Biomedical Engineering Reference
In-Depth Information
(2.168)
where
D
is the effective diffusion coefficient. According to
(2.40)
, the diffusion coefficient of the ion
with
V$
i
¼
0
;
j
z
þ
j¼j
z
j¼
1 is:
2
D
D
þ
D
¼
(2.169)
D
þ
þ
D
where
D
are the diffusion coefficients of the cations and anions, respectively. Since the
convection current is zero (
i
C
¼
and
D
þ
r
e
E
el
) due to electroneutrality, the electrical current in the electrolyte
is caused by electro-migration and diffusion (
i
O
>>
i
D
):
i ¼ i
O
þ i
D
¼ F
2
ðm
þ
þ m
ÞcE
el
FðD
þ
D
ÞVc;
(2.170)
where
E
is the electric field strength. Diffusive current
i
D
is usually much smaller than the electro-
migration current
i
O
; thus,
i
x
i
O
¼
s
el
E
el
. The conservation of ionic species and currents can be
formulated for the conductivity as:
D
s
D
t
¼ DV
2
s:
(2.171)
V$ð
s
el
E
el
Þ¼
0
:
(2.172)
With Gauss's law
Vð
3
E
el
Þ¼
r
el
;
the current continuity equation can be written as:
r
el
¼
3
E
el
$V
s
=
s
:
(2.173)
where
3
is the permittivity of the liquid and is assumed to be uniform.
2.6.2
Electrophoresis
Electrophoresis is the motion of a charged particle relative to the surrounding liquid in an electric field
(
Fig. 2.34
). Because of the small size and low Reynolds number involved, the Stokes model can be
assumed for the motion of the particle:
q
surf
E
el
¼
6
pu
ep
r
p
;
(2.174)
FIGURE 2.34
Electrophoretic motion of a positively charged sphere.
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