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Accurate determination of the Avrami exponent in phase transformations
Materials Science and Enghwering, A 149 ( 1991 ) L5-L8
L5
Letter
Accurate determination of the Avrami e x p o n e n t in p h a s e t r a n s f o r m a t i o n s
Ming Mao and Z. Altounian Centre Jor the l'hysics of Materials and Department of Physics, McGUl University, 3600 University Street, Montreal, Quebec lt3A 2T8 (Canada) (Received June 19, 1991 )
Abstract A method is proposed for the determination of the Avrami exponent in studies of phase transformations. In this method, the uncertainty associated with the estimation of the incubation time is greatly reduced. We have used this method in analysing the isothermal crystallization kinetics of a metallic glass Fe75VIB24. The results give an accurate Avrami exponent for this transformation. The uncertainty in the exponent is a factor of 6 smaller than that derived from a conventional Avrami plot.
1. Introduction In studies of phase transformations involving nucleation-and-growth processes, the Johnson-MehlAvrami (JMA) equation [1] has been extensively used [2-4] to give a phenomenological description of the time dependence of the transformed fraction x at constant temperature. The JMA equation is x =1 - e x p { - [ K r ( t - r ) ] " }
(1)
where r is the effective time lag or incubation time necessary for obtaining a population of critical-size nuclei characteristic of the annealing temperature [2], n is the Avrami exponent indicative of the transformation process and Kr depends on the temperature through the Arrhenius relation
where K 0 is a constant and E a is the activation energy of the transformation, n and Ea are required for characterizing the overall kinetics of the transformation process. To determine n from the data of an isothermal 0921-5093/91/$3.50
transformation, an Avrami plot is constructed, i.e. a plot of l n [ - l n ( 1 - x ) ] vs. In(t-r). The slope of such a plot gives the Avrami exponent n. It is worth noting that, since the JMA equation is an approximation to the true transformation rate equation [1], the Avrami plot may show deviation from linearity for x >/0.5. In spite of this, some workers [5] plot the first derivative of the Avrami plot (the so-called Calka plot) as a function of transformed fraction and attribute the variations in the local value of n to real changes in the transformation process. We shall show that the accuracy of the Avrami exponent determined from such plots is of the order of 20%. This error is far too large and causes ambiguities in determining the nature of the transformation process. The principal reason is the large uncertainty associated in determining the incubation time r. This subtle, yet crucial role of r on n is seldom recognized. To overcome these uncertainties, Thompson et al. [3] linearized the Avrami plot by treating r as an adjustable parameter for a given value of n. This method is not recommended as it assumes an a priori knowledge of n and no further changes are allowed in n throughout the transformation process. In this paper, we describe a new and simple method for an accurate determination of the Avrami exponent from a modified Avrami (MA) plot. The advantage of this method is that an exact knowledge of r is not necessary. To demonstrate our method, we have chosen a metallic glass of composition Fe75V~B24 which transforms by a simple polymorphous crystallization process.
2. Experimental methods Glassy ribbons (20 pm thick and 1 mm wide) of prepared by the meltspinning process. Isothermal and isochronal crystallizations were performed in a calibrated Perkin-Elmer differential scanning calorimeter. As expected, the crystallization occurred in a single exothermic stage at 76 7 K. Crystallization isotherms were taken in the temperature range 697-717 K. For comparison, a set of six isochronal scans at different heating rates were also performed. All the measurements were done under a steady flow of oxygen-free argon.
