Mathematical Sciences: Percolation in Composite Materials

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9622367 Golden We propose to develop new analytical and numerical techniques for investigating the behavior of a number of percolation models arising in the study of high contrast composite materials. Some of the directions for development include: (i) exploring the critical behavior of the random resistor network in light of a new, direct correspondence with the Ising model, (ii) analysis of continuum percolation models for porous media and matrix-particle composites, (iii) developing a diagnostic method for non-destructively telling whether one phase in a composite percolates or not, which could have applications to the detection of material failure or quality control for composite manufacturing, and (iv) modeling the transport of heat and brine through sea ice, a porous medium. %%% A broad range of problems in materials science involve composites which can be characterized by a high contrast in the properties of the components. For such media, the effective behavior is often dominated by the connectedness, or percolation properties of a particular phase. Examples include porous media, doped semiconductors, smart materials such as piezoresistors and thermistors, thin metal films (used in solar energy production), oil bearing sandstones, fracture patterns in rocks and metals, many structures in the human body, and sea ice (which plays a major role in climate and is an indicator of global change). In the proposed work we will study mathematical models of such materials. We anticipate that this project will further our capabilities to predict the effective behavior of naturally occurring or artificially made complex random materials, as well as aid the development of composites designed to achieve novel or optimal properties. ***