Correlations involving the Mooney—Rivlin C2 constant and the number of chain atoms between physical entanglements, Nc

Abstract

We start with two previously calculated semi-empirical equations: (1) C 2 C 1 = K 1A −2 (for 2 C 1 = 0.2 MPa). K 1 is a constant and A is the area per polymer chain; (2) N c = K 2 A 0.67 where K 2 is another constant. It follows from these two relationships that: C 2 C 1 (at 2 C 1 = 0.2) = K 3 N −3 c. N c, determined for bulk polymers, increases on dilution with a solvent according to N c( ø 2) = N c( ø 2 = 1) ø − r 2 where ø 2 is the volume fraction of elastomer and r is a constant for a given elastomer, usually unity. For vulcanizates prepared in the presence of diluents but tested in the bulk state, one predicts C 2 C 1 = K 4ø 3r 2 if entanglements are the sole source of the C 2 term. Plots of log C 2 C 1 against log ø 2 show slopes ranging from 0.5 to 3.0 but more commonly near unity. There is wide discrepancy between different authors. We conclude that chain entanglement cannot be the sole source of the C 2 term but that several factors probably operate. Different theories and ideas about the origin of the C 2 term are briefly reviewed. Excellent linear plots of log C 2 C 1 against log 2 C 1 are presented for trans-polypentenamer and for Hevea rubber. In general, sufficient data for the rigorous testing of the C 2 C 1 —ø 3r 2 relationship are not available. A correlation of f e f with area per chain (where f e is the energetic contribution to the total force, f) increases rapidly at small areas but levels off above ∼ 0.4 nm 2. All studies reported herein are based on published data.