Cocrystals combining order and correlated disorder via colloidal crystal engineering with DNA

Abstract

Colloidal cocrystallization enables the formation of multicomponent materials with unique physicochemical properties, yet the role of nanoparticle (NP) shape and specific ligand interactions to cocrystallize anisotropic and isotropic NPs, with order and correlated disorder, remains underexplored. Here, geometry-inspired strategies along with programmable DNA interactions are combined to achieve structural control of colloidal cocrystal assemblies. Coassembling polyhedral and spherical NPs with complementary DNA yields two classes of cocrystals: one where both components order, and another where polyhedral NPs form a periodic lattice, while spherical NPs remain disordered but spatially correlated with polyhedral edges and corners. The size ratio of the building blocks can be used to control the ordering of spherical NPs—smaller octahedral-to-sphere size ratios favor fully ordered cocrystals. Molecular dynamics simulations further elucidate the role of NP shapes and dimensions in the structural outcome of the cocrystal. This work provides a framework for deliberately targeting and accessing crystals with exotic multicomponent structures.

Cocrystals with programmable order and correlated disorder enable exotic multicomponent materials via DNA-guided assembly.

INTRODUCTION

Intermetallic alloys and certain biological or organic structures are a consequence of the cocrystallization of disparate building blocks (1–7), and therefore, the process of cocrystallization is of great fundamental and practical importance in chemistry and materials science. In the context of colloidal crystals, building block size, composition, shape, and ligand type are key parameters that can be used to direct cocrystallization (8–11). For example, mixtures of spherical nanoparticles (NPs) of different sizes, compositions, and ligand chemistries have been shown to form varying binary superlattices with different crystal symmetries (12–18). However, when the physical dimensions between two types of building blocks are markedly different (e.g., one order of magnitude), cocrystallization becomes challenging due to significant entropic penalties (19, 20). To this end, the work described here aims to understand how NP shape and specific ligand interactions can be used to control their coassembly, specifically into cocrystals where the components are interspersed, and at least one component is ordered.

This work uses colloidal crystal engineering with DNA, a versatile and powerful methodology for controlling the assembly of NPs into colloidal superlattices, in which sequence-specific interactions and hybridization events direct NP crystallization (7, 21–23). Colloidal crystal engineering with DNA has been used to assemble NP building blocks (isotropic and anisotropic) into a library of single crystalline superlattices (2, 9, 24–27). While crystalline matter is interesting and being able to program architecture with exquisite control using DNA is useful, a class of matter found in nature that has not been extensively realized or studied with this approach consists of ordered phases interspersed with a correlated disordered phase comprised of related NP building blocks, so-called cocrystals combining order and disorder. Correlated disordered materials found in nature such as alloys (28) and perovskites (29) have promising thermal and electrical conduction properties. Although primarily modeled thus far through simulations (30) due to synthetic challenges, these materials are promising for advancing the design of functional materials (28, 29).

We hypothesized that the relative sizes of the particles are important and that certain ones would favor AB cocrystallization (where A and B are the NP building blocks), while others would favor the formation of a lattice defined by A subunits surrounded by a disordered phase of B units. We designed the B units to bind the A units together; specific ordering of B is not required. Therefore, in this work, we studied the coassembly of anisotropic gold polyhedral NPs (i.e., cubes or octahedra) and isotropic spherical NPs in an attempt to identify the structural parameters that favor either of the two anticipated outcomes. We studied this for two different systems: cubes and spheres as well as regular octahedra and spheres. In both cases, we have identified size ratios that favor cocrystal formation both with and without correlated disorder in the arrangements of the spheres. Molecular dynamics (MD) simulations provide further insight into how the shapes and relative sizes of NPs affect structural outcomes.

This cocrystallization strategy, governed by particle dimensions and directed by DNA, provides exciting possibilities for the creation of multicomponent structures with hierarchical organization and advanced structural control. This study is fundamentally important because it provides a rationale for deliberately controlling the formation of cocrystals combining order and correlated disorder from the appropriate binary building blocks. Beyond expanding the classes of matter attainable through colloidal crystal engineering with DNA, it will allow researchers to explore how order and correlated disorder in these crystals affect their collective properties, thereby broadening our understanding of hierarchical materials.

