Regulating phase behavior of nanoparticle assemblies through engineering of DNA-mediated isotropic interactions

Significance

Revealing relationships between interaction potential and the resulting structure is a continuously developing area due to its importance for atomic, nanoscale, and microscale material systems, highly relevant for three-dimensional self-assembly of nanoscale systems. Typically, resulting phases of nanocomponents with isotropic interactions are limited to close-packed arrangements. Here, we propose, computationally validate, and experimentally realize an approach for assembly of spherical, isotropically interacting nanoparticles into different phases through tuning of the ratio between attraction and repulsion ranges. This was achieved by modulating the structure of the isotropic DNA shell of nanoparticles. The study shows that subtle differences in interaction potential can result in drastic changes of the phase behavior. The demonstrated approach opens new possibilities for generating diverse materials via self-assembly.

Abstract

Self-assembly of isotropically interacting particles into desired crystal structures could allow for creating designed functional materials via simple synthetic means. However, the ability to use isotropic particles to assemble different crystal types remains challenging, especially for generating low-coordinated crystal structures. Here, we demonstrate that isotropic pairwise interparticle interactions can be rationally tuned through the design of DNA shells in a range that allows transition from common, high-coordinated FCC-CuAu and BCC-CsCl lattices, to more exotic symmetries for spherical particles such as the SC-NaCl lattice and to low-coordinated crystal structures (i.e., cubic diamond, open honeycomb). The combination of computational and experimental approaches reveals such a design strategy using DNA-functionalized nanoparticles and successfully demonstrates the realization of BCC-CsCl, SC-NaCl, and a weakly ordered cubic diamond phase. The study reveals the phase behavior of isotropic nanoparticles for DNA–shell tunable interaction, which, due to the ease of synthesis is promising for the practical realization of non-close-packed lattices.

The ability to fabricate well-organized three-dimensional (3D) architectures from functional nanoscale materials has drawn considerable interest for uses in energy (1, 2), information storage (3, 4), photonics (5, 6), and biomedical applications (7, 8). Several self-assembly-based approaches, leveraging particle shapes, relative sizes, charges, and interactions (9–20) have been quite appealing in terms of their i) potential to be brought to commercial scale and ii) compatibility with a wide variety of material types, both inorganic and biological. In particular, the use of DNA-based approaches for the assembly of isotropic nanoparticles (NPs) into ordered superlattices have proven to be powerful and convenient strategies. DNA sequences grafted to the surfaces of NPs serve as molecular linkers where the self-assembly pathway of the NPs is dictated by the interplay between DNA interactions and entropic effects (21–25). Several approaches have been investigated for the assembly of ordered 3D structures by-design, including the use of mixtures of isotropic NPs of varying sizes to dictate the packing of the particles (26–29), but the resulting ordered phases are often limited to close-packed symmetries. It is also possible to modulate the design of the grafted DNA–shells to affect the size of the shell, grafting density, and shell design in order to induce the formation of specific crystallographic morphologies (30–32). Environmental conditions, such as ionic strength and pH can also play a role in the interparticle spacing based on the height of the DNA brushes and length of DNA linkages, for DNA-based assembled systems (33, 34).

The number of possible architectures that can be assembled using DNA-functionalized NPs (DNPs) can be further diversified by varying the shape and directionality of the interactions dictated by the DNA shells. The use of anisotropic core particles has been shown, both theoretically and experimentally, to facilitate the organization of particles beyond close-packed lattices and instead organize based on the symmetry of the particles (7, 35, 36). However, this strategy requires the use of new particle shapes for each new target crystal symmetry, as well as contributions from ligands bound to the surfaces that can contribute to the effective shape of the particles. To more precisely control the directionality of the interactions between isotropic particles, bonds with prescribed valency can be imprinted on the particle surface in the case of patchy particles for microscale particles (8, 37), and the use of this strategy for nanoscale particles has been demonstrated (38, 39) but remains fairly limited in the yield of correctly patterned NPs. Thus, there remains an unaddressed challenge in identifying a simple design parameter that can be used to tune the final crystal symmetry of assembled isotropic DNPs without additional fabrication steps. It has also been demonstrated that an ordered phase of self-assembled DNPs can be selectively transformed from a “mother” phase to several “daughter” phases through the post-assembly modification of the DNA shells, but this was only shown for a limited number of high-coordinated structures (40).

Considerable computational work has been done in parallel to investigate self-assembly strategies of nanoparticles with the goal of assembling those with isotropic interactions into complex crystal structures. Computational work has shown that isotropic pair potentials can be used to stabilize different types of low-coordinated crystal structures (41, 42); this gives hope of using isotropic particles to stabilize more complex crystal structures. However, the extracted pair potentials show extremely complex formats that require multiple distinct pair potential wells to stabilize those crystal structures (43, 44). It was also demonstrated computationally that a binary set of particles with soft-core repulsion can assemble into diverse morphologies, including a cubic diamond lattice (45). DNPs provide a possible experimental system in which to realize such a strategy, and thus, the identification of a simple, experimentally accessible strategy for DNA-grafted, isotropic nanoparticles is important for the fabrication of diverse, well-ordered crystal structures.

