Phase-field modeling of crystal nucleation II: Comparison with simulations and experiments

1. Phase-field modeling of crystal nucleation II:
Comparison with simulations and experiments
aWigner Research Centre for Physics, H-1525 Budapest, P. O. Box 49, Hungary
bBCAST, Brunel University, Uxbridge, Middlesex UB8 3PH, U.K.
L. Gránásy,a,b
Phase Field Workshop “Focus on Nucleation”,
26 Sep 2018, Center for Hierarchical Materials Design HQ, Northwestern
University, Evanston, IL, USA

2. The process to be modeled:
(MD simulation for the Lennard-Jones system by Frigyes Podmaniczky)
1
Nucleation:
Nuclei are defect-rich crystal-like domains forming on the nm scale:
Point defects, stacking faults, twin boundaries, capillary waves, etc.
Coloring: green – fcc-like, pink – hcp-like, liquid – transparent

3. I. Homogeneous nucleation in PF models:
(Solving the Euler-Lagrange equation (ELE))
- “standard” PF model (single component version of WB 1995)
- other models [g() and p() functions]
- comparison with experimental and MD results for Ni, LJ Ar, ice-water system
- binary crystal nucleation (ideal/regular solution)
- competing bcc/fcc nucleation in binary alloy (Fe-Ni)

4.    






  fdrIdrF
VV
2
2
33
2
...),,( 


)(Wgf Bulk free energy density:
2. Planar (1D equilibrium) interface:
(x) minimizes F  it satisfies
the Euler-Lagrange eq. i
i
y
x
yx
II
I
I
IIIF
























































 ][0
1D integral form of EL eq.:
(Cahn & Hilliard, JCP, 1958)









x
x
I
I

0
 22
1
4
1
)(  gQuartic double-well function:
1. Thermodynamics:
A. “Standard” phase-field theory
2
This form of g() can be obtained
from Ginzburg-Landau expansion
of the Helmholtz free energy for
BCC crystal symmetry.
(Shih et al. PRA, 1987)

5. Interfacial free energy:
 
26
)(2)(
2
21
0
22
W
d
d
dx
WgdxWg
dx
d
SL





 




















 


Interface profile:




























 %90%10
2
)9log(tanh1
2
1
22
1
tanh1
2
1
)(
d
x
x
W
x


Interface thickness:
)9log(
2
2
29.0
1.0
%90%10
W
d
d
dx
d









 
1D integral form of EL eq.:
(Cahn & Hilliard, JCP, 1958)









x
x
I
I

0
)9log(
3 %90%102 

dSL

%90%10
)9log(24


d
W SL
 )(
2
22


Wg
dx
d






3
To recover the HS behavior
(SL  T and d10%90% = const.), we replace
 2   2T
W  W T

6. 3. Properties of homogeneous nuclei: EL eq. in 3d
The free energy density:
Double-well & interpolation functions:
Model g() p()
Standard PFT (WB)  22
1
4
1
   23
61510  
fpWTgf  )()( 
4
With data of Ni:
From bottom to top:
T/Tf = 0.35, 0.31, 0.27, 0.23,0.19, 0.15 (solid lines)
corresponding to
T (K) = 604, 535, 466, 397, 328, and 256
T/Tf = 0 (dashed line)
𝑓 = 𝑓𝐿 − 𝑓𝑆 > 0

7. With data of Ni:
From left to right:
T/Tf = 0.35, 0.31, 0.27, 0.23,0.19, 0.15
Radial PF profiles:
Critical radius vs. Tr
Re – equimolar surface
Rp – surface of tension
CNT – classical theory
DIT – diffuse intf. theory
Euler-Lagrange eq.: Simplification: isotropic SL  spherical symmetry (good approximation for metals)































)(12
0 22
2
f
rrr
IIF
Boundary cond.: r = 0: /r = 0
r = :  = 0
To obtain W*, substitute num. solution
into expression of the free energy.
5
Nucleation rate:
 kTWZOKJ nnSS /*exp**0hom,  
*2
;
6
;*)(4 *2
3/2
*
kTn
g
Z
D
nO at
nn



