Joost de Graaf

Hierarchical self-assembly of suspended branched colloidal nanocrystals into superlattice structures

Self-assembly of molecular units into complex and functional superstructures is ubiquitous in biology. The number of superstructures realized by self-assembly of man-made nanoscale units is also growing. However, assemblies of colloidal inorganic nanocrystals are still at an elementary level, not only because of the simplicity of the shape of the nanocrystal building blocks and their interactions, but also because of the poor control over these parameters in the fabrication of more elaborate nanocrystals. Here, we show how monodisperse colloidal octapod-shaped nanocrystals self-assemble, in a suitable solution environment, on two sequential levels. First, linear chains of interlocked octapods are formed, and subsequently the chains spontaneously self-assemble into three-dimensional superstructures. Remarkably, all the instructions for the hierarchical self-assembly are encoded in the octapod shape. The mechanical strength of these superstructures is improved by welding the constituent nanocrystals together.

Low-dimensional semiconductor superlattices formed by geometric control over nanocrystal attachment.

Oriented attachment, the process in which nanometer-sized crystals fuse by atomic bonding of specific crystal facets, is expected to be more difficult to control than nanocrystal self-assembly that is driven by entropic factors or weak van der Waals attractions. Here, we present a study of oriented attachment of PbSe nanocrystals that counteract this tuition. The reaction was studied in a thin film of the suspension casted on an immiscible liquid at a given temperature. We report that attachment can be controlled such that it occurs with one type of facets exclusively. By control of the temperature and particle concentration we obtain one- or two-dimensional PbSe single crystals, the latter with a honeycomb or square superimposed periodicity in the nanometer range. We demonstrate the ability to convert these PbSe superstructures into other semiconductor compounds with the preservation of crystallinity and geometry.

Dense Regular Packings of Irregular Nonconvex Particles

We present a new numerical scheme to study systems of nonconvex, irregular, and punctured particles in an efficient manner. We employ this method to analyze regular packings of odd-shaped bodies, both from a nanoparticle and from a computational geometry perspective. Besides determining close-packed structures for 17 irregular shapes, we confirm several conjectures for the packings of a large set of 142 convex polyhedra and extend upon these. We also prove that we have obtained the densest packing for both rhombicuboctahedra and rhombic enneacontrahedra and we have improved upon the packing of enneagons and truncated tetrahedra.

Phase diagram of colloidal hard superballs: from cubes via spheres to octahedra

For hard anisotropic particles the formation of a wide variety of fascinating crystal and liquid-crystal phases is accomplished by entropy alone. A better understanding of these entropy-driven phase transitions will shed light on the self-assembly of nanoparticles, however, there are still many open questions in this regard. In this work, we use Monte Carlo simulations and free-energy calculations to determine the phase diagram of colloidal hard superballs, of which the shape interpolates between cubes and octahedra via spheres. 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 local curvature, i.e. rounded corners and curved faces, of superballs plays an important role in stabilizing 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.

Ordered Two-Dimensional Superstructures of Colloidal Octapod-Shaped Nanocrystals on Flat Substrates

We studied crystal structures in a monolayer consisting of anisotropic branched colloidal (nano)octapods. Experimentally, octapods were observed to form a monolayer on a substrate with a square-lattice crystal structure by drop-casting and fast evaporation of solvent. The experimental results were analyzed by Monte Carlo simulations using a hard octapod model consisting of four interpenetrating spherocylinders. We confirmed by means of free-energy calculations that crystal structures with a (binary-lattice) square morphology are indeed thermodynamically stable at high densities. The effect of the pod length-to-diameter ratio on the crystal structures was also considered and we used this to constructed the phase diagram for these hard octapods. In addition to the (binary-lattice) square crystal phase, a rhombic crystal and a hexagonal plastic-crystal (rotator) phase were obtained. Our phase diagram may prove instrumental in guiding future experimental studies.

Self-Assembly of Octapod-Shaped Colloidal Nanocrystals into a Hexagonal Ballerina Network Embedded in a Thin Polymer Film

