Chain contraction and nonlinear stress damping in primitive chain network simulations

Doi and Edwards (DE) proposed that the relaxation of entangled linear polymers under large deformation occurs in two steps: the fast chain contraction (via the longitudinal Rouse mode of the chain backbone) and the slow orientational relaxation (due to reptation). The DE model assumes these relaxation processes to be independent and decoupled. However, this decoupling is invalid for a generalized convective constraint release (CCR) mechanism that releases the entanglement on every occasion of the contraction of surrounding chains. Indeed, the decoupling does not occur in the sliplink models where the entanglement is represented by the binary interaction (hooking) of chains. Thus, we conducted primitive chain network simulations based on a multichain sliplink model to investigate the chain contraction under step shear. The simulation quantitatively reproduced experimental features of the nonlinear relaxation modulus $G(t,γ)$⁠. Namely, $G(t,γ)$ was cast in the time-strain separable form, $G(t,γ)=h(γ)G(t)$ with $h(γ)=damping$ function and $G(t)=linear$ modulus, but this rigorous separability was valid only at times $t$ comparable to the terminal relaxation time, although a deviation from this form was rather small (within ±10%) at $t>τR$ (longest Rouse relaxation time). A molecular origin of this delicate failure of time-strain separability at $t∼τR$ was examined for the chain contour length, subchain length, and subchain stretch. These quantities were found to relax in three steps, the fast, intermediate, and terminal steps, governed by the local force balance between the subchains, the longitudinal Rouse relaxation, and the reptation, respectively. The contributions of the terminal reptative mode to the chain length relaxation as well as the subchain length/stretch relaxation, not considered in the original DE model, emerged because the sliplinks (entanglement) were removed via the generalized CCR mechanism explained above and the reformation of the sliplinks was slow at around the chain center compared to the more rapidly fluctuating chain end. The number of monomers in the subchain were kept larger at the chain center than at the chain end because of the slow entanglement reformation at the center, thereby reducing the tension of the stretched subchain at the chain center compared to the DE prediction. This reduction of the tension at the chain center prevented completion of the length equilibration of subchains at $t∼τR$ (which contradicts to the DE prediction), and it forces the equilibration to complete through the reptative mode at $t⪢τR$⁠. The delicate failure of time-strain separability seen for $G(t,γ)$ at $t∼τR$ reflects this retarded length equilibration.