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# Broken flow symmetry explains the dynamics of small particles in deterministic lateral displacement arrays

Contributed by Robert H. Austin, May 14, 2017 (sent for review December 15, 2016; reviewed by Marcelo O. Magnasco, Cerbelli Stefano, and Chao Tang)

## Significance

Deterministic lateral displacement (DLD) is a technique for size fractionation of particles in continuous flow that has shown great potential for biological and clinical applications. Several theoretical models have been proposed to explain the trajectories of different-sized particles in relation to the geometry of the pillar array, but experimental evidence has demonstrated that a rich class of intermediate migration behavior exists, which is not predicted by models. In this work, we present a unified theoretical framework to infer the trajectory of particles in the whole array on the basis of trajectories in the unit cell. This framework explains many of the unexpected particle trajectories reported in literature and can be used to design arrays for the fractionation of particles, even at the smallest scales reaching the molecular realm. We also performed experiments that verified our predictions, even at the nanoscales. Using our model as a set of design rules, we developed a condenser structure that achieves full particle separation with a single fluidic input.

## Abstract

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.

Deterministic 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

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 *y* direction, with the pillars separated by a distance *y* direction. Adjacent rows of pillars are separated by a distance *x* direction and shifted a distance *y* direction. The shift between one row and the *x* direction. Therefore, the array geometry has a built-in angle *B*), such that *A*. Our results do not extend to the rotated square array layout.

Because of the invariance of *x* direction to a body of fluid in the array, and assuming point-like pillars where *x* axis. Each pillar would have a stagnation point in its upstream side and one in its downstream side (Fig. 1*F*). The streamlines that start at a stagnation point of one pillar and end at the pillar that is exactly

The original theory of operation for DLD devices (2, 12) is based on the assumption that, for pillars of finite size, the fluid streamlines are a smoothly deformed version of the streamlines, corresponding to an array of point-like pillars, inheriting the periodicity of the pillar array. Any sufficiently small passively advected particle starting at a position *y* direction, making a zigzag motion around the pillars. Particles larger than the critical diameter

A handful of theoretical models (15⇓–17) (summarized in *SI Appendix*) were developed to account for the nondichotomous particle trajectories in DLD arrays (Table 1). In their explanation, previous models relied either on array geometries with restricted conditions (e.g.,

We present a theoretical framework to more universally determine the dependence of the migration angle of particles in terms of DLD array geometry parameters. Our model aims at capturing the deterministic nature of the trajectories, as applied to high Pèclet number systems. We observed that the pillar array geometry distorts the fluid flow such that particle trajectories do not necessarily follow the periodicity or symmetry of the array, generating trajectories with migration angles that can have values between 0 and

## Results

### Fluid Dynamics in the Unit Cell.

We first study the velocity field in the unit cell of a 2D pillar array. Because the Reynolds number in typical DLD experiments is <**u** and *A*), where the component of the pressure gradient in the direction normal to the pillar surface is zero. Periodic boundary conditions require that the velocity and pressure fields on the lines BC and FG are identical. At the inlet (HA) and outlet (DE), we used normal velocity gradient (*C*). Note that we imposed no boundary conditions associated to the enclosing walls and treated our system as an unbounded infinite array.

The resulting fluid streamlines for different pillar diameters *C*–*F* (*Materials and Methods*). It will be convenient to categorize the streamlines starting at the inlet line into two types: (*i*) “veering flow” and (*ii*) “direct flow,” colored red and green, respectively, in Fig. 1*D*. We also define the “separatrix line” as a streamline dividing the veering and direct streamlines, which ends at a stagnation point on the bottom right pillar surface (Fig. 1*F*). Direct streamlines exit through the outlet of the same unit cell. In a unit cell, veering red streamlines below the separatrix line exit through the bottom segment BC to reenter through the top segment FG as the blue streamlines in the adjacent unit cell beneath, after which they proceed to exit at some point on the outlet line DE.

### Recurrence Map.

Every streamline represents the trajectory of a point-like, passively advected particle that enters the unit cell at some point *E*). The distances

This mapping is known in the theory of dynamical systems as a Poincare first recurrence map, or recurrence map for short. Fig. 2*A* shows one instance of the map *x* axis corresponds to the position *SI Appendix*).

The map can be iterated to track the trajectory of small particles over consecutive unit cells in the array for any system geometry. Fig. 2 *A* and *B* shows a particle originally at position 0 being transported to position 1 in the outlet line. Identifying the outlet line of this cell with the inlet line of the next cell, we identify the equivalent point 1 on the inlet line. Iterating this procedure, the particle ultimately lands at position 9 on the inlet line, which corresponds to a veering streamline. The particle now follows the red streamline and moves to the cell below, which is represented as reentering the cell from the top (blue streamline) to end up at position 10. This process simulates a single trajectory over multiple consecutive cells as shown in Fig. 2*C*, where the 10 streamline segments described above as entering and exiting from the same unit cell are now unfolded as they exist in the actual array. Because the trajectories of Fig. 2*C* are equivalent to the trajectories of Fig. 2*A*, the recurrence map can therefore be used to represent any single trajectory over multiple cells as the composition of a single function.

