chapter 5. diffusional transformation in solids

Nano & Flexible

Device Materials Lab.11

Chapter 5.

Diffusional Transformation in

solids

Young-Chang Joo

Nano Flexible Device Materials Lab

Seoul National University

Phase Transformation In Materials

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Contents

Homogeneous Nucleation in Solids

Heterogeneous Nucleation

Rate of Heterogeneous Nucleation

Precipitate Growth

Overall Transformation Kinetics: TTT Diagrams

Precipitation in Age-Hardening Alloys

Precipitation in Aluminum-Copper Alloys

Age Hardening

Spinodal Decomposition

Particle Coarsening


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Categories of Diffusion Phase Transformations

⇒ Long range diffusion is required


(a) Precipitation

(b) Eutectoid Transformation

𝛼′ → 𝛼 + 𝛽

Supersaturated solid solution

Precipitates

𝛾 → 𝛼 + 𝛽

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Categories of Diffusion Phase Transformations


(c) Order-Disorder Transformation

(d) Massive Transformation

(e) Polymorphic Transformation

⇒ without any composition change or long-range diffusion

𝛼(𝑑𝑖𝑠𝑜𝑟𝑑𝑒𝑟𝑒𝑑) → 𝛼′(𝑜𝑟𝑑𝑒𝑟𝑒𝑑)

𝛽 → 𝛼

The same composition as the parent phase

(ex) fcc Fe bcc Fe

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5.1 Homogeneous Nucleation in Solids

Free Energy Change:

SV GVAGVG

VGV 1) Volume Free Energy

SGV 3) Misfit Strain Energy


A2) Interface Energy

𝑖

𝛾𝑖𝐴𝑖 (𝑐𝑜ℎ, 𝑖𝑛𝑐𝑜ℎ, 𝑠𝑒𝑚𝑖 … )

𝑇1 → 𝑇2 𝛼′ → 𝛼 + 𝛽

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For spherical nucleation

23 4)(3

4rGGrG SV

)(

2*

SV GGr

2

3

)(3

16*

SV GGG


Misfit reduced the driving force of the transformation

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)/*(exp* 0 kTGCC

The concentration of critical size nuclei , 𝑪∗

(Co : # of atoms/unit volume)

Nucleation Rate


How frequently a critical nucleus can receive an atom ∝ 𝜔 · exp −∆𝐺𝑚

𝑅𝑇

Strong temp. dependency

2

3

)(3

16*

SV GGG

kT

G

kT

GCN

CfN

m *expexp

*

0hom

hom

Driving force for ppt

𝑁ℎ𝑜𝑚 = 𝑓0𝐶0exp −∆𝐺ℎ𝑜𝑚

∗

𝑘𝑇

Eq 4.12

kT

G

kT

GCN

CfN

m *expexp

*

0hom

hom

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𝑇1 → 𝑇2 𝛼′ → 𝛼 + 𝛽′

⇒ Driving force for nucleation

∴ At the initial stage of nucleation, composition 𝛼 does not change 𝑋0 constant

Total driving force for transformation

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Total free energy decrease per mole of nuclei


12 GGGn

eV XXXwhereXG 0

Driving Force for Nucleation

removedmolperXXG BBAA 1(𝑝𝑒𝑟 𝑚𝑜𝑙 𝛽 𝑟𝑒𝑚𝑜𝑣𝑒𝑑)

formedmolperXXG BBAA 2(𝑝𝑒𝑟 𝑚𝑜𝑙 𝛽 𝑓𝑜𝑟𝑚𝑒𝑑)

Volume free energy decrease associated

with Nucleation

ofvolumeunitperV

GG

m

nV

(𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝛽 )

For dilute solution,

• The driving force for precipitation increases with increasing undercooling

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(∝ ∆𝑇)

2

3

)(3

16*

SV GGG

Temperature dependence of Nucleation

kT

G

kT

GCN

CfN

m *expexp

*

0hom

hom

ofvolumeunitperV

GG

m

nV

Low undercooling

=> N negligible due driving force too small

High undercooling

=> N negligible due diffusion is too slow

Maximum nucleation rate at intermediate

undercooling!!!

