Evolution of Grain Boundaries

Abstract

Evolution and trend to equilibrium of a (planar) network of grain boundaries subject to curvature driven growth is established under the assumption that the system is initially close to some equilibrium configuration. Curvature driven growth is the primary mechanism in processing polycrystalline materials to achieve desired texture, ductility, toughness, strength, and other properties. Imposition of the Herring condition at triple junctions ensures that this system is dissipative and that the complementing conditions hold. We introduce a new way to employ the known Solonnikov-type estimates, which are only local in time, to obtain solutions that are global in time with controlled norm.

Self-similar solutions of a 2-D multiple-phase curvature flow

“…For the mean curvature flow for multiple phases (i.e. with triple junctions), most of the research is in the two space dimensional case; see [2,3,6,30,31,32,38,39,40,41].…”

“…A detailed derivation such as that in [30] demonstrates that, in an appropriate units of time and length, the grain boundary evolves according to the mean curvature flow…”

A generalization of the three-dimensional MacPherson-Srolovitz formula

“…That is to say, three interfaces meet along the triple junction with the angle 2π/3. For more discussion on the Herring condition, we refer to [1,7,13] and [15]. On the other hand, other conditions for junction lines may also be postulated to ensure that the system is dissipative.…”

A Theory and Challenges for Coarsening in Microstructure

“…Since their introduction by Mullins, [54], and Herring, [34], [35], a large and distinguished body of work has grown about these equations. Most relevant to here are [32], [20], [42], [55]. Curvature driven growth has old origins, dating at least to Burke and Turnbull [21].…”