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Amorphization of Ni–Al alloys by fast quenching from the liquid state: a molecular dynamics study
Journal of Non-Crystalline Solids 298 (2002) 60–66 www.elsevier.com/locate/jnoncrysol
Amorphization of Ni–Al alloys by fast quenching from the liquid state: a molecular dynamics study E.G. Noya, C. Rey, L.J. Gallego
*
Departamento de Fısica de la Materia Condensada, Facultad de Fısica, Universidad de Santiago de Compostela, E-15782 Santiago de Compostela, Spain Received 18 April 2001; received in revised form 3 October 2001
Abstract By means of constant temperature, constant thermodynamic tension molecular dynamics (MD) simulations using the embedded atom model proposed by Voter and Chen, we investigated the glass-forming ability of the alloys Ni3 Al, NiAl and NiAl3 under rapid quenching from the liquid state. Although the system Ni–Al does not form a glass when quenched from the liquid or vapour phases at the quenching rates achieved by conventional rapid quenching methods, at the very fast cooling rates used in this study the simulated system was amorphized from the liquid state at all compositions considered. This is consistent with the results of laser quenching experiments. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 31.15.Qg; 64.70.Pf
1. Introduction Molecular dynamics (MD) simulation is a powerful tool for investigating the properties of materials in general and metallic systems in particular. Though limited to small systems and short process times (a limitation that in some cases prevents the simulation of certain phenomena of interest), this technique can afford information on properties that would be very difficult or impossible to determine experimentally, or which simply have not yet been measured.
*
Corresponding author. Tel.: +34-981 563 100 x13995; fax: +34-981 520 676. E-mail address: [email protected] (L.J. Gallego).
The success of MD simulations depends on the accuracy of the model used to describe atomic interactions, which in the case of metallic systems is complicated by the important contribution of many-body effects. Semiempirical many-body potentials such as those based on the embedded atom model (EAM) [1] or the second moment approximation to the tight binding method (TBM-SMA) [2,3], which were developed to overcome the limitations of the pair-potential approximation for the description of metallic bonding, have successfully been used to analyze both bulk and surface properties of transition metals and their alloys. However, these semiempirical potentials are generally fitted to one-temperature solid-state properties, and it is not obvious a priori that the same potentials are equally capable of describing the
0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 1 0 4 8 - 1
E.G. Noya et al. / Journal of Non-Crystalline Solids 298 (2002) 60–66
properties of liquid and amorphous metals, with their quite different electron densities and interatomic separation distributions. Starting with the pioneering work of Foiles [4], several studies have shown that EAM potentials can in fact describe the structural, thermodynamic and atomic-transport properties of pure liquid transition and noble metals reasonably well (see, e.g., [5–13]). However, less work has been carried out on binary liquid alloys. The most extensive study in this area was recently performed by Asta et al. [14], who employed Monte Carlo and MD simulations to analyze the static and dynamic properties of liquid Ni–Al alloys using various interatomic EAM potentials. Their results showed that, although there are limitations to the application of EAM potentials for alloys rich in Al (partially due to the fact that no liquid data were used in their parameterization), the agreement between calculated and measured properties is reasonable. In particular, calculated partial structure factors were found to be in semiquantitative agreement with published neutron scattering measurements for Ni20 Al80 alloys [15], indicating that liquid-phase chemical short-range order (CSRO) is qualitatively well described. As is well known, CSRO effects generally play an important role in the formation of amophous alloys (see, e.g., [16]). In view of this versatility on the part of EAM potentials, in the work described here we used one optimized for Ni–Al in MD simulations performed to investigate the glass-forming ability of this system under rapid quenching from the liquid state (RQ). Ni–Al is interesting for several reasons. First, it is a borderline case as regards the formation of a metallic glass by RQ (see [17] and the discussion in Section 4). Secondly, it has a very rich phase diagram with several intermetallic phases [18], the nucleation and growth of which may compete with the formation of a metallic glass. Some of the results of this study can be compared with those obtained recently by Wang et al. [19] concerning the possibility of the amorphization of Ni3 Al by RQ (Wang et al. used Gao and Bacon’s potential [20], which is based on Finnis and Sinclair’s TBM-SMA scheme [21], in conjunction with constant presure, constant temperature (NPT) MD simulations).
