Frustrated self-assembly of non-Euclidean crystals of nanoparticles

Self-limiting states of polar misfits: frustrated assembly of warped-jigsaw particles

We study the ground state thermodynamics of a model class of geometrically frustrated assemblies, known as warped-jigsaw particles. While it is known that frustration in soft matter assemblies has the ability to propagate up to mesoscopic, multi-particle size scales, notably through the selection of the self-limiting domain, little is understood about how the symmetry of shape-misfit at the particle scale influences emergent morphologies at the mesoscale. Here we show that polarity in the shape-misfit of warped-jigsaw puzzles manifests at a larger scale in the morphology and thermodynamics of the ground-state assembly of self-limiting domains. We use a combination of continuum theory and discrete particle simulations to show that the polar misfit gives rise to two mesoscopically distinct polar, self-limiting ribbon domains. Thermodynamic selection between the two ribbon morphologies is controlled by a combination of the binding anisotropy along distinct neighbor directions and the orientation of polar shape-misfit. These predictions are valuable as design features for ongoing efforts to program self-limiting assemblies through the synthesis of intentionally frustrated particles, further suggesting a generic classification of frustrated assembly behavior in terms of the relative symmetries of shape-misfit and the underlying long-range inter-particle order it frustrates.

Octo-diamond crystal of nanoscale tetrahedra with interchanging chiral motifs

Despite their simplicity, tetrahedra can assemble into diverse high- and low-density structures. Here we report a low-density 'octo-diamond' structure formed by nanoscale solid tetrahedra with a 64-tetrahedron unit cell containing 8 cubic-diamond subcells. The formed crystal is achiral, but is composed of chiral bilayers with alternating handedness. The left- and right-handed chirality of the bilayers, combined with the plasmonic nature of the gold tetrahedra, produces chiroptical responses at the crystal surface. We uncover that the hydrophobic substrate facilitates the arrangement of tetrahedra into irregular ring-like patterns, creating a critical, uneven topography to stabilize the observed octo-diamond structure. This study reveals a potent way to affect colloidal crystallization through particle-substrate interactions, expanding the nanoparticle self-assembly toolbox.

Statistical Mechanics of Frustrated Assemblies and Incompatible Graphs

Geometrically frustrated assemblies where building blocks misfit have been shown to generate intriguing phenomena from self-limited growth, fiber formation, to structural complexity. We introduce a graph theory formulation of geometrically frustrated assemblies, capturing frustrated interactions through the concept of incompatible flows, providing a direct link between structural connectivity and frustration. This theory offers a minimal yet comprehensive framework for the fundamental statistical mechanics of frustrated assemblies, and connects it to tensor gauge theory formulations of amorphous solids. Through numerical simulations, the theory reveals new characteristics of frustrated assemblies, including two distinct percolation transitions for structure and incompatible flows, a crossover between cumulative and noncumulative frustration controlled by disorder, and a divergent length scale in their response.

Mixed-Valence Coordination Strategy Creating Ordered Ternary Ultrasmall Homo-/Hetero-structures Driven by Lattice Match for Advanced Photochromism and Encryption Applications

Overcoming the challenges of integrating disparate components in nanoarchitectures, this study introduces a straightforward strategy based on a mixed-valence coordination approach, creating an ordered ternary heterostructure integrated with ultrasmall homojunction. This singular ordered homojunction-heterostructure unites ultrathin 1D rutile TiO2 nanowires (NWs) and ultrathin anatase TiO2 NWs with 0D Prussian Blue Analogs (PBAs) nanoparticles (NPs), all exhibiting crystallographic oriented alignment with each other, forming a ternary mesocrystals. Experimental and theoretical insights disclose that the complex interplay between these dissimilar components is governed by a spontaneous lattice match effect, which not only optimizes but also directs the charge transfer, thereby enhancing both efficiency and stability. It also allows for tailoring the valence states of Fe within the PBA, fine-tuning of the composite's photochromic properties, and introducing abundant defect structures that foster strong interaction with oxygen molecules, enabling controllable color-switching dynamics. Consequently, the FeII 1-xFeIII x-PBA/TiO2 exhibits an optimized ternary structure of R-TiO2/A-TiO2/PBA, demonstrating exceptional photoelectronic properties, significantly enhancing photochromism and secure encryption capabilities. These insights establish a solid foundation for engineering sophisticated complex-ordered nanoarchitectures, advancing sustainable energy and environmental technologies.

