Cellular and Dendritic Solidification

Topic 7M.S Darwish

MECH 636: Solidification Modelling

preferred crystallographic direction

Cellular and Dendritic Solidification

Constitutional Undercooling

Co

T

T*

Cs

Cl

A BComposition

Temperature

*

*

Co

C1

CL(x)

DL/v

Solid Liquid

*

T

TLiquidus

xL

T1(co)

T*

T actual

0

x

constitutionalundercooling

Solid Liquid

criticalgradient

Gradient of Solute in the liquid at the interface

Gradient of Solute in the liquid at the interface�

dCl

dx '⎛ ⎝ ⎜

⎞ ⎠ ⎟ x'= 0

= − vDl

Cl∗ 1− k( )

if the temperature gradient in the liquid at the interface is equal or greater that

�

dTliquidusdx'

⎛

⎝ ⎜

⎞

⎠ ⎟ x '= 0

= kliquidusdCl

dx '⎛ ⎝ ⎜

⎞ ⎠ ⎟ x '= 0

�

dTldx '

⎛ ⎝ ⎜

⎞ ⎠ ⎟ x'= 0

≥dTliquidusdx '

⎛

⎝ ⎜

⎞

⎠ ⎟ x'= 0

no supercooling

�

dTldx '

⎛ ⎝ ⎜

⎞ ⎠ ⎟ x'= 0

v≥ −

kliquidusCs∗

kDl

1− k( ) no supercooling

Co

T

T*

Cs

Cl

A BComposition

Temperature

*

*

TTLiquidus

xL

T1(co)

T*

T actual

x

constitutionalundercooling

Solid Liquid

�

dTldx '

⎛ ⎝ ⎜

⎞ ⎠ ⎟ x'= 0

≥ T1 −T3D /v

Instability TheoryGrowth

Direction

!"!"

01

3

2

�

dTldx '

⎛ ⎝ ⎜

⎞ ⎠ ⎟ x'= 0

v+ρlΔH f

2Kl

≥ −kliquidusCs

∗ 1− k( )kDl

Kl + Ks

2Kl

S

Formation of Dendrites

When regular cells form and grow at relatively low rates, they grow perpendicular to the liquid-solid interface regardless of crystal orientation.

When the growth rate is increased crystallography effects begin to exert an influence and the cell growth direction deviates toward the preferred crystallographic growth direction. Simultaneously the cross section of the cell generally beings to deviate from its previously circular geometry owing to the effects of crystallography

preferred crystallographic direction

Heat

flow

Side View

Cross-Section

Cell to Dendrite Transition

Eutectic Growth

Regular and Irregular Eutectics

low entropy

High entropy

small VF large VF

Coupled Growth

Growth of plates depends on lateral diffusion

α

Soluterejection

β

Solventrejection

Eutectic Diffusion Field

Boundary Layer Lateral Mixing

Inter-lamellar Spacing

�

λ* =2γαβVmTeutLVΔT0

�

ΔG λ( ) =2γαβVm

λmin1− λmin

λ⎛ ⎝ ⎜

⎞ ⎠ ⎟

�

= − LvΔT0Te

1− λminλ

⎛ ⎝ ⎜

⎞ ⎠ ⎟

�

ΔCl = ΔClmax 1−λminλ

⎛ ⎝ ⎜

⎞ ⎠ ⎟

�

When λ = λ∞ ⇒ ΔGλ ismaximum⇒ Cα −Cβ( ) = ΔC ismaximum

�

Assume ΔCl∞ΔGλ�

When λ = λ∞ ⇒ ΔGλ ismaximum⇒ Cα −Cβ( ) = ΔCl ismaximum

αlC

βlC

l

�

ΔG λ( ) = −ΔGλ→∞ +2γαβVm

λ

�

ΔGλ→∞ = LVΔT0Teut

setting ∆G to 0

�

= −ΔGλ→∞ 1−λminλ

⎛ ⎝ ⎜

⎞ ⎠ ⎟

ΔCCα Cβ

v

λ

�

v ∝D dCdx

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = 2D

λCβ −Cα( )

�

v ∝D dCdx

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = 2D

λCβ −Cα( )�

= 2Dλ

ΔTC1mα

− 1mβ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

�

= 2Dλ

ΔTC1mα

− 1mβ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

�

ΔTC = λv2D

mαmβ

mβ −mα

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ = Kcλv

Curvature Effects

�

ΔTr = Γκ = Kr

λ

�

ΔTr = Γκ = Kr

λ

�

ΔTC = Kcλv

�

ΔT = Kcλv + Kr

λ

�

d ΔT( )dλ

= 0

�

d ΔG( )dλ

= 0

�

λ2v = Kr

KC

�

ΔTv

= 2 KrKC

�

ΔTλ = 2Kr

αlC

βlC

Interface Temperature

�

Jt = 2DΔCλ

�

Jr = vCeut 1− k( )under steady state

�

Jt = Jr

�

λv2D

= ΔCCeut 1− k( )

Interface Velocity

αlC

βlC

Competitive Growth of Dendritic and Eutectic

Phases