Kinetic Monte Carlo study of triplet-triplet annihilation ...
Kinetic Monte Carlo
description
Transcript of Kinetic Monte Carlo
![Page 1: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/1.jpg)
Kinetic Monte Carlo
Triangular lattice
![Page 2: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/2.jpg)
Diffusion
€
D = Θ ⋅DJ
€
DJ =1
2d( )t1N
Δr R i t( )
i=1
N
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
€
Θ=∂
μkBT
⎛
⎝ ⎜
⎞
⎠ ⎟
∂ ln x=
N
N 2 − N 2Thermodynamic
factor
Self DiffusionCoefficient
![Page 3: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/3.jpg)
€
Δr R i t( )
€
Δr R j t( )
€
DJ =1
2d( )t1N
Δr R i t( )
i=1
N
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
Diffusion
€
D* =1
2d( )t1N
Δr R i t( )2
i=1
N
∑
![Page 4: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/4.jpg)
Standard Monte Carlo to study diffusion
• Pick an atom at random
![Page 5: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/5.jpg)
Standard Monte Carlo to study diffusion
• Pick an atom at random• Pick a hop direction
![Page 6: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/6.jpg)
Standard Monte Carlo to study diffusion
• Pick an atom at random• Pick a hop direction• Calculate
€
exp −ΔEb kBT( )
![Page 7: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/7.jpg)
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( )
![Page 8: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/8.jpg)
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).
![Page 9: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/9.jpg)
![Page 10: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/10.jpg)
![Page 11: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/11.jpg)
![Page 12: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/12.jpg)
![Page 13: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/13.jpg)
![Page 14: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/14.jpg)
![Page 15: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/15.jpg)
![Page 16: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/16.jpg)
![Page 17: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/17.jpg)
€
Wi = ν * exp−ΔEikBT
⎛
⎝ ⎜
⎞
⎠ ⎟
For each potential hop i, calculate the hop rate
![Page 18: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/18.jpg)
€
Wi = ν * exp−ΔEikBT
⎛
⎝ ⎜
⎞
⎠ ⎟
For each potential hop i, calculate the hop rate
Then randomly choose a hop k, with probability
€
Wk
![Page 19: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/19.jpg)
€
Wi = ν * exp−ΔEikBT
⎛
⎝ ⎜
⎞
⎠ ⎟
For each potential hop i, calculate the hop rate
Then randomly choose a hop k, with probability
€
Wk= random number
€
ξ1
![Page 20: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/20.jpg)
€
Wi = ν * exp−ΔEikBT
⎛
⎝ ⎜
⎞
⎠ ⎟
For each potential hop i, calculate the hop rate
Then randomly choose a hop k, with probability
€
Wk= random number
€
ξ1
€
Wii=1
k−1
∑ <ξ1 ⋅W ≤ Wii= 0
k
∑
€
W = Wii= 0
Nhops
∑
![Page 21: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/21.jpg)
Then randomly choose a hop k, with probability
€
Wk= random number
€
ξ1
€
Wii=1
k−1
∑ <ξ1 ⋅W ≤ Wii= 0
k
∑
€
W = Wii= 0
Nhops
∑
![Page 22: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/22.jpg)
Time
After hop k we need to update the time
= random number
€
ξ 2
€
Δt = −1W
logξ 2
![Page 23: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/23.jpg)
Two independent stochastic variables:
the hop k and the waiting time
€
Wii=1
k−1
∑ <ξ1 ⋅W ≤ Wii= 0
k
∑
€
Δt = −1W
logξ 2 €
W = Wii= 0
Nhops
∑€
Wi = ν * exp−ΔEikBT
⎛
⎝ ⎜
⎞
⎠ ⎟
€
Δt
![Page 24: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/24.jpg)
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).
![Page 25: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/25.jpg)
Triangular 2-d lattice, 2NN pair interactions
€
Er
σ ( ) = Vo +Vpo intσ l +12
VNNpairσ l σ ii= NNpairs∑ +
12
VNNNpairσ l σ jj= NNNpairs
∑ ⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
l=1
N lattice−sites
∑
![Page 26: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/26.jpg)
Activation barrier
€
ΔEkra = Eactivated −state −12
E1 + E2( )
€
ΔEbarrier = ΔEkra +12
E final − Einitial( )
![Page 27: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/27.jpg)
Thermodynamics
€
Θ=∂
μkBT
⎛
⎝ ⎜
⎞
⎠ ⎟
∂ ln x=
N
N 2 − N 2
€
D = Θ ⋅DJ
![Page 28: Kinetic Monte Carlo](https://reader036.fdocuments.net/reader036/viewer/2022081507/56815e7e550346895dcd08e4/html5/thumbnails/28.jpg)
Kinetics
€
D = Θ ⋅DJ
€
DJ =1
2d( )t1N
Δr R i t( )
i=1
N
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2