Classification of complex local environments in systems of particle shapes through shape symmetry-encoded data augmentation.

Detecting and analyzing the local environment is crucial for investigating the dynamical processes of crystal nucleation and shape colloidal particle self-assembly. Recent developments in machine learning provide a promising avenue for better order parameters in complex systems that are challenging to study using traditional approaches. However, the application of machine learning to self-assembly on systems of particle shapes is still underexplored. To address this gap, we propose a simple, physics-agnostic, yet powerful approach that involves training a multilayer perceptron (MLP) as a local environment classifier for systems of particle shapes, using input features such as particle distances and orientations. Our MLP classifier is trained in a supervised manner with a shape symmetry-encoded data augmentation technique without the need for any conventional roto-translations invariant symmetry functions. We evaluate the performance of our classifiers on four different scenarios involving self-assembly of cubic structures, two-dimensional and three-dimensional patchy particle shape systems, hexagonal bipyramids with varying aspect ratios, and truncated shapes with different degrees of truncation. The proposed training process and data augmentation technique are both straightforward and flexible, enabling easy application of the classifier to other processes involving particle orientations. Our work thus presents a valuable tool for investigating self-assembly processes on systems of particle shapes, with potential applications in structure identification of any particle-based or molecular system where orientations can be defined.

Space-tiled colloidal crystals from DNA-forced shape-complementary polyhedra pairing.

Generating space-filling arrangements of most discrete polyhedra nanostructures of the same shape is not possible. However, if the appropriate individual building blocks are selected (e.g., cubes), or multiple shapes of the appropriate dimensions are matched (e.g., octahedra and tetrahedra) and their pairing interactions are subsequently forced, space-filled architectures may be possible. With flexible molecular ligands (polyethylene glycol-modified DNA), the shape of a polyhedral nanoparticle can be deliberately altered and used to realize geometries that favor space tessellation. In this work, 10 new colloidal crystals were synthesized from DNA-modified nanocrystal building blocks that differed in shapes and sizes, designed to form space-filling architectures with micron-scale dimensions. The insights and capabilities provided by this new strategy substantially expand the scope of colloidal crystals possible and provide an expanded tool kit for researchers interested in designing metamaterials.

Colloidal quasicrystals engineered with DNA.

In principle, designing and synthesizing almost any class of colloidal crystal is possible. Nonetheless, the deliberate and rational formation of colloidal quasicrystals has been difficult to achieve. Here we describe the assembly of colloidal quasicrystals by exploiting the geometry of nanoscale decahedra and the programmable bonding characteristics of DNA immobilized on their facets. This process is enthalpy-driven, works over a range of particle sizes and DNA lengths, and is made possible by the energetic preference of the system to maximize DNA duplex formation and favour facet alignment, generating local five- and six-coordinated motifs. This class of axial structures is defined by a square-triangle tiling with rhombus defects and successive on-average quasiperiodic layers exhibiting stacking disorder which provides the entropy necessary for thermodynamic stability. Taken together, these results establish an engineering milestone in the deliberate design of programmable matter.

Complex motion of steerable vesicular robots filled with active colloidal rods.

While the collective motion of active particles has been studied extensively, effective strategies to navigate particle swarms without external guidance remain elusive. We introduce a method to control the trajectories of two-dimensional swarms of active rod-like particles by confining the particles to rigid bounding membranes (vesicles) with non-uniform curvature. We show that the propelling agents spontaneously form clusters at the membrane wall and collectively propel the vesicle, turning it into an active superstructure. To further guide the motion of the superstructure, we add discontinuous features to the rigid membrane boundary in the form of a kinked tip, which acts as a steering component to direct the motion of the vesicle. We report that the system's geometrical and material properties, such as the aspect ratio and Péclet number of the active rods as well as the kink angle and flexibility of the membrane, determine the stacking of active particles close to the kinked confinement and induce a diverse set of dynamical behaviors of the superstructure, including linear and circular motion both in the direction of, and opposite to, the kink. From a systematic study of these various behaviors, we design vesicles with switchable and reversible locomotions by tuning the confinement parameters. The observed phenomena suggest a promising mechanism for particle transportation and could be used as a basic element to navigate active matter through complex and tortuous environments.

Tunable assembly of host-guest colloidal crystals.

Entropy compartmentalization provides new self-assembly routes to colloidal host-guest (HG) structures. Leveraging host particle shape to drive the assembly of HG structures has only recently been proposed and demonstrated. However, the extent to which the guest particles can dictate the structure of the porous network of host particles has not been explored. In this work, by modifying only the guest shape, we show athermal, binary mixtures of star-shaped host particles and convex polygon-shaped guest particles assemble as many as five distinct crystal structures, including rotator and discrete rotator guest crystals, two homoporous host crystals, and one heteroporous host crystal. Edge-to-edge alignment of neighboring stars results in the formation of three distinct pore motifs, whose preferential formation is controlled by the size and shape of the guest particles. Finally, we confirm, via free volume calculations, that assembly is driven by entropy compartmentalization, where the hosts and guests contribute differently to the free energy of the system; free volume calculations also explain differences in assembly based on guest shape. These results provide guest design rules for assembling colloidal HG structures, especially on surfaces and interfaces.

