First-principles investigation of strain effects on the stacking fault energies, dislocation core structure, and Peierls stress of magnesium and its alloys

Taking pure Mg, Mg-Al, and Mg-Zn as prototypes, the effects of strain on the stacking fault energies (SFEs), dislocation core structure, and Peierls stress were systematically investigated by means of density functional theory and the semidiscrete variational Peierls-Nabarro model. Our results suggest that volumetric strain may significantly influence the values of SFEs of both pure Mg and its alloys, which will eventually modify the dislocation core structure, Peierls stress, and preferred slip system, in agreement with recent experimental results. The so-called “strain factor” that was previously proposed for the solute strengthening could be justified as a major contribution to the strain effect on SFEs. Based on multivariate regression analysis, we proposed universal exponential relationships between the dislocation core structure, the Peierls stress, and the stable or unstable SFEs. Electronic structure calculations suggest that the variations of these critical parameters controlling strength and ductility under strain can be attributed to the strain-induced electronic polarization and redistribution of valence charge density at hollow sites. These findings provide a fundamental basis for tuning the strain effect to design novel Mg alloys with both high strength and ductility.

Figure 1

(a) Schematic illustration of the perfect lattice, basal stacking fault ${\mathrm{I}}_2$, and twin stacking fault ${\mathrm{T}}_2$ configurations generated via the alias shear. The letters A, B, and C represent different stacking sequences of (0001) planes, which are identified by blue, pink, and olive symbols, respectively, with the stacking fault being highlighted in red. The solid and open circles represent the atomic coordinates at 0 and ½ along $\left[11\overline{2}0\right]$ direction, respectively, and the dotted line indicates the stacking fault position. (b) The full γ surface of pure Mg for the basal (0001) plane. (c) The calculated SFE profiles of pure Mg, Mg-Al, and Mg-Zn along the $\left[11\overline{2}0\right]$ direction to indicate the stable SFEs (${\gamma }_{I_2}$ and ${\gamma }_{T_2}$) and unstable SFEs (${\gamma }_{U{I}_2}$ and ${\gamma }_{U{T}_2}$).

Figure 2

(a) A schematic to illustrate the dissociation of a perfect dislocation into two partials. The $b_1, b_2$, and $b_3$ are the Burgers vectors of perfect dislocation, and the $b_{p1}, b_{p2}$, and $b_{p3}$ are the Burgers vectors of partial dislocation. The $b_{p3,x}, b_{p3,z}, b_{-p1,x}$, and $b_{-p1,z}$ are the components of $b_{p3}$ and $b_{-p1}$ along $x$ and $z$ axes, respectively. (b–e) The dislocation core structure of pure Mg (black curve) and Mg alloys (pink curve for the Mg-Al alloy and blue curve for the Mg-Zn alloy) at equilibrium. The separation $d_x$ (or $d_z$) between the edge (or screw) components of the two partials is defined as the distance between the two peaks, and the width $w_x$ (or $w_z$) of the edge (or screw) components of a partial dislocation is the FWHM, approximately, as shown in this figure.

Figure 3

The calculated stable and unstable SFEs, the pressure, and the twinnability for (a, b) pure Mg and (c, d) Mg-Zn as a function of volumetric strain. It is found that the dependence of the stable and unstable SFEs on the volumetric stain can be well described by a polynomial fit.

Figure 4

The (a) distance ($d_x/b$) and (b) width ($w_x/b$) of the edge components of a dislocation under different volumetric strains. (c) The misfit density and (d) the pressure field (in GPa) produced by the dislocation under different volumetric strains. It is seen that under volumetric strain in dilatation or compression, the variation of $d_x/b$ with respect to the strain follows the exponential function, while the calculated value of $w_x/b$ is linearly dependent on the applied volumetric strain.

Figure 5

(a) A schematic to represent the dislocation movement from one symmetrical configuration to another equivalent symmetrical configuration. (b) The logarithm of Peierls stress under different volumetric strains for pure Mg and Mg alloys, which shows a linearly decreasing trend with increasing strain.

Figure 6

Contour plots of the valence charge density difference (VCDD) of pure Mg and Mg-Zn alloy under the volumetric strains: (a, d) $\mathrm{\Delta }V/V=-0.12$, (b, e) $\mathrm{\Delta }V/V=0.00$, and (c, f) $\mathrm{\Delta }V/V=0.13$, respectively. The unit of VCDD is $\mathrm{electrons}/\mathrm{Boh}{\mathrm{r}}^3$, and the thin black line corresponds to the isosurface of 0.001 8 $\mathrm{electrons}/\mathrm{Boh}{\mathrm{r}}^3$. It is illustrated that the charge density decreases with the increasing volumetric strain at hollow sites (indicated by the red dashed circles).

Figure 7

The relationship between the dislocation core structure, the Peierls stress, and the SFEs. The black, pink, and blue points correspond to pure Mg, Mg-Al alloy, and Mg-Zn alloy, respectively, and the red solid lines are calculated by the relationships Eqs. (10, 11, 12, 13), indicating that there are general exponential relationships between the dislocation core structure, the Peierls stress, and the stable or unstable SFEs.