First-Principles Investigation of the Diffusion of TM and the Nucleation and Growth of L1₂ Al₃TM Particles in Al Alloys

Abstract

The key parameters of growth and nucleation of Al₃TM particles (TM = Sc-Zn, Y-Cd and Hf-Hg) have been calculated using the combination of the first principles calculations with the quasi-harmonic approximation (QHA). Herein, the diffusion rate D_s of TM elements in Al is calculated using the diffusion activation energy Q, and the results show that the D_s of all impurity atoms increases logarithmically with the increase in temperature. With the increase in atomic number of TM, the D_s of 3–5d TM elements decreases linearly from Sc, Y and Hf to Mn, Ru and Ir, and then increases to Zn, Ag and Au, respectively. The interface energy γ_α/β, strain energy ΔE_cs, chemical formation energy variation ΔG_V and surface energy $E_{sur}^{ave}$ were further computed from the based interface and slab models, respectively. It was found that, with the increase in the atomic number of TM, the interface energies γ_α/β of Al/Al₃TM (TM = (Sc-Zn, Y-Cd)) decreased from Sc and Y to Mn and Tc and then increased to Zn and Cd, respectively (except for the (001) plane of Al/Al₃(Fe-Co), the (111) plane of Al/Al₃Pd and the (110) and (111) planes of Al/Al₃Cd). The strain energies ΔE_cs of Al/Al₃TM (TM = (Sc-Zn)) increased at first, and then decreased for all cycles. The chemical formation energy ΔG_V of all Al₃TM changed slightly in the temperature range of 0~1000 K, except that the ΔG_V of Al₃Sc, Al₃Cu, Al₃(Y-Zr), Al₃Cd, Al₃Hf and Al₃Hg increased nonlinearly. With the increase in atomic number at both 300 and 600 K, the ΔG_V of 3–5d TM elements increased from Sc, Y and Hf to Mn, Tc and Re at first, and then decreased to Co, Rh and Ir, respectively, and slightly changes at the end. With the increase in atomic number of TM, the variation trends of the surface energies of Al₃TM intermetallic compounds present similar changes for all cycles, and the (111) surface always has the lowest values.

1. Introduction

Al-based alloys have been widely applied in the electronics, aerospace and automotive industries due to their low density, high specific strength and welding strength [1]. Adding transition elements (TMs) can significantly improve the mechanical and thermodynamic properties of Al alloys [2,3,4,5,6]. For example, the existence of Sc (0.3%) in the Al matrix increases the ultimate rupture strength of annealed Al sheets from 55 to 240 MPa [7], and L1₂-Al₃Zr in the Al matrix is used as a grain refiner to improve the coarsening resistance and creep properties [8,9]. However, the high cost of Sc and Zr limits their applications in commercial Al alloys. Specifically, intermetallic compounds with TMs are suitable candidates for high temperature applications, as the crystal types in the Al matrix may be L1₂, D0₂₂, D0₂₃ or D0₁₉ structures [8,10,11,12], of which the L1₂ phase is an important intermetallic compound and has been widely studied [13,14,15,16]. Moreover, the TMs can be used to substitute the expensive Sc and Zr elements in L1₂-Al₃Sc and Al₃Zr.

The previous research proved that fixing the dislocations and grain boundaries can effectively refine the deformed and recrystallized grains, depending on the dispersed distribution of L1₂ Al₃TM particles during rising heat [17,18]. The diffusion rate of TM solute atoms in an Al matrix and the interfacial properties of Al₃TM/Al are important parameters for the investigation of nucleation, the growth of L1₂ Al₃TM phases [19,20,21], and the low-index bonds of particles to matrix [22,23]. However, the experimental exploration of appropriative substitution TMs is difficult because of the complex environment and the expensive cost [15,23,24,25,26,27,28]. Fortunately, in recent years, with the development of modern computer technologies, theoretical identification (e.g., first-principles (FP) calculations based on density functional theory (DFT) [15]) in the complicated systems (e.g., metals and ceramics) has become the most powerful method to accomplish this [29,30,31,32].

The stability and nucleation behavior of L1₂-Al₃Sc and Al₃Li binary phases have first been investigated using the framework of density functional theory (DFT) calculation by Mao et al. [15]. Their results showed that the L1₂-Al₃Sc and Al₃Li structures have lower formation energies than those of the corresponding D0₂₃, D0₁₉ and D0₂₂ structures. Furthermore, they found that the interface and strain energies of Al₃Sc are much higher than those of Al₃Li for all (001), (110) and (111) interfaces. Zhang et al. [33] have comprehensively studied the solubility of RE (RE = Y, Dy, Ho, Er, Tm and Lu) in Al based on the free energy difference between L1₂ bulk and Al solid solution matrix in the DFT theoretical framework. Their results indicated that the solubility of all rare earth (RE) (RE = Y, Dy, Ho, Er, Tm and Lu) elements increases with the increase in temperature (~1000 K). They also believed that Dy and Y elements can become better candidates for Sc due to the better stability of Al₃Dy and Al₃Y compounds and their almost identical solubility compared to the higher-cost Sc element. Sun et al. [34] have calculated low-index (001), (110) and (111) surface energies of L1₂-Al₃Sc particles adopting slab model with 15 Å vacuum region. Their results show that when the surface energies of non-stoichiometric (001) and (110) surfaces of Al₃Sc are calculated, their values should be considered as different under different Al chemical potentials, and in a wide range of Al chemical potentials, the surface energies of the (111) surface with AlSc-terminated have lower values, indicating that they are more stable than other surfaces.

However, up to date, the diffusion rates D_s of TMs in Al, the surface properties of L1₂-Al₃TM and the interface of Al₃TM/Al-matrix have not been systematically investigated. Specifically, the nucleation and growth of L1₂-Al₃TM (TM = Sc-Zn, Y-Ag, Hf-Au) particles at finite temperatures have not been obtained, and their relationship to the atomic number of TM hasn’t been described in detail due to the large computational cost required. In the present work, by combining the first-principles calculations with the quasi-harmonic approximation (QHA), the relationship between the particles’ nucleation/growth and atomic number/temperature are discussed. First, the diffusion rates D_s of TMs as a functional of atomic number and temperature have been researched. Then, the relationship between the driving force and the hindrance of particle nucleation and the atomic number of TM is explained based on the interface model. Finally, the effects of the surface stability of different intermetallic compounds with the change in atomic number based on the slab model are obtained.

