Synergistic Effect of Hetero Interstitial Atoms (C/N/O) on the Thermodynamic Stability in BCC Fe: A DFT Study

Abstract

Laser cladding rapid solidification technique is an effective strategy for manufacturing ultra-high-strength martensitic stainless steels (UHS-MSS). Due to super-saturation solution strengthening of interstitial atoms (IAs), martensitic stainless steels containing IAs exhibit excellent ultra-high strength and toughness and have high tolerance for oxygen impurities. Hence, studying the specific speciation and structural characteristics of IAs is of great significance for guiding laser cladding of ultra-high-strength steels. Herein, we use density functional theory (DFT) computations to analyze the stable occupancies of IAs and their interactions in body-centered cubic iron (BCC Fe). The findings show that single IAs prefer to occupy octahedral sites over tetrahedral sites. Therefore, octahedral sites are selected as the optimal sites for the following double IAs study. For homo IAs, C-C and N-N configurations exhibit greater stability at long-range distances, whereas O-O demonstrate optimal stability at intermediate distances. Crucially, hetero IAs configurations are more stable compared to single IAs and homo IAs, exhibiting a synergistic effect. Especially, the C-O combination shows the highest stability and strongest bonding character. Meanwhile, the dissociation behavior of O indicates that C-O and N-O have higher dissociation temperatures than single O, further verifying the synergistic effect of hetero IAs. This provides a theoretical basis for understanding the interstitial solution strengthening of laser cladding UHS-MSS.

1. Introduction

Ultra-high-strength martensitic stainless steels (UHS-MSS) have been increasingly used as lightweight structural component materials in aerospace, national defenses, and military industries recently, due to their outstanding comprehensive performances (such as ultra-high strength, excellent toughness, and fatigue properties) [1,2,3]. However, the traditional manufacturing process of UHS-MSS is very complex and costly, which limits its development and application [4,5]. Due to its unique advantages, laser cladding technology has become one of the main technologies for manufacturing various alloys [6,7,8,9,10] in recent years, especially UHS-MSS [11,12]. During laser cladding, interstitial atoms (IAs), carbon (C), and nitrogen (N) are usually considered as beneficial elements because of solution strengthening [13,14,15,16,17], while interstitial oxygen (O) as an impurity [18], is inevitably doped in alloys during the process. Nevertheless, the content of O requires control at an extremely low level in UHS-MSS, e.g., 0.0073 wt.% [19] or 0.001 wt.% [12] in AerMet100, greatly increasing manufacturing difficulty and cost. However, our previous experimental study found that MSS containing C, N, and O IAs, manufactured by laser cladding, have a high tolerance for O impurities and exhibit excellent strength and toughness properties [20,21,22]. Meanwhile, some studies revealed that C/N/O interstitials can form a supersaturated solid solution by laser cladding rapid solidification technique, which provides a highly efficient and low-cost strategy for manufacturing high-strength steel [23,24,25]. Furthermore, IAs form ordered interstitial complexes and play a key role in strengthening various alloy materials, which can be confirmed in the results of Zhao-ping Lu’s research team [26,27,28,29]. They reported that in high-entropy alloys and medium-entropy alloys, oxygen can form ordered oxygen complexes leading to interstitial strengthening, significantly increasing the strength and enhancing the ductility, and ordered nitrogen complexes can also be formed in high-entropy alloys to obtain a high strength and good ductility. Nevertheless, the specific speciation and structural characteristics of IAs remain unclear in UHS-MSS by laser cladding. Therefore, it is urgent to investigate the interstitial strengthening mechanism and thermodynamic behavior of C/N/O interstitials in UHS-MSS at the atomic scale.

First-principles calculation based on the density functional theory (DFT) provides an effective means to obtain the atomic effects of IAs and their interactions at the ab initio level [30,31]. Numerous studies have been dedicated to investigating the effects of IAs by using DFT calculations [13,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]. Previous studies mainly focused on the stability, solution, diffusion of single IAs, and their interaction with point defects in α-Fe [13,32,35,37,43,44,45]. Ye et al. [13], Shang et al. [35], and Luo et al. [45] studied the migration and occupation of O atoms; some of them believed that O is always stable in the octahedral interstice, but Luo et al. obtained that the tetrahedral site O atom can improve the stability of Fe atoms. Kadowak et al. [43], Ohtsuka et al. [44], and B. Kou et al. [32] investigated the occupation, stability, and diffusion of interstitial C atoms and found that the carbon atoms prefer to exist in the O-site and that interstitial C improves the stability of Fe-Mn-C system. Souissi et al. [37] studied the solution and migration energies of C, N, and O in BCC Fe, and the findings revealed that the calculated values of C and N octahedral interstitial sites are consistent with the experimental data, while the results of oxygen are inconsistent. From the above research results, it can be seen that there is an argument about the site preference of oxygen. Moreover, the stability of double IAs and their interactions also significantly impact solid solution strengthening [49].

