Transcript Stuff I work on down in the JHE basement

Stuff
Rapid
I work
Solidification
on down and
in the
Solute
JHE
basement
Trapping
Harith Humadi
Supervisors
Jeff Hoyt
Nikolas Provatas
1
Introduction
Planar Diffuse S/L Interface at
Thermodynamic Equilibrium
X
L
S
L
XL
Ti
S
XS
XS
XL
z0 z0+d
z
2
Introduction
Planar Diffuse L/S Interface During
Solidification
X
L
S
V
D/V
XL/S
Ti
L
~
S
XS
XS
XL
z0 z0+d
z
k  XS / XL /S
3
Rapid Solidification and Solute Trapping
X
L
S
V
L
XL/S
Ti
S
XL/S
XS
XS
XL
k  X S / X L / S 1
z0 z0+d
z
asV 
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Partition Coefficient Theories
Aziz (1982):
k e + V /VD
k(V ) 
1+ V /VD
VD
Jackson (2004):
k(V)  k
1/(1+AV )
e
Sobolev (1996), Galenko (2007):
ke [1 (V /VDB ) 2 ] + V /VD
k(V ) 
1 (V /VDB ) 2 + V /VD
V  VDB
k(V )  1
V  VDB
VDB
Interface
 Diffusion Speed

ke
Sobolev/Galenko Model
Bulk Diffusion Speed
Equilibrium Segregation
Coefficient


Aziz Model
Figure from Galenko, PRE (2007)
Experimental Data from Kittl, Aziz et al. (1995, 2000)
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Aziz Model
• Diffusion of solute across the
interface
• No correlation with the bulk
diffusive speed in the liquid
• Dilute system
k(V ) 
k e + V /VD
DX
VD 
1+ V /VD
d
V
Nishizawa, Thermodynamics of Microstructure, ASTM International (2008)
ke
Equilibrium Segregation
Coefficient


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Sobolev Model
Evolution Equation Maxwell Cataneo type
J
J +   DC
t
Mass Balance
C
  J VDB
t
C
 2C
+  2  D 2C
t
t
Bulk Diffusion Speed
General Fick’s Law


D
B2
D
V
Relaxation Time
In steady state regime, we use the solid-liquid interface as a reference plan:
2
B2
D
D(1 V /V
d 2C
dC
) 2 +V
 0 
dX
dX
(Sobolev,Phys rev,1996)
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
Sobolev Model Con’d
Solute concentration profile infront the moving interface




VX
B
(Ci  Co )exp


V  VD 
2 + C o

2
B
C(X)  

 D(1 V /VD ) 


Co
V  VDB  (Sobolev,Phys rev,1996)

Ci
Co
Solute concentration near the interface
Solute concentration in the liquid
Effective Diffusion Coefficient
D(1 V 2 /V B 2 ) V  V B 
D
D
D* (V /VDB )  

B
0
V  VD 

K (1 V 2 /V B 2 ) + V /V
D
D
 E
2
K(V )   (1 V 2 /V B ) + V /V
D
D

1

(Sobolev,Phys rev,1996)

V  V 


V  VDB 
B
D
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Motivation
• Different mechanical, electrical
and the thermal properties
• Splat cooled Al-35%Mg alloy
• Note sharp transition between
the two phases due to change in
solidification mechanism
• At finite solidification velocity,
partitionless transformations
occurred within alloy
• New supersaturation values in
alloys
Galenko, Herlach, Diffusionless Crystal Growth in Rapidly Solidifying
Eutectic Systems, Physical Review Letters, 96, 150602 (2006)
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Objective 1
Which theory is the most accurate description of solute trapping?
Aziz
Sobolev
k e + V /VD
ke [1 (V /VDB ) 2 ] + V /VD
k(V ) 
k(V ) 
OR
1 (V /VDB ) 2 + V /VD
1+ V /VD
k(V )  1
V  VDB
V  VDB

