Diffusional Transformations in Solids
Solid-state transformations taking place by thermally-activated
atomic movements - most common type of solid-state phase
transformations
Typical classification:
Precipitation Transformations: Generally expressed as α’→ α + β where α’ is a metastable supersaturated solid solution
β is a stable or metastable precipitate
α is a more stable solid solution with the same crystal structure as α’
but composition closer to equilibrium
Eutectoid Transformations: Generally expressed as γ→ α + β
Metastable phase (γ) replaced by a more stable mixture of α + β
Precipitation and eutectoid transformations require compositional
changes in the formation of the product phase and consequently
require long-range diffusion
Remaining three reactions do not require long-range diffusion.
Ordering Transformations: Generally expressed as α (disordered) →
α’ (ordered)
Massive Tranformations: Generally expressed as β→ α
Original phase decomposes into one or more new phases which have
the same composition as the parent phase but different crystal
structures
Polymorphic Transformations: Typically exhibited by single
component systems where different crystal structures are stable over
different temperature ranges. E.g. bcc-fcc transformation in Fe
1
Phase Transformations
Precipitation
Eutectoid
Ordering
Massive
Polymorphic
2
Phase Transformations
Homogeneous nucleation in solids
Consider the precipitation of B-rich β in an A-rich α matrix
B atoms have to cluster together in the α matrix to form a region of
composition close to β and then further atomic re-arrangements
might be required for the crystal structure to change to that of β.
Creation of α/β interface
Free energy change associated with nucleation has 3 contributions:
ΔG = −VΔGV + Aγ + VΔGS
Interfacial Strain
Volume free
energy reduction
energy
energy
€
Origin of strain energy: transformed volume will not fit perfectly in
the gap in the matrix - gives rise to misfit strain
Also, the interfacial energy is not isotropic anymore (unlike solid/
liquid interfaces) but varies with orientation - therefore summation
over all interfaces is required
For simplicity, assuming a spherical nucleus,
4
ΔG = − πr 3 ( ΔGV − ΔGS ) + 4πr 2γ
3
2γ
r* =
( ΔGV − ΔGS )
ΔG* =
16πγ 3
3( ΔGV − ΔGS )
2
€
3
Phase Transformations
Concentration of critical sized nucleus,
 −ΔG * 
f is the frequency with which
C* = C0 exp

atoms attach to critical sized
 kT 
Nhom = fC *
 −ΔGm 
 −ΔG * 
Nhom = ωC0 exp
 exp

 kT 
 kT 
€
nucleus
ΔGm is the activation energy
for atomic migration
ω related to vibration
frequency of atoms
Consider alloy of
composition X0,
quenched from T1
to T2.
Supersaturation in
α matrix
ΔG0 is the driving
force for the
transformation
from α to α + β
Not the driving
force for nucleation
- initial nucleus that
forms does not
change the α matrix
composition much
from X0
Free energy per mole of material of β nucleus composition but with α
α β
α β
structure
G1 = µ A X A + µ B X B
Free energy per mole of material of β nucleus composition with β
β β
β β
structure
G2 = µ A X A + µ B X B
Net driving force for nucleation,
ΔGn = G2 − G1
4
€
Phase Transformations
€
€
Volume free energy decrease associated with nucleation,
ΔGn
ΔGV =
Vm
For dilute solutions,
ΔGV ∝ ΔX
ΔX = X0 − Xe
€
Nhom can be evaluated as a function of temperature
€
ΔGV-ΔGS becomes effective
driving force and Te’
becomes effective
equilibrium temp.
Driving force increases with
decreasing T
ΔG* decreases with
decreasing T
As undercooling increases,
the concentration of nuclei
increases but the atomic
mobility (diffusivity)
decreases
No detectable nucleation
below a certain critical
undercooling
Highest nucleation rate at
intermediate undercoolings
Phase Transformations
€
 −ΔGm 
 −ΔG * 
Nhom = ωC0 exp
 exp

 kT 
 kT 
5
Influence of alloy composition on Nhom
Alloys containing less solute - lower temperature required in order to
reach critical undercooling - lower diffusivity - less nucleation rate
Assumptions of the above analysis:
Constant nucleation rate at a certain temp. - not in reality
Initially nucleation rate is low, then increases, and then finally
decreases when the existing nuclei grow and consume substantial
supersaturation of the matrix
Spherical nuclei - not in reality - nucleus shape determined by the
interfacial energy and the variations in this energy for different types
of interfaces - coherent (good atomic matching across interface) vs.
