 3, glassy relaxation processes are often associated with a fairly broad spectrum of relaxation times. A simple expression that describes this spectrum reasonably well over a wide range of time is the stretched exponential, or Kolrausch-Williams-Watts (KWW) expression (Kolrausch 1847 Williams and Watts 1970 Shlesinger and Montroll 1984) ... 

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... 

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)... 

Equation (79) is a general equation in which the recovery function R(z) and the shift factors a and a are unknown. Aa is unknown but easily measured. In the KAHR model, R(z) is a function which can be expressed by a spectrum of retardation times (equation 80). A more easily visualized function is the so-called Kolrausch -Williams-Watts (KWW) function introduced to describe glassy kinetics by Moynihan et... 

Analysis of the enthalpy relaxation the enthalpy relaxation time and the activation energy were calculated by KWW in accordance with the previous work (Kawai et al., 2004). The KWW theory was originally proposed in dielectric relaxation study by Williams and Watts (1970), then applied in the form of nonexponential function such as the enthalpy relaxation. In KWW theory, the enthalpy relaxation, AH eiax/ which corresponds to the peak area given from the enthalpy relaxation is expressed by the equation... 

To and p being relaxation parameters and 0 < / 1. This empirical function has found theoretical justification thanks to a general model for relaxation phenomena developed by Ngai [132a and b]. This function has been used by several authors to describe electric relaxation phenomena in ionic conductors [133] and also mechanical phenomena relaxation in polymers [134]. Using the Williams and Watts decay function it is possible to show that the expression of M (co ) becomes ... 

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... 

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