Fundamentals of Solidification

Outline

• Introduction

• Homogeneous nucleation

• Heterogeneous nucleation

• Growth and microstructure

• Summary

Introduction

• There are two types of solidification

– Glass formation

• Physical properties such as viscosity change

smoothly across the solidifying region

– Phase transition

• Some physical properties change abruptly,

such as viscosity, heat capacity

Introduction

• Solidification by phase transition is

modelled as two stage

– Nucleation

• Homogeneous nucleation

• Heterogeneous nucleation

– Growth

Nucleation and Grain Growth

• Nucleation;– Homogeneous nucleation: very pure metal, substantial

undercooling (0.2Tm)– Heterogeneous nucleation: nucleation agents (5ºC

undercooling)• Grain growth

– Planar: pure metal– Dendritic: solid solution

• Grain size – depends on number of nuclei and cooling rate.

Crystal Nucleation and Growth

“Manufacturing Processes for Engineering Materials,” by Serope Kalpakjian

7

Nucleation Rate

• The rate at which nucleation of a new phase occurs

is critical to the prediction of phase transformation

behavior.

• We will contrast homogeneous nucleation

(extremely rare!) with heterogeneous nucleation

(typical) rates.

• Why study homogeneous nucleation? Useful

foundation and simplest to understand.

• Bottom line: the quickest transformation wins!

8

Thermodynamics of nucleation

• How should we understand nucleation?

• The crucial point is to understand it as a balance

between the free energy available from the driving force,

and the energy consumed in creating new interface

(between parent and product phases). Once the rate of

change of free energy becomes negative, then an

embryo can grow.

• Parallel to the Griffith analysis: once the rate of (free)

energy change becomes negative with crack length

increase, then the crack can grow without limit.

9

Nucleation paths• It is important to remember that the actual outcome is

always the process that leads most rapidly to the change for

which a (thermodynamic) driving force exists.

• Anisotropy in the interfacial energy forces growing grains to

adopt anisotropic shapes in order to minimize high energy

orientations of the interface.

• Anisotropy in growth rates has a similar effect.

• Heterogeneous nucleation on surfaces, pre-existing

interfaces (grain boundaries), dislocations etc. is very

important.

• Elastic energy plays a major role in constraining nucleation.

10

Homogeneous Nucleation

• Assume that the new, product phase appears as spherical

particles.

• Free energy released by transformation is proportional to

the volume.

• Free energy consumed by creation of interface is

proportional to the surface area of particle and the

interfacial energy, g.

• Net change in free energy per particle, ∆Gr:

∆Gr = -4π/3 r3 ∆GV + 4πr2 g.

• Differentiate to find the stationary point (at which the rate of

change of free energy turns negative).

11

Critical radius, free energy

• d(∆Gr) = 0 =

-4π/ r*2 ∆G* + 8πr*g.

• From this we find the critical

radius and critical free energy.

r* = 2g/∆GV

∆G* = 16πg3/3∆GV2

• Crucial difference from

solidification: the role of elastic

energy!

Not at ∆Gr=0!!!

12

Elastic energy

• Why does elastic energy play such an important role in solid state phase transformations?

• Volume changes on transformation of order a few % are typical. Elastic energy is symmetric: net (hydrostatic) tension or compression leads to an increase in elastic energy. This elastic energy cost for creation of a new phase, ∆GS (= Ee2/2), must be subtracted in proportion to the volume of new phase.

∆Gr = -4π/3 r3 (∆GV - ∆GS) + 4πr2 g.d(∆Gr) = 0 = -4π r2 ∆G* + 8πr*g. r* = 2g / (∆GV - ∆GS);∆G* = 16πg3 / 3(∆GV - ∆GS)2.

Homogeneous nucleation

rr

Homogeneous nucleation

• No preferred nucleation sites

– Spontaneous

– Random

• Those of preferred sites

– Boundary

– Surface

– Inclusion, …

Local free energy change

1. Liquid to solid 2. Interface

Single nucleus

Critical radius

0/ drGd

SL

SL

GGr

2*

2

3

3

16*

SL

SL

GGG

(GL-GS) vs. supercooling

Free energy density vs. temperature

liquid

solid

temperature

Free energy density

19

Homogeneous Nucleation: examples

• Two examples of homogeneous nucleation in the solid

state are known.

1) Cu with 1-3% Co can be heat treated to precipitate Co

homogeneously. We will examine this case in a homework

aimed at predicting TTT diagrams.

2) Ni superalloys will precipitate Ni3Al homogeneously at

small undercoolings because of the small lattice misfit and

small interfacial energy.

• Why only these cases? Small interfacial energy, and small

elastic energy difference.

• Everything else: heterogeneous!

20

Elastic Anisotropy

• Remember that most crystalline solids are elastically

anisotropic: this means that the shape of a new phase is

likely to be anisotropic.

• If either the parent or the product phase is more compliant

in a particular direction, larger dimensions parallel to this

direction will be favored over stiffer directions. This is

offset by the interfacial energy term which must increase

as the surface-to-volume ratio increases.

• Example: Guinier-Preston zones in the Al-Cu system,

which are platelets on {100}.

21

Interfacial Energy

• Even in solidification, the anisotropy of the interfacial

energy matters. The energy of the solid-liquid interface

varies depending on which crystallographic surface is

involved. {111} surfaces tend to have the lowest energy in

fcc metals.

• In solid state precipitation, the anisotropy of the interface

matters even more! This is because there are two

crystalline surfaces involved in the interface. If the crystal

structures are different (often the case) then low energy

interfaces require good atomic matching between the two

planes. Sometimes this results from combining close-

packed interfaces.

