Hierarchical Modeling of Entangled Polymers

Summary : We demonstrate that it is possible to link multi-chain molecular dynamics simulations with the tube model using a single chain slip-links model as a bridge. This hierarchical approach allows significant speed up of simulations, permitting us to span the time scales relevant for a comparison with the tube theory. Fitting the mean-square displacement of individual monomers in molecular dynamics simulations with the slip-spring model, we show that it is possible to predict the stress relaxation. Then, we analyze the stress relaxation from slip-spring simulations in the framework of the tube theory. In the absence of constraint release, we establish that the relaxation modulus can be decomposed as the sum of contributions from fast and longitudinal Rouse modes, and tube survival. Finally, we discuss some open questions regarding possible future directions that could be profitable in rendering the tube model quantitative, even for mildly entangled polymers.     

Protruding chain at a sharp penetrable interface

A single protruding chain at a sharp discontinuity of the quality of the surrounding medium was shown by Monte Carlo (MC) simulations to be a good model representing semiquantitatively the major macroscopic features of polymer interfaces such as the interface thickness. This is in accord with early original theoretical suggestions. Compared to multichain simulations of a polymer/polymer interface it is clear that the protruding chain is not completely adequate to represent a diffuse multichain interface. The results on a protruding chain, however, indicate the proper interfacial thickness as obtained by MC simulations in multichain systems. The microscopic results (such as deformation, orientation of coils at the interface and chain end segregation into the interface) are similar to those obtained in multichain polymer interfaces. None of the investigated interface properties exhibits a dependence on the total chain length of the protruding chain.     

Molecular stress function theory and analysis of branching structure in industrial polyolefins

Despite substantial progress in analytical techniques for polymer characterization, a realistic picture of branching structure in industrial polymers still remains at large. Using a number of assumptions, structure-based constitutive models can distinguish between linear and branched structures in a qualitative sense. More detail on branching architecture, such as the number and length of side chains, the sequence in which they exist on the backbone and their contribution to polymer chain relaxation is more or less unknown. In the current study, elongational behavior of four commercial polyolefins is compared using the predictions of the MSF (molecular stress function) theory. The results will then be used to analyze the branching in a group of strain-hardening polypropylenes synthesized using single site catalyst.     

Viscoelasticity of stretched polymer chains: Analytical theory and computer-aided simulation

The viscoelasticity of stretched polymer chains has been studied by the method of collisional dynamics. To this end, time correlation functions of the fluctuations of the microscopic stress tensor are modeled and relaxation moduli are expressed. Before, for stretched polymer networks, correlation functions used to be calculated in terms of an approximate theory that allowed one to estimate the strain dependences of loss modulus. The calculated dependences are shown to agree qualitatively with the results of measurements performed over a wide interval of strains, including prefracture strains. This theory is verified by comparing the time correlation functions of stress tensor fluctuations for a single stretched chain; these functions are found by computer-aided simulation and calculated on the basis of the existing analytical theory. In this case, a simple theory is adopted according to which a polymer molecule represents a chain composed of N atoms connected by freely jointed elastic bonds. The first and Nth atoms of this chain are attached by harmonic springs to immobile points located at a fixed distance. The decay of time correlation functions under study can be resolved into three stages. After a short initial interval provided by local motions, one can observe a region of power-law decay, which is followed by monoexponential decay at long times. The results of computer-aided simulation generally agree with the predictions of analytical theory. Certain discrepancies primarily concern the dependences of the exponent of power-law relaxation on the degree of chain stretching.     

