Chemistry Reference
In-Depth Information
An extreme case of tumbling dominating the relaxation behaviour can be
seen with rigid rod-like species. The underlying concept is that the application
of a strain is relaxed by the rotary diffusion of the rod. This gives rise to a single
relaxation time. The precise shape of the molecule determines the form of the
storage and loss moduli with frequency, and there are a number of solutions.
The effect of branching, molecular weight distributions and unusual polymer
conformations can be predicted using the bead-spring model.
5.6.3 Undiluted and Concentrated Nonentangled Polymers
Above a concentration c
w
the polymer can be considered to be concentrated, as
shown by the phase diagram in Figure 5.23. At low molecular weights there is
no significant entanglement coupling. As there is little or no solvent present, the
hydrodynamic interactions can be considered to be negligible and the Rouse
model is appropriate for describing the chain viscoelasticity. The density of the
polymer controls the magnitude of the elasticity:
k
B
T
X
N
G
ðÞ¼
r
c
N
A
M
e
t
=
t
p
ð
5
:
95
Þ
p
¼
1
ot
p
2
k
B
T
X
N
G
0
ð
o
Þ¼
r
c
N
A
M
ð
5
:
96
Þ
2
1
þ
ot
p
p
¼
1
k
B
T
X
N
G
00
ð
o
Þ¼
r
c
N
A
M
ot
p
1
þ
ot
p
ð
5
:
97
Þ
2
p
¼
1
However, in such a high concentration regime we can no longer represent the
relaxation times (eqn (5.92)) in terms of the intrinsic viscosity. In the low-
frequency limit, because there is no permanent crosslinking present the loss
modulus divided by the frequency should equate with the low-shear-rate viscosity
k
B
T
X
N
Z
ðÞ
o
¼
G
00
ð
o
Þ¼
r
c
N
A
M
ot
p
ð
5
:
98
Þ
p
¼
1
If we perform the same substitutions made in eqns (5.93) and (5.94) we get
6Z
ðÞ
M
p
2
p
2
N
A
k
B
T
t
p
¼
ð
5
:
99
Þ
This relationship between the relaxation modes and the low-shear viscosity is an
important one. It indicates that the longest Rouse relaxation time, i.e. p
¼
1mode:
Z
ðÞ¼
t
1
p
2
N
A
k
B
T
6M
ð
5
:
100
Þ
determines the zero-shear viscosity. This applies to both undiluted and highly
concentrated polymers below their entanglement concentrations. We know that
in this concentration region of the polymer the viscosity has a near-linear
dependence on molecular weight (Figure 5.22). From eqn (5.92) we can see that
