Trends in Materials Simulation:
A Molecular Dynamics Miscellany
1
Florian Müller-Plathe (www.theo.chemie.tu-darmstadt.de)
Polymers: Scales & Methods
Review: FMP, Soft Materials 1, 1 (2003)
Processing
Morphology
Crystallisation,
rheology
time
Solvation,
permeation
Polymerisation
Quantum
chemistry
Finite
element
Soft fluid
Coarse-grained model
Atomistic
simulation
2
length
Polymers: Structure and Properties
• Chemistry
CH3
H3C
H3C
CH3
CH3
H3C
Cl
Cl
rubber
Cl
Cl
Cl
Cl
brittle
CH2OH
CH2OH
• Tacticity
HO
O
OH
HO
O
O
HO
insoluble
• Sequence
HO
HO O
CH2OH
S-S-S-S-B-B-B-B
engineering plastic
O
CH2OH
O
O
O
HO
O
O
HO
water-soluble
S-B-S-S-B-S-B-B
synthetic rubber
• Topology
3
shopping bag bullet-proof vest
Multiscaling and Dynamics
Transport properties
Multiscale Methods
Reverse nonequilibrium
molecular dynamics
Structure-based
coarse graining
Successes and failures
Successes and failures
Attempt of a synthesis:
Dynamics of coarse-grained models
4
Transport Coefficients
• Transport coefficient
J
E
driving force, field,
thermodynamic force
flux, flow, current, ...
• Non-equilibrium situation: steady-state, periodic, ...
• Linear response: small perturbation
• Isotropic medium: liquid, melt, ...
5
Linear response
• Mass
J1
Berthollet 1803
D ( c / z) Fick 1854
• Charge
I
Ohm 1826
(1 / R ) (
• Energy
JQ
/ z)
Fourier 1822
( T / z)
• Momentum
J(p )
( v / z)
[
xy
• Coupled energy and
mass
Soret
1879
J 1 Ludwig
D1 21856,
[( w
1 / z)
S T w 1 (1 w 1 ) ( T / z )]
]
• General
J
L X Onsager 1931
and mass
T
Jenergy
L
1
11
T
L1 Q
T
T2
6
Calculation of Transport
Coefficients by MD
Periodic
J( ,k)
Green-Kubo
Equilibrium
Einstein
Steady-state
J(
,k )
Non-equilibrium
Boundary-driven
Synthetic
F
-F
7
1st Ingredient:
Reverse cause and effect
Traditional NEMD
J
E
calculate impose
Examples:
• shear viscosity
• thermal conductivity
• Soret coefficient ST
Reverse NEMD
J
E
impose calculate
8
Thermal Conductivity
temperature
0.80
Heat flow
0.75
0.70
0.65
0.60 0
energy transport (unphysical)
2
4
6
8
10
slab number
Repeat periodically, wait for steady state
Fourier’s law:
JQ
dT / d z
9
F. Müller-Plathe, J. Chem. Phys. 106, 6082 (1997)
2nd Ingredient:
Unphysical velocity exchange
What we want
Hot
Find hottest particle in
the cold region and
coldest particle in the
hot region
1
Swap their
velocities v
1
z
2
2
Cold
If m1=m2:
no change in total linear momentum
no change in total kinetic energy
no change in total energy
10
Analyse
• Calculate temperature
340
profile T(z)
• Obtain gradient dT/dz
• Fourier’s law
330
rigid SPC/E
Temperature (K)
320
310
Jq
300
dT dz
290
280
flexible SPC/E
270
260
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
z (nm)
M. Zhang, E. Lussetti, L.E.S. de Souza, F. Müller-Plathe,
J. Phys.Chem. B 109, 15060 (2005).
Thermal conductivity
(Wm-1K-1)
rigid
0.81 0.01
flexible
0.95 0.01
experiment
0.607
11
Thermal conductivity
of molecular liquids
-1
-1
Experiment
Thermal conductivity / W K m
Ar
1
H2O
polyamide-6,6
n-hexane
0.1
0.1
benzene
c-hexane
1
Simulation
M. Zhang, E. Lussetti, L.E.S. de Souza, F. Müller-Plathe,
J. Phys.Chem. B 109, 15060 (2005).
12
Thermal Conductivity of Polyamide-6,6
Simulation
W/K m
amorphous (1.07 g/cm3)
(flexible model)
0.33-0.35
(semirigid model)
0.20
Experiment
W/K m
~0.25
stretched amorphous (0.95 g/cm3)
parallel to stretching 0.43
perpendicular
0.20
Enrico Lussetti, Takamichi Terao, FMP, J. Phys. Chem. B 111,
13
11516-11523 (2007); Phys. Rev. E 75, 057701 (2007).
