Valence-free open nanoparticle superlattices

Domain: physics
Published: 2026-01-17
Editorial outcome: minor_revision

Abstract
A cornerstone of advanced materials design is establishing a framework for assembling nanoparticle superstructures with tailored symmetries. A long-standing challenge has been assembling diamond-like superstructures for photonic devices. Traditionally, such open superstructures require functionalized nanoparticles with directional or anisotropic interactions, reminiscent of valence bonding in a diamond. Here, we present a robust strategy for assembling valence-free nanoparticles into a broad array of cubic superstructures. By grafting nanoparticles with oppositely charged, end-functionalized water-soluble polymers of adjustable molecular weight, we gain control over electrostatic interactions and conformational constraints. This unified approach yields lattices analogous to rock salt, CsCl, zinc-blende, diamond, and the rare simple cubic phase, with tunable lattice constants. Theoretical models and simulations elucidate the underlying interactions, providing a framework for engineering valence-free nanoparticle superlattices.

Introduction
Assembling colloidal nanoparticles (NPs) into cubic superstructures has long been a central challenge in nanoscience1–9. For example, the fabrication of diamond-like superstructures has traditionally relied on patchy or anisotropic NPs, with directional interaction sites10,11. The valence-like bonding mimics the directional bonding observed in carbon-based diamond structures. Various methods have been employed to achieve this goal, including the arrangement of NPs within tetrahedral DNA origami cages10 and the assembly of micron-sized tetrahedral clusters linked through DNA11. DNA origami tetrapods have also been used to create diamond lattices with photonic band gaps12. Low-coordination cubic lattices have also been realized through DNA-programmed NPs13. Only one report of a diamond superlattice, assembled without valence, exhibited broad X-ray diffraction peaks, signaling poor structural quality14. Given these challenges, we present a general strategy for assembling a diverse array of NP superlattices, including diamond, inspired by binary ionic compounds, achieved without relying on valence.

Binary ionic compounds exhibit a rich diversity of ordered structures dictated by ion size15,16. For example, similar-size ions such as Cs+ and Cl− tend to adopt simple cubic or pseudo-body-centered cubic (BCC)-like arrangements. For moderate size asymmetry, as seen with Na+/K+ and Cl−, the prevailing configuration is a rock salt or face-centered cubic (FCC)-like structure. Divalent compounds with higher size asymmetry, from oxides like CaO and BaO to chalcogenides such as ZnS, follow a similar trend17. ZnS can crystallize in a zinc-blende (FCC) or wurtzite (hexagonal) lattice, depending on external conditions18. Hypothetically, if the rock salt, cesium chloride, or zinc-blende structures were made with identical elements, they would correspond to simple cubic, BCC, and diamond lattices, respectively. Building on these structural insights, we now explore how similar principles apply to the assembly of spherical NPs. We functionalize NPs with poly(ethylene glycol) (PEG) chains terminated with carboxyl (–COOH) or amine (–NH2) end-groups, enabling control over their surface charge through adjustment of pH. This strategy is broadly applicable, as both the NP type and the terminal groups can be generalized to any system employing ligands with oppositely charged ends. Moreover, by varying the molecular weight of PEG, we tune the polymer chain length and, consequently, the hydrodynamic size of the grafted NPs. The resulting superlattices consist of NP cores embedded within a uniform polymeric matrix. When the NP cores are identical, the structures can adopt simpler forms such as BCC, simple cubic, or diamond arrangements, which are analogous to elemental binary ionic salts, depending on their hydrodynamic size. The key insight is that the hydrodynamic radius, tuned via polymer chain length, sets the effective ion size, and thus the structure and lattice constants are independent of the NP core diameter. Early experiments with micron-sized charged colloids in organic solvents yielded a range of ionic-like superlattices but did not produce ZnS superlattices, a precursor for assembling diamond superlattices19. However, theoretical work has shown that valence-free NPs, interacting via designed spherically symmetric potentials, can assemble into open lattices such as diamond and simple cubic20,21.

To further illustrate our assembly strategy, we model the grafted NPs as hard spheres with diameters DA and DB (with DA ≥ DB) and uniformly opposite surface charges (qA and qB). As shown in Fig. 1a, the key parameter is the size ratio;

γ = DB / DA ≤ 1: (1)

Results
Our approach begins with a binary system engineered to favor ZnS-like or NaCl-like superstructures. We use COOH-PEG5k-Au5 and NH2-PEG10k-Au10 NPs, which yield a size ratio of γ ≃ 0.6 optimal for a ZnS or NaCl-like superlattices assembly (see Fig. 1d). In addition, we define a parameter θ as

θ = MW COOH / MW NH2, (2)

representing the ratio of molecular weights (MWs) of the grafted PEGs with different end groups. This parameter is valuable for constructing the experimental phase diagram of the assembled structures described below.

Figure 2a shows the small-angle X-ray scattering (SAXS) diffraction pattern (I(Q) vs. Q) for a 4:1 mixture of COOH-PEG5k-Au5 and NH2-PEG10k-Au10 at pH 3. The diffraction pattern corresponds to a ZnS-like (zinc blende) structure, with the modeled intensity profile reproducing the experimental data. The ZnS-like structure can also form at different mixing ratios (1:1, 2:1, and 4:1), as shown in Fig. S7. Although ZnS-like assemblies consistently exhibit 1:1 stoichiometry, they can arise from mixtures with varying initial ratios of NPs, with any excess particles remaining unincorporated in solution. This behavior suggests that the assembly kinetics are not dependent on the initial particle ratios, as the system relaxes toward its thermodynamically favored stoichiometry, leaving excess particles unincorporated. A more detailed analysis, presented in the SI Fig. S25, shows that the experimental pattern is best reproduced when site-exchanged defects are included in the ZnS framework, consistent with electrostatic stabilization of the open lattice. However, larger gold nanoparticles (AuNPs) (20 and 10nm) (shown in Figs. S20 and S21) yield good agreement between experimental diffraction patterns and calculated ZnS structure factors, demonstrating that increased particle size enhances structural stability and promotes the formation of well-ordered superlattices.

To achieve a reliable and well-defined diffraction pattern, we initially employed core NPs with distinct sizes (5 and 10nm), thereby leveraging the electron density (ED) contrast between the two NP types to stabilize a ZnS-like lattice. For a binary system consisting of identical 10nm core NPs (γ = 0.68), the lack of ED contrast transforms the ZnS-like into a diamond-like superlattice. Our work focuses on the gold NP lattice, treating the polymer framework, whose low ED is virtually invisible in X-ray scattering, as negligible. In photonic applications, the NPs define the optical response, whereas the grafted polymers furnish the functional sites that organize and stabilize the active components25. This applies to the SAXS diffraction pattern shown in Fig. 2b, which is well described by a diamond structure with minor deviations. In the ideal diamond lattice, four tetrahedral interstitial sites per unit cell are available. Our detailed fits indicate that, in some cases, these interstitials are partially occupied. As discussed in the SI Fig. S26, partial interstitial occupancy within the diamond lattice (~0.7 NP per unit cell) provides an improved fit to the experimental diffraction pattern, suggesting minor deviations from ideal stoichiometry for enhanced structural stability. These combinations indicate that optimal ZnS or diamond assembly occurs at θ ≃ 0.5. To further validate the robustness of our self-assembly strategy, we extended our investigation to binary mixtures of 20 and 10nm AuNPs, functionalized with the same –COOH and –NH2 terminated PEG ligands. By maintaining similar γ and θ ratios, these larger particles also assemble into a well-ordered zinc-blende superlattice, as confirmed by the SAXS patterns in Figs. S20 and S21.

