Nucleation of L1₂-Al₃M (M = Sc, Er, Y, Zr) Nanophases in Aluminum Alloys: A First-Principles ThermodynamicsStudy

Abstract

High-performance Sc-containing aluminum alloys are limited in their industrial application due to the high cost of Sc elements. Er, Zr, and Y elements are candidates for replacing Sc elements. Combined with the first-principles thermodynamic calculation and the classical nucleation theory, the nucleation of L1₂-Al₃M (M = Sc, Er, Y, Zr) nanophases in dilutealuminum alloys were investigated to reveal their structural stability. The calculated results showed that the critical radius and nucleation energy of the L1₂-Al₃M phases were as follows: Al₃Er > Al₃Y > Al₃Sc > Al₃Zr. The Al₃Zr phase was the easiest to nucleate in thermodynamics, while the nucleation of the Al₃Y and Al₃Er phases were relatively difficult in thermodynamics. Various structures of Al₃(Y, Zr) phases with the radius r < 1 nm can coexist in Al-Y-Zr alloys. At a precipitate’s radius of 1–10 nanometers, the core–shelled Al₃Zr(Y) phase illustrated the highest nucleation energy, while the separated structure Al₃Zr/Al₃Y obtained the lowest one, and had thermodynamic advantages in the nucleation process. Moreover, the core–shelled Al₃Zr(Y) phase obtained a higher nucleation energy than Al₃Zr(Sc) and Al₃Zr(Er). Core–doubleshelled Al₃Zr/Er(Y) obtained a lower nucleation energy than that of Al₃Zr(Y) due to the negative ΔG_chem of Al₃Er and the negative Al₃Er/Al₃Y interfacial energy, and was preferentially precipitated in thermodynamics stability.

1. Introduction

Sc-containing aluminum alloys are ideal materials for key components in the aerospace, high-speed rail, and automobile industries due to their high strength, corrosion resistance, and formability [1,2,3]. L1₂-Al₃Sc nanoparticles precipitated in Sc-containing aluminum alloys can effectively inhibit the recrystallization process [4], thereby obtaining comprehensive properties such as high strength, toughness, and corrosion resistance. Moreover, Seidman et al. [5,6] developed a series of Al-Sc high-temperature aluminum alloys. However, due to the high diffusion rate of Sc atoms, L1₂-Al₃Sc nanoparticles are prone to coarsening, reducing their ability to inhibit recrystallization and high-temperature performance. On the other hand, Zr atoms can partially replace Sc atoms in the Al₃Sc nanophase, forming a core–shelled Al₃(Sc_1−x, Zr_x), namely an Al₃Sc core and Al₃Zr shell) [7], where the Al₃Zr shell improves the coarsening resistance of the Al₃(Sc_1−x, Zr_x) nanophase due to the low diffusion rate of Zr atoms in the aluminum matrix [8].

However, the high cost of Sc elements greatly limits the engineering application of Sc-containing aluminum alloys. As members of the Sc element family, Er, Yb rare earth elements and the Y element have been considered ideal substitutes for Sc elements. Er and Yb elements were reported to form a core–shell structure of Al₃(Er, Zr) [9,10] and Al₃(Yb, Zr) [11,12] nanophases, which also effectively inhibited the recrystallization of aluminum alloys. Based on high-throughput first-principles calculations of the nucleation and growth for the L1₂structure Al₃RE phases, Fan et al. [13,14] revealed the ΔG_V firstly decreased from Sc, Y to Ce, then increased linearly for RE elements, and the ΔG_V tended to increase linearly with the temperature. It was speculated that the Y element could replace the expensive Sc element.The investigation by Zhang et al. [15,16] showed that the precipitation phase was mainly the Al₃Y phase, which became the core of Al₃Zr and promoted the precipitation kinetics of solid solution Zr atoms, whereas a hybrid structure of Al₃(Zr, Y) rather than the typical core–shelled structure was observed after long-term homogenization, where the Y and Zr elements were uniformly distributed in Al₃(Y, Zr) nanoprecipitates through atom probe tomography (APT).

