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Isothermal decomposition kinetics of transformed-beta phase in a titanium-nickel alloy
ISOTHERMAL
DECOMPOSITION
KINETICS
OF TRANSFORMED-BETA
IN A TITANIUM-NICKEL D. H. POLONISt
and
J. GORDON
PHASE
ALLOY* PARRi
The decomposition kinetics of the transformed-8 phase during isothermal heat-treatment has been studied by measuring the rate of precipitation of TizNi. The reaction follows the type of equation suggested by Johnson and Mehl, and an activation energy of 84,000 Cal/mole has been determined. On the basis of the values of the coefficient “n” it is suggested that TirNi precipitates as plates at temperatures up to 525°C. Above this temperature there appears to be a tendency for the formation of spheroidal precipitates. A model for the reaction is proposed in which the rate controlling process is the self-diffusion of titanium. CINBTIQUE
DE
LA D&OMPOSITION ISOTHERME ALLIAGE TI-NI
DE LA PHASE-0
DANS
UN
La cinetique de la d&omposition de la phase-8 pendant un traitement isotherme est Ctudiee en mesurant la vitesse de precipitation de Ti2Ni. La reaction suit l’bquation de Johnson et Mehl et une energie d’activarion de 84.000 cal/mol a 6tt5 dCtermin6e. De la valeur de l’exposant “n,” il semble que Ti2Ni precipite en plaquettes jusqu’g 52.5”; au-dessus de cette temperature, il y aurait tendance $ une precipitation en sphhres. 11 est sugg&C que la reaction est contr816e par I’autodiffusion du titane. DIE
KINETIK
DER ISOTHERMEN BETA-PHASE EINER
ENTMISCHUNG DER UMGEWANDELTEN TITAN-NICKEL-LEGIERUNG
Die Kinetik der Entmischung der umgewandelten &Phase bei einer isothermen Wgrmebehandlung wurde durch Messung der .Ausscheidungsmenge von TisNi untersucht. Die Reaktion verlluft nach dem Gyp der Gleichung von Johnson und Mehl. Die Aktivierungsenergie wurde zu 84.000 cal/mol bestimmt. Auf Grund des Koeflizienten “lt” wird vermutet, dass sich Ti,Ni bei Temperaturen bis 525°C plattenfijrmig ausscheidet. Oberhalb dieser Temperatur eine Neigung zur Bildung von sph$rolitischen Ausscheidungen zu bestehen. Es wird ein Mode11 fiir die Reaktion vorgeschlagen, bei dem der massgebende Vorgang fiir den Reaktionsablauf die Selbstdiffusion des Titans ist.
INTRODUCTION
other titanium alloys. g,lo,ll The absence of any other phase was confirmed by X-ray diffraction studies. The 7.2 per cent nickel alloy was selected for the present study sin&, on tempering to produce equilibrium structures, phase-ratios could be assessed more accurately than with lower compositions. In addition, this alloy corresponds approximately to eutectoid composition. (See phase diagram, Fig. 2.) The phase diagram shows that the approach towards equilibrium during tempering yields the phase TizNi.
The tempering kinetics of some important ferrous alloys have been explored in detaillH5 by applying theories of diffusion and precipitation developed by Johnson and Meh16 and Zener.’ The work has included decomposition studies of retained and martensitically transformed austenite as well as graphitization in castirons. Most of the research has been concerned with interstitial alloying elements in materials of complex commercial composition. The present work is an analysis of the tempering kinetics of transformed-@ solid solution (hexagonal close-packed) in a 7.2 weighlper cent (6 per cent atomic) nickel alloy of titanium. It was possible to produce 100 per cent transformed-p (called a’) in titanium-nickel alloy powders (-200 mesh) containing up to 7.2 per cent nickel8 by quenching them from 1,OOO”C with a blast of purified argon gas. Specimens containing more nickel showed some retained 0 after quenching. Metallographic examination of completely transformed specimens revealed pronounced strain patterns (Fig. 1) which are typical of martensitic structures observed in * Received December 13, 19.54. t School of Mineral Engineering, University of Washington, Seattle, Washington. Formerly, Department of Mining and Metallurgy, The University of British Columbia, Vancouver, Canada. $ Department of Mining and Metallurgy, University of Alberta, Edmonton, Canada. Formerly, Department of Mining and Metallurgy, The University of British Columbia, Vancouver Canada. ACTA
METALLURGICA,
VOL.
