EMA5001 Lecture 17 Nucleation in Precipitation

© 2015 by Zhe Cheng

EMA5001 Lecture 17

Nucleation in Precipitation

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 17 Nucleation in Precipitation

Precipitation

Precipitation

Change in solubility with temperature

Often used for strengthening of Al alloys

Two different mechanisms for precipitation

Nucleation & Growth

− Need thermal activation

− Nucleus reach critical size

− With defined interface between new phase and matrix

Spinodal decomposition

− Need composition fluctuation

− No nucleation

− No defined interfaces

2

Foundation of Materials Science (Chinese), Pan, Tong, and

Tian, 1st Ed, 1998, p. 549-556


Homogeneous Nucleation in Solids (1)

Precipitation reaction α α + β

Local composition fluctuation

Re-arrangement atoms from α to β

Energetics for homogeneous nucleation

Driving force

− Volume free energy change Gv

Barriers

− Added α/β interface energy i for different interfaces (may not be isotropic)

− Volume strain energy Gs

Consider

− V is nucleus volume

− Ai is nucleus interface area for interface i

Total free energy change in nucleation

3

T

L

A

α β

B

α+β

siiv GVAGVG


Homogeneous Nucleation in Solids (2)

Continue from p.3

Assuming isotropic interface and spherical nucleus

Similar to homogeneous nucleation for solidification

Consider the beginning of precipitation, Gs is very small

4

siiv GVAGVG

23 43

4rGGrG sv

sv GGr

2*

23

3

16*

sv GGG

G

0 r

A r2

-V(Gv-Gs)

r3

G

-VGv

Tr

1*


Driving Force for Homogeneous

Nucleation in Solids

Molar free energy change for

the entire phase transformation

process is G0

G0 is NOT the driving force for

nucleation: i.e., the very first bit of β

precipitating out of α matrix

Driving force for nucleation (per mole

of precipitate)

G1 Molar free energy of material with

composition and structure of α

G2 Molar free energy of material with

composition and structure of β

5

12 GGGn

T

L

A

α

β T1

α+β T2

X0 Xeq B Xβ

G0

BX

BX

BBBABBAA XXXXG 11

BBBABBAA XXXXG 12

Gn

G1

G2

Xeq

T

A

Gα

Gβ

B X0 Xβ

A

A

B

B


Driving Force for Homogeneous

Nucleation in Solids

Continue from p.5

Driving force per unit volume (volume

free energy change)

For the very beginning,

The driving force is proportional to

supersaturation

The driving force is also is proportional to

undercooling

6

12 GGGn

m

nv

V

GG

eqn XXG 0

T

L

A

α

β T1

α+β T2

X0 Xeq

T

A

Gα

Gβ

B X0 Xeq

B Xβ

Xβ

G0 Gn

G1

G2

A

A

B

BTGn


Nucleation Rate in Homogeneous

Precipitation

Similar to discussion of nucleation rate for liquid solidification

C* Number density of nucleus with size r*

C0 Atom density

Homogeneous nucleation barrier, decreases with decreasing T

f Frequency of adding one more atom to a nucleus to make it “supercritical”

Nucleation rate for solid precipitation (m-3s-1)

For solid precipitation

ω Pre-exponential factor

Migration energy barrier (temperature independent)

7

kT

GCC

*

0

* exp

*

homG

kT

GfCfCN

*

hom0

*

hom exp

kT

Gf mexp

mG

kT

G

kT

GCN m

*

hom0hom expexp

ΔGm

ΔG

x

G


Nucleation Rate in Homogeneous

Precipitation

Continue from p. 7

Nucleation rate

8

kT

G

kT

GCN m

*

hom0hom expexp

T

0

α

Teq

α+β

Teq’

X0 XB

T

0

Te

Te’

ΔG

Gv

Gv - Gs

T

0

kTG /exp *

hom

kTGm /exp

T

0 N

decreases rapidly with decreasing T

constant mG

*

homG

*

homG


Other Considerations in Homogeneous

Precipitation

Impacts of solubility

/equilibrium

temperature

Nucleation rate changes

in precipitation process

due to

Change in supersaturation

Nucleus shape often NOT spherical

Preferred match of certain orientation to lower interfacial energy

Precipitation of meta-stable phase is common

Less driving force but also less barrier

9

T

0

α

α+β

Te (2)

X1 XB

T

0 N

Te (1)

X2

N (1) N (2)


Heterogeneous Precipitation

Defects sites in solids often have excess energy associated with it

compared with the defect-free bulks

If nucleation leads to removal of defects sites Release of excess

energy associated with the defects Lower barrier to nucleation

Heterogeneous nucleation at grain boundaries

Energy change (assuming no strain energy)

Critical nucleus size

Nucleation energy satisfy

10

dsv GAGGVG

β α

α

Cos2

AAGVG v

vGr

2*

SV

V

G

G hethet

*

hom

*

*

hom

*


Heterogeneous Precipitation

Heterogeneous nucleation at grain boundaries (continued)

Nucleation at grain edges (3 grains) and

grain corners (4 grains) reduces the

critical nucleation energy further

Matching of certain crystal plane could

reduce energy further

Similar for free surfaces and inclusion-

matrix interfaces

Nucleation at dislocations

Helps release strain energy

Solute segregation leading to local

enrichment

Nucleation at vacancies

Increases diffusion rate

Releases strain energy

11

Phase Transformations in Metals & Alloys, Porter, 3rd Ed,

2008, p. 272


Rate of Heterogeneous Nucleation in Solid

Heterogeneous nucleation rate depend on two factors

Heterogeneous nucleation rate

Assuming similar migration barrier

12

Homogeneous

sites Dislocations Vacancies

Stacking

faults

Grain

boundaries/

interphase

interfaces

Free

surface

Critical nucleation energy G* DECREASES

Nucleation sites (defects) density Cd DECREASES

kT

G

kT

GCN hetm

dhet

*

expexp

kT

GG

C

C

N

N hetdhet

**

hom

0hom

exp

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EMA5001 Lecture 17 Nucleation in Precipitation