simulations at nanoscales

Simulations at Nanoscales

Prof. Yan WangWoodruff School of Mechanical Engineering

Georgia Institute of TechnologyAtlanta, GA 30332, [email protected]

mailto:[email protected]

Multiscale Systems Engineering Research Group

Topics

Molecular Dynamics

Kinetic Monte Carlo


Molecular Dynamics

Center locations of atoms:

Inter-atomic potential function:

Force on atom i caused by N−1 other atoms:

+: repulsive force

−: attractive force

Momentum of atoms:

domain

{ }1, ,N

Nr r r= …

( )NrU

( )N

ii

rF

r

∂= −

∂U

{ } 11 1, , , ,N N

N N

dr drp p p m m

dt dt

⎧ ⎫⎪ ⎪= = ⎨ ⎬⎪ ⎪⎩ ⎭

… …


Molecular Dynamics

For an isolated system, Hamiltonian = total energy

MD simulation computes the phase-space trajectory

Newtonian Dynamics View Hamiltonian Dynamics ViewMotion is a response to an applied force

No explicit forces. Motion occurs to preserve the Hamiltonian function.

3N dimensional of configurationspace ri(t)’s

Phase space = 3N dimensional configuration space ri(t)’s + 3Ndimensional momentum space pi(t)’s

i i

ii i

p dr

p m dt∂

= =∂H

i

i

dp

r dt∂

= −∂H

2

2i

i ii

d rm F

rdt

∂= = −

∂U

( , )N Nr p const=H

11

1( , ) ( , , )

2

NN N

i i Ni i

r p p p r r Em=

= + =∑H Ui …


A simple example of phase-spaceone-dimensional harmonic oscillator

21( )

2x xγ=U

2 21 12 2

E p x constm

γ= + =

F xx

γ∂= − = −

∂U


Lennard-Jones Pair Potential

( )

n m

ijij ij

u r kr rσ σε

⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟= −⎢⎜ ⎟ ⎜ ⎟ ⎥⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

12 8

( ) 4ij

ij ij

u rr rσ σε

⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟= −⎢⎜ ⎟ ⎜ ⎟ ⎥⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

m

n mn nkn m m

−⎛ ⎞= ⎜ ⎟− ⎝ ⎠

where

- σ is the inter-atomic distance where u(σ)=0 - ε is the minimum energy

Lennard-Jones(12,6):


Reduce Computational Complexity

The most time-consuming part of a MD simulation is the calculation of the forces from pairwise potentials .

Truncated Potential: introducing the cut-off distance rc (where u(r)=0 for r> rc).

e.g. in L-J(12,6), rc=2.5σdiscontinuity at r=rc

error introduces small fluctuations in E

Shifted-Force Potential:

where

( ) ( )Nij

i j

r u r<

= ∑∑U

( )( )

( )0

cs

c

duF r r

F r drr r

∆⎧− + ≤⎪= ⎨⎪ >⎩

( )c

cr r

duF F r

dr∆

=

= − =


Implementation - Neighbor List

For each atom i, a list of neighboring atoms that lie within a distance rL of atom i is maintained. rL is slightly larger than rc , and typically rL = rc

+0.3σ. The force on atom i is only contributed by those listed atoms. The neighbor list is updated periodically over several (e.g. 10) consecutive time steps.For a 3D Lennard-Jones fluid at density ρσ3=0.8, each atom has about 75 neighbors lying within a radius rL=2.8σ. In average, 75/2≈40 storage locations per atom.


Verlet AlgorithmFrom Taylor series

(1)

(2)

Eq.(1)+Eq.(2)

Eq.(1)−Eq.(2)

Not self-starting: x(0) and v(0) are not enough to start

2 (3) 3 41 1( ) ( ) '( ) ''( ) ( ) ( )

2 ! 3 !r t t r t r t t r t t r t t O t∆ ∆ ∆ ∆ ∆+ = + + + +

2 (3) 3 41 1( ) ( ) '( ) ''( ) ( ) ( )

2 ! 3 !r t t r t r t t r t t r t t O t∆ ∆ ∆ ∆ ∆− = − + − +

2 4( ) 2 ( ) ( ) '( ) ( )r t t r t r t t v t t O t∆ ∆ ∆ ∆+ = − − + +

2( ) ( )( ) ( )

