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Field Method of Simulation of Phase Transformations in Materials
Alex Umantsev
Fayetteville State University, Fayetteville, NC
What do we need to account for?
Multi-phase states: thermodynamic systems may have multiple stable phase ‘coexisting’ at the same conditions.
Heterogeneous states: phase transformations may go along very complicated paths.
Dynamic structures: rate of transformation can make a difference for the final structure and properties
Nucleation: phase transformations are initiated through the process of overcoming of some kind of a potential barrier.
Phase Field Theory 101: Six Pillars of the Field Theory of Phase Transformations
Order para-meter
Free Energy
Hetero-geneity
Dyna-mics
Fluctua-tions
‘Hydrodynamics’
Field Theory
Landau theory
Ginzburg- Landau
Coarse-graining procedure
X
Y 1 23
ZnCu
blueZnredCu
ZnCu
Cu
NNNN
NNN
C
parameterorder
ionconcentrat
//1
C
R1 0.500 1.000
R2 0.486 0.971
R3 0.487 0.920
Pillar 1: Order Parameter: Symmetries of the System
--Zn atom --Cu atom
Order-disorder transformation in -brass (Cu-Zn alloy)
Pillar 2: Free Energy 432
0 41)(
31
21)(),( +T=Tg Wg
stable phase: ‘crystal’ metastable phase: ‘liquid’ W—potential barrier—energy scale -chemical potential difference /W-driving force
Landau potential
reaction coordinate
Pillar 3: Heterogeneous Systems: Gradient Energy
dv+W,g=G 2])()([21
Length scale:
WenergylInterfacia
WlengthnCorrelatio
:
1: – interface thickness
Order- Disorder Fe-Al Allen, Cahn Circa’82
Pillar 4: Kinetics
G=
dtd
WscaleTime
1:
The Gradient Flow Equation:
M.I. Mendelev, J. Schmalian, C.Z. Wang, J.R. Morris, and K.M. Ho, Phys. Rev. B, 74, 104206 (2006).
W
intV
Pillar 5: Internal fluctuations: Langevin force
TγkΓ
t)δ(t'x)Γδ(x')t',ζ(x't)ζ(x,
0t)ζ(x,
t)ζ(x,)δηδGγ(=
dtdη
B2
T,P
(x, t)—noise: Gaussian, white, additive
One more energy scale: thermal fluctuations-kBT
Pillar 6: ‘Hydrodynamic Modes’: Momentum Flow (Pressure or Stress) Diffusion (Concentration of species) Electromagnetic Field Variation (Waves) Heat Flow (Temperature Variation):
)()( t,Q+T=dtdTC x
dtdetQ 2
ETV
])[(),(
, x
General Heat-Equation
Heat Source
Latent Heat L
‘Thermodynamically consistent heat equation’ A. Umantsev and A. Roytburd. Sov. Phys. Solid State 30(4), 651-655, (1988)
Length Scales
– interfacial thickness
– capillary length
– kinetic length
– thermal length
– radius of curvature 1
2
K
CVl
Ll
L
CTl
Wl
n
E
T
C
I
Cl
l
RL
WCT
Il
lU
C
EC 2
dynamic
timekinetic
L
C2
timerelaxation
W 1
scalesTime
C TE
L W
Truly Multi-Scale Method
0 100 200 300 400
distance into the sample
liquid phase
--------solid phase
--------transition state
-----melting temp-----
temperature
order parameter
D. Danilov
lC l lT X
Å nm m mm Length scales
Tim
e sc
ale
Real Material Modeling Problem A: Convert thermodynamic functions of two or more phases into a continuous Landau-Gibbs Free Energy
Solution 1: LTPT: Symmetry expansion Solution 2: Speculate
G{i; T, P, C}
CALPHAD {T, P, C}
{i}
Problem B: Find the PFM parameters: {Wi, i, i}
Solution 1: Calculate from First Principles Solution 2: Calibrate from experiments or MD/MC simulations
Landau-Gibbs Free Energy
of Au-Sn Alloy
Au-Sn Binary (Bulk) Phase Diagram
Landau-Gibbs Free Energy of Carbon
-0.25 0.25 0.75 1.25-.025
0.250.75
1.25-820
-810
-800
-790
-780
-770
-760
-750
-740
crystallization OPcoordination OP
mola
r G
ibbs f
ree e
nerg
y [
kJ/m
ol]
liquid
graphite
diamond
Measurable Quantities
Interface energy:
Kinetic coefficient:
Wareaunitenergyfree ~
ETL
WgupercoolinsofKvelocitynterfacei
1
Interface thickness: W
lI
~
Equilibrium Fluctuations of OP ∆𝜂 𝑉 𝐤2=
𝑘𝐵𝑇
𝑉𝜕2𝑔𝜕𝜂2
𝜂 + 𝜅 𝐤 2
Non-Equilibrium Fluctuations of OP
time
Dynamic structure factor
Materials Genome Database of Interfacial Properties
Soldering 101
Cross-section
Solder
Cu
IMP
100m
5m
Sn-Solder
Cu-Plate
IMP
Soldering: InterMetallic Phase Growth
Sn-Solder
Cu-Plate
Where was the original interface between the tin and copper?
