Phase-field modeling of crystal nucleation I: Fundamentals and methods
1. Tamás Pusztai
Wigner Research Centre for Physics
Hungarian Academy of Sciences
Budapest, Hungary
Phase Field Workshop
ChiMaD Headquarters, Northwestern University
Sept 25-27, 2018
Phase-field modeling of crystal nucleation I:
Fundamentals and methods
2. Introduction
Solidification of an undercooled liquid
GrowthNucleation
200,000 time steps 220,000 time steps
160,000 time steps 180,000 time steps
Molecular dynamics simulation: By courtesy of R. S-Aga & J. R. Morris
Embryos of the new phase appear
via thermal fluctuation
Complex patterns evolve due to the
interplay of capillarity, diffusion,
and anisotropy.
3. Outline
• Homogeneous nucleation
• theory
• modeling
• problems
• Heterogeneous nucleation
• theory
• modeling of surfaces with adjustable wetting properties
• Greer’s athermal nucleation model
• theory
• modeling
4. Homogeneous nucleation
Classical nucleation theory (CNT):
assumptions:
• sharp interface
• bulk properties inside
• spherical shape (isotropic surface energy)
F3D(r) = 4r2
⇡ SL
4
3
r3
⇡ f F2D(r) = 2r⇡ SL r2
⇡ f
r⇤
3D =
2 SL
f
r⇤
2D =
SL
f
,
W⇤
3D =
16⇡ 3
SL
3 f2
W⇤
2D =
⇡ 2
SL
f
✓
Gibbs-Thomson: f = 2 sl, =
1
2
✓
1
R1
+
1
R2
◆
, 3D =
1
r
2D =
1
2r
◆
propertyofTamásPusztai(pusztai.tamas@wigner.mta.hu)-August22,201313:24
Homogeneous nucleation
4 × 10––15
2 × 10––15
––2 × 10––15
––4 × 10––15
1 × 10––8 2 × 10––8 3 × 10––8
R [m]
Volume term
Area term
DΔG(R)
Freeenergy[J]
4 × 10––8 5 × 10––8 6 × 10––8
0
0
DΔGn
homo
Rc
Fig. 7.1 Surface, bulk and total free energies of a spherical solid as function
radius for a fixed undercooling T = 5 K. Property data for Al are tabula
Table 7.1.
this occurs is determined by differentiating Eq. (7.2) with respect to
setting the result equal to zero:
Rc =
2 s`
⇢ sf T
=
2 s`
T
where s` = s`/(⇢s sf ) is the Gibbs-Thomson coefficient. Substi
this result into Eq. (7.2) gives
Ghomo
n =
4⇡ s`R2
c
3
=
16⇡
3
3
s`
(⇢ sf )2 T2
An embryo of radius Rc is called a critical nucleus, since it is energe
favorable for nuclei with R < Rc to melt, and for R > Rc to grow. Not
for T < 0, corresponding to temperatures above Tf , both terms o
Free energy of a spherical solid particle
of radius r: M. Rappaz: Solidification
2D3D
5. Homogeneous nucleation
J = J0 exp
✓
W⇤
kT
◆
Nucleation rate:
• Nucleation rates corresponding to time scales of typical experiments may
correspond to 10-100 molecules → the CNT fails miserably → a diffuse interface
model is needed!
• The nucleus is not necessarily spherical
extremely sensitive to W!
Limits of the classical theory:
milar bond-order pa-
ore said to be joined
” (26). Even in the
al-like bonds are not
particles with eight
onds are defined as
particles in a crys-
, hcp, rhcp, or bcc
recognized, whereas
umber of particles in
be crystal-like.
an experiment, sam-
astable liquid state,
m structural fluctua-
i of crystal-like par-
ticles were present. This is shown in two
early-time snapshots (Fig. 1, A and B),
where we represent crystal-like particles as
red spheres and liquid-like particles as blue
spheres, shown with a reduced diameter to
improve visibility. Typically, these sub-
critical nuclei contained no more than 20
particles and shrank to reduce their surface
energy. After a strongly -dependent peri-
od of time, critical nuclei formed and rap-
idly grew into large postcritical crystallites
(Fig. 1, C and D). By following the time
evolution of many crystallites, we deter-
mined the size dependence of the probabil-
ities pg and ps with which crystallites grow
or shrink (27). Because pg ϭ ps at the
critical size, we plot the difference pg – ps
as a function of crystallite radius and par-
ticle number M in Fig. 2 for a sample with
ϭ 0.47. We found an abrupt change from
negative to positive values of pg – ps (28),
allowing us to identify the critical size,
which is 60 Ͻ M Ͻ 160, in good agreement
with recent computer simulations (9). This
corresponds to rc Ϸ 6.2a, assuming a spher-
ical nucleus. The volume fraction of the
nuclei is larger than the value of the
fluid; above coexistence, the difference is
⌬ ϭ 0.012 Ϯ 0.003, independent of ,
where ⌬ increases slightly for M Ͼ 100.
