Density functional theory study of ω phase in steel with varied alloying elements

The presence of a long-abandoned hexagonal omega ( ω ) phase in steel samples is recently gaining momentum owing to the advances in transmission electron microscopy (TEM) measurements, even though it is already reported in other transition-metal alloys. The stabilization of this metastable phase is mainly investigated in presence of C, even though the formation of the ω phase is attributed to the combined effect of many factors, one among which is the enrichment of solute elements such as Al, Mn, Si, C, and Cr in the nanometer-sized regimes. The present study investigates the effect of the above alloying elements in ω -Fe using density functional theory (DFT) calculations. It is seen that the magnetic states of the atoms play a major role in the stability of ω - Fe. Cohesive energy calculations show that the alloying elements affect the energetics and stabilization of ω - Fe. Further, density of states calculations reveal the variation in d - band occupancy in the presence of alloying elements, which in turn affects the cohesive energy. Phonon band structure calculations show that only ω -Fe with substitutional C shows positive frequencies and hence possess thermodynamic stability. Finally, we con-firm the existence of ω - Fe using TEM measurements of a steel sample containing the same alloying elements. Our results can shed light on the stabilization of the ω in other transition-metal alloys as well, in the presence of minor alloying elements. phase in steel. Since the presence of the ω phase is reported in high-C steels that contain other alloying elements such as Mn, Si, Al, and Cr, [2] a detailed investigation can shed light on the stability and related properties. In this study, we address the energetics of ω -Fe using density functional theory (DFT) by incorporating alloying elements such as Al, Mn, Si, C, and Cr in substitutional positions and C in the interstitial position. Furthermore, the changes in electronic structure are analyzed with respect to different alloying elements to understand the relative stability and other properties. Previous investigations were mainly focused on the stabilization of ω -Fe by considering the presence of only C. [5] In the present investigation, our choice of alloying elements is motivated by the experimental observation of the ω phase in steel samples, which contain other alloying elements as well. [9] Our DFT predictions are further verified by transmission electron microscopy (TEM) analysis of the presence of the ω phase in a medium-carbon steel. This experimental steel sample contains the same alloying elements considered in the DFT simulation and has been processed by a thermomechanical rolling process combined with direct quenching and partitioning. [13] The predicted results were cross-checked via observation of the high-magnification structure in an ultrahigh-strength steel which had the same alloy constituents as employed in the DFT calculations. The studied material was an experimental medium-carbon steel with the chemical composition 0.3 C/0.6 Si/2.0 Mn/1.1 Al/2.2 Cr (wt%) and was produced using a thermomechanical rolling process combined with direct-quenching and partitioning. [22] Prior to rolling, vacuum-cast 110 × 80 × 60 mm blocks were annealed at 1200 (cid:1) C for 2 hours. Hot-rolling was done in a 1 MN Carl Wezel 2-HI, Reversing Laboratory Rolling Mill with four passes of about 0.2 strain per pass above the non-recrystallization temperature and four passes below it, resulting in a total reduction of ~80%. The finish rolling temperature was 825 (cid:1) C (±10 (cid:1) C), from where the plates were directly quenched in water down to a quench-stop temperature of 200 ± 25 (cid:1) C. Following direct quenching to the quench-stop temperature, the plates were immediately transferred to a furnace, which was preheated to the same temperature. The rolled samples were cooled very slowly over 40 to 50 hours by switching off the furnace in order to simulate the cooling of the coiled strips under actual industrial rolling. To observe and identify the ω - phase at high magnification, an FEI Helios Nanolab 600 focused-ion beam (FIB) was used for milling and lifting out to produce samples for TEM measurement. An FEI Helios Nanolab 600 with a Ga source was employed here. Rough milling was performed at a voltage of 30 kV, and fine millings at 5 kVand 2 kV, while the currents were adjusted in the range 90 pA to 22 nA during FIB operation. The resulting microstructure was investigated using a JEOL JEM-2200FS transmission electron microscope.

