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Translational diffusion in Monte Carlo simulations of polymer melts: Center of mass displacement vs. integrated velocity autocorrelation function
Abstract
Translational diffusion has been simulated in monodisperse melts of four linear alkanes, C2xH4x+2, x=6,30,50,158, and two cyclic alkanes, C2xH4x, x=30,50, at 473 K. The alkanes are expressed in a coarse-grained representation using x beads on a high coordination lattice, one bead for every two carbon atoms. Short-range intramolecular interactions are controlled by an adaptation of the rotational isomeric state model for unperturbed polyethylene, and the long-range interactions are controlled by a step-wise three-shell potential energy function derived from a continuous Lennard-Jones potential energy function. Acceptance of trial moves, each of which changes the coordinates of a single bead only, is governed by the Metropolis rule. Translational diffusion coefficients, D, are estimated from the mean square displacement of the center of mass and the integral of the velocity autocorrelation function. Both approaches yield the same value for D, which demonstrates that the velocity has been defined in a reasonable manner in the Monte Carlo simulation. A method is proposed for the estimation of D when the trajectory is not quite long enough to have achieved the behavior characteristic of the limit as time approaches infinity.
Many physicochemical processes depend on the diffusion of small molecules through solid materials. While crystallinity in polymers is advantageous with respect to structure performance, diffusion in such materials is difficult to predict. Here, we investigate the impact of crystal morphology and organization on the diffusion of small molecules using a lattice Monte Carlo approach. Interestingly, diffusion determined with this model does not depend on the internal morphology of the semi-crystalline regions. The obtained insight is highly valuable for developing predictive models for all processes in semi-crystalline polymers involving mass transport, like polymer degradation or drug release, and provide design criteria for the time-dependent functional behavior of multifunctional polymer systems.
Modeling the mechanical behavior of polymers and polymer-based materials is notoriously difficult, primarily due to the need to integrate physics taking place on multiple scales. In this chapter we review single-scale models used to represent polymers and their composites, including atomistic, coarse-grained, and continuum models. Each of these has limitations associated with either accuracy or efficiency. To combine their advantages while reducing the associated drawbacks, multiscale methods are desirable. Two strategies are presented, both belonging to the class of information-passing methods. In the first, the physics of (dielectric and stress) relaxation is studied on the molecular scale and relevant parameters are calibrated using single-scale molecular dynamics and Monte Carlo techniques. These parameters are then incorporated in constitutive laws whose functional form is physically motivated. Such constitutive laws can then be used in continuum models on larger scales. This strategy is exemplified for a class of model polymer nanocomposites. In the second approach, a system reduction technology is developed to coarse grain the structure and dynamics of atomistic models of dense polymers. The coarse-grained models are calibrated using an equilibrium fine-scale model of a monodisperse system and then are used to predict the behavior of other systems in equilibrium and nonequilibrium.
Polymer nanocomposites have been a central theme in nanotechnology over the last decade. This activity was determined primarily by expectations generated by early experiments showing significant property enhancement associated with the addition of a small fraction of nano-fillers in thermoplastics. A large number of experimental, numerical and theoretical investigations of the physical basis of these enhancements followed. Today it is generally accepted that, while the fillers do not modify chemically the matrix material, the mechanical properties of the polymer are nevertheless altered. This chapter reviews some of the numerical and experimental works aimed at developing this basic understanding. We focus on modifications of the chain structure and relaxation time in the vicinity of impenetrable fillers, the residence time of polymeric chains on filler surfaces in melts, and the formation of a network of chains connecting fillers. This molecular-scale physics has been incorporated in rheological models aimed at predicting the macroscopic response of the composite. One such model is presented here. Despite not reviewing the entire, large literature produced to date on this subject, the discussion provides insight into the essential physics of these systems, and a starting point for future investigations.
