Atomistic simulations of the interaction of alloying elements with grain boundaries in Mg

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Atomistic simulations of the interaction of alloying elements with grain boundaries in Mg Liam Huber a,⇑, Jo¨rg Rottler a, Matthias Militzer b b

a Department of Physics and Astronomy, The University of British Columbia, 6224 Agricultural Rd., Vancouver, BC V6T 1Z1, Canada The Centre for Metallurgical Process Engineering, The University of British Columbia, 309-6350 Stores Road, Vancouver, BC V6T 1Z4, Canada

Received 18 April 2014; received in revised form 19 June 2014; accepted 24 July 2014

Abstract Quantum density functional theory (DFT) is used to compute binding energies of important alloying elements (Ag, Zn, Ti, Al, Cd, Zr, Y, Ca, Nd, Ce and La) to a R7 grain boundary (GB) in Mg. In particular, quantifying the interaction of rare earth (RE) elements with Mg GBs is of significance given the strong effect of these solutes on modifying the texture of Mg alloys. For most alloying elements studied, the binding energy scales with the size of the solute and the local GB site volumes. Based on these trends, a model for the solute–GB binding energy is developed which accurately captures the behavior of the technologically important RE solutes and Ca. This model is then employed in conjunction with molecular statics calculations to predict solute segregation to general GBs not accessible by DFT. The predicted trends are found to be in qualitative agreement with available experimental data for GB segregation in Mg. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Magnesium; Rare earth; Grain boundary; Atomistic modeling

1. Introduction With its relative abundance and excellent strengthto-weight ratio, there has been a resurgence of interest in using Mg as a structural metal, particularly in the automotive sector [1,2], where concerns for fuel efficiency are driving manufacturers to reduce the weight of their vehicles. A major challenge to this application is the poor formability of Mg, which can be attributed to the development of a strong basal texture during processing [3,4]. Modifications of alloy composition can, in addition to optimizing processing techniques, be employed to weaken the texture, thereby improving formability and other material properties. In particular, additions of rare earth (RE) elements

⇑ Corresponding author.

E-mail address: [email protected] (L. Huber).

and Ca have been found to randomize texture and improve ductility in Mg alloys [5–8]. Studies directed at understanding the function and mechanisms that govern the role of RE elements on texture development in Mg are an area of active research [9–15]. The details of the underlying mechanisms may depend on the actual alloy composition. For example, Sandlo¨bes et al. [15] consider the role of REs on stacking fault energy as a critical aspect for improving the ductility of binary Mg alloys. Other researchers relate the beneficial role of RE additions to complex precipitation and segregation processes, both of which can significantly affect grain boundary migration rates during recrystallization due to pinning and solute drag. In particular, segregation of RE elements to grain boundaries is believed to be an important aspect of the texture modification [9,16,17]. Segregation of the RE elements Y [9] and Gd [18] to Mg grain boundaries has indeed been observed experimentally, as has the

http://dx.doi.org/10.1016/j.actamat.2014.07.047 1359-6454/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

L. Huber et al. / Acta Materialia 80 (2014) 194–204

segregation of Gd to coherent twin boundaries [19]. Robson [10] recently provided an analysis of Y grain boundary segregation using the Cahn–Lu¨cke–Stu¨we model [20,21] for solute drag to explain texture randomization through suppression of dynamic recrystallization. Basu and Al-Samman [14] investigated quaternary Mg alloys and suggested that solute drag and associated texture modifications due to Ce and Gd alloying can be augmented by the addition of the non-RE solutes Zn and Zr. Despite experimental evidence that RE grain boundary segregation may play an important role in texture randomization, there are presently very few theoretical studies available on grain boundary segregation in Mg. Nie et al. [19] confirmed their observations of Gd and Zn segregation to fully coherent twin boundaries with first-principles density functional theory (DFT) calculations. Zhang et al. [22] also used DFT to probe segregation to two sites of a twin boundary for many solutes at moderately high concentration (2:5%). Further, Sandlo¨bes et al. [15] and Shang et al. [23] used DFT to predict the effect of solutes on the stacking fault energy in binary Mg alloys, whereas Wolverton et al. [24] performed DFT calculations to evaluate the stability and elastic properties of precipitates involving RE elements. In light of the experimental significance of the interaction between RE and GBs, developing a reliable first-principles database will be of great use in the design of Mg alloys with improved mechanical properties. In this work, we employ DFT to study systematically the interaction of a variety of solutes, including a selection of the technologically relevant RE species, with a R7 coincident site lattice (CSL) GB in Mg. Due to the high computational expense of DFT calculations, studies of GBs are restricted to a small subset of boundaries with high symmetry and short periodic repeat distances. There have been numerous ab initio studies of solute–GB interaction (e.g. Refs. [25–28]), most of which use either Al (e.g. Refs. [29,30]) or Fe (e.g. Refs. [31–37]) as the host material. Apart from Ref. [19], we are not aware of any comprehensive first-principles studies of solute interaction with boundaries in Mg. By examining the behavior of multiple solute species at the same boundary, we are able to detect a common trend in solute–GB binding in Mg. We show that this trend can be understood in terms of the elastic strain energy between solute atoms and the host lattice, and develop a phenomenological model that captures this behavior quantitatively and more accurately than the commonly used Seah continuum model [38]. Our new model requires as principal input the local GB site volumes, which we obtain for more general GBs from molecular statics using an empirical embedded atom method (EAM) potential for pure Mg. This methodology opens the door to predicting binding energies at GBs that are beyond the reach of DFT. Finally, we estimate trends for solute segregation to several different types of GB using an extension of the Langmuir–McLean model [39] by White and Coghlan [40] and compare them qualitatively to experimental data.

