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For the reversible elongation of a polymer the combined first and second laws may be written in the form:
In equation above, there are two types of work: the usual PdV term, by which the
system performs work on the surroundings when it expands, and fdL, which
represents the work done on the system by an external force f when the
polymer is stretched an increment dL. Elongation is essentially a constant
volume process (|PdV| < 0.001 |fdL|) [1,2], and usually the pdV term can
be ignored. Furthermore, it has been observed that the isothermal work input
of elongation is almost exactly balanced by release of heat to the surroundings.
(A qualitative observation of this evolved heat may be made by holding a
rapidly stretched rubber band to the lips).
This means that the internal energy of isothermal elongation is
effectively zero. This result requires that 
be zero
for the reverse process
as well. Indeed, as mentioned above, when the rubber rebounds (no work), the
heat transfer is essentially zero. Recalling the condition that 
for an ideal gas, the requirement that
is used to define an ideal rubber[3].
To derive the important thermodynamic relationships for the elongation process, the Helmholtz free energy, A, which is applicable to constant volume processes, is written in differential form as follows:
(4)
Each of these may be further differentiated to give:
(5)
The equality of mixed partials shows that:
(6)
By equation 4a, the force is equal to the derivative of A with respect to L at constant T and V. Thus, equation 2 may be differentiated to give:
(7)
and because
(8)
This exercise in differential calculus has led to two very important results
given by equations 6 and 8. Considering equation 8 first, both the force and
the absolute temperature must be positive. This means that the entropy of
elongation at constant temperature,
,
must be negative. Although this result is based
strictly on
macroscopic observations, it may be related directly to the molecular
properties discussed above. When the elastomer is stretched, the randomly
coiled molecules must be straightened, and this decreases the disorder, and
hence the entropy, of the network.
According to equation 6,
can be obtained by measuring the
force as function of temperature with length and volume both held constant.
However, as indicated by Flory (2, pp 440-444 and 489-491), the proposed
experiment would not be simple, because normal thermal expansion would cause
the volume to increase as the temperature is raised. Flory presents a detailed
examination of this matter, and his treatment shows that equation 6 can be
rewritten as follows:
(9)
Equation 9 shows that the experiment can be carried out at constant pressure,
which is easily achieved by working at the ambient atmospheric conditions.
There is also a new variable, 
,
which is the relative
elongation of the
polymer:
(10)
where L and L0are, respectively,
the lengths of the rubber with and without the
applied
force. The quantity,
L/L0 is commonly called the strain
with the symbol
. Another
important quantity is the stress, often given the symbol
which
is the force per unit cross-sectional area of the sample.
=f/A0
Remember:
STRESS = NORMALIZED APPLIED FORCE(in analogy to the pressure of a gas)
and
STRAIN = NORMALIZED ELONGATION(in analogy to the volume of a gas)
Both sides of equation 9 may be divided by the area to give:
(11)
There is an obvious problem in the specification of the cross-sectional area,
because it decreases as the rubber is stretched. To obtain equation 11 the
unstressed area, AO,
was used. In later equations there will be occasion to use both the unstressed
and stressed dimensions, and the symbol 
will denote
the value of the stress obtained by dividing the force by the actual area at that
elongation,
A not A0, i.e.
=f/A;
=f/A0
Summarizing these important results, equation 9 shows that the derivative of
the entropy with respect to the elongation may be obtained by determining the
force needed to maintain a constant relative elongation,
,
as the temperature
is varied. Evaluation of this fundamental thermodynamic quantity is one of the
goals of this experiment. Another property of interest, especially to
engineers, is Young's modulus, Y which is defined as the derivative of
the stress with respect to the strain:
(12)
in which the corrected cross-sectional area is used to calculate the stress.
Young's modulus is also sometimes called the stiffness of the material;
a large value of Y indicates that a large force per unit area must be applied
to elongate the sample. Referring to equation 10, for a given force increment
the change in must equal the change in
.
Thus, Y can be obtained from
an isothermal experiment, by evaluating the slope of a plot of applied force
versus and dividing by the cross-sectional area.
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