2. Computed Methods

All calculations in this work were performed in Vienna ab initio Simulation Package (VASP) [34] with the 5.4.4 version, which adopts the framework of density functional theory (DFT) [35] calculations to solve the Kohn–Sham equation and obtain the total energy from different models. In the calculated processing of VASP, to relax all models to their most stable ground state, the electron–core interaction was described by the projector augmented wave (PAW) [36] method. The optimal choice of exchange–correlation functional was considered using the generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerh (PBE) version [31]. A 10 × 10 × 10 k-point sampling grid with the Gamma-centered Monkhorst–Pack method [37] in the first Brillouin zone was selected via strict convergence testing (see Figure 1) for bulk properties calculation. A cut off energy of the plane-wave basis of 500 eV was chosen for the whole calculated process. The energy and force tolerance were set to 10⁻⁷ eV and 0.01 eV/Å, respectively, by using conjugate gradient (CG) minimization and Broyden–Fletcher–Goldfarb–Shannon (BFGS) schemes [38].

Here, based on the slab model, we investigated three low-index surfaces, containing (100), (110) and (111) surfaces of Al and L1₂-Al₃TM, which adopted 14, 14 and 16 layers, respectively [24,39]. All interfaces of (100), (110) and (111) surfaces of Al/L1₂–Al₃TM are calculated by using the 18 layers interface model. A 10 × 10 × 1 k–mesh grid for both cases was tested to be suitable for this work. To simulate diffusion behavior, we constructed a 3 × 3 × 3 supercell with 4 × 4 × 4 k–mesh grids to obtain the diffusion barriers of the solute diffusion of TMs in the Al matrix based on the climbing-image nudged elastic band (CI-NEB) [40] method. Meanwhile, a spring force constant of 5 eV/Å was considered to keep all the images separated, and these CI-NEB iterations were continued until the forces on each atom were less than 0.05 eV/Å.

3. Conclusion Description

3.1. Diffusion

In the process of heating up, some atoms will detach from their original equilibrium positions and then diffuse to a new site while obtaining enough energy. Thus, the diffusion behavior is a common phenomenon in the field of material science and engineering. According to the Lifshitz and Slyozov and Wagner methods [41,42], the growth of particles is affected by the diffusion behavior of solute atoms, and the faster diffusion in the Al matrix is beneficial for the grain growth. In the current work, to investigate diffusion behavior, we first show a vacancy-substitution model, as depicted and visualized in Figure 2a by VESTA codes. The vacancy-substitution model can be divided into two types: the self-diffusion of the violet Al atom and the impurity diffusion of the TM pink ball [24,39]. The black arrow represents the diffusion path for the TM atom. To further investigate the diffusion behavior, the diffusion coefficient as a function of jump frequency I is expressed, which satisfies the Arrhenius equation as follows [40,43,44,45]:

$$D\left(T\right)=\frac{\lambda a^2}{2Z}I$$

(1)

where λ (λ = 2), Z (Z = 1) and a are the number of directions for atomic transitions, the dimension of diffusion and the corresponding atomic distance of diffusion, respectively.

Here, the jump frequency for both diffusions in solid-state was established using the classical transition state theory (TST) [46,47]:

$$I=\nu {exp}^{\left(-\frac{Q}{\kappa T}\right)}$$

(2)

where ν, Q, T and κ are the effective frequencies associated with the vibration of the transition atom, the diffusion activation energy, the special temperature and the Boltzmann constant, respectively.

According to Winter–Zener theory (WZT), the ν can be approximately expressed as [48]:

$$\nu {=\left(\frac{2E_{Diff}}{m{a}^2}\right)}^{\frac{1}{2}}$$

(3)

where m represents the atomic mass of transition atoms. Herein, two types of diffusion activation energies Q corresponding to self D₀ and impurity D_s diffusion coefficients are gained using first principles calculations. The Q for self-diffusion contains two separate energies: vacancy formation energy E_vac and the migration energy of Al atom E_m in Al matrix. For impurity diffusion activation energy, the activation energy Q consists of three parts: the substitutional solution energy E_s of a TM atom replacing a Al atom, vacancy formation energy E_f in the presence of TM in Al₁₀₇TM supercell, and the migration energy of diffusion E_m [49]:

$$E_s=E_{{Al}_{107}TM}-107E_{Al}-E_{TM}$$

(4)

$$E_f=E_{\left({Al}_{106}TM:Vac\right)}-E_{{Al}_{107}TM}+E_{Al}$$

(5)

$$E_b=E_s+E_f$$

(6)

where E_TM and E_Al are the energies of single TM and Al atoms in the stable bulk, respectively, and E_b is the binding energy of a TM atom substituting a vacancy in Al matrix.

To further investigate the physical mechanism of behaviors, the electron localized function (ELF) has been drawn using the VESTA code [50]. The ELF is defined as:

$$\mathrm{E}\mathrm{L}\mathrm{F}=\frac{1}{1+{\left(\frac{D_r}{{Dh}_r}\right)}^2}$$

(7)

where $D_r$ and ${Dh}_r$ are the true electron gas density and the pre-assumed uniform electron gas density, respectively.

The E_m, E_b (E_vac), Q and D_s (D₀) for TM and Al at 300 K with available experimental and theoretical values are summarized in Table 1 [51,52,53,54]. It can be seen that errors between the present and previous values in literatures for E_m, E_b (E_vac) and E_Diff are within 20%, and the current value of E_Diff of Sc element is only ~2% larger than that of the experiment value. To visually illustrate the regularity of the variations of activation energy Q as a function of the atomic number of TM, it is further plotted in Figure 2b. The result shows that the Q increases at first and then decreases as the atomic number increases (Sc-Zn, Y-Ag, Hf-Au) in the Al matrix (except for Cr of 2.23 eV), indicating that there is a correlation between the valence electron configuration of impurity elements and the activation energy Q. Additionally, the TM elements in the fourth cycle generally have lower diffusion activation energies Q_s, ranging from 0.35 to 2.60 eV. For Mn-Co, Tc-Rh and Re-Ir, they have larger Q_s in the Al matrix, which are 2.45~2.60, 3.82~3.94 and 3.95~4.26 eV, respectively, indicating that their diffusion abilities are relatively weak in the Al matrix. Meanwhile, for Cu-Zn, Ag and Au, the activation energy Q is very low, or even negative for a Cd of −0.12 eV and an Hg of −0.30 eV, as shown in Table 1, which shows they are easier to move in the Al matrix. In the undoped-Al system, self-diffusion activation energies Q₀ is lower compared to all Q_s in the doped system, except for the Q_s of Cu, Zn, Y and Ag, indicating that the diffusion of most TM atoms is more difficult than self-diffusion.