The structure and dynamics of linear, monodisperse polyethylene (PE) melts (C160H322 and C440H882) containing homogenously distributed spherical nanoparticles were investigated. The PE chains were simulated using a coarse grained model and a Monte Carlo algorithm. Two variables were considered: (i) the wall-to-wall distance between particles (d), and (ii) the interaction energy between monomers and particles. The various chain structures changed greatly with d while the monomer-particle interaction had little effect. The average size, shape, and orientation of PE chains did not differ significantly from those of a neat melt. Bridge segments were more stretched relative to segments in the neat melt and the stretch increased with increasing d. However, the number of bridge segments decreased markedly with increasing d. Chain dynamics was monitored by computing the Rouse relaxation modes and the mean-square displacement of the center of mass. The dynamics were slowed by both geometric effects (confinement by fillers) and energetic effects (monomer-particle energetic interaction).
We measure the forces confining the displacement of a single DNA molecule embedded within a concentrated solution of long relaxed circular DNA molecules (115 kbp at 1 mg/mL) using optical tweezers. We compare these measurements with our previous measurements of forces imposed by entangled linear DNA molecules of the same length and concentration. A tube-like confining field and three relaxation modes were observed, but the tube radius was 25% smaller (0.6 μm) and the longest relaxation time 3 times shorter (11 s) than with linear molecules, consistent with recent theoretical predictions by Iyer, Lele, and Juvekar. For displacements greater than the tube radius, the confining force imposed by circular polymers was substantially lower and shorter range than that measured with linear polymers.
Self-diffusion coefficients (D) of DNA molecules of varying length and concentration were measured by tracking the Brownian motion of individual fluorescently labeled tracer molecules. Four possible cases were examined: linear tracer molecules surrounded by linear molecules (L−L), circular tracers surrounded by linears (C−L), linear tracers surrounded by circles (L−C), and circles surrounded by circles (C−C). With 6 and 11 kilobasepair (kbp) DNA D was largely insensitive to topology and varied consistent with Rouse scaling (D L-1C-0.5). In contrast, with 25 and 45 kbp DNA topology had a strong influence. At 1 mg/mL we found DC-C > DL-C DL-L DC-L. In the L−L, L−C, and C−C cases a crossover from scaling consistent with the Rouse model to scaling consistent with the reptation model (D L-2C-1.75) was observed at 6 times the molecular overlap concentration. In contrast, DC-L decreased much more steeply with concentration, indicating that a process much slower than reptation governs that case.
Dynamics of attachment/detachment processes of chains to/from spherical fillers in a polymer nanocomposite is investigated by means of numerical simulations using a chemistry-specific model for the polymer. The effects of polymer-particle interaction, chain length, chain-to-filler distance, and filler radius on the attachment/ detachment processes are studied. It is found that the time a chain is in contact with a filler scales with the number of attached beads following a power law. Increasing the energetic interaction parameter between polymer and filler slows the detachment process, and the system average characteristic detachment time follows an Arrhenius law in which the activation energy is proportional to the polymer-particle energetic interaction parameter.
Models for nanofibers of amorphous polymers can be constructed by manipulation of the periodic boundary conditions commonly
employed in the simulations of polymer melts. After a strong extension of the periodicity in one direction, equilibration
causes the model to form a free-standing thin film. Repetition of this process with the boundary in a different direction
converts the free-standing thin film into a thin fiber, exposed to a vacuum. For a successful construction of the model for
the fiber, the system must be sufficiently robust, in terms of the degree of polymerization and number of the independent
parent chains. The length of the periodic box along the fiber axis can exceed the length of a fully extended parent chain
if a sufficiently large number of independent parent chains is employed in the simulation. The correlation length, ξ, in the
fit of the radial density profile to a hyperbolic tangent function is a useful indicator of the integrity of the model for
the thin fiber.
The crossover to Rouse-type behavior for the self-diffusion constant D, the viscosity η, and the equilibrium structural statistics of n-alkanes (6 ⩽ n ⩽ 66) is studied numerically. For small n the chains are non-Gaussian and the mean squared end-to-end distance 〈R2〉 is greater than 6〈RG2〉, where 〈RG2〉 is the mean squared radius of gyration. As n increases, 〈R2〉/〈RG2〉→6(1+a/n), where a depends on the interaction model. At constant density, the Rouse model is used to extract the monomeric friction coefficient ζ and the viscosity η independently from the diffusion constant D and the longest relaxation time τR. ζD extracted from D is nearly independent of chain length while ζτ obtained from τR is much larger than ζD for small n. The viscosity measured in a nonequilibrium molecular dynamics simulation is closely approximated by the value of η determined from τR while η inferred from D is smaller for small n. For n≳60, the two estimates for both ζ and η agree as predicted from the Rouse model. D calculated from three interaction models is studied for increasing n and compared to experimental data. © 1998 American Institute of Physics.
