Chain contraction and nonlinear stress damping in primitive chain network simulations

Doi and Edwards (DE) proposed that the relaxation of entangled linear polymers under large deformation occurs in two steps: the fast chain contraction (via the longitudinal Rouse mode of the chain backbone) and the slow orientational relaxation (due to reptation). The DE model assumes these relaxation processes to be independent and decoupled. However, this decoupling is invalid for a generalized convective constraint release (CCR) mechanism that releases the entanglement on every occasion of the contraction of surrounding chains. Indeed, the decoupling does not occur in the sliplink models where the entanglement is represented by the binary interaction (hooking) of chains. Thus, we conducted primitive chain network simulations based on a multichain sliplink model to investigate the chain contraction under step shear. The simulation quantitatively reproduced experimental features of the nonlinear relaxation modulus G(t,γ). Namely, G(t,γ) was cast in the time-strain separable form, G(t,γ)=h(γ)G(t) with h(γ)=damping function and G(t)=linear modulus, but this rigorous separability was valid only at times t comparable to the terminal relaxation time, although a deviation from this form was rather small (within ±10%) at t>τ(R) (longest Rouse relaxation time). A molecular origin of this delicate failure of time-strain separability at t～τ(R) was examined for the chain contour length, subchain length, and subchain stretch. These quantities were found to relax in three steps, the fast, intermediate, and terminal steps, governed by the local force balance between the subchains, the longitudinal Rouse relaxation, and the reptation, respectively. The contributions of the terminal reptative mode to the chain length relaxation as well as the subchain length/stretch relaxation, not considered in the original DE model, emerged because the sliplinks (entanglement) were removed via the generalized CCR mechanism explained above and the reformation of the sliplinks was slow at around the chain center compared to the more rapidly fluctuating chain end. The number of monomers in the subchain were kept larger at the chain center than at the chain end because of the slow entanglement reformation at the center, thereby reducing the tension of the stretched subchain at the chain center compared to the DE prediction. This reduction of the tension at the chain center prevented completion of the length equilibration of subchains at t～τ(R) (which contradicts to the DE prediction), and it forces the equilibration to complete through the reptative mode at t?τ(R). The delicate failure of time-strain separability seen for G(t,γ) at t～τ(R) reflects this retarded length equilibration.     

Motion of star‐branched chains in polymer networks: a Monte Carlo study

A simple model of branched polymers in confined space is developed. Star‐branched polymer molecules are built on a simple cubic lattice with excluded volume and no attractive interactions (good solvent conditions). A single star molecule is trapped in a network of linear polymer chains of restricted mobility. The simulations are carried out using the classical Metropolis algorithm. Static and dynamic properties of the star‐branched polymer are determined using various networks. The dependence of the longest relaxation time and the self‐diffusion coefficient on chain length and network properties are discussed and the proper scaling laws formulated. The possible mechanism of motion is discussed. The differences between the motion of star‐branched polymers in such a network are compared with the cases of a dense matrix of linear chains and regular rod‐like obstacles.     

Computer simulation of relaxation phenomena in star‐branched polymers. Temperature dependence

Dynamic Monte Carlo simulations of simple models of star‐branched polymers were conducted. A model star macromolecule consisted of f = 3 arms of equal length with a total number of polymer segments up to 800. The chain was confined to a simple cubic lattice with simple nearest neighbor attractive interactions. The relaxation phenomena were studied by means of autocorrelation functions in wide ranges of temperatures. Short‐time‐scale dynamic processes in the entire star‐branched chain were examined. It was found that under good solvent conditions the longest relaxation time of the end‐to‐center vector decreases with decreasing temperature. For low temperatures (below the Θ‐point) where the chain is collapsed, the dependence of the relaxation time on the temperature is opposite.     