2

r t t r t tv t O t

t

∆ ∆ ∆∆

+ − −= +


System of Units in MD ProgramsWith u*=u/ε and r*=r/σ, the unitless form of Lennard-Jones(12,6) is

Others:

12 61 1

* ( *) 4* *

u rr r

⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥= −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦


Truncation and Round-off Errors in Forward Finite Difference Method

Integration step size Δt

Total Global Error

Round-off error dominates

Truncation error dominates


Gear’s Predictor-Corrector Algorithm

Predict (by Tayler series)

2 3 4 5(3) (4) (5)( ) ( ) ( ) ( )

( ) ( ) '( ) ''( ) ( ) ( ) ( )2 ! 3 ! 4 ! 5 !i i i i i i i

t t t tt t t t t t t t t

∆ ∆ ∆ ∆∆ ∆+ ≈ + + + + +r r r r r r r2 3 4

(3) (4) (5)( ) ( ) ( )'( ) '( ) ''( ) ( ) ( ) ( )

2 ! 3 ! 4 !i i i i i i

t t tt t t t t t t t

∆ ∆ ∆∆ ∆+ ≈ + + + +r r r r r r2 3

(3) (4) (5)( ) ( )''( ) ''( ) ( ) ( ) ( )

2 ! 3 !i i i i i

t tt t t t t t t

∆ ∆∆ ∆+ ≈ + + +r r r r r2

(3) (3) (4) (5) ( )( ) ( ) ( ) ( )

2 !i i i i

tt t t t t t

∆∆ ∆+ ≈ + +r r r r

(4) (4) (5)( ) ( ) ( )i i it t t t t∆ ∆+ ≈ +r r r

(5) (5)( ) ( )i it t t∆+ ≈r r


Gear’s Predictor-Corrector AlgorithmEvaluate

• where u(rij) is the potential energy function that act between atoms i and j , and is the unit vector

Correct (by inter-atomic force)

( )ˆ( ) ( )ij

i ijj i ij

u rt t t t

r∆ ∆

≠

∂+ = − +

∂∑F r

ijr

''( ) ( ) /i i it t t t m∆ ∆+ = +r F

''( ) ''( )i it t t t∆ ∆= + − +D r r

0( ) ( )i it t t t α∆ ∆+ ← + +r r R

1'( ) '( )i it t t t

tα∆ ∆∆

+ ← + +R

r r

2 2

2 !''( ) ''( )

( )i it t t t

tα∆ ∆

∆+ ← + +

Rr r

(3) (3)3 3

3 !( ) ( )

( )i it t t t

tα∆ ∆

∆+ ← + +

Rr r

(4) (4)4 4

4 !( ) ( )

( )i it t t t

tα∆ ∆

∆+ ← + +

Rr r

(5) (5)5 5

5 !( ) ( )

( )i it t t t

tα∆ ∆

∆+ ← + +

Rr r

2( )

2 !

t∆=R D


Gear’s Predictor-Corrector Algorithm

Coefficients

αi order=3 order=4 order=5

α0 1/6 19/120 3/16

α1 5/6 3/4 251/360

α2 1 1 1

α3 1/3 1/2 11/18

α4 1/12 1/6

α5 1/60


Comparison by Energy Conservation( )21

_ (0) ( )M

kglobal error E E k t∆

== −∑

( )21

1_ _ _ (0) ( )

M

kglobal error per step E E k t

M∆

== −∑

Gear’s predictor-corrector algorithm

1-D collision of two L-J atoms


Macroscopic Properties

Macroscopic properties result from the collective behavior of individual atoms.Any measurable property A(rN(t),pN(t)) is a function of the phase point.

