Tetrahedral versus Octahedral Interstitials Sites for Oxygen Solutes in NbTiZr Medium-Entropy Alloy

doi:10.21203/rs.3.rs-4896503/v1

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Tetrahedral versus Octahedral Interstitials Sites for Oxygen Solutes in NbTiZr Medium-Entropy Alloy

https://doi.org/10.21203/rs.3.rs-4896503/v1

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Oxygen interstitial solutes have traditionally been thought to occupy only octahedral interstices in body-centered cubic (bcc) metals. However, there is ongoing debate about the competition between tetrahedral and octahedral interstitial sites for oxygen solutes in multi-principal element alloys, especially following recent experimental evidence obtained using deep-sub-angstrom-resolution electron ptychography. The driving force and atomistic mechanism behind the observed preference switch remain unclear. In this study, we systematically investigate the competition between tetrahedral and octahedral sites for oxygen, nitrogen, and carbon solute atoms in bcc NbTiZr alloys using density functional theory calculations. At dilute concentrations, these small interstitial solutes are observed to preferentially reside in octahedral interstices. Notably, we demonstrate that solutes can switch from octahedral to tetrahedral sites with increasing lattice expansion, interstitial solute concentration, and spatial heterogeneity. This transition is explained by a crossover in elastic strain energy associated with solutes in the two competing interstices. Our findings provide a deeper understanding of the behavior of small interstitial solutes and their solid solution strengthening effects in bcc multi-principal element alloys.

Physical sciences/Materials science/Structural materials/Metals and alloys

Physical sciences/Materials science/Theory and computation/Atomistic models

oxygen interstitial

tetrahedral interstices

multi-principal element alloy

body-centered cubic.

Alloying interstitial solutes (e.g., H, C, N, O, and B) into metals can significantly change the structural, thermal, electronic, and mechanical properties^1–4. Specifically, accommodating these small interstitial solutes in certain body-centered cubic (bcc) transition metal matrices influence the nucleation and gliding of screw dislocations and thus provides efficient solid solution strengthening^5–7. Small interstitial solutes have also been shown to drastically increase the ductile-to-brittle transition temperature of bcc metals⁸. Recently, this strategy of adding interstitial solutes has been adopted in the structural design of refractory multi-principal element alloys (MPEAs)^9–12. For instance, Lei et al.¹² reported that 2.0% oxygen doping in TiZrHfNb MPEAs led to the formation of ordered (O, Zr, Ti)-rich complexes, which altered the dislocation shear mode from planar to wavy slip, promoting double cross-slip and significantly increasing tensile strength and ductility. Lei et al.¹² also found that the addition of 3.0 at.% oxygen to bcc Ti-30Zr-14Nb MPEAs increased tensile yield strength by ~ 75% and ductility by ~ 164%. Furthermore, Lu et al.¹¹ addressed the room temperature brittleness of NbMoTaW refractory MPEAs by doping with light solutes such as B and C at the grain boundaries. Additionally, Liu et al.¹³ developed a massive interstitial solid solution alloy, TiNbZr-O-C-N, with an interstitial oxygen content of 12 at.%. This alloy exhibited an ultrahigh compressive yield strength of 4.2 GPa, approaching the theoretical limit, and large deformability at ambient temperature without localized shear deformation. These studies demonstrate that small interstitial solutes can dramatically alter the mechanical properties of MPEAs.

In bcc metals and alloys, interstitial solutes can occupy two distinct types of lattice sites: octahedral and tetrahedral interstices (denoted as O-sites or T-sites, respectively), as illustrated in Fig. 1a-b. O-sites are located at the centers of the edges of the bcc unit cell, surrounded by six host metal atoms. These sites create distorted space due to their coordination geometry, leading the positioning of solutes to yield asymmetric local strain fields and significant displacement of the nearest neighboring atoms. This can enhance yield strength by interacting effectively with dislocations. T-sites, on the other hand, are found at the quarter positions along the cube edges, being coordinated by four host atoms. These sites typically accommodate solutes in a distinct nature of the local atomic environment, resulting in a unique strain distribution. Understanding the preferential occupancy of these interstitial sites is crucial for tailoring the properties of bcc metals and alloys, as the site-specific interactions of solutes with the host lattice can significantly affect the material’s structural stability and mechanical strength.