incoherent (poor atomic matching)
Homogeneous nucleation practically impossible for incoherent high
energy interfaces
Therefore, most real systems with different crystal structures for α
and β phases do not exhibit homogeneous nucleation of β in α but
rather homogeneous nucleation of some metastable phase, closely
related to the α structure
Examples of homogeneous nucleation:
1. Cu + 1-3at% Co alloys - Cu and Co exhibit fcc structure with a 2%
mismatch in lattice parameters - γ ~ 200 mJ/m2, critical undercooling
for measurable homogeneous nucleation is ~ 40°C
2. Ni-Al system - precipitation of Ni3Al in Ni(Al) matrix, γ ~ 30 mJ/m2
6
Phase Transformations
Heterogeneous nucleation in solids
Typical heterogeneous sites are excess vacancies, dislocations, grain
boundaries, stacking faults, inclusions, free surfaces
Nucleation on defects will remove the defect - release the defect
energy (ΔGd)
ΔG = −V ( ΔGV − ΔGS ) + Aγ + ΔGd
€
Nucleation on grain boundaries
The nucleus shape will be the one that minimizes the interfacial
energy (ignoring misfit strain energy effects)
For incoherent grain boundary nucleus, this shape will be two
spherical caps joined at the grain boundary
cosθ =
γ αα
2γ αβ
ΔG = −VΔGV + Aαβ γ αβ − Aαα γ αα
*
ΔGhet
= S(θ )
*
ΔGhom
1
2
S(θ ) = ( 2 + cosθ )(1 − cosθ )
2
€
cosθ determines the effectiveness of the grain boundary nucleation
site for heterogeneous nucleation
Also, compared to grain boundary, grain edges and grain corners can
often act as more effective nucleation sites - Why?
7
Phase Transformations
High-angle grain boundaries are very effective nucleation sites - high
γαα
If the β precipitate can form low energy facets with the α matrix, on
one side of the grain boundary, then the critical nucleus size can be
further reduced
8
Phase Transformations
Nucleation on dislocations
Lattice distortion in the vicinity of dislocations assists in nucleation
Reduces ΔGs contribution to ΔG* - reduced strain energy of the
nucleus
e.g. coherent nucleus with a negative misfit - smaller volume than
matrix
favorable to form in a region of compressive stress - region of edge
dislocation with extra half plane
Dislocations can also assist in nucleation by acting as diffusion pipes short-circuit paths for diffusion - lower ΔGm
Dislocations are not very effective in reducing interfacial energy nucleation on dislocations usually requires formation of a low energy
coherent or semi-coherent interface can be formed between
precipitate and matrix (good matching on at least one plane between
ppt. and matrix)
Dislocation densities in annealed samples could still be ~ 1 µm-2 - can
lead to precipitate distribution as shown below
9
Phase Transformations
Nucleation on excess vacancies
Alloy quenched from a high temperature - excess vacancies are
retained (trapped)
Excess vacancies aid in nucleation by increasing diffusion rates
relieving misfit strain energy
These vacancies can either influence as individual vacancies or by
clustering
ΔGd for vacancies is relatively small - therefore are effective only
when a combination of conditions are met: low interfacial energy,
small volume strain energy, high driving force (similar to
homogeneous nucleation criteria)
Rate of Heterogeneous Nucleation
ΔGd in increasing order:
1. Homogeneous sites
2. Vacancies
3. Dislocations
4. Stacking faults
5. Grain boundaries / Interphase boundaries
6. Free surfaces
While nucleation will definitely be easier for sites at the bottom of list,
the relative concentration of these sites plays an important role in
determining their effectiveness as nucleation sites
% − ΔGm (
% − ΔG * (
N het = ωC1 exp'
* exp'
*
& kT )
& kT )
C1 is the concentration of the heterogeneous sites
€
10
Phase Transformations
Ratio of heterogeneous and homogeneous nucleation rates,
$ ΔG* − ΔG* '
N het C1
het
=
exp& hom
)
N hom C0
kT
%
(
€
€
Usually ΔG* is much smaller for heterogeneous vs. homogeneous, so
the exponential factor will be a large positive quantity
But the ratio of C1/C0 is also important
e.g. for grain boundary nucleation,
C1 δ
=
where δ is the grain boundary thickness and D is grain size
C0 D
For 50 µm grain size, taking δ = 0.5 nm, δ/D = 10-5
Site which gives highest volume nucleation rate depends on the
driving force (ΔGV)
At small undercoolings, low driving force, grain corner nucleation
favored - highest nucleation rate
As undercooling increases, other competing nucleation sites become
effective, grain edges, boundaries, .etc. - this is the case for isothermal
transformations
Continuous cooling situation changes - ΔGV is continuously changing
So nucleation starts at sites with lowest ΔG* and then as cooling
continues, other sites become more effective
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Phase Transformations
Precipitate Growth
Critical nucleus that forms is the one with the smallest nucleation
barrier - smallest critical volume
Ignoring strain energy effects the shape adopted is the one with
minimum interfacial energy
Usually nuclei will be bounded by a combination of coherent, semicoherent and incoherent interfaces
Coherent and semi-coherent interfaces - specific crystallographic
relationship - usually planar faceted interfaces
Incoherent interfaces - smoothly curved
Migration of semi-coherent interfaces - occurs by a ledge mechanism slow
Incoherent interfaces are highly mobile
Leads to a thin disc or plate shaped morphology - origin of the
Widmanstätten morphology
Growth of planar incoherent interfaces
Usually planar interfaces are not incoherent, but in some case like
nucleation of multiple precipitates at grain boundary can lead to a
continuous wetting layer at the grain boundary - planar incoherent
interface
Consider a slab of β phase growing from an α grain boundary with
an instantaneous growth rate of v
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Phase Transformations
Unit area of interface advances by distance dx - requires (Cβ - Ce)dx
moles of B to be transported to by diffusion through α
Equating this to the diffusive flux,
dx
D
dC
v=
=
.