Some research has already been conducted concerning the interactions between double IAs and their effects on BCC Fe [34,38,39,41,42]. Barouh et al. [34] reported that the X-X (X = C, N, O) interaction is strongly repulsive when the two solutes are too close, but as the distance increases, the interactions of N–N and C–C become less repulsive, while the interaction of O–O presents an obvious attractive effect at the 4th nn configurations (O–O distance 2.81 Å). He et al. [38] found that the interactions of C–B, B–B, and C–C are all repulsive, C–C is the most repulsive, and the repulsive effect decreases with increasing distance. Lv et al. [39] investigated that O–O and N–N increased the stability of the alloys; the formation energies of these alloys (with N or O) are all negative, with O having a stronger effect than N on the stability of the BCC Fe system. You et al. [41] found that the interstitial C atoms tend to be far away from each other; C–C, O–O, and C–O show repulsive interaction in various configurations of BCC Fe (the largest distance between two IAs is $\surd 2$ Å); the repulsion is largest for C–O, lowest for O–O, and middle for C–C, respectively. Ahlawat et al. [42] obtained that all IAs are repulsive to each other at shorter distances (two IAs distances within 3 Å), and N-N has the strongest repulsive effect than others, whereas at larger distances, two IAs show attractive interaction. According to their research results, it can be found that two homo IAs show repulsive interaction when they are too close, but there is a slight controversy over the O–O interactions and the repulsive strength between IAs at shorter distances, and the impact of double IAs on the stability of Fe supercells at larger distances is unclear. Moreover, research [38] has found that the equilibrium distance between IAs is around 5 Å. Therefore, it is necessary to comprehensively study the interactions between IAs within 5 Å. In summary, the site preference of single IAs and the influence and interaction of double IAs on the stability of BCC Fe supercells are not fully understood and still need more research, because of their great significance in explaining solid-solution strengthening.

In our work, first-principles calculations were conducted to comprehensively study the thermodynamic stability of IAs (C, N, O) in BCC Fe. Firstly, for single IAs, site preference via lattice parameters, magnetic moments, formation energy, and binding energy was analyzed. Then, diffusion pathways via the climbing-image nudged elastic band (CI-NEB) method were calculated to evaluate the migration characteristics of interstitial atoms. Subsequently, the thermodynamic stability of double homo interstitial pairs (C–C, N–N, O–O) with different configurations was investigated by energy (formation/binding energy) calculations. Next, the effects of hetero interstitial pairs (C–N, C–O, N–O) on lattice stability were discussed, compared with single interstitials and double homo IAs, through energy analysis (formation/binding) and electronic structure analysis (PDOS/charge density difference/Bader charge). Ultimately, the dissociation behavior of O from hetero interstitial pairs (C–O, N–O) was investigated by computing dissociation energies and dissociation temperatures.

2. Calculation Details and Structural Models

2.1. Methods

Density-functional-theory (DFT) calculations were performed using the plane-wave pseudopotential method [50,51] and the Vienna ab initio Simulation Package (VASP) [52,53,54]. The ion–electron interaction was characterized using projector-augmented wave (PAW) potentials [55,56], while the exchange-correlation functionals were provided by the generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) functional [57]. The Fe₅₄ supercell underwent an energy cutoff convergence test (200–800 eV) to ensure accuracy, as shown in Figure A1 in Appendix A. The results indicated that the system converges when the cutoff energy exceeds 500 eV. The cutoff energy was set at 600 eV to account for computing time and resource conservation. The VASPKIT 1.5.1 [58] software was used to generate K-point paths, and the Gamma center Monkhorst–Pack scheme was employed to sample the K-points in the Brillouin zone [59]. A 6 × 6 × 6 K-point grid was adopted for the BCC Fe lattice and its supercells after K-point convergence tests, as shown in Figure A2. The geometric optimization of supercells was conducted by fully relaxing the cell volume and atomic positions. Then, a single point energy calculation was performed to eliminate Pull stresses further and obtain more accurate energy. The convergence criteria of force and electronic self-consistence were set at −0.02 eV/Å per one atom and 10⁻⁶ eV per one atom, respectively.

The climbing image nudged elastic-band (CI-NEB) approach [60,61] was used to calculate the lowest energy paths and diffusion barriers of IAs (C/N/O) in the supercells [25]. Fe, being a transition metal element, is predicted to have a significant spin-polarization effect [42]. As a result, spin-polarized calculations for BCC Fe were performed in the current work since it is ferromagnetic.

