Generalized Stacking Fault Energy of {10-11}<11-23> Slip System in Mg-Based Binary Alloys: A First Principles Study

In this work, the generalized stacking fault energies (GSFEs) of {10-11}<11-23> slip system in a wide range of Mg-X (X = Ag, Al, Bi, Ca, Dy, Er, Gd, Ho, Li, Lu, Mn, Nd, Pb, Sc, Sm, Sn, Y, Yb, Zn and Zr) binary alloys has been studied. The doping concentration in the doping plane and the Mg-X system is 12.5 at.% and 1.79 at.%, respectively. Two slip modes (slip mode I and II) were considered. For pure magnesium, these two slip modes are equivalent to each other. However, substituting a solute atom into the magnesium matrix will cause different effects on these two slip modes. Based on the calculated GSFEs, two design maps were constructed to predict solute effects on the behavior of the {10-11}<11-23> dislocations. The design maps suggest that the addition of Ag, Al, Ca, Dy, Er, Gd, Ho, Lu, Nd, Sm, Y, Yb and Zn could facilitate the {10-11}<11-23> dislocations.


Introduction
Magnesium alloys show great potential for application in the automotive and aerospace industry. However, the use of magnesium alloys is limited by their poor ductility at ambient temperatures [1][2][3][4][5]. In hexagonal close-packed (HCP) alloys, the softest deformation mode is the basal slip, which provides only two independent slip systems. This is far from the von Mises criterion, in which five independent deformation modes are required for a compatible deformation [6]. Furthermore, basal slips cannot accommodate the strain along the c axis.
We believe this discrepancy arises from the insufficient consideration of simulation modes. As shown in Figure 1a,b, when the shear stress is applied along the <11-23> direction, there should exist two deformation modes. In practice, the weaker one should be activated. For pure magnesium, these two slip modes are equivalent. However, if a solute atom is substituted into the fourth lattice plane, slip mode I and II should exhibit different GSFEs. To the best of our knowledge, previous studies have not addressed this issue yet. In this work, the GSFEs of {10-11}<11-23> slip system in a wide range of Mg-X (X = Ag, Al, Bi, Ca, Dy, Er, Gd, Ho, Li, Lu, Mn, Nd, Pb, Sc, Sm, Sn, Y, Yb, Zn and Zr) binary alloys will be studied, aiming to provide a basis for the design of high-performance magnesium alloys. It should be noted the selected alloying elements have a maximum solubility of >1.0 at.%.
To this end, there must exist at least one stable stacking fault energy (SFE) on the {10-11} plane. The SFE for this dissociation could be searched out by the in-plane-relaxing permitted first principles simulations [19].
We believe this discrepancy arises from the insufficient consideration of simulation modes. As shown in Figure 1a,b, when the shear stress is applied along the <11-23> direction, there should exist two deformation modes. In practice, the weaker one should be activated. For pure magnesium, these two slip modes are equivalent. However, if a solute atom is substituted into the fourth lattice plane, slip mode I and II should exhibit different GSFEs. To the best of our knowledge, previous studies have not addressed this issue yet. In this work, the GSFEs of {10-11}<11-23> slip system in a wide range of Mg-X (X = Ag, Al, Bi, Ca, Dy, Er, Gd, Ho, Li, Lu, Mn, Nd, Pb, Sc, Sm, Sn, Y, Yb, Zn and Zr) binary alloys will be studied, aiming to provide a basis for the design of high-performance magnesium alloys. It should be noted the selected alloying elements have a maximum solubility of >1.0 at.%.

Computational Details
First principle simulations in this work were carried out by ABINIT code [24,25], accompanied by the projector augmented wave method (PAW) [26]. The exchange-correlation was described by the generalized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) form [27]. The Brillouin zone was sampled using a 4 × 7 × 3 with Monkhorst-Pack method [28]. The cut-off energy was set as 40 Ha and the total energy converged within 4 × 10 -6 Ha. A very recent study revealed that van der Waals interactions play a significant role in the deformation of magnesium, especially for (d) A slice view of electronic structure of the Mg55Y1. Two periodic images are shown.

