Interdisciplinary Applied Mathematics

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7.5.4 Charged Particle in a Pipe

Electrophoretic motion of a spherical colloidal particle moving along the axis of a pipe has been studied by (Keh and Anderson, 1985) using the method of reflections. Assuming thin EDL on the particle and the pipe surfaces, the electrophoretic velocity of the particle is given up to the О (-^) terms as

uEP = еЕ^р~ ^ ^1 — 1.28987    + 1.89632     1.02780

where d and D are the sphere and pipe diameters, respectively. The zeta potentials on the particle and wall surfaces are shown by Zp and Zw, respectively. The minus sign in front of Zw characterizes the electroosmotic flow. For example, for a negatively charged surface, the EDL will be positively charged, and the electroosmotic motion will be toward the cathode. At the same time, a positively charged particle will also move toward the cathode (in the    electric    field    direction),    and    the    electroosmotic    flow    will    enhance

the particle migration speed. The equation above can be utilized to characterize blockage effects of the pipe on particle velocity. To a leading order, = 0.5 gives 16% variation in the electrophoretic velocity. This becomes important in nanoflow applications. For example, the nanotube-Coulter-counter (Ito et al., 2003), utilized to determine the size and surface charge of the nanoparticles, should require such corrections when the particle size becomes comparable to the nanotube diameter. Another case that may need further attention is the blockage effects under finite EDL thickness, where the EDL from the pipe surface and that from the sphere overlap.

7.6 Dielectrophoresis

Dielectrophoresis is the motion of polarizable particles that are suspended in an electrolyte and subjected to a spatially nonuniform electric field (Pohl, 1978). The particle motion is produced by the dipole moments induced on the particle and the suspending fluid due to the nonuniform electric field.    When    the    induced dipole    moment    on    the    particles    is    larger    than

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