The topological state of entangled polymers has been analyzed recently in terms of primitive paths which allowed obtaining reliable predictions of the static (statistical) properties of the underlying entanglement network for a number of polymer melts. Through a systematic methodology that first maps atomistic molecular dynamics (MD) trajectories onto time trajectories of primitive chains and then documents primitive chain motion in terms of a curvilinear diffusion in a tubelike region around the coarse-grained chain contour, we are extending these static approaches here even further by computing the most fundamental function of the reptation theory, namely, the probability ψ(s,t) that a segment s of the primitive chain remains inside the initial tube after time t, accounting directly for contour length fluctuations and constraint release. The effective diameter of the tube is independently evaluated by observing tube constraints either on atomistic displacements or on the displacement of primitive chain segments orthogonal to the initial primitive path. Having computed the tube diameter, the tube itself around each primitive path is constructed by visiting each entanglement strand along the primitive path one after the other and approximating it by the space of a small cylinder having the same axis as the entanglement strand itself and a diameter equal to the estimated effective tube diameter. Reptation of the primitive chain longitudinally inside the effective constraining tube as well as local transverse fluctuations of the chain driven mainly from constraint release and regeneration mechanisms are evident in the simulation results; the latter causes parts of the chains to venture outside their average tube surface for certain periods of time. The computed ψ(s,t) curves account directly for both of these phenomena, as well as for contour length fluctuations, since all of them are automatically captured in the atomistic simulations. Linear viscoelastic properties such as the zero shear rate viscosity and the spectra of storage and loss moduli obtained on the basis of the obtained ψ(s,t) curves for three different polymer melts (polyethylene, cis-1,4-polybutadiene, and trans-1,4-polybutadiene) are consistent with experimental rheological data and in qualitative agreement with the double reptation and dual constraint models. The new methodology is general and can be routinely applied to analyze primitive path dynamics and chain reptation in atomistic trajectories (accumulated through long MD simulations) of other model polymers or polymeric systems (e.g., bidisperse, branched, grafted, etc.); it is thus believed to be particularly useful in the future in evaluating proposed tube models and developing more accurate theories for entangled systems.

Based on principles of nonequilibrium thermodynamics, we derive a generalized differential constitutive equation for polymer melts which incorporates terms that account for anisotropic hydrodynamic drag in the form suggested by Giesekus, finite chain extensibility with nonlinear molecular stretching, nonaffine deformation, and variation of the longest chain relaxation time with chain conformation. In the new equation, the expression for the Helmholtz free energy of deformation is defined such that the entropy remains bounded even at high deformation rates, as it should from a physical point of view. Key elements in the new constitutive model are the functions describing the dependence of the nonequilibrum free energy and the relaxation matrix on the conformation tensor. With suitable choices of these two functions, the new equation reduces to a number of well-known viscoelastic models. However, it is more general in the sense that it permits incorporating into a single constitutive differential equation more accurate expressions for the description of chain elasticity and relaxation. Restrictions on the parameters entering these two functions are obtained by analyzing the thermodynamic admissibility of the model. By analyzing the asymptotic behavior of the new constitutive equation at low and high shear rates in steady shear, one can fix all of its parameters from available rheological data for the conformation tensor except for one which should be fitted. We illustrate the procedure here where the new model is used to fit available rheological data for short polyethylene melts obtained through direct nonequilibrium molecular dynamics simulations in shear and planar elongation, with remarkable success.close161

State-of-the-art tube models for the dynamics of entangled polymer melts are usually validated on the basis of the agreement of their predictions for the linear viscoelastic properties (LVE data) of the system against experimentally measured data. We present here a more direct and fundamental test of these models based on their comparison against molecular dynamics (MD) simulation data for the dynamics of primitive paths (PPs) in the system under study. More precisely, we show how one can take advantage of a recently developed computational methodology (P. S. Stephanou, C. Baig, G. Tsolou, V. G. Mavrantzas and M. Kröger, J. Chem. Phys., 2010, 132, 124904) for calculating the most important function of all tube models, the segment survival probability ψ(s,t) and its average Ψ(t) (the overall tube survival probability), by projecting MD data of atomistically detailed samples onto the level of the primitive paths, to directly probe mechanisms proposed for chain relaxation, such as contour length fluctuation (CLF) and constraint release (CR). The simulation data for ψ(s,t) and Ψ(t) can be used next to evaluate refinements of the original Doi–Edwards reptation theory based on a modified diffusion equation for ψ(s,t) incorporating the terms proposed to account directly or indirectly for these effects (CLF and CR). The functions ψ(s,t) and Ψ(t) determined directly from the atomistic MD simulation data account automatically for all these relaxation mechanisms, as well as for any other mechanism present in the real melt. We present and discuss results from such an approach referring to model, strictly monodisperse cis- and trans-1,4-polybutadiene and polyethylene melts containing on average up to 6 entanglements per chain, simulated in full atomistic detail for times up to a few microseconds (that is, comparable to the chain disentanglement time τ_d). From the same simulations we also present results for two other measures of the PP dynamics in the framework of the reptation theory, the time auto-correlation function of the PP contour length L and the time auto-correlation function of the chain end-to-end vector R. Our methodology, which serves as a bridge between molecular simulations and analytical tube theories, helps quantify chain dynamics in entangled polymers and understand how it is influenced by factors like melt polydispersity and chain molecular architecture, or the presence of interfaces. It can also be straightforwardly extended to polymeric liquids under non-equilibrium conditions (e.g., subjected to a flow field) to understand the interplay between flow and entanglements.

Our experimental data unambiguously show (i) a damping behavior (the appearance of an undershoot following the overshoot) in the transient shear viscosity of a concentrated polymeric solution, and (ii) the absence of a corresponding behavior in the transient normal stress coefficients. Both trends are shown to be quantitatively captured by the bead-link chain kinetic theory for concentrated polymer solutions and entangled polymer melts proposed by Curtiss and Bird, supplemented by a non-constant link tension coefficient that we relate to the nematic order parameter. The observed phenomena are attributed to the tumbling behavior of the links, triggered by rotational fluctuations, on top of reptation. Using model parameters deduced from stationary data, we calculate the transient behavior of the stress tensor for this “tumbling-snake” model after startup of shear flow efficiently via simple Brownian dynamics. The unaltered method is capable of handling arbitrary homogeneous flows and has the promising capacity to improve our understanding of the transient behavior of concentrated polymer solutions.