Coarsening resistance of M2C carbides in secondary hardening steels: Part III. Comparison of theory and experiment

Coarsening Resistance of M2C Carbides in Secondary Hardening Steels: Part III. Comparison of Theory and Experiment HYUCK MO LEE and SAMUEL M. ALLEN A model for the coarsening resistance of multicomponent carbides was used to study the effect of Mo and Cr on the coarsening kinetics of M2C carbides in commercial AF1410 and experi- mental alloy steels. Experimental studies of coarsening behavior of the carbides in these steels have been made by using transmission electron microscopy (TEM), scanning transmission elec- tron microscopy (STEM), scanning electron microscopy (SEM), and X-ray diffraction (XRD). The measured coarsening rate constant agrees with model predictions within a factor of 2 to 3. The coarsening kinetics of M2C carbides in these alloys is found to be controlled by the volume diffusion of alloying element M. A Cr-Mo alloy steel with the predicted optimum composition showed the slowest coarsening kinetics and highest hardness at long tempering times. I. INTRODUCTION ONLY carbide structures capable of high coherency with body-centered cubic (bcc) iron will precipitate on a suf- ficiently fine scale, less than 5 nm in diameter, to pro- vide the high strength levels. Such carbides generally consist of the face-centered cubic (fcc) group (MC, M = Nb, Ta, Ti, V) and hexagonal close-packed (hcp) group (M2C, M = Fe, Cr, Mo, W). Coherency of the fast group is permitted by near-coincidence in the cube planes of both the carbide and bcc iron, favoring {100}~ platelets, while coherency of the second is allowed by near- coincidence of the carbide close-packed direction and the bcc iron cube direction, promoting (100)~ rods. In con- trast, less coherent (but often more stable) carbides, such as M6C , M7C3, and M23C 6, precipitate in coarser form with less strengthening and even embrittlement v/a inter- facial precipitation. When the strong carbide-forming elements Mo, Cr, and W are added to high Co-Ni steels, secondary hard- ening is accomplished by the precipitation of fine-scale M2C alloy carbides. In this case, the resistance of the M2C dispersion with respect to Ostwald ripening is an important matter. A model was introduced for the coars- ening resistance of multicomponent carbides, o] While the model treats the coarsening of shape-preserving parti- cles, it is applicable to nonspherical, in particular, rod- like particles. The alloy AF1410 I21 and four experimental aUoys [31 will be tested experimentally in terms of the secondary hard- ening reaction and coarsening kinetics of the M2C car- bides. Peak hardness is obtained when M2C particles are still coherent at 783 K in AF1410. t4j However, at this stage, particles are too small to be observed easily, and the effect of coherency strain energy on the coarsening HYUCK MO LEE, Assistant Professor, is with the Department of Materials Science and Engineering, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea. SAMUEL M. ALLEN, Associate Professor, is with the Center for Materials Science and Engineering~ Massachusetts Institute of Technology, Cambridge, MA 02139. Manuscript submitted January 30, 1991. behavior may not be negligible because the coherent equilibrium state is different from the incoherent equi- librium state, t5,61 Therefore, this study covers the tem- pering time range of 8 to 400 hours, where incoherent (but effective in hardening) M2C carbides are supposed to be in equilibrium with the matrix. The mechanism governing the coarsening kinetics of M2C carbides is also of interest. The experimental results are compared in de- tail with predictions of Thermo-Calc I3'7] and the modified coarsening theories, m II. EXPERIMENTAL PROCEDURES The chemical compositions of the materials used in this work, commercial AF1410 steel and experimental SRG1, SRG2, SRG3, and SRG4 steels, are listed in Table I (in weight percent). The chemical compositions of the experimental alloys are not exactly the same as those of calculations, t3J This slight change in composi- tion can have an effect on the previously calculated re- suits, such as coarsening rate constants and phase boundaries. Because of different temperature ranges where a sin- gle austenite phase field exists, the five alloys were given different homogenization and austenitization treatments. The AF1410 samples were homogenized at 1473 K for 8 hours, furnace cooled, then austenitized at 1116 K for 1 hour and water quenched. The SRG1 samples were homogenized at 1473 K for 8 hours, furnace cooled, then austenitized at 1473 K for 1 hour again and oil quenched. The SRG2, SRG3, and SRG4 samples were homoge- nized at 1473 K for 6 hours together then furnace cooled. The SRG2 samples were austenitized at 1423 K for 1 hour, the SRG3 samples at 1398 K for 1.5 hours, and the SRG4 samples at 1398 K for 1 hour. All of these samples were oil quenched after austenitization. Tem- pering was carried out at 783 K, where the thermo- dynamic calculations were done. The SRG1 samples were tempered at both 783 and 873 K so that the activation energy of coarsening kinetics might be obtained. Tem- pering time spanned the range from 8 to 400 hours. All of the samples were furnace treated after being sealed in METALLURGICAL