Characterizing Structure Through Shape Matching and Applications to Self-Assembly

Structural quantities such as order parameters and correlation functions are often employed to gain insight into the physical behavior and properties of condensed matter systems. Although standard quantities for characterizing structure exist, often they are insufficient for treating problems in the emerging field of nano- and microscale self-assembly, wherein the structures encountered may be complex and unusual. The computer science field of shape matching offers a robust solution to this problem by defining diverse methods for quantifying the similarity between arbitrarily complex shapes. Most order parameters and correlation functions used in condensed matter apply a specific measure of structural similarity within the context of a broader scheme. By substituting shape matching quantities for traditional quantities, we retain the essence of the broader scheme, but extend its applicability to more complex structures. Here we review some standard shape-matching techniques and discuss how they might be used to create highly flexible structural metrics for diverse systems such as self-assembled matter. We provide three proof-of-concept example problems applying shape-matching methods to identifying local and global structures and tracking structural transitions in complex assembled systems. The shape-matching methods reviewed here are applicable to a wide range of condensed matter systems, both simulated and experimental, provided particle positions are known or can be accurately imaged.