*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]:

$$E\_{\rm sur} = \frac{E\_{Al}^{slab} - N\mu\_{Al}^{bulk}}{2A} \tag{12}$$

where *Eslab Al* and *N* are the total energy and the number of Al atoms in the slab model, respectively. *μbulk Al* 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:

$$3\mu\_{Al}^{slab} + \mu\_{TM}^{slab} = \mu\_{Al\_3TM}^{bulk} \tag{13}$$

$$E\_{\rm sur} = \frac{E\_{Al\_3TM}^{slab} - N\mu\_{Al\_3TM}^{bulk}}{2A} \tag{14}$$

where *μslab Al* , *<sup>μ</sup>slab TM* and *<sup>μ</sup>bulk Al*3*TM* 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]:

$$E\_{sur} = \frac{E\_{Al\_3TM}^{slab} - N\mu\_{Al\_3TM}^{bulk} + n\mu\_{Al}^{slab}}{2A} \tag{15}$$

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

To obtain *μslab Al* in the systems, we first need to avoid Al and Sc bulk phases. Therefore, *μslab Al* and *<sup>μ</sup>slab Sc* are limited, as follows:

$$
\mu\_{Al}^{slab} - \mu\_{Al}^{bulk} < 0 \tag{16}
$$

$$
\mu\_{TM}^{slab} - \mu\_{TM}^{bulk} < 0 \tag{17}
$$

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

$$
\beta \mu\_{Al}^{bulk} + \mu\_{TM}^{bulk} + \Delta H\_f = \mu\_{Al\_3TM}^{bulk} \tag{18}
$$

Combined with Equations (12) and (15)–(17), two limit values of *μslab Al* are respectively given by:

$$
\mu\_{Al}^a = \mu\_{Al}^{slab} = \mu\_{Al}^{bulk} \tag{19}
$$

$$
\mu\_{Al}^b = \mu\_{Al}^{slab} = \mu\_{Al}^{bulk} + \frac{1}{3} \Delta H\_f \tag{20}
$$

The calculated surface energies of AlTM from different Al chemical potentials, *μ<sup>a</sup> Al* and *μb Al*, are summarized in Table 3 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]:

$$E\_{sur}^{\text{ave}} = \frac{1}{4A} \left[ E\_{slab}^{Al} + E\_{slab}^{AlTM} - \left( N\_{slab}^{Al} + N\_{slab}^{AlTM} \right) \times \mu\_{Al\_3TM}^{bulk} \right] \tag{21}$$

where *EAl slab*, *<sup>N</sup>Al slab* and *<sup>E</sup>AlTM slab* , *<sup>N</sup>AlTM slab* are the relaxed energy and total number of *TM* atoms in Al and AlTM-terminated surfaces, respectively.

**Table 3.** The calculated surface energy *Esur* (J·m<sup>−</sup>2) of the (001) and (110) surfaces from different Al chemical potential *μ<sup>a</sup> Al* and *<sup>μ</sup><sup>b</sup> Al* in Al3TM.


Figure 4a shows the slab model for calculating *Esur*, and the detail calculated results of *Eave sur* of Al3TM (RE = Sc-Zn, Y-Cd and Hf-Hg) are depicted in Table 4. 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 *Eave sur* 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 *Eave sur* with the atomic number of TM elements, and it can be found that the variation tendency of *Eave sur* of Al3TM intermetallic compounds presents similar characteristics for different cycles. For example, the *Eave sur* 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 *Eave sur* 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 *Eave sur* 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.


**Table 4.** The calculated surface energy *Esur* (J·m<sup>−</sup>2) in Al or Al3TM.

**Figure 4.** (**a**) The slab model. (**b**–**d**) The calculated average surface energy *Eave sur* with the change in atomic number. (**e**,**f**) The ELF on the (111) plane of Al3Sc and Al3Mo systems, respectively.
