Electrophoresis is used inlaboratories to separatemacromolecules based on their charges. The technique normally applies a negative charge calledcathode so anionic protein molecules move towards a positive charge calledanode.[3] Therefore, electrophoresis of positively charged particles or molecules (cations) is sometimes calledcataphoresis, while electrophoresis of negatively charged particles or molecules (anions) is sometimes calledanaphoresis.[4][5][6][7][8][9][10]
Liquid "droplet electrophoresis" is significantly different from the classic "particle electrophoresis" because of droplet characteristics such as a mobile surface charge and the nonrigidity of the interface. Also, the liquid–liquid system, where there is an interplay between thehydrodynamic andelectrokinetic forces in both phases, adds to the complexity of electrophoretic motion.[13]
Suspended particles have anelectric surface charge, strongly affected by surface adsorbed species,[17] on which an external electric field exerts anelectrostaticCoulomb force. According to thedouble layer theory, all surface charges in fluids are screened by adiffuse layer of ions, which has the same absolute charge but opposite sign with respect to that of the surface charge. Theelectric field also exerts a force on the ions in the diffuse layer which has direction opposite to that acting on thesurface charge. This latter force is not actually applied to the particle, but to theions 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 throughviscousstress. 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 thedrag on the moving particles due to theviscosity of the dispersant, in the case of lowReynolds number and moderate electric field strengthE, thedrift velocity of a dispersed particlev is simply proportional to the applied field, which leaves the electrophoreticmobility μe defined as:[18]
The most well known and widely used theory of electrophoresis was developed in 1903 byMarian Smoluchowski:[19]
The Smoluchowski theory is very powerful because it works fordispersed particles of anyshape at anyconcentration. It has limitations on its validity. For instance, it does not includeDebye 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 radiusa 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 mostaqueous systems, where the Debye length is usually only a fewnanometers. It only breaks for nano-colloids in solution withionic strength close to water.
The Smoluchowski theory also neglects the contributions fromsurface conductivity. This is expressed in modern theory as condition of smallDukhin 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[20] predicted the following relation for electrophoretic mobility:
.
This model can be useful for somenanoparticles and non-polar fluids, where Debye length is much larger than in the usual cases.
There are several analytical theories that incorporatesurface conductivity and eliminate the restriction of a small Dukhin number, pioneered byTheodoor Overbeek[21] and F. Booth.[22] Modern, rigorous theories valid for anyZeta potential and often anyaκ stem mostly from Dukhin–Semenikhin theory.[23]
In thethin double layer limit, these theories confirm the numerical solution to the problem provided by Richard W. O'Brien and Lee R. White.[24]
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 theStokes law,[25] while transport of different ions can be modeled using theNernst–Planck equation. This combined approach is referred to as the Poisson-Nernst-Planck-Stokes equations. It has been validated for the electrophoresis of particles.[26]
^Kastenholz, B. (2006). "Comparison of the electrochemical behavior of the high molecular mass cadmium proteins inArabidopsis thaliana and in vegetable plants on using preparative native continuous polyacrylamide gel electrophoresis (PNC-PAGE)".Electroanalysis.18 (1):103–6.doi:10.1002/elan.200403344.
^Lyklema, J. (1995).Fundamentals of Interface and Colloid Science. Vol. 2. p. 3.208.
^Hunter, R.J. (1989).Foundations of Colloid Science. Oxford University Press.
^Dukhin, S.S.; Derjaguin, B.V. (1974).Electrokinetic Phenomena. J. Wiley and Sons.
^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.arXiv:1303.2754.doi:10.1016/j.jeurceramsoc.2011.08.015.S2CID98812224.
^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.arXiv:1303.2742.doi:10.1016/j.jeurceramsoc.2010.12.017.S2CID98781292.
^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.
Barz, D.P.J.; P. Ehrhard (2005). "Model and Verification of Electrokinetic Flow and Transport in a Micro-Electrophoresis Device".Lab Chip.5 (9):949–958.doi:10.1039/b503696h.PMID16100579.
Jahn, G.C.; D.W. Hall; S.G. Zam (1986). "A Comparison of the Life Cycles of TwoAmblyospora (Microspora:Amblyosporidae) in the MosquitoesCulex salinarius andCulex tarsalis Coquillett".J. Florida Anti-Mosquito Assoc.57:24–27.
