Understanding how entanglements affect the behaviour of polymeric complex fluids is an open challenge in many fields. To elucidate the nature and consequence of entanglements in dense polymer solutions, we propose a novel method: a "dynamical entanglement analysis" (DEA) to extract spatio-temporal entanglement structures from the pair-wise displacement correlation of entangled chains. By applying this method to large-scale Molecular Dynamics simulations of linear and unknotted, nonconcatenated ring polymers, we find a strong and unexpected cooperative dynamics: the footprint of mutual entrainment between entangled chains. We show that DEA is a powerful and sensitive probe that reveals previously unnoticed, and architecture-dependent, spatio-temporal structures of dynamical entanglement in polymeric solutions. We also propose a mean-field approximation of our analysis which provides previously under-appreciated physical insights into the dynamics of generic entangled polymers. We envisage DEA will be useful to analyse the dynamical evolution of entanglements in generic polymeric systems such as blends and composites.
We present a comprehensive investigation of polymer diffusion in the semidilute regime by fluorescence correlation spectroscopy (FCS) and dynamic light scattering (DLS). Using single-labeled polystyrene chains, FCS leads to the self-diffusion coefficient while DLS gives the cooperative diffusion coefficient for exactly the same molecular weights and concentrations. Using FCS we observe a new fast mode in the semidilute entangled concentration regime beyond the slower mode which is due to self-diffusion. Comparison of FCS data with data obtained by DLS on the same polymers shows that the second…
In entangled polymer systems, there are several characteristic time scales, such as the entanglement time and the disengagement time. In molecular simulations, the longest relaxation time (the disengagement time) can be determined by the mean square displacement (MSD) of a segment or by the shear relaxation modulus. Here, we propose the relative fluctuation analysis method, which is originally developed for characterizing large fluctuations, to determine the longest relaxation time from the center of mass trajectories of polymer chains (the time-averaged MSDs). Applying the method to simulatio…
We investigate the viscoelastic properties of entangled networks of semiflexible polymers. At intermediate time scales the elastic response of these networks to shear deformation is described by the plateau modulus $G$. Different scaling laws with polymer concentration $c$ have been proposed based on the assumption that the deformation field is affine on all length scales. We develop a numerical approach that allows to calculate the modulus via free energy changes for both affine and non-affine deformations. The non-affine deformation field is obtained by a free energy minimization. Our findin…
Non-equilibrium tube length fluctuations of entangled polymers
我们研究了初始拉伸的缠结高分子链在松弛过程中的非平衡管长涨落。管长度的时间依赖性方差$σ^2$在早期状态下遵循起源于聚合物链段的扩散运动的简单通用幂律$σ^2 = A \sqrt{t}$。从标准蠕动理论和精确可解的格子气蠕动模型中解析计算了振幅$A$，并讨论了振幅与初始管长和平衡管长的关系。非普遍性表明测量的波动（例如，使用荧光显微镜）作为爬行动物模型的测试。
Some recent results on the rotational dynamics of polymers are reviewed and extended. We focus here on the relaxation of a polymer, either flexible or semiflexible, initially wrapped around a rigid rod. We also study the steady polymer rotation generated by a constant torque on the rod. The interplay of frictional and entropic forces leads to a complex dynamical behavior characterized by non-trivial universal exponents. The results are based on extensive simulations of polymers undergoing Rouse dynamics and on an analytical approach using force balance and scaling arguments. The analytical res…
To predict flow behavior of entangled polymer melt, we have developed multiscale simulation composed of Lagrangian fluid particle simulation and coarse-grained polymer dynamics simulation. We have introduced a particle deformation in the Lagrangian fluid particle simulation to describe elongation flow at a local point. The particle deformation is obtained to be consistent with the local polymer deformation.
We present a new reduced-order computational method for the molecular dynamics simulation of entangled polymer systems. The polymer chains are modeled as continuous Gaussian chains. Our algorithm is based on the application of the molecular dynamics simulation method to the pseudospectral representation of the Fourier modes of the chains.