Previous studies have consistently indicated that in bcc metals, O, C, and N atoms prefer O-sites over T-sites^14–16. However, a recent study by Lei¹² purported that individual O and N atoms in TiZrHfNb alloys occupy those two interstitial sites with equal probability, a finding derived primarily from first-principles calculations, contradicting all previous observations in pure metals. Experiments to directly observe interstitial solute occupancy in the host metal face formidable challenges, especially for imaging low concentrations of light solute atoms dissolved in the host metal. However, Liu et al.¹⁷ recently provided the first experimental evidence for the locations of individual oxygen atoms in a highly concentrated bcc solid solution—the NbTiZr medium-entropy alloy (MEA) with a 12 at% concentration of oxygen interstitials—using deep-sub-angstrom-resolution electron ptychography, observing oxygen in both octahedral and tetrahedral interstitial sites. Most notably, they offered quantitative evidence of an unusual preference for T-sites over O-sites in oxygen-enriched NbTiZr alloy. The mechanism behind this unexpected phenomenon need to be clearly understood.

In this work, we provide theoretical evidence for the preferential occupation of interstitial solutes in the bcc NbTiZr alloy through density functional theory (DFT) calculations. At dilute concentrations, oxygen (as well as carbon and nitrogen) atoms are observed to predominantly take octahedral interstices, whereas residence in tetrahedral interstices is relatively unstable. Remarkably, a transition from preferential octahedral to tetrahedral occupation can be induced by tuning lattice parameters, interstitial solute concentration, and spatial heterogeneity. This transition can be quantitatively explained by estimating the elastic strain energy penalties associated with small interstitial solutes.

The size effect of interstitial solutes

In general, the capacity of interstitial sites (both O-sites or T-sites) to accommodate small solutes is somewhat limited, as high concentrations of interstitial solutes tend to cause the formation of precipitated phases^18–22. Therefore, we began our investigation with a single solute accommodated in a bcc supercell, i.e., a single-solute model, to determine its favored site for occupancy. In contrast to the T-site, the O-site has a distinctive flat octahedral shape, as shown in Fig. 1c-d. For the six host metal atoms enclosing the O-site, the distance d_S, between the center of the O-site and the two closest adjacent host metal atoms is only a/2 (where a denotes the lattice parameter of the host metal), significantly smaller than the distance d_L = $\:a/\sqrt{2}$ between the center and the other four co-planar metal atoms. Consequently, when the octahedral site accommodates a single solute, the two nearest neighboring host metal atoms are displaced markedly away from their original lattice site, as illustrated schematically in Fig. 1d. Due to the limited open volume of octahedral or tetrahedral sites, various degrees of internal stress on the solute thus strongly depends on the interstitial solute size.

With strain considerations as our starting point, we estimated the elastic strain energy for an interstitial solute (atomic radius r) located at octahedral or tetrahedral sites within a bcc lattice (atomic radius R) as a function of r/R, as shown in Fig. 2a. We employed a hybrid empirical interatomic potential as a simplified model to represent the interactions of a single interstitial solute with Nb atoms sitting on their lattice sites (see METHODS for details). This strain consideration, a key component of the overall free energy, supports the prevailing belief that interstitials predominantly reside in the O-sites of a bcc metal²³. A cross-over is seen at a critical r/R value (~ 0.46), for the elastic strain energy of interstitial solute residing in the octahedral site versus the tetrahedral site. For interstitial solutes such as C, N, and O, which have r/R values within the range of 0.5 to 0.53, the elastic strain energy in the octahedral site is lower than that of the tetrahedral site, strongly suggesting that these solutes prefer the octahedral site in the Nb lattice, consistent with previous findings^14–16.

Figure 2a also demonstrates that in the region where r/R < 0.46, the T-site is more likely to accommodate interstitial solutes than the O-site. Consequently, for much smaller solutes such as H, for which r/R ≈ 0.24, the tetrahedral site is more stable, which is supported by previous studies on H accommodation in TiZrHfNbTa alloys¹⁰ as well as in simple bcc metals like Nb and Ta²⁴. Another way to demonstrate the size/strain effect is to place individual oxygen atoms at either octahedral or tetrahedral sites within the Nb lattice, while varying the degree of lattice expansion, (which corresponds to different r/R values, as detailed in the METHODS section). Due to the stress symmetry from neighboring metal atoms, the oxygen solute atom remains confined to its starting site in the Nb lattice without displacement. Figure 2b shows that as the Nb lattice expands (i.e., decreasing r/R), the total energy difference, $\:{E}_{O-site}-{E}_{T-site}$ gradually diminishes until the lattice expansion factor L₀ exceeds a critical value (~ 1.066), at which the total energy of $\:{E}_{T-site}$ becomes lower than that of $\:{E}_{O-site}$. Thus, lattice expansion can indeed cause oxygen solutes to transition from octahedral to tetrahedral sites. This analysis, different from the estimate in Fig. 1, was performed based on DFT-derived energies.

Interstitial residing in O-site versus T-site at dilute concentrations

In addition to pure bcc metals, the preferential occupancy of interstitial sites in a concentrated solid solution, i.e., MPEA, is the main focus of the present work. MPEAs typically exhibit a complex local atomic landscape characterized by significant chemical and structural inhomogeneity, including variations in lattice distortions and local chemical environment. These complexities have been shown to affect the mechanical properties of the materials. The predicted interactions between a single interstitial solute and host pure metal atoms (i.e., the simple model in Fig. 2) were subsequently studied in the more structurally complex bcc NbTiZr alloy, again based on energy evaluation using DFT. As depicted in Fig. 3a, our DFT results indicated that when a single interstitial solute was initially introduced into a tetrahedral site within an NbTiZr supercell, it spontaneously moved to the nearest octahedral site upon structural relaxation. Figure 3b illustrates that the host metals surrounding the oxygen solute undergo severe local lattice distortions. The oxygen solute entering the octahedral site, having hopped approximately 0.75 Å, significantly displaces the two closest metal atoms from their original sites. This is a more favorable accommodation because it incurs less strain energy compared to the case of tetrahedral site where four metal atoms need to be pushed apart.

To conduct a systematic survey that makes better statistical sense in reflecting the varying local atomic environment, we randomly inserted single O, (or C, or N) into different tetrahedral or octahedral sites in the NbTiZr supercell. For each solute species, 54 independent events were studied, and the atomic displacements of the interstitials were extracted during energy minimization. As shown in Fig. 4a, all O, N, and C atoms initially residing in octahedral sites exhibited very small average atomic displacements (0.09 Å for O/N and 0.08 Å for C), arising from the lattice distortion due to the varying local atomic environment. In contrast, atoms initially residing in tetrahedral sites displayed relatively larger atomic displacements (~ 0.75 Å for O, ~ 0.74 Å for N, and ~ 0.76 Å for C), which is close to the distance between an octahedral site and its adjacent tetrahedral site in the NbTiZr lattice (illustrated in Fig. 3b). Upon relaxation, all individual N, C, and O solutes in tetrahedral sites were found to be unstable (i.e., 0% of them remained in their original sites, as listed in Fig. 4b). These results imply that in NbTiZr alloys, individual small solutes tend to reside in octahedral sites rather than in tetrahedral sites, consistent with the general view that interstitial solutes prefer octahedral sites in bcc pure metals^{14, 15}.

Our findings may seem to contradict a previous study¹², which claimed that O and N solutes occupy tetrahedral and octahedral sites with equal probability in the TiZrHfNb alloy. Upon communicating with the authors¹², we learned that their conclusion was based on the observation that similar energies were reached when the two cases were compared, with solutes initially inserted in octahedral versus tetrahedral sites. However, they neglected to monitor the actual positions of the solute atoms after the relaxation. In the relaxed sample, actually, all solutes initially present in tetrahedral sites have relocated to octahedral sites to lower energy. That change in preferred solute residence was in fact the reason that no statistically meaningful energy difference was found between the two cases (the two eventually evolve into the same configuration).

Local atomic environment around interstitial solutes

Despite the relatively small atomic sizes of C, O, and N solutes compared to the host metals, they are still significantly larger than the open volumes of octahedral or tetrahedral sites in the NbTiZr lattice. To accommodate these small solutes, severe lattice distortions occur around the octahedral sites to release the internal lattice stresses caused by the extrusion of solutes into the host metal. Details of the lattice distortion calculations are provided in the METHODS section. Figure 5a illustrates that the larger the solute size (atomic size: O < N < C), the greater the lattice distortions (Δd) imposed on the surrounding host atoms, with average lattice distortions for O, N, and C being 0.273, 0.286, and 0.288 Å, respectively.

Figure 5b statistically shows the distribution of bond lengths of these small solutes with their adjacent host metal atoms. The bond lengths can be classified into two categories (i.e., d_S and d_L ), as labeled in Fig. 1d. For each type of solute (i.e., O, C, or N), the converged d_S and d_L were obtained by statistically averaging 108 and 216 independent bond lengths, respectively. Initially, in the case of octahedral sites without lattice relaxation, d_S and d_L are 1.70 Å (corresponding to $\:{d}_{S}=a/2$) and 2.39 Å (corresponding to $\:{d}_{L}=a/\surd\:2$), respectively. However, these bond lengths undergo significant changes after stress relaxation. Specifically, the average d_S values for O-, N-, and C-related bonds are 2.11, 2.09, and 2.12 Å, respectively, increasing by 0.41, 0.39, and 0.42 Å from their initial d_S. Conversely, the average d_L values for O-, N-, and C-related bonds are 2.21, 2.18, and 2.20 Å, respectively, yielding bond shortenings of 0.18, 0.21, and 0.19 Å. As shown in Fig. 5b, the average d_S remains shorter than d_L.

Figure 5c shows the atomic fraction of Nb, Ti, or Zr in the nearest neighbor shell of interstitial solutes N, C, and O by extracting the local atomic information of the 10% of samples with the lowest interstitial formation energies. All N, C, and O solutes exhibit strong chemical affinities with Ti. This chemical affinity between small solutes and the host metals has also been found in previous studies^{12, 25–27}. Interestingly, as the solute size increases (atomic size: O < N < C), the chemical affinity with Zr decreases while the affinity with Nb increases. This chemical affinity trend arises because, to fit into the open volume of the octahedral sites and release lattice stresses, the larger host metal Zr prefers to stay with the smaller solutes.

The case of highly concentrated interstitial solutes

A crucial question is under what conditions small solutes undergo an unusual switch from octahedral (O-site) to tetrahedral (T-site) occupancy. Earlier, we have shown that in bcc metals with dilute interstitial solute concentrations, the O-site is more favorable because it is less costly in energy for the O-site solute to push two nearest metal atoms away from their lattice positions. However, when the solutes are highly concentrated, the metal atoms become largely constrained by the surrounding interstitial solutes, as illustrated in Fig. 6a. This suppressed displacement would hamper effective release of elastic strain. To support this hypothesis, Fig. 6b illustrates the elastic strain energy for two small solutes (atomic radius r) located at O-sites or T-sites within a bcc Nb lattice (atomic radius R) as a function of r/R. A quantitative estimate of the strain energy is detailed in the METHODS section. Interestingly, the elastic strain energies of O-sites and T-sites exhibit a cross-over similar to the single-solute case (see Fig. 2a), but the critical r/R value in this high solute concentration case is shifted up to ~ 0.24. Notably, when the two O (or N, or C) solutes are located at T-sites, the elastic strain energy is significantly lower than at O-sites (see Fig. 6b). This is because the strain energy cost for an octahedral interstitial solute to displace the two neighboring metal atoms increases significantly at high solute concentrations, eventually becoming comparable to or exceeding that of T-sites. This causes an inversion in the strain energy levels for solutes residing in T-sites versus O-sites, leading to a gradual transition of interstitial solutes from O-sites to T-sites at elevated solute concentrations.

As such, our theoretical model posits that it is the confinement of host metal atoms by adjacent interstitial solutes that causes a shift in the preference from O-sites to T-sites (see illustration in Fig. 6a). These collective interactions will become more pronounced with increasing oxygen content and spatial heterogeneity of oxygen distribution. This explains the observation in recent experimental work¹⁷, which demonstrated that the distribution of oxygen is inhomogeneous within the (NbTiZr)₈₆O₁₂C₁N₁ alloy and the oxygen-enriched regions generally accommodate more tetrahedral interstitials. To verify the effect of oxygen concentration and heterogeneity in NbTiZr alloys, we employed DFT approaches. Among ten independent NbTiZr supercells, oxygen atoms with a range of concentrations and spatial heterogeneity were randomly incorporated into both T-sites and O-sites (see METHODS for details). We then counted the portion of interstitial solutes that remained at tetrahedral versus octahedral sites after structural relaxation. The parameter of oxygen heterogeneity, η, describes local regions enriched with oxygen compared to the whole sample (illustrated in Fig. 7a). The smallest value of η = 1 indicates homogeneous distribution, while larger η values indicate higher heterogeneity.

As depicted in Fig. 7b, oxygen solutes predominantly occupy O-sites when their concentration and heterogeneity are relatively small. Strikingly, the tendency to occupy T-sites gradually increases with rising oxygen concentration and heterogeneity, eventually exceeding the preference for O-sites. For instance, at an already high average oxygen concentration (e.g., 18 at.%), local regions with even higher oxygen concentrations show a stronger preference for the T-sites (Fig. 7b). This trend is consistent with the rationale illustrated in Fig. 6, where the strain energy required for an octahedral interstitial to displace neighboring metal atoms increases significantly with more interstitials in the neighborhood, becoming comparable to or exceeding that required for tetrahedral interstitials. As a result, O-sites become disfavored compared to the more spacious T-sites. A transition from O-sites to T-sites is even more likely when oxygen distribution becomes inhomogeneous, in which would render local regions even more crowded with oxygen interstitials.

Before closing, we revisit the experimental work¹⁷ that demonstrated the unusual T-site preference tendency using state-of-the-art adaptive propagator multislice electron ptychography. In experimental NbTiZr alloys with a high oxygen concentration (~ 12%), this elevated oxygen concentration inevitably leads to severe distortions in the local host lattice. To accommodate the extrusion of metal atoms caused by massive amounts of interstitial oxygen, the alloy matrix experiences significant volumetric strains (i.e., lattice expansion) driven by intra-atomic stresses. Additionally, atom probe tomography (APT) results indicate that oxygen solutes exhibit significant chemical inhomogeneity, with ~ 1 nm oxygen-rich regions alternating with those containing lower concentrations of oxygen. This chemical inhomogeneity partly originates from the stronger affinity of oxygen with Ti/Zr compared to Nb, resulting in local regions with oxygen concentrations above the already high sample average. Overall, lattice expansion, high interstitial solute concentration, and spatial heterogeneity are all present in these NbTiZr alloys, contributing to the unusual T-site preference of oxygen solutes. All of these are in line with our theoretical model, although we cannot numerically match the experimental results exactly (e.g., the average solute concentration in the sample, at which T-site oxygen becomes dominant, is about 12% in that experiment, while it is much higher in the present DFT study). This may be influenced by the local chemical inhomogeneities in the elemental distribution within experimental NbTiZr alloys²⁸.

This study provides a comprehensive theoretical and computational investigation into the preferential occupancy of octahedral (O) and tetrahedral (T) sites in body-centered cubic (bcc) NbTiZr alloys at both dilute and highly concentrated solute concentrations. Our predictions, grounded in the framework of elastic strain energies for single- and multi-solute models, have been substantiated using density functional theory (DFT) approaches. Our findings reveal that individual solutes such as oxygen (O), nitrogen (N), and carbon (C) tend to stably reside in O-sites within the NbTiZr lattice. A significant focus of this research was to elucidate the conditions under which a preference shift from O-sites to T-sites might occur in the bcc lattice. Our simulations provide evidence that a shift in preference from O-sites to T-sites at high oxygen concentrations can be induced by increasing oxygen heterogeneity or imposing lattice expansions. This trend can be explained by comparing the elastic strain energy cost for oxygen atoms residing in O-sites versus T-sites; a transition towards the latter is set off at sufficiently high oxygen concentrations when their competition is tilted to the point where the T-sites become more energetically favorable.

For other small solute elements such as N and C, similar stable occupancy in O-sites (except the grain boundary segregation) was observed in the NbTiZr alloy. A preference shift from O-sites to T-sites is also expected for these solutes at high concentrations. The increased likelihood of T-site occupation has significant implications for the compositional design and strengthening potential of solid solutions. This insight could serve as a reference for exploiting synergistic benefits by tailoring massive interstitial (high and undulating solute concentrations) in MPEAs for solid-solution hardening and potential irradiation tolerance^29–31. Overall, our study bridges the gap between theoretical predictions and experimental observations, providing a nuanced understanding of interstitial solute behavior in complex alloy solutions. These findings pave the way for innovative alloy design strategies aimed at enhancing mechanical properties through precise control of interstitial solute distributions.

Density Functional Calculations for Single-Solute NbTiZr System

For single-solute NbTiZr samples, initial equi-atomic NbTiZr supercells containing 128 atoms were generated using special quasi-random structures (SQSs)³² as implemented in the ATAT code³³. For each type of interstitial solute (i.e., O, N, or C), a single solute was randomly placed in either the O-site or T-site of the NbTiZr lattice to construct samples with dilute solute concentrations. A total of 54 individual O-site and T-site configurations were sampled. Structural relaxations and energy calculations were performed using the density functional theory (DFT)-based Vienna Ab Initio Simulation Package (VASP)³⁴. The generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE)³⁵ form was employed to describe the exchange-correlation function, and the valence electronic states of elements were handled by the projector-augmented wave (PAW) method³⁶. The valence-electron eigenstates were expanded using a plane wave basis set with a cutoff energy of 450 eV, and the energy criterion in each self-consistent field (SCF) iteration was set to 5◊10^− 5 eV. A k-mesh of 2◊2◊2 was adopted for all configurations.

Density Functional Calculations for Multi-Solute NbTiZr System

For multi-solute NbTiZr samples, initial equi-atomic NbTiZr supercells containing 128 atoms were generated using SQSs³² as implemented in the ATAT³³ code. Different amounts of oxygen atoms were then randomly placed at different O-sites and T-sites in the NbTiZr lattice to assemble samples accommodating highly concentrated interstitial solutes. By adjusting the degree of spatial dispersion of oxygen in the NbTiZr lattice at each specific solute concentration, the spatially heterogeneous distribution of oxygen was modeled (illustrated in Fig. 7a). To accurately capture the occupancy of O-site or T-site oxygen in the multi-solute model, structural information was statistically analyzed in ten independent NbTiZr supercells. Structural relaxation and energy calculations were performed using the VASP³⁴ code, with all computational parameters consistent with the single-solute model described above.

Density Functional Calculations for Oxygen-Nb System

We constructed a bcc pure Nb cubic supercell containing 54 atoms and placed a single oxygen atom in its O-site or T-site. The lattice expansion process was modeled by artificially enlarging the volume of the supercell with a lattice expansion factor L₀ = a/a₀, where a, a₀ denote the expanded lattice parameters and the equilibrium lattice parameter of the Nb lattice, respectively. Free energy calculations of the system were performed using the VASP³⁴ code, where the box was kept fixed during structural relaxation to avoid retraction of the expanding simulation box. The density of reciprocal space was sampled using a 3×3×3 k-grid mesh, and other computational parameters were kept consistent with those of the single-solute NbTiZr model.

Local lattice distortion calculations

The local lattice distortion of the first-nearest-neighbor (1nn) Nb, Ti, or Zr atoms around the interstitial solute can be expressed as:

$$\:\varDelta\:d=\varDelta\:{d}_{h}+\varDelta\:{d}_{s}=\frac{1}{N}\sum\:_{i}\sqrt{{({x}_{i}-{x}_{i}^{{\prime\:}})}^{2}+{({y}_{i}-{y}_{i}^{{\prime\:}})}^{2}+{({z}_{i}-{z}_{i}^{{\prime\:}})}^{2}}$$

Here, $\:\varDelta\:d$ consists of two components, i.e., the lattice distortion of the host elements itself, $\:\varDelta\:{d}_{h}$, arising from the mismatch of atomic sizes and chemical affinity among Nb, Ti and Zr, and the lattice distortion produced by the interstitial solutes on the host atoms, $\:\varDelta\:{d}_{s}$. ($\:{x}_{i}$, $\:{y}_{i}$, $\:{z}_{i}$) and ($\:{x}_{i}^{{\prime\:}}$, $\:{y}_{i}^{{\prime\:}}$, $\:{z}_{i}^{{\prime\:}}$) are the reduced coordinates of the unrelaxed and relaxed positions of atom $\:i\:$(1nn Nb, Ti or Zr), respectively. N denotes the number of 1nn Nb, Ti, or Zr atoms around the interstitial solutes. For each type of solute (i.e., O, C, or N), a total of 54 independent solute-interstice local conformational environments were used to count the lattice distortions caused by solutes on their adjacent host Nb, Ti, or Zr atoms.

Elastic strain energy calculations

The estimated elastic strain energy is a key component of the overall free energy. In the single-solute model, a single interstitial solute was placed in either the O-site or T-site of a cubic bcc Nb lattice containing 8192 atoms. For the multi-solute model, two solutes were introduced at the O-site or T-site on each side of the target Nb atom, respectively. The example of placing two solutes at the O-site is illustrated in Fig. 6a (i.e., along the direction of the short side of the flat O-site). Energy minimizations were implemented using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) ³⁷. To characterize the variations in the size of interstitial solutes, a simplified hybrid potential framework was employed. The interactions among metal Nb atoms are described by the embedded-atom method (EAM) potential developed by Zhou et al³⁸. Additionally, the Lennard-Jones (L-J) potential³⁹ was used to account for the trends in solute size by correcting for the equilibrium distance between the interstitial solute and Nb. The interatomic potential between the interstitial solute and Nb takes the form:

$$\:{E}_{s-Nb}^{LJ}\left({r}_{s-Nb}\right)=4{{\epsilon\:}}_{s-Nb}\left[{\left(\frac{{\sigma\:}_{s-Nb}}{{r}_{s-Nb}}\right)}^{12}-{\left(\frac{{\sigma\:}_{s-Nb}}{{r}_{s-Nb}}\right)}^{6}\right],{r}_{s-Nb}<{r}_{c}$$

where $\:{\epsilon\:}_{s-Nb}\:$and $\:{\sigma\:}_{s-Nb}$ are the attractive well depth and the diameter for the L-J potential describing the interactions between species$\:\:s$ ($\:s$ denotes the interstitial solute) and Nb. In our work, the L-J potential parameter $\:{\sigma\:}_{s-Nb}$ range from 1.7 to 2.6 Å, corresponding to the r/R range with 0.2 ~ 0.81 (see Fig. 2a and Fig. 6b). $\:{\epsilon\:}_{s-Nb}$ and L-J potential cutoff $\:{r}_{c}$ was set to 0.05 eV and 12 Å, respectively. The total energy was calculated using the conjugate gradient (CG) algorithm⁴⁰.

Data Availability

Data generated or analyzed during this study are included in this article.

Code Availability

All related codes are available from the corresponding authors on request.

Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant No. 52231001). J.D. and E.M. are indebted to Xi’an Jiaotong University for hosting their work at the Center for Alloy Innovation and Design (CAID). J.D. also acknowledges support from the National Natural Science Foundation of China (Grant No. 12004294), the National Youth Talents Program and the HPC platform of Xi’an Jiaotong University.

Author Contributions

J.D., and E.M. conceived and supervised the research. B.Z. performed the simulations and analyzed the results. All authors contributed to the discussions and writing of the manuscript.

Competing Interests

The authors declare no competing interests.

Nayak, S. K. et al. Insight into point defects and impurities in titanium from first principles. npj Comput. Mater. 4, 11 (2018).
Reynolds, C., Pollock, T. M. & Van der Ven, A. Solute-solute interactions in dilute Nb-X-O alloys from first principles. Acta Mater. 266, 119621 (2024).
Zhang, J. & Han, W.-Z. Oxygen solutes induced anomalous hardening, toughening and embrittlement in body-centered cubic vanadium. Acta Mater. 196, 122-132 (2020).
Song, Y. et al. Dynamic Homogenization of Internal Strain in Multi-Principal Element Alloy via High-Concentration Doping of Oxygen with Large Mobility. Small Methods 8, 2300871 (2024).
Ventelon, L. et al. Dislocation core reconstruction induced by carbon segregation in bcc iron. Phys. Rev. B 91, 220102(R) (2015).
Wang, Y., Wang, X., Li, Q., Xu, B. & Liu, W. Atomistic simulations of carbon effect on kink-pair energetics of bcc iron screw dislocations. J. Mater. Sci. 54, 10728-10736 (2019).
Lüthi, B., Ventelon, L., Rodney, D. & Willaime, F. Attractive interaction between interstitial solutes and screw dislocations in bcc iron from first principles. Comput. Mater. Sci 148, 21-26 (2018).
Ulitchny, M. G. & Gibala, R. The effects of interstitial solute additions on the mechanical properties of niobium and tantalum single crystals. J. less-common. met. 33, 105-116 (1973).
Ikeda, Y., Tanaka, I., Neugebauer, J. & Körmann, F. Impact of interstitial C on phase stability and stacking-fault energy of the CrMnFeCoNi high-entropy alloy. Phys. Rev. Mater. 3, 113603 (2019).
Moore, C. M. et al. Hydrogen accommodation in the TiZrNbHfTa high entropy alloy. Acta Mater. 229, 117832 (2022).
Wang, Z. et al. Solving oxygen embrittlement of refractory high-entropy alloy via grain boundary engineering. Mater. Today 54, 83-89 (2022).
Lei, Z. et al. Enhanced strength and ductility in a high-entropy alloy via ordered oxygen complexes. Nature 563, 546-550 (2018).
Liu, C. et al. Massive interstitial solid solution alloys achieve near-theoretical strength. Nat. Commun. 13, 1102 (2022).
Barouh, C., Schuler, T., Fu, C.-C. & Nastar, M. Interaction between vacancies and interstitial solutes (C, N, and O) in α-Fe: From electronic structure to thermodynamics. Phys. Rev. B 90, 054112 (2014).
Williamson, G. K. & Smallman, R. E. X-ray evidence for the interstitial position of carbon in α-iron. Acta Crystallogr. 6, 361-362 (1953).
Ahlawat, S., Srinivasu, K., Biswas, A. & Choudhury, N. First-principle investigation of electronic structures and interactions of foreign interstitial atoms (C, N, B, O) and intrinsic point defects in body- and face-centered cubic iron lattice: A comparative analysis. Comput. Mater. Sci 170, 109167 (2019).
Liu, C. et al. Direct Observation of Oxygen Atoms Taking Tetrahedral Interstitial Sites in Medium-Entropy Body-Centered-Cubic Solutions. Adv. Mater. 35, 2209941 (2023).
Astafurov, S. V. et al. The effect of solid-solution temperature on phase composition, tensile characteristics and fracture mechanism of V-containing CrMn-steels with high interstitial content C+N>1 mass. %. Mater. Sci. Eng. A 770, 138534 (2020).
Furuhara, T. et al. Sublattice alloy design of high-strength steels: Application of clustering and nanoscale precipitation of interstitial and substitutional solutes. Scr. Mater. 223, 115063 (2023).
Zhang, H. et al. Making a low-cost duplex titanium alloy ultra-strong and ductile via interstitial solutes. Acta Mater. 241, 118411 (2022).
Prasad, K. et al. Experimental evidence for segregation of interstitial impurities to defects in a near α titanium alloy during dynamic strain aging using energy filtered transmission electron microscopy. Mater. Sci. Eng. A 638, 90-96 (2015).
Jiao, M. et al. Manipulating the ordered oxygen complexes to achieve high strength and ductility in medium-entropy alloys. Nat. Commun. 14, 806 (2023).
Gottstein, G. Physical foundations of materials science. Springer (2004).
Sundell, P. & Wahnström, G. Self-trapping and diffusion of hydrogen in Nb and Ta from first principles. Phys. Rev. B 70, (2004).
Leveau, T., Ventelon, L. & Clouet, E. Interaction of C, N and O interstitial solute atoms with screw dislocations in HfNbTaTiZr high entropy alloy. Acta Mater. 275, 120062 (2024).
Zhang, B., Ding, J. & Ma, E. Chemical short-range order in body-centered-cubic TiZrHfNb high-entropy alloys. Appl. Phys. Lett. 119, 201908 (2021).
Wang, X. G. et al. Ultrabroad distribution of multiple anelasticities in O-doped refractory multiprincipal element alloys. Acta Mater. 271, 119911 (2024).
Xun, K. et al. Local chemical inhomogeneities in TiZrNb-based refractory high-entropy alloys. J. Mater. Sci. Technol. 135, 221-230 (2023).
Su, Z. et al. Enhancing the radiation tolerance of high-entropy alloys via solute-promoted chemical heterogeneities. Acta Mater. 245, 118662 (2023).
Zhang, Z. et al. Effect of local chemical order on the irradiation-induced defect evolution in CrCoNi medium-entropy alloy. Proc. Natl. Acad. Sci. U. S. A. 120, e2218673120 (2023).
Zhang, B. et al. Minimizing the diffusivity difference between vacancies and interstitials in multi-principal element alloys. Proc. Natl. Acad. Sci. U. S. A. 121, e2314248121 (2024).
Zunger, A., Wei, S.-H., Ferreira, L. & Bernard, J. E. Special quasirandom structures. Phys. Rev. Lett. 65, 353 (1990).
Van De Walle, A., Asta, M. & Ceder, G. The alloy theoretic automated toolkit: A user guide. Calphad 26, 539-553 (2002).
Kresse, G. & Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 15-50 (1996).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953 (1994).
Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. J. Comput. Phys. 117, 1-19 (1995).
Zhou, X. W. et al. Atomic scale structure of sputtered metal multilayers. Acta Mater. 49, 4005-4015 (2001).
Hansen, J.-P. & Verlet, L. Phase Transitions of the Lennard-Jones System. Phys. Rev. 184, 151-161 (1969).
Hestenes, M. R. & Stiefel, E. Methods of conjugate gradients for solving linear systems. NBS Washington, DC (1952).

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Tetrahedral versus Octahedral Interstitials Sites for Oxygen Solutes in NbTiZr Medium-Entropy Alloy

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Abstract

Figures

Introduction

Results

The size effect of interstitial solutes

Interstitial residing in O-site versus T-site at dilute concentrations

Local atomic environment around interstitial solutes

The case of highly concentrated interstitial solutes

DISCUSSIONS

CONCLUSIONS

Methods

Density Functional Calculations for Single-Solute NbTiZr System

Density Functional Calculations for Multi-Solute NbTiZr System

Density Functional Calculations for Oxygen-Nb System

Local lattice distortion calculations

Elastic strain energy calculations

Declarations

References

Additional Declarations

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