Biomedical Engineering Reference
In-Depth Information
Combining
(2.127)
and
(2.130)
results in the Poisson-Boltzmann equation:
sinh
ze
d
2
J
d
y
2
¼
2
zen
N
3
J
k
B
T
:
(2.131)
Under conditions such as a large characteristic length compared to the double layer thickness or
a high ion concentration in the electrolyte and small zeta potential relative to 25 mV, the right-hand
side of
(2.131)
can be linearized by the relation sinh(
x
)
¼
x
:
d
2
d
y
2
¼
l
D
;
J
(2.132)
where
l
D
is the double layer thickness, which is called the Debye length:
r
3k
B
T
2
z
2
e
2
n
N
l
D
¼
:
(2.133)
Solving
(2.132)
results in the potential distribution:
J ¼ J
wall
exp
y
l
D
:
(2.134)
2.6.1.2 Electroosmotic transport effect
The continuum models using mass and energy conservation equations can be used for describing the
electroosmotic transport effects. The conservation of momentum needs to consider the electrostatic
force created by the electric field:
r
Dv
2
v
2
v
2
Dt
¼V
p
þ
m
V
þ
r
el
E
el
¼V
p
þ
m
V
þ
3E
el
V
J:
(2.135)
If there is no pressure gradient applied to the flow,
(2.135)
has the one-dimensional form:
m
d
2
u
eo
d
y
2
¼ 3E
el
d
2
J
d
y
2
:
(2.136)
With the assumption of a thin Debye length compared to the channel diameter, the electrokinetic
velocity is:
3E
el
z
m
:
u
eo
¼
(2.137)
The velocity
u
eo
is also called the Smoluchowski velocity. If the Debye length is negligible
compared to other channel dimensions, the electrokinetic flow can be modeled with slip boundary
condition, where the slip velocity is the Smoluchowski velocity (
Fig. 2.30
).
With a constant viscosity
m
and a constant zeta potential
z
, the electroosmotic velocity is
proportional to the electric field strength
E
el
. The negative sign shows that the flow direction is
opposite to the field direction. The proportional factor is called the electroosmotic mobility:
u
eo
E
el
¼
3z
m
:
m
eo
¼
(2.138)
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