Generalized relaxation function

More General Relaxation Functions and Spectral Lineshapes... 

The above equations are generally valid for any isotropic material, including critical gels, as long as the strain amplitude y0 is sufficiently small. The material is completely characterized by the relaxation function G(t) and, in case of a solid, an additional equilibrium modulus Ge. 

The data has been superimposed by dividing the relaxation function G(t) by G(t = 0), the limiting short time value, and the time has been divided by the characteristic relaxation time Tr. The first feature to notice is that the stress relaxation function overshoots and shows a peak. This is an example of non-linear behaviour. It is related to both the material and the instrumental response (Section 4.5.1). The general shape of the curves (excluding the stress overshoot) can be described using two approaches. 

Other choices are possible for the relaxation function r. For these choices, expressions similar to Eq. (19) may be derived. Although it is awkward to deal with the absolute value in Eq. (14), a parabolic form meeting the principal requirements is more readily expressed in Fourier space. A simple quadratic (Willson, 1973) and its generalization to other powers have been used by Blass and Halsey (1981) ... 

Here n is an integral exponent. Frieden (1975), on the other hand, generalizes the original relaxation function [Eq. (14)] to treat the case of arbitrary upper bound B and lower bound A ... 

The Mittag-Leffler function [44-46] can be viewed as a natural generalization of the exponential function. Within fractional dynamics, it replaces the traditional exponential relaxation patterns of moments, modes, or of the Kramers survival. It is an entire function that decays completely monotoni-cally for 0 < a < 1. It is the exact relaxation function for the underlying multiscale process, and it leads to the Cole-Cole behavior for the complex... 

The fractions of protons decaying according to relaxation functions Rx and R2 are given by fj and f2. In molten polymers this relationship has long been exploited to provide a measure of the crosslink density in many polymer systems [64-83]. The form of the decay functions has been the subject of much discussion, however, it is often observed that Rj and R2 can be approximated by simple exponential decay functions. It is generally accepted that the protons with short relaxation times are those directly attached to or adjacent to crosslink points. As an example Figure 13.4 shows the decays of transverse... 

Due to the dead time, the correlation functions Fcos,sin(rm rp) for short tp cannot be measured with three-pulse sequences, but a further pulse is necessary to refocus the stimulated echo, leading to four-pulse sequences such as (tc/2)x — tp — (n/2) x - tm - (n/2)x - A - (rc/2)y, where the echo forms at a time tp + A after the last pulse.6,102 So far, we assumed x > tp so that molecular dynamics during the evolution time can be neglected. In the studies of relaxation processes in glasses, this assumption is not justified since very broad distributions of correlation times G( 1 n x) exist. Then, it is necessary to explicitly calculate the phases (h, tf) according to Eq. (8) so that, in general, correlation functions resulting from the above four-pulse sequence can be written as... 

Considering one chain segment characterized by a normahzed p stretching vector, and a number n of skeletal bonds, the transverse magnetic relaxation function of nuclear spins, attached to this segment, is generally expressed as a product... 

But, for large polydispersities, the Rouse process (B) of the long chains overlaps the reptation process of the short chains. Consequently, the most general expression of the relaxation function (or relaxation modulus) must include all the relaxation processes described in part 3.2. 

The function thus defined goes fi om 1 to 0 as / increases and is called the polarization relaxation function. We require t) = 0 for / < 0 to retain the proper causal sequence of events, so in general... 

In the time domain, the single-mode approximation, equation Eq. (191), is equivalent to assuming that the relaxation function Cy(t) as determined by the exact equation, Eq. (182) (which in general comprises an infinite number of Mittag-Leffler functions), may be approximated by one Mittag-Leffler function only, namely... 

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... 

In previous chapters phenomenological relaxation equations were used together with the Onsager regression hypothesis to compute time correlation functions. In this section we present a microscopic derivation of generalized relaxation equations (Zwan-zig, 1961 Berne, Mori, 1965 and 1971). These equations can be used to compute time-correlation functions under circumstances where the usual phenomenological equations do not apply. 

---