3. Phase transformations in solids
Mostly occurred by thermally activated
atomic movements
Two behaviors of atomic movements
Diffusion-controlled process of atoms
(diffusional transformation)
The other, the diffusion cannot
take place because of some
restrictions such as insufficient
time for atomic diffusion,
(diffusionless transformation)
3
4. Atomic movements
Diffusion-controlled
1 interatomic spacing
Atomic movements
Diffusionless
Atomic movements are less than one
interatomic spacing
1 interatomic spacing
Atomic movements
4
6. Precipitate reaction
A metastable supersaturated solid
solution of α’ transforms to two
phases of
More stable solid solution phase
of α (same crystal structure as α’)
and
Either stable or metastable α’ → α + β
precipitate phase of β
6
8. Supersaturated solid solution
α at To : A-8% B
When reached T1
α : 100 % at A-8% B
β : 0 %
When reached T2
Att = 0, α at A-8% B becomes unstable and
supersaturated with B solute atoms
The unstable and supersaturated α is
denoted as α’. 8
9. β precipitates
Equilibrium At T22,,
metastable supersaturated
solid solution α’ phase
Transforms to more
stable α phase with
composition of A-5% B
(same crystal structure as α’ phase) and
Allows precipitates of β phase to form with
composition of A-96% B
α’ → α +β α matrix
β precipitates 9
10. β precipitates
2 approaches of nucleation of solid β
Homogeneous nucleation
B atoms diffuse to form small volume of β
composition with a critical nucleus size of
precipitates β in a matrix of α.
Heterogeneous nucleation
Nucleation sites are non-equilibrium
defects such as excess vacancies,
dislocations, grain boundaries, stacking
faults, inclusions, and free surfaces.
10
11. Homogeneous nucleation of β
α solid solution passes solvus line
B atoms in α matrix diffuse to form
a small volume with β composition
Nucleation process
B atoms rearrange themselves to
form β crystal structure
11
12. Homogeneous nucleation of β
During nucleation process
α/β interfaces create → leading to an
activation energy barrier
α’
α
12 β
13. Homogeneous nucleation of β
During
nucleation process,
3 components of ∆G
Creationof β precipitates (driving force)
- Volume free energy reduction of V∆GV
Creation of α/β interfaces
- Increase of Aγ
Misfit strain energy between
α and β
- Increase of V∆GS α
∆Ghom = (–V∆GV) + Aγ + V∆GS β
13
14. Homogeneous nucleation of β
∆Ghom = (–V∆GV) + Aγ + V∆GS
Assuming a spherical nucleus with r
V = (4/3) πr3
A = 4 πr2
Critical radius
r* = 2γ / (∆GV – ∆GS)
Necessary free energy change
∆G* = 16πγ3 / 3(∆GV – ∆GS)2
14
15. Homogeneous nucleation of β
Critical radius
r* = 2γ / (∆GV – ∆GS)
Necessary free energy change
∆G* = 16πγ3 / 3(∆GV – ∆GS)2
For a given undercooling,
If r < r*, the system will lower its free energy
by dissolving the embryos back into solid
solution.
If r > r*, the system will lower its free energy
by allowing the nuclei to grow. 15
16. Homogeneous nucleation of β
Concentration of critical-sized
nuclei per unit volume
C*
= Co exp(–∆G*/kT) cluster/m3
where Co : initial number of atoms/volume
Homogeneous nucleation rate
N = f C* nuclei/m3 s
hom
where f = ω exp (–∆Gm/kT) : how fast a
critical nucleus can receive an atom from
α matrix (atomic migration) and ∆Gm :
activation energy for atomic migration
16
17. Homogeneous nucleation of β
Homogeneous nucleation rate
Nhom = ωCo exp(–∆Gm/kT) exp(–∆G*/kT)
where ∆G* = 16πγ3 / 3(∆GV – ∆GS)2
Assumption
∆Gm is constant, and ∆GS is ignored.
Consider ∆Gm and ∆GS
17
18. Homogeneous nucleation rate
Nhom = ωCo exp(–∆Gm/kT) exp(–∆G*/kT)
where ∆G* = 16πγ3 / 3(∆GV – ∆GS)2
Main factor controlling ∆G* is the
driving force for precipitation ∆GV .
