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Chapter 4. Solidification

Young-Chang Joo

Nano Flexible Device Materials Lab

Seoul National University

Phase Transfromation In Materials

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Contents

Nucleation in pure metals

Homogeneous nucleation

Homogeneous nucleation rate

Heterogeneous nucleation

Nucleation of melting

Growth of a pure solid

Continuous growth

Lateral growth

Heat flow and interface stability

Alloy solidification

Solidification of single-phase alloysPhase Transfromation In Materials

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Solidification

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Liquid Solid

Solidification

- Casting

- Single crystal growth

- Directional solidification

- Rapid solidification

- ….

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4.1 Nucleation in Pure Metals

Tm : GL = GS

Undercool ( supercool ) for nucleation : 250K ~ 1K

Homogeneous nucleation, Heterogeneous nucleation.

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4.1.1 Homogeneous Nucleation

Driving force for solidification

( Ch.1.2.3 )

GL = HL – TSL

GS = HS – TSSmT

TLG

L : ΔH = HL - HS

(Latent Heat)

T = Tm - ΔT

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L

VLS GVVG )(1 SLSLL

VL

S

VS AGVGVG 2

L

V

S

V GG ,

SLSL

S

V

L

VS AGGVGGG )(12

SLVr rGrG 23 4

3

4

: free energies per unit volume

for spherical nuclei (isotropic) of radius : r

4.1 Nucleation in Pure Metals

Phase Transfromation In Materials

0

mT

TLG

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SLVr rGrG 23 4

3

4

r < r* : unstable

(lower free E by reduce size )

r > r* : stable

(lower free E by increase size )

r* : critical nucleus size

4.1 Nucleation in Pure Metals

Phase Transfromation In Materials

TL

T

Gr

V

mSL

V

SL

122*

22

23

2

3

)(

1

3

16

)(3

16*

TL

T

GG

V

mSL

V

SL

r* , ΔG ↓ as ΔT ↑

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4.1 Nucleation in Pure Metals

Also from Gibbs-Thompson Eq. (Eq1.58)

r* : the radius of the solid sphere that is in (unstable) equilibrium

with surrounding liquid.

Solid sphere and liquid have the same energy.

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r

/ mol2γVmr

2γ

r/ unit volumeor

∆GV =2γSLr∗ r

∗ =2γSLΔGV

→

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4.1 Nucleation (Atomistic View)

Formation of atomic cluster

Liquid 2~4% larger volume than solid.

Instantaneously form many close-packed cluster of atoms

(same crystalline array of solid).

The average # of clusters of radius r

𝑇 > 𝑇𝑚 : holds for all value of r𝑇 < 𝑇𝑚 : 𝑟 ≤ 𝑟

∗

𝑛𝑟 exponentially decrease with Δ𝐺𝑟

𝑟이커지면 cluster를발견할확률은급속히감소

Phase Transfromation In Materials

TR

Gnn rr exp0

𝑛0 : total # of atoms.Δ𝐺𝑟 : excess free energy associated w/ the cluster (Eq. 4.4)

Increase rapidly with 𝑟

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4.1 Nucleation (Atomistic View)

Example

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1mm3 of Cu 𝑛𝑜 : ~ 1020 atoms

at Tm 0.3nm clusters (~10 atoms ) ~ 1014 clusters.

0.6nm clusters (~60 atoms) ~ 10 clusters

(cluster의수는 size에매우 sensitive)

practically, maximum cluster size 𝑟𝑚𝑎𝑥 (~100 atoms)

∆𝑇 = 𝑇𝑚 − 𝑇

Under cooling

∆𝑇가 증가하면 감소

∆𝑇𝑁: homogeneous nucleation이현실적으로가능한 supercooling temperature

TL

T

Gr

V

mSL

V

SL

122*

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4.1.2 Homogeneous Nucleation Rate

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Addition of one more atom to each cluster will convert to stable nuclei

)kT

Gexp(CC

*

hom 0𝐶𝑜 : atoms/unit volume

𝐶∗ : # of clusters with size of C* ( critical size )clusters/m3

)exp(*

hom0hom

kT

GCfN o

𝑓𝑜 : frequency ∝ (vibrational freq.), (Ea of diff. in liquid), (surface area of C*)

nuclei / m3∙s

})(

exp{20hom T

ACfN o

where 𝐴 =16𝜋𝛾𝑆𝐿

3𝑇𝑚

2

3𝐿𝑉2𝑘𝑇

← Relatively insensitive to temperature

𝑓0 ~ 1011

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4.1.2 Homogeneous Nucleation Rate

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exp(1

∆T2)

‘Explosion’ of nuclei

∆𝑇𝑁 ~ 0.2 𝑇𝑚 (i.e., ~200K) for most metals

})(

exp{20hom T

ACfN o

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4.1.3 Heterogeneous Nucleation

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22

23

)(

1

3

16*

TL

TG

v

mSL

SL

SMSLML cos SLSMML /)(cos

Nucleation becomes easy if

↓ by forming nucleus from mould wall

,

)(AAGVG MLSMSMSLSLvShet

)(43

4 23 SrGrG SLVhet

4/)cos1()cos2()( 2 S (S < 1)where

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4.1.3 Heterogeneous Nucleation

Phase Transfromation In Materials

)(3

16*

2*

2

3

SG

GandG

rV

SL

V

SL

homhet G)(SG

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4.1.3 Heterogeneous Nucleation

The effect of ∆𝑻 on ∆𝑮𝒉𝒆𝒕∗ & ∆𝑮𝒉𝒐𝒎

∗

Phase Transfromation In Materials

kT

Gnn het

*

1 exp*

𝑛 ∗: the number of nuclei, 𝑛1 : atoms in contact with the mold wall

kT

GCfN hethet

*

11 exp

𝑓1: frequency factor𝐶1: # of atoms in contact with hetero

nucleation sites/unit vol. of liquid

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4.1.3 Heterogeneous Nucleation

만일 mould wall이 microscopically flat하지않다면 cracks crevices

Phase Transfromation In Materials

Nucleation inside crevices

V

* GV*G 2

1

𝑉∗

: volume of critical nucleus

(cap or sphere)

Wetting angle이크더라도 nucleation이가능할수있다.

(그러나 crack의입구는 critical radius 보다커야함.)

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4.1.4 Nucleation of Melting

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SVLVSL (commonly)

일반적으로 wetting angle (𝜃)=0 → No superheating required!

Solid metal close to 𝑇𝑚

liquid/vapor + solid/vapor + solid/liquid