Cooperative dynamics in homopolymer melts: a comparison of theoretical predictions with neutron spin echo experiments

Cooperative Dynamics of Highly Entangled Linear Polymers within the Entanglement Tube

We present a quantitative comparison of the dynamic structure factors from unentangled and strongly entangled poly(butylene oxide) (PBO) melts. As expected, the low molecular weight PBO displays Rouse dynamics, however, with very significant subdiffusive center-of-mass diffusion. The spectra from high molecular weight entangled PBO can be very well described by the dynamic structure factor based on the concept of local reptation, including the Rouse dynamics within the tube and allowing for non-Gaussian corrections. Comparing quantitatively the spectra from both polymers leads to the surprising result that their spectra differ only by the contribution of classical Rouse diffusion for the low molecular weight melt. The subdiffusive component is common for both the low and high molecular weight PBO melts, indicating that in both melts the same interchain potential is active, thereby supporting the validity of the Generalized Langevin Equation approach.

Non-Rouse behavior of short ring polymers in melts by molecular dynamics simulations

Short ring polymers are expected to behave nearly Rouse-like due to the little effect of topological constraints of non-knot and non-concatenation. However, this notion is questioned because of several simulation and experiment findings in recent times, which requires a further more quantitative study. Therefore, we perform a deep investigation of statics and dynamics of flexible short ring polymers (N < 2Ne) in melts via molecular dynamics simulations by further taking linear analogues as well as all-crossing ring and linear polymers with switched off topological constraints for comparisons and demonstrate the noticeable deviations from the Rouse model in terms of local and global scales. Although the overall size is compact, the subchains are swollen, which is traced back to the deeper "segmental correlation hole" effect. The same scaling relationship of the non-Gaussian deviation of the static structure factor holds, but the deviation magnitude of rings is larger than that of linear analogues. By checking the non-Gaussian parameter and autocorrelation function of center-of-mass velocity, the physical origin of anomalous sub-diffusions of short rings is identified as unscreened viscoelastic hydrodynamic interactions and not correlation hole effects, like linear analogues.

The effect of chemical constitution on polyisoprene dynamics

Polyisoprene (PI) melts have been studied, with most reports focusing on systems with high 1,4-cis content. In contrast, 1,4-trans PI homopolymers or random copolymers have seldom been examined, despite a handful of investigations suggesting a distinct dynamic behavior. Herein, we employ all-atom simulations to investigate the effect of chemical architecture on the dynamics of cis and trans-PI homopolymers, as well as copolymers. We examine the thermodynamic, conformational, and structural properties of the polymers and validate the performance of the models. We probe chain dynamics, revealing that cis-PI presents accelerated translation and reorientation modes relative to trans as recorded by the mean square displacement of the chain center-of-mass as well as by the characteristic times of the lower modes in a Rouse analysis. Interestingly, progressing to higher modes, we observe a reversal with trans units exhibiting faster dynamics. This was further confirmed by calculations of local carbon-hydrogen vector reorientation dynamics, which offer a microscopic view of segmental mobility. To obtain insight into the simulation trajectories, we evaluate the intermediate incoherent scattering function that supports a temperature-dependent crossover in relative mobility that extends over separations beyond the Kuhn-length level. Finally, we analyzed the role of non-Gaussian displacements, which demonstrate that cis-PI exhibits increased heterogeneity in dynamics over short-timescales in contrast to trans-PI, where deviations persist over times extending to terminal dynamics. Our all-atom simulations provide a fundamental understanding of PI dynamics and the impact of microstructure while providing important data for the design and optimization of PI-based materials.

Simulational Tests of the Rouse Model

An extensive review of literature simulations of quiescent polymer melts is given, considering results that test aspects of the Rouse model in the melt. We focus on Rouse model predictions for the mean-square amplitudes ⟨(Xp(0))2⟩ and time correlation functions ⟨Xp(0)Xp(t)⟩ of the Rouse mode Xp(t). The simulations conclusively demonstrate that the Rouse model is invalid in polymer melts. In particular, and contrary to the Rouse model, (i) mean-square Rouse mode amplitudes ⟨(Xp(0))2⟩ do not scale as sin-2(pπ/2N), N being the number of beads in the polymer. For small p (say, p≤3) ⟨(Xp(0))2⟩ scales with p as p-2; for larger p, it scales as p-3. (ii) Rouse mode time correlation functions ⟨Xp(t)Xp(0)⟩ do not decay with time as exponentials; they instead decay as stretched exponentials exp(-αtβ). β depends on p, typically with a minimum near N/2 or N/4. (iii) Polymer bead displacements are not described by independent Gaussian random processes. (iv) For p≠q, ⟨Xp(t)Xq(0)⟩ is sometimes non-zero. (v) The response of a polymer coil to a shear flow is a rotation, not the affine deformation predicted by Rouse. We also briefly consider the Kirkwood-Riseman polymer model.