Entropy compartmentalization stabilizes open host-guest colloidal clathrates.

Clathrates are open crystals in which molecules are arranged in a hierarchy of polyhedral cages that encapsulate guest molecules and ions. As well as holding fundamental interest, molecular clathrates serve practical purposes, such as for gas storage, and their colloidal counterparts also appear promising for host-guest applications. Here using Monte Carlo simulations, we report the entropy-driven self-assembly of hard truncated triangular bipyramids into seven different host-guest colloidal clathrate crystals with unit cells ranging from 84 to 364 particles. The structures consist of cages that are either empty or occupied by guest particles, which can be different from or identical to the host particles. The simulations point to crystallization occurring through the compartmentalization of entropy between low- and high-entropy subsystems for the host and the guest particles, respectively. We use entropic bonding theory to design host-guest colloidal clathrates with explicit interparticle attraction, providing a route to realize such systems in the laboratory.

Inverse design of triblock Janus spheres for self-assembly of complex structures in the crystallization slot via digital alchemy.

The digital alchemy framework is an extended ensemble simulation technique that incorporates particle attributes as thermodynamic variables, enabling the inverse design of colloidal particles for desired behavior. Here, we extend the digital alchemy framework for the inverse design of patchy spheres that self-assemble into target crystal structures. To constrain the potentials to non-trivial solutions, we conduct digital alchemy simulations with constant second virial coefficient. We optimize the size, range, and strength of patchy interactions in model triblock Janus spheres to self-assemble the 2D kagome and snub square lattices and the 3D pyrochlore lattice, and demonstrate self-assembly of all three target structures with the designed models. The particles designed for the kagome and snub square lattices assemble into high quality clusters of their target structures, while competition from similar polymorphs lower the yield of the pyrochlore assemblies. We find that the alchemically designed potentials do not always match physical intuition, illustrating the ability of the method to find nontrivial solutions to the optimization problem. We identify a window of second virial coefficients that result in self-assembly of the target structures, analogous to the crystallization slot in protein crystallization.

Self-Assembly of Atomically Aligned Nanoparticle Superlattices from Pt-Fe3O4 Heterodimer Nanoparticles.

Multicomponent nanoparticle superlattices (SLs) promise the integration of nanoparticles (NPs) with remarkable electronic, magnetic, and optical properties into a single structure. Here, we demonstrate that heterodimers consisting of two conjoined NPs can self-assemble into novel multicomponent SLs with a high degree of alignment between the atomic lattices of individual NPs, which has been theorized to lead to a wide variety of remarkable properties. Specifically, by using simulations and experiments, we show that heterodimers composed of larger Fe3O4 domains decorated with a Pt domain at one vertex can self-assemble into an SL with long-range atomic alignment between the Fe3O4 domains of different NPs across the SL. The SLs show an unanticipated decreased coercivity relative to nonassembled NPs. In situ scattering of the self-assembly reveals a two-stage mechanism of self-assembly: translational ordering between NPs develops before atomic alignment. Our experiments and simulation indicate that atomic alignment requires selective epitaxial growth of the smaller domain during heterodimer synthesis and specific size ratios of the heterodimer domains as opposed to specific chemical composition. This composition independence makes the self-assembly principles elucidated here applicable to the future preparation of multicomponent materials with fine structural control.

Engineering the Thermodynamic Stability and Metastability of Mesophases of Colloidal Bipyramids through Shape Entropy.

We report several types of entropy-driven phase transition behaviors in hard bipyramid systems using Monte Carlo simulations. Bipyramidal nanoparticle shapes are synthesizable from gold and silver, with sizes ranging from tens to hundreds of nanometers. We report numerous colloidal crystalline phases with varying symmetries and complexities as the bipyramid aspect ratio and base polygon are varied. Some bipyramids are mesogenic and undergo either monotropic or enantiotropic phase transitions. We show that such mesophase behavior can be modulated by tuning the bipyramid aspect ratio. In addition, we report stepwise kinetic crystallization and melting pathways that occur via an intermediate mesophase as the system gains or loses order in successive stages. Our results demonstrate that complex phase transition behavior involving mesophases can be driven by entropy alone. Importantly, our results can guide the synthesis of bipyramid shapes able to assemble target structures and can be used to engineer the kinetic pathways to and from those structures to involve or avoid mesophases.

