Rouse Modes Research Articles

Single polymer confinement in a tube: Correlation between structure and dynamics

In this paper, we construct an effective model for the dynamics of an excluded-volume chain under confinement by extending the formalism of Rouse modes. We make specific predictions about the behavior of the modes for a single polymer confined to a tube. The results are tested against Monte Carlo simulations using the bond-fluctuation algorithm which uses a lattice representation of the polymer chain with excluded-volume effects.

Static and dynamic scaling of semiflexible polymer chains—a molecular dynamics simulation study of single chains and melts

A molecular dynamics study of the effect of increasing molecular chain stiffness on the dynamic properties of single linear chains and polymer melts is presented. The chain stiffness is controlled by a bending potential. By means of a normal mode analysis the systematic crossover behavior of coarse-grained linear polymer systems from Rouse modes to bending modes with increasing mode number p is presented systematically for the first time via simulation data. The magnitude and the onset of the region where crossover behavior occurs is investigated in dependence of a systematic variation of the chains’ persistence lengths. For long wavelength modes the well-known p−2 behavior is observed which represents the Rouse scaling, whereas for the bending modes one observes a distinct p−4 scaling of the mean square mode amplitudes and the relaxation times. Additionally, the effect of this crossover behavior on the monomer dynamics given by their mean square displacements is investigated. The findings are contrasted with previous simulation studies of semiflexible chain behavior and it is shown that those studies—due to too small persistence lengths Lp—were actually limited to the crossover regime where both, Rouse and bending modes contribute to the chain dynamics, and no distinct p−4 scaling can be observed. Using an expansion of the monomer position vector, a simple theory of the observed scaling behavior is proposed which allows for deriving analytic expressions that describe the presented simulation data very well.

Role of Internal Chain Dynamics on the Rupture Kinetic of Adhesive Contacts

We study the forced rupture of adhesive contacts between monomers that are not covalently linked in a Rouse chain. When the applied force (f) to the chain end is less than the critical force for rupture (f{c}), the reversible rupture process is coupled to the internal Rouse modes. If f/f{c}>1 the rupture is irreversible. In both limits, the nonexponential distribution of contact lifetimes, which depends sensitively on the location of the contact, follows the double-exponential (Gumbel) distribution. When two contacts are well separated along the chain, the rate limiting step in the sequential rupture kinetics is the disruption of the contact that is in the chain interior. If the two contacts are close to each other, they cooperate to sustain the stress, which results in an "all-or-none" transition.

Single-chain dynamics in a semidilute polymer solution under steady shear

We use Brownian dynamics computer simulations to investigate single-chain dynamics in a semidilute polymer solution undergoing a steady, uniform shear flow. In the presence of the shear flow, the system used in the present study exhibits anisotropic structure factors, often referred to as butterfly patterns, which rotate with increasing shear rate [P. P. Jose and G. Szamel, J. Chem. Phys. 127, 114905 (2007)]. The rotation of these patterns correlates with shear thinning of the solution. In order to elucidate the microscopic origin of this behavior, we have investigated the change in the single-chain dynamics in the solution: We have focused on the relaxation of the end-to-end vector, the Rouse modes, and the radius of gyration tensor. In equilibrium and for small shear rates, these quantities show double exponential relaxation. With increasing shear rate, they show oscillatory relaxation, which hints at the tumbling motion of the chain. In the high shear rate regime, the frequency of the oscillations of the end-to-end vector autocorrelation function shows a power law dependence on the shear rate. We have compared the single-chain dynamics in the semidilute solution with that in a dilute solution. An analysis of the instantaneous values of the radius of gyration tensor, the end-to-end distance, and the normal stress along the system's trajectory reveals a synchronization of the fluctuations of these quantities.

Entangledlike Chain Dynamics in Nonentangled Polymer Blends with Large Dynamic Asymmetry

We discuss simulations of a simple model for polymer blends in the framework of the Rouse model. At odds with standard predictions, large dynamic asymmetry between the two components induces strong nonexponentiality of the Rouse modes for the fast component. Despite chains being much shorter than the entanglement length, it also induces dynamic features resembling a crossover to entangledlike chain dynamics. This unusual behavior is associated with strong memory effects which break down the assumption of time uncorrelation of the external forces acting on the tagged chain.