Here, we combined simulations and experimental design strategies to determine and target practically achievable isotropic pair potential profiles to explore the formation of non-close-packed phases of spherical DNPs. Our computer simulations focused on tuning repulsive and attractive forces between the particle pairs, and we succinctly captured this behavior with a simple, but crucial, parameter: repulsion range over attraction range (r_rep/r_att). Simulations demonstrated that a wide range of structures can be assembled by changing this parameter, from close-packed structures [face-centered cubic (FCC-CuAu), body-centered cubic (BCC-CsCl), simple cubic (SC-NaCl)] to low-coordination cubic diamond and open-honeycomb (OHC) structures. Experimentally, we utilized a binary mixture of DNPs that bind via a dual linker approach with tailorable shell overlap to reach target r_rep/r_att values. DNA linker sequence length and identity provide powerful methods to explore a range of r_rep/r_att values. We experimentally investigated the phase diagram of the DNPs as a function of r_rep/r_att and interparticle hybridization energy, and we observed the formation of BCC and SC lattices, as well as weakly ordered cubic diamond-like networks. Additionally, the experimental work validates our computational findings regarding the universal nature of the r_rep/r_att parameter by achieving structural diversity through separate strategies of either increasing r_rep or decreasing r_att to reach target r_rep/r_att values.

Results and Discussion

We first present the general principle of how to utilize isotropic pair interactions of binary mixtures to assemble high-coordinated to low-coordinated crystal structures computationally (Fig. 1A) and the proposed design strategy of DNPs to realize such pair interactions experimentally (Fig. 1 B–D). In general, the critical parameter that is responsible for the final resultant crystal structures is the range of repulsion over the range of attraction of two particles denoted as r_rep/r_att. Molecular dynamics (MD) simulation (Materials and Methods) demonstrates that five different crystal structures, i.e., FCC-CuAu, BCC-CsCl, SC-NaCl, diamond, and open honeycomb OHC networks, can be obtained by increasing r_rep/r_att from 1.0 to 1.74. A small value of r_rep/r_att leads to the formation of close-packed structures of FCC-CuAu, BCC-CsCl, and SC-NaCl at r_rep/r_att = 1.0, 1.18, and 1.47, respectively. Remarkably, the self-assembly of low-coordination diamond and OHC networks is observed by simply increasing r_rep/r_att to 1.57 and 1.74, respectively. The identification of a simple parameter r_rep/r_att that controls the self-assembly behavior, from close-packed to open lattices, is a promising strategy for forming a greater diversity of lattices without changing shell design.

Fig. 1.

Achieving high-coordinated and low-coordinated lattices using DNPs by tuning r_rep/r_att. (A) Pair potential model used in MD simulations and the corresponding structures at different r_rep/r_att. The Right panel shows the snapshot of FCC-CuAu, BCC-CsCl, SC-NaCl, diamond, and open honeycomb (OHC) networks obtained at r_rep/r_att = 1.0, 1.23, 1.46, 1.57, and 1.74, respectively. (B) Conceptual illustration of DNPs showing r_att can be tuned by engineering DNA sticky sequences (S_A2 and S_B2) while r_rep remains inert. (C) Schematic of experimental design for DFPs. H refers to the double-stranded DNA region where complementary regions of the A and B linkers, 1 and 1′, are hybridized. Regions S_A1, S_A2, S_B1, and S_B2 are each single-stranded, noncomplementary regions of linkers A and B. The double-stranded regions of 2 and 2′, and 3 and 3′ are the regions where linkers A and B hybridize to NP_A and NP_B, respectively. Both NP_A and NP_B have a noncomplementary region of their DNA corona, T₁₀, a poly-T domain of 10 nucleotides. (D) Experimentally realizable range of calculated r_rep/r_att by tuning design of linker DNA sequences.

Motivated by these computational observations, we next sought to explore the possibility to realize these parameters physically. Fig. 1B illustrates schematically our DNPs for tuning r_rep/r_att by modifying DNA sequences. Conventionally, the attraction range r_att is comparable to the repulsion range r_rep when the complementary sticky sequence is put at the tail of DNA molecules (Fig. 1 B, Top). In this work, we tune r_rep/r_att by altering the sticky sequence location. Instead of putting sticky sequence at the tail of DNA molecule, it is moved from the tail to the middle of sequence (Fig. 1 B, Bottom). The attraction range ${\mathrm{r}}_{\text{att}}^{\ast }$ thus becomes shorter due to the increase in the overlap of the DNA shells between the two particles. In the meantime, however, the repulsive particles still preserve the same repulsion range r_rep no matter where the sticky sequence is located along the linker sequence, since two NPs cannot bind due to the DNA sequence mismatch. As a result, this will increase the relative ratio of r_rep/r_att accordingly.

