1 / 36

VAGELIS HARMANDARIS International Conference on Applied Mathematics Heraklion, 16/09/2013

Simulations of Soft Matter under Equilibrium and Non-equilibrium Conditions. VAGELIS HARMANDARIS International Conference on Applied Mathematics Heraklion, 16/09/2013. Outline. Introduction: Motivation, Length-Time Scales, Simulation Methods.

rich
Télécharger la présentation

VAGELIS HARMANDARIS International Conference on Applied Mathematics Heraklion, 16/09/2013

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Simulations of Soft Matter under Equilibrium and Non-equilibrium Conditions VAGELIS HARMANDARIS International Conference on Applied Mathematics Heraklion, 16/09/2013

  2. Outline • Introduction: Motivation, Length-Time Scales, Simulation Methods. • Multi-scale Particle Approaches: Atomistic and systematic coarse-grained simulations of polymers. • Application: Equilibrium polymeric systems. • Application: Non-equilibrium (flowing) polymer melts. • Conclusions – Open Questions.

  3. COMPLEX SYSTEMS: TIME - LENGTH SCALES: • A wide spread of characteristic times: (15 – 20 orders of magnitude!) -- bond vibrations: ~ 10-15 sec -- dihedral rotations: 10-12sec -- segmental relaxation: 10-9 - 10-12 sec -- maximum relaxation time, τ1: ~ 1 sec (for Τ < Τm)‏

  4. Hamiltonian (conserved quantity): Modeling of Complex Systems: Molecular Dynamics • Classical mechanics: solve classical equations of motion in phase space,Γ:=Γ(r, p). • In microcanonical (NVE) ensemble: The evolution of system from time t=0 to time t is given by : Liouville operator:

  5. Modeling of Complex Systems: Molecular Dynamics • Various methods for dynamical simulations in different ensembles. • In canonical (NVT) ensemble: -- Langevin (stochastic) Thermostat -- Nose-Hoover thermostat: [Nosé 1984; Hoover, 1985]: add one more degree of freedom ζ.

  6. -- stretching potential -- bending potential -- dihedral potential Van der Waals (LJ) Coulomb -- non-bonded potential Molecular Interaction Potential (Force Field) Molecular model: Information for the functions describing the molecular interactions between atoms. -- Potential parameters are obtained from more detailed simulations or fitting to experimental data.

  7. MULTI-SCALE DYNAMIC MODELING OF COMPLEX SYSTEMS Atomistic MD Simulations: Quantitative predictions of the dynamics in soft matter. Limits of Atomistic MD Simulations (with usual computer power): -- Length scale: few Å – O(10 nm)‏ -- Time scale: few fs - O(1 μs) (10-15 – 10-6 sec) ~ 107 – 109 time steps -- Molecular Length scale (concerning the global dynamics): up to a few Me for “simple” polymers like PE, PB much below Me for more complicated polymers (like PS)‏ Need: - Simulations in larger length – time scales. - Application in molecular weights relevant to polymer processing. - Quantitative predictions. Proposed method: - Coarse-grained particle models obtained directly from the chemistry.

  8. Systematic Coarse-Graining: Overall Procedure 1. Choice of the proper CG description. -- Microscopic (N particles) -- Mesoscopic (M “super particles”) -- Usually T is a linear operator (number of particles that correspond to a ‘super-particle’

  9. Systematic Coarse-Graining: Overall Procedure 2. Perform microscopic (atomistic) simulations of short chains (oligomers) (in vacuum) for short times. 3. Develop the effective CG force field using the atomistic data-configurations. 4. CG simulations (MD or MC) with the new coarse-grained model. Re-introduction (back-mapping) of the atomistic detail if needed.

  10. Effective (Mesoscopic) CG Interaction Potential (Force Field) CG Potential:In principle UCG is a function of all CG degrees of freedom in the system and of temperature (free energy): • CG Hamiltonian – Renormalization Group Map: • Remember: • Assumption 1:

  11. r Bonded CG Interaction Potential Bonded Potential • Degrees of freedom:bond lengths (r), bond angles (θ), dihedral angles () Procedure: • From the microscopic simulations we calculate the distribution functions of the degrees of freedom in the mesoscopic representation,PCG(r,θ,). • PCG(r,θ, ) follow a Boltzmann distribution: • Assumption 2: • Finally:

  12. q Non-bonded CG Interaction Potential: Reversible Work • Assumption 3: Pair-wise additivity Reversible work method [McCoy and Curro, Macromolecules, 31, 9362 (1998)] • By calculating the reversible work (potential of mean force) between the centers of mass of two isolated molecules as a function of distance: • Average < > over all degrees of freedom Γ that are integrated out (here orientational ) keeping the two center-of-masses fixed at distance r.

