Article pubs.acs.org/cm
Boron Suboxide and Boron Subphosphide Crystals: Hard Ceramics That Shear without Brittle Failure Qi An and William A. Goddard, III* Materials and Process Simulation Center, California Institute of Technology, Pasadena, California 91125, United States
Chem. Mater. 2015.27:2855-2860. Downloaded from pubs.acs.org by LA TROBE UNIV on 01/03/19. For personal use only.
S Supporting Information *
ABSTRACT: Boron suboxide (B6O), boron carbide (B4C), and related materials are superhard. However, they exhibit low fracture toughness, which limits their engineering applications. Here we show the shear deformation mechanism of B6O using density functional theory along the most plausible slip system (01̅11)/ . We discovered an unusual phenomenon in which the highly sheared system recovers its original crystal structure, which indicates the possibility of being sheared to a large strain without failure. We also found a similar structural recovery in boron subphosphide (B12P2) for shearing along the same slip system. In contrast, for components of B4C, we found brittle failure. These novel deformation mechanisms under high shear deformation conditions suggest that a key element to designing ductile hard materials is to couple the icosahedra via one- or two-atom chains that allow the system to shear by walking the intericosahedral bonds and chain bonds alternately to accommodate large shear without fracturing the icosahedra. identified γ boron phase (high pressure boron phase) along the (100)/⟨001⟩ slip system led to the discovery of three new boron phases. These phase transition paths in γ boron illustrate how changes in the bonding between the B12 icosahedra in various boron phases are related to material mechanical and electronic properties.17 Here we report investigations of the shear deformation of B6O along the amorphous slip system of (01̅11)/. We find that large shear deformations of single crystal B6O transform it without failure back to the original structure but with a different orientation. To examine the role of the chain structure in this novel deformation mechanism, we also examined the response of the similar structures of B12P2 and B12C3 for shear along the same slip system. Indeed, we find a similar structural recovery for B12P2, but not for B12C3, which instead exhibits brittle failure. We consider that these results on the deformation mechanisms provide hints on improving ductility of these hard materials.
1. INTRODUCTION Boron carbide (B4C), boron suboxide (B6O), boron subphosphide (B12P2), and related materials have been examined extensively with both experiment and theory because of their unique properties of low density, super hardness, high chemical inertness, and resistance to wear.1−10 B6O, with a Vickers hardness of 35 GPa,11 is considered as the third hardest material behind diamond and cubic boron nitride. It has also recently emerged as a promising high mobility p-type transparent conducting oxide with the direct band gap of ∼3.0 eV from theoretical calculations.12 However, these materials have not been commercialized because of their brittle behavior under pressure13−15 (and also the difficulty in synthesizing fully dense materials). To learn principles that might help the design of ductile superhard materials, it is essential to examine the bonding and structural transformations under shear deformations for these covalent solids of B6O, B4C, and related materials.16−20 Recent nanoindentation experiments18 observed amorphous shear band formation in B6O along the (011̅ 1)/ slip system [the rhombohedral representation is (001)r/⟨100⟩r, where r represents rhombohedra]. This amorphous band formation is similar to the amorphous band formation of B4C under pressures, which has been widely studied experimentally and theoretically19−23 because it is believed to be responsible for the brittle failure of B4C. Thus, we carried out quantum mechanics (QM) calculations to understand the atomistic mechanism of this amorphous band formation in B6O and other related materials. Another example of interesting mechanical properties in these systems was our observations17 that shearing the recent © 2015 American Chemical Society
2. METHODOLOGY All simulations used the VASP periodic code with plane wave basis sets.24−26 We used an energy cutoff of 600 eV in all the simulations since it gives excellent convergence on energy, force, stress, and geometries. The Monkhorst−Pack grid (8 × 8 × 8) in the k-space was used in the geometry optimization. Later on, a smaller grid (2 × 2 × 2) and larger supercell (2 × 2 × 2) were used in the shear deformations. We used the Received: December 19, 2014 Revised: April 1, 2015 Published: April 6, 2015 2855