Cyclic poly(dimethylsiloxanes) (PDMS) (CH3SiO)x
have been prepared1,2 in significant quantities up to molar masses of about 3 × 104 g mol−1. Both cyclic and linear PDMS with molar masses up to this value have freezing points in the range − 36 to − 86 °C.3 Hence, the physical properties which they exhibit at ambient temperatures are those characteristic of polymer melts. In addition, since the molar mass above which transient molecular entanglements become significant is around 2·4 × 104g mol−1 for PDMS melts,4 the physical properties of both cyclic and linear PDMS are determined largely by the configurational statistics of single molecules.
A three centre Lennard-Jones model of CS2 predicts time dependent properties of the liquid, along the orthobaric curve, in good agreement with experiment. The molecules diffuse more easily parallel to their original orientation. The angular velocity autocorrelation function (acf) is an excellent approximation to the first memory function of the reorientational acf. Simple models of the orientational motion fail because the torque and angular velocity acf's relax on similar time scales. Collective correlations of the orientational and centre of mass density fluctuations are calculated with increased accuracy at a number of small non-zero wavevectors.
New methods for coarse graining polymer chains onto lattice systems have been investigated. The mapping should take place in a rather local and exact manner to retain the specific properties of the considered polymer system. First a short overview is given of the types of lattice representations of polymer chains currently available. Secondly a method based on high-coordination lattices that has proven to be successful for mapping proteins is examined and shown to be unsuitable for interdependent bond directions. Thirdly, a new high-coordination lattice model is introduced and results of single-chain simulations are shown. These results of random walk chains, non-reversal walk chains and self-avoiding walk chains agree well with the expected theoretical values.
We present in this work results from atomistic molecular dynamics simulations of a n-C100H202 melt. This work represents a first effort to simulate a fully equilibrated ensemble of chains of sufficient length to follow Gaussian chains statistics and hence Rouse-like chain dynamics, employing well-validated, realistic potentials. In order to allow full equilibration the simulations were performed at experimental densities at high temperatures (≥450 K). Here we report on the static and dynamic properties of the melt obtained from two models, a united atom and an explicit atom model, and compare these results with experiment. These comparisons allow for a quantitative evaluation of the models and provide insight into the modeled system and the influence of the level of atomistic detail considered.
In support of our numerical simulations of self-diffusion D of chain molecules in the melt, we have used the pulsed-gradient spin-echo NMR method to make detailed D measurements in 15 liquid n-paraffins (8–60 carbons; molecular weight M = 114–844) at eight temperatures T from 30 to 170 °C. We find that D ∼ Mβ, with β changing approximately linearly from −2.72 to −1.85 as T increases. Thus the apparent activation energies also rises linearly with log M. In the absence of molecular entanglements, Rouse kinetics predicts β = −1, but Cohen–Turnbull–Bueche free-volume effects due to molecular chain ends add a further nonpower-law term, enhancing D increasingly at low M. The combined D vs M theory does, however, closely mimic a power law with the exponents observed at all our measurement temperatures. To obtain this result it is necessary to include in the free-volume term the dependence of the liquid’s density on M and T, analytically modeled based on the incomplete literature data. The fitted density model itself is useful as a guide for the molecular modeling effort. The success of this approach makes it possible to deduce a small intrinsic thermal activation energy contained within the larger, free-volume dominated, apparent values. © 1998 American Institute of Physics.
We have used the pulsed-gradient spin-echo proton nuclear magnetic resonance (NMR) method to measure two-component self-diffusion D in melts of two monodisperse polyethylenes, as well as in binary blends of n-dodecane and n-hexacontane, and of n-alkanes with the polyethylene (PE) specimens, over the full concentration range at various temperatures. In the n-alkane blends we observe ideal solution behavior at all concentrations, while in the alkane-PE blends the D ratio increases with rising PE content and molecular weight. This behavior is reproduced by standard free-volume theory including chain-end effects and measured variation of density with molecular weight and temperature. In PE melts and blends the theory must include the gradual transition from Rouse to reptative motion above an entanglement molecular weight which depends on PE concentration. Constraint release effects are included, and dilute solution behavior is modeled near infinite PE dilution. All free-volume parameters are taken from our earlier study of n-alkane melts; the extensions of that theory introduce only one additional adjustable parameter. © 1999 American Institute of Physics.
