From Pattern Formation to
Phase Field Crystal Model
吳國安 (Kuo-An Wu)
清華大學物理系
Department of Physics
National Tsing Hua University
3/23/2011
Pattern Formation in Crystal Growth
by Wilson Bentley (The snowflake man), 1885
Pattern Formation in Crystal Growth
At the nanoscale (atomistic scale)
Liquid-Solid interfaces
Anisotropy ↔ Morphology
Atomistic details ↔ Anisotropy?
Hoyt, McMaster
Solid-Solid interfaces
Grain boundary
Atomistic details ↔ growth?
Al-Cu dendrite, Voorhees Lab
Northwestern University
Schuh, MIT
Atomistic details ↔ Continuum theory at the nanoscale
Pattern Formation in Macromolecules
Polyelectrolyte Gels
Hexagonal phase in solvent rich region
Hex (-)
Hexagonal phase in polymer rich region
Competition between Enthalpy, Entropy, Elastic Network Energy,
Electrostatic energy, … etc
Hex (+)
Pattern Formation in Biology
Lincoln Park Zoo
Chicago
Rural Area, Wisconsin
Pattern Formation in Biology
Nuclear Lamina (核纖層)～30-100nm
In animal cells, only composed of 2 types lamins
Lamin (核纖層蛋白) A/C
Lamin B1, B2
Bleb Formation in Breast Cancer Cell Nucleus
Goldman Lab, Northwestern University
Confocal Immunofluorescence of a normal cell nucleus
Goldman Lab, Northwestern University
Crystal Growth at the Nanoscale
Solid-Liquid interface
Crystal growth from its
melt with interfacial
anisotropy
Solid-Fluid interface
under stress
Quantum dots
InAs/GaAs
Ng et al.,
Univ. of Manchester, UK
Solid-Solid interface
Grain boundaries
Schuh/MIT
Morphology vs. Anisotropy
tT  D2T
LVn  cD
  T 
TI  TM 
n
solid
  nT liquid
V
TM
TrS    n
L
  nˆ 
d 2
S   ij 
d  i d j
Anisotropy of 
Max(TI  T )  Min(TrS )
Gibbs-Thomson condition
1/TrS
(Max ΔT)
Phase-field simulations
of solidification
What causes the anisotropy?

Crystal growth – Solid-Liquid interface
Basal Plane
Anisotropy vs. Crystal structures

3
17  


4
4
2 2 2
  nˆ    o 1  1   ni     2  3 ni  66nx ny nz   
5
7 
 i
 i

  0   uK  eiK r
K 
K   K 
K   K 
110
BCC
111
FCC
uK  us
WHY?
fcc
uK  0
bcc
Ginzburg-Landau Theory
F (uk )
GL Theory for bcc-liquid interface
Free energy functional for a planar
solid-liquid interface with normal zˆ
 a2  cij ui u j 0, K  K  a3  cijk ui u j uk  0, K  K  K

i
j
i
j
k
 i, j

i , j ,k
n0 k BT
2
DF 
dr 

ui  z  

2
  a4  cijkl ui u j uk ul 0, Ki  K j  Kk  Kl  b ci z

  0   uK  eiK r
i
,
j
,
k
,
l
i


K 
Density Functional Theory of Freezing
DF   dr   r  ln
a2 
12
S  K110 
 r 
1
2   r  3   4 dr dr c  r , r  D  r  D  r 

dr
1
2
  a3u110 0 a4u
 1 2 1 2
0F a2u110
2110
c( K0 )  S ( K0 ) / S ( K0 )2
b  2c  KDF
110 


FS  FL
,
ci 
1 ˆ
K n  zˆ
4

2
FS
0
u110
Liquid structure factor
S(K)
a3 and a4 are
determined u110
Liquid
by equilibrium conditions
K0
Solid
K (Å-1)
Bcc-liquid interface profile
For the crystal face z  (110)
{110} is separated into three subsets
K1 10 , K 110
0
1
4
  0
  Kˆ  zˆ   1
 Kˆ  zˆ
2
1
2
K110 , K 1 10

