Modulus of Natural Rubber Cross-Linked by Dicumyl Peroxide III. Some Molecular Interpretations, Possible Refinements of Theory, and Conclusions

The shear modulus G = 5.925 × 10(−)(3)(fp − 0.45)T+G* (Part I), its energy component G* = 0.0684 (fp − 0.45)+ 2.70 (Part II), and the number of effective suh-chains per unit volume v(e)= (G – G*)/RT are given detailed molecular consideration. G is given in Mdyn cm(−2) for rubber cross-linked by adding p parts of dicumyl peroxide per hundred of rubber, and heating until a fraction f of the peroxide is decomposed. v(e) is found to be approximately twice the density of cross-links, after a correction for impurities and chain ends is made. It can not be computed as G/RT since only the entropy component of modulus is related to v(e). The sub-chains for the most highly cross-linked rubbers studied had a molecular weight of about 575 g mol(−1), corresponding to about 8 isoprene units. The modulus corresponding to no added cross-links is not zero. It is determined chiefiy by the energy component of the modulus; it does not arise from entanglements. The “front factor” is found to be unity. An extensive literature survey yields values of the quantity RTΨ(v(2)), where Ψ (v(2)) is the Flory- Rehner equation function of v(2), the equilibrium volume fraction obtained by swelling the cross-linked rubber. RTψ (v(2)) is found to be greater than G – G* but not as large as G itself.