Density functional theory-based investigations of solute kinetics and precipitate formation in binary magnesium-rare earth alloys: A review

Abstract Magnesium-rare earth (RE) alloys have received substantial attention over the last two decades for light-weight fuel efficient applications because of their low density of 1.74 gm/cc. However, it is only recently that density functional theory (DFT) based computations have been used to understand their microstructure, mechanical properties, and to successfully explain several experimental observations. Here, we have reviewed the current understanding of solute kinetics and precipitate formation in Mg-RE alloys. The current DFT studies on the formation of Mg-RE microstructures are broadly organized into three core concepts: (1) solute kinetics involving solute-vacancy interactions and solute diffusivities within the hexagonal-closed-pack (hcp) Mg lattice. (2) precipitation mechanisms, including thermodynamics, chemical bonding and elastic strain/distortion to explain solute clustering and subsequent precipitate phase formation within hcp-Mg. and (3) determination of precipitate/ hcp-Mg interfacial structure and energetics. These key aspects have been systematically investigated using nudged elastic band, high throughput convex hull computations, cluster variation methods, phonon calculations, and other techniques using DFT as an energy calculator. Studies on kinetics have been centered on determining the binding or interaction energies, migration energies, and quantifying vacancy-mediate diffusion of RE solutes such as La, Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. They have shown that these solutes behaved differently in terms of their interaction energies with both vacancies and the solvent, and showed reduced interaction tendencies with increasing ionic radius from La to Lu. Solute diffusivities were typically calculated using the 8-frequency model and Green function method, and salient results from these two theories are compared and contrasted here. Even though both theories differed in their quantitative predictions of activation energies, they revealed that, contrary to expectations, RE atoms like La, Ce and Nd, each with a larger ionic radius than solvent Mg, had faster diffusivity than the self-diffusion of Mg. In terms of precipitation in Mg-RE alloys, in this review, we have focused on delineating mechanisms in a model binary Mg-Nd alloy, because the availability of a large number of studies permitted us to present a near-unified picture of precipitation mechanisms from early to late stages of precipitate phase formation. These DFT investigations accurately described that precipitation initiates with ordering of Nd atoms in the hcp-Mg lattice (also called clustering in Mg literature) and such ordering is caused by the reduction in local elastic strains and enhanced chemical bonding. Subsequently, these ordered clusters directly result in the formation of GP zones and other ordered precipitate phases. DFT studies have cast doubt on the presence of ordered-hcp D019 phase (or the so called β ' ' phase). They also demonstrated that the orthogonal β ' phase does not exist as a structure with fixed Nd ordering sites but exist rather as a single precipitate containing a variation in Nd ordering sites inside the β ' structure. Experimental corroboration of DFT predictions of the early stages of precipitation are also presented via recent atomically resolved electron microscopy observations. We have also reviewed literature on the β 1 phase that forms during the intermediate stages of precipitation and is an important strengthening precipitate phase in commercial Mg-RE alloys such as WE43. Literature studies of bulk stabilities, elastic constant, chemical binding and high temperature stability from phonon dispersion curves have been summarized and compared for 0 K and finite temperature cases. We also found that the precipitation of the β 1 phase within hcp-Mg required a complex transformation strain tensor. We demonstrate that the individual components of the transformation strain tensor can be computed by applying DFT-based conjugate gradient minimization on supercells containing the crystallography of the β 1 / Mg orientation relationship. Finally, using the limited Mg-RE DFT literature available on interfaces, we qualitatively establish a correlation between interfacial energy, structure and the local chemical bonding, by using precipitate/matrix examples from binary Mg-Nd and ternary Mg-Nd-Zn. We show that descriptions of interfacial structure and stability, which rely solely on energetics, is self-limiting in that they ignore the bonding pertinent insights inherent in the valence charge density distribution available from DFT calculations. We present some interesting results that couple interfacial energies with local electron density to shine a new light on the stabilization of new precipitate phases due to additions of solutes. For example, addition of trace concentrations of Zn to the Mg-Nd stabilizes new hcp-γ” precipitates in the α-Mg matrix, which is not possible in Mg-Nd system alone. This is explained by correlating how the concentration of local electron density across this precipitate-matrix interface also lowers interfacial energy and limits distortion of interfacial bonds. This insight would not be available if we had considered only the interfacial energy values.