Multidirectional sorting modes in deterministic lateral displacement devices

Deterministic lateral displacement (DLD) devices separate micrometer-scale particles in solution based on their size using a laminar microfluidic flow in an array of obstacles. We investigate array geometries with rational row-shift fractions in DLD devices by use of a simple model including both advection and diffusion. Our model predicts multidirectional sorting modes that could be experimentally tested in high-throughput DLD devices containing obstacles that are much smaller than the separation between obstacles.

Figure 1

An array of posts (marked by black dots) with period $N=8$ and a shift of $M=3$ flow lanes per row, i.e., a row-shift fraction $ϵ=3∕8=0.375$. The flow $\mathbf{v}$ is directed along the $x$ axis from top to bottom. The dashed lines indicate the four possible separation directions. First, the two well-known modes: The straight mode $A$ with ${\theta }_A=0°$ for particles of radius $r$ with $r<(1∕N)\lambda$ (small dark gray circles starting in flow lane $R=1$), and the maximal displacement mode $C$ with ${\theta }_C=20°$ for $(3∕N)\lambda <r$ (large light gray circles starting in flow lane $R=4$). Additionally, the two separation modes: One $B_{+}$ towards the right with angle ${\theta }_{B_{+}}=2.4°$ for $(1∕N)\lambda <r<(2∕N)\lambda$ (small light gray circles starting in flow lane $R=2$), and another $B_{-}$ towards the left with angle ${\theta }_{B_{-}}=-7.1°$ for $(2∕N)\lambda <r<(3∕N)\lambda$ (large dark gray circles starting in flow lane $R=3$). The solid vertical lines indicate the flow lanes of width $\lambda ∕N$, while ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are the lattice vectors, and $\alpha$ is an aspect ratio.

Figure 2

An array with period $N=10$ and a shift of $M=3$ flow lanes per row, giving a row-shift fraction $ϵ=3∕10=0.3$. The flow $\mathbf{v}$ is directed along the $x$ axis from top to bottom. Here there are three sorting modes, delimited by two critical radii. Mode $A$ for particles of radius $r$ with $r<r_{c1}=(1∕10)\lambda$ (shown on the far left-hand side), and mode $C$, the maximal displacement mode for $(3∕10)\lambda =r_{c2}<r<(1∕2)\lambda$ (large light gray circles). A mode $B_{-}$ displaces particles with $r_{c1}<r<r_{c2}$ towards the left (large dark gray and intermediate, light gray circles).

Figure 3

(Color online) Composite image of numerically calculated spatial distributions for a device with $N=10$ and $M=3$. (a) Results for $D=0$ for particles with radii $r<r_{c1}$ (mode $A$), $r_{c1}<r⩽r_{c2}$ (mode $B$), and $r>r_{c2}$ (mode $C$). (b) Particles, with the same radii as in (a), moving through the array with a flow speed of $v=100\mu \mathrm{m}∕\mathrm{s}$, including the effect of diffusion. Broadening of all distributions due to diffusion can be seen and particles in mode $C$ are sorted less efficiently. Initial spatial distributions here are the same for all particle radii, and a narrow initial distribution is used for visual clarity.

Figure 4

(Color online) Sorting angles $\theta$ calculated for a device with $N=10$ and $M=3$ as described in Sec. 3 with $r_{c1}=1\mu \mathrm{m}$ and $r_{c2}=3\mu \mathrm{m}$. The initial position corresponds to the center of mass of the initial distribution between two posts and the final position corresponds to the center of mass after 10 rows. $\theta$ is plotted here versus particle radius $r$ with (open circles) and without (filled circles) diffusion with flow velocity $v=10\mu \mathrm{m}∕\mathrm{s}$. The negative sorting angles for $r_{c1}<r<r_{c2}$ indicate the presence of mode $B_{-}$ for this array. Inset shows the sorting angle $\theta$ around $r=r_{c2}=3\mu \mathrm{m}$ for a range of flow velocities (same $y$-axis range). Diffusion blurs the sharp transition between the sorting modes, as discussed in Sec. 4B.

Figure 5

(Color online) Distributions of three particle sizes: $r=0.90$ (dashed line), 2.00 [(red)gray], and $3.08\mu \mathrm{m}$ (black) after transport through 10 rows of the $N=10$, $M=3$ array. The total number of particles is the same in each case and each initial distribution (not shown) is a square distribution with a narrow width centered on $y=30\mu \mathrm{m}$. (a) No diffusion. (b) With diffusion and $v=1000\mu \mathrm{m}∕\mathrm{s}$. (c) With diffusion and $v=100\mu \mathrm{m}∕\mathrm{s}$. (d) With diffusion and $v=10\mu \mathrm{m}∕\mathrm{s}$. Panel (b) shows a case where mode $B_{-}$ could be detected experimentally. For the lower speed in panel (c), modes $A$ and $B_{-}$ cannot be resolved, but the combined distribution is broader than mode $A$ alone. For the even lower speed in panel (d) the distributions of particles in modes $A$ and $B_{-}$ are each wider than the separation between them and the two modes are completely unresolvable.