A model for the plateau in the relaxation modulus for linear chain polymer melts

A simple model is considered for the free energy associated with the relaxation of a linear chain polymer melt. The relaxation of the chain configurations results in an adjustment of the locations of the interchain contacts. The change in the distribution of the positions of the contacts between a pair of relaxing chains is hindered by the presence of other nearby chains in the melt. There is less configuration space available to the relaxing chains, due to the noncrossability of the chain backbones, than would be the case for phantom chains. This results in an increase in the free energy of the relaxing system. The analysis presented indicates that, given the free energy models considered, the extent of relaxation decreases as the length scale for relaxation increases. This results in a plateau in the relaxation modulus. This qualitative prediction of a plateau does not rely on the assumption of a specific mechanism for the chain dynamics, and is relatively insensitive to the form chosen for the terms in the free energy. If reasonable assumptions are made concerning the form of the free energy, then it is shown that the plateau which results depends on monomer length, Kuhn length, the monomer density for the melt, and, for solutions, the polymer concentration in a manner consistent with experimental data.     

Some phenomenological consequences of the Doi–Edwards theory of viscoelasticity

Doi and Edwards have recently proposed a molecular theory for the dynamics of entangled polymer liquids based on a tube model to represent the mutual constraints on configurational rearrangement of the chains. Expressions for diffusion coefficient, plateau modulus, zero-shear viscosity, steady-state recoverable compliance, and terminal relaxation time can be devloped, and relations among these properties that depend only upon observable quantities can be obtained. Several such relations are derived and are compared with experimental observations.     

Analysis of uncertainties in polymer viscoelastic properties obtained from equilibrium computer simulations

We critically evaluate the uncertainties in the stress autocorrelation function obtained from equilibrium molecular dynamics simulation of model polymer melts. This quantity is central to evaluating transport properties, e.g., the complex modulus and the viscosity. In contrast to the intuitive expectation that simulations have to be run five to six orders of magnitude longer than the chain relaxation time to reduce uncertainties to acceptable levels, our analysis shows that the majority of the uncertainty is associated with rapidly oscillating bonded interactions. These fluctuations occur on time scales which are approximately 10(4) times shorter than the relaxation time of a chain of length 80. Consequently, the effects of these oscillations on the stress autocorrelation function can be dramatically reduced by (i) conducting long simulations (typically 10(6) times longer than the bond relaxation times or only 10(2) chain relaxation times) and (ii) by performing running averages with time windows whose time scales are much longer than these oscillations. Conducting such long simulations also allows for the accurate determination of the melt viscosity and the low-frequency complex modulus, but performing running averages do not impact these quantities since they are time integrals of the stress autocorrelation function.     

Polymer chains in a soft nanotube: a Monte Carlo study

We study the equilibrium properties of flexible polymer chains confined in a soft tube by means of extensive Monte Carlo simulations. The tube wall is that of a single sheet six-coordinated self-avoiding tethered membrane. Our study assumes that there is no adsorption of the chain on the wall. By varying the length N of the polymer and the tube diameter D we examine the variation of the polymer gyration radius Rg and diffusion coefficient Ddiff in soft and rigid tubes of identical diameter and compare them to scaling theory predictions. We find that the swollen region of the soft tube surrounding the chain exhibits a cigarlike cylindrical shape for sufficiently narrow tubes with D相似文献   

Microscopic theory of rubber elasticity

A microscopic integral equation theory of elasticity in polymer liquids and networks is developed which addresses the nonclassical problem of the consequences of interchain repulsive interactions and packing correlations on mechanical response. The theory predicts strain induced softening, and a nonclassical intermolecular contribution to the linear modulus. The latter is of the same magnitude as the classical single chain entropy contribution at low polymer concentrations, but becomes much more important in the melt state, and dominant as the isotropic-nematic liquid crystal phase transition is approached. Comparison of the calculated stress-strain curve and induced nematic order parameter with computer simulations show good agreement. A nearly quadratic dependence of the linear elastic modulus on segmental concentration is found, as well as a novel fractional power law dependence on degree of polymerization. Quantitative comparison of the theory with experiments on polydimethylsiloxane networks are presented and good agreement is found. However, a nonzero modulus in the long chain limit is not predicted since quenched chemical crosslinks and trapped entanglements are not explicitly taken into account. The theory is generalizable to treat the structure, thermodynamics and mechanical response of nematic elastomers.     

