Exploring the zone of anisotropy and broken symmetries in DNA-mediated nanoparticle crystallization

Significance

Nanometer-sized materials (i.e., nanoparticles) can be used as building blocks to construct crystalline materials structured with high resolution. The specific arrangements that nanoparticles form can be controlled by their physical shape and size, as well as the molecules attached to their surfaces (i.e., ligands). In this work, DNA ligands are used as “bonds” whose sequence “encodes” which and how far apart nanoparticles interact with each other. Here, we study how the relative size of nanoparticles and DNA modulates the orientation and structural arrangement of nanoparticles within these crystalline materials and report the specific structural changes that occur for nanoparticles with different shapes. These results provide a roadmap to understand how to build nanoparticle-based materials with DNA.

Abstract

In this work, we present a joint experimental and molecular dynamics simulations effort to understand and map the crystallization behavior of polyhedral nanoparticles assembled via the interaction of DNA surface ligands. In these systems, we systematically investigated the interplay between the effects of particle core (via the particle symmetry and particle size) and ligands (via the ligand length) on crystallization behavior. This investigation revealed rich phase diagrams, previously unobserved phase transitions in polyhedral crystallization behavior, and an unexpected symmetry breaking in the ligand distribution on a particle surface. To understand these results, we introduce the concept of a zone of anisotropy, or the portion of the phase space where the anisotropy of the particle is preserved in the crystallization behavior. Through comparison of the zone of anisotropy for each particle we develop a foundational roadmap to guide future investigations.

Over the past decade, major advances in the control of nanoparticle interactions have led to powerful methods to assemble colloidal crystals (1–9). A high degree of structural control can be achieved in these methods if surface-bound ligands are used as nanoscale bonding elements to control the specificity, spacing, and strength of interactions. DNA has emerged as a particularly versatile ligand whose chemically and structurally defined nature can be used to program the symmetry, lattice parameters, and habit of colloidal crystals (1, 2, 7, 10–19). The shape of the underlying nanoparticle influences the directionality of DNA interactions, which can result in correlated nanoparticle orientations and predictable crystal symmetries based on geometric considerations (7, 13, 16, 19, 20). However, predictive control can be lost if the DNA shell does not preserve the anisotropy of the particle core (13), and thus key questions pertain to (i) the phase space over which predictable directional interactions persist and (ii) the nature of the phase transitions that occur as the anisotropy of the particle disappears. Identifying this “zone of anisotropy” and the broken symmetries that form are critical to establish design rules for work with nonspherical particles and to develop nanostructured materials with controlled properties.

Herein, we systematically investigate the phase space encoded by particle symmetry, particle size (L), and DNA length (D) to understand and map where directional interactions persist (Fig. 1). We show that particle symmetry dictates the crystalline states that can be accessed and how easily changes in L and D affect phase transitions between these states, which include transitions in Bravais lattice (i.e., the symmetry of how the particles are arranged within the unit cell) and particle orientation. The concepts introduced herein provide a roadmap to understand and predict particle crystallization behavior toward the construction of functional nanoparticle-based materials.

Fig. 1.

Polyhedral nanoparticles with different symmetries can be functionalized with DNA and used as building blocks to study nanoscale crystallization processes. (A) EM images of cube (green), octahedron (red), and rhombic dodecahedron (blue) nanoparticles used in this work shown in order of increasing particle symmetry. Below each image is a model of the particle shape and a corresponding ball-and-stick diagram to indicate the symmetry. Nanoparticle edge length (L) is indicated next to each shape. (B) The phase space encoded by DNA length (D) and nanoparticle surface area (SA) is depicted for each symmetry nanoparticle. The darker color region in each phase diagram indicates the zone of anisotropy (ZoA). (C) The phase transition depicted by the dashed line in B, wherein L is fixed as D is increased, is shown, wherein the anisotropy of the particle core is lost.

To map the zone of anisotropy in DNA-mediated nanoparticle crystallization, three common polyhedra were investigated: cubes, octahedra, and rhombic dodecahedra. These nanoparticles are primarily bound by a single crystallographic plane repeated across the structure 6, 8, or 12 times, respectively (Fig. 1A), and can be synthesized via a seed-mediated method that yields >95% of the desired shape with <5% variation in size (Fig. 1A) (21). Particle uniformity was rigorously analyzed with a recently reported and freely available program that algorithmically analyzes transmission electron microscopy (TEM) images to analytically determine nanoparticle structure (22). As-synthesized gold nanoparticles were densely functionalized with DNA, and subsequently DNA “linkers” of programmable length (tuned in rigid 12-nm, double-stranded segments) were hybridized to the surface-bound DNA. Each DNA linker possesses a short, single-stranded “sticky end” with a self-complementary sequence that extends into solution (Table S1). Together, this design yields polyvalent building blocks that connect to each other through the collective hybridization of many DNA sticky ends. To facilitate comparison between different shapes, surface area (SA) was used instead of L, because this number correlates with the number of DNA strands given a similar DNA density.

Table S1.

DNA sequences used for nanoparticle assembly

In the table, red represents the “anchor” strand, green represents the “linker” strand, and gray represents the “41-base complement” strand. MW, molecular weight.

Crystals were formed by slowly annealing DNA-functionalized nanoparticles from high to low temperature, which results in a slow increase in supersaturation that ensures nanoparticles crystallize into their lowest free energy configuration (23). These crystals were analyzed in solution with small angle X-ray scattering (SAXS; Fig. S1 and Tables S2–S4) and in the solid state, after encapsulation in silica, with EM (Fig. S2). Scattering data were modeled using the pseudolattice factor approach (SI Materials and Methods) (16, 24, 25) and compared with experimental data to determine crystal symmetry, lattice parameters, and particle orientation. Importantly, use of a slow crystallization procedure and highly uniform building blocks minimizes the formation of defects and kinetically trapped states, which enables increased crystalline domain sizes, crystalline formation over a wider range of conditions, and more definitive assignments of thermodynamic phase boundaries relative to previous reports (13).

Fig. S1.

SAXS patterns from crystallized rhombic dodecahedra (Left), octahedra (Center), and cubes (Right) with tunable interparticle distances. Experimental and simulated scattering patterns are offset for each sample in dark and light tones, respectively. Data are plotted on a log-linear scale, evenly spaced for each sample. Data are shown for different values of D, which indicates the number of base pairs in each DNA connection between particles, where greater values of D result in larger interparticle distances, indicated by scattering peaks that are shifted to smaller values of q. Values of D are color-coded for each sample: 62 (yellow), 144 (black), 226 (red), 308 (blue), and 390 (green).

Table S2.

Crystallization parameters for rhombic dodecahedra with different edge lengths (L) and DNA lengths (D) were determined from fits to experimental SAXS data

L, nm	D, no. of base pairs	Symmetry	a, nm	Gap, nm
20 (25)	62	FCC	89	22
	144	FCC	123	47
	226	FCC	150	66
	308	FCC	187	92
	390	FCC	215	112
28 (36)	62	FCC	115	23
	144	FCC	148	47
	226	FCC	181	70
	308	FCC	207	88
	390	FCC	239	111
36 (43)	62	FCC	133	24
	144	FCC	168	48
	226	FCC	200	71
	308	FCC	231	93
	390	FCC	259	113
43 (51)	62	FCC	153	24
	144	FCC	187	48
	226	FCC	214	67
	308	FCC	247	91
	390	FCC	277	112
50 (61)	62	FCC	171	21
	144	FCC	209	48
	226	FCC	240	70
	308	FCC	272	93
	390	FCC	302	114

For each particle size, two edge lengths are given, the first as determined from electron microscopy and the second as determined from fits to P(q) from SAXS experiments. Lattice symmetries for rhombic dodecahedra are given as FCC. For cubic lattices, a single lattice parameter a is needed to describe the unit cell. Gap refers to face-to-face distance for particles along the vector that particles are connected.

