We present a new quantitative development of the reptation picture of de Gennes−Doi−Edwards. It is well-known that the original reptation theory is unable to fit linear relaxation spectra (G‘ and G‘ ‘) as it misses several important physical processes:  (1) contour length fluctuations, (2) constraint release, and (3) longitudinal stress relaxation along the tube. All of these processes were treated theoretically before; however, the treatment used either uncontrolled approximations or failed to include all of them at the same time. The aim of this work is to combine self-consistent theories for contour length fluctuations and constraint release with reptation theory. First, we improve the treatment of contour length fluctuations using a combined theoretical and stochastic simulation approach. This allows us to obtain an expression for the single chain relaxation function μ(t) without any adjustable parameters and approximations. To include constraint release, we use the scheme proposed by Rubinstein and Colby, which provides an algorithm for calculating the full relaxation function G(t) from the single chain relaxation μ(t). Then longitudinal modes are added, and a detailed comparison with different experimental data is given. One of the conclusions is that polystyrene is described by theory very well, but polybutadiene shows problems, which may be a first indication of nonuniversality of polymer dynamics.

A refined version of the Doi and Edwards tube model for entangled polymer liquids is presented. The model is intended to cover linear chains in the full range of deformation rates from linear to strongly nonlinear flows. The effects of reptation, chain stretch, and convective constraint release are derived from a microscopic stochastic partial differential equation that describes the dynamics of the chain contour down to the length scale of the tube diameter. Contour length fluctuations are also included in an approximate manner. Predictions of mechanical stresses as well as the single chain structure factor under flow are shown. A comparison with experimental data is made in which all model parameters are fixed at universal values or are obtained from linear oscillatory shear measurements. With no parameter modification the model produces good agreement over a wide range of rheological data for entangled polymer solutions, including both nonlinear shear and extension.

The model presented in this paper describes simultaneously three different experimental techniques applied to different monodisperse polymer melts: neutron spin-echo (NSE), linear rheology, and diffusion. First, it shows that the standard tube model cannot be applied to NSE because the statistics of a one-dimensional (1D) chain in a three-dimensional (3D) random-walk tube become wrong on the length scale of the tube diameter, whereas all available NSE data are for scattering vectors in this range. Then a new single-chain dynamic slip-link model on the basis of a recent network model by Rubinstein and Panyukov is introduced. Instead of solving the model analytically, which would require uncontrolled approximations, the model is formulated in terms of stochastic differential equations, suitable for Brownian dynamics simulations. I perform these simple simulations, demonstrate that the model describes individual experiments well, and then compare the results with experiments on monodisperse polyethylene, polyethylene-propylene, polyisoprene, polybutadiene, and polystyrene. For all polymers, model parameters from one experiment are obtained, and the others are predicted without fitting. The results show some systematic discrepancies, suggesting possible inadequacy of the Gaussian chain model for some of the polymers, and possible inadequacy of time-temperature superposition.

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The linear viscoelastic (LVE) spectrum is one of the primary fingerprints of polymer solutions and melts, carrying information about most relaxation processes in the system. Many single chain theories and models start with predicting the LVE spectrum to validate their assumptions. However, until now, no reliable linear stress relaxation data were available from simulations of multichain systems. In this work, we propose a new efficient way to calculate a wide variety of correlation functions and mean-square displacements during simulations without significant additional CPU cost. Using this method, we calculate stress-stress autocorrelation functions for a simple bead-spring model of polymer melt for a wide range of chain lengths, densities, temperatures, and chain stiffnesses. The obtained stress-stress autocorrelation functions were compared with the single chain slip-spring model in order to obtain entanglement related parameters, such as the plateau modulus or the molecular weight between entanglements. Then, the dependence of the plateau modulus on the packing length is discussed. We have also identified three different contributions to the stress relaxation: bond length relaxation, colloidal and polymeric. Their dependence on the density and the temperature is demonstrated for short unentangled systems without inertia.

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Numerous papers have recently appeared in the literature presenting quantitative comparisons of experimental linear viscoelastic data to the most recent versions of "tube" models for entangled polymer melts and solutions. Since these tube models are now being used for quantitative, rather than just qualitative, predictions, it has become important that numerical pre-factors for the time constants that appear in these theories be evaluated correctly using literature data for the constants (i.e., density, plateau modulus, etc.) that go into the theories. However, in the literature two definitions of the entanglement spacing in terms of plateau modulus have been presented, and confusion between these has produced numerous errors in the recent literature. In addition, two different definitions of the "equilibration time," a fundamental time constant, have also appeared, creating additional potential for confusion. We therefore carefully review the alternative definitions and clarify the values of the pre-factors that must be used for the different definitions, in the hope of helping future authors to avoid such errors.

SynopsisWe develop a microscopic description of the contribution to stress relaxation in entangled polymer melts of convective constraint release, which is the release of entanglement constraints due to the effects of convective flow on chains surrounding a given chain. Our theory resolves three of the main shortcomings of the Doi-Edwards model in nonlinear rheology, in that it predicts ͑1͒ a monotonically increasing shear stress as a function of shear rate, ͑2͒ shear stress independent of molecular weight at sufficiently high shear rates, and ͑3͒ only modest anisotropies in the single chain scattering function, in agreement with experiment. In addition, our approach predicts that a stress maximum and resulting shear-banding instability would occur for living micelle solutions, as observed.