From MD simulations: K0  100 Wolde et al. JCP 1996
   SL ffpWTg
T
I  )()(
2
2
2


Free energy density:

8. 6
   SL ffpWTg
T
I  )()(
2
2
2


Different g() and p() functions:
B. Other Phase-Field type models

9. 1. Homogeneous nucleation in Ni (no adjustable parameters):
7
C. Applications of ELE & comparison with MC/MD/experiments
Complete set of data:
- W from MC umbrella sampling
- Experimental W: evaluated (J0,CNT) from undercooling
statistics obtained by chip calorimetry
- SL from MD (assumed to be isotropic)
- Thermodynamics from MD/experiment/Turnbull’s appr.
(Blokeloh et al. PRL, 2011)
Standard PF and
CNT with SL  T
work well
Gránásy et al., to be published

10. 2. Ice-water system (assumption on nucleation prefactor):
8
Input from experiment and MD:
- Experimental data for JSS(T) are used
- Experimental thermodynamics (accurate left of dashed line)
- SL from Hardy’s GB groove measurements
- Diffusion coefficient from experiment
- Nucleation prefactor 100J0,CNT is assumed (MD: Wolde et al. JCP 1996)
Standard PF and
CNT with SL  T
work well
Gránásy et al., to be published

11. 3. Modified Lennard-Jones system applied for Ar (no direct W* data):
9
Input from MD:
- Broughton-Gilmer type modified LJ system
(thermodynamics, SL, DL are known from MD)
(Broughton & Gilmer et al. JCP 1983, 1986)
- Nucleation prefactor from CNT was used
- Nucleation rates from MD
(Báez & Clancy JCP 1995)
Standard PF works,
CNT with SL=const.
fails
Gránásy et al. Phys. Rev. Lett. 2002

12. Notation:
CNT - classical nucleation theory (SL,eq)
SCCT - self-consistent classical theory (W* = W*CNT – W1,CNT)
PFT/S1 - PFT with standard g() & p() (single field)
PFT/S2 - PFT with standard g() & p() (two fields)
PFT/GL1 - PFT with GL g() & p() (single field)
PFT/GL2 - PFT with GL g() & p() (two fields)
4. Hard-sphere system: (complete set of data for test)
10
Assumption: isotropic interfacial free energy (spherical nucleus)
Standard PF and
CNT with SL  T
does not work!
Complete set of data:
- W from MC umbrella sampling (Auer & Frenkel, Nature, 2001)
- Thermodynamics from MD (EOS by Hall, JCP, 1970)
- Interfacial profiles from MD (Davidchack & Laird, JCP 1998)
10% - 90% thickness, d10%-90% ~ 3.2
- Interfacial free energy from MD (Davidchack & Laird, PRL, 2000)
orientation SL 2/kT
(111) 0.58  0.1
(100) 0.62  0.1
(110) 0.64  0.1
average 0.61
Revised value: SL 2/kT = 0.559 (Davidchack et al., JPCB, 2006)
SL& d10%-90%   2 & W
Tóth & Gránásy J. Phys. Chem. B 2009

13. 5. Comparison of PF type models in the case of Ni:
11
- W* is weakly dependent on g()
- W* is strongly dependent on p()
Gránásy et al., to be published
Further theoretical
work is needed !