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Molecular Dynamics
“…everything that living things do can be
understood in terms of the jigglings and
wigglings of atoms.”
Richard Feynman
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Model of Copper-Nickel Binary Alloy
ke = 0.5 at T=1750 K
• Provide driving force by
supersaturation and/or
undercooling
• Track the solid/liquid interface
• calculate solid and liquid
concentrations
Embedded-Atom Potential developed by S. M. Foiles (1985)
Phase-Diagram Calculation by Webb & Hoyt (2008)
Solid
Liquid
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Segregation Coefficient vs. Velocity
MD
MD
Sobolev
Sobolev
MD
MD
Aziz
Aziz
Jackson
Jackson
{100}
{110}
1.4(4)
1.6(4)
15(4)
21(10)
{100}
{110}
1.3(3)
1.5(3)
1.1(3)
1.0(2)
Yang, Humadi, Buta, Laird, Hoyt and Asta, Atomistic Simulations of Nonequilibrium Crysta-Growth Kinetics from Alloy Melts,
Physical Review Letters, 107, 025505 (2011)
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Objective 2
If Sobolev theory is correct, is it necessary to have that second
time derivative to get complete trapping?
 2C C
 2 +
 D 2C
t
t
Remember?

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Phase Field Crystal (PFC) Model
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• Use Helmholtz free energy to represent binary alloy system
ρ – Atomic Density
Ψ - Concentration
Cahn Hillard Formulation
• Phase Crystal Method allows:
• Faster computation times
• Ease in changing system parameters
• 2nd Order time scale dynamics
• Direct comparison to MD results
• Evolution of atomic density driven by
free energy minimization
Stefanovic, Density Functional Modeling of Mechanical Properties and Phase
Transformations in Nanocrystalline Materials, McMaster University (2008)
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PFC Dynamics
Dynamic Equations
 

2 dF
 M1  
t
d 


 

2 dF
 M1  
t
d 
ke = 0.89
Elder et al,2007
Density Wave Equation
 2dF 
 2 
 2 +  M1   Β is Spring constant
t
t
 d 
Concentration Wave Equation


 2dF  Wave Dynamics in the
 2 
 2 +
 M 2  
t
t
 d  concentration field
Eutectic like phase diagram
with rescaled temperature
(ΔBo) and concentration
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PFC Movies
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Segregation Coefficient vs. Velocity
 2dF 
 2 
 2 +
 M 2  
t
t
 d 
dF 

 M1 2 
t
d 

ke
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One more question?
How much does the wave term in the concentration (gamma)
affect the amount of trapping?
 2dF 
 2 
 2 +
 M 2  
t
t
 d 
Solid
Liquid

Interface
• As Gamma
increases the
amount of
trapping increases
• Beta is constant
• Same Velocity
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Conclusions
MD
Sobolev theory is a
better description
of solute trapping
The degree of
trapping depends
on the strength of
the wave term
Trapping
PFC
-2nd time derivative
is essential to obtain
complete trapping
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Halong Bay, Vietnam
By: Harith Humadi
Thank you
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Copper Solute Concentration Profile
High Velocity
Low Velocity
Liquidus
XCu
Solidus
Bulk liquid
-60
-40
Bulk liquid
Bulk solid
-20
0
20
40
Bulk solid
60
Distance (Å)
Distance (Å)
•Green and blue lines represent independent runs
•Black line present the solid/liquid interface
7/17/2015
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Breakdown of Free Energy Equation
ΔF – Helmholtz free energy
kB – Boltzmann constant
Bl, Bx – Isothermal compressibilities
ρ – Atomic density
ψ – Solute concentration
ρl – Atomic density of the liquid state
R – Length Scale
w,t,v,u – Materials parameters
K is an additional term for smoothing free energy in system
4K06 Interim Presentation
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Simulation and Analysis
3. Provide driving force through supersaturation and/or undercooling and monitor
interface position
Interface position from crystal-melt order
parameter (Hoyt et al. 2001)
4. Compute time-averaged density and temperature profiles
25
Examples of Interface Profiles
XL /S

Interface
temperature
XS

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Simulation and Analysis
5. Decide if there is partitioning or if growth is partition-less
L’
Average concentration
at time (t0 + L’/V)
L
Average concentration
at time t0
6. If partitioning, compute solid concentration from average of solidified crystal, and
liquid concentration as peak interface composition
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Copper Solute Concentration Profile
High Velocity
Low Velocity
Liquidus
XCu
Solidus
Bulk liquid
-60
-40
Bulk liquid
Bulk solid
-20
0
20
40
Bulk solid
60
Distance (Å)
Distance (Å)
•Green and blue lines represent independent runs
•Black line present the solid/liquid interface
7/17/2015
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