Introduction

Magnesium (Mg) alloys have tremendous potential as structural materials for automotive applications in engine block, transmission parts due to their lightweight. The concomitant fuel savings is however, thwarted by their unfavorable creep strength1,2,3,4. Mg-rare earth (RE) alloys are exceptions with demonstrated excellent creep resistance1,2,3. Their superior creep properties are correlated to the formation of high volume fractions of strengthening precipitates based on RE intermetallic compounds4,5. However, the scarcity and high cost of RE elements limits the development of creep–resistant Mg-RE alloys. This work is motivated by gaining an atomistic understanding of commonly found strengthening precipitates in creep-resistant Mg-RE alloys, and inform Mg-alloy design approaches by providing energetics and bonding characteristics associated with their presence within the hcp-Mg matrix. Therefore, we have investigated the structure of interface between intermetallic precipitate phase β1 (an ordered cubic structure) and hcp-Mg. This precipitate typically forms in Mg-Nd-based alloys like Mg-Nd, Mg-Nd-Y and Mg-Nd-Y-Zr, e.g. commercial WE43 and WE54 5,6,7. The coherent β1 phase has a plate-like morphology, and shares an (0001)Mg//(011)β1, [100]Mg//[21]β1 and [110]Mg//[11]β1 orientation relationship (OR) with hcp-Mg5,6,7,8,9,10,11,12. TEM results also indicate that plate-like β1 precipitates always forms with a high aspect ratio, and its broad-face {21}β1 parallel to {100}Mg 4,5,6,7,8,9,10,11. Furthermore, β1 nucleating within the Mg-matrix is associated with the orthorhombic β′ precipitate phase5,7, or as self-accommodating β1 triads5,11, and heterogeneously along dislocation lines via accommodation of β1 stress-free transformation strains associated with β1 formation7,8,9,13,14. Heterogeneous formation is particularly useful in restraining high-temperature creep deformation, because its dynamic nucleation arrests dislocation movement and enhances the creep-resistance of Mg-RE alloys4,9,10,11.

The formation tendencies of β1 precipitates have been characterized experimentally5,6,7,8,9,10,11, and its evolution within Mg matrix and on dislocations studied via phase-field models12,13,14. Despite extensive studies on β1 phase, basic thermodynamic quantities like β1/Mg interfacial energies are yet to be determined unequivocally. Quantitative understanding of β1/Mg interfaces is crucial for three reasons. First, atomic resolution transmission electron microscopy (TEM) studies have shown that the{112}β1/{100}Mg interfaces form both longer and shorter edges of β1 plates11. This is surprising. One would have expected at least an additional {111}β1/{110}Mg interface, because both {112}β1 and {111}β1 planes take part in forming the hcp-Mg/β1 orientation relationship. Second, a well-developed β1 plate shares a substantial interfacial-area with the parent Mg-matrix. Thus the interfacial energy, in conjunction with transformation strain energy, will likely determine the plate-like morphology12,13,14,15. Therefore, determination of hcp-Mg/β1 interfacial energies will provide valuable inputs to refine phase field models12,13,14,15. Third, this study allows us to rationalize whether these coherent interfaces have lower excess energies16.

Density functional theory (DFT)-based first principles calculations have been used in the past to determine interfacial energies of coherent precipitates in Mg-RE alloys14,15,17. So far, DFT studies on Mg-Nd have largely examined the stability of the β″ (ordered hcp structure5) and the β′ precipitate phases15,17; both precipitate phases form at early stages prior to β1 precipitation. For example, Issa et al. reported that β″ have positive and negative interfacial energies, when it joins with basal {0001}Mg and prismatic {100}Mg planes respectively15,17. Based on these interfacial energy calculations they proposed that β″ is unstable in Mg-Nd17, which was later corroborated with TEM observations18. The good agreement with experiments validates the available RE pseudopotentials with frozen-core f-electrons, and permits us to study other systems. However, the methodology used for determining negative interfacial energies needs further investigation because one intuitively associates an interface with positive excess energy. Therefore, the rationale for predicted negative interfacial energy was also investigated from the perspective of β1/Mg interfaces, especially since β1 is a key-strengthening phase in many commercial alloys17.