The stability of the zinc-blende structure offers a pathway for assembling diamond lattices. As shown in Fig. 1a, the effective NP hard sphere diameter is defined as the sum of its core and the PEG corona thickness26. By keeping the cores the same and tuning the PEG chain length to achieve γ values within γcZnS ≤ γ < γcNaCl, diamond superlattices are formed. In contrast, when γ falls within γcNaCl ≤ γ < γcCsCl, a simple cubic structure is obtained.

The phase diagram in Fig. 1d depends on the parameter

α = ΔqA / ΔqB, (3)

where ΔqA and ΔqB denote the amount of surface charge specified within a defined solid angle, where the charge needs to be considered explicitly to account for correlations (See Fig. 1a and a more detailed derivation in SI). These localized charge correlations introduce a short-range electrostatic correction to the hard-sphere model, shifting the critical γ values and thereby enhancing the relative stability of ZnS and NaCl lattices over CsCl. The dashed lines in Fig. 1d delineate a confined (α, γ) window corresponding to the optimal conditions for the assembly of the above cubic superstructures. This highlights that tuning the NP surface charge (and hence α) provides control over the resulting lattice symmetry.

A further essential knob for controlling the superstructure assembly is the effective surface charge of the NPs, which can be finely tuned by adjusting the pH. As shown in Fig. 2c, increasing the pH from 3 to 4 transforms the zinc-blende into a rock-salt superstructure. Similarly, the diamond-like superstructure observed at pH 3 (Fig. 2b) transitions to a simple cubic configuration at pH 4 (Fig. 2d). This simple cubic phase arises from NPs with identical 10nm cores, where the loss of ED contrast diminishes the (111) peak characteristic of a rock-salt lattice. Furthermore, the SAXS pattern of the simple cubic phase exhibits a broad peak, indicative of mesoscale order and structural defects. Detailed definitions of these ordering regimes are provided in the “Methods” section. This imperfect, simple cubic lattice sets the stage for our subsequent pursuit of a defect-free simple cubic structure. Thus, we explore alternative parameter combinations by maintaining a constant γ while varying θ (MWs of grafted PEG). Some combinations yield simple cubic short-range ordering (SRO). However, setting θ ≃ 5 and γ ≃ 0.8 (a condition that typically favors a rock-salt structure), the binary mixture of COOH-PEG10k-Au10 and NH2-PEG2k-Au10 produced a simple cubic analog. Fig. 3a shows the SAXS diffraction pattern of this assembly, which is accurately fitted to a simple cubic model.

By interchanging the terminal groups-swapping COOH and NH2, we stabilize a structurally well-defined diamond lattice. Figure 3b presents the SAXS diffraction pattern for a 1:1 mixture of COOH-PEG2k-Au10 and NH2-PEG10k-Au10 at pH 3 (θ ≃ 0.2), which is well-fitted to a diamond-like structure. This modification alters both the total and local charge on the NPs, as grafting density (and thus surface charge) depends on PEG chain length27,28. Moreover, the effective size ratio γ shifts from 0.80 (yielding a simple cubic lattice) to 0.64 (to a diamond lattice), in accordance with the calculated phase diagram in Fig. 1d. These findings emphasize that maintaining low θ and γ values is essential for stabilizing a diamond lattice, whereas higher values typically favor a rock-salt structure. We have successfully assembled a variety of diamond and zinc-blende lattices using these parameter regimes (see SI Figs. S13 and S15).

Our next binary system exhibits the phenomenon of particle fractionalization, a process in which NPs are selectively sorted by size from their inherent polydispersity (See SI for detailed discussions)29,30. Figure 3c shows the SAXS diffraction pattern for a binary mixture of COOH-PEG10k-Au10 and NH2-PEG5k-Au10 with θ ≃ 2 and γ ≃ 0.97, conditions that would typically yield a simple cubic pattern given the use of identical core NPs. However, the appearance of a weak (111) peak, characteristic of a NaCl-like structure (see Fig. 2b), indicates that the assembly process is selective, assigning distinct particle sizes to different lattice sites through fractionalization. This slight size disparity results in a weak (111) reflection analogous to the differences observed between the diffraction patterns of NaCl and KCl, where reduced ED contrast in KCl produces a similar effect despite both crystallizing in an FCC structure. We surmise that the particle fractionalization is likely due to the redistribution of NPs across multiple assemblies when the system lies near a phase boundary.

Based on the phase diagram in Fig. 1d, we identified a narrow range of parameters that yield a CsCl-like structure. By selecting a binary mixture of COOH-PEG5k-Au10 and NH2-PEG5k-Au5 NPs (γ ≃ 0.9 and θ ≃ 1), parameters that lie near the predicted coexistence region for NaCl and CsCl superstructures, and by tuning the pH to 3, we successfully assembled a CsCl-like superlattice. Figure 3d presents the SAXS diffraction pattern for this mixture at pH 3, which fits well to a CsCl-like structure. Small satellite peaks near the prominent (110) reflection indicate systematic defects or lattice elongation as discussed in detail in the SI. In contrast, using NPs with identical core sizes (10 and 10nm) under the same grafting conditions to achieve a BCC superstructure resulted only in low-quality, short-range BCC-like ordering, as shown in S18. The phase diagram in Fig. 4b justifies this finding by revealing that the CsCl/BCC phase is stable only within a very narrow parameter window. For completeness, the SI provides the detailed lattice parameters, nearest neighbor (NN) distances in Figs. S7–S19 and full width at half maximum (FWHM) in Table S8. FWHM values were used to estimate the sizes of the crystalline domains, which are on the order of 1μm or larger for the highest quality crystals, while lower quality domains are typically in the sub-micrometer range.

To rationalize and extend our experimental findings, we introduce the following semi-quantitative free energy expression, which incorporates electrostatic correlations:

E = P L + (N A e^2 q^2 / 4πε D) * (M αn / (1+γ) δ + c (γ - γc)) (4)

Here, γ, γc, α, and δ are defined in Fig. 1a; ML represent the Madelung energy of the corresponding lattice; c is a constant, and nAA denotes the number of NNs for an A NP. The second term in parentheses accounts for the short-range repulsion between B–B NPs as γ approaches γc. Increasing α enhances this B–B repulsion, thereby improving the stability of the ZnS structure by effectively shifting γc to a higher value (see Fig. 1d). A detailed discussion of this model is provided in the SI.

Understanding the assembly process requires elucidating the molecular-level mechanisms governing NP interactions, particularly the attraction of A–B NPs and the repulsion of B–B (and A–A) NPs described in Eq. (4). To this end, we performed all-atom molecular dynamics (MD) simulations, directly quantifying the electrostatic and hydrogen-bonding correlations that drive these forces. Our model system consists of planar walls (hypothetically NP surfaces) functionalized with PEG having COO− or NH3+ end groups, neutralized by electrolytes. The justification for this model and further supporting results are provided in the SI. Figure 4c shows the pair distribution function, g(r), for COO−-NH3+ pairs (solid black line), which exhibits a sharp peak indicating that roughly one-third of these end groups are bound. Additionally, almost half of these bound groups form bonds, with binding strength roughly independent of electrolyte concentration. This binding is enhanced by the formation of COO−···NH3+ hydrogen bonds. In contrast, Fig. 4d shows that equally charged end groups exhibit strong repulsion over distances up to 1nm, thereby validating the cutoff distance, rc, used in our effective model (see Fig. 1a). Snapshot images in Fig. 4d further illustrate that similarly charged end groups remain well separated, effectively overcoming potential screening by electrolytes or charge neutralization by counterions. Together, these simulations illustrate the B–B correlations, confirming the repulsive term assumed in Eq. (4), thus correcting the hard sphere model and leading to the phase diagram Fig. 4b.

