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Non-isothermal transformation kinetics: Application to metastable phases
OOOI-6160/R3503.00+ .cn, Pergmllon Press IA,1
~crn n~rral/. Vol. 31. No. 7. pp. 1053-1059. 1983 Printed in Greal Hrilain
NON-ISOTHERMAL APPLICATION
U.S.
TRANSFORMATION KINETICS: TO METASTABLE PHASES
L. V. MEISEL and P. J. COTE Army Armament Research and Development Command, Large Caliber Weapon Systems Laboratory, Benet Weapons Laboratory, Watervliet, NY 12189, U.S.A. (Reccir,ed
I5
July 1982; in r&red
,Jbrm I6 Dccetdwr
1982)
Abstract-Constant
heating rate differential scanning calorimetry is frequently employed to study the kinetics of transitions from metastdble phases and, in particular, crystallization of metallic glasses. Such data are analyzed by the Kissinger method. which was originally devised for the study of homogeneous reactions. The consensus in the literature, until recently. was that such applications (i.e. to heterogeneous solid state transformations) of the Kissinger method are not valid. We address the principal objections to these applications and provide alternative derivations of recent theoretical results which demonstrate that the Kissinger method is valid for heterogeneous reactions of the type described by the Johnson-Mehl-Avrami equation in the isothermal case. Isothermal and constant heating rate data on transformation of metastable NIP powders are presented. These experimental results and the discussions presented here help to clarify the effects of incubation times in non-isothermal transformation kinetics and provide a further demonstration of validity of the generalized Johnson-Mehl-Avrami theory for the description of heterogeneous solid state transformations. R&arm&-La calorimetric differentielle a balayage avec vitesse d%chauBage constante est utilisie souvent pour I’etude des cinctiques des transformations de la phase metastabile ct. en particulicr. la cristallisation des glaces metalliques. Ces donn&es sont analydes par la methode Kissinger, developpee originalement pour I’ttude des transformations homogenes. Le consensus dans la litterature jusqu’ a recemment, a et e que ces application (i.e. pour transformations hiterogtnes) de la methode Kissinger ne sont pas valides. Nous adressons les objections principales de ces applications et nous donnons une derivation alternative des resultats thloretiques recents qui dimontre que la methode Kissinger est valide pour des transformations hetirogtnes du type decrites par I’equation du Johnson-Mehl-Avrami pour le cas isotherme. Les donnees des transformations des pandres de NiP metastabile sont presenties. Les rtsultats exp&imentaux et les discours presentes ici aident a clarifier les effect des temps de I’incubation dans les transformations non-isothetmique et its dtmontrent la validitt de la thtorie giniralizb de Johnson-Mehl-Avrami pour la description des transformations hettrogines de l’etat solid.
Zusammenfasauag-Die Umwandlungskinetik metastabiler Phasen, besonders metallischer Glaeser wird oft durch differentielle Kalorimetrie mit einer konstanten Heizrate untersucht. Die dabei erhaltenen Daten werden nach der Methode von Kissinger analysiert, die urspruenglich fuer die Untersuchung homogener Reaktionen entwickelt wurde. Bis vor kurzem wurde im Schrifttum allgemein die Anscht vertreten, dass diese Anwendung der Kissinger-Methode feur heterogene Phasenumwandlungen nicht zulaessig sei. Wir betrachten die Haupteinwaende gegen diese Methode, und bieten alternative Ableitungen neuerer theoretischer Untersuchungenan, die zeigen, dass die Kissingermethode such fuer heterogene Reaktionen der Art, wie sie durch die Johnson-Mehl-Avrami gueltig ist. Es werden Daten geliefert, die durch
Gleichung
im isothennen
Fall beschrieben
werden,
isotherm&he Untersuchungen, sowie durch konstantes Autheizen von metastabilen Ni-P Pulvem erhalten wurden. Diese experimentellen Ergebnisse und die hier vorgebrachte Diskussion tragen zur Klaerung des Einglusses der lnkubationszeiten auf nicht-isothetme Umwandlungsvorgaenge bei, und liefem einen weiteren Hinweis feur die Gueltigkeit der verallgemeinerten
Johnson-Mehl-Avrami Koerpern.
Theorie
zur
Beschreibung
INTRODUCTION Differential thermal analysis techniques such as differential scanning calorimetry (DSC) provide convenient methods for obtaining information on reaction kinetics. A popular thermal analysis method developed by Kissinger [I,21 determines the kinetic parameters from graphs of the logarithm of the heating rate vs inverse temperature at the maximum of the reaction rate in constant heating rate experiments. This method was frequently used in studies of the crystallization of glassy metals [3-81 despite the
von
heterogenen
Phasenumwandlungen
in festen
fact that literature [9] on thermal analysis techniques reflected a consensus that application of the Kissinger method to solid state reactions is improper. However. the recent work of Henderson [lo] has provided a theoretical basis for the treatment of non-isothermal thermal analysis techniques and justifies the use of the Kissinger method for many solid state transformations. The three main objections to the use of this method for study of solid state reactions were: (I) thermal gradients are inherent in non-isothermal methods. Thus, it was claimed that significant inaccuracy will result from the application of the Kissinger
I053
II)? 4
MEISEL. ;tntl <‘OTE:
NON-ISOT11ERMAL
TRANSFORMATION
mc~hod which does not allow for the presence of
whcrc A is constant
tempcrnture gradients. (2) The reaction rate equation which is ilp~~lX~priillC liw iS0lllCrnlill cxpcrimcnts is
absolute
in the Kissinger
msunletl
analysis.
arpucd thaI ;I term involving derivative isothermal
must be included in the analysis cxpcrimcnts;
in the literature appropriate
partial of non-
this point has been debated
for over a decade. (3) The order of
reaction equation is
It is frequently
the temperature
assumed
for
in the Kissinger analysis
homogeneous
transformations,
(e.g. chemical reactions in a gas) but is not valid
for the heterogeneous transformations which generally occur in solid state reactions. Regarding the first objection, we describe simple procedures to reduce the influence of temperature gradients to negligible levels. The confusion in the literature surrounding the proper form of the reaction rate equation in the Kissinger analysis [objection (2)] results from the assumption that the progress of a .reaction can be described as a simple function of the time and temperature. The degree of reaction is clearly a functional [II], dependent on the temperature history, and not a simple function. (We address this question in the Appendix.) However, a result of the analysis presented here and the principle assumption in Henderson’s work is that the reaction rare is an ordinary function of the temperature and the degree of reaction. Our data show that the third objection is valid. Nevertheless, Henderson [IO] has shown that the Kissinger method can be applied to the analysis of many heterogeneous reactions. We provide an alternate treatment of non-isothermal transformation kinetics here, which indicates that the Kissinger method can be applied to any reaction of the type described by the Johnson-Mehl-Avrami (JMA) equation [12] in the isothermal case. Our treatment (as does that of Henderson) goes as follows: (i) generalize the JMA approach to deal with non-isothermal, heterogeneous reactions and (ii) demonstrate that in the constant heating rate case (within negligible errors) the Kissinger relationship obtains. Finally, isothermal and constant heating rate data on the transformation of metastable NiP powders are presented. Results on the influence of incubation times on constant heating rate experiments are included. The data are in excellent agreement with the theoretical results. GENERALIZATION OF THE JOHNSON-MEHLAVRAMI EQUATION
KINETICS pm-exponential
factor,
7‘ is the
temperature.
