EMA5001 Lecture 23 Models for Transformation Kinetics · Temperature – Time – Transformation...

© 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


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


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


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



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



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



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


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


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


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


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


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


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


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


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


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)

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EMA5001 Lecture 23 Models for Transformation Kinetics · Temperature – Time – Transformation...

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