The formation enthalpy (ΔH) of IAs X (X = C, N, O) in metals is defined as [62]:

$$\mathsf{\Delta }{H}_f=\mathsf{\Delta }{E}_f=E\left(F{e}_{54}{X}_n\right)-54\times E(F\mathrm{e})-nE\left(X\right)$$

(1)

where E(Fe₅₄X_n) represents the energies of the supercell with n X atoms, and X could be interstitial atom C, N, or O. E(Fe) represents energy per one atom of the perfect BCC Fe, E(C) represents the energy per one C atom in the graphite solid, with a value of −9.213 eV/atom, and E(O)/E(N) represents half energy of a gas-phase O2/N2 molecules in vacuum, with a value of −4.931 eV/−8.305 eV. At T = 0 K and p = 0 Pa, the formation enthalpy equals the calculated formation energy (ΔE_f), i.e., ΔH_f = ΔE_f, ignoring the zero-vibration contribution [63,64]. A positive formation energy denotes an endothermic process, whereas a negative value indicates an exothermic process, i.e., the formation of interstitial atomic configurations is unfavorable or favorable.

The binding energy is utilized to evaluate the stability of the IAs model. For a supercell containing C/N/O interstitial (X_i), the binding energy can be calculated by the equation:

$$E_{\mathrm{b}}\left(X_1,X_2,\cdots ,X_n\right)=\left[{\displaystyle \sum _{i=1}^{n}E(X_i)}\right]-\left[E(X_1+X_2+\cdots +X_n)+(n-1)E_0\right]$$

(2)

where E(X_i) represents the energy of the Fe supercell containing X_i alone, E (X₁ + X₂ + … + X_n) represents the energy of the Fe supercell having n IAs, and E₀ represents the energy of the Fe₅₄ supercell. Herein, a positive binding energy indicates attraction between these entities, whereas a negative binding energy denotes repulsion.

This work defines the charge density difference (Δρ(Fe₅₄X_n)) as the difference between the charge density of the n X-containing BCC Fe system, the BCC Fe system, and the X atom [45,65], in Equation (3):

$$\mathsf{\Delta }\rho =\rho (F{\mathrm{e}}_{54}{X}_n)-\rho (F{\mathrm{e}}_{54})-\sum _{i=1}^n\rho (X_i)$$

(3)

where ρ(Fe₅₄X_n) denotes the total charge density of the system, ρ(Fe₅₄) denotes the charge density of the Fe₅₄ supercell, and ρ(X_i) denotes the charge density of IAs X_i (X_i = C, N, O) at the corresponding position in the system.

2.2. Models

Martensitic stainless steels are conventionally modeled as body-centered cubic Fe (BCC Fe) for theoretical calculations [39,46,66]. Hence, the research model is conducted based on the BCC Fe. In our study, a body-centered cubic iron (BCC Fe) structure (Im3m space group) was established, with lattice constant a = b = c = 2.8664Å, and α = β = γ = 90°. A 54-atom supercell (3 × 3 × 3 unit cell) was adopted to calculate BCC Fe. It was used to investigate the stability of IAs (C, N, O) and the interactions in order to determine their influence on thermodynamic stability in BCC Fe. Because of its relatively small radii, C, N, and O tend to occupy the interstitial positions in the iron lattice [25]. One C/N/O atom was located on the interstice of Fe₅₄ to construct the supercell of Fe₅₄C/Fe₅₄N/Fe₅₄O. Figure 1 shows the structures of the Fe₅₄ supercell with C, N, and O atoms occupying octahedral sites (O-sites) and tetrahedral sites (T-sites), respectively.

3. Results and Discussion

3.1. Stability of Single Interstitial C/N/O Atoms in BCC Fe Supercell

3.1.1. The Analysis of the Formation Energy of Single Interstitial C/N/O Atoms in BCC Fe Supercell

Table 1. Formation energy (ΔE_f), lattice constant after optimization (a, b, c), supercell volume variation (∆V), average magnetic moment (M) of the BCC Fe (3 × 3 × 3) supercells.

Alloys	Formation Energy /eV	Lattice Constant/Å			Volume Variation/Å3	Average Magnetic Moment/μB
	ΔEf	a	b	c	∆V	M
Fe54	-	8.493	8.493	8.493	0	2.20
Fe54C-Oct	0.758	8.493	8.493	8.649	11.201	2.23
Fe54C-Tet	1.638	8.591	8.440	8.591	10.258	2.21
Fe54N-Oct	−0.518	8.517	8.517	8.639	11.698	2.27
Fe54N-Tet	0.225	8.574	8.506	8.574	10.786	2.22
Fe54O-Oct	−1.027	8.499	8.499	8.644	14.009	2.25
Fe54O-Tet	−0.511	8.578	8.473	8.578	12.635	2.21

shows the calculation of the formation energy (ΔE_f) for supercells with a single IA to assess their thermo-stability. The results indicate that single IAs (C/N/O) in BCC Fe exhibit lower formation energy and higher binding energy at O-sites compared to T-sites, which results in greater thermodynamic stability of single IAs at octahedral positions. Moreover, the positive formation energy of Fe₅₄C indicates that interstitial C is more difficult to form into a solid solution compared to N/O. Notably, the Fe₅₄O-Oct exhibits the lowest formation energy. The stability order for octahedral interstitial systems is O > N > C.