Phase-space trajectory

The measured value <A> at the macroscopic scale is an average over a time interval t:

( )0

0

1( ), ( )

t tN N

t

A A dt

τ τ τ+

< >= ∫ r p

( ){ }( ), ( ) , 1, ,N Nk t k t k M∆ ∆ =r p …

( )1

1( ), ( )

M N N

kA A k t k t

M∆ ∆

=< >= ∑ r p

or


Thermodynamic Properties

Average kinetic energy

Temperature

Pressure

1 1

3 1( ) ( )

2 2

M N

k i ik i

E NkT k t k tmM

∆ ∆= =

< >= = ∑∑ p pi

1 1

1( ) ( )

3

M N

i ik i

T k t k tmkNM

∆ ∆= =

= ∑∑ p pi

( )11

3ij

iji j ij

du rPr

kT NkT drρ <

= − ∑∑



Constant-volume heat capacity• fluctuation of internal energy

or

Adiabatic compressibility

vV

EC

T

⎛ ⎞∂= ⎜ ⎟∂⎝ ⎠

( ) ( )22 22

2 2 2

1 1 1vC E E E E E

kT kT kTδ ⎛ ⎞= = − = −⎜ ⎟

⎝ ⎠1

* *

*

3 11

2v

v

k

C NC N NT

NK E

−⎡ ⎤⎛ ⎞

= = − −⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Kinetic energy

1s

s

VV P

κ⎛ ⎞∂

= − ⎜ ⎟∂⎝ ⎠

( )1

* * 2* * * * *

3 * *

167 8

3s

s c

T NP U P

T

κ ε ρκ ρ δσ ρ

−⎡ ⎤

= = − − −⎢ ⎥⎣ ⎦

Internal energy per atom



Thermal Pressure Coefficient vV

PT

γ⎛ ⎞∂

= ⎜ ⎟∂⎝ ⎠3

* * * * *

*2

2 13

vv v k

C E Pk T

γ σγ ρ δ δ

⎡ ⎤= = −⎢ ⎥

⎣ ⎦


Issues in Property Estimation

How long a simulation run should be?Ideally, a measurement in thermodynamic equilibrium is

( )0

0

1lim ( ), ( )

t tN N

tt

A A dt

τ τ τ+

→∞< >= ∫ r p



Existence of metastable statese.g. one-dimensional harmonic oscillator

21 1

2 21 1 2 2 1

1( )

2( )3 1 1

( ) ( ) ( )4 2 2

x x xx

x x x x x

γ

γ γ

⎧<⎪⎪= ⎨

⎪ + − ≥⎪⎩

U



Not all properties are defined as time averages over some function of the phase-space trajectory, e.g.

entropy

free energies

chemical potential


Kinetic Monte Carlo (KMC)KMC is a type of discrete-event stochastic simulation methods KMC is extensively used in physics and chemistryKMC

defines a discrete set of states of the system (i.e. all possible configurations)simulates state transitions between states which are triggered by events (also called processes in chemistry-oriented literature) that cause state changes.

e.g. vapor deposition

e.g. 2H2O ↔ 2H2 + O2

(b) possible events

a1

a2 a3

a4a5

a7

a6

solid surface

(a) discrete states

particles

states: combinations of # of speciesH2O n n-2 n-4 …H2 0 2 4 …O2 0 1 2 …events: reactionsa1: → a2: ←

rate

propensity


How KMC Works – event selection

The inter-arrival time Tj of event j is assumed to be exponentially distributed as

The probability that the inter-arrival time of event j is the minimum among M independent events

Therefore, uniformly sample to choose the next event

( )j jT Exponential a∼

timeevent 1

timeevent j

timeevent M

Tj

1 1Pr[ min( , , )] / ( )

j M j MT T T a a a= = + +…

a1 a2 aj aM


How KMC Works – clock advancement

The inter-arrival time Tj of event j is

The earliest time T(1)=min(T1,…,TM) of any event occurs is

Therefore, exponentially sample to advance clock

timeevent 1

timeevent j

timeevent M

Tj

timeoverall next event

( ) ( )1 exp ( 1, , )j j

P T a j Mτ τ≤ = − − = …

( )( ) ( )( ) ( ) ( )1 1

1 11 1 1 exp

M

jj

M

jjaP T P T P Tτ τ τ τ

= =⎛ ⎞≤ = − > = − > = − −⎜ ⎟⎝ ⎠∏ ∑

( )1ln / where 0,1

M

jjT a Uniformρ ρ

== − ∑ ∼


Initiate regular lattice sites;Specify regular species on each regular site;

Define all possible events;

