Electronic origin of the anomalous solid solution hardening of Y and Gd in Mg: A first-principles study

Y and Gd demonstrate anomalous solid solution hardening efficiency, which cannot be understood using the elastic impurity-dislocation interaction theory. We performed first-principles calculations to investigate the effect of different alloying elements such as Al, Zn, Y, and Gd on the chemical bonding of Mg solid solutions. The present calculations clearly show that the anomalous solid solution hardening of Y and Gd in Mg may be understood based on the increased bonding strength of both Mg-Y (Gd) and Mg-Mg.

As the lightest metallic structural alloy with a good combination of castability, mechanical properties and ductility, magnesium alloys have attracted increasing attention in the automotive and aerospace industries [1]. It has been reported that the addition of rare-earth (RE) elements, such as Y, Gd, Ce, and Dy, significantly improves the mechanical properties of Mg alloys at ambient as well as at elevated temperatures [2,3]. The strengthening by the RE elements is more substantial than any other elements that have been added to Mg alloys [4,5]. One of the most effective strengthening mechanisms of RE in Mg is solid solution hardening, essentially of either elastic or electronic origin. The elastic origin has been quantitatively established based on the impurity-dislocation interaction theory by Fleischer [6] and Labusch [7], and is correlated with the atomic size factor (δ) and/or the modulus misfit parameter (η). For a number of alloying elements (Zn, Al, Pb, Li, Tl, Cd, In, etc.) in Mg the efficiency of solid solution hardening increases monotonically with the increase of |δ | or |η|, or a *Corresponding author (email: rongshichen@yahoo.com) combined misfit parameter of them [8,9]. This relation has been verified for aluminium [10], iridium [11], copper [12], nickel [13], and stainless steel [14] solid solution alloys. However, recent experimental studies [4,5,15,16] have demonstrated that Y and Gd exhibit anomalously higher solid solution hardening efficiency than those of Al and Zn in terms of the misfit parameters, which cannot be understood using the elastic interaction model. Such anomalous solid solution hardening of other RE elements, such as Ce [17] and Dy [18], was also reported. A few experimental and theoretical studies have been carried out to explore the solid solution hardening of RE in Mg. However, the nature of this anomalous behavior has not yet been explained in the literature. Miura et al. [18] evaluated the elastic interaction based on isotropic or anisotropic distortion strains by solute atoms, and it was found that the elastic strain by Zn atoms is higher than those of Y and Dy, while the solid solution hardening of basal slip by Y or Dy addition is much higher than that by Zn addition. Ninomiya et al. [19] have proposed another solid solution hardening model based on a discrete variational-Xα molecular orbital calculation, but the details of the solid solution hardening mechanism of Mg-RE alloys have not been elucidated yet.
In this paper, we have studied the solid solution hardening from the view point of chemical bonding by means of first-principles calculations based on the density functional theory. The aim of this work is to explain the anomalous solid solution hardening of Y and Gd in Mg from the electronic origin. Our findings are important for the further understanding of the alloying behavior in a simple metal, and are useful for the design of commercial magnesium alloys.

Computational methods
Magnesium has a hexagonal close-packed (hcp) lattice of space group P6 3 /mmc with two atoms in a unit cell. In the present work, Mg 53 X 1 solid solutions corresponding to a doping concentration of 1.85 at.% were modeled by a 3×3×3 supercell of magnesium (Mg 54 ), in which one magnesium atom was substituted by a solute atom X (X = Al, Zn, Y, and Gd) at the center site of the supercell. Primary test calculations demonstrated that the results were independent of the doped position. The Vienna Ab initio Simulation Package (VASP) [20] based on the first-principles density functional theory (DFT) was used. The projector augmented-wave pseudo-potentials [21,22] were used to depict the interactions between the valence electrons and the core. The generalized gradient approximations (GGA) in Perdew-Wang-91 (PW91) scheme [23] was adopted for describing the exchange-correlation interactions. An energy cut-off of 1.25 times the maximum potential energy of either Mg or X (whichever element has the higher default cut-off energy) as given in their pseudo-potential files was performed. The relaxation convergence for ions and electrons was 1×10 -6 eV. The integration in the Brillouin zone was made using Monkhorst-Pack [24] 5 × 5 × 5 k-points for the hcp supercell model. The convergence according to k-points has also been checked and the k-points of 5 × 5 × 5 are sufficient to give good results. The atomic arrangements were relaxed using the second-order Methfessel-Paxton technique [25] with a width of 0.2 eV. The tetrahedron method with Blöchl corrections [26] was used for the total energy calculation. The partial density of states (PDOS) and electron localization function (ELF) were obtained using the relaxed structure at the equilibrium volume.

