Lever Rule - feaweb.aub.edu.lb

Solute RedistributionTopic 6

M.S DarwishMECH 636: Solidification Modelling

dCL

d!s

kCo

Cs=kCo

CL

Introduction• Local Equilibrium

– Solidus and liquidus curves

– Equilibrium distribution coefficient, k0

– Tie–line “jump”

• Macrosegregation at Planar Interfaces

– Gulliver–Scheil equations

» Assumptions

» Mass balance

» Limitations

– Transport–limited segregation

» Initial Transient

» Steady–state solute distribution

» Terminal Transient

• Effects of Stirring & Convection

– Burton–Prim–Slichter Theory, kBPS

Lever Rule

Introduction

CS

CL

Distance

Composition

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 = Cs*

Cl*

Equilibrium at S/L

C

liquidus

solidus

T

Tie line “jump”

CBS L

Molecular scale

Continuum scale

Com

posi

tion

Distance0

CS = Co

End of Solidification

*

Com

posi

tion

Co

kCo

Distance0 L

CL = CL! Co

Start of Solidification

*

Equilibrium Solidification

Com

posi

tion

Co

CL

CSkCo

Distance0 L

At Temperature T

�

χ sCs + χ 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

Schiel Rule

Gulliver–Scheil Differential Mass Balance

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* −Cl

*( )d x = L − x( )dCl

�

Cl* Cs

*

Cl* −1

⎛

⎝ ⎜

⎞

⎠ ⎟ d x = L − x( )dCl

�

Cl = Cl*( )

�

Cl* k −1( )dx = L − x( )dCl

�

k −1( ) d ′ x L − ′ x ( )0

x

∫ = d ′ C l′ C lC0

Cl

∫

�

χ s = xLnote that

solid liquid

dxL0

x

Gulliver–Scheil Segregation Equation

• The Gulliver–Scheil mass balanceeventually leads to non–physicalpredictions as χs→1.

• The phase diagram provides limits to how high the concentrations, Cl or Cs can rise due to segregation. Secondphase formation (generally from eutecticreaction) then occurs.

Gulliver–Scheil is independent of the shape of the S/L interface!

Where

�

χ s = xL�

1− χ s( ) k0−1( ) = Cl

C0

No Solid Diffusion(Non-Equilibrium Lever Rule)

Com

posi

tion

Co

CL

CSkCo

Distance0 L

At Temperature T

*

Com

posi

tion

Co

kCo

Distance0 L

CL=CL! Co


*

kCo

Co

Csm

CE

Com

posi

tion

Distance0 L

End of Solidification

kCo Co Co/k

T

Csm Ceut

TLT

TS

CSCL k=CS/CL

C

A B

To

Composition

Temperature

�

Cl −Cs*( )dχ s = 1− χ s( )dCl

solidification of

�

dχ s

�

Cs* = kCo 1− χ s( )k−1

�

Cl = Co 1− χ s( )k−1�

dχ1− χ0

χ s

∫ = dCl

Cl 1− k( )Co

Cl

∫

Scheil relation

dCL

d!s

kCo

Cs=kCo

CLrejected solute from

solidsolute increase in

liquid

0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1

C l/Co

Fraction Solid, fs

0.1

k0=0.01

0.20.30.40.50.60.7 0.8

k0<1


�

1− χ s( ) k0−1( ) = Cl

C0

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

Cs/C

0

Fraction Solid, fs

k0=0.03

0.10.20.30.40.50.60.7

0.8

k0<1


�

k0 1− χ s( ) k0−1( ) = Cs

C0

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

C l/C0

Fraction Solid, fs

k0=10

5

3

2

1.2

1.5

1.05

k0>0


1�

1− χ s( ) k0−1( ) = Cl

C0

0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1

Cs/C

0

Fraction Solid, fs

k0=10

5

3

2

1.5

1.21.05

k0>1


�

k0 1− χ s( ) k0−1( ) = Cs

C0

Maximum solubilityof Cu in Al is 2.5 at.% or 4.5 wt%.

Al-Cu Phase Diagram

Application of Gulliver–Scheil Segregation

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10

Dendritic Microsegregation(Al-4.5 wt% Cu)

Conc

.ent

ratio

n, C

u (a

t.%)

Distance, X (microns)

Segregation limit set by solubility limit.

Al–2.5 at% Cu

Plane Front Solidification

Moving S/L

liquidus

solidus

T

C

xsolid liquid

CB

vC∞

Cs(x)

Cl(x)

Solute boundary layerdevelops ahead of the advancing S/L interface.

�

Cs* T( )

�

Cl* T( )

�

Cl* T( )

�

Cl* T( )

Steady State Solute Boundary Layer

!

Cl

*

!

Cs

*

!

Csx( )

!

Clx( )

!

