Chemistry Reference
In-Depth Information
A microstructural origin has been suggested by both Cates and Muthumkumar
38
in terms of a fractal cluster with dimension D (Section 6.3.5). The complex
viscosity was found to vary as:
Z
o
2
=ð
D
þ
2
Þ
B
ð
5
:
142
Þ
by Muthumkumar and Freed.
33
Using the above expressions to calculate the
complex viscosity and equating the powers we get a relationship between the
fractal dimension and the index m:
2m
1
m
D
¼
ð
5
:
143
Þ
This relationship indicates that for a stoichiometric gel the fractal dimension is 2.
This argument need not be restricted to polymer gels but any colloidal fractal
object. Another interesting feature to emerge from this modelling was the value of
S that did not reach a maximum value at r
¼
1 but reaches a plateau as r
2.
However, the fractal dimension changes little in this region giving a value of
D
¼
2; D only rises when the stoichiometry reduces to r
o
1. The network strength
increases past the gel point and the viscosity begins to diverge towards infinity.
The final network strength is an important reason for crosslinking polymers and
this is considered further in the following section.
-
5.7.2 Chemical Networks
The crosslink density of a polymer network determines the number of elasti-
cally effective chains. Some of the chains are tied to a network and will not be
able to relax the applied strain. This gives an elastic response at all frequencies
and times. In a stress-relaxation experiment the strain will decay due to some
Rousian relaxation processes of all the chains, but some energy will be ''per-
manently'' stored in the network. This long-time relaxation process was mod-
elled by the theory of rubber-like elasticity. The equilibrium modulus G
e
can be
expressed as:
n
n
k
B
T
R
e
R
2
G
e
¼
A
1
ð
5
:
144
Þ
where n
n
is the number of moles of networked chains per unit volume. The
mean squared end-to-end distance of the chain in the network is given by R
e
and that of the same molecular weight of an unconstrained polymer is R
2
. The
ratio of the terms is typically of the order of unity. The constant A
1
contains a
number of factors including terms for the functionality of the link for more
mobile crosslinks (Section 6.5.3). The variety and nature of the chemistry is
critical in determining some of these constants and it is difficult to ascertain
these without some experimental studies. Difunctional oligomers such as those
discussed in the previous section, with controlled crosslinking stoichiometry,
are the systems that are most amenable to modelling. Differing degrees of
crosslinking and changes in the functionality of a link site, i.e. the number of
