Particle analogs of electrons in colloidal crystals

Abstract

A versatile method for designing colloidal crystals involves the use of DNA as a particle-directing ligand. With such systems, DNA–nanoparticle conjugates are considered programmable atom equivalents (PAEs), and design rules have been devised to engineer crystallization outcomes. This work shows that when reduced in size and DNA grafting density, PAEs behave as electron equivalents (EEs), roaming through and stabilizing the lattices defined by larger PAEs, as electrons do in metals in the classical picture. This discovery defines a new property of colloidal crystals, metallicity, that is characterized by the extent of EE delocalization and diffusion. As the number of strands increases or the temperature decreases, the EEs localize, which is structurally reminiscent of a metal−insulator transition. Colloidal crystal metallicity, therefore, provides new routes to metallic, intermetallic, and compound phases.

One Sentence Summary:

A type of electron–atom duality, which establishes the concept of colloidal crystal metallicity, has been identified and characterized.

The interactions among electrons and atoms to form molecules and materials are foundational in physics and chemistry. However, in the science of colloidal crystals where particles are often analogized with atoms, a particle analog to electrons has not been invoked, despite the synthesis of hundreds of colloidal crystals (1–24) and the development of certain approaches into elaborate forms of crystal engineering (9–21). In particular, colloidal crystal engineering with DNA has led to the design of structures with diverse symmetries, lattice parameters, and crystal habits (12–22). However, to date, the particles modified with DNA that define such structures behave as programmable atom equivalents (PAEs) and have fixed particle positions at set stoichiometric ratios. Indeed, these systems are governed by a set of design rules and the complementary contact model (CCM) (14), the premise that they organize themselves to maximize contacts that lead to hybridization and structures like ionic compounds.

We now report on an electron–atom duality analog in colloidal crystal engineering with DNA, where the resulting colloidal assemblies are better classified as “metallic” structures. In such structures, small DNA-functionalized NPs become mobile and “electron-like” (or electron equivalents, or EEs), and they are essential for maintaining the positions of the larger PAE “atom” components. Mixtures of complementary DNA-functionalized nanoparticles that vary in size and DNA surface density were assembled and characterized by electron microscopy, synchrotron small-angle x-ray scattering (SAXS), and scale-accurate molecular dynamics (MD) simulations with explicit hybridization (17, 25, 26). Through a combination of theory, simulations, and experiments, we show that small particles grafted with low numbers of DNA strands (e.g., < 6), when mixed with complementary functionalized nanoparticles (Fig. 1A), form crystals but do not occupy specific lattice sites and diffuse through the crystal in a manner reminiscent of classical electrons in metals, as described by the original Drude model. The PAEs alone will not form crystals since they are almost solely repulsive. The delocalized EEs that move freely through the lattice are responsible for stabilizing it, a type of bonding more remiscent of metals than ionic compounds (Fig. 1B).

Fig. 1. Transition from PAE-PAE systems to PAE-EE systems.

(A) Illustrations of DNA-functionalized Au NPs behaving as programmable atom equivalents (PAEs) or electron equivalents (EEs) used in the MD simulation. (B) Snapshots from the MD simulation depicting “ionic” bonding behavior shown by PAE + PAE assemblies, and “metallic” bonding behavior shown by PAE + EE assemblies, where roaming EEs hold the crystal of repulsive PAEs together. (C, D) Four crystalline lattices assembled from 10-nm PAEs and complementary DNA-functionalized Au NPs (nominal core diameters of 10, 5, 2, and 1.4 nm, respectively): SAXS spectra (C), models, and cross-sectional LAADF images (D) for silica-encapsulated samples. All images in (D) share a scale bar of 25 nm. The 1.4-nm Au NPs in (D) (yellow arrows indicate visually identified ones) are dispersed randomly in the lattice and do not occupy specific lattice sites. (E) SAXS-determined distance between bonding 10-nm PAE pairs (same DNA type, defined in the inset according to CCM assumptions). (F) Quantification of linker DNA strands duplexed on 1.4-nm Au NPs (EEs) as a function of the input number of linkers/EE in the solution.

