Neutron Scattering

S (Q, t) = Sa (Q, t) S ß (Q, t) / S (Q)
S (Q)
(15 .7)
The approximation behind Eq .[15 .7] is called Vinyard approximation and approximates the
proper pair correlation fonction by its self counter part . The self correlation fonction for a
diffusive process relates directly to the mean square displacement .
2
Sse(f (Q,t) = exp 1-2
(r2
(t)"
(15 .8)
where (r 2(t» is the mean square displacement of the flowing particle . According to
Eq .[15 .1] this should be described by a stretched exponential with the consequence
(r2 (t)> =D tß
and
=QZ1aD-, lp
(15 .9)
the combination of Eq .[15 .8] and [15 .9] invokes sublinear diffusion of the polymer segments
as the underlying reason for the stretched exponential behaviour . Its signature is a power law
dependence ofthe Kohlrausch-William-Watts relaxation times T.,,, with an exponent 2/ß .
15.2 .2 Experimental results
15.2 .2 .1 Self correlation function
We commence with the secondaiy relaxation taking polyisobutylene as an example. Fig . 15 .4
presents the relaxation map of PIB . The solid line corresponds to the dielectric ß-relaxation,
the dashed line named S represerrts NMR results interpreted as a methyl group rotation. y and
y' are theoretically predicted relaxation mechanisms . In the dynamic regime, where the (Xrelaxation
is too slow to contribute, neutron backscattering has been employed, in order to
unravel details of the ß-process in this polymer. Fig .15 .5 presents as an example a spectrom
taken at T = 270K and Q = 1 .7Â-1 . The spectrum is characterized by a narrow peak, which
nearly coincides with the instrumental resolution function (dashed line) and a broad foot
revealing the relaxational behaviour. Such a spectral shape is typical for broad distributions of
relaxation times, where only a part of it is resolved in the spectrum .
15-7