Biomedical Engineering Reference
In-Depth Information
where:
2 2
z e n
k T
2
0
κ
=
2
(10.5)
ε ε
0
r B
k
-1
is an important length that describes the range of these electrostatic interactions:
it is known as the Debye length.
y
0
is the surface potential.
x
is the distance from
the surface.
Still in the framework of the Debye-Hückel approximation, the potential in the
case of spherical particles of radius
a
is [2]:
exp(
-
κ
(
x a
-
))
ψ ψ
=
a
(10.6)
0
x
Here,
x
is the distance from the center of the sphere.
It is good at this point to get an order of magnitude of the Debye length, which
obviously depends strongly on the salinity: for a monovalent ion (
z
= 1) at a 0.1M
concentration,
k
-1
~ 1 nm; if
c
~ 10
-7
M (which is the minimum that can be reached
in ultrapure water where the salinity is determined by the H
+
and OH
-
ions),
k
-1
~
1
m
m. At a distance from the surface larger than
k
-1
, the particles are neutral.
There are more elaborate models that describe this double layer more real-
istically. In particular, the Stern model considers a more complex profile of the
potential: a proximal zone where the counterions are immobilized and a diffuse
layer where the above discussion is valid again (Figure 10.2). It is then common to
introduce the
z
potential that gives the value of the potential at the point where the
shear around the particle becomes significant.
Practically,
z
is the only parameter that can be experimentally determined
either by electrophoresis for colloids or by other means: electro-osmosis for walls,
Figure 10.2
Profile of the double layer in the presence of a proximal immobile layer (Stern layer) for
a spherical particle. The potential takes the analytical form of (10.4) in the diffuse layer.