Pre-Averaged Sampling On the Entanglement Kinetics for Polymer Dynamics

A new model for entangled polymer dynamics based on pre-averaged sampling of the entanglement structure is proposed. Although it has been reported that sliplink simulations are powerful and promising to predict entangled polymer dynamics, it is still unpractical to calculate polymers with many entanglements. In the present study, a possible approach to achieve fast calculation is proposed by pre-averaged sampling of entanglement structure with skipping detail kinetics of entanglements dominated by chain ends in conventional sliplink models. To achieve time development of the chain conformation and entanglement structure, i) number of entanglement per chain and number of monomers for each segment are randomly obtained from the equilibrium distribution proposed by Schieber [J. Chem. Phys. 2003 , 118, 5162] and ii) the renewed entanglement structure is mechanically equilibrated. The established power-laws on molecular weight dependence of chain dimension, the longest relaxation time and self-diffusion coefficient were reasonably reproduced. Comparison on linear viscoelastic response is also discussed.     

Relaxation of a semiflexible grafted polymer

The relaxation of single grafted semiflexible chains freely rotating around the grafting point is investigated by means of two dimensional computer simulations and scaling arguments. Both free chains and chains surrounded by topological obstacles are considered. We compute the autocorrelation of the end-to-end vector for the whole chain and for terminal sections of various lengths. Our results are relevant for the relaxation of star polymers with stiff arms or branched semiflexible polymers moving in an array of obstacles.     

Influence of branch length asymmetry on the electrophoretic mobility of rigid rod-like DNA

The electrophoretic mobility of three-arm asymmetric star DNA molecules, produced by incorporating a short DNA branch at the midpoint of rigid-rod linear DNA fragments, is investigated in polyacrylamide gels. We determine how long the added branch must be to separate asymmetric star DNA from linear DNA with the same total molecular weight. This work focuses on two different geometric progressions of small DNA molecules. First, branches of increasing length were introduced at the center of a linear DNA fragment of constant length. At a given gel concentration, we find that relatively small branch lengths are enough to cause a detectable reduction in electrophoretic mobility. The second geometric progression starts with a small branch on a linear DNA fragment. As the length of this branch is increased, the DNA backbone length is decreased such that the total molar mass of the molecule remains constant. The branch length was then increased until the asymmetric branched molecule becomes a symmetric three-arm star polymer, allowing the effect of molecular topology on mobility to be studied independent of size effects. DNA molecules with very short branches have a mobility smaller than linear DNA of identical molar mass. The reason for this change in mobility when branching is introduced is not known, however, we explore two possible explanations in this article. (i) The branched DNA could have a greater interaction with the gel than linear DNA, causing it to move slower; (ii) the linear DNA could have modes of motion or access to pores that are unavailable to the branched DNA.     

星形聚苯乙烯本体粘弹性对结构的依赖性

用应力松弛法研完了一组4—12支链星形聚苯乙烯本体的粘弹性。从应力松弛主曲线测定了零切变粘度η_0、稳态柔量J_e~o和其它流变学参数。结果发现,零切变粘度和最大松弛时间τ_m强烈地依赖于聚合物的分子尺寸,并且可以用无扰特性粘度或折合分子量来描述。η_0和τ_m随支化度增大而增大。实验还直接证实了支链长度相同的星形聚合物的J_e~o与支链数目无关,但是星形聚合物的稳态柔量比具有和支链间分子量相同的线形聚合物的稳态柔量大。     

用~(13)C NMR弛豫研究星形聚苯乙烯在溶液中的分子运动(Ⅱ)

对不同支化度和不同支链链长的20％(W/V)星形聚苯乙烯溶液测定了~(13)C NMR弛豫参数,用1g-x~2分布、Cole-Cole分布和构象跳跃模型对主链的分子运动进行了分析讨论,并对芳环侧基的内旋转运动也进行了分析,求出了活化能和跳跃速率。结果表明,轻度化学交联对相关时间分布有一定影响,对链段运动的势垒没有明显影响。支链链长对~(13)C NMR弛豫的影响和对线形聚合物的影响是类似的。     