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2. DFT calculations at a R7 boundary 2.1. Computational method Solute–GB binding is the energy difference between the system when a solute, X, is located at a GB site, i, and when it exists in the bulk, isolated from the boundary. To construct these situations efficiently using periodic boundary conditions (PBCs) and conservation of particle numbers, we make four separate energy calculations: ;i EXbind ¼ ðEGB þ EXbulk Þ  ðEXGB;i þ Ebulk Þ

ð1Þ

where the subscripts “GB” and “bulk” denote the structure of the host–a R7 grain boundary or bulk crystal, respectively. Superscripts, where present, indicate the presence of a solute and, in the case of the GB structure, also a site label. We examine only substitutionally placed solutes and perform our calculations with DFT, which explicitly captures the chemical interaction of different atomic species from first principles. Unfortunately, DFT is computationally very expensive, and we are limited to relatively small system sizes of Oð102 Þ atoms. In this paper we consider the hexagonal close-packed (hcp) near-R7ð1 2 3 0Þ½0 0 1 21.8° symmetric tilt boundary. This boundary has been studied before for other metals [41–44], and multiple GB unit cell structures have been proposed. Following Ref. [41], we use two conformations of the boundary: A- and T-type, shown in Fig. 1(a) and (b), respectively. In both structures, atom a3 is created by merging two atoms that are symmetric neighbors across the GB plane, leaving only 13 atoms in the GB unit cell instead of the 14 atoms typically associated with an hcp R7 boundary. The site labeling in this figure is based on distance from the GB plane, and solute binding is calculated to only half of the off-plane sites since their mirror counterparts are equivalent by symmetry. In Fig. 1 and throughout the text, the simple three-index notation is used, which indicates direction by ½na1 na2 nc  multiples of the hcp lattice vectors a1 ; a2 and c. After relaxa˚ tion, the dimensions of the GB unit cell are 8:46 A ˚ in the ½2 1 0 direction and 5:12 A ˚ (5:14 A) ˚ in the (8:51 A) ½0 0 1 direction for the A-type (T-type) boundary – i.e. the unit cell structure of both conformations has the same dimensions to within less than 1%. We also find that ˚ 2 they have remarkably similar energies: 18.7 meV A 2 ˚ (18.6 meV A ) for the A-type (T-type) boundary (299 and 298 mJ m2, respectively). Note that this “grain boundary unit cell” made up of labeled sites repeats periodically only in the plane of the GB, and is aperiodic in the remaining dimension. To fulfill PBCs, we must therefore include a second GB in the simulation domain. This second boundary is never decorated with any solute in our calculations. All DFT calculations were performed using the Vienna Ab initio Simulation Package [45–48] (VASP). Projectoraugmented plane-wave pseudopotentials (PPs) [49,50] were used from the standard VASP library with the Perdew– Burke–Ernzerhof generalized gradient approximation

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L. Huber et al. / Acta Materialia 80 (2014) 194–204 Table 1 Atomic volumes in bulk Mg for investigated species, number of valence electrons used by the PP, and approximate orbital configuration in the solute bulk. Bold indicates the host material. 3

Atom

˚  Volume ½A

PP valency

Configuration

Ag Zn Ti Al Cd Zr Mg Y Ca Nd Ce La

10.9 12.3 13.2 14.4 19.0 20.7 22.8 35.0 39.4 40.3 43.7 44.8

11 12 10 3 12 12 2 11 8 11 11 11

4d 10 5s1 3d 10 4s2 3p6 3d 3 4s1 3s2 3p1 4d 10 5s2 4s2 5s1 4p6 4d 3 3s2 4s2 5s1 4p6 4d 2 3p6 4s2 5s2 6s2 5p6 5d 1 5s2 6s2 5p6 5d 1 5s2 6s2 5p6 5d 1

The bulk supercell of 150 atoms is made of 5  5  3 repetitions of the Mg unit cell vectors. GB supercells use two repetitions of the GB unit cell in the [2 1 0] and [0 0 1] directions. The supercell dimensions of the two conformations differ significantly only in the [4 5 0] direction, where ˚ and the T-type the A-type has a grain thickness of 15.4 A ˚ . This results in 232 (184) atoms in the A-type one of 12.3 A (T-type) supercell. The bulk supercell uses a k-point mesh with 5  5  5 divisions, while the A- and T-type supercells use 3  4  7 and 3  5  7 divisions, respectively. Using these supercells, we estimate an upper bound on error in binding energy due to cell size of approximately 0.06 eV. This is discussed in more detail in Appendix A. 2.2. Results and modeling

