Biomedical Engineering Reference
In-Depth Information
Integrate Equation 5.33 over the total pore length
L
:
(5.34)
J L
⋅ = −
K K D C
(
−
C
)
p
d
L
0
Comparing Equation 5.34 with Fick's first law, the flux equation obtained here is equivalent to
Fick's first law when the concentration gradient is linear. It is worth noting that Equation 5.34
was derived based on a single straight pore, with number of pores per unit area and channel
orientation in consideration, and ignore any long-range interaction (
E
(
β
) = 0). Equations 5.34
and 5.23 are in fact identical, and the coefficients in both equations share the same physical
meaning.
To estimate the effective diffusion coefficient in porous membranes, an expression of
K
d
is required. Different hydrodynamic dragging factors have been presented by Deen
[11]. An exact expression of the hydrodynamic dragging factor was given by Brenner and
Gaydos [17]:
9
8
9
4
2
2
K
d
= +
1
ln
+
0 461
.
+
ln
+
O
(
)
(5.35)
λ λ
λ
λ
λ
λ
average hydro-
dynamic dragging factor across the channel is assumed to be equal to the hydrodynamic
dragging factor along the centerline (centerline approximation). The centerline approxi-
Equation 5.15 was proved to be a special case of Equation 5.35 when the average hydro-
5.15 was proved to be a special case of Equation 5.35 when the average hydro-
was proved to be a special case of Equation 5.35 when the average hydro-
Equation 5.35 when the average hydro-
5.35 when the average hydro-
when the average hydro-
along the centerline (centerline approximation). The centerline approxi-
mation is a reasonable assumption because the viscosity of the solution is constant until
a distance from the wall of one solvent molecule (water molecule size = 2 Å) [19]. The
minimum distance of a solute to the pore side wall of the pore is defi ned by Bohor repul-
is deined by Bohor repul-
sion distance = 1.54 Å, which means the viscosity of the solvent in which a solute could
be found is equal to the viscosity in the centerline. Hence, the most sensitive parameter to
external potential energy is the partition coefficient (
K
p
).
At low solute concentration, the pore model presented above works well for neutral sol-
utes without interacting with the membrane. However, in practice, the interaction between
solute and membrane could be prominent, and more sophisticated models concerning the
solute-membrane interaction (
E
(
β
) ≠ 0) were recently summarized briefly by Ladero et al.
[19]. van der Waals interaction between solute and pore wall was formulated by Hamker
[20], and van der Waals interaction for different geometries were given by Tadmor [21].
Shao and Baltus [22] incorporated the van der Waals interaction energy with the pore
model, such that
the pore side wall of the pore is defi ned by Bohor repul-
pore side wall of the pore is defi ned by Bohor repul-
of the pore is defi ned by Bohor repul-
A
1
1
− −
− +
λ
β λ
λ
β λ
β λ
β λ
E
( )
=
+
+
ln
(5.36)
β
6
1
− −
1
− −
where
A
is the Hamaker constant (J), which varies from case to case. The Hamaker con-
stant
A
312
for the interaction between solutes 2 and 3 in medium 1 was given in terms of
the interaction between molecules of the same type separated by vacuum (
A
11
,
A
22
,
A
33
)
[23]:
(
)
(
)
0 5
.
0 5
.
0 5
.
0 5
.
A
=
A
=
A
−
A
A
−
A
(5.37)
312
33
11
22
11
