Ductility in Crystalline Boron Subphosphide (B12P2) - ACS Publications

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Ductility in Crystalline Boron Subphosphide (B P) for Large Strain Indentation Qi An, and William A. Goddard III J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b05429 • Publication Date (Web): 11 Jul 2017 Downloaded from http://pubs.acs.org on July 16, 2017

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The Journal of Physical Chemistry

Ductility in Crystalline Boron Subphosphide (B12P2) for Large Strain Indentation Qi An, 1,* and William A. Goddard III2 1

Department of Chemical and Materials Engineering,

University of Nevada, Reno, Reno, NV 89557, United States 2

Materials and Process Simulation Center,

California Institute of Technology, Pasadena, CA 91125, United States *

Corresponding author E-mail: [email protected]

Abstract: Our studies of brittle fracture in B4C showed that shear induced cracking of the (B11C) icosahedra leading to amorphous B4C regions induced cavitation and failure. This suggested that to obtain hard boron rich phases that are ductile, we need to replace the CBC chains of B4C with two-atom chains that can migrate between icosahedra during shear without cracking the icosahedra. We report here quantum mechanism (QM) simulation showing that under indentation stress conditions, superhard boron subphosphide (B12P2) displays just such a unique deformation mechanism. Thus, stress accumulated as shear increases is released by slip of the icosahedra planes through breaking and then reforming the P-P chain bonds without fracturing the (B12) icosahedra. This icosahedral slip may facilitate formation of mobile dislocation and deformation twinning in B12P2 under highly stress conditions, leading to high ductility. However, the presence of twin boundaries (TBs) in B12P2 will weaken the icosahedra along TBs, leading to the fracture of (B12) icosahedra under indentation stress conditions. These results suggest that crystalline B12P2 is an ideal superhard material to achieve high ductility.

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1. INTRODUCTION Icosahedral boron phosphide, also known as boron subphosphide, exhibits such novel properties as wide bandgap of 3.35 eV,1 high radiation resistance,2,3 chemical inertness, and high hardness.4−6 The combination of these properties makes it widely useful in radiation detection, electronics at high radiation environments, and in radioisotope batteries.1−3 One promising property of B12P2 is the “self-healing” phenomenon in which no visible damage is observed after prolonged exposure to high energy particles.7 Recent QM studies revealed that the B12P2 remains structurally stable in the presence of a vacancy or interstitial defect with an activation barrier for defect recombination as low as 3 meV.3 This potentially explains the self-healing mechanism under radiation conditions. Although B12P2 has potential important applications, the techniques of growing high quality crystals are still in the early stages.8 Thus, the poor material quality of current materials has limited its detailed characterization. In particular, the mechanisms of deformation and failure remain unknown. Previous studies showed that the similar compound boron carbide (B4C) suffers from the brittle failure under high pressure because of the amorphous shear band formation,9−12 which limits its engineering applications. We showed that these amorphous shear bands arise from the fracturing of the (B11C) icosahedra during shear, which increases the density of the amorphous bands compared to nearby crystalline regions, leading to tension that induces cavitation and then failure.13,14 However, our recent QM study4 showed that under pure shear deformation, single crystal of B12P2 can shear to a large strain without fracturing the (B12) 2 ACS Paragon Plus Environment

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icosahedra. This suggests that B12P2 may exhibit an alternative deformation mode that will not lead to the amorphous shear bands that induce brittle failure. Micro- and nano-indentation experiments provide the experimental means to validate our predicted properties of strength and hardness.15 Recent advances in nano-indentation have provided information about such new phenomena as local phase transformation, dissipative kink band formation, and strength enhancement.16−18 This has led to theoretical studies to elucidate the atomistic mechanics and mechanisms of materials deformation under indentation.19−22 In particular, QM simulations are essential to predict the stress-strain relationships and the atomistic bond transformations underlying these phenomena.21,22 Consequently, examining the structural changes and deformation mechanism under indentation stress conditions is essential to predict and validate indentation experiments. Our previous simulations showed that the icosahedra in B6O, B12P2 and (B11C)Si2 do not disintegrate as they are strained extensively under pure shear deformation.4,23 This contrasts with B4C, where the icosahedra deconstruct due to the coupling between the distorted C-B-C chain and the broken bond between adjacent icosahedra.13 These results suggest that replacing the C-B-C chain in B4C with 2-atom chain prevents the deconstruction of the icosahedra, thereby increasing ductility. However, our further study showed that testing uniform shear is not a sufficient conduction to ensure ductility. In particular, we found that the complex stress conditions under indentation caused the icosahedra in B6O to disintegrate.24 This suggests that to propose a ductile boride, we must also show that extensive deformation under indentation conditions avoids icosahedra deconstruction. Thus, to show that 3 ACS Paragon Plus Environment