Fig. 1. (a) A-type and (b) T-type near-R7ð1 2 3 0Þ½0 0 1 h ¼ 21:8 GB structures in Mg projected onto the ð0 0 0 1Þ plane. The structures pictured are intended to demonstrate the GB unit cell and not the full periodic system used in calculations. Atoms in different basal planes are distinguished by color and size. Ideal positions are shown, and relaxation is indicated by arrows. Basal unit vectors a1 and a2 in one of the grains are also shown. Dashed lines are a structural guide to the eye.

exchange correlation functional [51,52]. For RE solutes other than La, PPs with some f-electrons frozen into the core were used. Table 1 shows the number of valence electrons present in each PP and their ideal configuration in the bulk. Ground state structures were found by allowing all systems to fully relax with respect to both atomic positions and supercell shape and size using a conjugate-gradient ˚ 1. A planemethod with force convergence of 103 eV/A wave basis set with an energy cut-off of 300 eV and a k-point mesh of approximately 20,000 k-points per reciprocal atom (KPPRA) were both found to give binding energies converged to <0.01 eV.

The binding energy of solutes to different sites is shown in Fig. 2(a) and (b) for the A- and T-type boundaries, respectively. We see that all of the solutes larger than Mg (La, Ce, Nd, Ca and Y, red polygons) follow the same trend across various sites, highlighted by the red line that traces their mean binding across the sites. We call these solutes “La-like”. Similarly, many of the solutes smaller than Mg (Ag, Zn, Al and Cd, blue triangles) are also similar to each other across all the sites. The average binding of these “Ag-like” solutes at each site is traced by the blue line. The remaining two solutes, Ti and Zr (green + and ), do not clearly belong to either family. It can be seen in Table 1 that the solutes within a single family differ in their valency and electronic configuration while still following the same trend. Combined with the roughly mirrored trends of the large La-like and small Ag-like solutes, the observed grouping suggests that for many solutes size may be the most important characteristic. We conclude that elastic interaction with the GB dominates subtler electronic effects. Being the farthest away from the GB plane, sites A-f and T-f closely resemble bulk Mg and are expected to have very little binding. This hypothesis was tested for La and Ag, the largest and smallest solutes studied, and, while binding

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no solute, V Xbulk for the volume of the bulk supercell with a solute X, etc. Using these values, we can find the volume of each solute as it exists substitutionally in the Mg matrix relative to the Mg atom it has replaced: DV Xbulk ¼ V Xbulk  V bulk . In conjunction with the Mg atomic volume, V Mg ¼ V bulk =N bulk , where N bulk is the number of atoms, this also allows us to access the absolute solute size in Mg, V X ¼ DV Xbulk þ V Mg . Resulting atomic volumes can be found in Table 1. These volumes are qualitatively well correlated with atomic volumes of the different species. Individual GB sites cannot be isolated like chemical defects, so Voronoi analysis [56] is used to calculate the local volume of each site and this is compared to the bulk, DV GB;i ¼ V GB;i ðVoronoiÞ  V Mg . For a simple and intuitive estimate of the solute pressure, we begin with the definition of the bulk modulus for a material, B  VdP =dV , and isolate for the following first-order approximation of solute-induced pressure: PX  B

Fig. 2. Solute–GB binding energies for (a) A-type and (b) T-type at various GB sites. Lines connect average binding values for like-colored groups.

DV Xbulk V Mg

We calculate B ¼ 36:5 GPa using a unit cell of Mg with the same energy cut-off, KPPRA, etc. outlined in Section 2.1. This agrees well with other measurements and calculations [24,57,58]. The elastic energy of solute–GB interaction can then be written as: ;i EXmodel B

to these sites is not strictly zero, it was found to be small compared to other more favorable sites at the boundary. Thus, f-sites are not considered further in our DFT calculations. Some of the site–solute pairs undergo a structural transformation when allowed to relax, so that the solute ends at a different site than it started at. These transformations involve the GB changing from A- to T-type, or vice versa, as the GB plane shifts either closer to the solute (in the case of a change from an attractive site to more attractive site) or farther away from it (in the case of repulsive interactions becoming less repulsive). While the restructuring of boundaries due to solute segregation is also reported in other simulations [53,54] and observed experimentally [55] as a physical effect, here it could also be related to a finite size effect. More details on this transformation process along with a table of all site–solute binding energies can be found in Appendix B. To further analyze this data, we consider a simple model for the pressure gradient originating from the solute, P X , interacting with the volume difference (relative to the bulk) of the ith GB site, DV GB;i , in a pressure–volume term, DE ¼ P X DV GB;i . Since this analysis relies on various atomic volumes, it is necessary to first clarify how these volumes are calculated. In the process of calculating the component energies in Eq. (1), relaxing with respect to supercell size and shape also yields the total volume of each system, which we describe using a similar notation as the energies: V GB for the volume of the supercell with GB structure and

