Kohlrausch relaxation function

Figure 1. Electric modulus relaxation spectra (M") of the ionic liquid BMP-BOB at ambient pressure and 231 and 245 K are plotted as solid lines. High pressure M" data (0.5 GPa) at the temperatures that yield relaxation times similar to those of the ambient pressure data, 283 and 308 K, are included in the figure as squares. Data at 0.5 GPa data are slightly shifted in frequency to match perfectly the atmospheric peak frequencies. Long and short dashed lines are fits to a Kohlrausch relaxation function with fi=(l-n)= 0.56 and 0.50, respectively. The inset shows the good correspondence between the stretching parameter fi and the relaxation time at different temperatures and at atmospheric pressure and at 0.5 GPa.

An intimate connects exists between the shape of the relaxation function and steepness index [3,5,48,89,116,117], Strong liquids have less broad relaxation functions compared with fragile glass formers. The degree of nonexponentiality is reflected in the Kohlrausch exponent [1... 

Actually, up to the present time, many-body relaxation is still an unsolved problem in condensed matter physics. In his magical year of 1905, Einstein solved the problem of diffusion of pollen particles in water discovered in 1827 by the botanist, Robert Brown. In this Brownian diffusion problem, the diffusing particles are far apart and do not interact with each other and the correlation function is the linear exponential function, exp(-t/r). It is by far simpler a problem than the interacting many-body relaxation/diffusion problem involved in glass transition. It is a pity that Einstein in 1905 was unaware of the experimental work of R. Kohlrausch and his intriguing stretch exponential relaxation function, exp[-(t/r) ], published in 1847 and followed by other publications by his son, F. Kohlrausch. 

A basic feature of the response of fragile liquids to various perturbations is the pronounced nonexponential relaxation behavior. The relaxation function typically exhibits a two-step feature. The fast relaxation at short times is generally associated with vibrational degrees of freedom. The long-time decay of the relaxation function 4>(t), which is governed by the structural relaxation, can often be described by the stretched exponential or the Kohlrausch-Williams-Watts (KWW) function... 

Alvarez F, Alegria A, Cohnenero J (1991) Relationship between the time-domain Kohlrausch-Williams-Watts and frequency-domain Havriliak-Negami relaxation functions. Phys Rev B Condens Matter 44 7306-7312... 

Alvarez, F., Alegria, A., Colmenero, J. Relationship between the time-domain Kohlrausch-WUUams-Watts and frequency-domain HavriUak-Negami relaxation functions. Phys. Rev. B 44, 7306-7312 (1991)... 

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 ... 

Equation (7) is also only valid for a short-term creep experiment in which the time of creep is short relative to the time scale of aging such that the characteristic relaxation time is constant. It is noted that the relaxation function of equation (7) has the correct limits and differs from the Kohlrausch compliance function J = Js ) which was suggested initially by Struik (14) and which is not physically meaningful at long times (28). 

In practice, the Kohlrausch-Williams-Watts (KWW) or stretched-exponential relaxation function... 

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)... 

The molecular relaxation process has been studied by the autocorrelation function of normal modes for a linear polymer chain [177]. The relaxation spectrum can be analyzed by the Kohlrausch-Williams-Watts function [177,178] ... 

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)... 

The Mittag-Leffler function has interesting properties in both the short-time and the long-time limits. In the short-time limit it yields the Kohlrausch-Williams-Watts Law from stress relaxation in rheology 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).

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,... 

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