To experimentally realize a system of DNPs with tunable r_rep/r_att, a dual-linker approach was utilized for the design, as shown in Fig. 1C. Two sets of 10-nm diameter gold nanoparticles (AuNPs) were first grafted with noncomplementary DNA sequences (NP_A and NP_B). Each DNA-grafted AuNP was then incubated with its respective linker sequence. The overall length of the linkers was held constant at 100 nucleotides; however, two key features of the design were modulated to tune r_rep/r_att, i) the hybridization energy by varying the number of complementary base pairs (bps) (region H) between linkers A and B through regions 1 and 1′, and ii) the attraction range between NP_A and NP_B by decreasing the number of the poly-T bases S_A1 and S_B1 while simultaneously increasing the number of spacers S_A2 and S_B2, by transferring the number of nucleotides from S_A1 and S_B1 to S_A2 and S_B2, respectively. Linker designs with S_A1 set at the maximum value, result in no additional interpenetration of the shell of one particle into the other beyond the region of the sequences used for hybridization (H). As the number of nucleotides in the S_A1 (=S_B1) spacer decreases, the interparticle spacing decreases, and those nucleotides are transferred to the S_A2 (=S_B2) region, which further penetrates into the shell of the other particle type. This reduction of the S_A1 spacer and corresponding increase in the S_A2 spacer results in a decrease in the experimental attraction range and an increase in the r_rep/r_att. Using models (46, 47) to estimate the theoretical interparticle spacings and corona size of the DNA shells based on the experimental design parameters, H and S_A1, we can construct an experimental phase space of the relative ratio of r_rep/r_att (Fig. 1D). Notably, in the experimental design of the DNA linkers the total number of nucleotides in each linker always remains constant. This means that as design parameters change, the resultant size of each particle type does not change, as seen in light scattering data (SI Appendix, Figs. S5 and S18).

Next, we turn to the details of the simulations done as a part of this study. The impact of r_rep/r_att on mediating the assembly of crystal structures is investigated using MD simulations. Our DNP model uses an LJ-nm and Weeks-Chandler-Andersen (WCA) pair potential (Fig. 1A) to effectively capture the features of DNPs (see Materials and Methods and SI Appendix, Fig. S1 for more details) (13). We consider a binary system consisting of two different types of nanoparticles (i.e., A-type and B-type), with attractive interaction (i.e., LJ-nm) between unlike particles A-B (i.e., E_AB kept as −1.0 $\epsilon$ throughout the simulations) and purely repulsive interaction (i.e., WCA) between like particles A-A (i.e., E_AA) and B-B (i.e., E_BB). The order parameter that we investigated is the range of repulsion over the range of attraction denoted as r_rep/r_att. The r_rep is defined as the pair distance that U_rep goes to 0 while the r_att is defined as the pair distance with minimum binding interaction strength of U_att. Here, for convenience, we kept r_att the same throughout the simulations but only increase the range of repulsion r_rep to change r_rep/r_att. In a typical MD run, the system is initially equilibrated at a relatively high temperature (i.e., k_BT/E_AB = 1.0), where the nanoparticle mixtures are dispersed randomly in the dilute gas phase to mimic experimental conditions. Then, the system is slowly cooled from k_BT/E_AB = 0.2 to 0.05, at which particles can crystallize into large lattice domains.

The structural state of the systems with respect to r_rep/r_att are characterized by using pair correlation functions (PCFs). At the end of each simulation, the emergence of sharp peaks in the PCF indicates that particles are assembled into specific crystal structures. The crystalline structures were then confirmed by comparing the obtained PCF with that from predefined perfect crystalline structures. Fig. 2A summarizes the PCFs at four different r_rep/r_att. At r_rep/r_att = 1.0, DNPs assemble into an FCC-CuAu lattice, since the PCF peak positions overlap with the peaks of a perfect FCC-CuAu lattice. Once r_rep/r_att is increased to 1.18, 1.46, and 1.57, the PCF pattern shifts to that of BCC-CsCl, SC-NaCl, and diamond lattice, respectively. Note that a PCF can identify whether systems are crystallized into well-ordered distinct structures; however, it cannot clearly differentiate the details of the self-assembly process by itself, especially with the presence of polymorphisms. For instance, a PCF cannot differentiate between amorphous or polymorphic structures with partially crystallized diamond lattice at different times during the diamond lattice assembly (SI Appendix, Fig. S2).