Size-Dependent, Topology-Regulated, pH-Change-Tolerable, and Reversible Self-Assembly of Ultrasmall Nanoparticles

The multiscale ordering of colloidal nanoparticles (NPs) endows materials with diverse functions and performances. The controllable and predictable assembly of NPs is essential for the new generation of materials science. This study presents a topology-regulated self-assembly approach in an aqueous environment, utilizing polysorbate 20 (Tween-20) and ultrasmall gold nanoparticles (2, 4, and 8 nm AuNPs). The self-assembly process was governed by polyvalent hydrogen bonding interactions between the amphiphilic Tween-20 and tiopronin-capped NPs, with the amphipathic nature of Tween-20 primarily dictating the transformation from 1D to 3D structures. Notably, the NP size influences the assembly process, with the 2 nm particles demonstrating a well-regulated, pH-stable, and reversible assembly capability. Our findings provide a straightforward approach for controlling the assembly of simple nanoparticles and molecules into higher dimensional nano/microstructures, and close the knowledge gap in how NP size affects interactions within the assembly dynamics.

Geometrically Frustrated, Mechanical Metamaterial Membranes: Large-Scale Stress Accumulation and Size-Selective Assembly

We study the effect of geometric frustration on dilational mechanical metamaterial membranes. While shape frustrated elastic plates can only accommodate nonzero Gaussian curvature up to size scales that ultimately vanish with their elastic thickness, we show that frustrated metamembranes accumulate hyperbolic curvatures up to mesoscopic length scales that are ultimately independent of the size of their microscopic constituents. A continuum elastic theory and discrete numerical model describe the size-dependent shape and internal stresses of axisymmetric, trumpetlike frustrated metamembranes, revealing a nontrivial crossover to a much weaker power-law growth in elastic strain energy with size than in frustrated elastic membranes. We study a consequence of this for the self-limiting assembly thermodynamics of frustrated trumpets, showing a severalfold increase in the size range of self-limitation of metamembranes relative to elastic membranes.

Partially Bonded Crystals: A Pathway to Porosity and Polymorphism

In recent years, experimental and theoretical investigations have shown that anisotropic colloids can self-organize into ordered porous monolayers, where the interplay of localized bonding sites, so-called patches, with the particle's shape is responsible for driving the systems away from close-packing and toward porosity. Until now it has been assumed that patchy particles have to be fully bonded with their neighboring particles for crystals to form, and that, if full bonding cannot be achieved due to the choice of patch placement, disordered assemblies will form instead. In contrast, we show that by deliberately displacing the patches such that full bonding is disfavored, a different route to porous crystalline monolayers emerges, where geometric frustration and partial bonding are decisive process. The resulting dangling bonds lead to the emergence of effectively chiral units which then act as building blocks for energetically equivalent crystal polymorphs.

3D Nanofabrication via Directed Material Assembly: Mechanism, Method, and Future

Fabrication of complex three-dimensional (3D) structures at nanoscale is the core of nanotechnology, as it enables the creation of various micro-/nano-devices such as micro-robots, metamaterials, sensors, photonic devices, etc. Among most 3D nanofabrication strategies, the guided material assembly, an efficient bottom-up approach capable of directly constructing designed structures from precise integration of material building blocks, possesses compelling advantages in diverse material compatibility, sufficient driving forces, facile processing steps, and nanoscale resolution. In this review, we focus on assembly-based fabrication methods capable of creating complex 3D nanostructures (instead of periodic or 2.5D-only structures). Recent advances are classified based on the different assembly mechanisms, i.e., assembly driven by chemical reactions, physical interactions, and the synergy of multiple microscopic interactions. The design principles of representative fabrication strategies with an emphasis on their respective advantages, e.g., structural design flexibility, material compatibility, resolution, or applications are analyzed. In the summary and outlook, existing challenges, as well as possible paths to solutions for future development are reviewed. We believe that with recent advances in assembly-based nanofabrication strategies, 3D nanofabrication has achieved tremendous progress in resolution, material generality, and manufacturing cost, for it to make a greater impact in the near future.

Geometrically frustrated self-assembly of hyperbolic crystals from icosahedral nanoparticles

Geometric frustration is a fundamental concept in various areas of physics, and its role in self-assembly processes has recently been recognized as a source of intricate self-limited structures. Here we present an analytic theory of the geometrically frustrated self-assembly of regular icosahedral nanoparticle based on the non-Euclidean crystal {3,5,3} formed by icosahedra in hyperbolic space. By considering the minimization of elastic and repulsion energies, we characterize prestressed morphologies in this self-assembly system. Notably, the morphology exhibits a transition that is controlled by the size of the assembled cluster, leading to the spontaneous breaking of rotational symmetry.

