Interactions between Interstitial and Substitutional Elements of Solute Diatomic and Triatomic Clusters in α-Fe from First-principles Calculations

Abstract

The carburizing and nitriding, essential surface modification methods for steels, enhance wear, fatigue, and corrosion resistance by forming fine carbides, nitrides, and nanoclusters involving alloy elements. Understanding the interactions between interstitial X (C or N) and substitutional elements M is critical for optimizing these processes and tailoring the material properties to specific applications. This study investigates the interaction energies in diatomic and triatomic clusters involving C/N atoms and substitutional elements of Al, Si, Ti, V, Cr, Mn, Co, Ni, Cu, Zr, Nb, and Mo. Using the first-principles calculations, this study reveals the intricate balance of interactions within these clusters, highlighting how atomic arrangements and specific element combinations can lead to either repulsion or attraction. We found that the interaction energies for triatomic clusters can be represented using a linear combination of interaction energies for diatomic clusters. Stable triatomic clusters comprise the second nearest neighbor M–X interactions for Fe–Ti–N, Fe–V–N, and Fe–Nb–N alloys. This finding was consistent with experimental observations of the monolayer clusters. Our analysis using the multiple linear regression and stratified analysis reveals that the metallic radius of element M influences interaction in M–X clusters: a larger metallic radius causes repulsion in the first nearest neighbor clusters and attraction in the second and third nearest neighbor clusters due to strain relief.

1. Introduction

Carburizing and nitriding are widely utilized surface modification methods for steel, recognized for enhancing its wear, fatigue, and corrosion resistance.¹⁾ Through carburizing and nitriding, interstitial C and N atoms are incorporated into the steel’s surface layer, leading to the hardening of the surface. This hardening is understood to arise due to the precipitation hardening due to the formation of fine carbides or nitrides and nanoclusters formed between substitutional alloy elements (M).^1,2,3,4) These precipitates and nanoclusters are attributed to interactions with alloying elements such as Al, Ti, V, Cr, and Mo. From the analysis of elemental distribution and hardness distribution on the surface treated by carburizing and nitriding, it is suggested that the interactions between interstitial elements and substitutional elements contribute to surface hardening.^5,6,7) Understanding the interaction between these alloying elements and the introduced interstitial atoms can provide insights into optimizing the surface hardening process and tailoring the material properties for specific applications. However, a detailed understanding of the interaction energy between interstitial and substitutional elements has yet to progress sufficiently due to experimental challenges.^8,9) To optimize the utilization of alloy elements in nitriding and carburizing processes, a comprehensive understanding of these interactions is essential, necessitating insights from both experimental investigations and theoretical calculations.

Therefore, endeavors have been undertaken to elucidate the interactions between interstitial and substitutional elements by quantifying the interaction energy through the first-principles calculations.^{10,11,12,13,14,15,16,17)} For example, You et al. calculated the interaction energies for solute diatomic M–C and M–N clusters in α-Fe, considering C/N atoms and alloy elements M, including Al, Si, Ti, V, Cr, Mn, Co, Ni, Cu, Nb, and Mo, from the first to the fifth nearest neighbors (1nn to 5nn).¹⁰⁾ Furthermore, Liu et al. calculated the interaction energies between C atoms and 24 distinct transition metal elements for diatomic M–C clusters ranging from the first to the fourth nearest neighbors (1nn to 4nn).¹³⁾ Numerous first-principles calculations have also been reported concerning the interaction energies for solute diatomic M–M clusters involving only substitutional elements.^{18,19,20,21,22,23)} While these prior studies are available, they have predominantly been confined to investigating diatomic clusters.^{10,11,12,13,14,15,16,17,18,19,20,21,22,23)} Recently, Miyamoto et al. evaluated the interaction energies from the first-principles calculations for multi-atomic clusters in α-Fe, which include N atoms, Al atoms, and a tertiary additive element, namely Ti, V, or Cr.⁶⁾ Thus, in-depth studies concerning interaction energies in clusters comprising three or more solute atoms still need to be investigated.

Furthermore, the mechanisms through which solute atoms induce interactions still need to be more adequately understood. To address this, Liu et al. conducted a linear regression analysis between M–C interaction energies and the size factors of substitutional elements.¹³⁾ They found a strong correlation between the 1nn M–C interaction energies and substitutional solute element size, postulating that repulsive interactions arise due to the strain relief.¹³⁾ Conversely, You et al. pointed out that the 2s and 2p valence electrons of C and N atoms form hybridized orbitals with the valence electrons of substitutional elements, resulting in chemical interactions that involve the sharing or exchange of the valence electrons.¹⁰⁾ Gorbatov et al. observed that the M–M interaction energies exhibit a consistent pattern influenced by the atomic number, and this regularity is affected by the number of valence d electrons.¹⁸⁾

In this study, our objective is to clarify the relationship between interaction energies of solute diatomic clusters (M–C/M–N) and solute triatomic clusters (M–C–M/M–N–M) in α-Fe, which encompass C/N atoms and the substitutional element M (M = Al, Si, Ti, V, Cr, Mn, Co, Ni, Cu, Zr, Nb, and Mo). Although Zr has limited solubility in α-Fe and thus limited engineering relevance,²⁴⁾ its inclusion in our calculations is valuable for comparing with Ti, as both belong to the same group in the periodic table. The first-principles calculations determine these interaction energies and establish whether the triatomic cluster is an extension of the diatomic cluster or represents a distinct shift. We also calculate the interaction energies for diatomic M–M clusters of the substitutional element. Moreover, leveraging multiple linear regression and stratified analyses, we endeavor to discern and quantify the juxtaposed contributions of solute size and chemical interactions in the interaction energy.