Fig. 2.

Self-assembly of diamond lattice, FCC, BCC, and SC by varying r_rep/r_att. (A) Pair correlation functions (PCFs) as a function of pair distance at r_rep/r_att = 1.00, 1.18, 1.46, and 1.57, corresponding to the formation of FCC-CuAu, BCC-CsCl, SC-NaCl, and diamond lattice, respectively. Crystal structures are identified by comparing the PCF peak positions with those of predefined perfect FCC, BCC, and SC crystals (gray vertical lines). (B) Identified crystal particle number (red line) within the largest cluster (gray line) as a function of simulation time at r_rep/r_att = 1.46. (C) Identified crystal particle number (red and blue lines) within the largest cluster (gray line) as a function of simulation time at r_rep/r_att = 1.57. The sharp increase of identified diamond particles (red line) at t = 0.5 × 10⁸ confirms the formation of the diamond lattice. The crystalline particles are identified using fNGA (48). (D) The simulated snapshots showing the one-step crystallization towards a simple cubic lattice at r_rep/r_att = 1.46. (E) The simulated snapshots of the structural evolution during the self-assembly of diamond lattice via a two-step process at r_rep/r_att = 1.57. The zoomed in view of preformed amorphous clusters is provided in SI Appendix, Fig. S3.

Therefore, to further elucidate insights into the self-assembly process, we analyzed the identified crystalline particles within the largest cluster as a function of simulation time using fast neighborhood graph analysis (FNGA) method (48). Our simulation indicates that the crystallization pathways of close-packed structures all follow a one-step process—direct nucleation and growth of small to large crystals. Fig. 2B demonstrates an example of SC-NaCl lattice that self-assembles via a one-step process at r_rep/r_att = 1.46 along with visualized snapshots shown in Fig. 2D. In this case, first, a small crystalline nucleus with an SC-NaCl structure is formed. This small crystal nucleus then grows into a larger crystallite, and the final stabilized crystal is identical in structure with the initially formed nuclei.

Surprisingly, the simulation results reveal a two-step process for the self-assembly of diamond crystal. Particles initially dispersed in the dilute gas-phase aggregate locally into large amorphous clusters, followed by a transformation from amorphous clusters to a well-defined diamond lattice. Fig. 2 C and E shows the quantification and self-assembly process of diamond lattice at r_rep/r_att equals 1.57. The self-assembly starts from a disordered mixture of particles and ends with a well-defined diamond crystal. We note that a substantial amount of OHC-like particles form within the amorphous aggregates before they completely transform into the diamond lattice. This result provides direct evidence of a two-step self-assembly process of diamond lattices, suggesting that preformed amorphous clusters were assembled before the formation of well-defined diamond lattices (SI Appendix, Fig. S3).

We continue to experimentally investigate the impact of r_rep/r_att on the phase behavior of DNP assemblies. As shown previously, the concept of tuning r_rep/r_att is adopted as illustrated in Fig. 1C. Equimolar mixtures of NP_A and NP_B were mixed together after each was incubated with their respective linker (SI Appendix, Figs. S4 and S5) and then annealed to facilitate the self-assembly of the NPs to an equilibrium structure and then transferred to a capillary tube for scattering measurements. The structural state of the system was then investigated using small-angle X-ray scattering (SAXS) for H = 10 bp and as S_A1 was varied from 70 to 0 nucleotides poly-T in the linker design (SI Appendix, Fig. S6). The SAXS profiles for the NP superlattices could be modeled to confirm the structures of the experimental systems. The use of H = 10 bp showed the most dynamic range of realized structures. The H = 10 bp design has a maximum S_A1 length of 70 T, and assembly of 70 T system results in an ordered BCC-CsCl lattice (a = 51.34 nm). This sample and others were then imaged using optical microscopy, and it was possible to observe that the DNPs for the discussed designs were assembled into large, micron-sized domains (SI Appendix, Fig. S7). In order to probe the local ordering of the 10-nm particles, the DNP assemblies were then mineralized and encapsulated in silica (49, 50), followed by visualization using scanning electron microscopy (SEM), as described in SI Appendix.

The change of the SAXS pattern as a function of decreasing S_A1 and increasing r_rep/r_att is shown in Fig. 3B. The position of the first peak shifts from q = 0.01731 Å⁻¹ to q = 0.01768 Å⁻¹ as S_A1 goes from 70 to 55 T indicating that the NPs continue to assemble into a BCC-CsCl lattice with a smaller lattice parameter. Once S_A1 is reduced to 40 T, the profile of the SAXS pattern shifts to that of a SC-NaCl lattice (a = 38.81 nm) and continues to show the profile of a SC-NaCl lattice until S_A1 = 27 T. With further reduction in S_A1, we see the signal of a disordered state, and then at S_A1 = 10 T, there is a large shift of the first peak position to a lower q-space position, indicating the presence of a different structural state. We can also extract the average interparticle spacings of the particles in the various lattices from the experimental SAXS obtained structure factors, S(q), and then calculate an experimental value for the r_rep/r_att for each design if the particles successfully crystallized (as detailed in SI Appendix, section).

