In principle, electrophoresis is used in laboratories to separate
macromolecules based on charge.[11] The technique normally applies a
negative charge so proteins move towards a positive charge called
anode. It is used extensively in
DNA,
RNA and
protein analysis.[12]
Suspended particles have an
electric surface charge, strongly affected by surface adsorbed species,[16] on which an external electric field exerts an
electrostaticCoulomb force. According to the
double layer theory, all surface charges in fluids are screened by a
diffuse layer of ions, which has the same absolute charge but opposite sign with respect to that of the surface charge. 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. This latter force is not actually applied to the particle, but to the
ions in the diffuse layer located at some distance from the particle surface, and part of it is transferred all the way to the particle surface through
viscousstress. This part of the force is also called electrophoretic retardation force, or ERF in short.
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]
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:
.
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:
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. Booth.[21] Modern, rigorous theories valid for any
Zeta potential and often any aκ stem mostly from Dukhin–Semenikhin theory.[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. White.[23]
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,[24] 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.[25] This approach has been validated the electrophoresis of particles.[25]
^Kastenholz B. (2006). "Comparison of the electrochemical behavior of the high molecular mass cadmium proteins in Arabidopsis thaliana and in vegetable plants on using preparative native continuous polyacrylamide gel electrophoresis (PNC-PAGE)". Electroanalysis. 18 (1): 103–6.
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10.1002/elan.200403344.
^Malhotra, P. (2023). Analytical Chemistry: Basic Techniques and Methods. Springer, ISBN 9783031267567. p. 346.
^Tiselius, Arne (1937). "A new apparatus for electrophoretic analysis of colloidal mixtures". Transactions of the Faraday Society. 33: 524–531.
doi:
10.1039/TF9373300524.
^Michov, B. (1995). Elektrophorese: Theorie und Praxis. De Gruyter, ISBN 9783110149944. p. 405.
^
Hanaor, D.A.H.; Michelazzi, M.; Leonelli, C.; Sorrell, C.C. (2012). "The effects of carboxylic acids on the aqueous dispersion and electrophoretic deposition of ZrO2". Journal of the European Ceramic Society. 32 (1): 235–244.
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^Hanaor, Dorian; Michelazzi, Marco; Veronesi, Paolo; Leonelli, Cristina; Romagnoli, Marcello; Sorrell, Charles (2011). "Anodic aqueous electrophoretic deposition of titanium dioxide using carboxylic acids as dispersing agents". Journal of the European Ceramic Society. 31 (6): 1041–1047.
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^von Smoluchowski, M. (1903). "Contribution à la théorie de l'endosmose électrique et de quelques phénomènes corrélatifs". Bull. Int. Acad. Sci. Cracovie. 184.
^Hückel, E. (1924). "Die kataphorese der kugel". Phys. Z. 25: 204.
^Overbeek, J.Th.G (1943). "Theory of electrophoresis — The relaxation effect". Koll. Bith.: 287.
^Dukhin, S.S. and Semenikhin N.V. "Theory of double layer polarization and its effect on electrophoresis", Koll.Zhur. USSR, volume 32, page 366, 1970.
^O'Brien, R.W.; L.R. White (1978). "Electrophoretic mobility of a spherical colloidal particle". J. Chem. Soc. Faraday Trans. 2 (74): 1607.
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10.1039/F29787401607.
^Paxton, Walter F.; Sen, Ayusman; Mallouk, Thomas E. (2005-11-04). "Motility of Catalytic Nanoparticles through Self-Generated Forces". Chemistry - A European Journal. 11 (22). Wiley: 6462–6470.
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