Chain Confinement and Anomalous Diffusion in the Cross over Regime between Rouse and Reptation

By neutron spin echo (NSE) and pulsed field gradient (PFG) NMR, we study the dynamics of a polyethylene-oxide melt (PEO) with a molecular weight in the transition regime between Rouse and reptation dynamics. We analyze the data with a Rouse mode analysis allowing for reduced long wavelength Rouse modes amplitudes. For short times, subdiffusive center-of-mass mean square displacement ⟨r_com²(t)⟩ was allowed. This approach captures the NSE data well and provides accurate information on the topological constraints in a chain length regime, where the tube model is inapplicable. As predicted by reptation for the polymer ⟨r_com²(t)⟩, we experimentally found the subdiffusive regime with an exponent close to μ=12, which, however, crosses over to Fickian diffusion not at the Rouse time, but at a later time, when the ⟨r_com²(t)⟩ has covered a distance related to the tube diameter.

Anomalous Dynamics in Macromolecular Liquids

Macromolecular liquids display short-time anomalous behaviors in disagreement with conventional single-molecule mean-field theories. In this study, we analyze the behavior of the simplest but most realistic macromolecular system that displays anomalous dynamics, i.e., a melt of short homopolymer chains, starting from molecular dynamics simulation trajectories. Our study sheds some light on the microscopic molecular mechanisms responsible for the observed anomalous behavior. The relevance of the correlation hole, a unique property of polymer liquids, in relation to the observed subdiffusive dynamics, naturally emerges from the analysis of the van Hove distribution functions and other properties.

Cooperative Chain Dynamics of Tracer Chains in Highly Entangled Polyethylene Melts

By neutron spin echo spectroscopy, we have studied the center of mass motion of short tracer chains on the molecular length scale within a highly entangled polymer matrix. The center of mass mean square displacements of the tracers independent of their molecular weight is subdiffusive at short times until it has reached the size of the tube d; then, a crossover to Fickian diffusion takes place. This observation cannot be understood within the tube model of reptation, but is rationalized as a result of important interchain couplings that lead to cooperative chain motion within the entanglement volume ∼d^{3}. Thus, the cooperative tracer chain motions are limited by the tube size d. If the center of mass displacement exceeds this size, uncorrelated Fickian diffusion takes over. Compared to the prediction of the Rouse model we observe a significantly reduced contribution of the tracer's internal modes to the spectra corroborating the finding of cooperative rather than Rouse dynamics within d^{3}.

Insight into the Structure and Dynamics of Polymers by Neutron Scattering Combined with Atomistic Molecular Dynamics Simulations

Combining neutron scattering and fully atomistic molecular dynamics simulations allows unraveling structural and dynamical features of polymer melts at different length scales, mainly in the intermolecular and monomeric range. Here we present the methodology developed by us and the results of its application during the last years in a variety of polymers. This methodology is based on two pillars: (i) both techniques cover approximately the same length and time scales and (ii) the classical van Hove formalism allows easily calculating the magnitudes measured by neutron scattering from the simulated atomic trajectories. By direct comparison with experimental results, the simulated cell is validated. Thereafter, the information of the simulations can be exploited, calculating magnitudes that are experimentally inaccessible or extending the parameters range beyond the experimental capabilities. We show how detailed microscopic insight on structural features and dynamical processes of various kinds has been gained in polymeric systems with different degrees of complexity, and how intriguing questions as the collective behavior at intermediate length scales have been faced.

