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

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

COMPLEX SYSTEMS: TIME - LENGTH SCALES: A wide spread of characteristic times: (15 – 20 orders of magnitude!) -- bond vibrations: ~ sec -- dihedral rotations: sec -- segmental relaxation: sec -- maximum relaxation time, τ 1 : ~ 1 sec (for Τ < Τ m )‏

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

 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. Modeling of Complex Systems: Molecular Dynamics

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

MULTI-SCALE DYNAMIC MODELING OF COMPLEX SYSTEMS Limits of Atomistic MD Simulations (with usual computer power): -- Length scale:few Å – O(10 nm)‏ -- Time scale:few fs - O(1 μs)( – sec)~ 10 7 – 10 9 time steps -- Molecular Length scale (concerning the global dynamics): up to a few M e for “simple” polymers like PE, PB much below M e 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. Atomistic MD Simulations: Quantitative predictions of the dynamics in soft matter.

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’

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.

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

r 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, P CG (r,θ,  ).  P CG (r,θ,  ) follow a Boltzmann distribution: Assumption 2:  Finally: Bonded CG Interaction Potential

Non-bonded CG Interaction Potential: Reversible Work 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. q Assumption 3: Pair-wise additivity

 2:1 model: Each chemical repeat unit replaced by two CG spherical beads (PS: 16 atoms or 8 “united atoms” replaced by 2 beads). σ Α = 4.25 Å σ B = 4.80 Å 1) CHOICE OF THE PROPER COARSE-GRAINED MODEL  Chain tacticity is described through the effective bonded potentials.  Relatively easy to re-introduce atomistic detail if needed. 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)]  CG operator T: from “CH x ” to “A” and “B” description.  Each CG bead corresponds to O(10) atoms. 2) ATOMISTIC SIMULATIONS OF ISOLATED PS RANDOM WALKS

 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.] CG MD Simulations: Structure in the Atomistic Level after Re-introducing the Atomistic Detail in CG Configurations. g rem : total g(r) excluding correlations between first and second neighbors.

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. Configuration Free energy Atomistic CG

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

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. -- Exp. Data: NMR [Sillescu et al. Makromol. Chem., 188, 2317 (1987)] -- Rouse: D ~ M Reptation: D ~ M -2 Crossover region:-- CG MD: M e ~ gr/mol -- Exp.: M e ~ gr/mol

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

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

CG Polymer Simulations: Non-Equilibrium Systems  CG NEMD - Remember: CG interaction potentials are calculated as potential of mean force (they include entropy). In principle U CG (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? 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, W i = )  Study short atactic PS melts (M=2kDa, 20 monomers) by both atomistic and CG NEMD simulations. Method: [C. Baig and V. Harmandaris, Macromolecules, 43, 3156 (2010)]

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

CG Non-Equilibrium Polymers: Conformation Tensor  c yy, c zz : excellent agreement between atomistic and CG configurations.

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

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.

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.

 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. Conclusions

 Estimation of CG interaction potential (free energies): Check – improve all assumptions Ongoing work with M. Katsoulakis, D. Tsagarogiannis, A. Tsourtis Challenges – Current Work  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

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

EXTRA SLIDES

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]. CG PS configuration (50kDa) PP PS configuration (50kDa)  Calculate directly PP contour length L pp,, tube diameter: -- PP CG PS: N e ~ 180 ± 20 monomers

CALCULATION OF M e in PS: Comparison Between Different Methods  Several methods to calculate M e : broad spread of different estimates [V. Harmandaris and K. Kremer, Macromolecules, 42, 791 (2009)] MethodT(K)N e (mers)Reference Rheology ± 15 Liu et al., Polymer, 47, 4461 (2006) Self-diffusion coefficient Antonieti et al., Makrom. Chem., 188, 2317 (1984) Self-diffusion coefficient This work Segmental dynamics ± 30 This work Entanglement analysis ± 20 This work Entanglement analysis Spyriouni et al., Macromolecules, 40, 3876 (2007)

MESOSCOPIC BOND ANGLE POTENTIAL OF PS Distribution function P CG (θ,T)‏ CG Bending potential U CG (θ,T)‏

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

STATIC PROPERTIES : Radius of Gyration RGRG

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 SMOOTHENING OF THE ENERGY LANDSCAPE CG diffusion coefficient is larger than the atomistic one Rouse: Reptation:

Time Mapping Parameter: Translational vs Orientational Dynamics