Modulus of Natural Rubber Cross-Linked by Dicumyl Peroxide II. Comparison With Theory

Thermodynamics and molecular considerations are applied to an examination of the equation G = S(fp + B)T + H(fp + B) + A = 5.925 × 10^-3(fp - 0.45)T + 0.0684(fp - 0.45) + 2.70, found experimentally in Part I. G is the shear modulus in Mdyn cm^-2 at a temperature T for natural rubber cross-linked by adding p parts of dicumyl peroxide per hundred of rubber (phr) and heating until a fraction f of the peroxide is decomposed. G*, the energy component of the modulus, is H(fp + B) + A. The ratio G*/G decreases from 1.00 at the gel point (fp = 0.45 phr) to 0.5 near 2 phr and to 0.09 at 23.8 phr. The modulus G is related to ν , the number of moles of effective sub-chains per cm³, by the equation G - G* = v RT where R is the gas constant. If each molecule of decomposed dicumyl peroxide of molecular weight M produces one cross-link in the rubber of specific volume $$ , then it is predicted that $$ Mdyn cm^-2 phr^-1 K^-1, as compared with the experimental value 5.925 × 10^-3. Theory gives no prediction of the values of A, or of H. The gel point may be located experimentally as the point where the slope of the modulus-temperature relation is zero. The value of G at the gel point is the energy component G*. The experimental value of fp at the gel point permits a calculation of the molecular weight of the rubber before cross-linking as 193,000. The results afford a very satisfactory confirmation of the essential validity of the statistical theory of rubber elasticity in its simplest form, if due regard is paid to G*, the energy component of the modulus.