Nano & Flexible Device Materials Lab. 1 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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Chapter 5.
Diffusional Transformation in
solids
Young-Chang Joo
Nano Flexible Device Materials Lab
Seoul National University
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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
β Long range diffusion is required
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(a) Precipitation
(b) Eutectoid Transformation
πΌβ² β πΌ + π½
Supersaturated solid solution
Precipitates
πΎ β πΌ + π½
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Categories of Diffusion Phase Transformations
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(c) Order-Disorder Transformation
(d) Massive Transformation
(e) Polymorphic Transformation
β without any composition change or long-range diffusion
πΌ(πππ πππππππ) β πΌβ²(πππππππ)
π½ β πΌ
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
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A2) Interface Energy
π
πΎππ΄π (ππβ, ππππβ, π πππ β¦ )
π1 β π2 πΌβ² β πΌ + π½
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5.1 Homogeneous Nucleation in Solids
For spherical nucleation
23 4)(3
4rGGrG SV
)(
2*
SV GGr
2
3
)(3
16*
SV GGG
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Misfit reduced the driving force of the transformation
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5.1 Homogeneous Nucleation in Solids
)/*(exp* 0 kTGCC
The concentration of critical size nuclei , πͺβ
(Co : # of atoms/unit volume)
Nucleation Rate
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How frequently a critical nucleus can receive an atom β π Β· exp ββπΊπ
π π
Strong temp. dependency
2
3
)(3
16*
SV GGG
kT
G
kT
GCN
CfN
m *expexp
*
0hom
hom
Driving force for ppt
πβππ = π0πΆ0exp ββπΊβππ
β
ππ
Eq 4.12
kT
G
kT
GCN
CfN
m *expexp
*
0hom
hom
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5.1 Homogeneous Nucleation in Solids
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π1 β π2 πΌβ² β πΌ + π½β²
β Driving force for nucleation
β΄ At the initial stage of nucleation, composition πΌ does not change π0 constant
Total driving force for transformation
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5.1 Homogeneous Nucleation in Solids
Total free energy decrease per mole of nuclei
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12 GGGn
eV XXXwhereXG 0
Driving Force for Nucleation
removedmolperXXG BBAA 1(πππ πππ π½ πππππ£ππ)
formedmolperXXG BBAA 2(πππ πππ π½ ππππππ)
Volume free energy decrease associated
with Nucleation
ofvolumeunitperV
GG
m
nV
(πππ π’πππ‘ π£πππ’ππ ππ π½ )
For dilute solution,
β’ The driving force for precipitation increases with increasing undercooling
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5.1 Homogeneous Nucleation in Solids
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(β βπ)
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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5.1 Homogeneous Nucleation in Solids
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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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5.2 Heterogeneous Nucleation
Low Energy Interface
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5.2 Heterogeneous Nucleation
*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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5.2 Heterogeneous 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.
Ξ² forms at a certain sites within Ξ± (parent) during nucleation (interface created)
then the Ξ±/Ξ² interface βmigrateβ into the parent phase during growth.
Types of interfaces
1. Glissile: by γ glide β results in the shearing of parent lattice into the
product (Ξ²), 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
β Military transformation
Uncoordinated transfer of atoms across non-glissile interface
β Civilian transformation
Military transformation
The nearest neighbors of any atom are unchanged.
The parent product phases β 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 (Ξ± β Ξ³ 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 (Ξ²) 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 β 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
Ξ² ppt(almost pure B) grows behind a planar
interface into A-rich Ξ± of X0 composition.
β Ξ± near interface: Xi < X0 (bulk conc.)
β‘ Ξ² growth requires βππ΅(>0) driving force
β΅ the origin of the driving force for growth
β Xi > Xe
With net flux of B, the interface velocity
π = π΄ β π = π΄ ββππ©
π
π½π
π΄: interface mobility
π½π: molar volume of B
The corresponding flux across the interface
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5.3 Precipitate Growth (3.5 Interface Migration)
πͺπ© =πΏπ©
π½πBased on the conc. grad. in the Ξ± phase.
A flux of B atoms towards the interface =
π±π©πΆ = πͺπ©ππ = βπ΄βππ©
π πΏπ©/π½ππ [moles of B/m2 sec]
π±π©πΆ = βπ«
ππͺπ©ππ πππππππππ
At a s. state, those equations must be balanced.
π±π©π = π±π©
πΆ
β If M (interface mobility) is very high, (an incoherent interface), βππ΅π β
πΏπ = πΏπ Local equilibrium
The interface moves as fast as diffusion allows β diffusion controlled
Growth rate can be expressed as a function of time by solvingπͺπ© =
πΏπ©
π½π
π±π©πΆ = βπ«
ππͺπ©ππ πππππππππ
With boundary condition πΏπ = πΏπ πΏπ©(β) = πΏπ
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5.3 Precipitate Growth (3.5 Interface Migration)
β‘ If the mobility of the interface is low, it needs a chemical potential gradient
(βππ΅π ) and there will be a departure from local equilibrium at the interface.
