Symmetry considerations for the targeted assembly of entropically stabilized colloidal crystals via Voronoi particles

Nanocrystal Assemblies: Current Advances and Open Problems

We explore the potential of nanocrystals (a term used equivalently to nanoparticles) as building blocks for nanomaterials, and the current advances and open challenges for fundamental science developments and applications. Nanocrystal assemblies are inherently multiscale, and the generation of revolutionary material properties requires a precise understanding of the relationship between structure and function, the former being determined by classical effects and the latter often by quantum effects. With an emphasis on theory and computation, we discuss challenges that hamper current assembly strategies and to what extent nanocrystal assemblies represent thermodynamic equilibrium or kinetically trapped metastable states. We also examine dynamic effects and optimization of assembly protocols. Finally, we discuss promising material functions and examples of their realization with nanocrystal assemblies.

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).

Ligand Effects in Assembly of Cubic and Spherical Nanocrystals: Applications to Packing of Perovskite Nanocubes

We establish the formula representing cubic nanocrystals (NCs) as hard cubes taking into account the role of the ligands and describe how these results generalize to any other NC shapes. We derive the conditions under which the hard cube representation breaks down and provide explicit expressions for the effective size. We verify the results from the detailed potential of mean force calculations for two nanocubes in different orientations as well as with spherical nanocrystals. Our results explicitly demonstrate the relevance of certain ligand conformations, i.e., "vortices", and show that edges and corners provide natural sites for their emergence. We also provide both simulations and experimental results with single component cubic perovskite nanocrystals assembled into simple cubic superlattices, which further corroborate theoretical predictions. In this way, we extend the Orbifold Topological Model (OTM) accounting for the role of ligands beyond spherical nanocrystals and discuss its extension to arbitrary nanocrystal shapes. Our results provide detailed predictions for recent superlattices of perovskite nanocubes and spherical nanocrystals. Problems with existing united atom force fields are discussed.

Orientation-Controlled Ultralong Assembly of Janus Particles Induced by Bubble-Driven Instant Quasi-1D Interfaces

Constructing precisely oriented assemblies and exploring their orientation-dependent properties remain a challenge for Janus nanoparticles (JNPs) due to their asymmetric characteristics. Herein, we propose a bubble-driven instant quasi-1D interfacial strategy for the oriented assembly of JNP chains in a highly controllable manner. It is found that the rapid formation of templated bubbles can promote the interfacial orientation of JNPs kinetically, while the confined quasi-1D interface in the curved liquid bridge can constrain the disordered rotation of the particles, yielding well-oriented JNP chains in a long range. During the evaporation process, the interfacial orientation of the JNPs can be transferred to the assembled chains. By regulating the amphiphilicity of the JNPs, both heteraxial and coaxial JNP assemblies are obtained, which show different polarization dependences on light scattering, and the related colorimetric logic behaviors are demonstrated. This work demonstrates the great potential of patterned interfacial assembly with a manageable orientation and shows the broad prospect of asymmetric JNP assembly in constructing novel optoelectronic devices.

A route to hierarchical assembly of colloidal diamond

Photonic crystals, appealing for their ability to control light, are constructed from periodic regions of different dielectric constants. Yet, the structural holy grail in photonic materials, diamond, remains challenging to synthesize at the colloidal length scale. Here we explore new ways to assemble diamond using modified gyrobifastigial (mGBF) nanoparticles, a shape that resembles two anti-aligned triangular prisms. We investigate the parameter space that leads to the self-assembly of diamond, and we compare the likelihood of defects in diamond self-assembled via mGBF vs. the nanoparticle shape that is the current focus for assembling diamond, the truncated tetrahedra. We introduce a potential route for realizing mGBF particles by dimerizing triangular prisms using attractive patches, and we report the impact of this superstructure on the photonic properties.

