## 3 Results and Discussion  
In the present study, we only consider the DLD geometries with the periodicity \(N=8\), \(16\), and \(32\) (i.e., the row shift fraction \(\varepsilon=0.125\), \(0.0625\), and \(0.03125\)), and the saturation distance of the lubrication force \(h_{s}\) ranging from \(0.002R\) to \(0.01R\); here \(R\) being the cylinder radius. The particle trajectories for different particle sizes at \(h_{s}=0.01R\) and \(0.005R\) till \(t=30D/U_{m}\) in the DLD with the periodicity of \(N=8\) are plotted in Figure 4. We define the motion mode as \"lateral-displacement\" (referred to as LD) only if the particle keeps moving upwards along the same tilted channel, and otherwise as \"zigzag mode\". Figure 4 shows that at \(h_{s}=0.01R\) the motion mode undergoes a transition from zigzag for the particle of \(d=0.48D\) to LD for the particle of \(d=0.5D\). By contrast, the transition from zigzag to LD takes place from \(d=0.52D\) to \(d=0.56D\) for \(h_{s}=0.005R\), as shown in Figure 4b. The particle of \(d=0.52D\) moves with the LD mode for \(h_{s}=0.01R\), but moves with the zigzag mode for \(h_{s}=0.005R\). Figure 5 shows the particle trajectories for \(d=0.34D\),\(0.4D\),\(0.44D\), and \(0.48D\) at different \(h_{s}\) till \(t=30D/U_{m}\) in the DLD with the periodicity of \(N=16\). We see that the particle of \(d=0.4D\) experiences a mode transition from zigzag at \(h_{s}=0.005R\) to LD at \(h_{s}=0.01R\) in Figure 5b, and the particles of \(d=0.44D\) and \(0.48D\) experience such transition from \(h_{s}=0.002R\) to \(h_{s}=0.005R\) in Figure 5c,d. Therefore, the effect of the lubrication force saturation on the motion mode is significant. The lubrication force serves as a repulsive force when the particle approaches the cylinder and as an attractive force when the particle moves away from the cylinder. From our results, the lubrication force tends to make the particle more prone to the zigzag mode, consistent with the observation that the experimental critical particle diameter is larger than the theoretical prediction (i.e., twice the first flow lane width) [5; 10]. As the saturation distance \(h_{s}\) increases, the effect of the lubrication force decreases, resulting in the transition from zigzag to LD, as observed above. Figure 4: Particle trajectories for different particle sizes at (**a**) \(h_{s}=0.01R\) and (**b**) \(h_{s}=0.005R\) till \(t=30D/U_{m}\) in the DLD with the periodicity of \(N=8\). We search for the critical particle diameter of two motion modes with the smallest variation of the particle diameter being \(0.02D\). For example, for \(h_{s}=0.01R\) and \(N=8\) (i.e., \(\varepsilon=0.125\)), the mode transition takes place from \(d=0.48D\) to \(d=0.5D\), as shown in Figure 4a, and then we take the critical diameter as \(d_{c}=0.49D\) for this case. From our simulations, \(d_{c}/D=0.49\), \(0.35\), and \(0.27\) for \(N=8\), \(16\), and \(32\) at \(h_{s}=0.01R\), respectively, and \(d_{c}/D=0.55\), \(0.43\), and \(0.27\) for \(N=8\), \(16\), and \(32\) at \(h_{s}=0.005R\), respectively. The critical diameter is commonly normalized with the gap distance between vertical neighboring posts \(L_{g}\) (see Figure 1). Here, \(d_{c}/D=d_{c}/L_{g}\), since \(L_{g}=D\). Our results on the critical diameter are compared to the experiments of [PERSON] [10] in Figure 6, and good agreement between the two results can be observed. \(h_{s}\) may be regarded as the surface roughness in the experiments, and the surface roughness of \(0.01R\) may be reasonable for micro-posts in lab. From Figure 6, the critical diameter is higher at a lower \(h_{s}\) for \(N=8\) and \(16\), as observed in Figures 4 and 5; however, they are the same at \(h_{s}=0.01R\) and \(0.005R\) for \(N=32\).