Static Rouse modes and related quantities: Corrections to chain ideality in polymer melts

Following the Flory ideality hypothesis intrachain and interchain excluded-volume interactions are supposed to compensate each other in dense polymer systems. Multichain effects should thus be neglected and polymer conformations may be understood from simple phantom chain models. Here we provide evidence against this phantom chain, mean-field picture. We analyze numerically and theoretically the static correlation function of the Rouse modes. Our numerical results are obtained from computer simulations of two coarse-grained polymer models for which the strength of the monomer repulsion can be varied, from full excluded volume ("hard monomers") to no excluded volume ("phantom chains"). For nonvanishing excluded volume we find the simulated correlation function of the Rouse modes to deviate markedly from the predictions of phantom chain models. This demonstrates that there are nonnegligible correlations along the chains in a melt. These correlations can be taken into account by perturbation theory. Our simulation results are in good agreement with these new theoretical predictions.

Barrier crossing by a star polymer

We analyze the dynamics of a star polymer of F arms trapped in a double well potential. Initially the molecule is confined to one of the minima and can cross over the barrier to the other side. We use the continuum version of the Rouse-Ham model and calculate the rate of crossing using the multidimensional approach due to Langer [Ann. Phys. (N.Y.) 54, 258 (1969)]. Finding the transition state for the process is shown to be equivalent to the solution of Newton's equations for F independent particles, moving in an inverted potential. For each star polymer, there is a critical barrier top curvature, below which the star crosses over in coiled conformation. The value of the critical curvature is determined by the first Rouse mode of the star. If the curvature is greater than this critical value, the saddle point for the crossing is a stretched conformation of the star. For the coiled transition state, the activation energy is proportional to the total arm length of the star. For the stretched transition state, as one increases the length of an arm of the star, the activation energy at first increases and then decreases. This results from the fact that in the stretched state, only one arm of the polymer is stretched across the top of the barrier, while others need not be. We calculate the rate by expanding the energy around the saddle up to second order in the fluctuations. As we use the continuum model, there are infinite modes for the polymer and, consequently, the prefactor has infinite products. We show that these infinite products can be reduced to a simple expression, and evaluated easily. However, the rate diverges near NTc due to the multifurcation, which results in more than one unstable mode. The cure for this divergence is to keep terms up to fourth order in the expansion of energy for these modes. Performing this, we have calculated the rate as a function of the length of the star. It is found that the rate has a nonmonotonic dependence on the length, suggesting that longer stars may actually cross over the barrier faster.

Tests of mode coupling theory in a simple model for two-component miscible polymerblends

We present molecular dynamics simulations of the structural relaxation of a simplebead–spring model for polymer blends. The introduction of a different monomer sizeinduces a large timescale separation for the dynamics of the two components. Simulationresults for a large set of observables probing density correlations, Rouse modes, andorientations of bond and chain end-to-end vectors are analysed within the framework of themode coupling theory (MCT). An unusually large value of the exponent parameter isobtained. This feature suggests the possibility of an underlying higher-order MCT scenariofor dynamic arrest.

Molecular dynamics simulation of temperature and pressure effects on the intermediate length scale dynamics and zero shear rate viscosity of cis-1,4-polybutadiene: Rouse mode analysis and dynamic structure factor spectra