Self-limiting stacks of curvature-frustrated colloidal plates: Roles of intraparticle versus interparticle deformations

In geometrically frustrated assemblies local intersubunit misfits propagate to intra-assembly strain gradients, giving rise to anomalous self-limiting assembly thermodynamics. Here we use theory and coarse-grained simulation to study a recently developed class of "curvamer" particles, flexible shell-like particles that exhibit self-limiting assembly due to the build up of curvature deformation in cohesive stacks. To address a generic, yet poorly understood aspect of frustrated assembly, we introduce a model of curvamer assembly that incorporates both intraparticle shape deformation as well as compliance of interparticle cohesive gaps, an effect we can attribute to a finite range of attraction between particles. We show that the ratio of intraparticle (bending elasticity) to interparticle stiffness not only controls the regimes of self-limitation but also the nature of frustration propagation through curvamer stacks. We find a transition from uniformly bound, curvature-focusing stacks at small size to gap opened, uniformly curved stacks at large size is controlled by a dimensionless measure of inter- versus intracurvamer stiffness. The finite range of interparticle attraction determines the range of cohesion in stacks that are self-limiting, a prediction which is in strong agreement with numerical studies of our coarse-grained colloidal model. These predictions provide critical guidance for experimental realizations of frustrated particle systems designed to exhibit self-limitation at especially large multiparticle scales.

Exotic self-assembly of hard spheres in a morphometric solvent

The self-assembly of spheres into geometric structures, under various theoretical conditions, offers valuable insights into complex self-assembly processes in soft systems. Previous studies have utilized pair potentials between spheres to assemble maximum contact clusters in simulations and experiments. The morphometric approach to solvation free energy that we utilize here goes beyond pair potentials; it is a geometry-based theory that incorporates a weighted combination of geometric measures over the solvent accessible surface for solute configurations in a solvent. In this paper, we demonstrate that employing the morphometric model of solvation free energy in simulating the self-assembly of sphere clusters results, under most conditions, in the previously observed maximum contact clusters. Under other conditions, it unveils an assortment of extraordinary sphere configurations, such as double helices and rhombohedra. These exotic structures arise specifically under conditions where the interactions take multibody potentials into account. This investigation establishes a foundation for comprehending the diverse range of geometric forms in self-assembled structures, emphasizing the significance of the morphometric approach in this context.

Thermal stability and secondary aggregation of self-limiting, geometrically frustrated assemblies: Chain assembly of incommensurate polybricks

In geometrically frustrated assemblies, equilibrium self-limitation manifests in the form of a minimum in the free energy per subunit at a finite, multisubunit size which results from the competition between the elastic costs of frustration within an assembly and the surface energy at its boundaries. Physical realizations-from ill-fitting particle assemblies to self-twisting protein superstructures-are capable of multiple mechanisms of escaping the cumulative costs of frustration, resulting in unlimited equilibrium assembly, including elastic modes of "shape flattening" and the formation of weak, defective bonds that screen intra-assembly stresses. Here we study a model of one-dimensional chain assembly of incommensurate "polybricks" and determine its equilibrium assembly as a function of temperature, concentration, degree of shape frustration, elasticity, and interparticle binding, notably focusing on how weakly cohesive, defective bonds give rise to strongly temperature-dependent assembly. Complex assembly behavior derives from the competition between multiple distinct local minima in the free-energy landscape, including self-limiting chains, weakly bound aggregates of self-limiting chains, and strongly bound, elastically defrustrated assemblies. We show that this scenario, in general, gives rise to anomalous multiple aggregation behavior, in which disperse subunits (stable at low concentration and high temperature) first exhibit a primary aggregation transition to self-limiting chains (at intermediate concentration and temperature) which are ultimately unstable to condensation into unlimited assembly of finite-chains through weak binding beyond a secondary aggregation transition (at low temperature and high concentration). We show that window of stable self-limitation is determined both by the elastic costs of frustration in the assembly as well as energetic and entropic features of intersubunit binding.