Some research has been conducted on the formation mechanism of the hybrid structure of Al₃(Zr, Y). Zhang et al. [16] indicated that the hybrid structure of Al₃(Zr, Y) was attributed to the strong interactions between the Y and Zr atoms, resulting in their co-precipitation. Based on first-principles calculations, Wang et al. [17] indicated that the doping of the Y element and the Zr element decreased the interface energies of the FCC-Al(001)/FCC-Al₃Y(001) interface and formed a hybrid structure of Al₃(Y, Zr) instead of an Al₃Y core + Al₃Zr shell structure. The author’s previous research indicated that the interface energy of Al₃Zr/Al was lower than that of Al₃Zr/Al₃Y, and it was deduced that Al₃Zr tended to form a shell layer, while Al₃Y formed a core layer. However, the high coherent strain energy made the Al₃Y/Al₃Zr interface unstable, and it was difficult to form a stable cored Al₃Y/shelled Al₃Zr structure [18]. Although the author’s previous investigation elucidated the reason for Al₃Y/Al₃Zr not having a core–shelled structure based on the coherent strain energy, there were several issues that needed to be answered, such as whether the hybrid structure of Al₃(Zr, Y) was determined by atomic diffusion control or thermodynamic structure stability.

One view was that the formation of the core–shelled Al₃M phase was attributed to the differences in atomic diffusion rates. Al₃Sc and Al₃Er core structures were formed due to the fast diffusion rate of the Sc and Er atoms. The diffusion rate of the Zr element was slow, resulting in the formation of an Al₃Zr shell structure [19]. Furthermore, the core(Al₃Er)–double shell (Al₃Sc/Al₃Zr) structure of L1₂-Al₃(Sc, Er, Zr) was precipitated in Al-Sc-Er-Zr alloys after homogenization at 400 °C [20]. Seidman et al. [20] suggested that the core–double-shelled L1₂-Al₃(Sc, Er, Zr) can be attributed to their difference in diffusion rate, e.g., D_Er > D_Sc > D_Zr. Leibner et al. [21] found that there were two major groups of core–double-shelled L1₂-Al₃(Sc, Er, Zr) observed after aging at 600 °C/4 h, one having the usual core (Al₃Er)–double shell (Al₃Sc/Al₃Zr) structure and the other having an unusual core (Al₃Sc)–double shell (Al₃Er/Al₃Zr) structure. They suggested that the segregation of the Sc atom to dislocations and the interaction between the solid solution atoms and the Sc atom promoted the formation of the unusual core (Al₃Sc)–double shell structure. It should be noted that the hybrid structure of Al₃(Zr, Sc) [22] and Al₃(Er, Zr) [23] was also observed in aluminum alloys. Therefore, the difference in diffusion rates between atoms does not explain the formation of core–shelled structures well.

The thermodynamic analysis of nanophases’ nucleation based on first-principles calculations can provide insights into the phase transformation process of L1₂-Al₃M phases. Jiang et al. [24,25] used first-principles calculation methods to calculate the nucleation energies of the Al₃(Er, Zr) and Al₃(Sc, Zr) phases with different microstructures, revealing the thermodynamic stability of the Al₃(Er, Zr) and Al₃(Sc, Zr) phases during the homogenization precipitation. The nucleation properties calculated by Liu [26] showed that the core–shelled Al₃(Er1−x, Scx) obtained a highly stable structure due to its low nucleation energy, which was independent of the temperature and Sc/Er ratio. However, first-principles calculations of the nucleation and thermodynamic stability for the Al₃(Y, Zr) phase were rarely reported. Furthermore, the author’s investigation showed that Er atoms tended to segregate at the Al₃Y/Al₃Zr interface, and were inclined to form a core–double-shelled Al₃(Y, Er, Zr) structure with an Al₃Y core, an Al₃Er inner shell, and an Al₃Zr outer shell [18]. The nucleation and thermodynamic stability of core–double-shelled Al₃(Y, Er, Zr) needed to be evaluated to develop Al-Y-Zr series alloys.

Combining with the calculation results of interface energies and the coherent strain energy in the previous research [18], the total nucleation energies of various structures of L1₂-Al₃M (M = Sc, Er, Zr, Y)phases were calculated based on first-principles thermodynamic calculation and classical nucleation theory. The critical nucleation energies and nucleation radii of Al₃M phases were calculated to compare the nucleation differences of Al₃M nanophases. The nucleation energies of various structures of ternary L1₂-Al₃(Y, Zr) phases were investigated to reveal the formation mechanism of Al₃(Y, Zr) with a hybrid structure. The nucleation calculation result of core–shelled Al₃(Y, Zr) was also compared with that of core–shelled Al₃(Sc, Zr) and core–shelled Al₃(Er, Zr). Furthermore, based on first-principles calculations, the nucleation energy of the core–double-shelled Al₃(Er, Y, Zr) phase was investigated to evaluate its thermodynamic stability. This paper aimed to reveal the internal formation mechanism of L1₂-Al₃M with a core–shelled structure from the perspective of first-principles thermodynamic calculations, and provided guidance for the development of new Al-Y-Zr series alloys

