**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:


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 Δ*Ecs* 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 Δ*Ecs* for (110) and (111) interface systems are all located at Al/Al3Re, while the (001) interface system of Al/Al3Tc has the lowest value.


**Author Contributions:** Methodology, T.F.; Investigation, Z.R. and T.F.; Writing—original draft, Z.R.; Writing—review & editing, T.F.; Visualization, T.H., Z.R., K.W., K.H. and Y.W.; Project administration, T.H., K.W. and Y.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is supported by the Foshan Technology Project (1920001000409), the Scientific Research Project of Hunan Institute of Technology (HQ21016, 21A0564, HP21047, 21B0796), the Natural Science Foundation of China (52171115). The APC was funded by Foshan Technology Project (1920001000409).

**Data Availability Statement:** In this article, the key parameters of growth and nucleation of Al3TM particles based on the corresponding models (see Figures 2 and 3) have been discussed in our work. Figure 4 describe calculated surface energy of Al3TM and Al. Tables 1–4 present important data as a reference. All data can be found in the manuscript.

**Conflicts of Interest:** The authors declare that they have no known competing financial interest or personal relationships that have influenced the work reported in this paper.
