Chemistry Reference
In-Depth Information
but this means that the internal energy term (
@U/@l
)
T
cannot be identically zero,
however.
If the internal energy contribution to the force at constant length and sample
volume is
f
e
, its relative contribution is
f
e
f
5
T
f
@
f
1
2
(4-24)
@
T
v;l
Various measurements have shown that
f
e
/f
is about 0.1-0.2 for polybutadiene
and
cis
-polyisoprene elastomers. These polymers are essentially but not entirely
entropy springs.
4.5.2.3
Stress
Strain Properties of Cross-Linked Elastomers
Consider a cube of cross-linked elastomer with unit dimensions. This specimen is
subjected to a tensile force
f.
The ratio of the increase in length to the unstretched
length is the nominal strain
(epsilon), but the deformation is sometimes also
expressed as the extension ratio
Λ
(lambda):
ε
Λ5λ=λ
0
5
1
1ε
(4-25)
where
λ
0
are the stretched and unstretched specimen lengths, respectively.
With a cube of unit initial dimensions, the stress
λ
and
is equal to
f
. (Recall the defini-
tions of stress, normal strain, and modulus on page 24.) Also, in this special case
dλ5λ
0
dΛ5dΛ
τ
, and so
Eqs. (4-21) and (4-23)
are equivalent to
τ 52
T
ð@
S
=@ΛÞ
T;V
(4-26)
Statistical mechanical calculations
[6]
have shown that the entropy change is
given by
1
2
N
2
Δ
S
52
κðΛ
1 ð
2
=ΛÞ 2
3
Þ
(4-27)
where
N
is the number of chain segments between cross-links per unit volume
and
κ
(kappa) is Boltzmann's constant (
R/L
). Then, from
Eq. (4-26)
,
2
τ 5
N
κ
T
ðΛ2
1
=Λ
Þ
(4-28)
Equation (4-28)
is equivalent to
2
τ 5 ðρ
RT
=
M
c
ÞðΛ2
1
=Λ
Þ
(4-29)
is the elastomer density (gram per unit volume),
M
c
is the average molec-
ular weight between cross-links (
M
c
5ρ
ρ
where
L/N
), and
L
is Avogadro's constant.
Equation (4-29)
predicts that the stress
strain properties of an elastomer that
behaves like an entropy spring will depend only on the temperature, the density
