Decoding collective dynamics and complexity in nanoparticle assemblies using graph theory.
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| Title: | Decoding collective dynamics and complexity in nanoparticle assemblies using graph theory. |
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| Authors: | Hallstrom, Jonas (AUTHOR), Pan, Puquan (AUTHOR), Sia, Jayson (AUTHOR), Bae, Sangwok (AUTHOR), Qian, Dingwen (AUTHOR), Qian, Chang (AUTHOR), Liu, Sindy (AUTHOR), Yao, Lehan (AUTHOR), Truskett, Thomas M. (AUTHOR), Milliron, Delia J. (AUTHOR), Chen, Qian (AUTHOR), Mao, Xiaoming (AUTHOR), Bogdan, Paul (AUTHOR), Kotov, Nicholas A. (AUTHOR) |
| Source: | Science. 5/14/2026, Vol. 392 Issue 6799, p1-12. 12p. |
| Subjects: | Graph theory, Curvature, Nanostructured materials, Technological complexity, Plasmonics, Nanoparticles |
| Abstract: | 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. Editor's summary: Self-assembly of nanoparticles often produces structures that lie between crystalline order and complete disorder, but describing these states remains challenging. Hallstrom et al. applied graph theory to quantify the evolution of truncated gold nanocube assemblies imaged using electron microscopy. They showed that Ollivier-Ricci curvature and augmented Forman-Ricci curvature can track global structural complexity and local energetic stability, respectively. Moreover, they revealed that an intermediate, partially ordered network coincides with the strongest plasmonic responses. Finally, they confirmed that this theory is generalizable to other systems such as gold nanoprisms and indium tin oxide nanospheres. —Jack Huang INTRODUCTION: Nanoparticles (NPs) occupy a unique intermediate scale between molecules and colloids, exhibiting self-assembly behaviors that combine characteristics of both but remain fundamentally distinct, often resulting in complex structures that combine order and disorder. These partially organized states, including chains, clusters, pores, and low-density networks, are critical for technological applications yet remain challenging to quantify using traditional symmetry-based descriptors. A general methodology to quantitatively describe these low-order, low-density states is essential for advancing applications that exploit their optical, mechanical, and electrical properties. RATIONALE: We provide graph theory (GT) as a mathematical framework to characterize dynamic NP systems transitioning from dispersions of free particles to fully assembled colloidal crystals. Using wide-frame time-resolved liquid-phase transmission electron microscopy (LPTEM), we tracked >400 gold nanocubes in real time, supplemented by molecular dynamics (MD) and kinetic Monte Carlo (kMC) simulations of systems containing up to 10,000 particles. We focused on two new GT metrics: Ollivier-Ricci curvature (ORC), which captures global structural and transport patterns by quantifying how local interactions propagate through neighborhoods beyond nearest neighbors, and augmented Forman-Ricci curvature (AFRC), which quantifies local robustness through three-particle cluster dynamics. ORC provides a continuous measure of structural complexity, with negative values indicating bridging edges between communities and near-zero values representing regular lattices. AFRC tracks the formation and dissolution of energetically unfavorable three-particle cycles, serving as a proxy for system energy and reconfigurability. RESULTS: Analysis of gold nanocube assembly revealed three distinct stages characterized by ORC trajectories. Both the initial and final stages of the assembly showed near zero average ORC due to their near perfect disorder and order, respectively, whereas the intermediate stage exhibited strongly negative ORC, corresponding to the formation of dynamic mesocrystals (groups of 4 to 20 aligned NPs) interconnected by bridging edges. AFRC displayed inverse correlation with mean coordination number and system energy, with sigmoidal dependence capturing the progression from reconfigurable to stable states. Notably, we discovered a "Goldilocks regime" in which structural complexity maximizes functionality: The peak near infrared backscattering intensity (at 820 nm) occurred precisely when ORC reached its minimum at ~120 s, corresponding to interconnected mesocrystals with low symmetry. This intermediate state, combining order and disorder, produced 40% higher plasmonic response than either fully dispersed particles or perfectly ordered superlattices. The generalizability of the GT framework was demonstrated across three NP systems: gold nanocubes forming rhombic lattices, gold nanoprisms assembling into hexagonal superlattices through liquid-like nanocolumn intermediates, and indium tin oxide nanospheres forming either crystalline clusters or kinetically trapped gels depending on interaction strength. CONCLUSION: The GT metrics, such as ORC and AFRC, provide continuous measures of global connectivity patterns and local energetic states throughout dynamic self-assembly processes, capturing features invisible to conventional order parameters. Most importantly, ORC quantifies effective complexity in NP assemblies, enabling experimental validation of decades-old theoretical predictions that functionality peaks at intermediate complexity states combining order and disorder. The framework provides a path forward for designing and controlling complex nano- and microstuctured materials with properties that emerge from the interplay between local order and global connectivity, advancing applications in biosensing, photonics, and lightweight nanomaterials. Graph curvature measures quantify disorder and complexity for self-assembled particle systems.: NP assemblies can be viewed as a series of graphs or networks in which nodes represent NPs and edges represent strong interparticle interactions. ORC and AFRC can be found for every edge in these graphs, quantifying the structural order and complexity in the local area around those edges. Scale bars, 200 nm. [ABSTRACT FROM AUTHOR] |