Fig. 3.

Experimental self-assembly of DNPs by varying r_rep/r_att. (A) Schematic of the phase changes for DNPs with increasing a ratio r_rep/r_att. (B) Effect of shell design and r_rep/r_att on SAXS profiles, measured for H = 10 bp, as a function of increasing r_rep/r_att by decreasing S_A1(=S_B1). The table shows the correspondence between each shell design and r_rep/r_att values, where the colors match the measured SAXS profiles, aligned with the corresponding systems shown in the table (C–F). (C–F) SAXS results with experimental (markers and solid lines with colors), modeled structure factors (gray solid lines), S(q), with black stems indicating Bragg reflections, and 2D experimental SAXS images (circular scattering patterns). (C) S_A1=70 T, where the model unit cell is BCC-CsCl (a = 51.34 nm). Inset: Low- and high-magnification SEM images of silica-encapsulated lattices, scale bars are 100 nm and 50 nm, respectively, and the red circles indicate particle positions from the (110) plane of a BCC unit cell. (D) S_A1 = 40 T, where the model unit cell is SC-NaCl (a = 38.81 nm). The appearance of shoulder features in S(q) is associated with a BCC-SC transition. Inset: Low- and high-magnification SEM images of silica-encapsulated assemblies, scale bars are 100 nm and 50 nm, respectively, and the red circles indicate particle positions from the (100) plane of a SC unit cell. (E) S_A1 = 27 T, where the model unit cell is SC-NaCl (a = 37.76 nm). Inset: Low- and high-magnification SEM images of silica-encapsulated lattices, scale bars are 100 nm and 50 nm respectively, and the red circles indicate particle positions from the (100) plane of a SC unit cell. (F) S_A1 = 10 T, where the model unit cell is a cubic diamond (a = 73 nm). Inset: Low- and high-magnification SEM images of silica-encapsulated assemblies, scale bars are 100 nm and 50 nm respectively, and the red circles indicate particle positions from the (111) plane of a cubic diamond unit cell.

We also compared the experimental and modeled S(q) in order to reveal the structural state of the system. Fig. 3C shows that when S_A1 = 70 T, the system assembles into a BCC-CsCl lattice with a r_rep/r_att = 1.245. Based on the SEM imaging, it is possible to resolve NP positions at the surface of the silica-embedded lattice, and the high-magnification imaging identifies the (110) plane of a BCC unit cell of DNPs (Inset of Fig. 3 C, red circles and SI Appendix, Fig. S8). For S_A1 = 40 T, the scattering signatures of SAXS profile are more closely corresponding to that of a SC-NaCl lattice (Fig. 3D) where the strong features in the experimental data occur at the expected reflecting planes indicated. However, there is a strong shoulder at approximately q = 0.018Å⁻¹, most likely due to the relative ratio of r_rep/r_att (=1.411) being close to the boundary between BCC and SC phases. The SEM imaging (Inset of Fig. 3 D, red circles and SI Appendix, Fig. S8) shows a similar case where we resolve particles from the (100) plane of a SC unit cell, however, there appears to be distortion in the relative angles and distances in the particles which would not be present in a lattice with only SC crystallographic symmetry. As S_A1 is further decreased to 27 T, the S(q) (Fig. 3E) indicates a SC-NaCl scattering pattern, thus, revealing a formation of a well-ordered DNP array with a SC unit cell. For this design, the electron microscopy imaging (Inset of Fig. 3 E, red circles and SI Appendix, Fig. S8) shows a well-defined projection of the (100) plane, as well as smaller domains with cubic crystal habits. To access another region of the phase space, the length of S_A1 was reduced to 10 T, and the first S(q) peak shifted to a lower q-space value for the measured r_rep/r_att = 1.675. This experimental S(q) (Fig. 3F) was plotted against a model of a cubic diamond (CD) lattice (a = 73 nm) where disorder was introduced to the model using a Debye–Waller term or positional fluctuations of the NPs around their expected position (σ_DW = 0.05 or 5% of the lattice spacing) in the lattice. The SEM imaging of these assemblies shows smaller domain sizes, on the order of several hundred nanometers. However, a local organization of particles can be identified as a (111) plane of a cubic diamond unit cell (Fig. 3F), as indicated by particles on this ring-like pattern. This is consentient with SAXS results of weakly ordered cubic diamond organization. It is possible that to achieve a more ordered cubic diamond lattice, higher values of r_rep/r_att, which is difficult to access within this system design. Also, while we have explored different thermal annealing protocols, potentially a more complex annealing process might be required to achieve a higher degree of order for CD phase. Additional SEM images for each of the silica-embedded samples shown in Fig. 3 are provided in SI Appendix, Fig. S8.

