Session A49: Focus Session: Long-time, Entangled Dynamics in Polymers - Linear, Transient, Non-linear Rheology, Tubes

Microscopic theory of the tube confinement potential and relaxation of entangled needle liquids under stress

We have developed a first-principles theory of the transverse confinement potential in an entangled needle fluid based on exactly enforcing uncrossability at the two-rod level while self-consistently renormalizing many-particle effects [Sussman \& Schweizer PRL 107, 078102 (2011); J. Chem. Phys. 135, 131104 (2011)]. The predicted tube radius and long-time diffusion constant are consistent with the asymptotic reptation scaling laws under quiescent conditions, but in contrast with the usual tube model strong anharmonicities soften the confinement potential in a manner that quantitatively agrees with experiments on heavily entangled F-actin solutions. This weakening of entanglement constraints has multiple dramatic consequences under applied deformation: tube dilation, accelerated reptation, reduction of the transverse entropic barrier, and a critical stress or strain beyond which tube localization is destroyed. The degree-of-entanglement-dependent competition between reptative and transverse-hopping relaxation is established as a function of stress and strain. A mapping between rigid rods and flexible chain systems is also proposed, allowing predictions to be made for the tube diameter, entanglement onset, and transport properties of chain polymer liquids.   

Rheology of linear monodisperse polyethylene melts from atomistic Molecular Dynamics simulations

In our contribution we present results from very long, Molecular Dynamics simulations of linear monodisperse polyethylene (PE) chains with lengths ranging from 0.5 to 18 entanglements. We adopt a hierarchical modeling approach: in the first step we employ Monte Carlo simulations consisting of chain-connectivity altering algorithms to ensure full scale equilibration. Secondly, massive parallel MD simulations are conducted in the canonical ensemble. Besides the standard dynamical information, the stress relaxation curves are calculated for all PE systems. By bridging present atomistic results with the tube theory through the newly-introduced slip-spring model [Likhtman, Macromolecules 38, 6128 (2005)] we are able to calculate the plateau modulus and viscosity for well entangled, industrially relevant PE melts. In all cases, comparison between available experimental data and present simulation findings reveals a very good to excellent agreement. The proposed multi-scale methodology is generally applicable and can be extended to polymers of different molecular architecture and chemical constitution, as well as blends and more complex interfacial systems.   

Intermolecular constraints in the dynamics of semiflexible entangled polymer melts

We present a Langevin equation for the contemporary dynamics of a group of interpenetrating semiflexible entangled polymer chains. The theory explicitly accounts for the intermolecular intermonomer repulsion between a pair of chains, generated by their inability to cross each other, i.e. the phenomenon of entanglements. The ``effective'' potential experienced by the chains arises from the repulsion between two monomers belonging to different chains, propagating through the chain connectivity, and the dynamics of chain interdiffusion and relaxation. With time the local hard-core potential is overcome by the relative motion of the polymers and the system relaxes. The same formalism applies to both unentangled and entangled melts. Short chains do not experience entanglements, because their relaxation process is faster than the average time that is necessary for the chain to diffuse a distance comparable to the mesh size, or length between two entanglements. Finally no a priori hypothesis has to be made about the processes that drive relaxation as the formalism is simply the conventional Rouse approach, generalized to treat the motion of interacting macromolecules, whose chains cannot cross each other.   

Long-Time Dynamics in Polymers: Experimental Results

The long-time dynamics in several polymeric systems have been investigated with a focus on the relationship between the bulk and shear viscoelastic responses. Materials studied include polystyrene, a three-arm star polystyrene, and two polycyanurates of different crosslink densities. A custom-built pressurizable dilatometer has been used to measure the time-dependent bulk modulus, as well as the pressure-volume-temperature behavior in these materials. The temperature-dependent shift factors are used to test the TV$^{\gamma}$ thermodynamic scaling law proposed in the literature for segmental relaxation times. The thermodynamic scaling law successfully reduces the data for all of the materials; however, T - Tg scaling also successfully reduces the data and differences in implications of the two scaling approaches will be discussed. Comparison of the retardation spectra for the bulk and shear responses shows that at extremely long times, the chain mechanisms available to the shear response are not available to the bulk; for times related to the glassy dynamics, the two responses have similar slopes, indicating that they may have similar underlying molecular mechanisms, but the magnitudes are different, a finding that remains to be explained.   