We will call the span of a particle trajectory a “cycle,” which starts when the particle enters the unit cell after having zigzagged the previous pillar (i.e., right before entering the unit cell, the particle trajectory was a blue veering streamline in the previous cell), continues while the trajectory corresponds to direct streamlines, and completing the cycle when the particle escapes the unit cell as a red veering streamline. For example, the trace in Fig. 2*C* shows one cycle starting at position 0, corresponding to a trajectory that in the previous iteration had escaped the unit cell as a veering streamline (blue) and continued for 9 iterations as direct segments (green). In the next iteration, the trajectory exited the cell as a veering streamline (red) and started again as a direct segment (green).

These 10 segments constitute a cycle. We will call the number of segments in one cycle the “local periodicity” and denote it by *C*. This cycle can be written as follows:*A* and *B*, where the last point (10) does not match with the

Because of this mismatch between the *SI Appendix*. We defined the pseudoperiodicity

We calculated the recurrence map of pillar arrays with different geometries, as well as the local periodicity over multiple cycles (*SI Appendix*, Fig. S2 *A*–*D*). As the pillar diameter

### Geometric Effects on the Migration Angle.

The pseudoperiodicity is the average of the local periodicity of a particle trajectory over multiple cycles. Therefore, the migration angle of a trajectory can be calculated from its pseudoperiodicity. If the pseudoperiodicity is larger than *x* axis of the array (Fig. 3, red and blue line). In this case, we define the trajectory as in an “altered zigzag mode,” and a nonzero migration angle is given by

We computed the particle trajectories over multiple pillar arrays with the same structural periodicity, but different pillar diameters (*SI Appendix*, Fig. S2). As the pillar diameter increases, the pseudoperiodicity increases regardless of the initial position (*SI Appendix*, Fig. S3 *A*–*D*) and a corresponding increase in the migration angle results. This nonzero altered zigzag migration angle is solely the result of the geometry of the pillar array modifying the fluid flow, and not the interactions of the particle with the pillars.

In Fig. 4*A*, we plot the normalized migration angle

To test these predictions, we experimentally measured the migration angle of *Materials and Methods*). These experiments used full-width injection, in which particles are introduced across the entire width of the array inlet, and the migration angle is measured based on how much the particle flux is deflected (11). Beads were run at velocities *B*–*D* shows fluorescent microscope images of deflected bead flux for different *A* plots the experimental results along with our model predictions.

Although there is considerable agreement between theory and experiment, the agreement is not perfect. Potential reasons for this discrepancy include the fact that the pillar array in any real device is not infinitely periodic, so that boundary effects could influence

To verify the predicted dependence of the migration angle with *E*). Fig. 4*F* shows a fluorescence microscope image mosaic of 50-nm beads injected into the full width of the device with a first array of

Our model predicts an important phenomenon—the deflection of small particles with high *G* exploits this concept to achieve high-efficiency separation from a single fluidic input. In the first part of the device, which we refer to as a condenser,

### Pillar-Size Effect on the Critical Diameter.

We can use the recurrence map to model the trajectories of finite-size particles and study the dependence of the critical diameter on pillar geometry. As in Inglis et al. (12), our model uses two simplifying assumptions: (*i*) Flow is not modified due to the particle’s presence; and (*ii*) the hard walls and our finite size particles interact with perfect inelastic collision.

We simulated the trajectories of the particles’ center of mass with different diameters *A* and *B*. When the particle diameter is less than the critical diameter *A*), but with larger particles, only direct streamlines remain (green line in Fig. 5*B*). In this case, all trajectories collapse, leaving only direct streamlines, meaning that only a bump mode remains.

Fig. 5 *C* and *D* show the recurrence maps for particles with different diameters. With increasing particle size, the allowed region in the map (bright gray) shrinks due to geometric constraints of the particle and pillars. When the starting position is close to the streamline that leads to the stagnation point (

The critical diameter dependence on the geometry is shown in Fig. 5*E* for various

Fig. 5*F* shows the transition between the altered zigzag mode and bumping mode for diameters

## Conclusions

The original theory explaining the principles of operation of DLD arrays was formulated under the assumption that the symmetry of the fluid streamlines would match the symmetry of the pillar array. Our work shows that this assumption does not provide a complete picture of the underlying physics. By reformulating the principle of operation of DLD arrays, we have shown that particle trajectory dependencies on array geometry are stronger than in the original theory and that the particle trajectory behavior is considerably richer than originally thought. The nonzero migration angle of small particles discussed in this work stems from the anisotropic permeability of row-shifted parallelogram DLD array layouts, as has been discussed by other authors (15, 23). Indeed, we confirmed that the velocity field averaged over a unit cell has a nonzero lateral component and that the angle between the average velocity vector and the pressure gradient direction is almost identical to the migration angle of small particle trajectories calculated by using the recurrence map. (*SI Appendix*, Fig. S1).

