Electrophoresis

History

Theory

Suspended particles have an electric surface charge, strongly affected by surface adsorbed species, 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: : F_{\text{tot}} = 0 = F_{\text{el}} + F_{\mathrm{f}} + F_{\text{ret}}

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:

:\mu_e = {v \over E}.

The most well known and widely used theory of electrophoresis was developed in 1903 by Marian Smoluchowski:

:\mu_e = \frac{\varepsilon_r\varepsilon_0\zeta}{\eta},

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).

The Smoluchowski theory is very powerful because it works for dispersed particles of any shape at any concentration. It has limitations on its validity. For instance, it does not include Debye length κ−1 (units m). However, Debye length must be important for electrophoresis, as follows immediately from Figure 2, "Illustration of electrophoresis retardation". Increasing thickness of the double layer (DL) leads to removing the point of retardation force further from the particle surface. The thicker the DL, the smaller the retardation force must be.

Detailed theoretical analysis proved that the Smoluchowski theory is valid only for sufficiently thin DL, when particle radius a is much greater than the Debye length:

: a \kappa \gg 1.

This model of "thin double layer" offers tremendous simplifications not only for electrophoresis theory but for many other electrokinetic theories. This model is valid for most aqueous systems, where the Debye length is usually only a few nanometers. It only breaks for nano-colloids in solution with ionic strength close to water.

The Smoluchowski theory also neglects the contributions from surface conductivity. This is expressed in modern theory as condition of small Dukhin number:

: Du \ll 1

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:

: a \kappa .

Under this condition of a "thick double layer", Erich Hückel predicted the following relation for electrophoretic mobility:

:\mu_e = \frac{2\varepsilon_r\varepsilon_0\zeta}{3\eta}.

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 and F. Booth. Modern, rigorous theories valid for any Zeta potential and often any aκ stem mostly from Dukhin–Semenikhin theory.

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

For modeling more complex scenarios, these simplifications become inaccurate, and the electric field must be modeled spatially, tracking its magnitude and direction. Poisson's equation can be used to model this spatially-varying electric field. Its influence on fluid flow can be modeled with the Stokes law, while transport of different ions can be modeled using the Nernst–Planck equation. This combined approach is referred to as the Poisson-Nernst-Planck-Stokes equations. It has been validated for the electrophoresis of particles.

References

References

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