3.1.2. Effect of IAs on Lattice Constants and Magnetic Properties

To examine the effect of IAs on lattice constants, geometrically optimized BCC Fe with C/N/O IAs occupying O-sites and T-sites was used. The optimized lattice parameters are listed in Table 1. The lattice constants (a, b, c) and unit cell volume (V) relative to the pure Fe₅₄ supercell reference system (abbreviated as ΔV) for both interstitial configurations after optimization were compared. Positive ΔV values for both tetrahedral and octahedral interstitial configurations demonstrate that structures incorporating C/N/O IAs exhibit volumetric expansion compared to the pure BCC Fe system. In Fe₅₄ systems with interstitial C/N/O atoms, octahedral occupation induces significantly larger volumetric changes than tetrahedral occupation. Especially, the octahedral configuration displays large lattice constant deviations along the c-axis direction but approaches near-isotropic expansion behavior along the <001> crystallographic plane, leading to the transformation of the lattice into a body-centered tetragonal (BCT) lattice structure. It suggests that O-site occupation of C/N/O atoms represents an energetically favorable configuration compared to the T-site in BCC Fe, which is in accordance with the previous energy study in Section 3.1.1. Table 1 shows that IAs (C/N/O) increase the average magnetic moments of the supercells relative to the pure iron, reaching 2.2–2.3 μ_B, which agrees well with previous results [67]. Taking Fe₅₄C-O_CT as an example, we calculate the formation energy convergence uncertainty (±10⁻⁴ eV), as shown in

Table A1. Formation energy convergence uncertainty of Fe₅₄C-O_CT.

	Total Energy (eV)	Formation Energy (eV)	Convergence Uncertainty (eV)
1	−453.36276	0.7584	0
2	−453.36277	0.7583	−1 × 10−4
3	−453.36274	0.7586	0.0002

. Meanwhile, it can be seen that the influence of IAs occupying O-sites on the magnetic moment is greater than occupying T-sites. Furthermore, based on the analysis of formation energy, it can be concluded that IAs prefer to occupy O-sites. Therefore, we will focus solely on the effects of IAs occupying O-sites in the following studies.

From the top table of Figure 2, it can be seen that the magnetic moment of iron atoms adjacent to IAs changes drastically compared to the pure iron structure. As shown in Figure 2, for C and N interstitials, both 1NN and 2NN Fe atoms show marked magnetic moment reductions relative to the average value. In contrast, O interstitial displays a significant moment decrease in 1NN Fe atoms, while the 2NN Fe moments remain comparable to the average. This result can be further illustrated by electronic structure analysis.

3.1.3. Verifying Stability of O-Site Interstitial Atom at the Electronic Level

In this section, the electronic structures of the supercell Fe₅₄ with a single IA occupying the O-site are illustrated and compared, and the interaction between IAs and their neighboring iron atoms in BCC Fe is examined to clarify the stability of the supercells. The density of states (DOS), charge density difference, and Bader charge population are employed to describe the chemical bonding and charge transfer of the supercells.

To understand detailed bonding characteristics, the projected density of states (PDOS) of O-site IAs and their neighboring iron atom (1NN Fe and 2NN Fe) in Fe₅₄C, Fe₅₄N, and Fe₅₄O lattices is computed, as indicated in Figure 3. There are no energy gaps near the Fermi level, indicating the metallic properties of the supercells. PDOS shows that the hybridization between Fe (1NN Fe and 2NN Fe) 3d states and C/N/O 2p states is strong in both the Fe-lattices, which is in accordance with other calculations [34,42]. By contrast, it can be seen that the hybridization between Fe 3d and C 2p is stronger than that between N and O, and the Fe-C bond presents significant covalent bond properties. O 2p and Fe 3d exhibit obvious isolated peak hybridization, so the Fe-O bond has distinct ionic properties. The hybridization between C 2p and Fe 3d occurs at −6.4 eV in Fe₅₄C (Figure A3), the hybridization between N 2p and Fe 3d occurs at −7.5 eV in Fe₅₄N, and the hybridization between O 2p and Fe 3d occurs at −8.5 eV in Fe₅₄O. A significant asymmetry in the spin-up and spin-down electron densities reveals the ferromagnetic nature of BCC Fe supercells.

To gain further insight into the charge transfer between C/N/O and their neighbor iron atoms, charge transfer is examined with the aid of charge density difference and Bader charge population. As shown in Figure 4, there is an obvious charge accumulation around the C/N/O atoms, while the charge around their neighboring iron atoms is obviously depleted. Bader charge calculation results also show that the IAs gain electrons, while the neighboring iron atoms lose electrons. It indicates that the charge is transferred from iron to IAs. Because the Bader method is based on atomic volume division for charge calculation, it results in a certain deviation between the Bader charge calculation and the actual charge. The volume of O is the largest, and the calculated charge value is the lowest (−1.020). Moreover, the Bader method has poor correlation with electronegativity, and N charge calculation is overestimated (

Table A2. The Bader charge and atomic volume of IAs with their neighboring iron atoms in Fe₅₄C, Fe₅₄N and Fe₅₄O.