In summary, by leveraging pH-regulated surface charge and controlling the conformation of grafted polymers, we establish a generalizable, valence-free strategy for assembling open-framework cubic superlattices from binary mixtures of isotropic NPs. In these systems, a single set of design rules, encoded in the NP size ratio γ and the grafted ligand molecular weight ratio θ, directs crystallization into lattices analogous to rock salt, CsCl, zinc blende, diamond, and even the rare simple cubic phase. SAXS measurements indicate well-defined structural order, and our free-energy calculations and MD simulations provide quantitative support for the observed trends. Importantly, this approach is universally applicable to any NP-ligand system, enabling control over the lattice constant by tuning the core diameter and ligand molecular weight to adjust γ and θ. We demonstrated that our approach is scalable, as we have assembled a zinc-blende lattice with NPs of 5, 10, and 20nm in diameter, with a lattice constant ranging from 70 to 80nm. Our versatile strategy establishes a broadly applicable framework for realizing open architectures without directional bonding, offering a general route to designing functional NP superlattices for photonic to catalytic applications.

Methods
Materials
Unconjugated citrate-capped AuNPs of nominal core diameters of 5 and 10nm were purchased from Ted Pella Inc., and their size distributions were verified independently by transmission electron microscopy (TEM) (5.8±0.6 and 9.1±1, respectively, See Figs. S5 and S6) and by SAXS31. PEG with an average molecular weight of 2, 5, and 10 kDa, modified with a thiol (HS) group at one end and either a carboxyl (–COOH) or amine (–NH2) group at the other, was purchased from Creative PEGworks (NC, USA) and used without further purification. HCl was purchased from Fisher Scientific and used without further purification. Milli-Q water (resistivity 18.2MΩ cm at 25°C) was used in all experiments.

PEG grafting to AuNPs surfaces
AuNPs are functionalized with HS-PEG-COOH or HS-PEG-NH2 using a ligand exchange protocol28,32. PEG ligands were dissolved in water and thoroughly mixed to form a homogeneous suspension. This PEG solution, in molar excess, was added to AuNP suspensions at a 1:1500 AuNP-to-PEG chain ratio for 5nm AuNPs and a 1:6000 ratio for 10nm AuNPs. The mixture was rotated overnight at ~ 35RPM on a Roto-Shake Genie (Scientific Industries, NY, USA) under continuous mixing to facilitate ligand exchange. Unbound PEG was removed from the PEG-functionalized AuNPs by three rounds of centrifugation. For 5nm AuNPs, centrifugation was performed at 21,000×g for 90min per round, while for 10nm AuNPs, it was conducted at 20,000×g for 75min. After centrifugation, the resulting PEG-grafted AuNPs were resuspended in Milli-Q water to create a stock suspension. The final concentration of the PEG-AuNP suspensions was determined by measuring absorbance using ultraviolet–visible (UV–vis) spectroscopy (NanoDrop One Microvolume, Thermo Fisher Scientific). Concentrations were adjusted to approximately 20nM for 10nm AuNPs and 80nM for 5nm AuNPs, as described below.

The molarity of AuNP suspensions is calculated using the concentration data provided by the supplier (Ted Pella Inc.), which specifies the number of particles per milliliter of solution. The following equation is employed for the calculation:

Molarity (M) = (Number of particles per ml) / (NA * 10^-3 liters) (5)

where NA is Avogadro’s number.

For instance, the molarity of 10nm-sized bare AuNPs is determined from the particle concentration provided in the vendor’s technical data sheet, which is 5.7×10^12 particles/ml. Using Eq. (5), the molarity is calculated to be 9.46nM. After determining the molarity of the bare AuNPs, UV–vis spectrophotometry is utilized to determine their absorbance at a wavelength of 525nm. Assuming that the grafting of PEG does not influence the absorbance and that no aggregation occurs, the absorbance profiles of the PEG-grafted AuNPs can be compared with those of the bare AuNPs to determine the concentration of the grafted AuNPs in suspension. PEG grafting densities, determined by thermogravimetric analysis, are provided in the SI and are reproducible, independent of terminal group charge28,32–34.

In this study, PEG-AuNP refers to PEG-grafted AuNPs in general, while x-PEGy-AuNPz specifies the detailed characteristics of the NPs. In this labeling scheme, z represents the core diameter of the AuNPs in nanometers (e.g., 5 or 10nm), y refers to the molecular weight of the grafted PEG (2, 5, or 10 kDa), and x indicates the terminal functional group of the PEG chain (–COOH or –NH2). For example, COOH-PEG5k-Au10 describes 10nm AuNPs functionalized with 5kDa PEG terminated with a carboxyl group (–COOH). Similarly, NH2-PEG10k-Au5 denotes 5nm AuNPs grafted with 10kDa PEG terminated with an amine group (–NH2). The complete labeling scheme and detailed properties of all PEG-grafted AuNPs are provided in Table S1. The legend for each PEG-AuNPs is depicted as a hard sphere in Fig. 1b.

DLS and ζ−potential measurements
The DH of grafted NPs was measured by dynamic light scattering (DLS) on a Nano ZS90 with Zetasizer software (Malvern, UK). ζ−potential measurements, conducted on the same instrument, were used to quantify the surface charge of PEG-grafted AuNPs. These data are summarized in Table S1. Successful PEG grafting with distinct terminal groups (–COOH or –NH2) was confirmed by observing opposite surface charges and increased hydrodynamic sizes. Further details on PEG grafting and characterization are available in a recent publication23. While DH and the DLS-derived size ratio γ provide objective, experimentally measurable design parameters to guide NP assembly, they are not expected to exactly predict the resulting lattice constants, which are governed by collective packing effects, polymer deformation, and electrostatic interactions. The DH values show a systematic increase for –NH2-terminated particles compared to -COOH-terminated ones, particularly at higher PEG MWs (e.g., 10 kDa), reflecting the more hydrated and expanded corona of NH2-terminated PEG relative to the more compact conformation of COOH-terminated PEG.

Surface charge manipulation
To adjust the surface and overall charge of the grafted AuNPs, HCl was added to the suspensions, achieving final concentrations of 0.1, 1, and 10mM. Since the small sample volumes required for SAXS hinder direct pH measurement, HCl concentrations and corresponding estimated pH values are specified in parentheses. pH values were verified by bulk titration of representative samples, showing consistency within ±0.2 pH units, corresponding to the final pH values calculated from the amount of HCl added to the solutions. For SAXS, a calculated volume of AuNPs suspension was mixed in a glass scintillation vial at the desired mixing ratio. Mixing ratio are calculated in terms of NP number ratios. Stock HCl solutions were added to the AuNP mixture, thoroughly mixed, and incubated for ~20min. The mixture was then loaded into a quartz capillary (~1.5mm diameter) for SAXS measurements.