We can understand this ⌬ value as result-
ing from the higher osmotic pressure exert-
ed by the fluid on the nuclei (16), whereas
in the coexistence regime, ⌬ must reflect
the evolution of to the higher value,
ultimately attained by the crystallites,
where ⌬ ϭ m Ϫ f. The nucleation rate
densities were slower than 5 mmϪ3
sϪ1
for
Ͻ 0.45, as well as for Ͼ 0.53. Values
of the order of 10 mmϪ3
sϪ1
were found for
0.45 Ͻ Ͻ 0.53. However, for 0.47 Ͻ Ͻ
0.53, the average size of the nuclei began to
grow immediately after shear melting; thus,
there was little time for the sample to equil-
a
l
ϭ
e
A
d
-
e
3
e
e
e
s
-
R E P O R T S
disordered liquid, crystal-like bonds are not
uncommon; thus, only particles with eight
or more crystal-like bonds are defined as
being crystal-like. All particles in a crys-
tallite with perfect fcc, hcp, rhcp, or bcc
structure are correctly recognized, whereas
only an insignificant number of particles in
the liquid are found to be crystal-like.
At the beginning of an experiment, sam-
ples started in the metastable liquid state,
but because of random structural fluctua-
tions, subcritical nuclei of crystal-like par-
red spheres and liquid-like particles as blue
spheres, shown with a reduced diameter to
improve visibility. Typically, these sub-
critical nuclei contained no more than 20
particles and shrank to reduce their surface
energy. After a strongly -dependent peri-
od of time, critical nuclei formed and rap-
idly grew into large postcritical crystallites
(Fig. 1, C and D). By following the time
evolution of many crystallites, we deter-
mined the size dependence of the probabil-
ities pg and ps with which crystallites grow
Fig. 3. A snapshot of a
crystallite of postcritical
size in a sample with ϭ
0.47 is shown from three
different directions (A
through C). The 206 red
spheres represent crystal-
like particles and are
drawn to scale; the 243
extra blue particles share
at least one crystal-like
“bond” to a red particle
but are not identified as
crystal-like and are re-
duced in size for clarity.
(D) A cut with a thickness
of three particle layers
through the crystallite, il-
lustrating the hexagonal
structure of the layers.
Blue, red, and green
spheres represent parti-
cles in the different layers
(front to rear). This cut
was taken from the re-
gion that is indicated by
Gasser et al, Science, 2001
6. Phase-field modeling of homogeneous nucleation
• Method 1: Mimic nature. Add fluctuations (noise) and wait
Determine W* by finding the nucleus (saddle point solution) via solving the Euler-
Lagrange equations.
Sharp interface and bulk properties are not assumed any more
∂ϕ
∂t
= − M
δF
δϕ
+ξ
∂ϕ
∂c
= ∇
[
M ∇
δF
δc
+ ⃗ζ
]
• Method 2: Put supercritical seeds at random places at random times
J = J0 exp
(
−
W*
kT )
Nucleation rate:
Remark: quantitative results are difficult, to say the least…
where ξ and ⃗ζ are gaussian random variables with
zero mean that put the “right amount” of fluctuations into the system
7. Method 1: Homogeneous nucleation with noise
∂ϕ
∂t
= − M
δF
δϕ
+ξ( ⃗r, t) with
With the noise term added, the equation of motion becomes a stochastic PDE, the noise
amplitude is determined by the fluctuation-dissipation theorem
Hohenberg-Halperin classification:
(P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena. Reviews of Modern Physics, 49, 435–479, 1977)
Model A (Non-conserved field):
Model C (Non-conserved field coupled to a conserved field):
∂ϕ
∂t
= − Mϕ
δF
δϕ
+ξ( ⃗r, t) with < ξ( ⃗r, t)ξ( ⃗r′, t′) > = 2MϕkT δ( ⃗r − ⃗r′)δ(t − t′)
∂c
∂t
= ∇
[
Mc ∇
δF
δc
+ ⃗ζ ( ⃗r, t)
]
with < ζm( ⃗r, t)ζn( ⃗r′, t′) > = 2MckT δm,nδ( ⃗r − ⃗r′)δ(t − t′)
< ξ( ⃗r, t) > = 0
< ξ( ⃗r, t)ξ( ⃗r′, t′) > = 2MkT δ( ⃗r − ⃗r′)δ(t − t′)
8. Method 1: Homogeneous nucleation with noise
Discretization:
ϕt+1
x = ϕt
x + Δt M
(
ϵ2
ϕt
x+1 + ϕt
x−1 − 2ϕt
x
Δx2
− g′(ϕt
x) + p′(ϕt
x)
)
+ Δt
MkT
ΔxdΔt
ξ
In 2D or 1D simulation we still use 3D materials parameters → some thickness is
always implicitly assumed.
This is not a problem in the deterministic part, but results in a different scaling of the
stochastic part!
< ξ(r) ξ( ⃗r′) > = 2MkT δ( ⃗r − ⃗r′) =
2MkT
ΔxdΔt
δn,n′, where d = number of spatial dimensions
Complete equation of motion with finite difference and forward Euler:
Mind the dimensionality!!!
