Rheology and Microscopic Topology of Entangled Polymeric Liquids

Microscopic view of accelerated dynamics in deformed polymer glasses

A molecular level analysis of segmental trajectories obtained from molecular dynamics simulations is used to obtain the full relaxation time spectrum in aging polymer glasses subject to three different deformation protocols. As in experiments, dynamics can be accelerated by several orders of magnitude, and a narrowing of the distribution of relaxation times during creep is directly observed. Additionally, the acceleration factor describing the transformation of the relaxation time distributions is computed and found to obey a universal dependence on the strain, independent of age and deformation protocol.

Confining potential when a biopolymer filament reptates

Using single-molecule fluorescence imaging, we track Brownian motion perpendicular to the contour of tightly entangled F-actin filaments and extract the confining potential. The chain localization presents a small-displacement Hookean regime followed by a large amplitude regime where the effective restoring force is independent of displacement. The implied heterogeneity characterized by a distribution of tube width is modeled.

Combined molecular algorithms for the generation, equilibration and topological analysis of entangled polymers: methodology and performance

We review the methodology, algorithmic implementation and performance characteristics of a hierarchical modeling scheme for the generation, equilibration and topological analysis of polymer systems at various levels of molecular description: from atomistic polyethylene samples to random packings of freely-jointed chains of tangent hard spheres of uniform size. Our analysis focuses on hitherto less discussed algorithmic details of the implementation of both, the Monte Carlo (MC) procedure for the system generation and equilibration, and a postprocessing step, where we identify the underlying topological structure of the simulated systems in the form of primitive paths. In order to demonstrate our arguments, we study how molecular length and packing density (volume fraction) affect the performance of the MC scheme built around chain-connectivity altering moves. In parallel, we quantify the effect of finite system size, of polydispersity, and of the definition of the number of entanglements (and related entanglement molecular weight) on the results about the primitive path network. Along these lines we approve main concepts which had been previously proposed in the literature.

Fracture in glassy polymers: a molecular modeling perspective

Over the past 25 years, molecular modeling and simulations have provided important insights into the physics of deformation and fracture of glassy polymers. This review presents an overview of key results discussed in the context of experimentally observed polymer behavior. Both atomistic and coarse-grained polymer models have been used in different deformation protocols to study elastic properties, shear yielding, creep, physical aging, strain hardening and crazing. Simulations reproduce most of the macroscopic features of plasticity in polymer glasses such as stress-strain relations and creep response, and reveal information about the underlying atomistic processes. Trends of the shear yield stress with loading conditions, temperature and strain rate, and the atomistic dynamics under load have been systematically explored. Most polymers undergo physical aging, which leads to a history-dependent mechanical response. Simulations of strain hardening and crazing demonstrate the nature of polymer entanglements in the glassy state and the role of local plasticity and provide insight into the origin of fracture toughness of amorphous polymers.

Entanglement network in nanoparticle reinforced polymers

Polymer nanocomposites have been widely studied in efforts to engineer materials with mechanical properties superior to those of the pure polymer, but the molecular origins of the sought-after improved properties have remained elusive. An ideal polymer nanocomposite model has been conceived in which the nanoparticles are dispersed throughout the polymeric matrix. A detailed examination of topological constraints (or entanglements) in a nanocomposite glass provides new insights into the molecular origin of the improved properties in polymer nanocomposites by revealing that the nanoparticles impart significant enhancements to the entanglement network. Nanoparticles are found to serve as entanglement attractors, particularly at large deformations, altering the topological constraint network that arises in the composite material.

Entangled chain dynamics of polymer knots in extensional flow

We formulate a coarse-grained molecular-dynamics model of polymer chains in solution that includes hydrodynamic interactions, thermal fluctuations, nonlinear elasticity, and topology-preserving solvent mediated excluded volume interactions. The latter involve a combination of potential forces with explicit geometric detection and tracking of chain entanglements. By solving this model with numerical and computational methods, we study the physics of polymer knots in a strong extensional flow (Deborah number De=1.6 ). We show that knots slow down the stretching of individual polymers by obstructing via entanglements the "natural," unraveling, and flow-induced chain motions. Moreover, the steady-state polymer length and polymer-induced stress values are smaller in knotted chains than in topologically trivial chains. We indicate the molecular processes via which the rate of knot tightening affects the rheology of the solution.