WHILE stop criteria are not met Update a list of J active sites with sitePropensityj> 0 for site j

where is sum of all event propensitiesat site j

Update ;//choose a site for the next eventGenerate r1~Uniform(0,1);Find mth site where ;//choose an event from the chosen siteGenerate r2~Uniform(0,1);Find nth event where ;Fire event n at site m and update species at neighboring sites;//update event listFOR all events associated with site m

Add the event into the event list of site m;ENDUpdate propensities for site m and neighboring sites;//update system timeGenerate r3~Uniform(0,1);Advance system time T to T+Δt where Δt=−lnr3/totalProp;

END

1

J

j jtotalProp sitePropensity

== ∑

j jkksitePropensity propensity= ∑

1

11 1

m m

j jj jsitePropensity totalProp r sitePropensity

−

= =< × ≤∑ ∑

1

21 1

n n

mk kmk mkpropensity sitePropensity r propensity

−

= =< × ≤∑ ∑

Stochastic Simulation Algorithm (SSA)

Example: deposition

a1

a2 a3

a4a5

a7

a6


An Example of KMC for Chemical Reactions

PLac

RNAP

PLacRNAP

TrLacZ1

RbsLacZ

TrLacZ2TrLacY1TrLacY2

RbsLacY

Ribosome

RbsRibosomeLacZ

RbsRibosomeLacY

TrRbsLacZ

TrRbsLacY

LacY

LacZ

dgrLacZ

dgrRbsLacZ

dgrLacY

dgrRbsLacY

0.17

10

1

10.0151

0.36

0.17 0.45

0.17

0.45

0.4

0.4

0.036

0.015

6.42E-5

6.42E-5

0.3

0.3

lactose

LacZlactose

14

product

9.52E-5

431

Reactions of LacZ and LacY proteins in E. coli


Accelerating KMC: τ-Leap

Based on the relation between Exponential and Poisson distributionsHow to choose τ ?

Lower Bound:

Upper Bound (Leap Condition): τ is small enough such that the change in the state during [t,t+τ] is so slight that no propensity function will suffers an appreciable change.

How to choose aj(x) to generate Poisson variates kj’s to finish state change

1( )jja

τ >∑ x

1

M

j jj

k=

⎯⎯→ +∑x x v


Accelerating KMC: Binomial τ-Leap

In the original τ-leap, it is possible to have generate negative number of species, e.g.

S1 + S2 --> S3 with species sizes x1 = 1000, x2 = 1, and rate c=0.1. Propensity is a = cx1x2 = 100 There are two reactions: S1 + S2 --> S3, S1 --> S4 with a leap.

Instead of generating kj~Poisson(aj(x)τ), generating kj~Binomial(Nj, aj(x)τ/Nj) which explicitly considers the current size of species as Nj .

Nj = x1 for S1 --> S3Nj = min(x1, x2) for S1 + S2 --> S4Nj = x1/2 for S1 + S1 --> S5


Summary

Molecular Dynamics

Kinetic Monte Carlo


Further ReadingsMolecular Dynamics

J.M. Haile (1992) Molecular Dynamics Simulation: Elementary Methods. John Wiley & Sons, New York.

Kinetic Monte CarloChatterjee A. and Vlachos D.G. (2007) An overview of spatial microscopic and accelerated kinetic Monte Carlo methods. Journal of Computer-Aided Materials Design, 14:253-308 Gillespie D.T. (1976) A general method for numerically simulating the stochastic evolution of coupled chemical reactions. Journal of Computational Physics, 22: 403-434 Gillespie D.T. (2001) Approximate accelerated stochastic simulation of chemically reacting systems. Journal of Chemical Physics, 115: 1716-1733 Rathinam M., Petzold L.R., Cao Y., and Gillespie D.T. (2003) Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method. Journal of Chemical Physics, 119(24): 12784-12794 Tian T. and Burrage K. (2004) Binomial leap methods for simulating stochastic chemical kinetics, Journal of Chemical Physics, 121(21): 10356(1-9) Cai X. and Xu Z. (2007) K-leap method for accelerating stochastic simulation of coupled chemical reactions. Journal of Chemical Physics, 126(7): 074102(1-10)

simulations at nanoscales

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