A new model for Brownian dynamics simulations of entangled polymeric liquids is proposed here. Chains are coarse grained at the level of segments between consecutive entanglements; hence, the system is in fact a network of primitive chains. The model incorporates not only the “individual” mechanisms of reptation and tube length fluctuation, but also collective contributions arising from the 3D network structure of the entangled system, such as constraint release. Chain coupling is achieved by fulfilling force balance on the entanglement nodes. The Langevin equation for the nodes contains both the tension in the chain segments emanating from the node and an osmotic force arising from density fluctuations. Entanglements are modeled as slip links, each connecting two chain strands. The motion of monomers through slip links, which ultimately generates reptation as well as tube length fluctuations, is also described by a suitable Langevin equation. Creation and release of entanglements is controlled by the number of monomers at the chain ends. In a creation event, the partner chain segment is chosen randomly among those spatially close to the advancing chain end. To validate the model, equilibrium dynamics simulations were run for monodisperse linear chains containing up to Z=40 entanglements. The results show, in agreement with experiments, (i) a Z3.5±0.1 dependence of the longest relaxation time, (ii) a Z−2.4±0.2 dependence of the self-diffusion coefficient, and (iii) a relaxation modulus proportional to the square of the end-to-end vector correlation function, consistently with the dynamic tube dilation concept.

Well-entangled monodisperse linear polystyrene melts exhibit monotonic thinning of the steady state elongational viscosity with increasing the strain rate ε̇ even beyond the Rouse relaxation frequency, τR ‑1. This behavior is quite different from the thinning followed by hardening at ε̇ > τR ‑1 observed for entangled semidilute solutions. We attempt to elucidate the molecular origin of this difference by focusing on the concept of stretch/orientation-dependent monomeric friction ζ recently proposed by Ianniruberto and co-workers. Specifically, literature data of the stress relaxation after cessation of transient elongational flow, reported for both PS melts and solutions, are analyzed to evaluate the stretch/orientation-dependent decrease of ζ. In our working hypothesis, ζ is expressed as a function of the factor F so = λ̃2 S̅, where λ̃ is the normalized stretch ratio of entangled subchains defined with respect to the fully stretched state, and S̅ is an average orientational anisotropy of the components (polymer plus solvent if any) in the system. The factor F so was estimated from the stress decay data after flow cessation. The resulting functional form of ζ(F so) was then used in the primitive chain network (PCN) simulation including finite extensible nonlinear elasticity (FENE) to examine the elongational behavior of melts and solutions. For melts the simulation indicates that ζ decreases significantly under fast elongation because the entangled subchains are short and approach the fully stretched (and fully oriented) limit rather easily. Hence, the steady elongational viscosity ηE follows this decrease of ζ to exhibit the monotonic thinning even at ε̇ > τR ‑1. In contrast, for solutions, the simulated ηE exhibits thickening at ε̇ > τR ‑1 because the average anisotropy S̅ is governed by the solvent and remains small, thus overwhelming the increase of the subchain stretch λ̃. The simulated results proved to be in satisfactory agreement with the experiments.

We investigate the nonlinear shear and uniaxial extensional rheology of entangled polystyrene (PS) melts and solutions having the same number Z of entanglements, hence identical linear viscoelasticity. While experiments in extensional flows confirm that PS melts and solutions with the same Z behave differently, respective transient and steady data in simple shear over the largest possible range of rheometric shear rates (corresponding to Rouse−Weissenberg numbers from 0.01 to 40) demonstrate that melts and solutions exhibit identical behavior. Whereas the differences between melts and solutions in elongational flows are due to alignmentinduced friction reduction (more effective in melts than in solutions), in shear flows they disappear since the rotational component reduces monomeric alignment substantially. Recent molecular dynamics simulations of entangled polymers show that rotation induces molecular tumbling at high shear rates, and here a tube-based model involving tumbling effects is proposed in order to describe the response in shear. The main outcome is that tumbling can explain transient stress undershoot (following the overshoot) at high shear rates. Hence, the combination of tumbling in shear and friction reduction in extension successfully describes the whole range of experimental data and provides the basic ingredient for the development of molecular constitutive equations.

Recent data by Hassager and co-workers [Bach et al. Macromolecules 2003, 36, 5174] of elongational viscosity of nearly monodisperse polystyrene melts are interpreted by including in the classical tube theories for entangled polymer dynamics an interchain repulsive contribution. The proposed theory predicts the observed power law of ηel vs ˘in a straightforward way and qualitatively explains the observed scaling with the polymer molecular mass. Possible generalizations are discussed.

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It is known that polystyrene melts behave anomalously in fast elongational flows insofar as the steady-state elongational viscosity keeps decreasing with increasing stretching rate ε, without showing the typical upturn at ετ R ≈ 1, with τ R the Rouse time. The authors have recently suggested that such an anomalous behavior might be due to a decrease of the monomeric friction coefficient brought about by alignment of the Kuhn segments of the polymer to the stretching direction. Here we perform a quantitative analysis of such a possibility by first determining from existing stress−optical data how such a reduction should correlate to the order parameter of the Kuhn segments and then by performing nonequilibrium molecular dynamics (NEMD) simulations over a sequence of styrene oligomers. We have used NEMD not only to obtain diffusion coefficients of those oligomers but also, for what seems to be the first time, friction coefficients. Indeed, it is well-known that the Einstein relationship linking friction to diffusion does not hold true far from equilibrium. The friction coefficients so obtained correlate to the order parameter of the monomers in much the same way as in the polymeric case, and by increasing the length (or mass) of the oligomer, they appear to approach a similar characteristic curve.

Brownian dynamics simulations of the linear viscoelastic response of entangled polymers have been performed, and compared quantitatively with some existing solution data at a fixed concentration and variable molecular weight. The model is a three-dimensional network where the nodes are sliplinks connecting chains in pair. The simulations make use of Langevin equations both for the node motion in space, and for the one-dimensional monomer sliding through sliplinks. Comparison with data is very satisfactory, but the molecular weight between entanglements that emerges from the model is unconventionally small.

The extensional-flow data of Huang et al. [Macromolecules2015484158] of several polystyrene systems are here successfully compared with predictions of a recent model of Ianniruberto [J. Rheol.201559211], provided flow-induced friction-reduction effects are accounted for. For the case of solutions, friction reduction must include nematic interactions between the oligomeric solvent and the polymer molecules. It is here found that the coupling interaction parameter ε must be larger than that based on the mean orientation of the solvent molecules, as measured in mildly oriented systems; the larger ε found here is probably due either to the stronger orientation reached in the extensional flows, or to the fact that only the molecules close to the polymer matter in frictional effects. The model here suggested is the first multimode molecular model (aside from simulations) able to describe, qualitatively and even (almost) quantitatively, the nonlinear extensional rheology of entangled polymers, from melts to solutions.