as a result of the presence of large density of unstable nanoscale ω precursors, and the ω phase appears as a result of shuffling of the bcc lattice during the process. [8] Ping et al [1] have reported that the ω phase possesses 3.6% lesser volume and has 0.18 eV higher energy than bcc-Fe, with the atoms coupling anti-ferromagnetically in alternate layers. They emphasized that the increase in C concentration aids the formation of the ω phase starting from 4% and further stabilizes it as ω-Fe 3 C at 25%. It is suggested that the presence of the ω phase in martensitic steel contributes to strengthening under irradiation and acts as a sink for vacancies and H or He impurities. [1] Even though the initial studies emphasized only the role of C in stabilizing the ω phase in steel, a recent investigation has shown that N also helps in promoting the ω phase in high-N martensitic stainless steels. [9] Extant literature shows that the energetics of magnetic configurations also influence the stability of the ω phase, and it is seen that, apart from the ferromagnetic (FM) configuration, the +− magnetic state is also equally probable. [1,6,10] In addition, the presence of interstitial C is found to be detrimental to the stability of ω-Fe. Although the ω phase is reported to possess higher density, microhardness, and specific heat than the α phase in Ti, Zr, and their alloys, [11,12] these properties are not yet estimated in detail in ω-Fe in the presence of alloying elements other than C using energetics and electronic structure analysis. Nonetheless, the presence of other alloying elements may also influence the mechanical stability of the ω phase in steel. Since the presence of the ω phase is reported in high-C steels that contain other alloying elements such as Mn, Si, Al, and Cr, [2] a detailed investigation can shed light on the stability and related properties.
In this study, we address the energetics of ω-Fe using density functional theory (DFT) by incorporating alloying elements such as Al, Mn, Si, C, and Cr in substitutional positions and C in the interstitial position. Furthermore, the changes in electronic structure are analyzed with respect to different alloying elements to understand the relative stability and other properties. Previous investigations were mainly focused on the stabilization of ω-Fe by considering the presence of only C. [5] In the present investigation, our choice of alloying elements is motivated by the experimental observation of the ω phase in steel samples, which contain other alloying elements as well. [9] Our DFT predictions are further verified by transmission electron microscopy (TEM) analysis of the presence of the ω phase in a medium-carbon steel. This experimental steel sample contains the same alloying elements considered in the DFT simulation and has been processed by a thermomechanical rolling process combined with direct quenching and partitioning. [13] 2 | MATERIALS AND METHODS

| Computational methodology
The plane-wave-based pseudopotential code Vienna Ab initio Simulation Package (VASP) was employed for carrying out the DFT calculations. [14][15][16] The plane waves included in the basis set were expanded using a kinetic energy cut-off of 450 eV. Smearing was included within the Methfessel-Paxton scheme with a smearing width of 0.2 eV. [17] For the integration of the Brillouin zone, the Monkhorst Pack scheme was utilized.
A k-grid with dimensions of (12 × 12 × 20) was used for the 1 × 1 × 1 ω-Fe cell. This primitive 1 × 1 × 1 ω-Fe cell consists of three Fe atoms in such a way that two of them occupy the close-packed layers and one is in the loose-packed layer. A 3 × 3 × 2 super cell of ω-Fe with 54 atoms was also considered to compare the difference in properties with various alloy concentrations. For supercells corresponding to 3 × 3 × 2 dimensions, a k-grid of 2 × 2 × 5 was used. A corresponding bcc supercell containing 54 atoms (α phase) was also simulated for comparison purposes, and for which a corresponding k-grid of 2 × 2 × 2 was employed. The ω-Fe cells were relaxed without symmetry constraints and the corresponding energy and force tolerances of 10 −6 eV and 0.001 eV/Å, respectively, were achieved to ensure that the lattice parameters and internal atomic positions were optimized within zero external stress.
The generalized gradient approximation (GGA) was used to calculate the exchange and correlation functional. Projected augmented wave method (PAW) pseudopotentials were employed in the parameter-free Perdew-Burke-Ernzerhof (PBE) formalism. [18,19] The choice of the exchange and correlation functional as well as the pseudopotential was based on existing literature pertaining to calculations on magnetic materials. Previous studies have shown that the PBE approximation could reliably reproduce the energetics and magnetic properties of Fe-based systems including Fe-C alloys and ω-Fe in presence of alloying elements. [1,5,20] Moreover, the GGA-PBE formalism has been tested in relation to the intrinsic uncertainty in choosing the degree of localization of the exchange-correlation hole and proven to be accurate to describe the properties of 3d metals, including Fe. [13] Furthermore, to analyze the dynamical stability of ω-Fe in the presence of alloying elements, phonon frequencies were calculated employing the harmonic approximation for a lattice Hamiltonian using the finite-displacement method using the Phonopy program. [21] In order to calculate the phonon band structure, ω-Fe supercells of 2 × 2 × 5 dimensions with 24 atoms were employed to calculate the second-order force constants.
The segments on the band path mesh were sampled by using 101 points to get the phonon dispersion relations.

