Continuous Inertial Focusing and Separation of Particles by Shape

An effective approach to separating shaped particles is needed to isolate disease-causing cells for diagnostics or to aid in purifying nonspherical particles in applications ranging from food science to drug delivery. However, the separation of shaped particles is generally challenging, since nonspherical particles can freely rotate and present different faces while being sorted. We experimentally and numerically show that inertial fluid-dynamic effects allow for shape-dependent separation of flowing particles. (Spheres and rods with aspect ratios of $3:1$ and $5:1$ have all been separable.) Particle rotation around a conserved axis following Jeffery orbits is found to be a necessary component in producing different equilibrium positions across the channel that depend on particle rotational diameter. These differences are large enough to enable passive, continuous, high-purity, high-throughput, and shape-based separation downstream. Furthermore, we show that this shape-based separation can be applied to a large range of particle sizes and types, including small, artificially made $3\mathrm{\text{−}}\mu \mathrm{m}$ particles as well as bioparticles such as yeast. This practical approach for sorting particles by a previously inaccessible geometric parameter opens up a new capability that should find use in a range of fields.

Popular Summary

Separation of both natural and manmade particles from a background of other particles is essential in a variety of applications: It helps researchers isolate cells that cause disease and allows collection of particles with specific geometric properties in industrial processes. Filter separation using barriers with selected pore sizes is not effective for nonspherical particles, however, since they present different faces while passing through the openings. In our paper, we show with experiments and numerical simulations that inertial effects of a fluid flowing in a channel offer a way to achieve shape-dependent separation of particles with a variety of sizes and shapes.

When microparticles are placed in a channel containing moving liquid, the momentum of the fluid causes the particles to rotate and migrate across stream lines as they move along the channel. In this situation, rotating particles of different shapes eventually occupy different stable positions in the channel cross section. Rod-like particles stay closer to the center of the channel, while spherical particles tend to stay near the walls. These focusing effects are large enough to enable high-purity shape-based separation of large quantities of particles into separate outlets.

We have applied this technique to a large range of particle sizes and types, including spherical and ellipsoidal particles as small as 3 microns as well as biological particles such as budding yeast with different shapes during the cell life cycle. Our calculations predict stable focusing positions that depend on the particle’s largest cross-sectional dimension, and this prediction is confirmed by our imaging studies. This simple and practical approach for sorting particles opens up a new capability that should find use in a range of fields from preparing standardized anisotropic particles for composite materials to synchronizing the life cycles of yeast and bacteria populations for controlled experiments.

Figure 1

Focusing of ellipsoids of different aspect ratios to two sets of dynamic equilibrium positions. (a) In rectangular channels with a high aspect ratio, at moderate $Re$, randomly distributed particles are known to focus to two equilibrium regions centered at the long faces of the channels. (b) The particle shapes, stretching ratios, and ellipsoid dimensions evaluated in the current work. (c) The microfluidic device (upper drawing) used for shape-based separation consists of a simple straight 4-cm long channel, with $W=25$, 30, or $35\mu \mathrm{m}$, and $H=47\mu \mathrm{m}$. At the inlet (middle drawing, left), particles are initially randomly distributed within the fluid. Equilibrium positions ($X_{\mathrm{eq}}$) are measured at the channel outlet (middle drawing, right), 4 cm downstream of the inlet, where particles are assumed to be inertially focused due to the combined effect of $F_{{\mathrm{L}}_{\mathrm{w}}}$ (wall-effect lift) and $F_{{\mathrm{L}}_{\mathrm{s}}}$ (shear-gradient lift). Overlaid pictures ($\mathrm{\text{scale bar}}=10\mu \mathrm{m}$) illustrate particle distribution, respectively, at the inlet (right) and outlet (left). The lower images are multiple overlays of frames captured at the channel inlet (left) and outlet (right).

Figure 2

Rotational motion of ellipsoid particles in a microchannel. (a) Three modes of motion of $1:5$ rods are observed in a channel with aspect ratio ($A{R}_{\mathrm{c}}$) of 0.74: in-plane rotation, out-of-plane rotation, and no rotation. (b) As $R{e}_{\mathrm{p}}$ (calculated for a sphere of the same volume) is increased from 0.3 to 0.75, the frequencies of out-of-plane-rotation and no-rotation modes decrease, and most of the particles rotate in-plane around the vorticity axis [color legend shown in (a)]. The periods of rotation from our simulation and the Jeffery formula are plotted along with the experimental results for (c),(e) $1:3$ and (d),(e) $1:5$ rods in (c),(d) $A{R}_{\mathrm{c}}=0.64$ and (e),(f) $A{R}_{\mathrm{c}}=0.74$. Scale bars: $20\mu \mathrm{m}$. (g) The period of rotation $T$, normalized by average shear rate, increases as the particles get closer to the channel center line due to either an increase in particle length or a decrease in channel width. The normalized value of period is calculated by $T^{*}{U}_{\mathrm{avg}}/(W/2)$, where $T$ is the period of rotation, $U_{\mathrm{avg}}$ is the average fluid velocity, and $W$ is the channel width.