and E = k,O is an activation cncrgy for the tr;lnsli,rm;ltion process with k,, the Boltzmann
constant.
isothermal
processes so k(T)
Note that equation (I) dcscribcs is a constant
(which
depends on the temperature)
in equation (I). An expression for the reaction rate dx/dr can be derived directly from equation (I) and expressed as a simple function of the temperature and the degree of reaction. We are interested in generalizing equation (I) to treat experiments in which temperature is a function of time. If we assume that the transformation products and mechanism do not change with temperature, then it is reasonable to interpret k(T)1 in equation (I) as being proportional to the number of atomic jumps within the interval I at temperature T. If we assume that the progress of the transformation is determined by the number atomic jumps in the general (nonisothermal) case as well, then equations (I) and (2) generalize to x[r,T(r’)] = I - exp( - W”)
(3)
where W[f,T(f’)] - W, =
=
‘k[T(t’)]dr’ I0
k(T)(dT/dr’)-‘dT
(4)
and k(T) is still given by equation (2). T(t) is the temperature at t, n is the JMA exponent which depends on the nucleation and growth mechanisms, and W. and To are the temperature and W value at t = 0. Note that x[r,T(r’)] depends on r and the temperature history T(t’) for times r’ earlier than r and the same is true for W[r;T(r’)]. That is x[r,T(r’)] and W[t,T(r’)] are nor determined by r and the temperature at r as is clear from equation (3) and (4). Henderson [lo] assumed that the reaction rate in the non-isothermal case can be expressed as the same simple function of temperature and degree of reaction as in the isothermal case. He then integrates to obtain equation (3). Thus, it is seen that Henderson’s theory, which is based upon an assumption concerning the reaction rate, is equivalent to the present treatment which assumes that the progress of a reaction is determined by the number of atom movements. We now consider the constant heating rate case: dT/dt’ s T is constant and comes out of the integrand in equation (4). Thus T
The Johnson-Mehl-Avrami equation [12] describes a wide variety of isothermal solid state transformations and has the form s(r) = I - exp[-(k(T)/)“] k(T) = A exp( -O/T)
(1) (2)
W(T)-
W,=(A/T)
e-“/rdT s TII
~~[1-2(;)+6(;)1]
(5)
where we have integrated by parts as described in Reference [ 131and assumed that k( 7’,,)Ti is negligible
MEISEL
and COTE:
NON-ISOTHERMAL
in comparison to k(T)T*. Note that in the constant heating rate case we represent the functions x and W as simple functions of either the time or the temperature. Note also that equation (5). derived in the spirit of Kissinger’s work, is the “asymptotic expression” for the exponential integral as given in equation 5 1.51 of Abramowitz and Stegun [I41 while the approximate form given by Henderson is the “large n” approximation given in equation 51.52 of Ahromowitz and Stegun [14]. They are, for practical purposes, equivalent. In a constant heating rate DSC experiment, one measures a temperature rise which is assumed to be proportional to dx/dl as a function of T or t. From equation (5) and (4) one finds dx Z=e
- *” n W” dln W/dt
=(I -x)k(T)nW”-’ (equation (6b) wiih W expressed as a function of x is the starting point in Reference [lo]). We need to know how the temperature T, for the maximum reaction rate varies with the heating rate in order to employ the Kissinger method. The maximum reaction rate will occur for T such that d21nW -=n(W”dt2
1)
or, taking derivatives of equation (5) in the case that W, = 0, and keeping terms to order T,,@ -p+lu($)+ln(~)-Z~(l--$). m
(7)
Note that equation (7) reduces to the Kissinger expression for the n = 1 case as we might have anticipated since this corresponds to the homogeneous reaction case. Moreover, for metastable phases such as amorphous metals, the right hand side of equation (7) is generally negligible in comparison to the individual terms on the left hand side for common heating rates (2 l~‘C/min~. Thus, we see that the Kissinger method is appropriate for the anatysis not only of homogeneous reactions, but also for the analysis of heterogeneous reactions which are described by the Johnson-Mehl-Avrami equation in isothermal experiments. In particular, equation (7) implies dln(~/T~)/d(l/~~,)
z -8.
(8)
The approximation in equation (8) (RHS equation (7) =0) might introduce a 3% error in the value of 0 in the worst cases. (Typically, n > I,8 > 25 x lO’K, and T,, c 1.0 x 103K which suggests that the error introduced in 6 by setting the RHS of equation (7) = 0 is considerably less than I%.) Equation (7) also serves to determine the pre-exponential factor A from the intercept of a ln(f/Ti) vs (l/T,,,) plot. Equation (6), which describes the time (or temperature) de-
TRANSFORMATION
KlNETICS
I055
pendence of the reaction rate, and equation (7) and (8). which allow for the simple extraction of the parameters A and 0 by means of the Kissinger method, form the basis for the analysis of constant heating rate data. EXPERIMENTAL
PROCEDURES
AND RESULTS
The data were obtained with a DuPont 990 Thermal Analysis System and DSC module. Aluminum sample pans and an argon atmosphere were used in all runs. An empty sample pan served as a reference. The metastable NiP powders (< 325 mesh) were prepared by an electroless process. The powder composition (85 at.% Ni) was selected so that only one transformation [NiP (metastable F.C.C.)N&P + Ni] occurs; this avoids complications associated with the occurrence of several competing reactions. Isothermal transformation kinetics data for the metastable NiP powders were obtained by monitoring the time dependence of the DSC output for a series of temperatures. The DSC output IDsc is assumed to be proportional to the reaction rate dx/dt (IDsc = cdx/dt, c is constant) so the degree of reaction x(t) is given by x(t)=s(t)/S where I hE.c dT = c[x(t) -x(O)] 4~) = s0 and S = S(M)). The technique for extracting the time exponent n and k(T) in an isothermal experiment described by equation (I) is to ptot lnln(S/[S-s(t)]] vs lnt, so that n is the slope and k(T) = t;’ where to is the time in seconds when lnln (S/[S-s(t)]} = 0. Equation (8) shows that the constant heating rate method {Kissinger method) may be used to obtain values of the kinetic parameters, when the isothermal data are described by the JMA equation. Before obtaining the constant heating rate data, we examined the effects of thermal gradients in our DuPont DSC apparatus. The standard model [15] describing the DuPont DSC output Y contains the correction AY where
A Y = -c,JR, + R,) d Yjdt where c,~ is the sample’s heat capacity, R,, is the thermal resistance between the sample and the sample platform, and R, is the thermal resistance between the sample platform and heater. This term leads to a distortion of the DSC peaks and shifts the peaks to higher temperatures. If this model is valid, it suggests that errors can be avoided either by working with small sample masses (small c.~)or by repeating the experiments using different masses and extrapolating to zero mass (zero c,). In practice, this method does not work and the model apparently is inadequate. For example, we observed that in series of determinations for different sample masses, the melting point of Sn and the Curie temperature of Ni, did not extrapolate to the correct temperatures at zero mass
1036
MEISEL
(the correct temperature temperature
at zero
and COTE:
NON-ISOTHERMAL
TRANSFORMATION
KINETICS
is taken to be the transition
heating
rate).