A well-relaxed atomistic configuration of a 32-chain C 128 cis-1,4-polybutadiene ( cis-1,4-PB) system has been subjected to long (on the order of a few microseconds) molecular dynamics (MD) simulations in the NPT ensemble using the united-atom forcefield introduced by Smith et al. [G. Smith, D. Bedrov, W. Paul, A molecular dynamics simulation study of the alpha-relaxation in a 1,4-polybutadiene melt as probed by the coherent dynamic structure factor, J. Chem. Phys. 121 (2004) 4961–4967] on the basis of quantum chemistry calculations. This allowed us to study the temperature and pressure dependences of the Rouse-mode relaxation spectrum of cis-1,4-PB over a wide range of temperature (ranging from T = 430 K down to 165 K) and pressure (from P = 1 atm up to 3.5 kbar) conditions. Results are presented for: (a) the time decay of the autocorrelation function of the normal coordinates (Rouse modes), (b) the single chain intermediate coherent dynamic structure factor, S coh( q, t), and (c) the intermediate incoherent dynamic structure factor, S inc( q, t), for different values of the wavevector q. By mapping our MD simulation results onto the Rouse model, we have been able to extract a prediction for the zero shear rate viscosity of the simulated cis-1,4-PB system as a function of temperature and analyze its fragile character. In agreement with our previous MD simulation studies on the same system [G. Tsolou, V.A. Harmandaris, V.G. Mavrantzas, Atomistic molecular dynamics simulation of the temperature and pressure dependences of local and terminal relaxations in cis-1,4-polybutadiene, J. Chem. Phys. 124 (2006) 084906-1-11] and in contrast to what is experimentally observed [see, e.g., G. Floudas, T. Reisinger, Pressure dependence of the local and global dynamics of polyisoprene, J. Chem. Phys. 111 (1999) 5201–5204; C.M. Roland, R. Casalini, T. Psurek, S. Pawlus, M. Paluch, Segmental- and normal-mode dielectric relaxation of poly(propylene glycol) under pressure, J. Polym. Sci. Part B: Polym. Phys. 41 (2003) 3047–3052], we predict that pressure and temperature influence practically similarly all normal mode relaxation times along the simulated C 128 cis-1,4-PB chain. Furthermore, our MD simulation results predict a transition from a homogeneous to a heterogeneous dynamical behavior in the region of wavevectors near the first (intermolecular) peak in the static structure factor, consistently with recent neutron scattering (NS) measurements [see, e.g., B. Frick, G. Dosseh, A. Cailliaux, C. Alba-Simionesco, Pressure dependence of the segmental relaxation of polybutadiene and polyisobutylene and influence of molecular weight, Chem. Phys. 292 (2003) 311–323; A. Arbe, J. Colmenero, B. Farago, M. Monkenbusch, U. Buchenau, D. Richter, Intermediate length scale dynamics in glass forming polymers: coherent and incoherent quasielastic neutron scattering results on polyisobutylene, Chem. Phys. 292 (2003) 295–309] and previous simulation studies [see, e.g., J. Colmenero, F. Alvarez, A. Arbe, Self-motion and the alpha relaxation in a simulated glass-forming polymer: Crossover from Gaussian to non-Gaussian dynamic behavior, Phys. Rev. E 65 (2002) 041804-1-12].

Understanding Segmental Dynamics in Polymer Electrolytes: A Computer Study

AbstractWe study the segmental dynamics of poly(ethylene oxide) (PEO) from microscopic simulations in the neat polymer and a polymer electrolyte (PEO/LiBF4) by analyzing the normal modes. We verify the applicability of the Rouse theory, specifically for the polymer electrolyte where dynamic heterogeneities, arising from cation–polymer interactions, alter the mobility non‐uniformly along the chains. We find that the Rouse modes for both the systems are orthogonal despite the presence of non‐exponential relaxation of the modes and violation of the Gaussian self‐similarity of the chains. The slowdown of the segmental dynamics in the polymer electrolyte is rationalized by an order of magnitude increase in the friction coefficient for those monomers which are bound by cations. In general, for the electrolyte the Rouse predictions for the dynamics of segments (both free and/or bound) agree well except for very short times.magnified image