2. Computational Methods

Based on density functional theory (DFT) [27], first-principles calculations were carried out by VASP software [28]. The electron configuration was described by Al-3s²3p¹, Sc-3s²3p⁶4s¹3d², Zr-4s²4p⁶5s¹4d³, Er-6s²5p⁶5d¹, and Y-4s²4p⁶5s¹4d² valence states, respectively. The ion–electron interactions were described by the projection augmented wave (PAW) method withinthe frozen core approximation [29]. The exchange-correlation energy functional between electrons was described by the Perdew–Burke–Ernzerhof (PBE) [30,31] method of generalized gradient approximation (GGA). The kinetic energy cutoff of the plane wave basis and the size of the k-mesh for the Brillouin zone were tested for self-consistent convergence. The calculation of the bulk phase of L1₂-Al₃M (M = Sc, Er, Y, Zr) used conventional single cells. In each periodic direction of reciprocal space, the geometric structure was optimized by the Monkhorst–Pack k-point grids with linear k-mesh analytical values of less than 0.032π/Å. Using the linear tetrahedron method with the Blöchl correction, the total energy was calculated when the total energy converged to 10⁻⁴ eV/atom. The lattice constants (a) and bulk modulus (B) were predicted as fcc-Al (a = 4.042 Å and B = 78.2 GPa), L1₂-Al₃Sc (a = 4.103 Å and B = 86.4 GPa), L1₂-Al₃Zr (a = 4.108 Å and B = 102.3 GPa), and L1₂-Al₃Er (a = 4.232 Å and B = 78.5 GPa), respectively, which agreed well with Ref. [26].

Vibration entropy had a significant influence on the chemical formation energy ΔG_chem corresponding to the precipitation of the L1₂-Al₃M phase from the fcc-Al_nM solutionmatrix. The calculation of vibration entropy was based on the method of the density functional perturbation theory (DFPT) [32] under the simple harmonic approximation, and the phonon spectrum of Al₃M was calculated by using the 2 × 2 × 2 supercell model. The Al_nM was adopted by the 2 × 2 × 2 supercell Al matrix, and the M atom was doped and dissolved in the center. In this method, a small external disturbance was introduced, and the linear response of the system was calculated based on this disturbance. By calculating the response function, the perturbation expression of the vibration frequency can be derived, resulting in the vibration entropy difference of the Al₃M phase.

3. Results and Discussion

3.1. Nucleation of Binary L1₂-Al₃M Phases

According to the classical nucleation theory, the nucleation work consisted of two parts: the energy released by the precipitated phase from the Al matrix, and the energy from the new interface between the precipitated phase and the matrix. The precipitated phase was usually assumed to be a sphere with uniform density distribution. When the L1₂-Al₃M nanophases are precipitated from the Al matrix, their precipitation radius R and nucleation work ΔG can be expressed as:

$$\mathsf{\Delta }G=\frac{4\pi }{3}{R}^3·\mathsf{\Delta }{G}_V+4\pi R^2·\gamma$$

(1)

where γ is the interface energy per unit area after subtracting the coherent strain energy. The Al(001)/Al₃M(001)-contacting facet was the most energy-favored orientation [18,24,25], and the interface energy of Al(001)/Al₃M(001) was calculated to estimate the critical nucleation works and nucleation radius. ΔG_V is the volume-free energy per unit volume, which is defined as:

ΔG_V = ΔG_chem + G_s

(2)

where ΔG_chem is the chemical formation energy corresponding to the precipitation of the L1₂-Al₃M phase in the matrix; G_s is the coherent strain energy.