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B12P2 is an excellent candidate for a ductile boride we must extend the previous studies to indentation conditions. In this Article, we examine the deformation processes of crystalline B12P2 under indentation stress conditions. We find that crystalline B12P2 can shear to extended indentation strain without fracturing (B12) icosahedra because of the icosahedral slip between neighbor layers. These calculations suggest a deformation mode that leads to mobile dislocations and deformation twinning in B12P2 under highly stress conditions, which might lead to high ductility. However, we expect that introducing nanotwins in crystalline B12P2 will weaken the (B12) icosahedra along the twin boundaries (TBs), causing the fracture of (B12) icosahedra and thereby leading to brittle failure under indentation stress conditions. These results suggest that crystalline B12P2 is more ductile than other boron-rich compounds such as B4C and boron suboxide (B6O).

2. METHODOLOGY

All simulations were carried out using the Vienna Ab-initio Simulation Package (VASP) periodic code using the Perdew–Burke–Ernzerhof (PBE) functional form of the generalized gradient approximation for the electronic exchange-correlation interaction.25−28 We used the plane wave projector augmented wave (PAW) method for description of the ionic cores. The energy cutoff for the plane wave expansion was 600 eV, adequate for geometry optimization and shear deformation. Convergence is reached if the consecutive energy and force differences are within 10-6 eV for electronic iterations and 10−3 eV/Å for ionic relaxations, respectively. Brillouin zone integration was performed on Γ-centered symmetry reduced Monkhorst-Pack meshes with a fine 4 ACS Paragon Plus Environment

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resolution of 2π × 1/40 Å−1 for all calculations except for the shear deformation. For the shear deformation large 2 × 2 × 2 supercell was adopted with a more approximate 2 × 2 × 2 k-point grid mesh in the Brillouin zone.

To determine the ideal shear stress and the deformation process under pure shear deformation, we imposed the strain for a particular shear plane while allowing full structure relaxation for the other five strain components.29 To mimic the complex stress conditions under indentation experiments, we applied biaxial shear deformation where the ratio of the compressive stress beneath the indenter normal to the chosen shear plane has a fixed fraction of the tangential shear while the other four strain components are relaxed.21 This simulation aimed at mimicking deformation under the indenter by imposing the relations σzz=σzx×tanΦ where σzz is the normal stress, σzx is the shear stress and Φ is the centerline-to-face angle of the indenter. Here Φ = 68° for Vickers indenter, leading to the σzz = 2.48×σzx. Therefore, it is highly compressive under indentation stress conditions. The residual stresses after relaxing were less than 0.5 GPa for both pure shear and biaxial shear deformation. Since the shear strain is constrained in the deformation, the stress of system may become negative after the structure changes or fails. 3. RESULTS AND DISCUSSION The crystalline B12P2 structure has a rhombohedral unit cell with the (B12) icosahedron located at the corner and the P-P chain along the [111] rhombohedral direction, as shown in Fig. 1(a). Each (B12) icosahedron has 24 polyhedral electrons since it has 12 extra-icosahedral bonding with 6 neighboring chains and 6 other icosahedra. To satisfy the Wade’s rule (26 5 ACS Paragon Plus Environment


