UNIVERSITI TUNKU ABDUL RAHMAN (UTAR)

Solidification

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 Solidification of Metals

• Metals are melted to produce finished and

semi-finished parts.

• Two steps of solidification:

• Nucleation: Formation of stable nuclei
in the melt
• Growth of nuclei into crystals and the
formation of grain structure

• Thermal gradients define the shape of each

grain.

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 Solidification of Metals

Schematic illustration showing the several stages in

the solidification of metals: (a) formation of nuclei
into crystals, (b) growth of nuclei into crystals, and
(c) joining together of crystals to form grains and
associated grains boundaries

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 Phase Stability

• Solidification process involves changes of

phase
• For phase transformations that occur at
constant temperature and pressure, the
relative stability of a system is determined by
Gibbs free energy (G)
• The Gibbs free energy of a system:
G = H – TS
H = enthalpy, T = absolute temperature, S = entropy

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 Phase Stability

• Enthalpy is a measure of the heat content

of a system
H = E + PV
E = internal energy , P = pressure, V = volume

• Internal energy (E): total kinetic and

potential energies of the atoms in the
system

• Transformation occurs, the heat absorbed

or evolved will depend on the E and PV.

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 Phase Stability

• Condensed phases (liquid & solid), PV

term is small in comparison to E, H  E

• S is a measure of the randomness of the

system

• At constant temperature & pressure, a

closed system (fixed mass & composition)
will be in stable equilibrium if it has the
lowest G (or dG = 0)

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 Phase Stability

G = H – TS

• The state with the highest stability : best

compromise between low enthalpy and high
entropy.

• At low temperature, solid phase most stable

(strongest atomic bonding, lowest E & H)

• High temperature, -TS dominates, phases

with more freedom of atom movement
(liquid & gas) most stable.
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 Phase Stability

Arrangement of atoms

A schematic variation of Gibbs free energy with the

arrangement of atoms. Configuration A has the lowest
free energy and is therefore the arrangement when
the system is at stable equilibrium. Configuration B is
a metastable equilibrium.
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 Formation of Stable Nuclei in Liquid Metal

• Two main mechanisms nucleation of solid particles in

liquid metal: Homogeneous and Heterogeneous.

• Depending on the site at which nucleating events occur

Homogenous Nucleation:
• Homogenous nucleation in a liquid melt occurs
when the metal itself provides atoms to form nuclei
• Nuclei of the new phase form uniformly throughout
the parent phase
• When pure liquid metal significantly undercooled,
several slow moving atoms bond each other to form
nuclei

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 Homogeneous Nucleation

• Nucleus stable grow into crystal, must reach critical

size

• Cluster of atoms below critical size is called

embryo.

• If the cluster of atoms reach critical size, they grow

into crystals. Else get dissolved.

• Cluster of atoms that are greater than critical size

are called nucleus.

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 Homogeneous Nucleation

Energies Involved in Homogenous Nucleation

Two energy changes must be considered:

1.The volume (or bulk) free energy, Gv released by

the liquid-to-solid transformation

2.The surface energy, Gs required to form the new

solid surfaces of the solidified particles

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 Homogeneous Nucleation

Volume Free Energy, Gv

• The driving energy for liquid-to-solid
transformation is the difference in the
volume (bulk) free energy ΔGV of the liquid
and that of the solid Liquid

• Released by liquid to solid transformation

• ΔGv is change in free energy per unit

volume between liquid and solid

• Free energy change for a spherical nucleus

of radius r is given by

GV 4 3
 r Gv
3
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 Homogeneous Nucleation

Surface Energy, Gs
• Energy opposes to the formation of embryos and nuclei: the
energy to form the surfaces of these particles

• ΔGs is the energy required to form new solid surface

• γ is specific surface free energy of the particle

• Then, ΔGs is equal to the specific surface free energy of the

particle, γ, times the area of the surface of the sphere, or

Gs  4r 2

• ΔGs is retarding energy

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 Homogeneous Nucleation
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Total Free Energy (to form an embryo GT  r 3Gv  4r 2
or nucleus) 3
2
Since when r=r*, d(ΔGT)/dr = 0 r*  
Gv
ΔGs
+
Nucleus
ΔG*r
ΔGT
Free energy change (∆G)

