Entanglements of an End-Grafted Polymer Brush in a Polymeric Matrix

Characterizing the shear response of polymer-grafted nanoparticles

Grafting polymer chains to the surface of nanoparticles overcomes the challenge of nanoparticle dispersion within nanocomposites and establishes high-volume fractions that are found to enable enhanced material mechanical properties. This study utilizes coarse-grained molecular dynamics simulations to quantify how the shear modulus of polymer-grafted nanoparticle (PGN) systems in their glassy state depends on parameters such as strain rate, nanoparticle size, grafting density, and chain length. The results are interpreted through further analysis of the dynamics of chain conformations and volume fraction arguments. The volume fraction of nanoparticles is found to be the most influential variable in deciding the shear modulus of PGN systems. A simple rule of mixture is utilized to express the monotonic dependence of shear modulus on the volume fraction of nanoparticles. Due to the reinforcing effect of nanoparticles, shortening the grafted chains results in a higher shear modulus in PGNs, which is not seen in linear systems. These results offer timely insight into calibrating molecular design parameters for achieving the desired mechanical properties in PGNs.

Nanostructure Explains the Behavior of Slippery Covalently Attached Liquid Surfaces

Slippery covalently-attached liquid surfaces (SCALS) with low contact angle hysteresis (CAH, <5°) and nanoscale thickness display impressive anti-adhesive properties, similar to lubricant-infused surfaces. Their efficacy is generally attributed to the liquid-like mobility of the constituent tethered chains. However, the precise physico-chemical properties that facilitate this mobility are unknown, hindering rational design. This work quantifies the chain length, grafting density, and microviscosity of a range of polydimethylsiloxane (PDMS) SCALS, elucidating the nanostructure responsible for their properties. Three prominent methods are used to produce SCALS, with characterization carried out via single-molecule force measurements, neutron reflectometry, and fluorescence correlation spectroscopy. CO2 snow-jet cleaning was also shown to reduce the CAH of SCALS via a modification of their grafting density. SCALS behavior can be predicted by reduced grafting density, Σ, with the lowest water CAH achieved at Σ≈2. This study provides the first direct examination of SCALS grafting density, chain length, and microviscosity and supports the hypothesis that SCALS properties stem from a balance of layer uniformity and mobility.

Validation and Refinement of Unified Analytic Model for Flexible and Semiflexible Polymer Melt Entanglement

We combine molecular dynamics simulations and topological analyses (TA) to validate and refine a recently proposed unified analytic model [Hoy, R. S.; Kröger, M. Phys. Rev. Lett. 2020, 124, 147801] for the reduced entanglement length, tube diameter, and plateau modulus of polymer melts. While the functional forms of the previously published expressions are insensitive to the choice of the TA method and N _e -estimator, obtaining better statistics and eliminating all known sources of systematic error in the N _e -estimation alters their numerical coefficients. Our revised expressions quantitatively match bead-spring simulation data over the entire range of chain stiffnesses for which systems remain isotropic, semiquantitatively match all available experimental data for flexible, semiflexible, and stiff polymer melts (including new data for conjugated polymers that lie in a previously unpopulated stiffness regime), and outperform previously developed unified scaling theories.

Crosslinking of Pressure-Sensitive Adhesives with Polymer-Grafted Nanoparticles

Nanocomposite filler particles provide multiple routes to mechanically reinforce pressure-sensitive adhesives (PSAs), as their large surface area to volume ratios provide a means of effectively crosslinking multiple polymer chains. A major advancement could therefore be enabled by the design of a particle architecture that forms multiple physical and chemical interactions with the surrounding polymer matrix, while simultaneously ensuring particle dispersion and preventing particle aggregation. Understanding how such multivalent interactions between a nanoparticle crosslinking point and the PSA polymer affect material mechanical performance would provide both useful scientific knowledge on the mechanical structure-property relationships in polymer composites, as well as a new route to synthesizing useful PSA materials. Herein, we report the use of polymer-grafted nanoparticles (PGNPs) composed of poly(n-butyl acrylate-co-acrylic acid) chains grafted to SiO₂ nanoparticle (NP) surfaces to cohesively reinforce PSA films against shear stress without compromising their adhesive properties. The use of acrylic acid-decorated PGNPs allows for ionic crosslinking via metal salt coordination to be used in conjunction with physical entanglement, yielding 33% greater shear resistance and up to 3-fold longer holding times under static load. In addition, the effects of material parameters such as PGNP/crosslinker loading, polymer graft length, and core nanoparticle size on mechanical properties are also explored, providing insights into the use of PGNPs for the rational design of polymer composite-based PSAs.

