Solidification of material

SOLIDIFICATION OF METALS
(To be completed)
Prof. H. K. Khaira
Professor, Deptt. of MSME
M.A.N.I.T., Bhopal

Contents
• Solidification of Metals
• Cooling Curves

Solidification of a pure metal.

Solidification of Metals
1. During solidification, the liquid changes in to solid during cooling.
2. The energy of liquid is less than that of the solid above the melting point. Hence
liquid is stable above the melting point.
3. Below the melting point, the energy of liquid becomes more than that of the solid.
4. Hence below the melting point, the solid becomes more stable than the liquid.
5. Therefore at the melting point, liquid gets converted in to solid during cooling.
6. This transformation of liquid into solid below melting point is known as
solidification.

1. Thermodynamically, both liquid and solid have equal energy at melting point and
therefore both are equally stable at melting point.
2. Therefore, no solidification or melting will take place at the melting point. Liquid will
remain liquid and solid will remain solid.
3. Some under-cooling will be essential for solidification.
4. This transformation occurs by nucleation and growth.

Cooling curve for a pure metal showing
possible undercooling.
• The transformation
temperature, as shown on the
equilibrium
diagram, represents the point
at which the free energy of the
solid phase is equal to that of
the liquid phase.
• Thus, we may consider the
transition, as given in a phase
diagram, to occur when the
free energy change, ΔGV , is
infinitesimally small and
negative, i.e. when a small but
positive driving force exists
due to undercooling.

Nucleation and Growth of Crystals

Nucleation and Growth of Crystals
•

•

•
•
•

At the solidification
temperature, atoms from the
liquid, such as molten metal, begin to
bond together and start to form
crystals.
The moment a crystal begins to grow
is know as nucleus and the point
where it occurs is the nucleation
point.
When a metal begins to
solidify, multiple crystals begin to
grow in the liquid.
The final sizes of the individual
crystals depend on the number of
nucleation points.
The crystals increase in size by the
a)Nucleation of crystals, b) crystal growth, c)
progressive addition of atoms and
irregular grains form as crystals grow
grow until they impinge upon
together, d) grain boundaries as seen in a
adjacent growing crystal.
microscope.

Cooling Curve
• A cooling curve is a graphical plot of the
changes in temperature with time for a
material over the entire temperature range
through which it cools.

Cooling Curve for Pure Metals
• Under equilibrium conditions, all metals
exhibit a definite melting or freezing point.
• If a cooling curve is plotted for a pure metal, It
will show a horizontal line at the melting or
freezing temperature.

Cooling Curve of Alloys
• In this method, alloys with different compositions are melted and then the
temperature of the mixture is measured at certain time intervals while
cooling back to room temperature.
• A cooling curve for each mixture is constructed and the initial and final
phase change temperatures are determined.

Cooling Curve
• Then these temperatures are used for the construction of the
phase diagrams

Cooling curve for the solidification of
a pure metal.

Cooling curve for a solid solution.
• In case of alloys, the solidification does not
take place at a constant temperature.
• In such cases, the solidification occure in a
range of temperature.

Series of cooling curves for different alloys in a completely
soluble system. The dotted lines indicate the form of the phase
diagram

Phase Diagram of Solid Solution

Cooling Curves for Solid Solution

Mechanism of Solidification of Metals
• The solidification of metals occur by
nucleation and growth transformation.
• In nucleation and growth transformation, the
nuclei of the solid phase are formed and then
they grow.

Nucleation and Growth
Transformation
• Nucleation – The physical process by which a new phase is produced in a
material. In the case of solidification, this refers to the formation of tiny
stable solid particles in the liquid.
• Growth - The physical process by which a new phase increases in size. In
the case of solidification, this refers to the formation of a stable solid
particle as the liquid freezes.

Transformation
• The Nucleation and Growth Transformation
may be of two types
• 1. Homogeneous Nucleation
• 2. Heterogeneous Nucleation

Homogeneous Nucleation
• Homogeneous Nucleation – Formation of a
critically sized solid from the liquid by
clustering together of a large number of atoms
at a high undercooling (without an external
interface).

• The transformation temperature, as shown on
the equilibrium diagram, represents the point
at which the free energy of the solid phase is
equal to that of the liquid phase.
• Thus, we may consider the transition, as given
in a phase diagram, to occur when the free
energy change, ΔGV , is infinitesimally small
and negative, i.e. when a small but positive
driving force exists.

Cooling curve for a pure metal showing possible
undercooling.

Energy barrier separating structural
states.

Free Energy Changes
• The second phase has lower free energy than
the first phase
• Activation energy may be required for the
transformation to occur as shown above.