Phase separation in binary mixtures containing polymers: A quantitative comparison of single-chain-in-mean-field simulations and computer simulations of the corresponding multichain systems

We discuss a phenomenological, coarse-grained simulation scheme, single-chain-in-mean-field (SCMF) simulation, for investigating the kinetics of phase separation in dense polymer blends and mixtures of polymers and solvents. In the spirit of self-consistent-field calculations, we approximate the interacting multichain problem by that of a single chain in an external field, which, in turn, depends on the local densities of the components. To study the time evolution of the mixture, we perform an explicit Monte Carlo (MC) simulation of an ensemble of independent chains in the external field and periodically calculate the average densities and update the external field. Unlike dynamic self-consistent-field theory, these SCMF simulations do not assume that the chain conformations relax much more quickly than the density and incorporate the single-chain dynamics explicitly rather than via an Onsager coefficient. This allows us to study systems with large spatial inhomogeneities and dynamic asymmetries. To assess the accuracy and limitations of the simulation scheme, we compare the results of SCMF simulations using a discretized Edwards Hamiltonian with computer simulations of the corresponding multichain system for (1) the early stages of spinodal decomposition of a symmetric binary polymer blend in response to a quench from χN = 0.314 to χN = 5 (where χ is the Flory–Huggins parameter and N is the number of segments), for which the growth rate of composition fluctuations is compared with MC simulations of the bond fluctuation model and alternative dynamic self-consistent-field calculations, and (2) the evaporation of a solvent from a low-molecular-weight thin polymer film, for which a comparison is made with molecular dynamics (MD) simulations of a bead-necklace model with a monomeric solvent. In the latter case, the polymer conformations are extracted from MD simulations and modeled in the SCMF simulations by a discretized Edwards Hamiltonian augmented by a chain-bending potential. From the MD simulations of thin polymer films in equilibrium with its vapor, phase coexistence has been determined, and the second- and third-order virial coefficients in the SCMF simulations have been adjusted accordingly. Finally, MD simulations of bulk solutions of a polymer and a solvent over a range of compositions, as well as the pure solvent at various densities, have been performed to determine self-diffusion coefficients that enter the SCMF simulations in the form of density-dependent segmental mobilities. A comparison of the polymer and solvent profiles in a thin film as a function of time and the fraction of the solvent evaporating from a solvent-swollen film, as obtained from MD simulations and parameterized SCMF simulations, shows satisfactory agreement for this simple mapping procedure. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 934–958, 2005     

A time-integrated estimate of the entanglement mass in polymer melts in agreement with the one determined by time-resolved measurements

We make a critical examination of how the entanglement molecular mass Me is determined from various measurable quantities. We are guided by reptation theory, where it is assumed that characteristic relaxations abruptly change and become equal to those of a chain moving in a Gaussian tube, as soon as the corresponding length scales surpass the tube diameter d or similarly as soon as the corresponding mass surpasses a critical value. Taking this critical mass as a definition of the "reptational" entanglement mass, we observe that all methods based on time-resolved quantities, such as the single-chain dynamic structure factor S(q,t) and the zero-shear relaxation modulus G(t), give the same result. We observe that such a value differs, beyond error bars, from that obtained from the plateau modulus, which is a time-integrated quantity. We have investigated an alternative definition of entanglement mass in terms of time-integrated quantities and observe that the value of this specific entanglement mass is consistent with that obtained from the time-resolved observables. We comment on possible reasons for the plateau modulus discrepancy.     

