## 3 Results and Discussion
### The Phase Formation Mechanism and Mechanical Properties
The AlCu\({}_{2}\)Ti, Al\({}_{3}\)Cu\({}_{8}\)Ti\({}_{5}\), and Al\({}_{5}\)Cu\({}_{8}\)Ti\({}_{3}\) intermetallics are all thermodynamically and mechanically stable, which might be formed within a certain range of elemental composition. The elastic constants were applied to calculate the volume modulus (\(B\)), shear modulus (\(G\)), [PERSON]'s modulus (\(E\)), and Poisson's ratio (\(\upsilon\)). The hardness, brittleness, and toughness of these intermetallics were evaluated. The mechanical properties of AlCu\({}_{2}\)Ti, Al\({}_{3}\)Cu\({}_{8}\)Ti\({}_{5}\), and Al\({}_{5}\)Cu\({}_{8}\)Ti\({}_{3}\) are listed in Table 11. The calculated volume modulus is in good agreement with the literature [40; 41], and the hardness of the AlCu\({}_{2}\)Ti phase reached 6.75 GPa. According to the criterion for determining toughness and brittleness based on the G/B values, the AlCu\({}_{2}\)Ti, Al\({}_{3}\)Cu\({}_{8}\)Ti\({}_{5}\), and Al\({}_{5}\)Cu\({}_{8}\)Ti\({}_{3}\) were all ductile phases. As a ductile material with high elastic modulus and hardness, the AlCu\({}_{2}\)Ti phase is a good candidate in applications as a wear-resisting material. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Phase**} & \multicolumn{8}{c}{**Single-Crystal Elastic Constants (GPa)**} & \multirow{2}{*}{**Stability**} \\ \cline{2-2} \cline{4-9}  & \(C_{\rm 11}\) & \(C_{\rm 12}\) & \(C_{\rm 13}\) & \(C_{\rm 22}\) & \(C_{\rm 23}\) & \(C_{\rm 33}\) & \(C_{\rm 44}\) & \(C_{\rm 55}\) & \(C_{\rm 46}\) \\ \hline AlCu\({}_{2}\)Ti & 170 & 110 & & & & & 105 & & & yes \\  & 163  & 121  & & & & & 97  & & & & & 98  & & & \\ Al\({}_{3}\)Cu\({}_{8}\)Ti\({}_{5}\) & 149 & 120 & 121 & 151 & 121 & 152 & 95 & 95 & 95 & yes \\ Al\({}_{5}\)Cu\({}_{8}\)Ti\({}_{5}\) & 157 & 107 & 108 & 155 & 107 & 158 & 91 & 91 & 91 & yes \\ \hline \hline \end{tabular}
\end{table} 
Table 10: The calculated elastic constants for the Al-Cu-Ti intermetallics. Figure 11: The crystal structures of AlCu\({}_{2}\)Ti, Al\({}_{3}\)Cu\({}_{8}\)Ti\({}_{5}\), and Al\({}_{5}\)Cu\({}_{8}\)Ti\({}_{3}\), with the pink balls, the blue balls and the yellow balls representing Cu, Ti, and Al, respectively. The effect of N on the formation enthalpy of \(\beta\)-Cu\({}_{4}\)Ti, CuTi, Cu\({}_{3}\)Ti\({}_{2}\), and Ti\({}_{2}\)Cu is shown in Figure 12. It can be seen that the solid solution energy of the Cu-Ti intermetallics firstly dropped and then increased with further growth in N content. Therefore, the addition of N atoms affected system stability and had a significant influence on the formation enthalpy, which provided good theoretical guidance for the nitriding of Cu-Ti intermetallics. To evaluate the influence of the M atom (M = Al, Fe, Cu), the Ti\({}_{(1-x)}\)M\({}_{x}\)N system was also investigated. Table 12 shows the lattice parameters and formation enthalpy of Ti\({}_{(1-x)}\)M\({}_{x}\)N, which were set and calculated at the K-point of Ti\({}_{(1-x)}\)M\({}_{x}\)N in the Brillouin zone. The relationship between the phase formation enthalpy of Ti\({}_{(1-x)}\)M\({}_{x}\)N and the content of M is shown in Figure 12b. It can be seen that the Al, Fe and Cu in TiN had different influences on formation enthalpy, and high thermodynamic stability could be achieved for the Ti\({}_{(1-x)}\)M\({}_{x}\)N system when the content of element M was in a certain range. Specifically, when the Ti atom was totally replaced by the Fe atom, the formed FeN structure tended to be unstable. When the Cu atom replaced the Ti atom to form the Ti\({}_{(1-x)}\)M\({}_{x}\)N phase, the formation enthalpy increased rapidly, and the structure became unstable when x \(\geq\) 0.75. According to previous research works [42; 43], the TiN\({}_{x}\) grain boundary is isolated and the growth of TiN is hindered when the Cu content reaches a certain value. The calculated elastic constants \(C_{\rm{ij}}\) of Ti\({}_{(1-x)}\)M\({}_{x}\)N are shown in Table 13. It can be seen that all types of Ti\({}_{(1-x)}\)M\({}_{x}\)N compounds are mechanically stable except for FeN.