Furthermore, when the interactions are tailored by increasing the number of potential DNA bonds or lowering the temperature, these EEs condense into specific locations, yielding a transition akin to a metal−insulator transition. Finally, by taking advantage of this duality and the structural features of the DNA-modified particles that govern it, we realize three polymorphic crystal phases—body-centered cubic (bcc), face-centered cubic (fcc), and Frank-Kasper A15—and analyze the distribution and diffusion of the particles (EEs and PAEs) within them as a function of temperature and number of linkers/EE.

In a typical set of experiments, 10-nm-diameter Au nanoparticles (Au NPs) were densely modified with single-stranded propylthiolated DNA to yield conjugates with ~160 strands per NP. These modified NPs were hybridized with a complementary strand to form a rigid duplex region (18 bases) with a 6-base single-stranded overhang (Fig. S1). NPs with average diameters of 10, 5, 2, and 1.4 nm, respectively were modified in a similar manner but with a second type of complementary DNA overhang (Fig. 1A). The average number of DNA overhanging strands available for bonding (i.e., number of linkers/EE) is a function of input linker concentrations in the solution (Fig. 1F). For the first three combinations of NPs (10 + 10, 10 + 5, and 10 + 2 nm), all formed the expected CsCl lattice (space group: $\text{Pm}\overline{3}\text{m}$) (27) based upon the conventional CCM model and the description of them as ionic compound analogues (14).

However, with the 10 + 1.4 nm combination, the 10-nm NPs assumed a bcc lattice (space group: $\text{Im}\overline{3}\text{m}$), but the 1.4-nm NPs become invisible by SAXS (Fig. 1C). The lattice assignments were all verified by electron microscopy with low-angle annular dark field (LAADF) imaging after the structures were encased in silica (28), whereas the 1.4-nm NPs in the 10 + 1.4 nm combination did not appear at specific lattice sites (Fig. 1D). For the first three combinations, the NP positions are determined by the length of the DNA bonding elements that define them (Fig. 1E). However, for the 10 + 1.4 nm combination, there was a marked decrease in the interparticle distance compared with the expected value based upon CCM prediction (14).

To visualize the positions of the 1.4-nm NPs, we performed cryogenic transmission electron microscopy (cryo-TEM) on the as-synthesized bcc lattice formed from the 10 + 1.4 nm NPs and then stacked the repeating “unit cell” images and EE locations along the [111] zone axis (Fig. 2A). Cryo-TEM showed that the large NPs (PAEs) assumed a bcc lattice and the small NPs (EEs) were randomized throughout that lattice (Fig. 2B), in agreement with the MD simulations (Fig. 2D). In the simulations, crystalline structures were obtained for mixtures of complementary DNA-functionalized Au NPs at a fixed size ratio (10- to 2-nm diameter) but with variable EE:PAE ratios from 4:1 to 12:1 and number of linkers/EE from 4 to 8 (e.g., Fig. 2, D and E). Because the MD simulations were performed in the isobaric-isothermal ensemble (NPT) with pressure near zero (see Supplementary Materials), these complementary EEs, which are delocalized from specific lattice sites (Fig. 2D), were responsible for the attraction that holds the large PAEs in crystalline positions in these metal-like assemblies.

Fig. 2. Spatial probability distribution of EEs in PAE-EE assemblies in the bcc structure.

(A) Workflow for obtaining EE location-labeled “unit cell” images from cryo-TEM using image segmentation. ACF: auto-correlation function. (B, C) Overlay of averaged-intensity TEM images and identified EE locations in “unit cells” along the [111] direction. The input parameters refer to the ratio of each substance added to the solution, not within a crystal. (D, E) MD simulation snapshots of PAE-EE assemblies. (F) Measure of clustering tendency $\left(S_{\text{cl}}\right)$ of EEs in MD simulations. (G) Temperature-dependent trapping time $\left(\tau \right)$ of EEs in MD simulations. (H, I, J) Simulated Boltzmann volumes of EEs viewed along the [111] direction with EE:PAE = 4:1 and 4 (H) or 8 (I) linkers/EE, or with EE:PAE = 9:1 and 8 linkers/EE (J). Orange dashes approximate a repeating “unit cell” used in cryo-TEM image analysis.

