Relaxations in the Frequency Domain at Temperatures Slightly Higher than Tg

The a relaxation in the frequency domain can be interpreted in terms of a stretch mean relaxation time, as discussed below. According to Eq. (5.37), the response of a viscoelastic material to a shear stress a(t) is given by 

Here = Jra — Jua, where and are, respectively, the unrelaxed and relaxed compliance functions in the a relaxation process. The values of these quantities could in principle be obtained by extrapolation methods from complex plane plots of / ( ) versus / (co). If (j(t) = Gq Im exp(/o)r), then Eq. (12.1) can be written as 

Gua and being the unrelaxed and relaxed relaxation moduli at infinite and zero frequencies, respectively. 

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) 

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