Strong anisotropy in strength and toughness in defective hexagonal boron nitride

Using a combination of density functional theory calculations and molecular dynamics simulations we show that the strength and toughness of hexagonal boron nitride (hBN) containing isolated vacancy defects are strongly anisotropic, regardless of the size of the defect core. The degree of anisotropy is preserved for a number of defect structures including monovacancy, tetravacancy, tridecavacancy, triheptacontavacancy, or heptatriacontavacancy defects. The chirality-dependent effects are strongly nonlinear and well characterized by close-form mathematical equations indicating pronounced strength and toughness along the zigzag direction compared to the strength and toughness along the armchair direction. Also, the size dependence of the strength and toughness of the defective lattice shows an inverse relationship with the effective diameter of the defect core. An atomistic analysis of the deformation fields reveals that nonuniformity in bond length, bond strain, and force distribution in the nonlinear regime of mechanical deformation surrounding the defect cores forms the physical basis for the observed anisotropy. The anisotropic character of the lattice is governed primarily by the nearest-neighbor covalent interactions (dominated by the first-nearest neighbors). Consequently, as soon as a set of bonds rupture at the defect core, the entire lattice undergoes catastrophic failure underscoring the brittle nature of the fracture state in hBN. Results also suggest that chirality-dependent elastic effects are dominated by the third-order elastic modulus which stiffens the lattice at higher chiral angles, whereas the second- and fourth-order elastic moduli soften the lattice affecting the strength and toughness of the lattice in an intricate manner.

Figure 1

Interaction regime for an arbitrary atom $i$ is shown on the left and the corresponding energy (denoted as $V_2$) and force (denoted as $F$) diagrams are shown on the right as a function of the interatomic distance $r$. The shaded region indicates the total energy absorbed by a pair of B-N atoms in the bulk configuration over the strain window $0\le \epsilon \le {\epsilon }_n$, where ${\epsilon }_n$ is the strain at which bond rupture is initiated.

Figure 2

Stress-strain behavior of hBN under uniaxial-stress deformation: for loading along $0^{\circ }$ (armchair direction) and ${30}^{\circ }$ (zigzag direction). The linear regime covers the strain range of $0<\epsilon \le 2\%$, marked by regime $A$, and beyond this regime the mechanical behavior is governed by nonlinear elasticity. The vertical blue and red lines indicate the onset of anisotropic behavior of the lattice in the MD-SW and DFT-GGA simulations, respectively.

Figure 3

(a) hBN lattice with different radial vectors considered to create the vacancy defect configurations for loading along the armchair direction. (b) Schematic showing a comparison of the defect-core diameters. (c) Defect edge polygons created by connecting the locus of the nearest atoms from the defect core. (d) (From left to right): Monovacancy, tetravacancy, trideca-vacancy, heptatriacontavacancy, and triheptacontavacancy denoting 1, 4, 13, 37, and 73 missing atoms at the core, respectively. Their effective radii are marked by red circles and the atoms removed are shown in red.

Figure 4

Chirality-dependent elastic moduli ${\mathcal{C}}_1$ or the Young's modulus $E$, ${\mathcal{C}}_2$, ${\mathcal{C}}_3$, and ${\mathcal{C}}_4$ of hBN lattice containing monovacancy under symmetry-breaking uniaxial deformation. The solid lines represent fitted curves with the polynomial $C_i\left(\theta \right)=a_i+b_i\theta +c_i{\theta }^2$, where $a_i$, $b_i$, and $c_i$ are fitting parameters for the $i\mathrm{th}$ elastic modulus in units of GPa. Their values are shown in the plots.

Figure 5

The first plot shows the contribution of the individual elastic modulus ${\mathcal{C}}_i$ on the maximum differential stress $\mathrm{\Delta }{\sigma }^{\text{max}}$ as a function of applied strain, and the second plot shows the contribution of the combined effects of the odd-order elastic moduli ${\mathcal{C}}_1+{\mathcal{C}}_3$ and the even-order elastic moduli ${\mathcal{C}}_2+{\mathcal{C}}_4$ on the maximum differential stress $\mathrm{\Delta }{\sigma }^{\text{max}}$ as a function of applied strain. The stiffening regime $\mathrm{\Delta }{\sigma }^{\text{max}}>0$ and the softening regime $\mathrm{\Delta }{\sigma }^{\text{max}}<0$ are identified in the plots.