DOI: 10.1021/cm5046918 Chem. Mater. 2015, 27, 2855−2860
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For B12P2, the two chain phosphorus atoms are bonded to each other in addition to each being bonded to boron atoms in three icosahedra. Thus, each P atom can be thought of as P+, transferring one electron to the B12 icosahedron, leading to 26 electrons involved in intracage bonding. This can be written as (B12)2−(P−P)2+. Although (B12)(CCC) is not the ground state structure for B4C1,20 (its energy is 1.09 eV/unit cell higher than (B11Cp)(CBC)), it is of interest to examine and compare with B6O and B12P2 to illustrate how the different chain structures with the same B12 icosahedra affect the deformation mechanism. Since each terminal C makes bonds to boron atoms in three icosahedra, the intra chain C−C bonds must be single bonds. Thus, the middle C in the linear C−C−C chain must transfer two of its electrons, becoming C2+. Again, each icosahedron has 26 electrons involved in intracage bonding, which satisfies Wade’s rule. This can be represented as (B12)2−(C−C2+-C). To validate the accuracy of the density functional theory (DFT) methods with Perdew−Burke−Ernzerhof (PBE) functional, we first examined the mechanical properties of the B6O. PBE gives cell parameters of a = 5.15 Å and α = 63.11° for the rhombohedra unit cell, consistent with experimental values of a = 5.39 Å and α = 63.05°.11 The predicted elastic moduli are shown in Table S1 of the Supporting Information. This leads to a calculated bulk modulus B = 232.0 GPa and shear modulus G = 210.9 GPa using the Voigt−Reuss−Hill approximation.31 These mechanical properties are consistent with previous experimental values of B = 222 GPa32 and G = 204 GPa,33 respectively. The calculated cell parameters for B12P2 and B12C3 are a = 5.25 Å, α = 69.58°, and a = 5.19 Å, α = 65.89°, respectively. The shear simulation of B6O is shown in Figure 1, panel d where “a” is along ⟨100⟩r direction and “c” is along ⟨001⟩r direction. Figure 2 shows the stress−strain relation of the three
projector-augmented wave pseudopotentials for PBE exchangecorrelation functional for GGA calculations.27 The energy error for the termination electronic SCF and the force criterion for the geometry optimization were set equal to 10−6 eV and 10−3 eV/Å, respectively. To examine the shear failure of the modified materials, we imposed the strain for a particular shear plane while allowing full structure relaxation of the atoms for the other five strain components.28 The residual stresses after relaxing the five other strain directions were less than 0.5 GPa. For the shear simulations, we used a 2 × 2 × 2 supercell containing 112 atoms. The Monkhorst−Pack grid (2 × 2 × 2) in the k-space was used for the shear deformations. One shear simulation with (3 × 3 × 3) k-point mesh was performed, and the consistent results obtained were compared with the (2 × 2 × 2), which indicate that (2 × 2 × 2) k-point mesh is suitable for the cell changed shear simulation.
3. RESULTS AND DISCUSSION We first describe the structural and bonding characters of B6O, B12P2, and B12C3. They all have similar structures to that of the α-rhombohedral phase of boron. They are composed of B12 icosahedra plus chains having one to three atoms that interconnect the icosahedra. Figure 1, panels a−c show the
Figure 1. B6O, B12P2, and B12C3 structures and shear simulation model. (a) The B6O structure. (b) B12P2 structure. (c) The B12C3 structure. (d) The shear deformation along the (01̅11)/ for B6O. The boron, oxygen, phosphorus, and carbon atoms are represented by the green, red, thistle, and sienna balls, respectively.
rhombohedral structures of B6O, B12P2, and B12C3, respectively. All bonds in the chain and the extra-polyhedral bonds of the B12 unit of these structures can be considered as two-center− two-electron (2c−2e) bonds. In B6O, each oxygen atom is bonded to three icosahedral boron atoms, but it is not bonded to the other chain O, leaving a lone pair along ⟨111⟩r direction and transferring one electron to the B12 icosahedron. This is compatible with the B12 icosahedron that requires two additional electrons to stabilize the icosahedral structure (26 electrons involved in intracage bonding), which satisfies Wade’s rule.29,30 Thus, B6O can be written as (B12)2−O+O+.
