**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 [2–6]. 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 [8,10–12], of which the L12 phase is an important

**Citation:** Hu, T.; Ruan, Z.; Fan, T.; Wang, K.; He, K.; Wu, Y. First-Principles Investigation of the Diffusion of TM and the Nucleation and Growth of L12 Al3TM Particles in Al Alloys. *Crystals* **2023**, *13*, 1032. https://doi.org/10.3390/ cryst13071032

Academic Editors: Andrea Di Schino and Claudio Testani

Received: 5 June 2023 Revised: 26 June 2023 Accepted: 27 June 2023 Published: 29 June 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

intermetallic compound and has been widely studied [13–16]. 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 [19–21], 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 [15,23–28]. 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 [29–32].

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 *Ds* 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 *Ds* 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.
