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Solute RedistributionTopic 6
M.S Darwish
MECH 636: Solidification Modelling
dCL
d!s
kCo
Cs=kC
o
CL
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Local Equilibrium
Solidus and liquidus curves
Equilibrium distribution coefficient, k0
Tieline jump
Macrosegregation at Planar Interfaces
GulliverScheil equations
Assumptions
Mass balance
Limitations
Transportlimited segregation
Initial Transient
Steadystate solute distribution
Terminal Transient
Effects of Stirring & Convection
BurtonPrimSlichter Theory, kBPS
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CS
CL
Distance
Com
position
solid liquid
kCo Co Co/k
T
Cmax Ceut
TLT
TS
CSCL k=CS/CL
C
A B
To
Composition
Temperature
?
In practice equilibrium is
maintained at the solid liquid
interface where composition
agrees with the phase diagram
?
k=C
s
*
Cl
*
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C
liquidus
solidus
T
Tie line jump
CB
S L
Molecular scale
Continuum scale
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Composition
Distance0
CS
= Co
End of Solidification
*
Composition
Co
kCo
Distance0 L
CL
= CL! C
o
Start of Solidification
*
Composition
Co
CL
CSkCo
Distance0 L
At Temperature T
sCs + lCl =Co
Complete Diffusion in
Solid and Liquid
Respectively
kCo Co Co/k
T
Csm Ceut
TLT
TS
CSCL k=CS/CL
C
A B
To
Composition
Temperature
Lever Rule
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Although local interfacial equilibrium
generally holds during solidification, the
system cannot remain in global
equilibrium because diffusion rates in
the solid are too slow.
G. Gulliver (1921) and E. Scheil (1942)
both assumed:
Perfect mixing in liquid
No diffusion in solid
Local equilibrium at S/L interface
Cs
*C
l
*( )dx = L x( )dCl
Cl
* Cs*
Cl
*1
dx = L x( )dCl
Cl= C
l
*( )
Cl
*k1( )dx = L x( )dCl
k1( )d x
L x( )0
x
=d C
l
ClC0
Cl
s=
x
Lnote that
solid liquid
dxL0
x
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The GulliverScheil mass balanceeventually leads to nonphysicalpredictions as s1.
The phase diagram provides limits to
how high the concentrations, Clor Cscan rise due to segregation. Secondphase formation (generally from eutecticreaction) then occurs.
GulliverScheil is independentof the shape of the S/L interface!
Where
s=
x
L
1 s( )
k01( )=
Cl
C0
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Composition
Co
CL
CSkCo
Distance0 L
At Temperature T
*Composition
Co
kCo
Distance0 L
CL=
CL! C
o
Start of Solidification
*
kCo
Co
Csm
CE
Composition
Distance0 L
End of Solidification
kCo Co Co/k
T
Csm Ceut
TLT
TS
CSCL k=CS/CL
C
A B
To
Composition
Temperature
ClC
s
*( )ds = 1 s( )dClsolidification of
ds
Cs
*= kC
o1
s( )k1
Cl= C
o1
s( )k1
d
1 0
s
=dC
l
Cl1 k( )Co
Cl
Scheil relation
dCL
d!s
kCo
Cs=kC
o
CL
rejected solute fromsolid
solute increase inliquid
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0
1
2
3
4
5
0 0.2 0.4 0.6 0.8 1
Cl
/Co
Fraction Solid, fs
0.1
k0=0.01
0.2
0.3
0.4
0.5
0.6
0.70.8
k0
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0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1
Cs
/C0
Fraction Solid, fs
k0=0.03
0.10.2
0.30.4
0.50.6
0.7
0.8
k0
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0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Cl
/C0
Fraction Solid, fs
k0
=10
5
3
2
1.2
1.5
1.05
k0>0
1
1
s( )k01( )
=
Cl
C0
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0
1
2
3
4
5
0 0.2 0.4 0.6 0.8 1
Cs
/C0
Fraction Solid, fs
k0=10
5
3
2
1.5
1.2
1.05
k0>1
k0 1
s( )k01( )
=
Cs
C0
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Maximum solubilityof Cu in Al is 2.5 at.% or 4.5 wt%.
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0
0.5
1
1.5
2
2.5
3
0 2 4 6 8 10
Dendritic Microsegregation(Al-4.5 wt% Cu)
Conc.entration,
Cu(at.%)
Distance, X (microns)
Segregation limit setby solubility limit.
Al2.5 at% Cu
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liquidus
solidus
T
C
x
solid liquid
CB
vC
Cs(x
)
Cl(x)
Solute boundary layer
develops ahead of theadvancing S/L interface.
Cs
*T( )
Cl
*T( )
Cl
*T( )
Cl*
T( )
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Cl
*
Cs
*
Csx( )
Clx( )
C0
Solid Liquid
Solid LiquidJB
v
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Consider Geometry
Apply Ficks second law Find the general SS solution
Apply boundary conditions
Find solute gradient at theinterface
v
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vIn 1-D
dC
dt= DC D
2C
dC
dt= D
2C
z
dC
dt= D
2C
z= 0
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dz
v
Cs
Cl
dt
A
J =1
J1 J2
0
2
=
=
zdz
dcDJ
J1=J2
dCdz
= ClCs
*
( )dz
dt
= v Cl
Cs*
( )
v = D
ClC
s
*( )dC
dzz= 0
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C
C
Z
z
Z
vZz =
vdt
dz=
( )
+= z
D
vaazc exp10
Apply Boundary conditions to evaluate the constants:
( )0
Czc == ( ) Czc == 0
00Ca = 01 CCa =
( ) ( )
+= z
D
vCCCzc exp00
( )0
0
CCD
v
z
c
z
=
=
( )0=
=z
z
cDCCv
( ) ( ) = 0CC
D
vDCCv
( )1
0=
CC
CC=
Flux Balance at SS:
Evaluate dc/dz:
(Supersaturation)
Must equal 1 for SS!
