A Monte Carlo study of a tethered polymer chain in a uniform field

We study the non‐uniform stretching and relaxation of a long flexible end‐anchored polymer chain of N monomers (32 ≤ N ≤ 1 024) in a uniform field B by means of an off‐lattice bead‐spring Monte Carlo model. Our simulational results for the case of a Rouse‐like polymer in the good solvent regime confirm the existence of “trumpet”‐ and “flower”‐type chain conformations, predicted recently by scaling analysis based on the notion of Pincus tensile blobs. The observed elongation of the chain and the critical fields, separating three different regimes of chain deformation, are found to obey the predicted scaling behavior. The segment density distribution matches that of a DNA molecule pulled from one end at constant velocity in a good solvent. As expected, the relaxation of the stretch to coil transition of the polymer of length N is determined by the typical Rouse time τ ∝ N^2ν+1.     

Nonlinear stress relaxation of an entangled linear polystyrene in single step‐strain flow: A quantitative theoretical investigation

The nonlinear stress relaxation of a nearly monodisperse, moderately entangled polystyrene solution (i.e., roughly seven entanglements per chain at equilibrium) in single step‐strain flow is investigated quantitatively by a detailed comparison of an existing set of experimental data with a simulation based on the tube model. The proposed simulation enables the effects of primary nonlinear relaxation mechanisms other than chain retraction to be identified more clearly and investigated individually. Two peculiar nonlinear relaxation behaviors are observed in this experiment. One is concerned with an apparent enhancement in the stress relaxation at short times, and the other is responsible for a seeming slowdown of the stress relaxation at long times. These findings are discussed within the tube model, in view of the effects of convective constraint release, partial strand extension, and nonaffine deformation. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1281–1293, 2003     

Exploring rheological properties of aqueous polyaniline‐PVP dispersion

Stable aqueous dispersion of polyaniline (PAn) stabilized by a hydrophilic polymer poly(vinyl pyrrolidone) (PVP) exhibits interesting rheological properties different from its components. Shear thinning observed for both PVP and PAn–PVP colloid (PP) indicates partially entangled nature of the later. Linear viscoelastic response of PVP solution exhibit strong frequency dependence of elastic (G′) and viscous (G″) modulus over the whole frequency range (0.1–100 ras/s) where G′ never exceeds G″ indicating the applicability of the Rouse‐Zimm model to this system. On the other hand, there is a crossover of G′ and G″ in the rheological profile of PP dispersion so that a single relaxation time model can be applicable. Therefore, PVP presents an entangled polymeric system and supposed to have a spectrum of relaxation times, whereas PP resembles to a physically crosslinked system with a single relaxation time. Increasing the extent of hydrogen bonding within the system (by raising the fraction of PAn or by leaving the solution undisturbed for long) relaxation time also becomes longer. The large difference in values of steady and complex shear viscosity (η and η*) within LVE regime reflects that original Cox‐Merz rule is obviously inapplicable to these systems. But at larger strain amplitude, η and η* are satisfactorily coincident that indicates a broader applicability of the modified Cox‐Merz rule. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 2443–2455, 2008     

Shear damping function measurements for branched polymers

We present results for step‐strain experiments and the resulting damping functions of polyethylene blends of different structures, including solutions of linear, star and comb polymers. Remarkably, an entangled melt of combs exhibits a damping function close to that for entangled linear chains. Diluting the combs with faster‐relaxing material leads to a more nearly constant damping function. We find similar behavior for blends of commercial low density polyethylene LDPE. Our results suggest a simple picture: on timescales relevant to typical damping‐function experiments, the rheologically active portions of our PE combs as well as commercial LDPE are essentially chain backbones. When strongly entangled, these exhibit the Doi‐Edwards damping function; when diluted, the damping function tends toward the result for unentangled chains described by the Rouse model – namely, no damping. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 3117–3136, 2007     

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     

Coil‐stretch transition in deformation flows

We present a novel Monte‐Carlo lattice model for the study of the coil‐stretch transition for polymer chains in deformation flows. Our results indicate that elongational flows are much more effective than shear flows in stretching polymer chains, in full agreement with experimental observation. Our model data also show that the ε˙_c∼M^−1.5 powerlaw observed experimentally for the dependence of critical flow rate on polymer molecular weight can be fully explained through a nonuniform stretching of the chain by the flow. A higher powerlaw exponent is predicted in more affine deformation cases. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2422–2428, 2000     

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.     