22

Nucleation rate• To estimate the nucleation rate we need to know the population

density of embryos of the critical size and the rate at which such

embryos are formed.

• Population (concentration) of critical embryos, C*, is given by a

Boltzmann factor, where C0 is the number of atoms per unit

volume:

C* = C0 exp -(∆G*/kT)

• The rate at which a critical embryo is formed, f, depends on the

migration of atoms, i.e. diffusion, which is again given by a

Boltzmann factor, where ∆Gbulk is the activation energy for (bulk)

diffusion (∆Gm in P&E), and w is of the same order as the atomic

jump frequency:

f = w exp -(∆Gbulk/kT).

23

Nucleation rates, contd.

• Based on this approach,

we can now understand

the extremely strong

dependence of nucleation

rate on under cooling.

• Note that the net effect of

elastic energy is to offset

(decrease) the equilibrium

transformation

temperature.

24

Effect of undercooling

• The effect of undercooling on the nucleation rate is drastic,

because of the non-linear relation between the two quantities.

• By incorporating the previous expression into the nucleation rate

we obtain the following:

C* = C0 exp -(∆G*/kT) =

• Finally the nucleation

rate is the product

of C* and f:

N = f C*

C* C0 exp 163

3 GV GS 2kT

C0 exp16 3

3HT

Te

GS

2

kT

25

Nucleation Rate

• The combined equations are as follows.

• The nucleation rate is the product of

C* and f. Note that the product of w and C0 is a large

number because w is of the order of the atomic vibration

frequency, and C0 is the number of atoms per unit volume.

N f C * expGbulk

kT

C0 exp

163

3HT

Te

GS

2

kT

Heterogeneous nucleation

• Nucleation site

– Mold walls

– Inclusion

– Interface

– Surface

– Impurity

27

Heterogeneous Nucleation

• Heterogeneous nucleation must occur on some substrate:

grain boundaries

triple junctions

dislocations

(existing) second phase particles

• Consider a grain boundary: why is it effective? Answer:

by forming on a grain boundary, an embryo can offset its

“cost” in interfacial energy by eliminating some grain

boundary area.

Liquid

Inclusion

Nucleus IL

NL

IN

R

r

h

a

Heterogeneous nucleation

29

Grain boundary nucleation

• The semi-angle, q = cos-1gaa/2gab

• As for solidification, the radius of the spherical caps

depends only on the interfacial energy, so:

r* = 2gab/(∆GV-∆GS)

but a shape factor modifies the critical free energy:

∆G* = 16πgab3/3(∆GV-∆GS)2 S(q)

= 16πgab3/3(∆GV-∆GS)2 0.5(2 + cosq)(1 - cosq)2

Grainboundary in alpha

30

Other heterogeneous sites

• Other sites for

heterogeneous nucleation

have been listed.

• For the same contact angle,

grain corners (quadruple

points) are more effective

than grain edges (triple

lines), which are more

effective than grain

boundaries.

31

Heterogeneous nucleation rate

• The rate of heterogeneous nucleation, Nhet, is described by a

very similar equation as previously described for homogeneous

nucleation, Nhomo. The critical difference is in the critical free

energy, ∆G*, and the density of sites, C1.

• Homogeneous:

Nhomo = exp -(∆Gbulk/kT) C0 exp -(∆Ghomo*/kT)

• Heterogeneous:

Nhet = exp -(∆Gbulk/kT) C1 exp -(∆Ghet*/kT)

• For grain boundary nucleation, for example, the ratio of site

densities, C1/C0 = /D, where D is the grain size, and is the

boundary thickness.

Local free energy change

SLLSbeforeafter AGGVGGG

SLSL rGGrG 23 43

4

Spherical nucleus:

Thermodynamic barriers

Heterogeneous nucleation barrier

Homogeneous nucleation barrier

System free energy

• Ideal solution: Particle of different sizes

• ni particles with each contains i atoms

• n particles with each contains 1 atom

STGnG ic

ii

ii nn

nn

nn

nnkS lnln

Number of nuclei

• At equilibrium

0/ ic nG

i

i

nn

n

kT

Gln

inn

kT

Gnni exp

kT

Gnni

*exp*

when

Number of nuclei

Boltzmann formula:

Critical nuclei:

Inoculating agents

• Small interface energy

– Similar crystal structure

– Similar lattice distance

– Same physical properties

– Same chemical properties

Casting refinement

• Adding inoculating agents

– Overheat might melt the agents

• Surface refinement

– Coat agents on mold walls

• Pattern induced solidification

Growth and microstructure

T. F. Brower and M.C. Flemings, Trans. AIME, 239, 1620 (1967)

H.B. Dong and P.D. Lee, Acta Mater. 53 (2005) 659

Growth and microstructure

Outer chilled zones

Outer chilled zones

Outer chilled zones

Outer chilled zones

Pure metals: Formation of shell because temperature gradient is the key factor in grain growth.

Outer chilled zones

re-melted?

Pouring temperature

survived?

Microstructure of ingot

• Chilled zone

– Fine equiaxed grains.

– Pure substance: Continuous shell.

– Solution: Particles

– Particles flushed away from wall into the

central

• Re-melted

• Survived – nucleus

Intermediate columnar zone

Columnar grains grows

The grain is overtaken by neighbors.

Intermediate columnar zone

Growth and overtaken

Intermediate columnar zone

Columnar growth blocked

Central equiaxed zone

• Equiaxed grain

– Nucleation:

• Supercooling

• Falling particles

• Dendrite fragments– Elevated pouring

temperature:

• Larger equiaxed

grains

• More columnar zone

– Anisotropic properties

• Magnetic materials

• Turbo blade.

• More equiaxed zone

– Isotropic properties

– Less segregation

Structure and properties