Neutron Scattering - JUWEL - Forschungszentrum Jülich

8 O. Holderer, M. Zamponi and M. Monkenbusch
ζ v D R
3kT
l Δ 2 x
Fig. 4: Polymer chain in the Rouse model [3] as Gaussian chain with beads connected by
springs.
The Langevin equation can be solved and one can calculate with equation 8 the intermediate
scattering function, which is measured by NSE (for details, see the lecture on “Dynamics of
Macromolecules”):
I(Q, t) =exp(−Q 2 Dt)Iintern(Q, t) (12)
with a diffusive part with a relaxation rate proportional to Q2 and the part describing the internal
relaxation, which can be written for QRG >> 1:
Iintern(Q, t) = 12
Q2l2 with the relaxation rate
and
� ∞
0
h(u) = 2
�
π
� �
du exp(−u − (ΓQt)h(u/ (ΓQt))) (13)
ΓQ = kBT
12ζ Q4 l 2
dx cos(xu)(1 − e −x2
)/x 2
Note that the local relaxation rate depends on Q 4 . When I(Q, t)/I(Q, 0) is plotted against the
Rouse variable � ΓQt, all curves collapse onto a master curve if the Rouse model holds.
With this model e.g. the dynamic of short polymers in the melt can be described. With increasing
molecular weight other effects like the constraints imposed by mutual entanglements of the
polymer chains become important, which are described in the reptation model by DeGennes
(Nobel prize 1991). In this experiment polymers in solution, not in the melt, are considered.
The Rouse model then needs to be extended by hydrodynamic interactions as will be described
in the following section.
(14)
(15)