2. Calculation Procedures

The first-principles calculations were conducted using the Cambridge Serial Total-Energy Package (CASTEP),²⁵⁾ a plane wave ultra-soft pseudopotential code rooted in density functional theory (DFT).^26,27) The electronic exchange-correlation energy within DFT was determined using the generalized gradient approximation proposed by Perdew et al. (GGA-PW91).²⁸⁾ A cut-off energy of 350 eV, determined to be sufficient for ensuring accurate energetics for all elements in this study, was selected for the plane-wave basis. Furthermore, spin-polarized calculations were conducted, factoring in the ferromagnetic ordering.

Valence electrons for C and N were identified as the 2s and 2p electrons. For Al and Si, the 3s and 3p electrons were considered. The early 3d transition metals such as Ti, V, and Cr included the 3d and 4s electrons with the 3s and 3p electrons. For late 3d transition metals, namely Mn, Fe, Co, Ni, and Cu, only the 3d and 4s electrons were considered. The early 4d transition metals of Nb and Mo included the 4d and 5s electrons and the 4s and 4p electrons.

Ion relaxations for stable atomic configurations and lattice constants at zero pressure were determined using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method.²⁹⁾ The convergence parameters set included a total energy tolerance of 1 × 10⁻⁵ eV/atom, a maximum force tolerance of 0.3 eV/nm, a maximal displacement of 1 × 10⁻⁴, and a maximum stress of 0.05 GPa nm. Optimization was halted upon the fulfillment of all these criteria.

All the interaction energy calculations were based on a supercell of 4 × 4 × 4 bcc conventional unit cell, which contains 128 atoms. A 3 × 3 × 3 k-mesh grid, established via the Monkhorst-Pack scheme,³⁰⁾ was utilized for Brillouin zone sampling. Previous studies have proved these configurations large enough to calculate interaction energies between C/N and substituted atoms.^10,12) To double-check the convergence of the supercells, we compared the interaction energies of Zr–N–Zr triatomic clusters obtained from 3 × 3 × 3 supercells and 3 × 3 × 3 k-mesh; we chose Zr–N–Zr because it exhibits the strongest interaction of attraction in this study. The difference between the observed values under each condition was slight, less than 0.02 eV.

In our calculation for pure ferromagnetic α-Fe, the lattice parameter was 0.283 nm, and the local magnetic moment per Fe atom measured 2.21 μ_B. This lattice parameter is marginally smaller than the experimental value of 0.287 nm,³¹⁾ while our calculated magnetic moment for the Fe atom is slightly below the experimental result of 2.22 μ_B.³¹⁾

Consistent with published findings, the octahedral interstitial site was the most stable position of C and N in α-Fe. The configuration of the octahedral interstitial site of the C/N atom represented by X is shown in Fig. 1(a). The first to fifth nearest neighbor (1nn to 5nn) M atom sites from the X site are also shown in Fig. 1(a). The 1nn to 5nn interaction energy of a diatomic M–X cluster, ΔE_M–X, is calculated as follows:

  

$$\mathrm{\Delta }{E}_{\text{M}-\text{X}}=E\left[{\text{Fe}}_{127}\text{MX}\right]+E\left[{\text{Fe}}_{128}\right]-E\left[{\text{Fe}}_{127}\text{M}\right]-E\left[{\text{Fe}}_{128}\text{X}\right]$$(1)

where E[Fe₁₂₇MX] is the energy of the supercell containing 128 Fe atoms, the 1nn to 5nn M atom from an interstitial X atom, and the X atom, E[Fe₁₂₈] is the energy of a perfect supercell containing 128 Fe atoms, E[Fe₁₂₇M] is the energy of the supercell containing 127 Fe atoms and an isolated substitutional M atom, and E[Fe₁₂₈X] is the energy of the supercell containing 128 Fe atoms and an isolated interstitial X atom.

Fig. 1. (a) is the first to fifth nearest neighbor (1nn to 5nn) M atom sites from octahedral interstitial X atom. (b) is the first to fifth nearest neighbor (1nn to 5nn) M atom sites from another M atom. (c) is the configuration of M–X–M(1,2,1) triatomic cluster containing the first and second nearest neighbor (1nn and 2nn) M–X interactions and the first nearest neighbor (1nn) M–M interaction. Two substitutional atoms are denoted as M1 and M2, whereas X represents the interstitial C/N atom. (Online version in color.)

The first to fifth nearest neighbor (1nn to 5nn) M atom sites from another M atom are shown in Fig. 1(b). The 1nn to 5nn interaction energy of a diatomic M–M cluster, ΔE_M–M, is calculated as follows:

  

$$\mathrm{\Delta }{E}_{\text{M}-\text{M}}=E\left[{\text{Fe}}_{126}\text{MM}\right]+E\left[{\text{Fe}}_{128}\right]-2E\left[{\text{Fe}}_{127}\text{M}\right]$$(2)

where E[Fe₁₂₆MM] is the energy of the supercell containing 126 Fe atoms, the 1nn to 5nn M atom from another M atom, and this another M atom.

We computed the interaction energies of triatomic M–X–M (M–C–M/M–N–M) clusters, encompassing the first and first nearest neighbor (1nn and 1nn) M–X interactions, the first and second nearest neighbor (1nn and 2nn) M–X interactions, and the second and second nearest neighbor (2nn and 2nn) M–X interactions. Our decision to investigate these particular clusters was driven by the strong interactions observed at the 1nn and 2nn M–X interactions. Figure 1(c) shows the configuration of a M–X–M triatomic cluster containing the 1nn and 2nn M–X interactions. Notably, a substitutional M atom is the 1nn to another M atom. As a result, the M–X–M cluster shown in Fig. 1(c), characterized by the 1nn and 2nn M–X interactions and the 1nn M–M interaction, is termed M–X–M(1,2,1) in this work. Similarly, the M–X–M cluster, characterized by the 1nn and 1nn M–X interactions and the 2nn M–M interaction, is termed M–X–M(1,1,2). In the triatomic clusters M–X–M containing the 2nn and 2nn M–X interactions, one M atom can be the 2nn and 3nn to another M atom, resulting in M–X–M(2,2,2) and M–X–M(2,2,3), respectively.

