## 3 Results and Discussion
### The Phase Formation Mechanism and Mechanical Properties
The brittleness and toughness of the Ti-N compounds were also evaluated by the G/B ratio and Cauchy pressure \(C_{12}\)-\(C_{44}\) of Pettifor's principle [38; 39], which indicated that the TiN, Ti\({}_{2}\)N (P42/mm), and TiN\({}_{0.3}\) all belong to brittle compounds. Therefore, the surface modification layer of C61900 Cu alloy should be designed with the hard Ti-N compound as the outmost layer and the Cu-Ti intermetallics as the middle layer for the compromising hardness. The properties of the AlCu\({}_{2}\)Ti, Al\({}_{3}\)Cu\({}_{8}\)Ti\({}_{5}\), and Al\({}_{5}\)Cu\({}_{8}\)Ti\({}_{3}\) were also investigated by first-principle calculation. The crystal structures of the four phases are shown in Figure 11. An Al atom in the AlCu\({}_{2}\)Ti phase structure is replaced by a Ti atom to form the crystal structure of the Al\({}_{3}\)Cu\({}_{8}\)Ti\({}_{5}\) phase. Similarly, the formation mechanism of the Al\({}_{5}\)Cu\({}_{8}\)Ti\({}_{3}\) crystal structure is attributed to the replacement of the Ti atom with an Al atom in the AlCu\({}_{2}\)Ti crystal structure. Table 9 shows the calculated lattice parameters and the formation enthalpy of the AlCu\({}_{2}\)Ti, Al\({}_{3}\)Cu\({}_{8}\)Ti\({}_{5}\), and Al\({}_{5}\)Cu\({}_{8}\)Ti\({}_{3}\) phases, compared with other, related values in the literature [40; 41]. The calculated lattice constants are close to those in the literature with a deviation of less than 1%. The negative formation enthalpies of AlCu\({}_{2}\)Ti, \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \multirow{2}{*}{**Phase**} & \multirow{2}{*}{**Space**} & \multicolumn{8}{c}{**Single-Crystal Elastic Constants (GPa)**} & \multirow{2}{*}{**Stability**} \\ \cline{3-3} \cline{5-12}  & & & & & & & & & & & & \\ \cline{3-3} \cline{5-12}  & \multirow{2}{*}{**Group**} & \(C_{11}\) & \(C_{12}\) & \(C_{13}\) & \(C_{22}\) & \(C_{23}\) & \(C_{33}\) & \(C_{58}\) & \(C_{44}\) & \(C_{66}\) \\ \hline TiN\({}_{0.3}\) & P63/mmc & 265 & 83 & 92 & 270 & 93 & 288 & 95 & 96 & 80 & yes \\ Ti\({}_{2}\)N & P42/mmm & 227  & yes & & & & & & & \\  & & & & & & & & & & & \\  & & & & & & & & & & & \\  & & & & & & & & & & & \\ \end{tabular}
\end{table} 
Table 7: The calculated elastic constants for Ti-N compounds. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{2}{*}{**Phase**} & \multirow{2}{*}{**Space**} & \multirow{2}{*}{\(B\)**} & \multirow{2}{*}{\(G\)**} & \multirow{2}{*}{\(E\)**} & \multirow{2}{*}{\(v\)**} & \multirow{2}{*}{\(G\)/B**} & \multirow{2}{*}{\(C_{12}\)–\(C_{44}\)} & \multirow{2}{*}{\(H_{V}\)} \\  & & & & & & & & & & & & \\ \hline TiN\({}_{0.3}\) & P63/mmc & 150.75 & 90.92 & 227.10 & 0.25 & 0.60 & \(-\)13 & 12.49 \\  & & 145  & 199  & 0.27  & 0.54  & 45  & \\ Ti\({}_{2}\)N & P42/mmm & 203.98 & 139.36 & 340.53 & 0.22 & 0.68 & \(-\)1.21 & 20.00 \\  & & 203  & 119  & 299  & 0.25  & 0.26 & \(-\)1.21 & 20.00 \\ Ti\({}_{2}\)N & I41/amdz & 170.06 & 61.84 & 165.46 & 0.34 & 0.36 & \(-\)29.64 & 3.84 \\ TiN & Fm-3m & 276.65 & 176.89 & 437.44 & 0.24 & 0.64 & \(-\)90.13 & 21.47 \\  & & 277  & 188  & 460  & 0.223  & & & & & \\ \end{tabular}
\end{table} 
Table 8: The calculated bulk modulus \(B\) (GPa), shear modulus \(G\) (GPa), [PERSON]’s modulus \(E\) (GPa), [PERSON]’s ratio \(v\), [PERSON]’s ratio \(G\)/\(B\), Cauchy pressure \(C_{12}\)–\(C_{44}\), and Vickers hardness \(H_{V}\) for Ti-N compounds. Al\({}_{3}\)Cu\({}_{8}\)Ti\({}_{5}\), and Al\({}_{5}\)Cu\({}_{8}\)Ti\({}_{3}\) intermetallics indicate the thermodynamic stability of the above phases, where the lowest formation enthalpy of the AlCu\({}_{2}\)Ti phase indicates its best stability. In addition to the thermodynamic stability, the mechanical stability was also achieved to judge whether a phase is formed. The calculated elastic constants \(C_{\rm ij}\) of AlCu\({}_{2}\)Ti, Al\({}_{3}\)Cu\({}_{8}\)Ti\({}_{5}\), and Al\({}_{5}\)Cu\({}_{8}\)Ti\({}_{3}\) are shown in Table 10, which shows that all three phases have mechanical stability.