Chapter 4.

Solidification

Nucleation in pure metals

∆G=G2-G1=-V∆Gv+Aγsl
∆Gv=Lv∆T/Tm

For spherical shape,

γ
∆G

∆G*

0 r
*
r

∆Gr

2γ SL 2γ SLTm 1 -∆Gv
r* = =
∆Gv Lv ∆T
16πγ SL 16πγ SL Tm
3 3 2
1
∆G =*
=
3∆Gv
2
3Lv
2
∆T 2

G vs. T curve shows that the line for solid is enhanced by ∆Gv=2γ/r*. At Tm, r*
becomes infinite.

Gv GvL

∆Gv
r=r*
GvS
2γ/r*
∆T
r=∞

T
Tm

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Nucleation

nr=noexp(-∆Gr/kT)

r: in liquid
Cu at Tm, n=1014mm-3 if r=0.3nm, 10 mm-3 if r=0.6nm

In r vs ∆T curve, rmax, the maximum size of cluster with a reasonable probability of

occurring, increases (due to the decrease of ∆Gr with ∆T) whereas r* rapidly decreases
with ∆T. The point they meet is ∆TN, the undercooling required for hom. nucleation.

r
r*

rmax

∆T
- 0 ∆TN +

Homogeneous nucleation

C*=Coexp(-∆Ghom*/kT) clusters/m3
Nhom=foC* nuclei/m3s
fo=1011/s, Co=1029/m3, ∆G*=78kT for Nhom=1/cm3s
Nhom=foCoexp(-A/∆T2)

Therefore, Nhom vs ∆T curve shows an abrupt increase at ∆TN.

For most metals, ∆TN=200K

Heterogeneous nucleation

γSL
L
θ S
γML
γSM M

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From the balance of the forces,

γML=γSLcosθ+γSM

∆Ghet=-VS∆Gv+ASLγSL+ASMγSM-ASMγML

For a spherical cap,

∆Ghet=((-4πr3∆Gv/3)+4πr2γSL)S(θ)

S(θ)=(2+cosθ)(1-cosθ)2/4

S(θ) increases from 0 to 0.5 as θ varies from 0 to π/2.

r*=2γSL/∆Gv
∆G*=(16πγSL3/3∆Gv2)(S(θ))=∆GhomS(θ) F4.8

In the plot of ∆G* vs ∆T and N vs ∆T curve (F. 4.9), it can be shown that the two
nucleation rates require different amount of undercooling since ∆G* for heterogeneous
case is always lower than that of hom. nucleation.

Crevices in mold provides good heterogeneous nucleation sites since r* and

corresponding ∆G* are small although θ is large (π/2).

Nucleation of melting

Since γSL+γLV<γSV always holds, θ=0; ∆G*=0, ∆T=0

Growth of a pure solid

(1) Continuous growth
Rough surface (Diffuse interface) provides sites of easy nucleation. Conduction of
heat (Diffusion) controls the growth rate of pure solid. Diffusion of solute atoms
controls the growth rate of alloy.

Driving force:

∆G = L∆Ti/Tm
∆Ti = undercooling of the interface

v=M⋅∆G=k1∆Ti

k1 is very high; therefore, ∆Ti can be very small for normal growth.

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(2) Lateral growth
Smooth interface; interface mobility controls the growth rate.

Compared to nucleation on a flat surface, ledge growth requires less interfacial

energy.

Surface nucleation: Disc should be created on a flat surface.

Fig. 4.11, 4.12
Spiral growth: A screw dislocation terminating in the solid/liquid interface provides
a ledge. Due to the difference between the growth velocity at the dislocation core
and that at a remote location, a spiral forms. F4.13

In a plot of the v vs ∆T, (F. 4.14) the continuous growth is linear whereas others are
almost negligible up to a certain amount of undercooling. They eventually catch up
with the cont. growth rate at a large undercooling. Among the lateral growth, spiral
growth is faster than surface nucleation.

Growth from Twin intersections.

