Studies of Nano, Chemical, and Biological Materials by Studies of Nano, Chemical, and Biological Materials by Molecular SimulationsMolecular Simulations

Yanting Wang

Institute of Theoretical Physics, Chinese Academy of Sciences Beijing, China September 25, 2008

Institute of Theoretical Physics, Chinese Academy of Sciences

Atomistic Molecular Dynamics SimulationAtomistic Molecular Dynamics Simulation

i ij ijj

t tF F r

ii

i

tt

m

Fa

1i i it t t t v v a

1 1i i it t t t r r v

Empirical force fields 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)

ijF

Solving Newton’s Equations of Motion.

Quantifying Condensed Matter StructuresQuantifying Condensed Matter Structures

Bond-Orientational Order Parameters

Radial Distribution Function g(r)

• Capture the symmetry of spatial orientation of chemical bonds

• Non-zero values for crystal structures

• 0 for liquid

• 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)

Molecular electronicsIon detection

S. O. Obare et al., Langmuir 18, 10407 (2002)

R. F. Service, Science 294, 2442 (2001)

Electronic lithography

J. Zheng et al., Langmuir 16, 9673 (2000)

Both size and shape are important in experiments!

Chemical etchingGold nanowiresLarger Au particles change color

Some Applications of Gold NanomaterialsSome Applications of Gold Nanomaterials

Thermal Stability of Low Index Gold SurfacesThermal Stability of Low Index Gold Surfaces

Thermal stability of surface: {110} < {100} < {111}

Stable gold interior: FCC structure

Stability of Icosahedral Gold NanoclustersStability of Icosahedral Gold Nanoclusters**

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

Icosahedron at T=200K

Mackay Icosahedron with a missing central atom Asymmetric facet sizes

Simulated annealing from a liquid

* Y. Wang, S. Teitel, C. Dellago Chem. Phys. Lett. 394, 257 (2004)* Y. Wang, S. Teitel, C. Dellago J. Chem. Phys. 122, 214722 (2005)

Strained FCC interior All covered by stable {111} facets

Liquid at T=1500K

Cooling

First-Order Like Melting Transition First-Order Like Melting Transition

Potential energy vs. T

Surface

Interior

Cone algorithm* to group atoms into layers

Sub-layers

Heat to melt

Keep T constant for 43 ns T = 1075K for N = 2624 Magic and non-magic numbers

First-order like melting transition

* Y. Wang, S. Teitel, C. Dellago J. Chem. Phys. 122, 214722 (2005)

SurfaceInterior

Interior keeps ordered up to melting temperature Tm

Surface softens but does not melt below Tm

No Surface Premelt for Gold Icosahedral NanoclustersNo Surface Premelt for Gold Icosahedral Nanoclusters

N = 2624

Mean squared displacements (average diffusion)

All surface atoms diffuse just below melting Surface premelting?

Surface Atoms Diffuse Below MeltingSurface Atoms Diffuse Below Melting

N = 2624

t=1.075ns

4t

Movement

Movement

Average shape

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 !

““Premelt” of Vertices and Edges but not FacetsPremelt” of Vertices and Edges but not Facets

Mechanism

ConclusionsConclusions

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?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 Smeared Melting Transition for NN = 54 = 54**

Heat up sequentially

timestep 2.86 fs

108 steps at each T

Average potential energy per atom Heat capacity

Easy to disorder due to less binding energy

Melting transition from Ts ≈ 300 K to Te ≈ 1200 K

Ts

Te

* Y. Wang, S. Rashkeev J. Phys. Chem. C 113, 10517 (2009).

Snapshots at Different TemperaturesSnapshots at Different Temperatures

Both layers premelt below 560 K

No inter-layer diffusion below 560 K

Inter- and Intra- Layer DiffusionInter- and Intra- Layer Diffusion

Inter-layer diffusion starts at Ti ≈ 560 KAtomic self diffusion starts at Td ≈ 340 K

Td

Ti

22

1 1

1 M N

i j i jj i

r t t t tMN

r r

2 6r t Dt

Moved atoms: moving to the other layer at least once at each temperature

Ti

Liquid crystal-like structure between 340 K and 560 K

More Layers in Between: Approaching More Layers in Between: Approaching First-Order Melting TransitionFirst-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).

