ema5001 lecture 12 solidification via … 5001 physical properties of materials zhe cheng (2016) 12...

© 2016 by Zhe Cheng

EMA5001 Lecture 12

Solidification via

Homogeneous Nucleation

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 12 Solidification - Homogen Nucleation

Phase Transformation

Solid state phase transformation

Heterogeneous transformation

− Transformation occur at particular locations

− Always involve creation and migration of interface

− Often go through nucleation & growth

Homogeneous transformation

− Transformation everywhere in the materials

− May or may not involve creation of interface

Most phase transformation go through nucleation & growth

Creation of interface – nucleation

Migration of interface – growth

2


Classification of Nucleation & Growth

Type Phase Transformation

Classification

3

Type Military Civilian

Effect of temperature

Athermal Thermal activated

Interface type Glissile Non-glissile

Composition of precipitate and parent phase

Same composition

Same composition

Different composition

Nature of diffusion

No diffusion Short-range diffusion

Long-range diffusion

Rate limiting step

Interface control

Interface control

Interface control

Diffusion control

Mixed control

Examples Martensite, Twinning

Grain growth, α in Fe

Bainite Solidification Eutectoid


Applications

Casting metal parts (e.g., iron

and steel, bronze, aluminum)

Casting of semiconductors

(e.g., Si)

Growth of single crystals

(e.g., Al2O3)

Topics

Driving force & barrier

Critical conditions

Rate

Impacts of

− Composition

− Processing condition (e.g., T)

− Interfaces

Solidification & Crystallization

4

http://americas.kyocera.com/kicc/pdf/Kyocera%20Sapphire.pdf

Sapphire (Al2O3)

for advanced

display and lighting

http://pveducation.org/pvcdrom/manufacturing/multi-crystalline-silicon

Multi-crystalline Si for solar

cells

http://www.m0ukd.com/Solar_Panels/


Examples of a Process Thermodynamically Favorable yet not Readily

Occurring

Oxidation of carbohydrates at room temperature

Complete oxidation of stainless steel or aluminum

Liquid metal droplets undercooled below its

melting point

Origins of the “Barrier”

Carbohydrate oxidation - Splitting of oxygen molecules

Stainless steel or aluminum oxidation - Diffusion of oxygen or metal through the

oxide shell to sustain the reaction

Solidification of liquid - Creation of new surface or interfaces

− Nucleation of solid (or a new phase) and creation of the interface

− Growth of the nucleus and movement of the interface

Barriers in Reactions & Phase

Transformation

5

High energy

state Low energy

state

Activated state


Homogeneous Nucleation &

Heterogeneous Nucleation

Homogeneous Nucleation

Uniform throughout, no special location

Need large driving force (e.g., undercooling)

Less common

Heterogeneous Nucleation

Occurring at special locations (local defects: interfaces, impurities, etc.)

Small driving force (e.g., undercooling)

Depend on features and concentration of the defects (interfaces or impurities)

Common

6


Homogeneous Nucleation for Liquid

Solidification (1)

Energy change in homogeneous

nucleation

Define

VS Volume for solid nucleus

VL Volume for remaining liquid

Gibbs free energy per unit volume of liquid

Gibbs free energy per unit volume of solid

SL Solid-liquid Interfacial energy

ASL Solid-liquid Interfacial area

Total free energy of the liquid before nucleation

Total free energy of the after nucleation

Free energy change in this process:

7

T

GS

G GL

Tm

ΔGv

V=VS+VL

Liquid metal droplet

Solid

ΔGi=AγSL

Added interfacial energy Total volume

L

vLS GVVG 1

L

vGS

vG

SLSL

L

vL

S

vS AGVGVG 2

SLSL

S

v

L

vS AGGVGGG 12


Homogeneous Nucleation for Liquid

Solidification (2)

Continue from p.5

We have

Define

We have

is a function of temperature or undercooling

Lv is the latent heat of fusion per unit volume

If SL isotropic & the nucleus takes spherical shape

8

SLSL

S

v

L

vS AGGVG

S

v

L

vv GGG

SLSLvS AGVG

vG

SLv rGrG 23 43

4

G

r

Interfacial

energy r2

Volume free

energy r3

m

vv

T

TLG

T

GS

G GL

Tm

ΔGv

ΔT


Verification of Relationship between

Volume Free Energy Change & Undercooling

Proof

At melting point Tm:

Which means:

At a temperature below melting point, T

Assuming enthalpy and entropy change in solidification do NOT change with

temperature

9

m

vv

T

TLG

STLSTHG mvmv 0

STHGv

v

m

v

m

m

m

vvvv L

T

TL

T

TT

T

LTLSTLSTHG

STLH mv m

v

T

LS


Critical Radius & Free Energy Change

Continue from p. 6

Critical nucleus size r*

r < r*, nucleus shrinks; r > r*, nucleus grows

Knowing

We have

10

SLv rGrG 23 43

4

G

r

Interfacial

energy r2

Volume free

energy r3

r*

G*

m

vv

T

TLG

TL

Tr

v

mSL

12*

T

r

1

* When T 0,

r* and G* both ,

homogeneous nucleation

will NOT occur!

v

SL

Gr

2*0

dr

Gd

23

3

16*

v

SL

GG

22

23 1

3

16*

TL

TG

v

mSL

2

1*

TG


Verification of Nucleation Barrier

11

SLv rGrG 23 43

4

v

SL

Gr

2*

SL

v

SLv

v

SL

GG

GG

23

24

2

3

4

23

2

34

48

3

4

v

SL

v

SL

GGG

23

163

32

v

SL

GG

23

*

3

16

v

SL

GG


Critical Nucleus Size &

Maximum Cluster Size

Continue from p.7

The number of solid clusters with size r, nr

n0 The total number of atoms

Gr Excess energy for cluster of size r,

For a given T > 0 (Gv > 0), when r < r* :

r , Gr , nr ↓ dramatically

There is effectively a “maximum” cluster in the liquid

When T , Gv for a given r, Gr ↓, nr , rmax

T < TN, rmax < r*, no homogeneous nucleation

T >= TN, rmax > r*, homogeneous nucleation occurs

12

kT

Gnn r

r exp0

TL

Tr

v

mSL

12*

SLvr rGrG 23 43

4

r

T

Tr

1*

maxr

TN


Homogeneous Nucleation Rate

Define

C* Number density of nucleus with size r*

C0 Atom density

Homogeneous nucleation energy

f0 Frequency of adding one more atom to a nucleus

Nucleation rate (m-3s-1)

For homogeneous nucleation

We have

13

kT

GCC

*

hom0

* exp*

homG

kT

GCfN

*

hom00hom exp

22

23*

hom

1

3

16*

TL

TGG

v

mSL

200hom expT

ACfN

Nhom

T TN

Sudden increase of nucleation rate

TN ~ 0.2 Tm

ema5001 lecture 12 solidification via … 5001 physical properties of materials zhe cheng (2016) 12...

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