To summarize the triatomic clusters, we computed four types of M–X–M clusters: M–X–M(1,2,1), M–X–M(1,1,2), M–X–M(2,2,2), and M–X–M(2,2,3). The interaction energy of a triatomic M–X–M cluster, ΔE_M–X–M, is calculated as follows:

  

$$\begin{array}{l}\mathrm{\Delta }{E}_{\text{M}-\text{X}-\text{M}}=\\ E\left[{\text{Fe}}_{126}\text{MXM}\right]+2E\left[{\text{Fe}}_{128}\right]-2E\left[{\text{Fe}}_{127}\text{M}\right]-E\left[{\text{Fe}}_{128}\text{X}\right]\end{array}$$(3)

where E[Fe₁₂₆MXM] is the energy of the supercell containing 126 Fe atoms, two substitutional M atoms, and one interstitial X atom, composing a M–X–M cluster of M–X–M(1,2,1), M–X–M(1,1,2), M–X–M(2,2,2), or M–X–M(2,2,3).

To elucidate the contributions of individual factors to the interaction energy, we employed a multiple linear regression analysis using Python’s scikit-learn library.³²⁾ In the multiple linear regression model, the explanatory variables comprised the number of valence electrons and the atomic radius. The number of valence electrons was segmented by the s, p, and d orbitals as n_s, n_p, and n_d,³³⁾ reflecting that chemical interactions result from the shared valence s, p, and d electrons between atoms. Given the variability of atomic radii based on the bonding state, we examined a range of candidate atomic radii: ionic radius (r_i),³⁴⁾ covalent radius (r_c),³⁵⁾ and metallic radius (r_m).³⁶⁾ The most suitable radius was selected from these candidate atomic radii for inclusion in the final model. The specific values of these explanatory variables are presented in Table S1 (Supporting Information). The elements with matching n_s, n_p, and n_d values in this table, such as Ti–Zr and Cr–Mo, were utilized for stratified analysis.

The coefficient of determination (R²) was used as the criterion for selecting the suitable atomic radius is given by

  

$$R^2=1-\frac{{{\displaystyle \sum }}_{i=1}^n{\left(y_i-{\hat{y}}_i\right)}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(y_i-\overline{y}\right)}^2}$$(4)

where y_i is the true interaction energy, ${\hat{y}}_i$ is the predicted interaction energy, and $\overline{y}$ is the mean of the y_i dataset.

Each explanatory variable was adjusted to have a mean of zero and a standard deviation of one. This standardization, often referred to as z-score normalization,³⁷⁾ can be described as follows:

  

$$z_i=\frac{x_i-\overline{x}}{\sigma }$$(5)

where x_i represents the original value of the explanatory variables, $\overline{x}$ is the mean of the x_i dataset, and σ is its standard deviation. By employing this transformation, explanatory variables operate on a unified scale, enabling a more straightforward interpretation of which variables exert the most significant impact on the interaction energies.

3. Results

3.1. Diatomic Clusters

Figure 2 and Table S2 (Supporting Information) present the interaction energies for diatomic clusters formed between C/N atoms and alloying elements denoted as M, which include Al, Si, Ti, V, Cr, Mn, Co, Ni, Cu, Zr, Nb, and Mo, spanning from the 1nn to 5nn in α-Fe. Notably, the interaction energies at the 1nn and 2nn sites show varied values. However, the interaction energies at the 3nn, 4nn, and 5nn sites demonstrate minimal variation. Consequently, the pronounced interactions between alloying elements M and C/N atoms predominantly occur at the 1nn and 2nn sites.

Fig. 2. The interaction energies for diatomic clusters formed between (a) C or (b) N atoms and alloying elements M spanning from the first to the fifth nearest neighbors (1nn to 5nn) in α-Fe. (Online version in color.)

The interaction energies listed in Table S2 that follow commas are sourced from the prior first-principles calculations documented in the literature. Our calculations align closely with those reported in previous studies. However, when contrasted with earlier literature, most of our results are smaller values. The discrepancies between our results and those of others arise from various calculation conditions. These can be explained from three perspectives. The first perspective pertains to the supercell size and k-mesh. In this study, we utilized 4 × 4 × 4 supercells with 3 × 3 × 3 k-mesh. While other studies have employed values of 3 × 3 × 3 supercells with 3 × 3 × 3 k-mesh,^10,11) 3 × 3 × 3 supercells with 4 × 4 × 4 k-mesh,¹²⁾ 3 × 3 × 3 supercells with 5 × 5 × 5 k-mesh,¹³⁾ and 5 × 3 × 3 supercells with uncertain k-mesh.¹⁴⁾ It is worth noting that all the supercells mentioned in other studies are smaller than those adopted in this study. The second aspect relates to the cutoff energy of the wave function. In this study, we utilized a cutoff energy of 350 eV. While other studies have employed values of 500 eV,¹³⁾ 400 eV,^11,17) 300 eV,¹⁵⁾ and 286.7 eV,¹²⁾ some adopted the same 350 eV¹⁰⁾ as in our study. The third aspect relates to the pseudopotentials selected for the calculations. While some studies utilize the Projector Augmented Wave (PAW) method,^{11,13,14,15,17)} this study and others^10,12,14) employ the Ultrasoft pseudopotential (USPP). Kamminga et al. compared the interaction energies of M–N using the PAW and USPP methods and found a maximum difference of 0.15 eV between them.¹⁴⁾ The discrepancies in the outcomes of the first-principles calculations arise from the combined influences of these three aspects. Although there are subtle distinctions between our results and those from other studies, the overall trends observed from the calculations remain consistent.