Tube Dynamics of Mildly Entangled Polymers: Semiflexibility Effects

The prevailing theory of polymer rheology rests on a careful analysis of tube dynamics, tested by comparing predicted rheological response functions to experimental measurements. We provide a direct test of this theory by analyzing the tube dynamics of recently simulated mildly entangled polyethylene melt. The tube dynamics is obtained by defining the tube primitive path, \textit{i.e.}, the tube center line, as the short-time average of molecular dynamics trajectories, and by monitoring how the tangent-tangent correlations evolve with time. It was found that the tube is semiflexible, and that the tube relaxation rate obtained from simulation results cannot be accounted for by the prevailing theory, since the effect of contour length fluctuations built into the theory is too strong. This discrepancy is particularly relevant to the mildly entangled system. To fix this, we incorporated semiflexibility into the theory, which was originally designed for flexible tubes, and found that the corrected theory describes the simulation results nearly quantitatively.   

A critical analysis of typical assumptions in the theory of entangled polymer dynamics in elongational flows

The discrete slip-link model (DSM) was developed to describe the dynamics of entangled polymer melts of arbitrary chain architecture in arbitrary deformation. The model is able to predict linear viscoelasticity of monodisperse linear, polydisperse linear and star-branched systems. The model also shows good agreement with dielectric relaxation experiments. In this work we apply DSM to non-linear flows of monodisperse linear polystyrene and polyisoprene melts without any adjustable parameters. 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. We explore a number of simplifications of the model and their effect on flow predictions, including: finite extensibility, convective constraint release and activation of dangling ends. Only after discarding all approximations and assumptions as a source of discrepancy between DSM predictions and experimental data can we conclude whether additional physics concepts are necessary to describe non-linear rheology of entangled polymer melts.   

Exploring the role of long-chain branching in large deformation of entangled melts

Most of our past studies have focused on nonlinear responses to large deformation of entangled polymers made of linear flexible chains. Little is known about nonlinear rheological behavior of entangled polymers containing long-chain branching (LCB), apart from the literature work on low-density polyethylene (LDPE). In this work, we present a first study to compare linear polyisoprene with a well-defined dendritic polyisoprene. Consistent with the extensional rheological behavior of LDPE, we find LCB to impede shear yielding so that the entanglement network could extend significantly more before failure during uniaxial extension. This study also investigated its shear deformation behavior to explain the absence of necking like failure in uniaxial extension. The research is funded, in part, by a grant from the National Science Foundation (DMR-1105135).   

Similarity and difference between simple shear and uniaxial extension of entangled polymers

There is ample evidence to show that the essential physics governing yielding of entangled polymers is the same, independent of the mode of deformation, e.g., shear versus extension. In either of these two most commonly studied forms of deformation, the elastic retraction force associated with the chain deformation cannot grow without bound during continuous deformation. In practice, a transition from the initial dominantly elastic deformation to flow (irreversible deformation) inevitably takes place. Such yielding can produce strain localization in large deformation of well entangled polymer melts. Apart from the superficial difference related to the confusion about the ``strain hardening'' behavior, a true difference in the respective responses of entangled melts to shear and extension arises when the strain rate is sufficiently high. The entanglement network can still yield on its path to the eventual flow state upon startup shear. However, startup extension could cause the entanglements to lock in, and the melt undergoes rubber-like rupture instead of yielding. This presentation raises the question of whether shear is an intrinsically different deformation from uniaxial extension in the extremely high rate limit.   

Large-deformation and long-time behavior of entangled melts in complex geometries

Recent particle-tracking velocimetric (PTV) observations have revealed strain localization either during startup shear beyond the stress overshoot or after a large step shear of entangled polymers [e.g., Macromolecules, \textbf{42}, 6261 (2009)]. The physical pictures leading to these decohesion events have been put forward [J. Chem. Phys. \textbf{127}, 064903(2007); J. Rheol. \textbf{53}, 1389 (2009)]. In this presentation we apply the particle-tracking velocimetric method [Macromol. Mater. Engr. \textbf{292}, 15 (2007)] to study similar strain localization phenomena originating from yielding of the entanglement network in other forms of deformation including uniaxial extension, ``squeeze flow'' and extrusion of polymers from a wide open space into a narrow opening. The striking discontinuities in the velocity profile can all be understood in terms of a shear yielding criterion. The research is funded, in part, by a grant from the National Science Foundation (CMMI-0926522)   

Barrier to chain retraction: where we are six years after the first report of shear inhomogeneity in entangled polymers?