The variation in activation energy Q with the temperature increasing can be calculated from the above results by combining them with the quasi-harmonic vibration (QHA) [55]; by doing this, the change in diffusion rate D with the temperature can be obtained via Equations (1)–(3), and the results are summarized in Table 1 and Figure 2c,d. It should be noted that only the self-diffusion rate D₀ as a function of Q is presented in the inlet of Figure 2c, owing to the fact that that all activation energies Q of TM elements are nearly the same. The self-diffusion rate D₀ of 3.55 × 10⁻²⁸ m²·s⁻¹ for Al in this work is in general agreement with the experimental extension values from 1.76 × 10⁻²⁷~4.42 × 10⁻¹² m²·s⁻¹ in the range of 300~1000 K and 1.47 × 10⁻¹⁴~1.36 × 10⁻¹² in the range of 739~917 K in literature [56,57], seen from Table 1 and Figure 2c. Meanwhile, the theoretical predicted D₀ of 3.55 × 10⁻²⁸ m²·s⁻¹ of Al is lower than that of the experiment at 300 K. The reason for this may be that it is difficult to accurately determine the D₀ due to the influence of crystal structure defects, dislocations and grain boundaries in experiments. The D_s of all impurity atoms except for Cd and Hg increases logarithmically with the increase in temperature. A negative Q for Cd and Hg cases makes it impossible to theoretically calculate values according to Equations (1) and (2). Reasonably, the D indicate the inverse pattern to Q; higher barriers mean slower passage. Additionally, the larger the value at 300 K, the lower the increasing rate. This trend result is consistent with the variation trend of D_s for Mg, Si and Cu with temperature calculated by Mantina et al. [44]. Figure 2d further shows the diffusion rate D_s at 300 K as a function of the atomic number of TM, and it can be seen that the diffusion rate D_s first decreases linearly from 2.05 × 10⁻³⁷, 6.47 × 10⁻²⁴ and 2.79 × 10⁻⁴⁴ m²·s⁻¹ for Sc, Y and Hf to 2.43 × 10⁻⁵⁰, 6.77 × 10⁻⁷³ and 1.60 × 10⁻⁷⁸ m²·s⁻¹ for Mn, Ru and Ir and then increases with the increase in atomic number to 3.09 × 10⁻¹³, 9.17 × 10⁻¹⁷ and 2.93 × 10⁻²⁹ m²·s⁻¹ for Zn, Ag and Au, respectively (except for Cr of 3.36 × 10⁻⁴⁴ m²·s⁻¹).

From the above results, it can be seen that higher peaks occur for half- or near half-full d shells for all cycles considered. The reason for this may be that half- or near half-full d shells of the TM element in the Al matrix are more stable and more energy is required to force them to move from the stable site to the vacancy. Although the atomic diffusion barrier changes similarly with the increase in atomic number in the same period, TM with 3d shells present a faster diffusion behavior. To explore the underlying potential, the ELF of Sc and Ru doping systems on the (010) plane are presented in Figure 2e,f. The value of ELF, which is selected as 0 to 1, demonstrates the probability of finding an electron in the neighborhood space. To be specific, when it equals 0, it reflects a strongly delocalized electron area; when it equals 1, it corresponds to a strongly localized electron area. It can be seen that, when Sc and Ru are the first nearest neighbors of the vacancy, different values of ELF are exhibited. The Ru would make the surrounding electrons appear more likely than Sc, resulting in Ru being difficult to diffuse to the vacancy.

Figure 2. (a) The diffusion model. (b) The calculated diffusion barrier of a vacancy E_m, vacancy solute binding energy E_b (vacancy formation energy E_vac for self-diffusion in Al matrix) and diffusion activation energy E_Diff with the change in atomic number. (c) The diffusion rate D and E_Diff as a function of temperature. (d) The impurity diffusion rate D_s as a function of the atomic number of TM. (a,b) represent the experimental values from Murphy et al. [57] and Volin et al. [56], respectively. (e,f) The ELFs on the (010) planes of Sc and Ru doping systems, respectively.

Figure 2. (a) The diffusion model. (b) The calculated diffusion barrier of a vacancy E_m, vacancy solute binding energy E_b (vacancy formation energy E_vac for self-diffusion in Al matrix) and diffusion activation energy E_Diff with the change in atomic number. (c) The diffusion rate D and E_Diff as a function of temperature. (d) The impurity diffusion rate D_s as a function of the atomic number of TM. (a,b) represent the experimental values from Murphy et al. [57] and Volin et al. [56], respectively. (e,f) The ELFs on the (010) planes of Sc and Ru doping systems, respectively.

Table 1. The calculated diffusion barrier E_m (eV), vacancy–solute binding energy E_b (eV), diffusion activation energy Q (eV) and diffusion rate D_s (m²·s⁻¹) for TM atoms in Al matrix at 300 K. It should be noted that for pure Al, E_b and D_s are in fact E_vac and D₀, respectively. Note: A negative activation energy Q can’t meet calculating D_s according to Equations (1)–(3).