Previous DFT studies on Mg-Nd have not investigated β1 formation, its interfacial structure, and the associated bonding environment14,15,17. Furthermore, they did not examine thermodynamic quantities like surface excess energies - a difficult quantity to measure experimentally in the larger system limit14,15,17. Thus, guided by TEM studies5,6,7,8,9,10,11, we have investigated the energetics and structure of two types of β1/Mg interfaces, namely {112}β1/{100}Mg and {111}β1/{110}Mg, via DFT calculations.

Computational Methodology

First principles computations were carried out with the Vienna Ab-initio Simulation Package (VASP) employing the projector-augmented plane-wave (PAW) method18, with electron exchange and correlation described by the generalized gradient approximation (GGA)19,20. All calculations were performed at 0 K, with a cut-off energy of 360 eV, k-point spacing of 0.1 Å−1, 0.2 eV Methfessel-Paxton smearing width and Brillouin zone integration at 10−7 eV convergence threshold. Groundstate structures of supercells were obtained by relaxing ionic, shape, and volumetric degrees of freedom till global energy convergence was achieved to within 1 meV/atom, and the Hellmann-Feynman forces on the atoms were less than 1 meV/Å. Visualization and subsequent analysis of the relaxed structures was performed using Vesta21 and Ovito22.

Results

Construction of initial supercells

Figure 1a shows β1 Mg3Nd with a BiF3 structure, lattice parameter 7.391 Å, space group Fmm (#226, similar to fcc structures), and Pearson symbol cF16 (corresponding to bcc-ordered D03 strukturbericht)23. We constructed an orthorhombic β1 supercell whose edges are parallel to 〈011〉β1, 〈111〉β1 and 〈112〉β1. This choice corresponds to β1 - hcp-Mg orientation relationship seen in experiments5,6,7,8,9,10,11. Two orthogonal supercells containing 96 atoms, and lengths 35 (Fig. 1c) and 50 Å (Fig. 1d) were used in our analysis. In the resulting supercells (shown in Fig. 1c and d) 12 Nd atoms were replaced with Mg in one portion to create a “Mg-slab”, while retaining enough Nd atoms in the remainder to maintain Mg3Nd stoichiometry corresponding to β1. The resulting structures created {112}β1 (hereafter referred to as “{112}” supercell) and {111}β1 interfaces between Mg and the β1 precipitate as shown in Fig. 1c and d respectively. Note that in these initial supercells, Mg matrix has a “bcc” structure.

Figure 1
figure 1

Crystal structures of (a) cubic β1 and (b) β1 in an orthorhombic cell. Pre-relaxed supercells extended along: (c) 〈112〉β1 and without any prior vacancies, and (d) 〈111〉β1 and showing the locations of vacancy placed within the “bcc” Mg-region.

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The Mg-slab was further modified based on the β1/Mg interfacial structure proposed by Nie and Muddle5. They suggested that β1 formation is associated with vacancies near the β1/Mg interface along 〈111〉β1 5. Concomitantly, recent analytical calculations, involving stress-free transformations strains during β1 formation11,13, indicated the presence of a tensile strain along 〈111〉β1. Such tensile strains likely facilitate vacancy creation along 〈111〉β1 by expanding the volume near the interfaces. Therefore, to understand such effects a Mg vacancy was introduced near the β1/Mg interface (square, Fig. 1d) and at the center of Mg (diamond, Fig. 1d) in separate supercells. An Mg vacancy is placed inside “Mg” slab, rather than in the intermetallic β1 compound, was guided by β1/Mg interfacial structure proposed in literature6,11. Also, this reasonable because Mg vacancy formation energies in β1 (1.24 eV) is greater than in hcp-Mg (0.82 eV). The resulting vacancy concentration was ~1 at% for the entire supercell, and 2at% for the Mg slab. Hereon the vacancy containing supercells - near β1/Mg interface, and center of Mg slab - will be referred to as “{111}β1(Iv)” and “{111}β1(Cv)” supercells respectively.

