High-throughput search for RE (La, Ce and Y) contained stoichiometric compound in steels

The design of rare earth (RE) bearing steels requires a thorough understanding of the formation tendency of RE involved phases in steels, while searching for binary and ternary compounds with a wide variety of composition and permutation need a remarkable amount of experimentation which is nearly infeasible. In the present work, we perform a thorough search for the RE-contained compounds in steels by a data-driven high-throughput computational approach. The search results indicate that RE may react with O and N to form a large amounts of oxide and nitride inclusions, while only Y participate in the formation of sulfide inclusion Y2MnS4 and Y2CaS4. For the case of ternary compounds in Fe-based solid solution, it is found that RE prefers to form ternary phases with the non-metallic elements, i.e., B, C, O, P and Si, and only Y is found to combine with metal Cr to form YCr4Fe8. Finally, our screen suggests that RE can participate in the formation of the nano-scale precipitates of κ-carbides, L12 precipitates and B2 precipitates, but MC and M2C carbides.


Introduction
In recent years, rare earth (RE) elements addition has received increasing interest in steel metallurgy. A series of beneficial research for the development of RE-bearing steels have been focused on the purification and modification of inclusions since RE elements are characterized by significant negative free energy changes for compound formations [1,2]. It has been reported that RE elements exhibit a remarkable ability to reduce oxygen and sulfur content to a magnitude of 10 −6 , and the addition of RE elements results in a considerable change in inclusion composition and the formation of RE-contained oxides and sulfides [3,4]. RE elements can also dissolve in Fe matrix to form a solid solution, and thus affect the formation kinetics of precipitates and the phase transition due to their interactions with the defects, i.e. vacancy, dislocation and grain boundary, and other alloy elements [5,6]. Moreover, studies have shown that RE addition also improves the high-temperature oxidation resistance and corrosion resistance due to the reactive-element effect [7,8].
Although the effects of RE on the microstructure and properties have been investigated widely, the reported results of RE-involved compounds are so far focused in the inclusion formation during the steel-making process [9,10]. Further applications of RE elements in steels will require more acknowledgments of the existing state of RE elements and the discovery of RE-contained compounds in the steel system. However, a systematic search for the RE-contained compounds for a wide variety of solutes using brute force experimentation is infeasible.
The first-principles calculations have been employed in materials science for years, and have already exhibited the potential to greatly accelerate the design and prediction of new materials without having to synthesize them in advance. The past years have seen the emergence of many databases for materials properties from the first-principles calculations [11][12][13][14][15][16]. With advances in computational resource, the high-throughput computational method has been developed to increase computational efficiencies. So far, there are several remarkable efforts that use high-throughput first-principles calculations of compounds from large crystal structure databases for materials prediction and design, i.e. the Python Materials Genomics (Pymatgen) [17], Open Quantum Materials Database (OQMD) [18] and Automatic Flow (AFLOW) [19] software frameworks. Indeed, these frameworks have been successfully applied in the search for materials such as Li-ion batteries, alloys, intermetallics and inorganic compounds [20][21][22].
In the present work, to identify the formation preferences of the commonly used RE (La, Ce and Y) elements in steels, we perform a high-throughput search for the RE-contained compounds in steels refers to the inclusions, Fe-based solid solution and nano-sized precipitates.

Methodology
The objective of the present work is to identify compounds of being composed of RE (La, Ce and Y) elements that are likely to form in steels. We utilize the OQMD to search the corresponding compounds. The OQMD is a high-throughput DFT (Density functional theory) database which contains approximately 300,000 DFT total energy calculations of compounds from the Inorganic Crystal Structure Database (ICSD) and decorations of common crystal structures [18]. All the calculations were performed using the projector-augmented wave (PAW) method as implemented in the Vienna Ab initio Simulation Package (VASP) [23,24] with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation function of the generalized gradient approximation (GGA) [25]. Spin-polarized calculations were included for all calculations and the cutoff energy for plan-wave was 520 eV. The Brillouin zone integrations were performed using a Gamma scheme with at least 8,000 k-points per unit cell. Electronic minimization was done with Gaussian smearing of width 0.2 eV by setting the convergence criterion to 10 −5 eV atom −1 , whereas, ionic relaxation was performed using conjugate-gradient algorithm with a force criterion of 0.02 eV/Å.
To search the reasonable candidate compounds, these compounds must be stable or nearly stable. By calculating the formation energies of all the competing phases of a concerned chemical system, one can build a formation energy-composition phase diagram, in which a so-called convex hull is constructed by creating a bounding surface between the lowest formation energies of the corresponding compositions, and a phase with formation energy above the convex hull is metastable. The formation energy of a compound A m B n C l was calculated as [26]: where the formation energy is given by the energy of A m B n C l relative to the composition-weighted (x A , x B and x C ) average of the energies of the pure constituents each in their equilibrium crystal structures. E(A m B n C l ), E(A), E(B) and E(C) are the energies (per atom ) of the compound A m B n C l and constituents A, B and C, respectively. In the present paper, we consider the phases with formation energies on the convex hull (ΔH stab =0) and those within 25 meV atom −1 above the convex hull (nearly stable, ΔH stab 25 meV atom −1 ) are more likely to form in our system. In the present work, we concentrate on the search for RE-containing stoichiometric compounds, and the non-stoichiometric compounds which involve RE atoms that randomly distribute on the host lattice to form random substitutional phases are not explored.

