An entropy based theory of the grain boundary character distribution

Abstract

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

Recent Developments in Material Microstructure: a Theory of Coarsening

“…We have used this model to develop a statistical theory for critical events, [9], [10], [8]. It has been found to have its own GBCD as well, [4], [6], [5], [11], which we shall now review.…”

“…Our main idea in [4], [6], [5], [11] is that the GBCD statistic for the simplified model resembles the solution of a Fokker-Planck Equation via the mass transport implicit scheme, [20]. In [4], [6], [5], [11] the simplified model is formulated as a gradient flow which results in a dissipation inequality analogous to the one found for the coarsening grain network. Because of this simplicity, it will be possible to 'upscale' the network level system description to a higher level GBCD description that accomodates irreversibility.…”

“…As explained in [6], [5], [11], we can introduce now the idea of GBCD for the simplified 1D model. Let us consider a new ensemble based on the misorientation parameter α where we take Ω :…”

“…Finally, the first term in (3.8) can be estimated from below to represent energy lost due to frictional or viscous forces (for details cf. [11]). This results in the inequality, valid the pair v, ρ…”

“…Here we summarize our recent work developing an entropy based theory that suggests that the evolving GBCD satisfies a Fokker-Planck Equation, [5], [11], cf. also [6], [4], [12] to which we refer for a more complete exposition.…”