1.16 Boltzmann Maps From Grand Canonical Monte Carlo Simulations - Overview
Monte Carlo Simulation - Postech CMSE Lab. · 2013-02-19 · Monte Carlo – Grand Canonical...
Transcript of Monte Carlo Simulation - Postech CMSE Lab. · 2013-02-19 · Monte Carlo – Grand Canonical...
Monte Carlo SimulationMonte Carlo Simulation
ByeongByeong--Joo LeeJoo Lee
Byeong-Joo Leewww.postech.ac.kr/~calphad
ByeongByeong--Joo LeeJoo Lee
Dept. of MSEDept. of MSE
Pohang University Pohang University
of Science and Technologyof Science and Technology
(POSTECH)(POSTECH)
[email protected]@postech.ac.kr
BackgroundBackground -- PhasePhase SeparationSeparation inin GaInNGaInN
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Background Background –– Atomic distribution in Atomic distribution in SiGeSiGe nanowiresnanowires
0.6
0.8
1
X(G
e) 5nm
Surface Segregations
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0
0.2
0.4
0.6
0 0.5 1
normalized rX
(Ge
) 5nm
2nm
bulk
Why Monte Carlo ? Why Monte Carlo ?
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What is Monte Carlo ?What is Monte Carlo ?y
1
implicit integer (i-n)
implicit double precision (a-h,o-z)
call random_seed
i_circle = 0
do i = 1, 100000000
call random_number(x)
call random_number(y)
r2 = x*x + y*y
if(r2 .lt. 1.d0) i_circle = i_circle + 1
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xif(r2 .lt. 1.d0) i_circle = i_circle + 1
if(mod(i,1000000) .eq. 0) then
pi = 4.0d0 * dble(i_circle) / dble(i)
write(*,*) i, pi
endif
enddo
c
stop
end
Monte Carlo Monte Carlo –– Fundamentals (from Lecture Note of Prof. V. Fundamentals (from Lecture Note of Prof. V. VitekVitek, 2002), 2002)
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Monte Carlo Monte Carlo –– Distribution of states in real systemsDistribution of states in real systems
f(x)
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a b x
Monte Carlo Monte Carlo –– General Monte Carlo AlgorithmGeneral Monte Carlo Algorithm
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Monte Carlo Monte Carlo –– Canonical Ensemble (N,V,T)Canonical Ensemble (N,V,T)
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Monte Carlo Monte Carlo –– Metropolis MethodMetropolis Method
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Monte Carlo Monte Carlo –– Examples of Metropolis MethodExamples of Metropolis Method
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Monte Carlo Monte Carlo –– Magnetization vs. Temperature by Ising model of spinsMagnetization vs. Temperature by Ising model of spins
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Open System with a heat reservoir Open System with a heat reservoir –– Grand Canonical ensembleGrand Canonical ensemble
ii nnNNW ln~
ln~
ln ∑−=
0lnln max =−= ∑ ii ndnWd
0=∑dn 0=∑ nαδ
iri
ri
i EpEU ,
,
∑>=≡< iri
ri
EnUN ,
,
~∑=
ri
ri
nN ,
,
~∑=Number of µVT systems with Ei & Nr:
Average Energy of a system:
Average # of ptl. of a system:rri
ri
NpN ,
,
∑>=<rri
ri
NnNN ,
,
~∑>=<
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0, =∑ ridn 0, =∑ rinαδ
0ln , =−++ riri NEn γβα ri NE
ri eenγβα +−−=,
0, =∑ riidnE
ri
ri
NE
ri
NEri
rie
e
N
np
γβ
γβ
+−
+−
∑==
,
,
, ~
0, =∑ riidnEβ
ri NE
ri
e
Ne
γβ
α
+−
−
∑=
,
~
ri
ri
NE
ri
NE
i
ri
e
eE
Uγβ
γβ
+−
+−
∑
∑=
,
,
0, =∑ rirdnN 0, =∑ rirdnNγ
ri
ri
NE
ri
NE
r
ri
e
eN
Nγβ
γβ
+−
+−
∑
∑>=<
,
,
Monte Carlo Monte Carlo –– Grand Canonical Ensemble (Grand Canonical Ensemble (µµ,V,T),V,T)
The number of particles is variable
- fluctuations of the concentration are allowed
Generally, select randomly one of the procedures
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1) Change of the configuration
exp(-∆Ep/kT)
2) Creation of a particle
exp[-(∆Ep-µ)/kT]
3) Destruction of particles
exp[-(∆Ep+µ)/kT]
Monte Carlo Monte Carlo –– Modified Grand Canonical Ensemble (Modified Grand Canonical Ensemble (µµ,V,T),V,T)
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Monte Carlo Monte Carlo –– ApplicationsApplications
Segregation to an interface
Order-disorder transition in binary alloy
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Order-disorder transition in binary alloy
Morphology Evolution of an Alloy Particle
MD+MC+MS Hybrid Simulation
Phase Separation and Surface Segregation in CuPhase Separation and Surface Segregation in Cu--Ni Thin Films Ni Thin Films
298 K298 K 373 K373 K
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800 K800 K473 K473 K
Order/Disorder in CoOrder/Disorder in Co--Pt System Pt System -- S.I. Park et al., Scripta Mater., 2001.S.I. Park et al., Scripta Mater., 2001.
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Property Pt3Co PtCo PtCo3Cohesive Energy 5.500 5.215 4.873(eV/atom) 5.555±0.017 5.228±0.005
Lattice Constant a=3.833 3.754, c/a=.98 3.625(Å) a=3.831 3.745, c/a=.98 3.668
Transition 1070-1080 970-980 760-770Temperature (K) 1000 1100 840
Pure W W + 0.4wt% Ni Vacuum Annealing
Effect of Interface Energy Anisotropy on Morphological EvolutionEffect of Interface Energy Anisotropy on Morphological Evolution
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Pure W WPure W W--14at%Ni14at%Ni
Monte Carlo Monte Carlo –– Determination of Phase DiagramDetermination of Phase Diagram
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A Modified Embedded Atom Method Interatomic Potential for the Cu-Ni SystemByeong-Joo Lee and Jae-Hyeok Shim, CALPHAD 28, 125-132 (2004).
Monte Carlo Monte Carlo – Deposition or Etching
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