Heat flow and interface stability

Planar interface:

KsTs’=KLTL’+vLv

K: Js-1m-1K-1
Lv: Jm-3

Solid growing into superheated liquid: F4.15

TL’ increases; TS’ decreased; v decreases; protrusion melts and disappears.

e.g.: solidification from mould walls, cooler than liquid

Solid growing into supercooled liquid: F4.16

TL’ is negative and decreases more toward negative side; v increases; unstable interface
and protrusion development.

Thermal dendrites with their arms along <100> in cubic and <1-100> in hcp. F4.17

e.g.: nucleation from impurity particles in liquid during the beginning stage of
solidification.

Growth velocity at the tip of dendrite:

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 Tm
∆Tr
∆To
Ti
∆Tc

T∞

r
x

Assume TS’=0, then

− K L TL '
v=
Lv
K L ∆T
=
Lv r
K L ∆Tc
=
Lv r

∆Tr=Tm-Ti
∆Tc=Ti-T∞
∆To=∆Tr+∆Tc

Dependence of v on r: v increases with r

Dependence of ∆Tr on r: Gibbs-Thomson effect

∆Gv=2γ/r=Lv∆Tr/Tm

Minimum radius (r*) of the dendrite tip occurs when ∆Tr=∆To

∆Tr=∆Tor*/r

Therefore,

K L (∆To − ∆Tr )
v=
Lv r
 r *
∆To 1 − 
K
= L  r 
Lv r

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Max. v: r=2r*

Alloy Solidification: F. 4.19

Single Phase Alloy: Liquidus and Solidus assumed to be straight lines

Partition coefficient:
k=XS/XL

(1) Infinitely slow (equilibrium)

Starting liquid, Xo, Starting solid, kXo
Equilibrium is maintained throughout solidification: Complete diffusion in solid
Final solid: Xo

(2) No diffusion in solid, Perfect mixing in liquid, F4.21

Starting solid: kXo, Extra solute rejected into liquid.
Further solidification requires lower temperature.
Each solid retains initial concentration of solute due to insufficient diffusion: the
mean XS is lower than the solidus.
Lever rule should be between the XL and the mean XS, not the solidus.
At TE, liquid of XE still remains and undergoes eutectic solidification.
From this point on, the mean XS becomes Xo.

Non-equilibrium lever rule (Scheil Equation):

Small amount (dfS) of solid formation increases the solute content of liquid by dXL

(XL-XS)dfS=(1-fS)dXL

− d (1 − f S ) dX L dX L
= =
1− fS X L − X S X L (1 − k )
ln X L
− ln(1 − f S ) = +c
1− k
f S = 0; X L = X o
ln X o
0= +c
1− k
XL
ln
Xo
− ln(1 − f S ) =
1− k
X L = X o (1 − f S ) = X o f L
k −1 k −1

k −1
X S = kX L = kX o f L

The equation predicts that XL becomes infinity when fL=0.

(3) No diffusion in solid, Diffusional mixing in liquid, F4.22

No stirring in liquid; Solute rejected from solid only moves by diffusion.
Rapid buildup of solute ahead of the solid, XS quickly reaches Xo.

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 The liquid in contact with the solid attains Xo/k.
Steady state graph:
Solute diffusion through the liquid vs solute rejected due to solidification.

J=-DCL’=v(CL-CS)

Solution of the diffusion equation:

X S = X o for all x ≥ 0
dX L v
= − dx
XL − Xo D

ln ( X L − X o ) = −
v
x+c
D
x = 0; X L = X o / k
X 
c = ln o − X o 
 k 
X − Xo v
ln L =− x
1  D
X o  − 1
k 
1− k  − D
vx
XL − Xo = Xo e
 k 
 1 − k − Dx/ v 
X L = X o 1 + e 

 k 

The concentration gradient in liquid in contact with the solid:

cL’=dcL/dx=-vc/D=-(XL-XS)/(D/v)

All alloys have some of all the features described above.

(4) Celluar and Dendritic Solidification

Constitutional supercooling: F. 4.23

At the interface, TL=Te(not TE)=T3.
According to the temperature gradient in liquid, the liquidus temperature gradually
increases away from the interface into the liquid.

If the TL of inner liquid is below Te, then c.s. exists and protrusion develops.
The critical gradient for planar front solidification:

(5) Criterion for the planar interface:

TL’/v>(T1-T3)/D: the protrusion melts back.

