Electro-diffusion

This outline, as brief and superficial as it may be (for a more detailed description of basic electrochemical transport objects, the reader is referred to relevant texts, e.g., [1]—[3]) will permit a formulation of basic equations of electro-diffusion. A hierarchy of electro-diffusional phenomena will be sketched next, beginning with the simplest equilibrium ones. Subsequent chapters will be devoted to the study of some particular topics from different levels of this hierarchy. 

According to (1.3b), the nonconvectional electro-diffusion flux component j. is a superposition of the following two terms. The first is the diffusional Fick s component proportional to the concentration gradient VC. The second is the migrational component, proportional to the product of the ionic concentration Cj and the electric force —ZiFV

proportionality factor. Einstein s equality (1.3c) relates ionic mobility to diffusivity >. ... 

The classical equations (1.1)—(1.6) form the basis for the entire treatment of electro-diffusion in this book. 

As the first step in introducing some basic electro-diffusion notions, let us define the following dimensionless variables... 

In a hydrodynamically free system the flow of solution may be induced by the boundary conditions, as for example when a solution is fed forcibly into an electrodialysis (ED) cell. This type of flow is known as forced convection. The flow may also result from the action of the volume force entering the right-hand side of (1.6a). This is the so-called natural convection, either gravitational, if it results from the component defined by (1.6c), or electroconvection, if it results from the action of the electric force defined by (1.6d). In most practical situations the dimensionless Peclet number Pe, defined by (1.11b), is large. Accordingly, we distinguish between the bulk of the fluid where the solute transport is entirely dominated by convection, and the boundary diffusion layer, where the transport is electro-diffusion-dominated. Sometimes, as a crude qualitative model, the diffusion layer is replaced by a motionless unstirred layer (the Nemst film) with electrodiffusion assumed to be the only transport mechanism in it. The thickness of the unstirred layer is evaluated as the Peclet number-dependent thickness of the diffusion boundary layer. 

We will discuss next the ambipolar diffusion, that is, electro-diffusion of two oppositely charged ions in a solution of a univalent electrolyte with local electro-neutrality. Assume the dimensionless ionic diffusivities are constant. Then the relevant version of (1.9) is... 

A somewhat similar situation occurs in one-dimensional multi-ionic systems with local electro-neutrality in the absence of electric current. It will be shown in Chapter 3 that in this case again the electric field can be excluded from consideration and the equations of electro-diffusion are reduced to a coupled set of nonlinear diffusion equations. 

The next level is that of one-dimensional electro-diffusion with local electro-neutrality in the absence of an electric current. This is the realm of nonlinear diffusion to be treated in Chapter 3. A still higher level of the same hierarchy is formed by the nonlinear effects of stationary electric current, passing in one-dimensional electro-diflFusion systems with local electro-neutrality. A few typical phenomena of this type will be studied in Chapter 4. The treatment of Chapter 4 will lay the foundation for the discussion of the effects of nonequilibrium space charge characteristic of the fourth level to be treated in Chapter 5. 

The top level of the electro-diffusion hierarchy is formed by the electroconvection phenomena, of which electro-osmosis is in several respects the simplest one. Certain aspects of electro-osmosis will be treated in Chapter 6. The higher we climb the hierarchy outlined the less rigorous our mathematics will become and the more vague heuristic statements will appear. 

Finally, we point out that there is a close relation in description of ion electro-diffusion and the phenomenological theory of the electron and hole transport in semiconductors. In order to facilitate the reading we present below a brief ionics-semiconductor vocabulary. ... 

The Poisson-Boltzmann equation. Equilibrium is a steady state without macroscopic fluxes. As we pointed out in the Introduction, under these conditions the equations of electro-diffusion reduce to... 

The specific electro-diffusion phenomena, the field and force saturation and counterion condensation, as well as the corresponding features of the solutions to the Dirichlet problem for (2.1.2) to be addressed in this chapter, are closely related to those observed by Keller [7], [8] for the solutions of (2.1.3a) with f tp) positive definite, satisfying a certain growth condition. Keller considered f( 0, satisfying the condition... 

In this chapter we shall thus study two electro-diffusion equilibrium phenomena are related to the above feature. (Note that the right-hand side of (2.1.2) satisfies the condition (2.1.4) but is not positive definite.) Both phenomena reflect the peculiar response of the equilibrium ionic systems described by (2.1.2) to the increase of the electric charge carried by some of their parts. 

The aim of discussing these phenomena is to exemplify that, due to the strongly nonlinear nature of electro-diffusion, nontrivial effects may arise in... 

Locally Electro-Neutral Electro-Diffusion Without Electric Current... 

Here C is the concentration vector and D(C) is the diffusivity tensor defined by (3.1.15a). Thus, locally electro-neutral electro-diffusion without electric current is exactly equivalent to nonlinear multicomponent diffusion with a diffusivity tensor s being a rational function of concentrations of the charged species. 

Preliminaries. In the previous chapter we dealt with locally electro-neutral time-dependent electro-diffusion under the condition of no electric current in a medium with a spatially constant fixed charge density (ion-exchangers). It was observed that under these circumstances electrodiffusion is equivalent to nonlinear diffusion with concentration-dependent diffusivities. 

The equations of stationary one-dimensional electro-diffusion will be further integrated ( 4.2) for an arbitrary number of the transferred charged species of arbitrary valencies. This result will be applied next to a num-... 

In 4.4 the theory of 4.2 will be applied to study electro-diffusion of ions through a unipolar ion-exchange membrane, separating two electrolyte solutions. This will include the classical treatment of concentration polarization in a solution layer adjacent to an ion-exchange membrane under an electric current. The validity limits of this theory, set by the violations of local electro-neutrality and caused by the development of a macroscopic nonequilibrium space charge, will be indicated. (The effects of the nonequilibrium space charge are to be discussed at some length in Chapter 5.)... 

As was pointed out in the Introduction, with the above scaling the equations of stationary electro-diffusion assume the form ... 

Integration of the stationary electro-diffusion equations in one dimension. The integration of the stationary Nernst-Planck equations (4.1.1) with the LEN condition (4.1.3), in one dimension, for a medium with N constant for an arbitrary number of charged species of arbitrary valencies was first carried out by Schlogl [5]. A detailed account of Schlogl s procedure may be found in [6]. In this section we adopt a somewhat different, simpler integration procedure. 

Multiple steady states in one-dimensional electro-diffusion with local electro-neutrality [14]. This section is concerned with the construction and study of multiple steady states occurring in onedimensional ambipolar electro-diffusion with local electro-neutrality and... 

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