Spectral graph theory for characterization and homogenization of grain boundary networks

Abstract Grain boundary networks (GBNs) have a profound influence on the properties of both structural and functional materials. However, existing methods to characterize their complex structure have almost universally relied upon a binary classification of GBs as either “special” or “general”, which ignores the rich and continuous spectrum of GB types and properties. Furthermore, characterizing the aggregate network structure of GBs has proven complicated, with traditional methods focusing on local structure and also relying on a binary GB taxonomy, e.g. by evaluating how many “special” or “general” boundaries meet at triple junctions or quadruple nodes. Here we develop new structural metrics for GBNs, based on spectral graph theory, that encode both global network topology and the full spectrum of constituent GB properties, enabling high-fidelity characterization of arbitrary GBNs. Using these metrics, we derive an new structure-property relation for GBN diffusivity. The dominant term in this expression provides an efficient and accurate approximation, whose corresponding spectral metrics reveal the dominant microstructural features influencing the property of interest. The spectral index of this term serves as a type of global order parameter that reveals a fundamental structural transition in GBNs. This work provides a new framework to characterize the structure of GBNs in greater generality than previously possible and facilitates the development of new defect-sensitive structure-property models.