composition Fey5V1B24 were
© 1991 --Elsevier Sequoia, Lausanne
Letter
L6
From eqn. ( 1 ) we can write
3. Derivation of the modified Avrami plot
The Avrami plot is based on the following form of the JMA equation: l n [ - l n ( 1 - x)]=n l n K r + n I n ( t - r )
(3)
The major part of transformation occurs in the time interval between q (x=0.05) and t 2 (x=0.95), as shown in Fig. 1. Experimentally, the true incubation time r ° cannot be determined but is located between a lower (r') and an upper (r") bound, i.e. r' < r ° < r". On a natural logarithmic scale, the widths of the corresponding interval in the transformation process for r', r ° and r" are given by the following expressions: In ( t 2 - r',/
(4)
It is clear that, the smaller the r chosen, the faster will be the change in the transformed fraction, as shown in Fig. 2, resulting in a higher Avrami exponent. Conversely, a large r will result in a smaller Avrami exponent. Arbitrary values of the Avrami exponent can therefore be obtained from the Avrami plot. The situation can be corrected if the explicit form t - r that appears in eqn. (3) can be eliminated. Our new approach addresses this problem and uses the transformed fraction and its first derivative as the components of the fitting equation• The first derivative of the JMA equation is Ytt
- " = n K : ( t - "t')n-1(1 -- X) dt
(5)
K r ( t - r) = [ - In( 1 - x)]'/" which upon substitution in eqn. (5) leads to
dx/dt - nKT{ - In( 1 - x)]tn-l>/n 1-x
ln[-ln(1-x)]--
(8)
Selected differential scanning calorimetry crystallization isotherms for the metallic glass Fe75VlB24 are presented in Fig. 1. The time evolutions of the integral under the crystallization exotherm and the heat flow, normalized to the mass and the crystallization enthalpy of the sample, gives the transformed fraction x and its first derivative dx/dt respectively. Figure 3 shows Avrami plots for seven different but experimentally accepted values of the incubation time for the isothermal transformation at T= 697 K. Figure 4 shows the M A plot for the same seven values of r. It is evident that, unlike the Avrami plot, all the curves for
I 8.
•
t1
n ln(nK~)+ n ln(dX/d(/ n-1 n-1 \-~-x]
4. Results and discussion
O8
I0
6
_
The Avrami exponent can be evaluated from the slope of the MA plot of l n [ - l n ( 1 - x ) ] against ln[(dx/dt)/ ( l - x ) ] . In addition, the activation energy can be derived from the intercepts of the MA plot for different temperatures through the Arrhenius relation, eqn. (2).
I
2
(7)
Taking logarithms of both sides of eqn. (7) leads to our new fitting equation
"r' 7 ° 7 "
~ 7 1 4 K
(6)
I
I
T I
b. 7 ° c. "7""
t(min)
0 r=
0.6
K t P
+
.....
0
I
0.4
t2
I--I
10
I
1
20
I
L
30
L
40
0.2
i
i
6.8
7.2
t(min) Fig. 1. Selected differential scanning calorimetry isothermal transformations for the metallic glass Fe75VIB24:a.u., arbitrary units. The inset shows the beginning of the isothermal transformation for T= 697 K.
6.0
6.4
In(t-'r) Fig.2.The transformedfractionx calculated fordifferent values of r vs. ln(t - r) for the isothermaltransformationat T= 697 K.
Letter
-
.95
L7
T A B L E 1. The A. rami exponent n derived from the Avrami, Calka and modified Avrami plots, for T = 697 K, for different incubation times l"
~-1
x
H
(rain)
I
I
--3
-
7=1.76
-5
"7"=4.40
min
--
05
rain
4'.5
5.5
,
, 6.5
i
1.76 2.29 2.64 2.99 3.52 3.87 4.40
Avrami plot
Calka plot a
M A plot
3.32 3.16 3.05 2.96 2.85 2.80 2.71
3.49 3.35 3.24 3.16 3.05 3.01 2.92
2.91 2.95 2.99 3.00 3.01 2.98 2.97
"The local n values are calculated for x = 0.45.
7.5
In(t-'r) Fig. 3. Avrami plots, corresponding to T = 6 9 7 K, for different values of r.
I
I
I
I
I
T A B L E 2. The variation An in Avrami exponents derived from the Avrami, Calka and modified Avrami plots, for T = 697,702, 707 and 710 K with different values of r Temperature
I
.95
x
An
(I<) 697 702 707 710
Avrami plot
Calka plot ~'
M A plot
0.61 0.59 0.65 0.60
0.57 0.56 0.63 0.57
0.10 0.07 0.03 0.03
"The local n values are calculated for x = 0.45.
-'~ - 3
-5r//
.05
~
-4.5
t
~
-3.5
~
~
~
-2.5
ln((dX/dt)/(1
J
-1.5 -X))
Fig. 4. MA plots, for T = 697 K, for different values of t.
