Entropic colloidal crystallization pathways via fluid-fluid transitions and multidimensional prenucleation motifs

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.

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.

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.

In Situ X-ray Scattering Reveals Coarsening Rates of Superlattices Self-Assembled from Electrostatically Stabilized Metal Nanocrystals Depend Nonmonotonically on Driving Force

Self-assembly of colloidal nanocrystals (NCs) into superlattices (SLs) is an appealing strategy to design hierarchically organized materials with promising functionalities. Mechanistic studies are still needed to uncover the design principles for SL self-assembly, but such studies have been difficult to perform due to the fast time and short length scales of NC systems. To address this challenge, we developed an apparatus to directly measure the evolving phases in situ and in real time of an electrostatically stabilized Au NC solution before, during, and after it is quenched to form SLs using small-angle X-ray scattering. By developing a quantitative model, we fit the time-dependent scattering patterns to obtain the phase diagram of the system and the kinetics of the colloidal and SL phases as a function of varying quench conditions. The extracted phase diagram is consistent with particles whose interactions are short in range relative to their diameter. We find the degree of SL order is primarily determined by fast (subsecond) initial nucleation and growth kinetics, while coarsening at later times depends nonmonotonically on the driving force for self-assembly. We validate these results by direct comparison with simulations and use them to suggest dynamic design principles to optimize the crystallinity within a finite time window. The combination of this measurement methodology, quantitative analysis, and simulation should be generalizable to elucidate and better control the microscopic self-assembly pathways of a wide range of bottom-up assembled systems and architectures.

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.

A Review of Laser-Induced Crystallization from Solution

Crystallization abounds in nature and industrial practice. A plethora of indispensable products ranging from agrochemicals and pharmaceuticals to battery materials are produced in crystalline form in industrial practice. Yet, our control over the crystallization process across scales, from molecular to macroscopic, is far from complete. This bottleneck not only hinders our ability to engineer the properties of crystalline products essential for maintaining our quality of life but also hampers progress toward a sustainable circular economy in resource recovery. In recent years, approaches leveraging light fields have emerged as promising alternatives to manipulate crystallization. In this review article, we classify laser-induced crystallization approaches where light-material interactions are utilized to influence crystallization phenomena according to proposed underlying mechanisms and experimental setups. We discuss nonphotochemical laser-induced nucleation, high-intensity laser-induced nucleation, laser trapping-induced crystallization, and indirect methods in detail. Throughout the review, we highlight connections among these separately evolving subfields to encourage the interdisciplinary exchange of ideas.

Nanoparticle Assembly and Oriented Attachment: Correlating Controlling Factors to the Resulting Structures

Nanoparticle assembly and attachment are common pathways of crystal growth by which particles organize into larger scale materials with hierarchical structure and long-range order. In particular, oriented attachment (OA), which is a special type of particle assembly, has attracted great attention in recent years because of the wide range of material structures that result from this process, such as one-dimensional (1D) nanowires, two-dimensional (2D) sheets, three-dimensional (3D) branched structures, twinned crystals, defects, etc. Utilizing in situ transmission electron microscopy techniques, researchers observed orientation-specific forces that act over short distances (∼1 nm) from the particle surfaces and drive the OA process. Integrating recently developed 3D fast force mapping via atomic force microscopy with theories and simulations, researchers have resolved the near-surface solution structure, the molecular details of charge states at particle/fluid interfaces, inhomogeneity of surface charges, and dielectric/magnetic properties of particles that influence short- and long-range forces, such as electrostatic, van der Waals, hydration, and dipole-dipole forces. In this review, we discuss the fundamental principles for understanding particle assembly and attachment processes, and the controlling factors and resulting structures. We review recent progress in the field via examples of both experiments and modeling, and discuss current developments and the future outlook.

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.