Computational Details
First principle simulations in this work were carried out by ABINIT code [24,25], accompanied by the projector augmented wave method (PAW) [26]. The exchange-correlation was described by the generalized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) form [27]. The Brillouin zone was sampled using a 4 × 7 × 3 with Monkhorst-Pack method [28]. The cut-off energy was set as 40 Ha and the total energy converged within 4 × 10 −6 Ha. A very recent study revealed that van der Waals interactions play a significant role in the deformation of magnesium, especially for the slip systems with relatively large inter-plane space [29]. Thus, in our simulations, the GGA-PBE functional was corrected by Grimme's DFT-D3 functional to address the long-range electron-electron correlations [30]. Atomic models and electronic structures were visualized with VESTA [31].

Results and Discussions
As shown in Figure 2, there exist two unstable SFEs and one stable SFE on the {10-11}<11-23> GSFE curve of pure magnesium. However, Yin et al. demonstrated that there exist three stable points (SFE1, SFE2 and SFE3) on the {10-11} γ surface [18,32]. The authors believe this is caused by different relaxing procedures. To cover the whole γ surface, the relaxation along the x direction must be forbidden. In this case, the in-plane stress cannot be released, which will result in "wrong" GSFEs [15]. Paradoxically, if the relaxation along the x direction is permitted, the upper (or lower) half of the cell could relax into a local minimum. In other words, the GSFEs will fail to cover the whole γ surface if the in-plane-relaxing is permitted.
Let us continue our discussions with the Mg55Y1 alloy. The electronic structure of Mg55Y1 is shown in Figure 1c,d. In these figures, two periodic images are shown. Figure 1c,d reveal that there are more electrons between the fourth and fifth layers than between the third and fourth layers. This makes the slip mode I and II exhibit different GSFEs.
As listed in Table 1, for the slip mode I, the addition of Y decreases the GSFE0.3b from 175 to 147 mJ m −2 ; but increase the GSFE0.4b and GSFE0.7b from 164 and 318 mJ m −2 to 173 and 339 mJ m −2 , respectively. In this case, GSFE0.4b is larger than that of GSFE0.3b. In other words, the GSFE0.4b is not a local minimum (stable point) anymore. Consequently, the immobile {10-11}<11-23> transformation is preferred under slip mode I. Interestingly, for the slip mode II, the addition of Y increase the GSFE0.3b from 175 to 185 mJ m −2 ; but decrease the GSFE0.4b and GSFE0.7b from 164 and 318 mJ m −2 to 117 and 291 mJ m −2 , respectively.
Decreased GSFE0.7b means that the nucleation of the {10-11}<11-23> dislocations could get easier [33]. The stability of a metastable state is determined by the energy difference between the peak and trough. So, we suggest that a larger value of GSFE0.3b-GSFE0.4b might increase the stability of the stacking fault on the {10-11} planes and promote the mobile <c+a> transformation. To get a global view of solute effects on the behavior of the {10-11}<11-23> dislocations, two "design" maps were constructed, as shown in Figure 3. In these maps, the GSFE0.7b and the values of GSFE0.3b-GSFE0.4b are involved. From left to right the probability of the mobile <c+a> transformation gets increased, corresponding with high ductility. Alternatively, a relatively lower value of GSFE0.3b-GSFE0.4b might promote the immobile transformation. From top to bottom, the nucleation of the {10-11}<11-23> dislocations could get easier. It should be emphasized again the high-frequency <c+a> dislocation switching between the {10-11} and {11-22} planes is the key factor of high ductility [2]. To this end, the dissociated {10-11}<11-23> dislocations must be mobile.