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skeleton bonding electrons),30 each P atom in the linear chain transfers one electron to the icosahedra, becoming P+. This leads to a representation as (B12)2-(P+−P+) for the B12P2. The nanotwinned B12P2 structure shown in Fig. 1(b) is expected to have the {001} plane as the twin boundary (TB)4 as observed for other boron rich materials (e.g. B4C31, B6O24, β-B32). Here we focus on the nanotwinned structure that has two crystalline layers between TBs. For the B12P2 rhombohedral structure, PBE gives equilibrium lattice parameters a = 5.252 Å and α = 69.6°, leading to a density of ρ = 2.590 g/cm3, comparable to the experimental values of a = 5.249 Å, α= 69.6°, and ρ = 2.595 g/cm3.33 To examine the elastic properties and to validate our QM methods, we predicted the bulk modulus (B) and shear modulus (G) using Voigt−Reuss−Hill averaging.34 We found B = 199.1 GPa and G = 190.9 GPa for B12P2, which is consistent with previous experiments35,36 (B = 192 GPa35 and B = 207 GPa36). The elastic moduli are listed in the Table S1 of the Supporting Information (SI). The general measure of the materials strength is judged by indentation hardness, which measures the resistance of materials to deformation at a constant compression load. Our calculated Vickers hardness (Hv) for polycrystalline materials based on G/B37 leads to Hv = 38.1 GPa for B12P2 (listed in the Table S1), which agrees well with previous prediction38 of 32.9 GPa based on chemical bonding. The Hv for B12P2 is larger than B4C (32.9 GPa)39 and B6O (37.9 GPa)40. This suggests that B12P2 has a similar hardness as B4C and B6O. For the nanotwinned B12P2 structure, PBE gives equilibrium lattice parameters a = 5.257 Å, b = 19.004 Å, c = 5.257 Å, α = 90°, β = 119.4°, and γ = 90°, leading to a slightly lower density of ρ 6 ACS Paragon Plus Environment

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= 2.587 g/cm3 than crystalline B12P2 (ρ = 2.590 g/cm3). This nanotwinned structure has an electronic energy 59.6 meV/B12P2 above the crystalline B12P2, leading to a twin interfacial energy for B12P2 of 73.8 mJ/m2, which is comparable to B4C of 83.2 mJ/m2,31 and boron rich boron carbide (B13C2) of 40.6 mJ/m2.41 To examine how the nanotwins affect the mechanical properties, we used QM to predict the bulk modulus B = 198.7 GPa and shear modulus G = 190.1 GPa for nanotwinned B12P2 using the Voigt−Reuss−Hill averaging.34 These values are slightly lower than that of the crystalline B12P2, suggesting that the presence of nanotwins softens crystalline B12P2. In addition, we calculated the Vickers hardness of nanotwinned B12P2 based on G/B values, leading to 37.9 GPa which is slight lower than crystalline B12P2. We also calculated the Knoop hardness of these two structures. The Knoop hardness of single crystal may be significantly anisotropic.42 To predict the Knoop hardness of B12P2, we applied the method that takes into account the important chemical effects related to the strength of covalent bonding, degree of ionicity and directionality, and topology of the crystal structure.43 This method was previously applied to various boron phases such as α-B12, β-B106 and γ-B28, and leading to predicted Hardnesses that agree very well with experimental measurements.43 Our calculation leads to H = 29.54 and 29.48 GPa for crystalline B12P2 and twinned B12P2, respectively. This agrees well with the predicted Vickers hardness based on G/B. These values for nanotwinned B12P2 are listed in the Table S1 of SI. Formation of amorphous shear bands is the major brittle failure mechanism for superhard ceramics B4C9−12 and B6O24,44. However, our previous QM study4 showed that the B12P2 and B6O 7 ACS Paragon Plus Environment