Transformation
(1) The volume free energy ΔGV – free energy difference between the liquid and solid
Δ GV = 4/3πr3ΔGv
(2) The surface energy ΔGs – the energy needed to create a surface for the spherical
particles
ΔGs = 4πr2γ
γ → specific surface energy of the particle

Total free energy Change
ΔGT = ΔGV + ΔGs

Transformation
• Embryo - An embryo is a tiny particle of solid that
forms from the liquid as atoms cluster together.
The embryo is unstable and may either grow in to
a stable nucleus or re-dissolve.
• Nucleus – It is a tiny particle of solid that forms
from the liquid as atoms cluster together.
Because these particles are large enough to be
stable, nucleation has occurred and growth of the
solid can begin.

Critical Size of Nucleus
• The minimum size that must be formed by
atoms clustering together in the liquid before
the solid particle is stable and begins to grow.

r* : critical radius
(where ΔGT reaches the maximum)
• liquid metal is cooled below
freezing point, slow moving
atoms bond together to create
homogeneous nuclei
• Nucleus – larger than critical
size, can grow into a crystal
• Embryo – smaller than critical
size, continuously being
formed and redissolved in the
molten metal
• For Critical size of nucleus
d(ΔGT)/dr = 0 when r = r*
Or r* = - 2γ/ΔGv

critical nucleus size
• Critical nucleus size mainly determined by ΔGV
• Amount of undercooling increases, the critical
nucleus size decreases the relationship is
R* = 2γTm / ΔHf ΔT
Where

γ: surface free energy
Tm: freezing temperature
ΔHf : latent heat of fusion
ΔT: amount of undercooling

critical radius versus undercooling

critical nucleus size - Example
• calculate the critical radius of homogeneous
nucleus forms from pure liquid Cu.
• Assume
ΔT = 0.2ΔTm , γ = 1.77 × 10-7 J/cm2
Tm = 1083oC, Δ Hf = 1826 J/cm3
• calculate the number of atoms in criticalsized
nucleus at this undercooling

critical nucleus size - Solution
• ΔT = 0.2ΔTm = 1356 K × 0.2 = 271 K
2γTm
2(1.77 × 10-7 J/cm2)(1356 K )
• r* = ─── = ─────────────
ΔHf
ΔT (1826 J/cm3)(271 K)
• volume of nucleus

•
•
•
•

= 9.70 × 10-8 cm

= 4/3 π (9.70 × 10-8 cm) 3
= 3.82 × 10-21 cm3
Cu: FCC structure, unit length a = 3.61 × 10-8 cm
4 atoms per unit cell
volume of unit cell
= (3.61 × 10-8 cm) 3
= 4.70 × 10-23 cm 3
3.82 × 10-21 cm 3
number of atoms
= ───────
× 4 = 325 atoms
4.70 × 10-23 cm 3

critical nucleus size
• d(ΔGT)/dr = 0 when r = r*
• r* = - 2γ/ΔGv

(a) Effect of nucleus size on the free energy of nucleus
formation. (b) Effect of undercooling on the rate of
precipitation.

Homogeneous Nucleation
• Quantitatively, since ∆ Gv depends on the
volume of the nucleus and ∆ GS is proportional
to its surface area, we can write for a spherical
nucleus of radius r
∆ G = (4 π r3 /3) ∆ Gv + 4 π r2γ
• where ∆ Gv is the bulk free energy change
involved in the formation of the nucleus of
unit volume and γ is the surface energy of unit
area.

Critical Size of Nucleus
• When the nuclei are small the positive surface energy term predominates,
while when they are large the negative volume term predominates, so
that the change in free energy as a function of nucleus size. This indicates
that a critical nucleus size exists below which the free energy increases as
the nucleus grows, and above which further growth can proceed with a
lowering of free energy; ∆ Gmax may be considered as the energy or work
of nucleation W. Both rc and W may be calculated since

d ∆G/dr = 4 π r2∆Gv + 8 π rγ = 0 when r = rc and thus
rc = -2γ / ∆G v
• Substituting for rc gives
W =16 π γ 3/3 ∆Gv2

• The probability of an atom having sufficient
energy to jump the barrier is given, from the
Maxwell–Boltzmann distribution law, as
proportional to exp [Q/kT] where k is
Boltzmann’s constant, T is the temperature and
Q is usually expressed as the energy per atom in
electron volts.1
• The rate of reaction is given by
Rate = A exp [- Q/kT]
where A is a constant

• The surface energy factor is not strongly dependent on
temperature, but the greater the degree of undercooling or
supersaturation, the greater is the release of chemical free
energy and the smaller the critical nucleus size and energy
of nucleation.
• This can be shown analytically since
∆Gv = ∆H - T∆S,
• and at T = Te, ∆Gv = 0, so that ∆H = Te ∆S.
It therefore follows that
∆Gv =(Te -T) ∆S = ∆T∆S
• and because ∆Gv is proportional to ∆T, then
W is proportional to ∆S3 / ∆T2