Let us continue our discussions with the Mg55Y1 alloy. The electronic structure of Mg55Y1 is shown in Figure 1c,d. In these figures, two periodic images are shown. Figure 1c,d reveal that there are more electrons between the fourth and fifth layers than between the third and fourth layers. This makes the slip mode I and II exhibit different GSFEs.
As listed in Table 1, for the slip mode I, the addition of Y decreases the GSFE0.3b from 175 to 147 mJm −2 ; but increase the GSFE0.4b and GSFE0.7b from 164 and 318 mJm −2 to 173 and 339 mJm −2 , respectively. In this case, GSFE0.4b is larger than that of GSFE0.3b. In other words, the GSFE0.4b is not a local minimum (stable point) anymore. Consequently, the immobile {10-11}<11-23> transformation is preferred under slip mode I. Interestingly, for the slip mode II, the addition of Y increase the GSFE0.3b from 175 to 185 mJm −2 ; but decrease the GSFE0.4b and GSFE0.7b from 164 and 318 mJm −2 to 117 and 291 mJm −2 , respectively. In pure Mg, the value of GSFE0.3b-GSFE0.4b is relatively small (11 mJ m −2 ). This means that the stability of a {10-11} stacking fault could be relatively lower. In other words, {10-11}<11-23> dislocations could transform into immobile structures or slip onto the {11-22} planes [33]. In Mg55Y1 alloy, the value of GSFE0.3b-GSFE0.4b is 68 mJ m −2 for slip mode II. In this case, the probability of the mobile <c+a> transformation gets increased. This coincides with experimental results that the high ductility of Mg97Y3 (wt.%) alloy is facilitated by the high-frequency <c+a> dislocation switching between the {10-11} and {11-22} planes [2].
The values of GSFE0.3b-GSFE0.4b for Mg55X1 (X = Dy, Er and Ho) are 66, 64 and 65 mJ m −2 under slip mode II, respectively. Experimentally, Sandl‚ obes et al. demonstrated that the addition of 3 wt.% of these elements into pure Mg could increase the ductility in a great scale [34]. Under both slip mode I and II, Bi, Pb and Sn decrease GSFE0.7b dramatically. Moreover, our previous study revealed that Bi, Pb and Sn also decrease the maximum GSFE of {11-22}<11-23> slip systems obviously [17]. But, to the best of our knowledge, there is no evidence that Bi, Pb and Sn can dramatically increase the ductility of magnesium alloys. The authors suggest this is because, for both slip mode I and II, the values of GSFE0.3b-GSFE0.4b for Mg55X1 (X = Bi, Pb and Sn) are very small. Mg55Mn1 exhibits the largest value of GSFE0.3b-GSFE0.4b under slip mode I. Mg55Sc1 has a large value of GSFE0.3b-GSFE0.4b comparable to the value of Mg55ErY. However, the values of GSFE0.7b get increased by 38 and 35 mJ m −2 for Mg55Mn1 and Mg55Sc1, respectively. This means the nucleation of the {10-11}<11-23> dislocations could be relatively hard [33].

Conclusions
In summary, solute effects on the GSFEs of {10-11}<11-23> slip system were studied by means of first principle calculations. Generalized stacking fault energy (GSFEs) of {10-11}<11-23> slip system in a wide range of Mg-X (X = Ag, Al, Bi, Ca, Dy, Er, Gd, Ho, Li, Lu, Mn, Nd, Pb, Sc, Sm, Sn, Y, Yb, Zn and Zr) binary alloys were studied. The doping concentration in the doping plane and the Mg-X system is 12.5 at.% and 1.79 at.%, respectively. Two slip modes (slip mode I and II) were considered. For pure magnesium, these two slip modes were equal to each other. However, substituting a solute atom into the magnesium matrix will cause different effects on these two slip modes, as shown in Figure 2. These findings are also applicable to the GSFEs calculation when the slip plane exhibits a zigzag feature (such as {10-11} plane in Mg, Zr and Ti). Two design maps, as shown in Figure 3, were constructed to predict solute effects on the behavior of the {10-11}<11-23> dislocations. From left to right, the stability of the mobile structure or the probability of the mobile <c+a> transformation could get increased.