Alloy	Atom	Bader Charge	Atom Volume (Å3)
Fe54C	C	−1.048	7.388
	1NN Fe	0.056	10.677
	2NN Fe	0.147	11.079
Fe54N	N	−1.162	7.678
	1NN Fe	0.090	10.716
	2NN Fe	0.159	11.132
Fe54O	O	−1.020	7.965
	1NN Fe	0.088	11.000
	2NN Fe	0.169	11.291

). Therefore, the Bader charge results in this paper are used for the comparison of charge transfer in the same supercell between IAs and their neighboring iron atoms, not for the comparison between IAs. The charge density difference results show that the Fe–C bond indicates significant covalency between C and its neighboring Fe atoms, the Fe-N bond implies a strong polar covalent bond due to the large electronegativity difference, and the Fe–O bond shows a typical ionic bond characteristic of a spherical charge distribution. This implies stronger interaction in Fe₅₄O between IAs and Fe atom, but least in Fe₅₄C, which is consistent with the previous energy results in Section 3.1.1.

3.2. The Diffusion of a Single IA in Fe₅₄ Supercell

The diffusion behavior of IAs has a significant impact on the clustering, segregation, and precipitation of solutes within steel [36]. Therefore, it is essential to conduct an in-depth study of the diffusion behavior of IAs. To analyze the diffusion of single IAs in the BCC Fe lattice, the CI-NEB method was employed to optimize the best migration path for these atoms moving from one O-site to a neighboring O-site in the BCC Fe lattice, as illustrated in Figure 5a.

As shown in Figure 5b, in the migration path, the T-site performs as a saddle point. Moreover, according to the calculation of total energy, the IAs (C/N/O) atom is stable in the O-site, but metastable in the T-site. The result is similar to the conclusion drawn from the formation energy calculations (Table 1). The migration energies (E_m) for C, N, and O are 0.926 eV, 0.784 eV, and 0.557 eV, respectively. It indicates that O is easiest to migrate and that C is the least. It can be inferred that O tends to diffuse into interstitial sites both within the crystal structure and at its boundaries, thereby affecting the overall structural stability. Based on the above analysis, it can be obtained that the O-site occupation by C/N/O atoms in Fe₅₄ systems represents an energetically favorable configuration compared to T-site occupation.

3.3. The Occupying Sites and Stability of Double IAs in BCC Fe Supercell

3.3.1. Two Homo IAs in Fe₅₄ Lattice

Based on the discussion above (Section 3.1 and Section 3.2), a single IA tends to occupy the O-site. Taking Fe₅₄C₂ (with one C in an O-site and another C in a T-site) as an example, the stability of the T-site was tested. The results reveal that C is unstable in a T-site and inevitably reaches the adjacent O-site, as shown in Figure A4. Therefore, we only study the O-sites of double IAs. Moreover, O-sites are classified into two types: the octahedral central interstitial (OCI) site and the edge interstitial (OEI) site. When both C/N/O atoms occupy the OCI sites, there may be five possible configurations as shown in Figure 6a. For double C/N/O IAs, where one occupies an OEI site, there may be seven possible configurations within a single cell, as demonstrated in Figure 6b. The interaction becomes minimal at larger distances; hence the distance between two IAs is less than 5 Å and is considered here [34,42]. The specific energy calculation values and the distances between two homo IAs are shown in

Table A3. Formation energy (∆E_f), binding energy (E_b), and the distances (d) between interstitial atoms of different configurations in Fe₅₄C₂/Fe₅₄N₂/Fe₅₄O₂.

Configurations	∆Ef (eV)			Eb (eV)			d (Å)
	Fe54C2	Fe54N2	Fe54O2	Fe54C2	Fe54N2	Fe54O2	C-C	N-N	O-O
1	1.276	−0.577	−1.895	0.146	0.029	0.329	2.915	2.913	2.767
2	1.250	−0.638	−1.742	0.172	0.090	0.176	4.002	3.890	3.883
3	2.140	0.414	−1.231	−0.718	−0.962	−0.335	2.382	2.484	2.358
4	1.393	−0.481	−1.616	0.029	−0.067	0.050	3.483	3.499	3.522
5	3.106	1.120	−0.166	−1.684	−1.668	−1.400	3.469	3.423	3.434
6	2.140	5.530	−1.218	−0.718	−6.078	−0.348	2.382	2.407	2.354
7	3.276	−0.457	−1.052	−1.854	−0.091	−0.514	1.522	1.952	2.370
8	1.258	−0.425	−1.786	0.164	−0.123	0.220	2.864	2.466	2.779
9	1.391	−0.538	−1.588	0.031	−0.010	0.022	3.484	3.417	3.535
10	1.355	−0.523	−1.496	0.067	−0.025	−0.070	3.346	3.454	3.387
11	1.233	−0.640	−1.602	0.173	0.092	0.036	3.998	4.470	3.896
12	1.489	−0.366	−1.661	−0.067	−0.182	0.095	2.698	2.620	2.551

.