61
The essential technical details of the computational method used in this paper are given in Section 2. In Sections 3 and 4 we present and discuss our results. Finally, in Section 5, we summarize our main conclusions.
2. Computational procedure In the calculations on Ni–Al performed in this work atomic interactions were described using the EAM potential that Voter and Chen (VC) optimized for prediction of the elastic constants and vacancy formation energies of pure Ni and pure Al, the bond lengths and bond energies of the diatomic molecules Ni2 and Al2 , the lattice constant, cohesive energy, elastic constants, vacancy formation energy, antiphase boundary energies and superlattice intrinsic stacking fault energy of the compound Ni3 Al, and the lattice constant and cohesive energy of the compound NiAl [22]. The VC Ni–Al potential is one of the EAM versions that were used by Asta et al. [14] in their study of liquid Ni–Al alloys, and it has also been employed by Alemany et al. [23] to investigate the diffusion coefficient of Ni impurity in liquid Al. It is therefore reasonable to hope that this potential may also give a satisfactory description of the glass-forming ability of Ni–Al alloys. Unlike those of Wang et al. [19], our simulations were performed using the constant temperature, constant thermodynamic tension (TtN) MD technique [24], which combines the Nose canonical ensemble [25] with the Parrinello–Rahman variable shape and size ensemble [26,27]. TtN MD simulations, with quantum Sutton–Chen manybody potentials [28,29], have recently been carried out by Qi et al. [30] to investigate the glass-forming ability of the alloys Cu–Ag and Cu–Ni under RQ. We considered three different systems of 864 atoms in a cubic box with periodic boundary conditions, namely 648 Ni atoms and 216 Al atoms, 432 Ni atoms and 432 Al atoms, and 216 Ni atoms and 648 Al atoms, proportions which correspond to the compositions of the intermetallic compounds Ni3 Al, NiAl and NiAl3 , respectively. The equations of motion were solved using a fourth-order
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Gear predictor–corrector algorithm [31] with a time step of 2 104 ps. At each composition, the initial configuration was obtained by equilibrating the system in the liquid phase at 2000 K, a temperature much higher that the melting temperatures of the intermetallic compounds Ni3 Al, NiAl and NiAl3 (the experimentally determined melting temperatures of these phases are 1658, 1912 and 1127 K, respectively [18]). The system was then cooled to room temperature (300 K) in decrements of 100 K, being kept at each temperature for times of 10 or 2.5 ps corresponding to cooling rates of 1013 and 4 1013 K/s, respectively. These were the cooling rates considered by Wang et al. [19] in their NPT MD study of Ni3 Al using Gao and Bacon’s potential [20]. For each temperature, the volume V, total energy E and radial distribution function gðrÞ of the system were averaged over the full TtN MD trajectory.
3. Results Fig. 1 shows the variations of the volume and total energy as the Ni3 Al sample was cooled from 2000 to 300 K at rates of 1013 and 4 1013 K/s. The results for the two cooling rates virtually coincide, which contrasts with the (much more scattered) results of Wang et al. [19]. Both V and E vary continuously with T, but the slopes exhibit discontinuities that are indicative of a glass transition. For quantitative determination of the glass transition temperature Tg for a cooling rate of 1013 K/s we considered the EðT Þ data, ignoring the data close to the change of slope and fitting the remaining high-temperature and low-temperature data sets with separate Pade equations: EðT Þ ¼
a þ bT þ cT 2 : 1 þ dT
ð1Þ
Tg was taken to be the temperature at which these curves intersected each other, 910 K (nearly identical values were found using the V ðT Þ data or either the EðT Þ or V ðT Þ data for a cooling rate of 4 1013 K/s). This temperature is somewhat higher than the 855 K found by Wang et al. [19] using NPT MD calculations with Gao and
Fig. 1. Variations of the volume (upper panel) and energy (lower panel) of Ni3 Al cooled from the liquid state to T ¼ 300 K at rates of 1013 K/s (circles) and 4 1013 K/s (stars). The position of the glass transition temperature is shown by an arrow.