ð2Þ

DV Xbulk DV GB;i V Mg

ð3Þ

Fig. 3 shows a comparison of the solute–GB binding ;i energy, EXbind , to this elastic model. For site–solute pairs in which a structural transformation took place, the site volume of the end-structure site was used in the model. The model works well for the Ag- and La families, which are of particular technological relevance, but fails to describe the behavior of Ti and Zr, for which elasticity does not dominate interaction with the boundary. A linear fit to the Ag- and La families gives a slope of 1.04 and an

Fig. 3. Calculated solute–GB binding energy plotted against the model P X DV GB;i . The linear fit uses only La-like and Ag-like data.

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intercept of 0.08 eV. A fit with a slope of exactly one and an intercept of exactly 0 eV, along with a small standard deviation would indicate that we had accurately captured all the physics in our model. Indeed, our slope is very close to one. The small but non-negligible intercept warrants further consideration. Both the positive intercept (indicating favorable binding) and the noise around the origin – where at least one of DV Xbulk or DV GB;i is close to zero – may still be the result of elastic interactions, but of a type not captured by our simplified model. The Voronoi volume used to characterize the GB sites considers only the first nearest neighbors of the site and ignores its local neighborhood beyond this shell. However, this larger neighborhood will differ from the bulk as GBs typically have excess free volume. In particular, we found, using Voronoi analysis on relaxed structures, that the GB sites usually appear “soft” to the solutes: a small solute from the Ag family will typically contract its local neighbors more at the GB than in the bulk, while a larger atom from the La family will show a mirrored effect – expanding its immediate neighbors at a GB site farther than in the bulk. This softness makes even GB sites with the same or similar Voronoi volumes to the bulk more attractive, and likely contributes to the positive intercept in Fig. 3. In the bottom-left quadrant of Fig. 3, there are two data points for large solutes at compressive sites for which the model clearly underestimates the repulsive interaction. The very strong repulsive response of these small site–large solute pairs can be understood by considering the anisotropy in interatomic interaction, which must go to zero as the particles are separated, but is strongly repulsive as the atoms are brought together. Our model does not capture the highly non-linear interaction in the compressive spectrum well. For the most repulsive site–solute pairs of the Ag family, where small atoms are placed at very large sites, our model performs better. The details of repulsive sites, however, are of little interest as they are not likely to be occupied. Lastly, we note that Zr and Ti do not fit this model at all. These solutes were already observed to be anomalous in Fig. 2. In Table 1 we see that both of these solutes are smaller than Mg and have more electrons occupying partially filled shells than the other solutes investigated, including a large number of d-electrons. While the valence of the different PPs are often very similar, many of these electrons are members of filled shells. The orbital configurations given in this table are for the bulk configuration of the corresponding species, and we may expect some changes when the element is placed in Mg as a solute. In particular for Zr and Ti, s–d hybridization may be a strong factor. A more extensive study of the behavior of transition metal elements would be of interest, but is outside the scope of this work. While Y and the other lanthanides also have d-electrons, the electronic effects are small compared to the elastic effects for these large solutes. Our results are in good agreement with the recent work of Nie et al. [19], who studied segregation of Gd and Zn to coherent twin boundaries in Mg. In particular, they

observed that, for binary alloys, Gd (which is larger than Mg) consistently segregated to the boundary sites with the most free volume while Zn segregated to compressive sites. In ternary Mg–Gd–Zn alloy, they found Zn and Gd atoms occupying the same columns at the boundary. Their complementary first-principles calculations demonstrated that this observation was likely the result of Zn relieving strain from multiple Gd atoms occupying these sites in a columnar fashion. We observe similar mechanisms for solute–boundary interaction here. We also find good qualitative agreement with the recent work of Zhang et al. [22], who studied solute binding at a twin boundary. While they do not provide the volumes of the two twin sites, we find that the most favorable binding energies at our two boundaries are similar in magnitude for each solute to the most favorable binding energies they report. We note that their study uses a much smaller supercell, with 80 atoms, two of which are solutes, while we utilize at least 184 atoms in our GB supercells, with a single solute atom. Their effective solute concentration is therefore much higher. Compared to other elastic energy models for describing the solute–GB interaction, such as the continuum approximation of Seah [38,39], our model is simpler, makes no assumptions about the shape of the GB site, and does not require calculations of unstrained size and bulk modulus for each solute species. Further, when we applied the Seah model, we found that, while it predicts qualitative trends in binding for La- and Ag-like solutes reasonably well, it underestimates the magnitude of solute–GB interaction by a factor of 4.4 on average when compared to our DFT results. 3. General boundaries 3.1. Site volume fractions For the large solutes that are described well by our model, the only GB information we need to predict solute–GB binding energy is the local site volume. In principle, this can be obtained by modeling the GB using a well-constructed classical potential. Not only does this sidestep the most expensive component of the binding calculation, the solute–GB simulation, but the much larger scale of system sizes (easily 106 atoms) permits calculating solute interaction with GBs that are strictly inaccessible with DFT. In pursuit of this goal, we use the EAM potential for Mg by Mendelev et al. [59], rescaled so that the V Mg matches the bulk value from our DFT simulation. Molecular mechanical (MM) calculations were performed using the LAMMPS [60,61] code, allowing the system to be fully relaxed to a 0 K minima using conjugate gradient with a ˚ 1 . As a test, we force convergence criterion of 108 eV=A first replicated our A-type and T-type simulation cells in MM and compared the site volumes to the DFT result. The T-type boundary is found to be unstable using this

