Influence of Softness on the Stability of Binary Colloidal Crystals

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.

Compact Expansion of a Repulsive Suspension

Short-range repulsion governs the dynamics of matter from atoms to animals. Using theory, simulations, and experiments, we find that an ensemble of repulsive particles spreads compactly with a sharp boundary, in contrast to the diffusive spreading of Brownian particles. Starting from the pair interactions, at high densities, the many-body dynamics follow nonlinear diffusion with a self-similar expansion, growing as t^{1/4}; At longer times, thermal motion dominates with the classic t^{1/2} expansion. A logarithmic growth controlled by nearest-neighbor interactions connects the two self-similar regimes.

Theory and simulation of ligand functionalized nanoparticles - a pedagogical overview

Synthesizing reconfigurable nanoscale synthons with predictive control over shape, size, and interparticle interactions is a holy grail of bottom-up self-assembly. Grand challenges in their rational design, however, lie in both the large space of experimental synthetic parameters and proper understanding of the molecular mechanisms governing their formation. As such, computational and theoretical tools for predicting and modeling building block interactions have grown to become integral in modern day self-assembly research. In this review, we provide an in-depth discussion of the current state-of-the-art strategies available for modeling ligand functionalized nanoparticles. We focus on the critical role of how ligand interactions and surface distributions impact the emergent, pre-programmed behaviors between neighboring particles. To help build insights into the underlying physics, we first define an "ideal" limit - the short ligand, "hard" sphere approximation - and discuss all experimental handles through the lens of perturbations about this reference point. Finally, we identify theories that are capable of bridging interparticle interactions to nanoscale self-assembly and conclude by discussing exciting new directions for this field.

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).

The entropy-controlled strategy in self-assembling systems

Self-assembly of various building blocks has been considered as a powerful approach to generate novel materials with tailorable structures and optimal properties. Understanding physicochemical interactions and mechanisms related to structural formation and transitions is of essential importance for this approach. Although it is well-known that diverse forces and energies can significantly contribute to the structures and properties of self-assembling systems, the potential entropic contribution remains less well understood. The past few years have witnessed rapid progress in addressing the entropic effects on the structures, responses, and functions in the self-assembling systems, and many breakthroughs have been achieved. This review provides a framework regarding the entropy-controlled strategy of self-assembly, through which the structures and properties can be tailored by effectively tuning the entropic contribution and its interplay with the enthalpic counterpart. First, we focus on the fundamentals of entropy in thermodynamics and the entropy types that can be explored for self-assembly. Second, we discuss the rules of entropy in regulating the structural organization in self-assembly and delineate the entropic force and superentropic effect. Third, we introduce the basic principles, significance and approaches of the entropy-controlled strategy in self-assembly. Finally, we present the applications where this strategy has been employed in fields like colloids, macromolecular systems and nonequilibrium assembly. This review concludes with a discussion on future directions and future research opportunities for developing and applying the entropy-controlled strategy in complex self-assembling systems.

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.

Surface functionalization of inorganic nanoparticles with ligands: a necessary step for their utility

The importance of protecting inorganic nanoparticles with organic ligands and thus imparting the needed stabilization as colloidal dispersions for their potential applications is highlighted in this review.

Two-dimensional binary colloidal crystals formed by particles with two different sizes

The formation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {AB_2}$$\end{document}AB2 type two-dimensional binary colloidal crystals was studied by performing Monte Carlo simulations with two different size particles. The effect of interactions between particles and between particles and a wall, and the particles size ratios on the formation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {AB_2}$$\end{document}AB2 structure were examined. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {AB_2}$$\end{document}AB2 structures formed efficiently when the interaction between equivalently sized particles was smaller than that between differently sized particles. To create \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {AB_2}$$\end{document}AB2 on a wall, it was necessary to choose a suitable particles size ratios, and the attraction between the particles and the wall was greater than that between particles.

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.