Since both the computation and experiments revealed a similar structural diversity by merely increasing r_rep/r_att, it is now important to identify a more complete phase diagram and further validate the universality of such design parameters. Experimentally, a detailed phase diagram with four different linker designs (H varying from 8 to 15 bp) and experimental r_rep/r_att is shown in Fig. 4A and in detail in SI Appendix, Figs. S9–S14. In each case, regardless of the value of H used in the linker design, the lower values of r_rep/r_att would crystallize into BCC-CsCl superlattices. For H = 8 bp, as S_A1 goes lower than 55 T, the shells of the NPs are unable to interpenetrate the opposite shell and cannot assemble into ordered arrays. Otherwise, for H = 10, 12, and 15 bp, each design made the transition from BCC-CsCl to SC-NaCl, and to a more weakly ordered signal with three broad peaks that do appear to be similar to a cubic diamond unit cell. Notably, in some cases of the linker design, we can see a SAXS pattern that mostly corresponds to SC lattice with certain features that indicate that these designs exist near a phase transition (Fig. 4A). As the boundary for the structural transition from BCC-CsCl to SC-NaCl to a weakly ordered CD is similar despite the difference of linkers, we conclude that r_rep/r_att is the most responsible parameter for the observed range of obtained structures. The experimental phase diagram can also be formatted as a function of S_A1 to highlight the effect of the DNA design on the assembled structural state of the system (SI Appendix, Fig. S15). We also use the SAXS data to track the average center-to-center distances (D_cc) between NPs as a function of the length of S_A1. As shown in Fig. 4B, the interparticle spacings in the lattices do steadily decrease as the S_A1 decreases. The statistics for H = 10, 12, and 15 bp are plotted against a theoretical prediction of the D_cc based on a model (46, 47) for the size of DNA and the inorganic cores. Furthermore, to prove the generality of tuning interactions and a phase formation via r_rep/r_att, we performed an additional set of experiments with the implementation of a completely different linker design strategy using asymmetric DFP with a conceptually similar idea of tuning r_rep/r_att (SI Appendix, Figs. S16–S22). In the asymmetric version of the experimental design both the r_rep and r_att change as the linkers change. These designs yield similar results both computationally and experimentally. However, the observation of weakly ordered cubic diamond at a large value of r_rep/r_att from experiments is not consistent with computational prediction, which may be due to the change of the effective particle sizes in a binary mixture and different efficiency of DNA linkers grafting to both particles.

Fig. 4.

Phase diagrams. (A) Experimental phase diagram as a function of the number of complementary base pairs and experimental r_rep/r_att. The experimental r_rep/r_att is calculated based on the average center-to-center distances (D_cc) (B) D_{cc, experimental} as a function of S_A1 (error bars represent a SE based on three measurements). (C) Fraction of identified crystals in the self-assembly (i.e., BCC, FCC, SC, diamond, and OHCs) as a function of r_rep/r_att using fNGA. The fraction of BCC, FCC, SC, diamond, and OHCs were plotted from Top to Bottom panels (green, blue, orange, red and purple trajectories), respectively. (D) Probability of observing five crystal structures as a function of r_rep/r_att from the free-energy calculation at temperature k_BT/E_AB = 0.01 (SI Appendix).

Fortunately, simulations can provide additional insights and reveal much more detailed phase transition boundaries related to r_rep/r_att as well as the underlying reason for preventing observation of well-ordered cubic diamonds as a supplementary to experimental results. Fig. 4C shows a crystallization order diagram as a function of r_rep/r_att in a much finer grid obtained from MD simulations, indicating the corresponding range of r_rep/r_att that favors each of five different crystal structures (i.e., FCC, BCC, SC, diamond, and OHC networks). Each subpanel highlights the fraction of forming a single type of lattice among these five different lattices (X_lattice) as a function of r_rep/r_att from top to bottom. At r_rep/r_att between the range of 1.0 to 1.08, close-packed structure FCC-CuAu (green) forms, with the presence of more BCC-CsCl lattice with a continuous increase of r_rep/r_att. As r_rep/r_att further increases, a range where only BCC-CsCl (or SC-NaCl) forms can be identified from the order diagram at r_rep/r_att = 1.1 to 1.28 (or from 1.30 to 1.56). Notably, the diamond lattice (red) starts to appear at r_rep/r_att = 1.56. However, a single diamond lattice configuration without polymorphism is only observed around r_rep/r_att = 1.56. As further increases of r_rep/r_att, the fraction of diamond particles starts to decrease. In the meantime, the fraction of OHC particles (purple) becomes non-zero and gradually increases with increasing r_rep/r_att, suggesting that the diamond structures become more defective with the presence of more OHC particles. For r_rep/r_att > 1.7, the diamond structures diminish completely, and only OHC networks forms with different morphologies (SI Appendix, Fig. S23). The simulation results indicate that, around a specific r_rep/r_att = 1.57, a well-defined diamond structure can be formed using isotropic NPs. Note that the diamond symmetry crystal is a highly unexpected microstructure in NP mixtures. Surprisingly, in this work, we found a range of r_rep/r_att from 1.56 to 1.7, where diamond crystals can be self-assembled, though the resulting structures can be more defective at higher r_rep/r_att. Combining the observation of Fig. 2C, this may explain the observation of weakly ordered cubic diamond lattice experimentally at a larger value of r_rep/r_att since the defective structure is highly competitive not only during the self-assembly process but also sensitive to the selection of r_rep/r_att. It would interesting to explore in future studies whether the more precise tuning interaction potential shape, which is possible through DNA shell design, might stabilize CD with high correlation length in larger particle design space.