Figure 6

Chirality-dependent variation in elastic moduli of different orders: ${\mathcal{C}}_1$ or the Young's modulus $E$, ${\mathcal{C}}_2$, ${\mathcal{C}}_3$, and ${\mathcal{C}}_4$. The MD data points are well fit by a second-order polynomial, $C_i=a+b\theta +c{\theta }^2+d{\theta }^3+e{\theta }^4$, where $a$, $b$, $c$, $d$, and $e$ are fitting parameters in units of GPa. Their values are shown in the plots.

Figure 7

Chiral-angle-dependent (the left figure) strength anisotropy ${\chi }_s$ and (the right figure) toughness anisotropy ${\chi }_t$ for the five different defect configurations and for the defect-free lattice.

Figure 8

High symmetry vacancy configurations: monovacancy and tridecavacancy for a lattice with chirality $17.5^{\circ }$ at the highest elastic energy density state at the onset of fracture. The bonds are colored according to their lengths representing their strain states. The atoms are colored with the potential energy representing their energetic state. The dotted lines represent the missing bonds due to the absence of the atoms at the defect center, and the circle marks the deformed core structure. The fracture plane in indicated by the inclined solid line intersecting the highly deformed bonds. Force distribution is shown on the right.

Figure 9

Stress surface of ${\sigma }_{yy}$ in hBN surrounding the tetravacancy defect for loading along (a) the armchair direction ($0^{\circ }$), (b) $13.{89}^{\circ }$ with respect to the armchair direction, (d) $23.{41}^{\circ }$ with respect to the armchair direction, and (d) the zigzag direction ($0^{\circ }$). All are at 5% macroscopic strain. (e) Nonuniform distribution of bond lengths at the core for the conditions shown in panels (a)–(d), respectively. The red dashed line indicates the fracture plane(s), and the intersection of the black dotted lines indicate the center of the defect core.

Figure 10

The top panels show the stress-strain behavior of defective hBN for loading along the armchair and zigzag directions, respectively. The bottom panels show defect-size-dependent effective strength and toughness, where $d=0$ refers to the behavior of the defect-free lattice.

Figure 11

Symmetry-preserving stress surface of ${\sigma }_{yy}$ surrounding the defects representing (a) tetravacancy, (b) tridecavacancy, (c) heptatriacontavacancy, and (d) triheptacontavacancy at 5% macroscopic strain along the armchair direction. The color distribution refers to the stress range of 0.0 to 60.0 GPa. The average stress is well captured by the color of the far-field stress in the stress surface plots. Panel (e) shows the variation in maximum atomic stress and panel (f) shows the average atomic stress for the $V_1$, $V_4$, $V_{13}$, $V_{37}$, and $V_{73}$ configurations. They are well fit by first-order and second-order polynomials, respectively.

Figure 12

Normalized strength and toughness at ten different chiral angles for the $V_4$ configuration, containing one tetravacancy in the domain. The strength and toughness values at a given chiral direction are normalized by the respective strength and toughness values along the armchair direction, which are 56.168 and 60.52 GPa with the SW and BN-ExTeP potentials, respectively, and the corresponding toughness values are 2.47 and 3.1 ${\mathrm{Jm}}^{-2}$, respectively.

Figure 13

Comparison of atomic stress fields ${\sigma }_{yy}$ obtained from the SW and BN-ExTeP (Tersoff) potential simulations for $\theta =17.4^{\circ }$ at 8% macroscopic strain. The color bar represents stress in units of GPa. The uniaxial loading direction is shown by the double arrow.

Figure 14

Chirality-dependent strength and toughness of defective hBN lattice containing a tetravacancy at 300 K. The strength and toughness values at a given chiral direction are normalized by the respective strength and toughness values along the armchair direction. The strengths of the lattice along the armchair direction are 50.13 and 38.56 GPa with the SW and BN-ExTeP potentials, respectively, and the corresponding toughness values are 2.02 and 1.61 ${\mathrm{Jm}}^{-2}$.

Figure 15

Deformation state of the hBN lattice at 300 K with (a) the SW potential and (b) the BN-ExTeP potential, at 30% macroscopic strain. Local morphology is evident in the insets in panels (a) and (b) indicating intact defect structures. The fracture states of the lattice are shown in panels (c) and (d), which are obtained from the SW and BN-ExTeP potentials, respectively.

Figure 16

Stress-strain behavior of a defective lattice ($V_4$ with $\theta =17.4^{\circ }$) under one loading and unloading cycle at 1 and 300 K with the SW potential.