Integral and differential constitutive equations for entangled polymers with simple versions of CCR and force balance on entanglements

The theory of Doi and Edwards for entangled polymers has been recently modified for the case of fast flows to account for convective contributions to molecular dynamics. The flow-induced relative motion between neighboring chains removes constraints and speeds up relaxation. Convective constraint release (CCR) may thus explain why the shear stress is seen to approach a plateau at high shear rates instead of decreasing as predicted by the basic theory. In slow flows, as well as in step strain, another discrepancy between theory and observations can be found in the normal stress ratio in shear Ψ=−N2/N1. The theoretical value for Ψ at low deformations is 1/7 whereas measured values for well-entangled systems are systematically larger. We have recently considered the possibility that this discrepancy arises because force balance requirements at the entanglement nodes are ignored in the classical theory. Accordingly, we have proposed a change in the orientational tensor Q. Here, we sum up on these recent findings by proposing single-relaxation-time constitutive equations of the integral or rate type incorporating those concepts in a simple way. Such equations should be suitable for numerical simulation of complex flows.