Universal motion of mirror-symmetric microparticles in confined Stokes flow

Abstract

Comprehensive understanding of particle motion in microfluidic devices is essential to unlock additional technologies for shape-based separation and sorting of microparticles like microplastics, cells, and crystal polymorphs. Such particles interact hydrodynamically with confining surfaces, thus altering their trajectories. These hydrodynamic interactions are shape dependent and can be tuned to guide a particle along a specific path. We produce strongly confined particles with various shapes in a shallow microfluidic channel via stop flow lithography. Regardless of their exact shape, particles with a single mirror plane have identical modes of motion: in-plane rotation and cross-stream translation along a bell-shaped path. Each mode has a characteristic time, determined by particle geometry. Furthermore, each particle trajectory can be scaled by its respective characteristic times onto two master curves. We propose minimalistic relations linking these timescales to particle shape. Together these master curves yield a trajectory universal to particles with a single mirror plane.

Keywords: Hele–Shaw flow; microfluidics; particle-laden flow.

Fig. 1.

Mirror-symmetric particles in quasi-2D Stokes flow. (A–D) Stop-flow lithography (29) produces strongly confined microparticles with various shapes in a Hele–Shaw cell. We investigate particles with a single mirror plane, each one consisting of two or three simple building blocks such as disks, squares, or triangles, connected with rigid shafts. These particles are a useful toy system to study how the geometry of a particle determines its trajectory. (E) We demonstrate this strong shape dependence by comparing the trajectories of three particles with $R_1/R_2=1.5$: from top to bottom a trimer with $\varphi ={90}^{○}$ and a dimer and a trimer with $\varphi ={68}^{○}$. The small arrows denote the orientation of the particles. The trajectories are obtained via 3D finite-element calculations. (F) We assume a planar Poiseuille profile along the height of the channel and Couette flow in the thin lubrication gaps with height $h_{\mathrm{g}}$. Due to channel symmetry, we present only half of a Hele–Shaw cell with particle to scale and highlight the bottom confining wall. (G) Upon depth averaging, we arrive at the so-called Brinkman flow with steep velocity gradients near the side walls and constant velocity $u$ along most of the channel width. In this top view the particle is magnified 2.5 times. The streamlines in all three flow profiles are represented by horizontal blue arrows. (Scale bars, 50 $\mathrm{\mu }$m.)

Fig. 2.

(A and B) Particle-induced flow disturbances in a Hele–Shaw cell. As the particle thickness $H_{\mathrm{p}}=H-2h_{\mathrm{g}}$ is comparable to the channel height $H_{\mathrm{p}}/H\simeq 0.8$, the particle lags the surrounding flow, creating shape-specific velocity and pressure disturbances (cf. arrows and density plots in A and B). As the disturbances differ, so too do the hydrodynamic forces and torque acting on each particle differ. While the streamwise forces $F_x$ on a dimer and a trimer have similar magnitudes (horizontal blue arrows), the drift forces $F_y$ and torques $T_z$ acting on them differ (vertical red arrows and clockwise green arcs, respectively). (C–E) This shape dependence of the forces and torque results in distinct linear and angular velocities, which manifest themselves in the different trajectories followed by different particles (cf. C, D, and E). The orientation and scaled position $x/H$ as function of scaled time $t\times u/H$ are strongly dependent on particle shape. The disturbances to the pressure and velocity fields, as well as the forces and torques on the particles, are calculated using a 3D finite-element scheme (52). In all subfigures the flow is from left to right as denoted by the white arrow in A. (Scale bars, 50 $\mathrm{\mu }$m.)

Fig. 3.