Synthesizable nanoparticle eigenshapes for colloidal crystals

The gulf between the complexity and diversity of colloidal crystal phases predicted to form in computer simulation and that realized to date in experiment is narrowing, but is still wide. Prior work shows that many synthesized particles are far from optimal "eigenshapes" for target superlattice structures. We use digital alchemy to determine eigenshapes for possible target colloidal crystal structures for eight families of polyhedral nanoparticle shapes already synthesized in the laboratory. Within each family we predict optimal building block shapes to obtain several target superlattice structures, as a guide for future experiments. For three target crystal structures common to multiple families, we identify which of the optimal shapes is most optimal under the same thermodynamic conditions.

Unexpected Dependence of Photonic Band Gap Size on Randomness in Self-Assembled Colloidal Crystals

Using computer simulations, we explore how thermal noise-induced randomness in a self-assembled photonic crystal affects its photonic band gaps (PBGs). We consider a two-dimensional photonic crystal composed of a self-assembled array of parallel dielectric hard rods of infinite length with circular or square cross section. We find that PBGs can exist over a large range of intermediate packing densities and the largest band gap does not always appear at the highest packing density studied. Remarkably, for rods with square cross section at intermediate packing densities, the transverse magnetic (TM) band gap of the self-assembled (i.e., thermal) system can be larger than that of identical rods arranged in a perfect square lattice. By considering hollow rods, we find the band gap of transverse electric modes can be substantially increased while that of TM modes show no obvious improvement over solid rods. Our study suggests that particle shape and internal structure can be used to engineer the PBG of a self-assembled system despite the positional and orientational randomness arising from thermal noise.

Shape-driven entropic self-assembly of an open, reconfigurable, binary host-guest colloidal crystal

Entropically driven self-assembly of hard anisotropic particles, where particle shape gives rise to emergent valencies, provides a useful perspective for the design of nanoparticle and colloidal systems. Hard particles self-assemble into a rich variety of crystal structures, ranging in complexity from simple close-packed structures to structures with 432 particles in the unit cell. Entropic crystallization of open structures, however, is missing from this landscape. Here, we report the self-assembly of a two-dimensional binary mixture of hard particles into an open host-guest structure, where nonconvex, triangular host particles form a honeycomb lattice that encapsulates smaller guest particles. Notably, this open structure forms in the absence of enthalpic interactions by effectively splitting the structure into low- and high-entropy sublattices. This is the first such structure to be reported in a two-dimensional athermal system. We discuss the observed compartmentalization of entropy in this system, and show that the effect of the size of the guest particle on the stability of the structure gives rise to a reentrant phase behavior. This reentrance suggests the possibility for a reconfigurable colloidal material, and we provide a proof-of-concept by showing the assembly behavior while changing the size of the guest particles in situ. Our findings provide a strategy for designing open colloidal crystals, as well as binary systems that exhibit co-crystallization, which have been elusive thus far.

Aggregation in viscoelastic emulsion droplet gels with capillarity-driven rearrangements

Arrested, or partial, coalescence of viscoelastic emulsion droplets can occur when elastic resistance to deformation offsets droplet surface area minimization. Arrest is a critical element of food and consumer product microstructure and performance, but direct studies of structural arrest and rearrangement have been carried out using only two or three droplets at a time. The question remains whether the behavior of small numbers of droplets also occurs in larger, more realistic many-droplet systems. Here we study two-dimensional aggregation and arrested coalescence of emulsions containing ∼1000 droplets and find that the restructuring mechanisms observed for smaller systems have a large effect on local packing in multidroplet aggregates, but surprisingly do not significantly alter overall mass scaling in the aggregates. Specifically, increased regions of hexagonal packing are observed as the droplet solids level, and thus elasticity, is decreased because greater degrees of capillary force-driven restructuring are possible. Diffusion-limited droplet aggregation simulations that account for the restructuring mechanisms agree with the experimental results and suggest a basis for prediction of larger-scale network properties and bulk emulsion behavior.

