Cellular and Dendritic Solidification
Topic 7M.S Darwish
MECH 636: Solidification Modelling
preferred crystallographic direction
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
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
�
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