Relaxed structures

The relaxed structures of three supercells with β1/Mg interfaces are shown in Fig. 2. In the {112}, {111}β1(Iv) and {111}β1(Cv) supercells (Figs 1a and 2b,c respectively), the Mg slab with an initial bcc structure transforms to the correct hcp crystal structure indicated by hexagons in Fig. 2. The β1 however, retains its bcc-ordered structure as confirmed by common neighbor analysis (CNA)22,24. In this manner we created “equilibrium” supercells comprising of two types of β1/hcp-Mg interfaces: {112}β1//{100}Mg and {111}β1(Iv/Cv)//{110}Mg. The structural relaxation of {111}β1(Iv) and {111}β1(Cv) supercells (Fig. 2b and c), with an added vacancy (Fig. 1d), also created a region with vacancy-like excess free volume marked by dotted polygons in Fig. 2b and c. This occurs even when a single vacancy was initially placed at the center of the Mg slab (Fig. 1d). The excess volume of hcp-Mg in Fig. 2b and c measured by Voronoi tessellation21, was ~30 Å3, and ~23 Å3. In comparison, Mg slabs without prior vacancies inside the {111} supercell, did not have an excess volume regions and neither did their initial bcc Mg relax to a hcp structure (Supplementary Figure S1) – unlike those presented in Fig. 2b and c. Thus vacancies aid in the transformation of bcc-Mg to hcp.

Figure 2: Relaxed supercells showing β1/Mg interfaces.
figure 2

(a) {112}β1/{100}Mg interface, (b) {111}β1(Iv)/{110}Mg interface, with a vacancy that was initially placed near the interface, and (c) {111}β1(Cv)/{110}Mg interface with a vacancy was initially placed at the center of Mg-region.

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In the relaxed {112} supercell, Fig. 2a, certain crystallographic orientations of Mg and β1 were mutually coincident and exhibited the following orientation relationship (OR): (0001)Mg//(011)β1, [100]Mg//[21]β1, and [110]Mg//[11]β1. Electron diffraction patterns from literature have indicated that the β1-Mg OR causes a ~5.26° misorientation between (0001)Mg and (011)β1 planes5,6,7,8,9,10,11, which is in excellent agreement with DFT value of 5.3° indicated in Fig. 2a, and, also validates our approach. The misorientation can be quantified as the angle between [100]β1 and [110]Mg.

A comparison of β1 and Mg crystallographic directions within {111}β1(Iv) and {111}β1(Cv) supercells showed that, eventhough (0001)Mg and (011)β1 planes are parallel, the other crystallographic axes involved in β1-Mg OR are not coincident. For example, angle between [100]Mg and [21]β1 in {111}Iv and {111}Cv is ~15°, while those axes are coincident in case of the {112} supercell. In other words, DFT calculations suggest that a creation of only {111}β1(Iv/Cv)/{110}Mg interfaces may not yield the experimentally seen β1-Mg OR atleast near the interfacial regions. This result is surprising because, based on the β1-Mg orientation relationship, one would have expected a mutual alignment of β1 and Mg orientations in the {111}β1(Iv/C/v) supercells.

Implication of this “off β1-Mg OR” misalignment in the {111}β1(Iv/C/v) supercells was further probed by comparing it’s local atomic distortions with the {112} supercells. Mg-Mg, Mg-Nd and Nd-Nd bond-lengths measured from regions 3–4 atomic layers away from the interfaces were used to calculate the lattice strain , where aβ1/Mg-interface and abulk are the lattice parameters of each phase in the supercells, and their bulk equilibrium lattice respectively. Table 1 lists the β1 and Mg lattice parameters, and the lattice strains. Lattice strains, 5–10 times larger that the bulk systems, were found in {111}β1(Iv) and {111}β1(Cv) supercells along the a-axis of Mg or 〈110〉 and 〈100〉 of β1. We further note that placement of vacancy away from {111}β1(Iv)/{110}Mg interface (Fig. 1d) increases the β1 lattice strain by ~33% in {111}β1(Cv) in comparison to {111}β1(Iv). In other words, higher lattice strains are associated with the formation of {111}β1/{110}Mg interfaces than {112}β1/{010}Mg. Regardless, presence of vacancy-like excess volume in the {111}β1(Iv/Cv) supercells strains the Mg and β1 lattice, and will be shown to affect the β1/Mg interfacial energies.