The theoretical calculation can provide additional insights into the underlying driving force of the structural evolution and can decouple kinetics perspectives, which usually can be profound in experiments, from thermodynamics. To gain such insights, we calculated the absolute free energy of different lattices based on the Einstein Molecule approach (51, 52). Lattice free energy of predefined periodic perfect lattice with five different configurations of FCC-CuAu, BCC-CsCl, SC-NaCl, diamond, and OHCs lattices were calculated. Fig. 4D summarizes the probability of observing five different crystals as a function of r_rep/r_att from the lattice free-energy calculation using $P_j=\exp \left(-\frac{A_j}{k_{B}T}\right)/{\sum }_i\exp \left(-\frac{A_i}{k_{B}T}\right)$ . These results clearly indicate that the thermodynamic nature of structural diversity from FCC-CuAu to BCC-CsCl, SC-NaCl, diamond, and OHCs networks is in agreement with MD results in Fig. 4C. In addition, the free-energy results confirm that there exists a small range of r_rep/r_att (from 1.6 to 1.7) favoring the stabilization of diamond lattice, further suggesting that the formation of diamond lattice is indeed a thermodynamic product.

Finally, we address the applicability of achieving diamond lattice in experimentally realizable systems and show that the experimentally observed weakly ordered cubic diamond phase could be a polymorphic structure with a portion of the diamond lattice by comparing it with simulation results. To demonstrate such and to compare with experimental results, we conducted molecular simulation using a more realistic coarse-grained DNP model (53–55). We first demonstrate that the range of r_rep/r_att can be tuned in such DNP coarse-grained model by tuning DNA sequences directly like in experiments. Fig. 5A shows the calculated potential of mean force (PMF) as a function of pair distance from DNPs with three different sequence designs. As shown in the PMFs, the r_att of DNPs can be tailored from ~28 to 21σ by moving the location of complementary base pairs from the end to nearly the middle of the strands, while the r_rep is kept as same at ~35σ regardless of the location of complementary base pairs. Thus, to conceptually favor the formation of diamond lattice around r_rep/r_att = 1.6, A-type DNPs grafted with sequence T₆C₄T₄ and B-type DNPs grafted with T₆G₄T₄ are selected.

Fig. 5.

Comparison of disordered phase obtained from experiments and diamond lattice obtained from simulation. (A) Potential of mean force estimated from replica exchange molecular simulation for A-type particle coated with different DNA sequence T_nG₄T_m (i.e., T₆G₄T₄, T₈G₄T₂ and T₁₀G₄) and B-type particle coated with its complementary DNA sequence T_nC₄T_m (T₆C₄T₄, T₈C₄T₂ and T₁₀C₄). The dashed line shows the r_rep where interaction goes to 0. The Inset shows the repulsion barrier E_AA for different numbers of spacers n. (B) Structural factor S(q) obtained from experiments (blue), and simulation using implicit (green) and explicit (orange) models. The solid black lines show the theoretical Bragg reflections obtained from a perfect diamond lattice. (C) Final snapshot showing the self-assembled structures obtained from MD simulations using 216 explicitly modeled DFPs starting from a random configuration. (D) Another view of the final snapshot of Fig. 5C, highlighting the identified diamond particles (red) using fNGA and its bond topology between A–B particles. The unidentified particles are shown as transparent particles.