Liquid-state solder Solid-state solder
Onishi and Fujibuchi’75
Gagliano and Fine’00
baseline
PLC
IMP
Experimental results
Lord & Umantsev, J. Appl. Phys. 98, 063525 (2005) (Centennial Campus, 2004)
Total Free Energy
])()(
),,([22
cfdF r
))()()(
))()()(
)(),,(
(][
(][
s
s
cfcfBw
cfcfAw
cfcf
solidintermet
liquidsolid
liquid
Homogeneous part
normal
liquid
disordered
solid
inter
metallic molecular
liquid
crystallization
ord
eri
ng
Umantsev, JAP ‘07
0.0 0.2 0.4 0.6 0.8 1.0
concentration Cu
-35000
-30000
-25000
-20000
-15000
-10000
fre
e e
ne
rgy J
/mo
l
molecular liquid
atomic liquidsolution
fcc solidsolution
intermetallic compound
Thermocalc® G. Ghosh
1. fori=f()—not invariant against rotation of the reference
frame
2. fori=f(1,2,3,…)—orientation order parameter
3. fori=f(||)= s||+ q||2 —Kobayashi, Warren, Carter
f=+[1-s()]fliquid(c)+[s() -s()]fsolid(c)+ s()fintermet(c)
+(,) fori
0.0 0.2 0.4 0.6 0.8 1.0
order parameter
0
40
80
120
rela
xa
tio
n c
oe
ffic
ien
t
liquid crystall
KWC
G; this work
Ft
),(
Dynamics
Grain Boundary Contribution to Free Energy
Grain-boundary diffusion: Mobility: M=M(,,||)
Evolution Equations
F
t
F
t
)) ((
]),,([
soltinliqsolliq MMMMMM
cFcM
tc
Crystallization
Ordering
Diffusion
Parameters: 3 Diffusion
coefficients: solder
intermetallic substrate
+ 9 interfacial parameters
(interface energies thicknesses, and
mobilities)
Initial Conditions: slab, no nucleation
Scales: Length=0.25nm Time=0.25s
Ft
),(GB orientation
Solid State Soldering
Copper
Tin
Liquid state soldering
Copper
Tin
2D Modeling
solder
disordered
solid
inter
metallic
molecular
liquid
crystallization
ord
erin
g C-concentration
||-grad angle orient
Problem: rotation of grain orientation
Phase Field Simulation of Nucleation at Large Driving Forces
Numerical simulations x
x
t t
x, t are not just grid parameters. They are physical quantities–the noise correlation length and time.
Lifetime: time for a supercritical nucleus to appear in the system. Advantages:1. more reliable theory 2. free-energy landscape
3D as opposed to 2D Correlation properties of the fluctuations are very different.
10 20 30 40 50 60 70 80 90 100
10
20
30
40
50
60
70
80
90
100
Supercritical Nucleus
20 40 60 80 100
10
20
30
40
50
60
70
80
90
100 -0.2
0
0.2
0.4
0.6
0.8
10 20 30 40 50 60 70 80 90 100
10
20
30
40
50
60
70
80
90
100
10 20 30 40 50 60 70 80 90 100
10
20
30
40
50
60
70
80
90
100
X-Y plane
Y-Z plane
plane Z-X
Time Evolution of the System
escape
equilibrium fluctuations
________________ perturbation theory
xx dtV
tV ,1 10 20 30 40 50 60 70 80 90 100
10
20
30
40
50
60
70
80
90
100
-phase
transition state
rectification of fluctuations
Free Energy Barrier
10 20 30 40 50 60 70 80 90 100
10
20
30
40
50
60
70
80
90
100
3D Structure of Supercritical Nucleus
Shape characterization
1. Volumetric content
2. Eccentricity 3. Roughness 4. Probability
distribution
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