Icosahedral symmetry, which is not compatible with truly long-range order, can be found in many systems, such as liquids, glasses, atomic clusters, quasicrystals and virus-capsids. To obtain arrangements with a high degree of icosahedral order from tens of particles or more, interparticle attractive interactions are considered to be essential. Here, we report that entropy and spherical confinement suffice for the formation of icosahedral clusters consisting of up to 100,000 particles. Specifically, by using real-space measurements on nanometre- and micrometre-sized colloids, as well as computer simulations, we show that tens of thousands of hard spheres compressed under spherical confinement spontaneously crystallize into icosahedral clusters that are entropically favoured over the bulk face-centred cubic crystal structure. Our findings provide insights into the interplay between confinement and crystallization and into how these are connected to the formation of icosahedral structures.

Colloidal particles with site-specific directional interactions, so called "patchy particles", are promising candidates for bottomup assembly routes towards complex structures with rationally designed properties. Here we present an experimental realization of patchy colloidal particles based on material independent depletion interaction and surface roughness. Curved, smooth patches on rough colloids are shown to be exclusively attractive due to their different overlap volumes. We discuss in detail the case of colloids with one patch that serves as a model for molecular surfactants both with respect to their geometry and their interactions. These one-patch particles assemble into clusters that resemble surfactant micelles with the smooth and attractive sides of the colloids located at the interior. We term these clusters "colloidal micelles". Direct Monte Carlo simulations starting from a homogeneous state give rise to cluster size distributions that are in good agreement with those found in experiments. Important differences with surfactant micelles originate from the colloidal character of our model system and are investigated by simulations and addressed theoretically. Our new "patchy" model system opens up the possibility for self-assembly studies into finite-sized superstructures as well as crystals with as of yet inaccessible structures.anisotropic colloids | depletion interactions | Monte-Carlo simulations N ature has mastered the self-assembly of simple basic subunits into complex, functional structures with outstanding precision. Examples include biological membranes and viruses, which exhibit excellent control over the assembled structures with respect to their functionalities, shapes or sizes. However, the interactions between the building blocks, in the case of viruses, the protein subunits, are often complex and it remains challenging to identify the key elements for guiding and controlling the selfassembling process. By mimicking such self-assembly processes on a colloidal scale, insights into the paramount elements that control the assembly can be obtained in situ and applied to build up superstructures with new and desirable properties.Colloidal particles with site-specific directional interactions, so called "patchy particles", are promising candidates for bottom-up assembly routes towards such complex structures with rationally designed properties (1-3). The size and geometry of the patches together with the shape of the interparticle potential are expected to determine the formed structures and phases, which may range from empty liquids (4) and crystals (5-7) to finite-sized clusters (1, 2, 8-11), and lead to novel collective behavior (12).Recent experimental approaches to assemble colloidal particles at specific sites include hydrophobic-hydrophilic interactions (6,7,(13)(14)(15), and lock-and-key recognition mechanisms (16). With a wide variability of colloidal shapes available today, the ultimate challenge is to identify general methods to render specific areas of the colloids attractive or ...

All liquids (except helium owing to quantum effects) crystallize at low temperatures, forming ordered structures. The competition between disorder, which stabilizes the liquid phase, and energy, which leads to a preference for the crystalline structure, inevitably favours the crystal when the temperature is lowered and entropy becomes progressively less relevant. The liquid state survives at low temperatures only as a glass, an out-of-equilibrium arrested state of matter. This textbook description holds inevitably for atomic and molecular systems, where particle interactions are set by quantum-mechanical laws. The question remains whether it holds for colloidal particles, where interparticle interactions are usually short-ranged and tunable. Here we show that for patchy colloids with limited valence 1 , conditions can be found for which the liquid phase is stable even in the zero-temperature limit. Our results offer fresh cues for understanding the stability of gels 2 and the glass-forming ability of molecular network glasses 3,4 .The ability to control the selectivity and angular flexibility of interparticle interactions 5 makes it possible to provide valence to colloids. This can be done by means of chemical [6][7][8] or physical 9 patterning of the colloid surface, that is, locally changing either its chemical properties (for example, hydrophobicity), or the physical properties (for example, roughness). Exploiting valence offers enormous possibilities, some of which have been addressed in recent years, theoretically 10 , numerically 4,11,12 and experimentally 2,7,13,14 . As colloidal interactions are typically shortranged, valence provides an implicit quantization of the particle energy, which becomes essentially proportional to the (limited) number of formed bonds. In an optimal arrangement, all particles bind with their maximum number of neighbours and the system is in its lowest possible (ground) energy state. Typically, such an optimal arrangement is spatially ordered, defining the most stable crystal phase(s).In principle, when the density is not too high, it is possible to imagine disordered arrangements in which all particles are fully bonded, with exactly the same energy as the crystal. Such a state, which was recently shown to be stable in a model system for DNA-functionalized colloids 12 , is a liquid: a fluid, disordered phase at temperatures below the critical temperature. Under this unconventional condition, the stability of the system becomes controlled by its entropy, a case reminiscent of (purely entropydriven) hard-sphere particles. In this study we demonstrate that the flexibility of the bonds (encoded in the angular patch width)-a tunable quantity in the design of patchy colloidal particles-is the key element in controlling the entropy of the liquid as compared with that of the crystal. Large binding angles combined with limited valence give rise to a thermodynamically stable, fully bonded liquid phase.Department of Physics, Sapienza, Universitá di Roma, Piazzale Aldo Moro 2, I-00185, Roma,...