Carbon nanotubes
M. Alaghemandi, E. Algaer, M. C. Böhm, FMP,
Nanotechnology 20, 115704 (2009).

depends on tube length
 Breakdown of Fourier’s
law
 Possibly different
exponents for short
and long tubes.
Theory:
= 0.83 → 0.35
J. Wang, J.-S. Wang, Appl. Phys.
Lett. 88, 111909 (2006)
L0.54
-1
0.8
thermal conductivity
L0.5
-1
 power law
(Wm K )
1000
100
L0.77
10
10
 Chirality of tube unimportant
| FB Chemie | Prof. Dr. Florian Müller-Plathe | 14
(5,5)
(10,0)
(7,7)
(17,0)
(10,10)
(15,15)
(20,20)
100
tube length (nm)
400
Anisotropy of thermal conductivity
 CNT crystal
100 – 1000 W m-1K-1
(better than metal)
 : 0.14 W m-1K-1
(worse than polymer)

 Multiwall nanotube 10 nm


||:
21 W m-1K-1
: 0.24 W m-1K-1
||:
y
x
 CNT bundle


100 – 1000 W m-1K-1
: 0.09 W m-1K-1
||:
 Extreme anisotropies!
 Fast parallel mechanism (phonons) ↔
slow lateral mechanism (collisions between neighbouring tubes).
| FB Chemie | Prof. Dr. Florian Müller-Plathe | 15
Shear viscosity
z
A
-v x
z
m o m e n tu m
tr a n sfe r
(u n p h y sic al)
vx
j z(p x )
Lz
y
Lx
vx
Ly
m o m e n tu m flu x
(p h y sica l)
x
x
F. Müller-Plathe, Phys. Rev. E 59, 4894-4899 (1999).
16
Exchange of velocity component
What we want
Find 2 particles that
move against the
current
1
z
Swap their
vx
1
2
2
x
If m1=m2:
no change in total linear momentum
no change in total kinetic energy
no change in total energy
→ no thermostat!
17
Shear viscosity
• What have we done?
• Impose a flux of transverse linear momentum J(p )
= apply a shear force
• higher perturbation = swap velocities more often
• What still needs to be done?
• Measure shear rate:
Measure profile of flow velocity, determine dv /dz
• Viscosity is proportionality constant
J p
dv
dz
18
Velocity profiles
0.12
Weak
0.08
perturbation
Lennard-Jones liquid near triple point
Strong
perturbation
vx (m / )1/2
Velocity
0.04
0.00
1200
300
60
15
3
-0.04
-0.08
-0.12
0
3
6
9
z/
12
15
18
19
Linear response holds
NVE
NVT
slope 1
)]
3/
log10 [ jz(px) (
Transverse momentum flux
= shear force
0
-1
-2
Lennard-Jones liquid near triple point
-3
-4
-3
-2
-1
log10 [( vx / z) ( /m 2 )-1/2]
Velocity gradient = shear rate
0
20
Shear viscosity of the
Lennard-Jones fluid
4.0
constant energy
2(
m)-1/2
3.5
lit.
3.0
2.5
constant temperature
2.0
Lennard-Jones liquid near triple point
1.5
-3
-2
-1
log10 [ jz(px) ( 3 / )]
0
21
Viscous flow
without viscous heating?
Recall:
• Total energy is conserved: microcanonical (NVE)
But:
• Viscous flow friction heating !!!
Question:
• How is the heat removed?
Answer: Maxwell demon
• Undirected motion (heat) Directed motion (flow)
• Cooling in exchange regions
Proof: temperature profile
• Ekin(total) = Ekin(temperature) + Ekin(flow)
• Peculiar velocity ui = vi - v defines temperature
22
Temperature profile from
peculiar velocities
High shear
(Heat) / (Total Ekin)
1.05
1
0.95
0.9
W=3
W=15
0.85
0.8
0.75
0.7
0
5
10
15
20
z/sigma
23
Molecular Liquids
P. Bordat, F. Müller-Plathe,
J. Chem. Phys. 116, 3362 (2002)
24
Aqueous solutions of saccharose
Shear viscosity
(cP)
100
60%
10
30%
1
5%
10%
1%
2.6
2.8
3.0
3.2
3.4
3.6
25
Ionic liquids: Shear viscosity of [bmim][PF6]
Experiment:
0=204 cP at 303K
J z px
[Seddon et al., Pure Appl.