Theory of relaxation properties of two‐dimensional polymer networks, 2. Local dynamic characteristics

Using normal mode transformation obtained in Part 1 of this series^[1], the exact analytical expressions for the mean‐square displacements of junctions and non‐junction beads, the autocorrelation functions of the end‐to‐end chain vectors between neighboring junctions, and those of subchain vectors of a two‐dimensional regular network consisting of "bead and spring" Rouse chains are obtained. Contributions of intra‐ and interchain relaxation processes to the local dynamic characteristics considered are compared. The time behavior of dynamic quantities obtained is estimated for different scales of motions. The possibility of describing long‐time relaxation of a two‐dimensional network by a simplified coarse‐grained network model is demonstrated. It is shown that the local relaxation properties of a two‐dimensional polymer network (as well as a three‐dimensional network) on scales smaller than the average distance between cross‐links are very close to those of a single Rouse chain. The large‐scale collective relaxation of the polymer networks having a two‐dimensional connectivity differs considerably from that of the three‐dimensional networks.     

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.     

Crossover behavior in crystal growth rate from n‐alkane to polyethylene

A crystal growth rate equation, parameterized from molecular dynamics simulations of n‐alkanes, is compared to recent experiments on growth rates for polyethylene at high undercooling. The analysis reveals that the growth rate of alkanes and polyethylene can both be described by the same relationship. The appropriate relaxation time is used to describe the kinetic barrier to crystallization. For chains shorter than the entanglement length, this is the Rouse time. For chains longer than the entanglement molecular weight, kinetic limitations are modeled by the local relaxation of an entangled segment at the interface. This model supports a different mechanism for fast crystal growth at high undercooling than that usually inferred from slow growth data near the melting temperature. Use of the crystal growth rate model is illustrated for polyethylene crystallizing under conditions of slow cooling and fast cooling. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 2468–2473, 2005     

Stress relaxation under large step equibiaxial elongation for low‐density polyethylene

The stress relaxation under large step equibiaxial elongation for low‐density polyethylene with long‐chain branches revealed that the time‐strain separability holds in relaxation modulus G_B(t, ε_B), and damping function h_B(ε_B) exhibits weaker equibiaxial elongational strain ε_B dependence than that predicted by the Doi–Edwards theory without the independent alignment approximation. Dependencies of damping function h(γ) for step shear deformation and h_B(ε_B) on stretch ratio α of polymer contour length and orientation of a polymer chain in direction of the maximum orientation were evaluated, and it was found that the α dependencies of h(γ) and h_B(ε_B) are different, whereas dependencies of h(γ) and h_B(ε_B) on the orientation coincide fairly well. These results indicate that the damping is dominated by the chain orientation rather than α. This implies that withdrawal of long‐chain branches into tube of a backbone chain occurs when the orientation of the long‐chain branches is large and friction force against the branch point withdrawal is small. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 1275–1284, 2009     

