Embedded-atom potential for Ni-Al alloy

We construct a new embedded-atom potential for Ni-Al system. The lattice constants, cohesive energies, elastic constants, vacancy formation energies, stacking fault energies and equations of state of Ni and Al are included in the fitting process of potentials for pure Ni and pure Al. The cross-interaction potential is fitted to the lattice constants, cohesive energies, elastic constants of Ni3Al and NiAl and the energies of (100) APB and (111) SISF of Ni3Al. The potential accurately reproduces the various physical properties of Ni, Al, Ni3Al and NiAl phases. The results of the new embedded-atom potential and other embedded-atom potentials are compared and discussed in detail.


Introduction
Ni-Al intermetallic compounds mainly include five phases: Ni3Al, Ni5Al3, NiAl, Ni2Al3 and NiAl3. Among them, L12-type Ni3Al and B2-type NiAl are considered as high temperature structural materials because of their high melting point and low density [1][2]. Therefore, in recent years, people have extensively studied their physical properties in theory. The dislocation, creep and fracture behavior of the alloy can be better understood by atomistic simulation, and the reliability of atomic-level simulation depends on the interatomic potential. The most widely used potential in the atomistic simulation of intermetallic compounds is the many-body potential based on the embedded atom method (EAM).
Several EAM potentials for Ni-Al system have been proposed. The most famous of them are presented by Voter and Chen [3], and Mishin [4,5]. Among them, Voter-Chen (VC) EAM and Mishin-2004 (M04) EAM [5] are proposed for Ni3Al. The calculated lattice constants, cohesive energies, elastic constants, point defect properties and antiphase boundary (APB) energies of Ni3Al by these three EAM potentials agree with the experimental data. However, the calculated NiAl properties are often different from the experimental data, such as: the elastic constants of NiAl calculated by VC EAM and M04 EAM are about 50% larger than the experimental values, and the calculated point defect properties of NiAl deviate from the experimental results. Moreover, the (111) complex stacking fault (CSF) energies calculated by VC EAM and M04 EAM are less than the corresponding (111) APB energy, which is inconsistent with the experimental results. The Mishin-2002 (M02) EAM [4] is proposed for NiAl, the calculated properties of NiAl agree well with the experimental results. However, the maximum error between the calculated elastic constants of Ni3Al and the experimental data is about 50%, and the  [6].
In this paper, we first obtained a new EAM potentials of Ni-Al system by fitting the physical properties of Ni, Al, Ni3Al and NiAl. Then, the various properties of Ni, Al, Ni3Al and NiAl are calculated, and the results are discussed.

Development of EAM potential
The EAM model in this work within the framework of the original EAM theory [7]. The total energy of crystal can be written as Where i F is the embedding energy of atom i, i  is the host electron density at atom i, ij  is the pair potential between atoms i and j, ij r is the distance between atoms i and j, j f is the atomic electron density contributed by atom j. For a binary system A-B, seven potential functions, namely , c r and 2 h are adjustable parameters. Table 1. Parameters of the EAM potentials for Ni, Al and Ni-Al.
can be determined by fitting the vacancy formation energy, cohesive energy, lattice constant, elastic constants, energy difference between fcc and hcp structure and equations of state. The cross-interaction function ) (r AlNi  can be obtained by fitting the cohesive energies, lattice constants and elastic constants of Ni3Al and NiAl. Because the planar defects have an important influence on the mechanical properties of the alloy, we should consider the planar defects in the fitting. The (100) APB energy of Ni3Al is related to the energy difference of the L12 structure and the D022 structure [9], the (111) SISF energy of Ni3Al is related to the energy difference of the L12 structure and the D019 structure [5]. We also include the energy differences mentioned above when fitting the potential parameters.
The fitting results of potential parameters are listed in Table 1. In this work, all the calculations using the EAM potentials for the properties of elements and alloys are carried out by LAMMPS [10].

Properties of Ni and Al
The lattice constants (a0), cohesive energies (Ec), elastic constants (Cij), vacancy formation energies  4 obtained by the present EAM agree better with the the experimental data than those of VC EAM and M04 EAM. Therefore, the present EAM potential can effectively describe the properties of Ni and Al.  [11]. b Ref. [12]. c Ref. [13].

Lattice constant, cohesive energy and elastic constants.
The calculated values of the lattice constant, cohesive energy and elastic constants of Ni3Al and NiAl are listed in Table 3. For comparison, the results for the VC EAM, M02 EAM and M04 EAM and the experimental data are also shown in Table 3. It can be seen from Table 3 that the results of the present EAM agree well with the experimental data. The elastic constants of Ni3Al are not included in the fitting of the M02 EAM, and the elastic constants of NiAl are not included in the fitting of the VC EAM and the M04 EAM, so the corresponding calculated values have a large difference from the experimental data.  [14]. b Ref. [15]. c Ref. [3]. d Ref. [11]. e Ref. [16].

Structural stability.
The lattice constants and cohesive energies of several alternative structures of L12-Ni3Al and B2-NiAl are calculated by the present EAM. The lattice constants of the alternative structure and the energy differences between the alternative structure and the original structure are shown in Table 4. For comparison, the results of the first principles are also listed in the table. All of the first principles calculations in this work are based on the CASTEP software package [17], and the GGA-PBE exchange correlation functional is adopted [18]. It can be seen from Table 6  5 basically the same, and the ordering of cohesive energy in different structures is the same. The L12 structure is the lowest-energy structure of Ni3Al, and the B2 structure is the lowest-energy structure of NiAl. It is shown that the structural stability predicted by the present EAM is accurate and correct. Table 4. Lattice constants of the alternative structures of L12-Ni3Al and B2-NiAl, and energy differences (ΔEc) between the alternative structures and the original structure. If two lattice constant values are given, separated by commas, they denote a and c, respectively.
Where Erel is the energy of the system with point defect, Eper is the energy of the system with perfect lattice, E(Al) and E(Ni) are the energies of Al atom and Ni atom in perfect lattice respectively. The results are compared with the results of M02 EAM and M04 EAM in Table 5. If the formation energy of antisite defect is lower than that of the vacancy defect, it means that defects are mainly antisite defects; Otherwise, it means that defects are mainly vacancy defects. The experimental results show that: for Ni3Al, defects are mainly Ni antisite defects on the Ni-rich side and Al antisite defects on the Al-rich side; for NiAl, defects are mainly Ni antisite defects on the Ni-rich side and Ni vacancy defects on the Al-rich side [19]. The results of the three EAM potential are the same as the experimental results.   Table 6. For comparison, the experimental values and the results of M04 EAM are also listed in the Table 6. As can be seen from   [7]. b [20].

Summary
In the construction of the present EAM, the properties used for fitting include the lattice constants, cohesive energies, elastic constants, vacancy formation energies, stacking fault energies and equation of state of pure elements Ni and Al, the lattice constants, cohesive energies, elastic constants of Ni3Al and NiAl and the energies of (100) APB and (111) SISF of Ni3Al.
The lattice constants, cohesive energies, elastic constants, structural stability, point defects and planar defects of Ni, Al, Ni3Al and NiAl calculated by the present EAM agree well with the experimental results or the first-principles results. Compared with the VC EAM, M02 EAM and M04 EAM, the present EAM has improved the calculation results of the planar defects energies of Ni3Al and the elastic constants of NiAl. All these indicate that the present EAM potential can describe the properties of Ni-Al system more effectively.