TRANSACTIONS A VOLUME 22A, DECEMBER 1991--2877 Table I. Chemical Compositions of AFI410 and Four Experimental Alloy Steels (in Weight Percent) AF1410 SRG1 SRG2 SRG3 SRG4 C 0.16 0.23 0.24 0.24 0.24 Co 14.25 14.17 15.99 16.08 16.06 Ni 10.15 10.24 4.96 4.97 4.98 Mo 1.05 3.96 4.03 2.82 1.52 Cr 2.10 0.06 0.02 0.71 1.40 Fe bal. bal. bal. bal. bal. silica capsules under an Ar atmosphere in order to pre- vent decarburization and oxidation. To produce extraction replicas, specimens were mounted, polished conventionally, and etched with 5 pct nital. After coating with carbon, replicas were extracted in 10 pet nital. Scanning electron microscopy (SEM) was used to observe the microstructures of the alloys and all of the carbides, especially cementites and precipitated austenite particles. A Cambridge scanning electron microscope operating at 19 or 20 kV was employed. Specimens were mounted, polished conventionally, etched with 5 pct nital, demagnetized, and coated with Au. The volume fraction of precipitated austenite phase was determined by the standard ~direct comparison method. ~gj In this study, integrated intensities of (200) and (220) of austenite peaks were compared with (200) and (211) martensite peaks. Specimens were prepared by conven- tional polishing. A Rigaku 300 diffractometer with bent graphite monochromator was used. Because the mono- chromator was added and oriented to reflect only Cu K~ radiation, the background was reduced practically to zero, which made the analysis quite convenient. It was oper- ated at 50 kV and 200 mA. Cobalt, iron, and nickel have only small differences in atomic scattering factors. Therefore, compositional effects on the integrated inten- sities were not expected to arise from these elements. A DM 400 Microhardness Tester was used to measure Vickers hardness values at 200 g load. For comparison with other hardness values, these values were converted to Rockwell hardness values on the C scale with a 150 kg of major load according to a conversion table. [91 III. RESULTS A. Identification of M2C Phase In order to confirm whether the precipitated carbides are indeed M2C phase, selected-area diffraction patterns are obtained from many particles. As the particles lose their original orientation in extraction, they result in a ring diffraction pattern such as in Figure 1. The sample was AF1410 tempered at 783 K for 196 hours. This kind of ring pattern was a typical one for this study. It is indexed as hcp, and the lattice parameters of a and c are measured to be 0.290 and 0.455 nm, respec- tively, which shows the composition between Mo2C (a = 0.300 nm, c = 0.472 nm) and Cr2C (a = 0.279 nm, c = 0.446 nm). To measure the correct lattice constants, specimens were coated with 5-nm-thick AI as a calibra- tion standard. Fig. 1--Electron diffraction pattern from extraction replicas of AF1410 tempered at 783 K for 196 h. The brightest line of Figure 1 is indexed as (101). The lines are indexed from the center (002), (101), (102), and (110). Due to the brightness of the direct beam, a weak reflection (100) nearest to the center is obscured. B. Morphology and Size of MeC Particles The number of particles examined for each heat treat- ment is of the order of hundreds to ensure reliable data. The particles were rod shaped, as seen in Figures 2(a) through (f). Representative micrographs are shown for each alloy. 1. Length Length and time in Figure 3 are presented on a log- log plot. The slope at the lower right corner of the plot is one third, which is a slope for the volume diffusion- controlled coarsening kinetics. The slope can give a hint of how well the volume diffusion-controlled coarsening kinetics is observed. The length of M2C carbides in SRG1 samples tem- pered at 873 K is always greater than that of any other samples at 783 K due to high tempering temperatures. At 783 K, AF1410 samples show the largest length at any tempering time, and the next is SRG1 alloy. Among the alloys of the Fe-16Co-5Ni matrix, SRG2 has a greater length of M2C carbides than SRG3 and SRG4, even though the difference is not so significant. The length of carbides in SRG3 is similar to that of SRG4 up to 400 hours of tempering time. 2. Diameter Figure 4 shows the diameter of M2C rods vs time on a log-log scale. It is quite similar to the previous Figure 3 of length vs time, except for the case of SRG1 alloy. Diameters of M2C in SRG1 tempered at 873 K are greater than those of any other alloy tempered at 783 K. Among the alloys tempered at 783 K, AF1410 has the greatest dimension of diameter of M2C rods at any tem- pering time. The SRG2 has larger diameters of carbides than SRG3 and SRG4, with SRG3 and SRG4 competing 2878--VOLUME 22A, DECEMBER 1991 METALLURGICAL TRANSACTIONS A Fig. 2 - -TEM micrographs showing rod-shaped M2C carbides of tempered (a) AF1410 at 783 K for 196 h, (b) SRG1 at 783 K for 100 h, ic) SRG1-600 at 873 K for 16 h, (d) SRG2 at 783 K for 400 h, (e) SRG3 at 783 K for 196 h, and ( f ) SRG4 at 783 K for 400 h. I 00 .0 30.0 I0 .0 3.0 I I I I ....-,X SRGI -600 X / AF I410 ~ SRG4 ~ I /3 1.0 I I I I 5 I0 30 I00 300 T IME (HR) ~ig. 3--Average length of rod-type M2C carbides vs tempering time. dl of the samples were tempered at 783 K, except for SRGI-600 amples tempered at 873 K. 30.0 , I I I I I .