Glass transition-related thermorheological complexity in polystyrene melts

The relaxation-modulus G(t) functional forms covering the whole time range are given by incorporating a stretched exponentialfor the structural- (glassy-) relaxation process into the extended reptation theory (ERT; forentangled systems) or the Rouse theory (for entanglement-free systems). The creep complianceJ(t) curves of two entangled (A and B) and one entanglement-free (C)polystyrene samples (Plazek) as well as the viscoelastic spectraG*(ω) of fourentanglement-free polystyrene samples (Inoue et al) have been quantitatively analyzed in terms of thegiven G(t) functional forms. In such quantitatively successful analyses, the ERT or the Rouse theory works asthe frame of reference in both the line shape and timescale. The thermorheological complexity in theJ(t) curves is explained naturally and precisely by the temperature dependence of theenergetic-interaction-derived structural relaxation being stronger than that of theentropic ERT or Rouse dynamics in a simple way. Structural-relaxation timesτS (= 18s′K′) of all thestudied samples are equally well separated into two decoupled quantities: the structural-growth parameters′ and thefrictional factor K′ (for the Rouse–Mooney or Rouse modes of motion). The separation is fundamentally a clean-cut process:s′ is determined entirelyby the line shape of J(t) or G*(ω) while K′ is calculated from the timescale shifting factor obtained from the superpositionof the calculated curves onto the measured. The glassy-relaxation strengthAGf and the stretchingparameter β extracted from the J(t) and G*(ω) results over the glassy-relaxation region are in good agreement. The glass-transition temperatureTg is defined ascorresponding to τS = 1000 s for allthe studied samples. The τS, s′ and K′ data points of samples A, B and C extracted from theirJ(t) curves individually fall closely on the same curves when expressed as a function ofΔT = T−Tg, revealinga Tg-related universality within the polystyrene system, entangled or not. The revealeduniversality confirms the previously derived conclusion that the ERT and the Rouse theoryhave the same footing at the Rouse-segmental level. Representing important physicalfeatures of the universality, the length-scale of the structural relaxation increases asΔT diminishes andreaches the value of ∼3 nm at ΔT = 0 (or at Tg) for all three samples, A, B and C. Extracted from theG*(ω) results,the τS, s′ and K′ data of samples with molecular weights just below and well below entanglement molecular weightMe (13 500) are found to deviate more from the respective universal curves withdecreasing molecular weight. Deviation is estimated to start occurring atMw = 12 000.

Stress relaxation in polymer networks: Equilibrium behavior and dynamics

The elastic and relaxational properties of a polymer network have been calculated using a stress based formulation based on the Rouse mode expansion [W. L. Vandoolaeghe and E. M. Terentjev, J. Chem. Phys. 123, 34902 (2005)]. In this article, we propose an improved Rouse mode expansion incorporating appropriate boundary conditions. In contrast to the previous work, this improved formulation provides a smooth crossover from the classical equilibrium result of rubber elasticity to the shorter-time-scale Rouse relaxation of a polymer melt. Our results are compared with the classical phantom network approach in equilibrium, as well as both equilibrium and dynamic elongation experiments. The model captures the qualitative features of the data well and some of the quantitative aspects, such as the exponents seen in the dynamic modulus G(omega).

Excluded-volume corrections to the single-chain static properties of a polymer melt: Temperature, density and potential effects

Excluded-volume effects on single-chain statics are introduced by analytic corrections to the Rouse results. The final expressions do not depend on free parameters. They are compared with numerical simulations of a polymer melt for different values of the temperature, the density and the interaction potential. Density and interactions control the energy landscape of the system, whereas the temperature selects the accessible regions. The agreement between the theory and the Rouse modes does not depend markedly on the temperature with some improvement for the first modes (large length scales). Differently, increasing the packing and the stiffness of the monomer–monomer interaction reduces the deviations for the first modes, but it leaves the magnitude of the deviations for the high-indexes modes (short length scales) nearly unaffected or with some tendency to increase. The scaling properties of the corrections are briefly discussed.

Hierarchical Modeling of Entangled Polymers

AbstractSummary: 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.

Local Dynamics and Primitive Path Analysis for a Model Polymer Melt near a Surface

By applying local primitive path and Rouse modes analysis we study the chains conformations, local dynamics and viscosity of a model polymer melt in a polymer-wall interface. We establish that the presence of a repulsive wall leads to acceleration of the dynamics both for unentangled and weakly entangled melts and to a depletion in the entanglement density in the wall vicinity. When the surface bears some grafted chains, we show that the melt chains are accelerated in the unentangled regime and slowed down in the entangled regime. By analyzing the primitive paths we attribute the observed slowdown to an increase in the entanglement density in the interfacial layer. The presence of a relatively small density of grafting sites thus leads to improved mechanical properties (reinforcement) and decreases locally the entanglement length even if the surface is repulsive.