Spatially resolved structural order in low-temperature liquid electrolyte

Determining the degree and the spatial extent of structural order in liquids is a grand challenge. Here, we are able to resolve the structural order in a model organic electrolyte of 1 M lithium hexafluorophosphate (LiPF₆) dissolved in 1:1 (v/v) ethylene carbonate:diethylcarbonate by developing an integrated method of liquid-phase transmission electron microscopy (TEM), cryo-TEM operated at -30°C, four-dimensional scanning TEM, and data analysis based on deep learning. This study reveals the presence of short-range order (SRO) in the high-salt concentration domains of the liquid electrolyte from liquid phase separation at the low temperature. Molecular dynamics simulations suggest the SRO originates from the Li⁺-(PF₆^-)_n (n > 2) local structural order induced by high LiPF₆ salt concentration.

Collective Variables for Crystallization Simulations-from Early Developments to Recent Advances

Crystallization is an important physicochemical process which has relevance in material science, biology, and the environment. Decades of experimental and theoretical efforts have been made to understand this fundamental symmetry-breaking transition. While experiments provide equilibrium structures and shapes of crystals, they are limited to unraveling how molecules aggregate to form crystal nuclei that subsequently transform into bulk crystals. Computer simulations, mainly molecular dynamics (MD), can provide such microscopic details during the early stage of a crystallization event. Crystallization is a rare event that takes place in time scales much longer than a typical equilibrium MD simulation can sample. This inadequate sampling of the MD method can be easily circumvented by the use of enhanced sampling (ES) simulations. In most of the ES methods, the fluctuations of a system's slow degrees of freedom, called collective variables (CVs), are enhanced by applying a bias potential. This transforms the system from one state to the other within a short time scale. The most crucial part of such CV-based ES methods is to find suitable CVs, which often needs intuition and several trial-and-error optimization steps. Over the years, a plethora of CVs has been developed and applied in the study of crystallization. In this review, we provide a brief overview of CVs that have been developed and used in ES simulations to study crystallization from melt or solution. These CVs can be categorized mainly into four types: (i) spherical particle-based, (ii) molecular template-based, (iii) physical property-based, and (iv) CVs obtained from dimensionality reduction techniques. We present the context-based evolution of CVs, discuss the current challenges, and propose future directions to further develop effective CVs for the study of crystallization of complex systems.

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.

The Effect of Various Poly (N-vinylpyrrolidone) (PVP) Polymers on the Crystallization of Flutamide

In this study, several experimental techniques were applied to probe thermal properties, molecular dynamics, crystallization kinetics and intermolecular interactions in binary mixtures (BMs) composed of flutamide (FL) and various poly(N-vinylpyrrolidone) (PVP) polymers, including a commercial product and, importantly, samples obtained from high-pressure syntheses, which differ in microstructure (defined by the tacticity of the macromolecule) from the commercial PVP. Differential Scanning Calorimetry (DSC) studies revealed a particularly large difference between the glass transition temperature (T_g) of FL+PVPsynth. mixtures with 10 and 30 wt% of the excipient. In the case of the FL+PVPcomm. system, this effect was significantly lower. Such unexpected findings for the former mixtures were strictly connected to the variation of the microstructure of the polymer. Moreover, combined DSC and dielectric measurements showed that the onset of FL crystallization is significantly suppressed in the BM composed of the synthesized polymers. Further non-isothermal DSC investigations carried out on various FL+10 wt% PVP mixtures revealed a slowing down of FL crystallization in all FL-based systems (the best inhibitor of this process was PVP M_n = 190 kg/mol). Our research indicated a significant contribution of the microstructure of the polymer on the physical stability of the pharmaceutical—an issue completely overlooked in the literature.