Figure 2. Stress−strain relations for B6O, B12P2, and B12C3 shearing along the (01̅11)/ slip system.
materials along the same slip system of (01̅11)/. The stress−strain curve for B6O shows structural changes at points E (0.565 strain) and H (1.16 strain). However, the new structure does not fail but rather transforms back to the original structure with a new orientation with the “c” axis along the [111]r direction at point E and [101]r direction at point H. The shear strength for B6O is 43.5 GPa. Note that this second 2856
DOI: 10.1021/cm5046918 Chem. Mater. 2015, 27, 2855−2860
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Figure 3. Calculated isosurfaces of the ELF and the structures of B6O shear along (011̅ 1)/. (a) Initial structure. (b) The strain of 0.299 corresponds to the maximum shear strength. B58−B88 bonds between icosahedra are stretched, point B in Figure 2. (c) The strain of 0.322 where the B58−B88 bonds starts to break, point C in Figure 2. (d) The strain of 0.539 before structural changes, point D in Figure 2. (e) The strain of 0.565 where new structure forms where c axis along [111]r direction, point E in Figure 2. (f) The strain of 1.067 where continuous shear reach a maximum, point F in Figure 2. (g) The strain of 1.13 (point g in Figure 2) where B88−O9 is stretched but the second structure has not changed. (h) The strain of 1.16 (point H in Figure 2) where the second structure changes, but c is now along the [101]r direction. The boron and oxygen atoms are represented by the green and red balls, respectively. The blue dashed lines indicate the future broken and formed bonds.
However, the bond does not break, as indicated by the ELF shown in Figure 3, panel b. In the plastic deformation region from strain 0.322 to 0.539, the B58−B88 bond stretches continuously until it breaks as shown in Figure 3, panels c and d. The B58−B88 bond distance increases from 2.38 to 2.88 Å in Figure 3, panel c, leading to a broken bond broken and loss of strength. As the strain increases to 0.565, Figure 3, panel e and Figure S1b show that the previous B51−O9 and B57−O10 bonds are broken, with instead the O9 and O10 atoms bonding to the B88 and B94 atoms and the newly formed B51−B57 bond. Thus, this structural transformation takes it back to the original crystal structure, but in a new orientation, where the c axis is along the [111]r direction. As we continue to shear the newly formed structure to the strain of 1.07, the B88−O9 is stretched to 1.58 Å as shown in Figure 3, panel f. It stretches continuously to 1.13 strain before the second structural transformation occurs, as shown in Figure 3, panel g and Figure S1c. Finally, the second structural transformation occurs at 1.16 strain (Figure 3h and Figure S1d) where the B88−O9 breaks, followed by B63−O9 and B88−B70 bond formation. Now the c axis is along the [101]r direction. We did not continue to apply shear in the same slip plane to the second newly formed structure because the stress to form this structure is higher than the first structural change. For B12P2, Figure 4 shows the structures and the ELF at various important strains. Figure 4, panel a shows the intact
structural change requires a much higher shear strength, which indicates that it might be hard to proceed experimentally. The shear from point E to F is along the (111)r plane, which is not along the easiest slip system, making the larger shear strength than the slip system of (01̅11)/. The stress−strain curve for B12P2 is similar and shows structural transformations at points D′ and E′. The shear strength for B12P2 is 44.1 GPa at 0.322 strain. The structure recovers at E′, which leads to a deformation mechanism similar to B6O. In contrast, the stress−strain relation for B12C3 shows failure at 0.364 strain (shear stress 50.6 GPa) in which the icosahedron decomposes. The maximum shear stress for B12C3 is 51.0 GPa at 0.348 strain. To understand these structural transformation mechanisms, we examined how the bonding changes as B6O is sheared. Figure 3 shows the structures and the isoelectron surface (at 0.85) from the electron localization function (ELF) analysis at various critical strains. Figure S1 also shows the local atomic structures and atomic labels at key steps to illustrate the bond breaking and reforming process as O atoms walk around the icosahedra. The ELF, which ranges from 0 to 1, enables an effective and reliable analysis of covalent bonding and lone pair formation.34,35 Figure 3, panel a shows the intact structure, where ELF indicates the lone pair on each O atom along the [111]r direction. As the strain increases to 0.322, the B58−B88 bond between icosahedra is stretched from 1.70 to 2.38 Å. 2857
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Figure 4. Calculated isosurfaces of the ELF and the structures of B12P2 shear along the (01̅11)/ slip system. (a) Intact structure. (b) The strain of 0.322 corresponds to the maximum shear strength where P−P bonds and B1−B1 bonds are stretched, point B′ in Figure 2. (c) The strain of 0.392 where the P−P bonds are totally broken leading to formation of a lone pair on the phosphorus, point C′ in Figure 2. (d) The strain of 0.416 where the B1−B1 bonds between icosahedra are broken, leading to the abrupt stress decreases, point D′ in Figure 2. (e) The strain of 0.669 where the new P···P and B2···B2 distances decrease to 3.0 and 2.7 Å, point E′ in Figure 2. (f) The strain of 0.695 (point F′ in Figure 2) where new structure forms. The “c” axis of the new formed structure is along the [1̅1̅1]r direction that is different from the original direction of [001]r. The boron and phosphorus atoms are represented by the green and thistle balls, respectively. The red dashed lines indicate the future formed bonds.