Results and analysis
3.1. RE contained inclusions 3.1.1. Oxide inclusions For RE participating in the formation of oxide inclusions, we performed the search for that involved with the common oxide inclusions in steel-making process, i.e., aluminum oxide, silicon oxide, manganese oxide, chromium oxide, titanium oxide and iron oxide. Figure 1(a) presents the ternary phase diagram of Al-O-La, in which the red dots represent the stable compounds, and green dots represent the nearly stable compounds. As shown in figure 1(a), La may form LaO 2 and La 2 O 3 with oxygen, and these two oxides (or one of them) react with Al 2 O 3 (or LaAl 3 ) to form LaAlO 3 . The space group of LaAlO 3 is R3c and the corresponding crystal structure is illustrated in figure 1(b), and its calculated formation energy is 3.561 eV atom −1 as listed in the first line of  In the phase diagram, the stable and nearly stable compounds are denoted as red and green dots, respectively, and the stable phases are connected by black tie-lines to form the convex hull. Table 1. Oxide inclusions containing RE (La, Ce or Y) elements, in which ΔH f denotes the formation energy and ΔH stab represents the distance between the formation energy and the ground state convex hull.

Compound
Space group inclusions, LaAlO3 and CeAlO 3 inclusions have been confirmed in steel-making process by Wang and Yu et al [27,28]. Additionally, the R3c space group only observed in AlO-contained system, and Pnma space group is the most common structure for the RE-contained oxide inclusions.

Sulfide inclusions
For RE participating in the formation of sulfide inclusions, we performed the search for the compounds involved with manganese sulfide, iron sulfide and calcium sulfide. As shown in table 2, the results indicate that none of the three RE elements (La, Ce and Y) exhibits a preference for forming stable (or nearly stable) compounds with FeS. For MnS and CaS, only Y element is found to react with them and the products are Y 2 MnS 4 and Y 2 CaS 4 , respectively, in which Y 2 CaS 4 is nearly stable. Although it has been reported that La 2 Fe 2 S 5 with a space group of Cmc21 can be prepared from the reaction of La 2 S 3 and FeS at 1223 K [29], the calculations reveal that its formation energy is 0.131 eV atom −1 above the convex hull. Therefore, compared with Y 2 MnS 4 and Y 2 CaS 4 , it is considered that La 2 Fe 2 S 5 is difficult to be formed at 0 K. The phase diagrams of Y-Mn-S and Y-Ca-S, and the conventional crystal structures of the two phases Y 2 MnS 4 and Y 2 CaS 4 are presented in figure 2.

Nitride inclusions
For RE participating in the formation of nitride inclusions, we performed the search for the compounds involved with titanium nitride, niobium nitride, vanadium nitride, and aluminum nitride. As shown in table 3, TiN can only react with La to form La 3 Ti 2 N 6 with I4/mmm space group, and NbN can react with La and Ce to  form La 3 Nb 2 N 6 and Ce 3 Nb 2 N 6 with also I4/mmm space group. Both VN and AlN can react with the three RE elements, and the compounds formed by the former and REs are all I4/mmm structure while the compounds formed by the latter and REs are all Pm3m structure.