A high coordination lattice was recently introduced for the simulation of coarse-grained rotational isomeric state (RIS) model chains. This second nearest neighbor diamond (2nnd) lattice is formed by connecting every other site on a tetrahedral lattice. Monte Carlo simulations of polyethylene (PE) melts have recently been performed on the 2nnd lattice by incorporating the intramolecular short range interactions from the RIS model and the long range interactions using a potential derived from an averaging procedure of the Mayer function in the expansion for the second virial coefficient. In the present work, the reverse mapping of specific snapshots from the coarse-grained PE melt back to the fully atomistic representation in continuous space is demonstrated. Reverse mapping is, essentially, determining the location of intermediate backbone atoms that are not represented on the 2nnd lattice but exist on the diamond lattice. In certain situations, the new locations of two intermediate atoms may coincide, leading to an unrealistic local conformation with infinite energy. Although these collapse situations are rare in the case of PE, they still present a problem in finding an energetically acceptable snapshot, which can be further subject to energy minimization in continuous space. These local collapse phenomena can be avoided by taking into consideration the geometry of the underlying tetrahedral lattice during coarse-grained simulations. Consequently, specific snapshots from bulk PE simulations are reverse mapped and their energy is minimized. The cohesive energy densities of these energy-minimized snapshots are very close to the experimental values reported in the literature.
A new Monte Carlo simulation on a coarse-grained tetrahedral lattice was recently developed for rotational isomeric state polyethylene chains. The short-range interaction was included by extending the classical statistical weight matrix. To describe the cohesive nature of realistic polyethylene systems, nonbonded long-range interaction was considered. The second virial coefficient B2 of two chains under an interparticle potential was suitably written in a discretized form to be analyzed for this new lattice. Utilizing the vanishing B2 of a ϑ chain was previously suggested for the estimation of interaction parameters representing several nearest neighbors. This approach was shown to be sufficient for treatment of dilute solutions, where long-range interactions are relatively infrequent. This study presents an alternative method of parameter estimation which is more appropriate for the bulk state, where long-range interactions are abundant. Interaction parameters were defined from an averaging procedure of the Mayer function in the expression of B2. In the latter method, the widely used Lennard-Jones potential and associated parameters for the monomeric unit were used to calculated the average Mayer function. The estimated interaction parameters were used to simulate the average neighbor occupancy and the nonbonded energy per monomer. The nonbonded energy showed a minimum when it was plotted as a function of density. The latter method is considered better for the estimation of long-range interaction and yielded the density of minimum energy in a physically reasonable range. The calculated cohesive energy was shown to be close to the range defined by experiment.
Recently, a new high-coordination lattice model for rotational isomeric state (RIS) chains was introduced, with the motivation of eventually providing a method for the more efficient simulation of dense, multichain systems by using RIS chains. Here that new method has been extended to include short and long range interaction energies, using polyethylene as the example. The short range energies were included via an extended RIS model. The classical statistical weight matrix was expanded in size and coarse-grained. The conditional probabilities derived from this extended RIS model were the basis for the probabilities of the single-bead moves used in Monte Carlo simulations on the 2nnd lattice. The long range energies were introduced by using the lattice representation of the second virial coefficient. The coarse-grained linear chains have the proper mean-square dimensions, and the width of the distribution function for the squared dimensions has the proper value. Coarse-grained polyethylene macrocycles also have the proper mean-square dimensions.
