Chemistry Reference
In-Depth Information
chain composed of
N
freely oriented segments. The other dimensions of this polymeric chain are
y
)
0.5
.
and
z
. These are coordinates that become reduced as a result of stretching to 1/(
a
2
2
a
1
Þx
0
2
þða
1
Þr
0
2
Work done per chain
¼ð
3
kT=
2
Na
Þ½ða
1
where the chain ends are initially
r
0
distance apart [
16
],
r
0
2
2
2
2
¼ x
þ y
þ z
Bueche [
16
] also described the average energy per chain as
2
a
1
Þþða
1
Ave
:
energy/chain
¼ð
3
kT=
2
Þð
1
=
3
Þ½ða
1
Þ
, was shown to be related to strain in elongation of polymeric
elastomers. For up to 300% elongation, or more, the following relationship [
22
] applies:
The modulus of elasticity,
G
2
3
G ¼
2
mkTb
ð
1
þ
2
=g
Þ
Where
k
is Boltzmann's constant,
m
represents the number of polymeric chains in the sample, and
g
is
the strain. The relationship of stress to strain is:
2
3
¼
mkTb
ðg þ
=g
Þ
Stress
2
2
There are several molecular theories of rubber-like elasticity The simplest one is based on a
Gaussian distribution function for the end to end separation of the network chains: [
23
] (the
dimensions of the free chains as unperturbed by excluded volume effect are represented by (
2
)
0
)
r
2
3
=
2
exp
2
2
oðrÞ¼ð
3
=
2
pðr
Þ
0
Þ
ð
3
r
=
2
ðr
Þ
0
Þ
2
)
0
applies to the network chains both in the
unstretched and stretched state. The free energy of such a chain is described by a Boltzmann
relationship [
23
]:
Within this Gaussian distribution function (
r
2
2
FðTÞ¼KT
ln
oðrÞ¼CðTÞþð
3
kT=
2
ðr
Þ
0
Þr
C
(
T
) is a constant at a specified absolute temperature
T
. The change in free energy for a stretched
elastomer can be expressed as follows:
2
2
x
x
2
2
y
y
2
2
z
z
2
2
2
2
DF ¼½
3
kT=
2
ðr
Þ
0
½ða
þ a
þ a
Þðx
þ y
þ z
Þ
where
a
is the molecular deformation ratio of
r
components in
x
,
y
,
z
directions from the unstretched or
elastomer to one that was stretched and deformed. Additional discussions of this theory and other
theories of elasticity are not presented here because thorough discussions of this subject belong to
topics dedicated to physical properties of polymers.
2.2.3 Rheology and Viscoelasticity of Polymeric Materials
When an amorphous polymer possesses a certain amount of rotational freedom, it can be deformed
by application of force. Application of force will cause the polymer to flow and the molecules will