Table 1 Table of lattice parameters and strain in Mg and β1 slabs of the relaxed structures.
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β1/Mg interfacial energies

The geometry used for calculating β1/Mg interfacial energies (γβ1/Mg) is schematically shown in Fig. 3a as vacuum-space/bulk-β1/bulk-Mg/vacuum-space. Here the interacting β1/Mg surfaces are {111}β1(Iv/Cv)/{110}Mg and {112}β1/{010}Mg (see Supplementary Figure 2). This methodology was previously used by Liu et al. to examine the interface between an ordered intermetallic compound and the disordered matrix25. This approach further avoids the development of excess artificial interfacial strains by having vacuum on both sides of the unconstrained interfacial supercell. Thus, one need not include additional coherency strain energy terms (e.g. ref. 17) to evaluate the interfacial energies. Furthermore, since β1 and Mg are exposed to the vacuum on either ends (Fig. 3a), we are required to determine the “free” surface of the participating {111}β1, {110}Mg, {112}β1 and {100}Mg planes. The free surface energy, γS-p, for a phase “p” was calculated using the expression , where Ep(N) is the calculated ground state energy for a “N” atomic layer supercell slab of phase “p”, EBulk-P is the bulk energy of phase “p”, and A is the surface area25,26,27. The calculations involved a maximum of 24 (~35 Å) and 60 (~50 Å) layers for Mg and β1 respectively, and a minimum of ~12 Å of vacuum space. EBulk-P was determined from the slope of Ep(N) vs N plot. For pure metals, EBulk-P corresponds to their cohesive energy27, and our calculations yielded 1.53 eV/atom for Mg - in excellent agreement with literature26,27,28. Figure 3b shows that plot of γS-p vs. N for different β1 and Mg planes, and the results indicated that the γS-p values, for {111}β1(Iv/Cv), {110}Mg, {112}β1 and {100}Mg planes rapidly converge within the chosen range of N.

Figure 3
figure 3

(a) Configuration used for calculating β1/hcp-Mg interfacial energy. (b) Free surface energies (γS-p) of different β1 and Mg surface planes plotted as a function of number of atomic layers (N). (c) γS-p vs 1/N plots to determine γS-p in the larger size limit. The dotted line fits corresponding to R2 values of 0.94. The intercept of the linear fits, 1/N → 0 yield excess surface energy in the large system limit (N → ∞), which is a well-defined thermodynamic quantity. (d) Bar chart showing the interfacial energies of three β1/Mg interfaces reveal {112}β1/{100}Mg interfaces are energetically more stable than {111}β1/{110}Mg.

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The γS-p values for the Mg and β1 planes were calculated using linear fits (R2 ≈ 0.94) to γS-p vs. 1/N plots (Fig. 3c). We then extrapolated the fits for 1/N → 0, which correspond to N → ∞ layers or the “bulk”. Our surface energy γS-p of (0001)Mg was 0.52 J/m2, agreed well with literature reports of 0.52 and 0.55 J/m2 26,27. Figure 3b shows that β1 has higher excess surface energies (γS-p) than pure Mg. This indicates a stronger bonding within β1, and that it is difficult to cleave or fracture β1 during deformation. We further note that such surface excess energies have not been reported for any other precipitate phases in Mg-RE systems14,15,16.

