Deterministic lateral displacement (DLD) is a technique for size fractionation of particles in continuous flow that has shown great potential for biological applications. Several theoretical models have been proposed, but experimental evidence has demonstrated that a rich class of intermediate migration behavior exists, which is not predicted. We present a unified theoretical framework to infer the path of particles in the whole array on the basis of trajectories in a unit cell. This framework explains many of the unexpected particle trajectories reported and can be used to design arrays for even nanoscale particle fractionation. We performed experiments that verify these predictions and used our model to develop a condenser array that achieves full particle separation with a single fluidic input.nanofluidics | deterministic ratchet | particle tracking D eterministic lateral displacement (DLD) is an efficient technology used to sort and purify small particles (1). Since their introduction (2), DLD pillar arrays have been used in applications from cell sorting (3) to biosensors (4) and can efficiently sort, separate, and enrich a broad range of particles, including parasites (5), bacteria (6), blood cells (7-9), circulating tumor cells (10), and exosomes (11). The original theory (12) predicts that particle trajectories fall into one of two modes, bumping or zigzag, as determined by the critical diameter Dc defined by the array geometry (2, 12). However, experimental evidence is clear that a rich class of intermediate migration behavior exists between these modes (13,14). Although a few theoretical models (15-17) have been proposed to explain this behavior, a general framework to study how geometric symmetry caused by pillar array influences particle trajectories is still missing.The symmetry of the pillar array can be explained in a specifically chosen unit cell. A typical DLD pillar array and associated unit cell are schematically represented in Fig. 1 A and B. Rows of pillars with diameter D0 are located along the y direction, with the pillars separated by a distance Dy , leaving a gap G = Dy −D0 in between pillars in the y direction. Adjacent rows of pillars are separated by a distance Dx in the x direction and shifted a distance in the y direction. The shift between one row and the N -th nearest row is then N . If this shift is chosen to coincide with Dy , then the array is periodic, and invariant to a translation of NDx in the x direction. Therefore, the array geometry has a built-in angle θp (smallest angle in the red triangle in Fig. 1B), such that tan(θp) = 1/N when Dx = Dy . Note that two types of array designs have been studied in the literature: the row-shifted parallelogram array (or stretched array) and the rotated square lattice (18,19). Here, we only studied the stretched DLD array design as shown in Fig. 1A. Our results do not extend to the rotated square array layout.Because of the invariance of NDx translational transformation, the fluid streamlines are assumed to have the same symmetry. When a...