Jörg Baschnagel

Bridging the gap between atomistic and coarse-grained models of polymers : Status and perspectives

Recent developments that increase the time and distance scales accessible in the simulations of specific polymers are reviewed. Several different techniques are similar in that they replace a model expressed in fully atomistic detail with a coarse-grained model of the same polymer, atomistic → coarse-grained (and beyond!), thereby increasing the time and distance scales accessible within the expenditure of reasonable computational resources. The bridge represented by the right-pointing arrow can be constructed via different procedures, which are reviewed here. The review also considers the status of methods which reverse this arrow, atomistic ← coarse-grained. This “reverse-mapping” recovers a model expressed in fully atomistic detail from an arbitrarily chosen replica generated during the simulation of the coarse-grained system. Taken in conjunction with the efficiency of the simulation when the system is in its coarse-grained representation, the overall process Open image in new window permits a much more complete equilibration of the system (larger effective size of Δt) when that equilibration is performed with the coarse-grained replicas (II → III) than if it were attempted with the fully atomistic replicas (I → IV).

Growing range of correlated motion in a polymer melt on cooling towards the glass transition

Many liquids cooled to low temperatures form glasses (amorphous solids) instead of crystals. As the glass transition is approached, molecules become localized and relaxation times increase by many orders of magnitude. Many features of this ‘slowing down’ are reasonably well described by the mode-coupling theory of supercooled liquids. The ideal form of this theory predicts a dynamical critical temperature T c at which the molecules become permanently trapped in the ‘cage’ formed by their neighbours, and vitrification occurs. Although there is no sharp transition, because molecules do eventually escape their cage, its signature can still be observed in real and simulated liquids. Unlike conventional critical phenomena (such as the behaviour at the liquid–gas critical point), the mode-coupling transition is not accompanied by a diverging static correlation length. But simulation and experiment, show that liquids are dynamically heterogeneous, suggesting the possibility of a relevant ‘dynamical’ length scale characterizing the glass transition. Here we use computer simulations to investigate a melt of short, unentangled polymer chains over a range of temperatures for which the mode-coupling theory remains valid. We find that although density fluctuations remain short-ranged, spatial correlations between monomer displacements become long-ranged as T c is approached on cooling. In this way, we identify a growing dynamical correlation length, and a corresponding order parameter, associated with the glass transition. This finding suggests a possible connection between well established concepts in critical phenomena and the dynamics of glass-forming liquids.

Glass transition of polymer melts: test of theoretical concepts by computer simulation

Abstract Polymers are good glass formers and allow for the study of melts near the glass transition in (meta-)stable equilibrium. Theories of the glass transition imply such an equilibrium and can, hence, be tested by the study of polymer melts. After a brief summary of the basic experimental facts about the glass transition in polymers, the main theoretical concepts are reviewed: mode coupling theory (MCT), entropy theory, free-volume theory, the idea of a growing length describing the size of cooperative regions, etc. Then, two basic coarse-grained models of polymers are described, which have been developed aiming at a test of these concepts. The first model is the bond-fluctuation model on the simple cubic lattice; the second is a bead-spring model in the continuum. While the first model is studied by kinetic Monte Carlo methods, the second is studied by molecular dynamics simulation: the issue is addressed which aspects of the results are model-dependent and which aspects are generic and should universally apply, including real materials. Attempts to include chemical detail in order to describe real materials are discussed, too. It is shown that idealized MCT is a good description for the onset of slow relaxation (‘cage effect’) in the discussed model polymer melts, but the singularities at the critical temperature of MCT are rounded, and the correct description of the dynamics close to the calorimetric glass transition remains controversial. While the configurational entropy of the polymer melt does decrease strongly when the glass transition is approached, evidence is presented that the ‘entropy catastrophe’ of the theory of Gibbs and Di Marzio is an artifact of inaccurate approximations. Simulations of polymer melts confined in a thin film geometry are also reviewed, emphasizing the question to which extent they shed light on the issue of a growing glass correlation length. Finally, we discuss the implications of the glass transition on the motion of polymers at larger length scales. It is shown that for short (non-entangled) chains the Rouse model stays essentially valid, the friction coefficient reflecting the slowing down as described by the α-relaxation, while for small-scale motions (e.g. β-relaxation regime of MCT) the connectivity of the polymer is not very relevant. We conclude that the theories describe some aspects of vitrification phenomena correctly, but a unified description that puts all these phenomena into one coherent framework, does not yet exist.

