Biomedical Engineering Reference
In-Depth Information
sedimentation potential, or streaming potential (the electric potential developed by
the fall of the objects or by flowing water on them) for heavy or tormented particles
[3]. Note at this point that the simple Debye-Hückel description assumes that the
z
potential is located strictly on the surface of the particle (
z
=
y
0
). For a spherical
particle (10.6), the total charge q is then given by:
¥
2
ò
q
=
4
π ρ
r dr
=
4
πε ε
a
(1
+
κ ζ
a
)
(10.7)
0
r
a
Equation (10.7) gives a relationship between the surface charge density
s
and
z
:
σ ε ε ζ κ
=
(1
+
a a
) /
(10.8)
0
r
What are then the consequences of an electric field E on a charged particle? Al-
though it is tempting to express the resulting force acting on the particle in the form
F=qE, this is generally too simplistic an approach. Given the complex distribution
of charges described above, this is not so surprising.
Indeed, the electric field acts primarily on the charges of the double layer. For
simplicity, we will first describe the consequences of an electric field on an immobile
surface (electro-osmosis) before treating electrophoresis per se.
10.1.2 Electro-Osmosis
Let us consider a charged surface in an aqueous solution. The charge of the sur-
face is balanced with the Debye layer of counterions extending over a distance
k
-1
(10.4) and (10.5). An electric field E parallel to this surface, acts on this excess of
charges that move the fluid with them at a velocity
v(z).
We can then write the Na-
vier-Stokes equation (see Chapter 1):
2
ρ η
¶
v
E
+
=
0
(10.9)
2
¶
z
r
and
h
are respectively the charge density and the viscosity of the fluid. The other
equation is the Poisson equation (10.1) that gives
r
as a function of the electric
potential
y
.
2
ρ ε ψ
= - Ñ
l
e
l
=
e
0
.
e
r
is the fluid permittivity.
substituting into (10.9) gives
2
2
¶
ψ
¶
v
E
ε
=
η
(10.10)
l
2
2
¶
z
¶
z
On the surface, the usual boundary conditions apply: the velocity on the surface
is 0 (no slippage) and the electric potential velocity is
y
=
z
. Far from the surface,
¶
ψ
¶
v
.
=
=
0
¶
z
¶
z