Thermodynamic modelling of polymer solutions with a modified Staverman equation

A model describing the thermodynamic behaviour of polymer solutions is derived which explicitly accounts for the flexibility of the polymer chains. Based on computer simulations on various lattices it is shown that the flexibility of a polymer chain can be modelled by distinguishing different polymer conformations. Here each conformation is characterized by its corresponding number of external contact sites. The equilibrium between the different conformations is then solved for any polymer concentration and any combination of interaction energies utilizing a modified Staverman equation. The model predictions are in good agreement with the results of the computer simulations which were performed using the simple-sampling and the slithering-snake algorithm. Since the knowledge of the distribution of the conformations of a single polymer chain on an empty lattice is a prerequisite to perform the model calculations, Poisson distribution functions are fitted to the results of the corresponding computer simulations. The generalization of these distribution functions not only facilitates the use of the new model but also allows to model polymers of varying chain stiffness.     

Short‐time stretch relaxation of entangled polymer solutions investigated using full Rouse model predictions

We conduct a systematical investigation into the short‐time stretch relaxation behavior (i.e., shorter than the Rouse time but sufficiently longer than the glassy time) of entangled polymer liquid in single‐step strain flows, on the basis of theory/data comparisons for a broad series of type‐A entangled polymer solutions. First, within existing normal‐mode formulations, the Rouse model predictions on a full‐chain stretch relaxation in single‐step strain flows are derived for a popular 1‐D model proposed within the Doi–Edwards tube model, as well as for the original 3‐D model for nonentangled systems. In addition, an existing formula for the aforementioned 1‐D model that, however, rested upon a consistent‐averaging or the so‐called uniform‐chain‐stretch approximation is simultaneously examined. Subsequently, the previously derived formulas on chain stretch relaxation are directly incorporated into a reliable mean‐field tube model that utilizes the linear relaxation spectrum and the Rouse time constant consistently determined from linear viscoelastic data. It is found that the predictions of the 1‐D model differ substantially from that of the original 3‐D model at short times. Theory/data comparisons further indicate that the 1‐D model without approximations seems able to describe fairly well the nonlinear relaxation data under investigation. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1199–1211, 2006     

Characterization and gel point of randomly linked high cis-polybutadiene networks

Cis—Polybutadiene networks (98% cis), crosslinked in the bulk state by gamma irradiation were characterized in terms of their mechanical properties. Data from stress-strain analyses, swelling degrees, stress relaxation and random—linking theory are in accordance and the estimated gel points practically coincide. Trapped entanglements contributed up to 30% to the network modulus, from which also the plateau modulus was determined. The tensile data were obtained from Mooney—Rivlin representations. It was concluded that the assumption of additive contributions from chemical and entanglement crosslink densities in the Langley theory works well in the range of doses investigated here. A comparison with a tube—like approach of these networks is made.     

Linear viscoelastic properties of transient networks formed by associating polymers with multiple stickers

We have developed a single-chain theory that describes dynamics of associating polymer chains carrying multiple associative groups (or stickers) in the transient network formed by themselves and studied linear viscoelastic properties of this network. It is shown that if the average number N of stickers associated with the network junction per chain is large, the terminal relaxation time τ(A) that is proportional to τ(X)N(2) appears. The time τ(X) is the interval during which an associated sticker goes back to its equilibrium position by one or more dissociation steps. In this lower frequency regime ω<1/τ(X), the moduli are well described in terms of the Rouse model with the longest relaxation time τ(A). The large value of N is realized for chains carrying many stickers whose rate of association with the network junction is much larger than the dissociation rate. This associative Rouse behavior stems from the association/dissociation processes of stickers and is different from the ordinary Rouse behavior in the higher frequency regime, which is originated from the thermal segmental motion between stickers. If N is not large, the dynamic shear moduli are well described in terms of the Maxwell model characterized by a single relaxation time τ(X) in the moderate and lower frequency regimes. Thus, the transition occurs in the viscoelastic relaxation behavior from the Maxwell-type to the Rouse-type in ω<1/τ(X) as N increases. All these results are obtained under the affine deformation assumption for junction points. We also studied the effect of the junction fluctuations from the affine motion on the plateau modulus by introducing the virtual spring for bound stickers. It is shown that the plateau modulus is not affected by the junction fluctuations.     

