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Unformatted text preview: VII. Porous Media Lecture 32: Percolation MIT Student (and MZB) References: S. Torquato, Random Heterogeneous Materials (Springer 2002) D. Stauffer, Introduction to Percolation Theory In electrochemical energy systems, porous electrodes are generally used to maximize interfacial area to facilitate Faradaic reactions between the electron-conducting electrode matrix and the ion-conducting electrolyte. In batteries, energy density is also of concern, and can be augmented by increasing the volume fraction of active material. However, greater volume fraction means less porosity, and so lower conductivity and reduced power density. Understanding this balance requires a model of conductivity in composite materials. Here, “percolation” will refer to a connected conductive pathway in a system of multiple phases. 1. Random Microstructures 1.1 Lattice percolation The lattice model is the simplest description of connectivity in a random material with a given volume fraction, Φ . For discrete microstructures, p = probability of occupying a site (or bond) in a lattice = Φ when size ∞ . Clusters of connected occupied sites can be identified, and one can study statistics of the largest cluster and sites connected to boundaries as ‘p’ is varied. “Percolation” implies that a cluster spans to opposite boundaries, which is a prerequisite for nonzero conductivity. The first example of a percolation model was proposed by the polymer chemist, P. Flory, in 1941 to describe the sol-gel transition, in which the removal of solvent increases the monomer concentration and promotes the formation of large polymer clusters, purely by geometrical effects of random connectivity. We have all observed this process during the heating of an egg, as the clear liquid turns white. The basic concept, however, has much broader applications. Broadbent and Hammersley (1958) coined the term “percolation” for the phenomenon of forming a spanning cluster connecting two boundaries and developed a general mathematical theory, motivated by applications to flow in porous media. This is closer to our application of conduction in composite media and porous electrodes. Figure 1 - Random realization of site percolation on 2D square lattice Lecture 32: Percolation 10.626 (2011) Bazant Fig. 2 . Largest cluster in critical 2D site percolation – Berkeley.edu It is instructive to perform numerical experiments, as in the interactive online demo: http://fafnir.phyast.pitt.edu/myjava/perc/perc3.html This site allows you to visualize all clusters in gray and the largest cluster in red for a given realization with specific p in 2D site percolation. By sweeping p across the range 0 to 1 you can also clearly see the onset of percolation, where the largest cluster spans the system....
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- Spring '03
- Critical phenomena, Percolation, Bazant