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Phase field models
for crystal steps
Olivier Pierre-Louis
PRE 68 012604 (2003)
http://consoude.ujf-grenoble.frSpectro,
Grenoble, France
Solid on Solid
Steps
Continuum
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adatoms
Solid
c+c−
θ
φ
Phasefieldmodel
stepmodel
.
Discont.
Phase field model for crystal steps:
• Lui and Metiu (1994): surface relaxation
• Plapp and Karma (1998): spiral growth
Step kinetics?
• Schwoebel effect
• Step transparency
Phase field model with 1 global concentration field→ Fast step kinetics
P.f. model with 1 concentration field per terrace→ Arbitrary step kinetics
Discontinuous interface modelOn terraces
∂tc = D∇2c+ F − c/τ
At the step (Standard form)
D∂nc+ = ν+(c+ − c∗eq) + ν0(c+ − c−)
−D∂nc− = ν−(c− − c∗eq) + ν0(c− − c+)
c∗eq = ceq(1 + Γκ)
1
ΩV = (−D∂nc− − V c−) + (D∂nc+ + V c+)
νν+
−
ν0
zx
S
D
E
Re-formulation of boudary conditions:
Form I: Modified equilibrium concentration
D∂nc+ = ν+(c+ − ceq)
−D∂nc− = ν−(c− − ceq)
ceq = ceq(1 + Γκ) + βV
Ω
(i) Other physical meaning of transparency:modification of ceq on a moving step.
(ii) Link to solidificationlimit of strong transparency ν0 → ∞:
c+ = c− = ceq = ceq(1 + Γκ) + βV
Ω
Re-formulation of boudary conditions:
Form II: Global exchange
D∂nc+ = αV
Ω+ ν(c+ − c−)
−D∂nc− = (1 − α)V
Ω+ ν(c− − c+)
V
Ω=
1
β
[
α(c+ − c∗eq) + (1 − α)(c− − c∗eq)]
2 fields model
∂tc = ∇[M∇θ] + F −θ
τ− ∂th
τp∂tφ = W 2∇
2φ− fφ + λgφ(θ − θeq)
variational if g ∝ h.
F =
∫[
W 2
2(∇φ)2 + f +
λ
2(θ − θeq)
2
]
f
φ
Asymptotics narrow interface:W → 0, τp ∼ D/W 2.
Sharp interface limit: λ ∼Wweak coupling limit→ strong transparency
ν+ ∼ 1, ν− ∼ 1, ν0 ∼1
W
Thin interface limit: (θ − θeq) ∼Wsmall departure from equil at step→ Fast attachement kinetics
ν+ ∼1
W, ν− ∼
1
W, ν0 ∼
1
W
How many concentration fields?
φ
φ−φ+
φ
θ+
θ−
θ
1 conc field per terrace
Diffusion flux
Jd ∼ ∇θ
ψ∼ [ψ∇θ − θ∇ψ]
Model equations:
∂tθ+ + ∂th+ = D[φ+∇2θ+ − θ+∇
2φ+]
−1
τ(θ+ − θ∞+ )
−(B++θ+ − B+−θ−)
∂tθ− + ∂th− = D[φ−∇2θ− − θ−∇
2φ−]
−1
τ(θ− − θ∞− )
−(B−−θ− − B−+θ+)
τφ∂tφ = W 2∇
2φ− fφ
+ λ[g+φ(θ+ − θeqφ+) + g−φ(θ− − θeqφ−)]
variational if B±± ∼ φ±and g±φφ± ∼ h±φ + θeqφ±φ
Sharp interface limit: λ ∼W→ Arbitrary kinetics
ν± ∼ 1 ν0 ∼ 1
A numerical trick: the 3 fields model
n n+1n−1n−2 n+2
1 2 1 2 1θ
θ
x
x
10−1 100 101 102
d+ / W−0.15
−0.10
−0.05
0.00
0.05
V/V nu
m−1
10−1 100 101 102
d+ / W
0.01
0.02
0.03
0.04
V
Figure 1: Step veloc, 1 global conc field
100 101 102 103 104
d+/W
10−4
10−3
10−2
V
100 101 102 103 104
d+/W
−0.05
−0.03
−0.01
0.01
0.03
0.05
Figure 2: Step velocity, 1 conc field per terrace
−1 −0.5 0 0.5 1 α
0.006
0.007
0.008
0.009
0.01
0.011
0.012
V
Figure 3: Step velocity as a function of Schwoebeleffect (ν+ − ν−)/(ν+ + ν−)
10−2 10−1 100 101
ν 0
0.01
0.011
0.012
0.013
0.014
0.015
0.016
V
Figure 4: Velocity as a function of transparency ν0
adatom z
x
(+)(-)step
adatom
Figure 5: Bales and Zangwill instability
0 0.2 0.4 0.6q
−0.005
−0.003
−0.001
0.001
ω
Figure 6: Dispersion relation for step meandering
Nonlinear dynamics of an isolated step
Discontinuous Model→ multi-scale analysis (Bena et al 1994)→ Kuramoto-Sivashinsky Equation
∂th = −∂xxh− ∂xxxxh+ (∂xh)2
→ Spatio-temporal chaos
−100 −50 030
80
130
Figure 7: KS dynamics
−10 0 10x
50
70
90
110
130
time
Figure 8: Phase field model
−1 −0.5 0 0.5 1−10−4
0
10−4
Figure 9: Pairing rate during growth with invertedShwoebel effect
Conclusion
• Model with 1 global concentration field→ fast step kinetics
• Model with 1 concentration per terrace→ arbitrary step kinetics
Perspectives
• Link to microscopic dynamics
• Numerical simulations in complex geometry
• Nucleation, fluctuations
• Elasticity