We show that non-entangled polymers display an elastic-like behaviour at a macroscopic scale (probed at some 0.100 mm thickness) up to at least hundred degrees above the glass transition temperature. This observation, found under non-slippage conditions, both for side-chain liquid crystalline polymers and ordinary polymers, is in contradiction with the typically found flow behaviour of polymer melt. Our measurements were carried out with a conventional rheometer at thicknesses of several tenths millimetres. Thus, we were probing bulk properties. The observed elasticity supposedly implies that …
The tube model is a central concept in polymer physics, and allows to reduce the complex many-filament problem of an entangled polymer solution to a single filament description. We investigate the probability distribution function of conformations of confinement tubes and single encaged filaments in entangled semiflexible polymer solution. Computer simulations are developed that mimic the actual dynamics of confined polymers in disordered systems with topological constraints on time scales above local equilibration but well below large scale rearrangement of the network. We observe the statist…
Entanglement of polymer chains is ubiquitous in elastomers, gels, and biological tissues. While the effects of chain entanglement on elasticity and viscoelasticity of polymer networks have been intensively studied, it remains elusive how chain entanglement affects fracture and fatigue of polymer networks. In this paper, using polyacrylamide hydrogels as a model material, we systematically compare fracture toughness and fatigue threshold of polymer networks with various levels of chain entanglement. We find that the fracture toughness and fatigue threshold of an unentangled polymer network are …
Based on numerical modeling with the diffusive Giesekus constitutive equation, a recent Letter [Phys. Rev. Lett. 120, 138002 (2018), arXiv:1801.08798] claimed that the experimental reports of shear banding in entangled polymeric fluids were due to the disturbances from the sample edge. The conclusion that true bulk shear banding is precluded in the presence of edge effects is at odds with many past experimental, theoretical, numerical, and computer simulation studies. We briefly discuss some problems of this work.
Equilibration of polymer melts containing highly entangled long polymer chains in confinement or with free surfaces is a challenge for computer simulations. We approach this problem by first studying polymer melts based on the soft-sphere coarse-grained model confined between two walls with periodic boundary conditions in two directions parallel to the walls. Then we apply backmapping to reinsert the microscopic details of the underlying bead-spring model. Tuning the strength of the wall potential, the monomer density of confined polymer melts in equilibrium is kept at the bulk density even ne…
We investigated the elongational flows of the weakly entangled linear polymer melt using a coarse-grained molecular dynamics simulation. We extended the uniform extensional flow (UEF) method developed by Nicholson and Rutledge (D. A. Nicholson and G. C. Rutledge, J. Chem. Phys., 145, 244903 (2016)) for application to Langevin dynamics. We succeeded in observing the elongational viscosity of the weakly entangled linear polymer melt from the equilibrium state to the steady state using the extended UEF method, whereas the conventional rectangular parallelepiped shape technique for extensional flo…
The size of rings (also called cyclic polymers) in bidisperse blends of chemically identical rings is analyzed by computer simulations. Data of entangled ring blends and blends of interpenetrating rings are compared and it is shown that the compression of entangled rings can be explained by the changes in the penetrable fraction of the minimal surface bounded by the ring. Corrections for small rings can be approximated by a concatenation probability $1-P_{OO}$ that a ring entraps at least one other ring. Both results are in line with a previous work [1] to explain the compression of entangled …
The spatial correlations of entangled polymer dynamics are examined by molecular dynamics simulations and neutron spin-echo spectroscopy. Due to the soft nature of topological constraints, the initial spatial decays of intermediate scattering functions of entangled chains are, to the first approximation, surprisingly similar to those of an unentangled system in the functional forms. However, entanglements reveal themselves as a long tail in the reciprocal-space correlations, implying a weak but persistent dynamic localization in real space. Comparison with a number of existing theoretical mode…
The tube-like cages of stiff polymers in entangled solutions have been shown to exhibit characteristic spatial heterogeneities. We explain these observations by a systematic theory generalizing previous work by D. Morse (Phys. Rev. E 63:031502, 2001). With a local version of the binary collision approximation (BCA), the distribution of confinement strengths is calculated, and the magnitude and the distribution function of tube radius fluctuations are predicted. Our main result is a unique scaling function for the tube radius distribution, in good agreement with experimental and simulation data…
We analyze the capillarity-driven thinning dynamics of entangled polymer solutions described by the Doi-Edwards-Marrucci-Grizzuti (DEMG) model and the Rolie-Poly (RP) model. Both models capture polymer reptation, finite rates of chain retraction and finite extensibility of single polymer molecules, while differing slightly in their final form regarding to the convective constraint release. We calculate numerically the filament thinning profiles predicted by the two models with realistic entanglement densities, assuming cylindrical filament shapes and no fluid inertia. Both results reveal an ea…
The Langevin Equation for cooperative dynamics represents the dynamics of polymer melts with chains of increasing degree of polymerization, covering the full range of behavior from the unentangled to the entangled regime. This equation describes the motion of a group of interpenetrating polymers that are interacting through an effective potential resulting from the many-body coupling of the inter polymer potential inside the macromolecular liquid. The confinement of the dynamics due to the presence of entanglements is accounted for by an effective inter-monomer potential which is zero until th…
Starting from a given topological invariant, we argue that it is possible to construct a topological field theory with a finite number of Feynman diagrams and an amplitude of gauge invariant objects that is a function of that invariant. This is for example the case of the Gauss linking number and of the abelian BF models which has been already successfully applied in the statistical mechanics of polymers. In this work it is shown that a suitable generalization of the BF model can be applied also to polymer dynamics, where the polymer trajectories are not static, but change their shape during t…
The Monte Carlo carbyne model is modified to investigate the glass transition of the semi-flexible entangled polymer chains. The stochastic bombardment between monomers are monitored by Metropolis algorithm with help of the consideration of hard potential while the mobility of monomers is governed by its mass, scattering rate and temperature. Our model is capable to show that the glass transition temperature reduces with decreasing film thickness and the formation of critical voids in the thinner polymer contributing to the glass transition that is much easier than the bulk polymer.
