Nature's self-organized complex structures, such as viral capsids and bacterial flagella, exhibit efficiency, adaptability, and multi-functionality. This research investigates whether similar sophistication can be achieved in engineered materials by programming the self-assembly of three-dimensional (3D) complex structures using simple building blocks. The challenge lies in understanding how simple polyhedral nanoparticles (NPs) self-assemble into complex structures, a problem complicated by the interplay of NP-NP interactions and the high-dimensional phase space of assembly. Simulations struggle with the complexity of NP interactions and the rugged free-energy landscape, while real-time imaging techniques have only recently reached sufficient resolution to study these pathways. The answer is not only crucial for emerging technologies, but also provides insights into natural hierarchical structure formation. The paper proposes that this self-assembly phenomenon can be understood through the lens of non-Euclidean crystal structures of these polyhedra. While assembling most polyhedra in 3D Euclidean space is geometrically frustrated (leading to gaps or overlaps), they can form non-Euclidean crystals in curved spaces where these issues are eliminated by adjusting the Gaussian curvature. The paper uses this principle to construct an energy functional guiding the search for self-assembly morphologies, considering the competition between elastic frustration, electrostatic repulsion, boundary energy, and binding energy. The focus is on low-dimensional morphologies (2D sheets and 1D bundles) arising from competing attractive and repulsive interactions.

The paper reviews prior work on the self-assembly of chemically synthesized nanoparticles with diverse polyhedral shapes. It cites studies demonstrating the formation of various complex assemblies, including helices, curved platelets, and capsids, under attractive van der Waals forces, hydrogen bonds, and coordination bonds. Existing simulation challenges are highlighted, along with the recent advances in real-time imaging techniques to investigate self-assembly pathways. The authors draw a connection to the mathematical problem of packing regular polyhedra in 3D Euclidean space, emphasizing the geometric frustration inherent in such packings. The literature also informs the concept of frustrated self-assembly, involving both geometric frustration and repulsion-attraction frustration. Previous studies using non-Euclidean crystals in understanding complex structures of condensed matter, ranging from Frank-Kasper phases to metallic glasses and biological materials, are reviewed, laying groundwork for the proposed approach.

The core methodology involves developing a general model energy for frustrated NP self-assembly. This energy is composed of four terms: elastic frustration energy (Eelastic), electrostatic repulsion energy (Erepulsion), boundary energy (Eboundary), and binding energy (Ebind). The researchers use a continuum approach, modeling NPs and coordination bonds as a homogeneous continuum, even though the actual problem is discrete. The elastic energy is expressed in terms of the strain tensor (ε) and Lamé coefficients (μ, λ), characterizing the deviation of the actual metric (g) from the ideal stress-free metric (g) of the non-Euclidean crystal. Electrostatic repulsion is incorporated into Erepulsion. The boundary and binding energies are considered as functions of the chosen slice from the non-Euclidean crystal. The approach involves two steps: selecting an appropriate slice (M) from the non-Euclidean crystal, which depends on kinetic pathways and the competition between energy terms, and solving for the morphology of the assembled structure given the slice. The paper then specializes the model to the self-assembly of charged tetrahedral CdTe NPs using chiral surface ligands. The continuum approach is justified by the size of the NPs (3-5 nm) and coordination bridges (1 nm), allowing modeling of NPs and ligands as deformable tetrahedra and the assemblies as an elastic continuum. The 600-cell polytope, a regular tiling of the 3-sphere S³ by regular tetrahedra, is selected as the reference configuration because of its low curvature and minimal frustration. The authors derive the reference metric g from the 600-cell, parameterized with angular coordinates. They choose a specific slice M—a thin shell between toroidal surfaces—justified by kinetic pathways and energetic arguments, and analyze the energy of this shell. The thin shell approximation is used, expanding the metric around the mid-surface. The energy is minimized to find the actual first and second fundamental forms of the mid-surface, revealing a helicoidal structure whose chirality matches the chiral ligands. The pitch of the helicoid is shown to be tunable via electrostatic repulsion, controlled by charge density, screening length, and dielectric constant. Experiments using CdTe NPs with L- or D-cysteine ligands are conducted, and the results are compared to the theoretical predictions.

The key findings include the development of a theoretical model that successfully predicts the self-assembly of tetrahedral nanoparticles into helicoidal ribbons. This model leverages the concept of non-Euclidean crystals to describe the ideal, stress-free packing of nanoparticles, and it incorporates the effects of elastic frustration, electrostatic repulsion, and chiral surface ligands. The model accurately predicts the chirality of the assembled structures, and it demonstrates that the pitch of the helices can be tuned by adjusting the electrostatic repulsion between the nanoparticles. Experimental results using CdTe nanoparticles coated with L- or D-cysteine ligands confirm the theoretical predictions. Specifically, the experiments show that the pitch of the helicoidal ribbons increases with increasing methanol concentration in the solvent mixture, and the chirality of the ribbons matches the chirality of the ligands. Furthermore, numerical simulations of circular dichroism (CD) spectra demonstrate that the self-assembled helicoids exhibit a strong chiroptical response in the near-infrared region, which has potential applications in various fields including biomedical imaging and remote sensing. The analysis shows a monotonic increase of the CD peak with the pitch. This is validated by the agreement between experimental CD peak positions and theoretically predicted values. The amplitude of the CD spectra is significantly higher than that of typical biological molecules, further highlighting the potential for practical applications. The theory also suggests that the scalability of the assembly is a significant factor in achieving high yields, attributed to the translational invariance of the reference metric. This implies that smaller pieces of the sheet can merge and grow into larger structures. Finally, the study illustrates how chirality propagates from the molecular scale (ligands) to the micron-scale assembled helices.

The findings address the research question by demonstrating that a non-Euclidean crystal framework can accurately predict the complex self-assembly behavior of nanoparticles. The significance of the results lies in establishing a new design space for creating complex nano-structures with tunable properties. The ability to control the pitch of chiral helices through electrostatic repulsion opens up numerous possibilities for designing materials with specific optical and other functionalities. The agreement between theoretical predictions and experimental observations validates the approach and provides confidence in its predictive power for other nanoparticle systems. The high yield of the self-assembly process, attributed to the scalability of the system, further enhances the practical implications of this work. The propagation of chirality from the molecular to the macroscopic scale provides a deeper understanding of chirality transfer mechanisms in self-assembly. This research contributes to the broader field of self-assembly by introducing a new theoretical framework and demonstrating its utility for designing advanced materials with tailored properties.

This research introduces a non-Euclidean self-assembly theory for polyhedral nanoparticles, successfully explaining and predicting the formation of complex structures from simple building blocks. The application to tetrahedral NPs with chiral binding demonstrates the theory's ability to predict helicoidal structures, aligning with experimental observations. Electrostatic repulsion is identified as a crucial parameter for controlling final morphology. Future research could expand this framework to various NP types and explore diverse ways of selecting slices from non-Euclidean crystals to generate novel morphologies and materials capabilities.

While the study successfully models the self-assembly of tetrahedral nanoparticles, it simplifies the system using a continuum approach and thin shell approximation. The model might not capture all aspects of the complex interactions and kinetic processes, especially at larger length scales. Furthermore, the study focuses on a specific type of nanoparticle and ligand, limiting the direct generalizability of the findings to other systems. The focus on equilibrium morphologies might not fully account for non-equilibrium effects that can be significant in self-assembly processes.