Influence of Softness on the Stability of Binary Colloidal Crystals

Mixtures of two types of nanoparticles can self-assemble into a wide variety of binary colloidal crystals (also called binary nanoparticle superlattices), which are interesting for their structural diversity and potential applications. Although so-called packing models-which usually treat the particles as "hard" with only excluded volume interactions-seem to explain many reported dense crystalline phases, these models often fail to predict the right structure. Here, we examine the role of soft repulsive interparticle interactions on binary colloidal crystals comprising two sizes of spherical particles; such "softness" can arise due to ligand shells or screened electrostatics. We determine the ground state phase diagram of binary systems of particles interacting with an additive inverse power law potential using a basin hopping algorithm to calculate the enthalpy of an extremely large pool of candidate structures. We find that a surprisingly small amount of softness can destabilize dense packings in favor of less densely packed structures, which provides further evidence that considerations beyond packing are necessary for describing many of the observed phases of binary colloidal crystals. Importantly, we find that several of the phases stabilized by softness, which are characterized by relatively few interparticle contacts and a tendency for local icosahedral order, are more likely to be observed experimentally than those predicted by packing models. We also report a previously unknown dense AB₄ phase and conduct free energy calculations to examine how the stability of several crystals will vary with temperature. Our results further our understanding of why particular binary colloidal crystals form and will be useful as a reference for experimentalists working with softly repulsive colloids.

Design of Patchy Rhombi: From Close-Packed Tilings to Open Lattices

In the realm of functional materials, the production of two-dimensional structures with tunable porosity is of paramount relevance for many practical applications: surfaces with regular arrays of pores can be used for selective adsorption or immobilization of guest units that are complementary in shape and/or size to the pores, thus achieving, for instance, selective filtering or well-defined responses to external stimuli. The principles that govern the formation of such structures are valid at both the molecular and the colloidal scale. Here we provide simple design directions to combine the anisotropic shape of the building units-either molecules or colloids-and selective directional bonding. Using extensive computer simulations, we show that regular rhombic platelets decorated with attractive and repulsive interaction sites form specific tilings, going smoothly from close-packed arrangements to open lattices. The rationale behind the rich tiling scenario observed can be described in terms of steric incompatibilities, unsatisfied bonding geometries, and interplays between local and long-range order.

Periodic buckling and grain boundary slips in a colloidal model of solid friction

The intermittent 'stick-slip' dynamics in frictional sliding of solid bodies is common in everyday life and technology. This dynamics has been widely studied on a macroscopic scale, where the thermal motion can usually be neglected. However, the microscopic mechanisms behind the periodic stick-slip events are yet unclear. We employ confocal microscopy of colloidal spheres, to study the frictional dynamics at the boundary between two quasi-two-dimensional (2D) crystalline grains, with a single particle resolution. Such unprecedentedly-detailed observations of the microscopic-scale frictional solid-on-solid sliding have never been previously carried out. At this scale, the particles undergo an intense thermal motion, which masks the avalanche-like nature of the underlying frictional dynamics. We demonstrate that the underlying sliding dynamics involving out-of-plane buckling events, is intermittent and periodic, like in macroscopic friction. However, unlike in the common models of friction, the observed periodic frictional dynamics is promoted, rather than just suppressed, by the thermal noise, which maximizes the entropy of the system.

Phase behavior and design rules for plastic colloidal crystals of hard polyhedra via consideration of directional entropic forces

Plastic crystals - like liquid crystals - are mesophases that can exist between liquids and crystals and possess some of the characteristic traits of each of these states of matter. Plastic crystals exhibit translational order but orientational disorder. Here, we characterize the phase behavior in systems of hard polyhedra that self-assemble plastic face-centered cubic (pFCC) colloidal crystals. We report a first-order transition from a pFCC to a body-centered tetragonal (BCT) crystal, a smooth crossover from pFCC to an orientationally-ordered FCC crystal, and an apparent orientational glass transition wherein long-range order fails to develop from a plastic crystal upon an increase in density. Using global order parameters and local environment descriptors, we describe how particle shape influences the development of orientational order with increasing density, and we provide design rules based on the arrangement of facets for engineering plastic crystal behavior in colloidal systems.