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can recover the original crystal structure through icosahedral slip when they are sheared along the most plausible slip system (001)/ under pure shear deformation. Later, we showed that the (B12) icosahedra in B6O are deconstructed when it is sheared along (001)/ slip system under indentation stress conditions.24 For B12P2 we first examined the deformation process under indentation stress conditions by applying the biaxial shear deformation along the most plausible slip system (001)/. Then we examined how the nanotwins affect the deformation mechanism by applying both pure shear and biaxial shear deformation along the TB slip system which corresponds to the most plausible slip system in crystalline B12P2. Fig.2 displays the shear-stress−shear-strain relationships for crystalline and nanotwinned B12P2 under pure shear and biaxial shear deformation. For crystalline B12P2 under pure shear deformation, the P-P chain bond breaks at 0.416 strain without fracturing (B12) icosahedra, releasing the shear stress from 38.2 to 6.1 GPa.4 Relaxing the cell recovers the original cell and energy. Under biaxial shear deformation, the shear stress first increases monotonically to a maximum shear stress of 27.7 GPa as the shear strain increases to 0.166. Then it decreases slightly to 24.0 GPa at 0.254 strain and dramatically drops to 6.7 GPa at 0.276 strain, indicating structure transformation. After the shear stress decreases to -3.5 GPa at 0.299 strain, it starts to increase monotonically to 20.4 GPa until 0.465 strain. Finally, the second structure transformation occurs at 0.514 strain with the shear stress decreasing to 9.7 GPa. For nanotwinned B12P2, the stress-strain relationship under pure shear deformation displays a 8 ACS Paragon Plus Environment

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character similar to crystalline B12P2 (Fig 2(a)). The shear stress increases monotonically to 0.276 strain and then dramatically drops at 0.299 strain. Then it increases continuously with the same slope as crystalline B12P2, suggesting that the icosahedra are not deconstructed. The critical shear stress for nanotwinned B12P2 under pure shear deformation is 42.6 GPa which is slightly lower than that of crystalline B12P2 (44.2 GPa).4 For nanotwinned B12P2 under biaxial shear deformation, the shear stress increases monotonically to 0.166 strain and then becomes irregular after it drops at 0.187 strain (Fig. 2(b)), suggesting brittle failure of nanotwinned B12P2. The critical stress for nanotwinned B12P2 is 26.9 GPa which is slightly lower than the 27.7 GPa of crystalline B12P2. These results suggest that the presence of nanotwins softens the B12P2, which is consistent with the prediction from the elastic moduli and Vickers hardness. The details of the deformation process for crystalline B12P2 shearing along (010)/ slip system under indentation conditions are displayed in Fig. 3(a)-(e). 1.

0 to 0.166 strain: The system deforms elastically to leading to the maximum shear stress at

0.166 strain. No bonds broken as shown in Fig. 3(a). The P13-P14 chain bond decreases slightly from 2.25 to 2.19 Å because of the compressive stress. The B54-B32 icosahedral-icosahedral bond increases from original 1.74 to 1.83 Å because of this combination of compression and shear. 2.

0.166 to 0.254 strain: As the shear strain increases further, the shear stress decreases slightly

from 27.7 to 24.0 GPa. No bonds break as shown in Fig. 3(b). The P13-P14 chain bond and the 9 ACS Paragon Plus Environment


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B54-B32 icosahedral-icosahedral bond are slightly increased to 2.20 and 1.85 Å, respectively. 3.

0.254

to

0.276

strain:

Both

the

P13-P14

chain

bond

and

the

B54-B32

icosahedral-icosahedral bond break, leading to the formation of new P13-P6 bond shown in Fig. 3(c). This also leads to formation of the lone pair electrons on P13 atom which now bonds to P6 atom and two icosahedral B atoms through sigma bond. The lone pair electrons on P13 are displayed in Fig. 3(c) using the electron localization function (ELF) analysis.45 Meanwhile, two B atoms (B73 and B23) in icosahedron-1 (ICO-1) have no extra-icosahedral bonds. This leads to 10 extra-icosahedral bonds for half of the total icosahedra (e.g. ICO-1), leaving 26 polyhedral electrons (Wade’s rule). For the other half icosahedra (e.g. ICO-2), they still have 12 extra-icosahedral bonds and gain 2 electrons from the P-P chains to satisfy the Wade’s rule. This bond rearrangement process relieves the shear stress from 24.0 GPa to 6.7 GPa. However, no icosahedra are disintegrated in this process, as shown in Fig. 3(c). 4.

0.276 to 0.299 strain: The B94 atom reacts with the lone pair electrons on P13 atom to form

the new P13-B94 bond shown in Fig. 3(d). This recovers the structure to the original crystalline structure and releases the shear stress from 6.7 GPa to -3.5 GPa. 5.