~ I 0 .0 SRGI -600 ~ - - ~ / ~-~ SRGI ~ 1 / 3 1.0 0 .5 I I I I / 5 I0 30 I00 300 T IME (HR) Fig. 4 - - Average diameter of rod-type M2C carbides vs tempering time. All of the samples were tempered at 783 K, except for SRG1-600 samples tempered at 873 K. [ETALLURGICAL TRANSACTIONS A VOLUME 22A, DECEMBER 1991--2879 with each other. The SRG1 shows a different behavior compared with the previous figure. Its dimension in di- ameter is quite small enough to compete with SRG2, SRG3, and SRG4 samples, which means that the aspect ratio of rods in SRG1 alloys is greater than that of other alloys. 3. Effective radius In dealing with coarsening kinetics of nonspherical precipitates, the effective radius term was used before the introduction of the modified coarsening model. The effective radius is a radius of a sphere which has the same volume of the nonspherical particle. It is defined as follows for a rod-type particle: reef = defe/2 = [d2l]1/3/2 [1] Effective radius, as well as hardness, can be a good pa- rameter in comparing with other data on the growth ki- netics of M2C carbides. The effective radius of MzC particles vs the tempering time is reported in Figure 5. Its shape is almost the same as in Figures 3 and 4. Up to 48 hours of tempering time, the order of magnitude is increasing as follows: SRG4 < SRG3 < SRG2 < SRG1 < AF1410 and SRG 1-600 After that time, SRG1, SRG3, and SRG4 show a small value of the effective radius and are competing at the bottom of the plot, while AF1410 and SRG2 have larger magnitudes. 4. Aspect ratio All of the tested samples, except SRG1 alloy, show an aspect ratio between 2.5 and 5 in Figure 6, a result similar to those obtained previously. [4'1~ The only ex- ception in this study was the SRG1 alloy with the com- position of Fe-14Co-10Ni-4Mo-0.23C. The aspect ratio 30 l I I I I t SRGt -600 031 I I I I / 5 I0 30 I00 30O T IME (HR} Fig. 5--Effective radius of rod-type M2C carbides vs tempering time. All of the samples were tempered at 783 K, except for SRG1-600 samples tempered at 873 K. I0 I I I I ~ ne 6 ~4- 2 I I I I I 5 I0 30 I00 300 T IME (HR) Fig. 6--Aspect ratio of rod-type M2C carbides vs tempering time. All of the samples were tempered at 783 K, except for SRG1-600 samples tempered at 873 K. of M2C carbides in this alloy at 783 and 873 K is higher than that of other alloys observed, and its value is about 8 ~ 9, also similar to other results obtained previ- ously. [13-18] Generally speaking, the common features of systems which show small aspect ratios of M2C carbides are as follows. Heat treatment was done at low tempering tem- perature (approximately less than 823 K) or for short tempering time, and M2C carbides contained substantial amounts of Cr as well as Mo. The other groups of M2C carbides with higher aspect ratios have the characteris- tics that there was no addition of Cr and tempering was done at high temperatures or for lengthy times. Of course, these features are not absolute ones, and some of the systems violate these general guidelines. However, these general characteristics suggest that addition of Cr lowers the aspect ratio and long tempering, combined with high temperatures, raises the aspect ratio. As M2C particles tend to lose coherency with increased size, namely, with increased tempering time, incoherent M2C particles seem to favor a large aspect ratio and coherent M2C particles seem to favor a small aspect ratio. Nevertheless, this feature has not been conspicuously observed in Figure 6. It was observed to be comparatively a constant, as re- quired by the theory. C. Hardness Results of hardness measurements are presented in Figure 7. Maximum peak value increases in the follow- ing order: AFl410 < SRG4 < SRG3 < SRG1 < SRG2 Assuming that hardness increases with increased volume fraction of MEC phase, it is useful to check the calculated mole fraction of M2C phase for these alloys. Table II shows the prediction. The peak value is generally con- sistent with the calculated number of moles of the M2C phase, namely, the volume fraction of the M2C phase. 2880--VOLUME 22A, DECEMBER 1991 METALLURGICAL TRANSACTIONS A 6O 55 4O 35 I I I t _ _ o j j tO 30 Ioo 3oo TIME (HR) Fig. 7 - -Hardness (Re) of tempered alloy steels vs tempering time. All of the samples were tempered at 783 K, except for SRG1-600 samples tempered at 873 K. The SRG1 alloy holds a larger peak value than those of the SRG3 and SRG4 alloys, though the calculated num- ber of moles of the M2C phase in the SRG1 alloy is slightly smaller than those in the SRG3 and SRG4 alloys by 3.5 and 2.7 pct, respectively. However, the molar volume of the Mo2C phase is larger than that of the (Cr, Mo)2C phase. The actual volume fraction of the M2C phase in the SRG 1 alloy will be larger than those in the SRG3 and SRG4 alloys, though by a small margin. The relationship between the average diameter or length alone and hardness did not show a good correlation. When adopting an effective radius term due to slightly scat- tered aspect ratios between alloys, it gives a good correlation. The peak hardness of all of these alloys seems to be obtained when the effective radius of M2C particles is 1 - 3 nm, namely, when effective diam- eter is 2 -- 6 nm. Considering the previously published data, [4'I~ the above value is quite a consistent one for the maximum hardness obtained. There is an excep- tion in the system studied by Davies and Ralph.