The electronic origins of the “rare earth” texture effect in magnesium alloys

Although magnesium alloys are lightweight, recyclable and relatively cheap, they suffer from poor ductility. This can be improved by the addition of rare earth (RE) elements, and this is now a well-established criterion for wrought alloy design. It is notable that this behavior is largely restricted to the lanthanides, but no hypothesis is yet available to explain why other elements do not have the same effect. To answer this question, ab initio simulations of crystallographically complex boundaries have been undertaken to examine the electronic origin of the RE effect. While the electronic structure provided strong bonding between the RE elements and their Mg surroundings, local disruption in atomic arrangement at the grain boundaries was found to modify this effect. This work shows quantifiable changes in electronic structure of solutes resulting from grain boundary crystallography, and is suggested to be a contributing factor to the RE texture effect.


Experimental Background
The ductility [1] and formability [2][3][4] of magnesium alloys is known to be significantly improved by grain size refinement [5,6] and a change in the crystallographic preferred orientation [1]. The preferred orientation, commonly referred to as the "texture", is a statistical measure of the alignment of the grains within a material. The ductility of magnesium alloys is more sensitive to texture variations than other metals, with a randomization of the texture being one of the key goals of alloy design. This is achieved by the addition of one of the rare earth (RE) elements, as these elements are known to be potent texture modifiers. This is a general behaviour observed in the Lanthanide series of elements on the periodic table, along with Y and Ca. This is commonly referred to in the literature as the "rare earth effect".
It has been shown experimentally that RE elements segregate to dislocations [7] and grain boundaries [8][9][10][11][12][13][14][15][16][17], and it is the premise of the present work that the segregation of these solutes to the boundary are responsible for texture development [14]. Why such small concentrations of these elements, in the ppm range [18] can have such a potent effect still remains unclear.
It has also been found that Zn has a synergistic effect with the RE elements, modifying the texture even further. This can be seen in Fig 1 where the addition of Gd weakens the texture [19] (Figs 1a and 1b), but only when both Gd and Zn are present together orientation of maximum intensity moves from the centre of the stereographic projection (Figs. 1a and 1b) towards the transverse direction by a significant amount (Fig 1c) [20]. The same is found for Mg-Ce and Zn (Fig 1d to 1f) [21][22][23]. Although there has not been much speculation in the literature about why this synergistic effect may occur, co-segregation of Zn and RE elements have been observed at symmetrical boundaries [8], and also at general grain boundaries. There has been much speculation about the origin of this texture modification, and it has been suggested that the large atomic size of these elements causes the RE effect [18]. Although atomic size explains the energetic reasons for these elements to segregate to the grain boundary, it doesn't explain how their presence changes the preferential growth of certain boundary types.
Particularly perplexing is the restriction of this phenomena to the RE group within the periodic table, there is clearly something about the electronic structure of these elements that gives them different properties compared to others when alloyed with magnesium. Despite this clear indication that there are important chemical factors at play, the quantum mechanical behaviours of these elements in magnesium have not before been interrogated in depth.
To this end we now move to atomic scale grain boundary simulations to help us understand these experimental findings in more detail. In the present case we utilize first principles calculations, as this provides us with the ability to add different solute species into the simulation and provides a full electron description of the bonding and electronic structure of these elements. This does come with the drawback of the computational investment required 4 for such extensive calculations. For this reason, most studies (for example ref. [24]) restrict themselves to crystallographically simple tilt boundaries to reduce the size of the simulation to less than about one hundred atoms. However, the boundaries seen in experimentally measured materials are rarely simple or symmetrical (for example, those found in the textures shown in Fig 1). Therefore, in the present case, the authors push the size of the simulation to the largest computationally possible size in order to examine non-symmetrical "real" grain boundaries, and these are detailed in Fig 2. The crystallography of the three boundaries chosen for study were derived from the data in Fig 1. It can be seen from the pair distribution functions (PDF) for the three grain boundaries that there is a significant difference in the local atomic packing between the three boundaries, resulting in a co-ordination number for boundaries A, B and C of 7.83, 5.10 and 5.07 respectively. All are markedly lower than the co-ordination number of the bulk, 12. Another quantifiable difference between the three boundaries is the median interatomic distance. This parameter drops from 3.2 Å for boundary A to 2.7 Å for boundary B, and 3.0 Å for boundary C. These measurements of the local disorder at the boundary indicate that each atom at the boundary is surrounded by a smaller number of more closely spaced atoms compared to the bulk. These local surroundings are investigated further in the following section with a specific emphasis on how the local packing at the boundary effects the electronic structure of solutes that inhabit the grain boundary. The solutes investigated in most detail are Gd and Zn.