14. Thermodynamic properties:
Binary crystal-liquid
interface
Fields: (x) – structural order parameter
c(x) – concentration of species B
A. Regular solution:
Further simplification: Cp,L = Cp,S for all T (Turnbull’s linear approximation for f )
 )1log()1()log(
)1()(
)1( 100
,
0
, cccc
V
RT
V
ccT
cfcff
mm
LL
BLALL 


 )1log()1()log(
)1()(
)1( 100
,
0
, cccc
V
RT
V
ccT
cfcff
mm
SS
BSASS 


B. Ideal solution: 0and0 1100  SLSL
Free energy density for liquid (L) and solid (S) phases:
  Af
T
T
SpLpAfA HdTCCHTH
Af
,,,,
,
)(  
 
Af
Af
Af
T
T
SpLp
AfA
T
H
SdT
T
CC
STS
Af
,
,
,
,,
,
,
)(



 

Afm
AfAf
ASALA
TV
TTH
fff
,
,,0
,
0
,
)( 

12
D. Binary PF model for crystal nucleation in Ni-Cu system:

15. )()()()](1[)()(),( cfpcfpgcWcf SL  
C. Free energy surface:
Double-well &
interpolation functions:
 22
1
4
1
)(  g
 23
61510)(  p
1 Tf (K)
Hf
(GJ/m3)
Vm
(cm3/mol)
SL
(mJ/m2)
d10%-90%
(nm)
Cu 1358 1.78 7.4 228 1.596
Ni 1728 2.35 7.4 364 1
BA cWWccW  )1()(
)9log(
3 ,
%90%10
,
2
BABA
dSL 



BA
BA
SL
BA
d
W ,
%90%10
,
, )9log(24



A
B
B
A
d
d
SL
SL
%90%10
%90%10





13
Ideal solution: Application for the Cu-Ni system

16.   )()()()()](1[)()(
2
,
2
2
eLSL cccfpcfpgcWI  
Planar (1D equilibrium) interface :
Two fields: (x) & c(x) 
two Euler-Lagrange eqs.
c
c
I
c
I
c
I
c
F
IIF
andbetweeneq.implicit
0
































Boundary cond.:
x =  :  = 1 & c = cS,e
x =  :  = 0 & c = cL,e

Interfacial free energy:
  



 d
d
dx
cfdxcf
dx
d
eeSL  























1
0
22
)](,[2)](,[
2
Interface profile:
Interface thickness:





d
cf
d
d
dx
d
e
2/19.0
1.0
29.0
1.0
%90%10
)](,[ 






















d
cf
d
d
dx
xx
e
2/1
2
0
00
)](,[ 













  )()(ˆ)](ˆ,[)](ˆ,[)( ,,
,
eLLeL
c
L
cfcc
c
f
cfcff
eL



 
    )()(
)1/()(ˆ ,,,


pffgWW
RT
V
y
ecceccc
ABAB
m
y
eLeL
y
eL

 
  CfI  )(
2
2
2


eLc
L
c
f
,



 – Lagrange multiplier ensuring mass conservation
14

17. Two fields: (x) & c(x) 
two Euler-Lagrange eqs.
c
c
f
c
I
c
F
c
F
IIF
c
L
c
andbetweeneq.implicit
0
0,0,







  


















Boundary cond.:
r = 0:  = 0 & c = 0
r = :  = 0 & c = c
  )()(ˆ)](ˆ,[)](ˆ,[)(  




cfcc
c
f
cfcff L
c
L
















 )](ˆ,[2
2
2
2 cf
rrr
Crit. Fluctuation = Nucleus =
Extremum (saddle point) of F 
Solution of the Euler-Lagrange eqs.
    )()(
)1/()(ˆ


pffgWW
RT
V
y
ecceccc
ABAB
m
yy

 



3/12*
16
3





 



fW
eff
Boundary cond.: r = 0: /r = 0
r = :  = 0
15
Properties of nuclei: EL equation in 3D

18. - Multi Phase Field Theory: 3rd phase present at interfaces
- Folch & Plapp (2005): 3rd phase never present
- Physically motivated approach to free energy surface?
fcc
bcc
liquid






