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Atomic disorder–order phase transformation in nanocrystalline Fe–Al
Journal of Alloys and Compounds 334 (2002) 135–142
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Atomic disorder–order phase transformation in nanocrystalline Fe–Al S. Sarkar, C. Bansal* School of Physics, University of Hyderabad, Hyderabad 500 046, India Received 7 May 2001; accepted 3 July 2001
Abstract The atomic disorder to order (A2→DO 3 ) phase transformation was studied in nanocrystalline Fe–Al alloys prepared by mechanical alloying near Fe 3 Al stoichiometry. Important differences in the microstructure evolution as well as the kinetics of the transformation were observed as compared to the corresponding coarse grained alloys prepared by splat quenching. These were understood in terms of the small grain sizes and the presence of grain boundary regions in the nanophase alloy systems. 2002 Elsevier Science B.V. All rights reserved. ¨ spectroscopy Keywords: Transition metal compounds; Nanostructured materials; Mechanical alloying; X-ray diffraction; Mossbauer
1. Introduction The Fe–Al alloy system near Fe 3 Al stoichiometry shows very interesting order–disorder phase transformation behavior. The various ordered structures for b.c.c. alloys of A 3 B stoichiometry have been enumerated and analyzed by Cahn and his group [1]. Depending on the sign and strength of the first and second nearest neighbour interactions there are three possible ordered structures. These structures can be understood by considering a unit cell of lattice parameter twice that of the original b.c.c. unit cell. The A and B atoms can then occupy four possible sublattices a, b, g, and d (Fig. 1) in different ways to form these ordered structures which are designated as B32, B2, and DO 3 . The disordered b.c.c. phase is labeled as A2. In the B32 structure the adjacent a and g sites are occupied by Fe and the b and d sites are randomly occupied by Fe and Al. In case of B2 order two nonadjacent sublattices a and b are occupied by Fe atoms whereas the other two g and d are randomly occupied by Fe and Al atoms. For DO 3 order Fe atoms occupy three of the sublattices (b, g and d) whereas Al atoms occupy the fourth one (a). The probability distribution of Al atoms in the first nearest neighbour (1nn) shell of iron for these structures is also shown in Fig. 1. There is a distinct
*Corresponding author. Tel.: 191-40-301-0336; fax: 191-40-3010227. E-mail address: [email protected] (C. Bansal).
probability distribution for each of these ordered structures and the presence of these structures can be unambiguously ¨ observed using a technique such as Fe-57 Mossbauer spectroscopy, which is sensitive to the local environment around the Fe atom. The phase diagram [2] of the Fe–Al alloy system near Fe 3 Al stoichiometry shows the disordered A2 phase (a) at temperatures above 8008C, B2 ordered phase between 8008 and 5308C and the DO 3 ordered phase below 5308C. X-ray diffraction studies of the B2→DO 3 phase transformation showed the DO 3 long range order parameter in agreement with Bragg–Willams theory [3]. The transformation was observed to proceed by nucleation and growth process with a third degree order reaction kinetics. The temporal growth of DO 3 domains size (´) was found similar to the growth of grains in metals in the form ´ 5 kt n with n50.3 and k5a constant. In another set of measurements [4] the growth of DO 3 order was studied in alloys formed by piston-anvil quenching technique to retain the disordered A2 phase. In this case the alloys, heat treated at 3008C to develop DO 3 order, showed the transient formation of B32 order during the early stages of ordering when the DO 3 ordered domains were small. After long time annealing the equilibrium DO 3 order developed. This was interesting because there is no B32 ordered region in the equilibrium phase diagram and this observation was interpreted as a kinetic effect. Theoretical model calculations using the kinetic master equation in the point as well as the pair approximation formalism also showed formation of B2 order
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S. Sarkar, C. Bansal / Journal of Alloys and Compounds 334 (2002) 135 – 142
2. Experimental
Fig. 1. The four sublattices (a, b, g, and d) of the b.c.c. lattice required to construct the ordered structures. The arrows indicate nearest neighbour sublattices (e.g. a sub lattice atoms have atoms on g and d sublattices as 1nn atoms) (top). The calculated probabilities of Fe atoms having different no. of 1nn solute atoms for the A2, B32, B2, and DO 3 structures (bottom).
transiently during A2→DO 3 order transformation [5]. Monte Carlo simulations [6] also showed transient B32 phase formation in agreement with the above calculations and experiments. In the present work we study the ordering transformation behaviour of nanocrystalline Fe–Al alloys prepared by mechanical alloying in an initial disordered b.c.c. state far from thermodynamic equilibrium. These alloys are characterized by the presence of small grain sizes and large grain boundary regions which have been shown to have a significant influence on different phase transformations such as order-disorder, precipitation, and massive structural transformations [7–9]. In the present case, we find that for nanophase Fe–Al alloys the development of ordered structures as well as the ordering kinetics are different as compared to the coarse grained microcrystalline alloys.
