Spheres, electrophoretic

Electrophoresis of bubbles and drops is a story on its own. As long ago as 1861 Quincke ) observed the electrophoresis of small air bubbles in water. Such a motion is possible only when there is a double layer at the Interface, containing free ions. It is extremely difficult to keep oil-water or air-water Interfaces rigorously free from adsorbed ionic species. When these are present, especially for surfactants, Marangoni effects make the surface virtually inexten-slble then the drops or bubbles behave as solid spheres. Electrophoretic studies... 

Locke, BR Arce, P, Modeling Electrophoretic Transport of Polyelectrolytes in Beds of Non-porous Spheres, Separation Technology 3, 111, 1993. 

Lumpkin, O, Electrophoretic Mobility of a Porous Sphere Through Gel-Like Obstacles Hy-drodnamic Interactions, Journal of Chemical Physics 81, 5201, 1984. 

Radko, SP Chrambach, A, Electrophoretic Migration of Submicron Polystyrene Latex Spheres in Solutions of Linear Polyacrylamide, Macromolecules 32, 2617, 1999. 

Outer-sphere adsorption of Pb(ll)EDTA on goethite. Geochim. Cosmochim. Acta 63(19/20) 2957-2969 Bargar, J.R. Reitmeyer, R. Davis, J.A. (1999) Spectroscopic confirmation of uranium(Vl)-carbonato adsorption complexes on hematite. Environ. Sci. Techn. 33 2481-2483 Bargar, J.R. Reitmeyer, R. Lenhart, J.J. Davis, J.A. (2000) Characterization of U(Vl)-car-bonato ternary complexes on hematite EX-AFS and electrophoretic mobility measurements. Geochim. Cosmochim. Acta 64 ... 

It is apparent from the above sections that the understanding of electrophoretic mobility involves both the phenomena of fluid flow as discussed in Chapter 4 and the double-layer potential as discussed in Chapter 11. In both places we see that theoretical results are dependent on the geometry chosen to describe the boundary conditions of the system under consideration. This continues to be true in discussing electrophoresis, for which these two topics are combined. As was the case in Chapters 4 and 11, solutions to the various differential equations that arise are possible only for rather simple geometries, of which the sphere is preeminent. 

Henry187 derived a general electrophoretic equation for conducting and non-conducting spheres which takes the form... 

According to Booth and Henry188, the equation relating electrophoretic mobility with zeta potential for non-conducting spheres with large kq when corrected for surface conductance takes the form... 

The resulting force on the particle can be obtained by integrating the stress over the particle surface and expressed in terms of the particle velocity. Because the surface encloses a neutral body (particle s charge plus the ion charge in the double layer), the total force must be zero. Hence, the electrophoretic velocity of the sphere can be obtained as given by Eq. (1). Morrison [11] and Teubner [12] proved that the above expression is valid... 

When the applied electric field is not spatially homogeneous, the translational electrophoretic velocity of a sphere has a similar expression [8],... 

Analytical expression for the electrophoretic velocity of a sphere can be obtained for a thin but distorted double layer. Dukhin [6] first examined the effect of distortion of thin ion cloud on the electrophoresis of a sphere in a symmetric two-species electrolyte. Dukhin s approach was later simplified and extended by O Brien [7] for the case of a general electrolyte and a particle of arbitrary shape. Since 0(k 1) double layer thickness is much smaller than the characteristic particle size L, the ion cloud can be approximated as a structure composed of a charged plane interface and an adjacent diffuse cloud of ions. Within the double layer, the length scales for variation of quantities along the normal and tangential directions are k ] and L, respectively. 

O Brien s method was extended to study the electrophoresis of a nonuniformly charged sphere with thin but polarized ion cloud in a symmetric electrolyte [32]. The electrophoretic mobility depends on the charge distribution at the particle surface. It is found that the polarization effect of the ion could leads to different electrophoretic mobilities for particles with different zeta potential distributions but having an identical velocity for the limit of infinite Ka. This intriguing result is due to the fact that the theory for undistorted ion cloud is linear in the distribution of zeta potential, whereas the polarization effects are nonlinear. 

