Porous electrode theory developments

Pigeaud, A. Maru, H.C. Paetsch, L. Doyon, J. Bernard, R. Recent developments in porous electrodes for molten carbonate fuel cells. Proceedings of the Symposium on Porous Electrodes Theory and Practices, Maru, H.C., Katan, T., Klein, M.G., Eds. The Electrochemical Society, Inc. Pennington, NJ, 1984 234-259. 

The porous electrode theory was developed by several authors for dc conditions [185-188], bnt the theory is usually applied in the ac regime [92,100,101,189-199], where mainly small signal frequency-resolved techniques are used, the best example of which are ac theory and impedance spectra representation, introdnced in the previons section. The porous theory was first described by de Levi [92], who assumed that the interfacial impedance is independent of the distance within the pores to obtain an analytical solution. Becanse the dc potential decreases as a fnnction of depth, this corresponds to the assnmption that the faradaic impedance is independent of potential or that the porons model may only be applied in the absence of dc cnrrent. In snch a context, the effect of the transport and reaction phenomena and the capacitance effects on the pores of nanostructured electrodes are equally important, i.e., the effects associated with the capacitance of the ionic donble layer at the electrode/electrolyte-solntion interface. For instance, with regard to energy storage devices, the desirable specifications for energy density and power density, etc., are related to capacitance effects. It is a known fact that energy density decreases as the power density increases. This is true for EDLC or supercapacitors as well as for secondary batteries and fnel cells, particnlarly due to the distributed nature of the pores... 

A timeline of major developments in porous electrode theory that led to the current approaches in catalyst layer modeling is depicted in Figure 3.3. 

FIGURE 3.3 A timeline of developments in porous electrode theory and catalyst layer modeling. 

In this approach, originally developed by Fuller et al. [18, 53] based on the porous electrode theory [42], the active material is assumed to consist of spherical particles with a specific size, and solid phase diffusion in the radial direction is assumed to be the predominant mode of transport. The electrolyte phase concentration (Cg) and the potentials (4>s,4>e) are assumed to vary along the principal (i.e., thickness) direction only, and are henceforth referred to as the x direction. In other words, this model implicitly considers two length scales (1D + 1D), that is, the r direction inside the spherical particle and the x direction along the thickness. All other equations described earlier continue to remain valid except the solid phase diffusion. Equation 25.19 and the corresponding boundary/initial conditions. The solid phase diffusion equation now takes the following form ... 

This has led to a relatively profound development of theories for the current—voltage relationship of porous electrodes [105, 106]. The... 

Figure 38 identifies to some extent the possible cell designs in r.b.s. Conventional accumulators are composed of porous electrodes of the second kind [3, 11, 17] (1), but in the case of metal-free cells this is more or less the exception, and solid-state electrodes (A), (B) or (C) are combined, porous or not (2). The theory was developed by Atlung et al. [44-46]. (1) and (2) are based on electrochemical reactions. But electrodes with a high specific surface area, based on active carbon, carbon blacks, or other materials, allow for the special design of an ECDLC (3), where primarily electrochemical reactions are not involved. As indicated in Figure 38, the amount of electrolyte will be medium (i.e. between case (1) and (2)). 

The complexity of the system implies that many phenomena are not directly explainable by the basic theories of semiconductor electrochemistry. The basic theories are developed for idealized situations, but the electrode behavior of a specific system is almost always deviated from the idealized situations in many different ways. Also, the complex details of each phenomenon are associated with all the processes at the silicon/electrolyte interface from a macro scale to the atomic scale such that the rich details are lost when simplifications are made in developing theories. Additionally, most theories are developed based on the data that are from a limited domain in the multidimensional space of numerous variables. As a result, in general such theories are valid only within this domain of the variable space but are inconsistent with the data outside this domain. In fact, the specific theories developed by different research groups on the various phenomena of silicon electrodes are often inconsistent with each other. In this respect, this book had the opportunity to have the space and scope to assemble the data and to review the discrete theories in a global perspective. In a number of cases, this exercise resulted in more complete physical schemes for the mechanisms of the electrode phenomena, such as current oscillation, growth of anodic oxide, anisotropic etching, and formation of porous silicon. 

