Equation Kohlrausch-Watts-Williams

Molecular mobility in amorphous materials is related to the macromolecular properties like viscosity it is generally quantified in terms of mean relaxation time and it determines physical stability and reactivity. The relaxation time is defined as the time necessary for a molecule or chain segment to diffuse across the distance of one molecule or chain segment. The relaxation time varies with temperature and the typical relaxation times at Tg are estimated to be 100-200 s (Ediger et al. 1996). Molecular relaxation times can be characterized by the change of several bulk properties like enthalpy or volume or spectroscopic properties. The extent of relaxation is described empirically by the Kohlrausch-Williams-Watts equation (Hodge 1994) ... 

Kohlrausch-Williams-Watts Equation n This empirical equation aids designers of load-bearing products made of plastics and reinforced plastics. 

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]... 

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 has the familiar form of the Kohlrausch-Williams-Watts (KWW) equation [17], except that p and x are not empirical constants here, and they will be discussed in the next two sections. 

One of the features observed in many glass-forming liquids is the non-linear nature of any relaxation processes that occur around and below Tg. The relaxation rate is found to depend on the sign of initial departure of actual sample from the equilibrium state. The relaxation rate is described well by the Kohlrausch-Williams-Watts (KWl O empirical equation. ... 

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)... 

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).

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] ... 

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... 

At the lowest temperature (220 K, but above rp where the a- and p-relaxation processes merge, extremely long simulation times (1 is) were necessary to include even a portion of the time needed for structural (a) relaxation. Both the simulated and experiment results versus time (0 after 2ps could be fitted to the well-known Kohlrausch-Williams-Watts (KWW) stretched exponential equation ... 

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. 

A number of equations exist to calculate the molecular mobility or its reciprocal, the relaxation time, x (Andronis and Zografi 1997 Di Martino et al. 2000 Mao et al. 2007 Yoshioka et al. 1994). The two most commonly used equations that give an indication on the mobility within an amorphous sample are the Kohlrausch-Williams-Watts (KWW) equation and the Adam-Gibbs (AG) equation. 

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

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 ... 

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 ... 

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