Fraction transformed in isothermal process – Avrami analysisConsider a ® b transformation How do we determine the volume (or area) fraction transformed?
a
How do you deal with the overlap?
Mathematical device : extended volume fraction Xex º volume fraction
transformed disregarding overlap.
The actual volume fraction grows in a relative amount to the unconsumed
fraction, at the same rate the extended volume fraction does.:
( )/ /1 exdx dt
dX dtx
=-
Unconsumed fraction
exdXxdx
=-1
Integrate
( )exXx --= exp1 Avrami equation
Expand : ( ) ( ) !-+-= 32!31
21
exexex XXXx
dilute overlap of two overlap of three
or
Application to nucleation & growth : ( Johnson - Mehl)
Case (1) constant number of heterogeneous nuclei present from the beginning.
concentration: N growth rate of crystals : v
( )334 vtNXexp
=
úûù
êëé--= 3334exp1 tNvx p
x
t
Plot of ln t vs ln[-ln(1-x)]
should have slope of 3.
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 ( ) 43330 3
'34' tIvttvIdtX
t
expp
=-×= ò
3 413
x exp Iv tpé ù= - -ê úë û
Plot of ln t vs ln[-ln(1-x)] à slope of 4
These plots are called Johnson- Mehl –Arami plots
(JMA plots)
Calorimetry results
pow
er
DSCisothermals
Time (min) 20 40 60 80 100
329K 328K
327K
326K 325K
324K
Time
1/2
1
X
329K 328K 327K 326K 325K 324K
0
Fraction transformed
Case study : Devitrification of Au65Cu12Si9Ge14 glass
C. Thompson et. al., Acta Met., 31, 1883 (1983)
-16 -8 00 08 16
(b)
ln (1-t) 02 06 10 14 18-100-80-60-40-200204060
(a)
ln (t)
ln [-
ln(1
-x)]
JMA plot (327K)
must be introducedN = Iss(t -t)Slope = 4.0
ln [-
ln(1
-x)]
slope = 4
A B
α
l+ α
l+ β
α +β
β
xB→
T
l
Time-Temperature-Transformation CurvesTTT 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.
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.
x
log t
decreasing TT1 T4
T1
T4 trans start: 0 transformed
log time
Temp
Construction of TTT diagrams from Avrami Curves
50% transformed
100% transformed
50% transformed
Construction of TTT diagrams from Avrami Curves
Fe-C phase diagram
Fe-C phase diagram: Perlite
Fe-C TTT diagram example
