Atomistic modeling of the site occupancies of Ti and Cu in NiAl

Scripta mater. 42 (2000) 403– 408 www.elsevier.com/locate/scriptamat

ATOMISTIC MODELING OF THE SITE OCCUPANCIES OF Ti AND Cu IN NiAl Guillermo Bozzoloa,b, Ronald D. Noebeb and Jorge E. Garcesc a

Ohio Aerospace Institute, 22800 Cedar Point Rd., Cleveland, OH 44142, USA, bNASA Glenn Research Center at Lewis Field, Cleveland, OH 44135 USA, cCentro Atomico Bariloche, 8400 Bariloche, Argentina (Received July 13, 1999) (Accepted in revised form October 14, 1999)

Keywords: Computer simulation; Intermetallic; Structural behavior; Quantum approximate method; Defect theory and modeling

Introduction In a recent paper by Wilson and Howe (1), experimental evidence, regarding the site occupancy of two simultaneous alloying additions to NiAl, points to the possibility of a change in site occupancy behavior with concentration. They analyzed five B2 structured NiAl-Ti-Cu alloys to determine, via Atom Location by Channelling Enhanced Microanalysis (ALCHEMI) (2), the site occupancies of Ti and Cu in Al-deficient and stoichiometric NiAl alloys. Three alloys were produced with Cu replacing Al (Ni50Al(47-x)Ti3CuX with X ⫽ 1, 3, 6) and two were made with Cu added in place of Ni (Ni(50x)Al47Ti3CuX, with X ⫽ 1, 3). Their results showed that for Al-deficient alloys both Ti and Cu show a strong preference for Al sites (over 80% for both types of atoms) while much lower Cu concentrations are found in Al sites for Ni-deficient alloys. From this limited set of experimental results, interesting conclusions were drawn suggesting a close relationship between the site preference of the alloying additions and the Ni:Al ratio. The purpose of this paper is to show that even a limited amount of experimental data, coupled with economical atomistic simulations, can constitute a powerful tool for examining specific details on otherwise complex systems for which an exhaustive experimental analysis is prohibitive. The theoretical technique used for our purposes is the BFS method for alloys (3), which has been proven to be highly effective for the study of multicomponent systems. With the proper parameterization, it allows for an extremely economical, computationally simple, and physically sound description of large collections of atoms. The BFS method is a quantum approximate technique based on the assumption that the heat of formation, ⌬H, of a given collection of atoms is the sum of the contributions of each atom in the sample, ⌬H ⫽ ⌺␧ i . Each contribution ␧ i consists of two terms: a strain energy (␧ iS ) which accounts for the change in geometry with respect to a single monoatomic crystal of the reference i i ), linked by a coupling function ( g i ) so that ␧ i ⫽ ␧ iS ⫹ g i ␧ C . Three atom, and a chemical energy (␧ C parameters for each of the constituents atoms are needed (equilibrium lattice parameter, cohesive energy and bulk modulus) in order to calculate these terms. The chemical energy accounts for the corresponding change in composition, considered as a defect in an otherwise pure crystal. The chemical “defect” deals with pure and mixed bonds, therefore, two additional parameters are needed to describe these interactions. All the parameters used are determined via first-principles methods (3). We refer the 1359-6462/00/$–see front matter. Published by Elsevier Science Ltd. All rights reserved. PII: S1359-6462(99)00364-4

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Figure 1. Composition of the NiAl-Ti-Cu alloys modeled. Those for which experimental data exist (1) are denoted with filled circles. The numbers inside the circles indicate the concentration of Cu (in at. %). The horizontal and vertical axes indicate the concentration of Al and Ni, respectively. The Ti concentration was held constant at 3%.

reader to Ref. 3 for a detailed description of the BFS method, the parameters used in this work, and details on Monte Carlo-Metropolis simulations using BFS. Results and Discussion The BFS method is used for the calculation of the energetics of an alloy cell composed of a sufficiently large atom population (1024 atoms in this work). The temperature treatment is simulated by means of a Monte Carlo-Metropolis algorithm (3), where pairs of atoms at nearest-neighbor distance are allowed to switch places with a probability exp(⫺⌬E/kT), where k is Boltzmann’s constant, T is the temperature and ⌬E is the difference in energy of the cell before and after the switch. The changes in atomic distribution are allowed to continue until the total energy of the cell stabilizes. Fig. 1 depicts the range of alloy compositions modeled and also indicates the alloys studied experimentally in (1). Every computational cell has been subject to the same temperature treatment, where an initial high-temperature cell (i.e., random distribution of atoms) is steadily cooled to room temperature. One simple estimate of the site occupancy preference of each of the two alloying additions (Ti and Cu) can be obtained by computing the coordination matrix of the final, stable state for each cell, once the simulation is completed. The matrix element a mn represents the probability that an atom m has an atom n as a nearest-neighbor (NN). Table 1 displays the coordination matrices for all the alloy compositions studied. As long as the cell follows the basic B2 ordering (in this case, Ni and Al atoms occupying their own sublattices) then the matrix elements a mNi and a mAl can be taken as an approximate measure of the likelihood that an atom m occupies a site in the Al or Ni sublattice, respectively. If we denote with the symbol m(n) the probability of an atom m occupying a site in the n sublattice, then the previous statement can be written as a mNi ⬃ m( Al ) and a mAl ⬃ m(Ni). Clearly, clustering of atoms or the formation of precipitates can noticeably alter the meaning of a mn as a measure of site occupancy. However, general trends are, to a large extent, independent of such extended defects and can therefore be extracted from the coordination matrix of each cell, as discussed below. Moreover, a close examination of the coordination matrix yields sufficient information for a proper interpretation of the results. Large values of the diagonal elements in each matrix can indicate either antiphase boundaries or the presence of precipitates (i.e., a large value of a NiNi indicates that many Ni atoms are at NN distance, which would be highly unlikely in an NiAl alloy where, if perfectly ordered, a NiNi ⫽ 0). If the diagonal elements are small, then the off-diagonal elements can be taken as a good approximation

