Biomedical Engineering Reference
In-Depth Information
Equation (5.4) is sometimes called Fick's second law [3]. In every subdomain
where
D
is constant (does not depend on the spatial coordinates), (5.4) may be
rewritten as
¶
c
=
D c
D +
S
(5.5)
¶
t
showing that the diffusion equation of a parabolic nature. Equation (5.5) is a dif-
ferential equation with a solution that describes the concentration of a system as
a function of time and position. The solution depends on the boundary conditions
of the problem as well as on the parameter
D
. If the concentration
c
in diffusing
particles or molecules is small—which is usually the case—the diffusion coefficient
D
does not depend on
c
and (5.5) is linear. Note that the magnitude of
D
is 10
-9
m
2
/s for self-diffusion (diffusion of the molecules of the buffer fluid) and typically
10
-11
m
2
/s for colloidal substances.
5.3.2.2 Diffusion Coefficient
An expression of the diffusion coefficient of a particle in a carrier fluid was first
obtained by Einstein. This expression may be derived by two different approaches.
The first one is based on thermodynamics: the starting point is the Gibbs free energy
[3]. The magnitude of the driving force of diffusion is
1
¶
µ
F
= -
x
(5.6)
diffusion
N
¶
A
where
m
is the chemical potential of the of the diffusing species and
N
A
the Avoga-
dro number. Thermodynamics show that
µ µ
=
+
RT
ln(
γ
c
)
(5.7)
0
where
g
is the activity coefficient. For dilute systems,
g
= 1 and we obtain, after
substitution of (5.7) in (5.6)
k T c
¶
B
F
= -
x
(5.8)
diffusion
c
¶
Under stationary state conditions, the diffusion force is balanced by the viscous
resistance
k T c
¶
B
F
= -
=
F
=
C v
(5.9)
diffusion
friction
D
c
¶
x
where
C
D
is the friction factor and
v
the stationary velocity. Thus
k T c
¶
B
D
v
= -
C c x
(5.10)
¶
If we remark that the flux of material through a cross section is
�
�
J
=
cv