Oriented attachment of PbSe nanocubes can result in the formation of two-dimensional (2D) superstructures with long-range nanoscale and atomic order. This questions the applicability of classic models in which the superlattice grows by first forming a nucleus, followed by sequential irreversible attachment of nanocrystals, as one misaligned attachment would disrupt the 2D order beyond repair. Here, we demonstrate the formation mechanism of 2D PbSe superstructures with square geometry by using in situ grazing-incidence X-ray scattering (small angle and wide angle), ex situ electron microscopy, and Monte Carlo simulations. We observed nanocrystal adsorption at the liquid/gas interface, followed by the formation of a hexagonal nanocrystal monolayer. The hexagonal geometry transforms gradually through a pseudo-hexagonal phase into a phase with square order, driven by attractive interactions between the {100} planes perpendicular to the liquid substrate, which maximize facet-to-facet overlap. The nanocrystals then attach atomically via a necking process, resulting in 2D square superlattices.

The phase diagram of colloidal hard superballs, of which the shape interpolates between cubes and octahedra via spheres, is determined by free-energy calculations in Monte Carlo simulations. We discover not only a stable face-centered cubic (fcc) plastic crystal phase for near-spherical particles, but also a stable body-centered cubic (bcc) plastic crystal close to the octahedron shape. Moreover, coexistence of these two plastic crystals is observed with a substantial density gap. The plastic fcc and bcc crystals are, however, both unstable in the cube and octahedron limit, suggesting that the rounded corners of superballs play an important role in stablizing the rotator phases. In addition, we observe a two-step melting phenomenon for hard octahedra, in which the Minkowski crystal melts into a metastable bcc plastic crystal before melting into the fluid phase.PACS numbers: 82.70. Dd, 64.70.pv, 64.75.Yz, Recent breakthroughs in particle synthesis have resulted in a spectacular variety of anisotropic nanoparticles such as cubes, octapods, tetrapods, octahedra, icecones, etc. [1]. A natural starting point to study the selfassembled structures of these colloidal building blocks is to view them as hard particles [1]. Not only can these hard-particle models be used to predict properties of suitable experimental systems, but such models also provide a stepping stone towards systems where soft interactions play a role [2,3]. Moreover, the analysis of hard particles is of fundamental relevance and raises problems that influence fields as diverse as (soft) condensed matter [1,[4][5][6], mathematics [5,7], and computer science [8]. In this light the concurrent boom in simulation studies of hard anisotropic particles is not surprising [5][6][7][9][10][11][12][13][14][15][16].The best-known hard-particle system consists of hard spheres, which freeze into close-packed hexagonal (cph) crystal structures [8], of which the ABC-stacked cph crystal, better known as the face-centered cubic (fcc) crystal phase, is thermodynamically stable [17]. Hard anisotropic particles can form liquid-crystalline equilibrium states if they are sufficiently rod-or disclike [12,16], but particles with shapes that are close-to-spherical tend to order into plastic crystal phases, also known as rotator phases [14][15][16]. In fact, simple guidelines were recently proposed to predict the plastic-and liquid-crystal formation only on the basis of rotational symmetry and shape anisotropy of hard polyhedra [6]. In this Letter we will take a different approach, based on free-energy calculations, and address the question whether and to what extent rounding the corners and faces of polyhedral particles affects the phase behavior. Such curvature effects are of direct relevance to experimental systems, in which sterically and charged stabilised particles can often not be considered as perfectly flat-faced and sharp-edged [18]. For instance, recent experiments on nanocube assemblies show a continuous phase transformation between simple cubic and rhombohedral pha...

Using Monte Carlo simulations and free-energy calculations, we determine the phase diagram of a family of truncated hard cubes, where the shape evolves smoothly from a cube via a cuboctahedron to an octahedron. A remarkable diversity in crystal phases and close-packed structures is found, including a fully degenerate crystal structure, several plastic crystals, as well as vacancy-stabilized crystal phases, all depending sensitively on the precise particle shape. Our results illustrate the intricate relation between phase behavior and building-block shape, and can guide future experimental studies on polyhedral-shaped nanoparticles. DOI: 10.1103/PhysRevLett.111.015501 PACS numbers: 61.46.Df, 64.70.MÀ, 64.75.Yz, 82.70.Dd Recent advances in experimental techniques to synthesize polyhedron-shaped particles, such as faceted nanocrystals and colloids [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and the ability to perform self-assembly experiments with these particles [15][16][17][18][19][20][21][22], have attracted the interest of physicists, mathematicians, and computer scientists [23][24][25][26][27]. Additionally, predicting the densest packings of hard polyhedra has intrigued mathematicians since the time of the early Greek philosophers, such as Plato and Archimedes [28,29]. Modern computer platforms have made it possible to perform simulations of these systems, which has resulted not only in an improved understanding of the experimentally observed phenomenology in colloidal suspensions of such particles, but also in improved Ansätze for the morphology of their closepacked configurations [24,[30][31][32][33][34][35].The self-assembly of the basic building blocks at finite pressures may differ substantially from the packings achieved at high (sedimentation and solvent-evaporation) pressures. For instance, liquid-crystal, plastic-crystal, vacancy-rich simple-cubic, and quasicrystalline mesophases are stabilized by entropy alone under non-closepacked conditions of hard anisotropic particle systems [30][31][32][33][34]36,37]. Predicting the phase behavior from the shape of the building blocks alone is therefore a major challenge in materials science and is crucial for the design of new functional materials. It is thus not surprising that numerous studies have been devoted to providing simple guidelines for predicting the self-assembly from the particle shape alone [32][33][34].Recently, Henzie et al.[15] reported the shapecontrolled synthesis of truncated cubes. In their research, the close-packed crystals of these particles were studied using sedimentation experiments and simulations. They created exotic superlattices, and their results also tested several conjectures on the densest packings of hard polyhedra [23,[25][26][27]. However, Henzie et al. did not examine the finite-pressure behavior of the system. Mapping the full phase diagram for the system of truncated cubes is thus important, not only from a fundamental perspective but also to guide future experimental self-assembly studies to fabricate new func...

Melting in two-dimensional systems has remained controversial as theory, simulations, and experiments show contrasting results. One issue that obscures this discussion is whether or not theoretical predictions on strictly 2D systems describe those of quasi-2D experimental systems, where outof-plane fluctuations may alter the melting mechanism. Using event-driven Molecular Dynamics simulations, we find that the peculiar two-stage melting scenario of a continuous solid-hexatic and a first-order hexatic-liquid transition as observed for a truly 2D system of hard disks [Bernard and Krauth, Phys. Rev. Lett. 107, 155704 (2011)] persists for a quasi-2D system of hard spheres with out-of-plane particle motions as high as half the particle diameter. By calculating the renormalized Young's modulus, we show that the solid-hexatic transition is of the Kosterlitz-Thouless type, and occurs via dissociation of bound dislocation pairs. In addition, we find a first-order hexatic-liquid transition that seems to be driven by a spontaneous proliferation of grain boundaries.

We present a combined experimental, theoretical, and simulation study on the self-assembly of colloidal hexagonal bipyramid- and hexagonal bifrustum-shaped ZnS nanocrystals (NCs) into two-dimensional superlattices. The simulated NC superstructures are in good agreement with the experimental ones. This shows that the self-assembly process is primarily driven by minimization of the interfacial free-energies and maximization of the packing density. Our study shows that a small truncation of the hexagonal bipyramids is sufficient to change the symmetry of the resulting superlattice from hexagonal to tetragonal, highlighting the crucial importance of precise shape control in the fabrication of functional metamaterials by self-assembly of colloidal NCs.