RESULTS

Before assembly, anisotropic NPs (i.e., cubes and regular octahedra) and isotropic spheres of different sizes (fig. S1) were synthesized (31–33) and functionalized with complementary DNA (anchor A and linker A for anisotropic NPs; anchor B and linker B for spheres; Fig. 1, A and B, and table S1), respectively. These oligonucleotide-NP conjugates are referred to as “programmable atom equivalents (PAEs),” (34) and their sizes (L) are defined by their hydrodynamic dimensions, taking into account NP dimensions and DNA lengths (Fig. 1). A slow-cooling method was used for the cocrystallization that yields crystal structures that have reached equilibrium (22, 35). The resulting cocrystals were characterized in their as-prepared solution states through synchrotron small-angle x-ray scattering (SAXS). These crystals were then analyzed in the solid state using scanning electron microscopy (SEM) after silica embedding (36). The cross-sectional views of the crystals were characterized by analyzing the inner NP distribution using scanning transmission electron microscopy (STEM) after resin embedding and preparing cross sections with ultramicrotomy (36). Together, SAXS, S(T)EM, and tilted STEM characterization offer a complementary view of the ensemble and local ordering.

Fig. 1. Scheme showing the co-assembly of anisotropic nanoparticles (NPs) and spheres of different sizes with a controlled degree of ordering (order or correlated disorder).

(A) Cubes and spheres are functionalized with complementary DNA (the red sticky end is complementary with the green sticky end) and coassembled into simple cubic (sc) lattices with a controlled degree of ordering, where cubes are ordered, but spheres are disordered, dependent on relative particle dimensions. (B) Octahedra and spheres are functionalized with complementary DNA (the maize sticky end is complementary with the blue sticky end) and coassembled into cubic close-packed (ccp) lattices, where octahedra are ordered, but spheres can be ordered or disordered depending on the relative dimensions of the octahedra and spheres. Octahedral or cubic PAE edge length: L₁; spherical PAE diameter: L₂. (C) Zoom-in schematic of the proposed cocrystal combining order and correlated disorder assembled from large cubes and spheres in this work. The locations of the spheres, although random, are correlated with the positions of the corners and edges of the cubes. (D) Comparative schematic of the ordered crystal.

Coassembly of large cubes and small spheres

Cubes (edge length: ∼56 nm; linker A1; DNA length: ∼11 nm) and spheres (diameter: ∼5 or 10 nm; linker B1; DNA length: ∼11 nm) were assembled into cocrystals with simple cubic (sc) symmetry and cubic habits (Fig. 2 and table S1), where cubes are ordered, and spheres are distributed in the interstitial gaps between cubes in a disordered manner. Depending on the size ratio (r = L₁/L₂) between cubic PAEs (L₁) and spherical PAEs (L₂), a different number of spheres in each gap (on average) are observed. Specifically, more spheres are found in each gap when reducing the sphere sizes while keeping the same edge length of cubes. For example, there are more spheres in each gap when 5-nm spheres (r ∼ 2.9) are used compared to 10-nm spheres (r ∼ 2.4) (Fig. 2, D to F and J to L). Because small spheres minimally contribute to the measured SAXS signals, experimental SAXS profiles match simulations containing cubes arranged in a sc symmetry (Fig. 2, C and I). The lattice parameters of these two simple cubic lattices, composed of the same cubes but different spheres, are nearly the same (unit cell sizes are both around 85 nm based on SAXS profiles). This suggests a favorable corner or edge distribution of spheres around cubes because a face-favorable distribution of spheres would otherwise increase the gaps between cubes in the case of larger spheres and would therefore result in different lattice parameters. Tilted TEM images (figs. S2 and S3) confirm this hypothesis that spheres are mainly distributed around the corners and edges of the cubes. This could be attributed to the slight corner rounding of the synthesized cubes, coupled with both the entropic gain due to the spheres having access to more space near the corners and edges and the steric repulsion between spheres that would favor additional space. Together, these factors favor the location of spheres (more complementary DNA connections between cubes and spheres and more configurational entropy) around the corners and edges of the cubes.

Fig. 2. Simple cubic lattices constructed from cubes and spheres exhibit different degrees of ordering, dependent on particle size ratios.

(A to F) Model (A); corresponding SEM images at different magnifications (B); simulated and experimental SAXS profiles (C); and cross-sectional TEM images near the <100> (D), <110> (E), and <111> (F) crystal directions, showing the sc cocrystals composed of ordered cubes (edge length: ~56 nm) and correlatively disordered spheres (diameter: ~5 nm) within the lattice. (G to L) Model (G); corresponding SEM images at different magnifications (H); simulated and experimental SAXS profiles (I); and cross-section TEM images near the <100> (J), <110> (K), and <111> (L) crystal directions, showing the sc cocrystals composed of ordered cubes (edge length: ~56 nm) and correlatively disordered spheres (diameter: ~10 nm) within the lattice. Scale bars, 100 nm.

