Key Concepts of Percolation Theory

Percolation theory represents the most advanced and most widely used statistical framework to describe structural correlations and effective transport properties of random heterogeneous media (Sahimi, 2003 Torquato, 2002). Here, briefly described are the basic concepts of this theory (Sahimi, 2003 Stauffer and Aharony, 1994) and its application to catalyst layers in PEFCs. 

Let us consider a container with a large number, N 1, of randomly dispersed and noninteracting guest objects, embedded in a host medium. The characteristic size of individual objects must be very small, that is, microscopic, compared to the macroscopic size of the container. Objects could represent phase domains of a solid or liquid substance or they could represent pores in a porous medium. The roles of guest and host media are interchangeable and components of both media may be represented by spherical particles on a lattice. 

Individual objects possess a characteristic physical property like electronic conductivity, ion conductivity, magnetic susceptibility, gas diffusivity, or liquid permeability. Mixing of at least two distinct types of objects (of guest and host material) that 

At a critical value of the fraction of objects of one type, these objects would form an extended cluster that connects the opposite external faces of the sample. At this so-called percolation threshold, the corresponding physical property represented by the connected objects would start to increase above zero. Thereby percolation theory establishes constitutive relations between composition and structure of heterogeneous media and their physical properties of interest. For porous electrodes or catalyst layers in PEFC, these properties are electrical conductivities of electrons and protons, diffiisivities of gaseous reactants and water vapor, and liquid water permeability. 

Typical examples of percolation systems are (i) random mixtures of electronically conducting metallic spheres in an insulating host medium, for example, a polymer electrolyte, (ii) a network of gas pores providing high diffusivity in a porous matrix of a gas-tight material, or (iii) a porous electrode partially saturated with a liquid electrolyte. 

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