Multistep Crystallization of Dynamic Nanoparticle Superlattices in Nonaqueous Solutions

Crystallization is a universal phenomenon underpinning many industrial and natural processes and is fundamental to chemistry and materials science. However, microscopic crystallization pathways of nanoparticle superlattices have been seldom studied mainly owing to the difficulty of real-time observation of individual self-assembling nanoparticles in solution. Here, using in situ electron microscopy, we directly image the full self-assembly pathway from dispersed nanoparticles into ordered superlattices in nonaqueous solution. We show that electron-beam irradiation controls nanoparticle mobility, and the solvent composition largely dictates interparticle interactions and assembly behaviors. We uncover a multistep crystallization pathway consisting of four distinct stages through multi-order-parameter analysis and visualize the formation, migration, and annihilation of multiple types of defects in nanoparticle superlattices. These findings open the door for achieving independent control over imaging conditions and nanoparticle assembly conditions and will enable further study of the microscopic kinetics of assembly and phase transition in nanocolloidal systems.

Designing Enhanced Entropy Binding in Single-Chain Nanoparticles

Single-chain nanoparticles (SCNPs) are a new class of bio- and soft-matter polymeric objects in which a fraction of the monomers are able to form equivalently intra- or interpolymer bonds. Here we numerically show that a fully entropic gas-liquid phase separation can take place in SCNP systems. Control over the discontinuous (first-order) change-from a phase of independent diluted (fully-bonded) polymers to a phase in which polymers entropically bind to each other to form a (fully-bonded) polymer network-can be achieved by a judicious design of the patterns of reactive monomers along the polymer chain. Such a sensitivity arises from a delicate balance between the distinct entropic contributions controlling the binding.

Classifying soft self-assembled materials via unsupervised machine learning of defects

Unlike molecular crystals, soft self-assembled fibers, micelles, vesicles, etc., exhibit a certain order in the arrangement of their constitutive monomers but also high structural dynamicity and variability. Defects and disordered local domains that continuously form-and-repair in their structures impart to such materials unique adaptive and dynamical properties, which make them, e.g., capable to communicate with each other. However, objective criteria to compare such complex dynamical features and to classify soft supramolecular materials are non-trivial to attain. Here we show a data-driven workflow allowing us to achieve this goal. Building on unsupervised clustering of Smooth Overlap of Atomic Position (SOAP) data obtained from equilibrium molecular dynamics simulations, we can compare a variety of soft supramolecular assemblies via a robust SOAP metric. This provides us with a data-driven "defectometer" to classify different types of supramolecular materials based on the structural dynamics of the ordered/disordered local molecular environments that statistically emerge within them.

Influence on crystal nucleation of an order-disorder transition among the subcritical clusters

Studies of nucleation generally focus on the properties of the critical cluster, but the presence of defects within the crystal lattice means that the population of nuclei necessarily evolve through a distribution of precritical clusters with varying degrees of structural disorder on their way to forming a growing stable crystal. To investigate the role precritical clusters play in nucleation, we develop a simple thermodynamic model for crystal nucleation in terms of cluster size and the degree of cluster order that allows us to alter the work of forming the precritical clusters without affecting the properties of the critical cluster. The steady state and transient nucleation behavior of the system are then studied numerically, for different microscopic ordering kinetics. We find that the model exhibits a generic order-disorder transition in the precritical clusters. Independent of the types of ordering kinetics, increasing the accessibility of disordered precritical clusters decreases both the steady state nucleation rate and the nucleation lag time. Furthermore, the interplay between the free-energy surface and the microscopic ordering kinetics leads to three distinct nucleation pathways.

Quantifying Energetic and Entropic Pathways in Molecular Systems

When examining dynamics occurring at nonzero temperatures, both energy and entropy must be taken into account to describe activated barrier crossing events. Furthermore, good reaction coordinates need to be constructed to describe different metastable states and the transition mechanisms between them. Here we use a physics-based machine learning method called state predictive information bottleneck (SPIB) to find nonlinear reaction coordinates for three systems of varying complexity. SPIB is able to correctly predict an entropic bottleneck for an analytical flat-energy double-well system and identify the entropy- and energy-dominated pathways for an analytical four-well system. Finally, for a simulation of benzoic acid permeation through a lipid bilayer, SPIB is able to discover the entropic and energetic barriers to the permeation process. Given these results, we thus establish that SPIB is a reasonable and robust method for finding the important entropy, energy, and enthalpy barriers in physical systems, which can then be used to enhance the understanding and sampling of different activated mechanisms.

