Mesoscale Simulation of the Evolution of the Grain Boundary Character Distribution

A mesoscale, variational simulation of grain growth in two-dimensions has been used to explore the effects of grain boundary properties on the grain boundary character distribution. Anisotropy in the grain boundary energy has a stronger influence on the grain boundary character distribution than anisotropy in the grain boundary mobility. As grain growth proceeds from an initially random distribution, the grain boundary character distribution reaches a steady state that depends on the grain boundary energy. If the energy depends only on the lattice misorientation, then the population and energy are related by the Boltzmann distribution. When the energy depends on both lattice misorientation and boundary orientation, the steady state grain boundary character distribution is more complex and depends on both the energy and changes in the gradient of the energy with respect to orientation. Introduction With the availability of reliable large scale simulations of materials behavior, a principal challenge is the development of strategies for the extraction of information. For example, relative area histograms are stable statistics of bona fide simulations, but tend to discriminate poorly among input grain boundary energy functions [1]. Here, we use our variational approach to the mesoscale simulation of grain growth in large two dimensional systems to study the evolution of the grain boundary character distribution. Like relative area histograms, histograms of the grain boundary character distribution are stable during grain growth. Unlike the relative area histograms, they are sensitive to the assumed grain boundary energy anisotropy. In the two dimensional systems described here, the grain boundary character distribution, f(a,q), is the relative length of grain boundary with respect to lattice misorientation (a) and boundary orientation (q). More precisely, f(a,q) is the length of arc with lattice misorientation a and boundary orientation q divided by the total length of grain boundary in the configuration. Recent experimental studies have shown that the grain boundary character distribution is inversely correlated to the grain boundary energy [2,3]. Furthermore, the same low index and slow growing surface orientations that dominate the external forms of the materials are also the preferred grain boundary plane orientations [4-6]. Consistent with these observations, the results presented here show that during grain growth, low energy boundaries tend to populate the histogram more than higher energy boundaries. Furthermore, the simulations show that in the long term, energetic effects tend to dominate over kinetic effects. 2 Title of Publication (to be inserted by the publisher) Generally speaking, to understand a simulation requires a model or theory of it, as rigorous as possible. The feasibility of predicting relative area-type histograms by deriving master equations that accurately portray behavior of the simulation is an example of this [7]. Here we explore the challenge of extending these ideas to predict the grain boundary character distribution. Essentials of the simulation The thermodynamic theory of our simulation consists of the Mullins Equations, a system of evolution parabolic partial differential equations for each grain boundary curve. In two dimensions, grain boundaries meet at triple junctions, where an additional condition is required. We enforce the Herring force balance condition, which is the natural boundary condition for equilibrium, at these triple junctions. The resulting system is then dissipative for the energy [8]. The evolution of grain boundaries using the above model is thus viewed as a modified steepest descent method for the total grain boundary energy functional, where natural boundary conditions are imposed: this is the variational approach. This observation serves as the basis for our discretization method, and leads to stable semi-discrete schemes. In addition to using the gradient of the above functional to determine the grain boundary evolution we must consider certain critical events. During evolution, a grain boundary or a grain may shrink and disappear. This creates multiple junctions which are unstable. Such multiple junctions split into triple junctions in a way that is consistent with energy reduction. In summary, our implementation results in a globally dissipative system for grain growth of a polycrystalline network. Consider a collection of grain boundaries, which are curves in the plane, meeting at triple junctions. The energy per unit length of a curve G: x = x(s), 0 ≤ s ≤ s0, is given by g(q,a), where q denotes the angle of the normal to the boundary represented by G with respect to a reference axis and a denotes the lattice misorientation across G. The curve evolves according to the equation