Chem. 72, 2275(2000)]
d vx
dz
others: 173-450 cP
Simulation:
0→133 cP at 300K
Analogous: diffusion coefficients
Exper.
Sim.
Shear rate / ps-1
0.7 10-7 cm2/s
1.3 10-7 cm2/s
IONIC LIQUIDS DFG - SPP 1191
W. Zhao, F. Leroy, S. Balasubramanian, and F. MüllerPlathe, J. Phys. Chem. B 112, 8129 (2008).
26
Multiscaling and Dynamics
Transport properties
Multiscale Methods
Reverse nonequilibrium
molecular dynamics
Structure-based
coarse graining
Successes and failures
Successes and failures
Attempt of a synthesis:
Dynamics of coarse-grained models
27
Atomistic Model – Bridge Incomplete
28
Polymers: Scales & Methods
Review: FMP, Soft Materials 1, 1 (2003)
Processing
Morphology
Crystallisation,
rheology
time
Solvation,
permeation
Polymerisation
Quantum
chemistry
Finite
element
Soft fluid
Coarse-grained model
Atomistic
simulation
29
length
Coarse-grained Force Field
~10 atoms → 1 superatom
1000-100000 times cheaper
Effective interactions:
“bonds”, “angles”, non-bonded
Systematic coarse-graining:
• interactions as realistic as possible
• model is material-specific
• no generic “bead-and-spring”
model
30
Reviews: F. Müller-Plathe, ChemPhysChem 3, 754 (2002); Soft Materials 1, 1 (2003).
Coarse Graining
Objectives for coarse-grained model:
• Simpler than atomistic model: ~10 real atoms → 1 “superatom”
• Material-specific
• Reproduce structure of atomistic model
Atomistic simulation
Coarse-grained model
Structure
1.4
Torsion
1.2
1.0
RDF
0.8
0.6
0.4
0.2
0.0
0.2
Bond
Target
Best fit LJ 6-9
Best fit 6-8-10-12
Angle
31
0.4
0.6
0.8
1.0
Distance (nm)
1.2
1.4
1st Example: Poly(vinyl alcohol)
CH2
• Polymer in the melt
• 48 decamers
CH2 CH2 CH2
CH
CH
CH
OH
OH
OH
• Use:
• Packaging: O2 barrier
• Fishing nets: same refractive index as water
• Pervaporation membranes:
remove water from organic solvents
32
Coarse-Graining PVA: Angles
The easy part:
• Stiff degrees of freedom
• Direct Boltzmann inversion
0.030
0.025
Distribution from
atomistic simulation
0.020
0.015
0.010
V
kT
ln P( )
0.005
0.000
60
90
120
150
Angle (degrees)
OH
HO
OH
OH
Potential of mean force,
use as potential of
coarse-grained model
6
180
Potential of mean force (kT)
Distribution (arb. units)
0.035
5
4
3
2
1
0
60
90
120
Angle (degrees)
150
33
180
Coarse-Graining PVA: Nonbonded
1.4
1.2
1.0
RDF
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
Less easy:
• soft degrees of freedom
Target
• potential of mean force
Best fit LJ 6-9
≠ potential energy
Best fit 6-8-10-12
• cannot Boltzmann-invert
directly
0.8
1.0
Distance (nm)
1.2
1.4
34
Iterative Boltzmann Inversion
• Tabulated numerical potential V(r)
• Starting guess V0(r) RDF0(r) RDFtarget(r)
• Potential correction
V1 r
V0 r
• Iterate
Vn
1
r
Vn r
RDF0 r
kT ln
RDFtarget r
kT ln
RDFn r
RDFtarget r
until Vn(r) RDFn(r) = RDFtarget(r)
• Converges in few iterations
• Density/pressure correction can be added
D. Reith, H. Meyer, FMP, Comput. Phys. Commun. 148, 299 (2002).
35
Iterative Boltzmann Inversion (2)
36
[D. Reith, M. Pütz, F. Müller-Plathe, J. Comp.Chem. 24, 1624 (2003)]
2nd Example: Poly(acrylic acid)
Na+
Na+
Na+
• Polymer in solution
• 23-mer, Na salt, 2 wt.% in water
Na+
CO2-
CO2-
CO2-
CO2-
Na+
• Use:
• Additive in washing powders, detergents, etc.