The plots in Fig. 3b and c helps estimate the optimal value of “N” required for the interfacial energy calculations that reasonably accounts for the bulk phases along with the free-surface and interfacial energies. Thus, 12 (~15 Å) and 24 (~25 Å) layers was chosen for each phase making up the {112}β1/{100}Mg and {111}β1/{110}Mg interfaces respectively, and ~12 Å of vacuum was maintained above β1 and Mg slabs (Fig. 3a). β1/Mg interfacial energy () was estimated using the expression22:

where γS-β1 and γS-Mg are the free surface energies of β1 and Mg respectively, Esupercell is the ground state energy of the β1-Mg supercell while EBulk-β1 and EBulk-Mg are the bulk energies of β1 and Mg respectively, and A is the surface area. The γβ1/Mg values of these three interfaces are plotted in Fig. 3d as a bar chart. Their respective relaxed structures are also shown in Supplementary Figure 2. The calculated γβ1/Mg values for {112}β1, {111}β1(Cv) and {111}β1(Iv) supercells were 98, 282 and 762 mJ/m2 respectively. Thus, the {112}β1/{100}Mg interfaces are more energetically stable than {111}β1/{110}Mg. Also note that inclusion of excess surfaces energies for estimating interfacial energies (equation 1) will result in only positive excess energy values, which is thermodynamically reasonable.

Structure and bonding environment near β1/Mg interfaces

To explain the lower interfacial energy of {112}β1/{100}Mg in Fig. 3d, we have compared the bonding environment of these interfaces. The analysis correlates the interfacial atomic-coordination (Fig. 4a and c), with electron charge density distribution (Fig. 4b and d).

Figure 4
figure 4

(a,b) and (c,d) show structure and electron charge distributions of {112}β1/{100}Mg and {111}β1(Iv)/{110}Mg interfaces respectively. In (a) and (b) red and blue colors indicate atomic sites in hcp-Mg and β1 structures respectively, while green indicates “distorted” lattice sites. (e) shows the legend for electron charge densities in (b) and (d). (e) shows the structure of{111}β1(Cv)/{110}Mg interface. Interfacial width is indicated by translucent boxes, and the contours in the electron charge density plots engulf regions with lower charge densities compared to the bulk.

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In Fig. 4a and c the atom colors are assigned by CNA environment type18,20 and the bonds depict in-plane – i.e. (0001)Mg//(011)β1 – coordination around an atomic site: (i) red denotes hcp-Mg with 6 in-plane coordination, (ii) blue denotes bcc-ordered β1 with 4 in-plane coordination, and (iii) green depicts sites which were not identified as either hcp-Mg or β1. Thus, the green-colored bonds/atoms near the interfacial regions - indicated with shaded boxes across Fig. 4a–d - can be interpreted as distortion of the lattice sites, from bulk hcp-Mg and β1 structures.

A comparison of the interfacial regions, and their corresponding bonding environment reveal that the distorted lattice sites (shown by green colored bonds) are invariably located around regions with significant depletion of electron charge density distributions (marked by red arrows in Fig. 4b and d). In other words, interfacial lattice distortions correlate well with lower electron density or weak bonding29. Furthermore, the electron density distributions near the interfacial regions of {111}β1(Iv)/{110}Mg was discernibly lower (compare Fig. 4b and c with legend at the bottom), and spread over a wider region than {112}β1/{100}Mg. Correspondingly, lattice significantly distorted near the {111}β1(Iv)/{110}Mg interface. Therefore, considering the larger differences in the energies of {112}β1/{100}Mg (98 mJ/m2) and {111}β1(Iv)/{110}Mg (282 mJ/m2) interfaces, it is evident that smaller population of distorted lattice sites and narrow spatial extent near the interface correlate with lower interfacial energy. This is further confirmed for {111}β1(Cv)/{110}Mg system (762 mJ/m2) in Fig. 4e, which displayed a wider spatial extent of distorted interfacial lattice sites (overlapping both hcp-Mg and β1 structures) than the other two interfaces.

Discussion

One of our motivations was to rationalize the existence of only {112}β1/{100}Mg interfaces seen in Mg-Nd alloys, which was contrary to the exceptions based on β1/Mg OR11. The DFT calculations revealed that, unlike {112}β1/{100}Mg, formation of {111}β1/{110}Mg requires the presence of vacancy-like excess volume near the interfaces, which causes interfacial distortions, higher lattice strains and presumably strain energy within the mating phases. Such pronounced distortion/strain increases the interatomic spacings near the interface, reduces charge density between the atoms, and creates weaker interfacial bonding in {111}β1/{110}Mg. The relatively poor bonding increases the excess energy of {111}β1/{110}Mg incomparison to the {112}β1/{100}Mg interface, and makes the latter an energetically preferable interface.”

