*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 [40,43–45]:

$$D(T) = \frac{\lambda a^2}{2Z}I \tag{1}$$

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]:

$$I = \nu \varepsilon p^{(-\frac{Q}{kT})} \tag{2}$$

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]:

$$\nu = \left(\frac{2E\_{Diff}}{ma^2}\right)^{\frac{1}{2}}\tag{3}$$

where *m* represents the atomic mass of transition atoms. Herein, two types of diffusion activation energies *Q* corresponding to self *D*<sup>0</sup> and impurity *Ds* diffusion coefficients are gained using first principles calculations. The *Q* for self-diffusion contains two separate energies: vacancy formation energy *Evac* and the migration energy of *Al* atom *Em* in *Al* matrix. For impurity diffusion activation energy, the activation energy *Q* consists of three parts: the substitutional solution energy *Es* of a *TM* atom replacing a *Al* atom, vacancy formation energy *Ef* in the presence of *TM* in *Al*107*TM* supercell, and the migration energy of diffusion *Em* [49]:

$$E\_s = E\_{Al\_{107}TM} - 107E\_{Al} - E\_{TM} \tag{4}$$

$$E\_f = E\_{(Al\_{106}TM; NaCl)} - E\_{Al\_{107}TM} + E\_{Al} \tag{5}$$

$$E\_b = E\_s + E\_f \tag{6}$$

where *ETM* and *EAl* are the energies of single *TM* and *Al* atoms in the stable bulk, respectively, and *Eb* 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:

$$\text{ELF} = \frac{1}{1 + \left(\frac{D\_r}{D\bar{h}\_r}\right)^2} \tag{7}$$

where *Dr* and *Dhr* are the true electron gas density and the pre-assumed uniform electron gas density, respectively.

The *Em*, *Eb* (*Evac*), *Q* and *Ds* (*D*0) for *TM* and *Al* at 300 K with available experimental and theoretical values are summarized in Table 1 [51–54]. It can be seen that errors between the present and previous values in literatures for *Em*, *Eb* (*Evac*) and *EDiff* are within 20%, and the current value of *EDiff* 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 *Qs*, ranging from 0.35 to 2.60 eV. For Mn-Co, Tc-Rh and Re-Ir, they have larger *Qs* 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 *Q*<sup>0</sup> is lower compared to all *Qs* in the doped system, except for the *Qs* 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 *D*<sup>0</sup> 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 *<sup>D</sup>*<sup>0</sup> of 3.55 × <sup>10</sup>−<sup>28</sup> m2·s−<sup>1</sup> for Al in this work is in general agreement with the experimental extension values from 1.76 × <sup>10</sup><sup>−</sup>27~4.42 × <sup>10</sup>−<sup>12</sup> <sup>m</sup>2·s−<sup>1</sup> in the range of 300~1000 K and 1.47 × <sup>10</sup>−14~1.36 × <sup>10</sup>−<sup>12</sup> in the range of 739~917 K in literature [56,57], seen from Table 1 and Figure 2c. Meanwhile, the theoretical predicted *D*<sup>0</sup> of 3.55 × <sup>10</sup>−<sup>28</sup> m2·s−<sup>1</sup> 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 *D*<sup>0</sup> due to the influence of crystal structure defects, dislocations and grain boundaries in experiments. The *Ds* 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 *Ds* for Mg, Si and Cu with temperature calculated by Mantina et al. [44]. Figure 2d further shows the diffusion rate *Ds* at 300 K as a function of the atomic number of TM, and it can be seen that the diffusion rate *Ds* first decreases linearly from 2.05 × <sup>10</sup>−37, 6.47 × <sup>10</sup>−<sup>24</sup> and 2.79 × <sup>10</sup>−<sup>44</sup> <sup>m</sup>2·s−<sup>1</sup> for Sc, Y and Hf to 2.43 × <sup>10</sup><sup>−</sup>50, 6.77 × <sup>10</sup>−<sup>73</sup> and 1.60 × <sup>10</sup>−<sup>78</sup> <sup>m</sup>2·s−<sup>1</sup> for Mn, Ru and Ir and then increases with the increase in atomic number to 3.09 × <sup>10</sup>−13, 9.17 × <sup>10</sup>−<sup>17</sup> and 2.93 × <sup>10</sup>−<sup>29</sup> <sup>m</sup>2·s−<sup>1</sup> for Zn, Ag and Au, respectively (except for Cr of 3.36 × <sup>10</sup>−<sup>44</sup> m2·s<sup>−</sup>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.

**Table 1.** The calculated diffusion barrier *Em* (eV), vacancy–solute binding energy *Eb* (eV)*,* diffusion activation energy *<sup>Q</sup>* (eV) and diffusion rate *Ds* (m2·s<sup>−</sup>1) for TM atoms in Al matrix at 300 K. It should be noted that for pure Al, *Eb* and *Ds* are in fact *Evac* and *D*0, respectively. Note: A negative activation energy *Q* can't meet calculating *Ds* according to Equations (1)–(3).



**Table 1.** *Cont.*

**Figure 2.** (**a**) The diffusion model. (**b**) The calculated diffusion barrier of a vacancy *Em*, vacancy solute binding energy *Eb* (vacancy formation energy *Evac* for self-diffusion in Al matrix) and diffusion activation energy *EDiff* with the change in atomic number. (**c**) The diffusion rate *D* and *EDiff* as a function of temperature. (**d**) The impurity diffusion rate *Ds* 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.