Fe 12x Al x (x50.31 and 0.27) alloys were prepared by direct mechanical milling of the elemental Fe (99.99%1 purity, fine gray powder from Aldrich-Sigma chemicals) and Al (Aluminum shot, 3.2 mm m3N5 from Ventron-Alfa division) metal powders in a SPEX 8000 mixer mill using hardened steel balls and vial. A set of six balls (two of diameter 1 / 2 in. and four of diameter 1 / 4 in.) was used for milling. The ball:powder weight ratio was 5:1. The elemental powders were weighed according to the appropriate stoichiometric ratio and transferred to the vial and the milling subsequently was carried out under high purity Argon (IOLAR II grade from British Oxygen Company) atmosphere for 24 h. Fe contamination from the vial and the balls was found to be 2–3 at.% from the chemical analysis of the as-milled alloys which gave rise to final composition of the as-milled alloys as Fe 0.72 Al 0.28 and Fe 3 Al. The as-milled alloys were subjected to ordering heat treatments at 3008 and 4508C for various periods of times ranging from a few tens to a few hundred of hours. Heat treatments were carried out in borosilicate glass ampoules under Argon atmosphere and a pressure of 0.05 Torr. Powder X-ray diffraction patterns were recorded using a Phillips PW3710 based diffractometer with a Cu-target. Average grain sizes of the as-milled and the heat treated samples were determined from the line broadening of the (110) fundamental line of the b.c.c. alloys using the Scherrer formula [10] and by incorporating instrumental broadening which was determined by recording calibration spectra of pure bulk Fe powder. However no attempt was made to study the evolution of long range order from X-ray diffraction due to small intensities of the superlattice lines arising from partially ordered samples. ¨ Fe-57 Mossbauer spectroscopy was used to study the disorder–order phase transformation behavior of the alloy. The spectrometer was operated in a constant acceleration mode and transmission geometry was used for recording about 10 6 counts / channel in folded spectra for good ¨ statistics. The Mossbauer source was 25 mCi Co 57 in Rh matrix (from Du Pont Pharma). The g-ray detector was a gas filled (97%Xe13%CO 2 at 2 atm. pressure) propor¨ tional counter from LND Inc. (USA). Mossbauer data were ¨ and Dubois [11] to processed using the method of Le Caer evaluate the hyperfine magnetic field (hmf) distributions. A linear relationship between the Isomer shift (I) and hmf (H ) was considered in the form, I 5 aH 1 b, and the parameters a and b were optimized to get the minimum x 2.
3. Results and discussion Fig. 2 shows the typical X-ray diffraction patterns of the as-milled and heat treated Fe 3 Al alloy. The X-ray diffrac-
S. Sarkar, C. Bansal / Journal of Alloys and Compounds 334 (2002) 135 – 142
Fig. 2. Typical X-ray diffraction patterns for as-milled and heat treated Fe 3 Al alloys.
togram shows the formation of single disordered b.c.c. phase (A2) alloy. The average grain size of the as-milled alloy estimated from the linewidth of the fundamental (110) line was about 10 nm. Similar results were obtained ¨ from the X-ray study of Fe 0.72 Al 0.28 alloy. The Mossbauer spectra of the as-milled alloys are shown in Figs. 3 and 4, respectively for the 25 and 28 at.% Al alloys. These spectra are in excellent agreement with disordered Fe–Al alloys prepared by piston–anvil quenching [4]. The hmf distributions of the as-milled alloys as shown in Figs. 5–8 are also in good agreement with earlier reported results. The hmf seen by Fe in Fe–X alloys has been analyzed [12] in terms of the number of first and second neighbour solute atoms (X). The effect of the more distance neighbours was taken in terms of an average concentration dependence of all the fields. Ignoring the effect of second neighbours, which are very small in the Fe–Al system, the fields at Fe were found to be mainly dependent on the number of first neighbour Al atoms. The estimated fields for Fe atoms with different numbers of Al first neighbours are also shown in Fig. 5. The observed hmf distribution was fitted to a set of Gaussian functions centered at the field positions corresponding to different Al first neighbours configurations. The intensities of these Gaussian peaks corresponded very well to the probabilities of occurrence of these configurations as calculated using binomial distributions. This unambiguously shows that the initial state of the as-milled alloys is atomically disordered and the hmf assignments to
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¨ Fig. 3. Representative Mossbauer spectra for the as-milled and heat treated Fe 3 Al alloys and fits to data using model independent field distribution analysis (solid symbols, data; solid lines, fits).
¨ Fig. 4. Typical Mossbauer spectra for the as-milled and heat treated Fe 0.72 Al 0.28 alloys (solid symbols, data; solid lines, fits).
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¨ Fig. 7. hmf distributions evaluated from the Mossbauer spectra of asmilled, and 3008C heat treated samples of Fe 0.72 Al 0.28 composition. ¨ Fig. 5. hmf distributions evaluated from the Mossbauer spectra of asmilled and 3008C heat treated samples for Fe 3 Al stoichiometry alloy. The hyperfine field values of Fe atoms for different number of 1nn Al atoms are shown marked by arrows.
different number of Al first nearest neighbour (1nn) are justified. The evolution of short-range order in the system is reflected in the change in intensities of the various peaks in the hmf distributions. Figs. 5–8 show the evolution of the
peaks for the alloys heat treated at 3008 and 4508C for different periods of time. At 3008C the Fe 3 Al composition alloy shows a rapid increase in intensities of the 0 and 4 Al neighbour peaks and a decrease in the intensities of 2 and 3 Al neighbour peaks (Fig. 9). As seen from Fig. 1 the 0 and 4 Al peaks correspond to the growth of DO 3 ordered phase. The probability of 2 and 3 Al neighbour configurations is high for B32 order and we see from the evolution
¨ Fig. 6. hmf distributions evaluated from the Mossbauer spectra of asmilled and 4508C heat treated samples for Fe 3 Al composition alloy.
¨ Fig. 8. hmf distributions evaluated from the Mossbauer spectra of asmilled and 4508C heat treated samples of Fe 0.72 Al 0.28 composition.