Facile equilibration of well-entangled semiflexible bead-spring polymer melts

The widely used double-bridging hybrid (DBH) method for equilibrating simulated entangled polymer melts [Auhl et al., J. Chem. Phys. 119, 12718-12728 (2003)] loses its effectiveness as chain stiffness increases into the semiflexible regime because the energy barriers associated with double-bridging Monte Carlo moves become prohibitively high. Here we overcome this issue by combining DBH with the use of core-softened pair potentials. This reduces the energy barriers substantially, allowing us to equilibrate melts with N ≃ 40N_e and chain stiffnesses all the way up to the isotropic-nematic transition using simulations of no more than 100 × 10⁶ time steps. For semiflexible chains, our method is several times faster than the standard DBH; we exploit this speedup to develop improved expressions for Kremer-Grest melts' chain-stiffness-dependent Kuhn length ℓ_K and entanglement length N_e.

Effect of softness on glass melting and re-entrant solidification in mixtures of soft and hard colloids

We present a systematic investigation of the structure and dynamic properties of model soft-hard colloidal mixtures. Results of a coarse-grained theoretical model are contrasted with rheological data, where the soft and hard colloids are mimicked by large star polymers with high functionality as the soft component and smaller stars with ultrahigh functionality as the hard one. Previous work by us revealed the recovery of the ergodicity of glassy soft star solutions and subsequent arrested phase separation and re-entrant solid transition upon progressive addition of small hard depletants. Here, we use different components to show that a small variation in softness has a significant impact on the state diagram of such mixtures. In particular, we establish that rendering the soft component more penetrable and modifying the size ratio bring about a remarkable shift in both the phase separation region and the glass-melting line so that the region of restored ergodicity can be notably enhanced and extended to much higher star polymer concentrations than for pure systems. We further rationalize our findings by analyzing the features of the depletion interaction induced by the smaller component that result from the interplay between the size ratio and the softness of the large component. These results demonstrate the great sensitivity of the phase behavior of entropic mixtures to small changes in the molecular architecture of the soft stars and point to the importance of accounting for details of the internal microstructure of soft colloidal particles for tailoring the flow properties of soft composites.

Tailoring Antifouling Properties of Nanocarriers via Entropic Collision of Polymer Grafting

Polymer graftings (PGs) are widely employed in antifouling surfaces and drug delivery systems to regulate the interaction with a foreign environment. Through molecular dynamics simulations and scaling theory analysis, we investigate the physical antifouling properties of PGs via their collision behaviors. Compared with mushroom-like PGs with low grafting density, we find brush-like PGs with high grafting density could generate large deformation-induced entropic repulsive force during a collision, revealing a microscopic mechanism for the hop motions of polymer-grafted nanoparticles for drug delivery observed in experiment. In addition, the collision elasticity of PGs is found to decay with the collision velocity by a power law, i.e., a concise dynamic scaling despite the complex process involved, which is beyond expectation. These results elucidate the dynamic interacting mechanism of PGs, which are of immediate interest for a fundamental understanding of the antifouling performance of PGs and the rational design of PG-coated nanoparticles in nanomedicine for drug delivery.

Polymer Brush Coating and Adhesion Technology at Scale

Creating strong joints between dissimilar materials for high-performance hybrid products places high demands on modern adhesives. Traditionally, adhesion relies on the compatibility between surfaces, often requiring the use of primers and thick bonding layers to achieve stable joints. The coatings of polymer brushes enable the compatibilization of material surfaces through precise control over surface chemistry, facilitating strong adhesion through a nanometer-thin layer. Here, we give a detailed account of our research on adhesion promoted by polymer brushes along with examples from industrial applications. We discuss two fundamentally different adhesive mechanisms of polymer brushes, namely (1) physical bonding via entanglement and (2) chemical bonding. The former mechanism is demonstrated by e.g., the strong bonding between poly(methyl methacrylate) (PMMA) brush coated stainless steel and bulk PMMA, while the latter is shown by e.g., the improved adhesion between silicone and titanium substrates, functionalized by a hydrosilane-modified poly(hydroxyethyl methacrylate) (PHEMA) brush. This review establishes that the clever design of polymer brushes can facilitate strong bonding between metals and various polymer materials or compatibilize fillers or nanoparticles with otherwise incompatible polymeric matrices. To realize the full potential of polymer brush functionalized materials, we discuss the progress in the synthesis of polymer brushes under ambient and scalable industrial conditions, and present recent developments in atom transfer radical polymerization for the large-scale production of brush-modified materials.