18
19. Driving force for transformation
Initial composition Xo
Solution treated at T1
Then, quench to T2
Alloy is supersaturated with B
Alloy tries to precipitate β
When α → α+β completed,
free energy decreases by ∆Go
∆Go is a driving force for transformation.
19
20. β precipitation
Initially
First nuclei of β do not
change α composition from Xo
Small amount of materials
with nucleus composition βXB
(P) is removed from α phase
Free energy of the system
decreases by ∆G1
∆G1 = αµA βXA + αµB βXB (per mol β removed)
20
21. β precipitation
Rearranged into β crystals
Freeenergy of the system
decreases by ∆G2 (Q)
∆G2 = βµA βXA + βµB βXB
(per mol β formed)
Driving force for nucleation
∆G = ∆G2 – ∆G1 (per mol β)
n
21
22. β precipitation
Volume free energy
decrease
∆G = ∆Gn/Vm
V
(per unit volume of β)
For dilute solution
∆G ∝ ∆X
V
where ∆X = Xo – Xe
Drivingforce for precipitation increases
with increasing undercooling ∆T below Te.
22
23. When consider ∆Gm and ∆GS
∆G* = 16πγ3 / 3(∆GV – ∆GS)2
Taking the misfit
strain energy term
into account, the
effective driving force
become (∆GV – ∆GS)
Equilibrium temperature reduces from Te
to Te’ (effective equilibrium temperature)
23
25. Notice
Nhom = ωCo exp(–∆Gm/kT) exp(–∆G*/kT)
exp(–∆G*/kT) term
is zero until ∆Tc
(critical undercooling)
is reached – above, β
does not form .
exp(–∆Gm/kT) term
decreases rapidly as T decreases –
atomic mobility is too small. 25
26. Undercooling ∆T
If∆T < ∆Tc ,
N is negligible
(∆GV is too small).
N is maximum at
intermediate ∆T.
If ∆T >> ∆Tc ,
N is small or negligible
(Diffusion becomes too slow).
26
27. Real homogeneous nucleation
β precipitate is not always spherical.
The most effective way of minimizing
∆G* is to form nuclei with the smallest
total interfacial energy by
Form the same orientation relationship
with the matrix
Have coherent interfaces
Example is formation of metastable GP
zones in the Al-Cu alloys.
27
28. Al-Cu alloys
The equilibrium
consists of two
solid phases : α4, θ
Precipitate process
α → α1 + GP Zone
→ α2 + θ”
→ α3 + θ’
→ α4 + θ
28
29. Heterogeneous nucleation of β
Usually precipitate in matrix α
Nucleation sites of nonequilibrium defects
Excess vacancies
Nucleus creations
Dislocations
decrease some free
Grain boundaries
energy with an amount
Stacking faults of ∆Gd
Inclusions Therefore, help reducing
Free surfaces activation energy barrier
29
31. Heterogeneous nucleation of β
At α/α grain boundary
Assumption : no misfit strain energy at a
α/β interface
Optimum embryo shape for nucleation is
when total interfacial free energy is
minimized. 2 spherical caps
At the precipitate
∆Gd = Aααγαα
r* = 2γαβ/ ∆GV
31
33. Heterogeneous nucleation of β
∆G*het/∆G *hom = ∆V*het/∆V *hom = S(θ)
∆G* and ∆V*het can be reduced further by
het
nucleation on
Grain edge
Grain corner
1.0 Grain boundaries
∆ G*het /∆ G*hom = S(θ)
Grain edges
0.8 Grain corners
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6
cos θ
0.8 1.0
33
34. Heterogeneous nucleation of β
When matrix and precipitate are
compatible and allow formation of
lower energy coherent facets, nucleus
Will form whenever possible
Will have an orientation
relationship with matrix
34
36. Heterogeneous nucleation of β
Heterogeneous nucleation rate
Nhet = ωC1 exp(–∆Gm/kT) exp(–∆G*/kT)
where C1 is concentration of heterogeneous
nucleation sites per unit volume.
The rates can be obtained at very small driving forces.