Conformational free energy of melts of ring-linear polymer blends

The conformational free energy of ring polymers in a blend of ring and linear polymers is investigated using the bond-fluctuation model. Previously established scaling relationships for the free energy of a ring polymer are shown to be valid only in the mean-field sense, and alternative functional forms are investigated. It is shown that it may be difficult to accurately express the total free energy of a ring polymer by a simple scaling argument, or in closed form.

Topological analysis of polymeric melts: chain-length effects and fast-converging estimators for entanglement length

Primitive path analyses of entanglements are performed over a wide range of chain lengths for both bead spring and atomistic polyethylene polymer melts. Estimators for the entanglement length N_{e} which operate on results for a single chain length N are shown to produce systematic O(1/N) errors. The mathematical roots of these errors are identified as (a) treating chain ends as entanglements and (b) neglecting non-Gaussian corrections to chain and primitive path dimensions. The prefactors for the O(1/N) errors may be large; in general their magnitude depends both on the polymer model and the method used to obtain primitive paths. We propose, derive, and test new estimators which eliminate these systematic errors using information obtainable from the variation in entanglement characteristics with chain length. The new estimators produce accurate results for N_{e} from marginally entangled systems. Formulas based on direct enumeration of entanglements appear to converge faster and are simpler to apply.

Segment sizes of supramolecular polymers of N,N',N"-tris(3,7-dimethyloctyl)benzene-1,3,5-tricarboxamide in n-decane

The average segment sizes of rather flexible supramolecular polymers formed by a racemic mixture of N,N',N"-tris(3,7-dimethyloctyl)benzene-1,3,5-tricarboxamide (DO3B) in n-decane (C10) (DO3B/C10) and a mixture of DO3B and homochiral (S)-N,N',N"-tris(3,7-dimethyloctyl)benzene-1,3,5-tricarboxamide ((S)DO3B) in C10 (DO3B:(S)DO3B/C10) were determined using mechanical relaxation techniques, including linear and nonlinear viscoelastic experiments in addition to dielectric relaxation measurements. To evaluate the average sizes of segments for the supramolecular polymers formed in these systems, the conventional rubber elasticity theory was employed for the linear viscoelastic behavior and a non-Gaussian-type three-chain model assuming chains constructed with freely joined chain segments with finite sizes for nonlinear viscoelastic responses. Concentration-independent similar segment sizes for the supramolecular polymer formed in DO3B/C10 were evaluated using both the mechanical techniques. However, the segment size of the formed supramolecular polymer increased upon increasing the component of (S)DO3B to the total amount of DO3B and (S)DO3B in DO3B:(S)DO3B/Ci0o. The formed supramolecular polymer in the racemic DO3B/C10 system possessed threefold intermolecular hydrogen bonding aligned along its stiff columnar structure, with equimolar right- and left-handed helicities connected by defective portions, which were DO3B molecules containing defects in hydrogen bond formation. In the case of DO3B:(S)DO3B/C10, the addition of the chiral (S)DO3B component induced the majority rule effect for the helicity of the system to increase the major part of helicity, which led to the reduction of the number of defects and an increase in segment seizes. Because large macrodipoles are naturally generated by helical three-fold hydrogen bonding of amide groups in a head-to-tail arrangement, as in type-A polymers possessing parallel electric dipoles along their backbones, profound dielectric behavior provided a concentration-independent segment size for a supramolecular polymer formed in both the DO3B/ C10 and DO3B:(S)DO3B/C10 systems, the value of which was about one-third of that determined mechanically. A free-rotation chain model that is slightly more realistic than the freely jointed chain model was employed to consider the discrepancy in the evaluated segment sizes. Then, helical columns consisting of nine DO3B molecules and supramolecular polymer chain portions formed by a three-column connection behaved as equivalent freely jointed (Kuhn) segments.

