Formation of high-strength precipitates in Mg–RE alloys: The role of the Mg/ interfacial instability
Introduction
There is a significant demand for structural magnesium alloys due to their low density compared to aluminum and steel [1]. While two-thirds as dense as aluminum [1], magnesium alloys typically have lower yield strength than their aluminum counterparts, a fact that has inhibited more widespread use of Mg in automotive and aerospace applications. Therefore, a large amount of research is being directed at Mg alloy strengthening [1], [2], [3], [4], [5], [6]. Mg–rare earth (RE) alloys, either binary or with additional alloying elements, are of particular interest due to the high yield strengths achieved through precipitation hardening [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. Consequently, their microstructure, aging response and mechanical properties have been extensively documented experimentally (e.g. [11], [14], [15], [16], [17], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36]). In these systems, peak strength in the aged alloy often corresponds to the formation of and precipitates [14], [16], [17], [32], [33], [34], [35], [36], which form prismatic plates and are coherent or semicoherent with the Mg matrix.
Understanding the exact precipitation sequence and the fundamental driving forces giving rise to precipitation sequences is a critical part of the effort to intelligently design high-strength Mg alloys. Precipitate morphology is controlled by the strain and interfacial energy between the precipitate and the matrix, in addition to kinetic factors, e.g. diffusion [37]. Uncovering the controlling factor in the precipitate morphology may aid in designing alloy compositions, possibly without expensive RE elements, that yield a specific, tailored precipitate morphology. For example, in our previous work [38], we found that the formation of prismatic-plate-shaped precipitates observed experimentally in Mg–RE alloys can be attributed to the elastic strain energy anisotropy between basal and prismatic planes, indicating that potential RE replacements should exhibit such an anisotropy. In this work, we investigate Mg/ interfacial energies, another important contribution which furthers our understanding of the precipitate morphology and elucidates the relation between the and precipitates. The combination of anisotropic strain energy and interfacial energy dictates the morphology of the precipitates. Precipitates may adopt an equilibrium morphology dictated by the Wulff construction [39], [40] (when interfacial energies are dominant), thin plates [37] (when strain energy is dominant) or they may undergo a size-dependent change in morphology [41], [42], [43], [44], [45] (dictated by chemical interfacial energy at smaller length scales and strain energy at larger length scales). Quantifying both energetic contributions, strain and interfacial energies, thus provides a more complete understanding of the size- (and implicitly time-) dependent precipitate morphology in the Mg matrix.
Even though Mg–RE aging reactions have been extensively studied, some disagreement on the nature of the precipitates still exists, particularly concerning the early stages of the aging reaction. The disagreement centers around whether or not forms in Mg–RE systems. The various precipitation reactions are summarized in Table 1. In some studies, the Mg–(Ce, Nd, Pr) systems are observed to first precipitate out Guinier–Preston (GP) zones, followed by a phase in the D019 structure, then a phase [12], [46], [47], [48], [49]. The (D019) structure, which is a superstructure of hexagonal-close packed (hcp) form (i.e. a decoration of Mg and RE atoms on an hcp lattice), has dimensions and and Mg3RE composition, while the phase has a Mg7RE composition. More recent work, however, using high-angle annular detector dark-field scanning transmission electron microscopy (HAADF-STEM) [50] reported the formation of Nd pillars in a Mg–2.5 at.% Nd alloy but no . Another recent study [51], also using HAADF-STEM, reported the formation of planar GP zones followed by the phase. The structure and morphology of the phase has been less controversial, since it has been observed by most of the previous work [46], [47], [48], [49], [51] having a Mg7Nd composition in a body-centered orthorhombic structure with dimensions and . For reasons we explain below, we refer to this phase as -short.