Bacon’s potential [20], and considerably higher than the 795 K they found for a cooling rate of 4 1013 K/s. Fig. 2 shows the temperature dependence of the radial distribution function gðrÞ of the Ni3 Al system during cooling to 300 K at the two quenching rates considered. At both rates, the functions for the three lowest temperatures show the split second peak that is a well-known characteristic of metallic glasses (see, e.g., [30]). Fig. 3 shows the V ðT Þ and EðT Þ curves for the NiAl system. Fitting Pade equations again identifies a glass transition, this time at 820 K. The temperature dependence of the gðrÞ curves of NiAl is shown in Fig. 4. Finally, applying the same procedure to NiAl3 gave the V ðT Þ and EðT Þ curves shown in Fig. 5, a transition temperature of 510 K, and the gðrÞ curves (again with split second peaks at the two lowest temperatures) that are shown in Fig. 6.
E.G. Noya et al. / Journal of Non-Crystalline Solids 298 (2002) 60–66
Fig. 2. Temperature dependence of the radial distribution function of Ni3 Al when cooled from T ¼ 2000 K to T ¼ 300 K at rates of 1013 K/s (continuous curve) and 4 1013 K/s (dashed curve). The y-coordinates are correct for the 300 K curve; the curves for higher temperatures are shifted up by between 1 and 6 units.
63
Fig. 4. As for Fig. 2, but for NiAl.
Fig. 5. As for Fig. 1, but for NiAl3 .
4. Discussion
Fig. 3. As for Fig. 1, but for NiAl.
As was mentioned in Section 1, the Ni–Al system is a borderline case as regards the formation of
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Fig. 6. As for Fig. 2, but for NiAl3 .
a glassy state by RQ. This was first made explicit by Giessen [17], who took a sample of Ni-based alloys treated by the RQ method and plotted the maximum values of the heats of mixing of the liquid alloys, as determined by Miedema’s theory [32], against the ratios of the atomic radii of the components. On the map so constructed, a contour could be drawn which separated readily glassforming (RGF) systems (defined by their ability to form a glass under conventional quenching rates of 105 –106 K/s) from non-RGF systems. Ni–Al is a non-RGF alloy, but its representative point in this map is very close to the boundary between the RGF and non-RGF regions. Subsequently, free energy calculations performed by Yang and Bakker [33] using the semiempirical model developed by Miedema and coworkers [34,35] and L opez et al. [36] showed that the free energies of the solid solution and the amorphous phases are very close over the whole composition range. Thermodynamic conditions are not the only requisites that must be satisfied for amorphization to take place. In addition, the amorphization process must occur faster than the nucleation and growth of any equilibrium intermetallic compound, formation of which might be favoured by
the amorphization technique being used. The formation of high-melting-point intermetallic compounds is difficult to avoid at conventional RQ rates, which restricts the composition range over which amorphous alloys can be formed. By contrast, competition from such compounds is less intense in processes based on solid-state reactions (SSRs) [37,38], which accordingly, in general, allow wider glass-forming ranges. The phase diagram of the Ni–Al system features the line compound NiAl3 , which has a complex orthorhombic structure; the very stable compound NiAl, which has a CsCl (ordered bcc) structure; and the compound Ni3 Al, of L12 (ordered fcc) structure [18]. The formation of these compounds cannot be avoided when the system is treated by RQ at conventional quenching rates, but under ion beam bombardment of multilayered samples (a kind of SSR amorphization method), NiAl3 is amorphized, NiAl retains its crystalline structure and Ni3 Al becomes partially amorphous [39]. The differences among the three compounds as regards sensitivity to ion bombardment are due to their differences in crystal structure. Our MD results show that at all three compositions considered Ni–Al can be amorphized from the liquid state under very fast quenching. For Ni3 Al, this is in keeping with Wang et al.’s findings [19] in their NPT MD simulations based on Gao and Bacon’s potential [20]. The amorphization of the system with the composition of NiAl3 might also have been expected, given the very complex structure of the compound NiAl3 . However, the amorphization of the system with the composition of NiAl is striking given the possibility of competition between the amorphization process and the formation of the intermetallic compound NiAl, which has a simple crystal structure. Although the results presented in this paper correspond to cooling rates of the order of 1013 K/s, calculations performed at a cooling rate of 5 1012 K/s give quite similar results (not shown). It seems likely that qualitatively the same behaviour would have been found with cooling rates a few orders slower, such as the 1010 K/s estimated for quenching by pulsed laser irradiation [40] (in which an extremely thin molten layer
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formed on the surface of a solid by a short, intense laser pulse cools at a rate some four orders of magnitude faster than those of conventional techniques thanks to the absence of any poorly conducting interface between the liquid and solid phases). Such results would constitute simulated reproduction of the reported formation of glassy Ni–Al systems by the laser quenching method [41,42].