The electrophoretic velocity of the sphere is expressed in a power series of k up to 0(k6). The results for various boundary cases are given as follows ... 

FIG. 1. Dimensionless electrophoretic velocity of a sphere U/Uq vs. distance parameter A (1) parallel to a dielectric wall, (2) along the centerline between two parallel plates, (3) normal to a conducting plane, (4) along the axis of a cylindrical cylinder. 

Figure 3 presents the variation of the electrophoretic mobility with the dimensionless distance X when a sphere moves perpendicularly towards a conducting plane. The solid line represents the results from the bipolar coordinate method and the dash curve is the approximate results from the reflection method. A good agreement between the results from the both methods is attained. The electrophoretic velocity of the sphere decreases monotonically with increasing X and is expected to vanish as the particle... 

FIG. 3. Dimensionless electrophoretic velocity U/Uq of a sphere vs. distance parameter A for particle s motion normal to a conducting plane. Solid line is the exact numerical results and dashed line represents the approximate results from the method of reflections. 

This result is in good agreement with those for electrophoretic migration of a sphere in a cylindrical pore [8,42]. 

Figure 5 presents the variation of the electrophoretic velocities of the cylinder with X. It is shown that the translational velocity normal to the plane decreases with increasing X, whereas the parallel mobility increases with X. In addition to the translation, the cylinder rotates when the external electric field is applied parallel to the wall. The enhancement of the parallel migration results from the squeezed electric field lines in the small gap between the particle and wall surfaces. This velocity enhancement also occurs for the electrophoresis of a sphere parallel to a plane boundary when the gap width is sufficiently small [40]. The boundary effects on electrophoresis are stronger for a cylinder than for a sphere. 

It can be seen from the above expressions that no particle interaction and electrophoretic rotation arise when the two spheres have an identical zeta potential. In other words, each particle migrates at the same velocity as that of an isolated particle. For two spheres with different zeta potentials, the particle interaction and electrophoretic rotation do occur and the leading order components are proportional to X3, where X = a - -ai)lru- The interaction results from three effects ... 

Electrophoretic interactions between spherical particles with infinitely thin double layers can also be examined using the boundary collocation technique [16,54]. This method enables one not only to calculate the interactions among more than two particles, but also to deal with the case of particles in contact, for which the bispherical coordinate solution becomes singular. Analogous to the result for a pair of spheres, no interaction arises among the particles in electrophoresis as long as all the particles have an equal zeta potential. This important result is also confirmed by a potential-flow reasoning [10,55]. 

When the gap width between two particles becomes very small, numerical calculations involved in both the bispherical coordinate method and the boundary collocation technique are computationally intensive because the number of terms in the series required to be retained to achieve a desired accuracy increases tremendously. To solve this near-contact motion more effectively and accurately, Loewenberg and Davis [43] developed a lubrication solution for the electrophoretic motion of two spherical particles in near contact along their line of centers with the assumption of infinitely thin ion cloud. The axisymmetric motion of the two particles in near contact can be approximated as the pairwise motion of the spheres in point contact plus a deviation stemming from their relative motion caused by the contact force. The lubrication results agree very well with those obtained from the collocation method. It is shown that near contact electrophoretic interparticle... 

FIG. 6. Electrophoretic velocities of two spheres along their line of centers at various C2/C1 when a.2la =1 (a) dimensionless velocity of sphere 1, (b) dimensionless velocity of sphere 2. 

To evaluate the volume integrals in (84), the radial distribution function must be known. The pair distribution function affected by the Brownian motion and the relative electrophoretic velocity between a pair of particles is generally nonuniform and nonisotropic. When the particles are sufficiently small so that Brownian motion dominates, one can use a simple distribution function based on hard-sphere potential... 

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