Self-Similar Dynamics of Large Polymer Rings: A Neutron Spin Echo Study

This work clarifies the self-similar dynamics of large polymer rings using pulsed-field gradient nuclear magnetic resonance and neutron spin echo spectroscopy. We find center of mass diffusion taking place in three dynamic regimes starting (i) with a strongly subdiffusive domain ⟨r^{2}(t)⟩_{com}∼t^{α} (0.4≤α≤0.65); (ii) a second subdiffusive region ⟨r^{2}(t)⟩_{com}∼t^{0.75} that (iii) finally crosses over to Fickian diffusion. While the t^{0.75} range previously has been found in simulations and was predicted by theory, we attribute the first to the effect of cooperative dynamics resulting from the correlation hole potential. The internal dynamics at scales below the elementary loop size is well described by ring Rouse motion. At larger scales the dynamics is self-similar and follows very well the predictions of the scaling models with preference for the self-consistent fractal loopy globule model.

Polymer dynamics under confinement

We review recent neutron scattering work and related results from simulation and complementary techniques focusing on the microscopic dynamics of polymers under confinement. Confinement is either realized in model porous materials or in polymer nanocomposites (PNC). The dynamics of such confined polymers is affected on the local segmental level, the level of entanglements as well as on global levels: (i) at the segmental level the interaction with the surface is of key importance. At locally repulsive surfaces compared to the bulk the segmental dynamics is not altered. Attractive surfaces slow down the segmental dynamics in their neighborhood but do not give rise to dead, glassy layers. (ii) Confinement generally has little effect on the inter-chain entanglements: both for weakly as well as for marginally confined polymers the reptation tube size is not changed. Only for strongly confined polymers disentanglement takes place. Similarly, in PNC at higher NP loading disentanglement phenomena are observed; in addition, at very high loading a transition from polymer caused topological constraints to purely geometrical constraints is observed. (iii) On the more global scale NSE experiments revealed important information on the nature of the interphase between adsorbed layer and bulk polymer. (iv) Polymer grafts at NP mutually confine each other, an effect that is most pronounced for one component NP. (v) Global diffusion of entangled polymers both in weakly and strongly attractive PNC is governed by the ratio of bottle-neck to chain size that characterizes the 'entropic barrier' for global diffusion.

Solvent Effect on the Diffusion of Unentangled Linear Polymer Melts

We conducted molecular dynamics (MD) simulations to study how solvent chains affect the diffusion of linear polymers in the unentangled regime. For monodisperse solvent chains, the self-diffusivity of a tagged chain scales with its chain length. The solvent chain length affects both the prefactor and the exponent, the latter of which ranges from -0.79 to -0.85. The scaling exponent here deviates from -1 as predicted by the Rouse model, which may suggest that the friction coefficient increases with the solvent chain length. In addition, we carried out diffusion simulations on two polydisperse melts, one with the Flory-Schulz distribution and the other with the Gaussian distribution. The measured diffusivity as a function of the tagged chain length agrees with a simple proposed model accounting for the heterogeneous medium.

A new insight into the conformation and melt dynamics of hydrogenated polybutadiene as revealed by computer simulations

Extensive molecular dynamics simulations of the macromolecular conformation and the melt dynamics for model polymers of different molecular weights have been carried out. The selected models are hydrogenated polybutadienes with a 2% content of ethyl branches and linear polyethylene. It will be shown that the density and chain stiffness are clearly affected by both the molecular weight and the presence of ethyl branches. Furthermore, the results obtained from the simulations on the molecular size and, more remarkably, chain dynamics, perfectly match the neutron scattering experiments performed by Zamponi et al. in hydrogenated polybutadienes. We observe a clear chain contraction and a slow dynamics for the hydrogenated polybutadiene with respect to the linear chain of the same molecular length. Using the Likhtman-McLeish definitions, the obtained values of the entanglement relaxation time (τe) and the tube diameter (a) are found to be in agreement with the available experimental data (by rheology and neutron spin echo) as well as with those obtained by the simulations. Finally, a very good agreement of diffusion coefficients as a function of the molecular weight between simulations and experiments is observed. Therefore, there exists a clear difference between the results obtained for branched and linear polyethylene, accounting for a definitive effect of the short chain branching on the conformational properties and the melt dynamics of polyolefins.

