Local Structural Features Elucidate Crystallization of Complex Structures.

Complex crystal structures are composed of multiple local environments, and how this type of order emerges spontaneously during crystal growth has yet to be fully understood. We study crystal growth across various structures and along different crystallization pathways, using self-assembly simulations of identical particles that interact via multiwell isotropic pair potentials. We apply an unsupervised machine learning method to features from bond-orientational order metrics to identify different local motifs present during a given structure's crystallization process. In this manner, we distinguish different crystallographic sites in highly complex structures. Tailoring this order parameter to structures of varying complexity and coordination number, we study the emergence of local order along a multistep crystal growth pathway─from a low-density fluid to a high-density, supercooled amorphous liquid droplet and to a bulk crystal. We find a consistent under-coordination of the liquid relative to the average coordination number in the bulk crystal. We use our order parameter to analyze the geometrically frustrated growth of a Frank-Kasper phase and discover how structural defects compete with the formation of crystallographic sites that are more high-coordinated than the liquid environments. The method presented here for classifying order on a particle-by-particle level has broad applicability to future studies of structural self-assembly and crystal growth, and they can aid in the design of building blocks and for targeting pathways of formation of soft-matter structures.

Targeted Discovery of Low-Coordinated Crystal Structures via Tunable Particle Interactions.

Particles interacting via isotropic, multiwell pair potentials have been shown to self-assemble into a range of crystal structures, yet how the characteristics of the underlying interaction potential give rise to the resultant structure remains largely unknown. We have thus developed a functional form for the interaction potential in which all features can be tuned independently. We perform continuous parameter space searches by systematically changing pairs of parameters, controlling the various features of the interaction potential. By enforcing a repulsive first well (controlling particle interactions of the first neighbor shell), we stimulate the formation of low-coordinated assemblies. We report the self-assembly of 20 previously unknown crystal structure types, 14 of which have low coordination numbers. Despite limiting the search to a small region of the vast parameter space of possible particle interactions, a wealth of complexity and symmetry is apparent within these crystal structures, which include clathrates with empty cages and low-symmetry structures. Our findings suggest that an unknown number of previously undiscovered crystal structure configurations are possible through self-assembly, which can serve as interesting design targets for soft condensed matter synthesis.

Tuning assembly structures of hard shapes in confinement via interface curvature.

Assembly in confinement is a problem of great interest in colloidal structure design, plasmonics, photonics, and industrial packaging. Along with the range of design choices provided by particle shape and attraction or repulsion, confined systems add an additional layer of complexity through the interactions between particles and the container holding them. The range of possible behaviors produced by these systems remains largely unexplored, yet has profound consequences on the resultant assembled structure. Here, we address this problem by exploring how the assembly of hard tetrahedral particles is affected by a spherical container. We simulate particle assemblies in containers holding 4 to 10 000 particles and analyze the range of resultant structures. We find that the presence of a curved wall causes organization into distinct concentric shells in containers holding up to thousands of particles. In addition, we see that wall curvature affects structural motifs in systems as large as 10 000 particles, promoting local environments that maximally conform to the wall and providing a seed for the propagation of these motifs into the interior of the container. Through this work, we show how confining interfaces can be used to promote the assembly of structures markedly distinct from those seen in the more commonly studied bulk systems.

Multiscale hierarchical structures from a nanocluster mesophase.

Spontaneous hierarchical self-organization of nanometre-scale subunits into higher-level complex structures is ubiquitous in nature. The creation of synthetic nanomaterials that mimic the self-organization of complex superstructures commonly seen in biomolecules has proved challenging due to the lack of biomolecule-like building blocks that feature versatile, programmable interactions to render structural complexity. In this study, highly aligned structures are obtained from an organic-inorganic mesophase composed of monodisperse Cd37S18 magic-size cluster building blocks. Impressively, structural alignment spans over six orders of magnitude in length scale: nanoscale magic-size clusters arrange into a hexagonal geometry organized inside micrometre-sized filaments; self-assembly of these filaments leads to fibres that then organize into uniform arrays of centimetre-scale bands with well-defined surface periodicity. Enhanced patterning can be achieved by controlling processing conditions, resulting in bullseye and 'zigzag' stacking patterns with periodicity in two directions. Overall, we demonstrate that colloidal nanomaterials can exhibit a high level of self-organization behaviour at macroscopic-length scales.

Moving beyond the constraints of chemistry via crystal structure discovery with isotropic multiwell pair potentials.

The rigid constraints of chemistry-dictated by quantum mechanics and the discrete nature of the atom-limit the set of observable atomic crystal structures. What structures are possible in the absence of these constraints? Here, we systematically crystallize one-component systems of particles interacting with isotropic multiwell pair potentials. We investigate two tunable families of pairwise interaction potentials. Our simulations self-assemble a multitude of crystal structures ranging from basic lattices to complex networks. Sixteen of the structures have natural analogs spanning all coordination numbers found in inorganic chemistry. Fifteen more are hitherto unknown and occupy the space between covalent and metallic coordination environments. The discovered crystal structures constitute targets for self-assembly and expand our understanding of what a crystal structure can look like.

