# A Model for the Origin of Anisotropic Grain Boundary Character Distributions in Polycrystalline Materials

Abstract: A model is described for the development of anisotropic grain boundary character distributions from initially random distributions. The model is based on biased topological changes in the grain boundary network that eliminate and create boundaries during grain growth. The grain boundary energy influences the rates of these topological changes by altering the relative areas of the interfaces. The model predicts grain boundary character distributions that are inversely related to the grain boundary energy and are consistent with experimental observations.

## Summary (2 min read)

### INTRODUCTION

- The grain boundary character distribution (GBCD) is defined as the relative areas of grain boundaries as a function of lattice misorientation and grain boundary orientation.
- The only available comprehensive experimental data indicates that the logarithm of the population is approximately linear with the energy, which is consistent with the results of a three-dimensional computer simulation [6, 10, 11].
- This lengthening and shortening of boundaries enhances the relative areas of low energy grain boundaries.
- The purpose of this paper is to describe a model for the formation of anisotropic GBCDs from initially random GBCDs during normal grain growth.

### THE MODEL

- The authors begin by considering how the grain boundary energy influences the grain boundary area.
- In fact, in what follows, the authors assume that boundary area changes are the mechanism that biases topological changes and alters the number fractions of grain boundary types.
- Recent three dimensional grain growth simulations with anisotropic properties has shown that this is the case [15].
- To determine how the GBCD changes as critical events occur, the authors begin by assuming an initial random distribution and a functional form for the anisotropy of the grain boundary energy.
- With these initial conditions, all ∆ni are computed according to Eq. 3 and the populations, ni, are adjusted.

### RESULTS

- The number fractions of grain boundaries with the minimum energy and the maximum energy are plotted as a function of the number of critical events in Fig.
- Note that these changes in the numbers of boundaries do not account for the relative areas.
- The area distributions derived from the simulations are similar to those depicted in Fig.
- The results of these simulations were used to find a relationship between the relative grain boundary areas and the relative grain boundary energies.
- As illustrated in Fig. 7, the relationship between the logarithm of the population determined by the critical event model and the energy is nearly linear over small ranges of energy anisotropy.

### DISCUSSION

- The model described here reproduces, qualitatively, the principal features observed in anisotropic grain boundary character distributions.
- Second, it can produce MRD values greater than 3, as observed in real distributions.
- Finally, the logarithms of the areas of grain boundaries show an approximate inverse linear dependence on the grain boundary energy.
- The consistency between the observed distributions and those produced by the model indicates that the assumed mechanisms for the evolution of the distribution are plausible.
- The energy anisotropies have to have realistic ranges and exhibit variations over all crystallographic degrees of freedom.

### CONCLUSION

- This area anisotropy affects the probabilities with which different grain boundary types are eliminated during grain growth, and this leads to anisotropy in the numbers of boundaries of different types.
- A simple model based on these principles predicts a relationship between grain boundary energy and the GBCD that is consistent with experimental observations.

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