Impact of Vacancies on the Mechanical Properties of

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Impact of Vacancies on the Mechanical Properties of Ultraincompressible, Hard Rhenium Subnitrides: Re2N and Re3N Aria Mansouri Tehrani and Jakoah Brgoch* Department of Chemistry, University of Houston, Houston, Texas 77204, United States ABSTRACT: The development of ultraincompressible, superhard materials has recently focused on systems containing heavy transition metals and light main group elements. The combination of these atom types provides the high valence electron density electron and directional covalent bonding necessary to achieve the desired mechanical response. Rhenium subnitrides, Re2N and Re3N, are a perfect example of satisfying these criteria yielding high hardness (≈15 to 20 GPa) and an extreme bulk modulus (≈400 GPa). However, the volatility of nitrogen at the high temperatures required to synthesize these phases makes them prone to the formation of anion vacancies, which may alter the electronic structure and adversely affect the mechanical properties. Using first principle calculations, the possibility of vacancy formation in Re2N1−x (x = 0−0.25) and Re3N1−y (y = 0−0.25) is predicted as a function of applied pressure, while the effects of vacancies on the electronic structure and mechanical properties are investigated. These results suggest spontaneous vacancy formation occurs in Re2N1−x at relatively low pressures (20 GPa). This is in accordance with experiment where Re2N is generally synthesized at higher pressure than Re3N signifying the relatively energetically unfavorable nature of Re2N compared to Re3N.12 The synthesis of Re2N and Re3N has also been carried out using high-pressure solid-state metathesis reactions. At low pressure (≈5 GPa) Re metal is obtained over the desired subnitride product, whereas applying higher pressure yields nitrogen-rich phases.35 This highlights that pressure is a useful parameter for controlling composition in transition metal nitrides. To account for the occurrence of defects in the formation enthalpy calculations, supercells with different concentrations of nitrogen vacancies were modeled at a range of applied pressures. At low pressure (1 to 5 GPa) the calculations indicate the formation enthalpy of Re2N1−x decreases, i.e., becomes more energetically favorable, with an increasing defect

(4)

p

3Re + (1 − y)(γ − N) → Re3N1 − y

(5)

To first determine the possibility of forming vacancies in the respective crystal structures, Ev can be determined by constructing large supercells and removing a single nitrogen atom. The large supercells of Re2N (4 × 3 × 1; 71 atoms) and Re3N (5 × 5 × 1; 99 atoms) are necessary to limit vacancy− vacancy interactions. Table 1 shows the variation of Ev for Re2N Table 1. Vacancy Formation Energy (Ev) of Re2N and Re3N at Different Pressuresa pressure (GPa)

0

5

10

25

Ev,Re2N (eV/vacancy)

−0.504

−0.096

0.234

1.004

Ev,Re3N (eV/vacancy)

0.825

1.267

1.612

2.388

The results are for 4 × 3 × 1 (71 atoms) and 5 × 5 × 1 (99 atoms) supercells, respectively.

a

and Re3N at a range of pressures. The negative Ev determined for Re2N at low pressures (0 to 5 GPa) indicates the spontaneous formation of vacancies making a low pressure synthesis of the phase at the exact stoichiometry unlikely.33 There is also a striking difference between Re2N and Re3N with the latter having a positive vacancy formation energy at 0 GPa. As the pressure increases from 0 to 25 GPa, however, the vacancy formation energy of both phases increases suggesting a 2544

DOI: 10.1021/acs.chemmater.6b04408 Chem. Mater. 2017, 29, 2542−2549

Article

Chemistry of Materials

located above 300 cm−1 stem from the lighter N atoms with these vibrational modes well separated at higher energy, in agreement with previous calculations.37 Interestingly, the increasing vacancy concentration widens the gap between the low energy vibrational modes and the higher energy Re modes. This widening is caused by a shift in the high-frequency nitrogen-based phonon modes, which likely arises from an increase in anharmonicity due to the presence of nitrogen vacancies. Nonetheless, the dynamic stability of these defect structures supports the potential phase formation even with a high vacancy concentration. Crystal Chemistry and Electronic Structure. Vacancies in these subnitrides will also undoubtedly affect the crystal structure and subsequently the electronic structure. The relaxed (equilibrium), normalized unit cell volumes of the crystal structures are plotted as a function of vacancy concentration, illustrated in Figure 4. Overall, the calculated lattice parameters

concentration, as plotted in Figure 2a. This is in accordance with the negative vacancy formation energies calculated. The optimal vacancy concentration is achieved when x = 0.125. Increasing the vacancy concentration further leads to a precipitous rise in the formation enthalpy favoring decomposition to the starting materials at high x. Therefore, these calculations suggest ≈12.5% anion vacancies, or a composition of Re2N0.875, is the most energetically favorable composition ≤5 GPa. Increasing the applied pressure above 5 GPa shows that anion vacancies are no longer enthalpically favorable in this crystal structure and that a nearly stoichiometric Re2N is expected. The equilibrium vacancy concentration can be estimated using the vacancy formation energy values and an Arrhenius-type analysis.36 At pressures as high as 10 GPa and room temperature this compound is predicted to contain ≈0.01% of nitrogen vacancies. The trends are significantly different for Re3N1−y. The formation energy of this phase is shown to be favorable when reacting elemental starting materials even at ambient pressure. Increasing the applied pressure also shows a stabilization of the enthalpy with the products heavily favored at high pressure. Conducting similar supercell calculations on Re3N1−y shows a nearly linear increase in formation energy implying this phase is more likely to form a stoichiometric subnitride. In conjunction with the positive vacancy formation energy calculated for Re3N proves vacancies do not form spontaneously and that their formation must overcome an energy barrier to form. At ambient pressure and temperature Re3N is predicted to have an equilibrium vacancy concentration of 1.11 × 10−12%.36 In practice, however, the synthesis route will result in higher percentages of vacancies that are not necessarily at thermal equilibrium. Beyond the formation enthalpy of these rhenium subnitrides, their dynamic stability at different nitrogen vacancy concentrations is also considered. As illustrated in Figure 3, the disordered crystal structures are all dynamically stable up to the 25% vacancy concentration. The lower energy phonon modes in both crystal structures are primarily located between 0 and 300 cm−1 and dominated by the Re modes. The phonon modes

Figure 4. DFT optimized (normalized) volume plotted as a function of vacancy concentration for (a) Re2N1−x (x = 0, 0.042, 0.0625, 0.083, 0.125, 0.167, 0.25) and (b) Re3N1−y (y = 0, 0.04, 0.0625, 0.083, 0.111, 0.25).

of the stoichiometric compound (x = y = 0) show excellent agreement with the experimentally reported values differing by 0. Further, the Born stability criteria can be simplified following symmetry considerations with the hexagonal crystal system (mechanically) stable when39 (i) C11 > C12, (ii) 2C213 < C33(C11 + C12), and (iii) C44 > 0. 6

E = E0 +

1 V ∑ Cij ϵiϵj + O(ϵ3) 2 i=j=1

Figure 7. Calculated (a) bulk modulus, (b) shear modulus, and (c) predicted hardness42 plotted as a function of vacancy concentration for Re2N1−x. The calculated (d) bulk modulus, (e) shear modulus, and (f) predicted hardness42 plotted as a function of vacancy concentration for Re3N1−y. The dashed lines are provided as a guide for the eye.

(7)

For these calculations, pseudocubic 3 × 3 × 1 supercells were developed for both subnitrides. The effect of three vacancy concentrations was considered with Re2N1−x modeled as the stoichiometric crystal structure (x = 0) as well as a supercell containing 5.5% vacancies (Re2N0.945) and 11.1% vacancies (Re2N0.889), whereas Re3N1−y supercells contained 11.1% vacancies (Re3N0.889) and 22.2% vacancies (Re2N0.778). These models allow a comparison of the mechanical response as the vacancy concentration changes. The elastic constants of the stoichiometric and vacant structures from these calculations are listed in Table 2. When analyzing the elastic constants it is necessary to consider that vacancies perturb the local symmetry of the crystal structure generally causing a reduction in the symmetry. This is important when trying to analyze the structure’s mechanical stability using the simplified criteria as

that these materials are ultraincompressible with a B ≈ 400 GPa, in agreement with previous reports.13 Interestingly, the inclusion of vacancies in Re2N1−x and Re3N1−y both shows a linear deterioration of B decreasing by approximately 7.95 GPa (x = 0.125) and 6.08 GPa (y = 0.111). The shear modulus shows a more substantial change with the introduction of anion vacancies. For comparison, B decreases approximately 2.0% and 1.5% for Re2N0.875 and Re3N0.899, respectively, whereas G decreases by 8.5% (Re2N0.875) and 12.5% (Re3N0.899). This differences arises because B is primarily correlated to the valence electron density, which is mostly governed by the high valence density of Re, whereas G is related to the chemical 2547

DOI: 10.1021/acs.chemmater.6b04408 Chem. Mater. 2017, 29, 2542−2549

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Chemistry of Materials

higher ultimate strength and strain along the ⟨0001⟩ direction compared to the ⟨1̅21̅0⟩ direction emphasizing their highly anisotropic chemical bonding. The higher tensile strength along the ⟨0001⟩ direction originates from the presence of strong, covalently bonded Re−N layers. The addition of vacancies in both crystal structures leads to a discernible drop in the ultimate strength of Re2N1−x by as much as 30% (x = 0.25). Interestingly, Figure 8a shows the inclusion of vacancies does not influence tensile strength of Re2N1−x along the ⟨12̅ 10̅ ⟩ direction, which is likely due to the negligible changes of the Re−N in-plane bonding. Including vacancies appears to affect the mechanical properties of Re3N1−y more substantially, especially along the ⟨0001⟩ direction, which shows a decrease of 26% in the ultimate strength when y = 0.25. Because Re3N contains two independent rhenium sites, a large distortion of bond length occurs upon the creation of vacancy as opposed to negligible changes in the ⟨1̅21̅0⟩ strain of Re2N1−x. This is also supported based on the −COHP curves. The changes along the c-direction are nearly identical because the shear modulus and tensile strength are to a great extent due to the Re−N bonds, which are connected to stacked layers along the ⟨0001⟩ direction. Thus, minimizing the vacancies are necessary to maximize the mechanical response of these highly anisotropic transition metal subnitrides.

bonding. Considering the anion vacancies modeled here should only cause a minimal change in valence electron density, the small change in bulk modulus can be expected. The relative high shear strength of these subnitrides originates from the strong Re−N covalent bond, which is destroyed by the creation of anion vacancies. This is clearly represented by the 30% decrease in covalent bonding character as indicated by the −ICOHP analysis stemming from the antibonding nature of the Re−N bond in Re3N0.889. The elastic constants also allow a prediction of the hardness (Hv) for these materials. Despite the difficulties associated with the assessment of hardness, fitting functions based on the calculated B and G have proved capable of predicting intrinsic hardness of a wide range of materials.42 The model employs B and G following the parametrized relationship presented in eq 8 Hv = 2(κ 2G)0.585 − 3

(8)

where κ is Pugh’s ratio (G/B). It is evident from eq 8 that G provides a substantial contribution to the calculated Hv. As shown in Figure 7c and Figure 7f, an overall depletion of the hardness occurs for both subnitrides with the introduction of nitrogen vacancies. This can be rationalized through the general vacancy softening due to a decrease in the number of covalent bonds present in the crystal structure as well as the occupation of antibonding orbitals in the case of Re3N1−y. Moreover, the mechanical response of defective subnitride crystal structures can be examined by generating stress−strain curves for different anion vacancy concentrations, plotted in Figure 8. Two separate stress−strain curves must be analyzed

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CONCLUSION The development of transition metal nitrides for application as mechanically robust materials often requires high temperature and high pressure synthetic routes. Due to the high volatility of nitrogen under these conditions, these phases are prone to the formation of nitrogen vacancies if the reaction conditions are not optimized. The presence this atomic disorder in the crystal structure is anticipated to affect the physical properties, most notably the elastic moduli and hardness. Thus, the research here used DFT-based calculations to examine the changes in crystal chemistry and mechanical properties for a pair of rhenium subnitrides, Re2N and Re3N, when vacancies are present. Modeling the reaction conditions of forming the two subnitrides starting from Re metal and nitrogen under high pressure predicts that if the pressure is between 1 and 5 GPa then Re2N is enthalpically favorable, but it is prone to nitrogen vacancies with an energetic minimum occurring for Re2N0.875. Increasing the pressure above ≈5 GPa is sufficient to decrease the concentration of anion vacancies and that a nearly stoichiometric product is favored. Further, modeling the changes in the mechanical properties reveals that the presence of vacancies in the two subnitride crystal structures negatively impacts the elastic moduli and hardness, all of which decrease due predominantly to changes in the Re−N chemical bonding. These results highlight not only that first-principles calculations can be used to identify potential reaction conditions in the preparation of these materials but also that understanding the electronic structure is essential to know how vacancies will affect the chemical bonding and ultimately the mechanical response of these ultraincompressible and hard materials.

Figure 8. Depiction of stress−strain curve for (a) Re2N1−x at ⟨0001⟩ (the left panel) and ⟨1̅21̅0⟩ (the right panel) directions and (b) for Re3N1−y for the same directions.

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for these highly anisotropic structures with the left panels of Figure 8 corresponding to strain applied in c direction or ⟨0001⟩ (out-of plane), whereas the right panels correspond to distortion along the a direction or ⟨1̅21̅0⟩ (in-plane). The ultimate strength of the structure is defined at the point where the stress drops. Clearly, both compounds show significantly

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Jakoah Brgoch: 0000-0002-1406-1352 2548

DOI: 10.1021/acs.chemmater.6b04408 Chem. Mater. 2017, 29, 2542−2549

Article

Chemistry of Materials Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the generous financial support provided by the National Science Foundation through No. NSF-CMMI 15-62142, the R. A. Welch Foundation through the TcSUH Robert A. Welch Professorship in High Temperature Superconducting (HTSg) and Chemical Materials (E-0001), and the donors of the American Chemical Society Petroleum Research Fund (55625-DNI10) for supporting this research. Support for this work was provided by resources of the uHPC cluster managed by the University of Houston and acquired through NSF Award Number 15-31814. This research also used the Maxwell and Opuntia Clusters operated by the University of Houston and the Center for Advanced Computing and Data Systems.

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DOI: 10.1021/acs.chemmater.6b04408 Chem. Mater. 2017, 29, 2542−2549