Environmental Engineering Reference
In-Depth Information
particles experiencing strong attractive DEP forces are trapped at electrode edges
against flow drag [
6
].
To determine the electric field and then the dielectrophoretic forces, the electric
potential is solved for a defined space and set of boundary conditions that represent
the electrode array.
Using the phasor notation, an arbitrary electric potential oscillating at frequency
ˉ
can be defined as [
8
,
36
]:
n
o
Re
V
x
e
jˉt
V
x
ðÞ
¼
;
t
ðÞ
;
ð
:
Þ
14
1
1)
1/2
, x is the spatial coordinate,
t
is the time, Re{} indicates the real
part of the complex quantity, and
V
where
j
¼
(
jV
I
, with
V
R
and
V
I
the real and
imaginary part of the electric potential, respectively. The corresponding electric
field is Ex
¼
V
R
þ
n
o
, where E
Re
e
Ex
¼
∇
V
e
jˉ t
. For an
uncharged homogeneous medium, the electrical potentials satisfy the Laplace
equation:
ðÞ
¼
;
t
ðÞ
¼
∇
ð
V
R
þ
jV
I
Þ
2
V
R
¼
2
V
I
¼
∇
0
and
∇
0
:
ð
14
:
2
Þ
In the dipole approximation, the dielectrophoretic force acting on a dielectric
particle in a nonuniform electric field can be written as [
36
]:
Þ
E
¼ p
F
ð
∇
;
ð
14
:
3
Þ
and the time-averaged force on the particle is:
n
o
1
2
Re
Þ
E
∗
e
hi
¼
F
ð
p
∇
;
ð
14
:
4
Þ
is the induced dipole moment of the particle and “
*
” indicates the
complex conjugate. For a homogeneous dielectric sphere of radius
a
, the induced
dipole moment is given by [
8
]:
where
e
p
ðÞ
ε
m
K
ðÞ
E
;
a
3
p
ðÞ
¼
4
ˀ
ð
14
:
5
Þ
where
K
is the complex Clausius-Mossotti (CM) factor, which, for a spherical
particle, can be expressed as [
37
]:
ðÞ
ðÞ
¼
ε
p
ε
m
ε
p
þ
K
ε
m
;
ð
14
:
6
Þ
2
with
ε
p
and
ε
m
the absolute complex permittivity of the particle and the medium,
respectively.
