Numerical Comparison

We also solve the flow field and the electric potential within the device using the commercial flow solver COMSOL Multiphysics to provide a numerical comparison to the experimental results of the previous section. This numerical solution uses one-way coupling between the particle and the fluid. In other words, the fluid flow is solved ignoring the presence of any particles, and the particle motion is calculated afterward using post-processing. Since the largest particle size is 4.8 µm, the particles are small when compared to the channel height of 50 µm or the electrode width of 65 µm. Furthermore, the particles are in a dilute sample. For small particles in a dilute sample, the particles do not greatly influence the overall fluid flow and there is little particle-particle interaction. Using one-way coupling permits us to solve the flow field once and use this flow field to calculate all the particle trajectories afterward. Two-way coupling necessitates updating the fluid motion around the particle for each timestep of the particle trace, greatly increasing the computational time. Recall also that the classical DEP force expression (1.3) uses the background electric field without the presence of the particle, relying on a similar assumption of a small particle with respect to the length scale of the electric field.

Once the flow field and the electric potential field are solved with the same parameters from the experiment, we seed particles in the sample stream inlet and calculate their trajectories in the following manner. The particles are convected along with the same velocity as the fluid due to the drag force, until the DEP force acts on the particles to move them relative to the fluid. Since this is within the Stokes’ flow regime, we have

where µis the viscosity of the fluid, vp andvm are the velocities of the particle and

fluid medium, respectively, and fw(x, y, z) is a wall correction factor to the Stokes’

drag. The wall correction factor [20] for a sphere moving parallel to a plane wall is

fw = h 1− 9 16(a/h) + 1 8(a/h) 3₋ 45 256(a/h) 4₋ 1 16(a/h) 5i−1 (2.2)

and for a sphere moving normal to a wall is

fw = h 1− 9 8(a/h) + 1 2(a/h) 3i−1 . (2.3)

In (2.2) and (2.3) a is the particle radius andh is the distance between the center of the particle and the wall (i.e., h/a >1). From (2.1) we directly see that

∆xp=vp∆t =hvm+ FDEP 6πµafw(x, y, z)

i

∆t, (2.4)

where ∆xp is the change in the particle’s position over a timestep of ∆t.

Equation (2.4) allows us to calculate the trajectory for each individually seeded particle from its initial position within the sample stream inlet, and this calculation is done using COMSOL Multiphysics post-processing. Figure 8 shows a representative particle trace for each particle size with the flow field and electric potential solved numerically in COMSOL. The green, red, and blue trajectories correspond to a 1.0, 2.5, and 4.8 µm particle, respectively. As expected, PFF alone cannot sort this set of particles. Although the particles are aligned with the wall of the pinched segment so that they are arranged by size as they reach the expanded channel section, the distance between the particles’ trajectories as they exit the pinched segment is still not large enough to provide adequate sorting. Due to the PFF portion forcing the particles toward

Figure 8: Representative particle traces for each particle size with the flow field and electric potential solved numerically in COMSOL using the same parameters as in the experiment. The red, green, and blue trajectories correspond to the 1.0, 2.5, and 4.8 µm particles, respectively. Note: Particles enlarged to show their sizes relative to one another.

the top wall, the initial x- and y-positions of the particles have little effect on the final outlet channel. However, the initial z-position (perpendicular to the page in Figure 8) of a particle can still have some effect, since the pinched segment does not focus the particles in the vertical direction. The z-position of a particle determines how far it must be deflected vertically by the negative DEP force in order to reach the slower fluid flow near the top wall of the channel that allows it to be captured along an electrode. Figure 9 displays the particle trajectories for 100 particles of each size initially evenly distributed within the inlet channel. The spread in the trajectories among the same-sized particles is due mainly to the different initial z-positions of the particles.

After the pinched segment, the particles follow along the fluid streamlines, but as they reach the electrode array, the effects of the DEP force become apparent. A 4.8 µm particle is deflected from the horizontal direction of the fluid flow almost immediately as it encounters the first electrode and then is directed along it into Outlet 5. A 2.5µm particle takes longer to reach the top wall of the channel, and thus is not captured

Figure 9: Particle traces for each particle size with the flow field and electric potential solved numerically in COMSOL. Each particle size is seeded at 100 evenly distributed initial positions within the inlet channel. The red, green, and blue trajectories correspond to the 1.0, 2.5, and 4.8 µm particles, respectively. The spread in the outlet positions of the particles is mainly due to the different initial z-positions of the particles in the inlet channel. Note: Particles enlarged to show their sizes relative to one another.

by the DEP force until it passes over the second or third electrode, depending on its initial z-position. A 1.0 µm particle does not experience a DEP force strong enough to capture it completely, and thus is only deflected slightly even after passing over all four electrodes. These numerical results confirm what is seen in the experiments and illustrate how PFF and DEP combined can be used to sort particles by size.