Characterizing grain boundary network length features through a harmonic representation

When modeling and characterizing grain boundary networks (GBNs), there are situations where local descriptors such as triple junction fractions (TJFs) and special boundary fractions are identical between two microstructures, but the performance or properties between the two are distinct. These differences are caused by higher length-scale features that cannot be identified using local structural descriptors. Spectral graph theory (SGT) has been used previously to encode network length features, but did not enable direct quantitative comparisons between the structures that cause property differences. In this paper, we derive a harmonic representation of diffusion on GBNs based on SGT. This method enables direct quantitative comparisons between GBNs for microstructures with different morphologies, and identifies network length features responsible for structural and performance differences, which local descriptors cannot explain. We show an interpretation of the eigenmodes generated by this method that explains long-range structural causes of certain property differences. We apply this method to a large library of microstructures, and identify structural classes through clustering. We show that equal proportioned TJF and J 1 dominated microstructures are the most sensitive to network length differences because of boundary configurations, while J 2-J3 J 3 dominated structures are the least sensitive. This method also identifies network length features that result in anomalous percolation/non-percolation compared to predictions based on local correlations alone.