Similar
results
obtained with the DuPont instrument by Greer [16] were attributed to the heat capacity of the sample pan which he assumes is included in c,,.Our results do not support this explanation,:,however, since we find that thermal lag is reduced by only about 30% with the sample and reference pans removed. We therefore conclude that methods for thermal lag corrections based on this model are unreliable and we employ a more straightforward method similar to that used by Greer [ 171for the Perkins-Elmer DSC system; that is, we empirically determined the thermal lag vs heating rate by using the Curie temperature of Ni as well as the melting point of Sn. For solid samples in our DuPont DSC cells the lag AT is linear with heating rate F, insensitive to sample mass (provided we maintain a large contact area with the sample platform) and the ratio AT/f, using sample and reference pans, is measured to be 0.05 min. Powder samples exhibit effects which are distinct from those described above for solid samples: the observed thermal lags increase linearly with sample mass, presumably because of the inherent low thermal diffusivity of powder samples. We avoided this difficulty with powders by working with small samples (e.g. < 1 mg at lO”C/min.). Thus, although criticism 1 of the Kissinger method is valid, the method is still practical.
RESULT!3 AND DISCUSSION
The DSC measurements exhibit an “incubation time” riw, during which no observable transformation occurs. We find that the temperature dependence of ri, can be described as an activated process with E = 279 kJ/mol, near that for the phase transformation; this is often seen in studies of crystallization of metallic glasses [I 81. In the isothermal experiments, after fi”E,the DSC output (dx/dr) increases from zero, proceeds through a maximum, and returns to zero. This result is inconsistent with the predictions of the order of reaction equation assumed by Kissinger for homogeneous reactions, dx/df = k(l -x)”
P
I -
0h
T * -IL
k-2 c
-3 -4
-
I
I
I
1
I
I
4
5
6
7
a
9
II-I It-,,,,
Fig. I. Plots for extracting the JMA time exponent II in equation (I), as described in the text. Characteristic parameters are listed in Table I for the corresponding curve number.
value of 3.09. The deviation from linearity at long times may reflect breakdown of the theory for x(t) at the end of the transformation.
The logarithm of the rate constant k, obtained from Fig. I, is plotted against l/T in Fig. 2. The slope of the line, obtained from a least squares fit, gives E = kg0 = 221 kJ/mol for the activation energy and In A = 39.2, for A in s-‘. There is very little scatter about the line in Fig. 2 despite the inherent experimental difficulties in establishing accurate baselines with this method. The precautions and corrections described in the previous section were applied to constant heating rate data obtained for the same NiP powders used in the isothermal measurements. The resulting Kissinger plot is shown in Fig. 3. An excellent straight line with little scatter is given yielding E = 220 kJ/mol and pre-exponential factor A = exp(38.4)s-‘. Also shown in Fig. 3 are constant heating rate data obtained on a batch of these NiP powders which had been previously annealed through the incubation time (15.5 min at 295°C) as determined in the iso-
(9)
where m is called the order of reaction and k has been previously defined. The reaction rate given in equation (9) is maximum at t = 0 and decreases monotonically with increasing time. Thus, the present results support the third of the three principal criticisms of the application of the Kissinger method to heterogeneous solid state transformations as stated in the introduction. The DSC traces for a series of temperatures were integrated to yield plots of In[-ln( I -s/S)] vs In(t - r,,) which are shown in Fig. I. Reasonably good straight lines are obtained; the n value determined from Fig. I range from 2.8 to 3.2 with a mean
Fig. 2. The logarithm of the rate constant k deduced from the isothermal measurements is plotted against T ’ (k is in s- ‘).
MEISEL
and COTE:
NON-ISOTHERMAL
TRANSFORMATION
KINETICS
1057
annealed samples exhibit narrower and less skewed DSC peaks than given by the generalized JMA theory. Table I summarizes the predicted and observed values of T,, and FW H M for the heating rates employed
in this study.
SUMMARY
-
-prepared Preannealed
As
---.
I .70
1.60
I.50 103
K/T,
Fig.. 3. Kissinger plot for as-prepared and preannealed samples. t unit is K s-’ and T,,, unit is K.
thermal experiments. We assume that this preanneal eliminates the effect of incubation in the constant heating rate experiments so as to permit direct comparisons with theory. (Recall that incubation is not described by the JMA theory, although the subsequent reaction is in the isothermal case.) The resulting values for E and In.4 are 227 kJ/mol and 40.4 respectively. These compare well with the isothermal values E = 221 kJ/mol and InA = 39.2. As a further check on the adequacy of the generalized Johnson-Mehl-Avrami equation for describing constant heating rate experiments, the DSC peak positions and shapes were computed from equation (6b) using the values of E, A, and n determined from the isothermal measurements. Figure 4 shows typical computed and experimental results obtained in constant heating rate experiments including the efiect of preannealing through the incubation time. Generally. for preannealed samples the DSC peak positions are within 2K of predicted positions, the full width at half maximum (FWHM) is within IK, and the predicted asymmetry is observed. Un-
8 _Thaory
-
Ar-pn~nd Pnann*al*d
_
-- -. -----
6-
‘\
610
620
I 630
‘.. 640
‘\
a\
__ 650
T(K)
Fig. 4. Experimental and theoretical DSC trace for 50 K/min heating rate. Parameters used in the theoretical DSC trace for 50 K/min heating rate. Parameters used in the theoretical curve were obtained from isothermal data (E = 221 kJ/mol, InA = 39.2 and ii = 3.09). Area of thebretical c&e is. normalized to unity. Peai heights of the experimental curves were scaled to match the theoretical curve. Experimental data are shown for as-prepared and
preannealed samples.
AND CONCLIJSIONS
The JMA theory provides a satispdctory description of isothermal transformations as illustrated in Fig. 1. Thus, the JMA equation, equation (1) or (6b) in Ref. [IO], (rather than the order of reaction equation) was adopted as a basis for the study of nonisothermal transformations. A generalization. appropriate for an arbitrary temperature-time history was required. As described earlier and discussed in the Appendix, the degree of reaction s(t) must be a functional of the function r(t’) for all I’ < 1. (An analogous case is found in Cahn’s additivity criterion [ 191for analyzing the progress of a reaction along an arbitrary path in a T-T-T diagram.) Surprising results of this analysis (and that provided by Henderson [IO]) are that in the constant heating rate case, the descriptive equation, equation (7), is essentially independent of the JMA exponent n and that the Kissinger equation holds. Thus, although the basic equation in Kissinger’s analysis of homogeneous transformations is indeed inappropriate for heterogeneous solid state transformations, the Kissinger method can be applied to the analysis of heterogeneous transformations. Also, we can understand why the Kissinger method had previously been successfully applied to the fitting of constant heating rate data in many studies of heterogeneous solid state reactions. The validity of the theoretical description of the non-isothermal transformation case is indicated by the agreement seen in the kinetic parameters extracted from the isothermal and from the constant heating rate measurements in NiP powders (Table I). This agrccmcnl is pclrticularly good for llic samples in which the effects of incubation were removed by preannealing; the values for E and Inn agree within 3% which is within the limit of error inherent in the measurements. Another way of assessing the nonisothermal theory is to use the kinetic parameters deduced from isothermal studies to predict constant heating rate DSC curves. The close agreement between the observed and predicted DSC peak temperature T,,,, full width at half maximum FWHM, and asymmetry then stand as support for the correctness of the theoretical approach. We attribute the somewhat poorer results on the unannealed samples to the fact that the JMA equation does not describe incubation effects. An incubation time can be expected to inhibit the transformation on the low temperature side of the reaction rate peak in a constant heating rate experiment resulting in a narrower less skewed peak, as observed.