Hybrid method coupling fluctuating hydrodynamics and molecular dynamics for the simulation of macromolecules

We present a hybrid computational method for simulating the dynamics of macromolecules in solution which couples a mesoscale solver for the fluctuating hydrodynamics (FH) equations with molecular dynamics to describe the macromolecule. The two models interact through a dissipative Stokesian term first introduced by Ahlrichs and Dunweg [J. Chem. Phys. 111, 8225 (1999)]. We show that our method correctly captures the static and dynamical properties of polymer chains as predicted by the Zimm model. In particular, we show that the static conformations are best described when the ratio sigma/b=0.6, where sigma is the Lennard-Jones length parameter and b is the monomer bond length. We also find that the decay of the Rouse modes' autocorrelation function is better described with an analytical correction suggested by Ahlrichs and Dunweg. Our FH solver permits us to treat the fluid equation of state and transport parameters as direct simulation parameters. The expected independence of the chain dynamics on various choices of fluid equation of state and bulk viscosity is recovered, while excellent agreement is found for the temperature and shear viscosity dependence of center of mass diffusion between simulation results and predictions of the Zimm model. We find that Zimm model approximations start to fail when the Schmidt number Sc < or approximately 30. Finally, we investigate the importance of fluid fluctuations and show that using the preaveraged approximation for the hydrodynamic tensor leads to around 3% error in the diffusion coefficient for a polymer chain when the fluid discretization size is greater than 50 A.

Dynamics of free chains in polymer nanocomposites

The dynamics of entangled polymeric chains in a polymer filled with nanoparticles is studied by means of molecular dynamics simulations of a model system. The primary objective is to study to what extent the reptation of polymers not in direct contact with fillers is modified with respect to the neat material. To this end, two systems are considered: A regular filled material in which the filler-polymer affinity is controlled, and a system in which the beads in contact with the filler at the beginning of the production phase of the simulation are tethered to the filler surface. This second system represents the limit case of long polymer-filler attachment time. In this case attention is focused on the free chains of the melt. The dynamics in the two models is different. In the filled system uniform slowing down for all Rouse modes is observed. The effect varies monotonically with the filler-polymer affinity. Up to saturation, this behavior may be captured by usual models with an effective, affinity-dependent, friction coefficient. In the system with grafted chains, the free chain Rouse dynamics is identical to that in the neat material, except for the longest modes which are significantly slowed down. More interestingly, the dynamics of the free chains depends in a nonmonotonic way on the polymer-filler affinity, although the free chains do not come in direct contact with the filler. This effect is due to small changes in the structure of the polydisperse brush upon modification of the affinity. Specifically, the density of the brush and the amount of interpenetration of free and grafted chains depend on the filler-polymer affinity. The use of a reptation model with modified tube diameter to capture this behavior is discussed.

Cooperative polymer dynamics under nanoscopic pore confinements probed by field-cycling NMR relaxometry

Reptational dynamics of bulk polymer chains on a time scale between the Rouse mode relaxation time and the so-called disengagement time is not compatible with the basic thermodynamic law of fluctuations of the number of segments in a given volume. On the other hand, experimental field-cycling NMR relaxometry data of perfluoropolyether melts confined in Vycor, a porous silica glass of nominal pore dimension of 4 nm, closely display the predicted signatures for the molecular weight and frequency dependences of the spin-lattice relaxation time in this particular limit, namely T1 proportional M-1/2nu1/2. It is shown that this contradiction is an apparent one. In this paper a formalism is developed suggesting cooperative chain dynamics under nanoscopic pore confinements. The result is a cooperative reptational displacement phenomenon reducing the root-mean-squared displacement rate correspondingly but showing the same characteristic dependences as the ordinary reptation model. The tube diameter effective for cooperative reptation is estimated on this basis for the sample system under consideration and is found to be of the same order of magnitude as the nominal pore diameter of Vycor.

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.

Time domain effects in the single-chain dynamics of semidilute and concentrated polymer solutions

Single-chain dynamics in semidilute polymer solutions is Zimm-like on short time scales and Rouse-like on long time scales. The crossover is due to time-dependent screening of hydrodynamic interactions. We present a generalized Zimm theory for polymer dynamics in an elastic polymer network. We show that the coupled equations of motion of the polymer network–fluid system lead to a generalized Oseen tensor with a time-dependent hydrodynamic screening length. The time correlation function of the Rouse modes, monomer mean-square displacements and the single-chain dynamical structure factor are calculated.