L. Huber et al. / Acta Materialia 80 (2014) 194–204

potential, but the potential accurately reproduces A-type site volumes to within 2%. Next, a general tilt GB and a general GB were constructed and relaxed, using the same basic system setup with PBCs and two GBs in the simulation cell, and their site volume fractions were calculated. The tilt boundary is constructed by generating two new crystal lattices through rotations about the [0 0 1] axis and by translations modulo the lattice vectors of the two-atom Mg unit cell (the basal lattice vectors are shown in Fig. 1.) The GB cut is made in the (0 1  1 0) plane of the unrotated coordinates. Rotation and translation values were randomly generated [62], then the simulation domain was chosen such that both grains made a good approximation of repeating periodically: strains in the plane of the GB were <0.2% and the positions were rescaled so that each grain accommodated the existing strain equally in magnitude and the ˚ of their ideal positions in the atoms were within <0.1 A dimension normal to the GB plane. Our particular tilt boundary has grains rotated by 174.6° and 105.4°, uses only a single unit repetition in the [0 0 1] ˚ . This direction, and has a grain thickness of 30 A results in a rectangular supercell with dimensions ˚  582:7 A ˚  5:18 A ˚ prior to relaxation. Atoms from 60:0 A ˚ of each other the different grains which were within 2.0 A before relaxation were deleted, giving an initial GB expansion (the volume difference between the GB cell and a corresponding number of bulk atoms, normalized by the GB ˚ 3 =A ˚ 2 , which is similar to those observed area) of 0:62 A experimentally [63,64] for other systems. The last step before relaxation was common neighbor analysis [65,66] ˚ to identify GB sites. (CNA) with a cut-off distance of 3.85 A Part of this structure can be seen unrelaxed in Fig. 4, which gives the reader a sense of scale compared to the R7 boundary used in the DFT calculations. After relaxation, the ˚  581:4 A ˚  5:18 A. ˚ supercell dimensions are 58:8 A Construction of the completely general boundary is similar, but Bravais lattices for the two grains are rotated about the [2 1 0], [0 1 0] and [0 0 1] axes (in that order), and co-periodic dimensions for the grains are now non-trivial in both GB plane dimensions. Strains in the GB plane ˚ deviation was allowed from ideal are < 0:01%, but 0:3A positions in the direction normal to the plane. The same grain thickness was used such that for rotations of h½210 ; h½010 ; h½001 ¼ ð26:2 ; 120:7 ; 73:1 Þ and ð56:7 ; 83:1 ; 24:3 Þ, the unrelaxed (relaxed) supercell dimensions are ˚  574:9 A ˚  804:0 A ˚ ð59:4 A ˚  576:1 A ˚  801:6 AÞ. ˚ 60:0 A There is no convenient projection for two-dimensional viewing of this grain as there is with the tilt boundary. Site volume fractions predicted by the MM calculations for these two general boundaries and the R7 boundary can be seen in Fig. 5. Using CNA to identify GB sites yields ˚ – roughly the GB GBs with widths of approximately 6 A plane and a monolayer of atoms to either side. We see that the R7 boundary is reminiscent of the more general boundaries but, due to its much smaller number of unique sites, it does not have the extensive large-volume tail that the other

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Fig. 4. Unrelaxed general [0 0 1] tilt boundary with atoms projected onto the (0 0 0 1) plane. GB sites are shown in blue. Atoms in different basal planes are shown with different sizes. The image shows the entire periodic domain perpendicular to the GB plane, but shows only a small subsection along the boundary. The arrows show basal lattice parameters to indicate misorientation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

two boundaries have. For a given solute, the binding energies predicted by our elastic model can be obtained directly from these site volume fractions by subtracting the bulk volume, DV GB;i ¼ V GB;i ðMMÞ  V Mg , and multiplying by the solute pressure, P X  BDV Xbulk =V bulk . That is, solute– GB binding energy fractions have the same shape as the site volume fractions, needing only a change in the horizontal axis from volume to the appropriate energy scale. 3.2. Solute segregation We might suppose that since the R7 boundary samples the most populated volumes of the two general boundaries well, it provides a good representation of these boundaries. To gauge the physical consequence of the differences between the R7 GB and general boundaries, we use the simplest available model for solute segregation: the Langmuir– McLean theory [39] for the segregation of non-interacting solutes. Since the segregation relies exponentially on the solute–GB binding energy, we find that the large-volume tails (and thus strong binding for the La family) of the general boundaries result in qualitatively different behavior from the R7 boundary, which has only a few distinct large-volume sites. The Langmuir–McLean theory for segregation [39] of a solute to a GB gives the concentration at the GB, X GB , in terms of the nominal bulk concentration, X b , the solute– GB binding, E, and the temperature, T , as

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Fig. 5. Site volume fractions for (a) R7, (b) a general tilt and (c) a 3-D ˚ 3 bin size. Bulk site volume shown with general boundary using a 0:1 A dashed line.

