Indentation-strain stiffening in tungsten nitrides: Mechanisms and implications

We report on a systematic computational study of ideal strengths, i.e., the lowest stresses needed to destabilize a perfect crystal under a variety of loading conditions for six stable and metastable tungsten nitrides identified by advanced crystal structure search algorithms. We employ first-principles calculations to determine stress-strain relations and examine the corresponding atomistic bonding changes for a microscopic understanding of the structural deformation modes. The obtained results show that, in stark contrast to many previously studied transition-metal borides, carbides, and also most nitrides, the tungsten nitrides exhibit surprisingly broad and common patterns of strain-stiffening effects under indentation strains. These extraordinary behaviors are attributed to the favorable bonding arrangements in these materials, including the strong and well-connected W-W metallic bonds and $\mathrm{N}\equiv \mathrm{N}$ bonds in combination with W-N covalent bonds, which together create a strong three-dimensional bonding network that is capable of resisting large-indentation shear deformations, thereby greatly enhancing the corresponding mechanical strength that is directly responsible for the superior experimentally measured hardness results. These findings provide insights for elucidating unusual indentation strength enhancements in these materials and offer guidelines for exploring experimental synthesis and optimization of the presently identified and potentially additional very hard to superhard materials. These insights are also crucial for understanding the fundamental relationship between crystal and chemical bonding structures and the associated mechanical properties under a wide range of loading conditions in diverse application environments.

Figure 1

Crystal structures of six tungsten nitrides examined in this paper: (a) cP6-WN, (b) hP2-WN, (c) hP4-WN, (d) $\mathrm{hP}20-{\mathrm{W}}_2{\mathrm{N}}_3$, (e) $\mathrm{hP}22-{\mathrm{W}}_5{\mathrm{N}}_6$, and (f) $\mathrm{hP}6-{\mathrm{WN}}_2$.

Figure 2

Calculated phonon dispersion relations of the six tungsten nitrides studied in this paper. There is no imaginary phonon mode in the entire Brillouin zone in any of the six cases, indicating dynamic stability of these structures.

Figure 3

Calculated stress-strain relations and key structural snapshots of cubic cP6-WN in the ($1\overline{1}0$)[001] direction, which is equivalent to the (110)[001] direction.

Figure 4

Calculated stress-strain relations and key structural snapshots of cubic cP6-WN in the ($1\overline{1}0$)[110] direction, which is equivalent to the (110)[$1\overline{1}0$] direction.

Figure 5

Calculated stress-strain relations and key structural snapshots of cubic cP6-WN in the (001)[110] direction, which is equivalent to the (001)[$1\overline{1}0$] direction.

Figure 6

Calculated stress-strain relations and key structural snapshots of hexagonal hP2-WN in the (001)[$1\overline{1}0$] direction.

Figure 7

Calculated stress-strain relations and key structural snapshots of hexagonal hP2-WN in the ($1\overline{1}0$)[001] direction.

Figure 8

Calculated stress-strain relations and key structural snapshots of hexagonal hP2-WN in the ($1\overline{1}0$)[110] direction.

Figure 9

Calculated stress-strain relations and key structural snapshots of hexagonal hP4-WN in the (110)[001] direction.

Figure 10

Calculated stress-strain relations and key structural snapshots of hexagonal hP4-WN in the (001)[110] direction.

Figure 11

Calculated stress-strain relations and key structural snapshots of hexagonal hP4-WN in the (001)[$1\overline{1}0$] direction.

Figure 12

Calculated stress-strain relations and key structural snapshots of hexagonal $\mathrm{hP}6-{\mathrm{WN}}_2$ in the (110)[$1\overline{1}0$] direction.

Figure 13

Calculated stress-strain relations and key structural snapshots of hexagonal $\mathrm{hP}6-{\mathrm{WN}}_2$ in the (110)[001] direction.

Figure 14

Calculated stress-strain relations and key structural snapshots of hexagonal $\mathrm{hP}20-{\mathrm{W}}_2{\mathrm{N}}_3$ in the (110)[001] direction.

Figure 15

Calculated stress-strain relations and key structural snapshots of hexagonal $\mathrm{hP}20-{\mathrm{W}}_2{\mathrm{N}}_3$ in the (110)[$1\overline{1}0$] direction.

Figure 16

Calculated stress-strain relations and key structural snapshots of hexagonal $\mathrm{hP}20-{\mathrm{W}}_2{\mathrm{N}}_3$ in the ($1\overline{1}0$)[001] direction.

Figure 17

Calculated stress-strain relations and key structural snapshots of hexagonal $\mathrm{hP}20-{\mathrm{W}}_2{\mathrm{N}}_3$ in the ($1\overline{1}0$)[110] direction.

Figure 18

Calculated stress-strain relations and key structural snapshots of hexagonal $\mathrm{hP}22-{\mathrm{W}}_5{\mathrm{N}}_6$ in the ($1\overline{1}0$)[001] direction.

Figure 19

Calculated stress-strain relations and key structural snapshots of hexagonal $\mathrm{hP}22-{\mathrm{W}}_5{\mathrm{N}}_6$ in the ($1\overline{1}0$)[110] direction.

Figure 20

Calculated stress-strain relations and key structural snapshots of hexagonal $\mathrm{hP}22-{\mathrm{W}}_5{\mathrm{N}}_6$ in the (110)[001] direction.

Figure 21

Calculated stress-strain relations and key structural snapshots of hexagonal $\mathrm{hP}22-{\mathrm{W}}_5{\mathrm{N}}_6$ in the (110)[$1\overline{1}0$] direction.