When the M and X (either C or N) are the 1nn to each other, all the interaction energies are positive, which indicates repulsion between the M and X. The repulsions of 1nn M–C are greater than those of M–N for Al, Si, Ti, V, Cr, Mn, Co, Zr, Nb and Mo, and opposite results are given for Ni and Cu. The strongest repulsion was obtained with Nb–X and the weakest with Mn–X, irrespective of whether X is C or N.

When the M and X are the 2nn, the M–X interaction energies decrease from the 1nn M–X except for Si–C, Si–N, and Cr–N. The interaction energies between Ti and Zr with either C or N are negative, which indicates attraction between the M and X in the 2nn. When paired with Ti and Zr, the attractive force is stronger for N than for C. In addition, attractive interactions are observed for V, Mn, and Nb when paired with N in the 2nn.

When the M and X are positioned beyond the 3nn, their interactions steadily diminish, ultimately nearing insignificance as the separation between the M and X increases. The 3nn M–X interaction energies are negative for more M elements than the 2nn M–X. At the 3nn positions, Al, Ti, Cu, Zr, and Nb exhibit attractive interactions with either C or N. Only the Cu–C and Cu–N interactions present negative energy values when M and X are the 4nn. For the 5nn configurations, the attractive interactions are minimal but observed when Al, Ti, Ni, Cu, Zr, Nb, and Mo are paired with either C or N.

Figure 3 and Table S3 (Supporting Information) present the interaction energies of diatomic M–M clusters, spanning from the 1nn to 5nn in α-Fe. Table S3 showcases our interaction energies alongside those from the previous calculations. The subtle differences observed between our data and those by other researchers arise from the distinct computational conditions. Most studies,^19,20,21,22) including our own, have adopted a 4 × 4 × 4 supercell with a 3 × 3 × 3 k-mesh. In contrast, different research opted for 4 × 4 × 4 k-mesh.¹⁸⁾ While specific studies have employed the values of 300 eV²²⁾ and 240 eV²¹⁾ for the cutoff energy of the wave function, several investigations,^18,19,20) including ours, have adopted the 350 eV. In the first-principles calculation method, not only are USPP²¹⁾ as used in our study and PAW^18,19,22) employed, but methods like the full-potential Korringa-Kohn-Rostoker (FPKKR) Green’s function method²³⁾ and the Spanish Initiative for Electronic Simulations with Thousands of Atoms (SIESTA).²⁰⁾ Despite these methodological differences, the consistency of results among various studies is remarkable.

Fig. 3. The interaction energies for diatomic M–M clusters spanning from the first to the fifth nearest neighbors (1nn to 5nn) in α-Fe. (Online version in color.)

As shown in Fig. 3, the most significant interactions are observed at the 1nn M–M and the 2nn M–M. The magnitude of the interaction energy decreases rapidly as the distance between substitutional solute atoms increases. Interactions of M–M beyond the 3nn are comparatively minor. Mostly, the interaction energies between M atoms are positive, although a few exhibit negative interaction energies, indicating an attraction between these M atoms. The most significant attraction is recorded at −0.23 eV for the 1nn Cu–Cu, followed by −0.10 eV for the 3nn Nb–Nb. All other attractive interactions are weaker than −0.05 eV.

3.2. Triatomic Clusters

Figure 4 presents the interaction energies for triatomic clusters (M–C–M/M–N–M) in α-Fe, which encompass one C/N atom and two substitutional M atoms (M = Al, Si, Ti, V, Cr, Mn, Co, Ni, Cu, Zr, Nb, and Mo). The examined triatomic clusters are classified into four distinct M–X–M configurations: M–X–M(1,2,1), M–X–M(1,1,2), M–X–M(2,2,2), and M–X–M(2,2,3). The notation M–X–M(i,j,k) characterizes the triatomic clusters with the ith nearest neighbor M–X, the jth nearest neighbor M–X, and the kth nearest neighbor M–M interactions. The interaction energies for Zr–C–Zr(2,2,2)/(2,2,3), Zr–N–Zr(2,2,2)/(2,2,3), Ti–N–Ti(2,2,2)/(2,2,3), V–N–V(2,2,2)/(2,2,3), and Nb–N–Nb(2,2,2)/(2,2,3) were found to be negative. Negative values of the interaction energies for these triatomic clusters indicate that the M–X–M cluster is more stable relative to isolated solute atoms in α-Fe. All other computed interaction energies were positive in the calculated triatomic clusters. The configurations of triatomic clusters indicating attractive interactions are restricted to (2,2,2) and (2,2,3), signifying that stable triatomic clusters comprise the two 2nn M–X interactions.

Fig. 4. The interaction energies for triatomic clusters (a) M–C–M and (b) M–N–M encompassing one C/N atom and two substitutional M atoms in α-Fe. (Online version in color.)

When a diatomic cluster, such as the 2nn cluster of Ti–N, exhibits attractive interaction at the second nearest neighbor, triatomic clusters, such as Ti–N–Ti(2,2,2)/(2,2,3), similarly display attractive interactions. Such results suggest that the diatomic interactions relate to the interactions within the triatomic clusters. Consequently, the interaction energy of the triatomic clusters was theoretically computed based on the following linear combination:

  

$$\mathrm{\Delta }{\hat{E}}_{\text{M}-\text{X}-\text{M}}^{\left(i,j,k\right)}=\mathrm{\Delta }{E}_{\text{M}-\text{X}}^i+\mathrm{\Delta }{E}_{\text{M}-\text{X}}^j+\mathrm{\Delta }{E}_{\text{M}-\text{M}}^k$$(6)

where $\mathrm{\Delta }{\hat{E}}_{\text{M}-\text{X}-\text{M}}^{\left(i,j,k\right)}$ is predicted interaction energy of M–X–M(i,j,k) triatomic cluster, $\mathrm{\Delta }{E}_{\text{M}-\text{X}}^i$ is the interaction energy of the ith nearest neighbor (inn) M–X cluster, $\mathrm{\Delta }{E}_{\text{M}-\text{X}}^j$ is the interaction energy of the jth nearest neighbor (jnn) M–X cluster, and $\mathrm{\Delta }{E}_{\text{M}-\text{M}}^k$ is the interaction energy of the kth nearest neighbor (knn) M–M cluster.

