Biomedical Engineering Reference
In-Depth Information
Figure 2
. Schematic representation of electroosmosis (left) and electrophoresis (right)
Hence, the Debye length represents the 1/
e
decay distance for the potential as
well as the electric field at low potentials.
This potential can be added into the governing equation of fluid mechanics,
namely the Navier-Stokes equation, to calculate the flow produced by the elec-
troosmotic effect. Consider the geometry shown in Figure 2 where electroosmo-
tic flow is established in a long chamber of constant cross-section. Combining
the appropriate form of the Navier-Stokes equation with the potential distribu-
tion in Eq. [3], we get
F[
I
E
u
=
el
,
[4]
eof
where the component of the flow due to electroosmosis is denoted
u
eof
, the dy-
namic viscosity of the liquid is I, and [ is the zeta potential, or the potential at
the location of the shear plane just outside the Stern layer. This equation is
known as the
Helmholtz-Smoluchowski equation
and is accurate when the Debye
layer is thin relative to the channel dimension. Because of the typical low Rey-
nolds number behavior of electrokinetic flows, the velocity field can be directly
added to that obtained by imposing a pressure gradient on the flow to find the
combined result of the two forces. Obtaining solutions for the flow when the
Debye length is large generally requires resorting to numerical solutions because
the Debye-Hückel approximation is not valid when the Debye layer is an appre-
ciable fraction of the channel size.
For the purpose of comparing the effectiveness of several different elec-
troosmotic channel/solution combinations, the
electroosmotic mobility
, N
eo
is
defined as
u
E
N
=
eof
.
[5]
eo
el