Engineering entropy for the inverse design of colloidal crystals from hard shapes

Throughout the physical sciences, entropy stands out as a pivotal but enigmatic concept that, in materials design, typically takes a backseat to energy. Here, we demonstrate how to precisely engineer entropy to achieve desired colloidal crystals via particle shapes that, importantly, can be made in the laboratory. We demonstrate the inverse design of symmetric hard particles that assemble six different target colloidal crystals due solely to entropy maximization. Our approach efficiently samples 10⁸ particle shapes from 92- and 188-dimensional design spaces to discover thermodynamically optimal shapes. We design particle shapes that self-assemble into known crystals with optimized symmetry and thermodynamic stability, as well as new crystal structures with no known atomic or other equivalent.

Shape control of nanostructured cone-shaped particles by tuning the blend morphology of A-b-B diblock copolymers and C-type copolymers within emulsion droplets

An approach to blend AB-type block copolymers and C-type copolymers within the emulsion droplet is an efficient particle shape-engineering strategy.

Intermediate crystalline structures of colloids in shape space

We computationally study the thermodynamic assembly of more than 40 000 hard, convex polyhedra belonging to three families of shapes associated with the triangle groups 323, 423, and 523. Each family is defined by vertex and/or edge truncation of symmetric polyhedra with equal edge length, producing shapes for which the majority are intermediates of more symmetric polyhedra found among the Platonic, Archimedean, and Catalan solids. In addition to the complex crystals cI16 lithium, BC8 silicon, γ-brass, β-manganese, and a dodecagonal quasicrystal, we find that most intermediate shapes assemble distorted variants of four basic cubic crystals: face-centered cubic, body-centered cubic, simple cubic, and diamond. To quantify the degree of distortion, we developed an algorithm that extracts lattice vectors from particle positions and then evaluates closeness to the four reference cubic crystals. This analysis allows us to group together in shape space related intermediate structures that would otherwise be placed in different lattice systems had we followed the lattice systems' strict definitions for angles and lengths of lattice vectors. The resulting landscapes show, as a function of shape, regions where ordered structures assemble, what is assembled and at what density, locations of transitions between regions of ordered structures, and regions of disorder. Our results provide a guide to self-assembling a host of related colloidal crystals through systematic design, by careful tweaking of the particle shape.

Recent Advances in Multicomponent Particle Assembly

Particle assembly and co-assembly have been research frontiers in chemistry and material science in the past few decades. To achieve a large variety of intricate structures and functional materials, remarkable progress has been made in particle assembly principles and strategies. Essentially, particle assembly is driven by intrinsic interparticle interactions or the external control. In this article, we focus on binary or ternary particle co-assembly and review the principles and feasible strategies. These advances have led to new disciplines of microfabrication technology and material engineering. Although significant achievement on particle-based structures has been made, it is still challenging to fully develop general and facile strategies to precisely control the one-dimensional (1D) co-assembly. This article reviews the recent development on multicomponent particle co-assembly, which significantly increases structural complexity and functional diversity. In particular, we highlight the advances in the particle co-assembly of well-ordered 1D binary superstructures by liquid soft confinement. Finally, prospective outlook for future trends in this field is proposed.

Soft polyhedral particles based on cubic liquid crystalline emulsion droplets

Soft polyhedral particles based on variations of the cubic symmetry group are produced from a precursor emulsion by extracting solvent to grow facets on the droplets. The droplets transform into liquid crystals with solid-like rheology and controlled size and shape. Small-angle X-ray scattering confirms a bicontinuous cubic liquid crystalline phase forms from aqueous glycerol monoolein and is responsible for the particle faceting observed. Different polyhedra are produced by varying face growth rates through control of precursor droplet size, system temperature, and solubilization and adsorption of guest molecules. More exotic faceted shapes can be formed by the soft particles by applying asymmetric solvent removal gradients and by deforming the precursor droplets into, for example, ellipsoids before crystallization. The method is a powerful means to produce soft polyhedra, using continuous microfluidic or other approaches, or to act as templates for hard polyhedral particle synthesis.