Element	Em	Eb	Q	Ds
Al	0.68 0.55–0.70 [54] 0.57 [58]	0.63 0.60–0.80 [54] 0.63 [58]	1.31 1.15–1.50 [54] 1.20 [58] 1.31 [56]	3.55 × 10−28 1.76 × 10−27 [56]
Sc	0.85	0.97	1.82 1.79 [55]	2.05 × 10−37
Ti	1.43	0.84	2.27	6.04 × 10−45
V	1.90	0.42	2.32	1.09 × 10−45
Cr	2.14	0.09	2.23	3.36 × 10−44
Mn	2.11	0.49	2.60	2.43 × 10−50
Fe	1.90	0.63	2.53	3.10 × 10−49
Co	1.55	0.90	2.45	7.79 × 10−48
Ni	1.06	0.97	2.03	6.44 × 10−41
Cu	0.57	0.22	0.79	2.94 × 10−20
Zn	0.40	−0.04	0.35	3.09 × 10−13
Y	0.36	0.63	0.99	6.47 × 10−24
Zr	1.19	0.98	2.17	2.16 × 10−43
Nb	1.88	0.75	2.63	4.94 × 10−51
Mo	2.46	0.75	3.22	7.40 × 10−61
Tc	2.54	1.27	3.82	7.25 × 10−71
Ru	2.25	1.69	3.94	6.77 × 10−73
Rh	1.68	2.16	3.84	2.81 × 10−71
Pd	0.98	1.71	2.68	5.51 × 10−52
Ag	0.51	0.06	0.56	9.17 × 10−17
Cd	0.35	−0.46	−0.12	-
Hf	1.41	0.80	2.21	2.79 × 10−44
Ta	2.10	0.41	2.51	2.90 × 10−49
W	2.85	0.17	3.02	9.43 × 10−58
Re	3.09	0.86	3.95	2.93 × 10−73
Os	2.77	1.36	4.14	1.96 × 10−76
Ir	2.15	2.11	4.26	1.60 × 10−78
Pt	1.27	2.15	3.42	1.77 × 10−64
Au	0.53	0.78	1.31	2.93 × 10−29
Hg	0.21	−0.50	−0.30	-

Table 1. The calculated diffusion barrier E_m (eV), vacancy–solute binding energy E_b (eV), diffusion activation energy Q (eV) and diffusion rate D_s (m²·s⁻¹) for TM atoms in Al matrix at 300 K. It should be noted that for pure Al, E_b and D_s are in fact E_vac and D₀, respectively. Note: A negative activation energy Q can’t meet calculating D_s according to Equations (1)–(3).

Element	Em	Eb	Q	Ds
Al	0.68 0.55–0.70 [54] 0.57 [58]	0.63 0.60–0.80 [54] 0.63 [58]	1.31 1.15–1.50 [54] 1.20 [58] 1.31 [56]	3.55 × 10−28 1.76 × 10−27 [56]
Sc	0.85	0.97	1.82 1.79 [55]	2.05 × 10−37
Ti	1.43	0.84	2.27	6.04 × 10−45
V	1.90	0.42	2.32	1.09 × 10−45
Cr	2.14	0.09	2.23	3.36 × 10−44
Mn	2.11	0.49	2.60	2.43 × 10−50
Fe	1.90	0.63	2.53	3.10 × 10−49
Co	1.55	0.90	2.45	7.79 × 10−48
Ni	1.06	0.97	2.03	6.44 × 10−41
Cu	0.57	0.22	0.79	2.94 × 10−20
Zn	0.40	−0.04	0.35	3.09 × 10−13
Y	0.36	0.63	0.99	6.47 × 10−24
Zr	1.19	0.98	2.17	2.16 × 10−43
Nb	1.88	0.75	2.63	4.94 × 10−51
Mo	2.46	0.75	3.22	7.40 × 10−61
Tc	2.54	1.27	3.82	7.25 × 10−71
Ru	2.25	1.69	3.94	6.77 × 10−73
Rh	1.68	2.16	3.84	2.81 × 10−71
Pd	0.98	1.71	2.68	5.51 × 10−52
Ag	0.51	0.06	0.56	9.17 × 10−17
Cd	0.35	−0.46	−0.12	-
Hf	1.41	0.80	2.21	2.79 × 10−44
Ta	2.10	0.41	2.51	2.90 × 10−49
W	2.85	0.17	3.02	9.43 × 10−58
Re	3.09	0.86	3.95	2.93 × 10−73
Os	2.77	1.36	4.14	1.96 × 10−76
Ir	2.15	2.11	4.26	1.60 × 10−78
Pt	1.27	2.15	3.42	1.77 × 10−64
Au	0.53	0.78	1.31	2.93 × 10−29
Hg	0.21	−0.50	−0.30	-

3.2. Nucleation

According to the classical nucleation method (CNT) [44,56,57], the total energy of the nucleation process of second phases can be expressed as follows: $∆G_{tot}=\frac{4}{3}\pi R^3\left(∆G_V+{∆E}_{CS}\right)+4\pi R^2{\gamma }_{\alpha /\beta }$. Here, a positive strain energy contribution would be a hindrance when Al₃TM grains gradually form, while the difference in free energy in bulk between the matrix and particles and the interfacial free energy would promote particle nucleation.

Here, to calculate interface energy γ_α/β, we adopt a total energy of interface model that subtracts the total energy of the phases on either side of the interface in a two-phase system [23]:

$${\gamma }_{\alpha /\beta }=\frac{E_{\alpha /\beta }-\left(E_{\alpha }+E_{\beta }\right)}{2A}$$

(8)

where A is the area of the interface, E_α/β is the total internal energy of the relaxed α/β system containing an interface and E_α and E_β are the total internal energies of phases α and β from the strains of all directions, respectively.

The chemical formation energy difference ΔG_V of L1₂-Al₃TM precipitates can be expressed in dilute solid solution based thermodynamics, Al_nTM → Al₃TM + Al_n−3. It can be shown as [15]

$$∆G_V={∆G}_{{Al}_{3}TM}+\left(n-3\right){∆G}_{Al}-{∆G}_{{Al}_{n}TM}$$

(9)

where n (n = 31) and ΔG are the number of atoms and Gibbs free energy, respectively. To investigate the dependence of ΔG_V on temperature, the non-equilibrium free energy ΔG_V is derived as the following equation [15,59]:

$$G\left(V,P,T\right)=\mathrm{m}\mathrm{i}\mathrm{n}\left[F\left(V,T\right)\right]+PV$$

(10)

where $F\left(V;T\right)$ is the free energy computed by the sum of electronic internal energy and phonon Helmholtz free energy $F\left(V,T\right)=U_{el}+F_{vib}$. P is the circumstance pressure.