Table S4.

Crystallization parameters for cubes with different edge lengths (L) and DNA lengths (D) were determined from fits to experimental SAXS data

L, nm	D, no. of base pairs	Symmetry	a, nm	c/a	f	Gap, nm
42 (43)	62	SC	65	1	0.2	22
	144	BCT	88	1.70	0.2	45
	226	BCT	113	1.55	0.3	70
	308	BCT	136	1.51	0.4	93
	390	BCT	158	1.38	0.6	115
57 (58)	62	SC	78	1	0.3	20
	144	SC	103	1	0.2	45
	226	BCT	128	1.77	0.3	71
	308	BCT	148	1.72	0.2	91
	390	BCT	168	1.62	0.3	111
70 (64)	62	SC	86	1	0	22
	144	SC	109	1	0	45
	226	SC	132	1	0	68
	308	BCT	157	1.66	0.3	93
	390	BCT	173	1.68	0.5	110
88 (86)	62	SC	108	1	0	22
	144	SC	134	1	0	48
	226	SC	153	1	0.2	67
	308	SC	179	1	0.3	93
	390	SC	199	1	0.4	113

For each particle size, two edge lengths are given, the first as determined from electron microscopy and the second as determined from fits to P(q) from SAXS experiments. Lattice symmetries for cubes are given as either SC or BCT. For cubic lattices, a single lattice parameter a is needed to describe the unit cell, whereas for a tetragonal lattice, an additional c parameter is needed. Orientational order factors (f) are calculated for each sample and given on a scale from 0 to 1 (0 being perfectly ordered and 1 being completely random). Gap refers to face-to-face distance for particles along the vector that particles are connected.

Fig. S2.

Nanoparticle superlattices formed from cube nanoparticles were encapsulated, sectioned, and imaged to confirm particle symmetry and orientation. Cubes (L = 58 nm) with (A) n = 1, (B) n = 3, and (C) n = 4 DNA lengths are shown here as representative examples of the observed phase behavior. n = 1 DNA forms lattices with SC symmetry and correlated particle orientations, n = 3 DNA forms lattices with BCT symmetry and correlated particle orientations, and n = 4 DNA form lattices with BCT symmetry and uncorrelated particle orientations.

To elucidate the zone of anisotropy for each particle shape, five SA and five D were investigated, beginning with high-symmetry rhombic dodecahedra (Fig. 2 A and B, Fig. S1, and Table S2). Rhombic dodecahedra can pack with 100% efficiency when the 12 rhombus-shaped facets on a given nanoparticle align face-to-face with their neighbors in a crystalline phase (i.e., both translational and orientational order) with face-centered cubic (FCC) symmetry (20, 26). Soft interactions similarly predict an FCC crystalline phase, due to the greater number of DNA hybridization events that arise from face-to-face interactions (13). As SA decreases in these experiments (for a fixed D), one might expect that fewer DNA strands per facet would lead to a smaller enthalpic driving force for the crystalline phase and thus a transition to a plastic crystal phase (i.e., translational order, but orientational disorder) with FCC symmetry, as expected for spheres. Similarly, as D increases (for a fixed SA), one might expect that the greater free volume available to each sticky end would lead to an entropic driving force that decreases the directionality of interparticle interactions and results in a plastic crystal transition. Contrary to these expectations, FCC crystalline phases (Fig. 2 A and B) were observed for all but the sample with the smallest SA and largest D, and thus the zone of anisotropy encompasses nearly the totality of the investigated phase space. A linear increase in the lattice parameter (a) with D supports the crystalline phase assignment, and the slope of ∼0.28 nm per base pair agrees with previous work for face-to-face interactions (Table S2) (13, 16).

Fig. 2.

The crystallization behavior of high symmetry polyhedra exhibits limited complexity as a function of D and L. (A) An FCC crystalline phase is observed for the majority of the phase space investigated for rhombic dodecahedra. To-scale unit cells are shown at top, with lattice vectors, for small and large values of D. EM images corresponding to the unit cells are provided below, with an inset from the unit cell to show particles in the same lattice plane. Throughout this figure, turquoise represents the zone of anisotropy (i.e., a crystalline phase), and purple represents a plastic crystal phase. (Scale bars, 200 nm.) (B) The experimental phase diagram for rhombic dodecahedra is shown as a function of D and SA. The checkered region indicates an extrapolation for larger, more experimentally challenging sizes of rhombic dodecahedra. (C) MD simulations confirm the same phase transition observed for rhombic dodecahedra as experiments, for systems that have been scaled to smaller, computationally more accessible sizes in a manner consistent with experiments. The spatial distribution of DNA sticky ends is shown at right for each phase, where the DNA is localized between particles for the crystalline phase and more isotropically distributed for the plastic phase. (D) As D increases for the smallest L of octahedra investigated, a phase transition from a BCC crystalline phase to an FCC plastic crystal is observed. (E) Experimental phase diagram over the same range of D and L shows that the zone of anisotropy occupies a smaller area of the phase space relative to the rhombic dodecahedra. (F) MD simulations (for particles with similar SA as the rhombic dodecahedra) confirm the observed phase behavior.

To further understand how nanoparticle symmetry affects the zone of anisotropy, the crystallization of octahedra was similarly investigated (Fig. 2 D and E, Fig. S1, and Table S3). The dense packing of hard octahedra varies based on corner truncation or rounding (27–29), where body-centered cubic (BCC) or body-centered tetragonal (BCT) crystalline phases are favored for truncated octahedra (26, 29), and BCC or FCC plastic crystals are favored for rounded octahedra (28, 30). In experimental systems, the hydrodynamic particle shape deviates from an ideal octahedron in favor of a BCC crystalline phase (13). Similar to the rhombic dodecahedra described above, the zone of anisotropy for octahedra (i.e., a crystalline phase with BCC symmetry) fills the majority of the investigated phase space (Fig. 2E). Unlike the rhombic dodecahedra, however, a phase transition and lattice compression occurred for larger SA octahedra at shorter D—from a BCC crystalline phase to a FCC plastic crystal, as D increased (Fig. 2D).

Table S3.

Crystallization parameters for octahedra with different edge lengths (L) and DNA lengths (D) were determined from fits to experimental SAXS data

L, nm	D, no. of base pairs	Symmetry	a, nm	Gap, nm
41 (42)	62	BCC	64	21
	144	BCC	92	46
	226	BCC	120	70
	308	FCC	175	90
	390	FCC	200	107
65 (64)	62	BCC	84	20
	144	BCC	114	47
	226	BCC	140	69
	308	BCC	168	90
	390	BCC	187	107
90 (93)	62	BCC	114	23
	144	BCC	143	48
	226	BCC	171	72
	308	BCC	191	89
	390	BCC	215	110
110 (111)	62	BCC	131	22
	144	BCC	161	48
	226	BCC	185	69
	308	BCC	214	94
	390	BCC	238	115
125 (115)	62	BCC	136	24
	144	BCC	164	48
	226	BCC	190	71
	308	BCC	218	95
	390	BCC	241	115

For each particle size, two edge lengths are given, the first as determined from electron microscopy and the second as determined from fits to P(q) from SAXS experiments. Lattice symmetries for octahedra are given as either BCC or FCC. For cubic lattices, a single lattice parameter a is needed to describe the unit cell. Gap refers to face-to-face distance for particles along the vector that particles are connected.

