# Polymer Chemistry. The Basic Concepts - Himenz P.C.

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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.

Example 3.2

Calculate AS for a 100% elongation of a polymer sample at 25°C using

Eq.

(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)

gives

= -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

systems.

•

Recalling that this is an ideal elastomer, we apply Eq. (3.19) to Eq.

(3.37) to obtain

(3.37)

Solution

22 +1 -3

Entropy Elasticity

149

Dividing both sides of the equation by the cross-sectional area of the

sample gives

1

(3-39)

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

required.

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

L:

dat = kT

V

+ 2LqL 3 dL

Next we multiply and divide the right-hand side by L to get dat =

kT - (a + \ V a2

dL

L

(3.40)

(3.41)

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

written as

c _ RTp / 2

E'ìòÃ?

(3.42)

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

150

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

(3.44)

Lo

Applied to a single chain, v = 1 and L02 is replaced by Eq. (1.62)

to give 3kT

dF = Vt dL = kH dL

(3.45)

nl02

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

network.

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.

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