Kohlrausch stretched exponential function

This is none other than the time-temperature superposition principle. However, the exact shape of the function F(t) is not a generic feature of the theory. A good approximation is provided by the Kohlrausch stretched exponential function. In the frequency domain, Eqs. (23) define the shape of the susceptibility minimum usually observed in the gigahertz range, and an interpolation formula follows ... 

Because the material ages continuously, the creep tests performed as in Fig, 7.27 must be of short duration at each aging time so that the compliance result is representative of viscoelastic properties at that aging time. The individual curves and the shifted master curve (also called the momentary master curve) is typically well fit by a Kohlrausch stretched exponential function... 

A model having predictions that are consistent with the aforementioned experimental facts is the Coupling Model (CM) [21-26]. Complex many-body relaxation is necessitated by intermolecular interactions and constraints. The effects of the latter on structural relaxation are the main thrust of the model. The dispersion of structural relaxation times is a consequence of this cooperative dynamics, a conclusion that follows from the presence of fast and slow molecules (or chain segments) interchanging their roles at times on the order of the structural relaxation time Ta [27-29]. The dispersion of the structural relaxation can usually be described by the Kohlrausch-William-Watts (KWW) [30,31] stretched exponential function,... 

It is interesting to note that the stretched exponential function was first used by Kohlrausch in his studies on electrical relaxation of leyden jar (a glassy ionic material) in 1847 (cited by Ngai, 1996). 

Another way to extract information on ([)(/) is to assume a stretched exponential function, as described by the Kohlrausch-Williams-Watts (KWW) model [9,10] ... 

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... 

The response function is chosen, according to Moynihan et at. [50], in a manner of a Kohlrausch function [51], which is also called the stretched exponential function and is often used as a phenomenological description of relaxation in disordered systems. The value of p, which is called the stretching exponent and describes the nonexponential characteristic of the relaxation process, is defined as... 

The relaxation dynamics of junctions in polymer networks have not been well-known until the advent of solid-state NMR spin-lattice relaxation measurements in a series of poly(tetrahydrofuran) networks with tris(4-isocyanatophenyl)-thiophosphate junctions [100]. The junction relaxation properties were studied in networks with molecular weights between crosslinks. Me, ranging from 250 to 2900. The dominant mechanism for nuclear spin lattice relaxation times measured over a wide range of temperatures were fit satisfactorily by spectral density functions, /( ), derived from the Fourier transforms of the Kohlrausch stretched exponential correlation functions... 

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... 

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 ... 

S Yoshioka, Y Aso, S Kojima. Usefulness of the Kohlrausch-Williams-Watts stretched exponential function to describe protein aggregation in lyophilized formulations and the temperature dependence near the glass transition temperature. Pharm Res 18 256-260, 2001. 

In practice dielectric relaxation curves for polymers (1-5) and glass-forming liquids (eg. Fig. 1) are far broader than that for an SRT process. Numerous empirical relaxation fiinctions have been proposed for this behavior (1,2,5). One function is the stretched exponential function of Kohlrausch, Williams, and Watts (KWW) (4-6), that is widely apphed to dielectric and other relaxation data for amorphous polymers ... 

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 ... 

The dwell portion of the force versus time curves was fit to the Kohlrausch-Williams-Watts function (stretched exponential function) ... 

Several types of functional decay forms have been used to describe time-dependent processes in poled polymer systems. The Kohlrausch-Williams-Watts (KWW) [72] stretched exponential function has often been used to fit the orientation decay of chromophores in polymers ... 

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).

---