• Heterogeneous Nucleation – Formation of a
critically sized solid from the liquid on an
impurity surface.
• heterogeneous nucleation occurs in a liquid
on the surface of its container, insoluble
impurities and other structural materials that
lower the critical free energy required to form
a stable nucleus

Heterogeneous Transformation
• In practice, homogeneous nucleation rarely
takes place and heterogeneous nucleation
occurs either on the mould walls or on
insoluble impurity particles.
• A reduction in the interfacial energy would
facilitate nucleation at small values of ∆T.
• This occurs at a mould wall or pre-existing
solid particle

crystal growth and grain formation
•
•
•
•
•

nuclei → crystals → grains
polycrystalline – solidified metal containing many crystals
grains – crystals in solidified metal
grain boundaries – the surfaces between the grains
two major types of grain structures:
(1) equiaxed grains – crystals grow about equally in all
directions, commonly found adjacent to a cold mold
wall
(2) columnar grains – long, thin, coarse grains, created
when metal solidifies rather slow in the presence of a
steep temperature gradient. columnar grains grow
perpendicular to the mold surface

Transformation in solid solution

Transformation in solid solution
T(°C) L (liquid)
130 0
L: 35 wt% Ni
: 46 wt% Ni
32

L: 35wt%Ni

35

24
120 0

A
B
C
D36

46

43
L: 32 wt% Ni
: 43 wt% Ni

E

L: 24 wt% Ni
: 36 wt% Ni

(solid)
110 0
20

30

35
C0

40

50

wt% Ni

Transformation
• The factors which determine the rate of phase
change are:
• (1) the rate of nucleation, N (i.e. the number
of nuclei formed in unit volume in unit time)
and
• (2) the rate of growth, G (i.e. the rate of
increase in radius with time)

Dendrites
• In metals, the crystals that form in the liquid during freezing
generally follow a pattern consisting of a main branch with many
appendages. A crystal with this morphology slightly resembles a
pine tree and is called a dendrite, which means branching.
• The formation of dendrites occurs because crystals grow in defined
planes due to the crystal lattice they create.
• The figure to the right shows how a cubic crystal can grow in a melt
in three dimensions, which correspond to the six faces of the cube.
• For clarity of illustration, the adding of unit cells with continued
solidification from the six faces is shown simply as lines.
• Secondary dendrite arms branch off the primary arm, and tertiary
arms off the secondary arms and etcetera.

Dendrites
• During freezing of a polycrystalline material, many
dendritic crystals form and grow until they eventually
become large enough to impinge upon each other.
• Eventually, the interdendriticspaces between the
dendrite arms crystallize to yield a more regular crystal.
• The original dendritic pattern may not be apparent
when examining the microstructure of a material.
• However, dendrites can often be seen in solidification
voids that sometimes occur in castings or welds, as
shown in the next slide..

Computer simulated image of dendritic growth using a cellular automata technique.
Notice the branching on the dendrites. Photograph courtesy of the Institute of
Materials, based on the work of U. Dilthey, V. Pavlik and T. Reichel, Mathematical
Modelling of Weld Phenomena III, eds H. Cerjak and H. Bhadeshia, Institute of
Materials, 1997.

Steady-state patterns formed at the crystal–melt interface of a binary alloy of succinonitrile
and coumarin 152 during directional solidification.

Losert W et al. PNAS 1998;95:431-438

©1998 by National Academy of Sciences

(A) Time evolution of the interface morphology for SCN/rhodamine 6G at constant pulling
speed V (V = 3.11 μm/s, G = 2.8 K/cm, C∞ = 0.325 wt%).

Losert W et al. PNAS 1998;95:431-438

©1998 by National Academy of Sciences

Shrinkage
• Most materials contract or shrink during solidification and cooling.
Shrinkage is the result of:
– Contraction of the liquid as it cools prior to its solidification
– Contraction during phase change from a liquid to solid
– Contraction of the solid as it continues to cool to ambient temperature.

• Shrinkage can sometimes cause cracking to occur in component as it
solidifies.
• Since the coolest area of a volume of liquid is where it contacts a mold or
die, solidification usually begins first at this surface.
• As the crystals grow inward, the material continues to shrink.
• If the solid surface is too rigid and will not deform to accommodate the
internal shrinkage, the stresses can become high enough to exceed the
tensile strength of the material and cause a crack to form.
• Shrinkage cavitation sometimes occurs because as a material solidifies
inward, shrinkage occurred to such an extent that there is not enough
atoms present to fill the available space and a void is left.

Solidification of material

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