We determined the degree of delocalization of EEs in the MD simulations by discretizing the unit cell volumes $V_{\text{cell}}$ into ${\left(128\right)}^3$ voxels of equal volume $a^3$ such that $V_{\text{cell}}={\left(128a\right)}^3$ and then counting the EE visitation frequency in each voxel. This frequency gives a probability distribution $f_k$ in each voxel, $k$. A quantitative measure of clustering tendency, $S_{\text{cl}}$, is defined as:

$$S_{\text{cl}}=-{\sum }_k{f}_k\ln \left(f_k\right)+\ln \left(V_{\text{cell}}/a_0^3\right),$$ (Eq. 1)

where $a_0$ is the average of $a$ over all simulations used as a constant value to normalize the volume. The quantity $S_{\text{cl}}$ can be associated with an information entropy (29). This quantity is minimized if the EEs are localized and maximized if they are delocalized (when they have a uniform distribution). The resulting $S_{\text{cl}}$ shows a strong correlation with both EE:PAE ratio and number of linkers/EE (Fig. 2F): either an increase in the number of EEs in the lattice or a decrease in the number of duplexed DNA linkers on the EEs resulted in a more randomized density distribution of EEs in the lattice.

The EEs in the PAE-EE assemblies are classical particles and as such follow a Boltzmann distribution. Thus, the movement of EEs in time is related to free energy barriers that allow for an exponential relation between their trapping time and temperature (Fig. 2G), which is directly related to the degree of EE delocalization. To visualize the spatial distribution of EEs from MD simulations, we calculated the cumulative density of EEs by integrating $f_k$ from its maxima (where the density of EEs is the highest) and draw isosurfaces that separate the unit cell into equally-probable accessible volumes for the EEs, termed Boltzmann volumes (see Supplementary Information). The Boltzmann volumes for an assembly with low number of DNA linkers/EE were widely dispersed across the volume of the crystal (Fig. 2H). Increasing the number of linkers/EE resulted in the localization of EEs into a group of locations near the B sites in AB₆-type binary compounds (Fig. 2I and Table 1). This localized state is similar to the electron charge-density distributions in semiconductors and insulators (30). Furthermore, the Boltzmann volumes became more dispersed as the EE:PAE ratio increases (Fig. 2J; see also Fig. S33). Such a response was also experimentally observed by comparing the projected EE locations determined by cryo-TEM on samples of bcc assemblies with varying input EE:PAE ratios and EE linker concentrations in the solution. The EE local density in the crystal with a low EE:PAE ratio and high linker concentration (Fig. 2C) around the predicted localization sites was substantially higher than the uniform distribution baseline and than the local densities in the assemblies with higher EE:PAE ratios and lower linker concentrations (Figs. 2B and S17).

Table 1. Symmetry of PAE-EE assemblies in the fully localized state.

Wyckoff positions in square brackets have higher energies than the ground-state configurations.

PAE Lattice	Space Group	Localized EE Wyckoff Positions
BCC	$\text{Im}\overline{3}\text{m}$	12d
FCC	$\text{Fm}\overline{3}\text{m}$	8c, 32f
A15	$\text{Pm}\overline{3}\text{n}$	6c, 16i, 24k, [12f], [24j]

Compared to PAEs that reside on relatively fixed lattice sites, EEs in a PAE-EE assembly are macroscopically mobile beyond local vibrations. Time-series MD simulation snapshots showed that the EEs could diffuse between unit cells (Fig. 3A, S39, and Movie S4). To further probe the diffusion of EEs in experimentally realized assemblies, 10-nm PAEs and Cy5-DNA-labeled 1.4-nm EEs were assembled in the bcc structure. Subsequently, these crystals were incubated with a solution of Cy3-DNA-labeled EEs. To track any exchange of EEs between the crystalline assembly and EEs in solution, the ultraviolet-visible (UV-Vis) extinction spectra of the supernatant were measured over time (Fig. 3B, upper panel). A simultaneous increase of Cy5 signal and reduction of Cy3 signal suggested that the EEs were highly mobile and could diffuse macroscopically between the colloidal assemblies and solution. In comparison, no appreciable exchange events were observed for the bcc assemblies formed by 10 + 10 nm PAEs modified with identical dye-labeled DNA (Fig. 3B, lower panel), suggesting a much weaker diffusion of PAEs as compared to that of EEs. In both cases, the crystallinity of the assemblies was preserved post-exchange (Fig. S18).