crystal structure. However, now the c axis is along the [1̅1̅1]r direction as shown in Figure 4, panel f. For (B12)(CCC), a shear deformation of 0.381 strain leads to breaking the B12 icosahedron because the middle C2+ in the C− C−C chain bonds to the cage, as shown in Figure S2 of the Supporting Information. Thus, our single crystal deformations do not exhibit the amorphous shear band formation along the (01̅11)/ slip system as we found for (B11Cp)(CBC) in a previous study.20 However, nanoindentation experiments18,19 on B4C and B6O did observe amorphous bands along the (01̅11) crystal plane. Thus, we assume that the materials in these experiments have grain boundaries, defects, or other imperfections that cause local deformations to accommodate accumulated local stresses and thereby initiate amorphous shear bands. To illustrate how defects might modify the deformation mechanism in B6O, we constructed two vacancy models by either removing only a single O or B atom in the 2 × 2 × 2 extended crystal structure. These vacancy structures were relaxed prior to shear deformation. The stress−strain relations for these two vacancy systems and the comparison with pure crystal are displayed in Figure. 5. The ideal shear strength is 43.2 GPa for the O vacancy, essentially the same as the perfect crystal, 43.5 GPa. For the O vacancy, the structure changes at a much lower strain of 0.369 compared with 0.565 for the perfect
structure where ELF shows the P−P single bond formed in the chain. As the strain increases to 0.322, the P−P bonds stretches from 2.25 to 2.47 Å, while the B1−B1 bonds between icosahedra stretch from 1.74 to 2.28 Å, as shown in Figure 4, panel b. As the strain increases to 0.392, the P−P bond is broken as shown in Figure 4, panel c. This leads to a lone pair on each P atom, which also has single bonds to boron atoms in three icosahedra, so that each P becomes neutral. Thus, each B12 icosahedron has only 24 electrons for intracage bonding rather than the Wade’s rule number of 26. With further shear, the B1−B1 bonds increase to 2.45 Å without breaking. Then at 0.416 strain, the stress dramatically drops as the B1−B1 bonds break, as shown in Figure 4, panel d. At this point, each P atom forms four bonds to three icosahedra, with each icosahedron forming 12 extra-icosahedra bonds. Thus, again each P is formally P+ so that it transfers one electron to a B12 icosahedron, leading again to 26 electrons in intracage bonding, which again satisfies Wade’s rule. As the strain increases continuously to 0.669, the P···P distance increases to 3.02, while the B2···B2 distance increases to 2.70 Å, as shown in Figure 4, panel e. Finally, at the strain of 0.695, a new P−P bond forms, and a B2−B2 bond forms between icosahedra, leading to the structural transition that returns the system to the original 2858
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shear modulus, respectively. However, the calculated B/G = 1.09 for B6O, 1.00 for B12P2, and 1.37 for B12C3, whereas single crystal B6O and B12P2 are ductile, while B12C3 is brittle. Thus, one must be cautious when applying the Pugh criterion to these hard ceramics. Experimentally, it is hard to synthesize defect free crystalline B6O to test these ideas. However, it might be possible to synthesize very thin films of B6O or B12P2 through atomic layer deposition (ALD), where proper termination of the surface with hydrogen atoms through low energy electron etching (LE4) or microwave hydrogen plasma might lead to thin films with similar bonding. Such a thin film might provide an ideal structure for studying deformation and failure mechanisms in these ceramics since it might more easily be analyzed after deformations. To illustrate this, we constructed a thin film model with two layers of icosahedra deposited in the nonperiodic [001]r direction with all broken bonds terminated by hydrogen atoms. This is shown in Figure 6, panel a, which is
Figure 5. Stress−strain relations for perfect crystal, O vacancy, and B vacancy of B6O shearing along the (01̅11)/ slip system.
