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.


Supplementary Note 1 A Model Based Method for Tomographic Reconstructions of Nanoparticle Assemblies
Here we used a model based tomographic method that only relies on the projected centroids of the nanoparticles and bypasses the image intensity. This method only requires 5-10 tilt images and is useful for calibrating a TEM goniometer and field/scan distortions. This method is based on measuring the geometric centers ) )) of nanoparticles in the projected images as a function of the tilt angle, , where the superscript i is the label for each individual particle. The frame of reference/rotation axis can be shifted to a specific particle n by a pre-subtraction procedure, i.e., ( ) )) ( ) )) ( ) )). The three-dimensional (3D) positions of the particles at zero degree tilt in the particle n reference frame can be retrieved by a nonlinear least squares fitting of the below equation.
where, is the image/scan rotation with respect to the projected tilt axis, and is the offset angle between the 3D tilt axis and the projection plane (Fig. 2b). There are five unknowns-, , ) -in the above equation. At each tilt, two linearly independent equations can be provided-) )) are observables. Therefore, in principal, only three tilt images are needed to solve all the unknowns; in practice, however, additional images are needed to average out drift and scan distortions. In conjunction with the shape and size of each nanoparticle, which can be estimated from the projection images, a three dimensional model of the cluster can built.

Modeling of SAXS profiles from nanoparticle superlattices
In order to confirm the nanoparticle assemblies observed by SEM, we compared our experimental x-ray scattering structure factor data to the theoretical scattering for the nanoparticles on the proposed lattice. We use our recently-published scattering formalism, which simulates powder SAXS profiles for lattices of particles with arbitrary shape 1 . This formalism accounts for particle size, particle shape, and particle orientation within the unit cell. We also explicitly include disorder: particle size polydispersity, lattice disorder (Debye-Waller factor), and average grain size. The form factors, F(q), for spheres, cubes, and octahedra, are already described in the literature; we define P(q) to be the orientationally averaged form factor (which can be measured experimentally by heating the aggregates above the DNA melt temperature).
The structural packing is incorporated into a lattice factor (Z 0 (q)), assuming an isotropic distribution of grains. ( Polydispersity is accounted for by using a Gaussian distribution of particles sizes to compute an effective F(q) and P(q). The extinction of structural scattering leads to a corresponding increase in diffuse scattering. (3) In our model, the structure factor S(q) contains a diffuse scattering term (which accounts for particle polydispersity via β(q), and positional disorder via G(q)), and a structural term , where c is a scaling constant. The average aggregate size for the lattices is included in the peak width via the well-known Debye-Scherrer relation 2 .
The nanocubes used in this work are not perfect platonic solids; they have slightly rounded corners and edges as a result of the synthesis protocol. In order to account for this effect, the form factor for the cube-like nanoparticles was computed by using a 'superball': a mathematical equation which can be used to describe a rounded cube: The parameter p defines the shape of the object: For the cube-sphere assembly, an 'alternating simple cubic' lattice was simulated, where a cubic lattice is filled with alternating spheres and cubes in all three dimensions (analogous to the atomic NaCl system). The model we use explicitly includes the anisotropic form factor, properly accounting for particle size, shape, and orientation. The overall peak positions were fitted by varying the lattice parameter, a. We used the independently measured (from SEM) size and size polydispersity values as initial guesses in our model, but allowed the nominal size to vary slightly (<10%) to account for any systematic differences between SEM and SAXS probes of size.
For the octahedron-sphere system, we assumed a BCC-like lattice where the central particle in the unit cell is distinct from its neighbors at the corner sites (analogous to the atomic CsCl crystal); we again used SEM measurements of size and polydispersity in our model. Overall the model scattering curves for both systems reproduce the essential features of the corresponding experimental in-situ S(q), suggesting that the ordering observed by SEM is representative of the particle configuration in solution.

Supplementary Note 3 Calculation of interparticle distances from SAXS data
The nearest center-to-center interpartice distances (Dcc) between SNP and ANP in the assemblies were determined by the following equations: For NaCl lattice (the systems consisting of cubes and spheres), due to the similar form factor of cubes and spheres, the first diffraction peak q 1 arises from planes {200}, and thus and the lattice constant ; for CsCl lattice (the systems consisting of octahedrons and spheres), due to the similar form factor of octahedrons and spheres, q 1 is diffracted from planes {110} and we have √ and √ ) √ .
The correlation length (ξ) was used to estimate the average grain size. According to Scherrer analysis 5 , , where K is a dimensionless shape factor and has a typical value of about 0.9, λ is the x-ray wavelength, B and δ are accordingly the line broadening at half the maximum intensity (FWHM) in radians and in wave vector.

Supplementary Note 4 DNA modeling for the calculation of particles surface-to-surface separation distance (D ss )
The surface separation distance between cube (flat surface) and sphere (curved surface), D ss , was

Modeling and calculation of attraction potential energy between cube and sphere
The free energy of our DNA-NP systems mainly includes the pair attraction potential energy, , and the repulsion energy of DNA strands between same types of particles, e.g. cube and cube. Due to the small interaction area, the repulsion energy is negligible in comparison with is dominated by the hybridization energy of DNA bridges between a sphere (radius of R) and a cube (edge length of L cube ), with van der Waals (vdW) interactions contributing insignificantly, and thus is proportional to the number of hybridized DNA bridges formed between the their contradictory surfaces.
A cube-sphere pair is showed as a model for calculation of attraction potential energy. The left image is a side-view and the right one is a top-view.
When , the number of hybridized DNA bridges formed between sphere and cube is approximately proportional to the circle projection area of sphere on the square facet of cube, i.e., effective area, S eff , which can be obtained from simple geometry considerations as: (8) Where S ful is the surface area of the full sphere projection with radius of R; S ext is the surface area of the extruding projection that is excluded from square facet of cube with a distance of d ext .
As d dis is the sphere displacement from the cube central axis that crosses the center of the square face, the pair attraction potential energy can be approximated as follows:

( )
Considering compare convenience, the pair attraction potential energy can be normalized by| )|, as shown in Fig.5f.