Studies of Nano, Chemical, and Biological Materials by
Molecular Simulations
Yanting Wang
Institute of Theoretical Physics, Chinese Academy of Sciences
Institute of Theoretical Physics, Chinese Academy of Sciences
Beijing, China
September 25, 2008
Atomistic Molecular Dynamics Simulation
 Solving Newton’s Equations of Motion.
Fi  t    Fij rij  t  
j
ai  t  
Fi  t 
mi
vi  t  1  vi  t   ai  t   t
ri  t  1  ri  t   vi  t  1  t
 Empirical force fields Fij 
are determined by fitting experimental results or data from first principles
calculations
 Quality of empirical force fields has big influence on simulation results
 Capable of simulating up to millions of atoms (parallel computing)
Quantifying Condensed Matter Structures
 Bond-Orientational Order Parameters
• Capture the symmetry of spatial orientation of chemical bonds
• Non-zero values for crystal structures
• 0 for liquid
 Radial Distribution Function g(r)
• Appearance probability of other atoms with respect to a given atom
• Discrete values for solids
• Continuous waves for liquids
• 1 for ideal gas (isotropic structure)
Some Applications of Gold Nanomaterials
Ion detection
Larger Au particles change color
S. O. Obare et al., Langmuir 18,
10407 (2002)

Molecular electronics
Electronic lithography
Gold nanowires
R. F. Service, Science 294, 2442
(2001)
Chemical etching
J. Zheng et al., Langmuir 16,
9673 (2000)
Both size and shape are important in experiments!
Thermal Stability of Low Index Gold Surfaces

Stable gold interior: FCC structure

Thermal stability of surface: {110} < {100} < {111}
Stability of Icosahedral Gold Nanoclusters*
Simulated annealing from a liquid
 Empirical glue potential model
 Constant T molecular dynamics (MD)
 From 1500K to 200K with T=100K, and keep T constant for 21 ns
 thousands of atoms
Cooling
Liquid at T=1500K
 Strained FCC interior
 All covered by stable {111} facets
Icosahedron at T=200K
 Mackay Icosahedron with a missing central atom
 Asymmetric facet sizes
* Y. Wang, S. Teitel, C. Dellago Chem. Phys. Lett. 394, 257 (2004)
* Y. Wang, S. Teitel, C. Dellago J. Chem. Phys. 122, 214722 (2005)
First-Order Like Melting Transition
Heat to melt
 Keep T constant for 43 ns
 T = 1075K for N = 2624
 Magic and non-magic numbers
 Cone algorithm* to group atoms into layers
Potential energy vs. T
Surface
Interior

First-order like melting transition
* Y. Wang, S. Teitel, C. Dellago J. Chem. Phys. 122, 214722 (2005)
Sub-layers
No Surface Premelt for Gold Icosahedral Nanoclusters
Interior
Surface
N = 2624
 Interior keeps ordered up to melting temperature Tm
 Surface softens but does not melt below Tm
Surface Atoms Diffuse Below Melting
Mean squared displacements (average diffusion)
N = 2624
 All surface atoms diffuse just below melting
 Surface premelting?
“Premelt” of Vertices and Edges but not Facets
Movement
t=1.075ns
Movement
4t
Average shape
Mechanism
 Vertex and edge atoms diffuse increasingly with T
 Facets shrink but do not vanish below Tm=1075 K
 Facet atoms also diffuse below Tm because the facets are very small !
Conclusions
 First-order like melting transition for gold nanoclusters with
thousands of atoms
 Very stable {111} facets result in good thermal stability of
icosahedral gold nanoclusters
 Vertex and edge “premelt” softens the surface but no overall
surface premelting
Very Small Gold Nanoclusters?
 Smaller gold nanocluster has more active catalytic ability
 Debate if very small gold nanoclusters (< 2 nm ) are solid or liquid
 54 gold atoms (only two layers)
 Not an icosahedron
All surface atoms are on vertex or edge!
Smeared Melting Transition for N = 54*
 Heat up sequentially
 timestep 2.86 fs
 108 steps at each T
 Easy to disorder due to less binding energy
 Melting transition from Ts ≈ 300 K to Te ≈ 1200 K
Te
Ts
Average potential energy per atom
* Y. Wang, S. Rashkeev J. Phys. Chem. C 113, 10517 (2009).
Heat capacity
Snapshots at Different Temperatures
Both layers premelt below 560 K
 No inter-layer diffusion below 560 K
Inter- and Intra- Layer Diffusion
r 2  t  
2
1 M N


