Textbook: Phase transformations in metals and alloys

(Third Edition), By: Porter, Easterling, and Sherif (CRC

Press, 2009).

Diffusion and Kinetics

Lecture: Solidification and Growth Kinetics

Nikolai V. Priezjev

Solidification and Growth Kinetics

► Nucleation in Pure Metals

1) Homogeneous Nucleation, 2) Nucleation Rate, 3) Heterogeneous Nucleation

► Growth of a Pure Solid

1) Growth mechanisms: Continuous and Lateral

► Alloy Solidification

► Solidification of Ignots and Casting

► Rate of a phase transformation

Reading: Chapter 4 of Porter, Easterling, Sherif

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Nucleation

Heterogeneous – the new phase appears on the walls of the container, at impurity

particles, etc.

Homogeneous – solid nuclei spontaneously appear within the undercooled phase.

Let’s consider solidification of

a liquid phase undercooled

below the melting temperature

as a simple example of

a phase transformation.

Homogeneous nucleation

Is the transition from undercooled liquid to a

solid spherical particle in the liquid a

spontaneous one?

That is, does the Gibbs free energy decreases?

Reminder:

Driving

force for

solidification

(ΔGv)

Origin of the

interfacial

energy (γSL)

Homogeneous

nucleation

Homogeneous

nucleation

Gibbs-Thomson effect

r

Homogeneous

nucleation

Homogeneous

nucleation

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Homogeneous

nucleation

A two-dimensional representation of an instantaneous

picture of the liquid structure. Many close packed

crystal-like clusters (shaded) are present.

A number of spherical clusters

of radius r is given by

kT

Gnn r

r exp0

system in the atoms ofnumber 0 n

r allfor validwhen mTT

rrfor validwhen mTT

nuclei stable are

rr clusters when mTT

SLvr rGrG 23 43

4

Example: 1mm3 copper at Tm (~1020 atoms):

~1014 clusters of 0.3nm radius (~10 atoms)

~10 clusters with radius 0.6nm (~60 atoms)

Homogeneous

nucleation

A number of spherical clusters

of radius r is given by

kT

Gnn r

r exp0

system in the atoms ofnumber 0 n

r allfor validwhen mTT

rrfor validwhen mTT

nuclei stable are

rr clusters when mTT

The variation of r* and rmax with undercooling ΔT.

TH

Tr

m

mSL

12

SLvr rGrG 23 43

4

mT

Liquid Solid

Homogeneous

nucleation

kT

Gnn r

r exp0

Rate of

homogeneous

nucleation

Rate of

homogeneous

nucleation

kTH

TA

m

mSL

2

23

3

16

mT

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Heterogeneous nucleation

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Heterogeneous

nucleation

Shape factor

Heterogeneous

nucleation

Gibbs-Thomson effect

mT 5.0,90 S

Heterogeneous

nucleation

mT

Heterogeneous nucleation

Heterogeneous nucleation in mould-wall cracks, (a) The critical nuclei, (b) The

upper nucleus cannot grow out of the crack while the lower one can.

TH

Tr

m

mSL

12

vGVG

2

1

V* = volume of the critical nucleus (sphere or cap)

small

large

90

*

*

r

V

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Pre-melting

Growth mechanisms

iTkv 1Growth rate:

Growth mechanisms

2

2 )/exp( iTkv Growth rate:

Growth mechanisms

2

3 )( iTkv Growth rate:

Growth mechanisms

The influence of interface

undercooling (ΔTi) on growth rate

for atomically rough and smooth

interfaces.

mT

4.2.3 Heat Flow and Interface Instability (pure Metals)

(a) Temperature distribution for solidification when heat is extracted through the liquid,

(b) for a planar S/L interface, and (c) for a protrusion.

vLLSS vLTKTK

Thermal

conductivity Rate of growth

of the solidTemperature

gradient

Latent heat of

fusion per unit

volume

Supercooled

below Tm

4.2.3 Heat Flow and Interface Instability

4.2.3 Heat Flow and Interface Instability

Temperature

distribution at the

tip of a growing

thermal dendrite.