[ HI They found a peak hardness at a diameter of 7 -- 12 nm, and the aspect ratio was about 2, quite a small aspect ratio value. If the particles are smaller than the above-mentioned optimum value of 2 -- 6 rim, then they do not seem to Table l I . Calculated Mole Fraction of M2C Phase at 783 K Alloy Fe Database SGTE Database AF1410 0.0150 0.0151 SRG1 0.0219 0.0221 SRG2 0.0228 0.0230 SRG3 0.0227 0.0228 SRG4 0.0225 0.0226 be strong enough to hold the maximum hardness, sug- gesting that they will be cut by the dislocations. [2~ Op- timum particle sizes seem to be obtained when the precipitation-hardening mechanism that particles are sheared by the dislocations changes to the mechanism that dislocations bypass the particles. After that, parti- cles coarsen and the hardness drops, suggestive of the Orowan mechanism, because the intercarbide spacing increases with coarsening of particles for a constant volume fraction. 121] Lacking knowledge of the various parameters in hardening models, t2~ quantitative de- scriptions were not made. However, the SRG3 alloy with a minimum coarsening rate constant holds a large Rc value of 56 at 400 hours. This means that slow coarsening ki- netics maintains a high hardness value, even at long tem- pering times. D. Particle Size Distribution One of the main features of Lifshitz-Slyozov-Wagner (LSW) coarsening theory E22,23,24J is that the particle size distribution reaches the asymptotic one at long time. The maximum frequency is observed at about 1.135 times the mean particle size, and there is a cutoff size of 1.5 times the mean particle size. As the effect of volume fraction of the second phase is considered, the distri- bution becomes broader and more symmetric with in- creased volume fraction, t2sJ In this study, the maximum volume fraction of M2C is calculated to be less than 2.5 pet, which is too small a value to have a significant effect on the coarsening rate constant or particle size distribution. With this in mind, the following results will be analyzed. Figures 8(a) through (e) and 9(a) through (e) show the particle size distribution in terms of length and diameter, respectively, for five different alloys. The tempering times 2~- (o) AFI410 J ~-lS.9nm J ~ 29.1 nm 2[" (b) SRGI 0 I 2 0 I 2 "~ �9 25.4 nm 1 3.2nrn 0 I 2 0 I 2 .~- 15.2 am ! _ 0 1 2 Fig. 8 - -Sca led M2C particle size (length) distributions at 783 K vs x = 1 / [ , where f f (x ) dx = 1. (a) AF1410 tempered for 400 h, (b) SRG1 tempered for 100 h, (c) SRG2 tempered for 400 h, (d) SRG3 tem- pered for 196 h, and (e) SRG4 tempered for 400 h. METALLURGICAL TRANSACTIONS A VOLUME 22A, DECEMBER 1991--2881 ( .) I d 11.8 nm 0 I 2 d- IO.Onm I 0 I 2 (e) SRG4 2f (b) SRGI d-2.0nm 0 I 2 / (d) _SR63 0 I 2 0 I 2 Fig. 9 - -Sca led M2C particle size (diameter) distributions at 783 K vs x = d /~l , where f f (x ) dx = 1. (a) AF1410 tempered for 400 h, (b) SRG1 tempered for 100 h, (c) SRG2 tempered for 400 h, (d) SRG3 tempered for 196 h, and (e) SRG4 tempered for 400 h. itated from a supersaturated matrix. Therefore, this theory is based on a two-phase equilibrium. When more than two phases exist in the range of coarsening, it may or may not have an effect on coarsening, depending on the amount of that phase and how that phase is distributed. In the alloys examined in this study, there can be three kinds of phases which may exist besides the ferrite and the M2C phase. One of them is austenite. 1. Austenite phase With increased tempering time, the volume fraction of precipitated austenite is expected to increase from zero. Figures 10(a) and (b) show SEM micrographs of AF1410 and SRG1 alloy tempered at 783 K for 400 hours. The phase with light contrast is identified with X-ray dif- fraction (XRD) to be precipitated austenite. The volume fraction of precipitated austenite was measured using XRD techniques. In most cases observed, other combinations of diffraction peaks are found to yield similar results, indicating that texture was not an important factor in these measurements. Figure 11 clearly shows that AF1410 reaches the predicted equilibrium volume fraction of for which the distributions are plotted are the longest ones studied for determining the coarsening behavior for each alloy. Samples heat-treated for a shorter time behaved similarly, without any significant change in the particle size distribution. Some systems are reported to approach the LSW size distribution with increasing time, [26,27] but such behavior was not observed in this study. The measured size distribution is not exactly in agree- ment with the theoretical one of LSW theory. The ob- served cutoff ratio is between 1.5 and 2.5. Some of the samples, such as SRG2 and SRG4, are close to original LSW size distribution, while others are not. Considering the complexity of the alloys being studied, complete agreement with the theoretical size distribution would be quite unexpected. There are many factors which do not satisfy the assumptions of LSW theory: nonspherical morphology, multicomponent system, nonzero volume fraction (though small), and possible interference from coexisting phases, like carbides and the precipitated aus- tenite. All of these may have contributed to the observed particle