Plastic flow between nanometer-spaced planar defects in nanostructured diamond and boron nitride

The fundamental mechanisms of strengthening/hardening and toughening that may be modified by various nanometer-spaced planar defects in the ultrahard nanostructured diamond and boron nitride (BN), e.g., nanotwins, stacking faults, and coherent heterophase interfaces, are still far from understood. In the present work, by means of first-principles approaches to derive ideal strength and Peierls stress, we performed a comprehensive investigation on the effect of the nanometer-spaced planar defects on the strength and plasticity of nanostructured diamond and BN under both uniform and localized deformations. A profound strengthening under uniform strain is revealed to be closely dependent on the spacing of planar defects, yet differing from the disappearing dependence under localized strain. It is further shown that the breakage and reconstruction of covalent bonds occurs only for very small spacing of planar defects under uniform deformations, being inconsistent with the average spacing found in the experimentally prepared nanotwinned diamond and BN, thus casting a doubt on the feasibility of the previously proposed strengthening mechanism. Under localized deformations, only the planar defects of twin in c-diamond or c-BN and coherent heterophase interface in c-/h-diamond or c-/w-BN are found to increase the barrier for the parallel slip of both $1/2\langle 110\rangle$ shuffle-set full dislocation and $1/6\langle 112\rangle$ glide-set partial dislocation, resulting in the strengthening of nanostructured diamond and BN, which agrees to the experimental observation. These findings not only yield a physical insight in strengthening/toughening nanostructured diamond and BN, but highlight the importance to understand the synergetic effect of length scale and interface between planar defects in designing superhard nanostructured materials.

Figure 1

(a) Orientation of the Cartesian axes relative to the dislocation line with (b) shuffle-set and glide-set slip planes. (c) The GSFE profiles of diamond along the $\left[1\overline{1}0\right]$ and $\left[11\overline{2}\right]$ directions on the shuffle-set and glide-set planes, respectively. The unstable SFEs ${\gamma }_{\mathrm{US}}$ are highlighted by arrows. Example in determining the Peierls stress via Peierls-Nabarro model: (d) GSFE profile and the derived restoring force, (e) a least squares minimization of the difference between elastic resistance and restoring force, (f) the calculated profiles of disregistry and misfit density, and (g) the calculated dislocation energy and the derived stress profile as a function of dislocation position.

Figure 2

(a) The calculated GSFE profile for partial dislocation, and (b) as the shaded area is subtracted, the effective GSFE profile is obtained. Then the Peierls stress is determined using the effective GSFE profile as shown in Figs. 1–1.

Figure 3

Topological structure and formation energy of the equally spaced planar defects in the nanostructured diamond and BN. The formation energies of the planar defects of c-twin, c-SF, $I_1\text{−}\mathrm{SF},I_2\text{−}\mathrm{SF}$. and CHI are defined as $\left(E_{c\text{−}twin}\text{−}{E}_{c\text{−}Str.}\right)/A,\left(E_{c\text{−}SF}\text{−}{E}_{c\text{−}Str.}\right)/A,\left(E_{I1\text{−}SF}\text{−}{E}_{h\text{−}Str.}\right)/A,\left(E_{I2\text{−}SF}\text{−}{E}_{h\text{−}Str.}\right)/A$ and $\left(E_{CHI}\text{−}1/2E_{c\text{−}Str.}\text{−}1/2E_{h\text{−}Str.}\right)/A$, respectively, where $A$ is the area of planar defect. Light-blue regions correspond to the twinned region. Note that the interface spacing λ is defined as the distance between two planar defects in the supercell after optimization but before deformation.

Figure 4

(a) Topological structures of two types of planar defects with the same volume fraction of twinned region, i.e., the c-twin planar defect with different interface spacing λ and the CHI with different ratio of h-diamond or w-BN. Light-blue regions correspond to the twinned region. (b) The relative energy to c-diamond or c-BN as a function of volume fraction of twinned region.