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 T1-T3: Equilibrium freezing range of alloy: Large range promotes protrusions.

Sequence of planar solidification front into cells. F. 4.24

Solutes rejected from cells concentrate on cell walls, lowering the temp. to TE.
Second phase forms even for Xo<<Xmax
F4.25, 4.26

Dendrites
Transition from cellular microstructure to dendritic microstructure: F4.27, 4.28
Development of secondary arms and tertiary arms: <100>
Solute effects: low k enlarges T1-T3; promotes dendrites.
Cooling rate effects: Fast cooling makes lateral diffusion of the rejected solutes
difficult; Promotes cell formation; Smaller cell spacing.

Eutectic Solidification

Lamellar type F4.29, Rod type F4.30

Interdiffusion ahead of a eutectic front: F4.31

Gibbs energy diagram at ∆To below TE: F4.32

Growth of lamellar eutectics

2γVm
∆G (λ ) = −∆G (∞ ) +
λ
∆H∆To
∆G (∞ ) =
TE

Relationship between γ and V in lamellar structure:

Suppose lamellae of spacing λ in the slab of 1mx1mx1m,
No. of lamellae surfaces: 2/λ
Length of each surface: 1m
Total area of lamellae: 2/λ m2
Energy per unit vol.: 2γ/λ Jm-3
Energy per mole: 2γV/λ Jmole-1

T<TE; ∆To>0; ∆H>0; ∆G∞>0

Critical spacing, λ*:

∆G(λ∗)=0

2TEγVm
λ* = −
∆H∆To

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Free Energy Diagram for the Eutectic Solidification: F. 4.33
Separate curves for the liquid, solid α and solid β.
Note the shift of the G curves of the solids from λ=∞ to λ≠∞

There is a gap of ∆X between the equil. concentration of solute ahead of α and β phase.
Thus, concentration gradient of ∆X/λ develops. F4.34
Then,

J 1 dc  ∆X
v= =  − D  = k1 D
c c dl  λ

Dependence of ∆X on λ:

 λ *  λ *
∆X = ∆ X o  1 −  ∝ ∆To 1 − 
 λ   λ 

∆To  λ *
v = k2 D 1 − 
λ  λ 

Maximum growth rate at a fixed ∆To: λ=2λ*

Cellular solidification:
Existence of impurities in liquid, and the subsequent supercooling, results in an
instability of the planar eutectic front. Refer to the ‘fanning-out morphology’ in Fig.
4.35.

Off-Eutectic Alloys

F4.37
Primary α dendrites form at T1.
Rejected solute increases XL to XE; eutectic solidification follows.
Coring: primary α (low solute) at T1 and the eutectic (high solute) at TE.

Directional solidification
Overgrowth of eutectic/ Disappearance of primary dendrites:
High temp. gradient in liquid.

Peritectic solidification F. 4.39

L+α→β
Difficult to complete.
α dendrites first form at T1; Liquid reaches the composition ‘c’; β forms as the result of
the peritectic reaction; α coring is isolated from further reaction; finally β+γ eutectic
forms.

Directional solidification

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High TL’/v; Planar front solidification below T2

Solidification of Ingots and Castings

Castings: final shape

Ingots: To be worked (rolling for forging) later

Ingot structure: F. 4. 40

Temperature profile during cooing of ingot: Initially the mould wall is cool but just after
solidification it becomes hot. As a consequence there develops a supercooled region in
the liquid (Jena F.6.32)

Chill zone: Equiaxed – Outer surface

Columnar zone
Equiaxed zone – Center

Chill zone
Crystal formation on mould wall; Outgrowth of dendrites with primary arms normal to
the mould wall (parallel to the maximum temperature gradient) F4.41
Breakaway by melt turbulence
If T is low; complete equiaxed crystals fill the liquid at T<Tm; No chill zone
If TL is high; Breakaway crystals melt; Only those on the mould wall remain

Columnar zone
dT/dx decreases
<100> growth in cubic crystals: columnar grain F4.42
Mushy zone (pasty zone): Between the dendrite tips and the roots