ConclusionsConclusions

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

Increasing total E continuously to mimic laser heating

T=5K T=515K

T=1064K T=1468K

Experimental model

Z. L. Wang et al., Surf. Sci. 440, L809 (1999)

Pure FCC interior

Thermal Stability of Gold NanorodsThermal Stability of Gold Nanorods**

Two steps

* Y. Wang, C. Dellago J. Phys. Chem. B 107, 9214 (2003).

Surface-Driven Bulk Reorganization of Gold NanorodsSurface-Driven Bulk Reorganization of Gold Nanorods**Surface Second sub layer

Yellow: {111} Green: {100}Red: {110} Gray: other

Cross sections

Yellow: fcc Green: hcp Gray: other

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

* Y. Wang, S. Teitel, C. Dellago Nano Lett. 5, 2174 (2005).

ConclusionsConclusions

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) MethodMultiscale Coarse-Graining (MS-CG) Method** to Rigorously to Rigorously Build CG Force Fields from All-Atom Force FieldsBuild 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

Assuming central pairwise effective forces Minimizing force residual

2, ,1 13

I CGN NI AA I CG

ICG I

F FN N a a

a

Y = -å åv v

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).

Benifit: maller numbers of degrees of freedom and faster dynamics

( )̂CG CGF F r ra ab ab abb

=åv

Residual:

Each CG site:

Effective force: ( ) ( )dCG

DF fr r dd= -

2, ,

,,

1 13

12

3

I CGN NI AA I CG

ICG I

ddddd dCG

d dd

F FN N

G b cff fN

a aa

¢¢

¢

Y = -

æ ö÷ç ÷= - +ç ÷ç ÷çè ø

å å

å å

v v

( ) ( ) ( ),,

1 ˆ ˆI I I ID Ddd

II

G R R R d R dN ab ag ab ag

a b g

d d¢ ¢= - -å å å g

( ) ( ),1 ˆI I AA Id D

II

b R F R dN ab a ab

a b

d= -å å åvg

, ,1 I AA I AA

II

c F FN a a

a

= å åv v

g

Central pairwise, linear approximation

Multidimensional parabola

Obtained fromall-atom configurations

Multiscale Coarse-Graining by Force MinimizationMultiscale Coarse-Graining by Force Minimization

Residual: ,,

12

3 d dd ddddd dCG

ff fb cN ¢ ¢

¢

æ ö÷ç ÷Y = - +ç ÷ç ÷çè øå åG

Variational principle: 0dd

gf

¶Y= =¶ , d ddd

d

f b¢¢

¢=å G

Or finding the minimal solution by conjugate gradient minimization with Ψ and g

d

Only one minimal solution!

Ψ can be used to determine the best CG scheme

Subtract the Ewald Sum (long-range electrostatic) of point net charges

Match bonded and non-bonded interactions separately

Force Minimization by Conjugate Gradient MethodForce Minimization by Conjugate Gradient Method

Solving matrix directly

• Explicitly calculating pairwise atomic interactions between two groups All-atom MD to get the ensemble of relative orientations

• Very limited transferability: temperature, surface, sequence of amino acids Wrong pressure (density) without further constraint

* Y. Wang, W. Noid, P. Liu, G. A. Voth to be submitted.

Effective Force Coarse-Graining (EF-CG) MethodEffective Force Coarse-Graining (EF-CG) Method**

EF-CG non-bonded effective forces

Problems with MS-CG

1 1

1 1

ˆ

ˆˆ

M N

ij Di jD

D D

M N

ij ij ij Di j

D

r R RR R

F RR R R R

F r R R

R R

R

R

r R

1 1

M N

ij iji j

r r

ConclusionsConclusions

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 PolyglutaminesMS-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.

Ionic liquid = Room temperature molten salt Non-volatile High viscosity

Some Applications of Ionic LiquidsSome Applications of Ionic Liquids

Environment-friendly solvent for chemical

reactions Lubricant Propellant

* Y. Wang, S. Izvekov, T. Yan, and G. Voth, J. Phys. Chem. B 110, 3564 (2006).

Multiscale Coarse-Graining of Ionic LiquidsMultiscale Coarse-Graining of Ionic Liquids**

EMIM+/NO3- ionic liquid

64 ion pairs, T = 400 K Electrostatic and VDW interactions

Site-site RDFs (T = 400K)

Good structures No temperature transferability

Satisfactory CG Structures of Ionic LiquidsSatisfactory CG Structures of Ionic Liquids

Spatial Heterogeneity in Ionic LiquidsSpatial Heterogeneity in Ionic Liquids**

C1

C2

C4

C6

C8

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

* Y. Wang, G. A. Voth, J. Am. Chem. Soc. 127, 12192 (2005).