To understand the origin of these phase transitions, nanoparticles were modeled with molecular dynamics (MD) simulations. These simulations build upon previous work that has been shown to accurately predict the DNA-mediated crystallization behavior of spherical particles (23, 31–35); however, the broken symmetry of polyhedra requires additional considerations (SI Discussion and Tables S5–S7). Although the experimentally investigated samples are too large to model exactly, SA and D were scaled to a computationally accessible size regime in a manner consistent with experiments, and DNA density was set based on experiments (Fig. S3 and Tables S8 and S9). In these simulations, a collection of particles was fixed into a lattice with a specified symmetry then allowed to relax at a temperature slightly below the crystal melting temperature to evaluate stability (i.e., whether the structure reaches equilibrium). Through comparison of different lattices for each particle symmetry and size (e.g., a BCC crystalline phase and an FCC plastic crystal), one can evaluate the thermodynamically preferred phase and extract structural parameters from the system, including lattice parameters, the fraction of DNA strands hybridized per particle (F, enthalpic contributions), and the closest average distance between two adjacent DNA strands on a single particle (entropic contributions).

Table S5.

Values of d for all bead types

Type	d, nm
ssDNA	1.0
dsDNA	2.0
DNA backbone	1.0
Flanking beads	0.6
Sticky ends	0.6
Surfaces	2.0

Adapted from ref. 31.

Table S7.

Particle shapes, sizes, and lattices considered in the simulations

Particle shape	Edge sizes, nm	Lattices
Cube	12, 14, 16	SC, BCT (1.7), pBCT (1.7)
Octahedron	12, 14, 16	BCC, pFCC
Rhombic dodecahedron	4, 6, 8	pFCC, FCC

Lattices prefixed by p indicate plastic crystals.

Fig. S3.

Images from equilibrated MD simulations of octahedra (Top) and rhombic dodecahedra (Bottom) in different lattice symmetries at both the lattice (Left) and individual particle level (Right). For octahedra, particles are depicted in red and DNA sticky ends are depicted in light red. For the BCC data, L =16 nm and D = 16 bp, for the FCC data L = 14 nm and D = 32 bp, and for the pBCT data L = 12 nm and D = 28 bp. For rhombic dodecahedra, particles are depicted in blue and DNA sticky ends are depicted in light blue. For the FCC data, L = 8 nm and D = 12 bp and for the pFCC data L = 8 nm and D = 32 bp.

Table S8.

Statistical parameters from MD simulations calculated for rhombic dodecahedra particles with L = 4, 6, 8, and 10 nm

L, nm	D, no. of base pairs	Symmetry	a, nm	f	o (0–1)
4	12	FCC	28.5	0.78	0.81
	16	FCC	31.6	0.75	0.83
	20	FCC	35.2	0.72	0.86
	24	FCC	38.6	0.69	0.84
	28	FCC	41.1	0.66	0.84
	32	FCC	44.8	0.63	0.83
6	12	FCC	35.1	0.81	0.44
	16	FCC	38.2	0.78	0.76
	20	FCC	41.7	0.75	0.81
	24	FCC	45.2	0.73	0.81
	28	FCC	48.8	0.71	0.82
	32	FCC	52.4	0.68	0.80
8	12	FCC	40.9	0.83	0.10
	16	FCC	44.7	0.81	0.13
	20	FCC	48.4	0.79	0.19
	24	FCC	52.0	0.76	0.29
	28	FCC	55.5	0.74	0.49
	32	FCC	58.9	0.72	0.74
10	12	FCC	46.0	0.84	0.02
	16	FCC	49.9	0.82	0.02
	20	FCC	53.8	0.80	0.03
	24	FCC	57.6	0.78	0.06
	28	FCC	61.4	0.76	0.10
	32	FCC	65.0	0.74	0.17

L values of 4, 6, 8, and 10 nm correspond to 31, 69, 124, and 194 DNA strands, respectively.

Table S9.

Statistical parameters calculated for octahedra particles with L = 12, 14, 16, and 18 nm

L, nm	D, no. of base pairs	Symmetry	a, nm	f	o (0–1)
12	12	BCC	29.2	0.82	0.17
	16	FCC	39.8	0.79	0.70
	20	FCC	43.5	0.77	0.68
	24	FCC	47.0	0.74	0.91
	28	FCC	50.6	0.72	0.90
	32	FCC	54.2	0.70	0.95
14	12	BCC	33.1	0.81	0.04
	16	FCC	42.9	0.80	0.69
	20	FCC	46.5	0.78	0.66
	24	FCC	50.2	0.76	0.65
	28	FCC	54.0	0.74	0.77
	32	FCC	57.7	0.72	0.85
16	12	BCC	33.8	0.84	0.47
	16	BCC	36.9	0.82	0.50
	20	FCC	49.7	0.79	0.80
	24	FCC	53.3	0.77	0.75
	28	FCC	57.1	0.75	0.79
	32	FCC	60.8	0.73	0.73
18	12	BCC	36.0	0.85	0.05
	16	BCC	39.1	0.83	0.06
	20	BCC	42.2	0.80	0.07
	24	FCC	56.5	0.77	0.60
	28	FCC	60.1	0.76	0.57
	32	FCC	63.8	0.74	0.56

L values of 12, 14, 16, and 18 nm correspond to 85, 115, 151, and 191 DNA strands, respectively.

As observed with experiments, the crystalline phase for rhombic dodecahedra was preferred for smaller SA and larger D compared with octahedra (Fig. 2 C and F), with both shapes eventually transitioning to plastic FCC crystals. Furthermore, a linearly increased with D for all sizes of rhombic dodecahedra (Table S8) and lattice compression was observed for octahedra in plastic FCC crystals (Table S9), consistent with experiments. Calculations show that the observed transitions originate from ligand-based entropic and enthalpic contributions. In particular, an increased average inter-DNA spacing on the particle and lattice compression both indicate a wider range of less-oriented DNA states, and thus an increased free volume accessible to each DNA sticky end (32). This flexibility also allows for more DNA connections to be made between particles, as indicated by a greater F.

Based on these results, it was hypothesized that a further reduction in particle symmetry should lead to a smaller zone of anisotropy with a phase transition at larger SA and shorter D. More specifically, the increased sphericity of high-symmetry particles should result in a more diffuse DNA distribution (i.e., reduced number of DNA strands per facet) and a greater free volume available to each DNA strand. Thus, as D increases, the resultant increase in charge repulsion and free volume should not contribute as significantly as they do for lower-symmetry particles.