  13. CG MD DEVELOPMENT OF CG MODELS DIRECTLY FROM THE CHEMISTRY APPLICATION: POLYSTYRENE (PS)‏ [Harmandaris, et al. Macromolecules, 39, 6708 (2006); Macromol. Chem. Phys. 208, 2109 (2007); Macromolecules 40, 7026 (2007); Fritz et al. Macromolecules 42, 7579 (2009)] 1) CHOICE OF THE PROPER COARSE-GRAINED MODEL • 2:1 model: Each chemical repeat unit replaced by two CG spherical beads (PS: 16 atoms or 8 “united atoms” replaced by 2 beads). • CG operator T: from “CHx” to “A” and “B” description. • Each CG bead corresponds to O(10) atoms. σΑ = 4.25 Å σB = 4.80 Å • Chain tacticity is described through the effective bonded potentials. • Relatively easy to re-introduce atomistic detail if needed. 2) ATOMISTIC SIMULATIONS OF ISOLATED PS RANDOM WALKS

  14. CG MD Simulations: Structure in the Atomistic Level after Re-introducing the Atomistic Detail in CG Configurations. • Simulation data: atomistic configurations of polystyrene obtained by reinserting atomistic detail in the CG ones. • Wide-angle X-Ray diffraction measurements [Londono et al., J. Polym. Sc. B, 1996.] grem: total g(r) excluding correlations between first and second neighbors.

  15. CG Free energy Atomistic Configuration CG Polymer Dynamics is Faster than the Real Dynamics PS, 1kDa, T=463K Free Energy Landscape • --CG effective interactions are softer than the real-atomistic ones due to lost degrees of freedom (lost forces). • This results into a smoother energy landscape. • CG MD: We do not include friction forces.

  16. Time Scaling CG Polymer Dynamics – Quantitative Predictions CG dynamics is faster than the real dynamics. Time Mapping (semi-empirical) method: • Find the proper time in the CG description by moving the raw data in time. Choose a reference system. Scaling parameter, τx, corresponds to the ratio between the two friction coefficients. Time Mapping using the mean-square displacement of the chain center of mass • Check transferability of τx for different systems, conditions (ρ, T, P, …).

  17. Polymer Melts through CG MD Simulations: Self Diffusion Coefficient • Correct raw CG diffusion data using a time mapping approach. • [V. Harmandaris and K. Kremer, Soft Matter, 5, 3920 (2009)] • Crossover regime: from Rouse to reptation dynamics. Include the chain end (free volume) effect. -- Rouse: D ~ M-1 -- Reptation: D ~ M-2 Crossover region: -- CG MD: Me ~ 28.000-33.000 gr/mol -- Exp.: Me ~ 30.0000-35.000 gr/mol -- Exp. Data: NMR [Sillescu et al. Makromol. Chem., 188, 2317 (1987)]

  18. CG Simulations – Application: Non-Equilibrium Polymer Melts • Non-equilibrium molecular dynamics (NEMD): modeling of systems out of equilibrium - flowing conditions. • NEMD: Equations of motion (pSLLOD) simple shear flow • In canonical ensemble (Nose-Hoover) [C. Baig et al., J. Chem. Phys., 122, 11403, 2005] :

  19. ux Primary y x x CG Simulations – Application: Non-Equilibrium Polymer Melts • NEMD: equations of motion are not enough: we need the proper periodic boundary conditions. • Steady shear flow: Lees-Edwards Boundary Conditions simple shear flow

  20. CG Polymer Simulations: Non-Equilibrium Systems • CG NEMD - Remember: CG interaction potentials are calculated as potential of mean force (they include entropy). In principle UCG(x,T) should be obtained at each state point, at each flow field. Important question: How well polymer systems under non-equilibrium (flowing) conditions can be described by CG models developed at equilibrium? Method: [C. Baig and V. Harmandaris, Macromolecules, 43, 3156 (2010)] Use of existing equilibrium CG polystyrene (PS) model. • Direct comparison between atomistic and CG NEMD simulations for various flow fields. Strength of flow (Weissenberg number, Wi = 0.3 - 200) • Study short atactic PS melts (M=2kDa, 20 monomers) by both atomistic and CG NEMD simulations.

