Biomedical Engineering Reference
In-Depth Information
then by comparison with Fick's law, we obtain the important result
k T
B
D
D
=
(5.11)
C
The second approach [1] is based on Langevin's formula (similar to Newton's
law, but with a complementary term for the Brownian motion)
�
�
d v
�
(5.12)
m
= -
C v F t
+
( )
D
d t
where
m
is the mass of the particle, and
F(t)
a randomly fluctuating force. By mul-
tiplying (5.12) by
x
and taking the time average — symbolized by the brackets — we
can rewrite (5.12) under the form
2
æ
ö
æ
ö
d
d x
d x
d x
m
<
x
> - <
m
> = -
C
<
x
> + <
xF t
( )
>
ç
÷
ç
÷
D
d t
d t
d t
d t
è
ø
è
ø
However,
<
xF t
( )
> = <
x
> <
F t
( )
> =
0
The isotropic distribution of the energy yields
2
æ
ö
1
d x
1
m
<
> =
k T
ç
÷
B
2
d t
2
è
ø
then the Langevin equation is reduced to a differential equation
d
æ
d x
ö
d x
(5.13)
m
<
x
> =
k T C
-
<
x
>
ç
÷
B
D
d t
è
d t
ø
d t
The solution of (5.13) for instances larger than
C
D
/
m
is [1]
k T
2
B
D
<
x
> =
2
t
=
2
Dt
C
and
k T
B
D
D
=
C
5.3.2.3 Anisotropic Media
In free space, diffusion is isotropic. However, in a confined space, diffusion may be
anisotropic if the media containing the fluid is anisotropic [4]. Anisotropic media
have different diffusion properties in different directions. Some common examples
are textile fibers, polymer films and laminated microlayers in which the molecules
have a preferential direction of orientation (Figure 5.4). It is also shown [5] that at
the very vicinity of a surface, diffusion becomes anisotropic.
