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Colloidal gels assembled from nanoparticles (NPs) are a versatile class of soft network-based materials capable of rich dynamic, mechanical, and even optical or magnetic responses to stimuli. Their behaviors are governed by dynamics of heterogeneous structures coupled across multiple length and timescales. Observable dynamics range from nanoparticle diffusion and clustering to mesoscopic cluster dynamics and interactions to localized or collective network relaxations. Understanding how these hierarchically organized processes relate to macroscopic network properties remains a broad and unresolved problem in soft matter physics. The mechanisms of gel formation can depend sensitively on the pathway and the nature of NP interactions, thus far preventing a unified theoretical bridge between nanoscopic interactions, structural evolution, and network dynamics. Indirect measurement of dynamics using light-scattering techniques provides an experimental means to quantify underlying particle and network motion. The rich dynamic behavior of NP gels warrants consideration of a broad range of models to help interpret nonlinear relaxation phenomena such as anomalous diffusion, nonergodicity, and intrinsically nonequilibrium or mechanically driven dynamics. X-ray photon correlation spectroscopy (XPCS) has emerged as a powerful tool for probing nanoscopic motion in nanoparticle gels but alone cannot resolve the full spatiotemporal spectrum of dynamics that drive gelation, aging, and network mechanical properties. While rheo-XPCS enables simultaneous probing of nanoscale and bulk mechanical responses, complementary light scattering, microscopy, or simulations can extend spatiotemporal characterization and, consequently, understanding of NP gel network physics. Implementing a modular model platform with tunable primary nanoparticle features allows systematic variation of nanoscopic characteristics that drive emergent gel responses and inform the development of theoretical models for a wide range of soft, dynamic, nanostructured materials. Gels formed from particles with unique structural proxies, such as electromagnetic coupling in plasmonic NPs, provide additional metrics for model validation and offer opportunities to develop computational methods for the efficient and accurate replication of NP gel properties. The rapid expansion of XPCS capabilities at fourth-generation light sources, combined with complementary tools and robust model systems, positions the field to move beyond descriptive fundamental studies toward the design of nanoparticle gels with adaptive and programmable behaviors. 

Being intermediate in scale between molecules and colloids, nanoparticles combine characteristics of both. The structure of their self-assembled states combining order and disorder is difficult to quantify using traditional symmetry-based descriptors. Here, we applied graph theory (GT) to analyze assemblies of 400 to 10,000 nanoparticles across three material systems. We show that GT metrics, augmented Forman-Ricci curvature (AFRC) and Ollivier-Ricci curvature (ORC), capture local and global structural transitions from small clusters to extended networks. AFRC reflects the energetic state of the assembly, whereas ORC quantifies structural complexity and reveals a “Goldilocks” regime that maximizes plasmonic response. The generality of this approach is demonstrated for gold nanocubes, gold nanoprisms, and indium tin oxide nanospheres, providing a unified framework for describing and optimizing complex nanoparticle assemblies. 

Adding nonadsorbing polymers to hard microsphere dispersions generates osmotic depletion attractions that can be quantitatively predicted and designed to manipulate colloidal phase behavior. Whether depletion described by classical theories is the mechanism for polymer-mediated nanosphere attractions is less evident. Colloidal hard nanospheres and nonadsorbing polymers are challenging to realize given the diverse interactions typically present in nanoparticle dispersions. Here, we use small-angle x-ray scattering to assess whether the depletion mechanism holds at the nanoscale, leveraging a recent finding that uncharged, oleate-capped indium oxide nanocrystals exhibit near–hard-sphere interactions in toluene. Classical modeling of polystyrene depletant as penetrable spheres predicts depletion-induced phase boundaries, nanocrystal second osmotic virial coefficients, and colloidal structuring in agreement with experiments for polymer radii of gyration up to 80% of the nanocrystal radius. Experimentally observed weakening of depletion interactions for larger polymer-to-nanocrystal size ratios qualitatively follows theoretical predictions that account for how polymer physics influences depletant interactions.