Focusing and Sorting of Ellipsoidal Magnetic Particles in Microchannels

We present a simple method to control the position of ellipsoidal magnetic particles in microchannel Poiseuille flow at low Reynolds number using a static uniform magnetic field. The magnetic field is utilized to pin the particle orientation, and the hydrodynamic interactions between ellipsoids and channel walls allow control of the transverse position of the particles. We employ a far-field hydrodynamic theory and simulations using the boundary element method and Brownian dynamics to show how magnetic particles can be focused and segregated by size and shape. This is of importance for particle manipulation in lab-on-a-chip devices.

Figure 1

(a)–(b) 2D schematic showing the problem geometry where $\mathbf{m}=(m\cos {\varphi }_p,m\sin {\varphi }_p,0)$ is the magnetic moment of the ellipsoidal magnet and $\mathbf{B}=(B\cos {\varphi }_B,B\sin {\varphi }_B,0)$ is the applied uniform magnetic field. Circles on the particle show the direction of the magnetic moment. (c) Schematic of particle movement with different pinned orientations ${\varphi }_p^{*}$. (d) Stresslet $S_{yy}^{\infty }$ as a function of pinned orientation ${\varphi }_p^{*}$. The inset shows the magnitude of the coefficients of Eq. (4), and particle schematics show the direction of the transverse movement. (e) Comparison of far-field theory Eq. (3) and boundary element simulations for the lift velocity $U_y$ of an ellipsoidal particle with $\alpha =3$ and $\beta =100$.

Figure 2

Focusing under Poiseuille flow: (a)–(c) Stable angle ${\varphi }_p^{*}$ and transverse velocity $U_y$ as a function of the particle ($\alpha =3$) position $y$ for a magnetic field ${\beta }_w=60$ and ${\varphi }_B=$ (a) $\pi /2$, (b) 0 and (c) $-0.4\pi$ and channel width $H/a=20$. Lines: prediction from far-field theory; dots in (c): results from boundary element simulations. (d) Time history of particle position $y(t)$ from boundary element simulations under the conditions of (c). The gray dotted curves show trajectories in a rectangular channel of aspect ratio $H_z/H_y=4$ for different initial conditions $z_0=\{0.5H_z,0.8H_z\}$. The inset shows stable fixed point $y^{*}/H$ for ellipsoids ($\alpha =3$) as a function of the magnetic field ${\beta }_w$ and ${\varphi }_B$ obtained by far-field theory. White lines are isolines for every 0.1. (e) Distribution of focused particles under Poiseuille flow with ${\dot{\gamma }}_w=100{\mathrm{s}}^{-1}$ and channel width $H/a=20$ after 10 s, from Brownian dynamics simulations with particle size $a=10\mu \mathrm{m}$. Inset: theoretical predictions (lines) and Brownian dynamics simulation (dots) for the standard deviation of the distribution $\sigma /a$. Dotted horizontal line indicates separations $a\gtrsim 2\mu \mathrm{m}$.