Electrophoretic velocity equation

The movement of charged analytes is now considered as a consequence of the combination of their own individual electrophoretic mobilities (Equation [3.61]) and their participation in the bulk electro-osmotic flow (EOF). The net speed of motion of an analyte ion in the field direction (along the length of the capillary) is the vector sum of its electrophoretic velocity (Equation [3.61])... 

The 2eta potential (Fig. 8) is essentially the potential that can be measured at the surface of shear that forms if the sohd was to be moved relative to the surrounding ionic medium. Techniques for the measurement of the 2eta potentials of particles of various si2es are collectively known as electrokinetic potential measurement methods and include microelectrophoresis, streaming potential, sedimentation potential, and electro osmosis (19). A numerical value for 2eta potential from microelectrophoresis can be obtained to a first approximation from equation 2, where Tf = viscosity of the liquid, e = dielectric constant of the medium within the electrical double layer, = electrophoretic velocity, and E = electric field. 

FIG. 12.4 The domain within which most investigations of aqueous colloidal systems lie in terms of particle radii and 1 1 electrolyte concentration. The diagonal lines indicate the limits of the Hiickel and the Helmholtz-Smoluchowski equations. (Redrawn with permission from J. Th. G., Overbeek, Quantitative Interpretation of the Electrophoretic Velocity of Colloids. In Advances in Colloid Science, Vol. 3 (H. Mark and E. J. W. Verwey, Eds.), Wiley, New York, 1950.)... 

A simple expression for the electrophoretic velocity of a uniformly charged nonconducting particle is the Smoluchowski equation [1]... 

When an external electric field is applied, the equilibrium distributions of the ions will be distorted by both the imposed electric field and the induced motion of the particle. In general, one needs to solve the electrokinetic Eqs. (2)-(7) simultaneously by satisfying the above-mentioned boundary conditions to obtain the electrophoretic velocity of a colloidal particle. However, it is not an easy task to solve these coupled equations to arrive at a general expression for the particle mobility. In what follows, a number of approaches adopting various assumptions to simplify the governing equations will be presented and their corresponding results will be discussed. 

Electrophoresis of nonconducting colloidal particles has been reviewed in this chapter. One important parameter determining the electrophoretic velocity of a particle is the ratio of the double layer thickness to the particle dimension. This leads to Smoluchowski s equation and Huckel s prediction for the particle mobility at the two extrema of the ratio when deformation of the double layer is negligible. Distortion of the ion cloud arising from application of the external electric field becomes significant for high zeta potential. An opposite electric field is therefore induced in the deformed double layer so as to retard the particle s migration. 

The fundamental equations for the flow velocity of the liquid ii(r) at position r and that of the /th ionic species v,(r) are the same as those for the dilute case (Chapter 5) except that Eq. (5.10) applies to the region bb). The boundary conditions for u(r) and v,(r) are also the same as those for the dilute case, but we need additional boundary conditions to be satisfied at r = c. According to Kuwabara s cell model [4], we assume that at the outer surface of the unit cell (r = c) the liquid velocity is parallel to the electrophoretic velocity U of the particle,... 

There has been much controversy concerning the applicability of equation (22) to particles of different shapes according to Smoluchowski s treatment (1903) the equation for the electrophoretic velocity should be independent of the shape of the moving particle. On the other hand, Debye and Hiickel find that if the thickness of the double layer, i.e., 1/ic, is large in comparison with the radius of the particle, i.e., for small spherical particles, the velocity of electrophoresis is given by... 

The most common method for determining the zeta-potential is the microelectrophoretic procedure in which the movements of individual particles under the influence of a known electric field are followed microscopically. The zeta potential can be calculated from the electrophoretic velocity of the particles using the Helmholtz-Smoluchowski equation. 

Hiickel proposed an equation taking account of the influence of these factors upon the electrophoretic velocity of an ion considered as a sphere of radius r. 

Thus since the last term is zero for a closed cell by equation (8), the electrophoretic mobility is equal to the mean mobility uThe electrophoretic velocity may thus be obtained by making measurements at a series of levels, and obtaining a mean value from a graphical or other integration of the results with aid of equation 10. 

From Equation (5b) one can conclude that at equal electric field strength, the ratio of electrophoretic velocities at ionic strengths pi and p2 is equal... 

