Chemistry Reference
In-Depth Information
network where only two chains would be joined by the formation of the link. In
this latter case a value of N
i
42 is required for a network.) The potential
number of network springs is
n
s
¼
ð
N
i
1
Þ
2
N
A
c
M
n
ð
2
:
57
Þ
Here, the 2 in the denominator is to avoid double counting as it takes two
sites to form one link. However, closed loops and chain entanglements are both
possibilities and eqn (2.57) must be modified for these effects. We may write the
network modulus as
G
N
¼
n
s
k
B
T
ð
1
f
l
Þþ
Bc
2
ð
2
:
58
Þ
where the entanglement correction coefficient was assigned a value of
B
¼
0.25m
5
kg
-1
s
-2
in line with swollen rubber networks and f
l
is a correction
for the probability of closed loops being formed, which can be written
!
N
A
M
1
=
n
c
2b
3
=
2
N
i
1
1
=
N
i
f
l
¼
1
þ
ð
2
:
59
Þ
½
where b is dependent on the characteristic ratio of the polymer chain (6.38 for a
cellulose ether chain), the skeletal bond length, l
b
¼
0.187 nm, and molecular
weight per skeletal bond, M
b
¼
69 Daltons:
3M
b
2pC
N
l
b
b
¼
ð
2
:
60
Þ
The calculated curves from eqn (2.58) are shown in Figure 2.12 and give a
good fit to the data with between three and four hydrophobes per chain.
2.4.6 Noninteractive Fillers
There are many situations when polymer networks contain a filler. The particle
size is frequently chosen to be in the colloidal size range. This is partly for ease
of processing, but frequently fillers are used to produce colour and/or opacity
as well as improving the mechanical properties and it is for this reason also that
colloidal particles are used as they provide good light scattering at low addition
levels. However, we will concentrate on changes in mechanical properties
caused by fillers. The term ''noninteractive filler'' means that the filler does
not play a role in the crosslinking of the network. Even so, fillers can have a
marked effect on both the elastic properties and the wear resistance. Filler
particles are usually inorganic or organic particles with a high modulus. For
example, carbon particles are used in car tyres.
The simplest approach would appear to be to add the compliances of the two
phases in proportion to the volume fractions of each phase, j:
J
¼
1
G
N
ð
1
j
Þþ
1
G
f
j
ð
2
:
61
Þ