Superlattice Engineering with Chemically Precise Molecular Building Blocks

Correlating nanoscale building blocks with mesoscale superlattices, mimicking metal alloys, a rational engineering strategy becomes critical to generate designed periodicity with emergent properties. For molecule-based superlattices, nevertheless, nonrigid molecular features and multistep self-assembly make the molecule-to-superlattice correlation less straightforward. In addition, single component systems possess intrinsically limited volume asymmetry of self-assembled spherical motifs (also known as "mesoatoms"), further hampering novel superlattices' emergence. In the current work, we demonstrate that properly designed molecular systems could generate a spectrum of unconventional superlattices. Four categories of giant molecules are presented. We systematically explore the lattice-forming principles in unary and binary systems, unveiling how molecular stoichiometry, topology, and size differences impact the mesoatoms and further toward their superlattices. The presence of novel superlattices helps to correlate with Frank-Kasper phases previously discovered in soft matter. We envision the present work offers new insights about how complex superlattices could be rationally fabricated by scalable-preparation and easy-to-process materials.

Dynamics and phase behavior of two-dimensional size-asymmetric binary mixtures of core-softened colloids

The self-assembly of binary colloidal mixtures provides a bottom-up approach to create novel functional materials. To elucidate the effect of composition, temperature, and pressure on the self-assembly behavior of size-asymmetric mixtures, we performed extensive dynamics simulations of a simple model of polymer-grafted colloids. We have used a core-softened interaction potential and extended it to represent attractive interactions between unlike colloids and repulsions between like colloids. Our study focused on size-asymmetric mixtures where the ratio between the sizes of the colloidal cores was fixed at σ_Bσ_A=0.5. We have performed extensive simulations in the isothermal-isobaric and canonical (NVT) ensembles to elucidate the phase behavior and dynamics of mixtures with different stoichiometric ratios. Our simulation results uncovered a rich phase behavior, including the formation of hierarchical structures with many potential applications. For compositions where small colloids are the majority, sublattice melting occurs for a wide range of densities. Under these conditions, large colloids form a well-defined lattice, whereas small colloids can diffuse through the system. As the temperature is decreased, the small colloids localize, akin to a metal-insulator transition, with the small colloids playing a role similar to electrons. Our results are summarized in terms of phase diagrams.

Multifarious colloidal structures: new insight into ternary and quadripartite ordered assemblies

DNA-mediated assembly of colloidal particles can be utilized to produce a variety of structures which may have desirable phononic, photonic, or electronic transport properties. Recent developments in linker-mediated assembly processes allow for interactions to be coordinated between many different types of colloidal particles more easily and with fewer unique sequences than direct hybridization. However, the dynamics of colloidal self-assembly becomes increasingly more complex when coordinating interactions between three or more distinct interacting elements. In such cases particle pairs with similar binding energies are allowed to interact unpredictably, and enthalpically degenerate binding sites will be noticeably more present while numerous secondary phases may also result from the self-assembly process. Therefore, it is necessary to develop procedures for predicting feasible superstructure geometries for these systems before they can be implemented in material design. Here we investigate the formation of multifarious ordered structures through self-assembly of multiple types of spherically symmetrical colloidal particles with a variety of interaction matrices. We utilize Molecular Dynamics (MD) simulations to study the growth behavior of systems with different types of interacting elements and different particle sizes, and also predict the formation and stability of the target structures. We also study the phononic spectra of various ternary structures in order to identify the influence of key structural parameters on phonon bandgap frequencies and ranges. Our results provide direct guidelines for designing ternary and quadripartite multifarious colloidal structures, and motivate new directions for future experimental work to target formation of multi-component colloidal superstructures beyond the well-established binary symmetries studied in the past.

