Download - Diffusive Molecular Dynamics
2
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
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 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
4
( ) ( ) ( )
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
7
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=
9
log(D)
Atomic Environment-Dependent Diffusivity
Atomic coordination
number
12(perfect crystal)
9(surface)
10,11(dislocation core)
experimental or first-principles
diffusivities
17
coordination number coloring, showing edge dislocation
Dislocation Climb
vacancy occupation > 0.1
18
• 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.
19
Quasicontinuum - DMD?
image taken from Knap and Ortiz, Phys. Rev. Lett. 90 (2003) 226102.
DMDregion?
continuum diffusion
equation solver region,
with adaptive meshing?