< ξ > = 0
< ξ2
> = 1
deterministic part stochastic part
9. The 2D “toy model” used for illustrations
F[ϕ] =
∫ [
ϵ2
2
(∇ϕ)2
+ wg(ϕ) − Δf(T) p(ϕ)
]
dV
g(ϕ) = ϕ2
(1 − ϕ)2
p(ϕ) = ϕ3
(10 − 15ϕ + 6ϕ2
)
Simple phase-field model for a pure substance
Equilibrium solid-liquid interface:
ϕ(x) =
1 − tanh
(
x
2δ )
2
δ =
ϵ2
w
, γ =
ϵ2
w
3 2
Nucleation and growth in 2D:
T = Tm, Δf = 0
T < Tm, Δf > 0
r*2D
=
γ
Δf
=
ϵ2
w
3 2Δf
·
ϕ = − M
δF
δϕ
= M (ϵ2
∇2
ϕ − wg′(ϕ) + Δf p′(ϕ)) +
MkT
Δx2Δt
ξ
ϕ′(x) =
2w
ϵ2
ϕ(1 − ϕ)
10. 0 20 40 60 80
time
0
0.2
0.4
0.6
0.8
1
solidfraction
t 0
t 0
/2
t 0
/4
t 0
/8
t 0
/16
0 20 40 60 80
time
0
0.2
0.4
0.6
0.8
1
solidfraction
x 0
x 0
/2
x 0
/4
Solid seed growing without noise, convergence
Δx0 = 0.4, Δt0 = 0.01 Δx0 = 0.4, Δt0 = 0.01/16
Growth of a spherical seed:
Results are well converged!
ϵ2
= W = M = 1
Δf = 0.3, r*2D
= 0.79
11. Nucleation with noise, convergence with Δt
Δt0 Δt0/4 Δt0/16
0 5 10 15 20
time
0
0.2
0.4
0.6
0.8
1
solidfraction
t 0
t 0
/2
t 0
/4
t 0
/8
t 0
/16
ϵ2
= W = M = 1
Δf = 0.3
Δx0 = 0.4, Δt0 = 0.01
< ξ2
0 > =
0.015
Δx2
0Δt0
12. Nucleation with noise, convergence with Δx
Δx0 Δx0/2 Δx0/4
0 5 10 15 20
time
0
0.2
0.4
0.6
0.8
1
solidfraction
x 0
x 0
/2
x 0
/4ϵ2
= W = M = 1
Δf = 0.3
Δx0 = 0.4, Δt0 = 0.01
< ξ2
0 > =
0.015
Δx2
0Δt0
13. Generating Δx and Δt independent noise patterns
Special technique is used for generating random numbers that ensure similar
noise patterns independent of 𝛥t and 𝛥x:
• For reproducibility, the RNG seed was fixed
• The random numbers were always generated for the finest temporal and
spatial resolutions. The random numbers for the coarser simulations
were obtained from the finer simulations by averaging.
E.g.:
𝜉1 𝜉2
𝜉3 𝜉4
ξ =
1
4
4
∑
i=1
ξi
Var(ξi) = < ξ2
i > ∝
1
(Δx/2)2
Δx/2 Δx/2 Δx
Var(ξ) ∝
2
4
1
(Δx/2)2
=
1
(Δx)2
14. Why no convergence with Δx?
Increasing the spatial resolution when using white noise → new, higher frequencies
are added to the system → the energy of the system is changed (increased) →
nucleation is highly affected
L, N, Δx0 L, 2N, Δx0/2
In fact, for d ≥ 2 the total energy diverges as Δx → 0: ultraviolet divergence
Solution: use a filtered noise with cutoff λc > 2Δx0 before refining the grid
• “Top down” approach: it is just a necessity to have converged solutions, or even
just to avoid the ultraviolet divergence
• “Bottom up” approach: coarse graining with length 𝜆 → fluctuations below 𝜆 are
already included in the system → they should not be added again.
Justification:
15. Convergence, 𝛥x, filtered noise
Δx0 Δx0/2 Δx0/4
Δx0 Δx0/2 Δx0/4
0 5 10 15 20
time
0
0.2
0.4
0.6
0.8
1
solidfraction
x 0
x 0
/2
x 0
/4
0 5 10 15 20
time
0
0.2
0.4
0.6
0.8
1
solidfraction
x 0
x 0
/2
x 0
/4
λc = 2Δx0
λc = 4Δx0
16. approximation (second-order Taylor expansion) of the bare potential around
The result f ð Þ is a renormalised potential for .
These calculations can be readily verified numerically. As an example,
0.95 0.96 0.97 0.98 0.99 1
f
0
0.0002
0.0004
f
Simulation
Theory
Figure 3. Renormalised free energy density of the standard double-well potential as calcula
from Equation (44) and from numerical simulations, for T ¼ 0.05, Dx ¼ 0.5, Dt ¼ 0.005. O
the part close to one of the potential wells is shown. The zero of f was chosen at the minim
of the renormalised potential. The bin size for the histograms was D ¼ 0:01.
40 M. Plapp
ty]at20:5328July2013
0 5 10 15 20
time
0
0.2
0.4
0.6
0.8
1
solidfraction
t 0
t 0
/2
t 0
/4
t 0
/8
t 0
/16
Other issue: the renormalization of F by the noise
Adding noise renormalizes the phase-field equations.