## *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: Δ*Gtot* = <sup>4</sup> <sup>3</sup>*πR*3(Δ*GV* + <sup>Δ</sup>*ECS*) + 4*πR*<sup>2</sup>*γα*/*<sup>β</sup>*. 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 *γα/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]:

$$\gamma\_{\pi/\beta} = \frac{E\_{\pi/\beta} - \left(E\_{\pi} + E\_{\beta}\right)}{2A} \tag{8}$$

where *A* is the area of the interface, *Eα/<sup>β</sup>* is the total internal energy of the relaxed *α/β* system containing an interface and *E<sup>α</sup>* and *E<sup>β</sup>* are the total internal energies of phases *α* and *β* from the strains of all directions, respectively.

The chemical formation energy difference Δ*GV* of L12-Al3TM precipitates can be expressed in dilute solid solution based thermodynamics, *AlnTM* → *Al*3*TM* + *Aln*−3. It can be shown as [15]

$$
\Delta G\_V = \Delta G\_{Al\_3TM} + (n-3)\Delta G\_{Al} - \Delta G\_{Al\_nTM} \tag{9}
$$

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

$$G(V, P, T) = \min[F(V, T)] + PV \tag{10}$$

where *F*(*V*; *T*) is the free energy computed by the sum of electronic internal energy and phonon Helmholtz free energy *F*(*V*, *T*) = *Uel* + *Fvib*. *P* is the circumstance pressure.

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

$$
\Delta E\_{\rm CS} \left( \mathbf{x}, \hat{\mathbf{G}} \right) = \min\_{a\_{\boldsymbol{\theta}}} \left( \mathbf{x} \Delta E\_{\mathbf{a}}^{eqi} \left( a\_{\boldsymbol{\theta}}, \hat{\mathbf{G}} \right) + (1 - \mathbf{x}) \Delta E\_{\boldsymbol{\beta}}^{eqi} \left( a\_{\boldsymbol{\theta}}, \hat{\mathbf{G}} \right) \right) \tag{11}
$$

where *as* is the constrained superlattice parameter, Gˆ is the direction and *x* is the mole fraction of phase α. Δ*Eeqi <sup>α</sup>* and <sup>Δ</sup>*Eeqi <sup>β</sup>* 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, 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:


**Figure 3.** (**a**) The interface models. (**b**–**d**) The calculated interface energy *γα/<sup>β</sup>* and (**e**–**g**) strain energy Δ*Ecs* with the change in atomic number. (**h**) The chemical formation energy Δ*GV* with the change in temperature. (**i**,**j**) The chemical formation energy Δ*GV* as a function of atomic number under different constant temperature conditions. (**a**) represent the calculated result of Li et al. [23].


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

The calculated *γα/<sup>β</sup>* 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 *γα/β*, and the *γα/<sup>β</sup>* 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 *γα/<sup>β</sup>* 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 *γα/<sup>β</sup>* for Al/Al3TM (TM = (Hf-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 *γα/<sup>β</sup>* of Al/Al3TM (TM = (Hf-Hg) is larger than −250 mJ·m−2, and the Al/Al3TM with 3d64s2 has the lowest *γα/β*. Figure 3e shows the variation of strain energy Δ*Ecs* of Al/Al3TM (TM = (Sc-Zn) with the increase in atomic number. It can be seen that the Δ*Ecs* increases from 0.32~1.52 meV·atom−<sup>1</sup> for Sc to 12.06 meV·atom−<sup>1</sup> for Co on the (001) system, to 16.58 meV·atom−<sup>1</sup> for Fe on the (110) system, and to 28.62 meV·atom−<sup>1</sup> for Mn on the (111) system, respectively, and then they all decrease to 0.24 ~ 0.67 meV·atom−<sup>1</sup> for Zn (except for Al/Al3Mn, of the order of −0.14 meV·atom<sup>−</sup>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 Δ*Ecs* 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 <sup>Δ</sup>*Ecs*, being −14.84 meV·atom<sup>−</sup>1.

The trends of Δ*GV* 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 Δ*GV* of all Al3TM change slightly in the temperature range of 0~1000 K, except that the Δ*GV* 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−<sup>1</sup> to −24.38, 66.09, −88.46, −44.47, 71.60, −2.05 and ~88.51 meV·atom<sup>−</sup>1, respectively. Furthermore, the obtained Δ*GV* 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−<sup>1</sup> for Al3Sc and Al3Ti, respectively, at 600 K agree well with the value of −61.14 meV·atom−<sup>1</sup> at 350 ◦C (623 K) for Al3Sc and −66.15 meV·atom−<sup>1</sup> at 300 (573 K) for Al3Ti calculated by Li et al. [15]. From Figure 3i, one can see that the <sup>Δ</sup>*GV* at 300 K increases from −80.96 meV·atom−<sup>1</sup> for Sc, −120.46 meV·atom−<sup>1</sup> for Y and −66.82 meV·atom−<sup>1</sup> for Hf to 20.37 meV·atom−<sup>1</sup> for Mn, 53.89 meV·atom−<sup>1</sup> for Tc and 74.50 meV·atom−<sup>1</sup> for Re, and then decreases slightly to −11.72 meV·atom−<sup>1</sup> for Co, 9.62 meV·atom−<sup>1</sup> for Rh and 4.89 meV·atom−<sup>1</sup> for Ir, respectively. As a final step, they change slightly. At 600 K, the variation trends of Δ*GV* for 3–5d TMs are the same as those at 300 K.