C0

Solid Liquid

Solid LiquidJB

v

Computing the Steady-State Solute Profile

• Consider Geometry

• Apply Fick’s second law

• Find the general SS solution

• Apply boundary conditions

• Find solute gradient at the interface

v

v

Apply Fick’s Law and SS Condition

In 1-D

�

dCdt

= ∇ ⋅D∇C ≈ D∇2C

�

dCdt

= D ∂2C∂z

�

dCdt

= D ∂2C∂z

= 0

Apply a Flux balance at Interface

dz

v

Cs

Cl

dtAJ =1

J1 J2

02

=

−=zdz

dcDJJ1=J2

dCdz

�

= Cl −Cs*( ) dzdt

�

= v Cl −Cs*( )

�

v = − DCl −Cs

*( )dCdz z= 0

Example 1

Cα

Cγ

Z

z

Z

vtZz −=

vdtdz −=

dttcdZ

zcdc

∂∂+

∂∂=

tc

dtdZ

zc

dtdc

∂∂+

∂∂=

tcv

zc

dzcD

∂∂+

∂∂−=∂

2

2

zc

Dv

dzc

tc

D ∂∂+∂=

∂∂

2

21

02

2

=∂∂+∂zc

Dv

dzc

( ) ⎟⎠⎞⎜

⎝⎛−+= z

Dvaazc exp10

( ) ( )( )tzctZcc == ,

Cα

Cγ

Z

z

Z

vtZz −=

vdtdz −=

( ) ⎟⎠⎞⎜

⎝⎛−+= z

Dvaazc exp10

Apply Boundary conditions to evaluate the constants:

( ) 0Czc =∞= ( ) γCzc == 0

00 Ca = 01 CCa −= γ

( ) ( ) ⎟⎠⎞⎜

⎝⎛−−+= z

DvCCCzc exp00 γ

( )00

CCDv

zc

z−−=

∂∂

=γ

( )0=

⎟⎠⎞⎜

⎝⎛∂∂−=−

zzcDCCv αγ

( ) ( )⎥⎦⎤

⎢⎣⎡ −−−=− 0CC

DvDCCv γαγ

( )( ) 10 =

−−

αγ

γ

CCCC

Ω=

Flux Balance at SS:

Evaluate dc/dz:

(Supersaturation)

Must equal 1 for SS!

Cα

Cγ

Z

C0

Destabilization requires a certain growth distance.

Solute buildup increases with time(or growth distance).

Sequential Crystallization Concepts

• First step of crystallization:

• r th step of crystallization:

Where

At the interface

Limited Liquid DiffusionC

om

posi

tion

Co

Co/k

Distance0 L


kCo Co Co/k

T

Csm Ceut

TLT

TS

CSCL k=CS/CL

C

A B

To

Composition

Temperature

�

dCdt

= D d2Cdx 2

�

dCdx

dxdt

= v dCdx

�

−D dCdx

⎛ ⎝ ⎜

⎞ ⎠ ⎟ ∗

= v Cl∗ −Cs

∗( )

�

dCdx

⎛ ⎝ ⎜

⎞ ⎠ ⎟ ∗

= − vDCl

∗ 1− k( )

far from the interface

�

Cl = Co

�

Cl = Co /k

�

Cl = Co 1+ 1− kk

e− xD / v

⎛

⎝ ⎜

⎞

⎠ ⎟

Characteristic Distance

�

v dCdt

= D d2Cdx 2

�

D / v

InitialTransient

Com

posi

tion

Co

CEut

Distance0 L

Steady State

Csm

FinalTransient

After Solidification

InitialTransient

Com

posi

tion

Co

CEut

Distance0 L

Steady State

Csm

FinalTransient

After Solidification

�

Cl = Co 1+ 1− kk

e− xD / v

⎛

⎝ ⎜

⎞

⎠ ⎟

�

Cs∗ = Co 1+ 1− k( )e

−k xD / v

⎛

⎝ ⎜

⎞

⎠ ⎟

abfacdb

L

L

abf

L

T1ab

A B

c dT2e

f T3

Solute Boundary Layer

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/k0-C

0]

ζ=-(vX)/Dl

Steady-State Dimensionless Solute Field in Melt

1/e

1/e2

1/e3

Note the characteristicdiffusion length, ζ,

Diffusion Limited Steady State Segregation

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

[Cl(x

)-C0]/[

C0/k

0-C0]

ζ=-(vX)/Dl

Steady-State Dimensionless Solute Field in Melt

1/e

1/e2

1/e3

• Diffusion is negligible beyond 3 e-folding lengths from the S/L interface.• The linear boundary layer approximation is equal to 1 e-folding, ζ=1.

Terminal Solidification Transient

Terminal freezing was studied by V.G. Smith, W.A. Tiller, and J.W. Rutter, Canadian Journal of Physics, 33, p. 723, (1955). They obtained the following infinite series solution. Here Δx is the distance to the end of the ingot:

0

5

10

15

20

25

00.20.40.60.81

Scal

ed S

olid

Con

cent

ratio

n, C

s/C0

Dimensionless Distance from End, Δx/(Dl/v)

k0=0.05

0.2

0.5

3 1

Terminal Transient

0.1

• As a practical note, terminaltransients are seldom ofinterest in real crystal growthprocesses. The growth processis usually halted before thistransient begins.

