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This expression gives the average entropy change per chain; to get the
average for the sample, we multiply by the number v of subchains in the
sample. The total entropy change is
The quantity in parentheses is always positive for a > 1, the case of
elongation, making AS < 0 for stretching. Therefore AS is positive for
the opposite process, showing that entropy alone is sufficient to explain
the elastomer's snap. To get an idea of the magnitude of this entropy
effect, consider the following example.
Calculate AS for a 100% elongation of a polymer sample at 25°C using
(3.37) and compare AS with the value of ÄÍ calculated for the same
process in Example 3.1.
From Example 3.1, v = 3.4 X 10 s mol (expressed per cubic centimeter of
sample), and for a 100% elongation a = 2. Substitution into Eq. (3.37)
= -2.8 X 10"4 J K"1 cm'3 To compare with ÄÍ at 25°C, AS must be
multiplied by the temperature:
TAS = 298(-2.8 X 10"4) = -0.084 Jem-3
Comparison of this result with Example 3.1 shows that T AS and AH are of
the same order of magnitude. Example 3.1 stressed that the value of AH
estimated there was an upper limit. Experimental results show that the
assumption of ideality-while a slight oversimplification-generally
introduces an error of less than 10%. Comparison of the two examples
shows how much more difficult it is to deal with AH than with AS in these
Recalling that this is an ideal elastomer, we apply Eq. (3.19) to Eq.
(3.37) to obtain
22 +1 -3
Dividing both sides of the equation by the cross-sectional area of the
where the product of the cross section and L0 has been set equal to the
volume V of the sample. Note that at changes sign at a = 1, as physically
Although it is in this form that we compare theoretical predictions
with experiment in the next section, it is instructive to express Eq.
(3.39) in terms of L/L0 and then differentiate the result with respect to
dat = kT
+ 2LqL 3 dL
Next we multiply and divide the right-hand side by L to get dat =
kT - (a + \ V a2
Comparing this result with Eq. (3.1) shows that the quantity in brackets
equals Young's modulus for an ideal elastomer in a perfect network. Since
the number of subchains per unit volume, v/W, is also equal to pNA/Mc,
where Mc is the molecular weight of the subchain, the modulus may be
c _ RTp / 2
There are several things to notice about this result:
1. The modulus increases with temperature. This behavior is verified by
experiment. By contrast, the modulus of metals decreases with increasing
T. The difference arises from the fact that entropy is the origin of
elasticity in polymers but not in metals.
2. The modulus increases as Mc decreases. This effect on Mc is brought
about by increased crosslinking and is consistent with intuitive
expectations in this regard.
3. The modulus is not independent of a. For a > 1, the first term in
parentheses predominates and the modulus is directly proportional to a.
For a < 1, the first term is insignificant and E varies as a-2.
4. An interesting limit for small deformation (i.e., as 1) is
The Elastic and Viscoelastic States
showing the condition for Young's modulus to be constant and its
value when that condition is met.
5. Carrying out a parallel differentiation on Eq. (3.38) and examining
the case of a = 1 yields
dF = dL
Applied to a single chain, v = 1 and L02 is replaced by Eq. (1.62)
to give 3kT
dF = Vt dL = kH dL
This equation shows that at small deformations individual chains
obey Hooke's law with the force constant kH = 3kT/nl02 . This result may
be derived directly from random flight statistics without considering a
A typical cross-linked polymer at room temperature has p = 1 g cm-3 and
Mc = 104. According to Eq. (3.43), Young's modulus for such a polymer is
3(8.314 J Ê-1 mol-1 )(300 K)(103 kg m~3)/10 kg òîÃ1 = 7.5 X 10s Nm'2
which is on the order of the observed magnitude of this quantity for
polymers. By contrast, for metals E is on the order of 1011 N m-2.
3.5 Experimental Behavior of Elastomers
We have already observed that the entropy theory of elasticity predicts a
modulus of the right magnitude and possessing the proper temperature
coefficient. Now let us examine the suitability of Eq. (3.39) to describe
experimental results in detail.