The chemical reaction equation of the Al₃M nanophase precipitation can be written as: Al_nM = Al₃M + (n − 3)Al; so its chemical energy is expressed as [33]:

$$\begin{array}{c}\mathsf{\Delta }{G}_{\mathit{chem}}=G_{A{l}_{3}M}+(n-3){\mu }_{\mathit{Al}}-G_{A{l}_{n}M}\hfill \\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}=({\mathsf{\Delta }H}_{A{l}_{3}M}-{\mathsf{\Delta }H}_{A{l}_{n}M})-T({\mathsf{\Delta }S}_{A{l}_{3}M}-{\mathsf{\Delta }S}_{A{l}_{n}M})\hfill \end{array}$$

(3)

Here $\mathsf{\Delta }{H}_{{\mathit{Al}}_{3}M}$ and $\mathsf{\Delta }{H}_{{\mathit{Al}}_{n}M}$ are the formation enthalpies of L1₂-Al₃M and fcc-Al_nM, respectively, and the enthalpy can be approximately equal to the internal energy here because the volume–pressure term in the solid state can be ignored [33]; $\mathsf{\Delta }{S}_{{\mathit{Al}}_{3}M}$ and $\mathsf{\Delta }{S}_{{\mathit{Al}}_{n}M}$ are the formation entropy of L1₂-Al₃M and fcc-Al_nM, respectively.As the nucleation process of the Al₃M nanophases was sensitive to the temperature, the contribution of formation entropy should not be ignored. The contribution of formation entropy may become very important at high temperature. The entropy change in the alloy consisted of three parts: configuration entropy, hot electron entropy, and vibration entropy. In this calculation, the configuration entropy was generally negligible for a dilute alloy, which was clearly revealed for dilute Al-Sc-Zr alloys [24] and Al-Er-Zr alloys [25], and the hot electron entropy can be ignored for relatively low temperatures [34], so the vibration entropy was considered as contributing to entropy change.

According to the differentiation of Equation (1), the critical nucleation radius of R* and the critical nucleation work ΔG_V (R*) can be obtained as:

$$R*=\frac{-2\gamma }{\mathsf{\Delta }{G}_V}$$

(4)

$$\mathsf{\Delta }{G}_V(R*)=\frac{16\pi }{3}\frac{{\gamma }^3}{\mathsf{\Delta }{G}_V^2}$$

(5)

The corresponding differences in enthalpy and vibration entropy between the L1₂-Al₃M phases and the fcc-Al_nM solution matrix are shown in

Table 1. The corresponding enthalpy difference and vibration entropy difference of L1₂-Al₃M phase precipitation.

	$\mathsf{\Delta }{\mathit{H}}_{\mathit{f}}^{\mathit{A}{\mathit{l}}_{\mathbf{3}}\mathit{M}}\mathbf{-}\mathsf{\Delta }{\mathit{H}}_{\mathit{f}}^{\mathit{A}{\mathit{l}}_{\mathit{n}}\mathit{M}}$ (eV/Atom)		$\mathsf{\Delta }{\mathit{S}}_{\mathit{v}\mathit{i}\mathit{b}}^{\mathit{A}{\mathit{l}}_{\mathbf{3}}\mathit{M}}\mathbf{-}\mathsf{\Delta }{\mathit{S}}_{\mathit{v}\mathit{i}\mathit{b}}^{\mathit{A}{\mathit{l}}_{\mathit{n}}\mathit{M}}$ (kB/Atom)
Al3Sc-AlnSc	−0.718	−0.776 [24]	3.35	2.67 [24]
Al3Zr-AlnZr	−0.667	−0.831 [24]	4.01	2.72 [24]
Al3Er-AlnEr	−0.823	−0.867 [25]	5.18	3.53 [25]
Al3Y-AlnY	−0.902	-	5.72	-

. The corresponding differences in enthalpy ($\mathsf{\Delta }{H}_f^{A{l}_{3}M}-\mathsf{\Delta }{H}_f^{A{l}_{n}M}$) were −0.718 eV/atom, −0.667 eV/atom, −0.823 eV/atom, and −0.902 eV/atom for the Al₃Sc phase, Al₃Zr phase, Al₃Er phase, and Al₃Y phase, respectively. The enthalpy difference of Al₃Sc was in good agreement with the calculated values of −0.72 eV/atom in the literature [33], and was higher than the calculated values of −0.776 eV/atom in the literature [24]. The corresponding enthalpy differences of the Al₃Zr and Al₃Er phases were −0.667 eV/atom and 0.867 ev/atom, respectively, which were also slightly higher than that of the investigation [24,25]. The enthalpy difference of the Al₃Y phase has not yet been documented, but the calculation result was −0.902 eV/atom.