size distributions. When coarsening is controlled by interface reaction, grain-boundary diffusion, or dislocation pipe diffusion, the size distribution assumes a different shape. [28] How- ever, the fit between experimentally measured size dis- tributions of M2C particles and predicted ones by the above-mentioned mechanisms was not good, diminish- ing the possibility that coarsening in this work might have been controlled by mechanisms other than volume diffusion. E. Coexisting Phases Strictly speaking, the LSW coarsening theory models the later precipitation stage, after an approximate equi- librium volume fraction of the second phase has precip- Fig. 10 - -SEM micrographs showing precipitated (with light con- trast) austenite particles in (a) AFI410 and (b) SRG1 tempered at 783 K for 400 h. 2882--VOLUME 22A, DECEMBER 1991 METALLURGICAL TRANSACTIONS A 20 15 IO o~ 0 J'J" V OL UlME FRACTION I I l - AUSTENITE / ~ I I d l I I I I0 30 I00 300 IO00 TIME (HR) Fig. 11--Measured volume percent of precipitated austenite in AFI410 and SRG1 tempered at 783 K vs tempering time. The hatched area in the upper right comer shows expected values. austenite at about 400 hours and the SRG1 alloy has not yet reached that equilibrium value even at 400 hours. The hatched area in the upper right corner of the plot shows the expected volume fraction of austenite, 18.1 and 15.7 pct according to the Fe database and SGTE (Scientific Group Thermodata Europe) database, respec- tively, at 783 K. The equilibrium volume fraction of austenite phase in this Fe-16Co-5Ni matrix system at 783 K is calculated to be about 4 and 1.5 pct according to the Fe database and SGTE database, respectively. In SRG2, SRG3, and SRG4 samples, there is no evidence of austenite, either in SEM or XRD. Therefore, the effect of austenite on coarsening kinetics is absent in these alloys. In the AF1410 alloy alone, the amount of austenite is increasing with time. If the austenite phase is accelerating or retarding the coarsening kinetics o f M2C carbides, then the rate constant should increase or decrease with them or the t 1/3 law of the LSW theory should not be satisfied. How- ever, as will be seen in the Discussion, the AF1410 alloy shows a coarsening behavior with a rate constant which is in quite good agreement with the expected one. More- over, the measured compositions of the M2C phase are in good agreement with calculated compositions, t29j And calculated compositions of M2C phase were based on ferrite + M2C two-phase equilibrium. Therefore, the ef- fect of austenite on the coarsening kinetics of the M2C phase is concluded to be small. 2. Other carbide phases The precipitation of alloy carbides is usually preceded by the formation of cementite, which can precipitate through the carbon diffusion alone. As the size of ce- mentite is assumed to be of the order of 100 nm or more, they were easily seen with SEM at a magnification of 20,000 times. Photomicrographs taken with SEM are used in the image analyzer to get an area fraction of the pre- cipitated phase, assumed to be a carbide, and the results are tabulated in Table III. It is believed that any cementite phase precipitating early in tempering dissolves with the formation of alloy carbide, and thus, the volume fraction or area fraction of cementite phase will decrease with time. However, as seen in Table IH, the amount was not monotonically de- creasing. In SRG2 and SRG3, the amount is less than 2 pct, but it was fluctuating or decreasing followed by increase. Due to its small value, the effect on coarsening kinetics seems to be little, if any. However, the fact that the amount is increasing at a later stage may mean that more stable carbides, like M6C or M23C6, are precipi- tating at this stage. The SRG4 samples show a greater amount of precipitated carbide phase at all stages, and its amount reached up to 5 pct. In scanning transmission electron microscopy (STEM) analysis of SRG4 samples at 8 and 16 hours, submicron-sized Fe-rich cementites were seen, though at a low number density. However, an XRD technique showed no detectable carbide phase, probably due to the small size scale and low volume fraction. In considering the lower limit of carbides detectable by XRD, about 2 pct, and the reliability of this tech- nique, the maximum amount of precipitated carbide phase seems to be at most 2 pct, which is too small a value to exert a substantial effect on the coarsening kinetics of M2C carbides. Therefore, it may be concluded that the M2C size data reported in Figures 3 through 6 are ap- proximately free from the interference of other coexist- ing phases. IV. D ISCUSSION A. Coarsening Kinetics of M2C Phase It is not always straightforward to verify that the experiments were carried out completely in the coars- ening regime. Coarsening is supposed to dominate when an equilibrium volume fraction of a second phase has precipitated and the supersaturation has dropped to al- most zero. But, even according to the original LSW coarsening theory, supersaturation is still decreasing in Table IlL Measured Area Percent of Precipitated Carbides in SRG2, SRG3 and SRG4 Tempered at 783 K Using Scanning Electron Microscopy Time (h) Alloy 8 16 48 100 196 400 SRG2 0.91 0.81 1.85 0.85 2.03 0.92 SRG3 1.81 0.86 1.04 1.12 0.73 1.80 SRG4 5.08 