the M A plot are almost identical in the principal transformation range, 0.05~
Table 1, An is about 0.6 for the Avrami and Calka plots whereas for the M A plot we get a value of 0.1. This represents a sixfold improvement in the accuracy of n for the M A plot. The accuracy of the M A plot has also been tested in the analysis of the data taken at different temperatures. The results are shown in Table 2, which lists the values of An at different temperatures for each method. The range of allowable values for r gets narrower as T increases, but so does t 2 - - t I (see Fig. 1), and the consistently low values of An for the M A plot is again evident. The activation energy of the transformation process was calculated through the Arrhenius relation to be 2.53_+0.03 eV together with the pre-exponential factor K 0 of 1.7 x 1015 s- 1. The results are in excellent agreement with those obtained from the Kissinger [6] plot from isochronal data which are 2.50 + 0.07 eV and 0.9 x 1015 s- J respectively. Finally, once n and K r are obtained from the slope and the intercept of the MA plot, we can determine r from the peak position tp of the transformation. At the peak, dx/dt is a maximum and, from eqn. (5), one can write
l(n-ll'/" r = tp-K--T ~ T ]
(9)
L8
Letter
For isothermal transformations at T = 6 9 7 K and 710 K we get r = 2.78 min and 1.29 rain respectively. At higher temperatures, owing to the finite equilibration time to the annealing temperature, reliable values of r are not possible.
Acknowledgments We thank D. H. Ryan for helpful suggestions. This research was supported by grants from the National Sciences and Engineering Research Council of Canada and Fonds pour la Formation de Chercheurs et rAide la Recherche, Qu6bec.
References 5. Conclusions We have shown that analysing isothermal transformation data using the conventional Avrami plots give Avrami exponents which are rather arbitrary depending on the selection of the incubation time. Our new method presented in this paper greatly reduces this uncertainty by developing a new fitting equation which does not include the t - T term explicitly. Its direct application in analysing the isothermal crystallization data of the metallic glass Fe75VlB24 gives an accurate Avrami exponent and hence a definitive mechanism for the transformation process can be deduced.
1 M. E. Fine, Introduction to Phase Transformations in Condensed Systems, Macmillan,London, 1964. J. Burke, The Kinetics of Phase Transformations in Metals, Pergamon, Oxford, 1965. 2 M. G. Scott and P. Ramachandrarao, Mater Sci. Eng., 29 (1977) 137. 3 C. V. Thompson, A. L. Greer and F. Spaepen, Acta MetalL, 31 (1983) 1883. 4 Q. C. Wu, M. Harmelin,J. Bigot and G. Martin, J. Mater. Sci., 21 (1986)3581. 5 A. Calka and A. P. Radlinski, Proc. Fall Meet. Materials Research Society, Boston, MA, 1986, Materials Research Society,Pittsburgh, PA, 1987, p. 195. 6 H.E. Kissinger,Anal Chem., 29 (1957) 1702.
L5
Letter
Accurate determination of the Avrami e x p o n e n t in p h a s e t r a n s f o r m a t i o n s
Ming Mao and Z. Altounian Centre Jor the l'hysics of Materials and Department of Physics, McGUl University, 3600 University Street, Montreal, Quebec lt3A 2T8 (Canada) (Received June 19, 1991 )
Abstract A method is proposed for the determination of the Avrami exponent in studies of phase transformations. In this method, the uncertainty associated with the estimation of the incubation time is greatly reduced. We have used this method in analysing the isothermal crystallization kinetics of a metallic glass Fe75VIB24. The results give an accurate Avrami exponent for this transformation. The uncertainty in the exponent is a factor of 6 smaller than that derived from a conventional Avrami plot.
1. Introduction In studies of phase transformations involving nucleation-and-growth processes, the Johnson-MehlAvrami (JMA) equation [1] has been extensively used [2-4] to give a phenomenological description of the time dependence of the transformed fraction x at constant temperature. The JMA equation is x =1 - e x p { - [ K r ( t - r ) ] " }
(1)
where r is the effective time lag or incubation time necessary for obtaining a population of critical-size nuclei characteristic of the annealing temperature [2], n is the Avrami exponent indicative of the transformation process and Kr depends on the temperature through the Arrhenius relation
where K 0 is a constant and E a is the activation energy of the transformation, n and Ea are required for characterizing the overall kinetics of the transformation process. To determine n from the data of an isothermal 0921-5093/91/$3.50
transformation, an Avrami plot is constructed, i.e. a plot of l n [ - l n ( 1 - x ) ] vs. In(t-r). The slope of such a plot gives the Avrami exponent n. It is worth noting that, since the JMA equation is an approximation to the true transformation rate equation [1], the Avrami plot may show deviation from linearity for x >/0.5. In spite of this, some workers [5] plot the first derivative of the Avrami plot (the so-called Calka plot) as a function of transformed fraction and attribute the variations in the local value of n to real changes in the transformation process. We shall show that the accuracy of the Avrami exponent determined from such plots is of the order of 20%. This error is far too large and causes ambiguities in determining the nature of the transformation process. The principal reason is the large uncertainty associated in determining the incubation time r. This subtle, yet crucial role of r on n is seldom recognized. To overcome these uncertainties, Thompson et al. [3] linearized the Avrami plot by treating r as an adjustable parameter for a given value of n. This method is not recommended as it assumes an a priori knowledge of n and no further changes are allowed in n throughout the transformation process. In this paper, we describe a new and simple method for an accurate determination of the Avrami exponent from a modified Avrami (MA) plot. The advantage of this method is that an exact knowledge of r is not necessary. To demonstrate our method, we have chosen a metallic glass of composition Fe75V~B24 which transforms by a simple polymorphous crystallization process.