The relationship between the theoretical predictions based on the linear combination and the actual values of interaction energies obtained from the first-principles calculations is shown in Fig. 5. The diagonal lines in the figure represent a perfect agreement between the predicted and actual values. Figure 5 reveals that the linear combination values are in close agreement with the actual values from the first-principles calculations. The coefficients of determination, which quantify the relationship between the values obtained from the linear combination and the actual values, are shown in Table S4 (Supporting Information). All of these coefficients surpass the value of 0.915, and the average value is 0.944. A coefficient of determination surpassing 0.944 suggests that the linear combination can accurately approximate the interaction energies. Specifically, the linear combination accurately predicts over 94.4% of the variance observed in the actual data.

Fig. 5. The relationship of the interaction energies of the triatomic clusters between the theoretical predictions based on the linear combination and the actual values obtained from the first-principles calculations. (Online version in color.)

3.3. Multiple Linear Regression Analysis

We conducted a multiple linear regression analysis on the interaction energies of diatomic clusters to comprehensively understand the effects of solute atom size and chemical interactions resulting from the shared s, p, and d electrons among atoms. Our decision to focus on only diatomic clusters is justified, as the previous section demonstrated that triatomic clusters can be represented using the linear combination. Furthermore, as demonstrated earlier, the interaction energies of diatomic clusters beyond the third nearest neighbors show a substantial reduction in magnitude. As a result, the multiple linear regression analysis was limited to examining the interactions of M–X and M–M diatomic clusters within the 1nn, 2nn, and 3nn.

Table S5 (Supporting Information) presents the coefficient of determination for various diatomic clusters using the multiple linear regression analysis with three sets of explanatory variables. The explanatory variables under consideration include the number of valence electrons, n_s, n_p, and n_d, and the atomic radius, which is examined in three forms: ionic radius, r_i, covalent radius, r_c, and metallic radius, r_m. Regardless of the configurations of the diatomic clusters, the multiple linear regression models that utilize the metallic radius as an explanatory variable consistently yield the highest coefficients of determination. The average coefficients of determination are 0.657 when employing the ionic radius, 0.677 when using the covalent radius, and 0.857 when incorporating the metallic radius. In multiple linear regression analysis, it is usually for the coefficient of determination to increase with the number of explanatory variables. The result that the multiple linear regression models incorporating the metallic radius yield the highest coefficient of determination, even when the number of explanatory variables is held constant at four, indicates that this specific set of explanatory variables offers a statistically robust and precise representation of the interaction energy. Therefore, we employed the multiple linear regression model incorporating metallic radius as an explanatory variable.

The relationship between the predicted values from the multiple linear regression model incorporating metallic radius and the actual values of interaction energies obtained from the first-principles calculations is shown in Fig. 6. In the case of the 1nn M–C, M–N, and M–M, several data points deviate from the diagonal lines, denoting a discrepancy between the predicted and actual values. This discrepancy between the predicted and actual values is reflected in the relatively low coefficients of determination, with R² being 0.824 for 1nn M–C, 0.748 for 1nn M–N, and 0.745 for 1nn M–M. While the magnitude of these R² values is insufficient for accurately predicting interaction energies, it is adequately high to explain the factors related to the interaction energy.

Fig. 6. The relationship of the interaction energies of the diatomic (a) M–C, (b) M–N, and (a) M–M clusters between the predicted values from the multiple linear regression model and the actual values obtained from the first-principles calculations. (Online version in color.)

Figure 7(a) presents the standardized partial regression coefficients of the multiple linear regression model incorporating the metallic radius. These coefficients, which are either positive or negative, reflect the direction and magnitude of the influence each explanatory variable has on the interaction energy. Positive coefficients reflect a proportional relationship, where an increase in the explanatory variable corresponds to increased interaction energy, signifying repulsive interactions between solute atoms. On the other hand, negative coefficients indicate an inverse relationship, implying that an elevation in the explanatory variable is attractive interactions between solute atoms.

Fig. 7. (a) is the standardized partial regression coefficients for valence s (n_s), p (n_p), and d (n_d) electrons, as well as the metallic radius (r_m), in the multiple regression analysis. (b) is the coefficients solely for the metallic radius among all the coefficients. (Online version in color.)

The standardized partial regression coefficients for M–C and M–N clusters exhibit similar characteristics. In the case of the 1nn M–C/M–N clusters, all the coefficients are positive, except for the number of valence s electrons (n_s). Specifically, the coefficients corresponding to the metallic radius (r_m) are the most significant, underscoring its substantial influence on the interaction energies; a greater metallic radius exerts a repulsive effect. This analytical result is consistent with the findings of Liu et al., who noted a strong positive correlation between the interaction energies and the size factor of substitutional elements for the 1nn M–C clusters.¹³⁾ Following the r_m, an increase in the valence p electrons (n_p) induces a repulsive effect on the interaction energy.