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.

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.

Computational self-assembly of colloidal crystals from Platonic polyhedral sphere clusters.

We explore a rich phase space of crystals self-assembled from colloidal "polyhedral sphere clusters (PSCs)," each of which consists of equal-sized "halo" spheres placed at the vertices of a polyhedron such that they just touch along each edge. Such clusters, created experimentally by fusing spheres, can facilitate assembly of useful colloidal crystal symmetries not attainable by unclustered spheres. While not crucial for their self-assembly, the center of the PSC can contain a "core" particle that can be used as a scaffold to build the PSC. Using Brownian dynamics simulations, we show the self-assembly of eight distinct crystalline phases from PSCs that correspond to the five Platonic polyhedra, and that are made of spheres with purely repulsive interactions. Strong crystalline order is seen in the centers of mass of the PSCs, or equivalently the core particles. The halo particles also may organize into crystal structures, usually with weaker crystalline order than the core particles. Notably, however, in crystals assembled from the octahedral and icosahedral PSCs, the halo particles are also well ordered, nesting within the crystals formed by the cores. Interestingly, despite the rounded nature of the PSCs, in some cases we obtain structures similar to those of the corresponding faceted polyhedra interacting only via excluded volume. Only the tetrahedral PSCs fail to self-assemble into a crystal, but we demonstrate that a pre-assembled crystal - whose halo particles sit on a close-packed face-centered cubic lattice, and whose core particles form a diamond structure - is stable at high density and melts into a hexagonal phase at lower density.

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.

Inverse design of simple pair potentials for the self-assembly of complex structures.

The synthesis of complex materials through the self-assembly of particles at the nanoscale provides opportunities for the realization of novel material properties. However, the inverse design process to create experimentally feasible interparticle interaction strategies is uniquely challenging. Standard methods for the optimization of isotropic pair potentials tend toward overfitting, resulting in solutions with too many features and length scales that are challenging to map to mechanistic models. Here we introduce a method for the optimization of simple pair potentials that minimizes the relative entropy of the complex target structure while directly considering only those length scales most relevant for self-assembly. Our approach maximizes the relative information of a target pair distribution function with respect to an ansatz distribution function via an iterative update process. During this process, we filter high frequencies from the Fourier spectrum of the pair potential, resulting in interaction potentials that are smoother and simpler in real space and therefore likely easier to make. We show that pair potentials obtained by this method assemble their target structure more robustly with respect to optimization method parameters than potentials optimized without filtering.

Clusters of polyhedra in spherical confinement.

Dense particle packing in a confining volume remains a rich, largely unexplored problem, despite applications in blood clotting, plasmonics, industrial packaging and transport, colloidal molecule design, and information storage. Here, we report densest found clusters of the Platonic solids in spherical confinement, for up to [Formula: see text] constituent polyhedral particles. We examine the interplay between anisotropic particle shape and isotropic 3D confinement. Densest clusters exhibit a wide variety of symmetry point groups and form in up to three layers at higher N. For many N values, icosahedra and dodecahedra form clusters that resemble sphere clusters. These common structures are layers of optimal spherical codes in most cases, a surprising fact given the significant faceting of the icosahedron and dodecahedron. We also investigate cluster density as a function of N for each particle shape. We find that, in contrast to what happens in bulk, polyhedra often pack less densely than spheres. We also find especially dense clusters at so-called magic numbers of constituent particles. Our results showcase the structural diversity and experimental utility of families of solutions to the packing in confinement problem.

More statistics on intermetallic compounds - ternary phases.

How many different intermetallic compounds are known so far, and in how many different structure types do they crystallize? What are their chemical compositions, the most abundant ones and the rarest ones? These are some of the questions we are trying to find answers for in our statistical analysis of the structures of the 20,829 intermetallic phases included in the database Pearson's Crystal Data, with the goal of gaining insight into some of their ordering principles. In the present paper, we focus on the subset of 13,026 ternary intermetallics, which crystallize in 1391 different structure types; remarkably, 667 of them have just one representative. What makes these 667 structures so unique that they are not adopted by any other of the known intermetallic compounds? Notably, ternary compounds are known in only 5109 of the 85,320 theoretically possible ternary intermetallic systems so far. In order to get an overview of their chemical compositions we use structure maps with Mendeleev numbers as ordering parameters.

Some statistics on intermetallic compounds.

It is still largely unknown why intermetallic phases show such a large variety of crystal structures, with unit cell sizes varying between 1 and more than 20 000 atoms. The goal of our study was, therefore, to get a general overview of the symmetries, unit cell sizes, stoichiometries, most frequent structure types, and their stability fields based on the Mendeleev numbers as ordering parameters. A total of 20829 structures crystallizing in 2166 structure types have been studied for this purpose. Thereby, the focus was on a subset of 6441 binary intermetallic compounds, which crystallize in 943 structure types.

