Diffusive Molecular Dynamics

Ju Li, Bill Cox, Tom Lenosky,Ning Ma, Yunzhi Wang

Ohio State University

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Traditional Molecular Dynamics

• Traditional MD numerically integrates Newton’s equation of motion over 3N degrees of freedom, the atomic positions:

• It is difficult to reach diffusive time scales using traditional MD due to timestep (~ ps / 100), which needs to resolve atomic vibrations.

{ }, 1..i i N=x

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Diffusive MD: The Idea

Ferris wheel seen with long camera exposure time

Variational Gaussian Method

Lesar, Najafabadi and Srolovitz, Phys. Rev. Lett. 63 (1989) 624.

{ }, , 1..i i i Nα =x

DMD

ci: occupation probability(vacancy, solutes)

Define µi for each atom,to drive diffusion

{ }, , , 1..i i i i Nα =x c

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( ) ( ) ( )3 23 2 2 2

0 0 01

Gibbs-Bogoliubov Free Energy Bound:

1 exp exp | |2

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(| |, , )

Nji

i i i j j j i j i ji i j

B

i j i j

F F U U u d d

k T

w

αα α απ π

α α

∞ ∞

−∞ −∞= ≠

′ ′ ′ ′ ′ ′≤ + − = − − − − −

+

−

∑∑ ∫ ∫ x x x x x x x x

x x

2

1

ln

2Thermal wavelength:

Ni T

i

TB

e

mk T

απ

π=

Λ

Λ =

∑

Variational Gaussian Method

{xi,αi}true free energy

VG free energy

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Comparison with Monte Carlo

Lesar, Najafabadi and Srolovitz, Phys. Rev. Lett. 63 (1989) 624.

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DMD thermodynamics

( ) ( )2

1 1

1 3(| |, , ) ln ln 1 ln 12 2

N Ni

i j i j i j B i i i i ii i j i

F c c w k T c c c c ce

αα απ= ≠ =

Λ≤ − + + + − −

∑∑ ∑x x

{ }Add occupation order parameters to sites: , , , 1..i i i i Nα =x c

VG view DMD view

01

=

c

10

=

c

10

2

1 1

The chemical potential for each atom is easily derived:

1 3(| |, , ) ln ln2 2 1

N Ni i

i j i j i j Bi i j ii i

A cc w k Tc e c

αµ α απ= ≠ =

∂ Λ = = − + + ∂ − ∑∑ ∑x x

DMD kinetics

nearest-neighbor network

( )1

1 , if and are nearest neighbors2

0 otherwise

Ni

ij j ij

i j

ij

c kt

c ck i j

k

µ µ=

∂= −

∂

+ − =

∑

2B 0

calibrate against experimental diffusivity:Dk

k T a Z=

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• DMD is atomistic realization of regular solution model, with gradient thermo, long-range elastic interaction, and short-range coordination interactions all included.

• DMD kinetics is “solving Cahn-Hilliard equation on a moving atom grid”, with atomic spatial resolution, but at diffusive timescales.

• The “quasi-continuum” version of DMD can be coupled to well-established diffusion-microelasticity equation solvers such as phase-field method.

• No need to pre-build event catalog. Could be competitiveagainst kinetic Monte Carlo.