[3] [4] [5] [6] [7] [8] [9] [10] [11]
5. Conclusions Using the embedded atom model potential proposed for Ni–Al systems by Voter and Chen [22], who parameterized this potential on the basis of solid-state properties and those of the pure diatomic molecules, we performed TtN MD simulations to analyze the glass forming ability of liquid 3:1, 1:1 and 1:3 Ni–Al alloys under quenching at rates of the order of 1013 K/s. We found that all three alloys attained glassy states. Although Ni–Al alloys are classified as non-RGF systems because they fail to amorphize at conventional RQ rates (105 –106 K/s), our findings are in keeping with reports that they do amorphize when cooling rates of about 1010 K/s are achieved by laser techniques [41,42]. This illustrates the ability of EAM potentials to achieve, at least in certain cases, simulated reproduction of the behaviour of systems for which they were not specifically parameterized.
[12] [13]
[14] [15] [16] [17]
[18]
[19] [20] [21] [22]
[23]
Acknowledgements This work was supported by the CICYT, Spain (project PB98-0368-C02-02) and the Xunta de Galicia (projects PGIDT01PXI20605PR and PGIDT00PXI20611PN).
[24] [25] [26] [27] [28] [29] [30]
References [1] M.S. Daw, S.M. Foiles, M.I. Baskes, Mater. Sci. Rep. 9 (1993) 251. [2] D. Tom anek, A.A. Aligia, C.A. Balseiro, Phys. Rev. B 32 (1985) 5051.
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F. Cleri, V. Rosato, Phys. Rev. B 48 (1993) 22. S.M. Foiles, Phys. Rev. B 32 (1985) 3409. J.B. Adams, S.M. Foiles, Phys. Rev. B 41 (1990) 3316. L.M. Holzman, J.B. Adams, S.M. Foiles, W.N.G. Hitchon, J. Mater. Res. 6 (1991) 298. R. LeSar, R. Najafabadi, D.J. Srolovitz, J. Chem. Phys. 94 (1991) 5090. G.M. Bhuiyan, M. Silbert, M.J. Stott, Phys. Rev. B 53 (1996) 636. B. Sadigh, G. Grimvall, Phys. Rev. B 54 (1996) 15742. M.M.G. Alemany, C. Rey, L.J. Gallego, Phys. Rev. B 58 (1998) 685. M.M.G. Alemany, C. Rey, L.J. Gallego, J. Chem. Phys. 109 (1998) 3568. M.M.G. Alemany, C. Rey, L.J. Gallego, J. Chem. Phys. 109 (1998) 5175. M.M.G. Alemany, M. Calleja, C. Rey, L.J. Gallego, J. Casas, L.E. Gonzalez, J. Non-Cryst. Solids 250–252 (1999) 53. M. Asta, D. Morgan, J.J. Hoyt, B. Sadigh, J.D. Althoff, D. de Fontaine, S.M. Foiles, Phys. Rev. B 59 (1999) 14271. M. Maret, T. Pomme, A. Pasturel, P. Chieux, Phys. Rev. B 42 (1990) 1598. J.A. Alonso, N.H. March, Electrons in Metals and Alloys, Academic Press, London, 1989. B.C. Giessen, in: T. Masumoto, K. Suzuki (Eds.), Proc. 4th Int. Conf. on Rapidly Quenched Metals, vol. 1, Japan Inst. Metals, Sendai, 1982, p. 213. R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser, K.K. Kelley, Selected Values of the Thermodynamic Properties of Binary Alloys, American Society for Metals, Metals Park, OH, 1973. L. Wang, H. Liu, K. Chen, Z. Hu, J. Mater. Res. 13 (1998) 1497. F. Gao, D.J. Bacon, Philos. Mag. A 67 (1993) 275. M.W. Finnis, J.E. Sinclair, Philos. Mag. A 50 (1984) 45. A.F. Voter, S.P. Chen, in: R.W. Siegel, J.R. Weertman, R. Sinclair (Eds.), Characterization of Defects in Materials, MRS Symposia Proceedings No. 82, Materials Research Society, Pittsburgh, PA, 1987, p. 