1
)1(
L
bcc
fcc
fcc
bcc
liquid
GL expansion:  gij() & pij()
  )()()](1[)()(
),()(),()](1[),,(


fccbccLbccfccbccLfccfccbcc
LfccfccbccLbccfccbcc
gpppp
TfpTfpTf




F  dV cij ij  ji 
2
 f i ,T 
i j









 i
i
  1&
Following DFT (Shen & Oxtoby, 1996), two structural
order parameters:
 - solid-liquid phase field,
 - solid-solid phase field (for fcc-bcc: Bain’s distortion)
E. Competing FCC and BCC nucleation (GL approach):
16

19. Results for Fe-Ni: almost all input data are accessible with sufficient accuracy
Ginzburg-Landau double-well & interpolation functions:
Transition g() p()
BCC-L
FCC-L
FCC-BCC
 22
1
4
1
 
 22
1
4
1
 
  343

 222
1
6
1
   24
23  
  343

(Tóth at al. PRL 2011)
Exception:
FCC-BCC  [169, 672] mJ/m2
17

20. II. Simulating homogeneous nucleation in PF models:
(solving the equations of motion (EOMs))
- binary crystal nucleation (ideal/regular solution)
- nucleation of phase-separation in liquid Al-Bi

21. 1000  1000 grid
Phase field Concentration field
Equations of motion:
(Allen-Cahn + Cahn-Hilliard type)
fluxcfluxc
c
I
c
I
M
c
F
M
t
c
II
M
F
M
t


















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




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

)'()'(2),()','( ttkTMtt    rrrr )'()'(2),()','( 2
ttkTMtt cfluxflux   rrrr
 
)()()()()()](1[
)()(
2
00
2
2
cfcc
c
f
cfpcfp
gcWI
L
L
SL 







Periodic boundary cond.
18
A. Simulation of nucleation in
binary system I: Cu-Ni (ideal solution)

22. Phase separation in liquid miscibility gap: Model C + flow
1. Free energy functional:
2. PFT equations:
3. Balance laws: Mass conservation
Momentum conservation
Non-classical stress tensor
0


v
t



P


gvv
t
v 

 )(
 
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
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


 
),()()](),()][(1[)()(
22
2
2
2
2
3
TcfpfTcfpTgcw
cΓ
T
rdF
LoriS
c

















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F
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t
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f
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c
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F
Mv
t
j
m
jc
)()(
)1()(
)(



Coupling to hydrodynamics: a’ la Anderson et al.
ΠIP 












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


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
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
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

  HTpξΓTccΓTcccP cc )](1[)()(
2
1
)(
2
1 2222222

19
B. Simulation of nucleation in
binary system II: Al-Bi (regular solution) (Tegze et al. Mater. Sci. Eng. A 2005)

23. Marangoni motion
20
Phase separation in liquid miscibility gap:
Testing

24. J.Mainville et al. PRL (2001)
Liquid phase separation at the critical composition (Al-Bi):
21

25. T T
Flow accelerates droplet coagulation:
Left: No flow
Center: Flow + homogeneous T
Right: Flow + T towards the centerline
22
Liquid phase separation in (metastable) monotectic system (Al-Bi):

26. III. Nucleation vs. microstructure formation:
- quantitative PF modeling with nucleation
- PF modeling of Growth Front Nucleation

27. Complex patterns evolve
due to the interplay of
nucleation and growth.
23American Pale Ale Dirty Martini Vodka Tonic
Gin
Water Polycrystalline matter: Atmospheric sciences:
- technical alloys - aerosol formation (climate change)
- ceramics
- polymers
- minerals
- food products, etc.
In biology:
- bones, teeth
- kidney stone
- cholesterol in arteries
- amyloid plaques in Alzheimer’s disease
Also frozen drinks:
A. Complex polycrystalline structures:

28. 1. Impinging single crystals:
2. Polycrystalline
growth forms:
(Growth Front
Nucleation = GFN)
3. Impinging polycrystalline particles:
24
B. Classification of polycrystalline microstructures

29. 1. Diffusional instabilities:
2. Nucleation
- of growth centers
- homogeneous
- heterogeneous (on particles or walls)
- of new grains at the growth front (Growth Front Nucleation = GFN)
- heterogeneous (particle-induced)
- homogeneous (???)
with specific misorientation (fixed branching angle)
C. Contributing phenomena?
Crystal
Liquid
Mullins-Sekerka
instability
isotropic anisotropic
25

