A stochastic approach to coarsening in grain networks

Many of the high-tech devices we use today possess components that are made from polycrystalline materials. From interconnects in microprocessors with higher conductivity to materials that resist fracture, the engineering of these microstructures is crucial to our technological advancement. These macroscopic properties are affected by mesoscale characteristics like grain size, texture, and arrangement of the grains in their topological network of boundaries. A crucial problem in understanding this area of materials science is the modeling and simulation of the evolution of these networks. Simulation of these large, metastable networks involves solution of an average of 50,000 nonlinear partial differential equations. This approach to microstructural evolution can be replaced at a larger scale by a system of master equations for the evolution of grain size distribution functions. An earlier known phenomenological system of this type can be found in the paper of Fradkov et al [1].

Modeling of these equations, however, must be based on accommodating critical events into the deterministic rules for grain growth in order to obtain network properties. Such a model should be built empirically, from large scale simulation. And it must reproduce statistics of the simulation. These properties are the foundation of the master equations proposed by the authors BKLT [2]. In their work, they propose a system of equations that is internally consistent, and suggest that the reassignment of sides from deleted grains is random in nature.

In the work we present in this seminar, we will show that these master equations do in fact have a consistent probabilistic interpretation in terms of martingales. It follows that the empirical equations posed by BKLT have a life of their own as the 'upscale' equations of a stochastic process that describes the coarsening of the grain network . Finally, we show that the existence theory for solutions to these highly nonlinear integro-partial differential equations is symbiotically coupled to an existence theory for this stochastic process.

[1] V.E. Fradkov and D. Udler, Two-dimensional normal grain growth: topological aspects, Adv. Physics 43, 739 (1994)

[2] K. Barmak, D. Kinderlehrer, I. Livshits and S. Ta'asan, Remarks on a Multiscale Approach to Grain Growth in Polycrystals, Progress in Nonlinear Differential Equations and Their Applications, 68 (2006), pp.1-11.