Separation of Suspended Particles by Arrays of Obstacles in Microfluidic Devices

The stochastic transport of suspended particles through a periodic pattern of obstacles in microfluidic devices is investigated by means of the Fokker-Planck equation and numerical simulations. Asymmetric arrays of obstacles have been shown to induce the continuous separation of DNA molecules, with particles of different size migrating in different directions within the microdevice (vector separation). We show that the separation of tracer particles only occurs in the presence of a permeating driving force with a nonzero normal component at the surface of the solid obstacles, and arises from differences in the local Peclet number of the particles. On the other hand, finite-size particles also exhibit nonzero, but small, migration angles in the case of nonpermeating fields. Monte Carlo simulations for different driving fields agree with the solutions to the Fokker-Planck equation.

Figure 1

Schematic view of the microfluidic device investigated here and in Refs. 4, 6, 7, 8. The gray rectangles are solid obstacles. Suspended particles are transported through the system by a velocity (or force) field. The unit cell used in this work and the primitive cell of the system are also shown.

Figure 2

Contour plots for the steady state reduced probability density function ${\tilde{P}}^{\infty }(\mathbf{x})$ inside a unit cell. The external force is spatially uniform and oriented at 45°, $F_x=F_y$. The plots correspond to (a) ${\mathrm{Pe}}_{\ell }=1.12$ and (b) ${\mathrm{Pe}}_{\ell }=5.58$.

Figure 3

Probability distribution function in the unit cell, at lines $X$ and $Y$ shown in Fig. 1. The force is spatially uniform and oriented at 45°. ${\mathrm{Pe}}_{\ell }=5.58$. The solid lines correspond to the solution to Eqs. (4) presented in Fig. 2b. The open and solid points correspond to Monte Carlo simulations for two types of particles, which only differ on their mobility, and thus their diffusion coefficient, by a factor of 10.

Figure 4

Migration angle as a function of local Peclet number in a spatially uniform (permeating) field. The results correspond to tracer particles and finite-size particles ($a/d=1/3$ and $a/d=1/15$, where $a$ is the particle radius) for two different tilting angles ($\theta =0°$ and $\theta =7.2°$). The definitions of the tilt and migration angles are shown in Fig. 1. Symbols correspond to Monte Carlo simulations and lines correspond to predicted values from Eqs. (5) (note that due to the high computational cost of solving convection-diffusion equations at high ${\mathrm{Pe}}_{\ell }$ we were limited to ${\mathrm{Pe}}_{\ell }\lesssim 5$ [15]).

Figure 5

Migration angle as a function of the local Peclet number for finite-size particles. The results are analogous to those presented in Fig. 4 but for an electrostatic field (left) and a pressure driven flow (right).