Surface segregation of highly branched polymer additives in linear hosts

Entropy‐driven segregation of various branched and hyperbranched polymeric additives in chemically similar linear polymer hosts is studied using self‐consistent (SCF) mean‐field lattice simulations. The simulations account for the effect of molecular architecture on local configurational entropy in the blends, but ignores the effect of architecture on local density and blend compressibility. Star, dendrimer, and comb‐like additives are all found to be enriched at the surface of chemically identical linear host polymers. The magnitude of their surface excess increases with increased number of chain ends and decreases with increased segmental crowding near the branch point. Provided the number of arms and molecular weight of the branched additives are maintained constant, we find that the simplest branched architecture, the symmetric star, exhibits the strongest preference for the surface of binary polymer blends. We show that a single variable, here termed the “entropic driving force density,” controls the relative surface affinities of branched additives possessing a wide range of architectures. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 1788–1801, 2008     

Conformational properties and dynamics of molecular bottle‐brushes: A cellular‐automaton‐based simulation

Extensive computer simulation was performed using the bond‐fluctuation model and cellular‐automaton (CA)‐based simulation technique to probe the equilibrium structure and dynamical behavior of comb‐branched polymers in which the flexible side chains of a given length are placed regularly along the backbone and the number of branches increases linearly with total molecular weight. By applying very efficient CA algorithm – the “lattice molecular dynamics” (LMD) method – we have been able to study the properties of sufficiently large structures (up to 5880 monomeric units). Depending on the length of main and side chains as well as on interbranch spacing, we have calculated mean chain dimensions, local fractal dimensionalities, particle scattering functions, time autocorrelation functions, etc. The following main conclusions may be drawn from the results presented in our study: (i) The critical exponent, governing the mean size of the main chain, remains unchanged from its value known for a 3d self‐avoiding walk (SAW). On the other hand, two‐dimensional branched macromolecules with one‐sided branches are effectively in a collapsed state even under conditions of a good solvent, forming specific helical superstructures. (ii) Comparison of the simulated data with the predictions of the scaling model indicates that the latter is valid in describing the mean dimensions of the backbone as a function of side‐chain length and interbranch spacing. (iii) The excluded volume interaction between side chains dramatically slows down the relaxation of the backbone chain.     

A comparison of three polymer network models in current use

The relationship between three theories of polymer network deformation is explored. The theories are: the eight-chain model of Arruda and Boyce; the full network model of Wu and van der Giessen; and the crosslink–sliplink model of Edwards and Vilgis. All have a history of use as the network component in theories of solid polymer deformation. Given results from either the eight-chain or full network models, least-squares fitting of the stresses is used to derive optimal parameters of the Edwards–Vilgis model. Both the eight-chain and the full network models can be closely approximated by an Edwards–Vilgis model, provided the finite chain extensibility limit is not approached too closely. The eight-chain model is found to be equivalent to an Edwards–Vilgis model with a small number of sliplinks, whereas the full network model corresponds to an Edwards–Vilgis model with no sliplinks. The physical interpretation of these findings is discussed.     

Full‐chain dynamics of entangled linear and star polymers

The full‐chain dynamics and the linear viscoelastic properties of monodisperse, entangled linear and star polymers are simulated consistently via an equilibrium stochastic algorithm, based on a recently proposed full‐chain reptation theory 1 that is able to treat self‐consistently mechanisms of chain reptation, chain‐length fluctuations, and constraint release. In particular, it is the first time that the full‐chain simulation for star polymers is performed without subjecting to the great simplifications usually made. To facilitate the study on linear viscoelasticity, we employ a constraint release mechanism that resembles the idea of tube dilation, in contrast to the one used earlier in simulating flows, where constraint release was performed in a fashion similar to double reptation. Predictions of the simulation are compared qualitatively and quantitatively with experiments, and excellent agreement is found for all investigated properties, which include the scaling laws for the zero‐shear‐rate viscosity and the steady‐state compliance as well as the stress relaxation and dynamic moduli, for both polymer systems. The simulation for linear polymers indicates that the full‐chain reptation theory considered is able to predict very well the rheology of monodisperse linear polymers under both linear viscoelastic and flow conditions. The simulation for star polymers, on the other hand, strongly implies that double reptation alone is insufficient, and other unexplored mechanisms that may further enhance stress relaxation of the tube segments near the star center seem crucial, in explaining the linear viscoelasticity of star polymers. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 248–261, 2000     