Computer simulations of supercooled polymer melts in the bulk and in confined geometry

We survey results of computer simulations for the structure and dynamics of supercooled polymer melts and films. Our survey is mainly concerned with features of a coarse grained polymer model?a bead?spring model?in the temperature regime above the critical glass temperature Tc of the ideal mode-coupling theory (MCT). We divide our discussion into two parts: a part devoted to bulk properties and a part dealing with thin films. The discussion of the bulk properties focuses on two aspects: a comparison of the simulation results with MCT and an analysis of dynamic heterogeneities. We explain in detail how the analyses are performed and what results may be obtained, and we critically assess their strengths and weaknesses. In discussing the application of MCT we also present first results of a quantitative comparison which does not rely on fits, but exploits static input from the simulation to predict the relaxation dynamics. The second part of this review is devoted to extensions of the simulations from the bulk to thin films. We explore in detail the influence of the boundary condition, imposed by smooth or rough walls, on the structure and dynamics of the polymer melt. Geometric confinement is found to shift the glass transition temperature Tg (or Tc in our case) relative to the bulk. We compare our and other simulation results for the Tg shift with experimental data, briefly survey some theoretical ideas for explaining these shifts and discuss related simulation work on the glass transition of confined liquids. Finally, we also present some technical details of how to perform fits to MCT and give a brief introduction to another approach to the glass transition based on the potential energy landscape of a liquid.

Polymer-specific effects of bulk relaxation and stringlike correlated motion in the dynamics of a supercooled polymer melt

We analyze dynamical heterogeneities in a simulated “bead-spring” model of a nonentangled, supercooled polymer melt. We explore the importance of chain connectivity on the spatially heterogeneous motion of the monomers. We find that when monomers move, they tend to follow each other in one-dimensional paths, forming strings as previously reported in atomic liquids and colloidal suspensions. The mean string length is largest at a time close to the peak time of the mean cluster size of mobile monomers. This maximum string length increases, roughly in an exponential fashion, on cooling toward the critical temperature TMCT of the mode-coupling theory, but generally remains small, although large strings involving ten or more monomers are observed. An important contribution to this replacement comes from directly bonded neighbors in the chain. However, mobility is not concentrated along the backbone of the chains. Thus, a relaxation mechanism in which neighboring mobile monomers along the chain move predominant...

Molecular-dynamics simulation of a glassy polymer melt: Rouse model and cage effect

Abstract We report results of molecular-dynamics simulations for a glassy polymer melt consisting of short, linear bead-spring chains. The model does not crystallize upon cooling, but exhibits a glassy slowing down. The onset of this slowing down is brought about by the dense packing in the melt. It was shown in an earlier work that this onset is compatible with the predictions of the mode coupling theory of the glass transition. The physical process of “caging” of a monomer by its spatial neighbors leads to a distinct two step behavior in scattering functions and particle mean square displacements. In this work, we analyze the effects of this caging process on the Rouse description of polymer melt dynamics. The Rouse model is known, both from experimental and simulational work, to be a reasonable description of the dynamics of short chains in the melt. We show that the Rouse description is applicable for length and time scales above the typical scales for the caging process, and that the typical time scale of the Rouse model reflects the onset of freezing as described by mode coupling theory. The Rouse modes are eigenmodes of the chains in the supercooled state, and the relaxation times of the modes exhibit the same temperature dependence as the diffusion coefficient of the chains. The decay of the mode correlation functions is stretched and depends on the mode index. Therefore, there is no time-mode superposition of the correlation functions. However, they exhibit a time–temperature superposition at late times. At intermediate times, they decay in two steps for temperatures close to the dynamical critical temperature of mode coupling theory. The monomer displacement is compared with simulation results for a binary LJ-mixture to illustrate the differences which are introduced by the connectivity of the particles.