Gram-scale synthesis of two-dimensional polymer crystals and their structure analysis by X-ray diffraction.

The rise of graphene, a natural two-dimensional polymer (2DP) with topologically planar repeat units, has challenged synthetic chemistry, and has highlighted that accessing equivalent covalently bonded sheet-like macromolecules has, until recently, not been achieved. Here we show that non-centrosymmetric, enantiomorphic single crystals of a simple-to-make monomer can be photochemically converted into chiral 2DP crystals and cleanly reversed back to the monomer. X-ray diffraction established unequivocal structural proof for this synthetic 2DP, which has an all-carbon scaffold and can be synthesized on the gram scale. The monomer crystals are highly robust, can be easily grown to sizes greater than 1 mm and the resulting 2DP crystals exfoliated into nanometre-thin sheets. This unique combination of features suggests that these 2DPs could find use in membranes and nonlinear optics.

Polynuclear iron(II)-aminotriazole spincrossover complexes (polymers) in solution.

Polynuclear spincrossover (SCO) complexes prepared by the combination of [Fe(DMF)6](2+) and NH2trz (NH2trz = 4-amino-1,2,4-triazole) were studied (2ns(-) = counterion 2-naphthalenesulfonate). It is demonstrated that these [Fe(NH2trz)3](2ns)2 complexes can be dissolved-contrary to common reported experience-in N,N-dimethylformamide (DMF) and, therefore, can be conveniently processed by simple means. The resulting solutions were examined with UV/vis and X-ray absorption spectroscopy (XANES and EXAFS) as well as with small-angle X-ray scattering (SAXS). At a molar NH2trz/Fe(2+) ratio of 3/1, corresponding to the stoichiometric ratio of the ideal coordination compound, [Fe(NH2trz)3](2+) in the low-spin state was found to be in equilibrium with polynuclear species in the high-spin state. The equilibrium can be shifted virtually completely to the side of low-spin Fe(2+) by an excess of the ligand. The polymer therewith formed contains 100 or more Fe(2+) ions and is of a pronounced rigid-rod structure, with Fe-Fe distances around 3.32 Å (in comparison to 3.94 Å of the polynuclear species in the high-spin state). Reversible spin crossover takes place in solution upon a temperature increase to around 60 °C; this process is associated with a shift in equilibrium toward species shorter than the initial polynuclear species.

A new complex intermetallic phase in the system Al-Cu-Ta with familiar clusters and packing principles.

The structure of hP386-Al(57.4)Cu(3.6)Ta(39.0) was determined by single-crystal X-ray diffraction analysis. It can be described as a hexagonal close-packing of two types of endohedral fullerene-like clusters with different Frank-Kasper polyhedra filling the gaps. The description of the structure as a superstructure and as a layered structure illustrates other characteristic structural building principles. The diffuse scattering, which can be observed in some of the crystals, is qualitatively well reproduced by a disorder model. A comparison with the structures of the other complex intermetallics in the system Al-Cu-Ta indicates the decisive role that Cu plays in the constitution and packing of the clusters.

Structural building principles of complex face-centered cubic intermetallics.

Fundamental structural building principles are discussed for all 56 known intermetallic phases with approximately 400 or more atoms per unit cell and space-group symmetry F43m, Fd3m, Fd3, Fm3m or Fm3c. Despite fundamental differences in chemical composition, bonding and electronic band structure, their complex crystal structures show striking similarities indicating common building principles. We demonstrate that the structure-determining elements are flat and puckered atomic {110} layers stacked with periodicities 2p. The atoms on this set of layers, which intersect each other, form pentagon face-sharing endohedral fullerene-like clusters arranged in a face-centered cubic packing (f.c.c.). Due to their topological layer structure, all these crystal structures can be described as (p × p × p) = p(3)-fold superstructures of a common basic structure of the double-diamond type. The parameter p, with p = 3, 4, 7 or 11, is determined by the number of layers per repeat unit and the type of cluster packing, which in turn are controlled by chemical composition.

Large, larger, largest--a family of cluster-based tantalum copper aluminides with giant unit cells. I. Structure solution and refinement.

This is the first of two parts, where we report the structure determination of a novel family of cluster-based intermetallic phases of unprecedented complexity: cF444-Al(63.6)Ta(36.4) (AT-19), a = 19.1663 (1) A, V = 7040 A3, cF(5928-x)-Al(56.6)Cu(3.9)Ta(39.5), x = 20 (ACT-45), a = 45.376 (1) A, V = 93,428 A(3) and cF(23,256-x)-Al(55.4)Cu(5.4)Ta(39.1), x = 122 (ACT-71), a = 71.490 (4) A, V = 365,372 A3. The space group is F43m in all three cases. These cluster-based structures are closely related to the class of Frank-Kasper phases. It is remarkable that all three structures show the same average structure that resembles the cubic Laves phase.