X GB Xb ¼ expðE=k B T Þ 1  Xb  X GB

X 0GB

where F i is the fraction of GB sites which have binding Ei . Since the contribution to the total segregation from unfavorable sites is explicitly calculated and repulsive energies are permitted, X 0GB;i ¼ 1 for all i in this model. Except at very high temperatures, when all sites are far from full occupation, the contribution of repulsive sites will be exceedingly small compared to sites with favorable binding. As T ! 0 in the White–Coghlan model, we see that X GB goes to the ratio of attractive sites to total sites, as desired. While the single binding energy in the basic Langmuir– McLean equation produces a sigmoidal curve when the GB concentration is plotted against temperature, the White–Coghlan formulation results in a superposition of many of these curves. The total contribution of each term in the superposition is controlled by F i , and the temperature at which the term goes from unoccupied to saturated is controlled by Ei and X b . Both the Langmuir–McLean and White–Coghlan models ignore solute–solute interactions. White–Coghlan model predictions for Y segregation to our three GBs are shown in Fig. 6. These are calculated using site binding energies from our phenomenological model, which relies on DFT values for solute pressure (Eq. (2)) and MM relaxed GB site volumes. The contribution of individual terms of the White–Coghlan model superposition of Eq. (5) is best seen in the Y segregation to the R7 boundary in Fig. 6: we observe that the R7 boundary plateaus at a concentration of 24% at a temperature of 500 K. Referring back to Fig. 5, recall that this boundary has three sites with site volumes significantly larger than bulk volume (A-a3, A-d and A-d0 ), to which Y binds favorably. Since 24%  3=13, we infer that these three sites saturate very quickly (this can be confirmed by looking at the individual site concentrations prior to summing.) Following this, there is a period in which very little further segregation occurs until the temperature drops sufficiently for segregation at sites only slightly larger than the

ð4Þ

where X 0GB indicates the saturation of the GB at 0 K– i.e. the fraction of sites with favorable binding. This concept has been extended to segregation at a boundary with a distribution of binding energies by White and Coghlan [40], by considering segregation to sites with different binding energies, Ei , separately as    1 1  Xb X GB;i ¼ 1 þ ð5Þ expðEi =k B T Þ Xb then performing a weighted average of concentrations, X X GB ¼ F i X GB;i ð6Þ i

Fig. 6. Y concentration at the R7, general tilt and 3-D general boundaries as a function of temperature using Eq. (6) and nominal bulk concentration X b ¼ 0:75%.

L. Huber et al. / Acta Materialia 80 (2014) 194–204

bulk sites to commence. For the general boundaries, the much smoother distribution of site volumes results in a very different segregation profile, completely lacking the plateau of the R7 boundary. In all three cases, the concentration curves go to a 0 K saturation value that is the ratio of attractive to repulsive sites. One may expect that the inclusion of solute–solute interactions in the segregation model would decrease the Y segregation as the adjacent oversized solutes are likely to repel each other. While more complex models incorporating such interactions exist, they are not included in the White–Coghlan model. It is possible to probe these interactions using DFT, and some work has been done for solutes at high-symmetry boundaries in Fe [67], but such studies have not yet been performed in Mg. Nie et al. [19] observed entire columns in ð1 0  1nÞ twin boundaries (n ¼ 1; 2; 3) becoming extremely Gd-rich, which indicates that the prediction of high segregation levels is likely to be qualitatively correct – at least for REs like Y and Gd – despite omitting solute–solute interactions. Note that all of the input we are using for these models comes from DFT and EAM calculations at 0 K. In principle, the binding energy may change at higher temperatures as a result of harmonic and anharmonic effects, and even GB structures may change at very high T , i.e. near the melting temperature. Nevertheless, the binding energies at 0 K usually provide a good estimate of trends over different types of GBs for a wide temperature range. It is also interesting to note that the much smoother distribution of site–solute binding energies for the two general boundaries means that over any small temperature range there is some small fraction of sites which are going from unoccupied to saturated. This continuity gives rise to concentration–temperature profiles which are qualitatively distinct from the special R7 boundary. The vertical shift between the general GBs can be understood as a result of the larger average site volume and the longer large-volume tail of the general boundary we used when compared to our tilt boundary. For the general and to a lesser degree the special boundary cases, the sigmoidal curve of the Langmuir–McLean equation does not approximate the physics of a boundary with multiple binding energies well. For our general boundaries, there is no fit of the Langmuir–McLean equation to X GB that returns a single temperature-independent binding energy that produces segregation profiles qualitatively similar to the White–Coghlan model. The assumption that a boundary may have multiple favorable sites with different binding energies is reasonable and well supported by first-principles calculations [27,28,37], and unless the distribution is very narrow will result in segregation profiles which are very different from those predicted by the Langmuir–McLean model alone. In addition to the work of Nie et al. discussed earlier, Stanford et al. [18] have experimentally studied Gd segregation to GBs in Mg, and Hadorn et al. [9] have examined Y and Zn segregation. At similar temperatures, our work is

