## 3 Results and Discussion
### The Phase Formation Mechanism and Mechanical Properties  
The structure of \(\alpha\)-TiN\({}_{0.3}\) can be regarded as a solid solution of N atoms in the close-packed hexagonal structure of \(\alpha\)-Ti. A \(3\times 3\times 2\) supercell is built based on the microstructure of \(\alpha\)-Ti, where 11 nitrogen atoms occupy the octahedral space in the Ti\({}_{36}\) super crystal cell without changing the lattice geometry of Ti\({}_{36}\), forming Ti\({}_{36}\)N\({}_{11}\) to describe the \(\alpha\)-TiN\({}_{0.3}\) phase. The crystal structure models of the Ti-N system are shown in Figure 9.  
Figure 8: The scratching curves and typical surface morphology of C61900 Cu alloy after plasma nitriding at 650 \({}^{\circ}\)C for 4 h: (**a**) the friction force and acoustic signal curve of F1 film, (**b**) the surface morphology of the scratching grooves for F2 film, (**c**) the friction force and acoustic signal curve of F2 film, (**d**) the friction force and acoustic signal curve of F3 film. The Gibbs free energy of TiN, \(\epsilon\)-Ti\({}_{2}\)N, and TiN\({}_{0.3}\) between 900 K and 1100 K could be obtained, as shown in Figure 10. The negative Gibbs free energy of these three phases firstly indicated their possible formation; the calculated lattice parameters and the formation enthalpy of the Ti-N intermetallic compounds are listed in Table 6 and compared with the existing calculation results. The results showed that the higher the N content, the greater the formation enthalpy of the Ti-N compound and the better the thermodynamic stability, namely TiN \(>\) Ti\({}_{2}\)N(P42/mmm) \(>\) TiN\({}_{0.3}\). The calculated elastic constants C\({}_{\rm ij}\) of the Ti-N intermetallic compounds, as shown in Table 7, indicate that all Ti-N compounds are mechanically stable. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Phase** & **Space Group** & **Lattice Parameters** & **\(\Delta\)E (k) mol\({}^{-1}\) of Atom** \\ \hline TiN\({}_{0.3}\) & P63/mmc & \(a\) = 0.89152 & \(b\) = 0.89174 & \(c\) = 0.96274 & \(-\)104.40 \\  & & \(a\) = 0.59151 1 1 & \(b\) = 0.59887 1 & \(c\) = 0.97102 1 \\ Ti\({}_{2}\)N & P42/mmm & \(a\) = 0.49535 & \(c\) = 0.30321 & \(-\)148.99 \\  & & \(a\) = 0.4953 1 & \(c\) = 0.3033 1 & \\ Ti\({}_{2}\)N & I4I/amdz & \(a\) = 0.58868 & \(c\) = 0.86547 & \(-\)140.85 \\ TiN & Fm-3m & \(a\) = 0.42463 & & \(-\)192.92 \\  & & \(a\) = 0.4246 1  & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The calculated lattice parameters (\(a\), \(b\), and \(c\) in nm) of the Ti-N compounds by the first principle, compared with other theoretical results. Figure 10: Dependence of the Gibbs free energy (\(\Delta G\)) on the temperature (\(T\)) for the Ti-N compounds. Figure 9: The crystal structures of typical Ti-N compounds, with the gray balls and blue balls representing Ti and N, respectively. The elastic constants were applied to calculate the bulk modulus (\(B\)), shear modulus (\(G\)), Young's modulus (\(E\)), and Poisson's ratio (\(v\)) of the possible phases, as shown in Table 8. The Vickers hardness for the analyzed phases in the present work can also be obtained based on the following equation [37],  
\[H_{V}=2\cdot\left(K^{2}\cdot G\right)^{0.585}-3 \tag{1}\]  
where \(K=G/B\) is [PERSON]'s ratio. It can be seen that the calculated elastic modulus was in good agreement with the previous literature. The shear modulus was positively correlated with the phase stability, so Ti\({}_{2}\)N (41/amdz) had poor stability for the lowest shear modulus. The volume modulus (\(B\)), shear modulus (\(G\)), Young's modulus (\(E\)), and Vickers hardness (\(H_{V}\)) of other Ti-N compounds all met the relationship of TiN \(>\) Ti\({}_{2}\)N(P42/mm) \(>\) TiN\({}_{0.3}\). That is, with the increase in N content, both the modulus and Vickers hardness increased and were higher than those of the Cu-Ti intermetallics.