Exponential relaxation function

Fig. 5. The characteristic frequencies QR and time exponents (3 in the stretched exponential relaxation function obtained for the randomly labelled PDMS melt at 100 °C. (Reprinted with permission from [44]. Copyright 1989 Steinkopff Verlag, Darmstadt)...

Fig. 1. Response of membrane pressure sensor (Baratron 220, MKS, full pressure range 1000 mbar, resolution 0.15 % of full range) after a step-like increase in gas pressure at t=0. The full line represents a fit to the experimental curve ( ) with an exponential relaxation function. The dashed lines indicate the error bars at equilibrium pressure.

The long-pathway rearrangement processes expected for fragile materials at low temperatures are expected to be rare, to involve a local disruption of the otherwise well-structured amorphous medium, and to be relatively long-lived on the usual molecular time scale. These features all contribute to a substantial lengthening of the mean relaxation time /rci(7 ), Eq. (36), with declining temperature. Furthermore, the landscape diversity of deep traps and of the configuration space pathways that connect them should produce a broad spectrum of relaxation times, just as required by stretched-exponential relaxation functions, Eq. (34). 

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. 

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

It is tempting to inquire the necessary conditions which force a spin system to relax exponentially (we shall consider now the longitudinal relaxation only). The answer to this question is simple. A single exponential relaxation function is observed when a spin temperature can be defined. This problem is discussed in detail in reference 3 (p. 116) and we will only give here a brief summary of the arguments developed by Slichter. Let us consider a spin system characterized by a set of Zeeman energies E j, E, . .. at thermal equilibrium at temperature T. 

Of course, knowledge of the entire spectrum does provide more information. If the shape of the wings of G (co) is established correctly, then not only the value of tj but also angular momentum correlation function Kj(t) may be determined. Thus, in order to obtain full information from the optical spectra of liquids, it is necessary to use their periphery as well as the central Lorentzian part of the spectrum. In terms of correlation functions this means that the initial non-exponential relaxation, which characterizes the system s behaviour during free rotation, is of no less importance than its long-time exponential behaviour. Therefore, we pay special attention to how dynamic effects may be taken into account in the theory of orientational relaxation. 

At very short experimental times compared with tm the exponential term tends to 1. Under these circumstances the relaxation function tends to the value of the modulus of the spring. The response is simply that of the spring so that the initial stress divided by the strain gives the modulus of the spring. 

This is the stress relaxation function, so the slope plotted as a function of time provides us with G(t). Now in the limit of short times we find the exponential tends to unity ... 

The mathematics underlying transformation of the data from different experiments can be applied to simple models. In the case of the relationship between G (a>) and G(t) it is straightforward. To give an example, consider a Maxwell model. It has an exponentially decaying modulus with time. We have indicated that the relationship between the complex modulus and the relaxation function is given by Equation (4.117). So if we substitute the relaxation function into this expression we get... 

This is an extended exponential. It operates within the remit of linear viscoelastic theory. So for example for a simple exponential we can show that the integral under the relaxation function gives the low shear viscosity ... 

Finally, it is worthy of remark that, though the comparison between the timescales leads to an almost perfect agreement between the predictions of the Allegra and Ganazzoli model and the collective and self-motion results, it is evident that clear differences appear when comparing the spectral shapes of the respective functions. The model delivers close to exponential decays for both correlators while experimentally one observes significantly stretched relaxation function (j0=O.5). 

Fig. 6.23 a Comparison of the universal Zimm relaxation function F(x) to stretched exponentials. The dotted line is the residual error xlO to the best fit (a=1.354, =0.852). The p value of 2/3 only applies at large values of x with F(x)< 10 , which are irrelevant for NSE data. The dashed blue line in the right part of the figure corresponds to the asymptotic form given in [6] exp(-1.35x ). b Comparison of the integration result solid line), the approximation Eq. 6.48 dashed line) and the asymptotic form dashed-dotted line). Only for very large values of x>30 does the asymptotic value of p emerge... 

Using a simple kinetic model, Solomon demonstrated that the spin-lattice relaxation of the I and S spins was described by a system of coupled differential equations, with bi-exponential functions as general solutions. A single exponential relaxation for the I spin, corresponding to a well-defined Tu, could only be obtained in certain limiting situations, e.g., if the other spin, S, was different from I and had an independent and highly efficient relaxation pathway. This limit is normally fulfilled if S represents an electron spin. The spin-lattice relaxation rate, for the nuclear spin, I, is in such a situation given by ... 

A possible step in this direction can be made through use of earlier relaxation studies on other systems. Hunt and Powles,— when studying the proton relaxation in liquids and glasses, found the relaxation best described by a "defect-diffusion" model, in which a non-exponential correlation function corresponding to diffusion is Included together with the usual exponential function corresponding to rotational motion. The correlation function is taken as the product of the two independent reorientation pro-... 

To retain the analogy with a simple exponential function, it is considered in the cases described by Equations (1.8) and (1.9) that there is a distribution of barrier heights, g(G), each height corresponding to an exponential relaxation (Austin et al. 1975 Nagy et al. 2005). The concentration profile is in this case described by... 

Aging. If we assume independent exponential relaxations for the CRRs, we obtain the following expression for the two-times correlation function ... 

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