Consider the double well potential around the minima at 𝜙=0,1 → add fluctuations →
asymmetric restoring forces → the mean value vill be shifted from 𝜙=0,1.
Illustration:
my earlier transformation curves
Solution:
Renormalization of the potential
M. Plapp: Philosophical Magazine, 91, 25–44 (2011)
• The properties of our model are not what we think
• No 1 to 1 correspondence between the two nucleation methods (noise EL)
The renormalization is not significant for simulations with large cells (MS instability,
dendritic sidebranching), but cause problems with small cells (nucleation).
17. Method 2: Solving the Euler-Lagrange equations
nonlinear elliptic PDE
δF
δϕ
=
∂f
∂ϕ
− ϵ2
∇2
ϕ = 0
δF
δc
= ∇Mc ∇
∂f
∂c
= 0 →
∂f
∂c
= μ(ϕ, c) = const = μ0
F[ϕ, c] =
∫ [
ϵ2
2
(∇ϕ)2
+ wg(ϕ) + f(ϕ, c)
]
dV
The Euler-Lagrange equations:
Simple binary PF model
with no term(∇c)2
If μ(ϕ, c) is a simple function, then c(ϕ) can be obtained and plugged back into the first ELE
scalar equation
The binary problem is reduced to the single phase-field problem
Solution methods: relaxation methods, shooting methods, etc.
ϕ( ⃗r) → c(ϕ( ⃗r)) → W* = F[ϕ( ⃗r), c(ϕ( ⃗r))]
Further simplification: the spatial dimensions of the problem can be reduced if
spherical or cylindrical symmetry can be assumed
18. Homogeneous nucleation, summary
• Method 1: Add fluctuations (Gaussian random numbers) to the fields and wait…
• Works only on the nanoscale (large undercoolings)
• Add noise directly to non-conserved fields, or as flux noise to conserved fields
• Use the variance determined by the fluctuation-dissipation theorem (as a guide)
• Mind the dimensionality of the problem when considering the scaling with 𝛥x
• Use filtered noise for 𝛥x independent results
• Inhomogeneous systems, complex nuclei, non-trivial nucleation paths are
automatically handled and considered
• Method 2: Insert supercritical seeds at random times at random places
• Best used for small undercooling and larger nuclei
• For quantitative results, determine the solution and its properties via solving the
Euler-Lagrange equations
• Not practical if the system is inhomogeneous
• The consideration of different nuclei is not automatic
Pure homogeneous nucleation is extremely rare in nature!
Because of the renormalization by noise, the results by Method 1 and 2 are not directly
comparable, unless we are at large lengthscales with small noise
19. Heterogeneous nucleation
Classical nucleation theory (CNT):
assumptions:
• sharp interface, bulk properties inside
• isotropic surface energy, spherical
cap
γ
SL
γ
WS
γ
WL ψ
cos( ) =
wl ws
sl
W⇤
2D,het = F2D(r⇤
, ) = S2D( )W⇤
2D,hom
W⇤
3D,het = F3D(r⇤
, ) = S3D( )W⇤
3D,hom
r⇤
= r⇤
het = r⇤
hom
Young equation:
F2D(r, ) = 2r SL + 2r sin ( SW LW) r2
⇡ S2D( ) f
S2D( ) =
( cos sin )
⇡
F3D(r, ) = 2r2
⇡(1 cos ) SL + r2
⇡(1 cos2
)( SW LW)
4
3
r3
⇡ S3D( ) f
S3D( ) =
2 3 cos + cos3
4
S2D(ψ)
S3D(ψ)
0.2 0.4 0.6 0.8 1.0
ψ/π
0.2
0.4
0.6
0.8
1.0
S(ψ)
Results:
S
L
W
20. Phase field modeling of surfaces:
F[ (r), c(r)] =
Z
f( , c) +
✏2
2
(r )2
dV +
Z
Z( ) dS
Free energy functional including the Z(𝜙) surface function:
(Cahn JCP 1977)
At the extremum by 𝜙(r) and c(r), the variation of F should disappear for any
infinitesimally 𝜌(r) and 𝜒(r) compatible with the boundary conditions:
F = F[ (r) + ⇢(r), c(r) + (r)] F[ (r), c(r)] = 0
This leads to the Euler-Lagrange equations
in the volume
on the surface
Cases:
• 𝜙(r) is fixed along the boundary:
𝜌(r)≣0 on the surface, so the
surface EL eq. holds
• 𝜙(r) is not fixed along the
boundary:
the first part of the surface EL
eq. gives the b.c. to use
surface = boundary of the simulation domain → surface properties = boundary conditions
@f( , c)
@
✏2
r2
= 0
@f( , c)
@c
= µ
⇥
Z0
( ) ✏2
r · n
⇤
= 0
J.A. Warren et al. Phase field approach to heterogeneous crystal nucleation in alloys. Physical Review B, 79, 014204 (2009)
21. Model A
(not according to the Hohenberg-Halperin classification!!!)
isosurfaces of
r
surface
Goal: direct realization of the 𝜓 contact angle
(L. Gránásy)
r · n =
r
2w
✏2
(1 ) cos( )
Z0
( ) = ✏2
r · n = 6 SL (1 ) cos( )
Z( ) = SL(3 2
2 3
) cos( )
We need Z(𝜙) to calculate the free energy of the system
22. Model A
László Gránásy
Ni:
•d10-90% = 2 nm
• 𝛾 = 364 mJ/m2
•Δx = 2 Å (1 pixel ~ 1 atom)
•fluctuation-dissipation noise
•thermal feedback
= 45
= 60 = 90 = 120
23. Model A
Solving the PDEs in cylindrical coordinate system (Matlab PDE toolbox)
The work of formation compared to the
classical theory
24. Model B
Constant 𝜙=𝜙0 at the interface (Dirichlet b.c.)