MODELING POLYETHYLENE WITH THE BOND FLUCTUATION MODEL

This work presents an application of recently developed ideas about how to map real polymer systems onto abstract models. In our case the abstract model is the bond fluctuation model with a Monte Carlo dynamics. We study the temperature dependence of chain dimensions and of the self-diffusion behavior in the melt from high temperatures down to 200 K. The chain conformations are equilibrated over the whole temperature range, which is possible for the abstract type of model we use. The size of the chains as measured by the characteristic ratio is within 25% of experimental data. The simulated values of the chain self-diffusion coefficient have to be matched to experimental information at one temperature to obtain a scaling for the Monte Carlo time step. The melt viscosity from the simulations as determined by applying the Rouse model is then in good qualitative agreement with experimental data over the experimentally available temperature range. The activation energy as extracted from an Arrhenius fit is different because the simulations are done at constant volume. Both experimental data and the simulation, which covers a far greater temperature range, show Arrhenius behavior for the viscosity and no indication of a finite nonzero Vogel–Fulcher temperature. For one temperature (T=509 K) various time-dependent mean-square displacements are available from atomistic molecular-dynamics simulations, and are shown to be in excellent agreement with the results from the coarse-grained model.

Adsorption Transition of a Polymer Chain at a Weakly Attractive Surface: Monte Carlo Simulation of Off‐Lattice Models

A bead-spring model of a polymer chain with one end attached to a wall is studied by Monte Carlo simulations for chain lengths 16 ≤ N ≤ 256. Two types of adsorption potentials, 9-3 and 10-4 Lennard-Jones (LJ) potentials, between the effective monomers and the wall are assumed. For both cases the adsorption transition where the chain changes its asymptotic statistical properties from a three-dimensional to a two-dimensional configuration is located using a scaling analysis. It is shown that the crossover exponent φ = 0.50 ± 0.02 is the same for both LJ potentials. This value is compatible with recent theoretical predictions and simulation results for lattice models with short-range wall potentials. The results of our study support the expectation that the exponents describing the adsorption transition are universal, i.e., they are not influenced by the precise form and the long-range character of the adsorption potentials used. The technical aspects of the simulations (which use configurational bias methods as well as histogram re-weighting) are also carefully discussed.

Molecular-dynamics simulation of a glassy polymer melt: Incoherent scattering function

Abstract:We present simulation results for a model polymer melt, consisting of short, nonentangled chains, in the supercooled state. The analysis focuses on the monomer dynamics, which is monitored by the incoherent intermediate scattering function. The scattering function is recorded over six decades in time and for many different wave-vectors which range from the size of a chain to about three times the maximum position of the static structure factor. The lowest temperatures studied are slightly above Tc, the critical temperature of mode-coupling theory (MCT), where Tc was determined from a quantitative analysis of the β- and α-relaxations. We find evidence for the space-time factorization theorem in the β-relaxation regime, and for the time-temperature superposition principle in the α-regime, if the temperature is not too close to Tc. The wave-vector (q-) dependence of the nonergodicity parameter, of the critical amplitude, and the α-relaxation time are in qualitative agreement with calculations for hard spheres. For q larger than the maximum of the structure factor the α-relaxation time τq already agrees fairly well with the asymptotic MCT-prediction τq ∼ q -1/b. The behavior of the relaxation time at small q can be rationalized by the validity of the Gaussian approximation and the value of the Kohlrausch stretching exponent, as suggested in neutron-scattering experiments.

Investigating the influence of different thermodynamic paths on the structural relaxation in a glass-forming polymer melt

We present results from molecular dynamics simulations of the thermal glass transition in a dense polymer melt. In previous work we compared the simulation data with the idealized version of mode-coupling theory (MCT) and found that the theory provides a good description of the dynamics above the dynamical critical temperature. In order to investigate the influence of different thermodynamic paths on the structural relaxation (α-process), we performed simulations for three different pressures and are thus able to give a sketch of the critical line of MCT in the pressure-temperature plane, where, according to the idealized version of MCT, an ergodic-nonergodic transition should occur. Furthermore, by cooling our system along two different paths (an isobar and an isochor), with the same intersection point of the critical line, we demonstrate that neither the dynamical critical temperature nor the exponent γ depend on which path is chosen.