S. Sarkar, C. Bansal / Journal of Alloys and Compounds 334 (2002) 135 – 142
Fig. 9. Temporal evolution of fraction of Fe atoms with 0, 4 and (213) first near neighbor Al atoms at 3008C (symbols, experimental fractions evaluated from the hmf distributions; solid lines: fits to data using Eq. (7)).
of the intensities of these peaks that they saturate to a substantially high value after the initial fast decrease. For DO 3 order these intensities should be ideally zero. These observations show that there is a rapid growth of DO 3 order from the A2 matrix in the initial stages followed by the stabilization of B32 phase along with the DO 3 phase. However as seen from Fig. 1 the probability distributions for the disordered A2 and the B32 ordered phases are similar. To further confirm the presence of B32 order, we plotted the kinetic path followed by the system in the space spanned by two order parameters [4]. Fig. 10 shows such a plot wherein the intensity of the 0 Al 1nn peak is plotted against the intensity of the 4 Al 1nn peak. Also shown in the figure, are the calculated kinetic paths for DO 3 order alone, for (50% DO 3 150% B32) order and for (50% DO 3 150% B2) order. The slope of these paths are given by m 5 Dp(0) /Dp(4)
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Fig. 10. Kinetic path at 3008C for the Fe 3 Al alloy showing the evolution of the 0 Al vs. 4 Al 1nn peaks. Also shown in the figure are the calculated paths assuming growth of pure DO 3 order, (50% DO 3 150% B2) order, and (50% DO 3 150% B32) order from the disordered A2 matrix.
because pDO 3 ( j), pB 2 ( j), pB 32 ( j), and pA2 ( j) are constants, the slope of the kinetic path depends only on the fractions ( f ) of these structures. As seen from the figure there is a growth of DO 3 order initially followed by an admixture of both DO 3 and B32 ordered phases. Essentially a similar behaviour is observed
(1)
where Dp( j) is the change in the probability of Fe atoms with j Al atoms in their 1nn shell, w.r.t the probability in the disordered structure: Dp( j) 5 fDO 3 f pDO 3 ( j) 2 pA2 ( j) g 1 fB 2 f pB 2 ( j) 2 pA2 ( j) g 1 fB 32 f pB 32 ( j) 2 pA2 ( j) g
(2)
where fDO 3 , fB 2 , and fB 32 are the fractions of the three ordered structures that grow from the disordered A2 structure, and fDO 3 1 fB 2 1 fB 32 1 fA2 5 1
(3)
Fig. 11. Temporal evolution at 3008C of fraction of Fe atoms with different no. of 1nn Al atoms for the Fe 0.72 Al 0.28 alloy system.
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Fig. 12. Temporal evolution at 4508C of fraction of Fe atoms with 1, 4 and (213) 1nn Al atoms for Fe 3 Al (symbols, experimental fractions; solid lines, fit).
for the Fe 72 Al 28 alloy heat treated at 3008C (Fig. 11) and for both the alloys heat treated at 4508C (Figs. 12 and 13) except that the 0 Al peak is not resolved from the 1 Al peak. It is not possible to make a plot of the kinetic path similar to the one shown in Fig. 10, in these cases as the intensity of the 0 Al peak is very small and large errors are observed in the fits to the field distributions using Gaussian peaks at the field value corresponding to the 0 Al configu-
Fig. 13. Temporal evolution at 4508C of fraction of Fe atoms with different number of 1nn Al atoms for the Fe 0.72 Al 0.28 alloy system.
ration. This may be due to the presence of more B32 ordered phase, which results in larger intensity of 1 Al neighbour peak as compared to 0 Al neighbour peak. A comparison of the behaviour of nanocrystalline Fe–Al alloys with that of coarse grained Fe–Al alloys prepared by piston–anvil quenching [4] shows that for the nano grained alloy there is no transient B32 phase formation but a rapid growth of DO 3 order directly. This can be attributed to the fast diffusion paths provided by the grain boundary regions [7]. The growth of DO 3 ordered regions could however extend only up to the size of the nanocrystalline grains. Davies [13] also observed that in fine grained cold-rolled Fe–Al alloys the DO 3 domains stopped growing beyond the grain size of 15 nm. In our nanocrystalline alloys the grain growth as shown in Figs. 14 and 15 is very small and saturates to a maximum size of about 15 nm. This also explains the presence of substantial B32 order in later stages of the transformation. The B32 ordered regions are possible of arising in the anti-phase domain boundary (APDB) regions between DO 3 ordered domain [14]. This was also the explanation given for the transient formation of B32 order between small DO 3 ordered domains (approximately 3 nm) which grew initially in the coarse-grained alloy [4]. In the nanophase alloy the maximum growth of DO 3 domain size is limited by the grain size and the grain boundary regions also constitutes the B32 ordered APD boundaries for the system. Figs. 16 and 17 show the kinetic evolution of the short-range order parameter, S(4), defined in terms of the intensities p(4) of the 4 Al 1nn peaks, as follows: S(4) 5 f p expt (4) 2 p disord. (4) g / f p DO 3 (4) 2 p disord. (4) g
(4)
Fig. 14. Grain growth kinetics at 3008 and 4508C for the Fe 3 Al alloy (symbols, experimental grain size; solid line, fits to data as explained in text).
S. Sarkar, C. Bansal / Journal of Alloys and Compounds 334 (2002) 135 – 142
Fig. 15. Grain growth kinetics at 3008 and 4508C for the Fe 0.72 Al 0.28 alloy (symbols, experimental grain size; solid line, fit to data).
