Decoding Collective Dynamics and Complexity in Nanoparticle Assemblies via Graph Theory

Mathematically analyzing complex structures and pathways that emerge during the process of self-assembly, combining order and disorder, remains difficult with existing approaches, which typically focus on local order parameters and pairwise correlations. Considering nanoparticle (NP) dispersions as a complex system, we develop a generalized graph theory (GT)-based analysis of structures and dynamics over the assembly process. This enables us to parameterize the pathway from randomly distributed NPs to a network of interconnected communities that eventually reconfigure into a large colloidal lattice. Our GT framework utilizes the notion of graph curvature, in particular the formulation Ollivier-Ricci Curvature (ORC). Comparison of experimental pseudo-2D self-assembly trajectories captured by liquid-phase transmission electron microscopy and molecular dynamics simulations reveal that ORC is directly related to the evolving structural complexity of the NP system, peaking during the intermediate stages of community formation. These measures of complexity correlate with the functions of the assemblies, namely the plasmonic properties of the NP system where the blue-light scattering cross-sections of the assembly peak at the minimum ORC of the structure.