Diffusive Molecular Dynamics

Ju Li, William T. Cox, Thomas J. Lenosky, Ning Ma, Yunzhi Wang

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Traditional Molecular Dynamics

• Numerically integrate Newton’s equation of motion with 3N degrees of freedom, the atomic positions:

• Difficult to reach diffusive time scales due to timestep (~ ps/100) required to resolve atomic vibrations.

{ }, 1..i i N=x

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Diffusive MD: Basic Idea

Ferris wheel seen with long camera exposure time

Variational Gaussian Method

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

{ }, , 1..i i i Nα =x

DMD

ci: occupation probability(vacancy, solutes)

Define µi for each atomic site,to drive diffusion

{ }, , , 1..i i i i Nα =x c

Phase-Field Crystal: Elder, Grant, et al.Phys. Rev. Lett. 88 (2002) 245701Phys. Rev. E 70 (2004) 051605 Phys. Rev. B 75 (2007) 064107

change of basis: planewave → Gaussian

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( ) ( ) ( )

0 0 0

3 23 2 2 2

1

Gibbs-Bogoliubov Free Energy Bound:

1 exp exp | |2

(| |, , )

Nji

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

i j i j

F F U U

u d d

w

αα α απ π

α α

∞ ∞

−∞ −∞= ≠

≤ + −

′ ′ ′ ′ ′ ′= − − − − −

−

∑∑ ∫ ∫ x x x x x x x x

x x

2

1

3 2ln thermal wavelength 2

Ni T

B Ti B

k Te mk T

α ππ=

Λ+ Λ =

∑

Variational Gaussian Method

{xi,αi}true free energy

VG free energy

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Comparison with Exact Solution

Lesar, Najafabadi, 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

8

2

1 1

The chemical potential for each atomic site 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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log(D)

Atomic Environment-Dependent Diffusivity

Atomic coordination

number

12(perfect crystal)

9(surface)

10,11(dislocation core)

experimental or first-principles

diffusivities

10

Particleon surface

(largeparticle)

11

Particleon surface

(smallparticle)

12

Sinteringby hot

isostaticpressing

(porosityreduction in nanoparticlessuperlattice)

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Sinteringby Hot

IsostaticPressing

(randompowders)

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Nanoindentation

(only atomswith coordination

number ≠ 12are shown)

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Small Contact Radius, High Temperature

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Indenter accommodation by purely diffusional creep

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coordination number coloring, showing edge dislocation

Dislocation Climb

vacancy occupation > 0.1

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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 finite element method.

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

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Quasicontinuum - DMD?

image taken from Knap and Ortiz, Phys. Rev. Lett. 90 (2003) 226102.

DMDregion?

continuum diffusion

equation solver region,

with adaptive meshing?

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Stress-Induced Bain Transformation

FCC

BCC

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