K101 , K10 1 , K 101 , K 101
 Kˆ  zˆ
K 011 , K 01 1 , K 0 11 , K 0 11
 D F
0

  u110
 DF
0


u
 1 10
 D F
0

  u101

2

zˆ
1
4
us
K101
K110
0
K110
zˆ
Anisotropic Density Profiles
z  (1,1)
K1  10 ,  10 ,  01 ,  0 1 
2D Square Lattices
z  (1, 0)
K1  10  ,  10 
z  (3, 1)
K1  10  ,  10 
K    01 ,  0 1 
2
K    01 ,  0 1 
2
Symmetry breaks at interfaces → Anisotropy
Comparison with MD results
1
n( z ) 
Lx Ly
Lx Ly
  dxdy  (r )
0 0
  0   uK  eiK r
K 
BCC Iron
n x 10-23
(cm-3)
Comparison with MD results
Anisotropy
4 
 100   110
 100   110
 (erg/cm2)
Fe
100
110
111
4 (%)
MD (MH(SA)2)
177.0(11)
173.5(11)
173.4(11)
1.0(0.6)
GL theory
144.26
145.59
137.57
1.02
Predict the correct ordering of 
and weak anisotropy 1% for bcc crystals
Atomistic details (Crystal structures) matter!
Methodology for atomistic simulations
Molecular Dynamics (MD)
Mean field theory
Ginzburg-Landau theory
 1
 0
Rely on MD inputs
 Average out atomistic details
 Diffusive dynamics (ms)
 Larger length scale (m)
 Elasticity, defect structure, … etc?

Realistic physics
 Resolve vibration modes (ps)

Methodology for atomistic simulations
Mean field theory
Phase field crystal (PFC)
Molecular Dynamics (MD)
Realistic physics
 Resolve vibration modes (ps)

Average out vibration modes (ms)
 Atomistic details – elasticity, crystalline planes,
dislocations, … etc.

Formulation - Phase Field Crystal
(001) plane of
bcc crystals
(100)
(110)
Swift & Hohenberg, PRA (1977)
2D Patterns – Rolls, Hexagons
Elder et al., PRL (2002)
Propose a conserved SH equation
The Free Energy Functional
Equation of Motion
Capillary Anisotropy?
Elasticity?
PFC Model – Phase Diagram
Conserved Dynamics
Phase diagram
  

K {110}
 
AK eiK r
Multi-scale Analysis
Seek the perturbative solution
Assumption – interface width is much larger
than lattice parameter
Maxwell construction
The solid-liquid coexistence region
A weak first-order freezing transition
(The multi-scale analysis of bcc-liquid
interfaces will be carried out around c)
Multi-scale Analysis – Amplitude equation
Small  limit – diffuse interface
Multi-scale analysis
Ai0  Z 
Equal chemical potential in both phases
One of twelve stationary amplitude equations
eiKi r
Order Parameter Profile Comparison
For the crystal face
z  (110)
Determination of the PFC
model Parameter from density
functional theory of freezing
u110
  0.0923 for Fe with
MH(SA)2 potential (MD)
Comparison with MD results
Anisotropy
 100   110
4 
 100   110
 (erg/cm2)
Fe
100
110
111
4 (%)
MD (MH(SA)2)
177.0(11)
173.5(11)
173.4(11)
1.0(0.6)
GL theory
144.26
141.35
137.57
1.02
PFC
144.14
140.67
135.76
1.22
Predict the correct ordering of 
100 > 110 > 111
and weak anisotropy 1% for bcc crystals
What about Other Crystal Structures?
Phase diagram
  

K {110}
 
AK eiK r
y
(110)
x
11 1 
1 11
z
(011)
200 
GL theory of fcc-liquid
interfaces
(10 1)
2
2
F  a2u111
 a4u1411  b2u200
 b4u204 0
The principal reciprocal lattice vectror
F  a u a u a u
2
2 110
3
3 110
4
4 110
2

b
u
of fcc 31,111u1,200
1 cannot form triad
3

K ix  0
FCC-Liquid
BCC-Liquid
i 1
2
4
F  a2u111
 a4u111
F
F
u110
cannot form solid - liquid interfaces
FCC Model
The Two-mode
PFC modelfcc model
Twin Boundary
2
(1,1,1)
2
a
(2, 0, 0)
a
2
K 02 
1
1,1,1  1
3
2
K12 
S (K )
1
4
2,
0,
0



3
3
Phase Diagram
K
FCC Polycrystal
Design Desired Lattices
Elasticity
Example: Square Lattices
Multi-mode model
Single-mode model
Dictate interaction angle
(lattice symmtry)
Grain Boundary



Grain boundary is composed of
dislocations
Geometric arrangement of crystals
determines dislocation distribution
Distinct evolution for low and high
angle grain boundary
b
D
D
b : magnitude of Burgers vector
D : misorientation
D
Symmetric tilt planar grain boundary in gold
by STEM
GB sliding and coupling
GB Coupling – Low Angle GB
 /2
GB Sliding – High Angle GB
 /2
Dt
Well described by
continuum theory
Dn
Sutton & Balluffi, Interfaces in Crystalline Materials, 1995
Large Misorientations
  20
Curvature driven motion
G.B. sliding (fixed misorientation)
 remains constant
Well described by classical continuum theory
Small Orientations
Theory that only considers 
DF  2 R   GB  
Misorientation decreases?
Atoms at the center of the circular grain
 5
Misorientation increases!
Small Misorientations
G.B. coupling
For symmetric tilt boundaries
Misorientation increases
GB energy increases
Misorientation-dependent mobility:
(Taylor & Cahn)
Intermediate Misorientations – cont.
  10
v ~ C / rm
Area   r  C (1  at )
2
m  2.3
2
m1
Intermediate Misorientations
Faceted–Defaceted Transition
  10
 Frank-Bilby formula
 Tangential motion of dislocations
 Annihilation of dislocations
Intermediate Misorientations – cont.
Spacing d1 is a function of GB normal
2
nˆ
b1  b2
r
d1 