Celebrating Soft Matter's 10th Anniversary: Topology matters: structure and dynamics of ring polymers

Considering topology among all polymer architectures polymer rings are unique, as they are the simplest closed structures without ends. In this review we present recent experimental advances addressing the structure and dynamics of rings. We focus mainly on neutron scattering results that reveal experimental insight on a molecular scale. We first briefly reflect on the progress in ring chemistry that made the experimental access possible. Structural investigations characterizing rings as compact objects in the melts are put into theoretical context. In contrast to the plateau regime common for all other high molecular weight polymer systems, the dynamic modulus of pure ring systems is characterized by a power law decay, while the viscosity displays a much weaker molecular weight dependence as a corresponding linear melt. The dynamics of ring melts is uniquely addressed by neutron spin-echo spectroscopy. The sub-diffusive center of mass motion at short times agrees well with simulation as well as theoretical concepts. In the internal dynamics the basic length scale of the ring molecule, the loop size, manifests itself clearly. The experiments reveal strong evidence for loop motions and call for further theoretical work describing them. Finally, small fractions of ring molecules in linear melts turn out to be very sensitive probes in order to scrutinize the dynamics of the host with the potential to reveal fundamental aspects of the dynamics of branched polymer systems.

Chain end mobilities in polymer melts--a computational study

The Rouse model can be regarded as the standard model to describe the dynamics of a short polymer chain under melt conditions. In this contribution, we explicitly check one of the fundamental assumptions of this model, namely, that of a uniform friction coefficient for all monomers, on the basis of MD simulation data of a poly(ethylene oxide) (PEO) melt. This question immediately arises from the fact that in a real polymer melt, the terminal monomers have on average more intermolecular neighbors than the central monomers, and one would expect that exactly these details affect the precise value of the friction coefficient. The mobilities are determined by our recently developed statistical method, which provides detailed insights into the local polymer dynamics. Moreover, it yields complementary information to that obtained from the mean square displacement (MSD) or the Rouse mode analysis. It turns out that the Rouse assumption of a uniform mobility is fulfilled to a good approximation for the PEO melt. However, a more detailed analysis reveals that the underlying microscopic dynamics are highly affected by different contributions from intra- and intermolecular excluded volume interactions, which cannot be taken into account by a modified friction coefficient. Minor deviations occur only for the terminal monomers on larger time scales, which can be attributed to the presence of two different escape mechanisms from their first coordination sphere. These effects remain elusive when studying the dynamics with the MSD only.

Rouse Mode Analysis of Chain Relaxation in Homopolymer Melts

We use molecular dynamics simulations of the Kremer–Grest (KG) bead–spring model of polymer chains of length between 10 and 500, and a closely related analogue that allows for chain crossing, to clearly delineate the effects of entanglements on the length-scale-dependent chain relaxation in polymer melts. We analyze the resulting trajectories using the Rouse modes of the chains and find that entanglements strongly affect these modes. The relaxation rates of the chains show two limiting effective monomeric frictions, with the local modes experiencing much lower effective friction than the longer modes. The monomeric relaxation rates of longer modes vary approximately inversely with chain length due to kinetic confinement effects. The time-dependent relaxation of Rouse modes has a stretched exponential character with a minimum of stretching exponent in the vicinity of the entanglement chain length. None of these trends are found in models that allow for chain crossing. These facts, in combination, argue for the confined motion of chains for time scales between the entanglement time and their ultimate free diffusion.

Compact structure and non-Gaussian dynamics of ring polymer melts

We present a neutron scattering analysis of the structure and dynamics of PEO polymer rings with a molecular weight 2.5 times higher than the entanglement mass. The melt structure was found to be more compact than a Gaussian model would suggest. With increasing time the center of mass (c.o.m.) diffusion undergoes a transition from sub-diffusive to diffusive behavior. The transition time agrees well with the decorrelation time predicted by a mode coupling approach. As a novel feature well pronounced non-Gaussian behavior of the c.o.m. diffusion was found that shows surprising analogies to the cage effect known from glassy systems. Finally, the longest wavelength Rouse modes are suppressed possibly as a consequence of an onset of lattice animal features as hypothesized in theoretical approaches.