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Fig. 17. Evolution of short-range order parameter, S(4), with time at 3008 and 4508C for the Fe 0.72 Al 0.28 alloy (symbols, data; solid line, fit using Eq. (7)).
where p expt (4) is the experimentally observed intensity, p disord. (4) is the intensity for the disordered alloy calculated from the binomial distribution and p DO 3 (4) is the intensity for fully ordered alloy. The kinetics of order evolution in an alloy system can be treated in terms of the model proposed by Khachaturyan [15] which relates the kinetics of the transformation to the thermodynamics. According to this model the temporal rate of change of order parameter, h(t), is assumed proportional to the sensitivity of the Gibbs free energy change w.r.t the order parameter: dh(t) L dG(h ) ]] 5 2 ] ]] dt kT dh
(5)
where L is a mobility factor which depends on the kinetic mechanism of ordering. Expanding G(h ) in a Taylor series about equilibrium, h 5heq , and retaining only terms up to second order derivative of G (dG / dh 50 at h 5heq ) one gets dh(t) ]] 5 2 A(h 2 heq ) dt where
S D
L d 2G A 5 ] ]] kT dh 2
h 5h eq
(6)
The solution of this equation is Fig. 16. Evolution of the short-range order parameter, S(4), with time at 3008 and 4508C for the Fe 3 Al alloy (symbols, experimental data; solid lines, fits to data using Eq. (7)).
h(t) 5 (h0 2 heq ) exp(2At) 1 heq where h0 is the order parameter at time t50.
(7)
S. Sarkar, C. Bansal / Journal of Alloys and Compounds 334 (2002) 135 – 142
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Table 1 Values of different order parameters and relaxation times obtained by fitting the S(4) data of Figs. 16 and 17 to Eq. (7) Alloy stoichiometry
Temperature (8C)
Order parameter, S(4), at time t50
Equilibrium order parameter, S(4), attained
Ordering relaxation time (1 /A) (h)
Fe 3 Al
300 450
0.0 0.0
0.42 0.28
56 4.5
Fe 0.72 Al 0.28
300 450
0.02 0.02
0.68 0.72
27 12
The fits to the S(4) data using this equation (with h ;S(4)) are also shown in Figs. 16 and 17. The parameters obtained from the fits are shown in Table 1. As seen from the table the ordering relaxation time (1 /A) is small at higher temperatures, as expected. The equilibrium order parameter reached is seen to be less than unity. This also reflects the presence of residual B32 order as mentioned earlier. Another important point to note is that in polycrystalline coarse grained alloys the B2→DO 3 transformation was found to proceed as a third order reaction [3] whereas our equation to describe the order evolution corresponds to a first order reaction. This indicates that the ordering mechanism in the nanocrystalline alloys is different from the coarse grained alloys due to the presence of extensive grain boundary regions. As observed in our earlier measurements on kinetics of ordering in nanophase FeCo–Mo alloys [7] the grain boundary regions provide short circuited diffusion paths which give rise to faster diffusion relative to coarse grained systems. The kinetics of grain growth as shown in Figs. 14 and 15 also do not follow a conventional t 1 / 2 curvature driven grain growth mechanism as observed for bulk systems. The grain growth behavior in our system however, could also be fitted to Eq. (7) with h replaced by average grain size. This is not totally unexpected because the mechanism of grain growth as well as atomic ordering may be the same viz. diffusion through grain boundaries. Such correlation between grain growth and chemical ordering was also observed in nanocrystalline Fe 32x Mn x Si alloys [16] where the chemical ordering was seen to influence the grain growth behavior significantly. It was observed in these alloy systems that the atomic order proceeded very rapidly and this resulted in impeding of grain growth and once atomic ordering took place there was no further growth of grains. It is relevant to mention here that we have assumed a fixed mobility factor (L) in Eq. (5), instead of a time
varying L due to change in vacancy and defect concentration during annealing, i.e. a linear response theory, to analyze the temporal evolution of S(4) and grain growth. We feel this assumption is justified because in these nanocrystalline systems the atomic diffusion is taking place through grain boundary regions occupying a large volume fraction as compared to the coarse grained materials. This grain boundary diffusion gives rise to large L and therefore the change in L with time due to change in concentration of defects and vacancies with annealing time is expected to be only a small perturbation and hence this assumption of constant mobility factor is valid. In conclusion we observed that the nanophase nature of the Fe–Al alloys influenced the atomic order evolution as well as the kinetics of the ordering process very significantly. The observed behaviour was attributed to the nano size of the grains and the grain boundary regions in these nanocrystalline alloy systems.
Acknowledgements This work was supported by CSIR (India) under grant No. 3(795)96-EMR-II.