36
37. Heterogeneous nucleation of β
Heterogeneous nucleation rate
Nhet = ωC1 exp(–∆Gm/kT) exp(–∆G*/kT)
Homogeneous nucleation rate
Nhom = ωCo exp(–∆Gm/kT) exp(–∆G*/kT)
Differences in ω and ∆Gm are not significant
Nhet/Nhom = C1/Co exp{(∆G*hom–∆G*het)/kT}
where C1/Co is a ratio between the boundary thickness δ
and the grain size D
37
38. Heterogeneous nucleation of β
Nhet/Nhom = C1/Co exp{(∆G*hom–∆G*het)/kT}
Nucleation at grain boundary C1/Co
C1/Co = δ/D = 10-5
Nucleation at grain edge
C1/Co = (δ/D)2 = 10-10
Nucleation at grain corner
C1/Co = (δ/D)3 = 10-15
δ = 0.5 nm
D = 50 µm
38
40. Growth of precipitate
Solutes must migrate from deep within
the parent phase.
Solute from precipitate to
precipitate/parent interfaces
Long-range diffusion controlled
Time-dependent velocity
Interface controlled
Constant velocity
40
41. Growth of precipitate
Nuclei will grow on
Direction of smallest nucleation barrier
Smallest critical volume
Minimum total interfacial free energy
(in the case of strain-free)
41
42. Growth of precipitate
Ifnuclei are enclosed by a combination
of coherent and semicoherent facets,
Incoherent interface will have higher
mobility, so the interface can advance
faster.
42
43. Growth of planar incoherent
interfaces
Found at grain boundaries
Depletion of solute in α
from Co to Ce
Form β slab precipitate
Solute concentration is Cβ
Grow from zero thickness
at rate of v
43
44. Growth of planar incoherent
interfaces
β slab advances dx of
a unit area on i/c
Volume of β material (1·dx)
must be converted from
α (Ce) to β (Cβ)
Materials required per
unit area = 1·dx (Cβ – Ce)
Solute B atoms migrate = D (dC/dx)i/c dt
44
where D is interdiff. coef. = XBDA + XADB
45. Growth of planar incoherent
interfaces
Materials advancing=
Solute atom migration
1·dx (Cβ – Ce) = D (dC/dx)i/c dt
v = dx/dt
= [D/(Cβ – Ce)] (dC/dx)i/c
Approximation (dC/dx)i/c using
“conservation of solute”
α Area = β Area
½ L ∆Co = (Cβ – Co) x
45
46. Growth of planar incoherent
interfaces
½ L ∆Co = (Cβ – Co) x
∆Co/L = (∆Co)2/{2 x (Cβ – Co)}
∆Co/L ≈ (dC/dx)i/c
(dC/dx)i/c = (∆Co)2/{2 x (Cβ – Co)}
v = dx/dt
= [D/(Cβ – Ce)] (dC/dx)i/c
46
= [D/(C – C )] (∆C ) /{2 x (C – C )}
2
47. Growth of planar incoherent
interfaces
v = dx/dt = D(∆Co)2/{2 x (Cβ – Ce) (Cβ – Co)}
Assumption
Cβ – Ce ≈ C β – Co
Mole fraction : ∆X = ∆C
dx/dt = D(∆Xo)2/{2 x (Xβ – Xe)2}
x dx = D(∆Xo)2/{2 x (Xβ – Xe)2} dt
x2 = Dt(∆Xo)2/(Xβ – Xe)2
x = ∆Xo/(Xβ – Xe) √(Dt) → precipitate thickening
47
48. Growth of planar incoherent
interfaces
Precipitatethickening
x = ∆Xo/(Xβ – Xe) √(Dt)
x ∝ √(Dt)
v = dx/dt
v = ∆Xo/2(Xβ – Xe) √(D/t)
Supersaturation ∆Xo before
precipitation
∆Xo = Xo – Xe
48
49. Growth of planar incoherent
interfaces
x = ∆Xo/(Xβ – Xe) √(Dt)
v = ∆Xo/2(Xβ – Xe) √(D/t)
∆Xo = Xo – Xe
Growth rate is low when
Small ∆T → small ∆X
Large ∆ T → small diffusion 49
50. A sink for solute
Grain boundaries can act as a collector
plate of a sink for solute.
Volume diffusion of solute to grain boundary
Diffusion along the grain boundaries
Diffusion along the α/β interfaces
50
51. End of precipitate growth
Precipitates
stop advancing/growing
when the matrix composition reaches Xe
everywhere – there are no longer excess
solute supply for precipitation.
