Day 2: Monday morning, September 4, 2006

Session 2 Multiscale modeling and molecular simulations
Chair: T. Tzavaras

08:00  Atomistic simulation of polymers with a non-linear molecular architecture: Calculation of branch point friction and chain reptation time of an H-shaped polyethylene melt
N.Ch. Karayiannis^1,2, V.G. Mavrantzas^1,2
1 Department of Chemical Engineering, University of Patras, GR 26504, Greece
² Institute of Chemical Engineering and High-Temperature Chemical Processes (FORTH-ICE/HT), Patras GR 26504, Greece
A hierarchical simulation strategy is presented for simulating structure and dynamics in polymers characterized by a non-linear molecular architecture, such as the H-shaped macromolecules [1]. First, a novel Monte Carlo (MC) algorithm is employed to generate well equilibrated atomistic configurations of these highly non-linear chain structures. The new algorithm is built around state-of-the-art chain connectivity altering moves, like the end-bridging and double-bridging, and is many orders of magnitude more efficient than Molecular Dynamics (MD) in equilibrating these systems, even if multi-step time integration schemes are incorporated in the MD algorithm. In a second step, the equilibrated structures generated by the new MC algorithm are used as initial configurations in detailed NPT MD simulations of H-shaped polyethylene (PE) melts for very long times, on the order of microseconds. In our simulations, the average number of carbon atoms in the backbone has ranged from 48 up to 300 corresponding to both unentangled and entangled crossbars, while the average branch length was kept relatively small (it ranged from 24 up to 50 carbon atoms) corresponding always to unentangled arms. The MD simulation results provide convincing evidence for the different relaxation mechanisms exhibited by an H-polymer melt: the fast relaxation due solely to arm breathing (on the order of a few ns, for an H_300_50 melt), and the slow branch point diffusion which is accompanied by a sluggish backbone diffusion due to reptation (on the order of a few μs, for an H_300_50 melt). They have further demonstrated that the center-of-mass diffusivity in an H-polymer follows faithfully that of branch points, thus validating from first-principles the main assumption of the McLeish-Larson pom-pom theory that all friction in an H-molecule is concentrated at the branch points. For the longest H-polymers studied, logarithmic plots of the msd of the inner crossbar segments against time were seen to exhibit the four different regimes predicted by the reptation theory of Doi-Edwards for entangled linear polymer melts, with corresponding exponents remarkably close to those of the theory. This allowed us to extract the characteristic relaxation times τe, τR, and τd for each one of the simulated systems and their effective tube diameter.

[1] N.Ch. Karayiannis and V.G. Mavrantzas, Macromolecules 38, 8583 (2005).

08:25  Entanglements and Underlying Topology in Polymer Melts: from Atomistic Models to Entanglement Networks
C. Tzoumanekas^1,2, D.N. Theodorou^1
1 School of Chemical Engineering, National Technical University of Athens, 9 Heroon Polytechniou Street, Zografou Campus, Athens, 15780, Greece
² Dutch Polymer Institute (DPI), The Netherlands
Polymer melts of large molecular weight demonstrate complex rheological and dynamical properties which are very interesting both from a theoretical and a technological point of view. A successful conceptual framework for investigating this complex behavior at the molecular level is offered by the tube model. The tube model is based on the notion that the mutual uncrossability of polymer chains generates topological constraints which are generally called entanglements. Entanglements affect chain motion by effectively restricting individual chain conformations in a curvilinear tube-like region enclosing each chain. Accordingly, chain motion is confined laterally to the length scale of the tube diameter. Large-scale motion is promoted via de Gennes reptation, an effective one-dimensional diffusion of a chain along its tube axis which is called the Primitive Path (PP). PPs characterize the system topology by creating a large scale topological substructure, which is conceived as an entanglement network underlying the melt structure. We introduce a novel algorithm, referred to as CReTA (Contour Reduction Topological Analysis), which is capable of reducing a computer generated atomistic sample to a corresponding entanglement network of PPs. A topological analysis, based on scaled PP and entanglement network statistics of thermodynamically equilibrated Polyethylene, cis-1,4 Polybutadiene and Poly(ethylene terephthalate) (PET) melts, leads to a unifying microscopic description of the topology of flexible polymers. The distribution of the number of monomers between successive topological constraints along a chain reveals a non-uniform network mesh, while a dilute gas of entanglements is revealed by the network radial distribution functions. By mapping PP conformations to random walks, we predict an entanglement molecular weight and a tube diameter which are in accordance with experimental estimates from rheological measurements, for all the polymers under study. Moreover, we will provide a stochastic interpretation of our results in terms of a renewal point process model which treats topological constraints as stochastic events placed along the monomer sequence of a polymer chain.