3.49 3.86 1.17 2.72 4.60 METALLURGICAL TRANSACTIONS A VOLUME 22A, DECEMBER 1991--2883 proportion to t U3, and thus, the volume fraction must increase slightly as coarsening proceeds. This point makes it difficult to separate the growth regime and coarsening regime experimentally. When the equilibrium volume fraction is small, this becomes more difficult, because the value of measured volume fraction itself is subject to experimental error. Moreover, when the particles are inhomogeneously distributed, it is not easy to measure the volume fraction of particles using electron micrographs. If coarsening is dominated by vol- ume diffusion of solutes, then the size is expected to increase with cube root of time. When coarsening is dominated by other mechanisms, such as interface re- action, grain-boundary diffusion, and dislocation pipe diffusion, then size will increase with t ~/2, t 1/4, and t ~75, respectively. This requires us to first check the size in terms of time exponent to see which equation best fits the experimental data. Also, measured rate constants should be reasonable, and other relevant behavior should also be followed, such as activation energy result and size distribution, although the size distribution was al- ready mentioned as favoring the volume diffusion- controlled coarsening mechanism. As this study is focused on the coarsening kinetics of incoherent precipitates, all the data of the particle size cannot be used, irrespective of coherency and incoher- ency. From the partially measured lattice constants of the M2C phase in the AF1410 and SRG1 through SRG4 systems, [3~ the incoherent region was determined from after 100 hours of tempering for AF1410, 16 hours for SRG1, 48 hours for SRG2 and SRG3, and 100 hours for SRG4. A typical result for AF1410 is shown in the following figure. The cubes of the length and diameter of M2C carbides are plotted vs time in Figures 12(a) and (b). On a separate plot, the lattice constants of the hexagonal M2C phase are plotted vs tempering time (Figure 12(c)). t3~ According to the lattice constant data, MEC phase main- tains coherency until 8 hours of tempering and incoh- erency from after 100 hours and semicoherency between. All the data were shown in favor of the volume diffusion-controlled coarsening mechanism, namely, cubed size is proportional to time, because no other mechanism such as interface reaction, grain-boundary diffusion, or dislocation pipe diffusion gave a better fit- ring result than the volume diffusion-controlled coarsening. Before tabulating all the measured and calculated coarsening nite constants, the expected rate constant should be properly calculated here. According to Eq. [22],[u the rate constant in the form of cubed radius vs time is 4Kr/9. Therefore, the rate constant in the form of cubed diameter vs time becomes 32KJ9, namely, 9RT In (2A s) The term in the bracket can be calculated using the Thermo-Calc program and diffusivity data shown in Table IV. Now this term is to be multiplied by the pre- ceding term of 64 tr, V~m/9RT In (2As). Here, the known parameter is RT, and the rest are unknown yet. First, the aspect ratio, As, is not known clearly but was observed during the experiments and found to be comparatively a 25 I I ! I ~ '~ :'~ 1 0 I I I 0 I00 200 :300 400 Time (hr) (a) * ' ' t I I 0 o ioo 2o0 3oo 4o0 T ime (hr ) (b) COHERENT I : INCOHERENT . I I I 0 .488 , I ' I i , ! LATT ICE PARAI~. I r, Jrr3 nm 0 .4~ . . . . ~ - i O,2gl ' I ~ . . . . . 0.2119 - - - i 0.28"/' ' ' ' 0.1 0 .3 hO 3 .0 I0 30 I00 300 I000 T IME(HOURS) (:) Fig. 12--Variation of cubed average particle size and lattice param- eters of rod-type hexagonal M2C carbides in AF1410 tempered at 783 K vs tempering time, (a) length, (b) diameter, and (c) lattice parameters, t3q The dashed lines in (a) and (b) cover incoherent state. constant. Therefore, an average As for each alloy mea- sured separately in the incoherent regime was used. The molar volume of the M2C phase, V~, originated from the normalized Gibbs-Duhem equation which deals with a mole fraction, ~] which means that the molar volume for the hcp M2C phase is that for the molecular form of M2/3C1/3, where the total number of moles of all species 2884--VOLUME 22A, DECEMBER 1991 METALLURGICAL TRANSACTIONS A Table IV. Di f fus ion Coeff ic ient Data Used in Calcu lat ion o f Rate Constants Def ined by Equat ion [22] t~* Diffusivity, 10-4m2/s Reference Dcr = 8.52 exp ( -250,620/RT) 31 DMo = 13,000 exp ( -314,000/RT) 32 Dw = 25 exp ( -298,000/RT) 33 *Gas constant is in units of J/mole. becomes one. Even that value is changing with com- posit ion of M2C, though not by a large amount. Thus, it is calculated as fol lows: 3 NA Vm(M2/3CI/3) = 6 4 a2c • - - 9 = 7.3859 • 10-6m3/molar unit where the lattice constants of a and c were those of the Mo2C phase and N A was the Avogadro number. The last parameter is ors, interfacial energy. As this study is con- centrating on incoherent particles, it was assumed to be 0.7 J /m 2. In this way, all of the parameters were set up to cal- culate expected rate constants of diameter for rod-type M2C carbides, while those of length for rod-type M2C carbides are obtained by mult iplying As 3 to those of di- ameter. Table V summarizes the results mentioned just before, according to the Fe database