1. Fraction transformed in isothermal process – Avrami analysis
Consider    transformation How do we determine the volume (or area) fraction transformed? 
How do you deal with the overlap?
Mathematical device : extended volume fraction Xex  volume fraction transformed disregarding overlap.

2. The actual volume fraction grows in a relative amount to the unconsumed fraction, at the same rate the extended volume fraction does.: or
Unconsumed fraction
Integrate
Avrami equation
Expand :
dilute
overlap of two
overlap of three

3. Application to nucleation & growth : ( Johnson - Mehl)
Case (1) constant number of heterogeneous nuclei present from the beginning.
concentration: N
growth rate of crystals : v x t
Plot of ln t vs ln[-ln(1-x)] should have slope of 3.

4. Plot of ln t vs ln[-ln(1-x)]  slope of 4
Case (2) Assume a constant nucleation rate I, # of nuclei formed between t’
and t’ + dt’ ; concentration, N = I dt’ and at some later time ( t > t’ ) the “radius” of transformed phase is v (t – t’) so
Plot of ln t vs ln[-ln(1-x)]  slope of 4
These plots are called Johnson- Mehl –Arami plots
(JMA plots) c

5. Case study : Devitrification of Au65Cu12Si9Ge14 glass
C. Thompson et. al., Acta Met., 31, 1883 (1983)
Calorimetry results
Fraction transformed
power
DSC
isothermals
Time (min) 20 40 60 80
100
329K
328K
327K
326K
325K
324K
Time
1/2 1 X
329K
328K
327K
326K
325K
324K

6. JMA plot (327K) ln [-ln(1-x)] ln [-ln(1-x)] ln (1-t) ln (t)
slope = 4
ln (t)
must be introduced
N = Iss(t -)
Slope = 4.0

7. Time-Temperature-Transformation Curves
TTT curves” are a way of plotting transformation kinetics on a plot of temperature vs. time. A point on a curve tells the extent of transformation in a sample that is transformed isothermally at that temperature.
A TTT diagram shows curves that connect points of equal volume fraction transformed.
l+ β l T
l+ α α β
α +β A
xB → B

8. Time-Temperature-Transformation Curves
Curves on a TTT diagram have a characteristic “C” shape that is easily understood using phase transformations concepts.
The temperature at which the transformation kinetics are fastest is called the “nose” (•) of the TTT diagram
A TTT diagram shows curves that connect points of equal volume fraction transformed.

9. Construction of TTT diagrams from Avrami Curvesx
log t
100% transformed
decreasing T T1
50% transformed T4
50% transformed T4
trans start: 0 transformed T1
Temp
log time

10. Construction of TTT diagrams from Avrami Curves
Fe-C phase diagram

11. Fe-C phase diagram: Perlite

13. Fe-C TTT diagram example