In order to calculate the nucleation work of the Al₃M phase in the Al matrix, it was necessary to calculate the vibration entropy difference. The vibration entropy differences of the Al₃Sc phase, Al₃Zr phase, and Al₃Er phase were 3.35 k_B/atom, 4.01 k_B/atom, and 5.18 k_B/atom, respectively, which were higher than the calculated results in the literature (2.67 k_B/Sc [24], 2.72 k_B/Zr [24], and3.53 k_B/Er [25]). The vibration entropy difference of the Al₃Y phase was 5.72 k_B/atom.

The author’s previous research calculated the coherent strain energy of L1₂-Al₃M/Al [18], where the coherent strain energies were 0.0035 ev/atom for Al₃Sc/Al, 0.0023 eV/atom for Al₃Zr/Al, 0.0088 eV/atom for Al₃Er/Al, and 0.0094 eV/atom for Al₃Y/Al. Based on Equations (1)–(3), the computation result at 673 K illustrated that the interface strains contributed to only ~8.5% of the volumetric formation energy for the Al₃Sc phase, the Al₃Zr phase, the Al₃Er phase, and the Al₃Y phase in Al. It indicated that the coherent strain energy of Al₃M/Al had little influence on the precipitation of Al₃M nanophases.

Combining with the Al/Al₃M interface energy [18], the critical nucleation radius and critical nucleation work of each phase at 673 K are shown in

Table 2. Critical nucleation radius and critical nucleation work.

	Critical Nucleation Radius (Å)		Critical Nucleation Work (J)
	Present	Ref.	Present	Ref.
Al3Sc	5.95	6.6 [24]	2.01 × 10−19	2.9 × 10−19 [24]
Al3Zr	3.89	2.9 [24]	4.83 × 10−20	2.9 × 10−20 [24]
Al3Er	9.57	8.4 [25]	6.94 × 10−19	5.4 × 10−19 [25]
Al3Y	9.40	-	6.84 × 10−19	-

. For L1₂-Al₃M (M = Sc, Zr, Er, Y), the predicted critical nucleation radii were 5.95 Å, 3.89 Å, 9.57 Å, and 9.40 Å, for the Al₃Sc phase, the Al₃Zr phase, the Al₃Er phase, and the Al₃Y phase, respectively. The critical nucleation works were 2.01 × 10⁻¹⁹ J, 4.83 × 10⁻²⁰ J, 6.94 × 10⁻¹⁹ J, and 6.84 × 10⁻¹⁹ J for the Al₃Sc phase, the Al₃Zr phase, the Al₃Er phase, and the Al₃Y phase, respectively. Among them, the calculated value of the critical nucleation radius of the Al₃Sc phase was slightly lower than the literature value [24], but the critical nucleation radii of the Al₃Zr phase and the Al₃Er phase were slightly higher than the value in Jiang’s investigation [24,25]. On the other hand, the critical nucleation work of Al₃Sc was slightly less than the literature value [24], and the critical nucleation works of Al₃Zr and Al₃Er were slightly greater than the literature value [25]. The critical nucleation radius and critical nucleation work of the Al₃Y phase at 673 K has not been reported yet. The investigation of Fan et al. [14] showed that the critical nucleation radius of Al₃Y for the (100) plane was about 3 Å at 300K, which was lower than the calculation value in this research. The reason can be attributed to the different calculation methods of nucleation energy and the low temperature.

Among the various L1₂-Al₃M phases, the Al₃Zr phase obtained the smallest critical nucleation radius and lowest critical nucleation work, whereas the Al₃Er and Al₃Y phases obtained similar nucleation characteristics, and displayed the largest critical nucleation radius and highest critical nucleation work. The critical nucleation radius and nucleation work of Al₃Sc were lower than those of the Al₃Er and Al₃Y phases, which agreed well with Fan’s calculation [14]. It indicated that Al₃Zr had thermodynamic advantages in the nucleation process, while the Al₃Er and Al₃Y phases were relatively difficult to nucleate but had advantages in precipitation kinetics.