Flow induced crystallization of long chain branched polypropylenes under weak shear flow

In the present work long chain branched polypropylene (LCB PP) polymers were prepared by linear polypropylene and multi-functional monomer through melt grafting reaction. A quantitative rheological method was adopted to analysis the structure parameters of LCB PP. The effects of chain branched level on the crystallization kinetics of PP were investigated by rheology, differential scanning calorimetry, polarized optical microscope and wide-angle X-ray diffraction. The dynamic viscoelastic properties of LCB PP showed that the increase in the chain branched level caused a typical deviation from the terminal behavior and a different distribution of the melt relaxation spectrum in the long relaxation time regime. It was found that the chain branched level had a significant effect on the flow induced crystallization (FIC) process of PP melts. The crystallization of LCB PP was more sensitive to shear flow than that of linear PP during induced period at low shear rates. This result also indicated that the longer relaxation time of the polymer chains played an important role in the nucleation of PP under shear flow fields. LCB PP with high chain branched level showed accelerated crystallization kinetics in comparison with that with low chain branched level.     

Extent of branching from linear viscoelasticity of long‐chain‐branched polymers

We present new results and examine literature data concerning the linear viscoelastic behavior of polyethylene with sparse to intermediate levels of long‐chain branching (LCB). These branched polymers displayed a common rheological signature, namely, a region of frequency‐independent loss tangent along with the corequisite scaling of the storage and loss moduli to the same frequency exponent. This apparent power‐law response occurred within a finite frequency window and bore resemblance to the behavior of physical gels. The appearance of this region, however, was the consequence of the presence of two distinct, yet partially overlapping, terminal relaxation processes. After considering the analogous relaxation behavior of wholly linear polymers with bimodal molecular weight distributions, we considered the polymers with LCB as blends of linear and branched species to develop a simple method of quantifying the extent of LCB. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 1671–1684, 2004     

An NMR study of mobility in a crystalline side‐chain comblike polymer

Carbon‐13 spin–lattice relaxation times are measured for poly(octadecyl acrylate) above and below the melting point of the crystalline side chains. The chain backbone has long spin–lattice relaxation times below the melting point that shorten by more than an order of magnitude as the melting point range is traversed. Below the melting point, the backbone is nearly immobilized with spin–lattice relaxation changing very slowly with temperature. Above the melting point, the shorter spin–lattice relaxation times are typical of a rubber above the glass transition and decrease with increasing temperature. The methylene groups in the side chain are quite mobile well below the melting point, indicating fairly rapid anisotropic motion within the crystal. The methyl group at the end of the chain and the adjacent methylene group have longer spin–lattice relaxation times, indicating the greatest side‐chain mobility at the end, which is in the middle of the crystal structure. The side‐chain carbon adjacent to the carbonyl group is as mobile as the majority of the side‐chain carbon, indicating side‐chain mobility extends to all of the side‐chain CH₂ groups. The abrupt transition in the mobility of the backbone is not typical of the amorphous phase in a semicrystalline polymer where the backbone units can crystallize. The close proximity of every backbone segment to the crystalline domain locks backbone segmental motion below the melting point. Melting and crystallization of the side chains switch segmental motion of the backbone on and off. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 1548–1552, 2001     

Partitioning of star branched polymers into pores at three chromatography conditions

The partitioning of star branched polymers into a slit pore at three different chromatography conditions, namely, size exclusion chromatography (SEC), liquid chromatography at the critical condition (LCCC), and liquid adsorption chromatography (LAC) have been investigated with lattice Monte Carlo simulations. Two different chain models are used: random walks (RW) that have no excluded volume interaction and self-avoiding walks (SAW) that have excluded volume interaction. The simulation data obtained for the two chain models are compared to illustrate the effect of excluded volume interactions on the partitioning of star branched polymers. The two most outstanding effects observed due to the introduction of excluded volume interactions are: (i) stars with a high number of arms can be excluded from the pore at condition corresponding to the LCCC of the linear polymers; (ii) the partition coefficient of stars in LAC mode is not dependent only on the total number of monomers on the chain. These effects illustrated by the current study should be taken into account when interpreting experimental chromatography data for branched polymers.     