3, JULY
1955
FIG. 1. Magnification 2000X. Etchant 5 per cent HF in glycerin, followed by nitric acid rinse. 7.2 per cent Nialloy. Powder quenched in argon gas from 1ooO”C 307
ACTA
308
FIG. 2. Phase
diagram of the titanium-nickel after Margolin el a1.13
METALLURGICA,
system
During the formation of TizNi the (Y’phase is depleted of nickel until its composition conforms to that of equilibrium a! at the tempering temperature. It probably contains less than 0.1 per cent nickel at 500°C. EXPERIMENTAL
METHOD
The titanium alloy was prepared from iodide titanium and Johnson Matthey spectrographic standard nickel by levitation-melting.12 The method by which filed powders were quenched has been described elsewhere,g and the usual precautions against contamination were observed. Sufficient quenched powder of the 7.2 per cent nickel alloy was prepared for the subsequent isothermal heat-treatments which were made upon samples sealed in evacuated fine bore silica tubing. Temperature control during tempering was =F~‘C. Where heat-treatments were of less than 15 minutes’ duration, a thermocouple was attached to the specimen-tube and time was measured from the instant the thermocouple reached temperature. In all cases temperature was attained in 1.5 to 20 seconds. The extent of the tempering reaction after heattreatment was determined by computing from X-ray dam the amount of TizNi formed. In order to assess this quantitatively the ratio of the line intensities TiZNirJ33: (~~01was first obtained on Geiger-spectrometer plots for each of several Ti-Ni alloys ranging from 0 to 10 per cent nickel which had been heat-treated to produce equilibrium structures. The percentage TizNi for each alloy was determined from the phase diagram
FIG. 3. Graph
showing percentage by weight formed versus log10 time.
of TizNi
VOL.
3,
1955
and plotted against the corresponding ratio of intensities Ti2Nir,rP3 :01101.Hence, relative line intensities obtained after heat-treating the quenched alloy could be translated into phase ratios. It should be noted that both the 101 (Yline and 101 (Y’line are coincident about 8=20”. Consequently, it is necessary to assume that a! and (Y’ phases have equal structure factors. This introduces a small error since (Y’contains 6 atomic per cent nickel in the alloy used, and its structure factor differs from that of equilibrium (Y,which is practically pure titanium. However, calculations based on the assumption of random nickel distribution in (Y’reveal the error to be less than 2 per cent-a figure that is no greater than the inherent errors involved in the technique. EXPERIMENTAL
RESULTS
The tempering reaction in the alloy was followed at temperatures of 450°C, SOO”C, 525°C and 550°C. The weight percentages of TizNi formed during the reaction are plotted as a function of log10 time in Fig. 3. The curves possess the familiar ‘Y-shape (although the early
FIG. 4. Magnification 2000X. Etchant 5 per cent HF in glycerin, followed by nitric acid rinse. 7.2 per cent Ni alloy. Powder quenched and transformed 90 per cent at 500°C.
stages of the reaction could not be measured) and may be Ireasonably superimposed by translation along the log10 time axis. Once the reaction is initiated the amount of TisNi formed is approximately proportional to log10 time up to the concluding stages of the reaction (i.e., when more than 14 per cent TizNi has precipitated) at which juncture the rate decreases quite appreciably. Complete transformation would be at 18.9 per cent TizNi. Tempered specimens were examined metallographically in order to detect any changes due to diffusion and precipitation. The original strained structure appeared to be transformed to a Widmanstgtten precipitation type. Figure 4 shows a typical microstructure for 90 per cent transformation at 500°C. The structure is fine and acicular. Since TizNi is not resolvable at a magnification of 2,000 times, it is presumably very finely dispersed. At 550°C the structure is a little coarser
POLONIS
AND
PARR:
DECOMPOSITION
KINETICS
309
The coefficients k and n are constants for a particular process. If logarithms are taken in Eq. (2), j(t)) = - kt”
loge(l-
(4)
and k loglo--+a 2.3
1 log10 loglo-=
l-j(t)
loglot-
(5)
Cohen et al.’ have proposed the following equation which serves to define the specific rate constant K and is merely another form of Eq. (2) : df (0 --=K(ldt FIG. 5. Magnification 2000X. Etchant 5 per cent HF in glycerin, followed by nitric acid rinse. 7.2 per cent Ni alloy. Powder quenched and transformed 90 per cent at 55O’C.
j)l”,
(6)
where m=n1. Comparison of Eqs. (3) and (6) shows nk=K.
of TizNi can be seen. Microscopic examination of a specimen heated at 750°C for 15 minutes revealed a spheroidal precipitate of TizNi in a matrix of (Y(see Fig. 6). (Fig.
S), but no definite
precipitation
DISCUSSION
Substituting
--~.X~Ga*L4), (
where j(i) represents the fraction transformed A more general form of Eq. (1) is
K
1
loglo--+n
l-f(O
The general shape of the curves in Fig. 3 suggests a type of nucleation and growth process that was quantitatively analysed by Johnson and Mehl.6 They proposed an equation of the following form for a rate of nucleation ATand a rate of growth G:
j(t)=l-exp
Eq. (7) in (S),
log,, log10-=
OF RESULTS
(-Kin),
dt
(8)
I
I
(1) in time 1. --f(t)] oersu~ loglo time.
