Heuristic rule for binary superlattice coassembly: mixed plastic mesophases of hard polyhedral nanoparticles

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.

Effect of shape on liquid-vapor coexistence and surface properties of parallelepiped molecules

Liquid-vapor coexistence is calculated via molecular dynamics for a variety of parallelepiped shaped molecules. Models are constructed as an array of tangential hard spheres interacting with an attractive square-well potential. Each shape is formed by varying the number of spheres in their three sides. The initial density of the system is chosen close to the critical density of a SW fluid to obtain an equilibrated liquid-vapor coexistence curve by the process of spinodal decomposition. A pattern that relates the geometry of the molecular models and the existence or non-existence of a liquid-vapor orthobaric curve is shown.

Intermediate crystalline structures of colloids in shape space

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

Solid-phase nucleation free-energy barriers in truncated cubes: interplay of localized orientational order and facet alignment

The nucleation of ordered phases from the bulk isotropic phase of octahedron-like particles has been studied via Monte Carlo simulations and umbrella sampling. In particular, selected shapes that form ordered (plastic) phases with various symmetries (cubic and tetragonal) are chosen to unveil trends in the free-energy barrier heights (ΔG*'s) associated with disorder to order transitions. The shapes studied in this work have truncation parameter (s) values of 0.58, 0.75, 0.8 and 1. The case of octahedra (s = 1.0) is studied to provide a counter-example where the isotropic phase nucleates directly into a (Minkowski) crystal phase rather than a rotator phase. The simulated ΔG*'s for these systems are compared with those previously reported for hard spheres and truncated cubes with s = 0.5 (cuboctahedra, CO) and s = 2/3 (truncated octahedra, TO). The comparison shows that, for comparable degrees of supersaturation, all rotator phases nucleate with smaller ΔG*'s than that of the hard sphere crystal, whereas the octahedral crystal nucleates with a larger ΔG*. Our analysis of near-critical translationally ordered nuclei of octahedra shows a strong bias towards an orientational alignment which is incompatible with the tendency to form facet-to-facet contacts in the disordered phase, thus creating an additional entropic penalty for crystallization. For rotator phases of octahedra-like particles, we observe that the strength of the localized orientational order correlates inversely with ΔG*. We also observe that for s > 0.66 shapes and similar to octahedra, configurations with high facet alignment do not favor high orientational order, and thus ΔG*'s increase with truncation.

The stabilization of tubular crystals in mixtures of spherical particles

Novel crystal structures in binary atomic mixtures arise when the attractive well is wide enough to allow double occupancy by small particles. The resulting crystals consist of ordered packings of self assembled linear structures comprised of a cylindrical tube of large particles enclosing a close packed core of small particles that corresponds to a stacking of overlapping icosahedra. We show that the stability of these structures depends on two essential features of the spherically symmetric pairwise interactions: (i) a radius ratio between 0.414 and 0.588, and (ii) a width w of the attractive well in the interaction between unlike particles that satisfies w > σ_SS where σ_SS is the diameter of the small particle.

Self-assembly of a space-tessellating structure in the binary system of hard tetrahedra and octahedra

We report the formation of a binary crystal of hard polyhedra due solely to entropic forces. Although the alternating arrangement of octahedra and tetrahedra is a known space-tessellation, it had not previously been observed in self-assembly simulations. Both known one-component phases - the dodecagonal quasicrystal of tetrahedra and the densest-packing of octahedra in the Minkowski lattice - are found to coexist with the binary phase. Apart from an alternative, monoclinic packing of octahedra, no additional crystalline phases were observed.

Multi-component superstructures self-assembled from nanocrystal building blocks

More than three decades of intensive study to make high-quality nanocrystals have created a unique toolbox for building multi-component superstructures, which have been recognized as a new generation of metamaterials important to both fundamental sciences and applied technologies. This minireview summarizes recent advances in this exciting field. We will focus our discussion on the synthetic strategies and superstructures of this multi-component metamaterial, and highlight their novel properties and potential applications. Additionally, some perspectives on possible developments in this field are offered at the end of this review. We hope that this minireview will both inform and stimulate research interests for the design and fabrication of these nanocrystal-based multi-component metamaterials for diverse applications in the future.

Superlattices assembled through shape-induced directional binding

Organization of spherical particles into lattices is typically driven by packing considerations. Although the addition of directional binding can significantly broaden structural diversity, nanoscale implementation remains challenging. Here we investigate the assembly of clusters and lattices in which anisotropic polyhedral blocks coordinate isotropic spherical nanoparticles via shape-induced directional interactions facilitated by DNA recognition. We show that these polyhedral blocks--cubes and octahedrons--when mixed with spheres, promote the assembly of clusters with architecture determined by polyhedron symmetry. Moreover, three-dimensional binary superlattices are formed when DNA shells accommodate the shape disparity between nanoparticle interfaces. The crystallographic symmetry of assembled lattices is determined by the spatial symmetry of the block's facets, while structural order depends on DNA-tuned interactions and particle size ratio. The presented lattice assembly strategy, exploiting shape for defining the global structure and DNA-mediation locally, opens novel possibilities for by-design fabrication of binary lattices.

Entropic self-assembly of freely rotating polyhedral particles confined to a flat interface

The self-assembly of hard polyhedral particles confined to a flat interface is studied using Monte Carlo simulations. The particles are pinned to the interface by restricting their movement in the direction perpendicular to it while allowing their free rotations. The six different polyhedral shapes studied in this work are selected from a family of truncated cubes defined by a truncation parameter, s, which varies from cubes (s = 0) via cuboctahedra (s = 0.5) to octahedra (s = 1). Our results suggest that shapes with small values of s show square-like behavior whereas shapes with large values of s tend to show more disc-like behavior. At an intermediate value of s = 0.4, the phase behavior of the system shows both square-like and disc-like features. The results are also compared with the phase behavior of 3D bulk polyhedra and of 2D rounded hard squares. Both comparisons reveal key similarities in the number and sequence of mesophases and solid phases observed. These insights on 2D entropic self-assembly of polyhedral particles is a first step toward understanding the self-assembly of particles at fluid-fluid interfaces, which is driven by a complex interplay of entropic and enthalpic forces. A first-order analysis of the particle-surface energies associated with a fluid-fluid interface indicates that such enthalpic interactions will be particularly important in determining particle orientation behavior at low to intermediate concentrations.