Biomedical Engineering Reference
In-Depth Information
For weak electric fields, the velocity V of the particle can then be considered as
linearly dependent on the electric field.
(10.13)
V
= ×
E
where m is defined as the electrophoretic mobility.
Depending on the salinity and the radius of the particle, two extreme cases can
be analytically considered.
If the charge density is high or for large particles, we have k a >>1 and we can
use the “Smoluchowski” calculation on plane surfaces similar to the one derived
for electro-osmosis (see Section 10.1.2) [4, 5]. In this case, m can be derived from
the equilibrium of forces in the double layer, assuming that the charge profile is only
marginally modified by the electric field and we get:
εζ
µ
=
(10.14)
η
In the framework of the Debye-Hückel approximation, plugging (10.7) in (10.14)
gives:
q
µ
=
(10.15)
2
4
πηκ
a
The net result of (10.14) is that the mobility depends only on the z potential of
the particle but not on its size nor on its shape. This result is a direct consequence
of the confinement of the flow within the Debye layer. The physical mechanism
underlying this motion is thus very different from the usual Stokes law where the
bead directly submitted to an external force such as gravity experiences a viscous
drag.
We have so far focused on large particles or small Debye length, we can go to
the other extreme case of a very small particle or a very small salinity (large Debye
length). We then have k a <<1 and the particle can be seen as a pointlike object sur-
rounded by a very diffuse cloud of counterions that extends many radii away from
it. In that case, the flow lines extend far from the object over a characteristic length
now close to the bead radius. Not surprisingly given the above discussion, this case
is then described by the more familiar Hückel calculation and the mobility is ac-
curately given by:
µ
=
q
/ 6
πη εζ η
a
=
2
/ 3
(10.16)
by using (10.7).
Between these limiting results, the behavior of the particles is more complex
and best described by numerical simulations [5]. Furthermore, assumptions of
the previous calculations are often caught out. For instance, the Smoluchowski
calculation assumes that the charge distribution around the particle is only mar-
ginally disturbed by its motion. However, this is generally not true, the cloud
of counterions is deformed by the electric field, this shape in turn modifies the
friction. The same becomes true if the velocity of the particle becomes very
high [5].
 
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