Fig. 3. Diffusion of EEs in PAE-EE assemblies in the bcc structure.

(A) Trajectory of one EE over time in a lattice of PAEs (yellow) from the MD simulation. The color of the EE positions (red to green to blue) represents different timepoints. (B) The exchange of dye-labeled particles between crystalline lattices and solution was monitored by the change of light extinction in solution via UV-vis spectroscopy. Cy5-DNA-labeled EEs were exchanged from “metallic” PAE-EE assemblies (10 + 1.4 nm, bcc) by Cy3-DNA-labeled EEs in the supernatant over 24 h (upper panel), whereas no appreciable exchange was observed for PAE-PAE assemblies (10 + 10 nm, bcc, lower panel). (C, D) Predicted colloidal crystal metallicity $\left(M_{\text{cc}}\right)$ in BCC assemblies with different number of linkers/EE. The $M_{\text{cc}}$ value has a minimum against the EE:PAE ratio at 6:1 (C) but is monotonic against temperature (D).

The tunable spatial density distribution of EEs (based on number, temperature, and linker density) and their macroscale diffusion inside a crystalline PAE framework establish the concept of colloidal crystal metallicity, $M_{\text{cc}}$, as:

$$M_{\text{cc}}=S_{\text{cl}}-\ln \left(N_{\text{cell}}\right),$$ (Eq. 2)

where $N_{\text{cell}}$ is the number of EEs per unit cell, and it is used to make the metallicity independent of crystal unit cell and to reduce to the ideal entropy of a gas in the limit of non-interacting particles. In Eq. 1 the EEs are considered as a group (Fig. 2F), but in Eq. 2, $M_{\text{cc}}$ is a quantitative measure of the average degree of delocalization per EE, which can decrease when the EE:PAE ratio increases (Fig. 3C), even if $S_{\text{cl}}$ increases or remains nearly the same (Fig. 2F). Importantly, a metallicity minimum was attained at EE:PAE = 6:1, suggesting that the metal-like bonds in the colloidal system become saturated. Increasing the number of duplexed strands on EEs led to crystals with lower $M_{\text{cc}}$ values (Fig. 3C) because the EEs were more localized or trapped as the frequency of DNA binding events increased. When the system temperature increased, $M_{\text{cc}}$ also increases (Fig. 3D) because of the decrease in the number of hybridized sticky ends. These results show that the EEs can undergo a classical (non-quantized) process analogous to a metal?insulator transition observed in atomic solids (31), as the degree of EE delocalization and diffusion drastically changed when either the temperature, the number of linkers/EE, or the EE:PAE ratio was varied.

Colloidal crystal metallicity depends on the number of particles, the number of DNA strands that can engage in bonding, and the strength of the bonds formed. Because these parameters are difficult, if not impossible, to control in purely electrostatic systems because of electroneutrality requirements, metallicity has not been observed or defined in conventional ionic colloidal crystals (crystals formed from oppositely charged particles) (4, 9). The complementary DNA design on NPs ensures that a crystal will not form from PAEs alone (and vice versa from EEs alone), which distinguishes this system from a description of small particles doped in a crystalline solid formed from larger particles. Moreover, although this concept of metallicity superficially resembles “sublattice melting” in superionic conductors (32), where one sublattice of an ionic compound loses long-range order while the other is fixed (e.g., AgI), ion diffusion in such systems is mediated by Frenkel defects because of the constraint of charge balance. Thus, ionic systems have limited tunability and more strict stoichiometry requirements compared to the colloidal crystals formed through DNA-directed assembly events.