Employing neutron spin-echo spectroscopy we have studied the dynamic structure factors for the relaxation of a single. chain;in polymer melts. We have varied the molecular weights through the transition region from unrestricted Rouse dynamics to entanglement controlled behavior. Investigating the dependence of the dynamic structure factor on the momentum transfer and, it is possible to access the different relaxation modes separately. We found that, depending on their spatial extension in relation to the entanglement distance, larger scale relaxations are successively slowed down compared to Rouse relaxation. A comparison with macroscopic diffusion and viscosity data yields excellent internal consistency. Furthermore, we solve explicitly the generalized Rouse model by Hess(12) and compare its. predictions to our data. Fitting only two parameters, all the Q and molecular weight dependent structure factors can be well reproduced.
Polyethylene thin films are simulated on a high coordination lattice, which is formed by connecting every other site on a diamond lattice. The films present two surfaces to vacuum at 509 K. Each film contains 36 linear or cyclic chains of 50 beads on the coarse-grained lattice, where each bead represents two united-atom carbon atoms. Both linear and cyclic films exhibit bulk density and properties in the interior region, and hyperbolic density profiles at the interfaces. The mobility increases toward the interfaces on the scale of individual beads and chains due to the decrease in density. The increased mobility at the free surfaces is determined more strongly by the low density rather than the segregation of chain ends, since both films exhibit similar characteristics, and there are no chain ends in the film composed of cyclic chains.
Conformation energies of n-butane and n-pentane have been calculated as functions of their C-C bond rotation angles using semiempirical expressions for the repulsive and attractive energies between nonbonded atom pairs and an intrinsic threefold torsion potential having a barrier height of 2.8 kcal mole-1 for each C-C bond. Parameters for the nonbonded repulsions were adjusted to reproduce the experimental energy difference Eσ of 500 cal mole-1 between the gauche and the trans states. The same parameters yield barrier heights for rotations in propane, n-butane, isobutane, and neopentane which agree with experiment. The potential wells about gauche and trans minima in n-butane and n-pentane are similar in breadth. The former have been found to occur at ca. φg = ±112.5° from trans. Neighbor rotations of like sign g±g∓ in n-pentane are essentially unperturbed by one another. The energy of g±g∓ pairs of opposite sign is prohibitively large. These are superseded by pairs of shallow minima at ca. ±115, ∓80°, and ±80, ∓115°, with energies ca. 3.0 to 3.5 kcal mole-1 above it. The mean dimensions of polymethylene chains H(-CH2)n+1-H have been interpreted in light of these features of the conformation energy of first neighbor bonds in lower n-alkane homologs. The characteristic ratio (〈r2〉0/nl2)∞ and its temperature coefficient were computed for various values of the parameters involved. Experimental values of these quantities are well reproduced by taking Eσ = 400 ± 100 cal mole-1 and by assigning an energy of the order of 3.0 kcal mole-1 for the g±g∓ states cited above, in agreement with the conformation energy calculations.
A general method, suitable for fast computing machines, for investigating
such properties as equations of state for substances consisting of
interacting individual molecules is described. The method consists
of a modified Monte Carlo integration over configuration space. Results
for the two-dimensional rigid-sphere system have been obtained on
the Los Alamos MANIAC and are presented here. These results are compared
to the free volume equation of state and to a four-term virial coefficient
expansion. The Journal of Chemical Physics is copyrighted by The
American Institute of Physics.
A simple model of sphere packing has been investigated as an ideal model for long-range interactions for the packing of non-bonded residues in protein structures. By superposing all residues, the geometry of packing around a central residue is investigated. It is found that all residues conform almost perfectly to this lattice model for sphere packing when a radius of 6.5 A is used to define non-bonded (virtual) interacting residues. Side-chain positions with respect to sequential backbone segments are relatively regular as well. This lattice can readily be used in conformation simulations to reduce the conformational space.
Cyclic polymers, London: Else-vier
- H Hocker
- In
Hocker H. In: Semlyen JA, editor. Cyclic polymers, London: Else-vier, 1986. p. 205–6, chap. 6.
Introduction of short and long range R. Ozisik, P. Doruker / Computational and Theoretical Polymer Science 10 (2000) 411–418 energies to the simulation of real chains on the 2nd lattice
- Rapold Rf Mattice
- Wl
Rapold RF, Mattice WL. Introduction of short and long range R. Ozisik, P. Doruker / Computational and Theoretical Polymer Science 10 (2000) 411–418 energies to the simulation of real chains on the 2nd lattice. Macromolecules 1996;29:2457–66.