Modulus of Natural Rubber Crosslinked by Dicumyl Peroxide III. Some Molecular Interpretations, Possible Refinements of Theory, and Conclusions
Authors: Wood L. A. 
Published: February 2011

Abstract
The qualitative and quantitative agreement of predictions and results demonstrated in the previous sections is a strong confirmation of the essential validity of all the extremely simplified molecular considerations involved, including the general aspects of the statistical theory of rubber elasticity. We know of no previous experimental study extending over as wide ranges of crosslinking and temperature. In fact the crosslinking and temperature have been varied simultaneously on only a few occasions in previous work. An important advantage of the present work over many previous studies is the fact that measurements are made at very small deformations. Thus the results are expressed in terms of the modulus, defined as the limiting value of the ratio of stress to stain at zero deformation. Consequently, the results are independent of the stress-strain relation or equation of state. This means that no consideration needs to be given here, for example to the difference between the stress-strain relation predicted by the statistical theory of rubber elasticity and that given by the Mooney-Rivlin equation or the empirical equation of Martin, Roth, and Stiehler. The present study has shown that the modulus G includes a considerable component arising from internal energy changes as well as that arising from entropy changes. The energy component at room temperature is of the order of half the total when the degree of crosslinking is that normally used with dicumyl peroxide rubbers. It is concluded that the nonzero value of the modulus when extrapolated to zero crosslinking is due to the energy component of the modulus rather than to entanglements. Entanglements acting as pseudo-crosslinks would serve to increase only the entropy component. The gel point, defined as the minimum degree of crosslinking required to form a network, may be located experimentally as the crosslinking at which the slope of the modulus-temperature relation is zero. The value of the modulus G at the gel point is not zero, but is the energy component under this condition; the entropy component of G at the gel point is zero. The amount of dicumyl peroxide required to crosslink rubber to the gel point is the sum of that wasted by reaction with impurities in the rubber and that required to give one crosslink for each rubber molecule. The former quantity was about twice the latter in the work reported here. The entropy component of the modulus as determined from reported values of equilibrium swelling by the Flory-Rehner equation, is found to be significantly larger than that determined from mechanical measurements. However, the quantity computed is smaller than the sum of the entropy and energy components as determined from crosslinking considerations or from mechanical measurements. It increases linearly with crosslinking at a slightly greater rate than the modulus or the entropy component of the modulus. It is concluded that the “front factor” sometimes introduced in statistical theory considerations cannot differ from unity by more than about 7%. The difference is even less than this if allowance is made for entanglements functioning as pseudo-crosslinks.