Reports on the Mg–Gd aging reactions, also summarized in Table 1, disagree about whether or not the -D019 phase forms [9], [32], [36], [52], [53]. TEM studies [9], [32], [36] report the formation of a -D019 phase in the earliest stages of precipitation. On the other hand, also using TEM, D019 short-range ordering was observed in Mg matrix immediately following solution treatment, followed by the formation of the phase [52]. Similarly, using HAADF-STEM and high-resolution transmission electron microscopy (HRTEM), Nishijima et al. [53] report the formation of local, short-range order followed by, and temporally overlapping with, the formation of the phase. This phase, similar to that reported in Mg–Nd, is also body-centered orthorhombic with composition Mg7Gd, but with different dimensions: and . As the lattice parameter for this structure is twice as long as the structure reported in the Mg–Nd system above, we will refer to this structure as -long. Both structures, short and long, have been experimentally resolved [51], [53] and we discuss the relationship between them in the results section below. Knowing whether the -D019 phase forms and how it gives rise to the primary strengthener is critical to well-controlled precipitation hardening in Mg alloys.
First-principles calculations provide a valuable tool for studying precipitate formation and stability in Mg–RE systems. Density functional theory (DFT) calculations lend themselves to such an investigation for two reasons. First, it is possible to calculate quantities that may be difficult to measure experimentally, such as precipitate/matrix interfacial energy. Second, quantifying the two competing energetic contributions, strain and interfacial energy, provides an understanding of why certain precipitates form, and the morphology they adopt. First-principles calculations have provided valuable insight into precipitation in cubic, e.g. Al, alloys (e.g. [54], [55], [56], [57], [58], [59], [60], [61], [62]), as well as defects in Mg [63], [64], [65], [66], [67], [68], [69], and phase stability in Mg–RE–(Zn) systems [38], [70], [71], [72], [73], [74]. First-principles calculations have also been used to calculate short-range order in solid solutions, and critically analyze the connection (or lack thereof) between short-range order and the underlying low-temperature long-range order [75], [76], [77], [78], [79].
In this work, we use DFT calculations to elucidate the energetic competition between the metastable precipitate phases in Mg–RE alloys. We calculate the formation energies of the -short and -long phases across a series of Mg–RE systems and the interfacial energies of /Mg [0 0 0 1] and [1 0 0] interfaces. We find a surprising result: all the prismatic and some of the basal /Mg interfacial energies are negative. Because of negative interfacial energies, we predict that Mg/ interfaces are energetically favored to spontaneously form throughout the matrix, leading to ordered planes of Mg and . In other words, the Mg/ interfacial instability indicates that certain ordered arrangements of Mg and units are more energetically stable than Mg+ phase separation. We show, through DFT calculations, that the energetically favored arrangement of Mg and planes gives rise to phase formation. The Mg/ interfacial instability, therefore, reveals the formation of prismatic planes to be a precursor to the formation of precipitates.
Section snippets
DFT calculations
We perform first-principles DFT calculations using the Vienna Ab initio Simulation Package (VASP) [80], [81] and projector augmented wave potentials [82]. We utilized the PBE parameterization of the generalized gradient approximation (GGA-PBE) [83] for all calculations. All structures were relaxed with respect to all cell-internal and -external degrees of freedom at an energy cutoff of 350 eV. For each structure, gamma-centered k-point meshes were constructed to achieve at least 10,000 k-points
Energetic stability
The formation energy of a phase is determined by the energy of that phase relative to the composition-weighted energies of its constituent elements. To explore the energetic stability of the -short, and -long phases, we calculate the formation energy per atom of each phase, , using:In this equation, x is the atomic fraction of Mg, and E[A] is the total energy per atom of structure A. The formation energies of the and
Conclusions
We have used first-principles calculations to investigate and explain the experimentally reported Mg–RE metastable precipitates. We find the following:
- 1.
DFT calculations accurately predict energetic stability of three competing metastable structures: -short and -long. We find to be less stable than either phase for all Mg–RE systems calculated. Further, we accurately predict the structure competition between -short and -long, finding -short to be energetically preferred for the
Acknowledgments
We gratefully acknowledge the support of the Ford–Boeing–Northwestern (FBN) alliance, award No. 81132882. J.E.S. acknowledges support by the US Department of Energy, Office of Basic Energy Sciences through grant DE-FG02-98ER45721.
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