SAXS measurements
Figure 5 illustrates the procedure for creating binary mixtures, conducting the SAXS measurements, and their structural analysis. In-situ SAXS was employed to determine the structures formed in the suspensions at various pH levels and temperatures. Given the lower ED of the polymers compared to gold, the diffraction is dominated by the AuNPs. We employed AuNPs of two distinct sizes (10 and 5nm) to achieve scattering contrast in our binary mixtures. In contrast, when the binary system comprises of NPs with identical core sizes (10 or 5nm), the resulting superstructures exhibit a uniform, single-particle-like lattice (such as a diamond or simple cubic lattice). For a general binary system, we designate the constituent particles as A and B, where DA > DB, without specifying their precise sizes or the details of their PEG ligand terminations.

Synchrotron-based in-situ SAXS experiments were conducted at beamline 12-ID-B of the Advanced Photon Source, Argonne National Laboratory, and at the 11-BM CMS beamline of the National Synchrotron Light Source II (NSLS-II), Brookhaven National Laboratory. The incident X-ray energies were 13.3 and 13.5 keV, respectively. Bench-top in-situ SAXS was performed using a liquid-metal-jet Ga alloy source with an incident X-ray energy of 9.2 keV.

Incubated samples were transferred into ~1.5mm diameter quartz capillaries and positioned vertically in beamline-specific sample holders, normal to the incident beam. Data acquisition and processing were carried out following each beamline’s protocol. SAXS data were reduced to one-dimensional (1D) scattering profiles as a function of the wave vector magnitude Q, where Q = 4π sin θ / λ, with 2θ being the scattering angle and λ the X-ray wavelength. Detailed experimental setup, data acquisition, and analysis protocols are available in previous studies35–37.

The scattering profiles, I(Q), were background-corrected by subtracting signals from corresponding water samples, resulting in diffraction patterns resembling those of polycrystalline samples in suspension. The plotted scattering profiles are presented as raw I(Q) after only background subtraction; no form-factor correction P(Q) was applied owing to the dominance of long-range order scattering, and structural assignments are based on peak positions, with relative intensities noted but not used as the sole determinant. Data analysis followed standard procedures for X-ray diffraction of polycrystalline materials. For crystalline samples, peak positions were identified based on expected reflections for specific space groups, and structure factors were calculated and fitted to the data to determine the fundamental unit cell dimensions and particle occupancy. Based on structural organization, we classified the assemblies into three regimes. Assemblies are defined as long-range ordered (LRO) when the experimental diffraction peaks are resolution-limited, typically corresponding to crystalline domains larger than 10 unit cells. SRO is assigned when the peak width is at least twice the instrumental resolution, corresponding to crystalline domains smaller than 5 unit cells. Meso-range order describes intermediate cases, with crystalline domains of 5–10 unit cells23. The FWHM values used for these classifications are summarized in Table S8.