Structure and dynamics of cylindrical micelles at equilibrium and under shear flow

The dynamics and rheology of semidilute unentangled micellar solutions are investigated by Langevin dynamics mesoscopic simulations coupled to a microreversible kinetic model for scissions and recombinations. Two equilibrium state points, differing by the scission energy and therefore by the corresponding average micelle length, have been examined. The kinetic rates are tuned by an independent parameter of the model, whose range is chosen in such a way that the kinetics always strongly couple to the chain dynamics. Our results confirm, as predicted by Faivre and Gardissat, that the stress relaxation, as well as the monomer diffusion, is characterized by a time tauLambda, defined by the lifetime of a segment Lambda, whose Rouse relaxation time is equal to its lifetime. Moreover, the power-law dependence of the zero-shear viscosity versus tauLambda was evidenced. Under stationary shear, the chains are deformed and their average bond length is increased, which enhances the overall scission frequency. In turn, this induces an overall shortening of the chains in order to increase the overall corresponding chain-end recombination frequency, as required by the stationary conditions. Nonequilibrium simulations show that the chain deformation and orientation, as well as the rheology of the system, can be expressed as universal functions of a single reduced shear rate betaLambda=gammatauLambda (with gamma the bare shear rate). Furthermore, local analysis of the kinetics under stationary shear gives insights on the variation of the average length with shear rate.

Microphase separation in cross-linked polymer blends. Efficient replica RPA post-processing of simulation data for homopolymer networks

We investigate the behaviour of randomly cross-linked (co)polymer blends using a combination of replica theory and large-scale molecular dynamics simulations. In particular, we derive the analogue of the random phase approximation for systems with quenched disorder and show how the required correlation functions can be calculated efficiently. By post-processing simulation data for homopolymer networks we are able to describe neutron scattering measurements in heterogeneous systems without resorting to microscopic detail and otherwise unphysical assumptions. We obtain structure function data which illustrate the expected microphase separation and contain system-specific information relating to the intrinsic length scales of our networks.

Universal scaling, entanglements, and knots of model chain molecules

By identifying the maximally random jammed state of freely jointed chains of tangent hard spheres we are able to determine the distinct scaling regimes characterizing the dependence of chain dimensions and topology on volume fraction. Calculated distributions of (i) the contour length of the primitive paths and (ii) the number of entanglements per chain agree remarkably well with recent theoretical predictions in all scaling regimes. Furthermore, our simulations reveal a hitherto unsuspected connection between purely intramolecular (knots) and intermolecular (entanglements) topological constraints.

Structure and dynamics of interphase chromosomes

During interphase chromosomes decondense, but fluorescent in situ hybridization experiments reveal the existence of distinct territories occupied by individual chromosomes inside the nuclei of most eukaryotic cells. We use computer simulations to show that the existence and stability of territories is a kinetic effect that can be explained without invoking an underlying nuclear scaffold or protein-mediated interactions between DNA sequences. In particular, we show that the experimentally observed territory shapes and spatial distances between marked chromosome sites for human, Drosophila, and budding yeast chromosomes can be reproduced by a parameter-free minimal model of decondensing chromosomes. Our results suggest that the observed interphase structure and dynamics are due to generic polymer effects: confined Brownian motion conserving the local topological state of long chain molecules and segregation of mutually unentangled chains due to topological constraints.

