Local structure and bonding environment of intermetallic ${\beta }_1$ precipitate phase nucleus in binary Mg-Nd

Abstract

${\beta }_1$ is a key strengthening precipitate phase in Mg-Nd-based alloys, because it is know to improve high-temperature creep resistance. Here, using density functional theory-based first principles calculations, we have examined the structure and local environment of ${\beta }_1$ ($a_{{\beta }_1}~7.4\AA$) nucleus that forms within hcp-Mg lattice ($a_{M}g~3.2\AA$ and $c_{\mathit{Mg}}~5.2\AA$). For this purpose, we developed a crystallographically-informed embedded precipitate supercell approach, which allowed us to systematically examine several possible precipitate/matrix configurations, and determine energetically favorable structures. We learn that ${\beta }_1$ will initially form within hcp-Mg as smaller structural templates rather than the larger equilibrium structure. In essence, these templates are the “genetic imprint” of the equilibrium structure that contains only a portion of larger ${\beta }_1$ crystal structure, while retaining the stoichiometry and nominal symmetry of ${\beta }_1$. Such template-nucleus was stabilized by pockets of stiff covalent-bonded environment in hcp-Mg. Phonon density of states and dispersion wave spectra further corroborated such covalent character, and revealed that the predicted template-nucleus structure and its surroundings were stable at temperatures greater than 0 K. Broadly, our DFT results provide crucial insights into intermetallic phase nucleation, whose lattice parameters differ significantly from the host lattice.

Introduction

Ordered intermetallic compounds are ubiquitous in structural alloy microstructures either as the primary phase or secondary phase precipitates [1], [2]. Over the years, several structural transformation mechanisms have been proposed to explain their formation. Examples include, chemical ordering, e.g. $L{1}_2$ (strukturbericht) and $D{0}_{19}$ (strukturbericht), and mixed mode mechanisms like replacive-dispacive, e.g. ordered omega, or affine shear mediated displacive-diffusion, e.g. B2 (strukturbericht) in fcc lattice [1], [2]. Typically, mixed mode mechanisms produce daughter phases, whose space group symmetries differ from the parent phase.

Eventhough, such mechanisms provide critical insights into the formation of intermetallic compounds, they reveal little regarding the structure of the phase nucleus; specifically the atomic locations, and its immediate environment. Since the product phase is embedded within a parent matrix phase, stresses from the surrounding matrix (e.g. stresses around an Eshelby inclusion [3], [4]) and local relaxations will determine the final positions of atoms inside the nucleus and its surroundings [5], [6] (these positions change slightly as the nucleus grow in size [7]). One way to gain such lattice-level structural information is to monitor the nucleation process via in situ, atomically resolved, transmission electron microscopy (TEM), which is a difficult and time-consuming technique. Furthermore, large number of intermetallic phases have covalent character [5], [8], [9], and how their bonding character interacts with that of the host lattice/matrix, to influence the precipitation process, needs to be examined. Therefore, to better understand the structure of the intermetallic nucleus and it’s environment, one needs to know, (i) if there are preexisting pockets of covalent bonds in the host lattice that facilitates the nucleation by stabilizing the nucleus structure, and (ii) if such character persists in the immediate neighborhood after nucleation. In other words, the structure-bonding environment is intimately linked with nucleus formation, and necessitates a physics-based investigation via atomistic simulations.

Towards that end, using first principles-based density functional theory (DFT), we have investigated this matter by probing the nucleus, and it’s environment, of a prototypical intermetallic ${\beta }_1$ precipitate phase found in Mg-Nd and its commercial variants [10], [11]. In the precipitation sequence of Mg-Nd, ${\beta }_1$ forms at the intermediate stage after the formation of ordered phases like ${\beta }^{‴}$, i.e. SSSS $\to$ GP zones (N, V, hexagons) $\to$ ${\beta }^{‴}$ $\to$ ${\beta }_1$(${\mathit{Mg}}_3\mathit{Nd}$) $\to$ $\beta$(${\mathit{Mg}}_{12}\mathit{Nd}$) $\to$ ${\beta }_{\mathit{eq}}$(${\mathit{Mg}}_{41}{\mathit{Nd}}_5$) [12], [13]. In these alloys ${\beta }_1$ phase nucleates both homogeneously within hcp-Mg matrix and heterogeneously along dislocation lines, and the latter formation mechanism is known to improve high temperature creep resistance of Mg-Nd-based alloys [14], [15], [16], [7]. Based on TEM observations, Nie and Muddle concluded that ${\beta }_1$ formation was facilitated via affine transformational shear of the hcp-Mg lattice domains enriched in Nd atoms [10]. In contrast, ${\beta }^{‴}$ formation is facilitated by the ordering of the hcp-Mg lattice, which conveniently allows their examination through extant DFT methodologies involving hcp-coordinated supercells [13], [12], [17], [6]. Presently, DFT studies of ${\beta }_1$ are limited to probing ${\beta }_1$/Mg interfaces [18] and detailed examination of ${\beta }_1$ growth mechanism [7], which are more applicable to well-developed precipitates than it’s nucleus. Other studies include, determination of elastic and thermodynamic properties of bulk ${\beta }_1$ [19], [20], [21], and understanding its covalent character [20]. In other words, none of these studies have examined the structure of an embedded ${\beta }_1$ nucleus and it’s environment, which is the focus of this work.

Three vital constraints must be considered while conducting DFT examination of ${\beta }_1$ phase nucleus that is embedded inside hcp-Mg. First, the simulations must reproduced the experimentally observed ${\beta }_1$/hcp-Mg crystallographic orientation relationship (OR), i.e. ${(011)}_{{\beta }_1}//{(0002)}_{\mathit{Mg}},{[11\overline{1}]}_{{\beta }_1}//{[11\overline{2}0]}_{\mathit{Mg}}$, and ${[2\overline{1}1]}_{{\beta }_1}$ // ${[\overline{1}010]}_{\mathit{Mg}}$ [10]. Second, the symmetry and stoichiometry of the ${\beta }_1$ nucleus should be comparable to the equilibrium phase. This constraint maintains consistency with classical nucleation of ${\beta }_1$ in Mg-Nd, which was observed experimentally [7], [14]. Third, misfit strain tensor in the DFT computed structure should be comparable to those expected from ${\beta }_1$-hcp-Mg lattice correspondence [11], [15], [16], [7], because such strains are related to the affine shear of the hcp-Mg lattice [10]. It should be noted that these constrain are broadly applicable to intermetallic phase nucleation, because the initial nucleus will maintain OR with the parent and, in many cases, manifest a different space group symmetry than the host, e.g. ${\beta }_1$ ($\mathit{Fm}\overline{3}m$) v.s. hcp-Mg ($P{6}_3\mathit{mmc}$) [10], [22] and bcc-ordered B2 ($\mathit{Pm}\overline{3}m$) v.s. fcc ($\mathit{Fm}\overline{3}m$) [2].

In the past, other groups have employed DFT to examine precipitation in Mg-Nd; specifically to investigate GP zones and $\beta$ formation [13], [12], [17]. However, such “precipitation” occurs through simple ordering of the hcp-Mg lattice, and can be captured using standard DFT supercell; by replacing the solvent species with solute atoms to reproduce such ordering in the supercell [6], [23], [13], [12], [17]. In stark contrast, formation of ${\beta }_1$ requires affine shear of the solute-rich hcp-Mg domains instead of hcp-Mg ordering [10]. This mechanism results in a phase with substantially different space group symmetry than the host lattice structure (compared to ordering), and cannot be described by standard supercell approaches reported in literature [13], [12], [17]. Therefore, taking guidance from our past work [18], we have developed a novel computational approach for examining “embedded precipitate” within a reasonably sized DFT supercell, which “by design” takes into account the aforementioned affine transformational shear, ${\beta }_1$/hcp-Mg OR and their underlying difference in space group symmetries. We emphasize that the primary objective of this approach is to generate structures containing embedded ${\beta }_1$ nucleus, which will allow us to investigate energetics, chemical bonding and local strains via DFT, and not to examine ${\beta }_1$ morphology or it’s evolution.

Remainder of this manuscript is organized as follows: Section 2 describes embedded precipitate supercell approach and computational methods used in their analysis, Section 3 presents results and analysis, and in Section 4 we use these results to propose a qualitative physics-based mechanism for ${\beta }_1$ formation within hcp-Mg lattice, and present the implications of our findings on other structural alloys.