References [1] M.J. Richards, J.W. Cahn, Acta Metall. 19 (1971) 1263; S.M. Allen, J.W. Cahn, Acta Metall. 20 (1972) 423. [2] T.B. Massalski (Ed.), Binary Alloy Phase Diagrams, ASM International, Metals Park, OH, 1990. [3] K. Oki, M. Hasaka, T. Eguchi, Trans. Jpn. Inst. Met. 14 (1975) 8. [4] Z.Q. Gao, B. Fultz, Phil. Mag. B67 (1993) 787. [5] L. Anthony, B. Fultz, J. Mater. Res. 4 (1989) 1132; L. Anthony, B. Fultz, J. Mater. Res. 4 (1991) 1140. [6] L. Anthony, B. Fultz, J. Mater. Res. 9 (1994) 348. [7] S. Sarkar, C. Bansal, Acta Mater. 49 (2001) 1789. [8] S. Sarkar, C. Bansal, Solid State Phys. (India) 43 (2000) in press. [9] S. Sarkar, C. Bansal, A. Chatterjee, Phys. Rev. B62 (2000) 3218. [10] B.E. Warren, X-ray Diffraction, Dover, New York, 1990. ¨ J.M. Dubois, J. Phys. E 12 (1979) 1083. [11] G. Le Caer, ¨ [12] B. Fultz, in: G.J. Long, F. Grandjean (Eds.), Mossbauer Spectroscopy Applied to Magnetism and Materials Science, Plenum Press, New York, 1993, p. 1. [13] R.G. Davies, Trans. Met. Soc. AIME 230 (1964) 903. [14] D.A. Porter, K.E. Easterling, Phase Transformations in Metals and Alloys, Chapman and Hall, London, 1992. [15] A.G. Khachaturyan, in: B. Chalmers, J.W. Christian, T. Massalski (Eds.), Progress in Materials Science, Vol. 22, Pergamon Press, Oxford, 1975, p. 1. [16] C. Bansal, Z.Q. Gao, B. Fultz, Nanostruct. Mater. 5 (1995) 327.
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Atomic disorder–order phase transformation in nanocrystalline Fe–Al S. Sarkar, C. Bansal* School of Physics, University of Hyderabad, Hyderabad 500 046, India Received 7 May 2001; accepted 3 July 2001
Abstract The atomic disorder to order (A2→DO 3 ) phase transformation was studied in nanocrystalline Fe–Al alloys prepared by mechanical alloying near Fe 3 Al stoichiometry. Important differences in the microstructure evolution as well as the kinetics of the transformation were observed as compared to the corresponding coarse grained alloys prepared by splat quenching. These were understood in terms of the small grain sizes and the presence of grain boundary regions in the nanophase alloy systems. 2002 Elsevier Science B.V. All rights reserved. ¨ spectroscopy Keywords: Transition metal compounds; Nanostructured materials; Mechanical alloying; X-ray diffraction; Mossbauer
1. Introduction The Fe–Al alloy system near Fe 3 Al stoichiometry shows very interesting order–disorder phase transformation behavior. The various ordered structures for b.c.c. alloys of A 3 B stoichiometry have been enumerated and analyzed by Cahn and his group [1]. Depending on the sign and strength of the first and second nearest neighbour interactions there are three possible ordered structures. These structures can be understood by considering a unit cell of lattice parameter twice that of the original b.c.c. unit cell. The A and B atoms can then occupy four possible sublattices a, b, g, and d (Fig. 1) in different ways to form these ordered structures which are designated as B32, B2, and DO 3 . The disordered b.c.c. phase is labeled as A2. In the B32 structure the adjacent a and g sites are occupied by Fe and the b and d sites are randomly occupied by Fe and Al. In case of B2 order two nonadjacent sublattices a and b are occupied by Fe atoms whereas the other two g and d are randomly occupied by Fe and Al atoms. For DO 3 order Fe atoms occupy three of the sublattices (b, g and d) whereas Al atoms occupy the fourth one (a). The probability distribution of Al atoms in the first nearest neighbour (1nn) shell of iron for these structures is also shown in Fig. 1. There is a distinct
*Corresponding author. Tel.: 191-40-301-0336; fax: 191-40-3010227. E-mail address: [email protected] (C. Bansal).
probability distribution for each of these ordered structures and the presence of these structures can be unambiguously ¨ observed using a technique such as Fe-57 Mossbauer spectroscopy, which is sensitive to the local environment around the Fe atom. The phase diagram [2] of the Fe–Al alloy system near Fe 3 Al stoichiometry shows the disordered A2 phase (a) at temperatures above 8008C, B2 ordered phase between 8008 and 5308C and the DO 3 ordered phase below 5308C. X-ray diffraction studies of the B2→DO 3 phase transformation showed the DO 3 long range order parameter in agreement with Bragg–Willams theory [3]. The transformation was observed to proceed by nucleation and growth process with a third degree order reaction kinetics. The temporal growth of DO 3 domains size (´) was found similar to the growth of grains in metals in the form ´ 5 kt n with n50.3 and k5a constant. In another set of measurements [4] the growth of DO 3 order was studied in alloys formed by piston-anvil quenching technique to retain the disordered A2 phase. In this case the alloys, heat treated at 3008C to develop DO 3 order, showed the transient formation of B32 order during the early stages of ordering when the DO 3 ordered domains were small. After long time annealing the equilibrium DO 3 order developed. This was interesting because there is no B32 ordered region in the equilibrium phase diagram and this observation was interpreted as a kinetic effect. Theoretical model calculations using the kinetic master equation in the point as well as the pair approximation formalism also showed formation of B2 order
0925-8388 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0925-8388( 01 )01744-3
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S. Sarkar, C. Bansal / Journal of Alloys and Compounds 334 (2002) 135 – 142
2. Experimental
Fig. 1. The four sublattices (a, b, g, and d) of the b.c.c. lattice required to construct the ordered structures. The arrows indicate nearest neighbour sublattices (e.g. a sub lattice atoms have atoms on g and d sublattices as 1nn atoms) (top). The calculated probabilities of Fe atoms having different no. of 1nn solute atoms for the A2, B32, B2, and DO 3 structures (bottom).
transiently during A2→DO 3 order transformation [5]. Monte Carlo simulations [6] also showed transient B32 phase formation in agreement with the above calculations and experiments. In the present work we study the ordering transformation behaviour of nanocrystalline Fe–Al alloys prepared by mechanical alloying in an initial disordered b.c.c. state far from thermodynamic equilibrium. These alloys are characterized by the presence of small grain sizes and large grain boundary regions which have been shown to have a significant influence on different phase transformations such as order-disorder, precipitation, and massive structural transformations [7–9]. In the present case, we find that for nanophase Fe–Al alloys the development of ordered structures as well as the ordering kinetics are different as compared to the coarse grained microcrystalline alloys.