Densest ternary sphere packings

We present our exhaustive exploration of the densest ternary sphere packings (DTSPs) for 45 radius ratios and 237 kinds of compositions, which is a packing problem of three kinds of hard spheres with different radii, under periodic boundary conditions by a random structure searching method. To efficiently explore DTSPs we further develop the searching method based on the piling-up and iterative balance methods [Koshoji et al., Phys. Rev. E 103, 023307 (2021)2470-004510.1103/PhysRevE.103.023307]. The unbiased exploration identifies diverse 38 putative DTSPs appearing on phase diagrams in which 37 DTSPs of them are discovered in the study. The structural trend of DTSPs changes depending especially on the radius of small spheres. In case that the radius of small spheres is relatively small, structures of many DTSPs can be understood as derivatives of densest binary sphere packings (DBSPs), while characteristic structures specific to the ternary system emerge as the radius of small spheres becomes larger. In addition to DTSPs, we reveal a lot of semi-DTSPs (SDTSPs) which are obtained by excluding DBSPs in the calculation of phase diagram, and investigate the correspondence of DTSPs and SDTSPs with real crystals based on the space group, showing a considerable correspondence of SDTSPs having high symmetries with real crystals including Cu_{2}GaSr and ThCr_{2}Si_{2} structures. Our study suggests that the diverse structures of DBSPs, DTSPs, and SDTSPs can be effectively used as structural prototypes for searching complex crystal structures.

Intercluster Exchange-Stabilized Novel Complex Colloidal χc Phase

Designing complex cluster crystals with a specific function using simple colloidal building blocks remains a challenge in materials science. Herein, we propose a conceptually new design strategy for constructing complex cluster crystals via hierarchical self-assembly of simple soft Janus colloids. A novel and previously unreported colloidal cluster-χ (χ_c) phase, which resembles the essential structural features of α-manganese but at a larger length scale, is obtained through molecular dynamics simulations. The formation of the χ_c phase undergoes a remarkable two-step self-assembly process, that is, the self-assembly of clusters with specific size dispersity from Janus colloids, followed by the highly ordered organization of these clusters. More importantly, the dynamic exchange of particles between these clusters plays a critical role in stabilizing the χ_c phase. Such a conceptual design framework based on intercluster exchange has the potential to effectively construct novel complex cluster crystals by hierarchical self-assembly of colloidal building blocks.

Softness-Enhanced Self-Assembly of Pyrochlore- and Perovskite-like Colloidal Photonic Crystals from Triblock Janus Particles

It remains extremely challenging to build three-dimensional photonic crystals with complete photonic bandgaps by simple and experimentally realizable colloidal building blocks. Here, we demonstrate that particle softness can enhance both the self-assembly of pyrochlore- and perovskite-like lattice structures from simple deformable triblock Janus colloids and their photonic bandgap performances. Dynamics simulation results show that the region of stability of pyrochlore lattices can be greatly expanded by appropriately increasing softness, and the perovskite lattices are unexpectedly obtained at enough high softness. Photonic calculations show that the direct pyrochlore lattices formed from overlapping soft triblock Janus particles exhibit even larger photonic bandgaps than the ideal nonoverlapping pyrochlore lattice, and proper overlap arising from softness can also dramatically improve the photonic properties of the inverse pyrochlore and perovskite lattices. Our study offers a new and feasible self-assembly path toward three-dimensional photonic crystals with large and robust photonic bandgaps.

Kinetically Controlled Self-Assembly of Binary Polymer-Grafted Nanocrystals into Ordered Superstructures via Solvent Vapor Annealing

Polymer-inorganic nanocomposites based on polymer-grafted nanocrystals (PGNCs) are enabling technologically relevant applications owing to their unique physical, chemical, and mechanical properties. While diverse PGNC superstructures have been realized through evaporation-driven self-assembly, this approach presents multifaceted challenges in experimentally probing and controlling assembly kinetics. Here, we report a kinetically controlled assembly of binary superstructures from a homogeneous disordered PGNC mixture utilizing solvent vapor annealing (SVA). Using a NaZn₁₃-type superstructure as a model system, we demonstrate that varying the solvent vapor pressure during SVA allows for exquisite control of the rate and extent of PGNC assembly, providing access to nearly complete kinetic pathways of binary PGNC crystallization. Characterization of kinetically arrested intermediates reveals that assembly follows a multistep crystallization pathway involving spinodal-like preordering of PGNCs prior to NaZn₁₃ nucleation. Our work opens up new avenues for the synthesis of multicomponent PGNC superstructures exhibiting multifunctionalities and emergent properties through a thorough understanding of kinetic pathways.