In the case of the 2nn M–C/M–N clusters, the coefficients for the n_p show positive values and are also the most significant. This analytical result implies that the p-block representative elements like Al and Si, which include valence p electrons, exhibit a trend towards repulsive interaction energies for the 2nn M–C/M–N clusters. In contrast, the coefficients associated with the r_m are negative. This analytical result suggests that alloying elements possessing a greater metallic radius are attractive on the interaction energies for the 2nn M–C/M–N clusters.

In the case of the 3nn M–X/M–N clusters, all the coefficients are negative. The coefficients for r_m are the most significant and negative. This analytical result suggests that elements with a greater metallic radius are attractive on the interaction energies of the 3nn M–C/M–N clusters.

M–M clusters spanning from the 1nn to 3nn, the coefficients for the number of valence d electrons (n_d) are the most significant and negative. Increasing the number of d valence electrons exerts an attractive effect on the interaction energy of the M–M clusters. As the distance between the M atoms increases, the magnitude of these coefficients diminishes. This analytical result aligns with the finding of Gorbatov et al., who noted that M–M interactions exhibit a consistent pattern influenced by the atomic number, and this regularity is affected by the number of valence d electrons.¹⁸⁾

3.4. Stratified Analysis

The atomic radius is subject to change due to the interaction of an atom’s valence electrons with those of neighboring atoms during chemical bond formation. In covalent bonding, the sharing of valence electrons between atoms results in strong interatomic bonds and the subsequent expansion or contraction of the atom’s electron orbitals, consequently influencing its atomic radius. In ionic bonding, the transfer of valence electrons from one atom to another alters the atomic radius. Moreover, in metallic bonding, the free movement of valence electrons among atoms means that the valence electron density between these atoms influences the atomic radius. In this study, the multiple linear regression analysis was conducted using three types of atomic radii: covalent, ionic, and metallic. The highest coefficient of determination was achieved when employing the metallic radius. However, this result does not eliminate the influence of confounding factors. In statistical analysis, methods beyond a multiple linear regression, such as stratified analysis,³⁸⁾ are often utilized to remove the influence of confounding variables.

In stratified analysis, data is grouped based on specific characteristics, such as the number of valence electrons in this study, and each group is analyzed individually. This approach accurately assesses the relationship between the metallic radius and the interaction energy, excluding the influence of the number of valence electrons as the confounding variable. Specifically, in groups with identical valence electrons of Ti and Zr group and Cr and Mo group, the atomic radii of Zr and Mo are larger than those of Ti and Cr due to the increasing inner-shell electrons, respectively.

Figure 8(a) illustrates the interaction energy differences between Ti and Zr, while Fig. 8(b) shows the differences between Cr and Mo for the diatomic M–C, M–N and M–M clusters. Additionally, Fig. 7(b) presents the standardized partial regression coefficients for the metallic radius in the multiple linear regression analysis. The variations in the interaction energy differences observed in the stratified analysis align with the trends shown by the standardized partial regression coefficients. Not only the multiple linear regression but also the stratified analyses indicate that a larger metallic radius of alloying element M leads to repulsion in the 1nn M–X interactions and attraction in the 2nn and 3nn M–X interactions.

Fig. 8. The interaction energy differences (a) between Ti and Zr and (b) the differences between Cr and Mo for the diatomic M–C, M–N, and M–M clusters. (Online version in color.)

4. Discussions

4.1. Causes of Interactions

Both multiple linear regression and stratified analyses indicated that a larger atomic radius of alloying element M leads to repulsion in the 1nn M–X interactions and attraction in the 2nn and 3nn M–X interactions. While the multiple regression and stratified analyses indicate correlations between the metallic radius and interaction energies, these findings do not prove causation. It is essential to understand that finding a correlation does not automatically mean there is a cause-and-effect relationship. A critical step in establishing a causal relationship involves discussing the underlying physical causes of repulsion and attraction based on existing knowledge.

Figure 9 displays the average interatomic distances at various Fe sites near the interstitial X (C/N) atoms in Fe–C and Fe–N alloys, as obtained from the first-principles calculations. The schematic illustration of the various Fe sites near the interstitial X has been shown in Fig. 1(a). In a perfect crystal of pure α-Fe, the interatomic distance determined by the first-principles calculations is 0.245 nm, indicated by a horizontal line in Fig. 9. In Fe atom sites at the first and second nearest neighbors (1nn and 2nn) to the X atom, where the distances between Fe and X atoms are less than 0.245 nm, the coordination number (CN) increases from 8 to 9 due to the presence of the X atom. Consequently, at the 1nn and 2nn sites, the average interatomic distances were calculated by incorporating both the distances between Fe atoms and those between the Fe atom and the X atom.

Fig. 9. The average interatomic distances at various Fe sites near the interstitial X (C/N) atoms in Fe–C and Fe–N alloys. The horizontal line indicates the interatomic distance in a perfect crystal of pure α-Fe. The schematic illustration of the 1nn to the 5nn sites has been shown in Fig. 1(a). The coordination number (CN) increases from 8 to 9 at the 1nn and 2nn sites. (Online version in color.)

Figure 9 reveals that the 1nn and 4nn sites are tight, whereas the 2nn, 3nn, and 5nn sites are loose for the substitutional M atom. Significantly, the 1nn sites become exceptionally tight. Consequently, for the substitutional M atoms with a larger atomic radius than Fe, repulsive M–X interactions are experienced at the tight 1nn sites. In contrast, at the looser 2nn and 3nn sites, larger substitutional M atoms are subject to attractive M–X interactions. Therefore, the observed correlation between the M–X interaction energies and the metallic radius is attributed to the strain relief. While Liu et al. indicated that repulsive interactions in the 1nn M–C clusters result from the strain relief,¹³⁾ our study extends this understanding by revealing that strain relief also leads to attractive interactions at the 2nn and 3nn M–X clusters. Figure 9 also demonstrates that at the 4nn and 5nn sites, the average interatomic distances exhibit oscillations and gradually approach that of a perfect crystal of pure α-Fe. Consequently, this implies that the effects of strain relief at these 4nn and 5nn sites are relatively minor.

