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Directional locking in the deterministic transport of suspended particles

The transport of suspended particles through porous media is central to a wide range of separation techniques that rely on the principle that the trajectories followed by the suspended species depend on their interaction with the porous material. In many cases, it is the geometrical structure of the stationary phase that is the main factor influencing the transport of suspended species in the mobile phase. These geometric (or confinement) effects are most important when the pore dimensions are of the same order as those of the transported species. A clear example of a separation approach based on geometric effects is size-exclusion chromatography. Therefore, controlling the structure of the stationary phase is an important aspect in the development of an effective separation media. The advent of microfabrication techniques, and more specifically that of soft lithography, has led to the design of microfluidic devices with features in the micron and sub-micron scales. This allowed for the successful miniaturization of various separation techniques, including those based on steric and hydrodynamic effects.

In the present project we investigate the transport of a sphere suspended in a quiescent fluid and moving through a periodic array of solid obstacles under the action of a constant external force by means of Stokesian Dynamics simulations.

We were able to show that in the presence of non-hydrodynamic, short-range interactions between the solid obstacles and the suspended sphere, the moving particle becomes locked into periodic trajectories with an average orientation that coincides with one of the lattice directions and is, in general, different from the direction of the driving force. The locking angle depends on the details of the non-hydrodynamic interactions, and could lead to vector separation of different species for certain orientations of the external force. In fact, we have been able to explicitly show the presence of separation for the case of a mixture of suspended particles with different roughness moving through a square lattice of cylindrical obstacles.

Experimental evidence of deterministic separation in periodic systems

In this part of the project, we have been able to experimentally confirm the presence of directional locking in deterministic systems at low Reynolds numbers. The observed trajectories are periodic and as a consequence the migration angles always correspond to a lattice direction.  We have also demonstrated that different particles might move in different lattice directions for specific values of the forcing angle, resulting in size-based separation.  We have also demonstrated that the first critical angle shows a strong dependence on the particle size. The observed behavior is completely analogous to that observed in microfluidic systems that induce size-based separation by deterministic lateral displacement.

Our work indicates that non-hydrodynamic interactions are crucial in determining the observed directional locking behavior.  In fact, a two-particle model that includes the irreversible effects of non-hydrodynamic interactions not only reproduces the observed directional locking but also the entire dependence of the migration angle on the forcing direction.  The same two-particle model without non-hydrodynamic interactions will predict that the particles always move in the same direction as the external force independent of particle size.  The presence of non-hydrodynamic interactions between the moving particle and the obstacle disrupts the symmetry of the trajectory, inducing a net lateral displacement.  Even though the resulting lateral displacement is typically small, the periodic nature of the system allows such lateral displacement to accumulate, thus amplifying the effect into a macroscopic change in the migration angle.  Therefore, by controlling weak, short range, non-hydrodynamic forces, such as electrostatic forces, microfluidic devices employing periodic arrays should be able to enhance the separation of species.  In the future, we plan to explore the manipulation of these short-range, non-hydrodynamic forces in microfluidic devices employing similar periodic arrays.

In our experimental work we investigated the motion of stain-less steel balls falling through a periodic array of obstacles created with cylindrical LEGO pegs on a LEGO board and immersed in a transparent tank filled with glycerol. The simplicity of this system allows for the systematic variation of the underlying lattice structure in future work.