(J. Warren)
Obtaining the 𝜓 contact angle via Young’s law:
wl =
p
2✏2w
Z 0
0
2
(1 )2
= sl(3 2
0 2 3
0)
ws =
p
2✏2w
Z 1
0
2
(1 )2
= sl(1 3 2
0 + 2 3
0)
cos( ) =
wl ws
sl
= 2 2
0(3 2 0) 1
Setting 𝜙=𝜙0 at the interface: wetting layer
0 0.25 0.5 0.75 1
0
45
90
135
180
φ0
ψ
0 0.25 0.5 0.75 1
0
10
20
30
40
φ0
S
crit
There exist a critical value of 𝜙, below which the
interface can freely grow
25. Model B
Solving the PDEs in cylindrical coordinate system
The work of formation compared to the
classical theory
26. Model C
Constant ∇𝜙=h at the interface (Neumann b.c.)
(J. Warren)
wl = Z( 0) + sl(3 2
0 2 3
0)
ws = Z( 0) + sl(1 3 2
0 + 2 3
0)
cos( ) =
(1 4h)3/2
(1 + 4h)3/2
2
Obtaining the 𝜓 contact angle via Young’s law:
−0.15 −0.1 −0.05 0 0.05 0.1 0.15
0
45
90
135
180
h
ψ
r · n =
r
2w
✏2
h
@
@x
= ±
r
2w
✏2
(1 )
1,2,3,4 =
1 ±
p
1 ± 4h
2
27. Model C
Solving the PDEs in cylindrical coordinate system
The work of formation compared to the
classical theory
28. Comparison of Models A, B and C
0 10 20 30 40
0
0.2
0.4
0.6
0.8
1
S
W/WSC
Model A
Model B
Model C
All 3 models are in agreement with the classical nucleation theory the in the R → ∞ limit
29. Boundaries with diffuse interfaces
Solution:
• New, diffuse “wall” or “particle” field, 1 inside and 0 outside the excluded region
• New free energy functional and EOMs
Warren et al., Phys Rev. B 79, 014204 (2009)
Possible problems with sharp boundaries:
• Staircase-like approximation of complex geometries
• Sub-pixel movement of the interface is impossible
Justification:
• As the interface width of the particle function goes to zero, the “diffuse” free
energy functional falls back to the “sharp” one → the “diffuse” EOMs should be a
good approximation of the original, “sharp” EOMs
• The same EOMs can be obtained by the mathematically precise Diffuse Domain
Approach (S. Aland, Comput Model Eng Sci 57, 77 (2010))
F =
Z
f( , r ) (1 w) dV +
Z
Z( )|r w| dV
@
@t
= M
F
= . . .
33. Heterogeneous nucleation, summary
• Practically much more relevant, as in real systems, what we almost always observe
is heterogeneous (rather than homogeneous) nucleation
• Our approach to model heterogeneous nucleation is by setting the surface
properties (wetting) via appropriately chosen boundary conditions
• This treatment can be extended to surfaces with diffuse interface (meaning easy
treatment of more complex geometries)
• Nucleation by noise: the same issues as in the homogeneous case
• Nucleation via determining the EL solutions: more difficult than in the homogeneous
case, e.g. no radial symmetry can be utilized
• Lots of possible applications
34. Athermal nucleation
Heterogeneous nucleation: we assumed a large enough surface. But what if the surface
size is limited?
sub-
ryos
aller
i on
dius
ucle-
xist-
avity
cury
oge-
t he
tain-
ated
fore
per-
dent
her-
cata-
areas of nucleant can equally represent the surface
patches considered by Turnbull [15](Fig. 1(a)), or the
active faces of nucleant particles (Fig. 1(b)), for example
the {0001} faces of TiB2 inoculant particles used to
pendent of holding time at a given undercooling, but
proportional to the square of the undercooling DT. An
athermal nucleation law of this type is easy to imple-
ment in numerical modelling of solidification and is
widely used. As proposed by The´voz et al. [10], the
nucleation rate dn/dDT as a function of DT is commonly
taken to have a Gaussian form; this has been used in
probabilistic modelling of realistic grain structures [11].