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TABLE 1 Coordination Matrices for the 42 NiAl-Ti-Cu Alloys in Fig. 1, as a Function of Al (Horizontal Axis) and Ni (Vertical Axis) Concentration

All alloys contain 3 at. % Ti. Each matrix element amn, defined in the inset, is the probability (in percent) that an atom n has an atom m as a nearest neighbor. For example, the probability that Ti has Ni as a nearest-neighbor, a TiNi , for an alloy of composition 51Ni-40Al-3Ti-6Cu is 100%, as circled above, indicating that Ti resides exclusively on the Al sublattice. The bold matrices correspond to those alloys for which experimental results exist (1).

of the site preference. In other words, if a NiNi and a AlAl are small, the closer the cell is to a highly ordered state, which translates into a TiNi being a true measure of the likelihood of finding Ti in an Al site. In agreement with experimental results and the conclusions reached in Ref. 1, the simulations show that Cu does change site preference in NiAl, depending on the ratio of Ni to Al, as seen in the variations of a CuNi in Table 1. For every alloy shown, Ti(Al) ⬎ Ti(Ni), whereas Cu(Al) ⬎ Cu(Ni) for Ni-rich alloys and Cu(Ni) ⬎ Cu(Al) otherwise. These conclusions can be expanded in light of our numerical simulations and elucidate trends in general behavior that would otherwise be extrapolated from a necessarily small sample of experimental results. Fig. 2 shows a comparison of the computed site

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Figure 2. Comparison of the occupation probabilities Ti(Al) and Cu(Al), with the corresponding uncertainties, from Ref. 1 and the BFS results (arrows).

occupation probabilities Ti(Al) and Cu(Al) with those determined experimentally (1), clearly showing the agreement between experiment and theory and providing a great degree of confidence in our simulations. A quick glance at the coordination matrices shown in Table 1 provides information that is not immediately apparent from the limited experimental results. The results of these simulations are displayed in Fig. 3, where the site occupancy probabilities for Ti (Fig. 3a) and Cu (Fig. 3b) in Al sites are shown as a function of Al, Ni and Cu concentration. Comparing the two sets of curves, it is clear that Ti(Al) is always greater than zero. The minimum value obtained is 54% for Ni46Al51Ti3, as shown in Fig. 3a. In contrast, Cu(Al) decreases much more rapidly than Ti(Al) for increasing Al concentration and decreasing Ni and Cu concentration, eventually becoming zero, indicating a switch in site preference exclusively favoring Ni sites (a CuNi ⫽ 0). This change in site preference can be seen in Fig. 3b in the shape of a vaguely defined “boundary,” loosely defined by alloys with a 1:1 Ni:Al ratio, beyond which Cu(Al) (or the matrix element a CuNi ) displays a sharp drop for decreasing Ni:Al ratios, whereas the corresponding Ti site preference for Al sites (Ti(Al), or a TiNi ) in Fig. 3a decreases at a much slower rate and is mostly independent of the Cu concentration. In this sense, mapping of the

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Figure 3. Matrix elements (a) a TiNi and (b) a CuNi , taken as a measure of Ti(Al) and Cu(Al), respectively, as a function of Al and Cu concentration, for different values of x Ni . The projection of these curves onto the horizontal plane is shown with dashed lines, highlighting the fact that Cu(Al) becomes zero for high x Al and low x Ni and x Cu .