Computer simulations of self-assembly of anisotropic colloids

Computer simulations have played a significant role in understanding the physics of colloidal self-assembly, interpreting experimental observations, and predicting novel mesoscopic and crystalline structures. Recent advances in computer simulations of colloidal self-assembly driven by anisotropic or orientation-dependent inter-particle interactions are highlighted in this review. These interactions are broadly classified into two classes: entropic and enthalpic interactions. They mainly arise due to shape anisotropy, surface heterogeneity, compositional heterogeneity, external field, interfaces, and confinements. Key challenges and opportunities in the field are discussed.

A theory of entropic bonding

Entropy alone can self-assemble hard nanoparticles into colloidal crystals of remarkable complexity whose structures are the same as atomic and molecular crystals, but with larger lattice spacings. Molecular simulation is a powerful tool used extensively to study the self-assembly of ordered phases from disordered fluid phases of atoms, molecules, or nanoparticles. However, it is not yet possible to predict colloidal crystal structures a priori from particle shape as we can for atomic crystals from electronic valency. Here, we present such a first-principles theory. By calculating and minimizing excluded volume within the framework of statistical mechanics, we describe the directional entropic forces that collectively emerge between hard shapes, in familiar terms used to describe chemical bonds. We validate our theory by demonstrating that it predicts thermodynamically preferred structures for four families of hard polyhedra that match, in every instance, previous simulation results. The success of this first-principles approach to entropic colloidal crystal structure prediction furthers fundamental understanding of both entropically driven crystallization and conceptual pictures of bonding in matter.

Particle-Click-Particle: Colloidal Clusters from Click Seeded Emulsion Polymerization

The self-organization of building blocks in colloidal clusters and their architecture adjustment are crucial for colloid synthesis and design. We report on a click seeded emulsion polymerization via swelling-induced self-assembly...

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.

Frustrated self-assembly of non-Euclidean crystals of nanoparticles

Self-organized complex structures in nature, e.g., viral capsids, hierarchical biopolymers, and bacterial flagella, offer efficiency, adaptability, robustness, and multi-functionality. Can we program the self-assembly of three-dimensional (3D) complex structures using simple building blocks, and reach similar or higher level of sophistication in engineered materials? Here we present an analytic theory for the self-assembly of polyhedral nanoparticles (NPs) based on their crystal structures in non-Euclidean space. We show that the unavoidable geometrical frustration of these particle shapes, combined with competing attractive and repulsive interparticle interactions, lead to controllable self-assembly of structures of complex order. Applying this theory to tetrahedral NPs, we find high-yield and enantiopure self-assembly of helicoidal ribbons, exhibiting qualitative agreement with experimental observations. We expect that this theory will offer a general framework for the self-assembly of simple polyhedral building blocks into rich complex morphologies with new material capabilities such as tunable optical activity, essential for multiple emerging technologies.

Entropic formation of a thermodynamically stable colloidal quasicrystal with negligible phason strain

Quasicrystals have been discovered in a variety of materials ranging from metals to polymers. Yet, why and how they form is incompletely understood. In situ transmission electron microscopy of alloy quasicrystal formation in metals suggests an error-and-repair mechanism, whereby quasiperiodic crystals grow imperfectly with phason strain present, and only perfect themselves later into a high-quality quasicrystal with negligible phason strain. The growth mechanism has not been investigated for other types of quasicrystals, such as dendrimeric, polymeric, or colloidal quasicrystals. Soft-matter quasicrystals typically result from entropic, rather than energetic, interactions, and are not usually grown (either in laboratories or in silico) into large-volume quasicrystals. Consequently, it is unknown whether soft-matter quasicrystals form with the high degree of structural quality found in metal alloy quasicrystals. Here, we investigate the entropically driven growth of colloidal dodecagonal quasicrystals (DQCs) via computer simulation of systems of hard tetrahedra, which are simple models for anisotropic colloidal particles that form a quasicrystal. Using a pattern recognition algorithm applied to particle trajectories during DQC growth, we analyze phason strain to follow the evolution of quasiperiodic order. As in alloys, we observe high structural quality; DQCs with low phason strain crystallize directly from the melt and only require minimal further reduction of phason strain. We also observe transformation from a denser approximant to the DQC via continuous phason strain relaxation. Our results demonstrate that soft-matter quasicrystals dominated by entropy can be thermodynamically stable and grown with high structural quality--just like their alloy quasicrystal counterparts.