2. Experimental methods Glassy ribbons (20 pm thick and 1 mm wide) of prepared by the meltspinning process. Isothermal and isochronal crystallizations were performed in a calibrated Perkin-Elmer differential scanning calorimeter. As expected, the crystallization occurred in a single exothermic stage at 76 7 K. Crystallization isotherms were taken in the temperature range 697-717 K. For comparison, a set of six isochronal scans at different heating rates were also performed. All the measurements were done under a steady flow of oxygen-free argon.
composition Fey5V1B24 were
© 1991 --Elsevier Sequoia, Lausanne
Letter
L6
From eqn. ( 1 ) we can write
3. Derivation of the modified Avrami plot
The Avrami plot is based on the following form of the JMA equation: l n [ - l n ( 1 - x)]=n l n K r + n I n ( t - r )
(3)
The major part of transformation occurs in the time interval between q (x=0.05) and t 2 (x=0.95), as shown in Fig. 1. Experimentally, the true incubation time r ° cannot be determined but is located between a lower (r') and an upper (r") bound, i.e. r' < r ° < r". On a natural logarithmic scale, the widths of the corresponding interval in the transformation process for r', r ° and r" are given by the following expressions: In ( t 2 - r',/
(4)
It is clear that, the smaller the r chosen, the faster will be the change in the transformed fraction, as shown in Fig. 2, resulting in a higher Avrami exponent. Conversely, a large r will result in a smaller Avrami exponent. Arbitrary values of the Avrami exponent can therefore be obtained from the Avrami plot. The situation can be corrected if the explicit form t - r that appears in eqn. (3) can be eliminated. Our new approach addresses this problem and uses the transformed fraction and its first derivative as the components of the fitting equation• The first derivative of the JMA equation is Ytt
- " = n K : ( t - "t')n-1(1 -- X) dt
(5)
K r ( t - r) = [ - In( 1 - x)]'/" which upon substitution in eqn. (5) leads to
dx/dt - nKT{ - In( 1 - x)]tn-l>/n 1-x
ln[-ln(1-x)]--
(8)
Selected differential scanning calorimetry crystallization isotherms for the metallic glass Fe75VlB24 are presented in Fig. 1. The time evolutions of the integral under the crystallization exotherm and the heat flow, normalized to the mass and the crystallization enthalpy of the sample, gives the transformed fraction x and its first derivative dx/dt respectively. Figure 3 shows Avrami plots for seven different but experimentally accepted values of the incubation time for the isothermal transformation at T= 697 K. Figure 4 shows the M A plot for the same seven values of r. It is evident that, unlike the Avrami plot, all the curves for
I 8.
•
t1
n ln(nK~)+ n ln(dX/d(/ n-1 n-1 \-~-x]
4. Results and discussion
O8
I0
6
_
The Avrami exponent can be evaluated from the slope of the MA plot of l n [ - l n ( 1 - x ) ] against ln[(dx/dt)/ ( l - x ) ] . In addition, the activation energy can be derived from the intercepts of the MA plot for different temperatures through the Arrhenius relation, eqn. (2).
I
2
(7)
Taking logarithms of both sides of eqn. (7) leads to our new fitting equation
"r' 7 ° 7 "
~ 7 1 4 K
(6)
I
I
T I
b. 7 ° c. "7""
t(min)
0 r=
0.6
K t P
+
.....
0
I
0.4
t2
I--I
10
I
1
20
I
L
30
L
40
0.2
i
i
6.8
7.2
t(min) Fig. 1. Selected differential scanning calorimetry isothermal transformations for the metallic glass Fe75VIB24:a.u., arbitrary units. The inset shows the beginning of the isothermal transformation for T= 697 K.
6.0
6.4
In(t-'r) Fig.2.The transformedfractionx calculated fordifferent values of r vs. ln(t - r) for the isothermaltransformationat T= 697 K.