Monte Carlo simulations of stress relaxation of entanglement-free Fraenkel chains. I. Linear polymer viscoelasticity

Shear stress relaxation modulus GS(t) curves of entanglement-free Fraenkel chains have been calculated using Monte Carlo simulations based on the Langevin equation, carrying out both in the equilibrium state and following the application of a step shear deformation. While the fluctuation-dissipation theorem is perfectly demonstrated in the Rouse-chain model, a quasiversion of the fluctuation-dissipation theorem is observed in the Fraenkel-chain model. In both types of simulations on the Fraenkel-chain model, two distinct modes of dynamics emerge in GS(t), giving a line shape similar to that typically observed experimentally. Analyses show that the fast mode arises from the segment-tension fluctuations or reflects the relaxation of the segment tension created by segments being stretched by the applied step strain-an energetic-interactions-driven process-while the slow mode arises from the fluctuations in segmental orientation or represents the randomization of the segmental-orientation anisotropy induced by the step deformation-an entropy-driven process. Furthermore, it is demonstrated that the slow mode is well described by the Rouse theory in all aspects: the magnitude of modulus, the line shape of the relaxation curve, and the number-of-beads (N) dependence of the relaxation times. In other words, one Fraenkel segment substituting for one Rouse segment, it has been shown that the entropic-force constant on each segment is not a required element to give rise to the Rouse modes of motion, which describe the relaxation modulus of an entanglement-free polymer over the long-time region very well. This conclusion provides an explanation resolving a long-standing fundamental paradox in the success of Rouse-segment-based molecular theories for polymer viscoelasticity-namely, the paradox between the Rouse segment size being of the same order of magnitude as that of the Kuhn segment (each Fraenkel segment with a large force constant HF can be regarded as basically equivalent to a Kuhn segment) and the meaning of the Rouse segment as defined in the Rouse-chain model. The general agreement observed in the comparison of the simulation and experimental results indicates that the Fraenkel-chain model, while being still relatively simple, has captured the key element in energetic interactions--the rigidity on the segment--in a polymer system.     

Tube relaxation: A quantitative molecular model for the viscoelastic plateau of entangled polymeric media

The relaxation of an entangled polymeric medium in the viscoelastic plateau is investigated theoretically by using the slip-link representation of topological constraints. In addition to the chain retraction process introduced by Daoudi and investigated theoretically by Doi, we show that two processes contribute significantly to the relaxation: The first, “equilibration across slip-links,” is a longitudinal reequilibration between parts of the chain which have been differently extended or compressed, depending on their initial orientation relatively to the strain tensor. The second, “tube relaxation,” is a mean-field representation of the loss of topological constraints on one chain due to the retraction of the others. Closed analytical expressions for the stress accounting for these three processes are derived and compared with previous theories: the relaxation should be much more progressive than previously predicted, and the terminal time for retraction is reduced significantly by tube relaxation.     

Local Transient Jamming in Stress Relaxation of Bulk Amorphous Polymers

Bulk amorphous polymers become stretched and parallel-aligned under loading stress,and their intermolecular cooperation slows down the subsequent stress relaxation process.By means of dynamic Monte Carlo simulations,we employed the linear viscoelastic Maxwell model for stress relaxation of single polymers and investigated their intermolecular cooperation in the stress relaxation process of stretched and parallel-aligned bulk amorphous polymers.We carried out thermal fluctuation analysis on the reproduced Debye relaxation and Arrhenius fluid behaviors of bulk polymers.We found a transient state with stretch-coil coexistence among polymers in the stress relaxation process.Further structure analysis revealed a scenario of local jamming at the transient state,resulting in an entropy barrier for stretch-coil transition of partial polymers.The microscopic mechanism of intermolecular cooperation appears as unique to polymer stress relaxation,which interprets the hydrodynamic interactions as one of essential factors raising a high viscosity in bulk amorphous polymers.Our simulations set up a platform of molecular modeling in the study of polymer stress relaxation,which brought new insights into polymer dynamics and the related mechanical/rheological properties.