All-atom MD simulations
All-atom MD simulations using HOOMD-blue38 implemented through HOODLT39,40. HOODLT was used to generate the initial configurations, implement the force field, and analyze the results. The system consisted of two parallel plates composed of gold atoms, modeled as rigid bodies41 and grafted with PEG ligands terminating in charged end groups with the addition of salt (NaCl) in different concentrations, see justification of the model further below. The optimized potentials for liquid simulations force field42, implemented through Foyer43, was used. Three distinct systems were simulated: (a) a system with 98 PEG ligands in total 49 on each plate where one plate terminated in COO− and the other in NH3+ groups; (b) a system with 98 PEG ligands all terminating in COO− groups with Na+ as counterions; and (c) a system with 98 PEG ligands all terminating in NH3+ groups with Cl− as counterions. Each system was simulated at varying plate separations ranging from 4.6 to 11.6 nm, and at salt molalities of 0.5 and 1.1 mol/kg, with SPC/E water used as the solvent. All simulations were first equilibrated in the NPT (constant pressure and temperature) ensemble for 3ns using the MTTK thermostat and barostat to maintain constant temperature and pressure. This was followed by 40 nanoseconds of production run in the NVT (constant volume and temperature) ensemble with the same MTTK thermostat with a timestep of 1fs. All simulations were conducted on the Bridges-2 high-performance computing cluster at the Pittsburgh Supercomputing Center.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript presents a novel and significant approach for assembling open-framework cubic superlattices from spherical nanoparticles without relying on directional \"valence-like\" interactions. The authors demonstrate the assembly of various cubic structures including rock salt, CsCl, zinc-blende, diamond, and simple cubic phases by grafting gold nanoparticles with oppositely charged, end-functionalized PEG polymers and controlling electrostatic interactions through pH modulation. This work represents a significant advance in nanoparticle self-assembly with important implications for materials science and photonics. The core findings are sound and the experimental approach is well-designed. This manuscript may be accepted after minor modification.\nSpecific comments:\n1. The Methods section on pages 11-12 must replace HCl concentration estimates with direct pH measurements using microelectrodes, include detailed PEG grafting density quantification via NMR or XPS, and provide reproducibility data across multiple sample preparations.\n2. The manuscript requires addition of TEM/SEM images in Figure 2 and Figure 3 for direct structural visualization and selected area electron diffraction patterns to confirm SAXS assignments.\n3. Quantitative comparison between predicted phase boundaries in Figure 4b and experimental observations in Figure 4a is necessary, with explicit explanation of discrepancies. The \"particle fractionalization\" mechanism in Figure 3c requires detailed explanation.\n4. A comprehensive table comparing this method with DNA-origami, patchy particles, and other assembly approaches must include assembly time, yield, structural quality, and cost metrics. The manuscript needs specific discussion of scalability limitations, long-term stability testing over weeks to months, and thermal compatibility with device fabrication processes to establish practical viability beyond proof-of-concept demonstrations.\n5. Page 1: \"Only report of a diamond superlattice\" - grammatically incorrect, should be \"Only one report\"", "reviewer_2": "The authors report a strategy to create superlattices using valence-free nanoparticles (NPs). They thoroughly studied the diverse lattices formed by charged polymer-grafted NPs and demonstrated the effect of experimental parameters (e.g., the ratio of hydrodynamic sizes between two NPs) on the assembly structures. Specifically, they achieved a wide range of lattices, such as rock salt, CsCl, zinc-blende, diamond, and the exceptionally rare simple cubic phase, with tunable lattice constants. The lattice formation was confirmed through comprehensive in situ SAXS measurements and performing MD simulations. Although many previous works have demonstrated lattice formation in binary microparticle systems via controlled electrostatic interparticle interactions, relatively few studies have achieved this at the NP level. Therefore, I recommend the publication of this work in Nature Communications after revisions and addressing following questions/comments to clarify the manuscript.\n1. The authors emphasized the importance of using “valence-free” NPs for assembly by stating that “The fabrication of diamond-like superstructures has traditionally relied on patchy or anisotropic NPs, with directional interaction sites.” and “Only report of a diamond superlattice, assembled without valence, exhibited broad X-ray diffraction peaks, signaling poor structural quality.” However, it is unclear in the manuscript why using “valence-free” NPs is more important than utilizing “valence-like” bonding. Could the authors provide a more detailed explanation of the rationale behind employing “valence-free” NPs for assembly?\n2. The authors mentioned that “Although ZnS-like assemblies consistently exhibit 1:1 stoichiometry, they can arise from mixtures with varying initial ratios of NPs, with any excess particles remaining unincorporated in solution.” Does this mean that ZnS-like assemblies always form regardless of the NP ratios? Or is there a certain threshold ratio beyond which ZnS-like assemblies can no longer be formed? I have similar questions for other lattices, such as NaCl-like (Fig. S8) and diamond (Fig. S9) lattices.\n3. In the introduction, the authors mentioned that “Only report of a diamond superlattice, assembled without valence, exhibited broad X-ray diffraction peaks, signaling poor structural quality.” In this manuscript, the authors emphasize their SAXS signals are highly strong. Are the authors aiming to highlight that stronger SAXS signals can be obtained through an appropriate choice of NP systems or measurement techniques (i.e., from the perspective of the methodology)? Or are they emphasizing that their approach enables the formation of lattices with much longer-ranged order than the previously reported assemblies (i.e., from the perspective of sample quality)? In addition, do the authors have an estimate of the size of their lattices?\n4. The meaning of ΔQ in Fig. 1a is somewhat unclear. Could the authors clarify the meaning of ΔQ in more detail?\n5. In Fig. 2, the authors mention the zinc-blende and rock salt structures transform into diamond and simple cubic lattices, respectively, when the pH is changed from 3 to 4. Is this transformation reversible? For example, do the diamond and simple cubic lattices transform back into zinc-blende and rock salt structures, respectively, by lowering the pH from 4 to 3?\n6. The current SAXS measurement is performed in situ, meaning that the assemblies are formed in the solution state. What changes in the lattice structures would the authors expect if the assemblies are fully dried? Would the lattices be expected to remain unchanged? Alternatively, could the drying process induce lattice deformation, leading to changes in the lattice constant or transformations in the lattice type?\n7. Following question 6, could the authors provide some electron microscopy images of the lattices in the dried state? For example, the reference (Science 2006, 312, 420–424) cited by the authors in the manuscript also used similar “valence-free” NP systems and assembly strategies, and they provided several scanning electron microscopy images of the assembly structure. These are very helpful for readers to readily understand how the lattice structures appear.\n8. To me, it is somewhat difficult to intuitively understand why Fig. 3a and Fig. 3b show quite different γ values. According to the experimental description, only the terminal groups are interchanged, while the same PEG chain lengths (2k and 10k) are used. Therefore, I would have expected the γ values in Fig. 3a and Fig. 3b to be similar, but they are not. Could the authors clarify how simply changing the terminal groups leads to such a significant difference in the γ values?\nIn addition to the scientific questions and comments above, I also have some minor suggestions that could be addressed to further improve the clarity of the manuscript as below.\n1. The same symbol Q is used for two different meanings in the manuscript: (1) surface charge and (2) q-value in SAXS data. This dual usage could be confusing, and I suggest distinguishing the notations more clearly.\n2. Both terminologies “LRO” and “SRO” are used in Fig. 1c, but their definitions appear only in the Methods section, which is the end of the manuscript. For clarity, it would be helpful to define these terms earlier.\n3. Please double-check the indexing of the Supplementary Figures in the manuscript, as several inconsistencies have been noted. For example, the sentence “Notably, this ZnS-like structure can also form at different mixing ratios (1:1, 2:1, and 4:1), as shown in Fig. S6.” appears to reference Fig. S7 instead.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "In this work, the authors use AuNPs functionalized with PEG that have charged groups capped at the end, forming coronas that interact through electrostatic interactions. They use different lengths of PEG and vary AuNP core sizes to generate different interaction lengths. The size ratio between the effective interaction lengths yields a variety of phases, including zinc-blende, diamond, rock salt, KCl, CsCl, and simple cubic lattices, as validated by SAXS. The work is interesting, timely and well conducted. Overall, this is an exciting study about nanoparticle lattice control, and it is potentially suitable for publication in Nature Communications. However, I believe it requires some additional elements of discussion, clarification, and potentially analysis.\nMajor points:\n• SAXS analysis provides strong information about the lattice symmetry and unit cell, however, it would be good to have insight into the characteristic domain size of the resulting structures. This could potentially be addressed through additional SAXS analysis or/and with real-space characterization.