The correlation between the number of s, p, and d valence electrons and the interaction energies is also significant for specific elements and clusters. In the case of the 1nn and 2nn M–X clusters, increasing the number of valence p electrons exerts a repulsive effect on the interaction energy. Ito et al. demonstrated, using the crystal orbital Hamiltonian population (COHP) analysis, that a higher number of p electrons, which results in smaller extents of atomic orbitals and higher numbers of valence electrons, leads to stronger attractive at grain boundary sites with reduced coordination numbers, in the order of Al, Si, P, and S.³⁹⁾ From this reasoning, it can be understood that contrary to the grain boundaries, in the 1nn and 2nn sites to the X atom where the coordination number increases from 8 to 9, a higher number of valence p electrons induce greater repulsion.

4.2. Relationship to Experiments

Finally, we discuss the relationship between the calculated triatomic clusters and experimentally observed nanoclusters. We have demonstrated that the interaction energies of M–X–M triatomic clusters can be expressed as the linear combination of the interaction energies of two M–X and one M–M diatomic clusters. Although experimental observations are larger clusters than triatomic ones, we will attempt to discuss the stability of these larger clusters based on the linear combination’s extrapolations.

As previously demonstrated, the stable configurations for triatomic clusters are restricted to M–X–M(2,2,2) and M–X–M(2,2,3). Figure 10(a) displays the configurations of triatomic clusters M–X–M(1,1,2) and M–X–M(1,2,1), while Fig. 10(b) shows the configurations for M–X–M(2,2,2) and M–X–M(2,2,3).

Fig. 10. (a) is the configurations of triatomic clusters for M–X–M(1,1,2) and M–X–M(1,2,1). (b) is the configurations for M–X–M(2,2,2) and M–X–M(2,2,3). (c) and (d) are the monolayer M–X clusters along the {001}_α plane. (d) illustrates that the proportion of X atoms in the two Fe {001}_α layers increases. (e) is the B1-type MX compound in α-Fe. The solid lines indicate the relationships for the 1nn M–X, the 2nn M–X, and the 3nn M–X. (Online version in color.)

Sequentially arranging M–X–M(2,2,2) and M–X–M(2,2,3) clusters makes it possible to form a monolayer M–X cluster along the {001}α plane, as shown in Fig. 10(c). Our first-principles calculations have revealed that the formation of stable M–X–M(2,2,2) and M–X–M(2,2,3) triatomic clusters is restricted to specific alloy systems, namely Fe–Zr–C, Fe–Zr–N, Fe–Ti–N, Fe–V–N, and Fe–Nb–N. Due to the negligible solubility of Zr in Fe,²⁴⁾ it is impractical to confirm the existence of Zr–C/Zr–N clusters experimentally. The monolayer M–N clusters have been confirmed experimentally in the Fe–Ti–N,^5,40) Fe–V–N,⁵⁾ and Fe–Nb–N⁴¹⁾ systems. Hence, the stable triatomic clusters obtained from the first-principles calculations agree with experimentally observed monolayer M–N clusters.

A model has been proposed in monolayer Ti–N clusters where Ti and N atoms alternate in a single {001}_α layer, with N atoms included in the adjacent Fe {001}_α layers at a specific ratio.⁴⁰⁾ Figure 10(d) illustrates the schematic diagram of the proposed monolayer Ti–N cluster. As the proportion of N in the two adjacent Fe layers increases, the compositional ratio of Ti to N escalates from 1 to a maximum of 3.⁴⁰⁾ In such structures of the monolayer Ti–N cluster, an N atom in the adjacent Fe layers has the relationship of one 1nn M–X and four 3nn M–X. Following the principle of the linear combination, we computed the sum of interaction energies for one 1nn M–X cluster and four 3nn M–X clusters, with the results displayed in Fig. 11. Figure 11 indicates that the linear combination of one 1nn M–X and four 3nn M–X in attractive interactions for Cu–C, Cu–N, Ti–N, and Al–N. Consequently, the calculated results support the model where N is incorporated into the two adjacent Fe layers of the monolayer Ti–N cluster.

Fig. 11. The sum of the interaction energies for one 1nn M–X cluster ($\mathrm{\Delta }{E}_{\text{M}-\text{X}}^{1\text{nn}}$) and four 3nn M–X clusters ($\mathrm{\Delta }{E}_{\text{M}-\text{X}}^{1\text{nn}}$). (Online version in color.)

Monolayer M–X clusters, when stacked three-dimensionally along the [001]_α direction, can form B1-type MX compounds, as shown in Fig. 10(e). This precipitation is consistent with the Baker–Nutting orientation relationship,⁴²⁾ where (001)_α//(001)_MX and [110]_α//[100]_MX result in coherent precipitates. Recently, in Fe–Mo–N alloys, it has been observed that the thickness of Mo–N clusters increases from a monolayer to a bilayer and then to a trilayer, resulting in B1-type MoN.⁴³⁾ Such metastable layered M–N clusters and B1-type MN have been also observed in Fe–Cr–N^43,45) and Fe–Mn–N.^46,47) The linear combination of interaction energies does not account for forming metastable layered Mo–N, Cr–N, and Mn–N clusters.

In addition, while metastable layered clusters of Al–N, Ti–C, V–C, and Nb–C have not been observed, nanosized precipitates of B1-type AlN,^6,7,48) TiC,^2,49,50) VC,^2,50,51) and NbC^2,50) have been confirmed. In the case of Al–N, AlN does not precipitate unless dislocations are present, and the small amount of Cr, V, or Ti addition promotes the precipitation of AlN.^6,7,48) The formation mechanism of these B1-type MoN, CrN, MnN, AlN, TiC, VC, and NbC cannot be explained by the extrapolation from the first-principles calculations of triatomic clusters.

Shifting the focus of our discussion, we additionally performed the first-principles calculations on the formation energies of B1-type MX compounds from solute M and X in α-Fe, with the results shown in Fig. 12. The formation energy of a MX compound, ΔE_MX, was calculated as follows:

  

$$\begin{array}{l}\mathrm{\Delta }{E}_{\text{MX}}=E\left[\text{MX}\right]+\frac{127}{128}E\left[{\text{Fe}}_{128}\right]-E\left[{\text{Fe}}_{127}\text{M}\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}+E\left[{\text{Fe}}_{128}\right]-E\left[{\text{Fe}}_{128}\text{X}\right]\end{array}$$(7)

where E[MX] is the energy of a unit cell for a B1-type MX compound. Figure 12 indicates that the formation energies of MoN, CrN, MnN, AlN, TiC, VC, and NbC are negative, which is consistent with the experimental results of the precipitates of the B1-type MoN, CrN, MnN, AlN, TiC, VC, and NbC.

Fig. 12. The formation energies of B1-type MX compounds from solute M and X in α-Fe. (Online version in color.)

We focused on the linear combination of M–X and M–M interactions. However, this linear combination proved inadequate for B1-type MX compounds, as it led to a transition from repulsion to attraction in the interactions of MoN, CrN, MnN, AlN, TiC, VC, and NbC. During the transition from a monolayer cluster to the B1-type MX structure, there is an increase in tetragonal strain (c/a) within the clusters, complicating the application of a simple linear combination to the B1 structure. Additionally, while this study did not explore interactions between N atoms, the formation of body-centered tetragonal (bct) α″-Fe₁₆N₂ precipitates in Fe–N alloys, which lack M elements,⁵²⁾ indicates that N atoms might exhibit attractive interactions with other N atoms in α-Fe. This interaction could contribute to the overall attractive interaction in M–N clusters.

Future studies must address more interactions within larger clusters, particularly those involving interstitial elements X, using the first-principles calculations. This endeavor for future studies necessitates considering a variety of cluster configurations and interactions, significantly increasing the computational workload. Consequently, the development of more efficient and feasible computational strategies is crucial. These strategies include integrating machine learning techniques and the cluster expansion method.⁵³⁾ Moreover, collaborating with experimental approaches to validate and refine the accuracy of theoretical models is equally essential for advancing this field of study.

5. Conclusions

In this study, we aimed to elucidate the relationship between interaction energies of solute diatomic clusters (M–C/M–N) and solute triatomic clusters (M–C–M/M–N–M) in α-Fe, which encompass X (C or N) atoms and the substitutional element M (M = Al, Si, Ti, V, Cr, Mn, Co, Ni, Cu, Zr, Nb, and Mo). The following important results were obtained:

(1) The interaction energies are positive when M and X atoms are the first nearest neighbor (1nn), indicating repulsion. In contrast, specific element pairings, such as Ti–C/N and Zr–C/N, exhibit negative interaction energies at the second nearest neighbor (2nn), indicating attraction. The attractive interaction was also observed for V, Mn, and Nb when paired with N at the 2nn.

(2) The triatomic M–X–M clusters displaying attractive interactions are observed in specific alloy systems: Fe–Zr–C, Fe–Zr–N, Fe–Ti–N, Fe–V–N, and Fe–Nb–N. The configurations of the triatomic clusters, which demonstrate attractive interactions, were limited to (2,2,2) and (2,2,3). This finding indicates that stable triatomic clusters comprise the 2nn M–X interactions. We found that the interaction energies of triatomic clusters can be represented using a linear combination of M–X and M–M interaction energies.

(3) The multiple linear regression and stratified analysis indicate that a larger metallic radius of element M leads to repulsion in the 1nn M–X clusters and attraction in the 2nn and 3nn M–X clusters. This correlation between the M–X interaction and the metallic radius is attributed to the strain relief. Additionally, increased valence p electrons in the 1nn and 2nn M–X clusters lead to repulsion, while more valence d electrons in the M–M clusters result in attraction.

Finally, the stable M–X–M(2,2,2) and M–X–M(2,2,3) triatomic clusters in the Fe–Ti–N, Fe–V–N, and Fe–Nb–N systems were consistent with the experimentally observed monolayer clusters of Ti–N, V–N, and Nb–N. The linear combination of one 1nn M–X and four 3nn M–X interaction energies supported the model where N is incorporated into the two adjacent Fe layers of the monolayer Ti–N cluster. However, our linear combination approach for the M–X and M–M interactions was insufficient for explaining the formation of B1-type MX compounds like MoN, CrN, MnN, AlN, TiC, VC, and NbC. Therefore, further research is needed to understand interactions within larger clusters, necessitating more advanced computational strategies.

Supporting Information

Tables of the explanatory variables, the interaction energies from the first-principles calculations, and the coefficients of determination are available.

This material is available on the Website at https://doi.org/10.2355/isijinternational.ISIJINT-2024-062.

Acknowledgments

This work was supported by MEXT Program: Data Creation and Utilization Type Material Research and Development Project Grant Number JPMXP1122684766. The author (T.U.) gratefully acknowledges the financial support provided by the 29th ISIJ Research Promotion Grant, JSPS KAKENHI Grant Number JP23K04422, JST under Collaborative Research Based on Industrial Demand “Heterogeneous Structure Control: Towards Innovative Development of Metallic Structural Materials”, and the Light Metal Educational Foundation. The author (T.U.) greatly thank Prof. Goro Miyamoto, Prof. Masanori Enoki, and Prof. Hiroshi Numakura for sharing valuable information.

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