Letter
-
.95
L7
T A B L E 1. The A. rami exponent n derived from the Avrami, Calka and modified Avrami plots, for T = 697 K, for different incubation times l"
~-1
x
H
(rain)
I
I
--3
-
7=1.76
-5
"7"=4.40
min
--
05
rain
4'.5
5.5
,
, 6.5
i
1.76 2.29 2.64 2.99 3.52 3.87 4.40
Avrami plot
Calka plot a
M A plot
3.32 3.16 3.05 2.96 2.85 2.80 2.71
3.49 3.35 3.24 3.16 3.05 3.01 2.92
2.91 2.95 2.99 3.00 3.01 2.98 2.97
"The local n values are calculated for x = 0.45.
7.5
In(t-'r) Fig. 3. Avrami plots, corresponding to T = 6 9 7 K, for different values of r.
I
I
I
I
I
T A B L E 2. The variation An in Avrami exponents derived from the Avrami, Calka and modified Avrami plots, for T = 697,702, 707 and 710 K with different values of r Temperature
I
.95
x
An
(I<) 697 702 707 710
Avrami plot
Calka plot ~'
M A plot
0.61 0.59 0.65 0.60
0.57 0.56 0.63 0.57
0.10 0.07 0.03 0.03
"The local n values are calculated for x = 0.45.
-'~ - 3
-5r//
.05
~
-4.5
t
~
-3.5
~
~
~
-2.5
ln((dX/dt)/(1
J
-1.5 -X))
Fig. 4. MA plots, for T = 697 K, for different values of t.
the M A plot are almost identical in the principal transformation range, 0.05~
Table 1, An is about 0.6 for the Avrami and Calka plots whereas for the M A plot we get a value of 0.1. This represents a sixfold improvement in the accuracy of n for the M A plot. The accuracy of the M A plot has also been tested in the analysis of the data taken at different temperatures. The results are shown in Table 2, which lists the values of An at different temperatures for each method. The range of allowable values for r gets narrower as T increases, but so does t 2 - - t I (see Fig. 1), and the consistently low values of An for the M A plot is again evident. The activation energy of the transformation process was calculated through the Arrhenius relation to be 2.53_+0.03 eV together with the pre-exponential factor K 0 of 1.7 x 1015 s- 1. The results are in excellent agreement with those obtained from the Kissinger [6] plot from isochronal data which are 2.50 + 0.07 eV and 0.9 x 1015 s- J respectively. Finally, once n and K r are obtained from the slope and the intercept of the MA plot, we can determine r from the peak position tp of the transformation. At the peak, dx/dt is a maximum and, from eqn. (5), one can write
l(n-ll'/" r = tp-K--T ~ T ]
(9)
L8
Letter
For isothermal transformations at T = 6 9 7 K and 710 K we get r = 2.78 min and 1.29 rain respectively. At higher temperatures, owing to the finite equilibration time to the annealing temperature, reliable values of r are not possible.
Acknowledgments We thank D. H. Ryan for helpful suggestions. This research was supported by grants from the National Sciences and Engineering Research Council of Canada and Fonds pour la Formation de Chercheurs et rAide la Recherche, Qu6bec.
References 5. Conclusions We have shown that analysing isothermal transformation data using the conventional Avrami plots give Avrami exponents which are rather arbitrary depending on the selection of the incubation time. Our new method presented in this paper greatly reduces this uncertainty by developing a new fitting equation which does not include the t - T term explicitly. Its direct application in analysing the isothermal crystallization data of the metallic glass Fe75VlB24 gives an accurate Avrami exponent and hence a definitive mechanism for the transformation process can be deduced.
1 M. E. Fine, Introduction to Phase Transformations in Condensed Systems, Macmillan,London, 1964. J. Burke, The Kinetics of Phase Transformations in Metals, Pergamon, Oxford, 1965. 2 M. G. Scott and P. Ramachandrarao, Mater Sci. Eng., 29 (1977) 137. 3 C. V. Thompson, A. L. Greer and F. Spaepen, Acta MetalL, 31 (1983) 1883. 4 Q. C. Wu, M. Harmelin,J. Bigot and G. Martin, J. Mater. Sci., 21 (1986)3581. 5 A. Calka and A. P. Radlinski, Proc. Fall Meet. Materials Research Society, Boston, MA, 1986, Materials Research Society,Pittsburgh, PA, 1987, p. 195. 6 H.E. Kissinger,Anal Chem., 29 (1957) 1702.