Clustering of Entanglement Points in Highly Strained Polymer Melts

Polymer melts undergoing large deformation by elongation are studied by molecular dynamics simulations of bead-spring chains in melts. By applying a primitive path analysis to strongly deformed polymer melts, the role of topological constraints in highly entangled polymer melts is investigated and quantified. We show that the overall, large scale conformations of the primitive paths (PPs) of stretched chains follow affine deformation while the number and the distribution of entanglement points along the PPs do not. Right after deformation, PPs of chains retract in both directions parallel and perpendicular to the elongation. Upon further relaxation we observe a long-lived clustering of entanglement points. Together with the delayed relaxation time this leads to a metastable inhomogeneous distribution of topological constraints in the melts.

Modeling of Entangled Polymer Diffusion in Melts and Nanocomposites: A Review

This review concerns modeling studies of the fundamental problem of entangled (reptational) homopolymer diffusion in melts and nanocomposite materials in comparison to experiments. In polymer melts, the developed united atom and multibead spring models predict an exponent of the molecular weight dependence to the polymer diffusion very similar to experiments and the tube reptation model. There are rather unexplored parameters that can influence polymer diffusion such as polymer semiflexibility or polydispersity, leading to a different exponent. Models with soft potentials or slip-springs can estimate accurately the tube model predictions in polymer melts enabling us to reach larger length scales and simulate well entangled polymers. However, in polymer nanocomposites, reptational polymer diffusion is more complicated due to nanoparticle fillers size, loading, geometry and polymer-nanoparticle interactions.

Scaling and Interactions of Linear and Ring Polymer Brushes via DPD Simulations

Single and double layers of polymer coated surfaces are investigated by means of Dissipative Particle Dynamics (DPD), focusing on the difference between grafted ring and linear chains. Several different surface coverages σ , as well as chain lengths N and brush separations D, are analyzed for athermal, i.e., good solvent, conditions. The size in the form of the radius of gyration R g , the shape as asphericity δ ∗ , and orientation β ∗ , as well as density profiles as functions of distance from grafting plane ρ ( z ) , are studied. The effect of an added bond repulsion potential to suppress bond crossing in DPD is analyzed. Scaling laws of R g and its components R g ⊥ and R g ∥ are investigated. We find R g ∝ N ν , ν = 0.588 for surface coverages below the overlap surface concentration σ ∗ . For σ > σ ∗ we find R g ⊥ ∝ N ν ⊥ , ν ⊥ ≅ 1 and R g ∥ ∝ N ν ∥ , ν ∥ = 1 / 2 of ring brushes with the standard DPD model and ν ∥ ≅ 2 / 5 with added bond repulsion. The σ dependence of the radius of gyration was found to be R g ∝ σ μ with μ = 1 / 3 for surface coverages grater than σ ∗ . The perpendicular component R g ⊥ scales independent of the bond repulsion potential as R g ⊥ ∝ σ μ ⊥ , μ ⊥ = 1 / 3 , whereas the scaling of the parallel component exhibits a topological repulsion dependence R g ∥ ∝ σ μ ∥ , μ ∥ = - 1 / 12 for standard DPD and μ ∥ = - 1 / 6 for bond repulsion.

Polymers on nanoparticles: structure & dynamics

Grafting polymers to nanoparticle surfaces influences properties from the conformation of the polymer chains to the dispersion and assembly of nanoparticles within a polymeric material. Recently, a small body of work has begun to address the question of how grafting polymers to a nanoparticle surface impacts chain dynamics, and the resulting physical properties of a material. This Review discusses recent work that characterizes the structure and dynamics of polymers that are grafted to nanoparticles and opportunities for future research. Starting from the case of a single polymer chain attached to a nanoparticle core, this Review follows the structure of the chains as grafting density increases, and how this structure slows relaxation of polymer chains and affects macroscopic material properties.