1058
MEISEL
and CO-I-E.
NON-ISOTHERMAL
TRANSFORMATION A similar
constant
KINETICS heating
rate DSC
study was
performed
by Greer
amorphous
FeB which exhibits no incubation
Greer
(171 on the crystallization
[I71 made a numerical
extension
of the JMA
lines given here. consistent
with
His
(rather
isothermal numerical
our equations
excellent agreement
01
eliects.
than analytic)
theory
along
the
results,
which
are
(5) and (6), were in
with the data in a-FeB.
The principal results may be summarized as follows: (i) Henderson’s assumption that the JMA reaction rate equation holds for arbitrary temperature history (Ref.[ IO]) and our assumption that the degree of reaction is determined by the number of atom movements for arbitrary temperature history as described herein lead to equivalent theories and, in the constant heating rate case, indicate that the Kissinger method can be used to extract kinetic parameters. (ii) Our results obtained in samples exhibiting incubation effects in the isothermal case are seen to be approximately described by the theory. Moreover, it is shown that when incubation effects are eliminated (by preannealing) detailed agreement with theory is obtained. Experimental results of Greer in a-FeB [18] can be described in a direct way in terms of generalized JMA theory. In fact, in Greer’s work and in experiments in preannealed metastable our nickel-phosphorus alloys, the DSC peak positions, FWHM, and asymmetry obtained in constant heating rate measurements are in remarkable agreement with theoretical predictions based upon kinetic parameters deduced from isothermal experiments. Also the kinetic parameters [20] deduced from constant heating rate experiments and those deduced from isothermal experiments agree within experimental uncertainties. (iii) The commonly voiced objections to application of the Kissinger method can no longer be considered valid in view of the current theory of the constant heating rate case and the simple experimental techniques available to correct for thermal lag effects. Acknowledgements-We appreciate the careful efforts of MS Doris Quinn in obtaining the thermal analysis data and thank Mr George Capsimalis for providing the X-ray analysis. The assistance of Ellen Fogarty in preparing this manuscript is appreciated.
REFERENCES I. 2. 3. 4.
H. H. H. F.
F. E. 0. E.
Kissinger, J. f&s. natn Bur. Sld 57, 217 (1957). Kissinger, Ana/yr. Chem. 29, 1702 (1957). K. Kirchner, Muter. Sci. Eng. 23, 95 (1976). Luborsky and H. H. Liebermann, Appl. Phys.
Lerr. 33, 233 (1978). 5. J. C. Swartz, R. Kossowsky, J. J. Haugh and R. F.
Krause, J. appl. Phys. 52, 3324 (1981). 6. K. H. J. Bushow, J. appl. Phys. 52, 3319 (1981). 7. K. H. J. Bushow and N. M. Beekmans, Ph.vsicu .stutu.~ solidi (a) 60, 193 ( 1980). 8. M. Matsuura. S&l Sf. Cummun. 30, 231 (1979). 9. See. for example, W. W. Wendlant, Thermcd Mrrhods TV/ Au~/ysi.(. 2nd edn. Wiley. New York (1974) and J. H. Sharp in Dijji~renriul Thermui Anulysb (edited by R. C. Mackenzie). Vol. 2. Academic Press. New York (1972).
MEISEL and COTE:
NON-ISOTHERMAL
IO. D. W. Henderson, J. Non-Cr,r.rr. Solids 30, 301 (1979). II. See for example. L. Granasy and T. Kemeny, Thcrmochimica
Acta 42. 289 (1980).
12. W. A. Johnson and R. F. Mehl, Trans. Am. Inst. Min. Engrs 135, 416 (1939) and M. Avrami. J. chum. fhys. 7, I I03 (1939). 13. P. Murray and J. White. Tmns. Br. Ceram. SIC. 54, I51 (1955). 14. M. Abramowitz and 1. Stegun. Handbook q/’ Mathamafical Functions. Dover, New York. 15. R. A. Baxter. in Thermal Analysis (edited by R. F. Schwenker and P. D. Garn) Vol. I. Academic Press, New York (1969). 16. A. L. Greer, Thermochimica Acta 42, 193 (1980). 17. A. L. Greer, Acta metall. 30, I71 (1981). 18. Z. Altounian, Tu Guo-hua, J. Strom-Olsen and W. B. Muir, P/Y_I~s. Rec. B 24, 505 (1981) and references therein. 19. J. W. Cahn, Ac~n metall. 4, 572 (1956). 20. S. Ranganathan and M. von Heimendahl, J. mater. Sci. 16, 2401 (1981) and references therein. 21. J. R. MacCallum and J. Tanner, Nafure 225, 1127 (1970); P. Holba and J. Sestak, 2. Phys. Chem. Neue Folge 80, I (1972); J. Norwisz, Thermochimica Acfa 25, 123 (1978); A. Dutta and M. E. Ryan, Thermochimica Acla 33, 87 (1979) and J. R. MacCallum, Thermochemica
Acla
53, 375 (1982).
22. R. A. Felder and E. P. Stahel,
Nature
228, IO85 (1970).
APPENDIX The controversy
surrounding
the need for a temperature
TRANSFORMATION
KINETICS
1059
partial derivative in the order of reaction equation in the Kissinger analysis arose as follows: The order of reaction equation. equation (9). obtains under isothermal conditions and in equation (9) the reaction rate should properly be written &/&/r. As the temperature varies in a constant heating rdle cxperimcnt. it is often assumed [21] that the expression
for the reaction
rate should be
(Al) Kissinger argued that the second term on the right hand side of equation (I I) must vanish; inclusion of this term would invalidate Kissinger’s methods (Ref. 1221gives a discussion of these questions in Kissinger’s formulation of the homogeneous problem). Similar questions could be raised pertaining to our analysis and that of Henderson for hetergeneous solid state transformations so we provide a brief comment on this issue. (See references [IO,1 l,l7) for additional clarification of this point.) If we assume that equation (A I) is correct, then the degree of reaction x becomes a unique function of I and T given its value at 1 = 0, which implies that the degre-e of reaction at arbitrary time and temperature is independent of the path (independent of T(f ‘) in the interval 0 < f’ d I). This is an absurd result. It says, for example, that an amorphous sample heated to crystallization and then returned to room temperature in time interval t is in the same state as a sample kept at room temperature for time I. Our derivation illustrates clearly that one has x[~,T(f’)] not x(t,T(f)); Kissinger’s neglect of the second terin in equation (I 1) was appropriate as this term does not occur in a proper theoretical analysis.