(2)
which yields
df (0 ---~~~t”-‘.e-~l”=~n~“-l(1_j).
loglot.
2.3~~
FIG. 7. Graph of log10log&/l
j(t)=l-exp
(7)
(3)
From (8) it can be seen that if the tempering reaction is to obey the relationship, then a plot of log10 logloW - f (01 versus loglot will yield a straight line. The experimental values for the current investigation are plotted in Fig. 7 and produce reasonably straight lines up to the late stages of the reaction. The curves deviate from linearity at approximately the same ordinate values, suggesting the possibility of some structural feature that inhibits the growth process. Reference to Eq. (8) shows that the slopes of the curves in Fig. 7 are equivalent to the values of n for each temperature. The values are listed in Table I, TABLE I. Temy?ture 450
FIG. 6. Magnification 2ooOX. Etchant 5 per cent HF in glycerin, followed by nitric acid rinse. 7.2 per cent Ni alloy. Powder quenched
and transformed
100 per cent at 750°C.
zz 550
n 0.53 0.51 0.52 0.7
log1o(K/2.3n) -2.40 -1.45 -1.02 -0.8
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310
MET,4LLURGICA,
VOL.
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1955
where it is seen that, except for the value of 0.7 at SSO”C, n is fairly constant about 0.52. The values of loglo(K/2.3n) for each temperature have been obtained from the log time=0 intercepts of Fig. 7, and are listed in Table I. From basic rate theory, it is known that: &=Ae-QHRT,
(9)
where Kt is the specific rate constant in units time-l, Qt is the activation energy, and A is a frequency factor if the reaction is first order. If logKt is plotted against the reciprocal of the absolute temperature then a linear FIG. 9. Graph of log10 time zwsus l/T for 50 relation should exist and the slope will equal Qt/2.3R. per cent transformation. (This is the standard Arrhenius method of obtaining activation energy.) 1. The reaction is a growth process which can be From Eq. (7) the rate constant for a precipitation adequately described by the equation : reaction as defined by Eqs. (6) and (8) is f(t)= k=‘(. n Consequently, the values of logio(K/2.3n) have been plotted against l/T as shown in Fig. 8. An excellent linear
‘I-
1.3
VT
,
I*
,,p
FIG. 8. Graph of loglo(K/2.3n)
versus
l/T.
relationship yields an activation energy Qk= 43,500 cal/mole. From Eq. (8) it can be seen that the rate constant K is expressed in the units time-“, but the corresponding activation energy Qk must be evaluated in terms of K in units time-’ before any comparisons may be drawn with other rate processes. The conversion can be made automatically by plotting (l/n) log10(K/2.3n) versus l/T, or more simply by using the relationship Qt=QJn where Q t is the energy of activation for the controlling process. In this investigation, Qt= 43,500/n = 84,000 Cal/mole, where n is 0.5. Alternatively, the activation energy Qt for the tempering process may be obtained (5) from the slope of a graph of log time against l/T for a specific fraction of decomposition (e.g., 50 per cent) as shown in Fig. 9. This method gives the same value for Qf as the above calculation. DIFFUSION
l-exp(-_kP),
where n=O.S at 45O”C, 500°C and 525”C=O.7 at 550°C. 2. In the tempering treatments carried out at 450°C to 550°C the phase TizNi is not microscopically resolvable; however, at 750°C the precipitate appears clearly spheroidal. It is concluded that at the lower heat-treating temperatures, growth of many more dispersed nuclei occurs than at higher temperatures. 3. At the three lower tempering temperatures the coefficient n is approximately 0.5, which suggests that growth occurs by advancement of planar interfaces of TizNi. Cohen has indicated a similar process in his treatment of first stage tempering in steels.2 4. The later stages of the reaction do not comply with Eq. (2). This is perhaps due to the impingement of adjacent TipNi precipitates. 5. Diffusion in this system is by substitution. Growth of a TizNi precipitate requires that Ni atoms be transported through the CL’lattice to the interface. There is a great difference in nickel concentration between (Yand TizNi at the interface and consequently it is to be expected that the rate of movement of interface will be slow. In addition, a countercurrent diffusion of Ti atoms away from the interface, in the same direction as the TizNi growth, must occur in order to create vacancies for Ni atoms to form TLNi. 6. The activation energy for the controlling diffusion process has been determined as 84,000 cal/mole.
MODEL
The following experimental results and conclusions have been considered in proposing a model for the tempering of transformed-p in the 7.2 per cent nickel alloy :
FIG. 10. Proposed model for growth process.