175. M.M.G. Alemany, C. Rey, L.J. Gallego, J. Chem. Phys. 111 (1999) 9111. J.R. Ray, A. Rahman, J. Chem. Phys. 82 (1985) 4243. S. Nose, Mol. Phys. 52 (1984) 255. M. Parrinello, A. Rahman, Phys. Rev. Lett. 45 (1980) 1196. M. Parrinello, A. Rahman, J. Appl. Phys. 52 (1981) 7182. A.P. Sutton, J. Chen, Philos. Mag. Lett. 61 (1990) 139. H. Rafii-Tabar, A.P. Sutton, Philos. Mag. Lett. 63 (1991) 217. Y. Qi, T. C ß agin, Y. Kimura, W.A. Goddard III, Phys. Rev. B 59 (1999) 3527. M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford University, Oxford, 1990. A.R. Miedema, P.F. de Ch^atel, in: L.H. Bennett (Ed.), Theory of Alloy Phase Formation, TMS-AIME, Warrendale, PA, 1980. H. Yang, H. Bakker, J. Alloys Comp. 189 (1992) 113.
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[34] G.J. van der Kolk, A.R. Miedema, A.K. Niessen, J. LessCommon Met. 145 (1988) 1. [35] P.I. Loeff, A.W. Weeber, A.R. Miedema, J. Less-Common Met. 140 (1988) 299. [36] J.M. L opez, J.A. Alonso, L.J. Gallego, Phys. Rev. B 36 (1987) 3716. [37] R.B. Schwarz, R.R. Petrich, C.K. Saw, J. Non-Cryst. Solids 76 (1985) 281.
[38] K. Samwer, Phys. Rep. 161 (1988) 1. [39] L.S. Hung, M. Nastasi, J. Gyulai, J.W. Mayer, Appl. Phys. Lett. 42 (1983) 672. [40] C.G. Woychik, D.H. Lowndes, T.B. Massalski, Acta Metall. 33 (1985) 1861. [41] M. von Allmen, in: H. Beck, H.-J. G€ untherodt (Eds.), Topics in Applied Physics, Springer, Heidelberg, 1983, p. 261. [42] J.A. Alonso, J.M. L opez, Mater. Lett. 4 (1986) 316.
Amorphization of Ni–Al alloys by fast quenching from the liquid state: a molecular dynamics study E.G. Noya, C. Rey, L.J. Gallego
*
Departamento de Fısica de la Materia Condensada, Facultad de Fısica, Universidad de Santiago de Compostela, E-15782 Santiago de Compostela, Spain Received 18 April 2001; received in revised form 3 October 2001
Abstract By means of constant temperature, constant thermodynamic tension molecular dynamics (MD) simulations using the embedded atom model proposed by Voter and Chen, we investigated the glass-forming ability of the alloys Ni3 Al, NiAl and NiAl3 under rapid quenching from the liquid state. Although the system Ni–Al does not form a glass when quenched from the liquid or vapour phases at the quenching rates achieved by conventional rapid quenching methods, at the very fast cooling rates used in this study the simulated system was amorphized from the liquid state at all compositions considered. This is consistent with the results of laser quenching experiments. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 31.15.Qg; 64.70.Pf
1. Introduction Molecular dynamics (MD) simulation is a powerful tool for investigating the properties of materials in general and metallic systems in particular. Though limited to small systems and short process times (a limitation that in some cases prevents the simulation of certain phenomena of interest), this technique can afford information on properties that would be very difficult or impossible to determine experimentally, or which simply have not yet been measured.
*
Corresponding author. Tel.: +34-981 563 100 x13995; fax: +34-981 520 676. E-mail address: [email protected] (L.J. Gallego).