Mechanical Spectral Hole Burning in polymer solutions

Mechanical Spectral Hole Burning (MSHB) was previously developed to investigate dynamic heterogeneity for polymeric materials, which exhibit relatively weak dielectric responses. In our previous work, MSHB was applied to a densely entangled block copolymer and successfully distinguishes the heterogeneous from the homogeneous states. Here, a series of polystyrene (PS) solutions was chosen to further investigate the effect of different types of heterogeneity on mechanical spectral hole burning. The three types of heterogeneity of interest include the entanglement spacing, the entanglement density (or number of entanglements per chain), and chain end density. The heterogeneity was varied by changing either solution concentration or molecular weight of the PS. Different types of dynamics from close to the Rouse regime into the terminal region were also examined. Our results are consistent with a heterogeneous dynamics over the time scales from close to Rouse regime into the rubbery plateau regime and the rubbery plateau‐to‐terminal flow transition regime. Terminal relaxation dynamics, on the other hand, were found to be homogeneous for the PS/diethyl phthalate solutions investigated. The results also indicate the hole properties are dominated by the type of dynamics rather than the length scale of heterogeneity. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 2047–2062, 2009     

Long time dynamics of a polymer with rigid body monomer units relating to a protein model: Comparison with the rouse model

Simulation data from an off-lattice polymer model are compared with data from the Rouse model. The polymer model is built of sequentially connected rigid monomer units that represent the amide planes of a protein backbone. The time propagation of the dynamics of the polymer model is performed by a Monte Carlo method. The elementary Monte Carlo steps correspond to local confomational changes in a window of three consecutive monomer units. The time autocorrelation functions of end-to-end distances from segments within the linear chain molecule are considered in detail. Both models exhibit a stretched exponential decay pattern. A comparison of the data from the Rouse model and the computer simulation provide an estimate of the time unit of 15 ps for a full scan of the algorithm for local conformational changes along the chain. With a conservative estimate of the parameters governing the Rouse model this time unit is four orders of magnitude longer than the elementary time step of a conventional computer simulation of polymer dynamics based upon the classical equations of motion for all atoms. © 1992 by John Wiley & Sons, Inc.     

Static and dynamic scaling behavior of a polymer melt model with triple-well bending potential

We perform molecular-dynamics simulations for polymer melts of the coarse-grained poly(vinyl alcohol) model that crystallizes upon slow cooling. To establish the properties of its high temperature, liquid state as a reference point, we characterize in detail the structural features of equilibrated polymer melts with chain lengths 5 ≤ N ≤ 1000 at a temperature slightly above their crystallization temperature. We find that the conformations of sufficiently long polymers with N > 50 obey essentially the Flory's ideality hypothesis. The chain length dependence of the end-to-end distance and the gyration radius follow the scaling predictions of ideal chains and the probability distributions of the end-to-end distance, and form factors are in good agreement with those of ideal chains. The intrachain correlations reveal evidences for incomplete screening of self-interactions. However, the observed deviations are small. Our results rule out any preordering or mesophase structure formation that are proposed as precursors of polymer crystallization in the melt. Moreover, we characterize in detail primitive paths of long entangled polymer melts and we examine scaling predictions of Rouse and the reptation theory for the mean squared displacement of monomers and polymers center of mass. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2018 , 56, 1376–1392     

Dehydrogenation of poly(1,3‐cyclohexadiene)–polystyrene binary block copolymers

The dehydrogenation of poly(1,3‐cyclohexadiene)–polystyrene binary block copolymers obtained by anionic copolymerization with alkyllithium/amine systems was investigated for the first time. The dehydrogenation of the poly(1,3‐cyclohexadiene) block, which was composed of 1,2‐cyclohexadiene (1,2‐CHD) and 1,4‐cyclohexadiene (1,4‐CHD) units, was strongly affected by the polymer chain structure. The existence of 1,2‐CHD units prevented the dehydrogenation of the poly(1,3‐cyclohexadiene) block in the binary block copolymer. The rate of dehydrogenation was fast on a long sequence of 1,4‐CHD units, whereas it was relatively slow for 1,2‐CHD/1,4‐CHD (≈1/1) unit sequences. The bonding of the polystyrene block to the polymer chain effectively improved not only the rate of dehydrogenation of a long sequence of 1,4‐CHD units but also that of the polymer chain with a high content of 1,2‐CHD units. The dehydrogenation of a poly(1,3‐cyclohexadiene) block containing a small number of 1,2‐CHD units progressed via step‐by‐step reactions. The dehydrogenation of a long sequence of 1,4‐CHD units proceeded as the first step. Subsequently, in the second step, the 1,2‐CHD/1,4‐CHD (≈1/1) unit sequences remaining in the polymer chain were dehydrogenated. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 3526–3537, 2006     