and SGTE data- base. The first column in the data section represents the average aspect ratio for each al loy, and the next column is a term containing only the composit ions and diffusiv- ity. The data in the f'trst row for each al low were ob- tained using the Fe database, and those in the second row were obtained from the SGTE database. The third and fourth columns represent rate constants for length and diameter of rod-type M2C particles, respectively. According to the Fe database predict ions of KM (m2/s) in Table V, the coarsening rate constant of SRG3 is sl ightly larger than that of SRG2, while the rate constant of coarsening kinetics was predicted to be smallest for SRG3, TM because the composit ions of experimental al- loys studied are not exact ly the same as those of cal- culated ones. Nonetheless, the smaller aspect ratio of carbides in the former al loy makes the rate constant for the length smaller than that of the latter al loy. The SGTE database always predicted smaller coarsening rate con- stants for the SRG3 alloy than those of other alloys, which is quite consistent with the hardness data. However , as seen in this table, the SGTE database expects a lower coarsening rate constant than the Fe database. There are many ways of testing the fit between theory and experiment. One of them is to compare the calcu- lated value of interfacial energy and an inferred one, 0.7 J /m 2. Table VI compares exper imental ly measured and calculated coarsening rate constants for the length. As seen from the table, the Fe database gives better agreement with the experiments than the SGTE data- base, and the Fe database predict ion is in good agree- ment with the experimental observations, within a factor of 2 to 3. Only the SRG2 al loy showed a very large value of the interracial energy, though the reasons were not quite clear yet. I f all of the circumstances surrounding coarsening be- havior are true for assumptions made and all of the pa- rameters except interfacial energy are correct and experimental data are free from any kind of experimental error, then the calculated interfacial energy wil l be a cor- rect value. However , such ideal results are unlikely. And these data must be v iewed with some caution, as the al loys experienced some degree of decarburizat ion dur- ing heat treatment. Table V. Expected Coarsen ing Rate Constants of MzC Phase at 783 K Alloy As KM (m2/s) Kt (m3/s) gd (m3/s) 1.901 x 10 -22 1.448 • 10 -29 6.143 • 10 -31 AF1410 2.84 4.833 • 10 -23 3.682 • 10 -30 1.562 • 10 -31 2.868 x 10 -z3 3 .510 x 10 -29 5.716 • 10 -3" SRG 1 8.67 1.840 X 10 -23 2.251 • 10 -29 3.667 • 10 -32 2.096 • 10 -z3 3.041 x 10 -3~ 5.896 • 10 -32 SRG2 3.72 1.381 • 10 -23 2.002 • 10 -30 3.884 x 10 -32 2.125 x 10 -za 2.344 x 10 -a~ 6.324 x 10 -32 SRG3 3.41 6.452 • 10 -24 7.117 X 10 -31 1.921 X 10 -32 2.549 X 10 -24 3.692 X 10 -3~ 7.173 • 10 -32 SRG4 3.74 8.473 • 10 -24 1.227 • 10 -30 2.384 • 10 -32 Note: KM = (kM -- kFe) (kM -- 1) 64~sV ~. Kd KM (m3/s) 9RT In (2As) gt = A]K~ (m3/s) ~rs assumed equal to 0.7 J /m 2. 1st row = Fe database. 2nd row = SGTE database. (m2/s) METALLURGICAL TRANSACTIONS A VOLUME 22A, DECEMBER 1991--2885 Table VI. Comparison of Experimentally Measured and Predicted Coarsening Rate Constants in Terms of Length Alloy Measured Predicted Calculated Calculated (783 K) Kt (m3/s) Kt (m3/s) tr s (J/m 2) o- e (J/m 2) AF1410 1.364 • 10 -29 1.448 • 10 -~ 0.660 1.874 3.682 • 10 -30 2.593 7.364 SRG1 1.632 )< 10 -29 3.510 X 10 -29 0.325 2.818 2.251 x 10 -29 0.507 4.396 SRG2 1.243 • 10 -29 3.041 x 10 -a~ 2.862 10.646 2.002 x 10 -30 4.344 16.159 SRG3 3.629 x 10 -30 2.344 x 10 -a~ 1.084 3.696 7.117 x 10 -31 3.570 12.173 SRG4 2.404 >( 10 -30 3.692 x 10 -a~ 0.456 1.705 1.227 x 10 -30 1.372 5.131 Note: 1st row = Fe database. 2nd row = SGTE database. Overall, the coarsening model appears reasonable and could be applied to design of alloys with lower coars- ening rates for thermal softening resistance (as in tool steels); in structural steels for room-temperature appli- cations, it is more desirable to enhance the diffusional rate factor to accelerate secondary hardening at lower tempering temperatures. Moreover, the SRG3 system which was supposed to have the smallest coarsening rate constant showed the slowest coarsening kinetics and held the highest hardness value at long tempering times. These results demonstrate the utility of this approach to alloy design. B. Temperature Dependence of Coarsening Kinetics Every coarsening mechanism has its own activation energy, and this can be calculated by measuring coars- ening rate constants at different temperatures. In this case, a suitable range of temperature must be chosen. For the present study, low-temperature tempering does not pro- duce secondary hardening, and grain-boundary diffusion may be more dominant than volume diffusion. High temperature is suggested for volume diffusion-controlled coarsening. But more stable carbides are precipitating rapidly, and this makes observation of M2C carbides dif- ficult. And the wider the range of temperature, the smaller the calculation error for the same magnitude of the ex- perimental error. By taking these points into consider- ation, the temperature of 873 K was chosen for taking additional coarsening data accurately. It is not