The success of MD simulations depends on the accuracy of the model used to describe atomic interactions, which in the case of metallic systems is complicated by the important contribution of many-body effects. Semiempirical many-body potentials such as those based on the embedded atom model (EAM) [1] or the second moment approximation to the tight binding method (TBM-SMA) [2,3], which were developed to overcome the limitations of the pair-potential approximation for the description of metallic bonding, have successfully been used to analyze both bulk and surface properties of transition metals and their alloys. However, these semiempirical potentials are generally fitted to one-temperature solid-state properties, and it is not obvious a priori that the same potentials are equally capable of describing the
0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 1 0 4 8 - 1
E.G. Noya et al. / Journal of Non-Crystalline Solids 298 (2002) 60–66
properties of liquid and amorphous metals, with their quite different electron densities and interatomic separation distributions. Starting with the pioneering work of Foiles [4], several studies have shown that EAM potentials can in fact describe the structural, thermodynamic and atomic-transport properties of pure liquid transition and noble metals reasonably well (see, e.g., [5–13]). However, less work has been carried out on binary liquid alloys. The most extensive study in this area was recently performed by Asta et al. [14], who employed Monte Carlo and MD simulations to analyze the static and dynamic properties of liquid Ni–Al alloys using various interatomic EAM potentials. Their results showed that, although there are limitations to the application of EAM potentials for alloys rich in Al (partially due to the fact that no liquid data were used in their parameterization), the agreement between calculated and measured properties is reasonable. In particular, calculated partial structure factors were found to be in semiquantitative agreement with published neutron scattering measurements for Ni20 Al80 alloys [15], indicating that liquid-phase chemical short-range order (CSRO) is qualitatively well described. As is well known, CSRO effects generally play an important role in the formation of amophous alloys (see, e.g., [16]). In view of this versatility on the part of EAM potentials, in the work described here we used one optimized for Ni–Al in MD simulations performed to investigate the glass-forming ability of this system under rapid quenching from the liquid state (RQ). Ni–Al is interesting for several reasons. First, it is a borderline case as regards the formation of a metallic glass by RQ (see [17] and the discussion in Section 4). Secondly, it has a very rich phase diagram with several intermetallic phases [18], the nucleation and growth of which may compete with the formation of a metallic glass. Some of the results of this study can be compared with those obtained recently by Wang et al. [19] concerning the possibility of the amorphization of Ni3 Al by RQ (Wang et al. used Gao and Bacon’s potential [20], which is based on Finnis and Sinclair’s TBM-SMA scheme [21], in conjunction with constant presure, constant temperature (NPT) MD simulations).
61
The essential technical details of the computational method used in this paper are given in Section 2. In Sections 3 and 4 we present and discuss our results. Finally, in Section 5, we summarize our main conclusions.
2. Computational procedure In the calculations on Ni–Al performed in this work atomic interactions were described using the EAM potential that Voter and Chen (VC) optimized for prediction of the elastic constants and vacancy formation energies of pure Ni and pure Al, the bond lengths and bond energies of the diatomic molecules Ni2 and Al2 , the lattice constant, cohesive energy, elastic constants, vacancy formation energy, antiphase boundary energies and superlattice intrinsic stacking fault energy of the compound Ni3 Al, and the lattice constant and cohesive energy of the compound NiAl [22]. The VC Ni–Al potential is one of the EAM versions that were used by Asta et al. [14] in their study of liquid Ni–Al alloys, and it has also been employed by Alemany et al. [23] to investigate the diffusion coefficient of Ni impurity in liquid Al. It is therefore reasonable to hope that this potential may also give a satisfactory description of the glass-forming ability of Ni–Al alloys. Unlike those of Wang et al. [19], our simulations were performed using the constant temperature, constant thermodynamic tension (TtN) MD technique [24], which combines the Nose canonical ensemble [25] with the Parrinello–Rahman variable shape and size ensemble [26,27]. TtN MD simulations, with quantum Sutton–Chen manybody potentials [28,29], have recently been carried out by Qi et al. [30] to investigate the glass-forming ability of the alloys Cu–Ag and Cu–Ni under RQ. We considered three different systems of 864 atoms in a cubic box with periodic boundary conditions, namely 648 Ni atoms and 216 Al atoms, 432 Ni atoms and 432 Al atoms, and 216 Ni atoms and 648 Al atoms, proportions which correspond to the compositions of the intermetallic compounds Ni3 Al, NiAl and NiAl3 , respectively. The equations of motion were solved using a fourth-order
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Gear predictor–corrector algorithm [31] with a time step of 2 104 ps. At each composition, the initial configuration was obtained by equilibrating the system in the liquid phase at 2000 K, a temperature much higher that the melting temperatures of the intermetallic compounds Ni3 Al, NiAl and NiAl3 (the experimentally determined melting temperatures of these phases are 1658, 1912 and 1127 K, respectively [18]). The system was then cooled to room temperature (300 K) in decrements of 100 K, being kept at each temperature for times of 10 or 2.5 ps corresponding to cooling rates of 1013 and 4 1013 K/s, respectively. These were the cooling rates considered by Wang et al. [19] in their NPT MD study of Ni3 Al using Gao and Bacon’s potential [20]. For each temperature, the volume V, total energy E and radial distribution function gðrÞ of the system were averaged over the full TtN MD trajectory.
