Distribution functions and dynamical properties of stiff macromolecules

An analytically tractable model for chain molecules with bending stiffness is presented and the dynamical properties of such chains are investigated. The partition function is derived via the maximum entropy principle taking into account the chain connectivity as well as the bending restrictions in form of constraints. We demonstrate that second moments agree exactly with those known from the Kratky-Porod wormlike chain. Moreover, various distribution functions are calculated. In particular, the static structure factor is shown to be proportional to 1/q at large scattering vectors q. The equations of motion for a chain in a melt as well as in dilute solution are presented. In the latter case the hydrodynamic interaction is taken into account via the Rotne-Prager tensor. The dynamical equations are solved by a normal mode analysis. In the limit of a flexible chain the model reproduces the well-known Rouse and Zimm dynamics, respectively, on large length scales, whereas in the rod limit the eigenfunctions correspond to bending motion only. In addition, the coherent and incoherent dynamic structure factor is discussed. For melts we show that at large scattering vectors the incoherent dynamic structure factor is a universal function of only the combination q^8/3tp^1/3, where 1/(2p) is the persistence length of the macromolecules. The comparison of the theoretical results with quasielastic neutron and light scattering experiments of various polymers in solution and melt exhibits good agreement. Our investigations show that local stiffness strongly influences the dynamics of macromolecules on small length scales even for long and flexible chains.     

高分子动力学的单链模型

高分子单链模型是高分子稀溶液理论研究的基本模型.对其进行深入地分析,不仅有助于解决高分子稀溶液体系中溶液黏度和分子链扩散等基本问题,而且能够增进人们对高分子链结构与溶液性质间关联性的理解.虽然基于经典连续性介质力学的流体动力学理论可以定性,甚至半定量地获得稀溶液的一些重要性质,但是,随着科学技术的发展,人们从分子水平上建立了许多描述高分子稀溶液性质的模型和理论,期望能够定量地描述高分子稀溶液的性质.本文以高分子稀溶液中3个典型的单链模型为例(包括:不含流体力学相互作用的Rouse模型、含二体流体力学相互作用的Zimm模型和含多体流体力学相互作用的部分穿透球模型),综述高分子稀溶液的重要性质,并详细地给出其动力学方程的推导过程及其重要的研究进展.特别是,对于Rouse模型,本文还将其预言结果拓展到了短链高分子流体体系;此外,还介绍了这一领域的关键科学问题、发展前景和研究方向.     

Monte carlo study of the dynamics of star-branched polymers

A uniform star-branched polymer model with f = 3 arms based on a simple cubic lattice was studied by means of the dynamic Monte Carlo method. The model chain is athermal with excluded-volume interactions and it is flexible. A new type of local micromodification was introduced to make the branching point movable. Static properties of the star polymer are in accordance with other theoretical predictions and experimental evidence. Scaling of the self diffusion constant and the terminal relaxation times is close to those of the Rouse theory and to simulation results of linear chains.     

Hydrodynamic screening near planar boundaries: effects on semiflexible polymer dynamics

The influence of hydrodynamic screening near a surface on the dynamics of a single semiflexible polymer is studied by means of Brownian dynamics simulations and hydrodynamic mean field theory. The polymer motion is characterized in terms of the mean squared displacements of the end-monomers, the end-to-end vector, and the scalar end-to-end distance. In order to control hydrodynamic screening effects, the polymer is confined to a plane at a fixed separation from the wall. When gradually decreasing this separation, a crossover from Zimm-type towards Rouse (free-draining) polymer dynamics is induced. However, this crossover is rather slow and the free-draining limit is not completely reached--substantial deviations from Rouse-like dynamics are registered in both simulations and theory--even at distances of the polymer from the wall on the order of the monomer size. Remarkably, the effect of surface-induced screening of hydrodynamic interactions sensitively depends on the type of dynamic observable considered. For vectorial quantities such as the end-to-end vector, hydrodynamic interactions are important and therefore surface screening effects are sizeable. For a scalar quantity such as the end-to-end distance, on the other hand, hydrodynamic interactions are less important, but a pronounced dependence of dynamic scaling exponents on the persistence length to contour length ratio becomes noticeable. Our findings are discussed against the background of single-molecule experiments on f-actin [L. Le Goff et al., Phys. Rev. Lett. 89, 258101 (2002)].     