30. D. Possible solutions: multi-phase-field/multi-order-parameter theories
Separate field for individual grains …
Important works:
MOPT for grain coarsening:
L.Q. Chen & W. Yang, Phys. Rev. B (1994).
N. Moelans et al. PRL (2008).
MPFT for solidification:
I. Steinbach et al. Physica D (1996).
M. Plapp & R. Folch, PRE (2005).
P.C. Bollada et al. Physica D (2012).
H.K. Kim et al. Mod.Sim.Mater.Sci.Eng. (2012).
G.I. Tóth et al. PRB (2015).
M. Ohno et al. PRE (2017).
MPFT for solidification:
26
Advantages:
 All interfaces can be handled individually 
besides relative orientation, the inclination of the
interface can also be considered
Disadvantages:
 Thousands of fields might need to be handled
 Difficult to incorporate thermal fluctuations
 Not straightforward how to incorporate GFN

31. Further applications:
Miyoshi et al. npj Comput. Mater. (2017)MPFT: Grain coarsening
Hötzer et al. Acta Mater. (2016)MPFT: Spiraling eutectics
27
MOPT: Coarsening of 3-phase structure
Ravash et al. J. Mater. Sci. (2014)
MPFT: Geological problems Ankit et al. J. Geophys. Res. (2015)

32. E. Possible solutions: orientation-field theories (OFT)
Crystallographic orientations & grain boundaries???
Kobayashi, Warren, Carter: Physica D 2000:
- Non-conserved orientation field  to distinguish particles
of different cryst. orientation
fori  H
- Reasonable grain boundary dynamics
Gránásy, Börzsönyi, Pusztai: PRL 2002:
- Noise induced nucleation with orientation field in 2D
(orientation field in liquid fluctuates in time and space)
Pusztai, Bortel, Gránásy: EPL 2005:
- Noise induced nucleation with quaternion
representation of crystallographic orientation in 3D
(Equivalent formulation by Kobayashi & Warren, Physica D, 2005)
The | | theory:
28

33. Free energy (scalar):
- penalizes spatial change of 
- local functional [may depend on  , & derivatives, ( )2k ]
- invariant to rotation (explicit  dependence excluded)
Seek in form
n > 1 infinite broadening, unless one uses
n = 1 no such problem, BUT
Why this form? fori 
Hence our choice for the
orientational free energy density:
fori = HT p()
 Rotational invariance sacrificed!
“jello mould” potential
29

34. OFT for polycrystalline solidification: (Gránásy, Börzsönyi, Pusztai, PRL 2002)
Aim: - nucleation of grains with different orientation
We extend the orientation field  to liquid:
- constant  [0, 1] in solid
- fluctuates in time & space in liquid
New features:
- solid-type fluctuation in   orientational ordering
- orientational disorder can be trapped into solid (GFN)
 




 F
M
t
Free energy:
Time evolution:
(non-conserved dynamics)
where  = ,0 [1  p()]
  )(pHTfori
30

35. Molecular dynamics of liquid crystallization in 2D: (with Yukawa potential by Z. Donkó)
31
Structural analysis (complex bond order parameter):
- j : angle towards j-th neighbor in lab. frame
- |g6| :  degree of order
- phase:  local crystallographic orientation
Voronoi analysis: 4 - grey; 5 - blue; 6 - yellow; 7 - red
Orientation map Voronoi map ||
MD

36.    xksMM   /atanand)2cos(1),( y0
0

 )2cos(1),( 0   kss
Phase field
Concentration



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


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
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
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






...),()(
),()](1[)()(
)1(
Tc
c
f
p
Tc
c
f
pgWW
cDc
RT
v
c
S
L
AB
m

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
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22
)( 




 
s
TsHTpM
Orientation
(2D)
Equations of motion in 2D: (anisotropic, no SG term for c)
32