51
52. Growth of plates and needles
β precipitates have a constant thickness
and a cylindrically curved incoherent
edge of radius r
Growth governed by volume diffusion-
controlled process
52
53. Growth of plates and needles
Concentration gradient to drive
diffusion through the edge is ∆C/L
where L = kr, and k is a constant.
v = D∆C/{kr(Cβ – Cr)}
= D∆X/{kr(Xβ – Xr)}
where ∆X = Xo – Xr = ∆Xo(1-r*/r) and
∆Xo = Xo – Xe
v = D ∆Xo (1-r*/r) / {kr (Xβ – Xr)}
For spherical tip, X = X (1+2γV /RTr)
r e m
For cylindrical tip, X = X (1+γV /RTr)
53
r e m
54. Growth of plates and needles
For a constant thickness,
Lengthening rate v is constant with time;
therefore, x ∝ t.
Lengthening rate v is varied with D and r.
54
55. Plate-like precipitate
Observed by a ledge mechanism
Broad faces are semicoherent
Limit migration of atoms
Atoms will migrate and attach at the ledges
Their interfaces are incoherent.
Growth requires lateral motion of ledges
achieved by diffusion
v = uh/λ
= D∆Xo/{kλ(Xβ – Xe)}
= constant 55
57. Spinodal structure
Homogenous precipitates
of 2-phase mixtures
resulting from a phase
separation that occurs
under certain conditions
of temperature and
composition
57
58. Spinodal structure
Xois solution treated at To
Then, aged at TA
α tends to separate into 2-
o
phase mixture
Initially, G
o
Xo becomes unstable and try
to decreases its total free
energy by producing small
fluctuations in composition
resulting in A-rich and B-rich
regions 58
59. Spinodal structure
Xo
is unstable and try to
decreases its total free
energy
Up-hill
diffusion
Down-hill diffusion
until equilibrium phases
of α1 and α2 are reached at
compositions of X1 and X2
59
60. Spinodal structure
α1 and
α2 phase mixture
occurs by continuous
growth of initially small
amplitude fluctuations
Controlled by atomic
migration and diffusion
60
61. Spinodal structure
Xo will decay with time
∆X = ∆Xo exp(–t/τ)
where τ is a relaxing time
τ = λ2/(4π2D)
where λ is wavelength of
fluctuation and D is diffusion
coefficient.
61
62. Spinodal structure
TEM micrographs
2.5 – 10 nm in metallic system
Contrast comes mainly from
structure factor differences
Fe-28.5 wt.% Cr-10.6 wt.% Co
Aged at 600°C 4 h
Cu-33.5 at.% Ni-15 at.% Fe
Aged at 775°C 15 min, λ ≈ 25 nm ASM Handbook
62
63. Cellular precipitate
Precipitation
of a second phase from a
supersaturated solid solution
May occur through a reaction involving the
formation of colonies
Consisting of a 2-phase mixture
That grow and consume the matrix.
The transformation is very similar to the
eutectoid reaction.
63
64. Cellular precipitate
Morphology
Alternating lamellae
of precipitate phase
and depleted matrix
Duplex cells
Cooperative growth
of 2 phases
Originate from matrix
grain boundaries
64
65. Cellular precipitate
Cellular fronts advance
into the supersaturated
matrix and spatially partitions
the structure into transformed
and untransformed regions.
Composition and
orientation of α’ phase changes discontinuously
from Cα’ to Cα for α phase colony.
65
66. Cellular precipitate
Solutes to form β phase colony
diffuse from the neighboring α
colonies with a distance d = So/2
Solution redistribution occurs
by diffusion along the interface
at the composition
distribution region
Assumed
Solute distribution is
linear
66
67. Cellular precipitate
Amount of solute rejected from α’ to form β
plates with the rate of dm/dt
dm/dt = Jdiff A = R(Cβ – Cα’)
where R is the interface velocity
Jdiff = DB(∂C/∂x) and A = λ (2 sides) = 2λ
dm/dt = DB(∂C/∂x) 2λ = R(Cβ – Cα’)
DB{(Cα’ – Cα)/d} 2λ = R(Cβ – Cα’)
R = {2 DB λ/d} {(Cα’ – Cα)/(Cβ – Cα’)}
where d = (So – Sβ)/2, for small distance 67 = SoSβ/2
d
68. Cellular precipitate
Cellularor discontinuous
precipitation growing out
uniformly from the grain
boundaries
Fe-24.8Zn alloy
Aged at 600°C 6 min
(W.C. Leslie)
68