Nanoparticles with unconventional shapes may exhibit different types of assembly architectures that depend critically on the environmental conditions under which they are formed. Here, we demonstrate how the presence of polymer (polymethyl methacrylate, PMMA) molecules in a solution, in which CdSe(core)/CdS(pods) octapods are initially dispersed, affects the octapod-polymer organization upon solvent evaporation. We show that a fast drop-drying process can induce a remarkable two-dimensional (2D) self-assembly of octapods at the polymer/air interface. In the resulting structure, each octapod is oriented like a “ballerina”, that is, only one pod sticks out of the polymer film and is perpendicular to the polymer–air interface, while the opposite pod (with respect to the octapod’s center) is fully immersed in the film and points toward the substrate, like a ballerina performing a grand battement. In some areas, a hexagonal-like pattern is formed by the ballerinas in which the six nonvertical pods, which are all embedded in the film, maintain a pod–pod parallel configuration with respect to neighboring particles. We hypothesize that the mechanism responsible for such a self-assembly is based on a fast adsorption of the octapods from bulk solution to the droplet/air interface during the early stages of solvent evaporation. At this interface, the octapods maintain enough rotational freedom to organize mutually in a pod–pod parallel configuration between neighboring octapods. As the solvent evaporates, the octapods form a ballerina-rich octapod-polymer composite in which the octapods are in close contact with the substrate. Finally, we found that the resulting octapod-polymer composite is less hydrophilic than the polymer-only film.

In situ microscopy of the self-assembly of branched nanocrystals in solution

Solution-phase self-assembly of nanocrystals into mesoscale structures is a promising strategy for constructing functional materials from nanoscale components. Liquid environments are key to self-assembly since they allow suspended nanocrystals to diffuse and interact freely, but they also complicate experiments. Real-time observations with single-particle resolution could have transformative impact on our understanding of nanocrystal self-assembly. Here we use real-time in situ imaging by liquid-cell electron microscopy to elucidate the nucleation and growth mechanism and properties of linear chains of octapod-shaped nanocrystals in their native solution environment. Statistical mechanics modelling based on these observations and using the measured chain-length distribution clarifies the relative importance of dipolar and entropic forces in the assembly process and gives direct access to the interparticle interaction. Our results suggest that monomer-resolved in situ imaging combined with modelling can provide unprecedented quantitative insight into the microscopic processes and interactions that govern nanocrystal self-assembly in solution.

Crystal-structure prediction via the Floppy-Box Monte Carlo algorithm: Method and application to hard (non)convex particles

In this paper, we describe the way to set up the floppy-box Monte Carlo (FBMC) method [L. Filion, M. Marechal, B. van Oorschot, D. Pelt, F. Smallenburg, and M. Dijkstra, Phys. Rev. Lett. 103, 188302 (2009)] to predict crystal-structure candidates for colloidal particles. The algorithm is explained in detail to ensure that it can be straightforwardly implemented on the basis of this text. The handling of hard-particle interactions in the FBMC algorithm is given special attention, as (soft) short-range and semi-long-range interactions can be treated in an analogous way. We also discuss two types of algorithms for checking for overlaps between polyhedra, the method of separating axes and a triangular-tessellation based technique. These can be combined with the FBMC method to enable crystal-structure prediction for systems composed of highly shape-anisotropic particles. Moreover, we present the results for the dense crystal structures predicted using the FBMC method for 159 (non)convex faceted particles, on which the findings in [J. de Graaf, R. van Roij, and M. Dijkstra, Phys. Rev. Lett. 107, 155501 (2011)] were based. Finally, we comment on the process of crystal-structure prediction itself and the choices that can be made in these simulations.

Adsorption trajectories and free-energy separatrices for colloidal particles in contact with a liquid-liquid interface

We apply the recently developed triangular tessellation technique as presented by J. de Graaf et al. [Phys. Rev. E 80, 051405 (2009)] to calculate the free energy associated with the adsorption of anisotropic colloidal particles at a flat interface. From the free-energy landscape, we analyze the adsorption process, using a simplified version of Langevin dynamics. The present result is a first step towards understanding the time-dependent behavior of colloids near interfaces. This study shows a wide range of adsorption trajectories, where the emphasis lies on a strong dependence of the dynamics on the orientation of the colloid at initial contact with the interface. We believe that the observed orientational dependence in our simple model can be recovered in suitable experimental systems.

Triangular tessellation scheme for the adsorption free energy at the liquid-liquid interface : towards nonconvex patterned colloids

We present a numerical technique, namely, triangular tessellation, to calculate the free energy associated with the adsorption of a colloidal particle at a flat interface. The theory and numerical scheme presented here are sufficiently general to handle nonconvex patchy colloids with arbitrary surface patterns characterized by a wetting angle, e.g., amphiphilicity. We ignore interfacial deformation due to capillary, electrostatic, or gravitational forces, but the method can be extended to take such effects into account. It is verified that the numerical method presented is accurate and sufficiently stable to be applied to more general situations than presented in this paper. The merits of the tessellation method prove to outweigh those of traditionally used semianalytic approaches, especially when it comes to generality and applicability.