Due to lattice mismatch, both the harmonic and non-harmonic contributions were observed to calculate the strain energy ΔE_CS of the L1₂ precipitation phases [15]:

$${∆E}_{CS}\left(x,\mathrm{Ĝ}\right)=\underset{a_s}{\min }\left(x∆E_{\alpha }^{eqi}\left(a_s,\mathrm{Ĝ}\right)+\left(1-x\right)∆E_{\beta }^{eqi}\left(a_s,\mathrm{Ĝ}\right)\right)$$

(11)

where $a_s$ is the constrained superlattice parameter, $\mathrm{Ĝ}$ is the direction and x is the mole fraction of phase α. $∆E_{\alpha }^{eqi}$ and $∆E_{\beta }^{eqi}$ are the epitaxial deformation energies of phases α and β, respectively.

Figure 3a shows the interface model for calculating the interface properties in this work. The Al matrixes are highlighted in dashed rectangles, and different layer numbers are used for the calculation convenience. Comparing the present results with references [15,23] listed in

Table 2. The calculated interface energy γ_α/β (mJ·m⁻²) and strain energy ΔE_cs (meV·atom⁻¹) in Al/Al₃TM interface systems. (Note: A * symbol represents the calculated result from the vacuum slab model).

Dir.	(001)		(110)		(111)
Systems	γα/β	ΔECS	γα/β	ΔECS	γα/β	ΔECS
Al/Al3Sc	108.65 108.00 [15] 165.00 [23] 176.00 [23]	0.32 0.60 [23] - -	194.41 159.00 [15] 178.00 [15] 193.00 [23]	1.50	204.81 191.00 [23] 189.00 [15] 203.00 [15]	1.52
Al/Al3Ti	−38.48 61.85 * 52.00 [23]	0.20 0.30 [15]	−38.90 61.00	1.12	66.67 79.00 [15]	1.36
Al/Al3V	−147.83	1.77	−203.77	5.10	−75.76	7.56
Al/Al3Cr	−270.04	1.52	−379.95	9.92	−167.43	23.16
Al/Al3Mn	−468.86	−0.14	−429.65	13.80	−225.66	28.62
Al/Al3Fe	−291.22	10.64	−283.37	16.58	−113.54	23.69
Al/Al3Co	−200.04	12.06	−205.31	14.01	−49.49	22.43
Al/Al3Ni	−195.59	5.83	−176.89	6.20	−109.50	15.46
Al/Al3Cu	−143.59	0.59	−108.27	2.40	−48.53	8.43
Al/Al3Zn	−53.81	0.64	−88.33	0.67	−33.48	0.24
Al/Al3Y	93.37	5.13	159.60	9.72	181.29	14.30
Al/Al3Zr	20.15	0.65	1.39	2.43	86.32	2.48
Al/Al3Nb	−143.96	1.48	−160.59	2.07	−109.57	0.56
Al/Al3Mo	−309.65	1.87	−319.59	1.20	−201.04	18.33
Al/Al3Tc	−699.48	−14.84	−516.46	7.24	−201.10	20.79
Al/Al3Ru	−173.70	3.92	−228.40	8.52	−82.71	12.26
Al/Al3Rh	−138.07	2.92	−197.66	6.71	−42.72	9.98
Al/Al3Pd	−132.75	1.53	−153.88	1.94	−60.52	1.79
Al/Al3Ag	−142.47	0.86	−46.85	1.31	−5.67	0.40
Al/Al3Cd	−75.68	0.51	−261.59	4.33	−55.69	9.33
Al/Al3Hf	−37.53	1.14	−25.31	1.79	69.41	1.59
Al/Al3Ta	−169.68	0.50	−198.59	1.36	−124.38	1.26
Al/Al3W	−232.18	0.29	−467.97	4.35	−276.16	4.24
Al/Al3Re	−146.35	4.59	−1242.00	26.49	−396.57	33.54
Al/Al3Os	−243.80	4.34	−328.80	8.72	−174.06	13.36
Al/Al3Ir	−87.86	3.71	−173.40	6.26	−3.44	10.60
Al/Al3Pt	−190.37	0.38	−734.31	2.90	−246.39	17.81
Al/Al3Au	−118.52	0.58	−80.77	1.02	−33.30	0.67
Al/Al3Hg	−93.95	0.94	−341.48	3.64	−251.40	24.04

, there are larger errors compared by Mao and Li et al. [15,23], and these errors are further discussed. The main reasons are as follows:

• Li et al. [23] adopted the vacuum slab model for the calculation, resulting in the values of interface energies being affected by different terminal surfaces, and the interface energy of Al/Al₃Ti of 61.85 mJ·m⁻² calculated by the vacuum model is in a good agreement with that of Li et al. according to ${\gamma }_{\alpha /\beta }=\frac{E_{\alpha /\beta }^{*}-\left(E_{slab,\alpha }+E_{slab,\beta }\right)}{S}+E_{sur}^{\alpha }+E_{sur}^{\beta }$, where $E_{\alpha /\beta }^{*}$ is the total energy of the vacuum slab model system, $E_{slab}$ denotes the total energy of the fully relaxed surface slabs and $E_{sur}^{\alpha }$ and $E_{sur}^{\beta }$ represent the surface energies of the α and β surface slabs, respectively. Meanwhile, the strain energy caused by lattice mismatch in the vacuum slab model was not taken into account in the above equation.
• Mao et al. [15] had investigated interface properties in a periodic supercell and, considering the strain energy of interface model, they calculated interface properties with less accuracy, performed on a 0.13 (1/Å) spacing Monkhorst–Pack k-point mesh and an energy cutoff of 300 eV.

The calculated γ_α/β with the increase in atomic number is further depicted in Figure 3b–d. According to the CNT, the theoretical nucleation radius R^* cannot be calculated by a negative γ_α/β, and the γ_α/β of all Al/Al₃TM are less than 0 mJ·m⁻², except for the (111) of Al/Al₃Sc, Al/Al₃Ti, Al/Al₃(Y-Zr) and Al₃Hf systems.