POLONIS
AND
PARR:
DECOMPOSITION
On the basis of these observations, it is proposed that at 45O’C to 5.50°C (and probably below this range} growth of TizNi proceeds in plate-like form as shown in Fig. 10. TizNi is envisaged as advancing into the ar’ solid solution with a depletion of nickel ahead of the interface. The gradient of nickel concentration in QI’will be determined by the diffusion coefficient of the reaction and by the maximum and minimum nicke1 concentrations indicated by COand cl respectively. The line ‘au’ represents the centre-line between the midpoints of two growing Ti,Ni plates. Towards the end of the decomposition process the concentration COat line ‘au’ will start to diminish, resulting in a subsequent decrease in concentration gradient. In this model the product should consist of (Yregions surrounded by fine, well-dispersed plates of TieNi. At low tempering temperatures (less than 550°C) it is probable that the growth of many nuclei will proceed with event,ual impingement which results in the nonlinearity of the upper parts of the curves log10 log,o[l/l -S(t)] z’ersus log&. Thus, the Widmanstgtten type of a: precipitate seems reasonable on the basis of this model, and T&Ni formations would be manifested as an apparent thickening of a boundaries. Calculations based on cylindrical (Yneedles and uniform distribution of TizNi as surface layers around a! show that the thickness of the Ti,Ni layer is in the order of 0.05 microns-which would not be resolvable under the microscope. At 550°C the increased value of n (0.7) implies that precipitation occurs in thicker units, since as n approaches 1.5 spheroidal precipitates are expected. Although no measurements have been made at temperatures higher than 55O”C, the photomicrograph of a specimen treated at 750°C (Fig. 6) supports this view. The diffusion coefficient for the controlling process, as calculated from D=&a”v exp-Q/RT, (where a is interatomic spacing, and Y the thermal oscillation frequency) is in the order of 10-25 cm2 per set in the temperature range 450°C to 550°C. Such a small value implies a slow diffusion rate which will favour the formation of a very fine and well-dispersed precipitate. Since the proposed model requires countercurrent diffusion of Ti atoms and Ni atoms, the self-diffusion of Ti to create vacancies for the Ni may well be the rate-controlling factor. Although no figure exists for the activation energy of self-dsusion in titanium (Q8), an analysis of self-diffusion data for several metals indicates that QB is roughly proportional to the melting point r,. In Fig. 11, the available values of QB are plotted against Tf and a reasonably straight-line relationship is obtained. From Fig. 11 the activation energy of self-diffusion of titanium is estimated to be about 77 000 Cal/mole. This value corresponds remarkably well to the activation energy for the rate-controlling step of the tempering reaction in the titanium-nickel alloy.
311
KINETICS
FIG. 11. Graph showing activation zrersw melting temperature,
energy of self-diffusion, Tf, for severat metals.
Q8,
CONCLUSION
A model for the tempering kinetics of transformed-p in a 7.2 per cent nickel alloy of titanium has been proposed. This model is based on growth of plates of TinNi during isothermal heat-treatments between 450°C and 550°C. At temperatures above 525°C there is a tendency for the precipitation of thicker units and eventually spheroids. The reaction, to 85 per cent completion, is satisfactorily described by the rate equation based upon the work of Johnson and Mehl, Cohen and Zener. On the basis of the proposed model and the activation energy of the process, jt appears that the self-diffusion of titanium is the major rate controlling factor. ACKNOWLEDGMENTS
This work was carried out as part of a project (No. 425) sponsored by the Defence Research Board of Canada. A fellowship awarded to one of the authors (D. H. P.) by the International Nickel Company of Canada, Limited, is gratefully acknowledged. The authors wish to thank Professors F. A. Forward and W. Ml. Armstrong for their encouragement, their colIeagues for helpful discussion, and Mr. R. G. Butters for his assistance with certain aspects of the experimental work. REFERENCES 1. Fi94\)
Averbach
and
M. Cohen,
Trans.
A.S.M.
41,
1024
2. @?.S. koberts, B. L. Averbach, and N. Cohen, Trans. A.S.M. 45, 576 (1953). C. A. Wert, J. App. Phys. 20, 943 (1949). W. S. Owen, Trans. A.S.M. 46 (1954) preprint. fi9E@ke and W. S. Owen, J. Iron and Steel Inst. 176, 147 6. iv. A: Johnson and R. F’. Mehl, Trans. A.I.M.M.E. 135,416 (1939). C, Zener, J. App. Phys. 20, 9.50 (1949). x’* D. H. Polonis and J. G. Parr. To be published. 9: D. H. Polonis and J. G. Parr, Trans. A.I.M.M.E. 200 (1954); T. Metals. 1148 ioctober. 1954). 10. b. H. Poionis aid J. G. ‘Parr, 1. Inst. Metals, 162 (October, f9.54). (Letter to the Editor). 11. D. H. Polonis and J. G. Parr, J. Metals. To be published. 12. D. H. Polonis, R. G. Butters, and J. G. Parr, Research 7, 10, 5. (19.54). 13. ~~7M2~3golin, E. Ence and J. P. Nielsen, Trans. A.I.M.M.E.
f
.