Arrays of Colloidal Single Crystals Engineered with DNA in Lithographically Defined Microwells.

Lithographically defined microwell templates are used to study DNA-guided colloidal crystal assembly parameters, including superlattice position, habit orientation, and size, in an effort to increase our understanding of the crystallization process. In addition to enabling the synthesis of arrays of individual superlattices in arbitrary predefined patterns, the technique allows one to study the growth pathways of the crystals via ex situ scanning electron microscopy. Importantly, a Volmer-Weber (VM) (island formation)-like growth mode is identified, which has been reproduced via simulations. Notably, both experiment and simulation reveal that the crystallites merge and reorient within the microwells that defined the crystal growth to form single-crystalline structures, an observation not common for VM pathways. The control afforded by this platform will facilitate efforts in constructing metamaterials from colloidal crystals as well as their integration into optical devices and applications.

Rationalizing Euclidean Assemblies of Hard Polyhedra from Tessellations in Curved Space.

Entropic self-assembly is governed by the shape of the constituent particles, yet a priori prediction of crystal structures from particle shape alone is nontrivial for anything but the simplest of space-filling shapes. At the same time, most polyhedra are not space filling due to geometric constraints, but these constraints can be relaxed or even eliminated by sufficiently curving space. We show using Monte Carlo simulations that the majority of hard Platonic solids self-assemble entropically into space-filling crystals when constrained to the surface volume of a 3-sphere. As we gradually decrease curvature to "flatten" space and compare the local morphologies of crystals assembling in curved and flat space, we show that the Euclidean assemblies can be categorized as either remnants of tessellations in curved space (tetrahedra and dodecahedra) or nontessellation-based assemblies caused by large-scale geometric frustration (octahedra and icosahedra).

Entropically engineered formation of fivefold and icosahedral twinned clusters of colloidal shapes.

Fivefold and icosahedral symmetries induced by multiply twinned crystal structures have been studied extensively for their role in influencing the shape of synthetic nanoparticles, and solution chemistry or geometric confinement are widely considered to be essential. Here we report the purely entropy-driven formation of fivefold and icosahedral twinned clusters of particles in molecular simulation without geometric confinement or chemistry. Hard truncated tetrahedra self-assemble into cubic or hexagonal diamond colloidal crystals depending on the amount of edge and vertex truncation. By engineering particle shape to achieve a negligible entropy difference between the two diamond phases, we show that the formation of the multiply twinned clusters is easily induced. The twinned clusters are entropically stabilized within a dense fluid by a strong fluid-crystal interfacial tension arising from strong entropic bonding. Our findings provide a strategy for engineering twinning behavior in colloidal systems with and without explicit bonding elements between particles.

Symmetry-breaking in patch formation on triangular gold nanoparticles by asymmetric polymer grafting.

Synthesizing patchy particles with predictive control over patch size, shape, placement and number has been highly sought-after for nanoparticle assembly research, but is fraught with challenges. Here we show that polymers can be designed to selectively adsorb onto nanoparticle surfaces already partially coated by other chains to drive the formation of patchy nanoparticles with broken symmetry. In our model system of triangular gold nanoparticles and polystyrene-b-polyacrylic acid patch, single- and double-patch nanoparticles are produced at high yield. These asymmetric single-patch nanoparticles are shown to assemble into self-limited patch‒patch connected bowties exhibiting intriguing plasmonic properties. To unveil the mechanism of symmetry-breaking patch formation, we develop a theory that accurately predicts our experimental observations at all scales-from patch patterning on nanoparticles, to the size/shape of the patches, to the particle assemblies driven by patch‒patch interactions. Both the experimental strategy and theoretical prediction extend to nanoparticles of other shapes such as octahedra and bipyramids. Our work provides an approach to leverage polymer interactions with nanoscale curved surfaces for asymmetric grafting in nanomaterials engineering.

Curvature-controlled geometrical lensing behavior in self-propelled colloidal particle systems.

In many biological systems, the curvature of the surfaces cells live on influences their collective properties. Curvature should likewise influence the behavior of active colloidal particles. We show using molecular simulation of self-propelled active particles on surfaces of Gaussian curvature (both positive and negative) how curvature sign and magnitude can alter the system's collective behavior. Curvature acts as a geometrical lens and shifts the critical density of motility-induced phase separation (MIPS) to lower values for positive curvature and higher values for negative curvature, which we explain theoretically by the nature of parallel lines in spherical and hyperbolic space. Curvature also fluidizes dense MIPS clusters due to the emergence of defect patterns disrupting the crystalline order inside the clusters. Using our findings, we engineer three confining surfaces that strategically combine regions of different curvature to produce a host of novel dynamical behaviors, including cyclic MIPS on spherocylinders, directionally biased cyclic MIPS on spherocones, and position dependent cluster fluctuations on metaballs.