It can be seen from Figure 3b,c that the γ_α/β of Al/Al₃TM (TM = (Sc-Zn, Y-Cd)) decreases from Sc and Y to Mn and Tc, and then increases to Zn and Cd, respectively, except for the (001) of Al/Al₃(Fe-Co), the (111) of Al/Al₃Pd and the (110) and (111) of Al/Al₃Cd. These trends of γ_α/β for Al/Al₃TM (TM = (H_f-Hg)) in the (110) and (111) systems present two Al/Al₃Re and Al/Al₃Pt compound troughs in Figure 3d, and they show the same change with the increase in atomic number. For the (001) system, the γ_α/β of Al/Al₃TM (TM = (H_f-Hg) is larger than −250 mJ·m⁻², and the Al/Al₃TM with 3d⁶4s² has the lowest γ_α/β. Figure 3e shows the variation of strain energy ΔE_cs of Al/Al₃TM (TM = (Sc-Zn) with the increase in atomic number. It can be seen that the ΔE_cs increases from 0.32~1.52 meV·atom⁻¹ for Sc to 12.06 meV·atom⁻¹ for Co on the (001) system, to 16.58 meV·atom⁻¹ for Fe on the (110) system, and to 28.62 meV·atom⁻¹ for Mn on the (111) system, respectively, and then they all decrease to 0.24 ~ 0.67 meV·atom⁻¹ for Zn (except for Al/Al₃Mn, of the order of −0.14 meV·atom⁻¹). For the (110) and (111) systems of Al/Al₃TM (TM = (Y-Cd, Hf-Hg)), as seen in Figure 3f,g, respectively, the largest values of ΔE_cs for the (110) and (111) interface systems are all located at Al/Al₃Re, being 26.49 and 33.54 meV·atom⁻¹, respectively, while the (001) interface system of Al/Al₃Tc has the lowest value of ΔE_cs, being −14.84 meV·atom⁻¹.

The trends of ΔG_V as a function of temperature for all Al₃TM compounds have been calculated according to Equations (10) and (11), and results are shown as Figure 3h. The results show that the ΔG_V of all Al₃TM change slightly in the temperature range of 0~1000 K, except that the ΔG_V of Al₃Sc, Al₃Cu, Al₃(Y-Zr), Al₃Cd, Al₃Hf and Al₃Hg increase nonlinearly from −89.69, −1.44, −130.51, −93.86, −1.65, −72.35, and 0.65 meV·atom⁻¹ to −24.38, 66.09, −88.46, −44.47, 71.60, −2.05 and ~88.51 meV·atom⁻¹, respectively. Furthermore, the obtained ΔG_V as a function of the atomic number of TM is shown in Figure 3i,j, and the calculated values of −66.46 and −61.54 meV·atom⁻¹ for Al₃Sc and Al₃Ti, respectively, at 600 K agree well with the value of −61.14 meV·atom⁻¹ at 350 °C (623 K) for Al₃Sc and −66.15 meV·atom⁻¹ at 300 (573 K) for Al₃Ti calculated by Li et al. [15]. From Figure 3i, one can see that the ΔG_V at 300 K increases from −80.96 meV·atom⁻¹ for Sc, −120.46 meV·atom⁻¹ for Y and −66.82 meV·atom⁻¹ for Hf to 20.37 meV·atom⁻¹ for Mn, 53.89 meV·atom⁻¹ for Tc and 74.50 meV·atom⁻¹ for Re, and then decreases slightly to −11.72 meV·atom⁻¹ for Co, 9.62 meV·atom⁻¹ for Rh and 4.89 meV·atom⁻¹ for Ir, respectively. As a final step, they change slightly. At 600 K, the variation trends of ΔG_V for 3–5d TMs are the same as those at 300 K.

3.3. Surface Energy

In the framework of Peierls theory, a lower surface energy of bulk materials in comparison to an unstable stacking fault will cause metals to crack from material failure [42,60]. Thus, it is necessary to analyze surface energy for all Al₃TM particles and the Al matrix. The surface energy of Al is given by the following formula [61,62]:

$$E_{sur}=\frac{E_{Al}^{slab}-N{\mu }_{Al}^{bulk}}{2A}$$

(12)

where $E_{Al}^{slab}$ and N are the total energy and the number of Al atoms in the slab model, respectively. ${\mu }_{Al}^{bulk}$ represents the chemical potential of a single atom in bulk Al.

For stoichiometric surfaces (111) of the Al₃TM slab, the calculated formula is given as follows:

$$3{\mu }_{Al}^{slab}+{\mu }_{TM}^{slab}={\mu }_{{Al}_{3}TM}^{bulk}$$

(13)

$$E_{sur}=\frac{E_{{Al}_{3}TM}^{slab}-N{\mu }_{{Al}_{3}TM}^{bulk}}{2A}$$

(14)

where ${\mu }_{Al}^{slab}$, ${\mu }_{TM}^{slab}$ and ${\mu }_{{Al}_{3}TM}^{bulk}$ are the chemical potential of Al, AlTM-terminated and Al₃TM bulk, respectively. N and A are the number of Al₃TM cells and the surface area, respectively.

To further discuss the non-stoichiometric (001) and (110) surfaces of the Al3TM (3NTM ≠ NAl), we used the following the equation [24,63]:

$$E_{sur}=\frac{E_{{Al}_{3}TM}^{slab}-N{\mu }_{{Al}_{3}TM}^{bulk}+n{\mu }_{Al}^{slab}}{2A}$$

(15)

where n is the number of the rest (n < 0) and missing (n > 0) Al atoms.