3. Results Fig. 1 shows the variations of the volume and total energy as the Ni3 Al sample was cooled from 2000 to 300 K at rates of 1013 and 4 1013 K/s. The results for the two cooling rates virtually coincide, which contrasts with the (much more scattered) results of Wang et al. [19]. Both V and E vary continuously with T, but the slopes exhibit discontinuities that are indicative of a glass transition. For quantitative determination of the glass transition temperature Tg for a cooling rate of 1013 K/s we considered the EðT Þ data, ignoring the data close to the change of slope and fitting the remaining high-temperature and low-temperature data sets with separate Pade equations: EðT Þ ¼
a þ bT þ cT 2 : 1 þ dT
ð1Þ
Tg was taken to be the temperature at which these curves intersected each other, 910 K (nearly identical values were found using the V ðT Þ data or either the EðT Þ or V ðT Þ data for a cooling rate of 4 1013 K/s). This temperature is somewhat higher than the 855 K found by Wang et al. [19] using NPT MD calculations with Gao and
Fig. 1. Variations of the volume (upper panel) and energy (lower panel) of Ni3 Al cooled from the liquid state to T ¼ 300 K at rates of 1013 K/s (circles) and 4 1013 K/s (stars). The position of the glass transition temperature is shown by an arrow.
Bacon’s potential [20], and considerably higher than the 795 K they found for a cooling rate of 4 1013 K/s. Fig. 2 shows the temperature dependence of the radial distribution function gðrÞ of the Ni3 Al system during cooling to 300 K at the two quenching rates considered. At both rates, the functions for the three lowest temperatures show the split second peak that is a well-known characteristic of metallic glasses (see, e.g., [30]). Fig. 3 shows the V ðT Þ and EðT Þ curves for the NiAl system. Fitting Pade equations again identifies a glass transition, this time at 820 K. The temperature dependence of the gðrÞ curves of NiAl is shown in Fig. 4. Finally, applying the same procedure to NiAl3 gave the V ðT Þ and EðT Þ curves shown in Fig. 5, a transition temperature of 510 K, and the gðrÞ curves (again with split second peaks at the two lowest temperatures) that are shown in Fig. 6.
E.G. Noya et al. / Journal of Non-Crystalline Solids 298 (2002) 60–66
Fig. 2. Temperature dependence of the radial distribution function of Ni3 Al when cooled from T ¼ 2000 K to T ¼ 300 K at rates of 1013 K/s (continuous curve) and 4 1013 K/s (dashed curve). The y-coordinates are correct for the 300 K curve; the curves for higher temperatures are shifted up by between 1 and 6 units.
63
Fig. 4. As for Fig. 2, but for NiAl.
Fig. 5. As for Fig. 1, but for NiAl3 .
4. Discussion
Fig. 3. As for Fig. 1, but for NiAl.