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.     

Dynamics of Adsorbed Star‐Branched Polymer Chains: A Computer Simulation Study

The simple cubic‐lattice model of polymer chains was used to study the dynamic properties of adsorbed, branched polymers. The model star‐branched chains consisted of f = 3 arms of equal lengths. The chain was modeled with excluded volume, that is, in good solvent conditions. The only interaction assumed was a contact potential between polymer segments and an impenetrable surface. This potential was varied to cover both weak and strong adsorption regimes. The classical Metropolis sampling algorithm was used for models of star‐branched polymers in order to calculate the dynamic properties of adsorbed chains. It was shown that long‐time dynamics (diffusion constant) and short‐time dynamics (the longest relaxation time) were different for weak and strong adsorption. The diffusion of weakly adsorbed chains was found to be qualitatively the same as for free nonadsorbed chains, whereas strongly adsorbed chains behaved like two‐dimensional polymers. The time‐dependent properties of structural elements such as tails, loops, and trains were also determined.

The mean lifetimes of tails, loops, and trains versus the bead number for the chain with N = 799 beads for the case of the weak adsorption ε_a = −0.3.     

Dielectric Relaxation of Type‐A Polymers in Melts and Solutions

This article describes the dielectric relaxation behavior of flexible polymer chains having the so‐called type‐A dipoles parallel along the chain backbone. This behavior reflects the global chain motion. Viscoelastically well known features of this motion, such as the power‐law relationship between the relaxation time and molecular weight of entangled linear chains (τ₁ ∝ M^3.5), are also observed dielectrically. More importantly, the dielectric behavior of linear chains having once‐inverted type‐A dipoles enables us to find some detailed dynamic features such as changes in the eigenfunctions f_p of a local correlation function with the chain concentration in solutions. These changes are discussed in relation to motional coupling of concentrated chains. The dielectric properties detect the orientational correlation of two submolecules in the chain at two separate times, while the viscoelastic properties reflect the isochronal orientational anisotropy of individual submolecules. Thus the chain motion is differently averaged in the dielectric and viscoelastic properties, and comparison of these properties enables us to find novel dynamic features. Specifically, this comparison reveals the validity of the tube dilation molecular picture for entangled linear chains and weakening of the short‐time coherence of the submolecule motion due to the constraint release mechanism. Moreover, the dielectric method enables us to investigate the chain dynamics under strong flow and/or in a molecularly narrow space. In particular, the retarded dielectric relaxation found for homopolymers and block copolymers in such narrow spaces (in the microdomains for the latter) indicates important effects of the spatial and thermodynamic constraints on the global chain motion. All the above results in turn demonstrate the importance of the dielectric method in investigations of the polymer dynamics.     

Surface enrichment of branched polymers in linear hosts: effect of asymmetry in intersegmental interactions and density gradients

Variable density lattice treatment of surface enrichment of f-arm star-branched chains in star/linear polymer blends is compared with results of an analytical response theory proposed by Wu and Fredrickson [Macromolecules 29, 7919 (1996)]. We find that differences in treating the intersegmental interactions in the small interfacial region near a free surface lead to significant differences in the potentials by which polymer chain ends are attracted towards the surface. Consideration of an asymmetric relationship between segment potentials and density changes in polystyrene at 450 K and 0.1 MPa, for example, gives typically a threefold to fourfold enhancement in composition of star molecules at a vacuum interface. When contributions from gradients in density are included in the analysis even greater levels of surface enhancement (fivefold to sixfold increases) are observed. By appropriately estimating the attraction of chain ends and repulsion of branch points at a free surface, we show that concentration profiles of branched polymers predicted in the lattice model are consistent with results obtained in the analytical response theory.