Materials Science and Engineering A 500 (2009) 244–247
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Materials Science and Engineering A
journal homepage: www.elsevier.com/locate/msea
Short communication
Hardness based model for determining the kinetics of precipitation
Joy Mittra
∗
, U.D. Kulkarni, G.K. Dey
Materials Science Division, Bhabha Atomic Research Centre, Mumbai 400 085, India
article info
Article history:
Received 22 April 2008
Received in revised form
12 September 2008
Accepted 17 September 2008
Keywords:
Precipitation
Hardness
Kinetics
Martensitic steel
abstract
A theoretical model based on Johnson–Mehl–Avrami (JMA) formalism for determining kinetics and acti-
vation energy of a precipitation process is derived from the variation in hardness properties. Effectiveness
of the model for determining the kinetics of -NiAl precipitation in PH13-8Mo steel and for distinguishing
the kinetics between two temperatures is also demonstrated. The activation energy for -precipitation,
244.3 kJ/mol, determined over 808 K to 868 K is in good agreement with that of diffusion of Ni and Al in
-iron.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
Diffusion controlled phase transformation is a function of tem-
perature where, activation energy helps to comprehend the kinetics
of transformation. In the case of diffusion-controlled growth of pre-
cipitate through the formation of Guinier–Preston (GP) zone, the
strength of the material usually reaches a peak before falling due
to the loss of coherency [1,2]. In such cases, indirect determination
of activation energies through the measurement of physical prop-
erties, such as, strength and hardness becomes easy. In an earlier
attempt by Robino et al. [3], the possibility of using hardness data
in the Johnson–Mehl–Avrami (JMA) model [4,5] was worked out,
assuming the diffusion involving precipitation generates a volume
of soft impingement. As pointed out by Guo, Sha and Wilson [2,6],
JMA formalism represents a real precipitation process and gives a
close approximation in the range of initial stage to high volume
fraction and is seen to be successful in describing the precipita-
tion reaction and austenite reversion process during aging. Hence,
inability of the model by Robino et al. [3] to describe initial stage
of precipitation and questionable conclusion thereof that the JMA
equation might not be suitable to quantify the precipitate fraction
during the aging of PH13-8Mo steel has been discussed by Guo and
Sha [2,6]. Present work re-examines the possibility of using hard-
ness data in the JMA model with significant differences form the
earlier approach [3].
∗
Corresponding author. Tel.: +91 22 25590465; fax: +91 22 25505151.
E-mail address: joymit@barc.gov.in (J. Mittra).
The JMA equation of kinetics expresses the relationship between
volume fraction transformed, x, and the time of transformation, t,
in the following form.
x = 1 - e
-kt
n
(1)
Where, n and k are constants by which the kinetics are charac-
terized. The equation may also be written in the following linear
form.
log ln
1
(1 - x)
= log k + n log t (2)
To incorporate hardness data, we assume that the volume frac-
tion of precipitates, x, is with respect to that of maximum hardness
(H
F
) and is related to the hardness via flow properties of the material
in the following manner [3,7].
If H is the difference in hardness, then, H∝x
2/3
. If H
t
is
the hardness at any time t and H
0
is the hardness at t = 0, then,
H = H
t
- H
0
or, x∝(H
t
- H
0
)
3/2
. It is now expressed in the following
equation form.
x = C
0
+ C
1
(H
t
- H
0
)
3/2
(3)
where, C
0
and C
1
are constants. To find out the constants following
conditions may be set; when H
t
= H
F
, x = 1 and when H
t
= H
0
, x = 0.
So, C
1
=(H
F
- H
0
)
-3/2
and C
0
= 0. Hence, x =(H
t
- H
0
)
3/2
/(H
F
- H
0
)
3/2
is obtained. If x is substituted in Eq. (2), the following expression is
obtained
log ln
(H
F
- H
0
)
3/2
(H
F
- H
0
)
3/2
- (H
t
- H
0
)
3/2
= log k + n log t (4)
0921-5093/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.msea.2008.09.056