The form of dn/dDT is not intrinsic to the liquid, but
dependent on thermal history. In particular, as reviewed
by Turnbull [12], the extent to which a melt can be
undercooled may increase strongly with the degree to
which it was superheated above its liquidus temperature
TL. This can be explained as an effect of the survival of
embryos of the crystalline phase above TL in cavities
(conical or cylindrical, in a mould wall or other sub-
strate) [12]. If the superheat is greater, fewer embryos
survive and the survivors are in cavities with a smaller
mouth; the DT at which they become active nuclei on
cooling is inversely proportional to the mouth radius
[12]. (Similar analyses have been applied for the nucle-
ation of gas bubbles in supersaturated liquids. Pre-exist-
ing bubbles in cavities are active nuclei if the cavity
mouth exceeds a critical radius [13].)
Turnbull [14]showed that dispersions of mercury
droplets could be used to measure the rate of homoge-
neous nucleation under isothermal conditions, but he
found that the large DT required was not always obtain-
able. In some dispersions (presumed to be contaminated
with mercury oxide) the droplet undercoolings before
solidification were only 2–4 K [15]. For these disper-
sions, the fraction of droplets solidified was dependent
on DT but not on time. Turnbull attributed this to ather-
mal nucleation at surface patches acting as potent cata-
lysts. He noted that a embryo of the crystalline phase
formed on such a patch could become an active trans-
formation nucleus only when, on cooling, the critical
nucleation radius rÃ
becomes less than the radius of
the patch. On this basis, he was able to derive the size
distribution of patches from the distribution of DT val-
ues at which the droplets solidified.
Similar athermal heterogeneous nucleation occurs in
the solidification of inoculated aluminium alloys [16].
Inoculation with an Al–Ti–B master alloy contributes
particles of TiB2 to the melt, and nucleation on these
is dominant. The particles are hexagonal prisms and
rect predictions of final grain size as a function of refiner
addition level, alloy solute content and cooling rate [16].
The success of this free-growth model has prompted
studies of how the particle size distributions in inocu-
lants might be optimized [18].
In this paper we analyse further the heterogeneous
nucleation of solidification on nucleant substrates of a
defined size. The standard approach to finite-size effects,
taken by Fletcher [19–21], considers nucleant particles of
various shapes, but analyses only the rate of thermal
nucleation under isothermal conditions. We extend this
work by analysing athermal nucleation. For simplicity
we consider nucleant areas that are plane circles of ra-
dius rN, the analysis for more complicated shapes mostly
differing only by geometrical factors. Plane circular
areas of nucleant can equally represent the surface
patches considered by Turnbull [15](Fig. 1(a)), or the
active faces of nucleant particles (Fig. 1(b)), for example
the {0001} faces of TiB2 inoculant particles used to
Fig. 1. Examples (shaded in (a) and (b)) of circular nucleant areas of
the kind considered in this work: (a) a surface patch, (b) the active face
of a nucleant particle. The growth of solid from such a nucleant area
(c) involves an increase in the curvature of the liquid/solid interface
enabled by an increase in undercooling. The curvature is maximum
when the liquid/solid interface is hemispherical and there is free growth
beyond that point. The onset of free growth as the undercooling is
increased constitutes athermal heterogeneous nucleation of
Quested and Greer, Acta Mat. 2005
T = 2 sl
sl : Gibbs-Thomson coe cient
: mean curvatre
small undercooling: large radius
large undercooling: small radius
Athermal process: transformations which are observed not to proceed under isothermal
conditions, but only on cooling
35. Athermal nucleation
g bubbles in cavities are active nuclei if the cavity
outh exceeds a critical radius [13].)
Turnbull [14]showed that dispersions of mercury
oplets could be used to measure the rate of homoge-
ous nucleation under isothermal conditions, but he
und that the large DT required was not always obtain-
le. In some dispersions (presumed to be contaminated
th mercury oxide) the droplet undercoolings before
idification were only 2–4 K [15]. For these disper-
ns, the fraction of droplets solidified was dependent
DT but not on time. Turnbull attributed this to ather-
al nucleation at surface patches acting as potent cata-
ts. He noted that a embryo of the crystalline phase
med on such a patch could become an active trans-
mation nucleus only when, on cooling, the critical
cleation radius rÃ
becomes less than the radius of
e patch. On this basis, he was able to derive the size
tribution of patches from the distribution of DT val-
s at which the droplets solidified.
Similar athermal heterogeneous nucleation occurs in
e solidification of inoculated aluminium alloys [16].
oculation with an Al–Ti–B master alloy contributes
rticles of TiB2 to the melt, and nucleation on these
dominant. The particles are hexagonal prisms and
cleation of solid aluminium is on their flat {0001}
Fig. 1. Examples (shaded in (a) and (b)) of circular nucleant areas of
the kind considered in this work: (a) a surface patch, (b) the active face
of a nucleant particle. The growth of solid from such a nucleant area
(c) involves an increase in the curvature of the liquid/solid interface
enabled by an increase in undercooling. The curvature is maximum
when the liquid/solid interface is hemispherical and there is free growth
beyond that point. The onset of free growth as the undercooling is
increased constitutes athermal heterogeneous nucleation of
solidification.
good wetting properties, easy
heterogeneous nucleation
particle grows as far as the substrate
size allows, when growth must stop
further growth is possible if the radius
can be decreased, i.e., the
undercooling can be increased
critical radius (undercooling)?
