Monte Carlo simulations of stress relaxation of entanglement-free Fraenkel chains. II. Nonlinear polymer viscoelasticity.

The nonlinear viscoelastic behavior of the Fraenkel-chain model is studied with respect to the constitutive equation of the Rouse model. Distinctly different from the results of the Rouse model, the Fraenkel-chain model gives the following characteristic nonlinear behavior: (a) The two distinct dynamic modes in the relaxation modulus GS(t,lambda)--as observed in the linear region reported in Paper I [Y.-H. Lin and A. K. Das, J. Chem. Phys. 126, 074902 (2007), preceding paper]--or in the first normal-stress difference function GPsi1(t,lambda) are shown to have different strain dependences: strain hardening for the fast mode and strain softening for the slow mode. (b) The Lodge-Meissner relation GS(t,lambda)=GPsi1(t,lambda) holds over the whole time range, which has been shown both analytically and by simulation. (c) The second normal-stress difference is nonzero, being positive in the fast-mode region and negative in the slow-mode region. The comparisons between orientation and stress for all tensor components consistently confirm the strong correlation of the slow mode as well as its entropic nature with the segmental-orientation anisotropy as shown in the linear region studied in Paper I. A consequence of this correlation is the applicability of the stress-optical rule in the slow-mode region. This also leads to the expectation that the damping function h(lambda)=G(S)(t,lambda)/G(S)(t,lambda-->0) and the ratio between the first and second normal-stress differences, N2(t,lambda)/N1(t,lambda), are described by the orientation tensor which has the same form as that given by Doi and Edwards [J. Chem. Soc. Faraday Trans. 2 74, 1789 (1978); 74, 1802 (1978)] with independent-alignment approximation for an entangled system. The similarity between the slow mode of an entanglement-free Fraenkel-chain system and the terminal mode of an entangled polymer system as observed in the comparison of theory, simulation, and experiment suggests that the close correlation of the entropic nature of the mode with the orientation anisotropy--as of the Fraenkel segment or the primitive step in the Doi-Edwards theory--is a generally valid physical concept in polymer viscoelasticity.