Some Mesoscale Issues for Applied Mathematics

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Mesoscale is a term intended to convey an intermediate level description ofa physical system. Its function is to capture the interactions of thesystem at finer scales with interactions or influences from larger coarserscales, and the mesoscale level itself. Such systems, active acrossdisparate length and time scales, are inherently metastable. This featureis often revealed by hysteretic behavior or by a reluctance to evolvequickly to equilibrium. The focus in this proposal is on severalprototypes of these systems that occur in materials sceince. One is the role of interfaces, or grain boundaries, in determining or limiting the behavior of polycrystalline materials. The energy and mobility of grain boundaries depends on crystallography and geometry, according to established thermodynamic principles. Innovative new ways to determine these functions explicitly for important materials are the objective of the Mesoscale Interface Mapping Project. This involves developing automated microscopy to harvest large amounts of data from samples and then formulating and solving a complex inverse problem. One way to approach determination of mobility consists in the development of large scale simulations of grain boundary evolution. The second focus is the coarse grained descriptions of mesoscale systems, the functional analytic limit processes in the microstructure of solids or the averaging to distribution functions in systems with stochastic behavior. The new methods allow study of situations where kinetics arise directly in terms of thermodynamic state functions and naturally carry with them an appropriate topology. The issue of metastability has been under investigation in this context. There is now the opportunity to improve understanding, for example, of microstructural evolution in shape-memory materials and diffusion mediated transport in certain liquid crystal systems and in protein motors. This will include diagnostics for these systems.The challenge of the mesoscale in materials science is to understand how it constrains finer scale systems (at the molecular scale) and determines larger scale systems (at the scale of entire devices). This is accomplishedthrough coarse graining procedures. For example, many technologicallyuseful materials are polycrystalline, or granular, in nature. The aluminumskin of an aircraft and the copper or copper-aluminum interconnects incomputer chips are but two examples at vastly different size scales of suchgranular materials. It is widely understood that many aspects of thesematerials depend on the interfaces they contain, or their grain boundaries.Properties of grain boundaries determine the reliability as well as the mechanical strength. This project will exploit the exciting opportunities and challenges for mathematical science in this field and beyond. Specifically, coarse graining methods will be developed in order to better understand phenomena that occur in protein motors and in liquid crystals. A related problem of coarse graining arises when information is to be extracted from the immense amounts of data that can be produced with simulations ofcomplex systems, such as polycrystalline materials. This problem of coarse graining at an information scale rather than a physical scale will also be addressed in this project.