An Approach to the Mesoscale Simulation of Grain Growth

Abstract

The simulation of curvature driven growth in grain boundary systems is becoming an important tool in understanding the behavior of microstructure evolution and there is much distinguished work in this subject. Here we address the mesoscale simulation of large systems of grain boundaries subject to the Mullins equation of curvature driven growth with the Herring force balance equation imposed at triple junctions. We discuss several novel features of our approach which we anticipate will render it a flexible, scalable, and robust tool to aid in microstructural prediction. What is the result of the simulation? We discuss what such a simulation is capable of predicting, taking as a prototype the histogram of relative area population as it changes through the simulation. We do not use this data to seek the best distribution, like Hillert, Rayleigh, or lognormal. Instead we treat the set of distributions as the solution of an inverse problem for a time varying function and determine the equation they satisfy. This results in a coarse graining of the complex simulation to simpler system governed by a Fokker-Planck Equation. Even so, fundamental questions concerning the predictability of simulations of large metastable systems arise from these considerations.

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 Variational Approach to Modeling and Simulation of Grain Growth

“…[5] for a preliminary version). In [24], a first attempt to understand the evolution of grain growth statistics derived from this simulation is discussed.…”

“…There is an extensive and distinguished literature in the simulation of grain growth. These models can be divided into the following major groups: molecular dynamics (MD) [8], [16], [33], [34], [35], [19], [37], [39], [40], [42], Monte-Carlo (MC) [2], [3], [28], [29], [41], [42], front tracking [14], [15], vertex models [43], [44], [45], phase field models [6], [7], [25], [26], [42], partial differential equations (PDE) [9], [24], [27], and distribution function models [10], [11], [12], [13]. Each modeling approach is appropriate in studying grain growth on a particular length or time scale.…”

Grain Boundary Energy and Grain Growth in Highly-Textured Al Films and Foils: Experiment and Simulation

“…[10] For grain growth simulations both the Monte Carlo Potts (MC) model and a boundary tracking model termed the partial differential equation (PDE) model were used. [11][12][13][14] The PDE simulations portray the evolution of a network of two-dimensional curves governed by the Mullins Equation of curvature driven growth. The Herring Condition of force balance is imposed at each triple junction.…”