Environmental Engineering Reference
In-Depth Information
After some mathematical manipulation, it can be shown that the force in
Eq. (
14.4
) comprises two independent contributions (the subscripts
R
and
I
denote the real and imaginary parts of the corresponding complex quantities):
ˀε
m
a
3
K
R
∇
E
2
ˀε
m
a
3
K
I
∇
hi
¼
F
2
4
ð
E
R
E
I
Þ:
ð
14
:
7
Þ
The first term relates to an electric field which is nonuniform in magnitude but does
not exhibit any phase variation. Equation (
14.7
) suggests that dielectrophoresis can
be used as an effective method for separating particles, solely according to their
dielectric properties and sizes, an important step forward in nanoparticle manipu-
lation. In a dielectric medium, the direction of the DEP force is influenced by the
polarizability of the particle, which depends on the permittivities of the particle and
the suspending medium. Replacing the complex permittivity,
ε
¼
ε
j
˃=ˉ
, where
ε
the conductivity of the dielectric, in
Eq. (
14.6
), the CM factor can be expressed as [
15
,
38
]:
represents the electric permittivity and
˃
þ
ðÞ
¼
ε
p
ε
m
j
=ˉ ˃
p
˃
m
K
þ
:
ð
14
:
8
Þ
ε
p
þ
2
ε
m
j
=ˉ ˃
p
þ
2
˃
m
The variation in this factor is unique for a particular type of particle. According to
Eq. (
14.8
) the CM factor depends on the dielectric properties of the particle and
medium, and on the frequency of the applied field; at low frequencies the value and
the sign are determined by the electrical conductivities of the particle and the
medium and at higher frequencies by the permittivities. The high and low frequency
limits for the real part of the CM factor,
K
R
, are:
ˉ!
0
K
R
¼
˃
p
˃
m
lim
ð
14
:
9
Þ
˃
p
þ
˃
m
2
K
R
¼
ε
p
ε
m
ε
p
þ
lim
ˉ!1
ð
14
:
10
Þ
2
ε
m
Equations (
14.9
) and (
14.10
) show that the relative differences in ohmic losses
dominate the low frequency behavior of, while dielectric polarization effects are
more significant at high frequencies. These equations also show that
K
R
is bounded
(
1) regardless of frequency.
This behavior of
K
R
is crucial for the dielectrophoretic control of particles
because this factor determines the direction of the dielectrophoretic force: when
the sign of
K
R
is positive, the particle is more polarizable than its surrounding
medium and its movement is oriented towards regions with the highest field
strength. When
K
R
is negative, particles with polarizability less than that of the
medium move towards the region with the lowest field gradient. These phenomena
are known as
positive dielectrophoresis
(pDEP) and
negative dielectrophoresis
(nDEP), respectively. As
K
R
(
0.5
<
K
R
<
) has a complex dependence on the properties of
the particle (permittivity, conductivity) and the frequency of the applied field, a
ˉ
