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Chapter 5.
Diffusional Transformation in solids
Young-Chang Joo
Nano Flexible Device Materials Lab Seoul National University
Phase Transformation In Materials
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
Phase Transformation In Materials
(a) Precipitation
(b) Eutectoid Transformation
𝛼
′→ 𝛼 + 𝛽
Supersaturated solid solution Precipitates
𝛾 → 𝛼 + 𝛽
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:
S
V
A V G
G V
G
GV
V 1) Volume Free Energy
GS
V 3) Misfit Strain Energy
Phase Transformation In Materials
2) Interface Energy
A
𝑖
𝛾𝑖𝐴𝑖 (𝑐𝑜ℎ, 𝑖𝑛𝑐𝑜ℎ, 𝑠𝑒𝑚𝑖 … )
𝑇
1→ 𝑇
2𝛼
′→ 𝛼 + 𝛽
5.1 Homogeneous Nucleation in Solids
For spherical nucleation
3( ) 4 2
3
4 r G G r
G V S
) (
* 2
S
V G
r G
2 3
) (
3
* 16
S
V G
G G
Misfit reduced the driving force of the transformation
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5.1 Homogeneous Nucleation in Solids
) /
* (
exp
* C0 G kT
C
The concentration of critical size nuclei , 𝑪
∗(Co : # of atoms/unit volume)
Nucleation Rate
Phase Transformation In Materials
How frequently a critical nucleus can receive an atom ∝ 𝜔 · exp −∆𝐺𝑚
𝑅𝑇
Strong temp. dependency
2 3
) (
3
* 16
S
V G
G G
kT G kT
C G N
C f N
m *
exp exp
*
0 hom
hom
Driving force for ppt
𝑁ℎ𝑜𝑚 = 𝑓0𝐶0exp −∆𝐺ℎ𝑜𝑚∗ 𝑘𝑇 Eq 4.12
kT G kT
C G N
C f N
m
*
exp exp
*
0 hom
hom
5.1 Homogeneous Nucleation in Solids
𝑇
1→ 𝑇
2𝛼
′→ 𝛼 + 𝛽
′⇒ Driving force for nucleation
∴ At the initial stage of nucleation, composition 𝛼 does not change
𝑋 constant Total driving force for transformation
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5.1 Homogeneous Nucleation in Solids
Total free energy decrease per mole of nuclei
Phase Transformation In Materials 1
2 G
G
Gn
e V X where X X X
G
0
Driving Force for Nucleation
removed mol
per X
X
G A A B B
1 (𝑝𝑒𝑟 𝑚𝑜𝑙 𝛽 𝑟𝑒𝑚𝑜𝑣𝑒𝑑)
formed mol
per X
X
G A A B B
2 (𝑝𝑒𝑟 𝑚𝑜𝑙 𝛽 𝑓𝑜𝑟𝑚𝑒𝑑)
Volume free energy decrease associated with Nucleation
of volume unit
V per G G
m n V
(𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝛽 ) For dilute solution,
• The driving force for precipitation increases with increasing undercooling
5.1 Homogeneous Nucleation in Solids
(∝ ∆𝑇)
2 3
) (
3
* 16
S
V G
G G
Temperature dependence of Nucleation
kT G kT
C G N
C f N
m *
exp exp
*
0 hom
hom
of volume unit
V per G G
m n V
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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5.1 Homogeneous Nucleation in Solids
Phase Transformation In Materials
Effect of Alloy Composition
Dilute alloy has lower nucleation rate
5.2 Heterogeneous Nucleation
d S
V
het V G G A G
G
( )
/ 2
cos
A
A G
V
G
V
G
Vr * 2
/
)
* (
*
*
*
hom hom
V S
V G
G het het
) 2
cos 1
( ) cos 2
2 ( ) 1
(
S
Nucleation on Grain Boundaries
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5.2 Heterogeneous Nucleation
Low Energy Interface
Phase Transformation In Materials
5.2 Heterogeneous Nucleation
*
G
1 3 1
exp *
exp
nuclei m s
kT G kT
C G
Nhet m
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
G G
C C N
Nhet * *het
exp hom
0 1 hom
5 0
1 10
D C
C
for grain boundary nucleation
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5.2 Heterogeneous Nucleation
Phase Transformation In Materials
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)
Phase Transformation In Materials
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)
5.3 Precipitate Growth (3.5 Interface Migration)
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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5.3 Precipitate Growth (3.5 Interface Migration)
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 ∙ ∆𝝁
𝑩𝒊𝑽
𝒎𝑴: interface mobility 𝑽�𝒎: molar volume of B
The corresponding flux across the interface
Phase Transformation In Materials
5.3 Precipitate Growth (3.5 Interface Migration)
𝑪
𝑩= 𝑿
𝑩𝑽
𝒎Based on the conc. grad. in the α phase.