To gain insight into the behavior of the NPs in the presence of DNA-ligand shells, the coassembly of cubes and spheres of different sizes was performed by employing MD simulation techniques (Fig. 3). MD simulations were carried out using the open-source software HOOMD-blue (37) to investigate a parameterized model of NPs. Each NP in the model consists of a repulsive core and a DNA-ligand shell and accurately preserved the shapes of the PAEs. In the simulations, the DNA-ligand shells were designed with attractive interactions between different types of NPs and steric repulsive interactions between NPs of the same type. For example, the shells of cubes did not bind with other cube shells but did bind with sphere shells. The thickness of the DNA-ligand shells was determined on the basis of the relative length of the DNA compared to the size of the particles and parameterized using the experimental data. In addition, Yukawa potentials (38) were incorporated into the simulations to account for the general screened repulsion between negatively charged DNA-coated gold NPs in water. The strengths of these Yukawa potentials were parameterized to correlate with the surface area of the NPs.

Fig. 3. MD simulations of cocrystallization of cubes and spheres in different size ratios.

Core-shell models of the DNA-coated nanoparticles (NPs) [(A) r~2.56; (F) r~3.04]. BODs of the assembled cocrystal (right) and perfect crystal (left) [(B) r~2.56, three spheres per cube edge; (G) r~3.04, four spheres per cube edge]. Snapshot of the assembled cocrystals from cubes and spheres [(C) r~2.56; (H) r~3.04]. Heatmap of the number density probability of spheres in the sc lattice formed by cubes, computed from <100> slices near the gaps in the assembled cocrystal [(D) r~2.56; (I) r~3.04], showing the strong preference for spheres to be located at the corners and along the edges of the cubes for both large and small spheres. Cross-section images along the <100>, <110>, and <111> crystal directions (from left to right) [(E) r~2.56; (J) r~3.04], showing the sc cocrystals composed of ordered cubes and disordered spheres with different size ratios (r) within the lattice. Although the larger spheres appear relatively ordered in the <100> cross section, their disorder is apparent in the <110> and <111> cross sections.

In the simulations, it was observed that the cubes form a sc phase, while the spheres position themselves randomly but predominantly at the corners and edges of the cubes (Fig. 3, C and H). To further characterize the simulated cocrystal structures, bond order diagrams (BODs) were calculated from Steinhardt order parameters (39) using the open-source software freud (40). These BODs were compared to reference BODs calculated from the expected ideal structure (Fig. 3, B and G). The analysis of the BODs provides additional insights into the bonding patterns within the cocrystals. Table S2 summarizes the cocrystals assembled from cubes and spheres with different size ratios (r = L₁/L₂). Notably, for cocrystals assembled from cubes and spheres with r ∼ 2.56 there is a maximum of three spheres per cube edge. In contrast, for cubes and spheres with r ∼ 3.04, the preferred configuration has four spheres per cube edge. The simulations confirm the arrangement of cubes and spheres observed in the experiments and further confirm only a negligible change in lattice spacing, with an increased number of spheres in the corner and edge gaps in the case of smaller spheres.

To examine local ordering along different crystallographic planes of the cocrystals obtained in simulation, we captured cross-section snapshots along the <100>, <110>, and <111> crystal directions (Fig. 3, E and J). These snapshots provide a closer look at the arrangement of spheres and cubes within the cocrystals along specific crystallographic orientations. Ten random snapshots were extracted from the equilibrated MD trajectories, and the number density of spheres was averaged slice by slice and visualized as a heatmap (Fig. 3, D and I). In the heatmap, we observe that spheres exhibit a preference for certain locations within the cocrystals. Specifically, the heatmap demonstrates that spheres have a higher density at the corners and edges compared to the faces of the cubes. The number of nearest neighbors of spheres in these locations was found to be 8, 4, and 2, respectively, for the <100>, <110>, and <111> crystal directions. Therefore, the system allows the spheres to connect more polyhedra at the corners than at the edges and the faces. Overall, these results indicate that the corner positions are the most favored, followed by the edges and then faces of the cubes.

Notably, the sc cocrystals of cubes and spheres were only found when the anisotropy of cubic PAEs is maintained. For example, when cubes (edge length = 56 nm) and spheres (diameter = 20 nm) were functionalized with more flexible DNA linkers (linker A3; linker B3 in table S1), and the size ratio becomes smaller (r ∼ 1.86), they coassembled into face-centered cubic (fcc) lattices (fig. S4). In this cocrystal, both cubes and spheres are ordered, but there is no orientational ordering of the cubes because the anisotropy of the small cubes is masked when flexible linkers are used (41).