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in qualitative agreement with their observations that Gd and Y segregate strongly to general boundaries while Zn does not. Hadorn et al. also observed that segregation to different GBs varies, which we too find to occur. At present there is not sufficient experimental data over a wide enough temperature range to make a quantitative comparison. 4. Conclusion Using DFT calculations, we investigated the binding of solutes to R7 boundaries in Mg. We found for the large, technologically relevant solutes in the RE family and Ca that the binding is very favorable and scales well with the elastic interaction between the pressure on the host matrix induced by the solute and the volume difference between specific GB sites and the bulk. This is in agreement with, and expands upon, the experimental and computational observations of Nie et al. [19] of the segregation of Gd and Zn to extensive and compressive sites, respectively, in twin boundaries. Using MM, we constructed a general tilt and a completely general boundary, which, along with the construction of the same R7 boundary used in DFT, provided the GB site volumes necessary to calculate solute–GB binding with a phenomenological model. This allowed us to extend our study beyond the limited family of high-symmetry boundaries accessible by DFT. To the authors’ knowledge, these MM site volume distributions by themselves represent a novel investigation of the structure of special and general boundaries. We utilized the calculated binding energies in the White– Coghlan model [40] for solute segregation, which is an extension of the well-known Langmuir–McLean equation [39] incorporating multiple binding energies at a given boundary. Our results agree qualitatively with experiments for Y and Gd [18,9]. The White–Coghlan model does not account for solute–solute interactions, which could in principle be investigated in some detail using DFT. Results for segregation demonstrate that the Langmuir– McLean model is insufficient to capture the segregation behavior to a boundary with many distinct binding energies. Further, since the R7 boundary has relatively few distinct sites, its segregation profile as a function of temperature differs qualitatively from the more general boundaries. Over any small temperature change at these general boundaries, the large variety of different site volumes results in some small fraction of sites transitioning from unoccupied to occupied as binding energy becomes more important than entropy. This is true despite the strong overlap in the most common site volumes for all three boundary types studied. We expect to find such differences between any CSL boundary with few unique sites and general boundaries with a more continuous distribution. This underlines the need for efficient, reliable methods, which allow the study of grain boundaries whose periodicity lies far beyond the system sizes currently accessible by DFT.

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Acknowledgements The authors acknowledge the Natural Sciences and Engineering Research Council of Canada (NSERC) for providing financial support under its strategic network MagNET [68] and an Alexander Graham Bell Doctoral Canada Graduate Scholarship (L.H.). The authors are also grateful to WestGrid and Compute/Calcul Canada for providing computational resources. We would like to thank I. Elfimov, D. Trinkle and G. Leyson for stimulating discussions. Appendix A. Size convergence To converge with respect to supercell size, we must ensure that our two grain boundaries are effectively noninteracting and that the solutes do not interact with their own periodic images. To test the former, we simply check the GB energy – the energetic difference between the system with two GB planes and the same number of atoms in a bulk configuration, divided by the area of both planes – as a function of grain thickness. We find a value of ˚ 2 (18.6 meVA ˚ 2 ) for the A-type (T-type) 18.7 meV A boundary (299 and 298 mJ m2, respectively). Expanding ˚ to 30.1 A ˚ in the A-type the grains’ thickness from 15.4 A boundary changes this GB energy by less than 1% and we consider it well converged. To test the latter, we examine the changes in La–GB binding energy with changing system size. It should be noted that the bulk and GB portions can be decoupled such that the convergence of EGB  EXGB;i and EXbulk  Ebulk are studied separately. Table A.2 shows EXbulk  Ebulk using different multiples of the Mg two-atom unit cell. Convergence in different dimensions of the supercell are not independent, but, in order to avoid prohibitively large system sizes, each dimension was extended individually. The effects of changing GB unit cell repetitions and grain thickness are shown in Table A.3. In all cases, the binding of La to site A-a3 is used. La was chosen because its large size and strong elastic field make it most likely to interact with its own images at the largest distances. Performing error propagation under the simplifying assumption that convergence is independent in each dimension, we calculate an estimated size convergence error of 0:06 eV. Since La is the largest solute studied, we take this as an overestimate of error for other solutes. This issue of size convergence is discussed in more detail by Lejcˇek et al. [69] in the context of solute solid solubility. Despite using