08:50  Primitive Path Identification and Entanglement Statistics in Polymer Melts: Results from a Direct Topological Analysis on Atomistically Detailed Polyethylene Models
K. Foteinopoulou^1,2, N.Ch. Karayiannis^1,2, V.G. Mavrantzas^1,2, M. Kröger^3
1 Department of Chemical Engineering, University of Patras, GR 26504, Greece
² FORTH-ICE/HT, Patras GR 26504, Greece
³ ETH Zurich, Polymer Physics, Wolfgang-Pauli-Str. 10, CH-8093 Zurich, Switzerland
A large number of well equilibrated atomistic configurations of linear, strictly monodisperse polyethylene (PE) melts of molecular length ranging from C78 to C1000, obtained with the Double Bridging Monte Carlo algorithm, have been subjected to a detailed topological analysis with the Z code [Kröger, Comp. Phys. Comm., 2005]. The code constructs primitive paths that connect the two ends of a polymer chain (which in all cases are considered as fixed in space) geometrically under the constraint of no chain crossability, such that the multiple disconnected (coarse-grained) path has minimum contour length. When applied to a given, dense polymer configuration in 3-d space, it allows us to obtain the primitive path (PP) and the related number and positions of entanglements (kinks) for all chains in the simulation box, and extract information about the topological structure (the primitive path network) hidden in bulk PE. Results will be presented for the distribution and mean values of the number of entanglements per chain, the entanglement length, the tube diameter, the Kuhn step length and the contour length [1]. In particular, our analysis demonstrates that with increasing chain length, the entanglement molecular length reaches a plateau value characteristic of entangled polymeric behavior, which for the PE systems analyzed here comes out to be about 60 carbon atoms. We further validate recent predictions [Schieber, J. Chem. Phys., 2003] about the shape of the distribution of number of strands in a chain at equilibrium. At the same time we show, that the number of entanglements obtained by assuming random walk statistics [Everaers et al., Science, 2004] deviates significantly from these predictions which we regard as a clear sign of evidence that the direct counting of entanglements and related quantities, as proposed here, offers advantages for a quantitative analysis of the statistical nature of entanglements in polymeric systems.

[1] K. Foteinopoulou, N.Ch. Karayiannis, V.G. Mavrantzas, M. Kröger, Macromolecules 39, 4207 (2006) »

09:15  Multiscale Simulations of Chemically Complex Polymer Melts
K. Kamio¹, K. Moorthi¹, D.N. Theodorou^2
1 Mitsui Chemicals, Inc., Japan
² National Technical University of Athens, Greece
A new methodology for multiscale simulations of chemically complex polymers has been proposed and verified by applying it to poly(ethylene terephthalate) melt. A coarse-graining scheme has been applied to reduce the chemical complexity of the polymer. The resulting structurally simpler model facilitates the process of generalizing the connectivity-altering Monte Carlo method, which has been used to equilibrate the coarse-grained system, and accelerates computations due to the reduction of the degrees of freedom. The melt density, characteristic ratio and other conformational properties agree with experiment. The analysis of the subchain Kuhn length indicates that the melt is adequately equilibrated within Kuhn and chain length scale range. Topological analyses of the melt using CReTA and Z algorithms reveal that the melt system is also well equilibrated with respect to entanglement density.

