Post on 01-Jun-2020
transcript
© 2016 by Zhe Cheng
EMA5001 Lecture 23
Models for
Transformation Kinetics
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 22 Models for Phase Transform Kinetics
TTT Curve vs. Conversion Curve
Temperature – Time – Transformation
(TTT) Curve
“Nose” shaped
− Very high T: low driving force
− Very low T: Slow diffusion
For reactions happen during the
cooling process
Conversion – Time (CT) Curve
“S” shaped when time in log scale
For any reactions
Usually for isothermal reactions (can be
in undercooling or in superheating)
2
Exte
nt o
f
co
nve
rsio
n
Log (Time)
100%
Log (time)
Te
1% 99%
T
T1
T2
T1 T2
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 22 Models for Phase Transform Kinetics
Kinetics Under Very Low
Conversion Condition (1)
Assumptions
Spherical β nuclei
β nucleation rate constant of N
Growth rate of nucleus radius constant of v
Very low conversion and negligible “overlapping”
At time t, the volume of a β precipitate nucleated
at time τ (0 τ t) is
Number of β precipitates nucleated in a small time period of dτ (0 τ t) per unit
volume is
Total volume of transformed β per unit volume of matrix at time t is
3
Untransformed
volume
Transformed
volume 3
3
4 tvV
tt
nt dtNvNdtvdVV0
33
0
3
3
4
3
4
Nddn
Nuclei
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 22 Models for Phase Transform Kinetics
Kinetics Under Very Low
Conversion Condition (2)
Continue from p.3
Integrate, we have
For unit volume of starting matrix phase, the fractional of transformation (or extent of
conversion) will be
Note the above only applies when
Otherwise, the assumptions about constant nucleation rate, constant growth rate,
and no overlapping would not apply
4
t
t dtNvV0
33
3
4
43
3
1tNvVt
43
3
1
1tNv
Vf t
1f
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 22 Models for Phase Transform Kinetics
Kinetics Under Higher Conversion
JMA Equation (1)
Assumptions
Spherical β nuclei
β nucleation rate constant of N
Growth rate of nucleus radius constant of v
More precisely, the number of “real” β precipitates
nucleated in a small time period of dτ (0 τ t)
per unit volume is
f volume fraction of “real” precipitates
Introduce the concept of “imaginary (or extended) nucleation”, which includes
both “real” nuclei and “phantom” nuclei
The number of “imaginary” β precipitates nucleated in a small time period of dτ
at τ (0 τ t) per unit volume is
5
Untransformed
volume
Transformed volume
dfNdnr 1
Nddni
fdn
dn
i
r 1
Real
nuclei
Phantom
nuclei
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 22 Models for Phase Transform Kinetics
Kinetics Under Higher Conversion
JMA Equation (2)
Continue from p.5
We have
On the other hand, for “real” nuclei and “imaginary” nuclei,
fi volume fraction of “imaginary” (“real” + “phantom”) precipitates
Therefore,
Or
Integrate, we have
6
ii
r
i
r
df
df
dV
dV
dn
dn
fdn
dn
i
r 1
fdf
df
i
1
idff
df
1
ifef
1
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 22 Models for Phase Transform Kinetics
Kinetics Under Higher Conversion
JMA Equation (3)
Continue from p. 6
Remember that the “imaginary” fraction of conversion fi is essentially represented by
the fraction of transformation in low conversion (i.e., neglecting overlapping) case
We will have
known as the Johnson-Mehl-Avrami (JMA) equation
In general, n = ~1-4
The number of n depends on geometry of nuclei/precipitates and whether the
transformation involves only growth (i.e., “site saturation” scenario)
7
ifef
1
43
3
1tNvfi
43
3
1exp1 tNvf
nktf exp1
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 22 Models for Phase Transform Kinetics
Kinetics Under Higher Conversion
JMA Equation (4)
Continue from p.7
We have
Take log
Take log again
8
nktf exp1
nktf 1ln
Exte
nt o
f
co
nve
rsio
n
Log (Time)
100%
T1 T2
Log (Time)
T1
T2
tnkf lnln1lnln
f1ln
1ln
lnk1
lnk2
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 22 Models for Phase Transform Kinetics
Nucleation & Growth Model -
Instant Nucleation/Site Saturation Case
If all nucleation happen at the very beginning and only growth
afterwards
For 3D growth, dimension of a precipitate
under constant growth rate at time t
N0 is initial nuclei density,
the “imaginary” (or extended) fractional
conversion (real + phantom) will be:
Real fractional conversion:
We have: or
9
33
4vtVr
3
0
0
3
'3
4
tkV
Nvt
fi
ifef
1
3'exp1 tkf 3'1ln tkf ktf 3
1
1ln
Untransformed
volume Nuclei
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 22 Models for Phase Transform Kinetics
Nucleation & Growth Model -
Diffusion-Controlled Case
For diffusion-controlled 3D growth scenario for nucleation-growth:
Linear dimension (e.g., radius) of a precipitate would follow