N1 nˆ  b2  b1  b2 
 

Instability of tangential motion occurs when
R
3  sin  F   / 6  
3

G F


2  cos 2  F   / 6   2
G  F
0
F
/3
Dd
Three-Grain System
  5.2o
Grain Rotation?
GB wiggles
  0o
  5.2o
Grain Rotation
  5.2o
  0o
  5.2o
  0o
  5.2o
Grain Translation
5.2º
GB
Wriggles
0º
-5.2º
Dihedral angle
follows Frank’s formula
not the Herring relation
Self-Assembled Quantum Dots
Quantum-dot LEDs
Lee et al., Lawrence Livermore National Laboratory
Other Applications
- Tunable QD Laser
- Quantum Computing
- Telecommunication
- and more
Quantum dots
InAs/GaAs
Ng et al.,
Univ. of Manchester, UK
Stress Induced Instability – Asaro-Tiller-Grinfeld Instability
 DU
A
Cullis et al. (1992): 40 nm
Linear perturbation
calculation
DU
2 ˆ2
2
  thick
h  kSi
0.79
EGe
E , on
 ,0.21
 k (001) Si
A
substrate - Grown at 1023 K
2

c  (Defect-free
 2
 10 growth)
100 nm
kc
E

 hˆ exp  ik x x  ik y y 

kc
zˆ
h
Film
Misfit Parameter
a  as
 f
as
af
Substrate
 0
Schematic plot from Voorhees and Johnson
Solid State Physics, 59
as
xˆ
k
Later Stage Evolution - Cusp Formation - Dislocations
High stress concentration at the tip
Si0.5Ge0.5/Si(001)
Jesson et al., Z-Contrast,
Oak Ridge Natl. Lab.,
Phys. Rev. Lett. 1993
Various sizes
Simulation Parameters
0
The PFC model
F   dr

     1


2

F
 2
t

2
2
1
Ly
8
  1  4

4
Hexagonal
Phase
Simulation parameters
  0.10
Ly
Lx

Ny
Nx
Constant
Phase
o
a  2 A
3
2
N x  448 1280
N y  2048
o
Ly  1900 A
o
Lx  480 1360 A
# of atoms  15, 000
40, 000
7
Ly
8
Constant
Phase
(1+xx)Lx
Nonlinear Steady State for a Smaller k

k
Quantitative Comparison of Strain Fields
 xx  1%
F   dr

2




 1

2 
2
   1  4 ,      F

4
t

ˆxx , ˆyy
ˆxx
ˆyy
y
Correct elastic fields
 Elastic fields relax much
faster than the density field

     AK eiK r
K
  : conserved quantity

 AK : non-conserved quantity
Critical Wavenumber vs Strain
Linear perturbation theory
- Sharp Interface
- Homogeneous Materials
kc 
Classical Elasticity Theory E  xx
PFC simulations
Nonlinear Elasticity
PFC simulations
2E xx2

E : Young's modulus
 : Surface energy
W
c
 xx2

Linear Elasticity
kc ~ xx2 for small strains
 Nonlinear elasticity modifies length scale

1
Xie et al., Si0.5Ge0.5 films, PRL
PFC modeling of nonlinear elasticity
 xx ~ 0
Liquid
Inhomogeneous materials
 nonlinear elasticity

Solid
E ( xx )
 xx  2%
Finite Interface Thickness Effect
Eint erface  Esolid
Liquid, E=0
Upper bounds
c~1/2·xx-2
W~-1/2
E(x,y)
Solid, E=Eo

Interface thickness is no longer negligible at the nanoscale
Finite interface thickness W
Elastic constants vary smoothly
across the Interface region
Nonlinear Evolution for k ~ km

k
3D Island – BCC Systems
And More …
VLS nanowires
Nano-particles with defects
And More …
Pattern Formation - Examples
Agular et al, Oxford University
Ice Crystal
North Pole Hexagon on Saturn
Rock Formation in Ireland
Honeycomb
Graphene
Collaborators
Alain Karma
Northeatsern University
Mathis Plapp
Laboratoire de Physique de la Matière Condensée
Ecole Polytechnique
Peter W. Voorhees
Northwestsern University