\n• In Figure 3d, the SAXS profile is shown. The grey lines and black curve appear to display some features in between the (100) and (110) peaks. It would be good if these features can be commented on as they diverge from the modeled intensity profile.\n• While I believe this work is exciting, it is spiritually similar to the paper https://doi.org/10.1073/pnas.2302037120. This paper should be cited and discussed. Additionally, the first article cited 10.1126/science.121049 also reports an “open” crystal type of an NaCl crystal. Thus, the novelty claim that this work “introduces a new paradigm” should be toned down.\nMinor Comments:\n• The figures should clearly delineate between the experimental curves (in gray, I assume) and the experimental fit (the black curves). Right now, the figures only point out the blue curve as the model curve.\n• Figure 3c. I am unsure as to why the lattice spacing selected includes more than is shown for the simple cubic lattice. Additionally, since the lattice spacing is much higher, is it possible that the SAXS pattern is that of the simple cubic, just shifted to the left?\n• While it’s possible for the reader to calculate the nearest neighbor distances, it would be nice to have them calculated by the authors and displayed/compared against expected values, given the length of the PEG and core AuNP size."}, "author_response": "Response: We appreciate the Reviewer's recognition of our work.\nSpecific comments:\n1. The Methods section on pages 11-12 must replace HCl concentration estimates with\ndirect pH measurements using microelectrodes, include detailed PEG grafting density\nquantification via NMR or XPS, and provide reproducibility data across multiple sample\npreparations.\nResponse: We appreciate the reviewer's suggestions. Below, we provide a detailed\nresponse to the questions.\npH measurements: Given the small sample volumes used in this study (typically ~200\nμL), it is technically challenging to obtain accurate microelectrode measurements.\nNevertheless, we verified pH values by titrating a few representative suspensions with\nbulk addition of HCl and confirmed that the values are consistent within ±0.2 pH units. In\nthe revised manuscript, we have clarified this approach.\nAction taken: The Methods section has been revised to state that “pH values were verified by\nbulk titration of representative samples, showing consistency within ±0.2 pH units, corresponding\nto the final pH values calculated from the amount of HCl added to the solutions.”\nPEG grafting density: In our previous publications (cited), we have already reported PEG\ngrafting densities. For clarity, we now provide various values in the revised Supporting\nInformation. The grafting density was quantified using thermogravimetric analysis (TGA),\nwhich we find to be a robust and reliable method. Importantly, we observe that the grafting\ndensity remains consistent regardless of polymer charge (neutral or charged),\nunderscoring the reproducibility and robustness of our surface functionalization approach.\ndetermined by thermo gravimetric analysis (TGA), are provided in the Supporting Information and\nare reproducible, independent of terminal group charge.” Additional details are provided in the SI.\nReproducibility across preparations: To directly address reproducibility, we now provide\nadditional data in the Supporting Information. These data show that the most stable\nstructure (Zinc-Blende) is consistently reproduced across three independent facilities: the\nAdvanced Photon Source (Argonne National Laboratory), the National Synchrotron Light\nSource (Brookhaven National Laboratory), and an in-house SAXS instrument at the\nCenter for Functional Nanomaterials (CFN) at BNL. The reproducibility of the SAXS\ndiffraction patterns across different samples, instruments, and measurement times\ndemonstrates the robustness of our sample preparation and experimental methods.\nAction taken: We have added Figure R1 to the revised Supporting Information with the following\nstatement: “Experimental results are highly reproducible, as demonstrated by SAXS diffraction\npatterns collected at three independent facilities under identical sample conditions.”\nFigure R1. Small-angle X-ray scattering (SAXS) profiles [I(Q) vs Q] of binary mixtures of NH -\nPEG10k-Au10 and COOH-PEG2k-Au5 (1:4) at pH 3, collected from three independent sources,\n₂\ntimes, and sample batches. Intensities are normalized to a common maximum scale for\ncomparison.\n2. The manuscript requires the addition of TEM/SEM images in Figure 2 and Figure 3\nfor direct structural visualization and selected area electron diffraction patterns to\nconfirm SAXS assignments.\nwould be highly valuable. However, obtaining TEM images of these superstructures\ndirectly from dried solution samples remains a significant challenge in this area,\nespecially when dealing with water-soluble long-chain polymers such as PEG. Reported\nimages from other groups typically require specialized methods such as resin\nembedding of the assemblies, followed by microtoming of resin–particle composites into\n composites into\nultrathin sections suitable for TEM. Such methods are not currently available to us.\nWe have nevertheless made considerable efforts to obtain real-space images. We\nattempted multiple approaches, including cryogenic preparation, but these were\nunsuccessful because the very conditions that stabilize the superstructures in solution\nare disrupted during the blotting and drying processes. Similar problems arise with\nclassical drying-mediated preparation. Moreover, the long-chain polymer melts under the\nelectron beam, resulting in the destabilization of the assemblies. Over time, it has become\nclear that obtaining reliable TEM/SEM images of these systems is technically challenging\nand would require extensive method development. We are currently collaborating with\ninstrument specialists to develop more suitable preparation strategies, although this will\nlikely require additional time and effort.\nImportantly, these challenges are not unique to our work; experts in polymer-stabilized\nassembly systems broadly acknowledge that obtaining reliable TEM images under such\nconditions is a significant challenge. This is because aqueous electrolytic solutions\nundergo substantial changes (pH, salt concentration, dielectric constant) upon drying,\nwhich alter or deteriorate the assemblies.\nFinally, we emphasize that the most reliable and unbiased means of determining the\nstructures in our study is through X-ray diffraction methods such as SAXS, which probe\nthe assemblies in their native solution environment without perturbing them. For this\nreason, we rely on SAXS data as the primary and most trustworthy structural\ncharacterization tool in this work.\n3. Quantitative comparison between predicted phase boundaries in Figure 4b and\nexperimental observations in Figure 4a is necessary, with explicit explanation of\ndiscrepancies. The \"particle fractionalization\" mechanism in Figure 3c requires detailed\nexplanation.\nResponse: We appreciate the reviewer's attention to these issues. Figure 4a\nrepresents our best estimate of an experimental phase diagram, constructed from\nlimited but reliable data, while Figure 4b is derived from theoretical calculations using\nreasonable assumptions designed to capture the essential physics. Although the\ndiagrams do not match exactly, the correspondence between the major features of the\n3\ntwo is of high quality. We view this level of agreement as significant, given the complexity\nof the systems and the approximations required. Both diagrams should therefore be\nunderstood as broad guides to crystal growth as a function of polymer molecular weight\nand particle asymmetry, and as a framework for future refinements.\nRegarding the “particle fractionalization” mechanism in Figure 3c, this describes the\nredistribution of nanoparticles across multiple assemblies when the system lies near a\nphase boundary. Under such conditions (e.g., at specific polymer chain lengths or pH\nvalues), not all particles commit to a single ordered structure; instead, fractions reorganize\ninto competing phases, thereby lowering the system's free energy. This mechanism\nprovides a natural explanation for the coexistence and competition of phases observed\nexperimentally and is consistent with the theoretical framework we employ to interpret the\nSAXS data.\nAction taken: We have expanded the Supporting Information to include a detailed explanation\nof this phenomenon in the section “Particle fractionalization”. Also following sentence is added to\nthe main manuscript: “We surmise that the particle fractionalization is likely due to redistribution\nof nanoparticles across multiple assemblies when the system lies near a phase boundary.”\n4. A comprehensive table comparing this method with DNA-origami, patchy particles,\nand other assembly approaches must include assembly time, yield, structural quality,\nand cost metrics. The manuscript needs specific discussion of scalability limitations,\nlong-term stability testing over weeks to months, and thermal compatibility with device\nfabrication processes to establish practical viability beyond proof-of-concept\ndemonstrations.\nResponse: A direct quantitative comparison with DNA-origami and patchy particle\nassembly methods is not feasible, as we lack accessible information on their yield,\nstructural quality, or cost metrics. However, in the revised Supporting Information, we\nprovide these details for our method, allowing readers to evaluate it within a broader\ncontext.\nRegarding scalability, we note that this question falls beyond the scope of the current\nstudy. Currently, we can only speculate on how the method might be scaled up for\nindustrial applications, but we acknowledge that further optimization will be necessary.\nFor stability, we have monitored selected samples for up to six months and found them\nto remain structurally intact, indicating promising long-term stability. In terms of thermal\ncompatibility, the assemblies display stability over the range of 20–80 °C, consistent with\nthe temperature dependence of PEG–AuNP assemblies. We have clarified these points\nin the revised Supporting Information to make the practical implications of the method\nmore explicit.