Combined Experimental and Simulation Studies of Cross-Linked Polymer Brushes under Shear

We have studied the effect of cross-linking on the tribological behavior of polymer brushes using a combined experimental and theoretical approach. Tribological and indentation measurements on poly(glycidyl methacrylate) brushes and gels in the presence of dimethylformamide solvent were obtained by means of atomic force microscopy. To complement experiments, we have performed corresponding molecular dynamics (MD) simulations of a generic bead-spring model in the presence of explicit solvent and cross-linkers. Our study shows that cross-linking leads to an increase in friction between polymer brushes and a counter-surface. The coefficient of friction increases with increasing degree of cross-linking and decreases with increasing length of the cross-linker chains. We find that the brush-forming polymer chains in the outer layer play a significant role in reducing friction at the interface.

Perspective: Outstanding theoretical questions in polymer-nanoparticle hybrids

This topical review discusses the theoretical progress made in the field of polymer nanocomposites, i.e., hybrid materials created by mixing (typically inorganic) nanoparticles (NPs) with organic polymers. It primarily focuses on the outstanding issues in this field and is structured around five separate topics: (i) the synthesis of functionalized nanoparticles; (ii) their phase behavior when mixed with a homopolymer matrix and their assembly into well-defined superstructures; (iii) the role of processing on the structures realized by these hybrid materials and the role of the mobilities of the different constituents; (iv) the role of external fields (electric, magnetic) in the active assembly of the NPs; and (v) the engineering properties that result and the factors that control them. While the most is known about topic (ii), we believe that significant progress needs to be made in the other four topics before the practical promise offered by these materials can be realized. This review delineates the most pressing issues on these topics and poses specific questions that we believe need to be addressed in the immediate future.

Multiscale Molecular Simulations of Polymer-Matrix Nanocomposites: or What Molecular Simulations Have Taught us About the Fascinating Nanoworld

Following the substantial progress in molecular simulations of polymer-matrix nanocomposites, now is the time to reconsider this topic from a critical point of view. A comprehensive survey is reported herein providing an overview of classical molecular simulations, reviewing their major achievements in modeling polymer matrix nanocomposites, and identifying several open challenges. Molecular simulations at multiple length and time scales, working hand-in-hand with sensitive experiments, have enhanced our understanding of how nanofillers alter the structure, dynamics, thermodynamics, rheology and mechanical properties of the surrounding polymer matrices.

Monitoring the Switching of Single BSA-ATTO 488 Molecules Covalently End-Attached to a pH-Responsive PAA Brush

We describe a novel combination of a responsive polymer brush and a fluorescently labeled biomolecule, where the position of the biomolecule can be switched from inside to outside the brush and vice versa by a change in pH. For this, we grafted ultrathin, amino-terminated poly(acrylic acid) brushes to glass and silicon substrates. Individual bovine serum albumin (BSA) molecules labeled with fluorophore ATTO 488 were covalently end-attached to the polymers in this brush using a bis-N-succinimidyl-(pentaethylene glycol) linker. We investigated the dry layer properties of the brush-protein ensemble, and it is swelling behavior using spectroscopic ellipsometry. Total internal reflection fluorescence (TIRF) microscopy enabled us to study the distance-dependent switching of the fluorescently labeled protein molecules. The fluorescence emission from the labeled proteins ceased (out-state) when the polymer chains stretched away from the interface under basic pH conditions, and fluorescence recurred (in-state) when the chains collapsed under acidic conditions. Moreover, TIRF allowed us to study the fluorescence switching behavior of fluorescently labeled BSA molecules down to the single-molecule level, and we demonstrate that this switching is fast but that the exact intensity during the in-state is the result of a more random process. Control experiments verify that the switching behavior is directly correlated to the responsive behavior of the polymer brush. We propose this system as a platform for switchable sensor applications but also as a method to study the swelling and collapse of individual polymer chains in a responsive polymer brush.