Coassembly of large octahedra and small spheres

Regular octahedra (edge lengths = 46 to 75 nm) and spheres of different sizes (diameters = 20, 10, and 5 nm) were assembled using complementary DNA A and B (octahedra: linker A1 or A2; spheres: linker B1; table S1). It is worth noting that solid octahedra alone do not assemble into cubic close-packed (ccp) structures, which are characterized by their edge-to-edge interactions. Instead, self-complementary octahedra favor face-to-face interactions to maximize DNA interactions (42). Based on what we learned from the assembly of cubes and spheres as well as the shape complementarity between octahedra and tetrahedra, we hypothesized that spheres would form tetrahedral clusters and coassemble with octahedra into ccp crystals to maximize DNA hybridization, where spheres preferentially interact with corners and edges of octahedra and act as points of connection to link the edges of neighboring octahedra. Depending on the size ratios (r = L₁/L₂) of octahedral PAE (L₁) and spherical PAE (L₂), one or multiple spheres should form geometrically packed clusters that outline tetrahedra. Specifically, with the increase of r, more spheres would be able to populate each tetrahedral site.

To test this hypothesis, we gradually increased r by increasing L₁ (i.e., either by increasing octahedral edge length or increasing the DNA length attached to octahedral surfaces) or decreasing L₂ (i.e., decreasing sphere diameters). First, octahedra (edge length: ∼46 nm; DNA length: ∼11 nm; L₁: ∼68 nm) coassemble with spheres (diameter: ∼20 nm; DNA length: ∼11 nm; L₂: ∼42 nm; r: ∼1.62) into cocrystals with ccp symmetry and octahedral habits, where both spheres and octahedra are ordered, and each sphere occupies one tetrahedral void (Fig. 4, A to C, and fig. S5). As shown in Fig. 4B, the cross-section TEM images from three crystal directions, including <100>, <110>, and <111> directions, match well with the corresponding models. The crystal symmetry and structure were further confirmed by the experimental and simulated SAXS profiles, where ordered octahedra and spheres both contributed to the SAXS peaks (see fig. S6A for the unit cell of the simulated crystal). In addition, when r was increased to ~3 by increasing L₁ (octahedral edge length: ∼75 nm; DNA length: ∼11 nm; L₁: ∼97 nm) and decreasing L₂ (sphere diameter: ∼10 nm; DNA length: ∼11 nm; L₂: ∼32 nm), tetrahedral arrangements of ∼4 spheres coassemble with octahedra into ccp cocrystals (Fig. 4, D to F). This coassembly behavior is characterized by a pattern in the <110> direction, where three or four spheres, depending on the cross-section angles, are observed in one tetrahedral void of the lattice (Fig. 4E). Tilted cross-section TEM imaging (fig. S7) was also conducted to visualize the three-dimensional structures of the cocrystals. The symmetry and structure of the ccp cocrystals were also validated through comparison of experimental and simulated SAXS results, which showed that both the ordered octahedra and spheres contribute to the observed SAXS peaks (see fig. S6B for the unit cell of the simulated crystal). Note that due to the small size of the spheres, the SAXS peaks are primarily attributed to the much larger octahedra.

Fig. 4. ccp lattices composed of octahedra and spheres with varying size ratios, exhibiting a controlled degree of order/disorder.

(A to C) ccp cocrystals composed of ordered octahedra (edge length: ~46 nm; DNA length: ~11 nm) and ordered spheres (diameter: ~20 nm; DNA length: ~11 nm). (D to F) ccp cocrystals composed of ordered octahedra (edge length: ~75 nm; DNA length: ~11 nm) and ordered sphere (diameter: ~10 nm; DNA length: ~11 nm) clusters. Individual tetrahedral clusters are formed by four spheres (corners of tetrahedra). (G to I) ccp cocrystals composed of ordered octahedra (edge length: ~75 nm; DNA length: ~11 nm) and disordered sphere (diameter: ~5 nm; DNA length: ~11 nm) clusters. On average, individual clusters arranged on tetrahedral outlines are formed by approximately 10 spheres. (J to L) ccp cocrystals composed of ordered octahedra (edge length: ~75 nm; DNA length: ~17 nm) and disordered sphere (diameter: ~5 nm; DNA length: ~11 nm) clusters. On average, individual clusters arranged on tetrahedral outlines are formed by about 22 spheres. The number of spheres in each cluster arranged on tetrahedral outlines varies with the size ratio between octahedral PAEs and spherical PAEs. From left to right, each row shows the model [(A), (D), (G), and (J)]; cross-section TEM images in the vicinity of <100>, <110>, and <111> crystal directions [(B), (E), (H), and (K)]; and simulated and experimental SAXS profiles [(C), (F), (I), and (L)]. Scale bars, 100 nm.