To obtain ${\mu }_{Al}^{slab}$ in the systems, we first need to avoid Al and Sc bulk phases. Therefore, ${\mu }_{Al}^{slab}$ and ${\mu }_{Sc}^{slab}$ are limited, as follows:

$${\mu }_{Al}^{slab}-{\mu }_{Al}^{bulk}<0$$

(16)

$${\mu }_{TM}^{slab}-{\mu }_{TM}^{bulk}<0$$

(17)

Further, the thermodynamic stability of AlTM compounds should meet the equation given by:

$${3\mu }_{Al}^{bulk}+{\mu }_{TM}^{bulk}+∆H_f={\mu }_{{Al}_{3}TM}^{bulk}$$

(18)

Combined with Equations (12) and (15)–(17), two limit values of ${\mu }_{Al}^{slab}$ are respectively given by:

$${{\mu }_{Al}^a=\mu }_{Al}^{slab}={\mu }_{Al}^{bulk}$$

(19)

$${{\mu }_{Al}^b=\mu }_{Al}^{slab}={\mu }_{Al}^{bulk}+\frac{1}{3}∆H_f$$

(20)

The calculated surface energies of AlTM from different Al chemical potentials, ${\mu }_{Al}^a$ and ${\mu }_{Al}^b$, are summarized in

Table 3. The calculated surface energy E_sur (J·m⁻²) of the (001) and (110) surfaces from different Al chemical potential ${\mu }_{Al}^a$ and ${\mu }_{Al}^b$ in Al₃TM.

Systems	(001)				(110)
	Al-Ter.		AlTM-Ter.		Al-Ter.		AlTM-Ter.
	${\mathit{\mu }}_{\mathit{A}\mathit{l}}^{\mathit{a}}$	${\mathit{\mu }}_{\mathit{A}\mathit{l}}^{\mathit{b}}$	${\mathit{\mu }}_{\mathit{A}\mathit{l}}^{\mathit{a}}$	${\mathit{\mu }}_{\mathit{A}\mathit{l}}^{\mathit{b}}$	${\mathit{\mu }}_{\mathit{A}\mathit{l}}^{\mathit{a}}$	${\mathit{\mu }}_{\mathit{A}\mathit{l}}^{\mathit{b}}$	${\mathit{\mu }}_{\mathit{A}\mathit{l}}^{\mathit{a}}$	${\mathit{\mu }}_{\mathit{A}\mathit{l}}^{\mathit{b}}$
Al3Sc	1.10	1.69	1.42	0.84	1.19	1.61	1.63	1.22
Al3Ti	1.04	1.53	1.70	1.21	0.99	1.34	1.84	1.49
Al3V	1.03	1.24	1.61	1.40	0.86	1.01	1.77	1.63
Al3Cr	0.97	0.92	1.54	1.59	0.63	0.60	1.50	1.53
Al3Mn	0.93	0.99	1.55	1.49	0.42	0.47	1.19	1.14
Al3Fe	1.06	1.23	1.57	1.39	0.78	0.90	1.51	1.39
Al3Co	0.97	1.29	1.32	0.99	0.77	0.99	1.32	1.09
Al3Ni	0.78	1.10	1.03	0.71	0.73	0.96	0.90	0.68
Al3Cu	0.83	0.89	0.99	0.92	0.87	0.92	0.85	0.80
Al3Zn	0.84	0.82	0.78	0.80	0.93	0.91	0.75	0.77
Al3Y	1.13	1.63	0.96	0.46	1.12	1.47	1.34	0.98
Al3Zr	0.88	1.46	1.36	0.77	0.88	1.29	1.59	1.18
Al3Nb	0.81	1.16	1.32	0.97	0.70	0.95	1.61	1.37
Al3Mo	0.36	0.51	0.88	0.73	0.26	0.36	1.17	1.08
Al3Tc	1.02	1.29	1.62	1.35	0.00	0.18	0.77	0.59
Al3Ru	1.17	1.67	1.62	1.11	0.80	1.15	1.53	1.18
Al3Rh	0.85	1.54	1.01	0.31	0.66	1.15	1.04	0.55
Al3Pd	0.58	1.05	0.67	0.20	0.44	0.80	0.47	0.12
Al3Ag	0.68	0.63	0.53	0.58	0.72	0.68	0.50	0.53
Al3Cd	0.81	0.63	0.23	0.41	0.72	0.60	0.40	0.53
Al3Hf	0.96	1.45	1.63	1.14	0.94	1.29	1.78	1.43
Al3Ta	0.91	1.10	1.66	1.46	0.75	0.89	1.82	1.68
Al3W	0.64	0.54	1.45	1.54	0.57	0.50	1.50	1.56
Al3Re	0.90	0.95	1.76	1.70	0.00	0.04	0.93	0.89
Al3Os	1.08	1.40	1.81	1.49	0.59	0.82	1.52	1.29
Al3Ir	1.07	1.75	1.38	0.69	0.77	1.26	1.39	0.91
Al3Pt	0.66	1.30	0.71	0.07	0.49	0.97	0.55	0.08
Al3Au	0.55	0.70	0.35	0.20	0.46	0.57	0.31	0.20
Al3Hg	0.67	0.45	−0.15	0.07	0.35	0.20	0.19	0.34

as references. To solve the dependence of surface energy on Al chemical potential, the average surface energy of non-stoichiometric surfaces is obtained by two identical index surfaces of different termination [24,63]:

$$E_{sur}^{ave}=\frac{1}{4A}\left[E_{slab}^{Al}+E_{slab}^{AlTM}-\left(N_{slab}^{Al}+N_{slab}^{AlTM}\right){\times \mu }_{{Al}_{3}TM}^{bulk}\right]$$

(21)

where $E_{slab}^{Al}$, $N_{slab}^{Al}$ and $E_{slab}^{AlTM}$, $N_{slab}^{AlTM}$ are the relaxed energy and total number of TM atoms in Al and AlTM-terminated surfaces, respectively.

Figure 4a shows the slab model for calculating E_sur, and the detail calculated results of $E_{sur}^{ave}$ of Al₃TM (RE = Sc-Zn, Y-Cd and Hf-Hg) are depicted in

Table 4. The calculated surface energy E_sur (J·m⁻²) in Al or Al₃TM.