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Effect of Alloy Composition

Dilute alloy has lower nucleation rate

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5.2 Heterogeneous Nucleation

dSVhet GAGGVG )(

2/cos

AAGVG V

VGr /2*

)(*

*

*

*

homhom

SV

V

G

G hethet

2)cos1()cos2(2

1)( S

Nucleation on Grain Boundaries


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Low Energy Interface


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*G

13

1

*expexp

smnuclei

kT

G

kT

GCN m

het

5.2.1 Rate of Heterogeneous Nucleation

1) homogeneous sites

2) vacancies

3) dislocations

4) stacking faults

5) grain boundaries and interphase boundaries

6) free surfaces

Decreasing Order of

kT

GG

C

C

N

N hethet **exp hom

0

1

hom

5

0

1 10 DC

C for grain boundary nucleation


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5.3 Precipitate Growth

Growth can be categorized into diffusion-controlled growth and

interface-controlled growth

3.5 Interface Migration

Phase transformation occurs by nucleation growth process.

β forms at a certain sites within α (parent) during nucleation (interface created)

then the α/β interface “migrate” into the parent phase during growth.

Types of interfaces

1. Glissile: by ㅗ glide → results in the shearing of parent lattice into the

product (β), motion (glide) insensitive to temperature (athermal)

2. Non glissile (most of cases): migration by random jump of individual atoms

across the interface (similar to high angle grain boundary migration)


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5.3 Precipitate Growth (3.5 Interface Migration)

A. Heterogeneous Transformation

Classifying nucleation and growth transformation (=heterogeneous transformation)

Transformation by the migration of a glissile interface

→ Military transformation

Uncoordinated transfer of atoms across non-glissile interface

→ Civilian transformation

Military transformation

The nearest neighbors of any atom are unchanged.

The parent product phases – the same composition, no diffusion involved

(martensite transformation , mechanical twins)


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Civilian transformation: Diffusion of components between parent and products.

Interface controlled: if no comp. change (α → γ in Fe), the new phase grows as

fast as the atoms can cross the interface. (diffusion fast/interface reaction slow)

Diffusion controlled: if diffusion component growth will need 1-range diffusion.

if interfacial reaction is fast (easy transfer across the interface), the growth of

product (β) is controlled by diffusion of B and A

(diffusion slow/interface reaction fast)


If both process (diffusion and interface rxn rate): a

similar rate → mixed control

Non glissile interface includes s/l, s/v, s/s interfaces

(coh, incoh, semicoh)

B. Homogeneous Transformation

Spinodal decomposition, ordering transformation


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3.5.1 Diffusion-controlled and interface-controlled growth

β ppt(almost pure B) grows behind a planar

interface into A-rich α of X0 composition.

① α near interface: Xi < X0 (bulk conc.)

② β growth requires ∆𝜇𝐵(>0) driving force

∵ the origin of the driving force for growth

→ Xi > Xe

With net flux of B, the interface velocity

𝒗 = 𝑴 ∙ 𝑭 = 𝑴 ∙∆𝝁𝑩

𝒊

𝑽𝒎

𝑴: interface mobility

𝑽𝒎: molar volume of B

The corresponding flux across the interface


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𝑪𝑩 =𝑿𝑩

𝑽𝒎Based on the conc. grad. in the α phase.

A flux of B atoms towards the interface =

𝑱𝑩𝜶 = 𝑪𝑩𝒗𝒊 = −𝑴∆𝝁𝑩

𝒊 𝑿𝑩/𝑽𝒎𝟐 [moles of B/m2 sec]

𝑱𝑩𝜶 = −𝑫

𝝏𝑪𝑩𝝏𝒙 𝒊𝒏𝒕𝒆𝒓𝒇𝒂𝒄𝒆

At a s. state, those equations must be balanced.

𝑱𝑩𝒊 = 𝑱𝑩

𝜶

① If M (interface mobility) is very high, (an incoherent interface), ∆𝜇𝐵𝑖 ↓

𝑿𝒊 = 𝑿𝒆 Local equilibrium

The interface moves as fast as diffusion allows → diffusion controlled

Growth rate can be expressed as a function of time by solving𝑪𝑩 =

𝑿𝑩

𝑽𝒎

𝑱𝑩𝜶 = −𝑫

𝝏𝑪𝑩𝝏𝒙 𝒊𝒏𝒕𝒆𝒓𝒇𝒂𝒄𝒆

With boundary condition 𝑿𝒊 = 𝑿𝒆 𝑿𝑩(∞) = 𝑿𝟎


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② If the mobility of the interface is low, it needs a chemical potential gradient

(∆𝜇𝐵𝑖 ) and there will be a departure from local equilibrium at the interface.