The degree of ordering of spheres can be deliberately tuned by changing r. When r was further increased to ∼3.6 by decreasing L₂ (sphere diameter: ∼5 nm; DNA length: ∼11 nm; L₂: ∼27 nm), the symmetry of the coassembly remained unchanged. However, more spheres in each tetrahedral void are observed and the spheres appear to be disordered (Fig. 4, G to I), and the locations of the spheres are correlated with the positions of the corners and edges of octahedra. The average sphere number in each tetrahedral void can be geometrically estimated to be 10. Octahedra are ordered in the cocrystal, but there is no obvious ordering of spheres, as shown in the cross-section TEM images (Fig. 4H) collected from different crystal directions and tilted cross sections (fig. S8). The simulated SAXS profile of the crystal matches well with experimental SAXS results. Moreover, when we further increase L₁ to increase r to ∼4 (octahedral edge length: ∼75 nm; DNA length: ∼17 nm; L₁: ∼109 nm), ∼22 spheres, on average, are estimated in each tetrahedral void, and spheres are disordered in the ccp cocrystals shown in the TEM cross-section images (Fig. 4, J to L). Similarly, the SAXS profile of crystals composed of octahedra was simulated and aligned well with the experimental results.

Through systematic tuning of r by adjusting either L₁ or L₂, we synthesized a collection of ccp cocrystals with varying lattice parameters and sphere configurations. Our findings confirm that an increase in r results in a higher concentration of spheres at each tetrahedral site (figs. S9 to S16), and smaller r favors a cocrystal with two ordered phases. The distance between spheres and octahedra can also be adjusted by tuning DNA linker lengths without changing the sphere arrangements (figs. S10 and S11). Therefore, the degree of ordering (order or correlated disorder) and number of spheres in each tetrahedral site, along with the lattice parameters of the ccp cocrystals, can be deliberately tuned by changing NP size or DNA design using this method.

MD simulations were conducted to understand the cocrystallization of octahedra and spheres with various size ratios (r = L₁/L₂). Analysis of the equilibrated cocrystal snapshots revealed the formation of ccp symmetry by the octahedra, while the spheres occupy the tetrahedral voids between the octahedra. The number of spheres per void varies depending on the size ratios (r) in the mixtures. In a system with a size ratio of ∼1.62, one sphere is observed per tetrahedral void (Fig. 5, A to C). For a system with a size ratio of ∼3.03, tetrahedral clusters consisting of four spheres per void are observed (Fig. 5, D to F). In the case of a size ratio of ∼3.59, approximately 10 spheres form unions in a tetrahedral configuration within each void (Fig. 5, G to I). Furthermore, for a system with a size ratio of ∼4.04, where the shell of the octahedra is thicker by approximately 150%, tetrahedral clusters with five spheres per edge are observed in the voids of the equilibrated cocrystals (Fig. 5, J to L, and table S2). BODs were computed for all these systems (insets in Fig. 5, A, D, G, and J). In addition, the BODs of the octahedra alone were calculated to validate the ccp structure formed by octahedra (fig. S17). To further investigate local order/disorder within the cocrystals, cross-section snapshots were examined along crystallographic directions <100>, <110>, and <111> (Fig. 5, B, E, H, and K). These snapshots offer a detailed view of the arrangement of the spheres within the cocrystals.

Fig. 5. MD simulations on cocrystallization containing octahedra and spheres in different size ratios into ccp lattices.

The assembled cocrystals with different size ratios and the BODs [(A) r~1.62; (D) r~2.56; (G) r~3.03; (J) r~4.04]. Cross-section images along the <100>, <110>, and <111> (from left to right) crystal directions [(B) r~1.62; (E) r~2.56; (H) r~3.03; (K) r~4.04]. The number density probability heatmap of spheres in the ccp lattice formed by octahedra [(C) r~1.62; (F) r~2.56; (I) r~3.03; (L) r~4.04], computed from <111> slices near the octahedron facets in the assembled cocrystal.