0.299 to 0.489 strain: The shear stress monotonically increases to 20.4 GPa without

destroying the (B12) icosahedra as shown in Fig. 3(e). 6.

0.489 to 0.514 strain. We observe the second structure transformation and shear stress drops.

However, no icosahedra are disintegrated in this process, as shown in Fig. 3(f). This deformation mode of crystalline B12P2 suggests that the accumulated stress can be 10 ACS Paragon Plus Environment

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released by icosahedral slipping without fracturing icosahedra. This suggests that mobile dislocation and deformation twinning can be formed at high stress conditions, leading to a toughening mechanism in B12P2. To improve the ductility of the boron rich materials such as B4C, it is essential to prevent icosahedra disintegration under highly compressed states. The icosahedra in B4C and B6O deconstruct under biaxial shear deformation, but this deconstruction does not occur for crystalline B12P2. This suggests that crystalline B12P2 has much higher ductility than other boron rich compounds such as B4C and B6O. For the nanotwinned B12P2 under pure shear deformation, the detailed deformation mode shearing along the twin plane is displayed in Fig. 4(a)-(c). Fig. 4(a) indicates the intact nanotwinned structure. As the shear strain increases to 0.276, corresponding to the maximum shear stress of 42.6 GPa, the P9-P13 chain bond decreases slightly from 2.25 to 2.21 Å, while the P9-B12 and P13-B67 chain-icosahedral bonds both increase from 1.90 to 2.22 Å. No bonds break as shown in Fig. 4(b). After passing the critical strain of 0.276, both P9-B12 and P13-B67 bonds break, leading to the formation of new P9-B77 and P13-B15 bonds. This also breaks the original

B15-B29

icosahedral-icosahedral

bond

while

forming

a

new

B25-B19

icosahedral-icosahedral bond. This process rotates the P9-P13 chain bond and transforms the nanotwinned B12P2 structure to the crystalline B12P2 structure, as shown in Fig. 4(c). A similar structure transformation from the nanotwinned phase to the crystalline phase was also observed in B6O.24 The details of the deformation process for nanotwinned B12P2 under biaxial shear 11 ACS Paragon Plus Environment


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deformation is displayed in Fig. 4(d)-(f). As the strain increases to 0.166, corresponding to the maximum shear stress of 26.9 GPa, we observe that no icosahedral clusters deconstruct, as shown in Fig. 4(d). However, the B47-B93 bond within the icosahedron along the TB stretches from the original 1.96 to 2.38 Å. Meanwhile the distance between the P3 and P12 chain atoms decreases dramatically from 3.81 to 2.90 Å because of the compressive stress. As the shear strain increases to 0.187, the distance between P3 and P12 decreases to 2.26 Å, forming a new P-P chain bond and stretching the original P3-P7 and P12-P16 chain bonds to 2.41 Å, as shown in Fig. 4(e). This releases the shear stress from 26.9 to 17.7 GPa. However, no (B12) icosahedra deconstruct. As the shear strain further increases to 0.231 strain, the P3-P7 and P12-P16 chain bonds break and the icosahedra along the TBs deconstruct, which is expected to lead to brittle failure of nanotwinned B12P2. We expect that crystalline B12P2 has much higher ductility since the icosahedra do not deconstruct under indentation stress conditions. However, forming a TB in the crystalline structure leads to a high interfacial energy that weakens the icosahedra within the TBs. Thus, to preserve the ductility of B12P2 it is essential to suppress twin formation. We observe the icosahedral slip deformation mode in defect free B12P2. However, such defects as vacancies and grain boundaries that are ubiquitous in real materials, may significantly impact the icosahedral slip deformation mode. Such studies require cell sizes much larger than practical for QM. Thus, future study fitting the QM results in this paper to a new ReaxFF reactive force field will test whether and how such defects affect ductility. 12 ACS Paragon Plus Environment