The effect of boundary topology on electronic structure
The grain boundary is a region of localized disorder, the effect of this local disorder on the electronic structure of atoms located at the boundary has not before been studied. Here we quantify what changes occur in the electronic structure of solutes when located at grain boundaries of different crystallography. Figure 3 shows the projected density of states (pDOS) of Gd and Zn in four different locations: in the bulk, and at grain boundaries A, B and C. For reference, the behaviour of pure Mg is also shown for these same four locations. It can be seen that the effect of the local topology on Mg is small, with negligible difference in the pDOS for magnesium when the atom is located within the perfectly ordered bulk, or the complex grain boundary. If we next consider zinc, the pDOS in the bulk is basically flat for both s-and porbitals. However, when located at the complex grain boundary there is an increase in the pDOS at high energy levels, but these are far above the Fermi level and unlikely to have a large impact on behavior.
As for the Zn, the d-band is also a dominant bonding band, but in this case the Fermi level lies at the top of the d-band. Consequently, most electronic states in the pDOS lie ~7 eV below the Fermi level, Fig 3. The contribution to bonding is therefore markedly reduced as compared to, for example, Gd.
For the Gd s-and p-orbitals, there is negligible difference in pDOS near the Fermi level for the bulk as compared to the three grain boundaries. However, for the d-orbital of Gd, which dominates the total density of states, some fairly large differences can be observed in the pDOS at the Fermi level. Where boundaries A and B have similar pDOS to the bulk, grain boundary C develops a marked rise in pDOS near the Fermi level, indicating that for this solute in this particular boundary, different bonding behaviour can be expected. This is the first concrete evidence that electronic structure of Gd is different at grain boundaries of different crystallography, and this may begin to explain why this element can have such a significant effect on texture development during recrystallisation -the electronic structure (the valence band profile) of Gd is more sensitive to its surroundings than other elements. Gadolinium will therefore behave differently depending on the local crystallography. Nevertheless, for all boundaries, the outer shell d band stays partially occupied, with the Fermi level located at the bottom of the d-band as indicated by the large number of states located at ~0 eV.
It is also pertinent to note that Zn and Mg showed only small changes in pDOS of the bulk as compared to the different grain boundaries, indicating that these species are less likely to exhibit changes in bonding behavior depending on their surroundings.

The effect of solute species on bonding strength
The pDOS can be further analysed to quantify the differences in electronic structure between the solute species that were visually evident in Fig 3. The electronic band structure can be partitioned and the Hamiltonian weighted density of states (or crystal orbital Hamiltonian population, denoted here <-COHP> [26][27][28][29] can be computed. This parameter reveals the nature of chemical bonding, with negative and zero values of <-COHP> representing antibonding and non-bonding states respectively. Of particular importance is its value at the Fermi level, and closely below it, where bonding states imply system stability and antibonding states indicate instability. Table 1 summarises the bonding behvaiour of the solutes at each of the boundaries, and it is seen that although the absolute value of <-COHP> is different between the different boundaries, all show a stable bonding state due to their positive values. Once a stable bonding condition has been confirmed, we can move forward and examine the strength of this bond. In this case we use the integral of the crystal orbital Hamiltonian population, <-iCOHP>. Since the <-iCOHP> does not include unoccupied states above the Fermi level (which can only be occupied due to an external field or higher temperatures), the <-iCOHP> at the Fermi level can justifiably be taken as a representation of the strength of bonds [25,26,30,31].
The <-iCOHP> for each solute and boundary are detailed in Table 1. Let's firstly consider the change in bonding strength at the three different boundary types of crystallography and compare these to the bulk. The bonding strength of for all three species is lower at the boundary compared to the bulk, a result of the local disorder at the grain boundary. Notably, the magnitude of change is largest for Gd, consistent with the notion that this element is more sensitive than others to its local surroundings.
If we now consider the strength of bonding for the three species, we can see that that the strength of the bond between Mg and Gd is usually higher than it is for Mg and Zn. The <-iCOHP> values shown in Table 1 also indicate that the bond between Gd and Zn is larger than either Mg-Gd or Mg-Zn. This may be an explanation for the synergistic effect of these elements when co-located in the microstructure.