Influence of Chain Stiffness, Grafting Density and Normal Load on the Tribological and Structural Behavior of Polymer Brushes: A Nonequilibrium-Molecular-Dynamics Study

We have performed coarse-grained molecular-dynamics simulations on both flexible and semiflexible multi-bead-spring model polymer brushes in the presence of explicit solvent particles, to explore their tribological and structural behaviors. The effect of stiffness and tethering density on equilibrium-brush height is seen to be well reproduced within a Flory-type theory. After discussing the equilibrium behavior of the model brushes, we first study the shearing behavior of flexible chains at different grafting densities covering brush and mushroom regimes. Next, we focus on the effect of chain stiffness on the tribological behavior of polymer brushes. The tribological properties are interpreted by means of the simultaneously recorded density profiles. We find that the friction coefficient decreases with increasing persistence length, both in velocity and separation-dependency studies, over the stiffness range explored in this work.

A New Self-Consistent Field Model of Polymer/Nanoparticle Mixture

Field-theoretical method is efficient in predicting assembling structures of polymeric systems. However, it’s challenging to generalize this method to study the polymer/nanoparticle mixture due to its multi-scale nature. Here, we develop a new field-based model which unifies the nanoparticle description with the polymer field within the self-consistent field theory. Instead of being “ensemble-averaged” continuous distribution, the particle density in the final morphology can represent individual particles located at preferred positions. The discreteness of particle density allows our model to properly address the polymer-particle interface and the excluded-volume interaction. We use this model to study the simplest system of nanoparticles immersed in the dense homopolymer solution. The flexibility of tuning the interfacial details allows our model to capture the rich phenomena such as bridging aggregation and depletion attraction. Insights are obtained on the enthalpic and/or entropic origin of the structural variation due to the competition between depletion and interfacial interaction. This approach is readily extendable to the study of more complex polymer-based nanocomposites or biology-related systems, such as dendrimer/drug encapsulation and membrane/particle assembly.

Role of Thermal History and Entanglement Related Thickness Selection in Polymer Crystallization

Using molecular dynamics simulations and primitive path analysis, we show that hot entangled polymer melts can crystallize faster with higher crystallinities and larger crystalline stem lengths, as compared to cold melts under rapid quenching conditions or during cold-crystallization. This counterintuitive phenomenon similar to the so-called Mpemba effect observed for water can be explained by the temperature dependence of entanglements. Our results demonstrate the key role of the entanglement state for crystallization properties and provide a new approach to understand the role of thermal history and to the open question of thickness selection in polymer crystallization.

Control interfacial properties and tensile strength of glass fibre/PP composites by grafting poly(ethylene glycol) chains on glass fibre surface

The interfacial adhesion increased as the grafted PEG chains became longer.

Healing of polymer interfaces: Interfacial dynamics, entanglements, and strength

Self-healing of polymer films often takes place as the molecules diffuse across a damaged region, above their melting temperature. Using molecular dynamics simulations we probe the healing of polymer films and compare the results with those obtained for thermal welding of homopolymer slabs. These two processes differ from each other in their interfacial structure since damage leads to increased polydispersity and more short chains. A polymer sample was cut into two separate films that were then held together in the melt state. The recovery of the damaged film was followed as time elapsed and polymer molecules diffused across the interface. The mass uptake and formation of entanglements, as obtained from primitive path analysis, are extracted and correlated with the interfacial strength obtained from shear simulations. We find that the diffusion across the interface is significantly faster in the damaged film compared to welding because of the presence of short chains. Though interfacial entanglements increase more rapidly for the damaged films, a large fraction of these entanglements are near chain ends. As a result, the interfacial strength of the healing film increases more slowly than for welding. For both healing and welding, the interfacial strength saturates as the bulk entanglement density is recovered across the interface. However, the saturation strength of the damaged film is below the bulk strength for the polymer sample. At saturation, cut chains remain near the healing interface. They are less entangled and as a result they mechanically weaken the interface. Chain stiffness increases the density of entanglements, which increases the strength of the interface. Our results show that a few entanglements across the interface are sufficient to resist interfacial chain pullout and enhance the mechanical strength.