Equilibrium and dynamical properties of gaussian stiff chain molecules

We present a novel analytically tractable model for stiff chain molecules. The equilibrium distribution function of the chain is derived using the maximum-entropy principle. For that purpose, we first formulate a discrete chain model, where the connections of the points and the restriction on bending are taken into account via constraints. We then perform the limit to a continuous chain and show that the mean-square end-to-end distance and the radius of gyration of the continuous chain are identical with the same quantities of the Kratky-Porod wormlike chain. The dynamics of our chain is investigated in dilute solution without hydrodynamic interactions. The linear dynamical equation is solved by a normal mode analysis. We discuss the dependence of the relaxation times on the single parameter of the model, the persistence length. For small persistence lengths we obtain the well known relaxation times of the Rouse model. In the stiff-chain limit, we find the pure bending relaxation times and, in addition, the rotational relaxation time.     

Hydrodynamic effects on the extension of stars and dendrimers in external fields

We assess the influence of hydrodynamic interactions on the extension of branched polymers subjected to external forces. We envisage that the macromolecules move under external applied fields, exemplified by mechanical or electrical micromanipulations. We focus our attention on the difference of the results obtained using the Rouse and the Zimm models, both of which represent an extreme situation, so that the realistic behavior is encompassed by the two models. We focus on the mean displacement of a specified monomer, on the shape which the macromolecule attains and on the structural average of the displacements involved. We discuss how these dynamic properties depend on the underlying topology, such as the number of branches and their length for the stars and on the number of generations for the dendrimers. Interestingly, although there exist quantitative differences between the results of the Rouse and of the Zimm model, in both models there appear typical dynamical features which depend on the topology only. This stresses the role of the structure on the dynamics and offers the possibility of tracking it under realistic, experimental conditions.     

Length-scale dependent relaxation shear modulus and viscoelastic hydrodynamic interactions in polymer liquids

A quantitative theory of hydrodynamic interactions in unentangled polymer melts and concentrated solutions is presented. The study is focussed on the pre-Rouse transient time regimes (t < τ(R), the Rouse relaxation time) where the hydrodynamic response is governed mainly by the viscoelastic effects. It is shown that transient viscoelastic hydrodynamic interactions are not suppressed (screened) at large distances and are virtually independent of polymer molecular mass. A number of transient regimes of unusual and qualitatively different behavior of isotropic and anisotropic hydrodynamic response functions are elucidated. The regimes are characterized in terms of two main length-scale dependent characteristic times: momentum spreading time τ(i) ∝ r(4∕3) and viscoelastic time τ(?) ∝ r(4). It is shown that for t > τ(i) the viscoelastic hydrodynamic interactions can be described in terms of the time or length scale dependent effective viscosity which, for t < τ(R) and/or for r < R(coil), turns out to be much lower than the macroscopic "polymer" viscosity η(m). The theory also involves a quantitative analysis of the length-scale dependent stress relaxation in polymer melts. The general predictions for hydrodynamic interactions in thermostated systems with Langevin friction are obtained as well.     

Effect of molecular topology on the transport properties of dendrimers in dilute solution at Theta temperature: a Brownian dynamics study

Structure and transport properties of dendrimers in dilute solution are studied with the aid of Brownian dynamics simulations. To investigate the effect of molecular topology on the properties, linear chain, star, and dendrimer molecules of comparable molecular weights are studied. A bead-spring chain model with finitely extensible springs and fluctuating hydrodynamic interactions is used to represent polymer molecules under Theta conditions. Structural properties as well as the diffusivity and zero-shear-rate intrinsic viscosity of polymers with varied degrees of branching are analyzed. Results for the free-draining case are compared to and found in very good agreement with the Rouse model predictions. Translational diffusivity is evaluated and the difference between the short-time and long-time behavior due to dynamic correlations is observed. Incorporation of hydrodynamic interactions is found to be sufficient to reproduce the maximum in the intrinsic viscosity versus molecular weight observed experimentally for dendrimers. Results of the nonequilibrium Brownian dynamics simulations of dendrimers and linear chain polymers subjected to a planar shear flow in a wide range of strain rates are also reported. The flow-induced molecular deformation of molecules is found to decrease hydrodynamic interactions and lead to the appearance of shear thickening. Further, branching is found to suppress flow-induced molecular alignment and deformation.     