simulated results indicate much greater sensitivity of Cu(Al) with concentration than that seen for Ti(Al), which is mildly sensitive to whether the alloy is Ni-rich or Al-rich. The results in Ref. 1 are a clear manifestation of these general trends. It should be noted, however, that while the experimental results could suggest a sudden reversal in site preference for Cu at a specific stoichiometry, the theoretical results indicate that this is a more gradual change beginning for alloys with 1:1 Ni:Al ratio (i.e., the “surface” that could be built from the data displayed in Fig. 3 for either Ti(Al) or Cu(Al) would resemble a “cascade,” smoothly evolving from one regime to another, as opposed to a sharp “step” where there is no transitional region). Additional information can also be obtained from these simulations, regarding the interaction between Ti and Cu atoms. While a TiTi ⫽ 0 in every single alloy studied (indicating a strong repulsion between Ti atoms), a CuCu is (in most cases) finite, allowing the possibility of clustering of a small fraction of Cu atoms, particularly for Ni-rich alloys (for example, a CuCu ⫽ 3.0 for Ni50Al43Ti3Cu4, indicating that a Cu atom has a 3% probability of having another Cu atom as a nearest-neighbor). Moreover, the interaction matrix elements a TiCu and a CuTi are generally small for Ni-rich alloys, indicating the presence of few Ti and Cu NN pairs, therefore, a higher likelihood of finding both elements in Al sites. These probabilities slowly increase as the concentration of Ni decreases, consistent with the fact that for this range of concentrations Cu(Al) decreases much faster than Ti(Al), thus favoring the location of Ti and Cu atoms in different sublattices. This results, eventually, in a switch of site preference for Cu atoms from Al to Ni sites with a change in stoichiometry. Previous BFS work on the site substitution of ternary additions (4) indicates that Ti displays a strong preference for Al sites in Ni-rich NiAl alloys, with an energy gap (i.e., the difference between Ti in an Al sublattice site and Ti in an Ni-sublattice site of 0.14 eV/atom. This was substantially larger than the energy difference found for Cu in different sublattices, which was only 0.02 eV/atom. This noticeable difference in energy indicates that Ti will exhibit a strong tendency to occupy Al sites leaving Cu atoms whatever sites are available. Moreover, the small magnitude of the energy gap for Cu (which means that there is only a small energy cost for Cu(Ni) substitutions) is mostly independent of Ni, Al or Ti concentration (as reflected in this work by the negligible values of aTiCu or aCuTi). The strong ordering tendencies of NiAl alloys, coupled with the strong preference of Ti for Al sites, as well as the negligible

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Figure 4. Energy of formation (in eV/atom) for ternary Ni(Al,X) alloys (X ⫽ Ti, Cu) and Ni(Al,Ti,Cu) alloys when Ti occupies an Al site (Ti(Al)) or an Ni site, with the displaced Ni atom in an available Al site (Ti(Ni)Ni(Al)) (left column), similarly for Cu (right column) and when Ti and Cu are present simultaneously in the four combinations shown.

interaction between Cu and Ti atoms and the small energy difference for Cu atoms in Ni versus Al sites, all contribute to make Cu atoms the ones most likely to fill in for any deficiency on either side of stoichiometry. In the examples studied, this translates into a smooth transition from Al to Ni site occupancy as the change in composition becomes Ni-poor. The small energy cost for such change in behavior also explains the negligible number of Ni or Al antistructure atoms. This discussion is summarized in Fig. 4, where the energies of Ti, Cu and Ti ⫹ Cu substitutions in NiAl are shown. While Ti(Al) and Cu(Al) substitutions are preferred when each additive is considered separately, the energetics of the quaternary system (center column) indicate that the presence of Cu has negligible influence on the behavior of Ti (left column). More detail for the underlying reasons for such behavior will be forthcoming in a paper dealing with the various contributions (chemical and strain) to atomistic energetics. Conclusions The previously observed experimental behavior in (1) is actually part of a general trend for the site occupancy preference of Ti and Cu in NiAl, which responds primarily to the Ni:Al ratio because of negligible or weak interaction between the two alloying additions. It is seen that the site occupancy of various elements respond differently to changes in stoichiometry and in a more continuous fashion than that suggested experimentally. The behavior observed can be attributed to the strong tendency of Ti to occupy Al sites, almost regardless of composition, and the correspondingly “soft” preference of Cu for Al or Ni sites thus leaving Cu to accommodate readily available sites. Coupled with the rather weak interaction between Ti and Cu, when present in small quantities, it is to be expected that Cu will adapt its behavior so that no antistructural defects are created. Finally, due to the ease with which additional information and understanding of a system can be gained through simulations, we also propose the linkage between experimental and numerical simulation analysis as an integral process for the determination of the microscopic features of complex alloy systems. References 1. 2. 3. 4.

A. W. Wilson and J. M. Howe, Scripta Mater. 41, 327 (1999). C. J. Rossow, C. T. Forwood, M. A. Gibson, and P. R. Miller, Phil. Mag. A. 74, 57 (1996). G. Bozzolo, R. D. Noebe, J. Ferrante, and C. Amador, J. Comput.-Aided Mater. Design. 6, 1 (1999), and references therein. G. Bozzolo, R. D. Noebe, and F. Honecy, Intermetallics. in press.