~crn n~rral/. Vol. 31. No. 7. pp. 1053-1059. 1983 Printed in Greal Hrilain
NON-ISOTHERMAL APPLICATION
U.S.
TRANSFORMATION KINETICS: TO METASTABLE PHASES
L. V. MEISEL and P. J. COTE Army Armament Research and Development Command, Large Caliber Weapon Systems Laboratory, Benet Weapons Laboratory, Watervliet, NY 12189, U.S.A. (Reccir,ed
I5
July 1982; in r&red
,Jbrm I6 Dccetdwr
1982)
Abstract-Constant
heating rate differential scanning calorimetry is frequently employed to study the kinetics of transitions from metastdble phases and, in particular, crystallization of metallic glasses. Such data are analyzed by the Kissinger method. which was originally devised for the study of homogeneous reactions. The consensus in the literature, until recently. was that such applications (i.e. to heterogeneous solid state transformations) of the Kissinger method are not valid. We address the principal objections to these applications and provide alternative derivations of recent theoretical results which demonstrate that the Kissinger method is valid for heterogeneous reactions of the type described by the Johnson-Mehl-Avrami equation in the isothermal case. Isothermal and constant heating rate data on transformation of metastable NIP powders are presented. These experimental results and the discussions presented here help to clarify the effects of incubation times in non-isothermal transformation kinetics and provide a further demonstration of validity of the generalized Johnson-Mehl-Avrami theory for the description of heterogeneous solid state transformations. R&arm&-La calorimetric differentielle a balayage avec vitesse d%chauBage constante est utilisie souvent pour I’etude des cinctiques des transformations de la phase metastabile ct. en particulicr. la cristallisation des glaces metalliques. Ces donn&es sont analydes par la methode Kissinger, developpee originalement pour I’ttude des transformations homogenes. Le consensus dans la litterature jusqu’ a recemment, a et e que ces application (i.e. pour transformations hiterogtnes) de la methode Kissinger ne sont pas valides. Nous adressons les objections principales de ces applications et nous donnons une derivation alternative des resultats thloretiques recents qui dimontre que la methode Kissinger est valide pour des transformations hetirogtnes du type decrites par I’equation du Johnson-Mehl-Avrami pour le cas isotherme. Les donnees des transformations des pandres de NiP metastabile sont presenties. Les rtsultats exp&imentaux et les discours presentes ici aident a clarifier les effect des temps de I’incubation dans les transformations non-isothetmique et its dtmontrent la validitt de la thtorie giniralizb de Johnson-Mehl-Avrami pour la description des transformations hettrogines de l’etat solid.
Zusammenfasauag-Die Umwandlungskinetik metastabiler Phasen, besonders metallischer Glaeser wird oft durch differentielle Kalorimetrie mit einer konstanten Heizrate untersucht. Die dabei erhaltenen Daten werden nach der Methode von Kissinger analysiert, die urspruenglich fuer die Untersuchung homogener Reaktionen entwickelt wurde. Bis vor kurzem wurde im Schrifttum allgemein die Anscht vertreten, dass diese Anwendung der Kissinger-Methode feur heterogene Phasenumwandlungen nicht zulaessig sei. Wir betrachten die Haupteinwaende gegen diese Methode, und bieten alternative Ableitungen neuerer theoretischer Untersuchungenan, die zeigen, dass die Kissingermethode such fuer heterogene Reaktionen der Art, wie sie durch die Johnson-Mehl-Avrami gueltig ist. Es werden Daten geliefert, die durch
Gleichung
im isothennen
Fall beschrieben
werden,
isotherm&he Untersuchungen, sowie durch konstantes Autheizen von metastabilen Ni-P Pulvem erhalten wurden. Diese experimentellen Ergebnisse und die hier vorgebrachte Diskussion tragen zur Klaerung des Einglusses der lnkubationszeiten auf nicht-isothetme Umwandlungsvorgaenge bei, und liefem einen weiteren Hinweis feur die Gueltigkeit der verallgemeinerten
Johnson-Mehl-Avrami Koerpern.
Theorie
zur
Beschreibung
INTRODUCTION Differential thermal analysis techniques such as differential scanning calorimetry (DSC) provide convenient methods for obtaining information on reaction kinetics. A popular thermal analysis method developed by Kissinger [I,21 determines the kinetic parameters from graphs of the logarithm of the heating rate vs inverse temperature at the maximum of the reaction rate in constant heating rate experiments. This method was frequently used in studies of the crystallization of glassy metals [3-81 despite the
von
heterogenen
Phasenumwandlungen
in festen
fact that literature [9] on thermal analysis techniques reflected a consensus that application of the Kissinger method to solid state reactions is improper. However. the recent work of Henderson [lo] has provided a theoretical basis for the treatment of non-isothermal thermal analysis techniques and justifies the use of the Kissinger method for many solid state transformations. The three main objections to the use of this method for study of solid state reactions were: (I) thermal gradients are inherent in non-isothermal methods. Thus, it was claimed that significant inaccuracy will result from the application of the Kissinger
I053
II)? 4
MEISEL. ;tntl <‘OTE:
NON-ISOT11ERMAL
TRANSFORMATION
mc~hod which does not allow for the presence of
whcrc A is constant
tempcrnture gradients. (2) The reaction rate equation which is ilp~~lX~priillC liw iS0lllCrnlill cxpcrimcnts is
absolute
in the Kissinger
msunletl
analysis.
arpucd thaI ;I term involving derivative isothermal
must be included in the analysis cxpcrimcnts;
in the literature appropriate
partial of non-
this point has been debated
for over a decade. (3) The order of
reaction equation is
It is frequently
the temperature
assumed
for
in the Kissinger analysis
homogeneous
transformations,
(e.g. chemical reactions in a gas) but is not valid
for the heterogeneous transformations which generally occur in solid state reactions. Regarding the first objection, we describe simple procedures to reduce the influence of temperature gradients to negligible levels. The confusion in the literature surrounding the proper form of the reaction rate equation in the Kissinger analysis [objection (2)] results from the assumption that the progress of a .reaction can be described as a simple function of the time and temperature. The degree of reaction is clearly a functional [II], dependent on the temperature history, and not a simple function. (We address this question in the Appendix.) However, a result of the analysis presented here and the principle assumption in Henderson’s work is that the reaction rare is an ordinary function of the temperature and the degree of reaction. Our data show that the third objection is valid. Nevertheless, Henderson [IO] has shown that the Kissinger method can be applied to the analysis of many heterogeneous reactions. We provide an alternate treatment of non-isothermal transformation kinetics here, which indicates that the Kissinger method can be applied to any reaction of the type described by the Johnson-Mehl-Avrami (JMA) equation [12] in the isothermal case. Our treatment (as does that of Henderson) goes as follows: (i) generalize the JMA approach to deal with non-isothermal, heterogeneous reactions and (ii) demonstrate that in the constant heating rate case (within negligible errors) the Kissinger relationship obtains. Finally, isothermal and constant heating rate data on the transformation of metastable NiP powders are presented. Results on the influence of incubation times on constant heating rate experiments are included. The data are in excellent agreement with the theoretical results. GENERALIZATION OF THE JOHNSON-MEHLAVRAMI EQUATION
KINETICS pm-exponential
factor,
7‘ is the
temperature.