Systems	(001)	(110)	(111)
Al	0.79; 0.93 [23]	0.87; 0.98 [23]	0.68; 0.73 [39] 0.81 [23]
Al3Sc	1.26; 1.32 [24]	1.41; 1.45 [24]	1.18; 1.22 [24]; 1.17 [39]
Al3Ti	1.37	1.42	0.92; 0.93 [39]
Al3V	1.32	1.32	0.72; 0.65 [39]
Al3Cr	1.25	1.07	0.33
Al3Mn	1.24	0.81	0.33
Al3Fe	1.31	1.15	0.56
Al3Co	1.14	1.04	0.67
Al3Ni	0.91	0.82	0.57
Al3Cu	0.91	0.86	0.69
Al3Zn	0.81	0.84	0.73
Al3Y	1.05	1.23	1.06; 1.11 [39]
Al3Zr	1.12	1.23	0.80; 0.94 [39]
Al3Nb	1.06	1.16	0.54; 0.59 [39]
Al3Mo	0.62	0.73	−0.32
Al3Tc	1.32	0.39	−0.22
Al3Ru	1.39	1.17	0.44
Al3Rh	0.93	0.85	0.46
Al3Pd	0.62	0.46	0.21
Al3Ag	0.61	0.61	0.46
Al3Cd	0.52	0.56	0.47
Al3Hf	1.30	1.36	0.87
Al3Ta	1.28	1.29	0.66
Al3W	1.04	1.05	0.13
Al3Re	1.32	0.46	−0.32
Al3Os	1.45	1.06	0.41
Al3Ir	1.22	1.08	0.73
Al3Pt	0.69	0.52	0.30
Al3Au	0.45	0.38	0.35
Al3Hg	0.26	0.27	0.27

. It can be seen that for pure Al, Al₃(Sc-V) and Al₃(Y-Nb), the calculated values of this work are in good agreement with references [24,64,65]. The $E_{sur}^{ave}$ of low index surfaces of cubic Al follows in the sequence of (110) > (100) > (111), which follows the general law of surface energies for face-centered cubic metals [66]. Figure 4c–e illustrates the change in $E_{sur}^{ave}$ with the atomic number of TM elements, and it can be found that the variation tendency of $E_{sur}^{ave}$ of Al₃TM intermetallic compounds presents similar characteristics for different cycles. For example, the $E_{sur}^{ave}$ of Al₃TM for 3d elements firstly decreases from Sc to Mn, and then increases to Fe, and then decreases to Ni, and finally changes slightly in the (001) surface. The variation ranges of $E_{sur}^{ave}$ for the (001), (110) and (111) surfaces of Al₃TM are 0.25~1.44, 0.26~1.45 and −0.32~1.18 J·m⁻², respectively, and the (111) surface has the lowest surface energy for all elements. As can be seen from Figure 4e,f, we have calculated the values of ELF by using the Equation (7) on the (111) plane of Al₃Sc and Al₃Mo. It can be seen that the (111) plane of Al₃Sc has more strongly localized electron areas than the (111) plane of Al₃Mo, indicating that strongly localized electrons make the surface energy lower. Clearly, if the $E_{sur}^{ave}$ of Al₃TM is larger than that of Al, they would increase the toughness of Al alloys such as Al₃Sc. However, the complete toughness is not only determined by the above results but also by assessing the information of generalized stacking fault energy (GSFE) for particles in the Al matrix. The work needed for this is underway and will be published elsewhere.

4. Conclusions

In this work, we have calculated the diffusion rates of TM elements in Al and the key parameters of nucleation and growth of second phase particles Al₃TM (TM = Sc-Zn, Y-Cd and Hf-Hg) using the first principles combing with quasi-harmonic approximation in the theoretical framework of density functional theory. Firstly, as can be seen from the discussed results, the trends of the Q, chemical formation and surface energies with the change in atomic number in the same cycle are similar. The reason may be related to the valence electron configures (VECs) of TM elements because TM in the same cycle has the same VECs. Meanwhile, the calculation interface and strain energy of Al₃TM/Al composed of the same cycle of TM elements show a lack of similarity. The reason for this may be that the interface and strain energy are co-determined by the matrix and second phases. Here, the main conclusions are as follows:

• In the vacancy–substitution model, the diffusion activation energy Q first increases, and then decreases with the increase in atomic number (Sc-Zn, Y-Ag and Hf-Au) in the Al matrix, except for Cr; the TM elements in the fourth cycle generally have lower Q_s.
• Mn-Co, Tc-Rh and Re-Ir elements have larger activation energies Q_s in the Al matrix, while Cu-Zn, Ag and Au have lower activation energies Q_s; even Cd and Hg elements have negative activation energies. In the undoped-Al system, the self-diffusion activation energy Q₀ is lower compared to all Q_s in the doped system, except for the Q_s of Cu, Zn, Y and Ag.
• The diffusion rate D_s of all impurity atoms increases logarithmically with the increase in temperature. With the increase in atomic number, the diffusion rate D_s first decreases linearly from Sc, Y and Hf to Mn, Ru and Ir, and then increases to Zn, Ag and Au for 3–5d TM elements, respectively.
• With the increase in atomic number, the interface energy γ_α/β of Al/Al₃TM (TM = (Sc-Zn, Y-Cd)) decreases from Sc and Y to Mn and Tc, and then increases to Zn and Cd, respectively, except for (001) in Al/Al₃(Fe-Co), (111) in Al/Al₃Pd and (110) and (111) in Al/Al₃Cd. Meanwhile, the strain energy ΔE_cs increases from Sc to Co in the (001) system, to Fe in the (110) system, and to Mn in the (111) system, respectively, and then they all decreases to Zn, except for Al/Al₃Mn. The largest values of ΔE_cs for (110) and (111) interface systems are all located at Al/Al₃Re, while the (001) interface system of Al/Al₃Tc has the lowest value.
• The variation in chemical formation energy ΔG_V of all Al₃TM changes slightly in the temperature range of 0~1000 K, except that the ΔG_V of Al₃Sc, Al₃Cu, Al₃(Y-Zr), Al₃Cd, Al₃Hf and Al₃Hg increase nonlinearly. With the increase in atomic number at 300 K, the ΔG_V increases from Sc, Y and Hf to Mn, Tc and Re at first, and then decreases to Co, Rh and Ir, respectively, and finally, it slightly changes. The variation trends of the ΔG_V for 3–5d TMs are the same as those at 300 K.
• With the increase in atomic number, the trend of $E_{sur}^{ave}$ of Al₃TM intermetallic compounds presents a similar change in different cycles and the (111) surface always has the lowest surface energy in all surfaces of Al₃TM particles.