Pairwise interactions in inertially driven one-dimensional microfluidic crystals

In microfluidic devices, inertia drives particles to focus on a finite number of inertial focusing streamlines. Particles on the same streamline interact to form one-dimensional microfluidic crystals (or “particle trains”). Here we develop an asymptotic theory to describe the pairwise interactions underlying the formation of a one-dimensional crystal. Surprisingly, we show that particles assemble into stable equilibria, analogous to the motion of a damped spring. The damping of the spring is due to inertial focusing forces, and the spring force arises from the interplay of viscous particle-particle and particle-wall interactions. The equilibrium spacing can be represented by a quadratic function in the particle size and therefore can be controlled by tuning the particle radius.

Figure 1

(a) Particles on the same streamline form a 1D microfluidic crystal for $\mathrm{Re}=30$ and $\alpha =0.17$ and $\kappa =1.7$; scale bar represents $90\mu \mathrm{m}$ [8]. (b) Cross-streamline 2D microfluidic crystal; scale bar represents $50\mu \mathrm{m}$ [7]. (c) Streamlines around a single inertially focused particle simulated using FEM in Comsol Multiphysics (Los Angeles, CA) show stagnation points on the opposite side of the channel where particles can focus to form a stable crystal with particles alternating between streamlines. (d) Diagram for two particles focusing on opposite streamlines $h_1$ and $h_2$. The point $(x_c,y_c)$ (black dot) marks the center of the closed eddy formed by the particle focused at streamline $h_1$. (e) Diagram for two particles near the inertial focusing streamline and a single wall.

Figure 2

(a) Viscous P-P interactions in a shear flow predicts “bound” pairs of spheres with closed trajectories [17] and with $\dot{dy}<0$. Shown in the moving reference frame of one particle. (b) Viscous P-W interactions can be represented by image stresslets. The image on particle 1 acts on particle 2 and vice versa, creating a net $\dot{dy}>0$.

Figure 3

Analogy between nucleation and damped spring motion.

Figure 4

The separation of two particles $dx\left(t\right)$ as a function of time for $a=6\mu \mathrm{m}$, different initial separation lengths, and (a) $\mathrm{Re}=30$ or (b) $\mathrm{Re}=1$. (c) The equilibrium separation length $\lambda$ is a function of the relative particle size $\alpha =a/H$ and Eq. (15) captures this relationship well. Here the markers represent numerical solutions to Eqs. (12, 13), and the solid line is Eq. (15). Experimental measurements from Kahkeshani et al. [8] at ${\mathrm{Re}}_p=2.8$ (blue square) and Lee et al. [7] (red triangle) agree with our model. Error bars are standard deviations. (Inset) We observe that $\lambda$ is a linear function of $h$ and can be approximated by Eq. (14).

Figure 5

(a) A particle on the inertial focusing streamline acts on a second particle, drawing vortical path lines both upstream and downstream. The trailing vortex spirals outward while the leading vortex spirals inward. (b) If particles approach too closely, then vortices interfere constructively. (c) If particles are spaced further apart, the vortices interfere destructively.

Figure 6

Vortex interactions for two inertially focused particles. (a) $dx=3d<\lambda$, (b) $dx=4.17d=\lambda$, (c) $dx=6d>\lambda$.