Think of pushing a ball through a
hole of radius rN in the wall...
r⇤
= rN
r⇤
! T⇤
T⇤
= T⇤
(rN )
Quested and Greer, Acta Mat. 2005
36. cleant area of radius rN (Fig. 1(c)). The
erfacial energies underlying Eq. (3) does
(in contrast to earlier analyses [19–21]),
fined contact angle. Another consequence
overage of the nucleant area is that line
no roˆle. For the cap, the radius of curva-
quid/solid interface rLS is related to the
ap h by
ð6Þ
f the cap, Vcap is given by
2
À
h3
3
¼ p
r2
Nh
2
þ
h3
6
ð7Þ
f the liquid/solid interface ALS by
¼ p h2
þ r2
N
À Á
: ð8Þ
uired to form a cap of solid (Wcap) has
rom interfacial energies and from the free
associated with the solidification of the
Consistent with there being pre-existing
ence point for energy (Wcap = 0) is taken
n infinitesimally thin layer of solid coating
eant area. The only relevant interfacial en-
ALScLS. Consistent with the derivation of
ee energy of fusion per unit volume is ta-
DT, an excellent approximation for small
of cap formation,
S À pr2
N
Á
À V capDSVDT; ð9Þ
g from Eqs. (7) and (8), can be expressed
The form of Eq. (12) is plotted in Fig. 4 for five values of
dimensionless undercooling. For DT DTfg, the work of
cap formation passes through a minimum followed by a
maximum as h/rN is increased. These extrema occur at
h
rN
¼
DT fg
DT
Æ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
DT fg
DT
2
À 1
s0
@
1
A ð13Þ
and represent conditions of equilibrium across the li-
quid/solid interface. At these points the radius of curva-
ture of that interface has the critical value rÃ
given by
Eq. (1), expressed in dimensionless terms as
rÃ
rN
¼
DT fg
DT
: ð14Þ
-1
-0.5
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3 3.5 4
Dimensionless cap height ( /
N
)h r
∆ / ∆
fg
= 0.5T T
∆ / ∆
fg
= 0.625T T
∆ / ∆
fg
= 0.75T T
∆ / ∆
fg
= 0.875T T
∆ / ∆
fg
= 1T T
WW*cap
/Dimensionlesscapenergy,∆fg
T
W W*∆
cap
/ ∆ fg
T
Athermal nucleation
Quantitative analysis: calculation of the surface and volume contributions
vature with a plane such that the circle of intersection
has a radius equal to rN. The cap of smaller volume
(a) is in metastable equilibrium; the larger cap (b) is in
unstable equilibrium.
The work of formation and rLS are presented in Fig. 6
for the case of DT/DTfg = 0.5. Between the two equilib-
rium conditions shown in Fig. 5, rLS rÃ
. The work of
cap formation in this figure is normalized with respect
to W Ã
DT the critical work for homogeneous nucleation
at the actual undercooling:
W Ã
DT ¼
16pc3
LS
3DS2
VDT2
: ð15Þ
This normalization would not be useful in Fig. 4, since
W Ã
DT would vary from curve to curve, but it is useful
for the given undercooling in Fig. 6 in indicating how
the work of cap formation compares with the critical
work for homogeneous nucleation.
Fig. 4 shows that for DT/DTfg 1, there is an energy
barrier for nucleation. As the normalization is with re-
spect to the fixed quantity W Ã
DTfg
, the energies on the differ-
ent curves can be compared directly. As the dimensionless
undercooling is increased, the two extrema converge and
the barrier decreases. At DT/DTfg = 1, the work of forma-
tion as a function of cap height no longer exhibits
extrema; there is just the stationary point at
h ¼ rLS ¼ rÃ
¼
2cLS
DSVDT
ð16Þ
when the solid takes the form of a hemisphere and there
is no barrier to free growth (Fig. 4). Cooling through the
condition DT/DTfg = 1 gives athermal nucleation. At
DT/DTfg 1, there could be thermal activation over
the nucleation barrier, and we now assess the likelihood
of this pre-empting athermal nucleation on cooling.
3.2. The competition between thermal and athermal
nucleation
The initial, infinitesimally thin, coating of solid on the
nucleant area grows naturally to the metastable and dor-
mant condition shown in Fig. 5(a). From that condition,
the critical work of thermal nucleation DWcap is the dif-
ference in energy between the two extrema, which can be
expressed in dimensionless terms as
DW cap
¼
DTfg
2
À 1
# ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À
DT
2
s
ð17Þ
Fig. 5. The (a) metastable- and (b) unstable-equilibrium configura-
tions of the dormant-solid cap for DT DTfg. The radius of curvature
-0.4
-0.2
0
0.2
0.4
0
1
2
3
4
5
0 0.5 1 1.5 2 2.5 3 3.5 4
rr
Wcap
/Dimensionlesscapenergy,
Dimensionless cap height ( /
N
)h r
∆W W*
cap
/ ∆T
sionlessradiusofcurvature(
LS
/
N
)
Fig. 6. The dimensionless work of formation ðW cap=W Ã
Þ of the solid
cap and radius of curvature of the liquid/solid interface (rLS/rN) as a
function of the dimensionless cap height (h/rN) at a dimensionless
undercooling of 0.5 (Eq. (10)). The work of formation is normalized
with respect to the critical work for homogeneous nucleation at the
actual undercooling (Eq. (15)). A dimensionless undercooling of 0.5
results in a dimensionless critical radius of 2 (dashed line), which is the
value at the extrema on the work curve. The critical work for free
growth (DWcap) is the energy difference between the extrema and is less
than the critical work for homogeneous nucleation.