Observation of the neutron quasi-elastic broadening by polymers in soluttion

The quasi-elastic neutron scattering peaks obtained from polytetrahydrofuran and its perdeutero analogue have been observed in solution in CS₂. Thus the incoherent and coherent components can be separated. The results do not support the Rouse model of polymer chain dynamics but the Zimm model, which includes hydrodynamic interactions, may apply.     

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     

End-monomer Dynamics in Semiflexible Polymers

Spurred by an experimental controversy in the literature, we investigate the end-monomer dynamics of semiflexible polymers through Brownian hydrodynamic simulations and dynamic mean-field theory. Precise experimental observations over the last few years of end-monomer dynamics in the diffusion of double-stranded DNA have given conflicting results: one study indicated an unexpected Rouse-like scaling of the mean squared displacement (MSD) ?r(2)(t)? ~ t(1/2) at intermediate times, corresponding to fluctuations at length scales larger than the persistence length but smaller than the coil size; another study claimed the more conventional Zimm scaling ?r(2)(t)? ~ t(2/3) in the same time range. Using hydrodynamic simulations, analytical and scaling theories, we find a novel intermediate dynamical regime where the effective local exponent of the end-monomer MSD, α(t) = d log?r(2)(t)?/d log t, drops below the Zimm value of 2/3 for sufficiently long chains. The deviation from the Zimm prediction increases with chain length, though it does not reach the Rouse limit of 1/2. The qualitative features of this intermediate regime, found in simulations and in an improved mean-field theory for semiflexible polymers, in particular the variation of α(t) with chain and persistence lengths, can be reproduced through a heuristic scaling argument. Anomalously low values of the effective exponent α are explained by hydrodynamic effects related to the slow crossover from dynamics on length scales smaller than the persistence length to dynamics on larger length scales.     

Conformation and diffusion behavior of ring polymers in solution: a comparison between molecular dynamics, multiparticle collision dynamics, and lattice Boltzmann simulations

We have studied the effect of chain topology on the structural properties and diffusion of polymers in a dilute solution in a good solvent. Specifically, we have used three different simulation techniques to compare the chain size and diffusion coefficient of linear and ring polymers in solution. The polymer chain is modeled using a bead-spring representation. The solvent is modeled using three different techniques: molecular dynamics (MD) simulations with a particulate solvent in which hydrodynamic interactions are accounted through the intermolecular interactions, multiparticle collision dynamics (MPCD) with a point particle solvent which has stochastic interactions with the polymer, and the lattice Boltzmann method in which the polymer chains are coupled to the lattice fluid through friction. Our results show that the three methods give quantitatively similar results for the effect of chain topology on the conformation and diffusion behavior of the polymer chain in a good solvent. The ratio of diffusivities of ring and linear polymers is observed to be close to that predicted by perturbation calculations based on the Kirkwood hydrodynamic theory.     

Hydrodynamic screening of ring copolymer solutions

The relaxation frequency of the dynamic correlation function for a diblock cyclic copolymer in solution in the zero average condition is calculated in the presence of hydrodynamic interaction. The latter is introduced through the screened Oseen tensor where the hydrodynamic screening length is used as a parameter to determine the range of screening. Substantial differences with the case of linear homopolymer are found, especially in the low q range where the dynamics are much faster and correspond to nondiffusive processes as usually is the case for copolymer systems. As the screening length decreases the dynamics approach the Rouse behavior and the minimum of the relaxation frequency shifts to lower values. The proportionality constant of the relaxation frequency in the intermediate q range Γ(q) → q³/(k_BTη₀) is also investigated as a function of the screening length. © 1994 John Wiley & Sons, Inc.     

