Electrokinetic equation

The electrokinetic equation for this simple electrode reaction can be written as... 

Mohilner and Delahay [126] derived the electrokinetic equations for specific adsorption of reactant or product... 

For the hydrogen electrode reaction (HER) on different metals, it is predicted that the curve log ja vs. AG ds has a maximum with log j0 decreasing both at positive and negative values of AGads. This is due to the opposite effects of the free energy of adsorption on the geometric and exponential factors in the electrokinetic equation (volcano plots). 

During the past two decades, much attention has been drawn in this area and advances have been made in theoretical analysis concerning the applicability of Eq. (1) in a variety of systems. This chapter presents the state of understanding of the electrophoretic motion of colloidal particles under various conditions. We first introduce the basic concept and fundamental electrokinetic equations for electrophoretic motion. Then, we review some recent studies on the mobility of a single particle, the boundary effects and the particle interactions in electrophoresis. In addition, a few theoretical methods, which have been used to investigate the boundary effects and particle interactions, will be highlighted and demonstrated in the context. 

To study electrophoresis of particles subject to an external electric field, one needs to know the electrical potential, fluid flow and ion fluxes around the particle. In this section, we first present the fundamental electrokinetic equations for electrophoresis of colloidal particles. Previous studies on the electrophoresis of a single particle will then be reviewed, and important results will be stated. 

Goodenough and co-workers [10] made a detailed study of the solid state chemistry and electrochemistry of ruthenates of general formula Bi(2 2x)P-b2xRu20(7. v) with the pyrochlore structure and reported that the electroreduction of oxygen proceeds at low overpotentials according to the electrokinetic equation... 

Discuss Eq. (6.8.10) so as to provide insight into the meaning of the first electrokinetic equation. 

The fundamental electrokinetic equations for the liquid velocity u(r) at position r relative to the particle (u(r) —> —U as r = Irl oo) and the velocity of the tth ionic species v, are the same as those for rigid spheres except that the Navier-Stokes equations for u(r) become different for the regions outside and inside the surface layer, namely,... 

For a weak applied filed E, the electrokinetic equations are linearized to give... 

Consider an infinitely long cylindrical colloidal particle moving with a velocity C in a liquid containing a general electrolyte in an applied electric field E. By solving the fundamental electrokinetic equations, we finally obtain the following... 

Electrokinetic equations describing the electrical conductivity of a suspension of colloidal particles are the same as those for the electrophoretic mobility of colloidal particles and thus conductivity measurements can provide us with essentially the same information as that from electrophoretic mobihty measurements. Several theoretical studies have been made on dilute suspensions of hard particles [1-3], mercury drops [4], and spherical polyelectrolytes (charged porous spheres) [5], and on concentrated suspensions of hard spherical particles [6] and mercury drops [7] on the basis of Kuwabara s cell model [8], which was originally applied to electrophoresis problem [9,10]. In this chapter, we develop a theory of conductivity of a concentrated suspension of soft particles [11]. The results cover those for the dilute case in the limit of very low particle volume fractions. We confine ourselves to the case where the overlapping of the electrical double layers of adjacent particles is negligible. 

SEDIMENTATION POTENTIAL AND VELOCITY IN A SUSPENSION fundamental electrokinetic equations can be expressed in terms of h and as... 

Making a virtue of necessity, the appropriate electrokinetic equations can be written explicitly in terms of a and then subjected to experimental verification. For instance, [4.3.24] can be transformed to give... 

Relatively complete elaborations for the cylinder model have been given by for instance, Anderson and Koh and Levine et al. K In these two theories the solution Is assumed to contain (1-1) electrolytes with =u. Both theories fail to account for conduction behind the slip plane, and both solve the electrokinetic equations, taking double layer overlap into account. Anderson and Koh assume this overlap to take place at fixed surface charge (which, because of the implicit rigid particle model of the cylinder wall, comes down to fixed tr =cT ), whereas Levine et al. do so for constant surface potential (essentially fixed Anderson and Koh also considered capUlaries of other... 

Discuss the physical mechanism that gives rise to the first electrokinetic equation. 

O Brien, R. W., The solution of the electrokinetic equations for colloidal particles with thin double... 

In the presence of fluid flow, exact analytical solutions of the full electrokinetic equations are rare. An exceptional situation arises when the... 

If the solution to the problem of the equilibrium double layer is known, then the velocity is determined by the above formula. Thus, the solutions to the respective flow problems for the equilibrium situations considered in the previous subsection are readily written down. Solution to the electrokinetic equations is facilitated if the Debye layer thickness may be assumed small compared to the characteristic channel width Wq. This however is not usually the case in nanochannels since wq and are both on the order of nanometers. Exact analytical solutions... 

The first attempt to derive the relation between fi and was made by Von Smoluchowski [10] and Hiickel [11], and later by Henry [12]. Full electrokinetic equations determining electrophoretic mobility fi of spherical particles with arbitrary values of Ka and were derived independently by Overbeek [13] and Booth [14]. Wiersema et al. [15] solved the equations numerically. The computer calculation of the electrophoretic mobility was considerably improved by O Brien and White [16]. Approximate analytic mobility expressions have been proposed by several authors [17-19]. 

This review is organized as follows. The general electrokinetic equations that govern the flow of an electrolyte are presented in Sec. II along with a description of the infinite spatially periodic porous medium. 

In principle, both the potential and charge density at the surface of a porous solid can be calculated from electrokinetic data such as the electro-osmotic transfer rate or the conductivity of a porous plug. Interpretation of this type of experimental data is based on the solution of the so-called electrokinetic equations [10-13]... 

In the presence of fluid flow exact analytical solutions of the full electrokinetic equations are rare. An exceptional situation arises when the applied electric field and streamlines of the flow are both in a direction of homogeneity along which there is no variation of any physical quantity. In such cases, the effect of the external influence (such as the applied field) is to produce motion such that all movement of charges are along surfaces of constant charge density and therefore the equilibrium charge density is not altered by the flow. Examples of such a situation are... 

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