| Experimental methods
The predicted results were cross-checked via observation of the high-magnification structure in an ultrahigh-strength steel which had the same alloy constituents as employed in the DFT calculations. The studied material was an experimental medium-carbon steel with the chemical composition 0.3 C/0.6 Si/2.0 Mn/1.1 Al/2.2 Cr (wt%) and was produced using a thermomechanical rolling process combined with direct-quenching and partitioning. [22] Prior to rolling, vacuum-cast 110 × 80 ×60 mm blocks were annealed at 1200 C for 2 hours. Hot-rolling was done in a 1 MN Carl Wezel 2-HI, Reversing Laboratory Rolling Mill with four passes of about 0.2 strain per pass above the non-recrystallization temperature and four passes below it, resulting in a total reduction of~80%. The finish rolling temperature was 825 C (±10 C), from where the plates were directly quenched in water down to a quench-stop temperature of 200 ± 25 C. Following direct quenching to the quench-stop temperature, the plates were immediately transferred to a furnace, which was preheated to the same temperature. The rolled samples were cooled very slowly over 40 to 50 hours by switching off the furnace in order to simulate the cooling of the coiled strips under actual industrial rolling. To observe and identify the ω-phase at high magnification, an FEI Helios Nanolab 600 focused-ion beam (FIB) was used for milling and lifting out to produce samples for TEM measurement. An FEI Helios Nanolab 600 with a Ga source was employed here. Rough milling was performed at a voltage of 30 kV, and fine millings at 5 kVand 2 kV, while the currents were adjusted in the range 90 pA to 22 nA during FIB operation. The resulting microstructure was investigated using a JEOL JEM-2200FS transmission electron microscope.

| DFT analysis of the ω-Fe phase
DFT simulations were carried out on ω-Fe supercells in presence of the alloying elements Al, Mn, Si, C, and Cr. Initially, we had optimized the primitive 1 × 1 × 1 cell of ω-Fe and compared the lattice parameters with previous DFT investigations. The structure of 1 × 1 × 1 ω-Fe is presented in Figure 1 along with the interstitial position of the C atom. In accordance with extant literature, [1,5] we have considered five different magnetic orderings, namely ferromagnetic (FM), nonmagnetic (NM), and three configurations with specified spin orientations. Atomic spins were F I G U R E 1 A-E, Structure of the 1 × 1 × 1 ω-Fe unit cell with the magnetic states. F, 1 × 1 × 1 ω-Fe unit cell with interstitial C position. The blue and yellow atoms indicate Fe and C, respectively. The up (magenta) and down (black) arrows indicate spin-up and spin-down magnetic moments, respectively. The visualization was carried out using the software VESTA [23] initialized with positive and negative values to create ++−, +−+, and +−− magnetic states, where the + and − signs indicate that spins are oriented parallel and antiparallel, respectively. The initial magnetic state of the atoms considered in the present study is shown in Figure 1. The obtained results are presented in Table 1. The initial FM state of the atoms switches orientation to the −++ ferrimagnetic state after optimization. It can be seen from the energetics that both −++ and +−− ferrimagnetic states show equal stability among all configurations considered. The NM state is 80 meV/atom higher in energy than the most favorable FM/+−− energy states. This is in line with the report of Ikeda et al, [5] who obtained an NM state that is 90 meV higher in energy than the most favorable +−− energy state. However, Ping et al [1] derived that the ++− configuration had the highest cohesive energy. In another study, Ikeda et al [6] showed that the +−− magnetic state was more favored energetically by using a k-grid of 12 × 12 × 18 and a kinetic energy cut-off of 400 eV. We have used a slightly higher kinetic energy cut-off of 450 eV and a Monkhorst-Pack k-grid of 12 × 12 × 20 to ensure better accuracy, and obtained comparable results with the above investigations. The +−+ and ++− magnetic states, respectively, possess 30 meV per atom and 50 meV per atom higher energies compared to the most stable magnetic state. Interestingly, the parallelly oriented spins switch their orientation to the −++ configuration after optimization.
Initially, Al, Mn, Si, C, and Cr were introduced into the 1 × 1 × 1 ω-Fe unit cell (Figure 1) to understand the effect of the alloying elements at a higher concentration in the ω-Fe phase. Here, the substitutional Al, Mn, Si, Cr, and C atoms constitute a concentration of 33%, and the interstitial C corresponds to a concentration of 25%. In order to investigate the stability of the supercell with respect to the change in the alloying element, we calculated the cohesive energy (E c ) using Equation (1) and is shown in Figure 2.
where E total represents the total energy of the supercell consisting of n Fe or n M atoms, respectively. F I G U R E 2 Cohesive energy of a 1 × 1 × 1 unit cell of ω-Fe with C, Al, Mn, Si, and Cr (substitutional) and C interstitial elements FM and ++− configurations possess energies of the same magnitude for all alloying elements except Al. In general, the difference in the cohesive energies between FM and ++− configuration is negligible while +−− shows the lowest cohesive energy. This shows that for the ω phase of Fe, FM is the most stable configuration irrespective of the change in the alloying element.