Newtonian Event-Chain Monte Carlo and Collision Prediction with Polyhedral Particles

Polyhedral nanocrystals are building blocks for nanostructured materials that find applications in catalysis and plasmonics. Synthesis efforts and self-assembly experiments have been assisted by computer simulations that predict phase equilibria. Most current simulations employ Monte Carlo methods, which generate stochastic dynamics. Collective and correlated configuration updates are alternatives that promise higher computational efficiency and generate trajectories with realistic dynamics. One such alternative involves event-chain updates and has recently been proposed for spherical particles. In this contribution, we develop and apply event-chain Monte Carlo for hard convex polyhedra. Our simulation makes use of an improved computational geometry algorithm XenoSweep, which predicts sweep collision in a particularly simple way. We implement Newtonian event chains in the open-source general-purpose particle simulation toolkit HOOMD-blue for serial and parallel simulation. The speedup over state-of-the-art Monte Carlo is between a factor of 10 for nearly spherical polyhedra and a factor of 2 for highly aspherical polyhedra. Finally, we validate the Newtonian event-chain algorithm by applying it to a current research problem, the multistep nucleation of two classes of hard polyhedra.

Free energy surface of two-step nucleation

We test the theoretical free energy surface (FES) for two-step nucleation (TSN) proposed by Iwamatsu [J. Chem. Phys. 134, 164508 (2011)] by comparing the predictions of the theory to numerical results for the FES recently reported from Monte Carlo simulations of TSN in a simple lattice system [James et al., J. Chem. Phys. 150, 074501 (2019)]. No adjustable parameters are used to make this comparison. That is, all the parameters of the theory are evaluated directly for the model system, yielding a predicted FES, which we then compare to the FES obtained from simulations. We find that the theoretical FES successfully predicts the numerically evaluated FES over a range of thermodynamic conditions that spans distinct regimes of behavior associated with TSN. All the qualitative features of the FES are captured by the theory, and the quantitative comparison is also very good. Our results demonstrate that Iwamatsu's extension of classical nucleation theory provides an excellent framework for understanding the thermodynamics of TSN.

Low Interfacial Free Energy Describes the Bulk Ordering Transition in Colloidal Cubes

Many hard faceted nanoparticles are known to undergo disorder-to-order phase transitions following a classical nucleation and growth mechanism. In a previous study [J. Phys. Chem. B 2018, 122, 9264-9273], it was shown that hard cubes undergo a nonclassical phase transition with a bulk character instead of originating from consolidated nuclei. Significantly, an unusually high fraction of ordered particles was observed in the metastable basin of the disordered phase, even for very low degrees of supersaturation. This work aims to substantiate the conjecture that these unique properties originate from a comparatively low interfacial free energy between the disordered and ordered phases for hard cubes relative to other hard particle systems. Using the cleaving wall method to directly measure the interfacial free energy for cubes, it is found that its values are indeed small; e.g., at phase coexistence conditions, it is only one-fifth that for hard spheres. A theoretical nucleation model is used to explore the broader implications of low interfacial tension values and how this could result in a bulk ordering mechanism.

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.