Stress relaxation following deformation of an entangled polymeric liquid is thought to be affected by transient reforming of chain entanglements. In this work, we use single molecule techniques to study the relaxation of individual polymers in the transition regime from unentangled to entangled solutions. Our results reveal the emergence of dynamic heterogeneity underlying polymer relaxation behavior, including distinct molecular sub-populations described by a single-mode and a double-mode exponential relaxation process. The slower double-mode timescale $τ_{d,2}$ is consistent with a character…
Nonlinear extensional flows are common in polymer processing but remain challenging theoretically because dramatic stretching of chains deforms the entanglement network far from equilibrium. Here, we present coarse-grained simulations of extensional flows in entangled polymer melts for Rouse-Weissenberg numbers $Wi_R=0.06$-$52$ and Hencky strains $ε\geq6$. Simulations reproduce experimental trends in extensional viscosity with time, rate and molecular weight. Studies of molecular structure reveal an elongation and thinning of the confining tube with increasing $Wi_R$. The rising stress is quan…
It has been recently shown that the phase space trajectories for the anomalous dynamics of a tagged monomer of a polymer --- for single polymeric systems such as phantom Rouse, self-avoiding Rouse, Zimm, reptation, and translocation through a narrow pore in a membrane; as well as for many-polymeric system such as polymer melts in the entangled regime --- is robustly described by the Generalized Langevin Equation (GLE). Here I show that the probability distribution of phase space trajectories for all these classical anomalous dynamics for single polymers is that of a fractional Brownian motion …
The glass transition is a long-standing unsolved problem in materials science. For polymers, our understanding of glass-formation is particularly poor due to the added complexity of chain connectivity and flexibility; structural relaxation of polymers thus involves a complex interplay between intra- and inter-molecular cooperativity. Here we study how the glass transition temperature Tg varies with molecular weight M for different polymer chemistries and chain flexibilities. We find that Tg(M) is controlled by the average mass (or volume) per conformational degree of freedom, and that a `local…
We have developed a new multiscale simulation technique to investigate history-dependent flow behavior of entangled polymer melt, using a smoothed particle hydrodynamics simulation with microscopic simulators that account for the dynamics of entangled polymers acting on each fluid element. The multiscale simulation technique is applied to entangled polymer melt flow around a circular obstacle in a two-dimensional periodic system. It is found that the strain-rate history-dependent stress of the entangled polymer melt affects its flow behavior, and the memory in the stress causes nonlinear behav…
We present a computer simulation of entangled polymer solutions at equilibrium. The chains repel each other via a soft Gaussian potential, appropriate for semi-dilute solutions at the scale of a correlation blob. The key innovation to suppress chain crossings is to use a pseudo-continuous model of a backbone which effectively leaves no gaps between consecutive points on the chain, unlike the usual bead-and-spring model. Our algorithm is sufficiently fast to observe the entangled regime using a standard desktop computer. The simulated structural and mechanical correlations are in fair agreement…
In this paper, we employ Molecular Dynamics computer simulations to study and compare the statics and dynamics of linear and circular (ring) polymer chains in entangled solutions of different densities. While we confirm that linear chain conformations obey Gaussian statistics at all densities, rings tend to crumple becoming more and more compact as density increases. Conversely, contact frequencies between chain monomers are shown to depend on solution density for both chain topologies. Relaxation of chains at equilibrium is also shown to depend on topology, with ring polymers relaxing faster …
Single-filament tracing has been a valuable tool to directly determine geometrical and mechanical properties of entangled polymer networks. However, systematically verifying how the stiffness of the tracer filament or its molecular interactions with the surrounding network impacts the measurement of these parameters has not been possible with the established experimental systems. Here, we use mechanically programmable DNA nanotubes embedded in crosslinked and entangled F-actin networks, as well as in synthetic DNA networks, in order to measure fundamental, structural network properties like tu…
The classical rheological theories of entangled polymeric liquids are built upon two pillars: Gaussian statistics of entanglement strands and the assumption that the stress arises exclusively from the change of intramolecular configuration entropy. We show that these two hypotheses are not supported by molecular dynamics simulations of polymer melts. Specifically, the segment distribution functions at the entanglement length scale and below deviate considerably from the theoretical predictions, in both the equilibrium and deformed states. Further conformational analysis reveals that the intrac…
