Atomistic modeling of the γ and γ′-phases of the Ni–Al system
Introduction
Ni3Al is a technologically important intermetallic compound known for exhibiting an increase in the yield strength with temperature over a temperature range (yield stress anomaly) [1]. This compound also forms the basis of the γ′-phase which strengthens commercial Ni-base superalloys by the precipitation hardening mechanism [2], [3]. Small, usually cuboidal, precipitates of the γ′-phase embedded in Ni-base FCC solid solution (γ-phase) impose resistance to dislocation slip and increase the strength and creep resistance of superalloys. Atomistic computer simulations offer a means of gaining a better understanding of dislocation behavior in Ni3Al and in the γ/γ′ structure, see for example [4], [5], [6], [7], [8]. Point defects and diffusion in Ni3Al have also been examined by atomistic simulations [9], [10], [11], [12], [13], [14]. Diffusion is an important factor in the high-temperature behavior of Ni3Al and superalloys, as are the interphase boundaries γ/γ′. Diffusion controls the degradation rate of the γ/γ′ structure at high temperatures and the dislocation processes accompanying creep. The structure and energy of γ/γ′ interfaces influence the slip transfer between the two phases and the thermodynamics stability of γ′ particles.
The credibility of results delivered by atomistic simulations largely depends on the robustness of interatomic potentials employed in the simulations. Most simulations employ many-body potentials based on the Finnis–Sinclair method [15] or the embedded-atom method (EAM) [16]. Several EAM potentials have been proposed for Ni3Al (see [17] for a review), the best known of them being the potentials by Foiles and Daw (FD) [9] and by Voter and Chen (VC) [18]. All Ni3Al potentials developed so far have been fit to experimental values of the lattice parameter, cohesive energy, elastic constants and other experimental properties.
It has recently been recognized that the incorporation of first-principles data in the fitting database, together with experimental data, can significantly improve the reliability of interatomic potentials [19], [20], [21], [22], [23], [24], [25], [26]. First-principles data sample a larger area of configuration space than experimental data do, including configurations far away from equilibrium states. This broader sampling improves the transferability of potentials to various local environments that can be encountered during atomistic simulations. Even if the accuracy of fit to particular experimental numbers remains the same, potentials developed by this scheme tend to be more robust. This scheme has recently been applied to construct an accurate EAM potential for the B2–NiAl compound of the Ni–Al system [24]. Unfortunately, properties of Ni and Ni3Al predicted by that potential are not at a level of accuracy that would warrant its application to atomistic simulations of these phases.
Therefore, in this work we shift the focus to Ni and Ni3Al. We will apply our scheme [20], [23], [24], [26] to construct an accurate EAM potential describing Ni, Ni3Al, as well as the thermodynamic equilibrium between the γ and γ′-phases of the Ni–Al system. We are testing the new potential by calculating a large variety of properties ranging from phonon frequencies and thermal expansion to the Ni–Al phase diagram and γ/γ′ interfaces. Such a detailed characterization of a potential is essential for its future applications. We also discuss the issue of global transferability (or rather, lack thereof) of existing EAM potentials across the entire Ni–Al phase diagram.
Section snippets
Construction of the potentials
The EAM represents the total energy Etot of a collection of atoms in the form [16]Here Φij(rij) is the pair-interaction energy between atoms i and j separated by a distance rij, Fi is the embedding energy of atom i and is the host electron density induced by all surrounding atoms j at the location of atom i. The host electron density is given bywhere ρj(r) is the electron-density function assigned to atom j. The pair-interaction, electron-density
Properties of Ni and Al
Table 2 summarizes lattice properties of Ni and Al calculated with the present EAM potentials in comparison with experimental data [31], [32], [33], [34]. The table includes the phonon frequencies at the zone-boundary points X, L and K, but the calculations were actually performed all across the Brillouin zone. While the elastic constants are fit to experiment very accurately, the phonon frequencies (which were not included in the fit) tend to overestimate the experimental values in the
Lattice properties and structural stability of Ni3Al
The lattice parameter and elastic constants of Ni3Al are accurately fit to the experimental values [4], [44] (Table 5). The cohesive energy E0 is also quite close to the experimental value −4.62 eV [4], [44]. The cohesive energy was not used in the fit directly but was involved indirectly through the first-principles formation energy of Ni3Al and the experimental cohesive energies of Ni and Al.
The phonon dispersion curves in Ni3Al calculated with the new potential generally agree with
Planar faults in Ni3Al
Energies of planar faults in Ni3Al have a strong impact on the dislocation core structure and ultimately the deformation behavior of the material. The most common dislocations in Ni3Al are screw superdislocations dissociated into superpartials separated by an antiphase boundary (APB) on a {1 1 1} plane [1], [7], [47], [48]. Each superpartial, in turn, can dissociate into Shockley partials separated by a complex stacking fault (CSF). An alternative, although less common scenario, is
Point defects and diffusion in Ni3Al
Point defects accommodate off-stoichiometry and mediate atomic diffusion in intermetallic compounds. To calculate their formation free energies and equilibrium concentrations in Ni3Al, we will follow the scheme proposed in [60]. Neglecting interstitials, Ni3Al supports four types of point defect: vacancies VNi and VAl and antisites AlNi and NiAl (the subscript refers to the sublattice). So-called “raw” formation energies, entropies and volumes of individual point defects have been computed
Thermodynamic calculations
As a test of the ability of the present EAM potential to predict high-temperature properties of the γ and γ′-phases, the γ/γ′ equilibrium lines on the Ni–Al phase diagram have been calculated. This test is important in connection with possible applications of this potential to model γ′ particle precipitation and the γ/γ′ structure in Ni–Al alloys. The γ′/β lines of the phase diagram have also been calculated as a further test of transferability of the potential.
The first step of the calculation
Coherent Ni/Ni3Al interfaces
Using the new potential, the energies of coherent Ni/Ni3Al interfaces, which can be considered as a simple prototype of γ/γ′ interfaces, have been calculated for three different interface orientations. To this end, a rectangular supercell was created with desired crystal planes parallel to one of its faces. A plane dividing the supercell in two equal halves was chosen as an interface. In the planes lying on one side of the interface, all Al atoms were replaced by Ni, thus effectively creating a
Discussion and conclusions
The EAM potential developed in this work provides an accurate description of a spectrum of properties of Ni, Al and Ni3Al, including elastic constants, thermal expansion, point defects and planar faults. Furthermore, properties of the γ and γ′-phases formed by Ni and Ni3Al at elevated temperatures are also represented accurately. In particular, their phase fields on the Ni–Al phase diagram are reproduced in a reasonable agreement with experiment. It has been demonstrated that the potential is
Acknowledgements
I am grateful to P.M. Hazzledine for helpful discussions and to M.J. Mehl and D.A. Papaconstantopoulos for making their LAPW calculation results available for this work. This work was supported by the US Air Force Office of Scientific Research (Metallic Materials Program) under Grant No. F49620-01-1-0025.
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