As shown in Figure 6, for Fe₅₄C₂ and Fe₅₄N₂, configuration 11, with two homo interstitial C/N atoms located on the atomic close-packed plane <011> occupying the OEI-site, exhibits the lowest formation energy (1.233 eV/−0.640 eV) and the highest binding energy (0.173 eV/0.092 eV) and is the most stable among the 12 configurations. After structural optimization, the C–C distance is 4 Å, and the N–N distance is 4.5 Å. Configuration 2, with two homo IAs located on the atomic close-packed plane <011> occupying the diagonal OCI-site, is metastable. The positive binding energy of configuration 11 indicates an attractive C–C/N–N interaction, enhancing the stability of BCC Fe. For Fe₅₄O₂, configuration 1 possesses the lowest formation energy (−1.895 eV) and highest binding energy (0.329 eV), making it the most stable. After structural optimization, the O–O The distance is 2.8 Å. Configuration 8, where two homo interstitial O atoms occupy the opposite side OEI-site, is metastable. Compared to N–N and C–C, O–O interactions yield a lower formation energy and higher binding energy, signifying stronger attraction and greater system stability. To further probe the influence of IAs on the thermodynamic stability of the Fe₅₄ supercell, the stability of hetero interstitial atom pairs will be investigated in the following section.

3.3.2. Two Hetero IAs in Fe₅₄ Lattice

Figure 6 indicates that configurations 1, 2, 8, and 11 represent relatively stable structures for two homo IAs (C, N, or O). Consequently, these four positions were selected to investigate the occupational preference of two hetero IAs, as illustrated in Figure 7a. In configurations 1 and 2, both hetero IAs occupy OCI sites, while in configurations 8 and 11, both hetero IAs occupy OCI sites.

To assess the thermo-stability of hetero IAs, binding energies between the atoms were analyzed to determine the strength of their mutual interactions, apart from formation energies. Figure 7b shows that the C-O combination exhibits the highest binding energy, whereas C-N has the lowest E_b. The N-O binding energy lies between them. The binding energies of all hetero IAs are positive, indicating the attraction effect of two hetero IAs, enhancing the stability of supercells. Figure 7c shows that configuration 11, with two hetero IAs on the atomic close-packed plane <011>, occupying the OEI-site, demonstrates the lowest formation enthalpy and the highest binding energy, resulting in the highest stability. To further investigate the interactions between IAs and neighboring Fe atoms, electron property and local magnetic moment analyses were performed for configuration 11.

Previous studies indicate that C–O, C–N, and N–O combinations exhibit strong interactions with their neighboring Fe atoms [42]. Electronic structure analysis is conducted to investigate interactions between hetero interstitial pairs and neighboring Fe atoms further. Here, we analyze a typical set of seven neighbor Fe atomic sites that interact with the IAs (Figure 8), as well as the influences of C–O, C–N, and N–O combinations on the electronic distribution and magnetic properties.

The two-dimensional charge density difference plots for Fe₅₄CO, Fe₅₄CN, and Fe₅₄NO (Figure 9) and Bader charge calculation (

Table 2. Bader charge and local magnetic moment variations of IAs and their neighbor iron atoms in Fe₅₄CO, Fe₅₄CN, and Fe₅₄NO.

	Bader Charge			Local Magnetic Moment/μB
	Fe54CO	Fe54CN	Fe54NO	Fe54CO	Fe54CN	Fe54NO
C	−1.060	−1.058	-	−0.142	−0.142	-
N	-	−1.149	−1.157	-	−0.108	−0.103
O	−1.083	-	−1.097	0.072	-	0.073
Fe1	0.123	0.091	0.092	2.042	1.639	1.704
Fe2	0.170	0.163	0.174	2.460	2.181	2.219
Fe3	0.161	0.163	0.172	2.446	2.243	2.256
Fe4	0.285	0.351	0.378	2.332	2.230	2.272
Fe5	0.057	0.072	0.118	1.698	1.661	2.059
Fe6	0.168	0.157	0.163	2.242	2.225	2.439
Fe7	0.161	0.165	0.146	2.194	2.210	2.494

