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Recent Developments in Material Microstructure: A Theory of Coarsening

Saturday, October 24, 2015 - 1:45pm - 2:25pm
David Kinderlehrer (Carnegie-Mellon University)
Cellular networks are ubiquitous in nature. Most engineered materials are polycrystalline microstructures composed of a myriad of small grains separated by grain boundaries, thus comprising cellular networks. The recently discovered grain boundary character distribution (GBCD) is an empirical distribution of the relative length (in 2D) or area (in 3D) of interface with a given lattice misorientation and normal. During the coarsening, or growth, process, an initially random grain boundary arrangement reaches a steady state that is strongly correlated to the interfacial energy density. In simulation, under appropriate circumstances, the steady state GBCD and the energy are related by a Boltzmann distribution. This is among the simplest non-random distributions, corresponding to independent trials with respect to the energy. Why does such simplicity emerge from such complexity? Exploiting ideas from mass transport, we suggest that the evolution of the GBCD satisfies a Fokker-Planck Equation, an equation whose stationary state is a Boltzmann distribution.

Our view, then, is that this statistic from the coarsening process presents itself as a gradient flow in the mass transport sense. We may ask if there are other such statistics or processes `in the wild' and if there are systematic ways to discover them.

Although the subject of these investigations is most closely associated with another giant of continuum physics, the late W.W. Mullins, the intellectual process owes to Jerry Ericksen. Joint work with Katayun Barmak (Columbia), Eva Eggeling (T.U. Graz), Maria Emelianenko (George Mason). Yekaterina Epshteyn (Utah), Richard Sharp (Globys), and Shlomo Ta'asan (CMU) with the recent addition of Patrick Bardsley (Utah) and Xin Yang Lu (McGill)
MSC Code: 
74N15