diffusive molecular dynamics ju li, william t. cox, thomas j. lenosky, ning ma, yunzhi wang
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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 Nx
3
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 N x
DMD
ci: occupation probability(vacancy, solutes)
Define i for each atomic site,to drive diffusion
, , , 1..i i i i N x c
Phase-Field Crystal: Elder, Grant, et al. Phys. Rev. Lett. 88 (2002) 245701
Phys. 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:
1exp 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
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
Variational Gaussian Method
{xi,i}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 1
2 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
x x
Add occupation order parameters to sites: , , , 1..i i i i N x c
VG view DMD view
0
1
c
1
0
c
8
2
1 1
The chemical potential for each atomic site is easily derived:
1 3(| |, , ) ln ln
2 2 1
N Ni i
i j i j i j Bi i j ii i
A cc w k T
c e c
x x
DMD kinetics
nearest-neighbor network
1
1 , if and are nearest neighbors2
0 otherwise
Ni
ij j ij
i j
ij
ck
t
c ck i j
k
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
isostatic pressing
(porosityreduction in nanoparticlessuperlattice)
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Sinteringby Hot
Isostatic Pressing
(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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