Download - Kinetic Monte Carlo Triangular lattice. Diffusion Thermodynamic factor Self Diffusion Coefficient
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Kinetic Monte Carlo
Triangular lattice
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Diffusion
€
D = Θ ⋅DJ
€
DJ =1
2d( )t
1
NΔ
r R i t( )
i=1
N
∑ ⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
2
€
Θ=∂ μ
kBT
⎛
⎝ ⎜
⎞
⎠ ⎟
∂ ln x=
N
N 2 − N2
Thermodynamic factor
Self DiffusionCoefficient
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€
Δ r
R i t( )
€
Δ r
R j t( )
€
DJ =1
2d( )t
1
NΔ
r R i t( )
i=1
N
∑ ⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
2
Diffusion
€
D* =1
2d( )t
1
NΔ
r R i t( )
2
i=1
N
∑
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Standard Monte Carlo to study diffusion
• Pick an atom at random
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Standard Monte Carlo to study diffusion
• Pick an atom at random• Pick a hop direction
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Standard Monte Carlo to study diffusion
• Pick an atom at random• Pick a hop direction• Calculate
€
exp −ΔEb kBT( )
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Standard Monte Carlo to study diffusion
• Pick an atom at random• Pick a hop direction• Calculate • If ( >
random number) do the hop€
exp −ΔEb kBT( )
€
exp −ΔEb kBT( )
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Kinetic Monte Carlo
Consider all hops simultaneously
A. B. Bortz, M. H. Kalos, J. L. Lebowitz, J. Comput Phys, 17, 10 (1975).F. M. Bulnes, V. D. Pereyra, J. L. Riccardo, Phys. Rev. E, 58, 86 (1998).
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€
Wi = ν * exp−ΔEi
kBT
⎛
⎝ ⎜
⎞
⎠ ⎟
For each potential hop i, calculate the hop rate
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€
Wi = ν * exp−ΔEi
kBT
⎛
⎝ ⎜
⎞
⎠ ⎟
For each potential hop i, calculate the hop rate
Then randomly choose a hop k, with probability
€
Wk
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€
Wi = ν * exp−ΔEi
kBT
⎛
⎝ ⎜
⎞
⎠ ⎟
For each potential hop i, calculate the hop rate
Then randomly choose a hop k, with probability
€
Wk
= random number
€
ξ1
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€
Wi = ν * exp−ΔEi
kBT
⎛
⎝ ⎜
⎞
⎠ ⎟
For each potential hop i, calculate the hop rate
Then randomly choose a hop k, with probability
€
Wk
= random number
€
ξ1
€
Wi
i=1
k−1
∑ <ξ1 ⋅W ≤ Wi
i= 0
k
∑
€
W = Wi
i= 0
Nhops
∑
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Then randomly choose a hop k, with probability
€
Wk
= random number
€
ξ1
€
Wi
i=1
k−1
∑ <ξ1 ⋅W ≤ Wi
i= 0
k
∑
€
W = Wi
i= 0
Nhops
∑
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Time
After hop k we need to update the time
= random number
€
ξ 2
€
Δt = −1
Wlogξ 2
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Two independent stochastic variables:
the hop k and the waiting time
€
Wi
i=1
k−1
∑ <ξ1 ⋅W ≤ Wi
i= 0
k
∑
€
Δt = −1
Wlogξ 2 €
W = Wi
i= 0
Nhops
∑€
Wi = ν * exp−ΔEi
kBT
⎛
⎝ ⎜
⎞
⎠ ⎟
€
Δt
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Kinetic Monte Carlo
• Hop every time
• Consider all possible hops simultaneously
• Pick hop according its relative probability
• Update the time such that on average equals the time that we would have waited in standard Monte Carlo
€
Δt
A. B. Bortz, M. H. Kalos, J. L. Lebowitz, J. Comput Phys, 17, 10 (1975).F. M. Bulnes, V. D. Pereyra, J. L. Riccardo, Phys. Rev. E, 58, 86 (1998).
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Triangular 2-d lattice, 2NN pair interactions
€
Er
σ ( ) = Vo +Vpo intσ l +1
2VNNpairσ l σ i
i= NNpairs
∑ +1
2VNNNpairσ l σ j
j= NNNpairs
∑ ⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
l=1
N lattice−sites
∑
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Activation barrier
€
ΔEkra = Eactivated −state −1
2E1 + E2( )
€
ΔEbarrier = ΔEkra +1
2E final − Einitial( )
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Thermodynamics
€
Θ=∂ μ
kBT
⎛
⎝ ⎜
⎞
⎠ ⎟
∂ ln x=
N
N 2 − N2
€
D = Θ ⋅DJ
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Kinetics
€
D = Θ ⋅DJ
€
DJ =1
2d( )t
1
NΔ
r R i t( )
i=1
N
∑ ⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
2