Shape memory in self-adapting colloidal crystals.

Reconfigurable, mechanically responsive crystalline materials are central components in many sensing, soft robotic, and energy conversion and storage devices1-4. Crystalline materials can readily deform under various stimuli and the extent of recoverable deformation is highly dependent upon bond type1,2,5-10. Indeed, for structures held together via simple electrostatic interactions, minimal deformations are tolerated. By contrast, structures held together by molecular bonds can, in principle, sustain much larger deformations and more easily recover their original configurations. Here we study the deformation properties of well-faceted colloidal crystals engineered with DNA. These crystals are large in size (greater than 100 µm) and have a body-centred cubic (bcc) structure with a high viscoelastic volume fraction (of more than 97%). Therefore, they can be compressed into irregular shapes with wrinkles and creases, and, notably, these deformed crystals, upon rehydration, assume their initial well-formed crystalline morphology and internal nanoscale order within seconds. For most crystals, such compression and deformation would lead to permanent, irreversible damage. The substantial structural changes to the colloidal crystals are accompanied by notable and reversible optical property changes. For example, whereas the original and structurally recovered crystals exhibit near-perfect (over 98%) broadband absorption in the ultraviolet-visible region, the deformed crystals exhibit significantly increased reflection (up to 50% of incident light at certain wavelengths), mainly because of increases in their refractive index and inhomogeneity.

Tuning Stoichiometry to Promote Formation of Binary Colloidal Superlattices.

The self-assembly of binary nanoparticle superlattices from colloidal mixtures is a promising method for the fabrication of complex colloidal cocrystal structures. However, binary mixtures often form amorphous or metastable phases instead of the thermodynamically stable phase. Here we show that in binary mixtures of differently sized spherical particles, an excess of the smaller component can promote-and, in some cases, may be necessary for-the self-assembly of a binary cocrystal. Using computer simulations, we identify two mechanisms responsible for this phenomenon. First, excess small particles act like plasticizers and enable systems to reach a greater supersaturation before kinetic arrest occurs. Second, they can disfavor competing structures that may interfere with the growth of the target structure. We find the phase behavior of simulated mixtures of nearly hard spheres closely matches published experimental results. We demonstrate the generality of our findings for mixtures of particles of arbitrary shape by presenting a binary mixture of hard shapes that only self-assembles with an excess of the smaller component.

Evaporation-Driven Coassembly of Hierarchical, Multicomponent Networks.

Self-assembly is an increasingly popular approach to systematically control the formation of complex, multicomponent materials with structural features orders of magnitude larger than the constituent colloidal nanocrystals. Common approaches often involve templating via prefabricated patterns to control particle organization- or programming-specific interactions between individual building blocks. While effective, such fabrication methods suffer from major bottlenecks due to the complexity required in mask creation for patterning or surface modification techniques needed to program directed interactions between particles. Here, we propose an alternative strategy that aims to bypass such limitations. First, we design a ligand structure that can bridge two distinct nanocrystal types. Then, by leveraging the solvent's evaporative dynamics to drive particle organization, we direct a cross-linked, multicomponent system of nanocrystals to organize hierarchically into ordered, open-network structures with domain sizes orders of magnitude larger than the constituent building blocks. We employ simulation and theory to rationalize the driving forces governing this evaporation-driven process, showing excellent agreement across theory, simulations, and experiments. These results suggest that evaporation-driven organization can be a powerful approach to designing and fabricating hierarchical, multifunctional materials.

Structural Color Spectral Response of Dense Structures of Discoidal Particles Generated by Evaporative Assembly.

Structural color─optical response due to light diffraction or scattering from submicrometer-scale structures─is a promising means for sustainable coloration. To expand the functionality of structural color, we introduce discoidal shape anisotropy into colloidal particles and characterize how structural color reflection can be engineered. Uniaxial compression of spheres is used to prepare discoids with varying shape anisotropy and particle size. Discoids are assembled into thin films by evaporation. We find that structural color of assembled films displays components due to diffuse backscattering and multilayer reflection. As discoids become more anisotropic, the assembled structure is more disordered. The multilayer reflection is suppressed─peak height becomes smaller and peak width broader; thus, the color is predominantly from diffuse backscattering. Finally, the discoid structural color can be tuned by varying particle size and has low dependence on viewing angle. We corroborate our results by comparing experimental microstructures and measured reflection spectra with Monte Carlo simulations and calculated spectra by finite-difference time-domain simulation. Our findings demonstrate that the two tunable geometries of discoids─size and aspect ratio─generate different effects on spectral response and therefore can function as independent design parameters that expand possibilities for producing noniridescent structural color.