Chemistry Reference
In-Depth Information
This is proportional to the rate of change of occupancy of our tube with time.
Now, we can use this function to describe our shearing behaviour:
s
ðÞ¼
G
ðÞ
f
1
g
ðÞþ
Z
t
m
ð
t
Þ
f
1
g
ðÞ
dt
ð
6
:
105
Þ
0
This expression can describe the viscosity change with time at a fixed rate and,
in the limit of long times, provide the steady-state viscosity at a range of shear
rate. The term f
1
(g) is an integral function of the strain. Approximate forms are
available, for example:
f
1
ðÞ¼
5 g
5
g
4
þ
a
1
g
3
þ
b
1
g
5 g
6
þ
c
1
g
4
þ
d
1
g
2
þ
b
1
ð
6
:
106
Þ
ð
Þ
with a
1
¼
45.421, b
1
¼
1269.1, c
1
¼
10.942 and d
1
¼
343.57 and, while awkward
in form, contains no mathematical features that are not readily numerically
evaluated. This approach has been extended to include normal force and
elongational viscosity calculations both with time and the steady-state values.
The only change required is in the form of the strain function and an additional
strain term to determine the normal force. A comparison of the Graessley (G)
and Doi-Edwards (DE) models is shown in Figure 6.21 for the viscosity in
continuous shear. The reduction of viscosity with shear rate is much more rapid
with increasing rate with the DE model than the G model. The G model has
been well tested on near monodisperse molecular weight systems. In addition,
the DE model has the surprising result of predicting a reduction in stress with
increasing rate once a value a little greater than gt
d
is achieved. The reason for
this is yet to be resolved. It has been observed in pipe flow that with high
molecular weight polymers abrupt changes in shear rate occur. It is feasible that
the DE model is describing drag-reduction behaviour seen in pipe flow or
unsteady flow. If we include crudely the role of the Rouse relaxation mecha-
nisms within the tube there is a second process observed, shown as a kink on the
DE curve. This is also shown in Figure 6.21 with an arbitrarily selected value of
t
e
. The same data is shown as shear stress versus shear rate in Figure 6.22.
It is fairly clear that as t
e
approaches t
d
the role of Rouse relaxation is
significant enough to remove the dip altogether in the shear-stress-shear-rate
curve. As the relaxation process broadens, this process is likely to disappear
particularly for polymers with polydisperse molecular weight distributions. The
success of the DE model is that it correctly represents trends such as stress
overshoot. The result of such a calculation is shown in Figure 6.23.
The qualitative agreement between theory and experiment is very good.
24
The
DE approach can be used to provide qualitative information on the effects of
branching. In order for a simple unbranched polymer to relax an applied strain
we have seen there are two relaxation processes, a Rouse mechanism within the
tube and the disengagement of the polymer from the tube. For a branched
polymer the arm is tethered at one end so this restricts motion. In order for
disengagement to occur the arm has to retract itself down the tube. This is the
dominant timescale and determines the viscosity. We can think of this as akin to