Diverse densest binary sphere packings and phase diagram

We revisit the densest binary sphere packings (DBSPs) under periodic boundary conditions and present an updated phase diagram, including newly found 12 putative densest structures over the x-α plane, where x is the relative concentration and α is the radius ratio of the small and large spheres. To efficiently explore the DBSPs, we develop an unbiased random search approach based on both the piling-up method to generate initial structures in an unbiased way and the iterative balance method to optimize the volume of a unit cell while keeping the overlap of hard spheres minimized. With those two methods, we have discovered 12 putative DBSPs and thereby the phase diagram is updated, while our results are consistent with those of a previous study [Hopkins et al., Phys. Rev. E 85, 021130 (2012)]PLEEE81539-375510.1103/PhysRevE.85.021130 with a small correction for the case of 12 or fewer spheres in the unit cell. Five of the discovered 12 DBSPs are identified in the small radius range of 0.42≤α≤0.50, where several structures are competitive to each other with respect to packing fraction. Through the exhaustive search, diverse dense packings are discovered and, accordingly, we find that packing structures achieve high packing fractions by introducing distortion and/or combining a few local dense structural units. Furthermore, we investigate the correspondence of the DBSPs with crystals based on the space group. The result shows that many structural units in real crystals, e.g., LaH_{10} and SrGe_{2-δ} being high-pressure phases, can be understood as DBSPs. The correspondence implies that the densest sphere packings can be used effectively as structural prototypes for searching complex crystal structures, especially for high-pressure phases.

Ordered polymer composite materials: challenges and opportunities

Polymer nanocomposites containing nanoscale fillers are an important class of materials due to their ability to access a wide variety of properties as a function of their composition. In order to take full advantage of these properties, it is critical to control the distribution of nanofillers within the parent polymer matrix, as this structural organization affects how the two constituent components interact with one another. In particular, new methods for generating ordered arrays of nanofillers represent a key underexplored research area, as emergent properties arising from nanoscale ordering can be used to introduce novel functionality currently inaccessible in random composites. The knowledge gained from developing such methods will provide important insight into the thermodynamics and kinetics associated with nanomaterial and polymer assembly. These insights will not only benefit researchers working on new composite materials, but will also deepen our understanding of soft matter systems in general. In this review, we summarize contemporary research efforts in manipulating nanofiller organization in polymer nanocomposites and highlight future challenges and opportunities for constructing ordered nanocomposite materials.

Self-assembly and entropic effects in pear-shaped colloid systems. I. Shape sensitivity of bilayer phases in colloidal pear-shaped particle systems

The role of particle shape in self-assembly processes is a double-edged sword. On the one hand, particle shape and particle elongation are often considered the most fundamental determinants of soft matter structure formation. On the other hand, structure formation is often highly sensitive to details of shape. Here, we address the question of particle shape sensitivity for the self-assembly of hard pear-shaped particles by studying two models for this system: (a) the pear hard Gaussian overlap (PHGO) and (b) the hard pears of revolution (HPR) model. Hard pear-shaped particles, given by the PHGO model, are known to form a bicontinuous gyroid phase spontaneously. However, this model does not replicate an additive object perfectly and, hence, varies slightly in shape from a "true" pear-shape. Therefore, we investigate in the first part of this series the stability of the gyroid phase in pear-shaped particle systems. We show, based on the HPR phase diagram, that the gyroid phase does not form in pears with such a "true" hard pear-shaped potential. Moreover, we acquire first indications from the HPR and PHGO pair-correlation functions that the formation of the gyroid is probably attributed to the small non-additive properties of the PHGO potential.