The action of external electric field on the free disperse system results in particle motion (electrophoresis). The electrophoretic velocity, vE, is not a function of ( -potential only, but also depends on the particle radius, r, and the type of electrolyte present in the system. However, it turns out (see fine print further down) that all of these factors can be simultaneously accounted for by the numerical coefficient, kt, introduced into the Helmholtz-Smoluchowski equation (V.26). If the particles are spherical, k, changes from 2/3 for particles smaller compared to the ionic atmosphere thickness (kt 1) to 1 for large particles ( kt 1). Consequently, the particle flux due to the applied electric... 

An indirect way to obtain information about the potential at foam lamella interfaces is by bubble electrophoresis, in which an electric field is applied to a sample causing charged bubbles to move toward an oppositely charged electrode. The electrophoretic mobility is the measured electrophoretic velocity divided by the electric field gradient at the location where the velocity was measured. These results can be interpreted in terms of the electric potential at the plane of shear, also known as the zeta potential, using well-known equations available in the literature (29—31). Because the exact location of the shear plane is generally not known, the zeta potential is usually taken to be approximately equal to the potential at the Stem plane (Figure 11) ... 

From the above equation and the formula relating potential and charge (Eq. 7.1.11a), the electrophoretic velocity is... 

This is just the Helmholtz-Smoluchowski equation, as might have been expected, since electrophoresis is just the complement of electroosmosis. Its derivation shows that the electrophoretic velocity of a nonconducting particle is independent of the particle size and shape for a constant surface potential when the Debye length is everywhere small compared with the characteristic body dimension. Note that Eq. (7.2.6) differs from the Huckel large Debye length result (Eq. 7.2.2) only by the factor. ... 

Assume that the resistance to the cylinder motion is due to the shear stress associated with the electroosmotic flow that is generated, so that the Navier-Stokes equation reduces to a balance between viscous and electrical forces. Show that the solution for the electrophoretic velocity of the cylinder is the same as that for a sphere of the same zero potential with the Debye length small. 

In 1809, Reuss observed the electrokinetic phenomena when a direct current (DC) was applied to a clay-water mixture. Water moved through the capillary toward the cathode under the electric field. When the electric potential was removed, the flow of water immediately stopped. In 1861, Quincke found that the electric potential difference across a membrane resulted from streaming potential. Helmholtz first treated electroosmotic phenomena analytically in 1879, and provided a mathematical basis. Smoluchowski (1914) later modified it to also apply to electrophoretic velocity, also known as the Helmholtz-Smoluchowski (H-S) theory. The H-S theory describes under an apphed electric potential the migration velocity of one phase of material dispersed in another phase. The electroosmotic velocity of a fluid of certain viscosity and dielectric constant through a surface-charged porous medium of zeta or electrokinetic potential (0, under an electric gradient, E, is given by the H-S equation as follows ... 

Kinetic capillary electrophoresis is defined as capillary electrophoretic separation of species that interact during electrophoresis. " Thus, KCE involves two major processes affinity interaction of L and T, described by Equation 11.1, and separation of L, T, and C based on differences in their electrophoretic velocities, vl, vy, and vc, respectively. These two processes are described by the following general system of partial differential equations ... 

To analyze dispersion in TGF, we must modify our previous derivation to account for changes due to the temperature gradient. This includes an axially varying electrophoretic velocity and diffusivity. In its most general form, the transport equation becomes... 

An analytical solution for u was derived by Huber and Santiago [62]. Using a decomposition on the Helmholtz-Smolukowski equation similar to that performed by Ross for the electrophoretic velocity, they determined the nonuniform electroosmotic slip velocity to be... 

The approximate experimental determination of xl), is based on measurement of the velocity of a charged particle in a solvent subjected to an applied voltage. Such a particle experiences an electrical force that initiates motion. Since a hydrodynamic frictional force acts on the particle as it moves, a steady state is reached, with the particle moving with a constant velocity U. To calculate this electrophoretic velocity U theoretically, it is, in general, necessary to solve Poisson s equation (Equation 3.19) and the governing equations for ion transport subject to the condition that the electric field is constant far away from the particle. The appropriate viscous drag on the particle can be calculated from the velocity field and the electrical force on the particle from the electrical potential distribution. The fact that the sum of the two is zero provides the electrophoretic velocity U. Actual solutions are complex, and the electrical properties of the particle (e.g., polarizability, conductivity, surface conductivity, etc.) come into play. Details are given by Levich (1962) (see also Problem 7.8). 

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