06 solute redistribution

8/6/2019 06 Solute Redistribution

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Solute RedistributionTopic 6

M.S Darwish

MECH 636: Solidification Modelling

dCL

d!s

kCo

Cs=kC

o

CL


2/47

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

*


5/47

C

liquidus

solidus

T

Tie line jump

CB

S L

Molecular scale

Continuum scale


6/47

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


*

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%.


16/47

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( )


19/47

Cl

*

Cs

*

Csx( )

Clx( )

C0

Solid Liquid

Solid LiquidJB

v


20/47

Consider Geometry

Apply Ficks second law Find the general SS solution

Apply boundary conditions

Find solute gradient at theinterface

v


21/47

vIn 1-D

dC

dt= DC D

2C

dC

dt= D

2C

z

dC

dt= D

2C

z= 0


22/47

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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24/47

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!


25/47

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


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


28/47

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


29/47

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, ,


30/47

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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34/47

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( )


35/47

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

*( )


36/47

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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38/47

Plane Sphere Plate Needle


39/47


40/47

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)


41/47

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


42/47

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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46/47

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


47/47

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

06 solute redistribution

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