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Pugh proposed an empirical criterion that links the plastic properties of metals with their elastic moduli ration G/B.47 If G/B < 0.5, the material behaves in a ductile manner; otherwise it behaves in a brittle manner. Pugh’s criterion has been demonstrated and widely used in metallic system such as metal alloys47 and bulk metallic glasses.37 However, this criterion has not been established for superhard B4C based materials where mobile dislocations are rarely observed at room temperature. Thus, although G/B for B12P2 is 0.96 which is higher than that of B4C (0.83),23 we observed that the icosahedra in B12P2 do not deconstruct under both pure shear and indentation stress conditions, whereas they do deconstruct in B4C for both conditions, leading to amorphous band formation and brittle failure. This shows that B12P2 is much more ductile than B4C, violating the Pugh criterion. 4. SUMMARY Summarizing, we examined the deformation process of crystalline and nanotwinned B12P2 under both pure shear and indentation stress conditions. •

We found that the (B12) icosahedra in crystalline B12P2 do not deconstruct under both pure and biaxial shear deformation, with the shear stress relaxed by icosahedral slip between neighbor layers.

•

For nanotwinned B12P2 the (B12) icosahedra within the TBs deconstruct under indentation stress conditions. While the nanotwinned B12P2 does transform to crystalline B12P2 under pure shear deformation.

We observed that the icosahedra in B12P2 do not deconstruct under both pure shear and 13 ACS Paragon Plus Environment


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indentation stress conditions, whereas they do deconstruct in B4C for both conditions, leading to amorphous band formation and brittle failure. These results suggest that the crystalline B12P2 has the potential to form mobile dislocation and deformation twinning under complex stress condition, leading to higher ductility than other boron rich compounds such as B4C and B6O. But to achieve this ductility we must avoid nanotwinning in B12P2. ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website. Tables list the elastic moduli, Vickers hardness, ideal shear strength, critical stress under biaxial shear deformation, and the elastic constants for crystalline and nanotwinned B12P2. Acknowledgements This work was initiated with support by the National Science Foundation (DMR-1436985, program manager, John Schlueter). It was completed with support by the Defense Advanced Research Projects Agency (W31P4Q-13-1-0010 and W31P4Q-12-1-0008, program manager, John Paschkewitz) Q.A. was also supported by the Army Research Laboratory under Cooperative Agreement Number W911NF-12-2-0022 and by the U. S. Nuclear Regulatory Commission (NRC-HQ-84-15-G-0028).


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Figure 1

Figure 1. Icosahedral boron phosphide structures: (a) crystalline B12P2 structure with a rhombohedral unit cell; (b) nanotwinned B12P2 structure with two crystalline layers between TBs (solid line). The B and P atoms are represented by blue and orange balls, respectively. The TBs are represented by solid lines. The structures are plotted using VESTA.47



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Figure 2

Figure 2. The shear-stress−shear-strain relationships for crystalline and nanotwinned B12P2 shearing along the least shear slip system of (010)/ under pure shear (a) and biaxial shear (b) loading conditions.


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Figure 3

Figure 3. The structural changes for B12P2 shearing along (010)/ under biaxial shear loading: (a) structure at 0.166 strain, corresponding to maximum shear stress; (b) structure at 0.254 strain before the 1st shear stress significant drop; (c) structure at 0.276 strain after stress release, no (B12) icosahedra disintegrate. The ELF analysis within the parallelogram indicates that long pair electrons form on the P13 atom; (d) structure at 0.299 strain showing structure recovery to the original B12P2 structure; (e) structure at 0.489 strain before the 2nd shear stress significant drop; (f) structure at 0.514 strain after stress release, no (B12) icosahedra disintegrate. The key structural changes are shown in the parallelogram. The B and P atoms are represented by blue and orange balls, respectively.



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Figure 4

Figure 4. The structural changes for nanotwinned B12P2 under pure shear (a-c) and biaxial shear (d-f) deformation along the (001) twin plane: (a) intact structure; (b) structure at 0.276 strain corresponding to the maximum shear stress; (c) structure at 0.299 strain in which the twinned structure transforms to crystalline structure; (d) structure at 0.166 strain corresponding to the maximum shear stress; (e) structure at 0.187 where the shear stress is released by the formation of a P-P bond; (f) structure at 0.231 after failure where the icosahedral clusters near the TBs deconstruct. The B and P atoms are represented by blue and orange balls, respectively.


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Ductility in Crystalline Boron Subphosphide (B12P2) - ACS Publications

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