Isothermal

solid 0ST

vLLSS vLTKTK

r

T

L

K

L

TKv c

v

L

v

LL

Gibbs-Thomson effect:

m

v

T

TL

rG

2

r

rT

rL

TT

v

mr

2 0

0

2

TL

Tr

v

m

Driving force

for solidification

Minimum possible

critical nucleus radius

r

r

r

T

L

Kv

v

L 10

Interface temperature

*r2r max,

,0

,0

v

rv

rrv Gibbs-Thomson effect

Slow heat conduction

Maximum velocity

A hypothetical phase diagram, partition coefficient k = XS/XL is constant (independent of T).

XS = mole fraction of solute in the solid

XL = mole fraction of solute in the liquid at equilibrium.

Assumption: liquidus and

solidus are straight lines

composition of solidcomposition of liquid

4.3 Binary Alloy Solidification

4.3 Binary Alloy Solidification

Unidirectional solidification of alloy at X0. (a) A planar S/L interface and axial heat flow. (b)

Corresponding composition profile at T2 assuming complete equilibrium. Conservation of

solute requires the two shaded areas to be equal. Infinitely slow solidification.

No diffusion in solid, perfect mixing (stirring) in liquid.

Planar front solidification of alloy X0 assuming no diffusion in the solid, but complete mixing in the liquid. (a) As

before, but including the mean composition of the solid (dashed curve). (b) Composition profile just under T1.

(c) Composition profile at T2. (d) Composition profile at the eutectic temperature and below.

4.3 Binary Alloy Solidification

Liquid richer in solute.

X0

X0

4.3 Binary Alloy Solidification

No diffusion in solid,

diffusional mixing in liquid.

Liquid richer

in solute.

Steady state with v:

rate of solute diffusion

down concentration

gradient = rate solute

rejected from

interface.

)( SLL CCvCD

Development of microstructure in eutectic alloys (I)

Several different types of microstructure can be formed in slow cooling an different

compositions. Let’s consider cooling of liquid lead – tin system as an example.

Development of microstructure in eutectic alloys (II)

At compositions between the room temperature solubility limit and the maximum

solid solubility at the eutectic temperature, phase nucleates as the solid

solubility is

exceeded upon

crossing the

solvus line.

Development of microstructure in eutectic alloys (III)Solidification at the eutectic composition

No changes above the eutectic temperature TE. At TE all the liquid transforms to

and phases

(eutectic reaction).

Development of microstructure in eutectic alloys (IV)Solidification at the eutectic composition

Formation of the eutectic structure in the lead-tin system. In the micrograph, the

dark layers are lead-reach phase, the light layers are the tin-reach phase.

Compositions of and phases are very different eutectic reaction involves

redistribution of Pb and Sn atoms by atomic diffusion.

This simultaneous

formation of and

phases result in a

layered (lamellar)

microstructure

that is called

eutectic

structure.

Development of microstructure in eutectic alloys (V)

Compositions other than eutectic but within the range of the eutectic isotherm

Primary phase is formed in the + L region, and the eutectic structure that

includes layers of

and phases

(called eutectic

and eutectic

phases) is formed

upon crossing

the eutectic

isotherm.

Development of microstructure in eutectic alloys (VI)

Although the eutectic structure

consists of two phases, it is a

microconstituent with distinct

lamellar structure and fixed ratio

of the two phases.

4.3 Growth of Lamellar Eutectics

Interdiffusion in the liquid ahead of

a eutectic front.Molar free energy diagram at a temperature ΔT0

below the eutectic temperature, for the case = *.

Gibbs-Thomson effect:

mT

TH

rG

2 Driving force

for solidification

mVGG

2)()(

ET

THG 0

)(

0

2

TH

TV Em

=0

4.3 Growth of Lamellar Eutectics

(a) Molar free energy diagram at (TE - ΔT0) for the

case * < < ∞, showing the composition difference

available to drive diffusion through the liquid (ΔX).

(b) Model used to calculate the growth rate.

XDkv

1

Interdiffusion in the liquid ahead of

a eutectic front.

*

0 1XX

00 TX

*

02 1 1

TDkv

Homogeneous

nucleation

kT

Gnn r

r exp0

Rate of

homogeneous

nucleation

Gibbs-Thomson effect:

mT

TH

rG

2

Driving force

for solidification

5. TTT (Time Temperature Transformation) Diagrams

The percentage transformation versus time for different transformation temperatures.