Neutron Scattering

2
diameter d . At that time (ze) the chair has explored the lateral confinement r,-- Nz W ; with
(d2 - N, Ë 2 ) . For longer times t -< rR the Rouse modes relax along the tube (local reptation) .
Thereafter longitudinal creep governed by the Rouse diffusion coefficient DR along the tube
dominates .
This process takes place until the chair bas left its original confinement at a time
rd -
W
1N3 . Beyond that time normal diffusion takes over .
For the mean square segment displacement the reptation mechanism invokes a sequence of
power laws in the time variable . For short times t < re Rouse motion prevails and Are oc tii2
holds .
Then in the regime of local reptation we deal wich Rouse modes occurring along a
contorted Gaussian tube. The segment displacement along the tube follows a t1/2 law, in real
space considering the random walk nature of the tube, this transforms to a tli4 law. After all
Rouse modes have relaxed, Rouse diffusion along the contorted tube takes place .
A similar
argument as before leads to a power law Ar e ~c
lifetime of the tube constraints, Ar e cc t holds .
tv2 and only for times longer than rd,
the
The tube constraints also provoke a strong retardation for the single chain relaxations causing
a near plateau regime in the time dependent single chain conelation function. Neglecting the
initial free Rouse process de Gennes has formulated a tractable expression for the dynamic
structure factor which is valid for t > re, i .e . once confinement effects become important . In
the large Q limit the dynamic structure factor assumes the form
S(Q,t)
S(Q,0)
1-exp
CQd )2
exp(t/ro ) erfe W7
/ro ) (15 .22)
8
+ z
exp
(Qd) 2 )
~ 6
exp (-n2t / rd )
For short times S(Q, t)
decays mainly due to local reptation (first term), while for longer times
(and low Q) the second term resulting from the creep motion dominates . The two time scales
3 2
are given by ro = W36 4 and rd =
ßd2
. Since the ratio of these time scales is
proportional to N3 for long chains at intermediate times re < t < rd a pronounced plateau in
15-23