Fe 12x Al x (x50.31 and 0.27) alloys were prepared by direct mechanical milling of the elemental Fe (99.99%1 purity, fine gray powder from Aldrich-Sigma chemicals) and Al (Aluminum shot, 3.2 mm m3N5 from Ventron-Alfa division) metal powders in a SPEX 8000 mixer mill using hardened steel balls and vial. A set of six balls (two of diameter 1 / 2 in. and four of diameter 1 / 4 in.) was used for milling. The ball:powder weight ratio was 5:1. The elemental powders were weighed according to the appropriate stoichiometric ratio and transferred to the vial and the milling subsequently was carried out under high purity Argon (IOLAR II grade from British Oxygen Company) atmosphere for 24 h. Fe contamination from the vial and the balls was found to be 2–3 at.% from the chemical analysis of the as-milled alloys which gave rise to final composition of the as-milled alloys as Fe 0.72 Al 0.28 and Fe 3 Al. The as-milled alloys were subjected to ordering heat treatments at 3008 and 4508C for various periods of times ranging from a few tens to a few hundred of hours. Heat treatments were carried out in borosilicate glass ampoules under Argon atmosphere and a pressure of 0.05 Torr. Powder X-ray diffraction patterns were recorded using a Phillips PW3710 based diffractometer with a Cu-target. Average grain sizes of the as-milled and the heat treated samples were determined from the line broadening of the (110) fundamental line of the b.c.c. alloys using the Scherrer formula [10] and by incorporating instrumental broadening which was determined by recording calibration spectra of pure bulk Fe powder. However no attempt was made to study the evolution of long range order from X-ray diffraction due to small intensities of the superlattice lines arising from partially ordered samples. ¨ Fe-57 Mossbauer spectroscopy was used to study the disorder–order phase transformation behavior of the alloy. The spectrometer was operated in a constant acceleration mode and transmission geometry was used for recording about 10 6 counts / channel in folded spectra for good ¨ statistics. The Mossbauer source was 25 mCi Co 57 in Rh matrix (from Du Pont Pharma). The g-ray detector was a gas filled (97%Xe13%CO 2 at 2 atm. pressure) propor¨ tional counter from LND Inc. (USA). Mossbauer data were ¨ and Dubois [11] to processed using the method of Le Caer evaluate the hyperfine magnetic field (hmf) distributions. A linear relationship between the Isomer shift (I) and hmf (H ) was considered in the form, I 5 aH 1 b, and the parameters a and b were optimized to get the minimum x 2.
3. Results and discussion Fig. 2 shows the typical X-ray diffraction patterns of the as-milled and heat treated Fe 3 Al alloy. The X-ray diffrac-
S. Sarkar, C. Bansal / Journal of Alloys and Compounds 334 (2002) 135 – 142
Fig. 2. Typical X-ray diffraction patterns for as-milled and heat treated Fe 3 Al alloys.
togram shows the formation of single disordered b.c.c. phase (A2) alloy. The average grain size of the as-milled alloy estimated from the linewidth of the fundamental (110) line was about 10 nm. Similar results were obtained ¨ from the X-ray study of Fe 0.72 Al 0.28 alloy. The Mossbauer spectra of the as-milled alloys are shown in Figs. 3 and 4, respectively for the 25 and 28 at.% Al alloys. These spectra are in excellent agreement with disordered Fe–Al alloys prepared by piston–anvil quenching [4]. The hmf distributions of the as-milled alloys as shown in Figs. 5–8 are also in good agreement with earlier reported results. The hmf seen by Fe in Fe–X alloys has been analyzed [12] in terms of the number of first and second neighbour solute atoms (X). The effect of the more distance neighbours was taken in terms of an average concentration dependence of all the fields. Ignoring the effect of second neighbours, which are very small in the Fe–Al system, the fields at Fe were found to be mainly dependent on the number of first neighbour Al atoms. The estimated fields for Fe atoms with different numbers of Al first neighbours are also shown in Fig. 5. The observed hmf distribution was fitted to a set of Gaussian functions centered at the field positions corresponding to different Al first neighbours configurations. The intensities of these Gaussian peaks corresponded very well to the probabilities of occurrence of these configurations as calculated using binomial distributions. This unambiguously shows that the initial state of the as-milled alloys is atomically disordered and the hmf assignments to
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¨ Fig. 3. Representative Mossbauer spectra for the as-milled and heat treated Fe 3 Al alloys and fits to data using model independent field distribution analysis (solid symbols, data; solid lines, fits).
¨ Fig. 4. Typical Mossbauer spectra for the as-milled and heat treated Fe 0.72 Al 0.28 alloys (solid symbols, data; solid lines, fits).
