Continuity equation electrophoresis

In order to illustrate the effects of media structure on diffusive transport, several simple cases will be given here. These cases are also of interest for comparison to the more complex theories developed more recently and will help in illustrating the effects of media on electrophoresis. Consider the media shown in Figure 18, where a two-phase system contains uniform pores imbedded in a matrix of nonporous material. Solution of the one-dimensional point species continuity equation for transport in the pore, i.e., a phase, for the case where the external boundaries are at fixed concentration, Ci and Cn, gives an expression for total average flux... 

Equation 1 is the Poisson equation. This equation should be solved in order to obtain the electric potential distribution in the computational domain. On the right hand of this equation, the term F Z] iZ,c, shows the gradient influence of the co-ions and counterions on the electric potential inside the domain. The electric field is the gradient of the electric potential (Eq. 2). Equation 3 is the Nemst-Planck equation, where the definition of ionic flux is given by Eq. 4. On the right-hand side of this equation, (m c,), (D, Vc,), and (z,/t,c,V( ) represent flow field (the electroosmosis), diffusion, and electric field (the electrophoresis), respectively, which contribute to the ionic mass transfer. The ionic concentrations of each species can be found by solving these two equations. Equations 5 and 6 are the Navier-Stokes and the continuity equations, respectively, which describe the velocity field and the pressure gradient in the computational domain. 

A simple environment, in which the separation process in electrophoresis takes place, allows easy formulation of basic transport laws that describe electromigration with good exactness and enables the separation process to be understood well. For example, the approximate continuity equations that describe electromigration of n strong ions in free solution are... 

Equation 6.1 is valid for a macroscopic particle moving in a continuous medium. In electrophoresis where the analyte ion moves in the media where particle size is comparable with that of the analyte size, this is definitely not the case. Also, analyte ions are not spherical and the term of the ionic radius, the value of which is difficult to estimate, becomes ambiguous. Thus, even in... 

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. 

This last point needs some elaboration. Consider in Figure 7.1.29(c), any one of tbe two ports used to withdraw two different protein products continuously. If we bave three proteins present in the feed stream, then let the liquid coming to one of the ports be pure in one protein it is clear that the liquid passing over the other port will have two other proteins therefore the liquid stream withdrawn from this port cannot be pure in one protein. This is an inherent limitation of processes where the bulk flow is parallel to the direction of the force. Here, even though we have one force perpendicular to the bulk flow, in the presence of the electrical force parallel to bulk flow, the system in continuous operation is just like the counter-current electrophoresis of Figure 6.3.4 and equations (6.3.9a,b). However, in a hatch mode, one can have multi-component separation since each species will be focused to its own zone and therefore can be withdrawn at a later time. Multicomponent separation in electrochromatography needs a different flow vs. force configuration, to he discussed further in Section 8.2. 

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