To test this hypothesis, the crystallization of cubes with comparable SA and D was investigated (Fig. 3, Figs. S1 and S2, and Table S4). Cubes densely pack into a simple cubic (SC) crystalline phase, with slight distortions in orientation for mildly truncated or rounded corners, or a plastic or crystalline FCC phase for high degrees of imperfection (20, 27–30, 36–38). Indeed, a smaller zone of anisotropy was observed for cubes relative to octahedra and rhombic dodecahedra, consistent with the above hypothesis. Outside of this zone, two phase transitions were observed as SA decreases and D increases that involve a change of Bravais lattice: a first-order transition from an SC crystalline phase to a BCT crystalline phase, followed by a continuous transition to a BCT plastic crystal (Fig. 3). The presence of an intermediate crystalline phase with an interaction symmetry distinct from the nanoparticle shape represents an unexpected observation. Interestingly, despite the change from cubic to tetragonal for the first transition, a (i.e., the minor dimension in the square plane for the BCT phase and the only dimension for the cubic phases) increases linearly with D for all investigated SA, with the same rise per base pair as observed for rhombic dodecahedra and octahedra (Fig. 3). This result suggests that nanoparticles consistently hybridize to one another along two axes, the (100) and (010) as defined by the symmetry of the final lattice, whereas symmetry breaks along the third axis, the (001). The continuous nature of the latter transition can be seen from an orientational order factor (f, ranges from 0 for perfectly oriented to 1 for randomly oriented) determined from comparison between experimental and modeled SAXS data (Fig. 3C and Table S4). This change in f occurs as the c/a ratio for the tetragonal unit cell decreases from ∼1.7 to ∼1.4 (consistent with an FCC symmetry) (Fig. 3C). These results are consistent with our hypothesis that lower-symmetry particles would possess a smaller crystalline region of the phase space, but with more complexity than expected.

Fig. 3.

Experimental investigation into cube crystallization behavior shows two continuous phase transitions as a function of D and L. (A) As D increases for a given L (shown here as L = 57 nm), cubes crystallize with (from left to right) SC (crystalline), BCT (crystalline), and FCC (plastic) symmetries. For each symmetry, a unit cell is given, with the lattice parameters a and c indicated. Next to each unit cell is a TEM image of silica-embedded lattices sectioned along a particular crystallographic plane. Below each unit cell is an SEM image of silica-embedded lattices, with the lattice vectors indicated at bottom left. (Scale bars, 200 nm.) (B) Phase diagram for cubes as a function of D and SA, colored based on the symmetry— SC (turquoise) or BCT (purple)—where each box is centered on a single experimental data point. (C) For the smallest cube size investigated (L = 42 nm), the data for BCT phases are shown as a function of D. As D increases, c/a decreases and f increases. (D) The lattice parameter, a, increases linearly with D for all L investigated.

To understand the origins of these phase transitions, cubes were simulated with MD (Fig. 4, Figs. S4–S8, and Table S10). Importantly, phase transitions and lattice parameters consistent with experiments were observed as a function of SA and D (Fig. 4). Close inspection of the spatial distribution and angles of DNA engaged in hybridization shows that the symmetry breaking from SC to BCT phases originates from an unexpected symmetry breaking in the ligand distribution (Figs. S5 and S6). More specifically, the distances between DNA sticky ends are greater on the BCT (001), relative to the BCT (100) and (010), whereas all faces possess equivalent DNA distances in the SC and plastic BCT phases. Similar behavior is observed in the orientation angle of DNA molecules with respect to the nanoparticle surface, wherein larger angles on the BCT (001) indicate reduced face-to-face interactions. Interestingly, the broken symmetry of the DNA distribution becomes more pronounced away from the particle surface, where the DNA can explore a greater free volume. This occurs, in part, because the dense DNA shell near the surface of the particle restricts the position of the DNA. Simulations with and without a dense DNA shell near the surface indeed show that this dense shell is necessary to observe the transition to a BCT symmetry (Fig. S8).

Fig. 4.

MD simulations of cube crystallization reveal that phase transitions occur due to a novel symmetry breaking of the ligand distribution. (A) As D increases in these systems, the distribution of DNA ligands breaks symmetry along the (001). Specifically, the DNA originally on the (001) splits apart and begins to points toward the edges of the (001), such that it can attach to four cubes. As D is increased further, these four interactions become increasingly spaced apart due to electrostatic and free volume arguments. A model (Left) and image from an MD simulation (Right) are shown to indicate the distribution of DNA sticky ends. (B) MD simulations along particular crystallographic planes reveal the distribution of DNA sticky ends between particles. Planes shown are (100) for SC, (100) for BCT, and (110) for the plastic BCT. (C) A phase diagram based on simulation results confirms the same trends in phase and particle orientation as observed in experiment. The colors are consistent with Fig. 3 and the checkered region indicates an extrapolation. (D) DNA angle (θ) with respect to an ideal angle of 0 (perpendicular to the surface) shows a broken symmetry for the DNA on the (001) of the BCT crystalline phase only (red). (E) Distance between DNA sticky ends (d) similarly shows that symmetry breaking for the BCT phase occurs away from the particle surface (red). Comparison between the three symmetries further shows that as DNA length increases, the distribution of DNA distances also increases due to the greater free volume accessible.

Fig. S4.

Images from equilibrated MD simulations of cubes in different lattice symmetries at both the lattice (Top) and individual particle level (Bottom). The crystallographic plane for each image is listed to the left of each image, except for the pBCT, given the random orientation of particles. Particles are depicted in green and DNA sticky ends are depicted in light green. For the SC data, L = 14 nm and D = 12 bp, for the BCT data L = 12 nm and D = 12 bp, and for the pBCT data L = 12 nm and D = 28 bp.

Fig. S8.

MD simulations show that the density and location of DNA affect the phase behavior. Cubes are represented in green and DNA sticky ends are represented in blue. Cubes in the most front plane appear hollow, because they are cut in half. (A) The addition of excess anchor DNA to cubes results in a phase transition from triclinic to BCT. In these experiments, the particle size (L = 10 nm) and the linker density (0.17 strands per square nanometer) are kept constant, and additional anchor DNA is added to a total anchor DNA density of 0.34 strands per square nanometer. (B) Intermediate phases can be tuned based on L and D for systems without excess anchor DNA. (C) Intermediate phases can also be controlled by the spatial location of DNA. Two systems are shown here with L = 10 nm and D = 12 bp, with different δ. For low values of δ (Left), a FCC phase is observed, and for high values of δ (Right) a BCT phase is observed, albeit with a lower c/a ratio than observed experimentally (c/a = 1.32).

Table S10.

Statistical parameters calculated for cubic particles with L = 10, 12, 14, and 16 nm

L, nm	D, no. of base pairs	Symmetry	a, nm	c/a	f	o (0–1)
10	12	BCT	26.0	1.55	0.81	0.49
	16	BCT	28.9	1.45	0.79	0.69
	20	BCT	31.7	1.41	0.77	0.71
	24	BCT	34.5	1.40	0.75	0.81
	28	BCT	37.1	1.40	0.73	0.89
	32	BCT	39.8	1.39	0.71	0.87
12	12	BCT	28.3	1.63	0.81	0.42
	16	BCT	31.2	1.54	0.79	0.49
	20	BCT	34.2	1.47	0.78	0.62
	24	BCT	37.1	1.43	0.76	0.65
	28	BCT	39.9	1.42	0.75	0.70
	32	BCT	42.6	1.42	0.73	0.72
14	12	SC	30.5	1	0.79	0.15
	16	SC	33.3	1	0.76	0.15
	20	BCT	36.5	1.54	0.78	0.46
	24	BCT	39.4	1.48	0.77	0.49
	28	BCT	42.3	1.46	0.75	0.53
	32	BCT	45.1	1.44	0.74	0.60
16	12	SC	32.8	1	0.77	0.01
	16	SC	35.6	1	0.75	0.01
	20	SC	38.4	1	0.73	0.01
	24	SC	41.3	1	0.71	0.01
	28	SC	44.1	1	0.69	0.01
	32	SC	46.9	1	0.66	0.01

L values of 10, 12, 14, and 16 nm correspond to 102, 147, 200, and 261 DNA strands, respectively.