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¨ Fig. 7. hmf distributions evaluated from the Mossbauer spectra of asmilled, and 3008C heat treated samples of Fe 0.72 Al 0.28 composition. ¨ Fig. 5. hmf distributions evaluated from the Mossbauer spectra of asmilled and 3008C heat treated samples for Fe 3 Al stoichiometry alloy. The hyperfine field values of Fe atoms for different number of 1nn Al atoms are shown marked by arrows.
different number of Al first nearest neighbour (1nn) are justified. The evolution of short-range order in the system is reflected in the change in intensities of the various peaks in the hmf distributions. Figs. 5–8 show the evolution of the
peaks for the alloys heat treated at 3008 and 4508C for different periods of time. At 3008C the Fe 3 Al composition alloy shows a rapid increase in intensities of the 0 and 4 Al neighbour peaks and a decrease in the intensities of 2 and 3 Al neighbour peaks (Fig. 9). As seen from Fig. 1 the 0 and 4 Al peaks correspond to the growth of DO 3 ordered phase. The probability of 2 and 3 Al neighbour configurations is high for B32 order and we see from the evolution
¨ Fig. 6. hmf distributions evaluated from the Mossbauer spectra of asmilled and 4508C heat treated samples for Fe 3 Al composition alloy.
¨ Fig. 8. hmf distributions evaluated from the Mossbauer spectra of asmilled and 4508C heat treated samples of Fe 0.72 Al 0.28 composition.
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Fig. 9. Temporal evolution of fraction of Fe atoms with 0, 4 and (213) first near neighbor Al atoms at 3008C (symbols, experimental fractions evaluated from the hmf distributions; solid lines: fits to data using Eq. (7)).
of the intensities of these peaks that they saturate to a substantially high value after the initial fast decrease. For DO 3 order these intensities should be ideally zero. These observations show that there is a rapid growth of DO 3 order from the A2 matrix in the initial stages followed by the stabilization of B32 phase along with the DO 3 phase. However as seen from Fig. 1 the probability distributions for the disordered A2 and the B32 ordered phases are similar. To further confirm the presence of B32 order, we plotted the kinetic path followed by the system in the space spanned by two order parameters [4]. Fig. 10 shows such a plot wherein the intensity of the 0 Al 1nn peak is plotted against the intensity of the 4 Al 1nn peak. Also shown in the figure, are the calculated kinetic paths for DO 3 order alone, for (50% DO 3 150% B32) order and for (50% DO 3 150% B2) order. The slope of these paths are given by m 5 Dp(0) /Dp(4)
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Fig. 10. Kinetic path at 3008C for the Fe 3 Al alloy showing the evolution of the 0 Al vs. 4 Al 1nn peaks. Also shown in the figure are the calculated paths assuming growth of pure DO 3 order, (50% DO 3 150% B2) order, and (50% DO 3 150% B32) order from the disordered A2 matrix.
because pDO 3 ( j), pB 2 ( j), pB 32 ( j), and pA2 ( j) are constants, the slope of the kinetic path depends only on the fractions ( f ) of these structures. As seen from the figure there is a growth of DO 3 order initially followed by an admixture of both DO 3 and B32 ordered phases. Essentially a similar behaviour is observed
(1)
where Dp( j) is the change in the probability of Fe atoms with j Al atoms in their 1nn shell, w.r.t the probability in the disordered structure: Dp( j) 5 fDO 3 f pDO 3 ( j) 2 pA2 ( j) g 1 fB 2 f pB 2 ( j) 2 pA2 ( j) g 1 fB 32 f pB 32 ( j) 2 pA2 ( j) g
(2)
where fDO 3 , fB 2 , and fB 32 are the fractions of the three ordered structures that grow from the disordered A2 structure, and fDO 3 1 fB 2 1 fB 32 1 fA2 5 1
(3)
Fig. 11. Temporal evolution at 3008C of fraction of Fe atoms with different no. of 1nn Al atoms for the Fe 0.72 Al 0.28 alloy system.
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Fig. 12. Temporal evolution at 4508C of fraction of Fe atoms with 1, 4 and (213) 1nn Al atoms for Fe 3 Al (symbols, experimental fractions; solid lines, fit).
for the Fe 72 Al 28 alloy heat treated at 3008C (Fig. 11) and for both the alloys heat treated at 4508C (Figs. 12 and 13) except that the 0 Al peak is not resolved from the 1 Al peak. It is not possible to make a plot of the kinetic path similar to the one shown in Fig. 10, in these cases as the intensity of the 0 Al peak is very small and large errors are observed in the fits to the field distributions using Gaussian peaks at the field value corresponding to the 0 Al configu-
Fig. 13. Temporal evolution at 4508C of fraction of Fe atoms with different number of 1nn Al atoms for the Fe 0.72 Al 0.28 alloy system.
ration. This may be due to the presence of more B32 ordered phase, which results in larger intensity of 1 Al neighbour peak as compared to 0 Al neighbour peak. A comparison of the behaviour of nanocrystalline Fe–Al alloys with that of coarse grained Fe–Al alloys prepared by piston–anvil quenching [4] shows that for the nano grained alloy there is no transient B32 phase formation but a rapid growth of DO 3 order directly. This can be attributed to the fast diffusion paths provided by the grain boundary regions [7]. The growth of DO 3 ordered regions could however extend only up to the size of the nanocrystalline grains. Davies [13] also observed that in fine grained cold-rolled Fe–Al alloys the DO 3 domains stopped growing beyond the grain size of 15 nm. In our nanocrystalline alloys the grain growth as shown in Figs. 14 and 15 is very small and saturates to a maximum size of about 15 nm. This also explains the presence of substantial B32 order in later stages of the transformation. The B32 ordered regions are possible of arising in the anti-phase domain boundary (APDB) regions between DO 3 ordered domain [14]. This was also the explanation given for the transient formation of B32 order between small DO 3 ordered domains (approximately 3 nm) which grew initially in the coarse-grained alloy [4]. In the nanophase alloy the maximum growth of DO 3 domain size is limited by the grain size and the grain boundary regions also constitutes the B32 ordered APD boundaries for the system. Figs. 16 and 17 show the kinetic evolution of the short-range order parameter, S(4), defined in terms of the intensities p(4) of the 4 Al 1nn peaks, as follows: S(4) 5 f p expt (4) 2 p disord. (4) g / f p DO 3 (4) 2 p disord. (4) g
(4)
Fig. 14. Grain growth kinetics at 3008 and 4508C for the Fe 3 Al alloy (symbols, experimental grain size; solid line, fits to data as explained in text).
