Pavlos S. Stephanou

Quantifying chain reptation in entangled polymer melts: Topological and dynamical mapping of atomistic simulation results onto the tube model

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 psi(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 psi(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 psi(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.

Projection of atomistic simulation data for the dynamics of entangled polymers onto the tube theory: calculation of the segment survival probability function and comparison with modern tube models

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. Kroger, 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 τd). 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.

A generalized differential constitutive equation for polymer melts based on principles of nonequilibrium thermodynamics

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 m...

Effects of tube persistence length on dynamics of mildly entangled polymers

Recent atomistic simulations of polyethylene melt afford a close look at the entanglement dynamics of a real polymer. We analyze these results in a new way, by defining the tube primitive path as the mean path of consecutive molecular dynamics trajectories. The result suggests that tube is semiflexible, the persistence length being about half the entanglement length. The time dependent tangent–tangent correlation function of tube primitive path is then used to test the standard molecular model of tube dynamics. It is found that the effect of the semiflexibility is important in the mildly entangled system we studied and that incorporating its effect into the standard tube model improves the theory drastically.

How the flow affects the phase behaviour and microstructure of polymer nanocomposites.

We address the issue of flow effects on the phase behaviour of polymer nanocomposite melts by making use of a recently reported Hamiltonian set of evolution equations developed on principles of non-equilibrium thermodynamics. To this end, we calculate the spinodal curve, by computing values for the nanoparticle radius as a function of the polymer radius-of-gyration for which the second derivative of the generalized free energy of the system becomes zero. Under equilibrium conditions, we recover the phase diagram predicted by Mackay et al. [Science 311, 1740 (2006)]. Under non-equilibrium conditions, we account for the extra terms in the free energy due to changes in the conformations of polymer chains by the shear flow. Overall, our model predicts that flow enhances miscibility, since the corresponding miscibility window opens up for non-zero shear rate values.

Non-constant link tension coefficient in the tumbling-snake model subjected to simple shear

The authors of the present study have recently presented evidence that the tumbling-snake model for polymeric systems has the necessary capacity to predict the appearance of pronounced undershoots in the time-dependent shear viscosity as well as an absence of equally pronounced undershoots in the transient two normal stress coefficients. The undershoots were found to appear due to the tumbling behavior of the director u when a rotational Brownian diffusion term is considered within the equation of motion of polymer segments, and a theoretical basis concerning the use of a link tension coefficient given through the nematic order parameter had been provided. The current work elaborates on the quantitative predictions of the tumbling-snake model to demonstrate its capacity to predict undershoots in the time-dependent shear viscosity. These predictions are shown to compare favorably with experimental rheological data for both polymer melts and solutions, help us to clarify the microscopic origin of the observed phenomena, and demonstrate in detail why a constant link tension coefficient has to be abandoned.

A constitutive rheological model for agglomerating blood derived from nonequilibrium thermodynamics

Red blood cells tend to aggregate in the presence of plasma proteins, forming structures known as rouleaux. Here, we derive a constitutive rheological model for human blood which accounts for the formation and dissociation of rouleaux using the generalized bracket formulation of nonequilibrium thermodynamics. Similar to the model derived by Owens and co-workers [“A non-homogeneous constitutive model for human blood. Part 1. Model derivation and steady flow,” J. Fluid Mech. 617, 327–354 (2008)] through polymer network theory, each rouleau in our model is represented as a dumbbell; the corresponding structural variable is the conformation tensor of the dumbbell. The kinetics of rouleau formation and dissociation is treated as in the work of Germann et al. [“Nonequilibrium thermodynamic modeling of the structure and rheology of concentrated wormlike micellar solutions,” J. Non-Newton. Fluid Mech. 196, 51–57 (2013)] by assuming a set of reversible reactions, each characterized by a forward and a reverse rate ...

Tumbling-Snake Model for Polymeric Liquids Subjected to Biaxial Elongational Flows with a Focus on Planar Elongation

We have recently solved the tumbling-snake model for concentrated polymer solutions and entangled melts in the presence of both steady-state and transient shear and uniaxial elongational flows, supplemented by a variable link tension coefficient. Here, we provide the transient and stationary solutions of the tumbling-snake model under biaxial elongation both analytically, for small and large elongation rates, and via Brownian dynamics simulations, for the case of planar elongational flow over a wide range of rates, times, and the model parameters. We show that both the steady-state and transient first planar viscosity predictions are similar to their uniaxial counterparts, in accord with recent experimental data. The second planar viscosity seems to behave in all aspects similarly to the shear viscosity, if shear rate is replaced by elongation rate.

From intermediate anisotropic to isotropic friction at large strain rates to account for viscosity thickening in polymer solutions

The steady-state extensional viscosity of dense polymeric liquids in elongational flows is known to be peculiar in the sense that for entangled polymer melts it monotonically decreases-whereas for concentrated polymer solutions it increases-with increasing strain rate beyond the inverse Rouse time. To shed light on this issue, we solve the kinetic theory model for concentrated polymer solutions and entangled melts proposed by Curtiss and Bird, also known as the tumbling-snake model, supplemented by a variable link tension coefficient that we relate to the uniaxial nematic order parameter of the polymer. As a result, the friction tensor is increasingly becoming isotropic at large strain rates as the polymer concentration decreases, and the model is seen to capture the experimentally observed behavior. Additional refinements may supplement the present model to capture very strong flows. We furthermore derive analytic expressions for small rates and the linear viscoelastic behavior. This work builds upon our earlier work on the use of the tumbling-snake model under shear and demonstrates its capacity to improve our microscopic understanding of the rheology of entangled polymer melts and concentrated polymer solutions.