Combined Molecular Dynamics Simulation and Rouse Model Analysis of Static and Dynamic Properties of Unentangled Polymer Melts with Different Chain Architectures

Chain architecture effect on static and dynamic properties of unentangled polymers is explored by molecular dynamics simulation and Rouse mode analysis based on graph theory. For open chains, although they generally obey ideal scaling in chain dimensions, local structure exhibits nonideal behavior due to the incomplete excluded volume(EV) screening, the reduced mean square internal distance(MSID) can be well described by Wittmer' theory for linear chains and the resulting chain swelling is architecture dependent, i.e., the more branches a bit stronger swelling. For rings, unlike open chains they are compact in term of global sizes. Due to EV effect and nonconcatenated constraints their local structure exhibits a quite different non-Gaussian behavior from open chains, i.e., reduced MSID curves do not collapse to a single master curve and fail to converge to a chain-length-independent constant, which makes the direct application of Wittmer's theory to rings quite questionable.Deviation from ideality is further evidenced by limited applicability of Rouse prediction to mode amplitude and relaxation time at high modes as well as the non-constant and mode-dependent scaled Rouse mode amplitudes, while the latter is architecture-dependent and even molecular weight dependent for rings. The chain relaxation time is architecture-dependent, but the same scaling dependence on chain dimensions does hold for all studied architectures. Despite mode orthogonality at static state, the role of cross-correlation in orientation relaxation increases with time and the time-dependent coupling parameter rises faster for rings than open chains even at short time scales it is lower for rings.     

Oscillatory shear measurements on polystyrene melts in the terminal region

Oscillatory shear measurements have been made on a range of anionic polystyrene melts of molecular weights 1000--500,000. For M < 5000 the polymer chain is too short to act as a Gaussian coil and hence the compliance of the melt is very low. For 10,000 < M < 100,000 the compliance of the melt follows the Rouse model of the elasticity of isolated polymer molecules. It is necessary to use the Ferry, Landel and Williams extension of the Rouse theory for M > 40,000 to allow for the effect of entanglements on the complex modulus. For M > 200,000 the entangelment network dominates the compliance and the Rouse theory is no longer applicable.     

From caging to Rouse dynamics in polymer melts with intramolecular barriers: a critical test of the mode coupling theory

By means of computer simulations and solution of the equations of the mode coupling theory (MCT), we investigate the role of the intramolecular barriers on several dynamic aspects of nonentangled polymers. The investigated dynamic range extends from the caging regime characteristic of glass-formers to the relaxation of the chain Rouse modes. We review our recent work on this question, provide new results, and critically discuss the limitations of the theory. Solutions of the MCT for the structural relaxation reproduce qualitative trends of simulations for weak and moderate barriers. However, a progressive discrepancy is revealed as the limit of stiff chains is approached. This disagreement does not seem related with dynamic heterogeneities, which indeed are not enhanced by increasing barrier strength. It is not connected either with the breakdown of the convolution approximation for three-point static correlations, which retains its validity for stiff chains. These findings suggest the need of an improvement of the MCT equations for polymer melts. Concerning the relaxation of the chain degrees of freedom, MCT provides a microscopic basis for time scales from chain reorientation down to the caging regime. It rationalizes, from first principles, the observed deviations from the Rouse model on increasing the barrier strength. These include anomalous scaling of relaxation times, long-time plateaux, and nonmonotonous wavelength dependence of the mode correlators.     

