Directional Locking and the Role of Irreversible Interactions in Deterministic Hydrodynamics Separations in Microfluidic Devices

We performed macroscopic experiments on the motion of a sphere through an array of obstacles that highlight the deterministic nature of the lateral displacements that lead to particle separation in microfluidic systems. The motion of the spheres is irreversible and displays directional locking. The locking directions can be predicted with a single parameter that distinguishes between reversible and irreversible particle-obstacle collisions. These results stress the need to incorporate irreversible interactions to predict the movement of a non-Brownian sphere passing through a periodic array.

Figure 1

Trajectories for: (a) 3 mm and 6 mm spheres with directional locking into the (1,2) direction at a forcing angle of 30°; (b) a single period, obtained by translating all the periods of different 6 mm sphere trajectories; (c) trajectories of 6 mm spheres entering at different points in the array. All trajectories locked into the (1,2) direction for a forcing angle of 13.5°. (d) Trajectories of 3 mm (green dotted line), 6 mm (red dashed line), and 7.1 mm (grey solid line) spheres. The 3 mm sphere locked into the (1,4) direction, whereas the 6 mm and 7.1 mm spheres locked into the (0,1) direction. The forcing angle was 13°.

Figure 2

Effective angles as a function of the orientation of the driving force. The simulation results best fit the experiments with 3 mm (solid blue), 6 mm (dotted red), and 7.1 mm (dashed green) spheres when $b_c=2.30$ (3.5 mm), 1.36 (4.1 mm), and 1.48 (5.3 mm), respectively.

Figure 3

Schematic view of the different trajectories resulting from the collision of a freely suspended sphere (particle), on which a constant external force is applied, with a sphere that remains fixed at the center (obstacle). The symmetry of the particles trajectory is broken if the incoming impact parameter ($b_{\mathrm{in}}$) is less than the critical impact parameter ($b_c$) (see the text).

Figure 4

Migration angle vs forcing angle for particles of the same size ($a=3.18\mathrm{mm}$) but with different densities. The symbols correspond to the average migration angle. The lines correspond to the most probable angle.