For 3D growth with constant nucleation rate,
the imaginary (or extended) volume per unit volume of original phase will be:
The imaginary (real + phantom) fractional transformation is
Real fractional conversion:
We have or
10
5.00 utDtXX
Xx
e
tt
nri dtNuNdtudVV0
2/32/3
0
35.0
3
4
3
4
5.2'tkfi
ifef
1
5.2'exp1 tkf 5.2'1ln tkf ktf 5.2
1
1ln
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 22 Models for Phase Transform Kinetics
Diffusion-Based Model Under
Very Low Conversion – 1D Case
Diffusion-controlled growth of precipitate in 1D
Dimension change follows parabolic law
Fractional conversion (assuming low conversion)
We have
Or
11
β v
Xe
X0
Xi
XB
β α Interface control
Diffusion control
Xβ
tXX
XDx
e
2
2
02
tkx '2
2
1
00
't
L
k
L
xf
2
1
"tkf
ktf 2
x L0
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 22 Models for Phase Transform Kinetics
Diffusion-Controlled “Grow In” Model
with Thin Diffusion Field – 3D Case (1)
For diffusion-controlled reaction
Assuming
Spherical shape
Thickness of the product (transformed) layer
follow parabolic law
NO change in volume in reaction
Volume of untransformed portion
Fraction of transformation/conversion is given by
Consider we have
12
3
3
4rVr
0
0
V
VVf r
3
0
3
0
3
0
3
43
4
3
4
r
xrr
f
3
003
4rV
r0
x
Untransformed
volume
Transformed
volume
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 22 Models for Phase Transform Kinetics
Diffusion-Controlled “Grow In” Model
with Thin Diffusion Field – 3D Case (1)
Continue from p.2
We have
Re-arrange, we have
Remember that thickness of product layer
Another way to represent it
This is Jander’s equation:
Its limitations/assumptions are (i) low thickness of product (thin diffusion field), and
(ii) constant volume in reaction
More rigorous models such as Ginstling-Brounshtein and Carter’s Models exist
13
ktx 2
3
0
3
0
3
0
3
0
3
0
33
0 11
r
x
r
xrr
r
rrf
0
3
1
11r
xf
3
1
0 11 frx
ktrf
2
0
2
3
1
11 tr
kf
2
0
2
3
1
11
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 22 Models for Phase Transform Kinetics
Interface-Controlled “Grow In”/
“Shrinking-Core” Model – 3D Case (1)
For interface controlled “shrinking-core reaction
Examples: decomposition of solids
Assuming
Spherical shape
Constant rate of growth for the product layer thickness i.e.,
Volume of unreacted portion
Fraction of transformation/conversion is given by
Consider we have
14
3
3
4rVr
0
0
V
VVf r
ktrr 0
3
0
33
0
3
43
4
3
4
r
rr
f
3
003
4rV
r0
r
Untransformed
volume
Transformed
volume
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 22 Models for Phase Transform Kinetics
Interface-Controlled “Grow In”/
“Shrinking-Core” Model – 3D Case (2)
Continue from p.2
Remember that transformed portion radius
We have
Therefore,
Another way to represent it
Therefore,
Similarly, for interface controlled reaction for 2D case (such as decomposition of
cylinder shaped particles), we have
15
ktrr 0
3
0
3
0
3
0
3
0
3
0
3
0
33
0 1r
ktr
r
ktrr
r
rrf
3
0
11
t
r
kf
tr
kf
0
3
1
11
tr
kf
0
3
1
11
tr
kf
0
2
1
11
tkf '11 3
1
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 22 Models for Phase Transform Kinetics
Summary of Kinetic Reaction Models
Kinetic Model Equation Additional Notes
Power law model n = 1, 2, 3, etc.
Nucleation – Growth (grow out) model
Scenario could be by geometry (1D, 2D, 3D), nucleation (constant nucleation vs. site saturation), and rate limiting step (diffusion controlled vs. interface control)
Diffusion-controlled 1D model
Planar growth
Diffusion-controlled (shrinking-core) 3D model: Jander’s Equation
Simplification of constant reaction volume and thin diffusion layer thickness
Interface-controlled (shrinking-core) model
n=2 for 2D (shrinking area) n=3 for 3D (shrinking volume)
Reaction order model
n=1 for 1st order reaction n=2 for 2nd order reaction
16
ktf n 1
1ln
tr
kf
2
0
2
3
1
11
nfkdt
df 1
tr
kf n
0
1
11
ktf 2
nktf
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 22 Models for Phase Transform Kinetics
Summary of Physical Meaning for
n Factor in JMA Equation
JMA Equation
17
Growth Geometry
Nucleation rate Interface- controlled
Diffusion- controlled
3D
Constant nucleation rate 4 2.5
Zero nucleation rate (site saturation) 3 1.5
Decreasing nucleation rate 3-4 1.5-2.5
2D
Constant nucleation rate 3 2
Zero nucleation rate (site saturation) 2 1
Decreasing nucleation rate 2-3 1-2
1D
Constant nucleation rate 2 1.5
Zero nucleation rate (site saturation) 1 0.5
Decreasing nucleation rate 1-2 0.5-1.5
No growth Constant nucleation rate 1 0.5
nktf exp1
Hulbert, S. F., "Models for Solid-state Reactions in Powdered Compacts: A Review,“ Journal of the British Ceramic Society, Vol. 6, pp. 11 (1969)