\nsummarize the above points. Additionally, we have added Fig. S27 to the SI, which shows the\nthermal stability of our structure over a range of temperatures.\n5. Page 1: \"Only report of a diamond superlattice\" - grammatically incorrect, should be\n\"Only one report\"\nResponse: We have fixed this grammatical issue.\nReviewer #2 (Remarks to the Author):\nThe authors report a strategy to create superlattices using valence-free nanoparticles\n(NPs). They thoroughly studied the diverse lattices formed by charged polymer-grafted\nNPs and demonstrated the effect of experimental parameters (e.g., the ratio of\nhydrodynamic sizes between two NPs) on the assembly structures. Specifically, they\nachieved a wide range of lattices, such as rock salt, CsCl, zinc-blende, diamond, and\nthe exceptionally rare simple cubic phase, with tunable lattice constants. The lattice\nformation was confirmed through comprehensive in situ SAXS measurements and\nperforming MD simulations. Although many previous works have demonstrated lattice\nformation in binary microparticle systems via controlled electrostatic interparticle\ninteractions, relatively few studies have achieved this at the NP level. Therefore, I\nrecommend the publication of this work in Nature Communications after revisions and\naddressing the following questions/comments to clarify the manuscript.\nResponse: We thank the Reviewer for their thoughtful appreciation of our work in the\ncontext of nanoparticle self-assembly. We are also grateful for the constructive\ncomments and suggestions, which have helped us improve and clarify the manuscript.\n1. The authors emphasized the importance of using “valence-free” NPs for assembly by\nstating that “The fabrication of diamond-like superstructures has traditionally relied on\npatchy or anisotropic NPs, with directional interaction sites.” and “Only report of a\ndiamond superlattice, assembled without valence, exhibited broad X-ray diffraction\npeaks, signaling poor structural quality.” However, it is unclear in the manuscript why\nusing “valence-free” NPs is more important than utilizing “valence-like” bonding. Could\nthe authors provide a more detailed explanation of the rationale behind employing\n“valence-free” NPs for assembly?\nResponse: We thank the reviewer for this insightful question. The central message of\nour manuscript is not to position “valence-free” approaches as superior to “valence-like”\nbonding, but rather to demonstrate a robust and relatively simple alternative for\nassembling nanoparticles into well-ordered superlattices. Traditional strategies such as\n5\npatchy or anisotropic nanoparticles, rely on carefully engineered directional interactions\nand scaffolding, which can be powerful but also add significant complexity to the design\nprocess.\nIn contrast, our ionic-like assembly method, which utilizes valence-free nanoparticles,\nachieves diamond and other cubic superstructures without requiring directional\nspecificity. This demonstrates that directional bonding is not a strict requirement for\nforming these highly ordered lattices. Instead, the emergent interactions in our system\nspontaneously drive the organization of nanoparticles into multiple cubic structures, in a\nreproducible manner.\nWe believe this finding expands the design space for nanoparticle assembly by showing\nthat high-quality cubic lattices can be realized without the constraints of directional\ninteractions. Rather than replacing valence-like strategies, our approach complements\nthem by providing a simpler and more flexible pathway to complex structures.\n2. The authors mentioned that “Although ZnS-like assemblies consistently exhibit 1:1\nstoichiometry, they can arise from mixtures with varying initial ratios of NPs, with any\nexcess particles remaining unincorporated in solution.” Does this mean that ZnS-like\nassemblies always form regardless of the NP ratios? Or is there a certain threshold ratio\nbeyond which ZnS-like assemblies can no longer be formed? I have similar questions\nfor other lattices, such as NaCl-like (Fig. S8) and diamond (Fig. S9) lattices.\nResponse: We appreciate the reviewer’s request for clarification. Our results show that\nZnS-like assemblies are remarkably robust and can form over a broad range of initial\nNP ratios (typically 1:1 to 1:4, where the smaller NP core size is in excess). However,\nwe do not claim that ZnS structures arise homogeneously from any starting ratio.\nInstead, within this range, the assemblies consistently achieve an overall 1:1\nstoichiometry, with excess particles remaining unincorporated in solution. Beyond this\nrange (e.g., with ratios exceeding ~1:4), other types of assemblies may appear (but not\nbe observed), and the system no longer yields a purely ZnS-like phase.\nFor other lattices such as NaCl-like (Fig. S8) and diamond-like (Fig. S9), we observe\nsimilar robustness: the structures can form across a range of mixing ratios, with excess\nparticles excluded rather than incorporated into the lattice. To date, we have not observed\na sharp threshold ratio that prevents formation of these structures, but we recognize that\nexcessive imbalance may shift the system toward competing assemblies. We have\nclarified these points in the revised manuscript.\nAction taken: The following statement is added to the main manuscript: “This behavior\nsuggests that the assembly kinetics are not strongly dependent on the initial particle ratios, with\n6\nthe system robustly relaxing toward its thermodynamically favored stoichiometry and leaving\nexcess particles unincorporated.”\n3. In the introduction, the authors mentioned that “Only report of a diamond superlattice,\nassembled without valence, exhibited broad X-ray diffraction peaks, signaling poor\nstructural quality.” In this manuscript, the authors emphasize their SAXS signals are\nhighly strong. Are the authors aiming to highlight that stronger SAXS signals can be\nobtained through an appropriate choice of NP systems or measurement techniques (i.e.,\nfrom the perspective of the methodology)? Or are they emphasizing that their approach\nenables the formation of lattices with much longer-ranged order than the previously\nreported assemblies (i.e., from the perspective of sample quality)? In addition, do the\nauthors have an estimate of the size of their lattices?\nResponse: We appreciate the reviewer's point. The key distinction between our results\nand the previously reported “valence-free” diamond superlattice is not the absolute peak\nintensity, but rather the resolution and multiple diffraction peaks. In the earlier report,\nthe broad peaks merged several reflections, making it difficult to unambiguously assign\nlattice symmetry. In contrast, our SAXS data show highly resolved diffraction peaks; for\nexample, we resolve nine distinct Bragg peaks for the diamond-like lattice, compared to\nonly two broad peaks in the cited work. This level of resolution enables a clear structural\nassignment and provides strong evidence of long-range crystalline order.\nThe improvement arises from both the assembly approach, which yields higher-quality\nlattices, and the use of synchrotron X-ray radiation, which offers superior resolution\ncompared to in-house sources. Together, these factors allow us to demonstrate that\nvalence-free assembly can indeed produce diamond-like superlattices with long-range\norder and well-defined symmetry.\nBased on peak widths, we estimate the crystal domain sizes to be on the order of 1 micron\nor larger, since for many of these crystals, the peak width is resolution-limited. Since this\nwas a request from another reviewer, we have added the following clarifying statement\nto the main manuscript and added a Table in the SI (Table S8) for detailed crystal size\ndetermination.\nAction taken: The following statement is added to the main manuscript: “FWHM values were\nused to estimate the sizes of the crystalline domains, which are on the order of 1 µm or larger\nfor the highest quality crystals, while lower quality domains are typically in the submicrometer\nrange.”\n4. The meaning of ΔQ in Fig. 1a is somewhat unclear. Could the authors clarify the\nmeaning of ΔQ in more detail?\nis initially approximated as a point charge, but in reality, the charge is distributed across\nthe polymer shell. At large interparticle separations, Gauss’s theorem ensures that\ntreating the charge as either a point or a uniform surface distribution yields the same\ncontribution to the free energy. However, at short separations, the uniform\napproximation neglects the fact that charges are discrete and can interact through\nexplicit electrostatic correlations. To account for these effects, we introduce the\nparameters ΔqA and ΔqB (note: as per the reviewer's request, the notation ΔQ is\nchanged to Δq), which represent fractional charges distributed over spherical angles on\nthe nanoparticle surface. These terms capture local electrostatic interactions at the\nnearest-neighbor distance (rc), thereby incorporating correlation effects that are\nessential for understanding the dependence of the phase diagram on the size ratio γ.\nAction taken: We expanded the Fig. 1 caption to include the sentence: “∆qA and ∆qB are the\namount of charge that needs to be considered explicitly to account for electrostatic correlations,\nsee SI for a more detailed derivation.” In the main text, immediately after introducing the\nparameter α, we added: “where ∆qA and ∆qB denote the amount of surface charge specified\nwithin a defined solid angle, where the charge needs to be considered explicitly to account for\ncorrelations,” and we direct readers to the Supporting Information for the detailed derivation of\nΔq.\n5. In Fig. 2, the authors mention the zinc-blende and rock salt structures transform into\ndiamond and simple cubic lattices, respectively, when the pH is changed from 3 to 4. Is\nthis transformation reversible? For example, do the diamond and simple cubic lattices\ntransform back into zinc-blende and rock salt structures, respectively, by lowering the\npH from 4 to 3?