Localization of chain dynamics in entangled polymer melts

The dynamics of polymer melts in both the unentangled and entangled regimes is described by a Langevin equation for the correlated motion of a group of chains, interacting through both intra- and inter-molecular potentials. Entanglements are represented by an intermolecular monomer-monomer confining potential that has no effect on short chains, while interpolymer interactions, responsible for correlated motion and subdiffusive center-of-mass dynamics, are represented by an intermolecular center-of-mass potential derived from the Ornstein-Zernike equation. This potential ensures that the liquid of phantom chains reproduces the compressibility and free energy of the real samples. For polyethylene melts the calculated dynamic structure factor is found to be in quantitative agreement with neutron spin echo experiments of polyethylene melts with chain lengths that span both the unentangled and the entangled regimes. The theory shows a progressive localization of the cooperative chain dynamics at the crossover from the unentangled to the entangled regime, in the spirit of the reptation model.

Hydrodynamic and viscoelastic effects in polymer diffusion

We develop a fluctuating hydrodynamics approach to study the impact of the hydrodynamic and viscoelastic interactions on the motion of the center of mass of a polymer as well as on the relaxation of Rouse modes, either in a Θ solvent or in a melt of identical unentangled chains. We show that this method allows us to describe the effect of hydrodynamic interactions beyond the Zimm (for a single chain in a Θ solvent) or the Rouse models (for an unentangled melt). In the latter case, we recover the same important effect of the viscoelastic hydrodynamic interactions on the center-of-mass diffusion, first described in Farago et al (2011 Phys. Rev. Lett. 107 178301).

Mode-coupling approach to polymer diffusion in an unentangled melt. I. The effect of density fluctuations

We quantitatively assess the effect of density fluctuation modes on the dynamics of a tagged polymer in an unentangled melt. To this end, we develop a density-based mode-coupling theory (dMCT) using the Mori-Zwanzig approach and projecting the fluctuating force onto pair-density fluctuation modes. The effect of dynamical density fluctuations on the center-of-mass (c.m.) dynamics is also analyzed based on a perturbative approach and we show that dMCT and perturbation techniques yield identical results. The c.m. velocity autocorrelation function (c.m. VAF) exhibits a slow power law relaxation in the time range between the monomer time t_{1} and the Rouse relaxation time t_{N}. We obtain an analytical expression for the c.m. VAF in terms of molecular parameters. In particular, the c.m. VAF scales as -N^{-1}t^{-5/4} (where N is the number of monomer units per chain) in the relevant time regime. The results are qualitatively accounted for by the dynamical correlation hole effect. The predicted -t^{-5/4} dependence of the c.m. VAF is supported by data of non-momentum-conserving computer simulations. However, the comparison shows that the theory significantly underestimates the amplitude of the effect. This issue is discussed and an alternative approach is addressed in the second part of this series [Farago et al., Phys. Rev. E 85, 051807 (2012), the following paper].

Interchain coupled chain dynamics of poly(ethylene oxide) in blends with poly(methyl methacrylate): coupling model analysis

Quasielastic neutron scattering and molecular dynamics simulation data from poly(ethylene oxide) (PEO)/poly(methyl methacrylate) (PMMA) blends found that for short times the self-dynamics of PEO chain follows the Rouse model, but at longer times past t(c) = 1-2 ns it becomes slower and departs from the Rouse model in dependences on time, momentum transfer, and temperature. To explain the anomalies, others had proposed the random Rouse model (RRM) in which each monomer has different mobility taken from a broad log-normal distribution. Despite the success of the RRM, Diddens et al. [Eur. Phys. Lett. 95, 56003 (2011)] extracted the distribution of friction coefficients from the MD simulations of a PEO/PMMA blend and found that the distribution is much narrower than expected from the RRM. We propose a simpler alternative explanation of the data by utilizing alone the observed crossover of PEO chain dynamics at t(c). The present problem is just a special case of a general property of relaxation in interacting systems, which is the crossover from independent relaxation to coupled many-body relaxation at some t(c) determined by the interaction potential and intermolecular coupling/constraints. The generality is brought out vividly by pointing out that the crossover also had been observed by neutron scattering from entangled chains relaxation in monodisperse homopolymers, and from the segmental α-relaxation of PEO in blends with PMMA. The properties of all the relaxation processes in connection with the crossover are similar, despite the length scales of the relaxation in these systems are widely different.