3.2. Nucleation and Stability of Multicomponent L1₂-Al₃M Phases

As described in Section 3.1, the thermodynamic priority order of precipitation was: Al₃Zr > Al₃Sc > Al₃Er > Al₃Y. The lowest interface energy of Al₃Zr/Al suggested that the Al₃Zr phase always tended to wrap outside the precipitation phase during the precipitation process. Due to the low interfacial energy of L1₂-Al₃Zr/Al₃Sc and L1₂-Al₃Zr/Al₃Er, once a core–shell structure was formed, the core–shelled Al₃Sc (Zr) and Al₃Er (Zr) were stable structures. However, the previous research showed that Al₃(Y, Zr) transformed from a core–shelled structure into a hybrid structure during homogenization at high temperatures [15,16]. In this section, the nucleation of multicomponent L1₂-Al₃(N, Zr) (N = Y, Sc, Er) phases were investigated based on first-principles thermodynamic calculations. The nucleation of possible ternary L1₂-Al₃(Y, Zr) phases included the core–shelled structures (the Al₃Y-core + Al₃Zr-shell structure, denoted as L1₂-Al₃Zr(Y)), the hybrid structure (denoted as L1₂-Al₃(Zr_x, Y_1−x), and the separate nucleation of binary L1₂-Al₃Zr and L1₂-Al₃Y (denoted as L1₂-Al₃Zr/Al₃Y). Moreover, the nucleation calculation result of core–shelled Al₃Zr(Y) was also compared with that of core–shelled Al₃Zr(Sc,) and core–shelled Al₃Zr(Er).

Based on the classical nucleation theory, the structure stability of L1₂ nanoparticles with the different structures can be evaluated through the total nucleation energy ΔG_Al3(N,Zr) (N = Y, Sc, Er), and the expressions are given as [24]:

$$\mathsf{\Delta }{G}_{A{l}_{3}Zr(N)}=\frac{4\pi }{3}\left[\right(R^3-r^3)·\mathsf{\Delta }{G}_V^{A{l}_{3}Zr}+r^3·\mathsf{\Delta }{G}_V^{A{l}_{3}N}]+4\pi (r^2·{\gamma }_{A{l}_{3}Zr/A{l}_{3}N}+R^2·{\gamma }_{A{l}_{3}Zr/Al})$$

(6)

$$\mathsf{\Delta }{G}_{A{l}_{3}N+A{l}_{3}Zr}=\frac{4\pi }{3}(r^3\times \mathsf{\Delta }{G}_V^{A{l}_{3}N}+r^3\times \mathsf{\Delta }{G}_V^{A{l}_{3}Zr})+4\pi (r^2\times {\gamma }_{Al/A{l}_{3}N}+r^2\times {\gamma }_{Al/A{l}_{3}Zr})$$

(7)

$$\mathsf{\Delta }{G}_{A{l}_3(N_x,Z{r}_{1-x})}=\frac{4\pi }{3}{R}^3·\mathsf{\Delta }{G}_V^{A{l}_3(N_x,Z{r}_{1-x})}+4\pi R^2·{\gamma }_{A{l}_3(Nx,Zr1-x)}$$

(8)

Here R is the radius of ternary L1₂-Al₃(N, Zr), and r is the radius of the binary Al₃N. Assuming that all the solute atoms had completely precipitated from the Al matrix, the R and r values of ternary L1₂-Al₃(N, Zr) with a core–shelled structure depended on the relative precipitation amount of solute atoms N and Zr. $\mathsf{\Delta }{G}_V^{A{l}_{3}N}$ and $\mathsf{\Delta }{G}_V^{A{l}_{3}Zr}$ are the volumetric formation energy of the L1₂-Al₃N phase and the Al₃Zr phase in aluminum alloys. ${\gamma }_{A{l}_{3}Zr/A{l}_{3}N}$ and ${\gamma }_{A{l}_{3}Zr/Al}$ are the interface energies of the Al₃Zr/Al₃N and Al₃Zr/Al interface in aluminum alloys. The Al₃Zr(001)/Al₃N(001)-contacting facets were considered to be the most energy-favored orientation, and the interfaces’ energies were calculated in the authors’ previous investigation [18]. It should be noted that the interfaces’ energies were generally oerestimated at the actual precipitation temperature due to the density functional principles of the ground state. $\mathsf{\Delta }{G}_V^{A{l}_3(N_x,Z{r}_{1-x})}$ and ${\gamma }_{A{l}_3(N_x,Z{r}_{1-x})}$ are the volumetric formation energy and the interface energy of the hybrid structure of Al₃(N_x, Zr_1−x). However, it was difficult to directly calculate the value of $\mathsf{\Delta }{G}_V^{A{l}_3(N_x,Z{r}_{1-x})}$, which was estimated by the composition-weighted summation of $\mathsf{\Delta }{G}_V^{A{l}_{3}N}$ and $\mathsf{\Delta }{G}_V^{A{l}_{3}Zr}$ [24]. Similarly, ${\gamma }_{A{l}_3(N_x,Z{r}_{1-x})}$ was evaluated by the composition-weighted summation of ${\gamma }_{A{l}_{3}N/Al}$ and ${\gamma }_{A{l}_{3}Zr/Al}$.