Above critical Below critical

radius r* radius r*
ΔG
r
r*r* Radius of particle (r)
Energy Energy
lowered by Lowered by
growing into redissolving
crystals
- ΔGv
r* = critical nucleus: for r < r* nuclei shrink; for r >r* nuclei grow (to reduce energy)
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 Homogeneous Nucleation

• Magnitude of ΔGv depends on the temperature

∆Hf is the latent heat of fusion

Tm is the melting/solidification temperature (Kelvin)

• With lower temperature, both ΔG* and r* decrease accordingly

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 Homogeneous Nucleation

Critical Radius vs. Undercooling

• Greater the degree of undercooling, greater the change in
volume free energy ΔGv
• ΔGs does not change significantly with temperature
• As the amount of undercooling ΔT increases, critical
nucleus size decreases
• Critical radius (critical-sized nucleus) is related to
undercooling by relation
r* = Critical radius of nucleus
2Tm γ = Surface free energy
r*  ΔHf = Latent heat of fusion
H f T ΔT = Amount of undercooling (Tm - T)

Note: Hf and  are weakly dependent on T

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 Homogeneous Nucleation

Supercooling (Undercooling)
A phenomenon during the cooling of a liquid, an
appreciable nucleation rate (solidification) will begin only
after the temperature has been lowered to below the
equilibrium solidification (or melting) temperature (Tm).

Table: Values for the freezing temperature, heat of fusion, surface

energy, and maximum undercooling for selected metals

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 Heterogeneous Nucleation

• In practical situation, large amount of

undercooling is unlikely to occur for homogenous
nucleation

• Nucleation occurs in a liquid on the surfaces of

its container, insoluble impurities, grain
boundaries, dislocations, and other structural
material (nucleating agents) that lower the critical
free energy required to form a stable nucleus

• Nuclei form preferentially at structural

inhomogeneities

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 Heterogeneous Nucleation

• Since the surface energy is lower, the total

free-energy change for the formation of a
stable nucleus will be lower & the critical size
will be smaller

• Smaller amount of undercooling is required

to solidify or form stable nucleus

• Used excessively in industries

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 Heterogeneous Nucleation

• For heterogeneous to take place, solid nucleating

agent must be wetted by the liquid metal.
• The liquid should solidify easily on the nucleating
agent.
• Nucleating agent wetted by solidifying liquid, creating
a low contact angle  between the solid metal and
the nucleating angle
Solid-surface (SI), solid-liquid (SL), and
θ = contact angle liquid-surface (IL) interfacial energies
Liquid

SL
Solid
IL θ SI Nucleating
agent

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 Heterogeneous Nucleation

The energy barrier (ΔG*) for heterogeneous nucleation

is relatively lower than that of homogeneous nucleation
due to reduced surface free energy, 

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 Growth of Crystals and Formation of Grain Structure

• Nucleus grow into crystals in different orientations.

• Crystal boundaries are formed when crystals join
together at complete solidification.
• Crystals in solidified metals are called grains.
• Grains are separated by grain boundaries.
• More the number of
nucleation sites
available, more
the number of
grains formed.

Nuclei growing into grains

forming grain boundaries

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 Types of Grains

• Equiaxed Grains:
• Grains, smaller in size, grow equally in all directions.
• Formed at the sites of high concentration of the nuclei.
• Example: Cold mold wall

• Columnar Grains:
• Long thin and coarse.
• Grow predominantly in one direction.
• Formed at the sites of slow cooling
and steep temperature gradient.
• Example:Grains that are away from
the mold wall.

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 Kinetics of Solid-State Transformation

• Temperature dependence:
nucleation, growth,
transformation rates

• Time dependence rate

(kinetics of transformation)
is important in the heat
treatment of materials

• Fraction of reaction that

has occurred is measured
Plot of fraction reacted vs. the log of
as a function of time while time typical of many solid-state
temperature is maintained transformation which temperature is
held constant
as constant

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 Kinetics of Solid-State Transformation

• The fraction of transformation, y

y = 1 – exp(-ktn)
• Where k and n are time-dependent constants for the
particular reaction. This expression is referred to as
the Avrami equation

• The rate of a transformation is taken as the

reciprocal of time required for the transformation to
proceed halfway to completion, t0.5 or
1
Rate =
t0.5
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 Kinetics of Solid-State Transformation

Percent recrystallization as a function of time and at

constant temperature for pure copper

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