Ligand engineering of polymer nanocomposites: from the simple to the complex

One key to optimizing the performance of polymer nanocomposites for high-tech applications is surface ligand engineering of the nanofiller, which has been used to either tune the nanofiller morphology or introduce additional functionalities. Ligand engineering can be relatively simple such as a single population of short molecules on the nanoparticle surface designed for matrix compatibility. It can also have complexity that includes bimodal (or multimodal) populations of ligands that enable relatively independent control of enthalpic and entropic interactions between the nanofiller and matrix as well as introduce additional functionality and dynamic control. In this Spotlight on Applications, we provide a brief review into the use of brush ligands to tune the thermodynamic interactions between nanofiller and matrix and then focus on the potential for surface ligand engineering to create exciting nanocomposites properties for optoelectronic and dielectric applications.

Structure and Strength at Immiscible Polymer Interfaces

Thermal welding of polymer-polymer interfaces is important for integrating polymeric elements into devices. When two different polymers are joined, the strength of the weld depends critically on the degree of immiscibility. We perform large-scale molecular dynamics simulations of the structure-strength relation at immiscible polymer interfaces. Our simulations show that immiscibility arrests interdiffusion and limits the equilibrium interfacial width. Even for weakly immiscible films, the narrow interface is unable to transfer stress upon deformation as effectively as the bulk material, and chain pullout at the interface becomes the dominant failure mechanism. This greatly reduces the interfacial strength. The weak response of immiscible interfaces is shown to arise from an insufficient density of entanglements across the interface. We demonstrate that there is a threshold interfacial width below which no significant entanglements can form between opposite sides to strengthen the interface.

Molecular dynamics simulations of polymer welding: strength from interfacial entanglements

Large-scale simulations of thermal welding of polymers are performed to investigate the rise of mechanical strength at the polymer-polymer interface with the welding time t(w). The welding process is at the core of integrating polymeric elements into devices as well as in the thermal induced healing of polymers, processes that require the development of interfacial strength equal to that of the bulk. Our simulations show that the interfacial strength saturates at the bulk shear strength long before polymers diffuse by their radius of gyration. Along with the strength increase, the dominant failure mode changes from chain pullout at the interface to chain scission as in the bulk. The formation of sufficient entanglements across the interface, which we track using a primitive path analysis, is required to arrest catastrophic chain pullout at the interface. The bulk response is not fully recovered until the density of entanglements at the interface reaches the bulk value. Moreover, the increase of interfacial strength before saturation is proportional to the number of interfacial entanglements between chains from opposite sides.

Topological versus rheological entanglement length in primitive-path analysis protocols, tube models, and slip-link models

We show that the front factor appearing in the shear modulus of a phantom network, G(ph) = (1-2/f)(ρk(B)T)/N(s), also controls the ratio of the strand length, N(s), and the number of monomers per Kuhn length of the primitive paths, N(ph)(PPKuhn), characterizing the average network conformation. In particular, N(ph)(PPKuhn) = N(s)/(1-2/f) and G(ph) = (ρk(B)T)/N(ph)(PPKuhn). Neglecting the difference between cross-links and slip-links, these results can be transferred to entangled systems and the interpretation of primitive path analysis data. In agreement with the tube model, the analogy to phantom networks suggest that the rheological entanglement length, N(e)(rheo) = (ρk(B)T)/G(e), should equal N(e)(PPKuhn). Assuming binary entanglements with f = 4 functional junctions, we expect that N(e)(rheo) should be twice as large as the topological entanglement length, N(e)(topo). These results are in good agreement with reported primitive path analysis results for model systems and a wide range of polymeric materials. Implications for tube and slip-link models are discussed.

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.

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.

Systematic time-scale-bridging molecular dynamics applied to flowing polymer melts

We present a thermodynamically guided, low-noise, time-scale-bridging, and pertinently efficient strategy for the dynamic simulation of microscopic models for complex fluids. The systematic coarse-graining method is exemplified for low-molecular polymeric systems subjected to homogeneous flow fields. We use established concepts of nonequilibrium thermodynamics and an alternating Monte Carlo-molecular-dynamics iteration scheme in order to obtain the model equations for the slow variables. For chosen flow situations of interest, the established model predicts structural as well as material functions beyond the regime of linear response. As a by-product, we present steady-state simulation results for polymers in general flow situations, including simple, planar, and yet unexplored equibiaxial elongation. The method is simple to implement and allows for the calculation of time-dependent behavior through quantities readily available from nonequilibrium steady states.

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.