37. 1. Homogeneous nucleation (of growth centers):
Transient before final orientation established:
orientation
 = 0.5
Noise induced:
composition phase field orientation
solidus liquidus solidliquid
 color code
The nucleus is made entirely of interface 33
F. Nucleation modes in orientation-field theories

38. 2. Heterogeneous nucleation (of growth centers) with desired contact angle
L. Gránásy, T. Pusztai, D. Saylor, J. A. Warren, Phys. Rev. Lett. 2007
Note the capillary waves &
the corresponding
fluctuation of the contact angle!
Calculation for pure Ni :
- d10-90% = 2 nm
- exp = 364 mJ/m2
- x = 2 Å (1 pixel ~ 1 atom)
- fluctuation-dissipation noise
- thermal feedback
Boundary condition :
  )1()cos(
2
1
Su 

 





 n
 = 45 100  600
60
200  350
90
200  300
200  250120
34

39. Size dependence: 1 pixel 5 pixels 13 pixels 45 pixels
Orientation misfit:  = 0.1  = 0.2  = 0.3
Lateral disp. (pixels): x =  6 x =  3 x = 0 x = 3
 = 0
Experiment:
3. Heterogeneous Growth Front Nucleation:
Tip deflection at foreign particles
35

40. 4. Homogeneous Growth Front Nucleation I.
Reduced orientational mobility (M  Drot)
(new mechanism: trapped disorder in )
Complex undercooled liquids:
Drot/Dtr ( M/M ) decreases with increasing T
“decoupling”
M/30
36

41.  = 90 60 45 30
x = 0.10 0.15 0.20 0.25
 
 
 













otherwise1
4
1
for2sin
otherwise1
4
3
for2sin
)1(
2
00
1
00
0
10
0
n
n
F
m
m
F
FxxF
H
fori



5. Homogeneous GFN II. (branching with fixed misorientation)
37

42. Phenomena incorporated into the PF model in 2D & 3D:
isotropic anisotropic
composition phase field orientation
38

43. Input data? (models with orientation field)
- Thermodynamic data (free energy of all phases)
- S-L interfacial energy
- S-L interface thickness
- Grain boundary energy
- Diffusion coefficients: M  Dtransl
Mc  Dinterdiffusion
M  Drot
- Structure related data:
anisotropy of interfacial / grain boundary energies
anisotropy of mobilities
39

44. Different length- & time scales for , c, .
  1-2 nm  ~ 1 Å resolution (10,000 x = 1 m)
Compromise to enable computations :
broad interface is used ( enhanced solute trapping, etc.)
1. Use a broad interface:
Positive: quantitative simulations for a broad interface hypothetical system.
Problem: we are interested in real materials.
2. Staying on atomistic scale:
Positive: proper solute trapping/interface kinetics & nucleation.
Problem: only small computations (e.g., no dendrites), or enormous computation power is needed
(+ adaptive mesh), we may hope for ~ 1 µm3
3. Broad interface:
Positive: anti-trapping currents a’la Karma  proper growth kinetics
for large sizes (up to mm)
Problem: nucleation is wrong. (E.g., cell volume is larger than the nucleus)
Remedy I: hom. nucleation has to be incorporated by hand
(barrier from the Euler-Lagrange equations + physical ).
Remedy II: particle induced freezing a’la Greer
(different way of incorporating nucleation by hand).
,
40
G. Limitations & strategies for quantitative computing

45. 41
A. Impinging single crystals: Quantitative PF modeling of CET in Al45.5Ti54.5
IV. Applications
1500  300 grid number of particles ~ 200 / frame
0.75 mm  0.15 mm particle size (Gaussian) = (20 4) nm
- CALPHAD thermodynamics
- Anti-trapping current (Kim, Acta Mater., 2007)
- Greer’s free growth limited model
5 10 20 40
4
8
16
32
G (104 K/m)
4
8
16
32
V (104 m/s)
Comparison with Hunt’s model (2D)

