THE ORIGIN OF THE C2 TERM IN RUBBER ELASTICITY

A unifying stress-strain model for physical networks (polymer melts) a nd for permanent networks (rubbers) is presented. It is based on three assumptions: (1) the uncrossability condition of real chains can be m odeled by the tube concept; (2) the tube diameter is a function of the average stretch; and (3) the tube volume is invariant with respect to deformation. The model predicts that stress-strain behavior of dry an d swollen rubber networks is completely determined by four material co nstants: the equilibrium modulus G(infinity) of the bulk network; the critical entanglement modulus G(e) which is equivalent to the plateau modulus G(N) of the un-crosslinked parent melt; a finite extensibilit y parameter alpha; and a solvent-polymer interaction exponent beta. Pr edictions are compared with experimental data in elongation, and agree ment is excellent. The Mooney-Rivlin constant C2 of unswollen networks with high crosslink densities is limited to roughly G(N)/2, and the o rigin of the C2 term is shown to be due to nonaffine deformation of th e entanglement network. Nonaffine deformation of network strands is ca used by an increasing lateral restriction due to neighboring chains, w hile upon swelling, the nonaffine reduction of the microscopic length scale leads to the vanishing C2 value of highly swollen rubbers.