Mechanisms of Nucleation and Solid-Solid-Phase Transitions in Triblock Janus Assemblies

A model, including the chemical details of core nanoparticles as well as explicit surface charges and hydrophobic patches, of triblock Janus particles is employed to simulate nucleation and solid-solid phase transitions in two-dimensional layers. An explicit solvent and a substrate are included in the model, and hydrodynamic and many-body interactions were taken into account within many-body dissipative particle dynamics simulation. In order not to impose a mechanism a priori, we performed free (unbiased) simulations, leaving the system the freedom to choose its own pathways. In agreement with the experiment and previous biased simulations, a two-step mechanism for the nucleation of a kagome lattice from solution was detected. However, a distinct feature of the present unbiased versus biased simulations is that multiple nuclei emerge from the solution; upon their growth, the aligned and misaligned facets at the grain boundaries are introduced into the system. The liquid-like particles trapped between the neighboring nuclei connect them together. A mismatch in the symmetry planes of neighboring nuclei hinders the growth of less stable (smaller) nuclei. Unification of such nuclei at the grain boundaries of misaligned facets obeys a two-step mechanism: melting of the smaller nuclei, followed by subsequent nucleation of liquid-like particles at the interface of bigger neighboring nuclei. Besides, multiple postcritical nuclei are formed in the simulation box; the growth of some of which stops due to introduction of a strain in the system. Such an incomplete nucleation/growth mechanism is in complete agreement with the recent experiments. The solid-solid (hexagonal-to-kagome) phase transition, at weak superheatings, obeys a two-step mechanism: a slower step (formation of a liquid droplet), followed by a faster step (nucleation of kagome from the liquid droplet).

Self-Assembly of Patchy Colloidal Rods into Photonic Crystals Robust to Stacking Faults

Diamond-structured colloidal photonic crystals are much sought-after for their applications in visible light management because of their ability to support a complete photonic band gap (PBG). However, their realization via self-assembly pathways is a long-standing challenge. This challenge is rooted in three fundamental problems: the design of building blocks that assemble into diamond-like structures, the sensitivity of the PBG to stacking faults, and ensuring that the PBG opens at an experimentally attainable refractive index. Here we address these problems simultaneously using a multipronged computational approach. We use reverse engineering to establish the design principles for the rod-connected diamond structure (RCD), the so-called "champion" photonic crystal. We devise two distinct self-assembly routes for designer triblock patchy colloidal rods, both proceeding via tetrahedral clusters to yield a mixed phase of cubic and hexagonal polymorphs closely related to RCD. We use Monte Carlo simulations to show how these routes avoid a metastable amorphous phase. Finally, we show that both the polymorphs support spectrally overlapping PBGs. Importantly, randomly stacked hybrids of these polymorphs also display PBGs, thus circumventing the requirement of polymorph selection in a scalable fabrication method.

Two-step crystallization and solid-solid transitions in binary colloidal mixtures

Crystallization is fundamental to materials science and is central to a variety of applications, ranging from the fabrication of silicon wafers for microelectronics to the determination of protein structures. The basic picture is that a crystal nucleates from a homogeneous fluid by a spontaneous fluctuation that kicks the system over a single free-energy barrier. However, it is becoming apparent that nucleation is often more complicated than this simple picture and, instead, can proceed via multiple transformations of metastable structures along the pathway to the thermodynamic minimum. In this article, we observe, characterize, and model crystallization pathways using DNA-coated colloids. We use optical microscopy to investigate the crystallization of a binary colloidal mixture with single-particle resolution. We observe classical one-step pathways and nonclassical two-step pathways that proceed via a solid-solid transformation of a crystal intermediate. We also use enhanced sampling to compute the free-energy landscapes corresponding to our experiments and show that both one- and two-step pathways are driven by thermodynamics alone. Specifically, the two-step solid-solid transition is governed by a competition between two different crystal phases with free energies that depend on the crystal size. These results extend our understanding of available pathways to crystallization, by showing that size-dependent thermodynamic forces can produce pathways with multiple crystal phases that interconvert without free-energy barriers and could provide approaches to controlling the self-assembly of materials made from colloids.