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).

Define Heterogeneity order parameter (HOP)

hi exp( rij2 / 2 2)

j

L

N 1/3

• Invariant with box size L

• Average over all sites to get <h>

• For each site

Larger HOP represents more heterogeneous configuration.

Heterogeneity Order ParameterHeterogeneity Order Parameter**

Thermal Stability of Tail Domain in Ionic LiquidsThermal Stability of Tail Domain in Ionic Liquids**

* Y. Wang, G. A. Voth, J. Phys. Chem. B 110, 18601 (2006).

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?

Tail Domain Diffusion in Ionic LiquidsTail Domain Diffusion in Ionic Liquids Instantaneous LHOPs at T = 1230 KDefine Lattice HOP

Divide simulation box into cells

In each cell the ensemble average of HOP is

taken for all configurations

ci 1

Mhij

j1

M

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 LiquidsExtendable EF-CG Models of Ionic Liquids**

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).

CG force library Extendibility,

transferability, and manipulability

Disordering and Reordering of Ionic Liquids under Disordering and Reordering of Ionic Liquids under an External Electric Fieldan External Electric Field**

* Y. Wang J. Phys. Chem. B 113, 11058 (2009).

From heterogeneous to homogeneous to nematic-like due to the effective screening of the external electric field to the internal electrostatic interactions.

ConclusionsConclusions

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 MembranesPolymers for Gas-Separation Membranes

CO2 Capturer Air Dryer Air Mask

Environmental applications

Energy applications

Industrial applications

Military applications

…

UBE.com

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

Polybenzimidazole (PBI)

N

N N

NH

H

nH

HH

H

H H

HH

H H

Determining Crystalline Structure of PolymersDetermining Crystalline Structure of Polymers

Polybenzimidazole (PBI)

N

N N

NH

H

nH

HH

H

H H

HH

H H

Poly[bis(isobutoxycarbonyl)benzimidazole] (PBI-Butyl)

N

N N

N

n

O

O

OO

Kapton

N N O

O

O O

O

n

X-Z Plane Y-Z Plane

Infinitely-Long Crystalline Polymers at T = 300 KInfinitely-Long Crystalline Polymers at T = 300 K

N

N N

NH

H

nH

HH

H

H H

HH

H H

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

PBI + CO2

PBI + N2

Sizes along Y are expanded.Gas molecules can hardly get in between the layers.

Very stiff

CO2 and N2 inside PBICO2 and N2 inside PBI

PBI-Butyl + CO2

PBI-Butyl + N2

N

N N

N

n

O

O

OO

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

No dimension sizes are changed. Gas molecules are free to diffuse between layers.

Open up spaces

CO2 and N2 inside PBI-ButylCO2 and N2 inside PBI-Butyl

Kapton + CO2

Kapton + N2

Sizes along Z are expanded. Gas molecules change the crystal structure of Kapton.

Flexible

N N O

O

O O

O

n

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

CO2 and N2 inside KaptonCO2 and N2 inside Kapton

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.

ConclusionsConclusions

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 (I)Cracking of Crystalline PBI by Water (I)

Initial Middle Final

Water molecules stick together by hydrogen bonds

PBI crystal structure change slightly

16 water molecules

Cracking of Crystalline PBI by Water (II)Cracking of Crystalline PBI by Water (II)

Initial Final

Water molecules form hydrogen bonding network

PBI crystal structure change drastically

Cracking of Crystalline PBI by Water Cracking of Crystalline PBI by Water (III)(III)

160 water molecules

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.

ConclusionsConclusions

Fluctuation TheoremsFluctuation Theorems Jarzynski’s equality: ensemble average over all nonequilibrium trajectories

exp expB B

F Wk T k T

æ ö æ öD ÷ ÷ç ç÷ ÷- = -ç ç÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

C. Jarzynski Phys. Rev. Lett. 78, 2690 (1997)

Crook’s theorem: involving nonequilibrium trajectories for both ways

expF

R B

P W W F

P W k T

G. E. Crooks Phys. Rev. E 60, 2721 (1999)

Calculate free energy difference from fast nonequilibrium simulations. Transiently absorb heat from environment.

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