Dynamics of polyethylene melts studied by Monte Carlo simulations on a high coordination lattice

A detailed comparison is made between the experiment, prior simulations by other groups, and our simulation based on a newly designed dynamic Monte Carlo algorithm, on the dynamics of polyethylene (PE) melts. The new algorithm, namely, noncross random two-bead move has been developed on a high coordination lattice (the 2nnd lattice) for studying the dynamics of realistic polymers. The chain length (molecular weight) in our simulation ranges from C40 (562 Da) to C324 (4538 Da). The effects of finite chain length have been confirmed and significant non-Gaussian statistics evidently results in nonstandard static and dynamic properties of short PE chains. The diffusion coefficients scale with molecular weight (M) to the −1.7 power for short chains and −2.2 for longer chains, which coincides very well with experimental results. No pure Rouse scaling in diffusion has been observed. The transitional molecular weight to the entanglement regime is around 1500 Da. The detailed mean square displacements of middle bead (g1) are presented for several chain lengths. The reptation-like slowdown can be clearly observed only above M ∼ 2400 Da. The slope 0.25 predicted by the theory for the intermediate regime is missing; instead a slope close to 0.4 appears, indicating that additional relaxation mechanism exists in this transitional region. The relaxation times extracted by fitting the autocorrelation function of end-to-end vectors with reptation model scale with M to 2.5 for long chains, which seemingly conflicts with the scaling of diffusion. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2556–2571, 2006     

Higher order hydrodynamic interactions in the calculation of polymer transport properties

Higher order hydrodynamics interactions are short-range modifications to the Oseen tensor T _ij and its self-interaction counterpart T _ii. They differ from the Oseen tensor in having terms of higher order than the first in a/R, a being a bead radius and R being a bead-bead distance. Effects of higher order hydrodynamic interactions on whole chain–whole chain hydrodynamic interactions are here computed. Higher order hydrodynamic interactions are shown to lead to a concentration dependence of the diffusion and friction coefficients of a free monomer. However, while higher order interactions make contributions of the same nature to the drag coefficients of a monomer and of a whole chain, the contributions are not simply multiplicative, removing a justification for the common practice of correcting polymer solution transport data for “monomer friction effects” via a normalization with data on friction coefficients of free monomers. © 1993 John Wiley & Sons, Inc.     

Structural and dynamic single chain behaviour in a binary blend of low molecular mass poly(siloxanes) as studied by small angle neutron scattering and neutron spin echo spectroscopy

Elastic and quasi elastic neutron scattering investigations, using the small angle neutron scattering (SANS) and neutron spin echo (NSE) techniques, respectively, were performed in order to study the static and dynamic single chain behaviour in a binary blend of low molecular mass deuterated poly(dimethylsiloxane) (d-PDMS) and protonated poly(ethylmethylsiloxane) (p-PEMS) at the critical composition _c. Since the single chain observation requires that only a small amount of one of both components is labelled, the d-PDMS/p-PEMS system was modified in such a way that the major part of the protonated PEMS component was replaced by the corresponding deuterated material. Although the de-mixing of the PEMS isotopes occurs far below the de-mixing of the PDMS/PEMS system the resulting chemically binary d-PDMS/d-PEMS/p-PEMS blend with the volume composition 0.5/0.425/0.075 is strictly speaking a ternary system. This complication had to be taken into account, in particular with respect to the correct evaluation of the SANS data.The careful analysis of the SANS curves allows one to determine all three thermodynamic interaction parameters with reasonable reliability and gives evidence that the radii of gyration agree with those, which were determined in corresponding isotopic PDMS and PEMS blends. This is in contrast to the observation on real binary PDMS/PEMS blends at _c, where the collective conformational properties exhibit a considerable chain expansion. The NSE data of the ternary system follow completely the predictions of the Rouse model, which describes the dynamics of a dense low molecular mass polymeric system in a single chain approximation. The experimental observations are also in contrast to the results of former NSE measurements on binary PDMS/PEMS blends, where a transition from Rouse behaviour at short times to a much weaker relaxation at longer times became obvious.The results of the static and dynamic single chain behaviour presented here confirm the results of a computer simulation on a low molecular mass binary blend at the critical concentration, where explicitly the pure single chain behaviour was probed and no indications for chain expansion and deviations from the Rouse dynamics were found.Dedicated to Prof. Dr. Erhard W. Fischer on the occasion of the 75th anniversary of his birthday.