First-principle approach to rescale the dynamics of simulated coarse-grained macromolecular liquids

We present a detailed derivation and testing of our approach to rescale the dynamics of mesoscale simulations of coarse-grained polymer melts (I. Y. Lyubimov, J. McCarty, A. Clark, and M. G. Guenza, J. Chem. Phys. 132, 224903 (2010)). Starting from the first-principle Liouville equation and applying the Mori-Zwanzig projection operator technique, we derive the generalized Langevin equations (GLEs) for the coarse-grained representations of the liquid. The chosen slow variables in the projection operators define the length scale of coarse graining. Each polymer is represented at two levels of coarse graining: monomeric as a bead-and-spring model and molecular as a soft colloid. In the long-time regime where the center-of-mass follows Brownian motion and the internal dynamics is completely relaxed, the two descriptions must be equivalent. By enforcing this formal relation we derive from the GLEs the analytical rescaling factors to be applied to dynamical data in the coarse-grained representation to recover the monomeric description. Change in entropy and change in friction are the two corrections to be accounted for to compensate the effects of coarse graining on the polymer dynamics. The solution of the memory functions in the coarse-grained representations provides the dynamical rescaling of the friction coefficient. The calculation of the internal degrees of freedom provides the correction of the change in entropy due to coarse graining. The resulting rescaling formalism is a function of the coarse-grained model and thermodynamic parameters of the system simulated. The rescaled dynamics obtained from mesoscale simulations of polyethylene, represented as soft-colloidal particles, by applying our rescaling approach shows a good agreement with data of translational diffusion measured experimentally and from simulations. The proposed method is used to predict self-diffusion coefficients of new polyethylene samples.

Length scales and dynamics in the reorientational relaxation of tracers in molecular and polymeric glass formers via electron spin resonance spectroscopy

It is widely accepted that a temperature region exists above the glass transition temperature, playing a fundamental role in the physics of polymers and glass formers. In this region, several dynamic crossovers have been experimentally and numerically revealed in the past years and the onset of glassy behavior is generally located, because of cooperative and heterogeneous features exhibited by the dynamics. In this Article, the rotational dynamics of two different stiff molecular spin probes dissolved in poly(propylene glycol) has been investigated by electron spin resonance spectroscopy in a wide temperature range. Decoupling phenomena between macroscopic and microscopic transport properties have been observed in the crossover region. A comparison with previous studies carried out with the same tracers dissolved in different polymers and glass formers strongly supports the idea that the observed crossover signals the onset of spatial correlations of dynamics on the length scale probed by the tracers.

The thermodynamics of polymer mixing and compression in the semidilute regime

Polymers present a fascinating range of mechanical properties in the extension, compression, and flow of soft condensed matter. Entropy determines the physics of the elastic material behavior of polymeric systems in deformation. Classical models of polymer dynamics have assumed chain entanglement and resulting reptation in concentrated polymer solutions. Here, we present a thermodynamic treatment of interacting chains in solution with increasing concentration from the dilute to concentrated regimes. As the polymer chain concentration increases above the critical overlap, the chains must either compress or overlap and entangle, resulting in a decrease in chain configurational entropy. The free energy of chain entanglement is shown to be less favored than compression at concentrations above the critical overlap. Elastic forces act on the chains to reduce the dimensions to the ideal random walk size with increasing concentration. At significantly higher concentrations, the free energies reach an asymptote where chain compression and entanglement are simultaneously possible. Entanglement and reptation are shown to be statistically improbable in the semidilute regime, and it is concluded that the compression of polymer chains is favored at semidilute concentrations.

Combined molecular algorithms for the generation, equilibration and topological analysis of entangled polymers: methodology and performance

We review the methodology, algorithmic implementation and performance characteristics of a hierarchical modeling scheme for the generation, equilibration and topological analysis of polymer systems at various levels of molecular description: from atomistic polyethylene samples to random packings of freely-jointed chains of tangent hard spheres of uniform size. Our analysis focuses on hitherto less discussed algorithmic details of the implementation of both, the Monte Carlo (MC) procedure for the system generation and equilibration, and a postprocessing step, where we identify the underlying topological structure of the simulated systems in the form of primitive paths. In order to demonstrate our arguments, we study how molecular length and packing density (volume fraction) affect the performance of the MC scheme built around chain-connectivity altering moves. In parallel, we quantify the effect of finite system size, of polydispersity, and of the definition of the number of entanglements (and related entanglement molecular weight) on the results about the primitive path network. Along these lines we approve main concepts which had been previously proposed in the literature.