Biomedical Engineering Reference
In-Depth Information
Figure 5.6
Gaussian profiles of diffusion from a point source according to (5.19).
In the preceding example, one finds by using (5.20):
x
c
(
t
= 0.2) ~10
m
m;
x
c
(
t
=
1.0) ~20
m
m, and
x
c
(
t
= 10) ~60
m
m.
5.3.4 Semi-Infinite Space: Ilkovic's Solution
It is seldom that the diffusion equations (5.4) or (5.5) can be solved analytically.
There are some one-dimensional cases where an analytical solution may be found
(we have seen one in the preceding section); but usually, as soon as the geometry
of the diffusion problem is two-dimensional, or if the one-dimensional problem
presents complex boundary conditions, the use of a numerical approach is required
to solve the diffusion equation (5.4). We expose here the analytical solution of the
diffusion equation in the simple case of diffusion of species in a half space.
Suppose a half space with an initial concentration
c
0
. Suppose also that any
microparticle or macromolecule that contacts the solid wall limiting the half-space
domain is immediately immobilized. Then the concentration at the wall is zero at
any time.
The solution for the concentration equation is then
æ
ö
x
c
=
c erf
(5.21)
ç
÷
0
4
Dt
è
ø
where
x
is the distance from the wall, and the error function
erf
is defined by
x
2
2
u
ò
-
erf x
( )
=
e
du
π
0
This function has the characteristic values:
erf
(0) = 0
et
erf
(¥) = 1 and its
d erf
2
2
x
derivative is
-
. Thus, the derivative of (5.21) is
=
e
d x
π