ππ will satisfy π½π΅π = π½π΅
πΌ Then the interface will migrate under mixed control
β’ In the limit of a very low mobility, ππ = ππ, ππΆ
ππ₯ πππ‘ = 0 : interface controlled
- In a dilute or ideal solution, the driving force βππ΅π (composition vs. βππ΅)
π₯ππ΅π = (ππ΅
π -ππ΅0 ) = π πππ
ππ
ππ= π π ln 1 +
ππβππ
ππ=
π π
ππ(ππ β ππ) when ππ β ππ βͺ ππ
β΄ the rate of the interface that moves under interface control β ππ β ππ
π£ = π ββππ΅
π
ππβ ππ β ππ
Xeμ X0μ μ°¨μ΄κ° μλλ₯Ό κ²°μ
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5.3 Precipitate Growth
In the absence of strain energy effect, the shape of ppt determined to have
minimum Ξ³.
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 Ξ² 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
πΞΌ
πx= βD
ππΆ
ππ₯
For unit area of interface to advance
v =dx
dt=
ΰ·©π·
πΆπ½ β πΆπβππΆ
ππ₯
πΆπ½ β πΆπ β ππ₯ β 1 = ΰ·©π·(ππΆ
ππ₯) β ππ‘ β 1
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5.3 Precipitate Growth
As Ξ² grows, B has to come from a larger Ξ± region. β dc/dx
decreases with time
Simplified concentration profile
ππΆ
ππ₯=βπΆ0πΏ
βπΆ0 = πΆ0 β πΆπ
(πΆπ½βπΆπ)π₯ = πΏ β ΞπΆ0/2
ππ₯
ππ‘= π£ =
ΰ·©π·(βπΆ0)2
2(πΆπ½βπΆπ)(πΆπ½βπΆ0)If ππ is constant, the π = πΆππ with πΆπ½ β πΆπ = πΆπ½ β πΆ0
ππ₯
ππ‘= π£ =
ΰ·©π·(βπ0)2
2(ππ½βππ)2
ΰΆ±π₯ ππ₯ = ΰΆ±ΰ·©π·(βπ0)
2
2(ππ½βππ)2ππ‘ βπ0 = π0 β ππ
π₯ =βπ0
(ππ½ β ππ)ΰ·©π·π‘ π£ =
βπ02(ππ½ β ππ)
ΰ·©π·
π‘and
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5.3 Precipitate Growth
When diff. fields of separate ppt overlap, no valid. π£ =βπ0
2(ππ½βππ)
ΰ·©π·
π‘
Growth decelerate and finally cease when the matrix conc. become Xe
The approach for planar interface: applicable to curved interfaces
β any linear dimension of a spheroidal ppt ββ π·π‘
provided all interfaces migrate under vol. diff. control
The grain boundary ppt in the form of particle
grows faster than allowed by vol. diff.
β grain boundary fast diffusion path
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5.4 Overall Transformation Kinetics β 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 β TTT Diagram
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
tdΟ
Ο Ο+dΟ
specimenofvolume
dduringformed
nucleiofnumber
ttimeatmeasureddduring
nucleatedparticleoneofvol
df
.
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5.4 Overall Transformation Kinetics β TTT Diagram
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
4Λ
tvNf
dtvNfdftx
Cell volume :
β do not consider impingement & repeated nucleation
β only true for f βͺ 1
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5.4 Overall Transformation Kinetics β TTT Diagram
edffdf )1( fe : extended volume fraction
ignored impingement + repeated nucleation edf
dff 1
43
3exp1 tvNf
)(exp1 ntkf t
1
βt4
f
Β·Β·Β·Β·Β· 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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5.4 Overall Transformation Kinetics β TTT Diagram
)'(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
vβ: rate at which the individual sites are lost.
limiting case :
small β same as J-M eq.
large β
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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5.5 Precipitation in Age-Hardening Alloys
Transition phases
Phaseand ,
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5.5 Precipitation in Age-Hardening Alloys
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5.5 Precipitation in Age-Hardening Alloys
Growth of π½β²
in the expense of π½"
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5.5 Precipitation in Age-Hardening Alloys
5.5.3 Quenched in Vacancies
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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
π2πΊ
ππ2< 0, chemical spinodal
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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
~
D22 4/
D~
)exp(
t Within the spinodal < 0 & composition fluctuation
transf. rate β as Ξ» β
- For the Ξ» 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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- ΞT 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 Decompositionβπ2πΊ
ππ2>2πΎ
π2+ 2π2πΈβ²ππ
The limit for the decomposition mVEdX
Gd 2
2
2
2 Coherent Spinodal
For coherent Spinodal π2 > β2πΎ/(π2πΊ
ππ2+ 2π2πΈβ²ππ)
The min. possible wavelength β with ΞTβ below the coh. spinodal
0 SV GG
- Between incoh. & coh. miscibility gap,
- Large atomic size diff. β large
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5.5 Precipitation in Age-Hardening Alloys
5.5.6 Particle Coarsening
tkrr 3
0
3
eXDk where
ro : mean radius at time t=0
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5.5 Precipitation in Age-Hardening Alloys
)exp(0RT
QDD )exp(0
kT
QXX e , β΄
dt
rdβ rapidly with temp.
2r
k
dt
rd
Meaning : distribution of small ppts coarsen most rapidly.
Rate of coarsening
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