• Note, little if any interaction occurs when the S/L interfaceis more than a distance Dl/v from the end of the crucible.

Example 2: SS Spherical Growth

r

R

For spherical symmetry: 02

2

=∂∂=

∂∂=

∂∂

θθφccc

⎥⎦⎤

⎢⎣⎡

⎟⎠⎞⎜

⎝⎛

∂∂

∂∂=∇=

rcr

rrc

dtdc 2

22 1

For steady-state: 02 =⎟⎠⎞⎜

⎝⎛

∂∂

∂∂

rcr

rthus a

rcr −=∂∂2

and ( ) brarc +=

where a and b are constants.

Fick’s 2nd law: cDcDdtdc 2∇≈∇•∇=

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂+⎟

⎠⎞⎜

⎝⎛

∂∂

∂∂=∇ 2

2

22

22

sin1sin

sin11

θφφφ

φφcc

rcr

rrc

In spherical coordinates:

r

R where a and b are constants.

Boundary conditions:

General solution:

Particular solution:�

C r >> R( ) = C0 = ar

+ b

�

C r = R( ) = Cl* = a

R+ b

�

a = R Cl* −C0( )

�

C r( ) = C0 + RrCl* −C0( )

r

R Controlling solute gradient:

Flux balance: J1=J2

Substituting yields: or

NOTE: diffusivity

shape factorsupersaturation

�

dCdr

= − Rr2

Cl* −C0( )

�

dCdr r=R

= −Cl* −C0( )R�

C r( ) = C0 + RrCl* −C0( )

�

v = − DCl −Cs

*( )dCdz r=R

�

v = D 1R

⎛ ⎝ ⎜

⎞ ⎠ ⎟ Cl −C0( )Cl −Cs

*( )

�

RvD

=Cl −C0( )Cl −Cs

*( )

Example 3: SS Needle growth

Needle with hemispherical cap

r

Rv

Flux balance

A Ah

�

dCdr r=R

= −Cl* −C0( )R

�

A Cl* −C0( ) dzdt = −AhD

dCdr r=R

�

v = D Ah

AR⎛ ⎝ ⎜

⎞ ⎠ ⎟ Cl* −C0( )

Cl* −Cs

*( )

�

v = D 2R

⎛ ⎝ ⎜

⎞ ⎠ ⎟ Cl* −C0( )

Cl* −Cs

*( )

�

Ah

A= 2πR

2

πR2= 2

v

Left for homework…

The Shape Factor

Plane Sphere Plate Needle

Effect of Convection

Effect of Convection and Stirring

During crystal growth and plane–front solidification stirring occurs, which affects the segregation process. This was analyzed by Burton, Prim and Slichter (BPS Theory). They suggested three important zones: Solid zone (no diffusion) A “static” layer, δ, (diffusion transport) Liquid zone (complete mixing)

..

diffusionregion convective mixing region

0 x

melt

δ

C0

Cl^

solid

no diffusionCs^

Cs(x)

BPS Segregation Theory

BPS approximated the fluid dynamic momentum boundary layer as a “static zone,” and bulk liquid as fully mixed.They derived segregation curves (“inner” & “outer”) consistent with these approximations in the fluid mechanics.

Solid Liquid

δ

�

Cl x( ) = Cs* + Cl

* −Cs*( )e

− vDl

x⎛

⎝ ⎜

⎞

⎠ ⎟ , (x ≤ δ)

�

Cl x( ) = C0, (x > δ)

�

Cs*

�

Cl*

�

kBPS ≡Cs*

C0

BPS results (ko<1)

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

K BPS

δ/(Dl/v)

k0=0.005

0.020.01

0.30.5

0.15

0.05

Slower stirringRapid stirring

k0<1

0

20

40

60

80

100

0 2 4 6 8 10

k BPS/k o

δ/(Dl/v)

k0=0.01

0.0133

0.02

0.05

0.10.3

0.03


BPS results (ko>1)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

k BPS/k 0

δ/(Dl/v)

k0=5

3

2

1.5

1.2

1.1

k0>1


BPS Limits

i. no mixing: δ→∞ No natural

convectionCs C0

C0k0

δ→∞solid melt

Cs

Cs → C0

kBPS → 1

⎧⎨⎪

⎩⎪

solidCs → k0C0

kBPS → k0

⎧⎨⎪

⎩⎪

melt

Cl

CsCs

←δ ~0

C0ii. perfect mixing: δ→0 Very rapid stirring

BPS Summary

As summarized below, stirring provides kinetic control on the amount of macrosegregation segregation during steady–state plane front growth.

Faster stirring δ kBPS Cs

0 k0 k0 C0

0<δ<∞ k0< kBPS<1 k0 C0<Cs<0

∞ 1 C0

Key Points

• 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

Lever Rule - feaweb.aub.edu.lb

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