Biomedical Engineering Reference
In-Depth Information
2
x
Dt
-
c
¶
2
1
4
4
=
c
e
0
¶
x
π
Dt
With this in mind, the mass flux per surface unit, given by Fick's law, may be
written under the form
�
¶
c
2
1
D
ˆ
ˆ
J
= - Ñ
D c
= -
D
= -
D c
i
= -
c
i
(5.22)
0
0
wall
¶
x
π
t
π
4
Dt
x
=
0
where
i
is the unit vector perpendicular to the wall. This latter relation is called the
Ilkovic's solution to the diffusion problem. It shows that the mass flux is propor-
tional to the concentration far from the wall and to the square root of the diffusion
coefficient. Strictly, from (5.22) the initial mass flux is infinite. In fact, from a practi-
cal point of view, such a situation is not possible: there is always a transition time
during which the fluid with the concentration
c
0
is brought into contact with the
wall and the initial time for diffusion is always approximate.
5.3.5 Example of Diffusion Between Two Plates
In this section we show the limitation of the Ilkovic's solution for the problem of
diffusion between two plates. Suppose that we insert a small volume of liquid (
V
=
2
m
l) between two parallel horizontal glass plates (Figure 5.7) separated by a dis-
tance of 270
m
m. The liquid contains nanoparticles (hydrodynamic diameter
D
H
=
100 nm, diffusion coefficient
D
= 0.21
10
-11
m
2
/s) in concentration
c
0
.
At the beginning, the particles are uniformly dispersed in the liquid at rest; pro-
gressively the particles closer to the walls are immobilized by contact with the walls
under the action of the Brownian motion, and a concentration depletion progresses
from the walls towards the drop center.
It is possible to count the number of particles immobilized at any time on the
photographs of the upper plate taken under the microscope (Figure 5.8). It may
be shown that the particle size is so small that sedimentation can be completely
neglected and we assume that there is statistically the same number of particles im-
mobilized on the upper and lower plate.
Assuming that the drop is cylindrical (which is the equilibrium shape—see
Chapter 3), the diffusion process is governed by the two-dimensional axisymmetri-
cal equation
Figure 5.7
Schema of the drop and the two glass plates.