) reveal significant charge transfer between the C/N/O atoms and their nearest-neighbor Fe atoms. Charge depletion occurs around the neighboring Fe atoms, while charge accumulation occurs near the C, N, and O atoms. This is evidence of charge transfer from iron to IAs. Notably, the Fe4 atom, which is illustrated in Figure 8, situated between the IAs, exhibits pronounced charge transfer. Given the higher electronegativity of O, a distinct electric dipole forms between Fe and O (Figure 9). The PDOS results for Fe₅₄C in Figure 3a reveal strong hybridization between the C-2p and the surrounding Fe-3d. This hybridization consequently modifies the electron density distribution around neighboring Fe atoms, as demonstrated in Figure 9a. This behavior is distinct from the interactions of C-N and N-O, as presented in Figure 9c,e. A comparison of the bond lengths between C/N/O and their neighboring Fe atoms (Figure 9) and the Bader charge population (Table 2) reveals a negative correlation: the magnitude of charge transfer between C/N/O and their neighboring Fe atoms decreases with shorter bond lengths. Surprisingly, the smallest charge transfer occurs with the first nearest neighbors.

As shown in Table 2, compared to the charge transfer of a single IA in the Fe₅₄C, Fe₅₄N, and Fe₅₄O supercells (Figure 4), the charge transfer of O and C in the Fe₅₄CO, Fe₅₄CN, and Fe₅₄NO systems has increased to a certain extent. This demonstrates a synergistic effect relative to the single IA cases. Notably, the charge transfer magnitudes for both C and O increase simultaneously in the C–O combination, attributed to the exceptionally high electronegativity of O. For the C–N and N–O combinations, the charge transfer increases for C and O, while it decreases slightly for N. This suggests the existence of a charge transfer pathway between C–N and N–O, thereby enhancing bonding between IAs and improving the stability of BCC Fe alloys. Charge transfer and interactions between the IAs (C/N/O) and surrounding Fe atoms have a considerable impact on magnetic moments (Table 2). As a result of hybridization between the IAs 2p orbitals and the Fe atoms 3d orbitals, the magnetic moments of the first nearest-neighbor Fe atoms are significantly reduced compared to the average value of 2.2 μB. Conversely, the magnetic moments of other neighboring Fe atoms increase relative to the average, consistent with spin conservation compensation.

3.4. Synergistic Effects of Hetero IAs by Comparing with the Single and Homo IAs and Studying Oxygen Dissociation Behavior

From the previous analysis, it can be concluded that IAs tend to occupy O-sites, and two homo IAs tend to stabilize at a relatively long distance. The results of the stability study of two hetero IAs site preference indicate that configuration 11 (two hetero located on the close-packed plane <011> occupying the OEI-site) is the most stable (Figure 7). Based on the stable structures, we systematically compare thermodynamic stability and electronic structure across single IAs, homo IAs, and hetero IAs, thereby revealing how synergistic interactions between hetero interstitials enhance the stability of BCC Fe alloys.

The binding energies of two IAs supercells, occupying configuration 11, are all positive, indicating that there are interactions between the two IAs, which is confirmed by Section 3.3. This suggests that the combination of IAs helps to enhance the stability of the BCC Fe supercell. As shown in

Table 3. ΔE_f and E_b of single IA (occupying O-site) and double IAs (occupying configuration 11 as shown in Figure 7).

Alloys	∆Ef (eV)	Eb (eV)
Fe54C	0.758	-
Fe54N	−0.518	-
Fe54O	−1.027	-
Fe54C2	1.233	0.173
Fe54N2	−0.640	0.092
Fe54O2	−1.602	0.036
Fe54CO	0.042	0.189
Fe54CN	0.599	0.141
Fe54NO	−1.215	0.158

, compared to Fe₅₄C and Fe₅₄C₂, the formation energy of Fe₅₄CO and Fe₅₄CN is also negative, but it is significantly reduced, especially for Fe₅₄CO, which has a lower formation energy and a positive binding energy, making the structure more stable. This is owing to the strong interaction between C and O. Compared to Fe₅₄O, Fe₅₄NO has a lower formation energy and a higher binding energy, making the structure more stable. Combined with the previous analysis of the migration energy barriers of single IAs, where the migration energy barrier of O is the lowest and that of N is the highest, the strong C–O and N–O bonding not only make the structure more stable but also fix the easily migratory O within the crystal, preventing O from segregating at the grain boundaries.