Furthermore, the authors’ previous investigation indicated that Er atoms tended to segregate at the Al₃Y/Al₃Zr interface, and were inclined to form a core–double-shelled Al₃Y/Al₃Er/Al₃Zr structure [18], denoted as Al₃Zr/Er(Y), and its nucleation energy and thermodynamic stability can be evaluated as:

$$\begin{array}{l}\mathsf{\Delta }{G}_{A{l}_{3}Zr/Er(Y)}=\frac{4\pi }{3}[(R_2^3-R_1^3)\times \mathsf{\Delta }{G}_V^{A{l}_{3}Zr}+(R_1^3-r_{}^3)\times \mathsf{\Delta }{G}_V^{A{l}_{3}Er}+r_{}^3\times \mathsf{\Delta }{G}_V^{A{l}_{3}Y})\\ +4\pi (r_{}^2\times {\gamma }_{A{l}_{3}Y/A{l}_{3}Er}+R_1^2\times {\gamma }_{A{l}_{3}Zr/A{l}_{3}Er}+R_2^2\times {\gamma }_{A{l}_{3}Zr/Al})\end{array}$$

(9)

Here R₁ and R₂ are the radii of the first and second shells of the core–double-shelled Al₃Zr/Er (Y), respectively; r is the radius of the Al₃Y core layer.

Under the conditions of homogenization temperature (T = 673 K) and the equal solute atomic ratio (the atomic ratio of Y to Zr was 1), the total nucleation energies (ΔG) of the various possible structures for the L1₂-Al₃(Y, Zr) phase were calculated as a function of the precipitate radius (R), and the results are plotted in Figure 1. It showed that the nucleation energy of various structures of Al₃(Y, Zr) increased with the radius of the precipitated phase. At a radius of 0–1 nanometers, there was no significant difference in the free energy of each phase; thus, several structures of Al₃(Y, Zr) phases can coexist in the early stage of homogenization. To some extent, the 0–1 nanometer precipitation stage corresponded to the early aging stage of atomic clusters, and did not form a stable microstructure.

At a radius of 1–10 nanometers, the difference in the total nucleation energy among different structures became increasingly significant. The core–shelled Al₃Zr(Y) phase illustrated the highest nucleation work among various precipitate structures, indicating that the core–shelled Al₃Zr(Y) phase precipitated without advantage in thermodynamics. However, the separated nucleation of binary L1₂-Al₃Zr/Al₃Y obtained the lowest nucleation energy, suggesting that L1₂-Al₃Zr/Al₃Y had thermodynamic advantages in the nucleation process. Due to the low segregation energy of Zr elements at the Al₃Y interface, it was beneficial to drive the segregation of Zr elements at the Al₃Y interface [18]. Gao et al. [16] studied the early precipitation phase structure of Al-0.08Y-0.30Zr alloy at 350 °C for 10 min, and the results showed that the precipitation phase was mainly the Al₃Y phase, which became the core of Al₃Zr and promoted the precipitation kinetics of solid solution Zr atoms. However, it was difficult to form a stable core–shelled Al₃Y/Al₃Zr owing to the large coherency strain energy and high mismatch between Al₃Y and Al₃Zr [18]. Thus, the separated structure of L1₂-Al₃Zr/Al₃Y was considered to be the thermodynamically stable structure. The investigation by Gao et al. [16] showed that after isothermal aging at 400 °C for 200 h, the Y and Zr atoms in the Al-Y-Zr alloy were almost uniformly distributed within the precipitate phase, indicating a separated structure of L1₂-Al₃Zr/Al₃Y, and did not exhibit a clear core–shelled structure, which confirmed the first-principles calculation results in this paper.