(A and B) Universal motion of mirror-symmetric particles. Regardless of their detailed shape, all studied particles follow a universal trajectory. They exhibit the same quantitative behavior as long as we take into account two characteristic times, $\tau$ and ${\tau }_y$, scaling their modes of motion (52): exponentially decaying rotation $\theta \left(t\right)$ to orient with the big disk upstream (top curves) and bell-shaped translation in the lateral direction $y\left(t\right)-y\left(t_{\perp }\right)$ (bottom curves). The only geometrical element common to all studied particles is their single plane of mirror symmetry. In all cases, the error bars denoting experimental uncertainty are smaller than the symbols and are omitted. The particles and their motion are sketched in the middle. In the key disk, square, and triangle dimers are denoted with D, S, and F for brevity. Disk trimers are denoted as T. Angles, where given, are for the trimer angle $\varphi$ defined in Fig. 1C. The subscript “FEM” signifies that some trajectories are obtained through our finite-element scheme. We also demonstrate that this type of motion is present even at weaker confinement ($H_{\mathrm{p}}/H<0.8$) and is not limited to objects with constant thickness: A particle spanning half the channel height (${\mathrm{D}}_{\mathrm{F}\mathrm{E}\mathrm{M}}\hspace{0.17em}{H}_{\mathrm{p}}/H=0.5,1.54,-$) and a particle comprising two unequal spheres held together by a rigid cylinder (${\mathrm{s}\mathrm{p}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\hspace{0.17em}\mathrm{D}}_{\mathrm{F}\mathrm{E}\mathrm{M}},1.60,-$) follow the same universal path.

Fig. 4.

Relation of the characteristic timescales to particle geometry. The (A) rotation and (B) translation timescales needed to fully describe particle motion via Eqs. 2 and 3 are solely dependent on the geometry of the system. For identical flow parameters such as depth-averaged flow velocity $u$, gap thickness $h_{\mathrm{g}}$, and channel height $H$, the detailed shape of the particle determines $\tau$ and ${\tau }_y$. The rotational timescale depends on the area of the particle $S_{\mathrm{p}}$, its projected length when perpendicular to the flow $L_{\perp }$, and the distance $r_{\mathrm{a}\mathrm{r}\mathrm{m}}=|{\mathit{c}}_{\mathrm{p}}-{\mathit{c}}_0|$ spanning from the centroid ${\mathit{c}}_0=S_{\mathrm{p}}^{-1}{\int }_{S_{\mathrm{p}}}\mathit{r}\mathrm{d}{S}_{\mathrm{p}}$ to the center of perimeter ${\mathit{c}}_{\mathrm{p}}=P_{\mathrm{p}}^{-1}{\int }_{P_{\mathrm{p}}}{\mathit{r}}_{\mathrm{p}}\mathrm{d}{P}_{\mathrm{p}}$, where $\mathit{r}$ are the coordinates of the differential area elements $\mathrm{d}{S}_{\mathrm{p}}$ and ${\mathit{r}}_{\mathrm{p}}$ are points on the particle perimeter $P_{\mathrm{p}}$. Additionally, we need the area moment of inertia $I_{\mathrm{p}}={\int }_{S_{\mathrm{p}}}{\left(\mathit{r}-{\mathit{c}}_0\right)}^2\mathrm{d}{S}_{\mathrm{p}}$. We obtain the translational timescale via the area of the particle and its projected lengths $L_{\perp }$ and $L_{\parallel }$ when its mirror plane is perpendicular or parallel to the flow, respectively. The vertical error bars represent the SD of the timescales within an experimental series (SI Appendix, Table S1). The horizontal error bars are calculated from the uncertainty of the gap thickness $h_{\mathrm{g}}=2.5\pm 0.5\hspace{0.17em}\mathrm{\mu }\text{m}$. The dashed diagonal lines with a slope of unity are a guide to the eye and serve to assess the agreement between the experimental results and scaling relations. All particle shape parameters needed to estimate $\tau$ and ${\tau }_y$ are sketched in C. As the height of the particles $H_{\mathrm{p}}$ is reduced, their timescales begin to diverge, as shown numerically in D. Yet particles spanning half the channel height still exhibit a measurable cross-stream and rotational motion (gray dashed line).