A flux of B atoms towards the interface =
𝑱
𝑩𝜶= 𝑪
𝑩𝒗
𝒊= −𝑴∆𝝁
𝑩𝒊𝑿
𝑩/𝑽
𝒎𝟐 [moles of B/m2 sec]𝑱
𝑩𝜶= −𝑫 𝝏𝑪
𝑩𝝏𝒙
𝒊𝒏𝒕𝒆𝒓𝒇𝒂𝒄𝒆At a s. state, those equations must be balanced.
𝑱
𝑩𝒊= 𝑱
𝑩𝜶① If M (interface mobility) is very high, (an incoherent interface), ∆𝜇
𝐵𝑖↓ 𝑿
𝒊= 𝑿
𝒆 Local equilibriumThe interface moves as fast as diffusion allows → diffusion controlled Growth rate can be expressed as a function of time by solving
𝑪𝑩=𝑿𝑩 𝑽𝒎
𝑱𝑩𝜶 = −𝑫 𝝏𝑪𝑩
𝝏𝒙 𝒊𝒏𝒕𝒆𝒓𝒇𝒂𝒄𝒆
𝑿 = 𝑿 𝑿 (∞) = 𝑿
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5.3 Precipitate Growth (3.5 Interface Migration)
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 = 𝐽𝐵𝛼 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 = (𝜇𝐵𝑖 -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
Phase Transformation In Materials
5.3 Precipitate Growth
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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5.3 Precipitate Growth
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 = 𝐷 ෩
𝐶
𝛽− 𝐶
𝑒∙ 𝑑𝐶 𝑑𝑥 𝐶
𝛽− 𝐶
𝑒∙ 𝑑𝑥 ∙ 1 = ෩ 𝐷( 𝑑𝐶
𝑑𝑥 ) ∙ 𝑑𝑡 ∙ 1
Phase Transformation In Materials
5.3 Precipitate Growth
As β grows, B has to come from a larger α region. → dc/dx decreases with time
Simplified concentration profile 𝑑𝐶
𝑑𝑥 = ∆𝐶
0𝐿 ∆𝐶
0= 𝐶
0− 𝐶
𝑒(𝐶
𝛽−𝐶
𝑒)𝑥 = 𝐿 ∙ Δ𝐶
0/2 𝑑𝑥
𝑑𝑡 = 𝑣 = 𝐷(∆𝐶 ෩
0)
22(𝐶
𝛽−𝐶
𝑒)(𝐶
𝛽−𝐶
0)
If 𝑉𝑚 is constant, the 𝑋 = 𝐶𝑉𝑚 with 𝐶𝛽 − 𝐶𝑒 = 𝐶𝛽 − 𝐶0𝑑𝑥
𝑑𝑡 = 𝑣 = 𝐷(∆𝑋 ෩
0)
22(𝑋
𝛽−𝑋
𝑒)
2න 𝑥 𝑑𝑥 = න 𝐷(∆𝑋 ෩
0)
22(𝑋
𝛽−𝑋
𝑒)
2𝑑𝑡 ∆𝑋
0= 𝑋
0− 𝑋
𝑒𝑥 = ∆𝑋
0(𝑋
𝛽− 𝑋
𝑒) 𝐷𝑡 ෩ 𝑣 = ∆𝑋
02(𝑋
𝛽− 𝑋
𝑒)
𝐷 ෩
and
𝑡
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5.3 Precipitate Growth
When diff. fields of separate ppt overlap, no valid. 𝑣 =
∆𝑋02(𝑋𝛽−𝑋𝑒) 𝐷෩
𝑡
Growth decelerate and finally cease when the matrix conc. become X
e 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
Phase Transformation In Materials
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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5.4 Overall Transformation Kinetics – TTT Diagram
Johnson-Mehl-Avrami Equation
specimen of
Vol
phase new
of f Vol
.
.
Assumption :
Reaction produces by N + G
Nucleation occurs randomly throughout specimen
Reaction product grows radially until impingement
Define volume fraction transformed
f
t
dτ t
τ τ+dτ
specimen of
volume
d during formed
nuclei of
number t
time at
measured d
during
nucleated particle
one of
vol df
.
