Electro-diffusional

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. 

Of course, the borderline between the two pictures (the one-phase and the two-phase ones) becomes diffused when the second transforms into the first with the decrease of the typical pore radius. In fact, as will become clear in Chapter 6, 6.4, distinction between the phases becomes meaningless as soon as the typical pore radius becomes shorter than some typical electro-diffusional length scale—the Debye length—to be defined below. 

The LEN approximation will be employed in various electro-diffusional contexts in Chapters 3, 4, and 6. In particular, in Chapter 4 we shall elaborate upon the limits of applicability of the LEN approximation. It will be further treated as a leading approximation for the singularly perturbed system (1.9), (l.lld) in Chapter 5. 

The introductory information presented above allows us to outline a certain hierarchy of electro-diffusional phenomena and levels of description that form the backbone of this monograph. At the bottom, simplest level of this hierarchy lie the nonlinear equilibrium effects to be treated in Chapter 2. The entire treatment here will be based upon the Poisson-Boltzmann equation. 

Existence and uniqueness of solutions to (4.2.12) and to the appropriate algebraic equations for ji, B have to be studied separately in any specific electro-diffusional set-up. 

Besides, as was pointed out in the Preface, there exists a number of largely unexplained, practically relevant phenomena, occurring in purely electro-diffusional systems and potentially related to the multiplicity of steady states [19]—[21]. Finally, uniqueness results could be valuable for the numerical analysis of semiconductor models. 

Formulation. Consider two unity thick diffusion layers of a mixture of 1, 1-, and 1, z-valent3 electrolytes with a common anion, adjacent to a planar ideally permselective cation-exchange membrane. Direct the axis x normally to the membrane and let x = 0 coincide with the outer boundary of the diffusion layer. The diffusion layers will thus be located at 0 < x < 1 and 1 + A[Pg.139]

Effects of space charges (finite e). It is expected that the inclusion of space charges will eliminate current saturation (limiting currents) at the lower and middle branches. These branches are instead expected to meet at another turning point, due to punch through (in this purely electro-diffusional formulation without source terms).5... 

Uniqueness of electro-diffusional steady states is expected and should be proved for one-dimensional systems with less than three alterations of sign N. 

The space charge in the liquid junction [1]. By liquid junction or the liquid junction potential we mean the diffusion potential developing in an electrically insulated electrolyte solution with differing ionic diffu-sivities and an initial concentration discontinuity. Besides its conceptual importance as probably the simplest nonequilibrium electro-diffusional situation, the dynamics of liquid junction is important to understand for applications, such as salt bridges, etc. 

Equation (5.2.42) and the treatment that leads to it remind us that in accordance with the description in the Introduction it takes about 1 je as long for an electro-diffusional system to get from an arbitrary initial state to a macroscopically locally electro-neutral one. 

A Prototypical Convective Electro-Diffusional Phenomenon—Electro-Osmotic Oscillations... 

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 >. ... 

Considering their possible applications in fuel cells, hydrogen sensors, electro-chromic displays, and other industrial devices, there has been an intensive search for proton conducting crystals. In principle, this type of conduction may be achieved in two ways a) by substituting protons for other positively charged mobile structure elements of a particular crystal and b) by growing crystals which contain a sufficient amount of protons as regular structure elements. Diffusional motion (e.g., by a vacancy mechanism) then leads to proton conduction. Both sorts of proton conductors are known [P. Colomban (1992)]. 

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