Solidification, Lecture 4
NTNU
Three phase solidification
Eutectic growth
Types of eutectics
Peritectic growth
Segregation
Macro / microsegregation
Lever rule /Scheil segregation
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Three phase solidification
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Eutectic
l
α+β
Peritectic
l+α
β
Monotectic
l1
α+l2
Ce
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Eutectic solidification
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Fibrous
Lamellar
Regular
Al-Mg
Irregular
Al-Si
Reproduced from:W. Kurz & D. J. Fisher:
Fundamentals of Solidification
Trans Tech Publications, 1998
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Eutectic growth
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Growth
direction
•Simultaneous, cooperative
growth of 2 or more phases
•Diffusion parallel to growth front
•Isothermal growth front
•Characteristic lamellar spacing, 
determined by diffusion and curvature
 V  K1 T  K 2 T  K 3 V
2
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Irregular eutectics
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•One or both phases
grows facetted
•Non isothermal growth
•Grows at high undercooling
Al-Si
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Interface instabilities of eutectics
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Off-eutectic
Composition
3:rd element
Reproduced from:W. Kurz & D. J. Fisher:
Fundamentals of Solidification
Trans Tech Publications, 1998
6
Coupled Zone
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Al70Cu30
33%
Reproduced from:W. Kurz & D. J. Fisher:
Fundamentals of Solidification
Trans Tech Publications, 1998
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Off eutectic growth 1
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Al-30%Cu
Low temperature
gradient, high growth
rate
Dendrites + eutectic
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Off eutectic growth 2
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Al-30%Cu
High temperature
gradient, low growth
rate
Coupled eutectic
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Off eutectic growth 3
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Al-30%Cu
High temperature
gradient, high growth
rate
Cellular eutectic
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Decoupled, divorced eutectic
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•Major phase grows on
existing dendrites.
•Small amount of
eutectic
•No cooperative
growth
Al-5Cu
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Ternary & higher eutectics
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Cooperative growth
of 3 or more phases
Chinese script
Al-Mg-Si
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Peritectic solidification
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Primary: l→α
Peritectic: l + α →β
Eutectic: l → β+γ
Reproduced from:W. Kurz & D. J. Fisher:
Fundamentals of Solidification
Trans Tech Publications, 1991
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Peritectic solidification
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l + α →β
• Occurs at l/α interface
• Layer of β envelops α
• Further transformation requires
solid state diffusion through β
• Seldom to completion
α
β
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Segregation
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Macrosegregation
Scale: Casting
Microsegregation
Scale: Secondary dendrite arm spacing λ2
Reproduced from:M. C. Flemings
Solidification Processing
Mc Graw Hill, 1974
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Solute redistribution
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•Redistribution of solute since
liquid- and solid solubility are
different.
T
l
•Liquid enriched in solute (eutectic)
Cl
Cs
•Liquidus temperature decreases.
s
C0
k=Cs/Cl
C
•Equilibrium at s/l front but not
always in the solid or liquid due to
slow diffusion
Dl ~10-9 – 10-8 m2/s
Ds ~ 10-13 – 10-12 m2/s
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Complete equilibrium
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•
Valid only in special cases, slow
solidification, fast diffusion
•
Lever rule:
T
l
Cl
Cs
s
fs
fl (Cl - C0 ) = f s (C0 - Cs )
fl
C0
C0
Cl =
1- (1- k ) f s
C
•
Solid with uniform composition C0
k=Cs/Cl
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No solid diffusion, complete
mixing in liquid
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•
Reasonable assumption: slow diff. in solid,
faster diffusion, small diffusion distances,
and convection in liquid.
•
Gulliver-Scheil equation
T
l
Cl
Cs
s
Cl  C0 (1  f s )(k 1)
C0
C
•
Reasonable predictions in most cases
k=Cs/Cl
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Limited solid diffusion, full liquid
mixing
•
T
Back diffusion of solute into solid
Cl  C0 (1  (1  2k) f s )(k 1)(12k )
l
Cl
Cs
•

s
Fourier number determines amount
of diffusion

C0
k=Cs/Cl
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C
•
Ds t f
L2
Diffusion distance for dendritic
structures