metastable
unstabe
Quested and Greer, Acta Mat. 2005
Wcap
WΔTfg
= −
1
4
ΔT
ΔTfg (
h
rN )
3
+
3
4 (
h
rN )
2
−
3
4
ΔT
ΔTfg
h
rN
Vcap = π
(
r2
Nh
2
+
h3
6 )
ALS = π(h2
+ r2
N)
Wcap = γLS(ALS − r2
Nπ) − VcapΔSVΔT
For a given h, there are two equilibrium configurations
h
37. Athermal nucleation
and represent conditions of equilibrium across the li-
quid/solid interface. At these points the radius of curva-
ture of that interface has the critical value rÃ
given by
Eq. (1), expressed in dimensionless terms as
rÃ
rN
¼
DT fg
DT
: ð14Þ
-1
-0.5
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3 3.5 4
Dimensionless cap height ( /
N
)h r
∆ / ∆
fg
= 0.5T T
∆ / ∆
fg
= 0.625T T
∆ / ∆
fg
= 0.75T T
∆ / ∆
fg
= 0.875T T
∆ / ∆
fg
= 1T T
WW*cap
/Dimensionlesscapenergy,∆fg
T
W W*∆
cap
/ ∆ fg
T
Fig. 4. Dimensionless work of formation ðW cap=W Ã
DTfg
Þ of the solid cap
as a function of dimensionless cap height (h/rN) plotted for various
values of dimensionless undercooling (DT/DTfg) (Eq. (12)). The work
of formation is normalized with respect to the critical work for
homogeneous nucleation at the free-growth undercooling. The minima
(maxima) in these energy curves represent metastable (unstable)
equilibrium configurations.
ayer covering the nucleant would be energetically less
avourable than the formation of spherical-cap embryos
f solid (Fig. 2(a)).
Conventional analyses emphasise case II and classical
pherical-cap nuclei. Case I has been largely overlooked
ecause it appears to suggest that there is no nucleation
arrier to solidification, even though, as noted in Section
, a wetting layer of solid may be dormant with a free-
rowth barrier to its becoming a transformation nu-
leus. However, case I was considered in early work
n the effect of thermal history on undercooling of liq-
ids; Richards [34]suggested that a crystalline adsorbate
might exist on substrates. This idea was pursued in the
articular case of TiB2 inoculant particles in aluminium,
where the hypernucleation theory considered the condi-
ions for forming a quasi-solid nucleant layer on the sur-
ace of the particles above the liquidus temperature
35,36]. There is indeed some microscopical evidence
or a layer on these particles [17]. Transmission electron
microscopy has recently been used [37]to show that
here can be ordering in liquids at substrate surfaces,
ven above TL.
For the effective nucleation temperature to be given
y Eq. (2) according to the model depicted in Fig. 1(c),
here must be initial formation of solid on the nucleant
ubstrate and either of the cases identified above can ap-
ly. In case I the solid exists as a thin layer even above
TL and there is no nucleation barrier for formation of
his initial solid. It is likely to form even when the cool-
ng rate is very high. In case II, the solid must be nucle-
ted on the substrate and this occurs beyond a critical
ndercooling; accepting that there are geometrical prob-
ems for small h, the spherical-cap model is taken as the
fg
In regime (iii) rN rÃ
and there is no barrier to free
growth; in this case, the rate-limiting step for effective
nucleation (i.e., for the substrate to act as a transforma-
tion nucleus for solidification of the liquid) is the initial
formation of solid on the substrate.
0 0.5 1 1.5 2
ContactAngle,θ
∆ / ∆
fg
T T
0
π/2
(i) (iii)
(ii)
nucleation
not possible
no barrier to
free growth
Fig. 3. Regimes of nucleation behaviour as a function of reduced
undercooling (DT/DTfg) and contact angle h. In (i), the nucleant area is
too small to permit a spherical-cap nucleus to form. In (ii), the regime
of most interest in the present work, a nucleus can form but its growth
stops when it has spread over the nucleant area and thickened such
that the liquid/solid interface reaches the critical radius of curvature r*
.
The case of solid formed by adsorption or wetting can be represented
by h = 0. In (iii), rN r*
, there is no barrier to free growth, and the
rate-limiting step for effective nucleation is the initial formation of
Quested and Greer, Acta Mat. 2005
For ΔT ≃ ΔTfg normal nucleation (fluctuations) can also contribute
38. Athermal nucleation
Dormant embryos. Their size
depend on the undercooling.
T T⇤
(rN )
Dormant embryos start to
grow. No thermal activation is
required!
T = T⇤
(rN )
T T⇤
(rN )
Free growth limit
Use in phase-field models:
Distribute hypothetic particles with a given size
distribution. Induce nucleation (flip a pixel) for
those, that have larger than the critical
undercooling (corresponding to their size)
T = 17K
T = 18K