As was mentioned in Section 1, the Ni–Al system is a borderline case as regards the formation of
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Fig. 6. As for Fig. 2, but for NiAl3 .
a glassy state by RQ. This was first made explicit by Giessen [17], who took a sample of Ni-based alloys treated by the RQ method and plotted the maximum values of the heats of mixing of the liquid alloys, as determined by Miedema’s theory [32], against the ratios of the atomic radii of the components. On the map so constructed, a contour could be drawn which separated readily glassforming (RGF) systems (defined by their ability to form a glass under conventional quenching rates of 105 –106 K/s) from non-RGF systems. Ni–Al is a non-RGF alloy, but its representative point in this map is very close to the boundary between the RGF and non-RGF regions. Subsequently, free energy calculations performed by Yang and Bakker [33] using the semiempirical model developed by Miedema and coworkers [34,35] and L opez et al. [36] showed that the free energies of the solid solution and the amorphous phases are very close over the whole composition range. Thermodynamic conditions are not the only requisites that must be satisfied for amorphization to take place. In addition, the amorphization process must occur faster than the nucleation and growth of any equilibrium intermetallic compound, formation of which might be favoured by
the amorphization technique being used. The formation of high-melting-point intermetallic compounds is difficult to avoid at conventional RQ rates, which restricts the composition range over which amorphous alloys can be formed. By contrast, competition from such compounds is less intense in processes based on solid-state reactions (SSRs) [37,38], which accordingly, in general, allow wider glass-forming ranges. The phase diagram of the Ni–Al system features the line compound NiAl3 , which has a complex orthorhombic structure; the very stable compound NiAl, which has a CsCl (ordered bcc) structure; and the compound Ni3 Al, of L12 (ordered fcc) structure [18]. The formation of these compounds cannot be avoided when the system is treated by RQ at conventional quenching rates, but under ion beam bombardment of multilayered samples (a kind of SSR amorphization method), NiAl3 is amorphized, NiAl retains its crystalline structure and Ni3 Al becomes partially amorphous [39]. The differences among the three compounds as regards sensitivity to ion bombardment are due to their differences in crystal structure. Our MD results show that at all three compositions considered Ni–Al can be amorphized from the liquid state under very fast quenching. For Ni3 Al, this is in keeping with Wang et al.’s findings [19] in their NPT MD simulations based on Gao and Bacon’s potential [20]. The amorphization of the system with the composition of NiAl3 might also have been expected, given the very complex structure of the compound NiAl3 . However, the amorphization of the system with the composition of NiAl is striking given the possibility of competition between the amorphization process and the formation of the intermetallic compound NiAl, which has a simple crystal structure. Although the results presented in this paper correspond to cooling rates of the order of 1013 K/s, calculations performed at a cooling rate of 5 1012 K/s give quite similar results (not shown). It seems likely that qualitatively the same behaviour would have been found with cooling rates a few orders slower, such as the 1010 K/s estimated for quenching by pulsed laser irradiation [40] (in which an extremely thin molten layer
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formed on the surface of a solid by a short, intense laser pulse cools at a rate some four orders of magnitude faster than those of conventional techniques thanks to the absence of any poorly conducting interface between the liquid and solid phases). Such results would constitute simulated reproduction of the reported formation of glassy Ni–Al systems by the laser quenching method [41,42].
[3] [4] [5] [6] [7] [8] [9] [10] [11]
5. Conclusions Using the embedded atom model potential proposed for Ni–Al systems by Voter and Chen [22], who parameterized this potential on the basis of solid-state properties and those of the pure diatomic molecules, we performed TtN MD simulations to analyze the glass forming ability of liquid 3:1, 1:1 and 1:3 Ni–Al alloys under quenching at rates of the order of 1013 K/s. We found that all three alloys attained glassy states. Although Ni–Al alloys are classified as non-RGF systems because they fail to amorphize at conventional RQ rates (105 –106 K/s), our findings are in keeping with reports that they do amorphize when cooling rates of about 1010 K/s are achieved by laser techniques [41,42]. This illustrates the ability of EAM potentials to achieve, at least in certain cases, simulated reproduction of the behaviour of systems for which they were not specifically parameterized.
[12] [13]
[14] [15] [16] [17]
[18]
[19] [20] [21] [22]
[23]
Acknowledgements This work was supported by the CICYT, Spain (project PB98-0368-C02-02) and the Xunta de Galicia (projects PGIDT01PXI20605PR and PGIDT00PXI20611PN).
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