Asymmetric equilibrium core structures of pyramidal-II $\langle c+a\rangle$ dislocations in ten hexagonal-close-packed metals

The structures of pyramidal-II $〈c+a〉$ dislocations, one of the most important defects in structural hexagonal-close-packed (HCP) metals, have not been fully characterized for many of the HCP metals in use today. Here, we employ ab initio informed phase-field dislocation dynamics to determine the minimum energy structure of pyramidal $\left\{\overline{1}\overline{1}22\right\}〈11\overline{2}3〉$ dislocations in ten HCP metals, including Be, Co, Mg, Re, Ti, Zn, Cd, Hf, Y, and Zr. As input for the simulations, we calculate, using first-principles density functional theory, the $\left\{\overline{1}\overline{1}22\right\}$ generalized stacking fault energy (GSFE) curves for all ten metals. From these calculations, it is found that magnetism in Co is necessary for achieving a local minimum in the GSFE curve. We observe in simulations that edge and screw character dislocations split into two partials separated by a low-energy intrinsic stacking fault. The splitting distance is shown to scale inversely with the local minimum energy normalized by the product of its shear modulus and Burgers vector. Interestingly, some HCP metals exhibit an asymmetric structure, with either unequal partial Burgers vectors or widths, in contrast to the symmetric configuration expected from linear elastic dislocation theory. We explain these structures by properties of the local maxima in their GSFE curves. Metals with larger degrees of elastic anisotropy result in dislocations with larger splitting distances than would be expected under the commonly used assumption of elastic isotropy. These findings on the sizes and asymmetry in the structures of pyramidal-II $〈c+a〉$ dislocations are fundamental to understanding how these dislocations glide and interact or react with other defects when these metals are mechanically strained.

Figure 1

Comparison of GSFE profiles (a) for the pyramidal-II plane in ten HCP metals as determined by DFT, and (b) for Co with and without magnetism considered in the DFT simulations. In (a), the GSFE curve for Co is shown with the effect of ferromagnetism considered. The unstable SFEs, $U_{1,2}$, and intrinsic SFE, $I$, are labeled in (b).

Figure 2

Schematics showing (a) the initial simulation setup for the PFDD computational cell with two infinitely long parallel dislocations forming a dipole. In (b), the HCP unit cell is oriented (shown for an initial edge dislocation) such that the pyramidal-II glide plane lies parallel to the $xy$-plane in the simulation cell in (a). The simulation setup for a screw dislocation would have the same initialization shown in (a), while the HCP unit cell in (b) would be rotated around the $z$-axis by ${90}^{\circ }$. For clarity, the $x$-, $y$-, and $z$-axis in our simulation cell correspond to the $x$-, $y$-, and $z$-axis, respectively, in our HCP unit cell when initializing for an edge dislocation, and to the $y$-, $x$-, and $z$-axis, respectively, when initializing for a screw dislocation. Thus, the slip plane lies parallel to the simulation cell surface, the dislocation line sense lies parallel to the $y$-axis, and the Burgers vector (and unit cell) orientation is reflective of the desired dislocation character.

Figure 3

A schematic of the gradient of the order parameter, $d\zeta /dx$. The labels indicate the equilibrium SFW, $R_{\mathrm{e}}$, and their core widths, $w_{\mathrm{l}}$ and $w_{\mathrm{r}}$, respectively.

Figure 4

The order parameter, $\zeta$ (dotted lines), and its density, $d\zeta /dx$ (solid lines), along the $\left[11\overline{2}3\right]$ slip direction for initialized edge (a,b,c) and screw (d,e,f) dislocations in Be, Mg, and Y. Both isotropic (red) and anisotropic (blue) elastic considerations are compared. The vertical lines represent the location of the two partials before the dissociation, when they are one unstable perfect dislocation at zero position, and at the end of the simulation, when equilibrium is achieved.

Figure 5

The order parameter, $\zeta$ (dotted lines), and its density, $/dx$ (solid lines), along the $\left[11\overline{2}3\right]$ slip direction for an initialized edge and screw dislocation in Ti, Zr, and Hf. The coloring follows Fig. 4.

Figure 6

The order parameter, $\zeta$ (dotted lines), and its density, $d\zeta /dx$ (solid lines), along the $\left[11\overline{2}3\right]$ slip direction for an initialized edge and screw dislocation in Re and Co (FM). The coloring follows Fig. 4.

Figure 7

The order parameter, $\zeta$ (dotted lines), and its density, $d\zeta /dx$ (solid lines), along the $\left[11\overline{2}3\right]$ slip direction for initial edge and screw dislocations in Zn and Cd. The coloring follows Fig. 4.

Figure 8

The equilibrium partial separation distance $R_{\mathrm{e}}$, calculated using PFDD plotted against (a) the intrinsic SFE $I$ normalized by $Kb$, and (b) the unstable SFE $U_2$, also normalized by $Kb$. Expressions for $K$ are given in Ref. [23].

Figure 9

The ratio of the anisotropic to the isotropic equilibrium SFW $R_{\mathrm{e}}^{\mathrm{ani}}/R_{\mathrm{e}}^{\mathrm{iso}}$ plotted against (a) the anisotropic energy factor, $K$, normalized by the shear modulus, $\mu$, and (b) the logarithmic Euclidean anisotropy index, $A^L$, for both edge and screw dislocation dissociation.

Figure 10

The standard (a) and relaxed (b) GSFE surfaces for the pyramidal-II plane in Mg. The $x$- and $y$-axis values are normalized by the magnitude of a direct lattice translation vector along the $〈11\overline{2}3〉$ and $〈\overline{1}100〉$ directions, respectively.

Figure 11

The equilibrium SFW $R_{\mathrm{e}}$ plotted against the unstable SFE, $U_1$, normalized by the anisotropic energy factor, $K$, and the Burgers vector, $b$.

Figure 12

The ratio of the anisotropic to the isotropic equilibrium SFW $R_{\mathrm{e}}^{\mathrm{ani}}/R_{\mathrm{e}}^{\mathrm{iso}}$ plotted against the anisotropy indices (a) $\alpha$, (b) $\beta$, and (c) $\gamma$ for both edge and screw dislocation dissociation.