Coarse-grained molecular simulation starting from an initially disordered configuration was performed using the above design and compared with experimental results as shown in Fig. 5B. The snapshots of the final configuration of DNPs with explicitly modeled DNA strands (Fig. 5C) and only A–B bond topologies (Fig. 5D) obtained in MD simulation [and visualized using OVITO (56)] are provided. We identify a maximum of 158 diamond particles in MD simulation, corresponding to about 73% of the entire system (SI Appendix, Fig. S24), further highlighting the robustness of our design approach. Additionally, the results also illustrate the formed disordered phase illustrated previously in experiments are highly likely the polymorphic structures with both small diamond unit cells and other defective structures. Fig. 5B shows the comparison of the S(q) obtained from both explicit and implicit DNA model simulations, experiments, and theoretically perfect diamond lattices. We find that the S(q) curve obtained from explicit simulation matches well with the one obtained from the experimental weakly-ordered phase. Also, the general peak locations overlap with the one obtained from perfect diamond lattices despite the second and third peaks becoming less obvious. Even though the entirely periodic diamond structure is not observed in explicit DNA model–based MD simulation, within the simulation time we could complete, another set of simulations demonstrates that the system starting from a perfect cubic diamond structure does remain stable over a very long period of time (SI Appendix, Fig. S25). Considering the slow hybridization kinetics of DNP binding/unbinding, the observation of a less-ordered diamond phase could also be the result of kinetic traps. As we showed in Fig. 2C, the diamond lattice was transformed from amorphous precursors during the self-assembly process. The slow hybridization kinetics of DNA molecules grafted on NPs may lead to defective structures kinetically arrested for a long period of time. Notably, there are other possibilities that may promote disorder of diamond phase resulting in the difference between predicted diamond structures in MD simulations and experimental results. The experimental system usually has about 10% polydispersity in NP cores, which corresponds to about 3% polydispersity of our DNP. We further conduct a set of MD simulations using the implicit DNP model (Fig. 1A) and compare the impact of polydispersity on crystallization. We find that systems tend to form more amorphous-like structures with increasing polydispersity (SI Appendix, Fig. S26), in agreement with the previous literature (57, 58). Studying the assembly DNPs for the symmetric and asymmetric experimental designs in situ, using Dynamic Light Scattering measurements, shows the effect of the fundamental design and r_rep/r_att on the assembly melting temperatures of these systems (SI Appendix, Figs. S27–S29). Future guidance for forming a more ordered diamond phase may require improvement of the annealing protocol or fine-tuning of the design DNA shell strategy that minimizes polydispersity and fine-tunes interaction potential.

Conclusion

In conclusion, we report the self-assembly of isotropically functionalized NPs into low-coordination crystalline structures, opening up new avenues for bottom–up material design. Our results reveal that a critical parameter r_rep/r_att, which has long been ignored previously, can be adopted in binary isotropic NP systems to achieve such control over crystal structures. MD simulation and free-energy calculations provide an understanding of the fundamental nature of the resultant structures by increasing r_rep/r_att, demonstrating that five different crystal structures (i.e., FCC-CuAu, BCC-CsCl, SC-NaCl, diamond, and OHC networks) can be achieved during the self-assembly of NPs. The demonstrated key parameter r_rep/r_att to stabilize these crystals is much simpler to construct in the physical NP systems compared to other pair potential models predicted for isotropic particles (59, 60). We also elucidate that the self-assembly of close-packed structures is processed via a one-step process, while the self-assembly of diamond lattice is processed via a more complex two-step process.

More importantly, we have proposed experimental templates for generating such building blocks using many different designs of DNPs with modified DNA sequences. Experimentally, to achieve the control over r_rep/r_att and crystal structures, two sets of gold NPs are grafted with different designs of DNA sequences (NP_A and NP_B), capturing two key design features i) the hybridization energy by varying the complementary base pairs, and ii) the attraction range between NP_A and NP_B by decreasing the length of the poly-T spacers within sequences. The SAXS profiles for the NP superlattices confirm the general tendency of change from BCC-CsCl to SC-NaCl to diamond-like disordered phase by increasing r_rep/r_att, illustrating that it is responsible for the observed structural transitions. We also show the phase diagram details and the range of r_rep/r_att where these transitions occur. In the end, we validated that the diamond-like disordered phase observed from experiments can be a polymorphic phase or structures with defects, by comparing the experiential SAXS pattern with simulation results obtained from a coarse-grained DNP model, in which systems spontaneously crystallized into a polymorphic phase that large portion of diamond lattice can be identified.

We expect that the above results can pave the way for synthesizing many different crystal structures, especially opening up the possibility for forming low-coordination diamond crystals in NP systems in an easier manner. Lastly, the proposed parameter and the concept/modeling are not limited to DNPs. Other methods can also adopt the strategy we elucidated, which should result in similar results due to the fact that our MD simulation and free-energy results are largely generic.

Materials and Methods

Detailed materials, computational methods, and experimental procedures and details for data analysis are available in SI Appendix.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix.