Unexpected structural softening of interstitial boron solid solution WB_3+x Free

Boron-rich transition-metal borides (TMB_n) offer a cost-effective and versatile alternative to traditional superhard materials like diamond and cubic BN that require high pressure and high temperature synthesis conditions. Recent synthesis of ReB₂ at ambient pressure has ignited great interest in this class of metallic ultra-incompressible materials, leading to reports of many similar systems such as OsB₂, ReB₂, RuB₂, CrB₄, FeB₄, and WB₄.^1–10 Among the TMB_n family of materials, WB₄ exhibits an ultra high asymptotic (i.e., load independent) hardness of about 30 GPa,^3,8,9 which is attributed to its high content of boron that forms a three-dimensional (3D) covalent bonding network that enhances its resistance to shear deformation. The high content of inexpensive boron also reduces the material weight and cost, making WB₄ a promising ultrahard material for wide-ranging applications.

However, the structural assignment of WB₄ was called into question by recent theoretical studies^11,12 that reveal the structural instability of the assigned WB₄ structure due to phonon softening. An alternative WB₃ (P6₃/mmc) structural model was proposed based on the calculated formation energy and a comparison with X-ray diffraction (XRD) data.¹² Further calculations indicate, however, that WB₃ has a negative pressure dependence of the normalized lattice c/a ratio,¹³ while experiments show that it increases with pressure.¹⁰ Moreover, Vickers indentation measurement indicates that the synthesized WB₄ has an asymptotic hardness (32, 28, and 28 GPa)^3,8,9 higher than or similar to that of ReB₂ (30, 27, and 18 GPa),^2–4 while calculations show that the indentation strength of WB₃ (22 GPa) is lower than that of ReB₂ (28 GPa).¹³ A new WB₄ structure in P6₃/mmc–2u symmetry was recently predicted which has the lowest total energy for all stable WB₄ structures,¹⁴ but its formation energy is slightly higher than the combination of WB₃ and elementary α-B,¹⁵ thus, a tendency toward decomposition is expected. Very recently, based on a first-principles structural search and analysis of high-resolution TEM (HRTEM) images, an interstitial boron solid solution model WB_3+x (x < 0.5) was proposed, where boron atoms occupy interstitial positions in the WB₃ (P6₃/mmc) lattice.¹⁵ It is proposed that the interstitial boron will connect neighboring boron layers in WB₃ to form a 3D boron network and thus enhance the strength (and hardness) of WB_3+x. Such a solution strengthening effect has been experimentally observed in WB_n compounds with W substituted by Ta, Mn, and Cr elements.⁹ The WB_3+x model appears to explain all the XRD, HRTEM image, pressure dependence of the normalized lattice c/a ratio, and hardness experimental results of the synthesized WB₄.¹⁵ However, the key assumption that the interstitial doped boron will enhance the structural hardness or indentation strength of WB_3+x solid solution has not been verified. An accurate assessment of the (Vickers) indentation strength of the proposed WB_3+x structure in comparison with that of WB₃ would offer key insights into the structure of the synthesized (nominal) WB₄, which is essential to further study of this family of materials.

In this letter, we report first-principles calculations of the stress-strain relations of WB_3+x, in comparison with that of WB₃, in various shear deformation directions under the compressive pressure beneath an indenter^16–20 [see the supplementary material for details on computational methods²¹]. The lowest shear peak stress under an indenter (i.e., ideal indentation strength) gives the stress at which a perfect crystal becomes mechanically unstable in indentation. It provides a more accurate description for material strength under indentation hardness tests than pure ideal shear strength that is calculated neglecting the compressive pressure.^22–28 While material strength and hardness are controlled by many factors, such as defect nucleation and mobility, ideal (indentation) strength calculations can predict incipient plasticity in a crystal²⁹ and determine the lowest shear stress needed to destabilize a perfect crystal, thus setting an upper bound for material strength. Measured strength of high quality samples can actually approach the calculated ideal strength.³⁰ This makes ideal (indentation) strength a benchmark quantity in assessing material strength and hardness; it is especially useful in a comparative study of different materials.

We first assess the viability of the proposed WB_3+x structures.¹⁵ In Fig. 1(a), we plot the unit cell of WB₃ with four possible interstitial boron positions. On each W-layer, there exist two nearly equivalent interstitial boron positions. Various structures of WB_3.25 − i (i = 1,…,6) and WB_3.5 − j (j = 1, 2, 3) are obtained by doping one boron atom on each W-layer at different interstitial positions (see Fig. S2 in the supplementary material²¹). In Fig. 1(c), we show the average XRD spectra of WB_3.25 and WB_3.5 in comparison with those of WB₃ and WB₄. As the interstitial boron density increases, the XRD peak near 2θ = 34° splits due to the displacement of the tungsten atoms caused by the interstitial boron atoms. This peak splitting is not observed in the experimental XRD for the synthesized WB₄,^8–10 which casts doubt about the structural assignment. Moreover, the calculated phonon spectra show that imaginary phonon modes appear in WB_3.5, indicating dynamical instability of the WB_3.5 structures (see Fig. S3 in the supplementary material²¹).

We now evaluate the mechanical properties of the WB_3+x systems. The calculated stress-strain curves in various shear directions under pure and (Vickers) indentation shear deformation for WB₃, WB_3.25 − 1, which has the lowest energy among WB_3.25 structures, and WB₄ are given in Fig. 2. From these results, it is clear that the lowest peak stresses (around 10 GPa) of WB_3.25 − 1 in both pure and (Vickers) indentation shear deformation are considerably lower than those (37 GPa and 22 GPa) of WB₃. Instead of the expected solution strengthening, interstitial boron atoms cause an unexpected large structural softening in the WB_3+x structure in both the stress peaks and the initial slopes of the stress-stain curves that are related to its elastic constants. In Table I, we give the calculated elastic constants of WB₃ and WB_3.25 − 1. The main elastic constants, bulk, and shear modulus of WB_3.25 − 1 are all smaller than those of WB₃. The lowest (Vickers) indentation shear stress peak of WB₃ is 22 GPa, which is lower than that of ReB₂ (28 GPa).¹⁹ While the Vickers micro-indentation experiments show that the synthesized WB₄ has a measured load-independent asymptotic hardness (32, 28, and 28 GPa)^3,8,9 higher than that of ReB₂ (30, 27, and 18 GPa).^2–4 It was proposed that the interstitial boron atoms would enhance the hardness of WB_3+x by introducing three-center bonding between neighboring B-layers to form a robust 3D boron network in WB_3+x.¹⁵ Instead, the interstitial boron atoms reduce the indentation strength (to around 10 GPa for WB_3.25) of WB_3+x considerably. Our results indicate that the proposed WB_3+x structure is incompatible with a key property (i.e., hardness) of the synthesized (nominal) WB₄.^3,8,9

To understand the unexpected softening of the elastic constants and strength of WB_3+x solid solution, we first examine the total and partial density of states (DOS) of WB₃ and WB_3.25 − 1, as shown in Fig. 3. The strong interactions among B_p states between B atoms and hybridization among W_d and B_p states between W and B atoms form strong B-B and W-B covalent bonds in WB₃, which can also be seen in Fig. 4(a) where we plot the charge distributions on different crystalline planes in WB₃. Such strong orbital interactions split the electronic states into the bonding and anti-bonding states separated by a wide and deep pseudogap, where the bonding states are almost fully occupied by the electrons in WB₃, producing a very stable structure [see Figs. 3(a)–3(c)]. While in WB_3.25 − 1, the interstitial boron atoms not only weaken the hybridization between W and B atoms by moving the W atoms further away from the B atoms but also reduce the interactions among the B atoms in the hexagonal B-layer. This can be seen in Fig. 4(b) where the charge distribution between B₂–B₃ and B₅–B₆ on the (001) plane are greatly reduced, and so is the charge directly between B₁-W₁ on the (010) plane of WB_3.25 − 1. Such reduced interactions among B_p–B_p and W_d–B_p orbital weaken the stability of the WB_3.25 − 1 structure, resulting in a narrow and shallow pseudogap with the Fermi energy moving away from its bottom [see Figs. 3(d)–3(f)]. Such correlations among pseudogap, electron band filling (shift of Fermi level), and structural stability have been found in other transition metal borides.³¹ 

In Fig. 5, we plot the calculated electron localization function (ELF) that gives a local measurement of electron paring³² on the $( 1 1 ¯ 0 )$ plane passing through B₀ atom together with its structural snapshot for WB_3.25 − 1 at equilibrium (ϵ = 0) and ϵ = 0.075 and 0.09 under Vickers indentation shear in $( 1 1 ¯ 0 ) [ 001 ]$ direction. The results in Fig. 5(a) show that the interstitial boron atom does form three-center bonding among ΔB₀B₁B₄ and ΔB₀B₇B₈ connecting the neighboring B-layers into a 3D network in WB_3.25 − 1. However, as the boron three-center bonding can easily change into other bonding, for instance, a new two-center bonding,²⁸ such 3D boron network is not robust under structural deformation. The results in Figs. 5(b) and 5(c) clearly show the weakening and eventual breaking up of the three-center bonding among ΔB₀B₁B₄ and ΔB₀B₇B₈, giving rise to a low shear stress peak of about 10 GPa, under the (Vickers) indentation shear in the $( 1 1 ¯ 0 ) [ 001 ]$ direction [see Fig. 2(d)]. The same behavior was seen in other WB_3.25 structures (see Fig. S4 of the supplementary material²¹), where all the WB_3.25 − i (i = 2,…,6) exhibit (Vickers) indentation strength of about 10 GPa, and pseudogap narrowing and shallowing with the Fermi energy moving away from their bottoms.

In summary, our first-principles calculations show that the ideal indentation strength of the proposed solid solution WB_3+x (10 GPa) is considerably lower than that of ReB₂ (28 GPa), suggesting that the hardness of WB_3+x should be well below that of ReB₂. This result is in direct contradiction to the experimental findings that experimentally synthesized (nominal) WB₄ has a Vickers hardness higher than that of ReB₂. Moreover, the calculated XRD spectra of WB_3+x exhibit peak splitting near 2θ = 34° as boron solution content x increases, which is not observed in the XRD of the synthesized WB₄ compound. These contrasting results offer compelling evidence demonstrating that the proposed interstitial boron solid solution WB_3+x structure is incompatible with key properties of the synthesized WB₄ samples. The structural determination of the synthesized W–B compounds is thus reemerging as an open question that deserves further investigation, and the resolution of this issue will have an important impact on the further development of this promising family of ultrahard materials. Among all the structural models (including WB_3+x) proposed so far to explain the synthesized (nominal) WB₄ sample,^11–15 they are either (dynamically or thermodynamically) unstable or with key parameters, such as hardness, inconsistent with the experimental findings. One possible explanation might be that the real synthesized (nominal) WB₄ sample is a mixture of WB₃ and elemental boron clusters with nanometer sizes. Since boron does not scatter X-ray strongly, the XRD spectrum of the sample is determined mainly by WB₃, and this spectrum agrees well with the calculated results,¹² while elemental boron nano-clusters enhance the hardness of the synthesized sample over that of WB₃. This suggests the possibility that WB₄ may not actually exist in the synthesized sample. More experimental investigations are needed to clarify this issue. Our results also show that increasing boron content in TMB_n compounds by (interstitial) doping may not always enhance their hardness.

Work supported by NNSF of China (No. 11174200) at SJTU and DOE (DE-NA0001982) at UNLV.