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C
C
Z
C0
Destabilizationrequires acertain growthdistance.
Solute buildupincreases with time(or growth distance).
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At the interface
Composition
Co
Co/k
Distance0 L
Start of Solidification
kCo Co Co/k
T
Csm Ceut
TLT
TS
CS
CL k=CS/CL
C
A B
To
Composition
Temperature
dC
dt= D
d2C
dx2
dC
dx
dx
dt= v
dC
dx
DdC
dx
= v Cl
Cs
( )
dC
dx
= v
DC
l
1 k( )
far from the interface
Cl= C
o
Cl= C
o/k
Cl=C
o1+
1 k
k
e
x
D / v
Characteristic Distanc
vdC
dt= D
d2C
dx2
D / v
Initial
Transient
Composition
Co
CEut
Distance0 L
Steady State
Csm
FinalTransient
After Solidification
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InitialTransient
Composition
Co
CEut
Distance0 L
Steady State
Csm
FinalTransient
After Solidification
Cl= C
o1+
1 k
k
e
x
D / v
C
s
=C
o
1+ 1
k
( )e
kx
D / v
ab
f
ac
d
bL
L
ab
f
L
T1ab
A B
c d
T2 e f T3
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0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
[Cl(x
)-
C0
]/[C
0/k
0-
C0
]
=-(vX)/Dl
Steady-State Dimensionless Solute Field in Melt
1/e
1/e2
1/e3
Note the characteristicdifusion length, ,
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0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
[Cl(
x)-
C0
]/[C
0/k
0-
C0]
=-(vX)/Dl
Steady-State Dimensionless Solute Field in Melt
1/e
1/e2
1/e3
Difusion is negligible beyond 3 e-folding lengths from the S/Linterface. The linear boundary layer approximation is equal to 1 e-folding,
=1.
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0
5
10
15
20
25
00.20.40.60.81
ScaledSolidConcentr
ation,
Cs
/C0
Dimensionless Distance from End, x/(Dl/v)
k0=0.05
0.2
0.5
31
Terminal Transient
0.1
As a practical note, terminaltransients are seldom ofinterest in real crystal growthprocesses. The growth process
is usually halted before thistransient begins.
Note, little if any interactionoccurs when the S/L interfaceis more than a distance Dl/v
from the end of the crucible.
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r
R where a and b are constants.
Boundary conditions:
General solution:
Particular solution:
C r >> R( ) = C0 =a
r+ b
C r = R( ) = Cl*=
a
R+ b
a = R Cl
*C
0( )
C r( ) = C0 +R
rC
l
*C
0( )
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r
RControlling solute gradient:
Flux balance: J1=J2
Substituting yields: or
NOTE: difusivity
shape factor
supersaturation
dC
dr=
R
r2C
l
*C
0( )
dC
drr=R
=
Cl
*C
0( )R
C r( ) = C0 +R
rC
l
*C
0( )
v=
D
Cl
Cs
*( )
dC
dzr=R
v = D1
R
C
lC
0( )
ClC
s
*( )
Rv
D=
ClC
0( )
ClC
s
*( )
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Needle with hemispherical cap
r
R
v
Flux balance
A Ah
dC
drr=R
=
Cl
*C
0( )R
A Cl
*C
0( ) dzdt
= AhD dC
drr=R
v = DA
h
AR
C
l
* C0( )
Cl
* Cs
*( )
v = D
2
R
Cl
* C0( )
Cl
* Cs
*( )
Ah
A=
2R2
R2
= 2
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Plane Sphere Plate Needle
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During crystal growth and planefront solidification stirring occurs, whichaffects the segregation process. This was analyzed by Burton, Prim andSlichter (BPS Theory). They suggested three important zones: Solid zone (no diffusion) A static layer,, (diffusion transport) Liquid zone (complete mixing)
diffusionregion convective mixing region
0 x
melt
C0
Cl^
solid
no diffusionCs
^
Cs(x)
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BPS approximated the fluid dynamic momentum boundary layer as a staticzone, and bulk liquid as fully mixed.
They derived segregation curves (inner & outer) consistent with these
approximations in the fluid mechanics.
Solid Liquid
Cl x( ) =Cs*
+ Cl*
Cs*
( )e
v
Dl
x
, (x )
Clx( ) =C0, (x >)
Cs*
Cl
*
kBPS
Cs
*
C0
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0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
KBPS
/(Dl/v)
k0=0.005
0.020.01
0.3
0.5
0.15
0.05
Slower stirringRapidstirring
k0
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0
20
40
60
80
100
0 2 4 6 8 10
kBPS
/ko
/(Dl/v)
k0=0.01
0.0133
0.02
0.05
0.1
0.3
0.03
Slower stirringRapid stirring
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As summarized below, stirring provides kinetic control on the
amount of macrosegregation segregation during steadystate
plane front growth.
Faster stirring kBPS Cs
0 k0 k0C0
0
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Equilibrium (lever-law) at S/L interfaces requires:
Tie-line jumps
Solidus & liquidus slopes (ms & ml)
Segregation coefficient, k0
Motion of the interface causes mass transport
Diffusion boundary layer
Solute rejection
Gulliver-Scheil behavior leads to macrosegregation.
Transients arise during solidification:
Initial
Steady-state
Terminal
Bouyancy convection treated with Burton-Prim-Schlichter theory