and E = k,O is an activation cncrgy for the tr;lnsli,rm;ltion process with k,, the Boltzmann
constant.
isothermal
processes so k(T)
Note that equation (I) dcscribcs is a constant
(which
depends on the temperature)
in equation (I). An expression for the reaction rate dx/dr can be derived directly from equation (I) and expressed as a simple function of the temperature and the degree of reaction. We are interested in generalizing equation (I) to treat experiments in which temperature is a function of time. If we assume that the transformation products and mechanism do not change with temperature, then it is reasonable to interpret k(T)1 in equation (I) as being proportional to the number of atomic jumps within the interval I at temperature T. If we assume that the progress of the transformation is determined by the number atomic jumps in the general (nonisothermal) case as well, then equations (I) and (2) generalize to x[r,T(r’)] = I - exp( - W”)
(3)
where W[f,T(f’)] - W, =
=
‘k[T(t’)]dr’ I0
k(T)(dT/dr’)-‘dT
(4)
and k(T) is still given by equation (2). T(t) is the temperature at t, n is the JMA exponent which depends on the nucleation and growth mechanisms, and W. and To are the temperature and W value at t = 0. Note that x[r,T(r’)] depends on r and the temperature history T(t’) for times r’ earlier than r and the same is true for W[r;T(r’)]. That is x[r,T(r’)] and W[t,T(r’)] are nor determined by r and the temperature at r as is clear from equation (3) and (4). Henderson [lo] assumed that the reaction rate in the non-isothermal case can be expressed as the same simple function of temperature and degree of reaction as in the isothermal case. He then integrates to obtain equation (3). Thus, it is seen that Henderson’s theory, which is based upon an assumption concerning the reaction rate, is equivalent to the present treatment which assumes that the progress of a reaction is determined by the number of atom movements. We now consider the constant heating rate case: dT/dt’ s T is constant and comes out of the integrand in equation (4). Thus T
The Johnson-Mehl-Avrami equation [12] describes a wide variety of isothermal solid state transformations and has the form s(r) = I - exp[-(k(T)/)“] k(T) = A exp( -O/T)
(1) (2)
W(T)-
W,=(A/T)
e-“/rdT s TII
~~[1-2(;)+6(;)1]
(5)
where we have integrated by parts as described in Reference [ 131and assumed that k( 7’,,)Ti is negligible
MEISEL
and COTE:
NON-ISOTHERMAL
in comparison to k(T)T*. Note that in the constant heating rate case we represent the functions x and W as simple functions of either the time or the temperature. Note also that equation (5). derived in the spirit of Kissinger’s work, is the “asymptotic expression” for the exponential integral as given in equation 5 1.51 of Abramowitz and Stegun [I41 while the approximate form given by Henderson is the “large n” approximation given in equation 51.52 of Ahromowitz and Stegun [14]. They are, for practical purposes, equivalent. In a constant heating rate DSC experiment, one measures a temperature rise which is assumed to be proportional to dx/dl as a function of T or t. From equation (5) and (4) one finds dx Z=e
- *” n W” dln W/dt
=(I -x)k(T)nW”-’ (equation (6b) wiih W expressed as a function of x is the starting point in Reference [lo]). We need to know how the temperature T, for the maximum reaction rate varies with the heating rate in order to employ the Kissinger method. The maximum reaction rate will occur for T such that d21nW -=n(W”dt2
1)
or, taking derivatives of equation (5) in the case that W, = 0, and keeping terms to order T,,@ -p+lu($)+ln(~)-Z~(l--$). m
(7)
Note that equation (7) reduces to the Kissinger expression for the n = 1 case as we might have anticipated since this corresponds to the homogeneous reaction case. Moreover, for metastable phases such as amorphous metals, the right hand side of equation (7) is generally negligible in comparison to the individual terms on the left hand side for common heating rates (2 l~‘C/min~. Thus, we see that the Kissinger method is appropriate for the anatysis not only of homogeneous reactions, but also for the analysis of heterogeneous reactions which are described by the Johnson-Mehl-Avrami equation in isothermal experiments. In particular, equation (7) implies dln(~/T~)/d(l/~~,)
z -8.
(8)
The approximation in equation (8) (RHS equation (7) =0) might introduce a 3% error in the value of 0 in the worst cases. (Typically, n > I,8 > 25 x lO’K, and T,, c 1.0 x 103K which suggests that the error introduced in 6 by setting the RHS of equation (7) = 0 is considerably less than I%.) Equation (7) also serves to determine the pre-exponential factor A from the intercept of a ln(f/Ti) vs (l/T,,,) plot. Equation (6), which describes the time (or temperature) de-
TRANSFORMATION
KlNETICS
I055
pendence of the reaction rate, and equation (7) and (8). which allow for the simple extraction of the parameters A and 0 by means of the Kissinger method, form the basis for the analysis of constant heating rate data. EXPERIMENTAL
PROCEDURES
AND RESULTS
The data were obtained with a DuPont 990 Thermal Analysis System and DSC module. Aluminum sample pans and an argon atmosphere were used in all runs. An empty sample pan served as a reference. The metastable NiP powders (< 325 mesh) were prepared by an electroless process. The powder composition (85 at.% Ni) was selected so that only one transformation [NiP (metastable F.C.C.)N&P + Ni] occurs; this avoids complications associated with the occurrence of several competing reactions. Isothermal transformation kinetics data for the metastable NiP powders were obtained by monitoring the time dependence of the DSC output for a series of temperatures. The DSC output IDsc is assumed to be proportional to the reaction rate dx/dt (IDsc = cdx/dt, c is constant) so the degree of reaction x(t) is given by x(t)=s(t)/S where I hE.c dT = c[x(t) -x(O)] 4~) = s0 and S = S(M)). The technique for extracting the time exponent n and k(T) in an isothermal experiment described by equation (I) is to ptot lnln(S/[S-s(t)]] vs lnt, so that n is the slope and k(T) = t;’ where to is the time in seconds when lnln (S/[S-s(t)]} = 0. Equation (8) shows that the constant heating rate method {Kissinger method) may be used to obtain values of the kinetic parameters, when the isothermal data are described by the JMA equation. Before obtaining the constant heating rate data, we examined the effects of thermal gradients in our DuPont DSC apparatus. The standard model [15] describing the DuPont DSC output Y contains the correction AY where
A Y = -c,JR, + R,) d Yjdt where c,~ is the sample’s heat capacity, R,, is the thermal resistance between the sample and the sample platform, and R, is the thermal resistance between the sample platform and heater. This term leads to a distortion of the DSC peaks and shifts the peaks to higher temperatures. If this model is valid, it suggests that errors can be avoided either by working with small sample masses (small c.~)or by repeating the experiments using different masses and extrapolating to zero mass (zero c,). In practice, this method does not work and the model apparently is inadequate. For example, we observed that in series of determinations for different sample masses, the melting point of Sn and the Curie temperature of Ni, did not extrapolate to the correct temperatures at zero mass
1036
MEISEL
(the correct temperature temperature
at zero
and COTE:
NON-ISOTHERMAL
TRANSFORMATION
KINETICS
is taken to be the transition
heating
rate).
