Biomedical Engineering Reference
In-Depth Information
Figure 10.1
Distribution of charges around a spherical particle of radius
a
. The gray halo represents
the cloud of counterions.
This classical calculation starts by writing the Poisson equation that relates the
potential
y
to charge density
r
:
ρ
2
Ñ
ψ
= -
(10.1)
ε ε
0
r
where
e
0
is the vacuum permittivity (» 8.85·10
-12
C
2
J
-1
m
-1
) and
e
r
the relative per-
mitivity (» 80 for water).
At equilibrium,
r
is given by summing the number of charges in the solution.
The number
n
i
of ions of each species is given by a Boltzmann distribution:
n
=
n
0
exp(
-
z e
ψ
/
k T
)
(10.2)
i
i
i
B
z
i
is the valency of the ions, e is the elementary charge (1 eV = 1.6·10
-19
C),
k
B
the
Boltzmann constant (~ 1.38·10
-23
J.K
-1
) and
T
the temperature.
n
i0
is the number of
these ions far from the surface. The charge density is then given by:
å
ρ
=
n z e
=
n
0
(
-
ze
×
exp(
-
ze
ψ
/
k T
)
+
ze
×
exp(
ze
ψ
/
k T
))
(10.3)
i
i
B
B
i
in the case of symmetric electrolytes.
The Gouy-Chapman model combines (10.1) and (10.3), the boundary con-
ditions being given by the global electro-neutrality condition and by the surface
charge.
In a plane geometry, this equation can be solved analytically but the case
ey<<k
B
T
is instructive (Debye-Hückel approximation). The linearization of (10.3)
then leads to the classical expression:
ψ ψ
=
0
exp(
×
- ×
κ
x
(10.4)