Fig. S5.

Orientational distribution of DNA linkers on the surface of cubes in the SC, BCT, and pBCT crystal structures. Deviation angles (θ) between the end-to-end vector and the facet normal show a stronger trend for DNA chains to point away from the facet normal along the z-facet (x–y edges) compared with other facets and edges for the BCT. All facets and edges are similar for the pBCT and SC phases. An example to-scale end-to-end vector is displayed for each lattice, where the red, orange, and yellow segments correspond to the single-stranded anchor, double-stranded, and sticky end segments, respectively.

Fig. S6.

Analysis of the conformation of each segment of the DNA linkers located near the facet midpoints of cubes in the BCT structure. (A) The angle of the DNA linker relative to the facet normal and (B) the distance between DNA both show a narrow distribution of values near the particle surface but increase in magnitude and breadth away from the particle surface. Further from the particle surface symmetry breaks along the z-facet, as the DNA deforms and spreads out away from the particle surface to increase the conformational entropy of the DNA.

Collectively, these deviations allow each cube in the crystalline BCT lattice to hybridize to four neighbors in a face-to-face fashion along the BCT (100) and (010), similar to SC lattices, but to four neighbors above and four below along the BCT (001). Simulations indicate that the observed transitions predominantly originate from the enthalpy and entropy of the DNA, rather than the packing entropy of the particle shape (Fig. 4) (20, 30). In particular, we hypothesize that the DNA shell breaks symmetry as D increases to (i) reduce the electrostatic repulsion associated with a longer DNA backbone, (ii) increase the free volume available to the DNA, especially at the sticky ends (as discussed above), and (iii) ultimately increase the number of DNA hybridization events that connect nanoparticles within the lattice. The lack of a corollary symmetry in the nanoparticle, or preferred orientation of the (001) with respect to the experimental environment or crystallization conditions, suggests that symmetry breaks upon addition to the lattice, driven by the local DNA environment. These hypotheses are consistent with the transition from crystalline BCT to plastic BCT, and the decrease in the plastic BCT c/a ratio, as D further increases—both of which coincide with an increase in the distance between DNA strands and DNA angle. This unique behavior highlights the importance of considering both particle and ligand effects in nanoparticle crystallization.

This work defines the phase space where directional interactions persist for programmable atom equivalents consisting of anisotropic nanoparticle cores as “atoms” and DNA ligands as “bonds.” The understanding gained from this work establishes the range of conditions where design rules based on geometric considerations can be used to predict crystal structure. If these results were extended to lower-symmetry particles (e.g., tetrahedra), one would imagine an even smaller zone of anisotropy and richer phase diagram, including the possibility of quasicrystal or diamond lattices (20, 30, 36, 39). Due to the highly modular and programmable nature of nucleic acids, the location of these phase transitions can likely be tuned depending on the desired structure, via modulation of the design (e.g., flexibility) (40) or the type of nucleic acid used (e.g., locked nucleic acid or RNA) (41). Beyond defining the zone of anisotropy, this work also establishes a strategy to assemble nonspherical building blocks into a rich set of different phases based on the interplay of the particle and ligand structure.

SI Materials and Methods

Anisotropic Nanoparticle Synthesis and Characterization.

Uniform gold nanoparticles were synthesized via a seed-mediated method that yields >95% of the desired shape with <5% variation in size (21). In this method, a solution of gold nanoparticles is chemically refined through iterative reductive growth and oxidative dissolution reactions. Refinement leads to uniform spherical nanoparticles, which can subsequently be used as precursors, or “seeds,” to template the growth of anisotropic nanoparticle products. Particle uniformity was rigorously analyzed with a recently reported and freely available program that algorithmically analyzes TEM images to analytically determine nanoparticle structure (22).

Edge length (L) and corner rounding (a) were measured to facilitate comparison between different sizes of a given shape. L and a were used to calculate surface area (SA). For a constant surface density of DNA, SA enables one to more directly compare size between different shapes (i.e., two shapes with similar surface areas should have comparable numbers of DNA strands).

For cubes and rhombic dodecahedra, nanoparticle dimensions were determined by the automated analysis of TEM images according to Laramy et al. (22). The 2D projections of 3D shapes used were squares and hexagons for cubes and rhombic dodecahedra, respectively. To use a hexagonal fit for rhombic dodecahedra, regions of the samples were selected with the appropriate orientation. While this may bias the sample toward greater uniformities than reported, previous investigations on samples produced from the same method showed comparable coefficients of variation. Radius of curvature was also measured via automated analysis; however, it should be noted that these values may measure edge rounding due to the angle of analysis. For octahedra, nanoparticle dimensions were determined by the method outlined in O’Brien et al. (21).

Four cube samples were analyzed, with dimensions L, a, and SA as follows:

• $42\pm 2\hspace{0.17em}\text{nm},3\pm 1\hspace{0.17em}\text{nm},1.0\mathrm{\times }{10}^4\hspace{0.17em}{\text{nm}}^2$

• $57\pm 2\hspace{0.17em}\text{nm},3\pm 1\hspace{0.17em}\text{nm},1.9\mathrm{\times }{10}^4\hspace{0.17em}{\text{nm}}^2$

• $70\pm 3\hspace{0.17em}\text{nm},4\pm 1\hspace{0.17em}\text{nm},2.9\mathrm{\times }{10}^4\hspace{0.17em}{\text{nm}}^2$

• $88\pm 4\hspace{0.17em}\text{nm},5\pm 1\hspace{0.17em}\text{nm},4.6\mathrm{\times }{10}^4\hspace{0.17em}{\text{nm}}^2$

Five octahedra samples were analyzed, with dimensions L, a, and SA as follows:

• $41\pm 2\hspace{0.17em}\text{nm},3\hspace{0.17em}\text{nm},0.58\mathrm{\times }{10}^4\hspace{0.17em}{\text{nm}}^2$

• $65\pm 3\hspace{0.17em}\text{nm},3\hspace{0.17em}\text{nm},1.5\mathrm{\times }{10}^4\hspace{0.17em}{\text{nm}}^2$

• $90\pm 4\hspace{0.17em}\text{nm},3\hspace{0.17em}\text{nm},2.8\mathrm{\times }{10}^4\hspace{0.17em}{\text{nm}}^2$

• $110\pm 6\hspace{0.17em}\text{nm},4\hspace{0.17em}\text{nm},4.2\mathrm{\times }{10}^4\hspace{0.17em}{\text{nm}}^2$

• $125\pm 7\hspace{0.17em}\text{nm},4\hspace{0.17em}\text{nm},5.4\mathrm{\times }{10}^4\hspace{0.17em}{\text{nm}}^2$

Five rhombic dodecahedra samples were analyzed, with dimensions L, a, and SA as follows:

• $20\pm 1\hspace{0.17em}\text{nm},0.45\mathrm{\times }{10}^4\hspace{0.17em}{\text{nm}}^2$

• $28\pm 1\hspace{0.17em}\text{nm},0.89\mathrm{\times }{10}^4\hspace{0.17em}{\text{nm}}^2$

• $36\pm 1\hspace{0.17em}\text{nm},1.5\mathrm{\times }{10}^4\hspace{0.17em}{\text{nm}}^2$

• $43\pm 2\hspace{0.17em}\text{nm},2.1\mathrm{\times }{10}^4\hspace{0.17em}{\text{nm}}^2$

• $50\pm 2\hspace{0.17em}\text{nm},2.8\mathrm{\times }{10}^4\hspace{0.17em}{\text{nm}}^2$

DNA Synthesis, Purification, and Design.

