Proposal to Solve the Time−Stress Discrepancy of Tube Models

Recently, Liu et al. (Macromolecules 2006, 39, 3093) showed a systematic discrepancy of tube model predictions for describing the apparent plateau modulus of short (weakly entangled) linear chains while predicting very accurately their terminal relaxation times. In the present article, we investigate the origin of this problem, which we call “time-stress discrepancy”, by confronting our time-marching algorithm (TMA) with experimental viscoelastic data of nearly monodisperse linear polymers. We show that the contour length fluctuations of the outer molecular segments are overestimated, not taking into account the fact that a chain needs a short but essential time to be considered in equilibrium in its tube. Indeed, tube models consider that stress relaxation by reptation or contour length fluctuations starts at time t = 0 after an imposed small deformation, whereas in reality, there is a proceeding fast Rouse relaxation, which is especially important for shorter chains. Therefore, we propose to use a new segment coordinate system for describing the contour length fluctuations process, which restores the consistency with the tube definition and ensures that the necessary time for reaching an equilibrated system is equal to the relaxation time of an entanglement segment, τe. Results obtained with the so-corrected TMA model show a very good agreement with experimental data. In particular, the molecular-weight dependence of the apparent plateau modulus, the zero-shear viscosity, and the terminal relaxation time are now correctly predicted.