Chemistry Reference
In-Depth Information
So, by equating the two network moduli we can obtain our tube parameters. If
we include all the necessary numerical constants we get
Nb
2
a
2
M
e
¼
4
M
ð
6
:
100
Þ
5
This relationship allows us to make a link with experimentally determined
parameters such as entanglement density. This should apply in the melt state.
In practice, this relationship is not completely satisfactory and to an extent a
must be regarded as an adjustable parameter. It should be recalled throughout
this modelling that we have not included the nature of any specific interactions
and in real systems these are likely to be determining factors. The low-
shear-rate viscosity can be determined from the integral in the stress-relaxation
function. As it is dominated by the longest time process, it is quite satisfactory
to calculate it from the reptation motion alone:
Z
ðÞ¼
Z
N
G
N
c
ðÞ
dt
¼
p
2
12
G
N
t
d
ð
6
:
101
Þ
0
The tube-disengagement time is the longest relaxation process. Now, suppose
we were to apply a large strain to the sample so that we are no longer in
the linear regime. When the chain is deformed the contour length increases
and the tube is deformed. At a time t equivalent to t
R
this part of the strain is
relaxed and the contour length is that in the quiescent state. After time t
d
the primitive chain has disengaged itself from the surrounding tube by repta-
tion. The deformation of the tube and the subsequent relaxation is a
complex feature to model. In its simplest approach the Doi-Edwards solution
results in a decoupling between the linear relaxation function and the
strain dependence. We have seen such a decoupling approach before (Section
6.2.2):
Gt
; ðÞ¼
G
ðÞ
f
1
ðÞ
g
ð
6
:
102
Þ
The function f
1
(g) represents the nonlinear strain dependence of the deformed
tube. The nonlinear stress-relaxation function in the reptation zone is thus
N
p
2
p
2
exp
tp
2
t
d
G
ðÞ¼
G
N
c
ðÞ
f
1
ðÞ
g
¼
G
N
f
1
ðÞ
g
8
ð
6
:
103
Þ
p;odd
This follows the form of phenomenological descriptions of polymer systems
such as the BKZ model,
10
which is encouraging. Very good fits to experiments
have been found using this approach. In order to take this idea forward the
most convenient method is to use the memory function:
m
ðÞ¼
G
N
dc
ðÞ
dt
ð
6
:
104
Þ