dt Cβ − Ce dx
Simplified approach proposed by Zener - assume a linear dC/dx
€
dC ΔC0 C0 − Ce
=
=
dx
L
L
( Cβ − C0 ) x = LΔ2C0 Solute
conservation
DΔC02
v=
2( Cβ − Ce )( Cβ − C0 ) x
€
13
Phase Transformations
Integrating,
x=
v=
€
Effect of alloy composition and temperature on growth
ΔX 0
Dt
( Xβ − Xe )
ΔX 0
D
2( Xβ − Xe ) t
Important points:
x ∝ Dt
Precipitate thickening obeys parabolic growth rate
v ∝ ΔX 0
€
v∝
Growth rate is proportional to supersaturation
D
t
€
€
The growth rates predicted
above change once the
diffusion fields of separate
precipitates start
overlapping
Growth rates slow down
and continue slowing down
till the equilibrium
composition Ce is achieved
throughout the matrix
14
Phase Transformations
Equations were derived basically for planar interfaces, but situation
does not change drastically for non-planar interfaces as long as growth
is controlled by volume diffusion of solute through the matrix
It can be shown that if there is volume diffusion then any part of the
interface of a spherical precipitate grows by Dt
Growth of grain boundary precipitate usually occurs at a rate much
faster than that allowed by volume diffusion - faster short circuit
€
diffusion through the grain boundary region
Growth of grain boundary allotriomorphs involves 3 steps:
1. Volume diffusion of solute to grain boundary
2. Diffusion of solute along grain boundary
3. Diffusion along α/β interfaces to cause thickening of β precipitate
15
Phase Transformations
Diffusion-controlled lengthening of Plates or Needles
Consider β precipitate shaped as a plate of constant thickness with
cylindrically-shaped incoherent edge of radius r
Due to the curvature at the tip of precipitate, the equilibrium
composition in the matrix adjacent to the growing tip is increased to
Cr (Gibbs-Thompson)
Concentration gradient ahead of tip = ΔC/L where ΔC = C0 - Cr
Diffusion problem more complex due to radial diffusion
Lengthening rate (growth rate) is constant and x proportional to t linear growth
Equation holds for growth of plates and needles by volume diffusion
control - reasonable for curved ends of needles, but for plates the
edges are often faceted and migrate by a ledge mechanism
16
Phase Transformations
Thickening of Plate-like Precipitates
Earlier treatment for planar incoherent interfaces - not for broad
faces of plate-like precipitates - usually semi-coherent interfaces
Migrate by the lateral movement of ledges
Assume plate-like precipitate thickening by lateral movement of
linear ledges of constant spacing λ and height h
€
Half thickness of plate increases at a rate (u is rate of lateral
migration),
v = uh
λ
Ledge migration is similar to plate lengthening problem - long range
diffusion required to and from ledges
Edge of ledges are incoherent, lateral growth is governed by volume
diffusion and therefore u follows a linear growth law
17
Phase Transformations
Provided there is no overlap of concentration fields, the rate of
thickening is inversely proportional to the interledge spacing λ
Important requirement for validity of the equation of v - constant
supply of ledges
Mechanisms of ledge generation - repeated surface nucleation, spiral
growth, nucleation at precipitate edges, .etc.
In situ hot stage TEM studies done under high-resolution indicate
that the plate thickening process is intermittent - there are
appreciable intervals of time in which there is no perceptible increase
of plate thickness followed by sudden large increases in thickness
Discontinuous thickening of plates indicates that supply of ledges is
the rate-limiting factor for plate thickening
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Phase Transformations