Electrophoresis

Also, the liquid–liquid system, where there is an interplay between the hydrodynamic and electrokinetic forces in both phases, adds to the complexity of electrophoretic motion.

The electric field also exerts a force on the ions in the diffuse layer which has direction opposite to that acting on the surface charge.

When the electric field is applied and the charged particle to be analyzed is at steady movement through the diffuse layer, the total resulting force is zero: Considering the drag on the moving particles due to the viscosity of the dispersant, in the case of low Reynolds number and moderate electric field strength E, the drift velocity of a dispersed particle v is simply proportional to the applied field, which leaves the electrophoretic mobility μe defined as:[17] The most well known and widely used theory of electrophoresis was developed in 1903 by Marian Smoluchowski:[18] where εr is the dielectric constant of the dispersion medium, ε0 is the permittivity of free space (C2 N−1 m−2), η is dynamic viscosity of the dispersion medium (Pa s), and ζ is zeta potential (i.e., the electrokinetic potential of the slipping plane in the double layer, units mV or V).

Increasing thickness of the double layer (DL) leads to removing the point of retardation force further from the particle surface.

This is expressed in modern theory as condition of small Dukhin number: In the effort of expanding the range of validity of electrophoretic theories, the opposite asymptotic case was considered, when Debye length is larger than particle radius: Under this condition of a "thick double layer", Erich Hückel[19] predicted the following relation for electrophoretic mobility: This model can be useful for some nanoparticles and non-polar fluids, where Debye length is much larger than in the usual cases.

There are several analytical theories that incorporate surface conductivity and eliminate the restriction of a small Dukhin number, pioneered by Theodoor Overbeek[20] and F.

[22] In the thin double layer limit, these theories confirm the numerical solution to the problem provided by Richard W. O'Brien and Lee R.