Ternary compounds in Fe-based solid solution
For the formation of ternary phases in Fe matrix, these phases need to be in equilibrium with the host lattice [22]. In other words, if a phase is not in equilibrium with the host phase, it will be excluded from consideration even if it is stable. The equilibrium state between a ternary compound and host phase (in this case, it refers to Fe) is determined by calculating the convex hull around the compound and the host phase, and examining if there is a tie-line between these two phases. This approach is illustrated in figure 3. Figure 3(a) shows that CeFe 4 B 4 is a stable phase but not in equilibrium with Fe matrix because there is not a tie-line between CeFe 4 B 4 and Fe, the blue dashed line refers to a non-existent tie-line. On the other hand, figure 3(b) indicates that CeFeC 2 is connected with Fe by a tie-line which suggests that the addition of small amounts of C and Ce to Fe should give rise to the formation of a two-phase equilibrium between CeFeC 2 and Fe matrix. Table 4 lists the ternary precipitation compounds (RE-Fe-X) associated with the common non-metallic elements (X) in steels which are predicted to be formed in Fe-based alloy system. It can be seen that N has no tendency to form stable second phases with RE and Fe. Among the rest of the concerned non-metallic elements, i.e., B, C, O, P and Si, O only forms precipitation phase with La, B can react with La and Y to form La 2 Fe 14 B and Y 2 Fe 14 B with P42/mnm crystal structure which are the common phase in rare earth permanent magnet materials [30], and C, P and Si exhibit a tendency to form precipitates with all the three RE elements. For Ce-Fe-Si ternary system, the stable phases CeFeSi and CeFe 2 Si 2 have also been confirmed by Berthebaud et al [31] in their phase diagram experiments. Additionally, in the search for the common metallic elements involved ternary precipitations, our results reveal that only Cr can combine with Y to produce YCr 4 Fe 8 with I4/mmm crystal structure.  Table 3. Nitride inclusions containing RE (La, Ce or Y) elements, in which ΔH f denotes the formation energy and ΔH stab represents the distance between the formation energy and the ground state convex hull.

Nano-Scale Precipitates
Precipitation hardening is one of the effective techniques to design advanced steels with superior strength characteristics. In recent years, nano-sized precipitation strengthened steels have received increasing interest because of their remarkable mechanical behaviors, such as excellent creep resistance, high specific strength to weight ratio, outstanding combination of strength and ductility [32][33][34]. In this section, we explore the possibility of RE (La, Ce and Y) participating in the formation of nano-scale intermetallic precipitates in steels. As presented in the first column of table 5, the recently investigated intermetallic precipitates in steels that exhibit fascinating strengthening and toughening effects at the nanometer scale are κ-carbides [35], MC carbides [36], M 2 C carbides [37], L1 2 precipitates [38], B2 precipitates [39] and L2 1 precipitates [40]. So, To identify REcompounds that can strengthen steel effectively by precipitation strengthening effect, we searched based on the above precipitation protopypes. In the case of solid phase transitions in metal, it is often found that a metastable phase is formed and maintained for a relatively long period, such as γ' precipitation phase (space group P3m1) in Mg-Y-Zn system which is 168 meV atom −1 above the convex hull [41]. Therefore, we raise the stability threshold from 25 meV atom −1 to 300 meV atom −1 in this section. As shown in table 5, the search results indicate that these three RE elements don't prefer to form MC and M 2 C type carbides. Y can combine with Al and C to form a κ-carbides precipitate Y 3 AlC (ΔH stab =0 eV atom −1 ), and with Ni to form L1 2 type precipitate Ni 3 Ti (ΔH stab =0.183 eV atom −1 ). On the other hand, La, Ce and Y all show the tendency to participate in the formation of B2 and L2 1 precipitates with the corresponding stoichiometric ratio. Based on the reported experimental investigations in metals [42,43], we may conclude that RE elements are more  inclined to form the predicted precipitates with a site occupancy, e.g., partially disordered B2 La 1-x-y Ni y Al x instead of ordered B2 LaAl or LaNi, while this site preference of RE in the precipitates need more detailed investigations in future.

Conclusions
In the present work, we present a comprehensive search for compounds involved with RE (La, Ce and Y) elements in steels through a high-throughput computational approach. The candidate compounds were determined based on their stability which is associated with the corresponding distance of formation energy from the ground state convex hull, and the predicted results provide the compounds in stable equilibrium, or in nearly stable state which are likely to form as metastable phases. The results indicate that RE may react with O and N to form a large amounts of oxide and nitride inclusions, while for sulfide inclusions only Y participate in the formation of Y 2 MnS 4 and Y 2 CaS 4 . For the search for ternary compounds in Fe-based solid solution, we take into account the two-phase equilibrium between the considered compounds and Fe matrix, and the results reveal that RE prefers to form ternary phases with the non-metallic elements, i.e., B, C, O, P and Si, while only Y is found to combine with metal Cr to form YCr 4 Fe 8 . Finally, our screen suggests that RE can participate in the formation of the nano-scale precipitates of κ-carbides, L1 2 precipitates and B2 precipitates, but MC and M 2 C carbides.