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Device Materials Lab.77

5.4 Overall Transformation Kinetics – TTT Diagram

TTT Diagram :

the fraction of Transformation (f) as a function of Time and Temperature

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Device Materials Lab.88

5.4 Overall Transformation Kinetics – TTT Diagram

Johnson-Mehl-Avrami Kolmogorov Equation

specimenofVol

phasenewofVolf

.

.

Assumption :

Reaction produces by nucleation and growth

Nucleation occurs randomly throughout specimen

Reaction product grows radially until impingement

Define volume fraction transformed

f

t

tdτ

τ τ+dτ

specimenofvolume

dduringformed

nucleiofnumber

ttimeatmeasureddduring

nucleatedparticleoneofvol

df

.

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𝛼 → 𝛽

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Device Materials Lab.99

5.4 Johnson-Mehl-Avrami Kolmogorov Equation

0

0

3 )()]([3

4

V

dNVtv

df

v : growth rate (assumed const. )

N : nucleation rate (const. )

33

33

)(3

4

)(3

4

3

4

tvV

vtrV

43

0

33

0

3

3

4

tvNf

d)t(vNf̂df

e

tx

e

volume :

→ do not consider impingement & repeated nucleation

→ only true for f ≪ 1

Phase Transformation In Materials

fe : extended volume fraction

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Device Materials Lab.1010

Johnson-Mehl-Avrami Kolmogorov Equation

edffdf )1( edf

dff 1

43

31 tNvexpf

)tkexp(f n 1 t

1

∝t4

f

····· 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, t0.5

exp (-0.7) = 0.5

i.e.

Example above.

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Device Materials Lab.1111

)pt(expII o

pI)pt(exppIdt

dI)t(N o

pt)ptexp(

p

vIexpf o 1

81

3

3

pt

6

33tp

33

31 tvNexpf o

Other example:

1. Fixed nucleation site and active at the beginning.

2. Nucleation site depends on the time

p: rate at which the individual sites are lost.

limiting cases : small → same as J-M eq.

large → I quickly goes to zero.

Phase Transformation In Materials

Johnson-Mehl-Avrami Kolmogorov Equation

(Io : number of active nucleation site / unit volume)

pt

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Device Materials Lab.1212

5.5 Precipitation in Age-Hardening Alloys

5.5.1 Precipitation in Aluminum-Copper Alloys

Transition phases

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43210 '"zonesGP

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Device Materials Lab.1313

5.5 Precipitation in Age-Hardening Alloys

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Device Materials Lab.1414

5.5 Precipitation in Age-Hardening Alloys

Growth of 𝜽′

in the expense of 𝜽"

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Device Materials Lab.1515

5.5.3 Quenched in Vacancies

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Precipitation Free Zone: PFZ

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Device Materials Lab.1616

5.5.4 Age Hardening

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Device Materials Lab.1717

5.5.5 Spinodal Decomposition

No barrier for nucleation

Phase diagram with a miscibility gap

𝑑2𝐺

𝑑𝑋2< 0, chemical spinodal

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Ω > 0

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Device Materials Lab.1818

The Effect of ∆Hmix and T on ∆Gmix

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From Ch1

Ω < 0

Ω > 0

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Device Materials Lab.1919

5.5.5 Spinodal Decomposition

Composition profiles in an alloy

quenched into the spinodal region

Composition profiles in an alloy

outside the spinodal points

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Device Materials Lab.2020

Spinodal decomposion

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Device Materials Lab.2121

5.5.5 Spinodal Decomposition

The Rate of Transformation

Rate controlled by interdiffusion coefficient D~

D22 4/

D~

)exp(

t Within the spinodal < 0 , composition fluctuation

transf. rate ↑ as λ ↓

- : characteristic time constant

- λ : wavelength of the composition modulation (1-D assumed)

There is a minimum value of below which spinodal decomposion cannot occur

To calculate λ, it is need to take care of 1) interfacial energy 2) coherency strain energy

Phase Transformation In Materials