• Flocculation agent in water treatment
• Disposable nappies
37
Na+
Na+
Na+
Na+
CO2-
CO2-
Na+
CO2-
CO2-
Atomistic:
209 atoms
3000 H2O
counterions
~ 10000 atoms
Mapping:
Water:
Poly(acrylic acid)
Coarse-grained:
23 particles
no H2O
coarse-grained superatom
centre of mass of atomistic monomer
effective interaction, viscous medium
38
[D. Reith, B. Müller, FMP, S. Wiegand, J.Chem. Phys. 116, 9100 (2002).]
Comparison with Experiment
1
RH
1
N2
i, j
1
Rij
Hydrodynamic radius (nm)
100
Simulation
Dynamic Light Scatt.
10
Parameterisation
1
100
1000
10000
Molecular weight (monomers) 39
[D. Reith, B. Müller, F. Müller-Plathe, S. Wiegand, J. Chem. Phys. 116, 9100 (2002).]
Systems coarse-grained
Polymer melts:
• Poly(vinyl alcohol)
D. Reith, H. Meyer, FMP, Macromolecules 34, 2235 (2001).
• Polyisoprene
trans: R. Faller, FMP, Polymer 43, 621 (2002); cis: T. Spyriouni, C. Tzoumanekas,
D. Theodorou, FMP, G. Milano, Macromolecules 40, 3876 (2007).
• Amorphous cellulose
S. Queyroy, S. Neyertz, D. Brown, FMP, Macromolecules 37, 7338 (2004).
• Atactic polystyrene
G. Milano, FMP, J. Phys. Chem. B 109, 18609 (2005).
• Amorphous polyamide-6,6
P. Carbone, H. A. Karimi Varzaneh, X.Y. Chen, FMP, J. Chem. Phys. 128, 064904 (2008)
Polymer Solutions
• Poly(acrylic acid)
D. Reith, B. Müller, F. Müller-Plathe, S. Wiegand, J. Chem. Phys. 116, 9100 (2002).
• Poly(ethylene oxide)
40
R. Cordeiro, FMP, in preparation.
Systems coarse-grained
Non-polymers
• Liquid: Diphenyl carbonate
H. Meyer, O. Biermann, R. Faller, D. Reith, FMP, J. Chem. Phys. 113, 6265 (2000).
• Liquid: Ethylbenzene
H.-J. Qian, P. Carbone, X. Chen, H. A. Karimi-Varzaneh, C. C. Liew, FMP,
Macromolecules 41, 9919 (2008).
• PAMAM dendrimers
P. Carbone, F. Negri, FMP, Macromolecules 40, 7044 (2007).
• Ionic liquid: [bmim][PF6]
W. Zhao H.A. Karimi Varzaneh, FMP, in preparation.
• Carbon nanotubes
G. Illya, H.A. Karimi Varzaneh, FMP, in preparation.
• Phospholipid membranes
G. Illya, T.J. Müller, H.A. Karimi Varzaneh, FMP, in preparation.
41
Robust Bridge?
42
Multiscaling and Dynamics
Transport properties
Multiscale Methods
Reverse nonequilibrium
molecular dynamics
Structure-based
coarse graining
Successes and failures
Successes and failures
Attempt of a synthesis:
Dynamics of coarse-grained models
43
Challenge #1: Dynamics
• One known case:
polycarbonate
• Temperature dependence
(Vogel-Fulcher)
reproduced after
empirical curve shift
[W. Tschöp, K. Kremer, J. Batoulis, T. Bürger, O. Hahn, Acta Polym. 49, 61
(1998); ibid. 49, 75].
44
Dynamics
Known feature of coarse-grained potentials
• repulsion (excluded volume) softer than atomistic
• less friction
• faster dynamics: higher D, lower
Questions
• Can coarse-grained models be constructed with correct
dynamics?
• Is there a constant scale factor?
45
Zero-shear Viscosity (cP)
Viscosity: MW dependence
1
0.1
500 K
520 K
540 K
Polystyrene
coarse-grained model
slope ~ 1.0-1.1
below entanglement
1000
10000
Molecular Weight (g/mol)
46
X. Chen, P. Carbone, W.L. Cavalcanti, G. Milano, F. Müller-Plathe, Macromolecules 40, 8087 (2007).
Viscosity: shear-rate dependence
100 Monomers:
• zero-shear limit
not yet reached
• melt is not
Newtonian
9 Monomers:
zero-shear limit
47
X. Chen, P. Carbone, W.L. Cavalcanti, G. Milano, F. Müller-Plathe, Macromolecules 40, 8087 (2007).
Chain structure under shear
Atomistic structure
after re-insertion of
the atoms:
“backmapping”
n scattering, PS-30
parallel to
stretching
perpendicular
to stretching
48
G. Santangelo, A. di Matteo, F. Müller-Plathe, G. Milano, J. Phys. Chem. B 111, 2765 (2007).
X. Chen, P. Carbone, G. Santangelo, A. di Matteo, G. Milano, FMP, Phys. Chem. Chem. Phys. 11, 1977 (2009).
1
216
PS-9
91
116
155
0.1
500
510
520
530
540
Temperature (K)
550
Zero-shear Viscosity(cP)
10
experiment
Self-diffusion
Zero-shear Viscosity (cP)
Viscosity: Comparison with Experiment
PS-100
100
336
285
10
1
560
520
525
530
535
540
Temperature (K)
simulation
49
But…
Polyamide-6,6 above 500 K:
coarse-grained dynamics
→ atomistic dynamics
“finer coarse-graining”
6, 7, or 9 atoms/superatom
(polystyrene: 18)
Dcoarse-grained/Datomistic
Chain diffusion coefficient
Explanation???
Interaction details unimportant at high T?
Only excluded volume matters?
H.A. Karimi Varzaneh, X. Chen and F. Müller-Plathe, J. Chem. Phys. 128, 064904 (2008)
50
Challenge #1: Dynamics
• RNEMD works technically also for coarse-grained models
• Reproduce qualitative features: MW dependence,
microstructure, …
• Deviation from experiment: 1-2 orders
• Coarse-grained model reproduces simultaneously:
• structure
• density
• absolute dynamics
• temperature dependence
• Instead: try the equations of motion
51
Dynamics: Cure the Symptoms (2)
[H.-J. Qian, C. C. Liew, FMP, Phys. Chem. Chem. Phys. 11, 1962 (2009).]
Newtonian dynamics → Dissipative particle dynamics
rand
f ij f ij interactio n
v ij e ij e ij
e ij
12


 
Δt 


friction
random
friction forces and random forces act between pairs of atoms
– (ffriction+ frandom)
eij
ffriction+ frandom
no net momentum is added/lost
DPD is momentum-conserving → can be used with RNEMD
52
Dynamics: Cure the Symptoms (3)
[H.-J. Qian, C. C. Liew, FMP, Phys. Chem. Chem. Phys. 11, 1962 (2009).]
1. Determine time scaling factor between coarse-grained
dynamics and reference dynamics (atomistic or experiment),
e.g.
D CG , MD D ref , MD
2. Empirical rule for optimum
DPD noise strength
[ N s1/ 2 ] 7.2( 0.9) 0.35 3.4
Developed for ethyl benzene (298 K);
Holds for:
• ethyl benzene (238 K – 380 K)
• polystyrene melts
• ionic liquid [bmim] PF6
Analogous rule for Lowe-Anderson dynamics
53
Not a Bridge yet!
54
Summary
Reverse non-equilibrium MD
• Works robustly for thermal conductivity and
shear viscosity
• Is as accurate as the force field
• Cannot beat the inherent relaxation times
• Proliferation of RNEMD
Multiscale simulation
• Iterative Boltzmann Inversion works robustly
• Structure-based coarse graining for many systems
• proliferation of IBI
Coarse-grained dynamics
• Cannot fix the potential
• EOM with friction works phenomenologically
55
• Needed: friction contribution of removed degrees of freedom
Thanks!
2002-2005
< 2002
>2005
BASF, Rhodia, BMBF, EU (Marie-Curie), DFG,
Max-Planck-Gesellschaft (IMPRS-PMS), Humboldt56
Stiftung, Fonds der Chemischen Industrie