46. color code
Particles represented by orientation pinning centers:
areas of random but fixed orientation
L. Gránásy, T. Pusztai, T. Börzsönyi
Research Institute for Solid State Physics
and Optics, Budapest, Hungary, 2002
Experiment: PEO/PMMA + clay
Simulation: 3000  3000 grid
Ferreiro et al., PRE (2002)
B. “Dizzy” dendrites
L. Gránásy et al. Nature Materials, Febr. 2003
42

47. C. Polycrystalline spherulites
Category 1
spherulite
Spherulites are almost everywhere
- Se
- cast iron (nodular)
- polymers/biopolymers
- metallic/oxide glasses
- eutectic systems
- urine (kidney stone)
- cholesterol
- insulin
- chocolate
43
Category 2
spherulite

48. Formation of Category 1 spherulites:
(a) Gradual transition from single crystal nucleus to Category 1 spherulite:
MD for hard-spheres:
(O’Malley & Snook, PRL 2003)
(b) Growth starts from polycrystalline nuclei:
[e.g., TMPS = poly-(tertramethyl-p-silphenylene)-siloxane
Magill, J. Appl. Phys. (1964)]
Interface breakdown
Polycrystalline nucleus
Experiment 44

49. Experiment
Simulation
Experiment
Simulation
Description of
morphology with a
few model params.
(anisotropies, branching
angle, MS well depth, …)
45

50. S = 1.5 1.8 1.9 1.95 2.0 2.1 2.2
200200400 grid
Triclinic crystal symmetry
Ellipsoidal symmetry of
kinetic anisotropy
Coloring: Inclination relative to
nucleated direction in deg.
S = 0.75 0.85 0.90 0.95 1.00 1.10
2D
46
From needle crystals to polycrystalline spherulites:

51. Experiments on orientation:
PF simulation:
Polarized transmission optical
microscope
Gatos et al. Macromol. (2007)
47

52. Spherulitic growth in channel:
Scratch:
Courtesy of V. Ferreiro
Holes:
Courtesy of M. Ferguson
Channel, scratch, holes:
D. Manipulating crystallization:
48
Lee et al. Adv. Mater. (2012)

53. 200200400 grid
Orientation selector
Dendrite in toroidal shell
600200600 grid
400400400 grid
Dendrite in spherical shell
Confined space:
49

54. 23p
IV. Summary:
I. Single-field PF models:
 “Standard” PF model works for Ni, water & Ar (LJ),
fails for the HS system
 Ginzburg-Landau model is accurate for HS, does
not work for the others
FURTHER WORK IS NEEDED!!!
III. Nucleation mechanisms in OF models:
 Homogeneous/heterogeneous
 Growth Front Nucleation: particle induced /
homogeneous/fixed-angle branching
II. Two- & three-field PF models:
 For Cu-Ni and Fe-Ni systems
reasonable agreement with
undercooling experiments
IV. Nucleation vs microstructure:
 Quantitative simulations with OF models
 Polycrystalline growth:
- particle-induced,
- random/fixed angle branching

55. Institute for Solid State Physics and Optics
WIGNER RESEARCH CENTRE FOR PHYSICS
Hungarian Academy of Sciences
H-1121 Budapest, Konkoly-Thege u. 29-33
Computational Materials Science Group in WRCP:
László Gránásy Prof. - team leader nucleation, PF, DFT, …
Tamás Pusztai Sci. Adv.. - nucleation, PF, topological defects
György Tegze Sen. Sci. - CFD, num. methods
Frigyes Podmaniczky PhD student - DFT, anisotropy, nucleation
László Rátkai PhD student - eutectics, LB flow
László Gránásy
Sci. Advisor
Tamás Pusztai
Sci. Advisor
Frigyes
Podmaniczky
PhD student
László Rátkai
PhD student
György Tegze
Senior Scientist
Computational Materials Science Group