Following the analysis conducted for the cube-sphere cocrystals, a similar approach for sampling simulated octahedron-sphere cocrystals was used. Ten equilibrium snapshots were selected randomly from the MD trajectory, and the mean values of the number density probability of spheres were calculated. These values were then visualized as a heatmap (Fig. 5, C, F, I, and L). Consistent with the previous observations, the preference for sphere locations remained, with corners being the most favored locations for spheres, followed by edges, and faces being the least preferred due to more accessible connections near corner and edge sites. This can also be explained by the orientations of octahedra assembled in the ccp lattice because each geometrical configuration of octahedral NPs leads to one unique final structure. For instance, corner-to-corner contact, face-to-face alignment, and edge-to-edge alignment result in sc, bcc, and ccp structures, respectively. In addition, the connections and BODs of the spheres alone were calculated to validate that both the local tetrahedral features and the global symmetry of the tetrahedral clusters are formed by multiple spheres (fig. S18). These BOD analyses provide additional insights into the structural characteristics and ordering within the cocrystals formed by octahedra and spheres with different size ratios.

DISCUSSION

This work is important because it shows how particle dimensions and DNA ligand design can be used to deliberately direct the formation of two different but related classes of cocrystals. There is a well-defined design space that can yield either cocrystals with one or two components in crystalline states, and the degree of correlated disorder (spatial correlations within the disordered phase) can be deliberately programmed into the targeted architecture by tuning the geometry and size of the two NP building blocks. In principle, the lessons learned can be extended to other NP systems, markedly expanding the scope of architectures that can be accessed. Notably, it should be possible to use the technique as a framework to engineer light scattering and diffraction, which may find utility in optics and catalysis (43–45).

MATERIALS AND METHODS

Materials

Spherical Au NPs in different sizes were purchased from Ted Pella. All DNA synthesis reagents were purchased from Glen Research. Ascorbic acid (99%), cetyltrimethylammonium bromide (99%), hydrogen tetrachloroaurate-(III) (HAuCl₄•3H₂O, 99.9+%), sodium chloride (NaCl; 99.5%), trisodium citrate dihydrate (sodium citrate; 99.9%), sodium borohydride (NaBH₄; 99.99%), cetylpyridinium chloride (99+%), SDS (99+%), and triethoxysilane were purchased from Sigma-Aldrich. EMbed-812, dodecenyl succinic anhydride, methyl-5-norbornene-2,3-dicarboxylic anhydride, and DMP-30 were purchased from Electron Microscopy Sciences. All the chemicals were used as purchased without further purification.

Synthesis of cubes and octahedra

Cubic and octahedral Au NPs were synthesized via literature methods (31, 32, 46). In this strategy, iterative oxidative dissolution and reductive growth reactions were used to control NP seed structural uniformity. Subsequently, these seeds were used to template the growth of different anisotropic NPs: cubes and octahedra.

DNA synthesis and purification

Oligonucleotide sequences were carefully designed for all experiments before synthesis (table S1). Oligonucleotides were synthesized on a Mermade 12 DNA synthesizer. After synthesis, the oligonucleotides were cleaved from the controlled pore glass beads using a solution containing a 1:1 volume mixture of 30% ammonium hydroxide and 40% aqueous methylamine solution (incubation at 55°C for 40 min). After evaporation, all of the oligonucleotides were purified using reverse-phase high-performance liquid chromatography (RP-HPLC) on a Varian Microsorb C18 column (10 μm, 300 mm by 10 mm). Then, the oligonucleotides were treated with acetic acid and ethyl acetate solutions to remove the dimethoxytrityl (DMT) functional groups. After synthesis and purification, all oligonucleotides were characterized by matrix-assisted laser desorption ionization–time-of-flight–mass spectrometry (MALDI-TOF-MS) to confirm their molecular mass and purity.

Cocrystallization of anisotropic and spherical NPs

All the NPs were functionalized with anchor DNA following a literature procedure (34, 47). First, 3′-propylthiol-terminated anchor strands were incubated with 100 mM dithiothreitol (DTT) for 1 hour to cleave the disulfide end. Then, the DTT was removed via size exclusion chromatography with a NAP25 Column (GE Healthcare). Afterward, the anchor strands were added to the NP suspensions (∼8 nmol of DNA/ml of NPs), and 1 weight % (wt %) SDS and 1 M sodium phosphate (pH = 7.5) were added to reach final concentrations of 0.01 wt % SDS and 10 mM sodium phosphate, respectively. Next, stepwise additions of 5 M NaCl solution (with each aliquot raising the total NaCl concentration by approximately 0.1 M) were added to the solution until it reached a final concentration of 0.5 M NaCl; each addition was followed by 30 s of sonication. This solution was shaken overnight to maximize DNA loading. Excess DNA strands were removed by three rounds of centrifugation/supernatant removal/resuspension. After the final centrifugation step, the anchor-coated NPs were redispersed in 0.5 M NaCl buffer (with 0.01 wt % SDS and 10 mM sodium phosphate buffer). Then, 10 nmol of DNA linkers was added to 200 μl of NP solution [∼10 optical density (OD)]. The crystallization was performed through slow-cool annealing with a ProFlex PCR system (Applied Biosystems) (48). Specifically, the temperature was slowly decreased from 66° to 25°C at a rate of 0.01°C/min.