09:40  Hierarchical Modeling of Polystyrene: From Atomistic to Coarse-Grained Simulations
V. Harmandaris, N. van der Vegt, K. Kremer
Max-Planck-Institute for Polymer Research, D-55021 Mainz, Germany
We present a hierarchical approach that combines atomistic and mesoscopic simulations which can generally be applied to vinyl polymers. As a test case the approach is applied to atactic polystyrene (PS). First a specific model for atactic PS is chosen. The bonded parameters in the coarse-grained force field, based on data obtained from atomistic simulations of isolated PS dimers, are chosen in a way which allows to differentiate between meso- and racemic dyads. This approach allows to study isotactic and syndiotactic melts. Nonbonded interactions between coarse-grained beads were chosen as purely repulsive. The proposed mesoscopic model reproduces both the local structure and the chain dimensions properly. An explicit time mapping is performed, based on the atomistic and CG mean square displacements of short chains, demonstrating an effective speed up of about three orders of magnitude compared to brute force atomistic simulations. Finally the equilibrated coarse-grained chains are back mapped onto the atomistic systems. This opens new routes for obtaining well equilibrated high molecular weight polymeric systems and also providing very long dynamic trajectories at the atomistic level for these polymers.

10:05  Coffee break

10:20  AdResS for hybrid atomistic/coarse-grained molecular dynamics simulations
M. Praprotnik, L. Delle Site, K. Kremer
Max Planck Institute for Polymer Research, Mainz, Germany
A new Adaptive resolution scheme (AdResS) for efficient hybrid atomistic/coarse-grained particle-based molecular dynamics (MD) simulations is presented. The key feature of this approach is that it allows for a dynamical change of the number of molecular degrees of freedom during the course of a MD simulation by an on-the-fly switching between the atomistic and mesoscopic levels of detail.

[1] M. Praprotnik, L. Delle Site, and K. Kremer, J. Chem. Phys. 123, 224106, 2005.
[2] M. Praprotnik, L. Delle Site, and K. Kremer, Phys. Rev. E. 73, 2006.

10:45  Bridging time scale is amorphous glassy polymers
D. Tsalikis, G.C. Boulougouris, L. Peristeras, D.N. Theodorou
School of Chemical Engineering, National Technical University of Athens, Greece
Polymer glasses are complex non-equilibrium materials. Although significant advances have been achieved in modelling deformation, yield and fracture of polymer glasses at the macroscopic level [1], connecting these properties to the chemical constitution and to the formation and processing history of a glass is still a challenge. This is because of the extremely broad spectra of characteristic times that govern molecular motion in the glassy state. Conventional atomistic simulation techniques, such as molecular dynamics (MD), employ an integration time step of approximately 10-15 s to track fast vibrational motions and thus can only address time scales up to 100 ns and length scales up to 10 nm with currently available computational means. As a consequence, they face two very serious challenges: (a) It is impossible to obtain a computer glass with a formation history that is both well-defined and realistic: MD vitrification experiments necessarily impose cooling rates of at least 108 K s-1, nine orders of magnitude higher than the rates of typical laboratory experiments. (b) Even if one is able to form molecular configurations that are truly representative of a real glass, an MD deformation experiment has to be performed at a strain rate of at least 106 s-1, which is much higher than those encountered in most applications. In this work we discuss a strategy for addressing challenge (b) described above leading to a better understanding of yield and strain softening phenomena, as well as physical ageing. It is based on the idea that the local configuration of a glass is trapped in the vicinity of a local minimum of the energy, undergoing infrequent transitions to neighbouring minima across free energy barriers that may vary widely in height. This .energy landscape. picture focusses on the determination of representative energy minima and of the transition paths leading from those to neighbouring minima in the multidimensional configuration space of the glass. Thermodynamic properties and elastic constants in the individual energy minima are estimated by invoking a quasiharmonic approximation for the energy, and the corresponding properties of the glass are obtained through arithmetic (.quenched.) averaging over all minima. The rate constants for transitions from a minimum to neighbouring minima are estimated using the principles of multidimensional transition-state theory and the temporal evolution of the system, in the presence or absence of external stress, is tracked by Kinetic Monte Carlo (KMC) simulation as a succession of transitions between the minima. Such .quasi-dynamics. simulations can deal with arbitrarily slow transition rates and thus overcome the long-time problems of .brute-force. MD. The steps used in our approach could be described as follows: We focus on a small region of a polymer glass containing a few hundreds of atoms. We envision that the configuration of this region fluctuates in the vicinity of a local minimum of the potential energy, or .inherent structure. [2]. Transitions between minima are largely inhibited by the presence of energy barriers that are high relative to kBT. Configuration space is thus partitioned into .basins of attraction.. The measured volumetric properties and elastic constants of the glass are shaped by the restricted probability distributions of configurations within individual basins. Physical ageing brings about a gradual redistribution among the basins through infrequent transitions across the energy barriers, and therefore a gradual change in the properties of the glass. We invoke a quasi-harmonic approximation (QHA) by assuming that the potential energy of a glassy region of given spatial extent, while it fluctuates in the vicinity of its inherent structure, is well approximated by a Taylor expansion to second order around the inherent structure. At given stress, the glassy region will adopt that strain which minimizes G. For the simulation of time-dependent plastic deformation and physical ageing phenomena, it is essential to determine all relevant transition pathways out of a given minimum. We begin by finding as many as possible saddle points around the minimum in the multidimensional configuration space of the polymer using the .dimer method. of Henkelman and J�on [5], which does not require second derivatives. In this procedure it is essential to recognize already visited minima on the basis of their energy and configuration. For each transition path between energy minima A and B, the rate constant is estimated according to transition-state theory, computed via the QHA as described above. Tracking the temporal evolution of the system thus reduces to a KMC simulation of a sequence of elementary transitions between basins, their rate constants being computed .on the fly.. Even at this level of description, there is a wide distribution of time scales (as one would expect in a glassy system), resulting from the wide distribution of free energy barriers connecting the .inherent structures.. This makes the use of classical KMC schemes highly inefficient. For this reason we have developed a novel KMC scheme that allows us to sample efficiently over a wide range of time scales.