Limiting the valence: advancements and new perspectives on patchy colloids, soft functionalized nanoparticles and biomolecules

Artistic representation of limited valance units consisting of a soft core (in blue) and a small number of flexible bonding patches (in orange).

Self-Assembly of Colloidal Nanocrystals: From Intricate Structures to Functional Materials

Chemical methods developed over the past two decades enable preparation of colloidal nanocrystals with uniform size and shape. These Brownian objects readily order into superlattices. Recently, the range of accessible inorganic cores and tunable surface chemistries dramatically increased, expanding the set of nanocrystal arrangements experimentally attainable. In this review, we discuss efforts to create next-generation materials via bottom-up organization of nanocrystals with preprogrammed functionality and self-assembly instructions. This process is often driven by both interparticle interactions and the influence of the assembly environment. The introduction provides the reader with a practical overview of nanocrystal synthesis, self-assembly, and superlattice characterization. We then summarize the theory of nanocrystal interactions and examine fundamental principles governing nanocrystal self-assembly from hard and soft particle perspectives borrowed from the comparatively established fields of micrometer colloids and block copolymer assembly. We outline the extensive catalog of superlattices prepared to date using hydrocarbon-capped nanocrystals with spherical, polyhedral, rod, plate, and branched inorganic core shapes, as well as those obtained by mixing combinations thereof. We also provide an overview of structural defects in nanocrystal superlattices. We then explore the unique possibilities offered by leveraging nontraditional surface chemistries and assembly environments to control superlattice structure and produce nonbulk assemblies. We end with a discussion of the unique optical, magnetic, electronic, and catalytic properties of ordered nanocrystal superlattices, and the coming advances required to make use of this new class of solids.

Controlling Chirality of Entropic Crystals

Colloidal crystal structures with complexity and diversity rivaling atomic and molecular crystals have been predicted and obtained for hard particles by entropy maximization. However, thus far homochiral colloidal crystals, which are candidates for photonic metamaterials, are absent. Using Monte Carlo simulations we show that chiral polyhedra exhibiting weak directional entropic forces self-assemble either an achiral crystal or a chiral crystal with limited control over the crystal handedness. Building blocks with stronger faceting exhibit higher selectivity and assemble a chiral crystal with handedness uniquely determined by the particle chirality. Tuning the strength of directional entropic forces by means of particle rounding or the use of depletants allows for reconfiguration between achiral and homochiral crystals. We rationalize our findings by quantifying the chirality strength of each particle, both from particle geometry and potential of mean force and torque diagrams.

Shape allophiles improve entropic assembly

We investigate a class of "shape allophiles" that fit together like puzzle pieces as a method to access and stabilize desired structures by controlling directional entropic forces. Squares are cut into rectangular halves, which are shaped in an allophilic manner with the goal of re-assembling the squares while self-assembling the square lattice. We examine the assembly characteristics of this system via the potential of mean force and torque, and the fraction of particles that entropically bind. We generalize our findings and apply them to self-assemble triangles into a square lattice via allophilic shaping. Through these studies we show how shape allophiles can be useful for assembling and stabilizing desired phases with appropriate allophilic design.

Preparation of highly permeable BPPO microfiltration membrane with binary porous structures on a colloidal crystal substrate by the breath figure method

A highly permeable brominated poly(phenylene oxide) (BPPO) microfiltration membrane with binary porous structures was fabricated by combination of the breath figure and colloidal crystal template methods. The pore size in the bottom layer of the membrane was adjusted by the diameter of SiO2 microspheres in the colloidal crystal template, while the pore size in the top layer of the membrane was adjusted by varying the BPPO concentration in the casting solution. The permeability of the membrane cast on the colloidal crystal substrate was much higher than that of the membrane cast on a bare silicon wafer. The binary porous BPPO membrane with high permeability and antifouling property was used for microfiltration applications.