r
t

t

r
t
 i  j  i  j 
MN j 1 i 1 
r 2  t   6 Dt
Moved atoms: moving to the other
layer at least once at each temperature
Ti
Ti
Td
Atomic self diffusion starts at Td ≈ 340 K
Inter-layer diffusion starts at Ti ≈ 560 K
Liquid crystal-like structure between 340 K and 560 K
More Layers in Between: Approaching
First-Order Melting Transition*
Onset Temperature Ts and Complete Temperature Te of Melting Transition, Self
Diffusion Temperature Td, and Interlayer Diffusion Temperature Ti
atoms
layers
Ts
Te
Ti
Td
54
2
300
1200
560
340
146
3
350
1000
300
450
308
4
400
900
400
500
560
5
550
850
500
600
 Melting temperature region narrows down for more layers
 Only two-layer cluster has intra-layer diffusion first
* Y. Wang, S. Rashkeev J. Phys. Chem. C 113, 10517 (2009).
Conclusions
 Smeared melting transition for two-layer gold nanocluster
 Mechanism consistent with icosahedral gold nanoclusters
 Liquid-crystal like partially melted state for two-layer gold
nanocluster: intra-layer diffusion without inter-layer diffusion
 Approaching well-defined first-order melting transition for gold
nanoclusters with more layers
 Very small gold nanoclusters have abundant phase behavior
that can not be predicted by simply extrapolating the behavior of
larger gold nanoclusters
Thermal Stability of Gold Nanorods*
 Increasing total E continuously to mimic laser heating
Experimental model
T=5K
T=1064K
T=515K
T=1468K
Pure FCC
interior
Z. L. Wang et al., Surf. Sci. 440, L809 (1999)
Two steps
* Y. Wang, C. Dellago J. Phys. Chem. B 107, 9214
(2003).
Surface-Driven Bulk Reorganization of Gold Nanorods*
Surface Second sub layer
Cross sections
 Temperature by
temperature step heating
 Minimizing total
surface area
Surface changes to all
{111} facets
 Interior changes
fcc→hcp→fcc by
sliding planes, induced
by surface change
 Interior fcc reorients
Yellow: {111} Green: {100}
Red: {110} Gray: other
Yellow: fcc Green: hcp Gray: other
* Y. Wang, S. Teitel, C. Dellago Nano Lett. 5, 2174 (2005).
Conclusions
 Thermal stability of gold nanoclusters and gold nanorods is
closely related to specific surface structures (not only surface
stress matters)
 Shape change of gold nanorods comes from the balance
between surface and internal free energetics
Multiscale Coarse-Graining (MS-CG) Method* to Rigorously
Build CG Force Fields from All-Atom Force Fields
• Pioneer work by Dr. Sergey Izvekov with block-averaging
• Theory by Prof. Will Noid (Penn State U), Prof. Jhih-Wei Chu (UC-Berkeley), Dr. Vinod Krishna, and Prof. Gary Ayton
• Help from Prof. Hans C. Andersen (Stanford)
• I implemented the force-minimization approach
Benifit: maller numbers of degrees of freedom and faster dynamics
Assuming central pairwise effective forces
 Minimizing force residual
N
IN
C
G
v
v
2
11
I
,
A
A I
,
C
G
Y
=
F
F
å
å
a
a
3
N
N
C
G
II a
 Well rebuild structural properties
 Can eliminate some atoms at CG level
 Does NOT consider transferability!
* W. Noid, P. Liu, Y. Wang et al. J. Chem. Phys. 128, 244115 (2008).
Multiscale Coarse-Graining by Force Minimization
CG
Effective force: F (r ) = fd dD (r - d )
Each CG site:
Central pairwise, linear approximation
v
C
G
C
G
ˆ
F
r
r
)
åF
a =
a
b(
a
b
a
b
b
Residual:
C
G v
v
2
1 1 NI N
IA
,A
IC
,G
Y
=
F
-F
åå
a
a
3
N
N
I
a
C
G I
ö
1æ
÷
ç
÷
=
G
ff
2
b
f
+
c
ç
å
å
¢
¢
d
d
d
÷
d
,
d
d
ç
÷
3
Nç
è¢
ø
,d
C
G d
d
Multidimensional parabola
1
Iˆ
I
I
I
ˆ
¢
G
=
R
g
R
d
R
d
d
R
d
(
)
(
)
(
)
å
å
å
¢
a
b
a
g
D
a
b
D
a
g
d
,
d
N
Ia
b
,
g
I
v
1
Î I
,
A
A
I
b
=
R
g
F
d
R
d
(
)
(
å
å
å
d
a
b
a)
D
a
b
N
I I ab
v
v
1
I
,
A
A I
,
A
A
c
= å
F
F
å
a g
a
N
I I a
Obtained from
all-atom configurations
Force Minimization by Conjugate Gradient Method
Residual:
æ
ö
1
÷
ç
÷
 Solving matrix directly
Y
= ç
G
f
f
2
b
f
+
c
å
å
¢
¢
d
d
d ÷
d
,
d
d
ç
÷
ç
3
N
è
ø
¢
d
,
d
d
C
G
Variational principle:

gd =
¶Y
=0
¶fd
åG f
d¢
=b
d
dd
, ¢d¢
Or finding the minimal solution by conjugate gradient minimization with Ψ and gd
Subtract the Ewald Sum (long-range electrostatic)
of point net charges

Match bonded and non-bonded interactions
separately


Only one minimal solution!

Ψ can be used to determine the best CG scheme
Effective Force Coarse-Graining (EF-CG) Method*
 Problems with MS-CG
• Very limited transferability: temperature, surface, sequence of amino acids
 Wrong pressure (density) without further constraint
 EF-CG non-bonded effective forces
• Explicitly calculating pairwise atomic interactions between two groups
 All-atom MD to get the ensemble of relative orientations



r
i
j 
i
j



r

M
N
i

1
j1
ˆ

R



RR
r

ij
D
MN



RR

R


D
F
R








RR
D
i
1j
1


RR
D
ˆ ˆ
F

rRR
rR


MN
i
1j
1

ij
ij
ij
D


RR
D
* Y. Wang, W. Noid, P. Liu, G. A. Voth to be submitted.
Conclusions
 CG methods enable faster simulations and longer effective simulation time
 MS-CG method rebuilds structures accurately but has very limited
transferability
 MS-CG method can eliminate some atoms (e.g., implicit solvent)
 EF-CG method has much better transferability by compromising a
little accuracy of structures
MS-CG MD Study of Aggregation of Polyglutamines*
 Polyglutamine aggregation is the clinic cause
of 14 neural diseases, including Huntington’s,
Alzheimer's, and Parkinson's diseases
 All-atom simulations have a very slow
dynamics that can not be adequately sampled
 Water-free MS-CG model
 CG MD simulations extend from nanoseconds
to milliseconds
CG MD results consistent with experiments:
 Longer chain system exhibits stronger aggregation
 Degrees of aggregation depend on concentration
 Mechanism based on weak VDW interactions and fluctuation nature
* Y. Wang, G. A. Voth to be submitted.
Some Applications of Ionic Liquids
 Ionic liquid = Room temperature molten salt
 Non-volatile
 High viscosity
Environment-friendly
solvent for chemical
reactions
Lubricant
Propellant
Multiscale Coarse-Graining of Ionic Liquids*
 EMIM+/NO3- ionic liquid
 64 ion pairs, T = 400 K
 Electrostatic and VDW interactions
* Y. Wang, S. Izvekov, T. Yan, and G. Voth, J. Phys. Chem. B 110, 3564 (2006).
Satisfactory CG Structures of Ionic Liquids
Site-site RDFs (T = 400K)
Good structures
 No temperature transferability

Spatial Heterogeneity in Ionic Liquids*
With longer cationic side chains:
 Polar head groups and anions retain local structure due to electrostatic interactions
 Nonpolar tail groups aggregate to form separate domains due to VDW interactions
C1
C2
C4
C6
C8
* Y. Wang, G. A. Voth, J. Am. Chem. Soc. 127, 12192 (2005).
Heterogeneity Order Parameter*
Define Heterogeneity order parameter (HOP)
• For each site
hi   exp(r / 2 )
2
ij
j
2

L
N 1/3
• Average over all sites to get <h>
• Invariant with box size L
 Larger HOP represents more heterogeneous configuration.
 Quantifying degrees of heterogeneous distribution by a single value
 Detecting aggregation
 Monitoring self-assembly process
* Y. Wang, G. A. Voth J. Phys. Chem. B 110, 18601 (2006).
Thermal Stability of Tail Domain in Ionic Liquids*
 Heat capacity plot shows a second order transition at T = 1200 K
 Contradictory: HOP of instantaneous configurations do not show a transition at T =
1200 K?
* Y. Wang, G. A. Voth, J. Phys. Chem. B 110, 18601 (2006).
Tail Domain Diffusion in Ionic Liquids
 Instantaneous LHOPs at T = 1230 K
Define Lattice HOP
 Divide simulation box into cells
 In each cell the ensemble average of HOP is
taken for all configurations
1
ci 
M
M
h
ij
j 1
Mechanism
 Heterogeneous tail domains have fixed positions at low T
(solid-like structure)
 Tail domains are more smeared with increasing T
 Above Tc, instantaneous tail domains still form (liquid-like
structure), but have a uniform ensemble average
Extendable EF-CG Models of Ionic Liquids*
 CG
force library
 Extendibility,
transferability, and
manipulability