Table A.2 Size convergence for La in bulk Mg. System dimensions given in N a1 ; N a2 ; N c multiples of the two-atom unit cell vectors. N a1  N a2  N c

k-Mesh

ELa bulk - Ebulk [eV]

553 663 554

555 446 554

3.262 3.220 3.283

Table A.3 Size convergence for La at site A-a3 as a function of relaxed supercell parameters (directions d ½4 5 0 ; d ½2 1 0 and d ½0 0 1 can be seen in Fig. 1). ˚ d ½4 5 0 ½A

˚ d ½2 1 0 ½A

˚ d ½0 0 1 ½A

k-Mesh

ELa;Aa3  EGB ½eV GB

30.8 60.2 30.8 30.8 30.8 30.8 30.9

8.5 8.4 17.0 25.4 8.5 8.5 16.9

5.1 5.2 5.1 5.1 10.3 15.4 10.3

3  7  19 2  7  13 3  5  13 3  3  13 397 385 347

3.979 3.997 3.922 3.905 3.740 3.710 3.795

system sizes moderately larger than discussed in that work, we still see that good convergence (<0.01 eV) is difficult to obtain for poorly soluble solutes like La in Mg. Appendix B. Restructuring The A- and T-type conformations of the R7 GB are topologically close and, through rearrangement of the atoms near the GB plane, it is possible to transition from one conformation to the other when a solute is placed at the boundary. While our solute concentration in GB supercells is small (< 0:5%), the concentration at the GB is moderate, at 1:9%, one atom out of the 52 in our GB plane being made from 2  2 repetitions of the 13 atom GB unit cell. Because we use PBCs, placing a solute at a particular site creates a superlattice where this site is occupied in all periodic images of the supercell. As a result, if a solute is placed at a site where it facilitates GB restructuring, all its periodic images have the same effect. Thus, we do not have a 1:9% GB concentration of randomly distributed solutes, but of organized solutes. While structural transformations due to solute segregation have been reported elsewhere [53–55], it is likely that the transformations we observe here are finite size effects resulting from the superlattice GB occupation just described. When these A ! T/T ! A transitions occur, the GB plane must shift slightly. When the solute is sitting at an attractive site and it undergoes a transition, the GB plane moves towards the solute and it ends at a site that is even more attractive. Conversely, if a solute at a repulsive site induces the GB to restructure, then the GB plane moves so that the solute sits farther away at a more bulk-like position. All the site–solute pair binding energies can be found in Table B.4, with the pairs that underwent a transition highlighted in bold. In a small number of cases, the presence of the solute destabilized the system and the two GBs in the supercell annihilated each other, leaving a bulk-like structure. No energies are reported for these cases. As a useful example, we look at La family binding to site A-a1. This site is compressive and we expect a repulsive interaction. Referring to Table B.4, we see that this is indeed the case. Here Y and Ca, the two smallest members of the La family, exhibit the most repulsive interactions, but do not precipitate any GB restructuring. The three

L. Huber et al. / Acta Materialia 80 (2014) 194–204

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Table B.4 DFT solute–GB binding energies in eV. Bold values indicate that a structural transition took place, and in all cases the “Site” column refers to the initial site the solute is placed at. A value of N/A indicates that the boundaries annihilated. Site

Ag

Zn

Al

Cd

Y

Ca

Nd

Ce

La

Ti

Zr

A-a1 A-a2 A-a3 A-b A-c A-d A-e

0.31 0.01 0.00 0.31 0.03 0.08 0.29

0.33 0.02 0.03 0.28 0.01 0.07 0.32

0.17 0.04 0.01 0.20 0.00 0.10 0.17

0.15 0.01 0.03 0.17 0.02 0.05 0.14

0.13 0.06 0.20 0.01 0.12 0.71 0.02

0.10 0.04 0.35 0.04 0.12 0.76 0.05

0.04 0.08 0.38 0.06 0.17 0.94 0.01

0.01 0.10 0.49 N/A 0.21 1.07 0.00

0.02 0.08 0.53 0.02 0.23 1.13 0.01

0.15 0.06 0.27 0.17 0.02 0.03 0.09

0.16 0.07 0.18 0.01 0.03 0.21 0.05

T-a1 T-a2 T-a3 T-b T-c T-d T-e

0.28 0.03 0.42 0.27 0.03 0.02 0.28

0.26 0.01 0.34 0.30 0.01 0.03 0.19

0.18 0.02 0.33 0.15 0.02 0.03 0.12

0.15 0.02 0.18 0.12 0.01 0.03 0.09

0.43 0.09 0.65 0.05 0.04 0.14 0.11

0.31 0.11 0.73 0.07 0.02 0.30 0.11

N/A 0.14 0.88 0.05 0.06 0.31 0.19

N/A 0.18 1.00 0.02 0.08 0.42 0.22

N/A 0.20 1.06 N/A 0.06 0.47 0.24

0.19 0.01 0.17 0.00 0.04 0.29 N/A

0.38 0.00 0.15 N/A 0.05 0.22 N/A

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