Size-dependent particle migration and trapping in three-dimensional microbubble streaming flows

Acoustically actuated sessile bubbles can be used as a tool to manipulate microparticles, vesicles, and cells. In this work, using acoustically actuated sessile semicylindrical microbubbles, we demonstrate experimentally that finite-sized microparticles undergo size-sensitive migration and trapping toward specific spatial positions in three dimensions with high reproducibility. The particle trajectories are successfully reproduced by passive advection of the particles in a steady three-dimensional streaming flow field augmented with volume exclusion from the confining boundaries. For different particle sizes, this volume exclusion mechanism leads to three regimes of qualitatively different migratory behavior, suggesting applications for separating, trapping, and sorting of particles in three dimensions.

Figure 1

Schematic (not to scale) experimental setup: A PDMS microchannel with width $W=500\mu \mathrm{m}$, height $H=72\mu \mathrm{m}$, and total length $\sim 30$ mm is filled with liquid and a sessile cylindrical bubble is formed in the blind side channel of width $w=90\mu \mathrm{m}$ and depth $l=350\mu \mathrm{m}$. By switching the valve (indicated in gray), inlet and outlet of the microchannel are connected, leading to a uniform pressure in the system that stops the flow. The size of the bubble is controlled by adjusting the static pressure of the liquid with a pressure controller. A piezo transducer attached to the glass slide excites surface oscillations at the bubble interface generating a vortical acoustic streaming flow (in red) in the microchannel.

Figure 2

Streaming flow structures near an oscillating semicylindrical microbubble. (a) Trajectories of 2-$\mu \mathrm{m}$-diameter particles. The plot shows one trajectory in each of the equivalent quadrants. (b) Sketch of the flow structures and symmetry planes as observed in experimental investigations. The flow field is set up in four toroidal flow structures. These structures are symmetric to the midplane $z=0$ (spanning in the $xy$ direction) and the sagittal plane at $x=0$ (spanning in the $yz$ direction). (c) A reconstruction of three toroidal flow surfaces reveals the nested structure of these tori.

Figure 3

Experimentally determined $z$ position of particles with $d_p=$ 2, 5, 10, and 15 $\mu m$ in microbubble streaming flow as a function of time. After the particles pass the perigee for the first time (closest position to the bubble's interface, set to $t=$ 0), they tend to migrate to a certain $z$ position at different rates depending on their size. Dashed lines depict the coreline position $Z_{\mathrm{core}}=\pm 18\mu \mathrm{m}$. A red circle illustrates the size of the particle and dotted lines are drawn at one particle radius from the top and bottom walls. (a) $z$ position for particles with $d_p=$ 2 $\mu \mathrm{m}$, they slowly migrate toward inner tori of the flow structure. Focusing at a defined $z$ position is not observed. (b) $z$ position for particles with $d_p=$ 5 $\mu \mathrm{m}$; they migrate within $\sim 1$ second toward $Z_{\mathrm{core}}=\pm 18\mu \mathrm{m}$. (c) particles with $d_p=$ 10-$\mu \mathrm{m}$ migrate to different final $z$ position within $\sim 0.3$ s. (d) particles with $d_p=$ 15 $\mu \mathrm{m}$ migrate within $\sim 0.3$ s and focus at the top and bottom walls of the microchannel. Dashed line: midplane; dotted lines: closest position of the particle center to top and bottom walls.

Figure 4

Schematic of the excluded volume interaction introduced in the numerical model for trajectories of differently sized particles. While being advected in the streaming flow of the oscillating bubble, particles approach the bubble (black line) along streamlines (gray lines). A particle that would penetrate the bubble in a certain timestep of the simulation is radially displaced and forced onto another streamline.

Figure 5

Numerical $z$ positions of particles with $d_p=$ 2, 5, 10, and 15 $\mu \mathrm{m}$ in microbubble streaming flow as a function of time. After the particles pass the perigee for the first time (closest position to the bubble's interface, set to $t=0$), larger particles start to migrate to a certain $z$ position. Dashed lines depict the coreline position $Z_{\mathrm{core}}=\pm 18\mu \mathrm{m}$. A red circle illustrates the size of the particle and dotted lines are drawn at one particle radius from the top and bottom walls. (a) $z$ positions for particles with $d_p=$ 2 $\mu \mathrm{m}$, some slowly migrate toward the inner tori of the flow structure, others span the whole volume of the microchannel, but none focus at a particular $z$ position. (b) Particles with $d_p=$ 5 $\mu \mathrm{m}$ clearly migrate toward $Z_{\mathrm{core}}$. While the migration times obtained from simulations are comparable to the experimental ones (Fig. 3) for most of the particle sizes, the migration of 5-$\mu \mathrm{m}$-diameter particles is somewhat slower and therefore a longer time axis is shown in comparison to Fig. 3. (c) Particles with $d_p=$ 10 $\mu \mathrm{m}$ quickly migrate in the vicinity of the microchannel walls. (d) Particles with $d_p=$ 15 $\mu \mathrm{m}$ quickly migrate toward the microchannel walls, with frustrated attempts to penetrate the wall.

Figure 6

Final particle positions $Z_{\mathrm{end}}$ as a function of particle diameter $d_p$. Black circle markers show the average $Z_{\mathrm{end}}$ for experimental trajectories of a certain particle size and the red bar is the respective standard deviation. Small blue dots show $Z_{\mathrm{end}}$ for many simulated particle trajectories and the blue line the corresponding mean. The PDMS channel is indicated in gray with an exclusion zone that particles cannot enter due to their finite size (dotted black line). Due to the channel symmetry experimental results are mirrored to positive values at the midplane ($z=0$), while simulations use only initial particle positions with $z>0$.

Figure 7

Typical particle trajectories computed from simulations in the three regimes introduced in Fig. 6 for identical initial conditions. Particles follow their computed streamline until displaced by excluded volume interactions, as described in Sec. 3c. A tracer particle trajectory with the same initial condition (defined as a particle with $d_p=0\mu \mathrm{m}$) is plotted in semitransparent gray. Semitransparent spheres indicate the particle's initial position and opaque spheres indicate the last particle position computed in the simulation. (a) Particle with $d_p=$ 3 $\mu \mathrm{m}$ showing an initial displacement and then passive advection on a flow torus. (b) The 8-$\mu \mathrm{m}$-diameter particles follow a part of a torus, cut short by interactions with the bubble near the coreline. (c) Typical trajectory of 15-$\mu \mathrm{m}$-diameter particles, which quickly migrate toward the bottom wall and remain trapped there.