Solute Segregation and Pinning Effect on Lateral Twin Boundary in Magnesium

Abstract

Deformation twinning creates a three-dimensional twin domain via the migration of forward, normal and lateral twin boundaries (TBs) with respect to twin shear direction, normal to twin plane and twin lateral direction. Solute segregation and pinning effect on the forward and normal TBs have been experimentally observed and demonstrated via atomistic simulations. Here, we conducted a comprehensive study of solute segregation and the pinning effect on the lateral TBs in Mg. First-principles density functional theory calculations were used to obtain the segregation and formation energies of 19 alloying elements in coherent regions of lateral TBs. Alloying elements with greater difference in atomic radius from Mg generally show more negative segregation energy. Moreover, alloying elements with good solubility are selected to demonstrate the pinning effect on a coherent interface. Ge, Ga, Y, Gd, La and Ca show negative segregation energy and solubility energy, indicating that these elements can form stable segregation and have a strong pinning effect at the lateral boundary. Molecular dynamics simulations revealed that solutes in coherent regions are more effective in pinning lateral TBs than those in misfit regions. The results provide insight into the selection of solute atoms for tailoring twinning behavior.

1. Introduction

The easy activation of deformation twinning in Mg and Mg alloys results in strong plastic anisotropy and is responsible for limited room temperature formability [1,2]. Suppressing or retarding the nucleation and propagation of deformation twins can be achieved through alloy addition [3,4,5], grain refinement [6,7,8] and texture weakening [9,10]. A lot of experiments have demonstrated that alloying solute elements can modify the relative activity of different plastic deformation modes via the synergetic effect of solid-solution hardening on dislocations and solute segregation pinning on twin boundaries (TBs), improving the strength and ductility of Mg and Mg alloys [11]. The pinning effect of solute atoms on TBs has attracted attention recently [12,13], especially the segregation of solute atoms at coherent twin boundaries (CTBs) [14]. In practice, deformation twinning creates a three-dimensional twin domain embedded in a matrix. Corresponding to the crystallography of a twin, twin boundaries are categorized into forward, normal and lateral TBs (Figure 1a), with respect to η₁ along twin shear direction, K₁ normal to the twin plane, and λ along the twin lateral direction [15]. In addition to normal TB (usually referred to as a coherent twin boundary) which is related to twin thickening, twin propagation in the twinning direction and the lateral direction is accomplished through the migration of the forward and lateral TBs [15].

The segregation of alloying elements at TBs [14,16,17] and their pinning effect on TBs [12] have been revealed by extensive studies, most of which focus on behaviors of forward [17] and normal TBs [12,14,18,19]. In terms of crystallography, forward and normal TBs are tilt boundaries about the λ direction [20] and thus contain excess volume, which is defined as the Voronoi volume difference between the atoms close to the interface and in the matrix. The excess volume $∆{\mathrm{V}}_i$ can be calculated by

$$∆{\mathrm{V}}_x=V_i^x-{\mathrm{V}}_m$$

(1)

where $V_i^x$ is the Voronoi volume of atoms close to the interface and ${\mathrm{V}}_m$ is the Voronoi volume of atoms in the matrix. Solutes with a smaller/larger atomic radius than the matrix generally prefer the TBs’ location with negative/positive excess volume to lower the local stress field [14,21,22]. Atomistic simulations revealed that most of the alloying elements possess negative formation energy [21] and segregation energy [23], suggesting strong segregation along normal and forward TBs and a pinning effect on the migration of the two types of TBs. Meanwhile, lateral TBs are twist boundaries about the λ direction, possessing slight excess volume with respect to the matrix. Therefore, the segregation behaviors of lateral TBs cannot be easily predicted based solely on the excess volume. Based on a recent statistical analysis of electron back-scatter diffraction results [15] and atomistic simulation results [24], deformation twins in Mg propagate faster along the lateral direction than along the forward propagation direction as the twin is smaller than 40 μm during the twin growth. More details about the twin growth can be found in the Supplementary Materials. Therefore, at the beginning of twinning, it is more effective to suppress twin propagation/growth via the pinning migration of lateral TBs, further avoiding materials’ strong plastic anisotropy. In this sense, it is worth understanding solute segregation and the pinning effect on the lateral TBs.

Figure 1. Structural characteristics of lateral TBs. (a) Schematic showing a cube-shaped 3D twin domain. (b₁) Plane view of a T-PP₂ interface showing a coherent dichromatic complex (CDC) of two prismatic planes in twin (red empty symbols) and matrix (black solid symbols). In the unit hexagonal cells in twin (red rectangle) and matrix (black rectangle), the shadowed region indicates that the two atoms will occupy the same position after atomic shuffling. (b₂) The disclination characteristics are revealed in the rotated coherent dichromatic pattern (RCDP). (b₃) The coherent dichromatic pattern (CDP) indicates the 90° twist rotation of the T-PP₂ interface. The disclination twist angle is φ = 3.71°, reprinted from Ref. [25]. (c₁) The atomic structure of a twin domain enclosed by normal TBs (CTB) and lateral TBs (T-PP₂), observed along η₁ direction. (c₂) The atomic structure of a T-PP₂ interface observed along the λ direction.

Figure 1. Structural characteristics of lateral TBs. (a) Schematic showing a cube-shaped 3D twin domain. (b₁) Plane view of a T-PP₂ interface showing a coherent dichromatic complex (CDC) of two prismatic planes in twin (red empty symbols) and matrix (black solid symbols). In the unit hexagonal cells in twin (red rectangle) and matrix (black rectangle), the shadowed region indicates that the two atoms will occupy the same position after atomic shuffling. (b₂) The disclination characteristics are revealed in the rotated coherent dichromatic pattern (RCDP). (b₃) The coherent dichromatic pattern (CDP) indicates the 90° twist rotation of the T-PP₂ interface. The disclination twist angle is φ = 3.71°, reprinted from Ref. [25]. (c₁) The atomic structure of a twin domain enclosed by normal TBs (CTB) and lateral TBs (T-PP₂), observed along η₁ direction. (c₂) The atomic structure of a T-PP₂ interface observed along the λ direction.

In this work, we conducted first-principles density functional theory (DFT) calculations and molecular dynamics (MD) simulations to investigate solute segregation and the pinning effect on lateral TBs. The atomic structures of the lateral TBs in Mg were, for the first time, simulated and characterized by Liu et al. [25]. As shown in Figure 1(c₁,c₂), the lateral TBs of a {$\stackrel{-}{1}012$} twin exhibit a typical semi-coherent twist boundary with misfit dislocations [25]. The coherent interface is parallel to prismatic planes (referred to as twist prismatic/prismatic (T-PP₂) interface) in both the twin and matrix. The formation of the misfit dislocations cancels the φ = 3.71° twist disclination angle (Figure 1(b₁–b₃)) associated with the pile up of TDs on the T-PP₂ interface. In the research [25], Y. Liu, N. Li, S. Shao, et al. used high-resolution transmission electron microscopy and atomic simulations to characterize the lateral boundary of the {10$\overline{1}$2} deformation twins in magnesium. They found that the structure of the lateral boundary is a semi-coherent T-PP₂ boundary. In the research [26], J.P. Hirth, J. Wang and C.N. Tome used topological model, simulation and characterization methods to analyze the structure and defects of twin interfaces. They also demonstrated that the structure of the interface along the η direction contained screw dislocation. DFT calculations revealed that the coherent T-PP₂ interface possesses slight excess volume but obvious bonding distortion and enabled to obtain the segregation and formation energies of 19 alloying elements in the coherent T-PP₂ interface. Alloying elements with a greater difference in atomic radius to Mg generally show more negative segregation energy. Moreover, alloying elements with good solubility are selected to demonstrate the pinning effect on coherent T-PP₂ interfaces. MD simulations revealed that solutes in coherent regions are more effective in pinning lateral TBs than those in misfit regions. The results provide an insight into the selection of solute atoms for tailoring twin response.

2. Methodology

The DFT calculations are performed with the Vienna Ab-initio Simulation Package (VASP) [27,28] to investigate the solubility and pinning effect of different alloying elements on a coherent region of lateral TBs. The generalized gradient approximation (GGA) with Perdew–Burke–Ernzerhof (PBE) parametrization [29,30] is used for the exchange and correlation functions. The core electrons are replaced by the projector augmented wave (PAW) pseudopotentials [31] with the valence states shown in

Table 1. Details of first-principles calculations on bulk materials.

Element	Valency	K-Mesh	Structure	Lattice (Å)	ECoh (eV)
Zn	12	30 × 29 × 27	Hexagonal	a = 2.61, c = 4.87	−1.10
Ge	14	7 × 12 × 23	Cubic	a = 5.67	−4.51
Ga	13	9 × 9 × 29	Orthorhombic	a = 4.43, b = 7.60, c = 4.56	−2.89
Al	3	7 × 12 × 23	Cubic	a = 4.04	−3.75
Cd	12	27 × 25 × 22	Hexagonal	a = 2.96, c = 5.89	−0.74
Li	3	9 × 15 × 30	Cubic	a = 4.33	−1.91
Mg	2	25 × 24 × 26	Hexagonal	a = 3.17, c = 5.14	−1.51
Zr	12	24 × 23 × 25	Hexagonal	a = 3.24, c = 5.17	−8.52
Sc	11	9 × 14 × 28	Cubic	a = 4.65	−6.20
Na	7	10 × 16 × 31	Cubic	a = 4.21	−1.31
Dy	9	22 × 21 × 24	Hexagonal	a = 3.64, c = 5.58	−4.53
Y	11	8 × 13 × 26	Cubic	a = 5.10	−6.40
Yb	8	10 × 15 × 31	Cubic	a = 4.24	−1.46
Gd	9	22 × 21 × 22	Hexagonal	a = 3.64, c = 5.59	−4.57
Pm	11	22 × 21 × 11	Hexagonal	a = 3.68, c = 11.80	−4.68
Ca	10	7 × 12 × 24	Cubic	a = 5.58	−1.93
K	9	8 × 12 × 24	Cubic	a = 5.40	−1.04
Ce	12	9 × 14 × 28	Cubic	a = 4.67	−5.92
La	11	21 × 20 × 11	Hexagonal	a = 3.80, c = 12.04	−4.88
Ba	10	9 × 8 × 18	Hexagonal	a = 4.63, b = 8.02, c = 7.29	−1.88

. The plane-wave energy cut-off of 500 eV and a Methfessel–Paxton smearing of 0.01 eV are used. For all DFT calculations, the self-consistent iteration is stopped when the change in total energy is smaller than 10⁻⁵ eV. The convergence criterion of the geometry optimizations is that the forces acting on each atom are smaller than 0.01 eV/Å. The 80-atom 0.54 × 6.39 × 0.54 nm³ models containing coherent T-PP₂ interfaces with/without a solute element (X) are constructed in the coordinates where the x-, y- and z-axes are along the [$1\stackrel{-}{2}10$]m||[$1\stackrel{-}{2}10$]t, [$0001$]m||[$\stackrel{-}{1}010$]t and [$\stackrel{-}{1}010$]m||[$000\stackrel{-}{1}$]t directions. With a 7 × 1 × 7 Monkhorst–Pack (M-P) K-point grid, all the models are relaxed.

The simulations of the migration of the T-PP₂ interfaces, both in the absence and presence of solute segregation, are conducted using a bi-crystal model of the dimensions of 247.69 × 28.51 × 0.76 nm³ with a ($\stackrel{-}{1}012$) twinning relationship in the coordinates where the x-, y- and z-axes are along the λ, K₁ and η₁ directions. This orientation ensures that the simulated interfaces accurately mimic the structural characteristics and migration behavior of real-world T-PP₂ interfaces. The modified embedded atom method (MEAM) potential for the Mg-Ca system developed by Kim et al. [32] is employed in a Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [33]. In the simulation, periodic boundary conditions are applied along the y- and z-directions. Additionally, a fixed 1 nm wide layer is introduced perpendicular to the x-direction. Prior to simulating the migration process, the model is subjected to a relaxation process to bring it to a near-equilibrium state. This relaxation is performed at a temperature of 10 K for 20 ps in order to obtain the most stable structure and allow for a more controlled investigation of the migration behavior. During the relaxation process, the atomic positions are allowed to adjust until the system reaches a stable configuration, characterized by minimal changes in energy and atomic positions.

3. Results and Discussion

Figure 2a manifests the atomic structure of a coherent T-PP₂ interface. Although atoms near the interface have negligible excess volume ($∆{\mathrm{V}}_B$: 0.138, $∆{\mathrm{V}}_C$: 0.2343, $∆{\mathrm{V}}_D$: 0.2342), the bonds between them and their neighboring atoms are significantly distorted. Figure 2(b₁–b₄) displays the twelve first nearest neighbor (1NN) bonds of atoms A-D; these are labeled in Figure 2a. The bond-i (i = 1, 2, …, 12) associated with atom-x (x = A, B, C, D) can be defined. Figure 2c compares the lengths of bond-i in Figure 2(b₁–b₄). All bonds of atom A that are far from the interface possess a 3.1~3.3 Å length. For the atoms B-D that are near the interface, their bond-7 is significantly extended; these measure 3.53, 3.87 and 3.85 Å. The length of the other bonds is still within 3.1~3.3 Å. Generally, bond strength becomes weaker when its length is larger than its equilibrium [34]. According to the research by Zhao, the type and length can significantly affect the elastic stiffness of the metallic glasses [35,36]. So, one can reasonably speculate that the region near the coherent T-PP₂ interface is less stiff than the single crystal.

The segregation energy of a solute ($∆{\mathrm{E}}_{\mathrm{s}}$), which is the energy difference between solutes on and far from the interface, evaluates the pinning effect, and is calculated by

$$∆{\mathrm{E}}_{\mathrm{s}}={\mathrm{E}}_{\mathrm{tw}}\left({\mathit{Mg}}_{79}{X}_{1^C}\right)-{\mathrm{E}}_{\mathrm{tw}}\left({\mathit{Mg}}_{79}{X}_{1^A}\right)$$

(2)

where ${\mathrm{E}}_{\mathrm{tw}}\left({\mathit{Mg}}_{79}{X}_{1^C}\right)$ and ${\mathrm{E}}_{\mathrm{tw}}\left({\mathit{Mg}}_{79}{X}_{1^A}\right)$ are the system energies of the models with solutes X substituting atoms C and A, defined in Figure 2a. The segregation energy for 19 alloying elements is plotted in Figure 3a and summarized in

Table 2. Segregation and formation energies of different alloying elements on coherent T-PP₂ interface.

Solute	$∆{\mathbf{E}}_{\mathbf{s}}$ (eV)	$∆{\mathbf{E}}_{\mathbf{f}}$ (eV)
Zn	−0.055	−0.082
Ge	−0.047	−0.230
Ga	−0.045	−0.281
Al	−0.013	0.095
Cd	−0.008	−0.290
Li	−0.031	−0.177
Mg	-	-
Zr	0.034	0.611
Sc	−0.014	−0.044
Na	−0.043	0.302
Dy	−0.039	−0.129
Y	−0.048	−0.224
Yb	−0.064	−0.032
Gd	−0.043	−0.197
Pm	−0.061	−0.259
Ca	−0.074	−0.127
K	−0.129	1.106
Ce	0.049	0.832
La	−0.087	−0.281
Ba	−0.212	0.256

. The 19 elements studied in reference [21] are selected to compare the segregation behavior between lateral TBs and normal/forward TBs. Most of the 19 elements are used to prepare the corresponding alloys in experiments [37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53]. It is found that the segregation energy becomes more negative with an increasing difference in the atomic radius between Mg (R_Mg) and solute X (R_X), no matter whether R_X is smaller or larger than R_Mg. This is different from the solute segregation behaviors on normal and forward TBs, on which a smaller/larger R_X than R_Mg only results in negative segregation energy at a location with negative/positive excess volume to lower the associated elastic energy [21]. The unique solute segregation behaviors on lateral TB can still be explained from a mechanical point of view. Solutes with R_X different from R_Mg prefer to locate in less stiff regions near the interface to lower the associated elastic energy [14]. It should be noted that Ce solute does not follow the trend because substituting Ce in a single crystal does not trigger an obvious strain field (Figure 3b).

In addition to negative segregation energy, ideal alloying elements that inhibit the migration of the coherent T-PP₂ interface should have great solubility on the interface. The formation energy of a solute ($∆{\mathrm{E}}_{\mathrm{f}}$) can evaluate the solubility, and is estimated by

$$∆{\mathrm{E}}_{\mathrm{f}}={\mathrm{E}}_{\mathrm{tw}}\left({\mathit{Mg}}_{79}{X}_{1^C}\right)-{\mathrm{E}}_{\mathrm{tw}}\left({\mathit{Mg}}_{80}\right)-{\mathrm{E}}_{\mathrm{coh}}\left(X\right)+{\mathrm{E}}_{\mathrm{coh}}\left(\mathit{Mg}\right)$$

(3)

where ${\mathrm{E}}_{\mathrm{tw}}\left({\mathit{Mg}}_{80}\right)$ is the system energy of the models without a solute on the interface. ${\mathrm{E}}_{\mathrm{coh}}\left(X\right)$ and ${\mathrm{E}}_{\mathrm{coh}}\left(\mathit{Mg}\right)$ are cohesive energies of bulk materials, as listed in Table 1. The formation energy with respect to the alloying element is summarized in Table 2. Ga, Cd, Pm and La solutes possess the highest solubility, while Ge, Li, Dy, Y, Gd and Ca solutes have a relatively lower solubility. The solubility energy associated with Zn, Al, Sc and Tb solutes is nearly zero and is positive when associated with the Zr, Na, K, Ce and Ba solutes. As shown in Figure 4, Ge, Ga, Y, Gd, Pm, La and Ca solutes with more negative formation and segregation energies have relatively greater solubility and can subsequently pin a coherent T-PP₂ interface. Zr and Ce solutes with more positive formation and segregation energies should be less effective. Generally, when this method is used to measure the segregation behavior of solutes, the lower the segregation energy and the formation energy are, the more effective the segregation of solute atoms can be formed; that is, the solute can form a stable segregation and strong pinning effect on the interface. It is worth noting that this method is not only applicable to the T-PP₂ interface, but can also be used to evaluate the segregation of solute atoms on any interfaces. The results may explain previously made experimental observations such as Y being more effective in reducing twin activity than Al [54], La addition significantly reducing the twin fraction in Mg-Ce alloys [55] and Gd being more effective than Al and Zn in delaying twinning [56]. Meanwhile, some numerical results also show that Y segregation can significantly reduce the twin volume fraction [57], while Gd segregation can hinder the motion of a twin tip and delay the propagation of a deformation twin [58].

The relaxed T-PP₂ interface as shown in Figure 5a is semi-coherent. Misfit dislocations with the Burgers vector b_mis = 0.47[$\stackrel{-}{1}01\stackrel{-}{1}$] periodically locate on the interface [59]. The distance between two nearby misfit dislocations is 3.12 nm. The migration process of the T-PP₂ interface without Ca segregation under σ₁₃ = 1100 MPa at 300 K is shown in Figure 5(b₁–b₃). It should be noted that the high shear stress is applied to enable the migration of lateral TBs within the time-scale (~nano-second) of MD simulations. So, the results from MD simulations are only capable to show the trend in solute segregation along boundaries. At the beginning (10 ps, Figure 5b₁), the coherent regions move leftward while the misfit dislocations remain steady. Then, the misfit dislocations can either move leftward (28 ps, Figure 5b₂) or rightward (32 ps, Figure 5b₃). The leftward motion of misfit dislocations is activated via the dragging of the migrating coherent regions, while the rightward motion is driven using the Peach–Koehler force [60] under external load. A rapid migration of the T-PP₂ interface is observed after the depinning of the coherent regions from the misfit dislocations. As shown in Figure 5c, the T-PP₂ interface without a Ca solute migrates 29.64 nm in 40 ps. Figure 5d,e illustrate the difference in the pinning effect between Ca solutes (0.16 atom/nm² areal density) in coherent and misfit regions. The Ca solutes in misfit regions exhibit a slight pinning effect. The T-PP₂ interface migrates 16.41 nm in 40 ps under the same loading condition (Figure 5d). Meanwhile, the Ca solutes in coherent regions strongly pin the migration of the T-PP₂ interface because they significantly retard the motion of the coherent region that initiates the migration of the entire T-PP₂ interface. Consequently, the T-PP₂ interface only migrates 2.11 nm in 40 ps (Figure 5e). The MD results suggest that solutes in coherent regions are more effective in pinning lateral TBs than those in misfit regions. In this sense, alloying elements that can strongly segregate in and obstruct coherent regions of lateral TBs are desired for the inhibition of the lateral growth of deformation twins.

4. Conclusions

Deformation twins, as three-dimensional structural entities, are bounded by three kinds of interfaces, encompassing normal, forward, and lateral TBs. These boundaries play pivotal roles in governing the propagation and evolution of twinning. Among these, the rapidly migrating lateral TBs have emerged as critical factors influencing the growth of deformation twins. By effectively pinning the fast-moving lateral TBs, it becomes feasible to significantly impede the twin growth, thereby modulating the microstructural evolution and mechanical properties of the material. In this work, solute segregation and its pinning effect on lateral TBs (T-PP₂ interface) have been investigated by MD modeling and DFT calculations. The results provide an insight into the selection of solute atoms for controlling twin activity.

• The coherent T-PP₂ interface of the lateral TBs with minor excess volume and obvious extended bonds should be less stiff than the region far from the interface. Therefore, alloying elements with a significant difference in atomic radius with regard to Mg possess more negative segregation energy.
• Combined with the data of formation energy, DFT calculations enable the selection of alloying elements with a good solubility and pinning effect on a coherent T-PP₂ interface. Ge, Ga, Y, Gd, Pm, La and Ca can form stable segregation on the T-PP2; meanwhile, they also have a strong pinning effect on the interface.
• Along the semi-coherent T-PP₂ interface, solute atoms segregated in coherent regions are more effective at pinning TB migration than those segregated in misfit regions.