## 5 Results and Discussion
### Effect of the Diffusion of Submicron Particles  
To explore the changes in particle trajectories and the gathering effect in the fully expanded flow field, submicron particles evenly released at the inlet were tracked. Figure 10 shows the particle trajectory of Al with different submicron particle sizes and with \(S_{f}\) = 0.5. The streamline of the flow field is illustrated in Figure 11. The gas continues to expand and accelerate between the divergent section and outlet of the micronozzle, promoting submicron particle focusing. This implies that the drag force further increases with the gas flow as a result of the increasing particle velocity, weakening particle diffusion and enhancing the dynamic focusing of particles. As mentioned in the reference paper by [PERSON] et al. [3], when particle diameters increase, the particle trajectory undergoes the processes of repeated changes: the divergence process(I-a), the collimation process(I-b), a focused beam(I-c), the intersection process(II), a focused beam(III-c), the collimation process(III-b) and the divergence process (III-a). These processes can be divided into three stages. The divergence process(I-a), collimation process(I-b) and a focused beam(I-c) are included in the first stage; the intersection process(II) is included in the second stage; the divergence process(III-a), collimation process(III-b) and a focused beam(III-c) are included in the third stage. Figure 10: Al particle trajectory in the micronozzle with \(S_{f}\) = 0.5 for: **(a)**\(d_{p}\) = 0.3 \(\upmu\)m (divergence process(I-a)); **(b)**\(d_{p}\) = 0.6 \(\upmu\)m (collimation process(I-b)); **(c)**\(d_{p}\) = 0.7 \(\upmu\)m (a focused beam(I-c)); **(d)**\(d_{p}\) = 0.8 \(\upmu\)m (intersection process(II)); **(e)**\(d_{p}\) = 1.0 \(\upmu\)m (intersection process(II)); **(f)**\(d_{p}\) = 1.4 \(\upmu\)m (intersection process(II)); **(g)**\(d_{p}\) = 1.6 \(\upmu\)m (a focused beam(III-c)); **(h)**\(d_{p}\) = 2.2 \(\upmu\)m (collimation process(III-b)); **(i)**\(d_{p}\) = 4.5 \(\upmu\)m (divergence process(III-a)). Figure 9: Velocity in the \(y\)-direction of the particle released from the same initial position with variable densities. In Figure 10, the particles diffuse slightly for \(d_{p}\) = 4.5 \(\upmu\)m owing to their large inertia. When \(d_{p}\) is about 0.6-0.7 \(\upmu\)m or 2.2-4.5 \(\upmu\)m, a collimated or focused particle beam forms, both of which are desirable in the field of cold spraying. When \(d_{p}\) = 0.7 \(\upmu\)m and \(d_{p}\) = 1.6 \(\upmu\)m, aerodynamic focusing is observed, which is the best working state. Some relatively large particles converge so fast that their trajectories cross inside the nozzle, as shown in Figure 10d-f. If the particle trajectories cross, the nozzle exit is not in the best focus position. Moreover, the critical diameter of Al particles for the occurrence of these processes is also different under various shape factors, as indicated in Figure 12. As \(S_{f}\) increases, the critical diameter becomes smaller. When \(S_{f}\) = 0.7, 0.9 and 1.0, a collimation process(I-b) occurs when \(d_{p}\) is about 0.5-0.6 \(\upmu\)m and an intersection process(II) occurs for \(d_{p}\) = 0.7 \(\upmu\)m. This explains why \(u_{p,y}\) is a relatively larger positive value at the nozzle exit for \(d_{p}\) = 0.7 \(\upmu\)m in Figure 8, indicating that the beam is defocusing. When \(S_{f}\) = 0.4 and 0.5, the collimation process(I-b) occurs when \(d_{p}\) is approximately 0.6-0.8\(\upmu\)m, which corresponds to the value of \(u_{p,y}\), which is near zero in Figure 8, forming a focused particle beam. Figure 13 demonstrates the diffusion range of the submicron particles of the nozzle outlet with \(S_{f}\) = 0.5 of particles when \(d_{p}\) = 0.2 \(\upmu\)m to 2.4 \(\upmu\)m under three different densities. The exit radius of particle diffusion is represented as \(R\). All three materials have two minimum particle diffusion radii which were marked with triangles, and at this moment particles are focused.