𝑋𝑖 will satisfy 𝐽𝐵𝑖 = 𝐽𝐵

𝛼 Then the interface will migrate under mixed control

③ In the limit of a very low mobility, 𝑋𝑖 = 𝑋𝑒, 𝜕𝐶

𝜕𝑥 𝑖𝑛𝑡 = 0 : interface controlled

- In a dilute or ideal solution, the driving force ∆𝜇𝐵𝑖 (composition vs. ∆𝜇𝐵)

𝛥𝜇𝐵𝑖 = (𝜇𝐵

𝑖 -𝜇𝐵0 ) = 𝑅𝑇𝑙𝑛

𝑋𝑖

𝑋𝑒= 𝑅𝑇 ln 1 +

𝑋𝑖−𝑋𝑒

𝑋𝑒=

𝑅𝑇

𝑋𝑒(𝑋𝑖 − 𝑋𝑒) when 𝑋𝑖 − 𝑋𝑒 ≪ 𝑋𝑒

∴ the rate of the interface that moves under interface control ∝ 𝑋𝑖 − 𝑋𝑒

𝑣 = 𝑀 ∙∆𝜇𝐵

𝑖

𝑉𝑚∝ 𝑋𝑖 − 𝑋𝑒

Xe와 X0의 차이가 속도를 결정


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In the absence of strain energy effect, the shape of ppt determined to have

minimum γ.

Ledge

mechanism

5.3.1 Growth behind planar incoherent interfaces

Normally planar interface- semi- or coh. Interface in a matrix. But after grain

boundary nucleation, planar incoherent interface possible

(the formation of incoherent nuclei on a grain boundary : a slab of β ppt)


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The growth of incoherent ppt on grain boundary

A slab of solute-rich ppt

Since incoherent diffusion controlled growth. Local

equilibrium assumed.

v = f(dC

dx) J = cv = M

𝜕μ

𝜕x= −D

𝜕𝐶

𝜕𝑥

For unit area of interface to advance

v =dx

dt=

෩𝐷

𝐶𝛽 − 𝐶𝑒∙𝑑𝐶

𝑑𝑥

𝐶𝛽 − 𝐶𝑒 ∙ 𝑑𝑥 ∙ 1 = ෩𝐷(𝑑𝐶

𝑑𝑥) ∙ 𝑑𝑡 ∙ 1


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As β grows, B has to come from a larger α region. → dc/dx

decreases with time

Simplified concentration profile

𝑑𝐶

𝑑𝑥=∆𝐶0𝐿

∆𝐶0 = 𝐶0 − 𝐶𝑒

(𝐶𝛽−𝐶𝑒)𝑥 = 𝐿 ∙ Δ𝐶0/2

𝑑𝑥

𝑑𝑡= 𝑣 =

෩𝐷(∆𝐶0)2

2(𝐶𝛽−𝐶𝑒)(𝐶𝛽−𝐶0)If 𝑉𝑚 is constant, the 𝑋 = 𝐶𝑉𝑚 with 𝐶𝛽 − 𝐶𝑒 = 𝐶𝛽 − 𝐶0

𝑑𝑥

𝑑𝑡= 𝑣 =

෩𝐷(∆𝑋0)2

2(𝑋𝛽−𝑋𝑒)2

න𝑥 𝑑𝑥 = න෩𝐷(∆𝑋0)

2

2(𝑋𝛽−𝑋𝑒)2𝑑𝑡 ∆𝑋0 = 𝑋0 − 𝑋𝑒

𝑥 =∆𝑋0

(𝑋𝛽 − 𝑋𝑒)෩𝐷𝑡 𝑣 =

∆𝑋02(𝑋𝛽 − 𝑋𝑒)

෩𝐷

𝑡and


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When diff. fields of separate ppt overlap, no valid. 𝑣 =∆𝑋0

2(𝑋𝛽−𝑋𝑒)

෩𝐷

𝑡

Growth decelerate and finally cease when the matrix conc. become Xe

The approach for planar interface: applicable to curved interfaces

→ any linear dimension of a spheroidal ppt ↑∝ 𝐷𝑡

provided all interfaces migrate under vol. diff. control

The grain boundary ppt in the form of particle

grows faster than allowed by vol. diff.

