Chemistry Reference
In-Depth Information
f
0
stretched to a
where
k
is the Boltzmann's constant. The retractive force for a single polymer chain,
length d
r
at a temperature
T
is, therefore,
f
0
¼Tð
2
d
S=
d
rÞ¼
2
kTb
r
It is generally assumed that the total retractive force of a given sample of an elastomer is the sum of
all the
f
0
forces for all the polymeric chains that it consists of. This is claimed to be justified in most
cases, though inaccurate in detail [
22
].
Tobolsky wrote the equation for the entropy change of an unstretched to a stretched elastomer as
depending upon the number of configurations in the two states [
12
]:
S S
u
¼ k
ln
O=O
u
where
O
u
represent the number of configurations. The evaluation of these configurations by
numerous methods allows one to write the equations for the change in entropy as:
O
and
1
2
N
0
k½ðL=L
u
Þ
2
S S
u
¼
þ
2
L=L
u
3
where
L
u
are the relative lengths of the unstretched and stretched elastomer. Tobolsky
derived the tensile strength as being [
12
]:
N
0
and
L
and
2
X ¼ðN
0
kT=L
u
Þ½ðL=L
u
ÞðL=L
u
Þ
By dividing both sides of the equation by the cross-sectional area of the sample, one can obtain the
stress-strain curve for an ideal rubber.
The retractive force of an elastomer, as explained above, increases with the temperature. In other
words, the temperature of elastomers increases with adiabatic stretching [
21
,
22
]. The equation for the
relationship was written by Kelvin back in 1857 [
22
]:
ð@f =@TÞ
p
;
l
¼ðC
p
=TÞð@T=@LÞ
p
;
adiabatic
where
again is the change in length of the elastomer. Experimental evidence
supports this, as the temperature of elastomers, like rubber, rises upon stretching. This equation can
also be written in another form:
C
p
is specific heat and
∂
l
ð@T=@f
0
Þ
p
;
adiabatic
¼ T=C
p
ð@l=@TÞ
p
; f
0
In actual dealing with polymers, stretching rubber and other elastomers requires overcoming the
energy barriers of the polymeric chains with the internal energy of the material depending slightly on
elongation, because
ð@H=@LÞ
T
;
p
¼ð@E=@lÞ
T
;
p
þ pð@V=@LÞ
T
;
p
∂
V
is the change in volume. At normal pressures the second term on the right becomes negligible. It
represents deviation from ideality. The contribution of the internal energy
E
to the force of retraction is
f ¼ð@E=@LÞ
T
;
p
Tð@S=@LÞ
T
Bueche [
16
] expressed differently the work done on stretching an elastic polymeric body.
It describes deforming an elastomer of
x
length, stretched to an increase in length
a
in a polymeric
