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

1. Phase-field modeling of crystal nucleation:
Comparison with simulations and experiments
(summary of 26 Sep 2018 talk)
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”,
22 Apr 2019, 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. Homogeneous nucleation in single-field PF models:
(Solving the Euler-Lagrange equation (ELE))
- “standard” PF model (single component version of WB 1995)
- comparison with experimental and MD results for Ni, LJ Ar, ice-water system
- other models [g() and p() functions]

4. A. “Standard” phase-field theory (single-component WB theory)
2
)(Wgf Bulk free energy density in eq.:  22
1
4
1
)(  g
Quartic double-well function:
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)
1. Structural order parameter (phase field)
PF theory:  (r, t)
CDFT: number density n(r, t)
2. Free energy:
   






  fdrIdrF
VV
2
2
33
2
...),,( 



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 integrated form of ELE:
(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
2. Planar (1D equilibrium) interface:

6. 3. Properties of homogeneous nuclei: ELE in 3d
Free energy density in the undercooled state:
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.
   SL ffpWTg
T
I  )()(
2
2
2


Free energy density:
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
4. Nucleation rate
5

8. 1. Homogeneous nucleation in Ni (no adjustable parameters):
6
B. 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 PFT and
CNT with SL  T
work well
Gránásy et al., to be published

9. 2. Ice-water system (assumption on nucleation prefactor):
7
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 PFT and
CNT with SL  T
work well
Gránásy et al., to be published

10. 3. Modified Lennard-Jones system applied for Ar (no direct W* data):
8
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 PFT works,
CNT with SL=const.
fails
Gránásy et al. Phys. Rev. Lett. 2002

11. Notation:
CNT - classical nucleation theory (SL,eq)
sPF - standard PFT
GL - PFT with g() & p() from GL
4. Hard-sphere system: (complete set of data for test)
9
Assumption: isotropic interfacial free energy (spherical nucleus)
Standard PFT 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 et al., JPCB, 2006)
SL 2/kT = 0.559
SL& d10%-90%   2 & W
Tóth & Gránásy J. Phys. Chem. B 2009
GL
CNT
sPF

12. 10
   SL ffpWTg
T
I  )()(
2
2
2


Different g() and p() functions:
C. Different Phase-Field type models

13. 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.    
















fpgWM
t
F
M
t
)()(22
Freezing of undercooled
single component liquid
- Overdamped relaxation dynamics (no 2nd order time derivative); non-linear diffusion eq.
- The EOM reduces the free energy with elapsing time
- Time scale is determined by the mobility coefficient M
- A steady-state traveling front solution exists for all undercoolings
 relate M to measurable quantities
- With specific p() an analytic solution of tanh profile exists (same as the eq. profile)
- Thermal fluctuations can be represented by adding fluctuation-dissipation noise to EOM
- Discretization  filtering the noise
)'()'(2),()','( ttkTMtt    rrrr
tx
kTM
A d

 2
Amplitude of discretized noise:Derivation of M:
W
fMvPFT
2
12
2


RT
G
a
D
RT
G
a
D
v FrenkelWilson











 

00
exp1









RTda
DV
Mvv m
FrenkelWilsonPFT
%90%1003
)9log(

0),( tr
12
D. Dynamics – non-conservative EOM (Allen-Cahn type)

15. Summary:
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 THEORETICAL WORK IS NEEDED!!!
ELE: Cu, 2D, 1100 K
EOM: Gauss filtered noise ELE sol. + EOM, no noise
ELE sol.+ EOM, with Gauss filtered noise
13