Figure 5

The shear strengths along the (111)$\left[11\overline{2}\right]$ slip system as a function of interface spacing λ and the calculated stress-strain curves for several planar defects in nanotwinned diamond: (a),(b) c-twin, (c),(d) c-SF, (e),(f) $I_1\text{−}\mathrm{SF}$, (g),(h) $I_2\text{−}\mathrm{SF}$, and (i),(j) CHI in c-/h-diamond.

Figure 6

The shear strengths along the (111)$\left[11\overline{2}\right]$ slip system as a function of interface spacing λ and the calculated stress-strain curves for several planar defects in nanotwinned BN: (a),(b) c-twin, (c),(d) c-SF, (e),(f) $I_1\text{−}\mathrm{SF}$, (g),(h)$I_2\text{−}\mathrm{SF}$ and (i),(j) CHI in c-/w-BN.

Figure 7

(a) The calculated stress-strain curve, and (b)–(m) the snapshots of deformed bond structure, the VCDD (in electrons $/\mathrm{Boh}{\mathrm{r}}^3)$ and the variation of bond lengths (in Å) under various shear strains for (b)–(e) the $I_1\text{−}\mathrm{SF}$ in nanostructured diamond with an interface spacing $\lambda =0.82\mathrm{nm}$, and the CHI in (f)–(i) c-/h-diamond and (j)–(m) c-/w-BN with an interface spacing $\lambda =0.82\mathrm{nm}$. The same isosurface level of ±0.017 electrons $/\mathrm{Boh}{\mathrm{r}}^3$ is used. Blue and yellow regions signify the states of charge depletion (negative) and accumulation (position), respectively.

Figure 8

The variations of bond lengths in (a) c-twin, (b) c-SF, (c) $I_1\text{−}\mathrm{SF}$, (d) $I_2\text{−}\mathrm{SF}$, and (e) CHI in c-/h-diamond. The glide-set planes are highlighted by blue dashed lines to indicate the boundary between cubic and hexagonal structures. (f)–(i) The unstable GSFE ${\gamma }_{\mathrm{US}}\left(\mathrm{in}\mathrm{J}/{\mathrm{m}}^2\right)$ and Peierls stress ${\tau }_{\mathrm{P}}$ (in GPa) of shuffle-set plane along $\left[1\overline{1}0\right]$ direction and glide-set planes along $\left[11\overline{2}\right]$ direction. The unstable GSFEs and Peierls stresses of c-diamond and h-diamond are illustrated by the orange and blue solid lines, respectively.

Figure 9

The variations of bond lengths in (a) c-twin, (b) c-SF, (c) $I_1\text{−}\mathrm{SF}$, (d) $I_2\text{−}\mathrm{SF}$, and (e) CHI in c-/w-BN. The glide-set planes are highlighted by blue dashed lines to indicate the boundary between cubic and hexagonal structures. (f)–(i) The unstable GSFE ${\gamma }_{\mathrm{US}}\left(\mathrm{in}\mathrm{J}/{\mathrm{m}}^2\right)$ and Peierls stress ${\tau }_{\mathrm{P}}$ (in GPa) of shuffle-set plane along $\left[1\overline{1}0\right]$ direction and glide-set planes along $\left[11\overline{2}\right]$ direction. The unstable GSFEs and Peierls stresses of c-BN and w-BN are illustrated by the orange and blue solid lines, respectively.

Figure 10

The determined weakest positions for the parallel slip of $1/2\langle 110\rangle$ shuffle-set perfect dislocation (highlighted by origin arrows) and $1/6\langle 112\rangle$ glide-set partial dislocation (highlighted by blue arrows) in the nanostructured diamond and BN with nanometer-spaced planar defects based on the calculated Peierls stress. The weakest positions for nanostructured BN are same as those of nanostructured diamond except that of $I_2\text{−}\mathrm{SF}$ in w-BN for 1/6⟨112⟩ glide-set partial dislocation, for which the weakest position is inside the nanolayer of w-BN (highlighted by hollow blue arrow).