Self-assembly of uniform polyhedral silver nanocrystals into densest packings and exotic superlattices

Journal name:
Nature Materials
Year published:
Published online

Understanding how polyhedra pack into extended arrangements is integral to the design and discovery of crystalline materials at all length scales1, 2, 3. Much progress has been made in enumerating and characterizing the packing of polyhedral shapes4, 5, 6, and the self-assembly of polyhedral nanocrystals into ordered superstructures7, 8, 9. However, directing the self-assembly of polyhedral nanocrystals into densest packings requires precise control of particle shape10, polydispersity11, interactions and driving forces 12. Here we show with experiment and computer simulation that a range of nanoscale Ag polyhedra can self-assemble into their conjectured densest packings6. When passivated with adsorbing polymer, the polyhedra behave as quasi-hard particles and assemble into millimetre-sized three-dimensional supercrystals by sedimentation. We also show, by inducing depletion attraction through excess polymer in solution, that octahedra form an exotic superstructure with complex helical motifs rather than the densest Minkowski lattice13. Such large-scale Ag supercrystals may facilitate the design of scalable three-dimensional plasmonic metamaterials for sensing14, 15, nanophotonics16 and photocatalysis17.

At a glance


  1. Self-assembly of dense polyhedron lattices.
    Figure 1: Self-assembly of dense polyhedron lattices.

    a, Schematic representation of polyhedral shapes accessible using the Ag polyol synthesis. b, Microfluidic reservoir used to assemble the Ag nanocrystals. c, Dark-field micrograph showing the growing Ag supercrystal. dh, SEM micrographs of the colloidal lattices (left) and the corresponding diagrams of their densest known lattice packings (right): d, cubes; e, truncated cubes; f, cuboctahedra; g, truncated octahedra; h, octahedra. Colour in g and h is used to indicate different layers of the crystal. Scale bars are 500 nm unless otherwise noted.

  2. Long-range order of assembled lattices.
    Figure 2: Long-range order of assembled lattices.

    Large-scale SEM images of an octahedron lattice on a Si substrate. a, The imaged region is a cross-section of a large supercrystal that extends millimetres below the plane of the image, and was exposed by cleaving the substrate and superlattice into two pieces. bd, Inspection of particle orientation in the magnified sections reveals that long-range order is maintained across the entire imaged sample. Insets: Fourier transforms of these sections match closely. e, SEM image of a large Minkowski facet. f, A magnified section of the experimental lattice with a superimposed rendering of the ideal Minkowski lattice shows excellent agreement. g, Cross-polarized optical micrograph of a millimetre-sized superlattice slab. The uniform colour of the reflected light from top to bottom indicates that the crystal is composed of a single domain. All scale bars are 2 μm unless otherwise noted.

  3. Gravitational driving force and packings near surfaces.
    Figure 3: Gravitational driving force and packings near surfaces.

    a, Snapshot of a Monte Carlo simulation (Supplementary Methods) of hard octahedra under the influence of gravity (left). Zoom-in images (right), from top to bottom, show a dilute fluid, a dense fluid and a growing solid of the Minkowski type. Octahedra are coloured according to orientation. b, Packing fraction as a function of Monte Carlo sweeps and height along the z axis in the elongated box. A density gradient develops in the initially homogenous system, resulting in a fluid–solid transformation at the bottom. Biased by a flat surface, octahedra form a simple hexagonal lattice (space group P6/mmm). c, Small patches of this structure, retrieved early in the assembly process. d, Top-down view (left), with inset showing a vacancy in the lattice, and side view (right). e, Zero-gravity, isobaric Monte Carlo simulations with a hard bottom wall and weak interparticle attractions show coexistence between the layered structure and the Minkowski lattice, consistent with experiment. f, Comparison between a flat layer of the simple hexagonal lattice (left) and corresponding staggered layers in the Minkowski lattice (right).

  4. Polymer-mediated nanocrystal interactions.
    Figure 4: Polymer-mediated nanocrystal interactions.

    a, Contributions to the free energy of two octahedra aligned face to face, as a function of the separation X of surface-adsorbed polymer layers (see inset; Supplementary Methods). The entropic repulsion (orange) of densely packed surface polymers (radius of gyration, Rg=10 nm) dominates the vdW attraction (red) of the silver cores. The resulting potential (green) is only weakly attractive, enabling the nanocrystals to behave as quasi-hard particles. Unadsorbed polymer coils act as depleting agents, inducing effective attractions (blue) that can lead to self-assembly of different structures. b, Potentials of mean force for two hard octahedra in a bath of hard 20 nm spheres (see inset), as a function of the distance x between nearest midpoints of octahedron faces (see inset, interaction centres are marked in yellow), averaged over all rotational degrees of freedom. Legend values correspond to the sphere packing fraction  φ.

  5. Ag octahedra assemble into a previously unknown lattice in the presence of excess PVP.
    Figure 5: Ag octahedra assemble into a previously unknown lattice in the presence of excess PVP.

    a, SEM micrograph of a new octahedron supercrystal. b,c, The lattice consists of tetramer motifs (b) and forms spontaneously in Monte Carlo simulations of octahedra with depletion attractions (c). d, The tetramers can be decomposed into lines and counter-rotating helices, where the helices in this plane consist of particles belonging to three other sets of interlocking lines. e, The lattice has been coloured to show these four equivalent sets of lines (yellow, blue, green and red) that are oriented in different [111] directions. Lines belonging to the same [111] direction are parallel and shifted relative to one another in alternating layers (green particles on left; right, enlarged SEM micrograph from the corresponding experimental lattice). When viewed end on (yellow particles), the particle lines stack hexagonally.


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Author information


  1. Department of Chemistry, University of California, Berkeley, California 94720, USA

    • Joel Henzie,
    • Michael Grünwald,
    • Asaph Widmer-Cooper,
    • Phillip L. Geissler &
    • Peidong Yang
  2. Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

    • Asaph Widmer-Cooper,
    • Phillip L. Geissler &
    • Peidong Yang
  3. School of Chemistry, University of Sydney, Sydney, New South Wales 2006, Australia

    • Asaph Widmer-Cooper
  4. Department of Materials Science and Engineering, University of California, Berkeley, California 94720, USA

    • Peidong Yang


J.H. and M.G. both contributed extensively to this work. J.H. initiated the study, conceived and conducted the experiments, analysed results and cowrote the paper. M.G. identified all crystal structures, conceived and implemented the simulations and theoretical models, analysed results and cowrote the paper. A.W-C. suggested simulations and experiments, analysed results, helped prepare figures and cowrote the paper. P.L.G. suggested simulations and experiments, analysed results and cowrote the paper. P.Y. initiated the study, suggested experiments, analysed results and cowrote the paper.

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