[1] Meijer, H.E.H. and Govaert, L.E. Prog. Polym. Sci. 2005, 30, 915-938.
[2] Stillinger, F.H. Science 1995, 267, 1935-1939.
[3] Kopsias, N.P. and Theodorou, D.N. J. Chem. Phys. 1998, 109, 8573-8582.
[4] Lyulin, A.V. Europhys. Lett. 2005, 71, 618-624.
[5] Henkelman, G. and Jonsson, H. J. Chem. Phys. 1999, 111, 7010-7022.

11:10  Computer Simulation of Colloidal Electrophoresis
B. Dünweg, V. Lobaskin, K. Seethalakshmy-Hariharan, C. Holm
Max Planck Institute for Polymer Research, Mainz, Germany
We study the motion of a charged colloidal sphere surrounded by solvent, counterions, and salt ions, under the influence of an external electric field. The ions are modeled as particles which interact dissipatively with a lattice Boltzmann background, such that hydrodynamic interactions are taken into account. Similarly, the colloid is modeled as a spherical array of such point particles. Finite concentration values are taken into account by simulating the system in a box with periodic boundary conditions. In terms of dimensionless reduced parameters, the results compare favorably with experimental data. As a complementary approach, we solve the electrokinetic equations by a finite element method.

11:35  Discussion
T. Tzavaras

12:05  Lunch

   Day 2: Monday afternoon, September 4, 2006

Session 3 Non-equilibrium thermodynamics and Molecular Dynamics
Chair: M. Kröger

14:00  A molecular dynamics study of the stress-optical behavior of a linear short-chain polyethylene melt under shear
C. Baig, B.J. Edwards, D.J. Keffer
Department of Chemical Engineering, University of Tennessee, Knoxville, TN 37996-2200, USA
In this study, we investigate details of the stress-optical behavior of a linear polyethylene melt under shear using a realistic potential model. We demonstrate the existence of the critical shear stress, above which the stress-optical rule (SOR) begins to fail. The critical shear stress of the SOR of this melt turns out to be approximately 5.5 MPa, which is fairly higher than 3.2 MPa at which shear thinning starts. This indicates that the SOR is valid up to a point well beyond the incipient point of shear thinning. Furthermore, contrary to conventional wisdom, the breakdown of the SOR turns out not to be exactly correlated with the saturation of chain extension and orientation: it is observed to occur at shear rates well before maximum chain extension is obtained. In addition to the stress and birefringence tensors, we also compare two important coarse-grained second-rank tensors, the conformation and orientation tensors. The birefringence, conformation, and orientation tensors display nonlinear relationships to each other at high values of the shear stress, and the deviation from linearity begins at approximately the critical shear stress for the breakdown of the SOR.