DECOMPOSITION
KINETICS
OF TRANSFORMED-BETA
IN A TITANIUM-NICKEL D. H. POLONISt
and
J. GORDON
PHASE
ALLOY* PARRi
The decomposition kinetics of the transformed-8 phase during isothermal heat-treatment has been studied by measuring the rate of precipitation of TizNi. The reaction follows the type of equation suggested by Johnson and Mehl, and an activation energy of 84,000 Cal/mole has been determined. On the basis of the values of the coefficient “n” it is suggested that TirNi precipitates as plates at temperatures up to 525°C. Above this temperature there appears to be a tendency for the formation of spheroidal precipitates. A model for the reaction is proposed in which the rate controlling process is the self-diffusion of titanium. CINBTIQUE
DE
LA D&OMPOSITION ISOTHERME ALLIAGE TI-NI
DE LA PHASE-0
DANS
UN
La cinetique de la d&omposition de la phase-8 pendant un traitement isotherme est Ctudiee en mesurant la vitesse de precipitation de Ti2Ni. La reaction suit l’bquation de Johnson et Mehl et une energie d’activarion de 84.000 cal/mol a 6tt5 dCtermin6e. De la valeur de l’exposant “n,” il semble que Ti2Ni precipite en plaquettes jusqu’g 52.5”; au-dessus de cette temperature, il y aurait tendance $ une precipitation en sphhres. 11 est sugg&C que la reaction est contr816e par I’autodiffusion du titane. DIE
KINETIK
DER ISOTHERMEN BETA-PHASE EINER
ENTMISCHUNG DER UMGEWANDELTEN TITAN-NICKEL-LEGIERUNG
Die Kinetik der Entmischung der umgewandelten &Phase bei einer isothermen Wgrmebehandlung wurde durch Messung der .Ausscheidungsmenge von TisNi untersucht. Die Reaktion verlluft nach dem Gyp der Gleichung von Johnson und Mehl. Die Aktivierungsenergie wurde zu 84.000 cal/mol bestimmt. Auf Grund des Koeflizienten “lt” wird vermutet, dass sich Ti,Ni bei Temperaturen bis 525°C plattenfijrmig ausscheidet. Oberhalb dieser Temperatur eine Neigung zur Bildung von sph$rolitischen Ausscheidungen zu bestehen. Es wird ein Mode11 fiir die Reaktion vorgeschlagen, bei dem der massgebende Vorgang fiir den Reaktionsablauf die Selbstdiffusion des Titans ist.
INTRODUCTION
other titanium alloys. g,lo,ll The absence of any other phase was confirmed by X-ray diffraction studies. The 7.2 per cent nickel alloy was selected for the present study sin&, on tempering to produce equilibrium structures, phase-ratios could be assessed more accurately than with lower compositions. In addition, this alloy corresponds approximately to eutectoid composition. (See phase diagram, Fig. 2.) The phase diagram shows that the approach towards equilibrium during tempering yields the phase TizNi.
The tempering kinetics of some important ferrous alloys have been explored in detaillH5 by applying theories of diffusion and precipitation developed by Johnson and Meh16 and Zener.’ The work has included decomposition studies of retained and martensitically transformed austenite as well as graphitization in castirons. Most of the research has been concerned with interstitial alloying elements in materials of complex commercial composition. The present work is an analysis of the tempering kinetics of transformed-@ solid solution (hexagonal close-packed) in a 7.2 weighlper cent (6 per cent atomic) nickel alloy of titanium. It was possible to produce 100 per cent transformed-p (called a’) in titanium-nickel alloy powders (-200 mesh) containing up to 7.2 per cent nickel8 by quenching them from 1,OOO”C with a blast of purified argon gas. Specimens containing more nickel showed some retained 0 after quenching. Metallographic examination of completely transformed specimens revealed pronounced strain patterns (Fig. 1) which are typical of martensitic structures observed in * Received December 13, 19.54. t School of Mineral Engineering, University of Washington, Seattle, Washington. Formerly, Department of Mining and Metallurgy, The University of British Columbia, Vancouver, Canada. $ Department of Mining and Metallurgy, University of Alberta, Edmonton, Canada. Formerly, Department of Mining and Metallurgy, The University of British Columbia, Vancouver Canada. ACTA
METALLURGICA,
VOL.
