Mesoscale Simulation of Grain Growth

Simulation is becoming an increasingly important tool, not only in materials science in a general way, but in the study of grain growth in particular. Here we exhibit a consistent variational approach to the mesoscale simulation of large systems of grain boundaries subject to Mullins Equation of curvature driven growth. Simulations must be accurate and at a scale large enough to have statistical significance. Moreover, they must be sufficiently flexible to use very general energies and mobilities. We introduce this theory and its discretization as a dissipative system in two and three dimensions. The approach has several interesting features. It consists in solving very large systems of nonlinear evolution equations with nonlinear boundary conditions at triple points or on triple lines. Critical events, the disappearance of grains and and the disappearance or exhange of edges, must be accomodated. The data structure is curves in two dimensions and surfaces in three dimensions. We discuss some consequences and challenges, including some ideas about coarse graining the simulation. Introduction We discuss the mesoscale simulation of large networks of grains or interfaces in two and three dimensions. We give a brief introduction to the format and explain our algorithm. Evolution is governed by the Mullins Equations of curvature driven growth, discussed below, which consist of a system of evolution partial differential equations for each boundary curve, in two dimensions, or facet, in three dimensions. Grain boundaries typically meet at triple junctions, in two dimensions, or on triple lines, in three dimensions, where a boundary condition is required. Here we enforce the Herring Condition, a force balance. This is the natural boundary conditon for equilibrium of the Journal Title and Volume Number (to be inserted by the publisher) 2 Mullins Equation, a fact that may not be well known. The resulting system is dissipative for the energy and the coarsening process may be viewed as a modified steepest descent for the total grain boundary energy. Certain critical events, such as grain disappearance and the exchange or disappearance of facets, must be accomodated. We describe our strategy for this which conserves the dissipative character of the process. A special feature of our approach is that the data structure consists only of curves, in two dimensions, and surfaces in three dimensions, which offers an opportunity to work with large systems. It offers the opportunity to employ experimentally derived energy densities and mobilities. Initial configurations may have statistically representative properties derived from experimentally characterized microstructures. For general perspectives, references, and methods relative to the area, we refer to [1]. Since, generally, the result of such a simulation must be interepreted in some statistical terms, we are led to the companion issue of coarse graining in mesoscale simulation. By this we mean understanding what distributions are reliable properties of the computed ensemble and what equations they themselves satisfy. For reasons of space, we defer discussion of this to a future work, but we present some results in this direction. We also refer to [2], where this simulation is implemented to investigate anisotropy. Mullins Equation and Herring Condition The form of the Mullins Equation and the Herring Condition useful for algorithmic implementation in large scale simulation may be derived by a variational procedure. Consider a network of grains with facets which meet in triple lines. To begin, suppose given three facets represented as graphs over an x = (x1,x2) plane in (x1,x2,x3) space meeting along a triple line Γ, where Γ' denotes the projection of Γ onto the x-plane, Ω denotes the regions above and below Γ'. The energy of just a single facet,