Williams-Watts function, equation

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 is an experimentally demonstrated fact that the a relaxation in the time domain fits the stretch exponential decay function (0 or the Kohlrausch-Williams-Watts (KWW) equation (7,8)... 

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... 

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).

In the time domain, the a relaxation process of amorphous polymers is successfully described by the Kolrausch-Williams-Watts (KWW) relaxation function (equation (15)) where the parameter Pkww that takes in account the non-Debye character of the time decay leads to an asynunetric broadening of < )( ) at short times, which typically vary between 0.2 and 0.5. 

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] ... 

To obtain estimates of diffusivity from the simulations, Xiang and Anderson [24b] adapted the Kohlrausch-Williams-Watts (KWW) stretched exponential function [94] to fit the Dj decay profiles using the following equation ... 

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. 

At low hydrations, the decay of the autocorrelation function Fi may be described by the Kohlrausch-Williams-Watts stretched exponential equation [650, 651] ... 

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

Jd in equation (6) is equal to Jg - Jg, where Jg is the steady state compliance which is equal to the long-time limiting value of the recoverable compliance for a non-cross-linked system. i/(t) is the normalized memory function for the compliance, and it goes from i/ (0) = 0.0 to (oo) = 1.0. The normalized memory function can often be described using the generalized Kohlrausch-William-Watts (KWW) (37,38) or stretched exponential function ... 

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... 

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 ... 

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]... 

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... 

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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