# Interdisciplinary Applied Mathematics

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that of the fluid, the particles move toward regions of high electric field density. This is known as positive dielectrophoresis. In the case of the fluid being more polarizable than the particles, the particles move away from the high electric field density, which is known as negative dielectrophoresis (Cummings and Singh, 2000).

The time-averaged dielectrophoretic force is given by

Fdep = 2пг3етЩК (w)]V||£rms ||2,    (7.49)

where Erms is the rms electric field, r is the particle radius, em is the dielectric permittivity of the medium, ш is the electric field frequency, and Ж[К(ш)] indicates the real part of the Clausius-Mossotti factor (К(ш)), which is a measure of the effective polarizability of the particle, given by (Morgan et al., 1999)

К(ш)

(4 ~ 4)

(e* + 2e*m)

(7.50)

where    ep    and    e*m    are    the    complex    permittivities    of    the particle    and    the

medium, respectively. The complex permittivity is defined by

e

*

##### л/—Ia

ш

where e is the permittivity, and a is the conductivity of the dielectric medium.

Ignoring the Brownian motion, the buoyancy force, and the motion of the buffer solution, the equation of motion for a suspended particle can be written as

where Fd is the instantaneous drag force acting on the particle. For particle sizes smaller than 10 pm in buffer solutions with viscosity close to that of water, the Reynolds number based on the particle size is smaller than unity. Hence, the inertial effects on particle motion can be neglected. If we assume a dilute solution, so that particles do not interact and spherical particles with radius    r,    we    can    use    Stokes’s formula    for    the    drag    force:

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