Screening of hydrodynamic interactions in Brownian rod suspensions

We present the details and results of a simulation study addressing the dynamics and rheology of rod suspensions over a wide regime of concentrations ranging from dilute to concentrated systems. Our study compares the results of two complementary simulation methods. The first method adapts a recently proposed explicit solvent simulation strategy and incorporates both hydrodynamical effects and steric interactions between the rod units. We compare the results of such a method with those obtained from a Brownian dynamics simulation approach which retains the steric interactions but neglects the effects of hydrodynamic interactions. Overall, our results in the context of the translational and rotational diffusivities are in agreement with the hydrodynamical predictions in the dilute regime and the corresponding results of the tube model and its extensions thereof in the semidilute regimes. The latter results suggest that effects of hydrodynamic interactions on the translational and rotational diffusivities are secondary relative to the steric interactions and at best lead only to a small correction to the results of the classical tube model. Our results in the context of linear viscoelasticity also broadly confirms the predictions of the tube model for the storage and loss moduli and allows us to extract for the first time the independent hydrodynamic and Brownian contributions to the zero shear viscosity. While the relative magnitudes of these contributions are consistent with the theoretical predictions, the quantitative magnitudes are quite different from the theoretical predictions. Overall, these results confirm the validity of the hydrodynamic "screening" hypothesis and ratify the neglect of hydrodynamical stresses in quantifying the linear rheology of Brownian rod suspensions.

Entangled dynamics of a stiff polymer

Entangled networks of stiff biopolymers exhibit complex dynamic response, emerging from the topological constraints that neighboring filaments impose upon each other. We propose a class of reference models for entanglement dynamics of stiff polymers and provide a quantitative foundation of the tube concept for stiff polymers. For an infinitely thin needle exploring a planar course of point obstacles, we have performed large-scale computer simulations proving the conjectured scaling relations from the fast transverse equilibration to the slowest process of orientational relaxation. We determine the rotational diffusion coefficient of the tracer, its angular confinement, the tube diameter, and the orientational correlation functions.

Analytic expressions for the statistics of the primitive-path length in entangled polymers

An analytic expression is proposed for the primitive-path length of entangled polymer chains. The expression is derived from statistical mechanics of a chain that is a random walk with randomly scattered entanglements. The only parameters are the number of Kuhn steps in the chain and a dimensionless parameter beta that contains information about the entanglement density and Kuhn step size. The expression is found to compare very favorably with numerical results recently found from examining topological constraints in microscopic simulations. The comparison also predicts well the plateau modulus of polyethylene, suggesting that the slip-link model is a viable intermediate in the search for true ab initio rheology predictions. Since the expression is analytic, it can be used to make predictions where the simulations cannot reach, and hence is applicable for coarse graining.

Strain hardening of polymer glasses: entanglements, energetics, and plasticity

Simulations are used to examine the microscopic origins of strain hardening in polymer glasses. While stress-strain curves for a wide range of temperature can be fit to the functional form predicted by entropic network models, many other results are fundamentally inconsistent with the physical picture underlying these models. Stresses are too large to be entropic and have the wrong trend with temperature. The most dramatic hardening at large strains reflects increases in energy as chains are pulled taut between entanglements rather than a change in entropy. A weak entropic stress is only observed in shape recovery of deformed samples when heated above the glass transition. While short chains do not form an entangled network, they exhibit partial shape recovery, orientation, and strain hardening. Stresses for all chain lengths collapse when plotted against a microscopic measure of chain stretching rather than the macroscopic stretch. The thermal contribution to the stress is directly proportional to the rate of plasticity as measured by breaking and reforming of interchain bonds. These observations suggest that the correct microscopic theory of strain hardening should be based on glassy state physics rather than rubber elasticity.

Equilibrium statistical distributions for subchains in an entangled polymer melt

In de Gennes-Doi-Edwards theory for entangled polymeric melts, a length scale r(0) is introduced, giving the equilibrium mesh size of the physical network of chains. Each polymer molecule is then represented as a random walk, with a step size r(0) (a "subchain", made up of n(0) Kuhn segments) dictated by the existence of entanglements. Progressing from this simple picture, an issue that has been constantly overlooked so far, despite its potential relevance, is that of finite-size effects at the de Gennes-Doi-Edwards characteristic length scale. Actually, since a subchain in a melt is a "small", nonmacroscopic system, fluctuations of both its length and its number of Kuhn segments are certainly nonnegligible. An ad hoc theoretical treatment from nonstandard (nano) statistical mechanics and thermodynamics seems then required, to find the anticipated equilibrium statistical distributions of the subchain population. In this contribution, we carefully discuss this topic. Some predictions from the nonstandard fluctuation-inclusive approach on the statistics of subchains are here obtained, and compared with existing simulations, even down to the atomistic level.