A conceptual change in crystallisation mechanisms of oxide materials from solutions in closed systems

Atomic and molecular level interactions in solutions dictate the structural and functional attributes of crystals. These features clearly dictate the properties of materials and their applicability in technologies. However, the microscopic phenomena of particle formation-nucleation and growth-in real systems are still not fully understood. Specifically, crystallisation occurring in closed systems are largely unproven. Combining coherent experimental data, we here demonstrate a fundamental nucleation-growth mechanism that occurs in a model zinc oxide system when particles are formed under continuous, rapid heating under closed reaction conditions. Defying all previous reports, we show that the nucleation commences only when the heating is terminated. A prenucleation clusters pathway is observed for nucleation, followed by crystallite assembly-growth. We show that the nucleation-growth processes result from temporal and dynamic activity of constituent ions and gaseous molecules in solution and by the irreversible expulsion of the dissolved gaseous molecules. We suggest that this nucleation process is generic to most closed systems that go through precipitation, and, therefore, important for the crystallisation of a variety of metal oxides, composites and minerals. We anticipate that the work may be a platform for future experimental and theoretical investigation promoting deeper understanding of the nucleation-growth phenomena of a variety of practical systems.

Assembly of nanocrystal clusters by solvent evaporation: icosahedral order and the breakdown of the Maxwell regime

We carry out molecular dynamics simulations of N gold alkylthiolated nanocrystals (0 ≤ N ≤ 29) contained in liquid droplets of octane, nonane and decane coexisting with its vapor. The equilibrium structures that result when all the solvent dries up consist of highly symmetric nanocrystal clusters with different degrees of icosahedral order that are thoroughly characterized. We show that the relaxation times follow two regimes, a first for small nanocrystal packing fraction, dominated by the diffusion of vapor molecules (Maxwell regime, relaxation times independent of N) and another, for larger packing fractions, where the solvent diffuses through the cluster (with relaxation times growing like N2/3). We discuss the connection to the assembly of superlattices, prediction of lattice constants and evaporation models.

Nucleation-Elongation Dynamics of Two-Dimensional Covalent Organic Frameworks

Homogeneous two-dimensional (2D) polymerization is a poorly understood process in which topologically planar monomers react to form planar macromolecules, often termed 2D covalent organic frameworks (COFs). While these COFs have traditionally been limited to weakly crystalline aggregated powders, they were recently grown as micron-sized single crystals by temporally resolving the growth and nucleation processes. Here, we present a quantitative analysis of the nucleation and growth rates of 2D COFs via kinetic Monte Carlo (KMC) simulations using COF-5 as an example, which show that nucleation and growth have second-order and first-order dependences on monomer concentration, respectively. The computational results were confirmed experimentally by systematic measurements of COF nucleation and growth rates performed via in situ X-ray scattering, which validated the respective monomer concentration dependencies of the nucleation and elongation processes. A major consequence is that there exists a threshold monomer concentration below which growth dominates over nucleation. Our computational and experimental findings rationalize recent empirical observations that, in the formation of 2D COF single crystals, growth dominates over nucleation when monomers are added slowly, so as to limit their concentrations. This mechanistic understanding of the nucleation and growth processes will inform the rational control of polymerization in two dimensions and ultimately enable access to high-quality samples of designed two-dimensional polymers.

The entropic bond in colloidal crystals

A vast array of natural phenomena can be understood through the long-established schema of chemical bonding. Conventional chemical bonds arise through local gradients resulting from the rearrangement of electrons; however, it is possible that the hallmark features of chemical bonding could arise through local gradients resulting from nonelectronic forms of mediation. If other forms of mediation give rise to "bonds" that act like conventional ones, recognizing them as bonds could open new forms of supramolecular descriptions of phenomena at the nano- and microscales. Here, we show via a minimal model that crowded hard-particle systems governed solely by entropy exhibit the hallmark features of bonding despite the absence of chemical interactions. We quantitatively characterize these features and compare them to those exhibited by chemical bonds to argue for the existence of entropic bonds. As an example of the utility of the entropic bond classification, we demonstrate the nearly equivalent tradeoff between chemical bonds and entropic bonds in the colloidal crystallization of hard hexagonal nanoplates.