Similar
results
obtained with the DuPont instrument by Greer [16] were attributed to the heat capacity of the sample pan which he assumes is included in c,,.Our results do not support this explanation,:,however, since we find that thermal lag is reduced by only about 30% with the sample and reference pans removed. We therefore conclude that methods for thermal lag corrections based on this model are unreliable and we employ a more straightforward method similar to that used by Greer [ 171for the Perkins-Elmer DSC system; that is, we empirically determined the thermal lag vs heating rate by using the Curie temperature of Ni as well as the melting point of Sn. For solid samples in our DuPont DSC cells the lag AT is linear with heating rate F, insensitive to sample mass (provided we maintain a large contact area with the sample platform) and the ratio AT/f, using sample and reference pans, is measured to be 0.05 min. Powder samples exhibit effects which are distinct from those described above for solid samples: the observed thermal lags increase linearly with sample mass, presumably because of the inherent low thermal diffusivity of powder samples. We avoided this difficulty with powders by working with small samples (e.g. < 1 mg at lO”C/min.). Thus, although criticism 1 of the Kissinger method is valid, the method is still practical.
RESULT!3 AND DISCUSSION
The DSC measurements exhibit an “incubation time” riw, during which no observable transformation occurs. We find that the temperature dependence of ri, can be described as an activated process with E = 279 kJ/mol, near that for the phase transformation; this is often seen in studies of crystallization of metallic glasses [I 81. In the isothermal experiments, after fi”E,the DSC output (dx/dr) increases from zero, proceeds through a maximum, and returns to zero. This result is inconsistent with the predictions of the order of reaction equation assumed by Kissinger for homogeneous reactions, dx/df = k(l -x)”
P
I -
0h
T * -IL
k-2 c
-3 -4
-
I
I
I
1
I
I
4
5
6
7
a
9
II-I It-,,,,
Fig. I. Plots for extracting the JMA time exponent II in equation (I), as described in the text. Characteristic parameters are listed in Table I for the corresponding curve number.
value of 3.09. The deviation from linearity at long times may reflect breakdown of the theory for x(t) at the end of the transformation.
The logarithm of the rate constant k, obtained from Fig. I, is plotted against l/T in Fig. 2. The slope of the line, obtained from a least squares fit, gives E = kg0 = 221 kJ/mol for the activation energy and In A = 39.2, for A in s-‘. There is very little scatter about the line in Fig. 2 despite the inherent experimental difficulties in establishing accurate baselines with this method. The precautions and corrections described in the previous section were applied to constant heating rate data obtained for the same NiP powders used in the isothermal measurements. The resulting Kissinger plot is shown in Fig. 3. An excellent straight line with little scatter is given yielding E = 220 kJ/mol and pre-exponential factor A = exp(38.4)s-‘. Also shown in Fig. 3 are constant heating rate data obtained on a batch of these NiP powders which had been previously annealed through the incubation time (15.5 min at 295°C) as determined in the iso-
(9)
where m is called the order of reaction and k has been previously defined. The reaction rate given in equation (9) is maximum at t = 0 and decreases monotonically with increasing time. Thus, the present results support the third of the three principal criticisms of the application of the Kissinger method to heterogeneous solid state transformations as stated in the introduction. The DSC traces for a series of temperatures were integrated to yield plots of In[-ln( I -s/S)] vs In(t - r,,) which are shown in Fig. I. Reasonably good straight lines are obtained; the n value determined from Fig. I range from 2.8 to 3.2 with a mean
Fig. 2. The logarithm of the rate constant k deduced from the isothermal measurements is plotted against T ’ (k is in s- ‘).
MEISEL
and COTE:
NON-ISOTHERMAL
TRANSFORMATION
KINETICS
1057
annealed samples exhibit narrower and less skewed DSC peaks than given by the generalized JMA theory. Table I summarizes the predicted and observed values of T,, and FW H M for the heating rates employed
in this study.
SUMMARY
-
-prepared Preannealed
As
---.
I .70
1.60
I.50 103
K/T,
Fig.. 3. Kissinger plot for as-prepared and preannealed samples. t unit is K s-’ and T,,, unit is K.
thermal experiments. We assume that this preanneal eliminates the effect of incubation in the constant heating rate experiments so as to permit direct comparisons with theory. (Recall that incubation is not described by the JMA theory, although the subsequent reaction is in the isothermal case.) The resulting values for E and In.4 are 227 kJ/mol and 40.4 respectively. These compare well with the isothermal values E = 221 kJ/mol and InA = 39.2. As a further check on the adequacy of the generalized Johnson-Mehl-Avrami equation for describing constant heating rate experiments, the DSC peak positions and shapes were computed from equation (6b) using the values of E, A, and n determined from the isothermal measurements. Figure 4 shows typical computed and experimental results obtained in constant heating rate experiments including the efiect of preannealing through the incubation time. Generally. for preannealed samples the DSC peak positions are within 2K of predicted positions, the full width at half maximum (FWHM) is within IK, and the predicted asymmetry is observed. Un-
8 _Thaory
-
Ar-pn~nd Pnann*al*d
_
-- -. -----
6-
‘\
610
620
I 630
‘.. 640
‘\
a\
__ 650
T(K)
Fig. 4. Experimental and theoretical DSC trace for 50 K/min heating rate. Parameters used in the theoretical DSC trace for 50 K/min heating rate. Parameters used in the theoretical curve were obtained from isothermal data (E = 221 kJ/mol, InA = 39.2 and ii = 3.09). Area of thebretical c&e is. normalized to unity. Peai heights of the experimental curves were scaled to match the theoretical curve. Experimental data are shown for as-prepared and
preannealed samples.
AND CONCLIJSIONS
The JMA theory provides a satispdctory description of isothermal transformations as illustrated in Fig. 1. Thus, the JMA equation, equation (1) or (6b) in Ref. [IO], (rather than the order of reaction equation) was adopted as a basis for the study of nonisothermal transformations. A generalization. appropriate for an arbitrary temperature-time history was required. As described earlier and discussed in the Appendix, the degree of reaction s(t) must be a functional of the function r(t’) for all I’ < 1. (An analogous case is found in Cahn’s additivity criterion [ 191for analyzing the progress of a reaction along an arbitrary path in a T-T-T diagram.) Surprising results of this analysis (and that provided by Henderson [IO]) are that in the constant heating rate case, the descriptive equation, equation (7), is essentially independent of the JMA exponent n and that the Kissinger equation holds. Thus, although the basic equation in Kissinger’s analysis of homogeneous transformations is indeed inappropriate for heterogeneous solid state transformations, the Kissinger method can be applied to the analysis of heterogeneous transformations. Also, we can understand why the Kissinger method had previously been successfully applied to the fitting of constant heating rate data in many studies of heterogeneous solid state reactions. The validity of the theoretical description of the non-isothermal transformation case is indicated by the agreement seen in the kinetic parameters extracted from the isothermal and from the constant heating rate measurements in NiP powders (Table I). This agrccmcnl is pclrticularly good for llic samples in which the effects of incubation were removed by preannealing; the values for E and Inn agree within 3% which is within the limit of error inherent in the measurements. Another way of assessing the nonisothermal theory is to use the kinetic parameters deduced from isothermal studies to predict constant heating rate DSC curves. The close agreement between the observed and predicted DSC peak temperature T,,,, full width at half maximum FWHM, and asymmetry then stand as support for the correctness of the theoretical approach. We attribute the somewhat poorer results on the unannealed samples to the fact that the JMA equation does not describe incubation effects. An incubation time can be expected to inhibit the transformation on the low temperature side of the reaction rate peak in a constant heating rate experiment resulting in a narrower less skewed peak, as observed.
