Materials Science Forum, Vol.715-716 (2012), pp. 279-285

Mesoscale experiment and simulation permit harvesting information about both geometric features and texture in material microstructures. The 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 grain boundary normal. During the growth process, an initially random texture distribution reaches a steady state that is strongly correlated to the interfacial energy density. In simulation, it is found that if the given energy depends only on lattice misorientation, then 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 a simple distribution arise from such a complex system? We derive an entropy based theory which suggests that the evolution of the GBCD satisfies a Fokker-Planck Equation. Cellular structures coarsen according to a local evolution law, curvature driven growth, and are limited by space filling constraints. The interaction between the evolution law and the constraints is governed primarily by the force balance at triple junctions, the natural boundary condition associated to curvature driven growth, and determines a dissipation relation. A simplified coarsening model is introduced which is driven by the boundary conditions and reflects the network level dissipation relation of the grain growth system. It resembles an ensemble of inertia-free spring-mass-dashpots. Critical application is made of the recent characterization of Fokker-Planck kinetics as a gradient flow for a free energy in deriving the theory. The theory predicts the results of large scale 2D simulations and is consistent with experiment.

Materials Science Forum, Vol.715-716 (2012), pp. 473-479

We revisit grain growth and the puzzle of its stagnation in thin metallic films. We bring together a large body of experimental data that includes the size of more than 30,000 grains obtained from 23 thin film samples of Al and Cu with thicknesses in the range of 25 to 158 nm. In addition to grain size, a broad range of other metrics such as the number of sides and the average side class of nearest neighbors is used to compare the experimental results with the results of two dimensional simulations of grain growth with isotropic boundary energy. In order to identify the underlying cause of the differences between these simulations and experiments, five factors are examined. These are (i) surface energy and elastic strain energy reduction, (ii) anisotropy of grain boundary energy, and retarding and pinning forces such as (iii) solute drag, (iv) grain boundary grooving and (v) triple junction drag. No single factor provides an explanation for the observed experimental behavior.

Physical Review B, Vol.83 (2011), 13, pp. 134117-1 - 134117-12

Mesoscale experiment and simulation permit harvesting information about both geometric features and texture in polycrystals. The 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 growth process, an initially random distribution reaches a steady state that is strongly correlated to the interfacial energy density. In simulation, it is found that if the given energy density depends only on lattice misorientation, then 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 a simple distribution arise from such a complex system? We derive an entropy based theory which suggests that the evolution of the GBCD satisfies a Fokker-Planck Equation. Cellular structures coarsen according to a local evolution law, curvature driven growth, and are limited by space filling constraints. The interaction between the evolution law and the constraints is governed primarily by the force balance at triple junctions, the natural boundary condition associated to curvature driven growth, and determines a dissipation relation. A simplified coarsening model is introduced which is driven by the boundary conditions and reflects the network level dissipation relation of the grain growth system. It resembles an ensemble of inertia-free spring-mass-dashpots. Critical application is made of the recent characterization of Fokker-Planck kinetics as a gradient flow for a free energy in deriving the theory. The theory predicts the results of large scale 2D simulations and is consistent with experiment.

Discrete and continuous dynamical systems / Series A, Vol.30 (2011), 2, pp. 427-454

Cellular networks are ubiquitous in nature. They exhibit behavior on many different length and time scales and are generally metastable. Most technologically useful materials are polycrystalline microstructures composed of a myriad of small monocrystalline grains separated by grain boundaries. The energetics and connectivity of the grain boundary network plays a crucial role in determining the properties of a material across a wide range of scales. A central problem in materials science is to develop technologies capable of producing an arrangement of grains a texture appropriate for a desired set of material properties. Here we discuss the role of energy in texture development, measured by a character distribution. We derive an entropy based theory based on mass transport and a Kantorovich-Rubinstein-Wasserstein metric to suggest that, to first approximation, this distribution behaves like the solution to a Fokker-Planck Equation.

Rendiconti di matematica e delle sue applicazioni, Vol.29 (2009), 1, pp. 65-81

Cellular networks are ubiquitous in nature. They exhibit behavior on many different length and time scales and are generally metastable. Most technologically useful materials are polycrystalline microstructures composed of a myriad of small monocrystalline grains separated by grain boundaries. The energetics and connectivity of the grain boundary network plays a crucial role in determining the properties of a material across a wide range of scales. A central problem in materials science is to develop technologies capable of producing an arrangement of grains - a texture - that provides for a desired set of material properties. Here we discuss briefly the role of energy in texture development, measured by a character distribution, and how this is different from the evolution of geometric features, which we term geometric coarsening. For this purpose we present a critical event model to deepen our understanding of the topological reconfigurations that occur during the growth process.