3, JULY
1955
FIG. 1. Magnification 2000X. Etchant 5 per cent HF in glycerin, followed by nitric acid rinse. 7.2 per cent Nialloy. Powder quenched in argon gas from 1ooO”C 307
ACTA
308
FIG. 2. Phase
diagram of the titanium-nickel after Margolin el a1.13
METALLURGICA,
system
During the formation of TizNi the (Y’phase is depleted of nickel until its composition conforms to that of equilibrium a! at the tempering temperature. It probably contains less than 0.1 per cent nickel at 500°C. EXPERIMENTAL
METHOD
The titanium alloy was prepared from iodide titanium and Johnson Matthey spectrographic standard nickel by levitation-melting.12 The method by which filed powders were quenched has been described elsewhere,g and the usual precautions against contamination were observed. Sufficient quenched powder of the 7.2 per cent nickel alloy was prepared for the subsequent isothermal heat-treatments which were made upon samples sealed in evacuated fine bore silica tubing. Temperature control during tempering was =F~‘C. Where heat-treatments were of less than 15 minutes’ duration, a thermocouple was attached to the specimen-tube and time was measured from the instant the thermocouple reached temperature. In all cases temperature was attained in 1.5 to 20 seconds. The extent of the tempering reaction after heattreatment was determined by computing from X-ray dam the amount of TizNi formed. In order to assess this quantitatively the ratio of the line intensities TiZNirJ33: (~~01was first obtained on Geiger-spectrometer plots for each of several Ti-Ni alloys ranging from 0 to 10 per cent nickel which had been heat-treated to produce equilibrium structures. The percentage TizNi for each alloy was determined from the phase diagram
FIG. 3. Graph
showing percentage by weight formed versus log10 time.
of TizNi
VOL.
3,
1955
and plotted against the corresponding ratio of intensities Ti2Nir,rP3 :01101.Hence, relative line intensities obtained after heat-treating the quenched alloy could be translated into phase ratios. It should be noted that both the 101 (Yline and 101 (Y’line are coincident about 8=20”. Consequently, it is necessary to assume that a! and (Y’ phases have equal structure factors. This introduces a small error since (Y’contains 6 atomic per cent nickel in the alloy used, and its structure factor differs from that of equilibrium (Y,which is practically pure titanium. However, calculations based on the assumption of random nickel distribution in (Y’reveal the error to be less than 2 per cent-a figure that is no greater than the inherent errors involved in the technique. EXPERIMENTAL
RESULTS
The tempering reaction in the alloy was followed at temperatures of 450°C, SOO”C, 525°C and 550°C. The weight percentages of TizNi formed during the reaction are plotted as a function of log10 time in Fig. 3. The curves possess the familiar ‘Y-shape (although the early
FIG. 4. Magnification 2000X. Etchant 5 per cent HF in glycerin, followed by nitric acid rinse. 7.2 per cent Ni alloy. Powder quenched and transformed 90 per cent at 500°C.
stages of the reaction could not be measured) and may be Ireasonably superimposed by translation along the log10 time axis. Once the reaction is initiated the amount of TisNi formed is approximately proportional to log10 time up to the concluding stages of the reaction (i.e., when more than 14 per cent TizNi has precipitated) at which juncture the rate decreases quite appreciably. Complete transformation would be at 18.9 per cent TizNi. Tempered specimens were examined metallographically in order to detect any changes due to diffusion and precipitation. The original strained structure appeared to be transformed to a Widmanstgtten precipitation type. Figure 4 shows a typical microstructure for 90 per cent transformation at 500°C. The structure is fine and acicular. Since TizNi is not resolvable at a magnification of 2,000 times, it is presumably very finely dispersed. At 550°C the structure is a little coarser
POLONIS
AND
PARR:
DECOMPOSITION
KINETICS
309
The coefficients k and n are constants for a particular process. If logarithms are taken in Eq. (2), j(t)) = - kt”
loge(l-
(4)
and k loglo--+a 2.3
1 log10 loglo-=
l-j(t)
loglot-
(5)
Cohen et al.’ have proposed the following equation which serves to define the specific rate constant K and is merely another form of Eq. (2) : df (0 --=K(ldt FIG. 5. Magnification 2000X. Etchant 5 per cent HF in glycerin, followed by nitric acid rinse. 7.2 per cent Ni alloy. Powder quenched and transformed 90 per cent at 55O’C.
j)l”,
(6)
where m=n1. Comparison of Eqs. (3) and (6) shows nk=K.
of TizNi can be seen. Microscopic examination of a specimen heated at 750°C for 15 minutes revealed a spheroidal precipitate of TizNi in a matrix of (Y(see Fig. 6). (Fig.
S), but no definite
precipitation
DISCUSSION
Substituting
--~.X~Ga*L4), (
where j(i) represents the fraction transformed A more general form of Eq. (1) is
K
1
loglo--+n
l-f(O
The general shape of the curves in Fig. 3 suggests a type of nucleation and growth process that was quantitatively analysed by Johnson and Mehl.6 They proposed an equation of the following form for a rate of nucleation ATand a rate of growth G:
j(t)=l-exp
Eq. (7) in (S),
log,, log10-=
OF RESULTS
(-Kin),
dt
(8)
I
I
(1) in time 1. --f(t)] oersu~ loglo time.
