(14)(back to Table of Contents)
Although molecular properties have been used to explain the equations derived so far, the thermodynamic relationships themselves have involved no assumptions about the microscopic behavior of the elastomer; all parameters (temperature, force, elongation, and volume) can be measured directly on the bulk sample. The molecular treatment of rubber elasticity involves the use of statistical mechanics, and the macroscopic observations can also be used to verify the predictions of the theoretical model. Several of the references (1-4) provide very readable derivations of the theory, which will be described in an abbreviated form here.
Consistent with the molecular description presented in the Introduction, the theory assumes that there is no internal energy contribution to the elongation process. The elongation work is needed solely to overcome the entropy effect, specifically the unfavorable change in the conformational entropy, Scon, which is related to the many ways that the chains can be arranged spatially by rotations about single bonds. The conformational entropy is given by the Boltzmann formula:
where k is Boltzmann's constant, and W is the total number of conformations available to the system. The goal, therefore, is to derive expressions for Scon in both the unstretched and the stretched rubber.
In its simplest form, the model treats a chain segment as a series of n uniform links (corresponding to the chemical bonds), which are assumed to be freely jointed; i.e. they are not restricted by bond angles or the volume that they occupy. In the polymer network a chain segment corresponds to the portion of the polymer extending from one cross-link to the next. Provided that a segment does not approach full extension, the probability that it extends randomly from an arbitrary point (x,y,z) to another point (x+dxu,y+dyu,z+dzu) is given by a simple Gaussian distribution in which the subscript u refers to the unstressed polymer. In the exponential the sum
(14)
is equal to 
the squared distance between
the two ends of the
chain segment. This is divided by 
the mean square end-to-end distance, given by:
(15)
where N0 is the total number of possible conformations available to the chain
segment. The total number of ways, W, that the segment ends can be
separated by 
is the probability given by equation 14 multiplied by the total number of
conformations:
(16)
where C0 replaces the term 
which will
remain constant throughout the
derivation.
The mean square end-to-end distance, is
shown in the complete treatment to equal nl2, where l
is the length of each link, so that the RMS distance
itself is just n1/2l. If the freely-jointed chain were to achieve full
extension, its length would be nl. Because n1/2l is much less than nl for a
large value of n (i.e. many links), this result justifies the assumption that
the segment does not reach its fullest extension.
The polymer network is assumed to consist of v identical segments. For
this assembly the total number of conformations, Wup in which the unstressed
segment ends are separated by is
(17)
Thus, the conformational entropy of the unstressed rubber is
(18)
In equation 18 each of the 
can be replaced by the mean-square average for the assembly to give:
(19)
The expression for S, the conformation entropy for the stretched sample, is derived in the same manner, but with the additional assumption that polymer elongation is an ?affine? deformation. This means that the changes in the x, y, and z components of each chain segment are proportional to the corresponding changes in the bulk material:
where 
correspond to the respective
relative changes in the
bulk sample (i.e., 
, etc.).
Thus, the entropy of the stressed rubber
is given by:
(21)
and the difference in entropies of the stressed and unstressed sample is:
(22)
For an isotropic network all directions are equally probable, so that on the average
(23)
With these substitutions the difference in entropies reduces to
(24)
This equation simplifies further, because elongation is a constant volume process. If the x axis is specified as the direction of elongation, the following relations must hold:
(25)
Substitution of these conditions into equation 24 gives the desired expression for the entropy of the stretched polymer:
(26)
Equation 26may be differentiated with respect to 
to produce
(27)
Combining this result with Equation 8 and the definition of
gives
(28)
or
(29)
where Ne is the number of elastically active chain segments per unit
volume.
Thus, a plot of the stress (using the uncorrected cross-sectional area) versus
at constant
temperature should be linear with slope equal to NekT.
The relationship fails
to hold for greater than about 4 (elongations of more than 300%),
because the
assumption that a polymer chain has an end-to-end distance considerably less
than the length at full extension fails to hold at large elongations. Also, as
discussed above, the elastomer may
develop crystalline regions as the polymer chains are extended and adopt a more
ordered arrangement.
The form of equation 29 is very similar to that of the ideal gas law, which may be written as
(30)
where N is the number of molecules of gas, V0 is the initial volume, and the term in parentheses is the expansion factor. The stress, which has units of force per unit area, is analogous to the pressure, and the number of elastically active chain segments replaces the number of gas molecules. The differences in equations 29 and 30 occur in the terms in parentheses. First of all, the quotients are inverted (L/L0 versus V0/V). For an ideal gas the pressure acts to decrease the volume, but the stress on a rubber band increases the length. Also, the strain factor in equation 29 is somewhat more complicated than the ideal gas expansion factor, due to the near incompressibility of the polymer (3). The reasons for the analogy relate to the fundamental thermodynamics of the systems. For an ideal gaslideal rubber there is no internal energy change for an isothermal expansion/elongation; the processes are entirely entropy-driven. The gas expands to maximize its entropy, whereas the molecules in the rubber retract to their disordered random-coil state, when the elongation stress is removed.
Equation 29 may be differentiated to give Young's modulus:
(31)
where the second term in parentheses is needed to correct the cross-sectional
area to the value in the elongated sample. An average value for Ne
may be obtained from
plots 
versus
Then, at any specified temperature and elongation a
theoretically predicted value of Y may be calculated from equation 31.
The theoretically
predicted Y should agree with the value obtained directly from the f
versus plot at the same temperature.
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