The "rare earth" effect
The observation of the differences between Zn and Gd begins to explain the differences in

Closing remarks
It has been argued here that the change in properties that result from RE elements located at the grain boundary is the result of the interplay between the electronic structure of the lanthanide solutes and the local atomic arrangement in the vicinity of a given grain boundary.
On the one hand the outer shell of the lanthanide solute is partially occupied, and the Fermi level lies at its bottom leading to a stable solid solution phase in Mg, and strong chemical bonding with the matrix. On the other hand, the marked disruptions in the local atomic packing of atoms at the disordered grain boundaries, quantified by local PDF's, have been shown to significantly modify the electronic structure of solutes located at those boundaries. This effect was found to varying significantly from one boundary to another, and the change in bonding strength (as quantified by <-iCOHP>) was mainly found to be larger for the RE elements, and lower for those elements that do not modify texture.
It is commonly argued that it is the atomic size of the RE elements that is the underlying feature that gives them unique properties in magnesium. The present quantum mechanical calculations are not inconsistent with this assertion. The large atomic radius of the RE elements make it energetically favorable for RE elements to be located at the grain boundaries, while the sensitivity to the boundary crystallography and enhanced bonding can be traced back to the electronic structure, most notably, of the outer orbital.

Materials and methods
The simulations were carried out using the Vienna ab initio software package (VASP) [33] implementing the projector augmented wave method to represent the combined potential of core electrons and nuclei [34]. The Perdew-Burke-Ernzerhof gradient approximation was implemented to represent the exchange-correlation functional [35]. A cut-off energy of 400 eV was chosen for the plane wave basis and the self-consistent electronic optimization was converged to 10 -6 eV. The mesh of -centered k-points to sample the Brillouin zone were chosen such that their density per reciprocal space is at least 50,000 Å -3 . The atomic configuration is optimized using the conjugate gradient method until the mean atomic forces are less than 0.02 eV/ Å -1 .
The simulation cells were designed to comprise two grains and a large enough vacuum layer on top to prevent interference from the out-of-plane images due to periodic boundary conditions [24]. In order to meet the required periodic boundary condition in the grain boundary plane, without rendering the ab initio simulation unfeasible due to an overly large number of atoms in the cell, the top grain was strained in the boundary plane if required. Details about the simulation cell are given in Table 2 Table 2 -Details of the orientation, size and crystallography of the three grain boundaries. The grain boundary energy (γGB) is also shown.
While the atomic representation of twin boundaries comprises a supercell containing two misoriented grains with a large enough vacuum layer on top [36][37][38][39], the general grain boundary can be considered as a general interface between two grains (G1 and G2) represented by two slabs. As a consequence, the simulation cell total energy is the contribution of the bulk energy of grains ( , 1 and , 2 ), two surface energies ( 1+ 2 ) and the grain boundary energy ( ) itself. Thus, the latter can be expressed as = / − ( 1 + 2 ) − ( , 1 − , 2 )/ and rearranging the right hand side of this equation will give [40]: where WSEP and S are the work of separation defined as the reversible work needed to separate the grain boundary into two free surfaces [41] and boundary area respectively.
When solute is added to the grain boundary, the segregated grain boundary energy can be expressed as = + ∆ Γ where ∆ and Γ are the change in segregation energy of the grain boundary and the conversion factor of segregation energy into energy per area respectively [42].