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Fig. 15. Grain growth kinetics at 3008 and 4508C for the Fe 0.72 Al 0.28 alloy (symbols, experimental grain size; solid line, fit to data).
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Fig. 17. Evolution of short-range order parameter, S(4), with time at 3008 and 4508C for the Fe 0.72 Al 0.28 alloy (symbols, data; solid line, fit using Eq. (7)).
where p expt (4) is the experimentally observed intensity, p disord. (4) is the intensity for the disordered alloy calculated from the binomial distribution and p DO 3 (4) is the intensity for fully ordered alloy. The kinetics of order evolution in an alloy system can be treated in terms of the model proposed by Khachaturyan [15] which relates the kinetics of the transformation to the thermodynamics. According to this model the temporal rate of change of order parameter, h(t), is assumed proportional to the sensitivity of the Gibbs free energy change w.r.t the order parameter: dh(t) L dG(h ) ]] 5 2 ] ]] dt kT dh
(5)
where L is a mobility factor which depends on the kinetic mechanism of ordering. Expanding G(h ) in a Taylor series about equilibrium, h 5heq , and retaining only terms up to second order derivative of G (dG / dh 50 at h 5heq ) one gets dh(t) ]] 5 2 A(h 2 heq ) dt where
S D
L d 2G A 5 ] ]] kT dh 2
h 5h eq
(6)
The solution of this equation is Fig. 16. Evolution of the short-range order parameter, S(4), with time at 3008 and 4508C for the Fe 3 Al alloy (symbols, experimental data; solid lines, fits to data using Eq. (7)).
h(t) 5 (h0 2 heq ) exp(2At) 1 heq where h0 is the order parameter at time t50.
(7)
S. Sarkar, C. Bansal / Journal of Alloys and Compounds 334 (2002) 135 – 142
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Table 1 Values of different order parameters and relaxation times obtained by fitting the S(4) data of Figs. 16 and 17 to Eq. (7) Alloy stoichiometry
Temperature (8C)
Order parameter, S(4), at time t50
Equilibrium order parameter, S(4), attained
Ordering relaxation time (1 /A) (h)
Fe 3 Al
300 450
0.0 0.0
0.42 0.28
56 4.5
Fe 0.72 Al 0.28
300 450
0.02 0.02
0.68 0.72
27 12
The fits to the S(4) data using this equation (with h ;S(4)) are also shown in Figs. 16 and 17. The parameters obtained from the fits are shown in Table 1. As seen from the table the ordering relaxation time (1 /A) is small at higher temperatures, as expected. The equilibrium order parameter reached is seen to be less than unity. This also reflects the presence of residual B32 order as mentioned earlier. Another important point to note is that in polycrystalline coarse grained alloys the B2→DO 3 transformation was found to proceed as a third order reaction [3] whereas our equation to describe the order evolution corresponds to a first order reaction. This indicates that the ordering mechanism in the nanocrystalline alloys is different from the coarse grained alloys due to the presence of extensive grain boundary regions. As observed in our earlier measurements on kinetics of ordering in nanophase FeCo–Mo alloys [7] the grain boundary regions provide short circuited diffusion paths which give rise to faster diffusion relative to coarse grained systems. The kinetics of grain growth as shown in Figs. 14 and 15 also do not follow a conventional t 1 / 2 curvature driven grain growth mechanism as observed for bulk systems. The grain growth behavior in our system however, could also be fitted to Eq. (7) with h replaced by average grain size. This is not totally unexpected because the mechanism of grain growth as well as atomic ordering may be the same viz. diffusion through grain boundaries. Such correlation between grain growth and chemical ordering was also observed in nanocrystalline Fe 32x Mn x Si alloys [16] where the chemical ordering was seen to influence the grain growth behavior significantly. It was observed in these alloy systems that the atomic order proceeded very rapidly and this resulted in impeding of grain growth and once atomic ordering took place there was no further growth of grains. It is relevant to mention here that we have assumed a fixed mobility factor (L) in Eq. (5), instead of a time
varying L due to change in vacancy and defect concentration during annealing, i.e. a linear response theory, to analyze the temporal evolution of S(4) and grain growth. We feel this assumption is justified because in these nanocrystalline systems the atomic diffusion is taking place through grain boundary regions occupying a large volume fraction as compared to the coarse grained materials. This grain boundary diffusion gives rise to large L and therefore the change in L with time due to change in concentration of defects and vacancies with annealing time is expected to be only a small perturbation and hence this assumption of constant mobility factor is valid. In conclusion we observed that the nanophase nature of the Fe–Al alloys influenced the atomic order evolution as well as the kinetics of the ordering process very significantly. The observed behaviour was attributed to the nano size of the grains and the grain boundary regions in these nanocrystalline alloy systems.
Acknowledgements This work was supported by CSIR (India) under grant No. 3(795)96-EMR-II.
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