All oligonucleotides used in this work were synthesized on a solid-support MM48 synthesizer (BioAutomation) with reagents purchased from Glen Research. Oligonucleotides <70 bases were synthesized with a 5′ trityl group and purified via reverse-phase HPLCy (HPLC; Agilent), followed by standard deprotection procedures. MALDI-TOF-MS was used to confirm the molecular weight of HPLC-purified oligonucleotides. Oligonucleotides >70 bp were synthesized without a 5′ trityl group and purified via PAGE with 8% polyacrylamide gel stock and 8 M urea at 350 V for 1 h, 15 min. Analytical PAGE was used to confirm the length of the oligonucleotides through comparison with a 10-base DNA ladder (Qiagen). The OligoAnalyzer tool from Integrated DNA Technologies was used to determine an extinction coefficient for each DNA strand and UV-visible (UV-Vis) spectrophotometry was used to determine concentrations.

The DNA design used here (16) possesses two components:

• i)An “anchor strand” that attaches to the nanoparticles. The anchor strand is composed of three regions: a 3′ hexylthiol that connects the DNA to the gold nanoparticle surface, an A₁₀ region designed to distance the functional portion of the DNA from the nanoparticle surface, and an 18-base recognition sequence.
• ii)A “linker strand” that hybridizes to the anchor strand. The linker strand is composed of three regions: an 18-base sequence complementary to the anchor strand; a duplexed “spacer” region with n = 0, 1, 2, 3, or 4 40-bp segments designed to control the effective “DNA bond” length; and a terminus, or “sticky end,” designed to link two nanoparticles together through complementary base pair interactions. Each 40-base segment is separated from the neighboring regions by a single A “flexor” base and is duplexed with a separate, complementary 40-base sequence. Specific sequences used are located in Table S1.

Nanoparticle Functionalization and Assembly.

Nanoparticles were functionalized with 3′ thiol oligonucleotides according to literature precedent (16) to a NaCl concentration of 0.5 M. UV-Vis measurements of the nanoparticle localized surface plasmon resonance position before and after functionalization showed minimal change (<2 nm) for all samples and indicated that no significant corner truncation or shape transformation occurred as a result of this ligand replacement process. After DNA functionalization, nanoparticles were suspended in a 0.2 M NaCl, 0.01 M sodium phosphate buffer (pH 7.4), and 0.01% SDS solution. To each nanoparticle solution, 25,000 linker strands were added per nanoparticle (in excess of the number of thiolated strands for all nanoparticles used). The n = 1–4 linker strands were prehybridized with 40-base complements and annealed at 40 °C for 30 min then allowed to cool to room temperature. After the addition of linker to each nanoparticle solution samples were annealed at 40 °C for 30 min then allowed to cool to room temperature.

Nanoparticle solutions were prepared such that the final concentration of each nanoparticle was 20 pM in a total volume of 100 µL. Each solution was then pipetted into 200-µL PCR eight-tube strips (Life Technologies) and placed into a thermal cycler (Life Technologies). The temperature of the thermal cycler was slowly cooled from 55 °C (where particles are fully discrete) to 20 °C (where particles are fully associated) for all samples at a rate of 0.1 °C/10 min. The slow speed of this temperature change ensures that the system possesses sufficient time to reach equilibrium at each temperature. Importantly, the interparticle spacings investigated were sufficient (>15 nm) to neglect short-ranged, material-dependent effects (e.g., van der Waals) and thus DNA hybridization dominates interparticle interactions.

SAXS.

SAXS experiments were performed at the DuPont–Northwestern–Dow Collaborative Access Team beamline of the Advanced Photon Source at Argonne National Laboratory. X-rays with λ = 1.24 Å (E = 10 keV) were used. The sample angle was calibrated with a silver behenate standard. Two sets of slits were used to define and collimate the beam and parasitic scattering was removed with a pinhole. The X-ray beam cross-section measured 200 μm and exposure times varied from 0.05 to 0.3 s. Scattered radiation was detected with a CCD area detector. Dark current frames were subtracted from all data.

One-dimensional SAXS data were obtained from an azimuthal average of 2D scattering patterns. SAXS data in the main text and SI Discussion are presented as scattering intensity, I(q), as a function of the scattering vector, q:

$$\text{q}=4\mathrm{\pi }\text{sin}\left(\mathrm{\theta }\right)/\mathit{\lambda },$$

where q is half of the scattering angle 2θ and λ is the wavelength of X-ray radiation. Due to the orders of magnitude difference in the scattering associated with gold relative to the DNA and solvent, scattering from these sources was assumed to be negligible. I(q) includes scattering contributions from the discrete nanoparticles described as a form factor, P(q), and from lattice effects (i.e., ordered nanoparticles) described as a structure factor, S(q). One-dimensional experimental and simulated SAXS patterns are shown in Fig. S1 and Tables S2–S4, as are the structural parameters determined from the simulated data.

Full scattering profiles for crystalline assemblies of polyhedral particles were modeled via the pseudolattice factor approach, based on powder diffraction theory and the decoupling approximation. Details of this approach can be found in two recent publications from Senesi and Lee (24) and Li et al. (25). Modeling considerations specific to this work are described at length in the supplementary information of O’Brien et al. (21), including the effects of particle orientation, particle size dispersity, microstrain, and the preferred orientation of crystals. Modeled scattering profiles are used to determine lattice symmetry, lattice parameters, and particle orientation factors, all of which are given in SI Discussion.

EM of Nanoparticle Superlattices.

Nanoparticle superlattices were encapsulated in silica, embedded in a polymeric resin, and microtomed into sections 100–300 nm thick (on the order of one to three unit cells), according to literature precedent (Fig. S2). These methods preserve the crystal symmetry and lattice parameters of the lattice but allow transfer to the solid state for imaging. Thin sections allow one to evaluate particle orientation, which is typically difficult for the full micrometer-sized crystal. Due to the small thickness and uncontrolled orientation of sections, the majority of the crystals are not cut along a particular crystallographic plane and thus some nanoparticles are damaged in this process. This results in the distorted shapes and sizes seen in SI Discussion. All images collected for sectioned superlattices were collected on a Hitachi HD2300 STEM in Z-contrast mode. All other images of encapsulated superlattices were collected on a Hitachi U8030 SEM.

MD Simulations.

Molecular dynamics simulations were performed to predict and understand experimental phase behavior (Figs. S3 and S4) (31, 32). The coarse-grained approach used in these simulations was first described by Knorowski et al. (33) and adapted for dsDNA by Li et al. (31). This approach has been further modified here to account for the broken symmetry of polyhedral nanoparticles and to remove self-hybridization behavior.