crystal, at which point the structure recovers to itself, as shown in Figure S3 of the Supporting Information. The O vacancy leads to formation of a new bond (bond1 in Figure S3b) between icosahedra of the top two layers, which prevents the relative shift of the top two layers. Instead, the bottom two layers must shift with respect to each other so that the structure recovers to itself with the c axis in the new structure along the [102]r direction. Thus, we expect that the O vacancy need not lead to an amorphous band. For deformation with a B vacancy structure, the ideal shear strength is 41.5 GPa, 4.6% smaller than the perfect crystal. Here we observe breaking of the icosahedron at a shear strain of 1.007, as shown in Figure S4 of the Supporting Information. This indicates that a B vacancy in the B12 cage of B6O leads to failure of B12 icosahedron, which likely would lead to amorphous band formation. In contrast, a single O vacancy in the chain does not lead to this failure for our idealized system. Our deformation mechanism of perfect crystal indicates that twins can form, with the (001)r//(111)r structure shown in Figure S5 of the Supporting Information. This twinning formation mechanism is consistent with the experimental observation that twinning of B6O occurs along the (001)r plane.11 Since our defect-free B6O only forms twins or transforms back into itself, the amorphous band formation observed in B6O must be related to the defect structures, such as vacancies, or grain boundaries. For B12P2, we find similar structural changes that might lead to the (001)r//(11̅ 1̅ )r twin structure. This could also be tested experimentally. Our results show that the two-atom chain structures (either the P−P chain or two isolated O atoms) play key roles in the structural transformations, accommodating shear deformations without breaking the icosahedra. This indicates that a critical design element for ductile hard materials is to incorporate twoatom or one-atom chains rather than the three-atom chains in boron carbide. The three-atom chain does not do this because the central atom is very reactive, and as the system is sheared, the chain bends allowing the central atom to attack an icosahedral atom, which causes it to disintegrate and leads to amorphous shear failure.20 The Pugh’s criterion for metals36 indicates that materials with B/G > 1.75 tend to be ductile, while materials with smaller B/G tend to be brittle. Here, B and G are the bulk modulus and
Figure 6. Proposed B6O thin film structure (terminated with H atoms) and sheared along the (100)r direction. (a) The intact structure. (b) At 0.08 strain, where the B−B bonds between icosahedra are broken. (c) At 0.12 strain, where the B−O bonds between chain and icosahedra break and reform. (d) At 0.2 strain, where the structure recovers the original structure. The strain is calculated as △x/x along the (100)r direction. The boron, oxygen, and hydrogen atoms are represented by the green, red, and white balls, respectively.
a 2 × 2 supercell along periodic [100]r and [010]r directions. Then we sheared the top layer along the [100]r direction by fixing the atomic coordinate along the [100]r direction and relaxing the atoms along other two directions. The structural changes are shown in Figure 6. No icosahedra break during the shear process, and the structure is recovered after a strain of 0.2.
4. SUMMARY In summary, we report here the shear deformations of B6O, B12P2, and B12C3 crystals along the (011̅ 1)/ slip systems. For B6O and B12P2, we observed recovery of the original crystal structure. No broken icosahedra are observed during the whole shear simulation process. However, the same deformations of the B12C3 and (B11Cp)CBC (which are the components of B4C boron carbide) were observed with structural failure in which the central atom of the chain bends to attack an atom of the icosahedron, which causes brittle failure. These results suggest that two-atom chains can enable large shear deformations without breaking the icosahedra, leading to a ductile but hard material. We suggest that these ideas could be tested experimentally through ALD deposition 2859
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Chemistry of Materials and stabilization of thin film of B6O or B12P2. Since the experiments on B6O find amorphous band formation, we suggest that this may be related to the defect structures in multigranular B6O. We also propose the twinning formation mechanisms for B6O and B12P2.
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ASSOCIATED CONTENT
S Supporting Information *
Table of elastic moduli (GPa) for the boron suboxide. Figures of the local atomic structures for B6O shearing along the (01̅11)/ slip system, the deformation mechanism of (B12)(CCC) crystal, the deformation mechanism of B6O crystal with O vacancy, the deformation mechanism of B6O crystal with B vacancy, and the proposed atomistic twining structure in boron suboxide. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the Defense Advanced Research Projects Agency (W31P4Q-13-1-0010, program manager, Judah Goldwasser).
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REFERENCES
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