(2)
which yields
df (0 ---~~~t”-‘.e-~l”=~n~“-l(1_j).
loglot.
2.3~~
FIG. 7. Graph of log10log&/l
j(t)=l-exp
(7)
(3)
From (8) it can be seen that if the tempering reaction is to obey the relationship, then a plot of log10 logloW - f (01 versus loglot will yield a straight line. The experimental values for the current investigation are plotted in Fig. 7 and produce reasonably straight lines up to the late stages of the reaction. The curves deviate from linearity at approximately the same ordinate values, suggesting the possibility of some structural feature that inhibits the growth process. Reference to Eq. (8) shows that the slopes of the curves in Fig. 7 are equivalent to the values of n for each temperature. The values are listed in Table I, TABLE I. Temy?ture 450
FIG. 6. Magnification 2ooOX. Etchant 5 per cent HF in glycerin, followed by nitric acid rinse. 7.2 per cent Ni alloy. Powder quenched
and transformed
100 per cent at 750°C.
zz 550
n 0.53 0.51 0.52 0.7
log1o(K/2.3n) -2.40 -1.45 -1.02 -0.8
ACTA
310
MET,4LLURGICA,
VOL.
3,
1955
where it is seen that, except for the value of 0.7 at SSO”C, n is fairly constant about 0.52. The values of loglo(K/2.3n) for each temperature have been obtained from the log time=0 intercepts of Fig. 7, and are listed in Table I. From basic rate theory, it is known that: &=Ae-QHRT,
(9)
where Kt is the specific rate constant in units time-l, Qt is the activation energy, and A is a frequency factor if the reaction is first order. If logKt is plotted against the reciprocal of the absolute temperature then a linear FIG. 9. Graph of log10 time zwsus l/T for 50 relation should exist and the slope will equal Qt/2.3R. per cent transformation. (This is the standard Arrhenius method of obtaining activation energy.) 1. The reaction is a growth process which can be From Eq. (7) the rate constant for a precipitation adequately described by the equation : reaction as defined by Eqs. (6) and (8) is f(t)= k=‘(. n Consequently, the values of logio(K/2.3n) have been plotted against l/T as shown in Fig. 8. An excellent linear
‘I-
1.3
VT
,
I*
,,p
FIG. 8. Graph of loglo(K/2.3n)
versus
l/T.
relationship yields an activation energy Qk= 43,500 cal/mole. From Eq. (8) it can be seen that the rate constant K is expressed in the units time-“, but the corresponding activation energy Qk must be evaluated in terms of K in units time-’ before any comparisons may be drawn with other rate processes. The conversion can be made automatically by plotting (l/n) log10(K/2.3n) versus l/T, or more simply by using the relationship Qt=QJn where Q t is the energy of activation for the controlling process. In this investigation, Qt= 43,500/n = 84,000 Cal/mole, where n is 0.5. Alternatively, the activation energy Qt for the tempering process may be obtained (5) from the slope of a graph of log time against l/T for a specific fraction of decomposition (e.g., 50 per cent) as shown in Fig. 9. This method gives the same value for Qf as the above calculation. DIFFUSION
l-exp(-_kP),
where n=O.S at 45O”C, 500°C and 525”C=O.7 at 550°C. 2. In the tempering treatments carried out at 450°C to 550°C the phase TizNi is not microscopically resolvable; however, at 750°C the precipitate appears clearly spheroidal. It is concluded that at the lower heat-treating temperatures, growth of many more dispersed nuclei occurs than at higher temperatures. 3. At the three lower tempering temperatures the coefficient n is approximately 0.5, which suggests that growth occurs by advancement of planar interfaces of TizNi. Cohen has indicated a similar process in his treatment of first stage tempering in steels.2 4. The later stages of the reaction do not comply with Eq. (2). This is perhaps due to the impingement of adjacent TipNi precipitates. 5. Diffusion in this system is by substitution. Growth of a TizNi precipitate requires that Ni atoms be transported through the CL’lattice to the interface. There is a great difference in nickel concentration between (Yand TizNi at the interface and consequently it is to be expected that the rate of movement of interface will be slow. In addition, a countercurrent diffusion of Ti atoms away from the interface, in the same direction as the TizNi growth, must occur in order to create vacancies for Ni atoms to form TLNi. 6. The activation energy for the controlling diffusion process has been determined as 84,000 cal/mole.
MODEL
The following experimental results and conclusions have been considered in proposing a model for the tempering of transformed-p in the 7.2 per cent nickel alloy :
FIG. 10. Proposed model for growth process.