→ grain boundary fast diffusion path


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5.4 Overall Transformation Kinetics – TTT Diagram

TTT Diagram :

the fraction of Transformation (f) as a function of Time and Temperature


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Johnson-Mehl-Avrami Equation

specimenofVol

phasenewofVolf

.

.

Assumption :

Reaction produces by N + G

Nucleation occurs randomly throughout specimen

Reaction product grows radially until impingement

Define volume fraction transformed

f

t

tdτ

τ τ+dτ

specimenofvolume

dduringformed

nucleiofnumber

ttimeatmeasureddduring

nucleatedparticleoneofvol

df

.


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0

0

3 )()]([3

4

V

dNVtv

df

v : cell growth rate ( assumed const. )

N : nucleation rate ( const. )

33

33

)(3

4

)(3

4

3

4

tvV

vtrV

43

0

33

0

3

)(3

4ˆ

tvNf

dtvNfdftx

Cell volume :

→ do not consider impingement & repeated nucleation

→ only true for f ≪ 1


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edffdf )1( fe : extended volume fraction

ignored impingement + repeated nucleation edf

dff 1

43

3exp1 tvNf

)(exp1 ntkf t

1

∝t4

f

····· J-M-A Eq.

k : sensitive to temp. ( N. v )

n : 1 ~ 4

7.05.0 ntk

nkt

/15.0

7.0 4/34/15.0

9.0

vNt

For the case of 50% transform,

Exp(-0.7) = 0.5

t0.5 :

i.e.Example above.


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)'(exp' tvNN o

'')()'(exp''

vNtNtvvNdt

dNN o

tvtv

v

vNf o '1)'exp(

'

8exp1

3

3

tv'tv'

6

' 33 tv

33

3exp1 tvNf o

Other examples.

I : depends on the time

No : no. of active nucleation site/unit volume

v’: rate at which the individual sites are lost.

limiting case :

small → same as J-M eq.

large →

N quickly goes to zero.


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5.5 Precipitation in Age-Hardening Alloys

5.5.1 Precipitation in Aluminum-Copper Alloys

GP Zones


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Transition phases

Phaseand ,


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Growth of 𝜽′

in the expense of 𝜽"


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5.5.3 Quenched in Vacancies


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5.5.4 Age Hardening


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5.5.5 Spinodal Decomposition

No barrier to nucleation

𝑑2𝐺

𝑑𝑋2< 0, chemical spinodal


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Composition profiles in an alloy

quenched into the spinodal region

Composition profiles in an alloy

outside the spinodal points


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The Rate of Transformation Rate controlled by interdiffusion coefficient D

~

D22 4/

D~

)exp(

t Within the spinodal < 0 & composition fluctuation

transf. rate ↑ as λ ↓

- For the λ of the comp. fluctuation, need to take care of

Free Energy change for the decomposition 2

2

2

2

1X

dX

GdGC

1) interfacial energy 2) coherency strain energy

Interfacial energy (gradient energy )

2

XKG

Coherency strain energy

)1(/1

)(

/)/(

22

2

EEdX

da

awhere

VEXG

aXdXdaEG

mS

S

2

)(2

2 22

22

2 XVE

K

dX

GdG m

Total free energy change


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- ΔT between the coh. and incoh. Miscibility

gap , or the chemical and coh. Spinodal :

dependent of


Condition for Spinodal Decomposition−𝑑2𝐺

𝑑𝑋2>2𝐾

𝜆2+ 2𝜂2𝐸′𝑉𝑚

The limit for the decomposition mVEdX

Gd 2

2

2

2 Coherent Spinodal

For coherent Spinodal 𝜆2 > −2𝐾/(𝑑2𝐺

𝑑𝑋2+ 2𝜂2𝐸′𝑉𝑚)

The min. possible wavelength ↓ with ΔT↑ below the coh. spinodal

0 SV GG

- Between incoh. & coh. miscibility gap,

- Large atomic size diff. → large


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5.5.6 Particle Coarsening

tkrr 3

0

3

eXDk where

ro : mean radius at time t=0


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)exp(0RT

QDD )exp(0

kT

QXX e , ∴

dt

rd↑ rapidly with temp.

2r

k

dt

rd

Meaning : distribution of small ppts coarsen most rapidly.

Rate of coarsening


chapter 5. diffusional transformation in solids

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