L
2
2
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Segregation with back diffusion
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Reproduced from:W. Kurz & D. J. Fisher:
Fundamentals of Solidification
Trans Tech Publications, 1998
20
Freezing point during segregation
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Reproduced from:W. Kurz & D. J. Fisher:
Fundamentals of Solidification
Trans Tech Publications, 1998
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Microsegregation
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Freezing range, ΔT*, larger than
equilibrium freezing range, ΔT0
Liquidus concentration Cl higher
than C0/k, often reaches eutectic
conc. Ce
T
l
C0
T0
Reproduced from:W. Kurz & D. J. Fisher:
Fundamentals of Solidification
Trans Tech Publications, 1998
s
C0/k
ΔT*
Ce
C
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Example Segregation in Al-1%Si
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How much eutectic?
C0
Assume full equilibrium.
660
No eutectic
577
1.65
11.7
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Example Segregation in Al-1%Si
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How much eutectic?
Assume no diff in solid, full mixing
in liquid
C0
660
Assume straight lines
k=Cs/Cl=1.65/11.7=0.14
577
1.65
11.7
Cl  C0 (1  f s )(k 1)
Fraction eutectic, fe= fl=(1-fs)when
Cl=Ce
fe= 0.06
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Example Segregation in Al-1%Si
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How much eutectic?
Assume some diffusion in solid
Assume cooling rate 1K/s
Assume λ2=60 μm
Assume D=10-12m2/s
C0
660
Cl  C0 (1  (1  2k) f s )(k 1)(12k )

( k - 1) /(1- 2 a k )
Cl = C0 [1- (1- 2a k ) f s ]
577
1.65
11.7

Ds t f
L2
m=dT/dCl=577-660/11.7=-7.1
Liquidus temperature, Tl=Tf+mC0=653C

Freezing
range, ΔT* : 653-577=76K
tf= ΔT* /(dT/dt)=76s
L= λ2/2=30μm
α=0.08
fe=0.035
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Summary/ Conclusions
NTNU
• Eutectic structures can be regular/irregular, depending on facetted or
non-facetted growth.
• Eutectic structures can be lamellar or fibrous, depending on relative
amounts of the phases.
• Eutectic grows with phases side by side in a coupled way with diffusion
parallel to the front. The front is macroscopically flat and isothermal.
• Eutectics are characterized by a lamellar spacing, λ, which is controlled
by diffusion and curvature and is a function of growth rate.
• Irregular eutectics grow at a higher undercooling and a non-isothermal
front.
• Alloys with off-eutectic compositions can grow in a coupled way
depending on temperature gradient and growth rate
• Small amounts of residual eutectics often solidify in a decoupled,
“divorced” way
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Summary / Conclusions
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• Peritectic reactions, l + α →β, occur at the interface between α and the
melt. This means that α becomes isolated and further reaction can only
occur by solid state diffusion through β which is a slow process.
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Summary/ Conclusions
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• Redistribution of solute since liquid- and solid solubility are different
leads to segregation.
• Microsegregation occurs as concentration variations on a scale of the
secondary dendrite arm spacing
• Equilibrium solidification only occurs in special cases, fast solid
diffusion, slow cooling.
• Assumption of no solid diffusion, complete mixing in liquid often
realistic assumption
• Fourier number used to assess degree of back diffusion into solid
during solidification
• Non-equilibrium segregation causes freezing range, ΔT* to be larger
than equilibrium freezing range, ΔT0 and liquidus concentrationCl to
increase beyondC0/k, often up to Ce.
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NTNU
Lars Arnberg
Dept of Materials Science and Engineering
Norwegian University of Science and Technology
7491 Trondheim, Norway
arnberg@ntnu.no
Tel: +47 930 03 212 (India: 948 2965 476)
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