Thermodynamic Limitations of the Reptation Model

Summary: The paper deals with the question whether the tube/reptation model of polymer chain dynamics is compatible with general laws of statistical physics. Based on a relation between the mean squared fluctuation of the number of segments in a given volume element and the isothermal compressibility of the polymer system, it follows straightforwardly that the tube/reptation model predicts fluctuations larger than permitted by thermodynamics on the time scale t ≳ τ_R, where τ_R is the Rouse relaxation time.     

Why is understanding glassy polymer mechanics so difficult?

In this Perspective, I describe recent work on systems in which the traditional distinctions between (i) unentangled versus well‐entangled systems and (ii) melts versus glasses seem least useful, and argue for the broader use in glassy polymer mechanics of two more dichotomies: systems which possess (iii) unary versus binary and (iv) cooperative versus noncooperative relaxation dynamics. I discuss the applicability of (iii–iv) to understanding the functional form of strain hardening. Results from molecular dynamics simulations show that the “dramatic” hardening observed in densely entangled systems is associated with a crossover from unary, noncooperative to binary, cooperative relaxation as strain increases; chains stretch between entanglement points, altering the character of local plasticity. Promising approaches for future research along these lines are discussed. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2011     

Stress overshoot of polymer solutions at high rates of shear

Overshoot of shear stress, σ, and the first normal stress difference, N₁, in shear flow were investigated for polystyrene solutions. The magnitudes of shear corresponding to these stresses, γ_σm and γ_Nm, for entangled as well as nonentangled solutions were universal functions of γ˙τ_eq, respectively, and γ_Nm was approximately equal to 2γ_σm at any rate of shear, γ˙. Here τ_eq = τ_R for nonentangled systems and τ_eq = 2τ_R for entangled systems, where τ_R is the longest Rouse relaxation time evaluated from the dynamic viscoelasticity at high frequencies. Only concentrated solutions exhibited stress overshoot at low reduced rates of shear, γ˙τ_eq < 1. The behavior at very low rates, γ˙τ_eq < 0.2, was consistent with the Doi–Edwards tube model theory for entangled polymers. At high rates, γ˙τ_eq > 1, γ_σm and γ_Nm were approximately proportional to γ˙τ_eq. At very high rates of shear, the peak of σ is located at t = τ_R, possibly indicating that the polymer chain shrinks with a characteristic time τ_R in dilute solutions. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 1917–1925, 2000     

Synthesis and characterization of a helical step‐ladder polyarylene

A helical step‐ladder polyarylene incorporating chiral (R)‐2,2′‐dioctoxy‐1,1′‐binaphthyl units was synthesized for the first time. The first step involved the preparation of a precursor poly(arylene ketone) via a palladium‐mediated Suzuki‐type cross‐coupling reaction with the aid of microwave heating. Two polymer‐analog reaction steps, the reduction of the keto groups to tertiary alcohol functionalities and subsequent intramolecular Friedel–Crafts cyclization, gave a step‐ladder polymer ( 6 ) in good yields with reasonable mean average molecular weights greater than 13,000. The regioselective cyclization pattern in the α position of the naphthalene core was confirmed by a comparison of the NMR data of the polymer with those of the corresponding model ladder oligomers, 12 and 13 , and also a single‐crystal structure of 13 . The optical spectra of the oligomers and polymers indicated that there was little electronic interaction across the binaphthyl units. The circular dichroism spectrum of 6 exhibited a strong bisignate Cotton effect in the π–π* absorption region of the planar chromophores, which reflected the strong exciton coupling within the helical polymer chain. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 5533–5545, 2006