To comprehend the energetics and stability in diluted alloy concentrations, a 3 × 3 × 2 ω-Fe supercell comprising of 54 atoms was simulated and the cohesive energies were calculated in the presence of Al, Mn, Si, C, and Cr, respectively. The substitutional position was considered for all elements whereas the interstitial position was taken only for the C atom. Since we had obtained a ferrimagnetic state after initializing with ferromagnetic moments for the 1 × 1 × 1 cell of ω-Fe, the initial magnetic state considered for the calculations were FM, as optimization would render the most favorable magnetic orientation. A detailed analysis by initializing with all possible magnetic states for various supercell sizes is reserved for a future investigation. Our aim is to see whether the presence of alloying elements aids in the stabilization of the ω-Fe phase. A systematic study of the interstitial C concentration in ω-Fe by varying the supercell dimensions carried out by Ping et al [1] has shown that the segregation tendency of C atoms is higher in the ω phase than in the α phase of Fe. They have postulated that the increase in concentration of C atoms helps in stabilizing the ω phase in steel, which can later transform to martensite or bainite structures depending on factors such as the cooling rate, external stress, and the existence of other elements. Hence, it is important to study the effect of these alloying elements, as their presence is unavoidable in steel.
For comparison purposes, we have also simulated a bcc-Fe supercell consisting of 54 atoms and added the corresponding alloying elements to calculate the energies. The composition of alloying elements corresponds to about 1.8% in this case. The obtained cohesive energies for supercells of ωand α-Fe containing 54 atoms are plotted in Figure 3. Interestingly, the cohesive energy is higher for ω-Fe than for the corresponding α-Fe supercell with the same number of atoms. The enhanced cohesive energies indicate that the presence of alloying elements helps in stabilizing the ω-phase of Fe. Among the elements considered in the present study, ω-Fe supercell containing interstitial C and substitutional Si showed high cohesive energies relative to the other elements. A comparison with α-Fe shows that the change in cohesive energies are similar across different elements, as only the substitutional C shows slight variation. However, this trend may change with the variation in concentration of the alloying F I G U R E 3 Cohesive energies of ω-Fe and α-Fe in the presence of alloying elements calculated for a 54-atom supercell. For comparison, the cohesive energies of ω-Fe and α-Fe cells without alloying elements are also shown F I G U R E 4 Density of states of ω-Fe 1 × 1 × 1 with interstitial c atom, plotted with A, GGA + U and B, GGA approximations elements. It is also important to note that the coexistence of more than one alloying element may lead to complex interactions and different outcomes. Still, these results give a qualitative understanding of the cohesive property of ω-Fe in the presence of alloying elements. Since the alloying elements enhance the cohesive energy of the ω phase compared to the α phase, ω-enriched steel may present better corrosion resistance, and therefore can be a promising alternative for use in environmentally challenging applications. Hence, our study confirms the importance of other alloying elements apart from C in stabilizing the ω phase in steel, in concurrence with other experimental studies. [2] This trend has also been observed in DFT calculations of the cohesive properties of Ru and Pd in the ω-Fe supercell. [1] To gain further insight into the observed energy trends, we have calculated the spin-polarized density of states (DOS) of ω-Fe supercells. It is known that Fe-based oxides and oxyhydroxides are mostly semiconductors exhibiting complex magnetic states and their electronic structures are underestimated by GGA. To overcome this, on-site Coulomb interactions are included in calculations to describe the strongly correlated 3d electrons. [24] In order to check the effect of electronic correlations in ω-Fe, we have conducted a calculation by including the Hubbard parameter and compared with standard GGA calculation. The Coulomb repulsion U = 1.8 eV and Hund's exchange J = 0.9 eV is used, from existing reports, as these values have given accurate results. [25][26][27][28] We have compared the DOS of a 1 × 1 × 1 ω-Fe cell containing interstitial C atom, calculated with F I G U R E 5 Density of states of ω-Fe with alloying elements  Figure 4A,B, respectively. It is evident from the DOS that the spin polarization and metallic character are retained with the inclusion of U and J parameters as well. Except the slight shift of the d orbitals of Fe toward lower energy regions in GGA + U approximation, the electronic properties are not altered significantly. Hence, we show that GGA can successfully describe the electronic properties of ω-Fe, and further calculations are conducted without the inclusion of the U parameter.
