Biomedical Engineering Reference
In-Depth Information
Hamaker constant (
A
water-water
= 3.51 × 10
-20
J) of some commonly used solute and solvent
could be found in literature [19, 22]. As a result of introduction of van der Waals interac-
aals interac-
tion, when the solute is very close to the pore wall, the interaction energy is blooming
up infinitely, which implies deposition of the solute on the pore wall. For nonabsorbing
solutes, it is necessary to define a region in which van der Waals interaction is present
(region I), and in the region that is very close to the pore wall, van der Waals interaction
is absent (region II). Typically, the region I extend from centerline to
z
=
r
- 0.05
a
, so that
partition coefficient is
Waals interac-
1 1 05
−
.
1
−
∫
λ
λ
C
C
E
kT
( )
β
∫
K
=
b
b
=
2
exp
−
d
+
2
d
(5.38)
β β
β β
p
0
1 1 0
−
.
5
λ
The effects of the electrostatic double-layer and the acid-base interaction between solute
and pore wall have been added to the external potential function [24,25]. Experimental
results [19] suggested that the model proposed by Bhattacharjee and Sharma [24] fitted for
protein solution (BSA) the best. In the same report [19], an empirical correction of Hamaker
constant dramatically improved the performance of Shao's model dramatically.
The last but not the least factor that controls diffusion coefficient of macromolecules
through fine pores is the solute concentration. It has been reported that the partition coef-
solute concentration. It has been reported that the partition coef-
ficient was shown to increase with bulk solution concentration. This could be explained
by the change in molecular size with concentration. For a dilute solution in a good sol-
the solute concentration. It has been reported that the partition coef-
dilute solution in a good sol-
vent, coiled molecules expand; however, as the concentration increases, the repulsion force
between solute molecules pushes the entangled molecule segments back. Therefore, the
molecule becomes smaller. At high concentrations, the dimensions approach their unper-
turbed values [26]. However, literature results show the inverse for some flexible solutes,
such as BSA [27] and polyethylene glycol (PEG) [28]. Pioneer work by Batchelorz [29], who
eliminated the solution constraint, offers a concentration dependent expression of the dif-
the change in molecular size with concentration. For a dilute solution in a good sol-
change in molecular size with concentration. For a dilute solution in a good sol-
a dilute solution in a good sol-
the dif-
dif-
fusion coefficient.
K
( )
d
d
∏
φ
π
(5.39)
D
( )
=
φ
6
a
n
where
→
is the volume fraction of the particles (
φ
=
C
NV
MW
), and
n
is the mean number den-
d
d
Π
n
sity of the particles,
is the derivative of chemical potential per particle, which serves
as the thermodynamic driving force for diffusion.
K
(
→
) is the sedimentation coefficient.
The first term of the above equation is in fact a concentration dependent friction dragging
coefficient
f
(
→
) similar to
f
in the Stokes-Einstein equation. For hard spheres in bulk solu-
tion, in which only two body interactions are considered, the effective diffusion coefficient
is given:
( )
D
D
φ
2
= +
1 1 45
.
+
(
)
(5.40)
φ
O
φ
0
D
0
is the diffusion coefficient at an infinite dilution. According to Equation 5.40, diffusion
coefficient increases with increases in concentrations. In the case of BSA, coefficient of
O
(
→
)