Phase Diagram and Structure Map of Binary Nanoparticle Superlattices from a Lennard-Jones Model

A first-principles prediction of the binary nanoparticle phase diagram assembled by solvent evaporation has eluded theoretical approaches. In this paper, we show that a binary system interacting through the Lennard-Jones (LJ) potential contains all experimental phases in which nanoparticles are effectively described as quasi hard spheres. We report a phase diagram consisting of 53 equilibrium phases, whose stability is quite insensitive to the microscopic details of the potentials, thus giving rise to some type of universality. Furthermore, we show that binary lattices may be understood as consisting of certain particle clusters, i.e., motifs, that provide a generalization of the four conventional Frank-Kasper polyhedral units. Our results show that metastable phases share the very same motifs as equilibrium phases. We discuss the connection with packing models, phase diagrams with repulsive potentials, and the prediction of likely experimental superlattices.

Spontaneous Crystallization in Systems of Binary Hard Sphere Colloids

Computer simulations of the fluid-to-solid phase transition in the hard sphere system were instrumental for our understanding of crystallization processes. But while colloid experiments and theory have been predicting the stability of several binary hard sphere crystals for many years, simulations were not successful to confirm this phenomenon. Here, we report the growth of binary hard sphere crystals isostructural to Laves phases, AlB_{2}, and NaZn_{13} in simulation directly from the fluid. We analyze particle kinetics during Laves phase growth using event-driven molecular dynamics simulations with and without swap moves that speed up diffusion. The crystallization process transitions from nucleation and growth to spinodal decomposition already deep within the fluid-solid coexistence regime. Finally, we present packing fraction-size ratio state diagrams in the vicinity of the stability regions of three binary crystals.

Unravelling three-dimensional adsorption geometries of PbSe nanocrystal monolayers at a liquid-air interface

The adsorption, self-organization and oriented attachment of PbSe nanocrystals (NCs) at liquid-air interfaces has led to remarkable nanocrystal superlattices with atomic order and a superimposed nanoscale geometry. Earlier studies examined the NC self-organization at the suspension/air interface with time-resolved in-situ X-ray scattering. Upon continuous evaporation of the solvent, the NC interfacial layer will finally contact the (ethylene glycol) liquid substrate on which the suspension was casted. In order to obtain structural information on the NC organization at this stage of the process, we examined the ethylene glycol/NC interface in detail for PbSe NCs of different sizes, combining in-situ grazing-incidence small-and-wide-angle X-ray scattering (GISAXS/GIWAXS), X-ray reflectivity (XRR) and analytical calculations of the adsorption geometry of these NCs. Here, we observe in-situ three characteristic adsorption geometries varying with the NC size. Based on the experimental evidence and simulations, we reveal fully three-dimensional arrangements of PbSe nanocrystals at the ethylene glycol-air interface with and without the presence of rest amounts of toluene.

Kinetics-controlled design principles for two-dimensional open lattices using atom-mimicking patchy particles

The design and discovery of new two-dimensional materials with desired structures and properties are always one of the most fundamental goals in materials science. Here we present an atom-mimicking design concept to achieve direct self-assembly of two-dimensional low-coordinated open lattices using three-dimensional patchy particle systems. Besides honeycomb lattices, a new type of two-dimensional square-octagon lattice is obtained through rational design of the patch configuration of soft three-patch particles. However, unexpectedly the building blocks with thermodynamically favoured patch configuration cannot form square-octagon lattices in our simulations. We further reveal the kinetic mechanisms controlling the formation of the honeycomb and square-octagon lattices. The results indicate that the kinetically favoured intermediates play a critical role in determining the structure of obtained open lattices. This kinetics-controlled design principle provides a particularly effective and extendable framework to construct other novel open lattice structures.