  We analyze the trajectory of suspended spherical particles moving through a
square array of obstacles, in the deterministic limit and at zero Reynolds
number. We show that, in the dilute approximation of widely separated
obstacles, the average motion of the particles is equivalent to the trajectory
followed by a point particle moving through an array of obstacles with an
effective radius. The effective radius accounts for the hydrodynamic as well as
short-range repulsive non-hydrodynamic interactions between the suspended
particles and the obstacles and is equal to the critical offset at which
particle trajectories become irreversible. Using this equivalent system we
demonstrate the presence of directional locking in the trajectory of the
particles and derive an inequality that accurately describes the Devil's
staircase type of structure observed in the migration angle as a function of
the forcing direction. Finally, we use these results to determine the optimum
resolution in the fractionation of binary mixtures using deterministic lateral
displacement separation microfluidic systems.
