Atomistic Determination of Anisotropic Surface Energy-Associated Growth Patterns of Magnesium Alloy Dendrites.

Because of the existence of anisotropic surface energy with respect to the hexagonal close-packed (hcp) lattice structure, magnesium alloy dendrite prefers to grow along certain crystallographic directions and exhibits a complex growth pattern. To disclose the underlying mechanism behind the three-dimensional (3-D) growth pattern of magnesium alloy dendrite, an anisotropy function was developed in light of the spherical harmonics and experimental findings. Relevant atomistic simulations based on density functional theory were then performed to determine the anisotropic surface energy along different crystallographic directions, and the corresponding anisotropic strength was quantified via the least-square regression. Results of phase field simulations showed that the proposed anisotropy function could satisfactorily describe the 3-D growth pattern of the α-Mg dendrite observed in the experiments. Our investigations shed great insight into understanding the pattern formation of the hcp magnesium alloy dendrite at an atomic level.

Introduction

Magnesium alloys are extensively applied in the industry because of their light weight, high specific strength, and environmental friendliness.[1,2] The α phase or α-Mg dendrite is one of the most important microstructures;[3,4] the dendritic size, orientation, and morphological distributions significantly influence the mechanical properties of the magnesium alloys.[5−7] Therefore, understanding the dendritic growth pattern, in particular determination of the preferred growth orientation-related three-dimensional (3-D) morphology, is of crucial importance in controlling the solidification microstructure and the associated properties of the alloys. The α-Mg dendrite with an hexagonal close-packed (hcp) lattice structure usually prefers to grow along the highly symmetrical directions,[8−10] resulting in a 6-fold symmetry on the basal plane. In this respect, recent experimental investigations have confirmed that the α-Mg dendrite exhibits an 18-primary-branch pattern in 3-D, with 6 along the ⟨112̅0⟩ directions on the basal plane and the other 12 along the ⟨112̅x⟩ directions on the nonbasal planes, such as ⟨112̅3⟩ and ⟨224̅5⟩ (i.e., ⟨112̅ 2.5⟩).[11−13]

During solidification, both thermodynamic and kinetic effects at the solid–liquid interface play crucial roles in determining the microstructure pattern formation.[14−20] For a dendritic microstructure, however, the growth orientation and pattern formation are mostly determined by the thermodynamic factors related to the anisotropic surface energy in light of the underlying lattice structure.[4,6,8,9] Meanwhile, the atomic stacking based on the lattice configuration can also be interpreted and understood in terms of the anisotropic surface energy[21] and accordingly understanding the 3-D growth pattern formation of the α-Mg dendrite requires a quantitative investigation of the anisotropic surface energy at the atomic level. Relevant studies have revealed that additional elements could also affect the anisotropic surface energy and thus the dendritic growth pattern, leading to complex dendritic microstructures that are not fully understood.[22,23]

It is well established that dendrite tends to grow along those directions with the maximum surface energy or the minimum surface stiffness.[24−26] To quantify this, an explicit surface energy function must be formulated as a prior. For the metallic alloy dendrite with an face-centered-cubic (fcc) lattice structure, such formulation has been well established and extensively employed.[5,25] However, this is not the case for the magnesium alloy dendrite with an hcp lattice structure. One of the available models for the hcp α-Mg dendrite was[17,27]where γ0 is the average surface energy and ε20, ε40, ε60, and ε66 measure the anisotropic strength along different directions. Equation could perfectly describe the 6-fold symmetry of the α-Mg dendrite at the basal plane, but it could not characterize the dendritic pattern at the nonbasal planes. Consequently, the simulated dendritic growth pattern deviated significantly from that observed in the experiment.[12,22] Therefore, it is necessary to develop an anisotropic function to describe the practical pattern formation of the hcp magnesium alloy dendrite. According to the experimental findings on the 3-D growth pattern of the α-Mg dendrite, in this work, we first presented an anisotropy function by combining certain terms of the spherical harmonics. Relevant atomistic simulations based on the hcp lattice structure and density functional theory (DFT)[28] were then performed for the first time to quantify the associated anisotropic strength along different directions. The influence of both solute type and solute concentration on the anisotropic parameters was also investigated and discussed.