Despite the remarkable agreement between theory and experiment shown in Fig. 4*A*, we should note the assumptions and limitations of our theory. First, we assume an infinite lattice and neglect the effect of the array-bounding walls. Second, the fact that the mean direction of the streamlines follows an angle not in the direction of the pressure gradient imposes an anisotropic permeability on the fluid flow. In finite arrays, this off-axis flow would create a depletion of fluid from one wall and a buildup of fluid on the other wall, modifying the pressure boundary conditions assumed in our model. Both of these limitations will be less severe if the array is sufficiently extended. A third limitation is the assumption that the unit cell has constant pressure at the input and output. Simulations done in systems of two or more unit cells show that this assumption is approximately correct. Finally, we assume that the particle presence does not affect the streamlines and that the particle–pillar interaction can be reduced to a hard-wall repulsion. These assumptions can be justified by using Maxey–Riley equations and the fact that the Reynolds number in the system is small (supplement in ref. 11). (Note that some of these approximations are also made in the original theory.)

Our theory provides rules on how to modify the pillar array geometry to tune not only the critical diameter, but also the migration angle of small particles in the altered zigzag mode. These results can be used as guiding principles for DLD design to control the separation of particles and biocolloids from the microscale to the nanoscale. The condenser adaptation of these design rules also provides a path toward simplified DLD systems that can achieve full separation of particles by using a single fluidic inlet.

## Materials and Methods

### Continuum Simulation.

We calculated the 2D velocity and pressure fields in a unit cell by solving the Navier–Stokes equation using the finite volume method. Several techniques, such as the finite element method (COMSOL) and the Boltzmann lattice method have been used to simulate particle motion in a pillar array (15, 24). In this work, we used OpenFOAM together with its Lagrangian particle tracking library (25). In addition, gmsh was used for a mesh generation tool. By using it, the geometric components (

A discrete element method was used to calculate the particle trajectories in the fluid, taking into account the necessary particle–pillar interaction. For the latter, we used a simple hard-wall repulsion model with homemade code modifying OpenFOAM’s Lagrangian particle library. Other particle–pillar interactions such as soft-wall models are available, but the simplified model can grab the essence of the dynamics (21). We calculated and mapped the trajectories of a

### Chip Fabrication.

The nanoDLD device arrays were fabricated on 40- ×40-mm chips on *i*) optical contact lithography was used to define the microchannel features followed by a reactive-ion etch (RIE) to transfer these features into the *ii*) after stripping the resist from (*i*), a combination of electron-beam (e-beam) and deep-ultraviolet (DUV) lithography were used sequentially to pattern the nanoDLD pillar array features by adjoining the microchannels, transferring these features into the same HM by using a similar RIE process. With both sets of patterns etched into the HM down to bulk Si, all open features were simultaneously etched into

The 40- × 40-mm chips were bonded with

### Priming Chip.

To wet the chips a 100-mL solution of 2% (vol/vol) Tween 20 (Sigma-Aldrich) in deionized water was prepared in a 500-mL beaker. A custom glass holder was used to position the chips vertically, with the inlet ports down, in the Tween 20 solution to allow capillary wetting of the fluidic channels. Typical wetting times were

### Bead Displacement Experiments.

NanoDLD device chips with gap sizes ranging from

## Acknowledgments

We thank Stacey Gifford, Navneet Dorgra, and Pablo Meyer from the IBM Nanobiotechnology program, and Prof. Howard A. Stone from Princeton University for fruitful discussions. We also thank the IBM Microelectronics Research Laboratory staff for processing the microfluidic chips used in this study, with special thanks to Simon Dawes and Markus Brink for e-beam lithography assistance.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: austin{at}princeton.edu, kimsung{at}us.ibm.com, or gustavo{at}us.ibm.com.

Author contributions: S.-C.K., B.H.W., H.H., J.T.S., R.H.A., and G.S. designed research; S.-C.K., B.H.W., H.H., J.T.S., and G.S. performed research; S.-C.K., B.H.W., H.H., J.T.S., and G.S. contributed new reagents/analytic tools; S.-C.K., B.H.W., H.H., J.T.S., and G.S. analyzed data; and S.-C.K., B.H.W., H.H., J.T.S., R.H.A., and G.S. wrote the paper.

Reviewers: M.O.M., Rockefeller University; C.S., University of Rome; and C.T., Peking University.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1706645114/-/DCSupplemental.

Freely available online through the PNAS open access option.

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