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| Database: | Psychology and Behavioral Sciences Collection |
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| Abstract: | 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. Editor's summary: Self-assembly of nanoparticles often produces structures that lie between crystalline order and complete disorder, but describing these states remains challenging. Hallstrom et al. applied graph theory to quantify the evolution of truncated gold nanocube assemblies imaged using electron microscopy. They showed that Ollivier-Ricci curvature and augmented Forman-Ricci curvature can track global structural complexity and local energetic stability, respectively. Moreover, they revealed that an intermediate, partially ordered network coincides with the strongest plasmonic responses. Finally, they confirmed that this theory is generalizable to other systems such as gold nanoprisms and indium tin oxide nanospheres. —Jack Huang INTRODUCTION: Nanoparticles (NPs) occupy a unique intermediate scale between molecules and colloids, exhibiting self-assembly behaviors that combine characteristics of both but remain fundamentally distinct, often resulting in complex structures that combine order and disorder. These partially organized states, including chains, clusters, pores, and low-density networks, are critical for technological applications yet remain challenging to quantify using traditional symmetry-based descriptors. A general methodology to quantitatively describe these low-order, low-density states is essential for advancing applications that exploit their optical, mechanical, and electrical properties. RATIONALE: We provide graph theory (GT) as a mathematical framework to characterize dynamic NP systems transitioning from dispersions of free particles to fully assembled colloidal crystals. Using wide-frame time-resolved liquid-phase transmission electron microscopy (LPTEM), we tracked >400 gold nanocubes in real time, supplemented by molecular dynamics (MD) and kinetic Monte Carlo (kMC) simulations of systems containing up to 10,000 particles. We focused on two new GT metrics: Ollivier-Ricci curvature (ORC), which captures global structural and transport patterns by quantifying how local interactions propagate through neighborhoods beyond nearest neighbors, and augmented Forman-Ricci curvature (AFRC), which quantifies local robustness through three-particle cluster dynamics. ORC provides a continuous measure of structural complexity, with negative values indicating bridging edges between communities and near-zero values representing regular lattices. AFRC tracks the formation and dissolution of energetically unfavorable three-particle cycles, serving as a proxy for system energy and reconfigurability. RESULTS: Analysis of gold nanocube assembly revealed three distinct stages characterized by ORC trajectories. Both the initial and final stages of the assembly showed near zero average ORC due to their near perfect disorder and order, respectively, whereas the intermediate stage exhibited strongly negative ORC, corresponding to the formation of dynamic mesocrystals (groups of 4 to 20 aligned NPs) interconnected by bridging edges. AFRC displayed inverse correlation with mean coordination number and system energy, with sigmoidal dependence capturing the progression from reconfigurable to stable states. Notably, we discovered a "Goldilocks regime" in which structural complexity maximizes functionality: The peak near infrared backscattering intensity (at 820 nm) occurred precisely when ORC reached its minimum at ~120 s, corresponding to interconnected mesocrystals with low symmetry. This intermediate state, combining order and disorder, produced 40% higher plasmonic response than either fully dispersed particles or perfectly ordered superlattices. The generalizability of the GT framework was demonstrated across three NP systems: gold nanocubes forming rhombic lattices, gold nanoprisms assembling into hexagonal superlattices through liquid-like nanocolumn intermediates, and indium tin oxide nanospheres forming either crystalline clusters or kinetically trapped gels depending on interaction strength. CONCLUSION: The GT metrics, such as ORC and AFRC, provide continuous measures of global connectivity patterns and local energetic states throughout dynamic self-assembly processes, capturing features invisible to conventional order parameters. Most importantly, ORC quantifies effective complexity in NP assemblies, enabling experimental validation of decades-old theoretical predictions that functionality peaks at intermediate complexity states combining order and disorder. The framework provides a path forward for designing and controlling complex nano- and microstuctured materials with properties that emerge from the interplay between local order and global connectivity, advancing applications in biosensing, photonics, and lightweight nanomaterials. Graph curvature measures quantify disorder and complexity for self-assembled particle systems.: NP assemblies can be viewed as a series of graphs or networks in which nodes represent NPs and edges represent strong interparticle interactions. ORC and AFRC can be found for every edge in these graphs, quantifying the structural order and complexity in the local area around those edges. Scale bars, 200 nm. [ABSTRACT FROM AUTHOR] |
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| ISSN: | 00368075 |
| DOI: | 10.1126/science.aeb5134 |