In this model, the DNA and nanoparticles are represented by discrete beads, which possess a diameter, d, chosen to represent physical parameters (Table S6). In particular, the DNA is split up into five different types of beads, chosen to mimic the experimental design: ssDNA, dsDNA, phosphate backbone, DNA sticky ends, and flanking beads. Flanking beads are used surrounding each sticky end to reflect the restricted confirmations of DNA base-pairing interactions. Each bead is connected by springs with a characteristic stiffness and flexibility, based on the specific type. Given these parameters for each bead and connection, the characteristic interaction distance between each different type of bead, σ, is then taken into account and evaluated using a Lennard-Jones (LJ) potential (Table S6).

Table S6.

Values of σ for all repulsive pair interactions

Pair	σ, nm
DNA backbone–DNA sticky end	1.2
Flanking bead–DNA sticky end	0.86
dsDNA–DNA sticky end	0.91
Flanking bead-flanking bead	0.8
DNA sticky end–DNA sticky end	2.0
Nanoparticle–nanoparticle	1 × 10−5
All other pairs	(d1 + d2)/2

Adapted from ref. 31.

To mimic the experimentally used DNA linker design, we used the following bead design: one ssDNA bead for the single-stranded portion of the anchor strands (yellow), three beads (corresponding to six bases) for the DNA sticky ends (orange), and three to eight beads (corresponding to 12–32 bp) for the double-stranded portions of the linker strands (blue). The density of the linker DNA was modeled with the experimentally measured density of 0.17 strands per square nanometer. To mimic the anchor strands not hybridized to a linker strand, a chain of two ssDNA and four dsDNA base pairs (one of each kind of bead) were used. The density of the surface DNA was set to 0.34 strands per nanometer (2), consistent with experimental data obtained from spherical particles.

It should be noted that smaller nanoparticles and shorter DNA lengths are used in simulation compared with experiment due to the limited amount of memory available on current graphical processing units (GPUs).

Interaction parameters.

The interaction parameters used in this work originated from work from Li et al. (31), with slight modifications (Tables S5 and S6). This model only considers pairwise interactions between beads, which is typical of MD. In particular, all nonbonded particles interact through LJ potential (U_LJ):

$$\begin{array}{c}{U}_{LJ}=4\mathit{\in }\left({\left(\left(\mathit{\sigma }\right)/\left(r\right)\right)}^{12}-{\left(\left(\mathit{\sigma }\right)/\left(r\right)\right)}^6\right)-U_{LJ}\left(r_c\right),r<r_c\\ U_{LJ}=0,r>r_c,\end{array}$$

where ϵ represents an energy scale factor, r represents the distance between two objects, r_c represents the maximum distance at which we consider interactions, and σ represents the distance where U_LJ is zero. For all repulsive interactions, r_c = 2^1/6σ, ϵ = 1 (Weeks–Chandler–Andersen potential) and for all attractive interactions r_c = 4 nm, ϵ = +7, σ = 2.0 nm (only complementary sticky ends are attractive).

The connections between each bead of a DNA chain are modeled by a harmonic distance potential along the chain, U = (1/2)k(r − r₀)², where k = 82.5 nm⁻² and r₀ = 1.68 nm. A harmonic angular potential is also added between two consecutive bonds, U_θ = (1/2)k_θ(θ − θ₀)², where k_θ = 30 rad⁻² for dsDNA and k_θ = 2 rad⁻² for the bond between dsDNA and the sticky end, as well as the ssDNA–dsDNA angle. Very stiff potentials were used to keep the backbone and flanking beads in their position: k_θ = 100 rad⁻² for the flanking–backbone–flanking angle, k_θ = 120 rad⁻² for the dsDNA–DNA backbone–DNA sticky end angle, and k_θ = 50 rad⁻² for the DNA sticky end–DNA sticky end–flanking bead angle.

Due to software constraints in constant pressure simulations, we represented the nanoparticle surface with a shell of beads. This shell contains 5N_s springs, where N_s is the number of beads on the surface and each spring was designed to be highly rigid with k = 2,000 nm⁻². An additional bead in the center is linked to all surface beads with the same spring constant. Each shape was assembled by symmetry and all operations of the corresponding point group are valid. The number of beads on the surface of each shape was designed to be 25–50% larger than the number of DNA strands, to ensure a sufficient number of surface sites. The DNA was randomly located across these locations. The DNA chains initially point outward of the nanoparticle and a repulsive potential between the shell and the chains keep them out of unphysical configurations such as a chain penetrating the particle core. We did not see any such configurations in our simulations. Python scripts to generate initial configurations for HOOMD-blue (34, 35) as well as example scripts are available online (https://github.com/NUMGirard/hoobas/).

Equilibration of crystal structures.

To investigate the stability of a particular lattice, we first estimated lattice parameters on the basis of L and D (Table S7). Nanoparticles were then placed at lattice positions with the appropriate crystal symmetry but slightly smaller lattice parameters than our estimations. Each simulation contains 64 nanoparticles, where the range of nanoparticle sizes was chosen such that we could observe the desired transitions. For cubes, SC, BCT, and plastic BCT (pBCT) symmetries were examined; for octahedra, BCC and plastic FCC (pFCC) symmetries were investigated; and for rhombic dodecahedra, FCC and plastic FCC (pFCC) symmetries were investigated. For BCT lattices, we also set an initial value of the $\stackrel{\sim }{c}a$ crystal axis ratio. We originally considered other lattices such as Minkowski lattice and a BCT (c/a = 1.2) interpolant for octahedra, but these configurations were not stable at any of the tested points and were thus not systematically calculated. An initially random configuration was also tested for cubes, but lattices exhibited a large number of defects. Within those simulations, we only observed local SC or BCT symmetries, and thus these data were not included due to the overall uncertainty of the symmetry.

Each system was equilibrated in NPT ensemble (constant number of particles, pressure, and temperature), using a Nosé–Hoover barostat, with a timestep Δt = 3 × 10⁻³δt. In these calculations, δt = √(mσ²/ϵ), where m is a unit of mass. A constant pressure of P = 10⁻⁴δp was used, where δp = ϵ/σ³. An equilibration time of t_eq = 30 × 10⁶Δt was used, which results in individual calculations that take between 1–4 d on a single GPU, depending on the nanoparticle size and DNA length.

The temperature (T) of the calculations was determined based on the melting temperature (T_m) of the system. T_m is a function of many parameters, including the DNA sticky end sequence, nanoparticle size and shape, and DNA stiffness. To determine T_m, simulations were run at multiple values of T, where T > T_m results in particles that move off lattice (i.e., the crystal melts, where nanoparticles dissociate from one another). The calculations show that T_m > 1.6, consistent with previous results. To favor crystalline arrangements, we chose a value below the T_m of T = 1.2, where the particles remained on lattice.

To evaluate each lattice, we looked at the lattice parameters as a function of simulation time. These data contain three sections: (i) an initial thermal equilibration at constant volume, (ii) a metastable state where the symmetry is entirely determined by the initial configuration of the system, and (iii) a completely relaxed state, where the final symmetry may be different from the initial one. The phase diagram was built by following which symmetry was obtained from the third section, the completely relaxed state. It is worth noting that the NPT integration scheme used for these simulations is a fully deformable triclinic box, which tends to varies in shape and volume to release local stresses if a specific crystal lattice is unstable relative to another crystal structure. Effectively, this allows us to correctly determine the crystal lattice obtained from simulation and to compare relative free energies between competing phases. Although these simulations are theoretically able to stay trapped in kinetic metastable states, we have not observed any data points where two different crystal lattices were stable.

Self-hybridization behavior.