14:25  Microscopic chaos in shear and elongational flows
F. Frascoli¹, D.J. Bernhardt (nee Searles)², B.D. Todd^1
1 Centre for Molecular Simulation, Swinburne University of Technology, PO Box 218, Hawthorn, Victoria 3122, Australia
² School of Science, Griffith University, Brisbane, Queensland 4111, Australia
The simulation of atoms and molecules under planar elongational flow in a nonequilibrium steady state for arbitrarily long times has recently been made possible by an appropriate implementation of nonequilibrium molecular dynamics (NEMD) with suitable periodic boundary conditions. We address some fundamental questions regarding the chaotic behaviour of this type of flow and compare its chaotic properties with the behaviour of fluids under planar shear flow. We analyse the spectra of Lyapunov exponents for a number of state points where the energy dissipation is the same for both flows, simulating a nonequilibrium steady state for isoenergetic and isokinetic constrained dynamics. We test the conjugate pairing rule and confirm its validity for planar elongation flow, as is expected from the Hamiltonian nature of the adiabatic equations of motion. Discussion about the chaoticity of the convective part of the flows, the link between Lyapunov exponents and viscosity and phase space contraction will also be presented.

14:50  Simulations of shear banding in paints
W.J. Briels, A. van den Noort
University of Twente, The Netherlands
A coarse grain model to simulate the rheological properties of paints will be presented. The model shows extreme shear thinning at shear rates as low as several reciprocal seconds. In several cases shear banding was found along the gradient direction. Not only were shear rates different in different bands, but also their densities. Special attention will be paid to the evolution of the stress during the period when the systems settles from its initial homogeneous state to the final banded state.

15:15  Flow and Motion of Polymer Droplets on Polymer Brushes
M. Müller¹, C. Pastorino², K. Binder^2
1 Institut für Theoretische Physik, Georg-August Universität, D37077 Göttingen, Germany
² Institut für Physik, WA331, Johannes Gutenberg Universität, D55099 Mainz, Germany
Polymer brushes are soft, elastically deformable substrates giving rise to a rich wetting behavior and additional molecular dissipation mechanisms for the motion of droplets. We study thin polymer films and droplets on flat brush-covered substrates by non-equilibrium molecular dynamics simulation of a coarse-grained bead-spring model. The brush consists of identical polymers as the droplets. Upon increasing the grafting density the free polymers are expelled from the brush and a brush-melt interface gradually builds up. Molecular conformations and the overlap between brush and melt are studied in equilibrium and under shear. The velocity profile across a thin polymer film is investigated. The slip length adopts large positive values (perfect slip) for low grafting densities, but decreases and becomes negative for densely grafted, autophobic brushes. At high grafting density the polymer melt dewets from the brush and forms droplets. Nanoscopic polymer droplets driven by volume forces are investigated in their steady state.

15:40  A Generalized Hamiltonian-Based Algorithm for Rigorous Equilibrium Molecular Dynamics Simulation in the NVT, NpT, and μVT Ensembles
J. Santiago, D.J. Keffer, B.J. Edwards, C. Baig
University of Tennessee, Knoxville, TN 37996-2200, USA
We provide a methodical procedure for generating equations of motion for rigorous simulation in three different statistical ensembles, the canonical ensemble (NVT), the isothermal-isobaric ensemble (NpT), and the grand canonical ensemble (μVT) under equilibrium conditions. The procedure begins with a Hamiltonian in terms of laboratory coordinates in a mathematical frame of reference where time and/or mass is dilated. The equations of motion are derived relying on the symplectic relationship between the Hamiltonian and the equations of motion. We define a non-canonical transformation from the laboratory coordinates in the mathematical frame of reference to laboratory coordinates in the physical frame of reference, in much the same way as the original NVT development of Nose and Hoover. However, the new equations are completely general, unlike their predecessors, in that they are valid whether or not an external force field is present. Several illustrations of simulations involving these ensembles will be presented which validate the new algorithms.

16:05  Coffee break

16:20  Discussion
M. Kröger