One of the most controversial hypotheses for explaining the origin of the thermodynamic anomalies characterizing liquid water postulates the presence of a metastable second-order liquid-liquid critical point [1] located in the “no-man’s land” [2]. In this scenario, two liquids with distinct local structure emerge near the critical temperature. Unfortunately, since spontaneous crystallization is rapid in this region, experimental support for this hypothesis relies on significant extrapolations, either from the metastable liquid or from amorphous solid water [3, 4]. Although the liquid-liquid transition is expected to feature in many tetrahedrally coordinated liquids, including silicon [5], carbon [6] and silica, even numerical studies of atomic and molecular models have been unable to conclusively prove the existence of this transition. Here we provide such evidence for a model in which it is possible to continuously tune the softness of the interparticle interaction and the flexibility of the bonds, the key ingredients controlling the existence of the critical point. We show that conditions exist where the full coexistence is thermodynamically stable with respect to crystallization. Our work offers a basis for designing colloidal analogues of water exhibiting liquid-liquid transitions in equilibrium, opening the way for experimental confirmation of the original hypothesis.

We examine the effect of vacancies on the phase behavior and structure of systems consisting of hard cubes using event-driven molecular dynamics and Monte Carlo simulations. We find a firstorder phase transition between a fluid and a simple cubic crystal phase that is stabilized by a surprisingly large number of vacancies, reaching a net vacancy concentration of approximately 6.4% near bulk coexistence. Remarkably, we find that vacancies increase the positional order in the system. Finally, we show that the vacancies are delocalized and therefore hard to detect.hard polyhedra | colloids | free energy T he free energy of crystal phases is generally minimized by a finite fraction of point defects like vacancies and interstitials. However, the equilibrium number of such defects in most colloidal and atomic/molecular crystals with a single constituent is extremely low. Exemplarily, for the face-centered cubic crystal of hard spheres, one of the few colloidal systems where the vacancy and interstitial fractions have been calculated, the fraction of vacancies and interstitials is on the order of 10 −4 and 10 −8 , respectively, near coexistence (1). As such, neither the vacancies nor the interstitials strongly affect the phase behavior, and so most studies of crystals ignore the effect of these defects. Nevertheless, vacancies and interstitials have a significant effect on the dynamics in an otherwise perfect crystal, as the main mechanism for particle diffusion is hopping of particles from filled to empty sites or between interstitial sites.In this paper, we examine a system of hard cubes where, as we will demonstrate, one cannot ignore the presence of vacancies. Arguably, a cube is one of the simplest nonspherical shapes and the archetype of a space-filling polyhedron. Surprisingly, despite the simplicity of this system, we find that the stable ordered phase is strongly affected by the presence of vacancies, so much that vacancies actually increase the positional order and change the melting point. Remarkably, the fraction of vacancies in this system is more than two orders of magnitude higher than that of hard spheres or any other known typical, experimentally realizable, single-component atomic or colloidal system, reaching 6.4% near coexistence. Additionally, while purely hard (not rounded) colloidal cubes are yet to be realized, colloidal cubes with various interactions are now a reality (2-7), and it is likely that hard cubes will be realized in the future.Here, we use Monte Carlo (MC) and event-driven molecular dynamics (EDMD) (8) simulations to examine in detail the effect of vacancies on the equilibrium phase behavior of hard cubes. The model we study consists of N perfectly sharp hard cubes with edge length σ in a volume V . Aside from hard-core interactions that prevent configurations of overlapping cubes, the particles do not interact. In both types of simulation (MC and EDMD), overlaps are detected using an algorithm based on the separating axis theorem (e.g., ref. 9). More information on the EDMD impleme...