Rudi J. A. Steenbakkers

We evaluate the thermodynamic consistency of the anisotropic mobile slip-link model for entangled flexible polymers. The level of description is that of a single chain, whose interactions with other chains are coarse grained to discrete entanglements. The dynamics of the model consist of the motion of entanglements through space and of the chain through the entanglements, as well as the creation and destruction of entanglements, which are implemented in a mean-field way. Entanglements are modeled as discrete slip links, whose spatial positions are confined by quadratic potentials. The confinement potentials move with the macroscopic velocity field, hence the entanglements fluctuate around purely affine motion. We allow for anisotropy of these fluctuations, described by a set of shape tensors. By casting the model in the form of the general equation for the nonequilibrium reversible-irreversible coupling from nonequilibrium thermodynamics, we show that (i) since the confinement potentials contribute to the chain free energy, they must also contribute to the stress tensor, (ii) these stress contributions are of two kinds: one related to the virtual springs connecting the slip links to the centers of the confinement potentials and the other related to the shape tensors, and (iii) these two kinds of stress contributions cancel each other if the confinement potentials become anisotropic in flow, according to a lower-convected evolution of the confinement strength or, equivalently, an upper-convected evolution of the shape tensors of the entanglement spatial fluctuations. In previous publications, we have shown that this cancellation is necessary for the model to obey the stress-optical rule and the Green-Kubo relation, and simultaneously to agree with plateau modulus predictions of multichain models and simulations.

We consider four criteria of acceptability for single-chain mean-field entangled polymer models: consistency with a multi-chain level of description, consistency with nonequilibrium thermodynamics, consistency with the stress-optic rule, and self-consistency between Green–Kubo predictions and linear viscoelastic predictions for infinitesimally driven systems. Each of these topics has been considered independently elsewhere. However, we are aware of no molecular entanglement model that satisfies all four criteria simultaneously. Here we show that an idea from Ronca and Allegra, generalized to arbitrary flows, can be implemented in a slip-link model to create a model that does satisfy all four criteria. Aside from the direct benefits of agreement, the result modifies the relation between the initial relaxation modulus G(0) and the entanglement molecular weight Me. If this implementation is correct, current estimates for Me would require modification that brings their values more in line with estimates based on topological analysis of molecular dynamics simulations.

The discrete slip-link model (DSM) was developed to describe the dynamics of flexible polymer melts. The model is able to predict linear viscoelasticity of monodisperse linear, polydisperse linear, and branched systems. The model also shows good agreement with dielectric relaxation experiments, except for the single data set available for bidisperse linear systems with a small volume fraction of long chains. In this work, both shear and elongational flow predictions obtained using the DSM without parameter adjustment are shown. Model predictions for shear flow agree very well with experimental results. The DSM is able to capture the transient response as well as the steady-state viscosity. However, for elongational flow, agreement is unsatisfactory at large strains. The DSM captures the onset of strain hardening, but after a Hencky strain between 2 and 3, it predicts transient strain softening, whereas experiments show only monotonic growth. We explore a number of assumptions and approximations of the model and their effect on flow predictions. The approximations are related to the neglect of these phenomena, which are expected to be more sensitive in elongational flow: finite extensibility, convective constraint release, and deformation of dangling ends. We indeed find that shear flow predictions are insensitive to these approximations, but elongational flow is affected. However, none of these effects is able to bring prediction in line with experiments. We conclude that the currently accepted view of entanglement dynamics is incomplete.The discrete slip-link model (DSM) was developed to describe the dynamics of flexible polymer melts. The model is able to predict linear viscoelasticity of monodisperse linear, polydisperse linear, and branched systems. The model also shows good agreement with dielectric relaxation experiments, except for the single data set available for bidisperse linear systems with a small volume fraction of long chains. In this work, both shear and elongational flow predictions obtained using the DSM without parameter adjustment are shown. Model predictions for shear flow agree very well with experimental results. The DSM is able to capture the transient response as well as the steady-state viscosity. However, for elongational flow, agreement is unsatisfactory at large strains. The DSM captures the onset of strain hardening, but after a Hencky strain between 2 and 3, it predicts transient strain softening, whereas experiments show only monotonic growth. We explore a number of assumptions and approximations of the mod...

We present the free energy of a single-chain mean-field model for polymer melt dynamics, which uses a continuous (tube-like) approximation to the discrete entanglements with surrounding chains, but, in contrast to previous tube models, includes fluctuations in the number density of Kuhn steps along the primitive path and in the degree of entanglement. The free energy is obtained from that of the slip-link model with fluctuating entanglement positions [J. D. Schieber and K. Horio, J. Chem. Phys. 132, 074905 (2010)]10.1063/1.3314727 by taking the continuous limit of (functions of) the discrete Kuhn-step numbers and end-to-end vectors of the strands between entanglements. This coarse-graining from a more-detailed level of description has the advantage that no ad hoc arguments need to be introduced. Moreover, the thermodynamic consistency of the slip-link model [J. D. Schieber, J. Non-Equilib. Thermodyn. 28, 179 (2003)]10.1515/JNETDY.2003.010 can be preserved. Fluctuations in the positions of entanglements lead to a harmonic bending term in the free energy of the continuous chain, similar to that derived by Read et al. [Macromolecules 41, 6843 (2008)]10.1021/ma8009855 starting from a modified GLaMM model [R. S. Graham, A. E. Likhtman, T. C. B. McLeish, and S. T. Milner, J. Rheol. 47, 1171 (2003)]10.1122/1.1595099. If these fluctuations are set to zero, the free energy becomes purely Gaussian and corresponds to the continuous limit of the original slip-link model, with affinely moving entanglements [J. D. Schieber, J. Chem. Phys. 118, 5162 (2003)]10.1063/1.1553764. The free energy reduces to that of Read et al. under their assumptions of a homogeneous Kuhn-step number density and a constant degree of entanglement. Finally, we show how a transformation of the primitive-path coordinate can be applied to make the degree of entanglement an outcome of the model instead of a variable. In summary, this paper constitutes a first step towards a unified mathematical formulation of tube models. The next step will be to formulate the dynamics of the primitive-path conformation and the entanglement density along the primitive path. Now that the free energy is known, statistical mechanics can be employed for this purpose.

A model for flow‐enhanced nucleation is presented based on the concept of a polymer melt containing a fixed number of nucleation precursors with a fixed size distribution. Depending on the size, precursors can either be active (i.e. susceptible to nucleation, the characteristic time scale of which is governed by the deformation rate) and grow into a spherulite or remain dormant. The size distribution of precursors is derived by combining nucleation theory and experimentally determined quiescent spherulite number densities. Longitudinal precursor growth, causing activation of dormant precursors, is a function of molecular deformation: the stretch of high molecular weight chains. Both the eXtended Pom‐Pom and the Rolie‐Poly model are tested to calculate the molecular deformation. A quantitative agreement is found between simulations and experimental results. magnified image