1058
MEISEL
and CO-I-E.
NON-ISOTHERMAL
TRANSFORMATION A similar
constant
KINETICS heating
rate DSC
study was
performed
by Greer
amorphous
FeB which exhibits no incubation
Greer
(171 on the crystallization
[I71 made a numerical
extension
of the JMA
lines given here. consistent
with
His
(rather
isothermal numerical
our equations
excellent agreement
01
eliects.
than analytic)
theory
along
the
results,
which
are
(5) and (6), were in
with the data in a-FeB.
The principal results may be summarized as follows: (i) Henderson’s assumption that the JMA reaction rate equation holds for arbitrary temperature history (Ref.[ IO]) and our assumption that the degree of reaction is determined by the number of atom movements for arbitrary temperature history as described herein lead to equivalent theories and, in the constant heating rate case, indicate that the Kissinger method can be used to extract kinetic parameters. (ii) Our results obtained in samples exhibiting incubation effects in the isothermal case are seen to be approximately described by the theory. Moreover, it is shown that when incubation effects are eliminated (by preannealing) detailed agreement with theory is obtained. Experimental results of Greer in a-FeB [18] can be described in a direct way in terms of generalized JMA theory. In fact, in Greer’s work and in experiments in preannealed metastable our nickel-phosphorus alloys, the DSC peak positions, FWHM, and asymmetry obtained in constant heating rate measurements are in remarkable agreement with theoretical predictions based upon kinetic parameters deduced from isothermal experiments. Also the kinetic parameters [20] deduced from constant heating rate experiments and those deduced from isothermal experiments agree within experimental uncertainties. (iii) The commonly voiced objections to application of the Kissinger method can no longer be considered valid in view of the current theory of the constant heating rate case and the simple experimental techniques available to correct for thermal lag effects. Acknowledgements-We appreciate the careful efforts of MS Doris Quinn in obtaining the thermal analysis data and thank Mr George Capsimalis for providing the X-ray analysis. The assistance of Ellen Fogarty in preparing this manuscript is appreciated.
REFERENCES I. 2. 3. 4.
H. H. H. F.
F. E. 0. E.
Kissinger, J. f&s. natn Bur. Sld 57, 217 (1957). Kissinger, Ana/yr. Chem. 29, 1702 (1957). K. Kirchner, Muter. Sci. Eng. 23, 95 (1976). Luborsky and H. H. Liebermann, Appl. Phys.
Lerr. 33, 233 (1978). 5. J. C. Swartz, R. Kossowsky, J. J. Haugh and R. F.
Krause, J. appl. Phys. 52, 3324 (1981). 6. K. H. J. Bushow, J. appl. Phys. 52, 3319 (1981). 7. K. H. J. Bushow and N. M. Beekmans, Ph.vsicu .stutu.~ solidi (a) 60, 193 ( 1980). 8. M. Matsuura. S&l Sf. Cummun. 30, 231 (1979). 9. See. for example, W. W. Wendlant, Thermcd Mrrhods TV/ Au~/ysi.(. 2nd edn. Wiley. New York (1974) and J. H. Sharp in Dijji~renriul Thermui Anulysb (edited by R. C. Mackenzie). Vol. 2. Academic Press. New York (1972).
MEISEL and COTE:
NON-ISOTHERMAL
IO. D. W. Henderson, J. Non-Cr,r.rr. Solids 30, 301 (1979). II. See for example. L. Granasy and T. Kemeny, Thcrmochimica
Acta 42. 289 (1980).
12. W. A. Johnson and R. F. Mehl, Trans. Am. Inst. Min. Engrs 135, 416 (1939) and M. Avrami. J. chum. fhys. 7, I I03 (1939). 13. P. Murray and J. White. Tmns. Br. Ceram. SIC. 54, I51 (1955). 14. M. Abramowitz and 1. Stegun. Handbook q/’ Mathamafical Functions. Dover, New York. 15. R. A. Baxter. in Thermal Analysis (edited by R. F. Schwenker and P. D. Garn) Vol. I. Academic Press, New York (1969). 16. A. L. Greer, Thermochimica Acta 42, 193 (1980). 17. A. L. Greer, Acta metall. 30, I71 (1981). 18. Z. Altounian, Tu Guo-hua, J. Strom-Olsen and W. B. Muir, P/Y_I~s. Rec. B 24, 505 (1981) and references therein. 19. J. W. Cahn, Ac~n metall. 4, 572 (1956). 20. S. Ranganathan and M. von Heimendahl, J. mater. Sci. 16, 2401 (1981) and references therein. 21. J. R. MacCallum and J. Tanner, Nafure 225, 1127 (1970); P. Holba and J. Sestak, 2. Phys. Chem. Neue Folge 80, I (1972); J. Norwisz, Thermochimica Acfa 25, 123 (1978); A. Dutta and M. E. Ryan, Thermochimica Acla 33, 87 (1979) and J. R. MacCallum, Thermochemica
Acla
53, 375 (1982).
22. R. A. Felder and E. P. Stahel,
Nature
228, IO85 (1970).
APPENDIX The controversy
surrounding
the need for a temperature
TRANSFORMATION
KINETICS
1059
partial derivative in the order of reaction equation in the Kissinger analysis arose as follows: The order of reaction equation. equation (9). obtains under isothermal conditions and in equation (9) the reaction rate should properly be written &/&/r. As the temperature varies in a constant heating rdle cxperimcnt. it is often assumed [21] that the expression
for the reaction
rate should be
(Al) Kissinger argued that the second term on the right hand side of equation (I I) must vanish; inclusion of this term would invalidate Kissinger’s methods (Ref. 1221gives a discussion of these questions in Kissinger’s formulation of the homogeneous problem). Similar questions could be raised pertaining to our analysis and that of Henderson for hetergeneous solid state transformations so we provide a brief comment on this issue. (See references [IO,1 l,l7) for additional clarification of this point.) If we assume that equation (A I) is correct, then the degree of reaction x becomes a unique function of I and T given its value at 1 = 0, which implies that the degre-e of reaction at arbitrary time and temperature is independent of the path (independent of T(f ‘) in the interval 0 < f’ d I). This is an absurd result. It says, for example, that an amorphous sample heated to crystallization and then returned to room temperature in time interval t is in the same state as a sample kept at room temperature for time I. Our derivation illustrates clearly that one has x[~,T(f’)] not x(t,T(f)); Kissinger’s neglect of the second terin in equation (I 1) was appropriate as this term does not occur in a proper theoretical analysis.