POLONIS
AND
PARR:
DECOMPOSITION
On the basis of these observations, it is proposed that at 45O’C to 5.50°C (and probably below this range} growth of TizNi proceeds in plate-like form as shown in Fig. 10. TizNi is envisaged as advancing into the ar’ solid solution with a depletion of nickel ahead of the interface. The gradient of nickel concentration in QI’will be determined by the diffusion coefficient of the reaction and by the maximum and minimum nicke1 concentrations indicated by COand cl respectively. The line ‘au’ represents the centre-line between the midpoints of two growing Ti,Ni plates. Towards the end of the decomposition process the concentration COat line ‘au’ will start to diminish, resulting in a subsequent decrease in concentration gradient. In this model the product should consist of (Yregions surrounded by fine, well-dispersed plates of TieNi. At low tempering temperatures (less than 550°C) it is probable that the growth of many nuclei will proceed with event,ual impingement which results in the nonlinearity of the upper parts of the curves log10 log,o[l/l -S(t)] z’ersus log&. Thus, the Widmanstgtten type of a: precipitate seems reasonable on the basis of this model, and T&Ni formations would be manifested as an apparent thickening of a boundaries. Calculations based on cylindrical (Yneedles and uniform distribution of TizNi as surface layers around a! show that the thickness of the Ti,Ni layer is in the order of 0.05 microns-which would not be resolvable under the microscope. At 550°C the increased value of n (0.7) implies that precipitation occurs in thicker units, since as n approaches 1.5 spheroidal precipitates are expected. Although no measurements have been made at temperatures higher than 55O”C, the photomicrograph of a specimen treated at 750°C (Fig. 6) supports this view. The diffusion coefficient for the controlling process, as calculated from D=&a”v exp-Q/RT, (where a is interatomic spacing, and Y the thermal oscillation frequency) is in the order of 10-25 cm2 per set in the temperature range 450°C to 550°C. Such a small value implies a slow diffusion rate which will favour the formation of a very fine and well-dispersed precipitate. Since the proposed model requires countercurrent diffusion of Ti atoms and Ni atoms, the self-diffusion of Ti to create vacancies for the Ni may well be the rate-controlling factor. Although no figure exists for the activation energy of self-dsusion in titanium (Q8), an analysis of self-diffusion data for several metals indicates that QB is roughly proportional to the melting point r,. In Fig. 11, the available values of QB are plotted against Tf and a reasonably straight-line relationship is obtained. From Fig. 11 the activation energy of self-diffusion of titanium is estimated to be about 77 000 Cal/mole. This value corresponds remarkably well to the activation energy for the rate-controlling step of the tempering reaction in the titanium-nickel alloy.
311
KINETICS
FIG. 11. Graph showing activation zrersw melting temperature,
energy of self-diffusion, Tf, for severat metals.
Q8,
CONCLUSION
A model for the tempering kinetics of transformed-p in a 7.2 per cent nickel alloy of titanium has been proposed. This model is based on growth of plates of TinNi during isothermal heat-treatments between 450°C and 550°C. At temperatures above 525°C there is a tendency for the precipitation of thicker units and eventually spheroids. The reaction, to 85 per cent completion, is satisfactorily described by the rate equation based upon the work of Johnson and Mehl, Cohen and Zener. On the basis of the proposed model and the activation energy of the process, jt appears that the self-diffusion of titanium is the major rate controlling factor. ACKNOWLEDGMENTS
This work was carried out as part of a project (No. 425) sponsored by the Defence Research Board of Canada. A fellowship awarded to one of the authors (D. H. P.) by the International Nickel Company of Canada, Limited, is gratefully acknowledged. The authors wish to thank Professors F. A. Forward and W. Ml. Armstrong for their encouragement, their colIeagues for helpful discussion, and Mr. R. G. Butters for his assistance with certain aspects of the experimental work. REFERENCES 1. Fi94\)
Averbach
and
M. Cohen,
Trans.
A.S.M.
41,
1024
2. @?.S. koberts, B. L. Averbach, and N. Cohen, Trans. A.S.M. 45, 576 (1953). C. A. Wert, J. App. Phys. 20, 943 (1949). W. S. Owen, Trans. A.S.M. 46 (1954) preprint. fi9E@ke and W. S. Owen, J. Iron and Steel Inst. 176, 147 6. iv. A: Johnson and R. F’. Mehl, Trans. A.I.M.M.E. 135,416 (1939). C, Zener, J. App. Phys. 20, 9.50 (1949). x’* D. H. Polonis and J. G. Parr. To be published. 9: D. H. Polonis and J. G. Parr, Trans. A.I.M.M.E. 200 (1954); T. Metals. 1148 ioctober. 1954). 10. b. H. Poionis aid J. G. ‘Parr, 1. Inst. Metals, 162 (October, f9.54). (Letter to the Editor). 11. D. H. Polonis and J. G. Parr, J. Metals. To be published. 12. D. H. Polonis, R. G. Butters, and J. G. Parr, Research 7, 10, 5. (19.54). 13. ~~7M2~3golin, E. Ence and J. P. Nielsen, Trans. A.I.M.M.E.
f
.