Polydispersity is inevitable in industrially produced polymers. Established theories of polymer dynamics and rheology, however, were mostly built on monodisperse linear polymers. Dynamics of polydisperse polymers is yet to be fully explored -- especially how chains of different lengths affect the dynamics of one another in a mixture. This study explored the dynamics of bidisperse polymer melts using molecular dynamics and a bead-spring chain model. Binary mixtures between a moderately entangled long-chain species and an unentangled or marginally entangled short-chain species were investigated.…
Cooperative polymer dynamics at the crossover between the unentangled and the entangled regimes: theoretical predictions of scattering and linear shear relaxation for polyethylene melts
Polymer melts represent the flow-history-dependent behavior. To clearly show this behavior, we have investigated flow behavior of an entangled polymer melt around two cylinders placed in tandem along the flow direction in a two dimensional periodic system. In this system, the polymer states around a cylinder in downstream side are different from the ones around another cylinder in upstream side because the former ones have a memory of a strain experienced when passing around the cylinder in upstream side but the latter ones do not have the memory. Therefore, the shear stress distributions arou…
We derive an effective Maxwell-London equation for entangled polymer complex under the topological constraint, borrowing the theoretical framework from the topological field theory. We find that the transverse current flux of the test polymer chain, surrounded with entangled chains, decays exponentially from its average position with finite penetration depth, which is analogous to the magnetic-field decay in a superconductor (SC). Like as the mass acquirement of photons in SC is the origin of the magnetic-field decay, the polymer earns uncrossible intersections along the chain due to the prese…
Based on non-equilibrium molecular dynamics simulations of entangled polymer melts, a recent Letter [Phys. Rev. Lett. $\textbf{121}$, 047801 (2018), arXiv:1806.09509] claims that the rising extensional stress is quantitatively consistent with the decreasing entropy of chains at the equilibrium entanglement length. We point out that exactly the opposite is true: the intrachain entropic stress arising from individual entanglement strands generally does not agree with the total "macroscopic" stress. The conclusion of the Letter is based on an incomplete and questionable analysis of a limited rang…
We use computer simulations to study the relaxation of strongly deformed highly entangled polymer melts in the non-linear viscoelastic regime, focusing on anisotropic chain conformations after isochoric elongation. The Doi-Edwards tube model and its GLaMM extension, incorporating contour length fluctuation and convective constraint release, predict a retraction of the polymer chain extension in all directions, setting in immediately after deformation. This prediction has been challenged by experiment, simulation, and other theoretical studies, questioning the general validity of the tube conce…
In this work we developed a theoretical model to describe the polymerization of very long macromolecules with simultaneous precipitation. As a reference system for our model we consider UHMWPE polymerization process, via homogeneous catalysis and with molecular weight above $10^6$ monomers. We derived time dependency of the entanglement length in the system, and assessed how polymerization rate and fraction of initiators affects this dependency. We show that decrease of the fraction of initiators leads to increase of the average entanglement length in the melt after complete polymerization. Th…
Effect of small additive molecules on the structural relaxation of polymer melts is investigated via molecular dynamics simulations. At a constant external pressure and a fixed number concentration of added molecules, the variation of particle diameter leads to a non-monotonic change of the relaxation dynamics of the polymer melt. For non-entangled chains, this effect is rationalized in terms of an enhanced added-particle-dynamics which competes with a weaker coupling strength upon decreasing the particle size. Interestingly, cooling simulations reveal a non-monotonic effect on the glass trans…
We report observations of a remarkable scaling behavior with respect to concentration in the passive microbead rheology of two highly entangled polymeric solutions, polyethylene oxide (PEO) and hyaluronic acid (HA). This behavior was reported previously [Hill et al., PLOS ONE (2014)] for human lung mucus, a complex biological hydrogel, motivating the current study for synthetic polymeric solutions PEO and HA. The strategy is to identify, and focus within, a wide range of lag times $τ$ for which passive micron diameter beads exhibit self-similar (fractional, power law) mean-squared-displacement…
In this paper we study a system of entangled chains that bear reversible cross-links in a melt state. The cross-links are tethered uniformly on the backbone of each chain. A slip-link type model for the system is presented and solved for the relaxation modulus. The effects of entanglements and reversible cross-linkers are modelled as discrete form of constraints that influence the motion of the primitive path. In contrast to a non-associating entangled system the model calculations demonstrate that the elastic modulus has a much higher first plateau and a delayed terminal relaxation. These eff…