Phase Transformation In Materials
5.4 Overall Transformation Kinetics – TTT Diagram
0
0
3 ( )
)]
( 3 [
4
V
d NV t
v df
v : cell growth rate ( assumed const. )
N : nucleation rate ( const. )
3 3
3 3
) 3 (
4
) 3 (
4 3
4
t v V
vt r
V
4 3
0
3 3
0
3
) 3 (
ˆ 4
t v N f
d t
v N f
d
f x t
Cell volume :
→ do not consider impingement & repeated nucleation
→ only true for f ≪ 1
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5.4 Overall Transformation Kinetics – TTT Diagram
dfe
f df (1 )
f
e: extended volume fraction
ignored impingement + repeated nucleation
dfef df 1
3 4
exp 3
1 N v t
f
) (
exp
1 kt n
f t
1
∝t4 f
··· J-M-A Eq.
k : sensitive to temp. ( N. v ) n : 1 ~ 4
7 .
5
0
. 0n
t
k
k
nt
0.5 0
1.
/7
0.5 1/4 3/49 . 0
v t N
For the case of 50% transform,
Exp(-0.7) = 0.5
t
0.5:
i.e.
Example above.Phase Transformation In Materials
5.4 Overall Transformation Kinetics – TTT Diagram
) ' ( exp
' N v t
N o
' ' )
( )
' ( exp ' '
v N t
N t
v v
dt N
N dN o
v t vt
v v
f No exp( ' ) 1 '
' exp 8
1 3
3
t v'
t
v'
6 '3t3 v
3 3
exp 3
1 N v t
f o
Other examples.
I : depends on the time
N
o: 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
Phase Transformation In Materials
5.5 Precipitation in Age-Hardening Alloys
Transition phases
Phase and
,
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5.5 Precipitation in Age-Hardening Alloys
Phase Transformation In Materials
5.5 Precipitation in Age-Hardening Alloys
Growth of 𝜽′ in the expense of 𝜽"
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5.5 Precipitation in Age-Hardening Alloys
5.5.3 Quenched in Vacancies
Phase Transformation In Materials
5.5 Precipitation in Age-Hardening Alloys
5.5.4 Age Hardening
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5.5 Precipitation in Age-Hardening Alloys
5.5.5 Spinodal Decomposition
No barrier to nucleation
𝑑2𝐺𝑑𝑋2
< 0
, chemical spinodalPhase Transformation In Materials
5.5 Precipitation in Age-Hardening Alloys
Composition profiles in an alloy quenched into the spinodal region
Composition profiles in an alloy outside the spinodal points
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5.5 Precipitation in Age-Hardening Alloys
The Rate of Transformation
Rate controlled by interdiffusion coefficient
D~2
D
2
/ 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
22
1 X
dX G GC d
1) interfacial energy 2) coherency strain energy
Interfacial energy (gradient energy )
2
K X G Coherency strain energy
) 1 ( 1 /
) (
/ )
/ (
2 2
2
E dX E
da where a
V E X G
a X dX
da E
G
m S
S
2 ) 2 (
2 2 2
2 2
2 X
V K E
dX G
G d m
Total free energy change
Phase Transformation In Materials
- ΔT between the coh. and incoh. Miscibility gap , or the chemical and coh. Spinodal : dependent of
5.5 Precipitation in Age-Hardening Alloys
Condition for Spinodal Decomposition
− 𝑑
2𝐺
𝑑𝑋
2> 2𝐾
𝜆
2+ 2𝜂
2𝐸
′𝑉
𝑚The limit for the decomposition E Vm
dX G
d2 2 22 Coherent Spinodal For coherent Spinodal
𝜆
2> −2𝐾/( 𝑑
2𝐺
𝑑𝑋
2+ 2𝜂
2𝐸
′𝑉
𝑚)
The min. possible wavelength ↓ with ΔT↑ below the coh. spinodal
0
GV GS
- Between incoh. & coh. miscibility gap,
- Large atomic size diff. → large
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5.5 Precipitation in Age-Hardening Alloys
5.5.6 Particle Coarsening
r
3 r
03 k t
X
eD k
where
ro : mean radius at time t=0
Phase Transformation In Materials
5.5 Precipitation in Age-Hardening Alloys
)
0 exp(
RT D Q
D 0 exp( )
kT X Q
Xe
, ∴
dt r
d ↑ rapidly with temp.
r
2k dt
r d
Meaning : distribution of small ppts coarsen most rapidly.
Rate of coarsening