To further elucidate the bonding characteristics of IAs and their influence on the electronic structure, PDOS analysis is conducted on both single IAs and double IAs supercells, as shown in Figure 3 and Figure 10. Contrasting with single IAs supercells, among the two IAs supercells at configuration 11 (Figure 7), Fe₅₄C₂ exhibits a higher DOS near and at the Fermi level, making it less stable. However, multiple hybridization peaks are observed in the −7 to −5 eV range between C 2p and Fe 3d (Figure A3), implying strong covalent bonding characteristics. For Fe₅₄N₂ and Fe₅₄O₂, N 2p and Fe 3d orbitals form hybridization peaks near −4.5 eV. While O 2p and Fe 3d orbitals form isolated hybridization peaks near −8 eV in Figure 10. Additionally, the DOS at the Fermi level is lower than that of the single-interstitial alloys Fe₅₄N and Fe₅₄O, resulting in a more stable system. Comparing Figure 3 and Figure 10, a significant hybridization phenomenon is observed between C–O, C–N, and N–O, and hybridization is also evident between Fe–C, Fe–N, and Fe–O. It suggests that bonding occurs between hetero IAs, and charge transfer exists between the IAs and their neighboring Fe atoms. This finding is supported by the Bader charge analysis in Table 2. Compared to the effect of single and homo IAs, hetero IAs exhibit lower DOS near and at the Fermi level, implying higher electrochemical stability, which indicates that hetero IAs have a synergistic effect on the stability of the BCC Fe supercell.

According to the investigations on IAs diffusion in Section 3.2, O is more easily diffused compared to C and N. Experimental results indicate that O as an impurity is more easily dissociated from the complexes under high temperature, leading to grain boundary segregation, thereby affecting the mechanical properties of alloy steels [68].

Therefore, the dissociation behavior of O is studied to verify the synergistic effect of hetero IAs (C–O and N–O) by calculating the dissociation energy [25] and dissociation temperature, which is estimated by the Polanyi–Wigner equations [69]. As illustrated in

Table 4. The binding energy (E_b), migration energy (E_m), dissolution energy (E_dis), and dissolution temperature (T_dis) of single O (occupying O-site), C–O, and N–O (occupying configuration 4 as shown in Figure 7). Note that E_dis = E_m + E_b is adopted for IA, and the heating rate during dissociation calculation is 1 K/s.

	Eb (eV)	Em (eV)	Edis (eV)	Tdis (K)
single O	-	0.557	-	202
CO → C + O	0.189	0.557	0.746	266
NO → N + O	0.158	0.557	0.715	255

, the dissolution energy of both C–O and N–O combinations is higher than the migration energy of a single O, indicating that O is more difficult to dissolute from them. To common sense, a higher dissociation energy corresponds to a higher dissociation temperature [48]. The dissociation temperature of C–O (266 K) is the highest among them. This implies that the combination of C–O enhances the thermal stability of the system, verifying the synergistic effect of hetero IAs on the stability of BCC Fe again. In addition, the migration energy of O calculated by DFT is quite different from the experimental value [35]. The reason is that the larger attraction of Fe vacancies to O makes the migration energy of O larger. This paper focuses on the interaction between interstitial atoms and their interaction with neighboring irons and does not consider the effect of vacancy on oxygen migration. The calculated dissociation temperature is used to analyze the synergistic effect trend of hetero IAs. However, the effect of vacancy should be considered in the actual dissociation of oxygen. The migration energy of oxygen with vacancy in reference [37] is calculated, and the corresponding dissociation energy and dissociation temperature are shown in

Table A5. Calculation of dissociation energy (E_dis) and dissolution temperature (T_dis) of single O (occupying O-site), C-O, and N-O considering vacancy effect. Note that E_dis = E_m + E_b is adopted for IA, and the heating rate during dissociation calculation is 1 K/s.

	Eb (eV)	Em (eV)	Edis (eV)	Tdis (K)
single O	-	0.897 *	-	318
CO → C + O	0.189	0.897 *	1.086	384
NO → N + O	0.158	0.897 *	1.055	373

* Ref. [37].

.

4. Conclusions

The study systematically investigated the thermodynamic stability and atomic interaction mechanisms of interstitial C, N, and O atoms in BCC Fe using DFT calculations. Key findings are summarized as follows:

(1) For single IAs, they prefer occupying O-sites over T-sites. The octahedral configuration displays large lattice constant deviations along the c-axis, resulting in a body-centered tetragonal (BCT) lattice structure. The migration energy barrier of O is the lowest, and that of N is the highest, and the transition state is the tetrahedral T-sites during migration.

(2) For homo IAs, C–C and N–N configurations exhibit greater stability at long-range distance (4.0 Å and 4.5 Å) and obtain the most stable structure on the close-packed plane <011>, whereas O–O demonstrate optimal stability at intermediate distances (2.8 Å). For hetero IAs, C–O, C–N, and N–O all exhibit greater stability at long-range distances (about 4.0 Å). Configuration 11 (two hetero atoms located on the close-packed plane <011> occupying the OEI-site) is the most stable.

(3) The positive binding energy of double IAs systems shows that two IAs have a mutual attraction, particularly C–C, C–O, and N–O. Compared with single and homo IAs, hetero IAs systems exhibit lower DOS near and at the Fermi level, indicating higher electrochemical stability, especially, the C–O combination displays the highest stability and strongest bonding character. In addition, the dissociation temperature of C–O (266 K) and N-O (255 K) is higher than single O. All of the above reveal that the synergistic effect of hetero IAs helps to enhance the stability of the BCC Fe supercell.