Despite the fact that excess volume results in an energetically unstable {111}β1/{110}Mg interface. However, the presence of such excess volume near the interfaces may explain the growth of plate-like β1 within Mg-matrix and along dislocation lines5,6,7,8,9,10,11. Nie and Muddle have proposed that shear transformation strains are associated with β1 formation, because of which excess vacancy concentration is created at the short ends oriented along 〈110〉Mg. It is likely β1 growth can occur via vacancy migration to the interfaces. However, to maintain elongated plate-like morphology during the growth process, β1 needs a stable interface in its long edge. Our DFT results suggest that a low lattice strain and interfacial energy of {112}β1/{100}Mg interfaces allows β1 precipitates to acquire a plate-like morphology with a high aspect ratio and possess a broad {112}β1-face parallel to {100}Mg as seen in experiments4,5,6,7,8,9,10,11,12. In the case of dislocation assisted β1 form as linear chains along 〈110〉Mg 10,13,30, which are also observed in our creep-tested microstructures10. We postulate that such linear growth of β1 along dislocation lines occur via transfer of vacancies to metastable interfaces like {111}β1(Iv)/{110}Mg (Fig. 2b), which tends to have lower interfacial energy than vacancy formation away form the β1/Mg interface (i.e. {111}β1(Cv)/{110}Mg in Figs 2c and 3d). Since dislocations can rapidly supply vacancies via dislocation pipe diffusion, the diffusion assisted movement of {111}β1(Iv)/{110}Mg type interfaces rapidly consumes a large portion of the host dislocation line, and orients β1 along 〈110〉Mg. Finally, since {111}β1(Iv)/{110}Mg interfaces are energetically unfavorable, interfacial atomic rearrangements, e.g. via shear process5,6, produces {112}β1/{100}Mg interfaces in the final microstructure.

From the perspective of alloy design approaches; the contribution of this report is two fold. We demonstrate that in order to develop alloys with advantages associated with “β1-like” precipitation one needs to (i) retain a well bonded coherent interface at the precipitate broad faces, e.g. {112}β1/{100}Mg, while simultaneously (ii) form mobile yet unstable interfaces on the short edges to allow precipitate growth along dislocation lines. However, finding non-RE substitutions, which fits these two criteria, is our next challenge, and is currently being tackled by coupling evolutionary algorithms with first principle calculations.

Conclusion

Formation of nanoscale β1-Mg3Nd precipitates in hcp-Mg matrix is known to enhance the creep resistance of Mg-alloys because of their tendency to form on dislocation lines and sluggish coarsening kinetics when they form in the hcp alloy matrix. Using first principles calculations, we have compared the excess energy and the structure of {112}β1/{100}Mg and {111}β1/{110}Mg interfaces. Our calculations revealed that the formation of {112}β1/{100}Mg interfaces, compared to {111}β1/{110}Mg interface, is associated with significant reduction in both the interfacial energy and lattice strains in the adjacent β1 and Mg matrix. The favorable formation energetics of the {112}β1/{100}Mg interface, in conjunction with small lattice strains, influences β1 acquiring a plate-like morphology with its broad-face {112}β1 parallel to {100}Mg. Electronic structure of these interfaces revealed that a lower interfacial energy also correlates with a smaller population of distorted lattice sites near the interfacial regions. Additionally, our DFT investigation informs that creep-resistant Mg alloys will benefit from “β1-like” precipitation because of two key features (i) retain a well bonded coherent interface at the precipitate broad faces, e.g. {112}β1/{100}Mg, and (ii) destabilize other interfaces (corresponding to precipitate-Mg OR) to promote growth along a desired direction, e.g 〈110〉Mg.

Additional Information

How to cite this article: Choudhuri, D. et al. Interfacial structures and energetics of the strengthening precipitate phase in creep-resistant Mg-Nd-based alloys. Sci. Rep. 7, 40540; doi: 10.1038/srep40540 (2017).

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