Results and discussion
The calculated lattice parameters a and c/a, equilibrium volume V 0 , and cohesive energy E coh of pure Mg together with the available experimental data [27][28][29] and previous calculations [30][31][32] are shown in Table 1. It is clear that the present calculations agree well with the experimental values and previous calculations, with the difference being less than 3% for all the results. Figure 1 shows the theoretical equilibrium volume of Mg-X solid solutions as a function of solute composition. The values of the atomic size factor (δ) for each element can be determined from Figure 1 according to the method developed by King [28], as also shown in Table 1. The calculated atomic size factor of Al and Zn are -12.91% and -18.13%, respectively. Reasonable agreement between our calculations and the experimental data of King [28] (-13.73% and -20.00%, respectively) as well as with earlier calculations by Uesugi [30] on Al (-12.40%) is obtained. The calculated atomic size factor of Y and Gd are +15.63% and +17.29%, respectively. No experimental or theoretical data of δ for Y and Gd has been reported so far. Noting that the atomic size factor (absolute value) of Y and Gd are smaller than that of Zn, which cannot explain the prominent difference in strengthening efficiency between Mg-Y (Gd) and Mg-Al (Zn) alloys. Therefore, this anomalous solid solution hardening behavior may be due to the specific chemical bonding, as will be discussed in the following section using the Mg 53 X 1 solid solutions as an example.  In order to gain better understanding of the chemical bonding in Mg 53 X 1 , we have calculated the partial density of state (PDOS), which is shown in Figure 2. It is seen that the s-and p-electron states of alloying elements are spread in a relatively wide energy range. It is also interesting to note that the PDOS of Mg 53 X 1 can be divided into two groups, one consisting of Mg 53 Al 1 and Mg 53 Zn 1 , and the other consisting of Mg 53 Y 1 and Mg 53 Gd 1 . For Mg 53 Al 1 and Mg 53 Zn 1 (Figure 2 states of Y (Gd). Strong hybridization between the p-orbital of Mg and the d-orbital of Y (Gd) atoms is also observed. This hybridization presents covalent bonding characteristics identified by the electron localization function (ELF) as shown in Figure 3 and plays a major role in the solid solution hardening in Mg-Y (Gd) alloys. The results clearly show similar chemical bonding characters between Mg 53 Al 1 and Mg 53 Zn 1 , and that between Mg 53 Y 1 and Mg 53 Gd 1 .
A rather quantitative understanding on the chemical bonding of Mg 53 X 1 solid solutions can be gained by analyzing the electron localization function (ELF). Figure 3 shows the ELF contours in the (0001) plane for Mg 53 X 1 , as well as that of Mg 54 for comparison. For Mg 54 (Figure 3(a)), a typical metallic bonding feature with ELF of 0.5 (green background) is observed together with weak covalent bonding (ELF = 0.7) between the adjacent Mg atoms. Significant difference in the chemical bonding caused by the introduction of a solute atom can be observed. Like the above analysis of PDOS, ELF of Mg 53 X 1 can also be divided into two similar groups. For Mg 53 Al 1 and Mg 53 Zn 1 (Figure 3(b) and (c)), electrons at around Al (Zn) extend to two neighboring Mg atoms forming quite weak covalent bonds, for example, the Mg1-Al-Mg2 and Mg1-Zn-Mg2 covalent bonds, with the later bond being stronger than the former. Furthermore, the introduction of Al (Zn) results in the increased bonding strength between some neighboring Mg atoms, for example, between Mg2 and Mg3 in Figure  3(b) and (c), which is clearly seen by the large ELF values. For Mg 53 Y 1 and Mg 53 Gd 1 (Figure 3(d) and (e)), the covalent interaction among the Mg-Y(Gd)-Mg bonds is much stronger, suggesting the increased bonding strength. This enhanced bonding strength is attributed to the p-d covalent interactions, according to the PDOS analysis (Figure 2(c) and (d)). This further indicates that the hybridization of the valence electrons between the Mg and Y (Gd) atoms plays a decisive role in solid solution hardening in the Mg-Y (Gd) alloy. Furthermore, compared with Mg 54 , the covalence bonding among some Mg atoms which are close to Y (Gd) is also increased, for example, between Mg3 and Mg4 in Figure 3(d) and (e), which is not the case in Mg 53 Al 1 and Mg 53 Zn 1 . Therefore, the substitution of Y or Gd in Mg results in significant enhancement in the chemical bonding not only between the Y (Gd) and Mg atoms, but also between the Mg atoms. The covalent bond interactions in the Mg-Y (Gd) and Mg-Mg bonds are responsible for the anomalously higher solid solution hardening efficiency of RE elements in Mg.

Conclusions
In summary, we have studied the chemical bonding of Mg 53 X 1 solid solutions, where X = Al, Zn, Y, and Gd, by means of first-principles calculations. Reasonable agreement between calculated and measured data is observed, thus validating the first-principles methodology employed. The addition of Y (Gd) to Mg affects the atomic binding states not only between Y (Gd) and Mg atoms, but also between Mg atoms. The mechanism of the anomalous solid solution hardening of RE elements in Mg may be understood based on the increased bonding strength of both Mg-Y (Gd) and Mg-Mg. This suggests the important role of the electronic origin of the solid solution hardening.