Resin embedding of silica-embedded crystals and ultramicrotomy

The samples were embedded in a silica/resin following literature methods (34, 36). Briefly, the silica-embedded colloidal crystals were added to 0.2 ml of 4% gelatin. The gelatin sample was dehydrated upon immersion in anhydrous ethanol solutions of increasing concentration (30% → 50% → 70% → 80% → 90% → 100%). Next, the sample immersed in 100% ethanol was solvent-exchanged with acetone twice for 10 min. In acetone, the gelatin was embedded in EMBed-812 resin (Electron Microscopy Sciences) following the standard protocol provided by the manufacturer. The samples were held at 65°C for 24 hours to polymerize and solidify the resin, and the resin was then sectioned into 80-nm and 120-nm slices (Leica EM UC7).

SEM and TEM characterization

These crystals were analyzed in the solid state, after encapsulation in silica (36), with SEM and TEM. This encapsulation method allows one to transfer the cocrystals in the assembled state and preserve crystal symmetry for imaging. To characterize the inside distribution of the crystals, the silica-stabilized crystals were subsequently embedded in resin and sliced into 80 and 120 nm thin sections with a Leica UC7 ultramicrotome and diamond knife and collected on TEM grids. The tilt series high-angle annular dark field (HAADF) images were obtained using a JEOL ARM 200CF microscope, which was equipped with a probe corrector and operated at 200 kV. The spatial resolution in scanning TEM mode is around 0.8 Å. The convergence angle used was 30 mrad, and the collection angle for HAADF imaging ranged from 90 to 250 mrad. The tilt series were collected manually from −70° to +70° at 5° increments.

SAXS characterization

SAXS data were collected in beamlines 5ID-D (DuPont-Northwestern-Dow Collaborative Access Team) and 12-ID-B of the Advanced Photon Source at Argonne National Laboratory. Using synchrotron x-rays with radiation energies of 10 to 13 keV, the samples were exposed for 0.1 to 1 s, and the scattered beam was collected on a charge-coupled device detector. The data were azimuthally averaged to yield one-dimensional spectra, which were plotted as scattering intensities, on a logarithmic scale, against the scattering vector 𝑞.

SAXS indexing and simulation

One-dimensional SAXS spectra were indexed to identify crystallographic symmetries and lattice parameters (49). To simulate the SAXS spectra, unit cell models were constructed with anisotropic Au NPs arranged in the specified phase symmetry and lattice parameters, such that all SAXS peaks are matched. The particle size, polydispersity factor, average crystalline domain size, Debye-Waller factor, and micro-strain parameter were empirically adjusted to match the widths and intensities of the peaks across the spectrum. Modeling codes and models for the SAXS simulation are available on Zenodo (DOI: 10.5281/zenodo.15122760). When spheres are disordered, they do not contribute to SAXS peaks and therefore are not included in the simulated models.

Coarse-grained model for DNA-coated NPs

We developed a model for DNA-coated NPs by using the aWCA (anisotropic Weeks-Chandler-Anderson) framework, as outlined in the work by Ramasubramani et al. (50). The aWCA is a mean-field theoretical framework, which is implemented into the open-source HOOMD-blue python package to simulate the assembly of interacting particle shapes. Adapted to our NPs, the NPs are characterized by polyhedral cores that retain $V_{\text{WCA},0}$, and the $V_{\text{WCA},\mathrm{c}}$

$$V_{\text{WCA}}\left(r\right)=V_{\text{WCA},0}\left(r\right)+V_{\text{WCA},\mathrm{c}}\left(r\right)$$

The isotropic interaction component, $V_{\text{WCA},0}\left(r\right)$, is repulsive and depends on the radial distance, denoted as r, between the centers of mass of the polyhedra. It can be expressed as

$$V_{\text{WCA},0}\left(r\right)=4{\mathrm{\epsilon }}_{\text{WCA},0}[{\left(\frac{{\mathrm{\sigma }}_{\text{WCA},0}}{r}\right)}^{12}-{\left(\frac{{\mathrm{\sigma }}_{\text{WCA},0}}{r}\right)}^6]$$

