Macrohomogeneous approach

On the other side of the scale are the macroscopic models, which are in line with the macrohomogeneous approach taken in this chapter. There are two main schools of thought regarding membrane models, those that assume the membrane is a single phase and those that assume it is two phases. The former usually leads to a diffusion model [11, 12] and the latter to a hydraulic one [13, 14]. Both models can be made to agree with experimental data, but neither describes the full range of data nor all of the observed effects, like Schroder s paradox. 

Functional models of such a complex structure are based on the idea of macrohomogeneous approach. This means that the microscopic details of a CL structure are ignored and CL is considered as a continuum with prescribed transport properties for ions, electrons and neutrals. These transport properties are usually taken from experiments or from structural models. The latter are beyond the scope of this book the reader is referred to (Eikerling et ah, 2007) for a review of structural models. 

So far, however, some challenging questions about the CL structure-function relations remain unanswered. The macrohomogeneous approach utilizes effective transport parameters of CL obtained experimentally. Generally, this limitation needs to be relaxed. Ideally, a CL model should generate all the necessary transport data self-consistently. The DNS model is the step toward this goal. 

Combination of the macrohomogeneous approach for porous electrodes with a statistical description of effective properties of random composite media rests upon concepts of percolation theory (Broadbent and Hammersley, 1957 Isichenko, 1992 Stauffer and Aharony, 1994). Involving these concepts significantly enhanced capabilities of CL models in view of a systematic optimization of thickness, composition, and porous structure (Eikerling and Komyshev, 1998 Eikerling et al., 2004). The resulting stmcture-based model correlates the performance of the CCL with volumetric amounts of Pt, C, ionomer, and pores. The basis for the percolation approach is that a catalyst particle can take part in reaction only if it is connected simultaneously to percolating clusters of carbon/Pt, electrolyte phase, and pore space. Initially, the electrolyte phase was assumed to consist of ionomer only. However, in order to properly describe local reaction conditions and reaction rate distributions, it is necessary to account for water-filled pores and ionomer-phase domains as media for proton transport. 

The final simple macrohomogeneous porous-electrode models are the ones that are more akin to thin-film models. In these models, the same approach is taken, but instead of gas diffusion in the catalyst layer, the reactant gas dissolves in the electrolyte and moves by diffusion and reaction. The... 

Figure 48. Kenjo s ID macrohomogeneous model for polarization and ohmic losses in a composite electrode, (a) Sketch of the composite microstructure, (b) Description of ionic conduction in the ionic subphase and reaction at the TPB s in terms of interpenetrating thin films following the approach of ref 302. (c) Predicted overpotential profile in the electrode near the electrode/electrolyte interface, (d) Predicted admittance as a function of the electrode thickness as used to fit the data in Figure 47. (Reprinted with permission from refs 300 and 301. Copyright 1991 and 1992 Electrochemical Society, Inc. and Elsevier, reepectively.)...

As the macrohomogeneous electrode theory has proven its worth in electrode diagnostics and design, so the finer details of electroactive layer structure and elec-trocatalytic mechanisms are moving to the fore. A useful concept is to consider agglomerates as structural units of the electroactive layer. Ideal locations of electroactive particles are at the true two- or even three-phase boundary. This approach is capable of and vital for showing that micropores inside agglomerates are filled with liquid water to keep the particles active. Even for well defined and extensively... 

Membrane operation in the fuel cell is affected by structinal characteristics and detailed microscopic mechanisms or proton transport, discussed above. However, at the level of macroscopic membrane performance in an operating fuel cell with fluxes of protons and water, only phenomenological approaches are feasible. Essentially, in this context, the membrane is considered as an effective, macrohomogeneous medium. All structures and processes are averaged over micro-to-mesoscopic domains, referred to as representative elementary volume elements (REVs). At the same time, these REVs are small compared to membrane thickness so that non-uniform distributions of water content and proton conductivities across the membrane could be studied. 

In this chapter the scope of our discussion was restricted by the macrohomogeneous model of CL performance and its derivatives. The first numerical macrohomogeneous models of CCL for a PEM fuel cell were developed by Springer and Gottesfeld (1991) and by Bernard and Verbrugge (1991). These models included the diffusion equation for oxygen transport, the Tafel law for the rate of ORR and Ohm s law for the proton transport in the electrolyte phase. A similar approach was then used by Perry, Newman and Cairns (Perry et al., 1998) and by Eikerling and Kornyshev (1998) for combined numerical and analytical studies. 

The previous model erases aU electrical fields and interfacial barriers in the mesostructure, which is viewed in effect as a homogeneous medium. However, in semiconductor mesostructures, filled with an HTM, one can also allow for the presence of an electrical field and semiconductor barrier at the internal interface ETM/HTM. The prevalence of one approach or the other, i.e., a macrohomogeneous model that only contemplates the Fermi level or the explicit presence of internal interface barriers, depends on doping densities, size of semiconductor particles or wires, and Debye length both in the semiconductor nanostructure and in the HTM [95-97]. 

Simple pore models (Srinivasan et al., 1967), thin-film models (Srinivasan and Hurwitz, 1967), macrohomogeneous models, and refined variants of agglomerate models are still being applied and further developed for the present generation of ionomer-bound composite catalyst layers in PEFCs (Gloaguen and Durand, 1997 Jaouen et al., 2002 Karan, 2007 Kulikovsky, 2002a, 2010b Sun et al., 2005). Effectiveness factor approaches have been elaborated as quantitative tools to compare... 

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