Figure 2 shows the total nucleation energies (ΔG) for three kinds of core–shelled structures, Al₃Zr(Sc), Al₃Zr(Er), and Al₃Zr(Y), under the condition of homogenization at 673 K and the complete precipitation of Sc, Y, Er, and Zr in equal proportion, respectively. The nucleation energies of core–shelled Al₃Zr(Y) and Al₃Zr(Sc) increased with the radius of the precipitated phase, whereas the nucleation energies of Al₃Zr(Er) were negative, and decreased with the radius of the precipitated phase. The calculations of Al₃Zr(Sc) and Al₃Zr(Er) were similar to the investigation by Jiang et al. [24,25]. The order of the nucleation energies was: Al₃Zr(Y) > Al₃Zr(Sc) > Al₃Zr(Er). The core–shelled Al₃Zr(Y) phase obtained the highest nucleation energy, indicating that it was inclined to form a separated structure, L1₂-Al₃Zr/Al₃Y, which was very consistent with the experimental observation [16]. The core–shelled Al₃Zr(Sc) and Al₃Zr(Er) were thermodynamically stable structures owing to their low nucleation energies, which were confirmed by the experimental observation in Al-Sc-Zr alloys [8] and Al-Er-Zr alloys [9]. In comparison with Al₃Zr(Y) and Al₃Zr(Sc), although the Al₃Er/Al₃Zr interface had a higher coherent strain energy than that the of Al₃Sc/Al₃Zr interface [18], core-shelled Al₃Zr(Er) obtained a low nucleation energy due to its low chemical energy ΔG_chem and the Al₃Er/Al₃Zr interface energy. Thus, the nucleation energy of Al₃M nanophases depended on their chemical energy ΔG_chem and the interface energy.

The nucleation energies (ΔG) of core–double-shelled Al₃Zr/Er(Y) were carried out under the condition of homogenization at 673 K and the complete precipitation of Y, Er, and Zr in equal proportion. The nucleation energy of Al₃Zr/Er(Y) was negative and significantly decreased with the precipitation radius, as shown in Figure 2. The nucleation energy of Al₃Zr/Er(Y) was far lower than that of core–shelled Al₃Zr(Y), and obtained high thermodynamic stability, preferentially precipitating in thermodynamics. Core–double-shelled Al₃Zr/Er(Y) was inclined to form in Al-Y-Er--Zr alloys, as the Er atom tended to segregate at the Al₃Y/Al₃Zr interface [18]. The segregation of the Er atom dramatically decreased the nucleation energy due to the decrease in ΔG_chem and strain energy G_S, as illustrated in Al₃Zr(Er), although the high interfacial energy of Al₃Y/Al₃Er replaced the relatively low interface energy of Al₃Y/Al₃Zr. Interestingly, the nucleation energy of Al₃Zr/Er(Y) was even lower than that of Al₃Zr(Er) due to the addition of the Y atom, which can be attributed to the negative interface energy of Al₃Er/Al₃Y and low coherent strain energy G_s. Similarly, in the Al-Sc-Zr aluminum alloy, the addition of the Er atom formed a core–double-shelled Al₃Zr/Sc (Er) instead of forming separated Al₃(Sc, Zr) and Al₃(Er, Zr) [20], which was attributed to the decreased nucleation energy of Al₃(Sc, Zr) nanoparticles by its low chemical energy ΔG_chem. Therefore, the design of the core–double-shelled Al₃Zr/Er(Y) nanophase can provide guidance for the development of new Al-Er-Y-Zr alloys.

4. Conclusions

Based on the first-principles thermodynamic calculation, the nucleation energies of the L1₂-Al₃M (M = Sc, Zr, Er, Y) nanophases in aluminum alloys were studied combined with classical nucleation theory. The conclusions were as follows:

(1)

The critical radius and nucleation work of the L1₂-Al₃M precipitate phase were as follows: Al₃Er > Al₃Y > Al₃Sc > Al₃Zr. The Al₃Zr phase was the easiest to nucleate in thermodynamics, while the nucleation of the Al₃Y and Al₃Er phases were relatively difficult in thermodynamics.

(2)

Various structures of Al₃(Y, Zr) phases with the radius r < 1 nm can coexist in Al-Y-Zr alloys. At a precipitate’s radius of 1–10 nanometers, the core–shelled Al₃Zr(Y) phase illustrated the highest nucleation energy, while the separated structure, Al₃Zr/Al₃Y, obtained the lowest one, and had thermodynamic advantages in the nucleation process.

(3)

Core–double-shelled Al₃Zr/Er(Y) obtained a lower nucleation energy than that of Al₃Zr(Y) due to the negative ΔG_chem of Al₃Er and the negative Al₃Er/Al₃Y interface energy, and preferentially precipitated in thermodynamics stability.