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: ~ 10-15 sec

-- dihedral rotations: 10-12 sec

-- segmental relaxation: 10-9 - 10-12 sec

-- maximum relaxation time, τ1: ~ 1 sec (for Τ < Τm)

Classical mechanics: solve classical equations of motion in phase space, Γ:=Γ(r, p).

In microcanonical (NVE) ensemble:

( ) exp (0)t iLt

1

,N

i ii i i

iL H

r Fr p

gK

: i ii

i

r pr

t m

: ii i

i

p Up F

t r

2

( )2

iNVE

i i

pH K U U

m r rHamiltonian (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.

i

p

Q

2

3 iB

i i

pp Nk T

m

ii

i

pr

m

i i

i

pUp p

r Q

22

( ) 3 ln2

iNVT NVE s s B

i i

ppH H K V V Nk T

m Q r

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.

21 ( )2 obend bendV k -- bending potential

21 ( )2 ostr strV k l l -- stretching potential

-- dihedral potential5

0cos ( )i

tors ii

V c

-- non-bonded potential12 6

4LJVr r

Van der Waals (LJ) Coulomb

ri j

qij

q qV

ε

( ) bonded non bonded str bend tors LJ qU V V V V V V V r

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) (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, PBmuch 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.

Atomistic MD Simulations: Quantitative predictions of the dynamics in soft matter.

Systematic Coarse-Graining: Overall Procedure

1. Choice of the proper CG description.

1

, 1, 2,...,N

i i ij

c i M

q r

-- Microscopic (N particles)

: TQ R

1 2: , ,..., MQ q q q 1 2: , ,..., NR r r r

-- 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.

( , ) ( , ) ( , )CG CG CGbonded non bondedU T U T U T Q Q Q

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):

1 2

1 2 1 1 2( , ,..., , ).... exp , ,..., ,... | , ,...,

: ,CG

M

ATN N MU T CG

N

U dP T

Ze

q q qr r r r r q q q

Q

Remember:

Assumption 1:

1

N

i i ij

c

q r

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, PCG(r,θ,).

PCG(r,θ, ) follow a Boltzmann distribution: ( , , ), , exp

CGCG bondedU r

P rkT

Assumption 2:

( , ) ln , , ( , , )CG CGbonded BU x T k T P x T x r

, ,CG CG CG CGP r P r P P

Finally:

Bonded CG Interaction Potential

( , ) ( , , , )CG CGbonded bondedU T U T Q r

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:

exp ,( , ) lnCG ATnb UU T rq

,

,AT ATij

i j

U U r r

Average < > over all degrees of freedom Γ that are integrated out (here orientational ) keeping the two center-of-masses fixed at distance r.

1 2

1 2 1 1 2( , , ).... exp , ,..., ,... | ,CG

nb

ATN NU T

N

U d

Ze

q q

r r r r r q q

q

Assumption 3: Pair-wise additivity

1

1

( , ) , ,M M

CG CGnon bonded nb i j

i j i

U T U T

Q q q

2:1 model: Each chemical repeat unit replaced by two CG spherical beads (PS: 16 atoms or 8 “united atoms” replaced by 2 beads).

σΑ = 4.25 Å σB = 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 “CHx” 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.

grem: 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

ene

rgy

Atomistic

CG

AT CG

CG Polymer Dynamics – Quantitative Predictions

Check transferability of τx 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, τx, 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-1

-- Reptation: D ~ M-2

Crossover region: -- CG MD: Me ~ 28.000-33.000 gr/mol -- Exp.: Me ~ 30.0000-35.000 gr/mol

Non-equilibrium molecular dynamics (NEMD): modeling of systems out of equilibrium - flowing conditions.

CG Simulations – Application: Non-Equilibrium Polymer Melts

: ii t

q

p

p

t Q

2

3iB

i i

Nk Tt m

p p

NEMD: Equations of motion (pSLLOD)

ii i i im

t

p

F p u q u u

In canonical ensemble (Nose-Hoover) [C. Baig et al., J. Chem. Phys., 122, 11403, 2005] :

0 0

0 0 0

0 0 0

u

simple shear flow

ii i i i i

pm

t Q

p

F p u q u u p

Primary

L

L

y

ux

x

y

x

ux

tL

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

0 0

0 0 0

0 0 0

u

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 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?

( , ) ln ,CG CGBU T k T P TQ Q

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) Wi

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

2

3

eq

R Rc

R

Wi Properties as a function of strength of flow (Weissenberg number)

Conformation tensor

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.

R

CG Non-Equilibrium Polymers: Conformation Tensor

cyy, czz: excellent agreement between atomistic and CG configurations.

CG Non-Equilibrium Polymers: Dynamics

Translational motion

2( ) (0)

lim6

cm cm

G t

R t RD

t

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

2( ) (0) ( ) expeq

r

tR t R R t

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 assumptionsOngoing work with M. Katsoulakis, D. Tsagarogiannis, A. Tsourtis

Challenges – Current Work

Quantitative prediction of dynamics based on statistical mechanicse.g. Mori-Zwanzig formalism (Talk by Rafael Delgado-Buscalioni)

Parameterizing CG models under non-equilibrium conditionse.g. Information-theoretic tools (Talk by Petr Plechac)

Application of the whole procedure in more complex systemse.g. Multi-component biomolecular systems, hybrid polymer based

nanocompositesOngoing 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)

pp

ppL

NRa

)(2

Calculate directly PP contour length Lpp,, tube diameter:

)(2 NR

NaN monpp

e -- PP CG PS: Ne ~ 180 ± 20 monomers

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)]

Method T(K) Ne (mers) Reference

Rheology 423 140 ± 15 Liu et al., Polymer, 47, 4461 (2006)

Self-diffusion coefficient 458 280-320 Antonieti et al., Makrom. Chem., 188, 2317 (1984)

Self-diffusion coefficient 463 240-300 This work

Segmental dynamics 463 110 ± 30 This work

Entanglement analysis 463 180 ± 20 This work

Entanglement analysis 413 124 Spyriouni et al., Macromolecules, 40, 3876 (2007)

eN M

RTG

5

40

MESOSCOPIC BOND ANGLE POTENTIAL OF PS

Distribution function PCG(θ,T)

CG Bending potential UCG(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 Gyration22

1

N

G i CMi

R

R RRG

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

AT CG

CG diffusion coefficient is larger than the atomistic one

AT CGD D

Bk T

ND

Rouse:

2

23 Bk T a

N RD

Reptation:

Time Mapping Parameter: Translational vs Orientational Dynamics