The contact interaction component, $V_{\text{WCA},\mathrm{c}}$, arises between the closest pairs of facets and depends on the contact distance, denoted as $r_{\mathrm{c}}$. This interaction is described by the equation

$$V_{\text{WCA},\mathrm{c}}\left(r\right)=4{\mathrm{\epsilon }}_{\text{WCA},\mathrm{c}}[{\left(\frac{{\mathrm{\sigma }}_{\text{WCA},\mathrm{c}}}{r_{\mathrm{c}}}\right)}^{12}-{\left(\frac{{\mathrm{\sigma }}_{\text{WCA},\mathrm{c}}}{r_{\mathrm{c}}}\right)}^6]$$

Here, ${\mathrm{\sigma }}_{\text{WCA},0}$ represents the insphere diameter of the polyhedron core, while ${\mathrm{\sigma }}_{\text{WCA},\mathrm{c}}$ controls the relative strength of the contact interaction and influences the accuracy of representing the desired shape. The energies associated with the contact and isotropic interactions are denoted as ${\mathrm{\epsilon }}_{\text{WCA},0}$ and ${\mathrm{\epsilon }}_{\text{WCA},\mathrm{c}}$, respectively. These energy terms depend on various factors, including ${\mathrm{\sigma }}_{\text{WCA},0}$, ${\mathrm{\sigma }}_{\text{WCA},\mathrm{c}}$, the radial distance r, and the orientations of the polyhedra. The cutoff distances for $r_{\text{WCA},0,\text{cut}}$ and $r_{\text{WCA},\mathrm{c},\text{cut}}$ are determined by $r_{\text{WCA},\mathrm{c},\text{cut}}={\mathrm{\sigma }}_{\text{WCA},0}\times 2^{1/6}$ and $r_{\text{WCA},\mathrm{c},\text{cut}}={\mathrm{\sigma }}_{\text{WCA},\mathrm{c}}\times 2^{1/6}$, respectively. Detailed information about the potential parameters can be found in tables S3 to S8.

We model oligonucleotides and single-stranded DNA by patches located on the polyhedral cores. The attractive forces arising from the complementary “sticky” ends of DNA are accounted for using a Gaussian potential with a shift distance. The potential function, denoted as $V_{\text{sg}}\left(r\right)$, is given by

$$V_{\text{sg}}\left(r\right)={\mathrm{\epsilon }}_{\text{sg}} \text{exp}[-\frac{1}{2}\left(\frac{r-r_0}{{\mathrm{\sigma }}_{\text{sg}}}\right)]$$

Here, r represents the distance between the centers of the patches, while $r_0$ denotes the distance at which DNA hybridization occurs. The width of the attractive well, ${\mathrm{\sigma }}_{\text{sg}}$, determines the flexibility of the patch-patch interaction, while ${\mathrm{\epsilon }}_{\text{sg}}$represents the energy associated with this interaction. We consider three types of attractive patches, vertex patch, edge patch, and facet patch, which correspond to attractive interactions near vertices, edges, and facets, respectively. The positions of these patches within the polyhedra and spheres are listed in tables S9 and S10. Specific values for the potential parameters can be found in tables S3 to S8.

Because of the generally negative charge of DNA-coated gold NPs, we also account for electrostatic repulsion in our model. This repulsion is described using the Yukawa potential

$$V_{\text{Yukawa}}\left(r\right)={\mathrm{\epsilon }}_{\text{Yukawa}}\frac{\text{exp}(-\mathrm{\kappa }r)}{r}$$

Here, ${\mathrm{\epsilon }}_{\text{Yukawa}}$ represents the amplitude of the potential, r is the radial distance to the particle, and κ is the inverse of the Debye screening length. In our model, we neglect long-range repulsion among small spheres due to their size difference and the grafting density of DNA. Specific values for the potential parameters associated with the Yukawa potential can be found in tables S3 to S8.

MD simulations

All MD simulations were performed using resources from XSEDE (51). We performed MD simulations with a fixed number of cubes or octahedra and varied the number of spheres based on the experimentally determined stoichiometry. Periodic boundary conditions were applied to avoid finite size effects. To achieve assembly, we initialized the system with the polyhedra and spheres equilibrated in a mixed, disordered fluid phase at a volume fraction $\mathrm{\varphi }=V_{\text{particles}}/V_{\text{box}}=0.3$. We then increased the volume fraction ϕ by increments of 0.01 over 1 × 10⁵ steps, annealing for 1 × 10⁶ steps after each compression, until the nucleation of an ordered phase was detected. The temperature was held constant at k_BT = 1.0 (in units of ε) in all simulations.

Supplementary Materials

This PDF file includes:

Figs. S1 to S18

Tables S1 to S10