Building blocks of non-Euclidean ribbons: size-controlled self-assembly via discrete frustrated particles

Geometric frustration offers a pathway to soft matter self-assembly with controllable finite sizes. While the understanding of frustration in soft matter assembly derives almost exclusively from continuum elastic descriptions, a current challenge is to understand the connection between microscopic physical properties of misfitting "building blocks" and emergent assembly behavior at the mesoscale. We present and analyze a particle-based description of what is arguably the best studied example for frustrated soft matter assembly, negative-curvature ribbon assembly, observed in both assemblies of chiral surfactants and shape-frustrated nanoparticles. Based on our particle model, known as saddle wedge monomers, we numerically test the connection between microscopic shape and interactions of the misfitting subunits and the emergent behavior at the supra-particle scale, specifically focussing on the propagation and relaxation of inter-particle strains, the emergent role of extrinsic shape on frustrated ribbons and the equilibrium regime of finite width selection. Beyond the intuitive role of shape misfit, we show that self-limitation is critically dependent on the finite range of cohesive interactions, with larger size finite assemblies requiring increasing short-range interparticle forces. Additionally, we demonstrate that non-linearities arising from discrete particle interactions alter self-limiting behavior due to both strain-softening in shape-flattened assembly and partial yielding of highly strained bonds, which in turn may give rise to states of hierarchical, multidomain assembly. Tracing the regimes of frustration-limited assembly to the specific microscopic features of misfitting particle shapes and interactions provides necessary guidance for translating the theory of size-programmable assembly into design of intentionally-frustrated colloidal particles.

Chiral assemblies of pinwheel superlattices on substrates

The unique topology and physics of chiral superlattices make their self-assembly from nanoparticles highly sought after yet challenging in regard to (meta)materials^1-3. Here we show that tetrahedral gold nanoparticles can transform from a perovskite-like, low-density phase with corner-to-corner connections into pinwheel assemblies with corner-to-edge connections and denser packing. Whereas corner-sharing assemblies are achiral, pinwheel superlattices become strongly mirror asymmetric on solid substrates as demonstrated by chirality measures. Liquid-phase transmission electron microscopy and computational models show that van der Waals and electrostatic interactions between nanoparticles control thermodynamic equilibrium. Variable corner-to-edge connections among tetrahedra enable fine-tuning of chirality. The domains of the bilayer superlattices show strong chiroptical activity as identified by photon-induced near-field electron microscopy and finite-difference time-domain simulations. The simplicity and versatility of substrate-supported chiral superlattices facilitate the manufacture of metastructured coatings with unusual optical, mechanical and electronic characteristics.

Multifunctional Hyphae Carbon Powering Lithium-Sulfur Batteries

Biotechnology can bring new breakthroughs on design and fabrication of energy materials and devices. In this work, a novel and facile biological self-assembly technology to fabricate multifunctional Rhizopus hyphae carbon fiber (RHCF) and its derivatives on a large scale for electrochemical energy storage is proposed. Crosslinked hollow carbon fibers are successfully prepared by conversion of Rhizopus hyphae, and macroscopic production of centimeter-level carbon balls consisting of hollow RHCFs is further realized. Moreover, the self-assembled RHCF balls show strong adsorption characteristics on metal ions and can be converted into a series of derivatives such as RHCF/metal oxides. Notably, the designed RHCF derivatives are demonstrated with powerful multifunctionability as cathode, anode, and separator for lithium-sulfur batteries (LSBs). The RHCF can act as the host material to combine with metal oxide (CoO) and S, Li metal, and a polypropylene (PP) separator to form a new RHCF/CoO-S cathode, an RHCF/Li anode, and an RHCF/PP separator, respectively. Consequently, the optimized LSB full cell presents excellent cycling performance and superior high-rate capacity (881.3 mA h g^-1 at 1 C). This work provides a new method for large-scale preparation of hollow carbon fibers and derivatives for advanced energy storage and conversion.

Cumulative geometric frustration in physical assemblies

Geometric frustration arises whenever the constituents of a physical assembly locally favor an arrangement that cannot be realized globally. Recently, such frustrated assemblies were shown to exhibit filamentation, size limitation, large morphological variations and other exotic response properties. While these unique characteristics can be shown to be a direct outcome of the geometric frustration, some geometrically frustrated systems do not exhibit any of the above phenomena. In this work we exploit the intrinsic approach to provide a framework for directly addressing the frustration in physical assemblies. The framework highlights the role of the compatibility conditions associated with the intrinsic fields describing the physical assembly. We show that the structure of the compatibility conditions determines the behavior of small assemblies and in particular predicts their superextensive energy growth exponent. We illustrate the use of this framework to several well-known frustrated assemblies.