In simulations, the DNA sticky ends are modeled as a X-Y-Z chain, where Y attracts Y and X attracts Z. Due to the force field used, two DNA sticky ends from the same particle can hybridize the X and Z beads of their chains with only a small misorientation. This represents unphysical bonding and thus has to be removed from the simulation. Because the experimentally used DNA length is much shorter than the persistence length of DNA, we do not expect to see this type of self-hybridization and therefore we can safely disable them in simulations. We do note, however, that if we studied significantly longer DNA strands or ssDNA, which is much more flexible, we could not remove this safely. This self-hybridization behavior was not seen in our previous models, because the particles were coated with complementary sequences instead of self-complementary sequences. To remove these interactions, we added charged beads to the sticky ends, where each bead has a charge that is specific to the nanoparticle it is bound to (i.e., numbered from q = 1 to q = N, where N is the number of nanoparticles in the system). All other particles are set to a charge q = 0. We have then modified the source code of HOOMD to only compute LJ interactions if one of the charges is 0 or if both are different.

Analysis of DNA hybridization behavior.

We computed the number of DNA hybridization events per particle by counting the number of sticky ends that are located within a distance r < 1.5σ′, where σ′ = 1.2 nm is the length of a typical hydrogen bond.

We also computed the average distance between DNA linkers engaged in hybridization from a single nanoparticle. This distance is given by

$$d_h=\langle \min \left|r_i\mathrm{⃗}-r_j\mathrm{⃗}\right|\rangle ,$$

where the index j runs over all beads of DNA chains, whereas i ≠ j runs over all beads of DNA chains originating from the nanoparticle where j is located. We then compared this to the average inter-DNA distance from all chains on a single nanoparticle, computed in the same way, except i runs over all chains, denoted d_i. The average is performed over all simulation domain and all time steps.

To compare simulations to the experiment orientational order factor (f) we calculated the orientational order, o, as

$$o=<\max \left({\text{s}}_{\text{i},\text{k}}.{\text{s}}_{\text{j},1},{\text{s}}_{\text{i},\text{k}}.{\text{s}}_{\text{j},2},\mathrm{\dots },{\text{s}}_{\text{i},\text{k}}.{\text{s}}_{\text{j},\text{N}}\right){>}_{\text{i},\text{j}\mathrm{\ne }\text{i},\text{k}},$$

where i, j≠i indicate the nanoparticle in the system possessing N facets. The vector s_i,k is then the vector normal to the kth facet of the ith particle. For perfectly oriented nanoparticles, this value averages to one. For completely random orientations, this value is smaller than one and is dependent on the particle shape, because no two $s_i$, $s_j$ can differ in orientation by more than half the dihedral angle of the polyhedron To compare this value to experimental values, we rescaled this parameter to o′ = (1 − o)/(1 − min o), where min o is the shape-dependent value for a lattice with perfectly random orientation. Although this definition is not strictly equivalent to the experimental orientational order parameter, it should show correlation with it. New parameter o′ then ranges from 0 (oriented) to 1 (nonoriented).

Equilibrated lattice data.

Data from equilibrated MD simulations are shown in Tables S8–S10.

Analysis of DNA linker conformation.

To understand the conformation of the DNA linkers in each of the experimentally observed crystal structures, we performed a statistical analysis of chain deformation. To measure deformation we ascribed vectors to each of the segments of the DNA and proceeded to collect the deviation angle between this vector and the vector normal (Figs. S5–S7). Vectors were assigned to six individual segments along the DNA, as well as to the DNA as a whole, to understand the overall angle and the angle along each part of the DNA. These six segments were (i) from the DNA anchor to the first double-stranded segment, (ii) from the first double-stranded segment to the second double-stranded segment, (iii) from the second double-stranded segment to the third double-stranded segment, (iv) from the third double-stranded segment to the first sticky end segment (effectively the flexor before the sticky end), (v) from the first sticky end segment to the second sticky end segment, and (vi) from the second sticky end segment to the third sticky end segment. Each sticky end has three segments corresponding to two bases each. To identify the angle of the whole DNA strand, an end-to-end vector was assigned from the DNA anchor to the final sticky end segment. This analysis was done for DNA linkers attached to faces and edges. After collecting the deviation angle data for each DNA strand, we calculated the ensemble average of these tilt angles per chain.

Fig. S7.

Analysis of the conformation of each segment of the DNA linkers located near the facet midpoints of cubes in the SC structure. (A) The angle of the DNA linker relative to the facet normal and (B) the distance between DNA both show a narrow distribution of values near the particle surface but increase in magnitude and breadth away from the particle surface. No symmetry breaking is observed at any point along the DNA.

Additional analysis was performed to determine the distance between neighboring DNA linkers (d). This analysis directly correlates to the free volume available to the DNA and thus relates to the conformational entropy of the DNA. To evaluate this distance, each segment of the DNA was separated into its own layer. Within each layer, a single DNA was chosen and evaluated to find the minimum distance between a neighboring chain, which we have described as d in this work. This was then repeated for all segments on a given facet in the system to understand the distribution of DNA conformations.

SI Discussion

To understand why an intermediate crystal symmetry was observed between SC and plastic FCC for cubes, we modeled a range of different DNA designs (Fig. S8). In each of the different DNA designs we investigated, the two limits (large cube sizes, short DNA and small cube sizes, long DNA) always crystallized with SC and pFCC crystal symmetries, respectively (note that a pBCT with c/a = sqrt(2) is equivalent to a pFCC]. However, the intermediate phases could be changed or altogether removed depending on the DNA surface configuration, which indicates that the observed symmetry breaking is due to these DNA effects.

To begin with, we modeled the effect of DNA surface density. In particular, previous experiments have shown that not all surface strands on a nanoparticle are able to hybridize linker strands. As a result, the density of the linker DNA will not be equal to the density of anchor DNA. To understand how this difference in density affects intermediate phases, we modeled nanoparticles with and without excess anchor DNA at a constant linker density (0.17 strands per square nanometer). Interestingly, for the nanoparticles with low anchor DNA (0.17 strands per square nanometer), a crystalline triclinic phase was observed. This phase looks similar to a BCT structure, except the particle orientations are shifted such that DNA on cube faces seems to be engaged in edge–edge interactions. When excess anchor DNA is added (to a total anchor DNA density of 0.34 strands per square nanometer) the BCT phase is recovered, consistent with experiments. This result suggests that the excess surface DNA is necessary to restrict the free volume available to linker strands and thus confine the DNA on faces to a more limited set of edge–edge configurations.

To further understand how a lack of excess anchor DNA affects intermediate phase behavior, we explored the effect of L and D in these systems. Specifically, for small particle sizes (L = 12 nm), we found that shorter DNA lengths favored a crystalline HCP phase, whereas longer DNA lengths favored a plastic monoclinic phase. If the particle size is increased, the monoclinic crystal symmetry can be stabilized with regular particle orientations.

We additionally looked at how the spatial distribution of linker DNA on the nanoparticle surface, without excess surface DNA, affects the phase behavior. To model this, we attributed a potential dependence to the distance between attachment points and the cube in the form exp(−δ²/r_edge²), where δ is some characteristic distance we varied from 1 nm to 8 nm for a cube with L = 10 nm. Interestingly, this modification led to a recovery of pBCT symmetries, but with different c/a ratios compared with experiments: from 1.41 at δ = 1 nm to 1.32 at δ = 8 nm. The deviation of these simulations from experimental results suggests that there is not a spatial distribution of DNA centered on the cube faces.