It may be noticed that Eq. (206) is formally identical with that developed for the Gouy-Chapman theory of the double layer. Substitution of Eq. (189) into (205) gives the expression for the steady-state current on porous electrodes ... 

The importance of structural effects in gas diffusion electrodes was realized long before the development of the current generation of CLs for PEFCs. The basic theory of gas diffusion electrodes, including the interplay of reactant transport through porous networks and electrochemical processes at highly dispersed electrode I electrolyte interfaces, dates back to the 1940s and 50s [13, 14]. Later work realized the importance of surface area and utilization of electrocatalysts in porous electrodes [15]. A series of seminal contributions by R. De Levie opened... 

Major contributions to the development of the macrokinetic or macrohomogeneous theory of porous electrodes were made by Yu, A, Chizmadzhev and Yu, G, Chirkov [25-27], In these works the importance of the interplay of oxygen diffusion and interfacial kinetics had already been realized. For oxygen reduction electrodes, a large electrocatalytically active surface area per unit volume,, has to compensate for the smallness of the intrinsic activity per unit... 

Major contributions to the development of the macrokinetic or macrohomoge-neous theory of porous electrodes were made by Yu. A. Chizmadzhev, Yu.G. Chirkov,... 

MSU. The studies at MSU were supervised by Frumkin much more intently than those at the Institute. On the other hand, Frumkin always thoroughly read all the papers before submitting them for publication. These studies had made fundamental contributions to the theory and practice of electrocatalysis, especially with regard to the dissociative adsorption of organic compounds, the nature of adsorbates, and the rate-determining step of electrooxidation processes. Methods were also developed for optimizing the structure of porous electrodes. The method of standard contact porosimetry was widely used to quantify various materials. Finally, the book by W. Vielstich Brennstojfelemente was translated into Russian with a preface written by Frumkin. 

Oscillatory behavior observed as periodic potential transients at constant current or periodic current transients at constant potential is found frequently when more than two parallel electrode reactions are coupled. Usually, an upper and a lower current-potential curve limit the oscillation region. These two curves represent stable states [139] according to the theory of stability of electrode states [140]. Oscillatory phenomena occurring during the oxidation of certain fuels on solid electrodes are discussed in this section. The discussion is not extended to porous electrodes because the theory of the diffusion electrode has not been developed to the point to allow an adequate description of the complex coupling of parallel electrode reactions and mass transport processes in the liquid and gaseous phase. 

Garcia etal. [41] developed a two-dimensional porous electrode model and accounted for potential and charge distributions in the electrolyte. They employed transport equations derived from dilute solution theory, which is generally not adequate for LIB systems. The stress generation effect is built into the 2D DNS modeling framework with a simplified, sphere-packed electrode microstmcture description. 

The main hypotheses for developing the EHD impedance theory are that the electrode interface is uniformly accessible and the electrode surface has uniform reactivity. However, in many cases, real interfaces deviate from this ideal picture due, for example, either to incomplete monolayer adsorption leading to the concept of partial blocking (2-D adsorption) or to the formation of layers of finite thickness (3-D phenomena). These effects do not involve the interfacial kinetics on bare portions of the metal, which, for simplification, will be assumed to be inherently fast. The changes will affect only the local mass transport toward the reaction sites. Before presenting an application of practical interest, the theoretical EHD impedance for partially blocked electrodes and for electrodes coated by a porous layer will be analyzed. 

Another problem in application of the basic theories is associated with surface geometry. Most theories are developed to describe the relationships among the area-averaged quantities such as charge density, current density, and potentials assuming a uniform electrode surface. In fact, the silicon surface may not be uniform at the micrometer, nanometer, or atomic scales. There can be great variations in the distribution of reactions from extremely uniform, for example, in electropolishing, to extremely nonuniform, for example, in the formation of porous silicon. 

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