Results and Discussion

According to the experimental observations, the α-Mg dendrite exhibits an 18-primary-branch pattern in 3-D.[12,13] A typical way of characterizing such a growth pattern is by formulating an anisotropic function on the basis of spherical harmonics[17,29]where f is the expansion coefficient, y(θ, φ) is the spherical harmonics function, and θ ∈ [0, π] and φ ∈ [0, 2π] are the spherical coordinate angles. We employed the y20-square and y53-square terms to describe the 18-primary-branch pattern of the α-Mg dendritewhere α20 and α53 are the anisotropic parameters andwhere n, n, and n are the components of surface normal unit vector n⃗ along the x, y, and z directions, respectively, as shown in Figure a. It is worth stressing that the y53-square term could satisfactorily illustrate the 18-primary-branch pattern (including both basal and nonbasal directions) of the magnesium alloy dendrite observed in the experiments.[12] Accordingly, eqs and 5 can be written asOn substituting eqs and 8 into eq , the resultant surface energy formulation is expressed aswhere ε’s (i = 1, 2, and 3) are the anisotropic parameters weighing the anisotropic strength along different directions. In particular, ε1 relates to dendritic growth along the direction normal to the basal plane (i.e., ⟨0001⟩), whereas ε2 and ε3 relate to that along both the basal plane and nonbasal plane directions (i.e., ⟨112̅x⟩ and ⟨112̅0⟩).

Figure 1

Schematic illustration of the spherical and Cartesian coordination systems (a) and the atomic lattice structure of hcp magnesium (b), showing the correlation between the indexes of crystallographic planes and the corresponding orientations.

The magnitudes of ε1, ε2, ε3, and γ0 in eq were quantitatively determined via the DFT-based atomistic simulations by constructing a slab model with a periodic boundary condition. The surface unit cell repeats along the crystallographic orientation normal to crystallographic plane {hkil}.[28,30] For the hcp lattice structure,[31,32] the indexes of the crystallographic plane and the perpendicular orientation (e.g., ⟨112̅k⟩ ⊥ ⟨112̅x⟩) are correlated via k = 2x(c/a)2/3 (see Figure b). As shown in Figure a, this model includes a vacuum region and a slab with several atomic layers; the bottom atomic layers in the slab were fixed, whereas the top ones were relaxed during optimization. Accordingly, the surface energy along 12 crystallographic directions related to the preferred growth orientations of the magnesium alloy dendrite, including {0001}, {101̅0}, {101̅1}, and {112̅k} with k ranging from 0 to 8 (see Supporting Table SI), was first determined. These values were then used to calculate the magnitude of ε1, ε2, ε3, and γ0 on the basis of the least-square regression method.[8,33] The atomic structure of binary Mg-based alloys was simulated using the solid solution model, where a certain number of solvent atoms was replaced by the solute atoms.[34,35] It has been confirmed that the position of additional atoms does not affect significantly the variation trend of the calculated surface energy with respect to the surface orientations.[36,37] The concentration of the additional elements for the magnesium alloys was chosen on the basis of their solubility in the solid or the maximum solute concentration in the Mg matrix.[36] We optimized the atomic structure of bulk materials and then constructed the surface slab model with respect to the growth orientations of the α-Mg dendrite. In terms of an n-layer surface slab model with respect to the {hkil} plane, the surface energy can be obtained via[30,33]where Eslab is the total energy per primitive slab unit cell, Ebulk is the total energy per primitive bulk unit cell, S is the surface area per primitive slab unit cell, and the factor of 2 accounts for two equivalent surfaces of a particular slab model. Details on the construction of the atomic structure model and surface energy calculations for these magnesium alloys can be found elsewhere.[37] Accordingly, the surface anisotropic parameters were deduced from these calculated data of orientation-dependent surface energy.