At APS2006, we reported the first PTV observations of macroscopic motions after shear cessation from step strain on an entangled polybutadiene solution (Macromolecules \textbf{2007}, $40$, 8031). Since then we have shown that the classical polystyrene solutions display similar non-quiescent relaxation, invalidating the agreement between the data based on PS solutions and the Doi-Edwards damping function. Based on polymer melts we found that this network breakup phenomenon also occurs after a step strain produced with a rate that according to the tube model is too low to generate chain stretching (Macromolecules \textbf{2009}, $42$, 6261). Does the current tube model possess the necessary ingredients to depict these findings? Here we present new experimental data that further supports the concept of a finite cohesion level for the entanglement network: There is a finite confining force that keeps chains engaged in the network, ensures the structural integrity and allows linear response behavior to take place. In contrast, the tube model perceives barrier-free chain retraction on the Rouse time for any amount of imposed strain, which would necessarily lead to destruction of the original network. Our experiments show that this does not appear to be the case.   

Neutron Reflectivity Study of Interdiffusion of Ionomers into Van der Waals Polymer Thin Films

The slow dynamic processes in amorphous ionic polymers are affected by physical cross-links resulting from clustering of the ionic groups. Therefore in addition to entanglement barriers, the motion of the polymers is coupled to the dynamics of the ionic clusters where the resulting dynamics is an interplay between the effects of the two types of barriers. Using neutron reflectometry we have probed a model system where interfacial diffusion of a Van der Waals polymer, polystyrene, into its sulfonated analogs. Results controlling the molecular weights that determine the overall number of entanglements as well as the degree of sulfonation which affects the strength and number of the ionic clusters will be presented. Comparison to the diffusion of polystyrene into polystyrene will resolve the effects of the ionic clusters from those of entanglements. The presence of the physical cross-links slows down the dynamics significantly with respect to that of polystyrene and an asymmetric process where the non-ionic blocks migrate into the ionic one is observed. Further rearrangements take place at a later stage.   

Development of Interfacial Strength and Entanglements During Welding of Polymers

Thermal welding is a common means of joining polymer parts. Interfacial strength increases with welding time $t_w $ as polymer chains diffuse across the interface. The microscopic origin of this interfacial strength enhancement was investigated with large scale molecular simulations employing a coarse-grained bead-spring model. Polymer surfaces were held together at a temperature well above the glass transition temperature $T_g$. States at $t_w $ up to $10^9$ time steps were then quenched to a temperature below $T_g $ for mechanical tests. We test the interfacial strength by shearing the weld along a direction parallel to the interface. The maximum shear stress $\sigma _{\max} $before failure is used to characterize the interfacial strength. We find that $\sigma _{\max} $ increases as $t_w ^{1/4}$ before saturating to its bulk value. This agrees with previous experiments by a lap-joint shear method [1]. In addition, our analysis shows that the dominant shear failure mode changes from chain pull-out at the interface for small $t_w $, to chain scission for large $t_w $. We examine the average contour length $$ of chains that have diffused across the interface. As predicted by the reptation dynamics, $$ increases as $t_w ^{1/4}$. We also track the evolution of entanglements using the \textit{Primitive Path Analysis} (PPA) algorithm [2]. Our results show that the total number of interfacial entanglements is directly related to interfacial strength and also increases as $t_w ^{1/4}$. \\[4pt] [1] D. B. Kline, R. P. Wool,\textit{ Polym. Eng. Sci. }1988, 28(1), 52--57.\\[0pt] [2] R. Everaers \textit{et al}. , \textit{Science}, 303, 823-826, 2004   

Using the parallel plates geometry for nonlinear rheological measurements

Conventional wisdom dictates that studying the mechanical response of viscoelastic materials in the nonlinear regime should be done either with a cone-plate or a Couette geometry, where the applied strain is homogenous in the measuring volume. However, the use of parallel plates would have important advantages in a wide range of applications. For instance solid-like hydrogel materials can often be processed readily into flat films. We show that the nonlinear viscoelastic behavior can also be obtained from measurements in a parallel plate geometry. By tracing the torque response and its derivative with respect to the applied strain, we obtain a general stress strain relation, which indeed captures the proper material behavior. The approach does not require any assumptions for the material's viscoelastic behavior. We show practical examples different classes of soft materials to illustrate that our approach enables access to the full nonlinear response of these materials, including the detailed shape of the stress response in large amplitude oscillatory shear measurements. Our approach should be applicable to a wide range of soft materials, including hydrogels, colloidal suspensions, or biological tissues.