Interestingly, new phases were accessed by adjusting the input EE:PAE ratio in solution and the total DNA coverage on EEs. For example, as the input EE:PAE ratio was progressively increased, an fcc lattice (space group: $\text{Fm}\overline{3}\text{m}$) emerged (Fig. 4, A and C). The increase in EE:PAE ratio resulted in stronger cumulative bonding interactions (there are more DNA bonding connections under such circumstances), which was reflected by the increase in the crystal melting temperature, $T_{\text{m}}$, from 31° (bcc) to 41 °C (fcc) (Fig. 4B). In addition, if both the total DNA coverage on EEs (characterized by the total number of duplexed and non-duplexed strands) and the input EE:PAE ratio in solution increase, the Frank-Kasper A15 phase (space group: $\text{Pm}\overline{3}\text{n}$) emerged (Fig. 4, D and E). The formation of the A15 phase may be associated with its tendency to minimize the contact area between repulsive PAEs, similar to the trend observed in dendrimer assemblies (33) (see also Fig. S27) but different from the Cr₃Si structure in binary superlattices (14, 34). These phases were also observed in the simulations (Table S9). We note that the three phases realized in this system mimic the metal tungsten, in which bcc, fcc, and A15 structures have all been realized either in the bulk or in thin films (35) (see also Table S6).

Fig. 4. Equilibrium phases realized by PAE-EE assemblies.

(A, B) Schematic representation of the equilibrium conditions of bcc, fcc, and A15 phases (A), and their corresponding thermal melting transitions (B). (C) SAXS spectra showing the equilibrium phase transition from BCC (red, input EE:PAE = 10:1) to a bcc/fcc mixture (purple, input EE:PAE = 20:1), and then to a majority fcc phase (blue, input EE:PAE = 40:1). (D) Experimental (green) and simulated (black) SAXS spectra of A15 assemblies. (E) Cryo-TEM image of an A15 assembly. Inset: A15 lattice model along the [001] direction. (F, G) Simulated Boltzmann volumes of FCC (F) and A15 (G) phases as a function of the number of linkers/EE and EE:PAE ratio at a constant temperature $\left(k_{\text{B}}T=1.30\right)$. A whole graph is reconstructed by 1/8 of the unit cells from each combination. (H, I) Predicted colloidal crystal metallicity $\left(M_{\text{cc}}\right)$ in FCC (H) and A15 (I) assemblies as a function of the number of linkers/EE and EE:PAE ratio. Two low-metallicity configurations (gray shades) are present in (I).

Critically, the MD simulations allowed us to identify the positions that the EEs reside in at low $M_{\text{cc}}$ values. For example, when the number of DNA linkers/EE was maximized and the EE:PAE ratio was decreased, the EEs settled into distinct locations in the fcc and A15 assemblies, as shown by the Boltzmann volumes (Fig. 4, F and G) and the corresponding $M_{\text{cc}}$ values (Fig. 4, H and I). The localized lattice for the FCC structure resembled a fully filled high-temperature Cu₂Se lattice (Fig. S34, A₄B₄₀), whereas the A15 shows two possible configurations, either clathrate type I (Fig. S35A, A₈B₄₆) (36) or an unreported lattice (Fig. S35B, A₈B₈₂). Indeed, the latter configuration, in which $M_{\text{cc}}$ reached a local minimum, contained all sites in the clathrate structure that were fully occupied, and the additional EEs that could occupy the lowest energy positions began to fill the higher-energy 12f and 24j positions (Table 1).

Taken together, this work makes the case for describing certain classes of colloidal crystals in a fundamentally new way, where in the case of mobile particles (EEs) the concept of metallicity becomes important. By understanding the factors that govern EE diffusion and delocalization, we have a better understanding of the structures and phases that can be accessed through colloidal crystals, potentially including metals, intermetallics, and complex metal alloys. It also challenges the colloidal science community to identify exotic new properties that arise from the PAE-to-EE transition and structures that exhibit high degrees of metallicity, as well as to develop theoretical models that capture the effects that lead to metallicity.

Data and materials availability:

All data needed to evaluate the conclusions in this manuscript are present in the main text or the supplementary materials. Additional data or codes are available upon request to the corresponding authors.