Figure 2

Graphic illustration of the slab model for surface energy calculation (a) and typical atomic structures corresponding to the surface orientations ⟨101̅0⟩ (b) and ⟨112̅0⟩ (c) of the Mg–1.56 atom % Al alloy. (Note that the red spheres and the blue spheres denote the aluminum atoms and the magnesium atoms, respectively.)

The atomistic simulations in this work were performed using the Vienna ab initio simulation package within the DFT framework.[28,38] The exchange and correlation interaction was described by local density approximation with projector-augmented wave potentials.[39] After convergence tests, a plane-wave cutoff energy of 450 eV was used for pure Mg and 420 eV for binary Mg-based alloys, including Mg–Al, Mg–Ba, Mg–Sn, Mg–Ca, Mg–Y, and Mg–Zn. The k-point separation in the Brillouin zone of the reciprocal space was set as 0.01 Å–1 for each surface unit cell.[40] Meanwhile, the total energy was converged to 5 × 10–7 eV/atom with respect to electronic, ionic, and unit cell degrees of freedom. The anisotropic surface energy was obtained from accurate calculations in accordance with our representative slab models. Figure b,c shows the typical atomic structures corresponding to the ⟨101̅0⟩ and ⟨112̅0⟩ surface orientations of Mg–1.56 atom % Al alloy, respectively. The calculated surface energy along different orientations is listed in Supporting Table SI. The estimated surface energy of the {0001} basal plane, that is, the most close-packed crystallographic plane, was minimum, which agreed well with the previous reported results.[17,27] It is worth mentioning that the method for the anisotropic surface energy calculation of hcp Mg-based alloys could be employed for other alloy systems as well. However, a specific slab model must be developed to accommodate the influence of the lattice structure for other alloys, for example, the fcc lattice structure for Ni-based alloys.

Depending on the DFT-based calculations and the least-square regression according to eq , the γ0, ε1, ε2, and ε3 were evaluated and the corresponding values are listed in Table , together with those from the literature. The results indicated that for all considered alloys, ε1 was always negative, indicating that the α-Mg dendrite preferred not to grow along ⟨0001⟩, which agreed well with the experimental observations.[11,12,23] On the contrary, ε2 was positive, signifying that the α-Mg dendrite preferred to grow along both ⟨112̅0⟩ and ⟨112̅x⟩. Moreover, the value of ε3 was either negative or positive, indicating that the growth tendency of the α-Mg dendrite along ⟨112̅0⟩ changed according to local conditions. Figure a,b shows the scope and distribution of anisotropic parameters ε1, ε2, and ε3 for binary Mg–Al and Mg–Zn alloys, respectively, and the insert maps show the corresponding polar diagrams of anisotropic surface energy. The orientation-dependent surface energy of the other Mg-based alloys, including Mg–Ba, Mg–Sn, Mg–Ca, and Mg–Y, exhibited similar behavior as that of the Mg–Al and Mg–Zn alloys, as shown in Supporting Figure S1. The average value of γ(θ, φ) for these Mg-based alloys was ∼0.8 J/m2 and changed as the additional solute elements.

Table 1

Average Value of Surface Energy γ0 (J/m2) and Anisotropy Parameters ε1, ε2, and ε3 of Mg-Based Alloys, Obtained by Least-Square Regression according to eq and the Resultant Data from DFT-Based Calculations

composition	γ0	ε1	ε2	ε3	(γ112̅0 – γ101̅0)/2γ0	(γ112̅0 – γ0001)/2γ0	reference
Mg–1.56 atom % Sn	0.7919	–0.0552	0.0741	–0.0770	0.1114	0.134	this work
Mg–1.56 atom % Ba	0.7605	–0.0457	0.0956	–0.084	0.0888	0.1265	this work
Mg–1.56 atom % Ca	0.7989	–0.059	0.0422	–0.5843	0.0460	0.0962	this work
Mg–1.56 atom % Y	0.8232	–0.0594	0.044	–0.3839	0.0664	0.1086	this work
Mg–1.56 atom % Al	0.8011	–0.0548	0.0805	0.0594	0.1237	0.1431	this work
Mg–3.12 atom % Al	0.8003	–0.0508	0.0812	–0.0345	0.1248	0.1429	this work
Mg–6.25 atom % Al	0.8067	–0.0479	0.0736	–0.1688	0.1216	0.1353	this work
Mg–7.0 atom % Al	0.8202	–0.0629	0.0742	–0.1046	0.0790	0.1273	this work
Mg–11.5 atom % Al	0.8188	–0.0467	0.1344	0.0401	0.0785	0.1619	this work
Mg–2.0 atom % Zn	0.8094	–0.0317	0.0757	–0.4284	0.0873	0.1396	this work
Mg–1.33 atom % Zn	0.81	–0.033	0.0765	–0.4631	0.0849	0.1365	this work
Mg–0.67 atom % Zn	0.8108	–0.0331	0.0715	–0.5014	0.0813	0.1335	this work
100 atom % Mg	0.7986	–0.0056	0.0593	–0.3174	0.0534	0.0728	this work
	0.1222						ref (17)
	0.0899				0.18 ± 0.08	1.2 ± 0.7	ref (27)

Figure 3

Scope and distribution of the determined anisotropic parameters ε1, ε2, and ε3 for (a) Mg–Al alloys and (b) Mg–Zn alloys. Inset maps show the corresponding polar diagram of the anisotropic surface energy.

For all considered Mg–Al alloys, ε1 varied from −0.0467 to −0.0629, ε2 from 0.0736 to 0.1344, and ε3 from −0.1688 to 0.0594, as shown in Figure a. This result signified that the Mg–Al alloy dendrite tended to grow along ⟨112̅0⟩ in the basal plane and along ⟨112̅x⟩ in the nonbasal planes and the growth tendency along the ⟨112̅x⟩ direction was dependent on the magnitude of Al concentration. For different Mg–Zn alloys, ε1 changed from −0.0331 to −0.033, ε2 from 0.0715 to 0.0765, and ε3 from −0.5014 to −0.4284, as shown in Figure b. In comparison to that for the Mg–Al alloy dendrite, the value of ε3 for the Mg–Zn alloy dendrite tended to be more negative, indicating a weaker growth tendency along ⟨112̅0⟩ in the basal plane. The growth tendency along ⟨112̅x⟩ in the nonbasal planes of the Mg–Zn alloy dendrite was comparable to that of the Mg–Al alloy dendrite. The underlying reason behind such difference can be understood by comparing the atomic interaction between the solvent and solute, as indicated by the enthalpy of mixing, (ΔH).[41] The predicted growth tendency of the α-Mg dendrite according to eq in terms of the anisotropic parameters was in agreement with previous experimental findings.[13,22]

Figure a,b shows typical graphic illustrations of the surface anisotropy function of the Mg–1.56 atom % Al alloy and the Mg–2.0 atom % Zn alloy dendrites, respectively. In Figure a, ε1 = −0.0547, ε2 = 0.0805, ε3 = 0.0594, and γ0 = 0.8011 J/m2, and in Figure b, ε1 = −0.0317, ε2 = 0.0757, ε3 = −0.4284, and γ0 = 0.8094 J/m2, that is, according to the DFT-based calculations and the least-square estimated values shown in Figure a,b, respectively. Then, the 3-D growth pattern of the hcp α-Mg dendrite can be quantitatively determined with the evaluated anisotropic parameters. However, the present surface anisotropy function is still limited in terms of evaluating the angle between the ⟨112̅0⟩ and ⟨112̅x⟩ directions, that is, it could never exceed 45°, which is different from some of the values measured in the experiments.[12,13] Further investigations are thus required to resolve this uncertainty. Nevertheless, the proposed anisotropy function is still able to capture the 18-primary-branch pattern of the hcp α-Mg dendrite observed in the experiments, which has not been achieved by other models.

Figure 4

Schematic illustration for the surface anisotropy function of the (a) Mg–1.56 atom % Al alloy dendrite and (b) Mg–2.0 atom % Zn alloy dendrite. The anisotropic parameters determined via the DFT-based calculations and the least-square regression are (a) ε1 = −0.0547, ε2 = 0.0805, and ε3 = 0.0594 and (b) ε1 = −0.0317, ε2 = 0.0757, and ε3 = −0.4284, respectively.

The DFT-based atomistic simulations in the present work were performed under equilibrium conditions, which is different from the practical experimental cases. During solidification, the way the atoms are absorbed from the liquid phase to the solid phase is highly dependent on the lattice structure of the formed crystal, and the atoms tend to attach these locations with either the minimum surface stiffness or the maximum surface energy,[42−45] both being more concerned with the lattice structure rather than the kinetic behavior at the liquid–solid interface. Accordingly, we believed that the crystallographic lattice structure is the most fundamental factor that determines dendritic pattern formation. Remarkably, even with such difference, the predicted results on the growth pattern of the magnesium alloy dendrite agreed quite well with the experimental findings.[36,37] However, more investigation, in particular relevant atomistic simulations at elevated temperatures, should be performed to explore the exact influence of temperature on the dendritic growth pattern.

On the basis of the quantified anisotropic parameters, the growth pattern of the α-Mg dendrite was simulated by incorporating eq into the phase field model. Figure shows the simulated morphology of a Mg–Sn alloy dendrite and a comparison with those reported in ref (12) where the anisotropic parameters were guessed values. The results confirmed that the proposed anisotropy function could reasonably describe the 3-D growth pattern of the α-Mg dendrite. It is noted that these guessed anisotropic parameters in ref (12) were significantly larger than the ones determined on the basis of DFT calculations, and such difference could generate artificial effects on the predictions of the 3-D dendritic morphology. Besides the anisotropy function, meaningful simulations of the hcp α-Mg dendrite also require a thorough investigation of other important parameters, including partition coefficient, supercooling, release of heat of fusion, and so on. Relevant studies along these directions are still in progress, and the detailed results will be reported in a forthcoming paper.

Figure 5

Growth patterns of an α-Mg dendrite simulated using the phase field model on the basis of the quantified anisotropy function in light of relevant DFT-based calculations (a0–a3) and a comparison with those simulated in a previous work (b0–b3).[12] (a0) and (b0) show the 3-D dendritic morphology, (a1–a3) and (b1–b3) show the dendritic morphology viewed from the ⟨0001⟩, ⟨101̅0⟩, and ⟨112̅0⟩ directions, and (a4, a5) and (b4, b5) show the patterns by cutting the dendrite along the {0001} and {101̅0} sections, respectively.

Conclusions

In summary, an anisotropy function based on the spherical harmonics and experimental findings was proposed to describe the 3-D growth pattern of the hcp magnesium alloy dendrite. Relevant atomistic simulations in light of density functional theory and the hcp lattice structure were performed to quantify the surface energy along different growth directions. Accordingly, the anisotropic parameters in the surface energy formulation and the average values of surface energy (∼0.8 J/m2) for different Mg-based alloys were predicted via the least-square regression method. The results showed that the proposed surface anisotropy function in terms of the anisotropic parameters could satisfactorily describe the practical 3-D morphology of the magnesium alloy dendrite observed in the experiments. The present work offers an important insight for a better understanding on the 3-D growth pattern of the hcp magnesium alloy dendrite and thus provides a theoretical guidance for further exploring the microstructure formation of those metallic alloy dendrites with a hexagonal symmetrical structure.