Figure 3

Focusing distributions depend on particle shape, channel aspect ratio ($A{R}_{\mathrm{c}}$), and the Reynolds number of the flow. (a),(b),(d),(e),(g),(h) Histograms of $X_{\mathrm{eq}}$ for spheres, $1:3$ rods, and $1:5$ rods, in different channel cross sections and at different flow rates, indicate that equilibrium positions vary for different shapes. (a),(b) $A{R}_{\mathrm{c}}=0.74$, and $Q=20$ and $110\mu \mathrm{L}/\min $, respectively; (d),(e) $A{R}_{\mathrm{c}}=0.64$, and $Q=30$ and $40\mu \mathrm{L}/\min $, respectively; (g),(h) $A{R}_{\mathrm{c}}=0.53$, and $Q=20$ and $50\mu \mathrm{L}/\min $, respectively. (c),(f),(i) Averaged $X_{\mathrm{eq}}$ is plotted for all channel geometries and flow conditions tested, with error bars indicating the standard deviation obtained from at least 100 measurements.

Figure 4

Simulations allow accurate prediction of focusing positions for shaped particles. Simulation result and streamlines are shown for flow in the reference frame of (a) a focused rod and (b) a spherical particle in a straight channel. (c) Simulations are performed considering particle shape and aspect ratio with (i) the particle rotating (rotating rods) and (ii) the rod constrained from rotation (aligned rods), and also considering spheres with diameters that correspond to the rods’ (iii) longest dimension (spheres $D=b$) and (iv) smallest dimension (spheres $D=a$). (d),(e),(f) Comparison of equilibrium positions away from the wall, obtained by experiments and numerical simulations, for (d) $A{R}_{\mathrm{c}}=0.53$, $Q=50\mu \mathrm{L}/\min $; (e) $A{R}_{\mathrm{c}}=0.64$, $Q=40\mu \mathrm{L}/\min $; and (f) $A{R}_{\mathrm{c}}=0.74$, $Q=110\mu \mathrm{L}/\min $.

Figure 5

Separation results for $6\mathrm{\text{−}}\mu \mathrm{m}$ spherical and ellipsoid particles. The collected ratio of particles is shown at each outlet, considering the extraction yield EY and the enrichment ratio ER (g),(h),(i), and the extraction purity EP (j),(k),(l). Each value shows the mean $\pm \mathrm{SD}$ from three independent experiments. Three configurations of the SAPS device are considered (a),(b),(c), and stacked pictures of the separation are shown for each configuration (d),(e),(f). Scale bar: $50\mu \mathrm{m}$. (a) $A{R}_{\mathrm{c}}=0.53$, $Q=40\mu \mathrm{L}/\min $, five outlets with equal resistances (R); (b) $A{R}_{\mathrm{c}}=0.64$, $Q=80\mu \mathrm{L}/\min $, five outlets (O1 to O5) with ${\alpha }_{1:2}=3/4$ and ${\alpha }_{1:3}=1/2$; (c) $A{R}_{\mathrm{c}}=0.64$, $Q=70\mu \mathrm{L}/\min $, seven outlets (O1 to O7) with ${\alpha }_{1:2}=3/4$, ${\alpha }_{1:3}=1/2$, ${\alpha }_{1:4}=1/4$. In (i) and (l), since no particles were captured from outlets O1 and O7, only the collected ratio of particles at outlets O2 to O6 appear in the graphs.

Figure 6

Yeast-cell sorting in SAPS device C ($A{R}_{\mathrm{c}}=0.64$, seven outlets with ${\alpha }_{1:2}=3/4$, ${\alpha }_{1:3}=1/2$, ${\alpha }_{1:4}=1/4$) at $60\mu \mathrm{L}/\min $. (a) A picture of the cells in the inlet. Cells are categorized into five groups: small single (blue), large single (red), budded (orange), doublet (green), and aggregate (purple). (b) Singles have a high extraction yield in outlet 2, while  in outlet 3, the purity of budded cells increases. (c) The collected ratio of particles is shown at each outlet, considering the extraction yield EY and enrichment ratio ER and (d) extraction purity EP. Each value shows the mean $\pm \mathrm{SD}$ from three independent experiments. Since no particles were captured from outlets O1 and O7, only the collected ratio of particles at outlets O2 to O6 appear in the graphs.