  21. CG Non-Equilibrium Polymers: Conformations • Properties as a function of strength of flow (Weissenberg number) • Conformation tensor R • Atomistic cxx: asymptotic behavior at high Wi because of (a) finite chain extensibility, (b) chain rotation during shear flow. • CG cxx: allows for larger maximum chain extension at high Wi because of the softer interaction potentials.

  22. CG Non-Equilibrium Polymers: Conformation Tensor • cyy, czz: excellent agreement between atomistic and CG configurations.

  23. CG Non-Equilibrium Polymers: Dynamics • Is the time mapping factor similar for different flow fields? • [C. Baig and V. Harmandaris, Macromolecules, 43, 3156 (2010)] Translational motion • Purely convective contributions from the applied strain rate are excluded. • Very good qualitative agreement between atomistic and CG (raw) data at low and intermediate flow fields.

  24. CG Non-Equilibrium Polymers: Dynamics Orientational motion • Rotational relaxation time: small variations at low strain rates, large decrease at high flow fields. • Good agreement between atomistic and CG at low and intermediate flow fields.

  25. CG Non-Equilibrium Polymers: Dynamics • Time mapping parameter as a function of the strength of flow. • Strong flow fields: smaller time mapping parameter  effective CG bead friction decreases less than the atomistic one. Reason: CG chains become more deformed than the atomistic ones.

  26. Conclusions • Hierarchical systematic CG models, developed from isolated atomistic chains, correctly predict polymer structure and dimensions. • Time mapping using dynamical information from atomistic description allow for quantitative dynamical predictions from the CG simulations, for many cases. • Overall speed up of the CG MD simulations, compared with the atomistic MD, is ~ 3-5 orders of magnitude. • System at non-equilibrium conditions can be accurately studied by CG NEMD simulations at low and medium flow fields. • Deviations between atomistic and CG NEMD data at high flow fields due to softer CG interaction potentials.

  27. Challenges – Current Work • Estimation of CG interaction potential (free energies): Check – improve all assumptions • Ongoing work with M. Katsoulakis, D. Tsagarogiannis, A. Tsourtis • Quantitative prediction of dynamics based on statistical mechanics • e.g. Mori-Zwanzig formalism (Talk by Rafael Delgado-Buscalioni) • Parameterizing CG models under non-equilibrium conditions • e.g. Information-theoretic tools (Talk by Petr Plechac) • Application of the whole procedure in more complex systems • e.g. Multi-component biomolecular systems, hybrid polymer based nanocomposites • Ongoing work with A. Rissanou

  28. ACKNOWLEDGMENTS Prof. C. Baig [School of Nano-Bioscience and Chemical Engineering, UNIST University, Korea] Funding: ACMAC UOC [Regional Potential Grant FP7] DFG [SPP 1369 “Interphases and Interfaces ”, Germany] Graphene Research Center, FORTH [Greece]

  29. EXTRA SLIDES

  30. APPLICATION: PRIMITIVE PATHS OF LONG POLYSTYRENE MELTS • Describe the systems in the levels of primitive paths [V. Harmandaris and K. Kremer, Macromolecules, 42, 791, (2009)] • Entanglement Analysis using the Primitive Path Analysis (PPA) method [Evereaers et al., Science 2004, 303, 823]. PP PS configuration (50kDa) CG PS configuration (50kDa) • Calculate directly PP contour length Lpp,, tube diameter: -- PP CG PS: Ne ~ 180 ± 20 monomers

  31. CALCULATION OF Me in PS: Comparison Between Different Methods • Several methods to calculate Me: broad spread of different estimates [V. Harmandaris and K. Kremer, Macromolecules, 42, 791 (2009)]

  32. MESOSCOPIC BOND ANGLE POTENTIAL OF PS Distribution function PCG(θ,T)‏ CG Bending potential UCG(θ,T)‏

  33. CG Simulations – Applications: Equilibrium Polymer Melts • Systems Studied: Atactic PS melts with molecular weight from 1kDa (10 monomers) up to 50kDa (1kDa = 1000 gr/mol). • NVT Ensemble. • Langevin thermostat (T=463K). • Periodic boundary conditions.

  34. STATIC PROPERTIES : Radius of Gyration RG

  35. SMOOTHENING OF THE ENERGY LANDSCAPE Qualitative prediction: due to lost degrees of freedom (lost forces) in the local level  Local friction coefficient in CG mesoscopic description is smaller than in the microscopic-atomistic one Rouse: Reptation: CG diffusion coefficient is larger than the atomistic one

  36. Time Mapping Parameter: Translational vs Orientational Dynamics

More Related