The orbital and atom-resolved DOS corresponding to the ω-Fe supercells with different alloying elements are presented in Figure 5. The change in band width of the d orbitals of Fe atom is detected by the influence of alloying elements, as they seem to vary in each case. This is due to the influence of hybridization of the respective p and/or d orbitals of the alloying element. We observe that, in general, the narrowing of the d band width is related to the increase in cohesive energy, and vice versa, which is in line with previous studies. [12] The increase in d-band occupancy and the after effect of the reduction in ω-phase stability have also been observed in group IV transition metals and their alloys. [12] In order to investigate the dynamical stability, we have calculated the phonon band structure of ω-Fe in the presence of the alloying elements C, Al, Mn, Si, and Cr using the Phonopy package [21] and presented in Figure 6. Both the interstitial and substitutional positions are considered for C, and other elements are considered in the substitutional positions only. The phonon frequencies can give a qualitative estimate of the dynamical stability of the structure; for example, if some phonon frequencies are especially small, a distortion along this normal coordinate can happen with little expense of energy. On the other hand, if the energy curvature along a coordinate turns out to be negative, it signifies that the material is thermodynamically unstable against the normal mode displacements. [21] The calculated phonon band structures show that only the ω-Fe supercell with substitutional C is devoid of imaginary frequencies and hence exhibits dynamical stability when compared to other configurations. The presence of an imaginary frequency or a negative eigenvalue in the phonon band structure of ω-Fe with the alloying elements signifies that the atomic displacements reduce the potential energy of the system in the locality of the equilibrium atomic positions. A structure possessing imaginary phonon modes can be changed to alternative structures by employing continuous atomic displacements and lattice deformations. [21] This also indicates that these structures can only be formed under special crystal symmetry constraints, and our study confirms the existence of ω-Fe as the lowest energy phase that can occur in the fcc-bcc pathway. [7] Therefore, our findings reconfirm the experimental observations that the non-close-packed ω structure in steel can be formed under special atomic constraints at twin boundaries or other interfaces. This happens as a result of the combined effect of factors such as chemistry, processing conditions, or the presence of alloying elements. Our investigations emphasize the role of the presence of minor alloying elements in stabilizing the ω phase in steel.

| Microstructure of the DQP steel
To check our DFT predictions of the ω phase of Fe, we have analyzed the structure of a steel sample via TEM investigations. The microstructure of the direct-quenched and partitioned (DQP) steel with the discussed alloying elements is bimodal and consists of three different crystallographic phases: bcc-ferritic phase, fcc-austenite, and hexagonal ω phase (Figures 7 and 8). The bcc structure is in the form of conventional lath-martensite with essentially fine films of layers of inter-lath residual austenite that has stabilized owing to carbon partitioning ( Figure 7). Elsewhere, the bcc structure is nano-twinned, and between these nano-twins there exist extremely fine precipitate-like layers of the ω phase ( Figure 8). This presence of ω phase with nano-twinned bcc is in good accordance with the previous results of Ping et al [1,9,10] The proportions of these two F I G U R E 8 A, TEM image of a nano-twinned region. B, Dark-field image of the intra-twin ω phase between the nano-twins. C, Diffraction pattern from [113] bcc zone axis, where both bcc (112) plane sharing twin structure and ω phase with its (1/3) (112) bcc and (2/3) (112) bcc peaks are visible morphologies are estimated to be roughly equal. Hence, our DFT prediction of the existence of the ω phase along with minor alloying elements is established by the TEM observations in a steel sample containing the same alloying elements.

| CONCLUSION
In summary, we have systematically studied the effect of the alloying elements C, Mn, Al, Si, and Cr on the stability of the ω phase of Fe by employing DFT simulations and confirmed the presence of it on a steel sample containing same elements using TEM observations. We found that the presence of alloying elements alters the d-band width of ω-Fe and subsequently the cohesive energy increases. This indicates that ω-enriched steel may possess improved corrosion resistance and could therefore be suitable for applications in more detrimental environments. Our phonon calculations indicated that the C atom in the substitutional position possesses thermodynamic stability in ω-Fe compared to other alloying elements, indicating the importance of special crystal symmetry constraints and grain boundaries in stabilizing the ω phase. Our results can shed new light on the experimental observations on the origin and energetic stability of the ω phase found in steel samples in the presence of minor alloying elements. Moreover, our findings can give insights into the stabilization of the ω phase in other transition-metal systems in the presence of minor alloying elements.