enough to calculate accurately the activation energy of coars- ening using just two different temperatures, but it can give an indication of how large or small it will be. The systems of AF1410, SRG3, and SRG4 have al- loying elements of both Cr and Mo, and according to this theory, both of them take part in coarsening kinetics no matter what the amount is. Therefore, both activation energies of Cr and Mo diffusion make the whole acti- vation energy of coarsening kinetics difficult to compare with experimental result. To make the result meaningful, SRG1 alloy was chosen for this study, as it involves just Mo2C. When diffusivity of Mo is expressed in Arrhenius form as follows: DMo = D~ exp (--QMo/RT) [2] where D~ and QMo are the constant and activation en- ergy of species Mo in diffusivity, respectively, and the coarsening rate constant takes the following form: /3 3 64 orsVmA s D~ exp (-QMo/RT) Kt = [3] 9RT In (2As) (kMo -- kFe) (kMo -- 1)X~o ~ When this term is differentiated with 1/T, its form is expressed as follows: d In Kt QMo d _ _ - - 71 - - - d(1/T) R d(1/T) ,hi ] Tin (2As)" (kMo --kFr (kMo - 1)X~o ~ [4] Necessary information such as K~ and As is tabulated in Table VII. The values of Kt and As are measured val- ues, and the rest are calculated ones. As the activation energy of the coarsening process itself is -Rd In KJd(1/T) in Arrhenius form, it is measured as 355.36 kJ/mole. There are four factors which contribute to this value ac- cording to Eq. [4]. The temperature term, QMo, the as- pect ratio term, and the compositional term are among them, as seen in Table VII. When all those values are considered, the activation energy of diffusion of Mo is calculated to be 354.56 kJ /mole, while it is 331.53 kJ /mole when temperature and aspect ratio terms are ne- glected, as in References 34 and 35. This value is surely much greater than the activation energy of carbon dif- fusion in iron, which is about 80 kJ /mole, and that of grain boundary diffusion, which usually has a smaller activation energy than that of volume diffusion. The same is true for the dislocation pipe diffusion. [361 From Table IV, the activation energy of volume diffusion of Mo is seen as 314 kJ /mole, according to Borisov et al. [321 The difference of 17.53 kJ is not large. Borisov et al. reported that the addition of 0.7 pct Mo to pure Fe increased activation energy of Mo by 9 k J /mole from 305 kJ/mole. Davies and Ralph tnJ measured it as 290 kJ /mole in a coarsening experiment of Mo2C car- bides, while Murphy and Whiteman t~9] obtained a value of 347 kJ /mole in the precipitation kinetics of Mo2C. 2886-- VOLUME 22A, DECEMBER 1991 METALLURGICAL TRANSACTIONS A Table VII. Measured Coarsening Rate Constants and Aspect Ratio and Calculated Concentration Parameter and Partitioning Coefficients in SRGI Tempered at 783 and 873 K SRG1 K1 (m3/s) As kMo kFe X~o.| 783 K 1.632 • 10 -29 8.67 317.3 0.0217 0.00204 873 K 4.536 • 10 -27 7.87 223.4 0.0410 0.00282 d In Kt Note: Q~t = - R - - d( l /T) = 355.36 kJ/mole = QMo + temperature term + As term + compositional term = 354.56 + (-6.87) + (-16.16) + (23.83) QMo = 331.53 without the effect of temperature and As term Lacking knowledge of the Mo diffusivity in ferro- magnetic Fe at this relatively low temperature, a decisive conclusion is hard to make. However, the observed value of 354.56 or 331.53 kJ/mole is in satisfactory agree- ment with reported ones. This also implies that coars- ening, at least in the SRG1 alloy, is governed by volume diffusion of Mo. V. CONCLUSIONS 1. M2C carbides precipitate as intralath rod-type parti- cles in these alloys, and their length is less than 30 nm and diameter is about 10 nm even after 400 hours of tempering at 783 K. The aspect ratio is ob- served to be 2.5 to 4.5 in AF1410, SRG2, SRG3, and SRG4 and about 8 in SRG1 alloy for all tem- pering stages. 2. The rank of expected and measured Kt is as follows: Fe database: SRG3 < SRG2 < SRG4 < AF1410 < SRG1; SGTE database: SRG3 < SRG4 < SRG2 < AF1410 < SRG1; measured value: SRG4 - 24. C. Wagner: Z. Electrochem., 1961, vol. 65, pp. 581-91. 25. P.W. Voorhees: d. Star. Phys., 1985, vol. 38, pp. 231-52. 26. K.M. Vedula and R.W. Heckel: Metall. Trans., 1970, vol. 1, pp. 9-18. 27. D. Ramakrishna and S.P. Gupta: Mater. Sci. Eng., 1987, vol. 92, pp. 179-91. 28. R.D. Vengrenovitch: Acta Metall., 1982, vol. 30, pp. 1079-86. 29. H.M. Lee, A.J. Garratt-Reed, and S.M. Allen: Scripta Metall., 1991, vol. 25, pp. 685-88. 30. J.S. Montgomery and G.B. Olson: Proc. 34th Sagamore Army Materials Research Conf., G.B. Olson, M. Azrin, and E.S. Wright, eds., U.S. Army Materials Technology Laboratory, Watertown, MA, 1990, pp. 147-78. 31. A.W. Bowen and G.M. Leak: MetaU. Trans., 1970, vol. 1, pp. 1695-1700. 32. V.T. Borisov, V.M. Golikov, and G.V. Sherbedinskiy: Phys. Met. MetaUogr., 1966, vol. 22, pp. 175-76. 33. F.H. Wolhbier: Diffusion Data, Gulf Publishing Co., Diffusion Information Center, Aedermannsdorf, Switzerland, 1969, vol. 3 (3). 34. S.K. Bhattacharyya and K.C. Russell: Metall. Trans., 1972, vol. 3, pp. 2195-99. 35. S.K. Bhattacharyya and K.C. Russell: Metall. Trans. A, 1976, vol. 7A, pp. 453-62. 36. N.A. Gjostein: in Diffusion, H.I. Aaronson, ed., ASM, Metals Park, OH, 1973, pp. 241-74. 2888--VOLUME 22A, DECEMBER 1991 METALLURGICAL TRANSACTIONS A