Extendable CG models correctly rebuild spatial heterogeneity
features
CG RDFs do not change much for C12 from 512 (27,136) to 4096
ion pairs (217,088 atoms)
Proving spatial heterogeneity is truly nano-scale, not artificial
effect of finite-size effect
* Y. Wang, S. Feng, G. A. Voth J. Chem. Theor. Comp. 5,
1091 (2009).
Disordering and Reordering of Ionic Liquids under
an External Electric Field*
From heterogeneous to
homogeneous to nematic-like due
to the effective screening of the
external electric field to the
internal electrostatic interactions.
* Y. Wang J. Phys. Chem. B 113, 11058 (2009).
Conclusions
 Spatial heterogeneity phenomenon was found in ionic liquids,
attributed to the competition of electrostatic and VDW interactions
 Solid-like tail domains in ionic liquids go through a second order
melting-like transition and become liquid-like above Tc
 EF-CG method was applied to build extendable and transferable CG
models for ionic liquids, which is important for the systematic design of
ionic liquids
 Ionic liquid structure changes from spatial heterogeneous to
homogeneous to nematic-like under an external electric field
Polymers for Gas-Separation Membranes
UBE.com
CO2 Capturer
Air Dryer
 Environmental applications
 Energy applications
 Industrial applications
 Military applications
…
Air Mask
Determining Crystalline Structure of Polymers
Polybenzimidazole (PBI)
H
H
H
H
H
N
H
H
N
N
N
H
H
H
H
H
n
 AMBER force field
 Put one-unit molecules on lattice positions
 Relax at P = 1 atm and T = 10 K
 Measure lattice constants in relaxed configuration
Infinitely-Long Crystalline Polymers at T = 300 K
X-Z Plane
Polybenzimidazole (PBI)
H
H
H
H
H
N
H
H
N
H
N
N
H
H
H
n
H
Poly[bis(isobutoxycarbonyl)benzimidazole] (PBI-Butyl)
O
O
N
N
N
N
O
O
n
Kapton
O
O
N
N
O
O
O
n
Y-Z Plane
CO2 and N2 inside PBI
PBI + CO2
Very stiff
H
H
H
H
H
N
H
H
N
N
N
H
H
H
H
n
H
PBI + N2
Sizes along Y are expanded.
Gas molecules can hardly
get in between the layers.
System
X (Å)
Y (Å)
Z (Å)
Volume (nm3)
PBI
75.09 ± 0.11
28.38 ± 0.09
25.83 ± 0.12
55.05 ± 0.19
PBI + CO2
75.08 ± 0.05
29.97 ± 0.11
25.82 ± 0.08
58.10 ± 0.17
PBI + N2
75.09 ± 0.06
29.57 ± 0.18
26.03 ± 0.10
57.68 ± 0.17
CO2 and N2 inside PBI-Butyl
PBI-Butyl + CO2
Open up spaces
O
O
N
N
N
N
O
O
PBI-Butyl + N2
n
No dimension sizes are
changed. Gas molecules
are free to diffuse
between layers.
System
X (Å)
Y (Å)
Z (Å)
Volume (nm3)
PBI-Butyl
75.55 ± 0.05
52.35 ± 0.18
30.25 ± 0.07
119.51 ± 0.31
PBI-Butyl + CO2
75.52 ± 0.05
52.05 ± 0.20
30.30 ± 0.08
119.08 ± 0.39
PBI-Butyl + N2
75.52 ± 0.05
52.41 ± 0.20
30.22 ± 0.08
119.61 ± 0.37
CO2 and N2 inside Kapton
Kapton + CO2
Flexible
O
O
N
N
O
O
O
Kapton + N2
n
Sizes along Z are expanded.
Gas molecules change the
crystal structure of Kapton.
System
X (Å)
Y (Å)
Z (Å)
Volume (nm3)
Kapton
84.77 ± 0.10
27.50 ± 0.04
27.02 ± 0.07
62.97 ± 0.15
Kapton + CO2
84.91 ± 0.06
27.80 ± 0.09
28.36 ± 0.10
66.94 ± 0.16
Kapton + N2
84.65 ± 0.08
26.63 ± 0.11
30.43 ± 0.14
68.58 ± 0.18
Conclusions
 PBI forms a very strong and closely packed crystalline structure.
 CO2 and N2 can hardly diffuse in PBI crystal.
 Crystal structure of PBI-Butyl is rigid, but the butyl side chains make
the interlayer distances larger.
 CO2 and N2 can freely diffuse between the layers.
 Kapton crystal structure is also closely packed, but the interlayer
coupling is weaker than in PBI.
 CO2 and N2 can be accommodated between the layers which
increases the interlayer distances.
 CO2 and N2 behave similar in these three crystalline polymers.
Cracking of Crystalline PBI by Water (I)
Water
PBI
Initial
Final
 Water molecules are attracted to PBI surface
 Water molecules do not penetrate inside PBI
 Water cluster suppresses the collective thermal vibration of PBI crystal
Cracking of Crystalline PBI by Water (II)
Initial
16 water
molecules
Middle
 Water molecules stick together by hydrogen bonds
 PBI crystal structure change slightly
Final
Cracking of Crystalline PBI by Water (III)
Initial
160 water
molecules
Final
 Water molecules form hydrogen bonding network
 PBI crystal structure change drastically
Conclusions
 To crack the crystal structure, PBI must have defects.
 Strong binding of water molecules by hydrogen bonding network is
possible to destroy local PBI crystal structures, thus to crack the crystal.
Fluctuation Theorems
 Jarzynski’s equality: ensemble average over all nonequilibrium trajectories
C. Jarzynski Phys. Rev. Lett. 78, 2690 (1997)
æ DF ÷
ö
ç
÷=
exp çç÷
çè kBT ÷
ø
æ W ö
÷
÷
exp ççç÷
÷
çè kBT ø
 Crook’s theorem: involving nonequilibrium trajectories for both ways
G. E. Crooks Phys. Rev. E 60, 2721 (1999)
PF W 
 W  F 
 exp 

PR W 
k
T
 B

 Calculate free energy difference from fast
nonequilibrium simulations.
 Transiently absorb heat from environment.
高级研究生课程
分子建模与模拟导论：2009年秋季
星期三下午15:20 – 17:00 S102教室