Equiaxed zone
Nuclei (seed): secondary dendrite arms detached from the primary arms.
Concentration high: Equiaxed zone enlarged
Concentration low: Columnar zone enlarged

Shrinkage effects
Narrow freezing range; narrow mushy zone; central shrinkage pipe formation
Wide freezing range; broad mushy zone; no shrinkage pipe; liquid level decreases
across the width of ingot; dendrite links up; small voids or pores

Segregation

macro-: whole ingot

micro-: secondary dendrite arm spacing

Causes of Macro-segregation:

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Shrinkage: Inverse segregation, flow-back of the liquid of high solute concentration to
compensate shrinkage (k<1); Center low, Outer high. e.g. Al-Cu, Cu-Sn, wide
freezing range.
Density difference in the inter-dendritic liquid: Al-Cu case, Cu sinks down
Density difference between solid and liquid
Convection current

Negative segregation at bottom for k<1.

General pattern of segregation: F4.43

Treatment of segregation: Homogenization

Continuous casting

Cu-mould
Dynamic process
Analogy to welding

Heat flow in welding

Mould: Work piece, the same composition as liquid

q: rate of heat input
v: velocity of arc movement
Ks: thermal conductivity of weld metal
t: thickness of plate

The heat flow eq.:

∂T
2K s = −∇ 2 q
∂ ( x − vt )

where Ks is the thermal conductivity in the unit of wm-1K-1 and q is the heat transfer
rate in the unit of w.

Distance between isotherms: F. 4. 46

q
λ∝
K s vt

Solidification of Fusion Welds

Composition of melt changed due to dilution

Surface oxide layer melted
Melt is cooled down

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Base metal, heat sink and nucleation site with θ=0
Epitaxial growth

TIG: thin plates, steeper thermal gradients

Submerged Arc Welding: thick plates, higher heat input
Grains with <100> orientations predominate solidification
Coarse grain structure in the weldment

Growth directions of grains in weldment: Max. temp. grad. + <100>, F. 4.48

Influence of the welding speed

Weld Pool shape: elliptical shape is changed to pear shape with increasing weld speed,
F.4.49

Geometry of crystal growth

TIG (Low speed) vs Submerged Arc Weld (High speed) F4.50

R = vcosθ F4.51
R: crystal growth rate vector
v: weld pool velocity vector (speed of the movement of the isotherms)

R is greatest at the center line: θ=0.

Low R: planar interface (Columnar)
High R: cellular, dendritic (Equiaxed), F.4.52

Solidification during quenching from the melt

Rapid solidification: of 104 to 107K/s.

Liquid metal atomization, melt spinning, roller-quenching, plasma spraying, laser or
electron beam surface treatment

Amorphous metallic glass: In case of metal, it requires a cooling rate higher than 106K/s
to produce the glassy state. The glass transition (second order pahse transition) can be
monitored by a discontinuity in the heat capacity (Jena F6.38).

Case studies

Casting of carbon and Low-alloy Steels

F4.53
Low alloy steels: C 0.1-1.0%, Si, Mn, Cr, Ni, Mo
Peritectic δ, Eutectoidal Pearlite
Inter-dendritic residual phases: γ/Fe3P/Fe3C, Retained austenite, FeS (hot cracking),
MnS F4.54

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Casting of High-speed steels

C: 0.5-1.0%, Cr, Mo, W, V

Pseudo-binary phase diagram F4.55
At high temp. γ field, Cr, V and W have low solubility.
MxC formation: γ/MxC eutectic at inter-dendritic region
Plastic working to break up the eutectic; austenitizing; double tempering F4.56

Stainless steel weld metal

Cr: 17-19%, Ni: 8-10%, C: 0.05-0.1%, Si, Mn

Pseudo-binary phase diagram, F. 4.57a
Schaeffler diagram, F.4.57b
δ or γ solidifies first depending on composition.
Effect of N contamination during welding: hot cracking
Harmful effect of P in Cr-rich ferrite and S in inclusions

P is dissolved in δ ferrite whereas S is dissolved in inclusion. F.458

The role of Mn:

If the weld structure is predominantly γ, then Mn remains in γ and no effective role is
played.
If the weld structure is fine γ+δ then Mn helps making S and P harmless.

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