Graph theoretical description of phase transitions in nanoscale assemblies of surfactants

Abstract Phase transitions are typically quantified using order parameters, such as crystal lattice distances, Legendre polynomials, and radial distribution functions (RDF). These parameters can identify even subtle changes in packing distances and symmetries being ideally suited for crystalline materials or high-contrast phases with large structural differences. However, the structural changes are more difficult to quantify when the organizational pattern of constituent molecules becomes less ordered and more complex. This is the case for phases with high asymmetry and multiscale organization as well as for the structural fluctuations preceding phase transitions, which are essential for understanding the system pathways between phases. Here, we show that graph theoretical (GT) descriptors can successfully characterize different phases and pathway between them for hierarchically organized nanoscale systems exemplified by binary surfactant/water systems. We found that the GT centrality parameters and node-based fractal dimension can identify complex phases more accurately than traditional analyses and quantify the system behavior preceding the transitions. The trajectories for GT parameter of closeness centrality reflect (1) fluctuation-induced breakup of micelles and (2) long-range cooperative interactions in the network of surfactant molecules. GT parametrization can be generalized for a wide range of nanoscale colloidal systems, which can facilitate identification of complex phases.