\nResponse: We thank the reviewer for this question. In principle, the transformation\nbetween zinc-blende/rock-salt (at pH ~3) and diamond/simple cubic (at pH ~4) is\nreversible, as it is driven by changes in protonation/deprotonation state of the terminal\ngroups and thus by electrostatic interactions. However, in practice, achieving full\nreversibility is experimentally challenging. Lowering the pH is straightforward by the\naddition of HCl, but raising the pH afterward requires the addition of alkaline agents\nsuch as NaOH, which produces NaCl as a byproduct. The presence of NaCl disrupts\nthe delicate ionic interactions responsible for lattice stability, preventing a clean reversal.\nTherefore, while the assemblies are highly responsive to pH changes and the mechanism\nis theoretically reversible, in our current system, the chemical constraints of adjusting pH\n(particularly the accumulation of salts) limit our ability to demonstrate reversible cycles.\nDeveloping salt-free strategies to tune pH would be a promising avenue for future work.\n8\n6. The current SAXS measurement is performed in situ, meaning that the assemblies\nare formed in the solution state. What changes in the lattice structures would the authors\nexpect if the assemblies are fully dried? Would the lattices be expected to remain\nunchanged? Alternatively, could the drying process induce lattice deformation, leading\nto changes in the lattice constant or transformations in the lattice type?\nResponse: We thank the reviewer for this important question. Most nanoparticle\ncrystallization studies, including ours, are performed in either aqueous suspensions or\norganic solvents, and one of the long-standing challenges in the field is how to extract\nintact crystals from such suspensions. Drying is not a trivial process because it\ninherently alters the conditions that stabilize the assemblies, such as pH, ionic strength,\nand hydration. In our case, the assemblies rely on soluble polymers (PEG) in aqueous\nenvironments, similar to DNA-programmed assemblies, where drying can induce\nconformational changes of the polymer or DNA strands.\nAs a result, drying may indeed alter lattice parameters and introduce defects. However,\na complete transformation from one lattice to another is unlikely. Instead, the more\nprobable outcomes are deformation of the lattice constants and the introduction of\nstructural defects, which can disrupt long-range order. For this reason, in-solution SAXS\nprovides the most reliable structural information for these assemblies in their native\nstabilized state.\n7. Following question 6, could the authors provide some electron microscopy images of\nthe lattices in the dried state? For example, the reference (Science 2006, 312, 420–424)\ncited by the authors in the manuscript also used similar “valence-free” NP systems and\nassembly strategies, and they provided several scanning electron microscopy images\nof the assembly structure. These are very helpful for readers to readily understand how\nthe lattice structures appear.\nResponse: We thank the reviewer for this suggestion. This point overlaps with our\ndetailed response to Reviewer 1, Question 2, where we extensively discuss the\ntechnical challenges associated with obtaining reliable TEM/SEM images for polymer-\nstabilized nanoparticle superlattices. In brief, unlike the system reported in Science\n2006, 312, 420–424, which involved solvent-evaporated assemblies stabilized by short-\nchain ligands and thus inherently stable in the dry state, our superlattices are stabilized\nin aqueous solution by pH-dependent electrostatic interactions and polymer coronas.\nDrying or electron-beam exposure alters these stabilization conditions, often leading to\nlattice deformation or melting of the polymer matrix. Consequently, in-situ SAXS\nprovides the most reliable and unbiased structural determination under native solution\nconditions.\n9\n8. To me, it is somewhat difficult to intuitively understand why Fig. 3a and Fig. 3b show\nquite different γ values. According to the experimental description, only the terminal\ngroups are interchanged, while the same PEG chain lengths (2k and 10k) are used.\nTherefore, I would have expected the γ values in Fig. 3a and Fig. 3b to be similar, but\nthey are not. Could the authors clarify how simply changing the terminal groups leads\nto such a significant difference in the γ values?\nResponse: We thank the Reviewer for this valid observation, which we partially\naddressed in the original manuscript. In the revised Methods section, we now\nemphasize that the hydrodynamic size of grafted AuNPs is strongly influenced by the\nnature of the terminal charge group. Specifically, for COOH-terminated AuNPs, the PEG\ncorona tends to fold back at neutral pH, leading to a more compact effective size (the\nvalues are listed in Table S1). In contrast, NH2-terminated AuNPs carry higher charge\ndensity and are surrounded by a hydration shell, which increases their apparent\nhydrodynamic diameter. This difference in effective size explains the observed variation\nin gamma values between Figures 3a and 3b. We have also clarified this point in the\nrevised text and note that a detailed discussion can be found in our previous publication\n(https://doi.org/10.1016/j.jcis.2023.07.095)\nAction taken: We have revised the Figure caption for 3(b) to include “The observed fluctuations\nin γ arise from changes in the hydrodynamic diameter (DH) of the NH2-terminated\nnanoparticles” and added text below to the methods section. “The DH values show a systematic\nincrease for –NH –terminated particles compared to –COOH–terminated ones, particularly at\nhigher PEG molecular weights (e.g., 10 kDa), reflecting the more hydrated and expanded\n₂\ncorona of -NH -terminated PEG relative to the more compact conformation of COOH–\nterminated PEG.”\n₂\nIn addition to the scientific questions and comments above, I also have some minor\nsuggestions that could be addressed to further improve the clarity of the manuscript as\nbelow.\n1. The same symbol Q is used for two different meanings in the manuscript: (1) surface\ncharge and (2) q-value in SAXS data. This dual usage could be confusing, and I suggest\ndistinguishing the notations more clearly.\nResponse: Thank you for the suggestion. We have changed the notation of the NP\ncharge from “Q” to “q” to avoid confusion.\n2. Both terminologies “LRO” and “SRO” are used in Fig. 1c, but their definitions appear\nonly in the Methods section, which is the end of the manuscript. For clarity, it would be\nhelpful to define these terms earlier.\nwas originally defined in the Methods section, following the journal’s format. In the\nrevised manuscript, we have added a clarifying sentence in the main text that introduces\nthese terms and refers the reader to the Methods section for their detailed definitions.\nAction Taken: The Following statement was added to the main text to guide the readers:\n“Detailed definitions of these ordering regimes are provided in the Methods section.”\n3. Please double-check the indexing of the Supplementary Figures in the manuscript,\nas several inconsistencies have been noted. For example, the sentence “Notably, this\nZnS-like structure can also form at different mixing ratios (1:1, 2:1, and 4:1), as shown\nin Fig. S6.” appears to reference Fig. S7 instead.\nResponse: We thank the reviewer for pointing out the referencing. We have fixed this\nissue in the revised manuscript.\nReviewer #3 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports.\nThis is part of the Nature Communications initiative to facilitate training in peer review\nand to provide appropriate recognition for Early Career Researchers who co-review\nmanuscripts.\nResponse: Thank you for your positive and constructive comments.\nReviewer #4 (Remarks to the Author):\nIn this work, the authors use AuNPs functionalized with PEG that have charged groups\ncapped at the end, forming coronas that interact through electrostatic interactions. They\nuse different lengths of PEG and vary AuNP core sizes to generate different interaction\nlengths. The size ratio between the effective interaction lengths yields a variety of\nphases, including zinc-blende, diamond, rock salt, KCl, CsCl, and simple cubic lattices,\nas validated by SAXS. The work is interesting, timely and well conducted. Overall, this\nis an exciting study about nanoparticle lattice control, and it is potentially suitable for\npublication in Nature Communications. However, I believe it requires some additional\nelements of discussion, clarification, and potentially analysis.\ngreatly helped us improve the manuscript.\nMajor points:\n• SAXS analysis provides strong information about the lattice symmetry and unit cell,\nhowever, it would be good to have insight into the characteristic domain size of the\nresulting structures. This could potentially be addressed through additional SAXS\nanalysis or/and with real-space characterization.\nResponse: We appreciate the Reviewer's valuable suggestion. In response, we have\nadded a table in the Supporting Information that summarizes the full width at half\nmaximum (FWHM) values of the diffraction peaks for the structures presented in the\nmain text, along with the corresponding estimated crystal sizes. As shown in Table S8,\nthe estimated crystalline domain sizes, based on the linewidths of Bragg reflections, are\ntypically on the order of slightly larger than 1 µm for the highest-quality crystals. We\nnote, however, that these estimates represent lower bounds, as some diffraction peaks\nare resolution-limited, indicating that certain crystallites may be significantly larger than\nour reported values. As far as real-space visualization concerned, please refer to our\ndetailed response Reviewer 1 Question 2.\nAction taken: We have added the following statement to the manuscript: “For completeness,\nthe Supporting Information provides the detailed lattice parameters, nearest-neighbor distances\nin Figs. S7–S20, and peak line widths (FWHM) in Table S8. The FWHM values were used to\nestimate the sizes of the crystalline domains, which are on the order of 1 μm or larger for the\nhighest-quality crystals, while lower-quality domains are typically in the submicrometer range.”\nand estimated number of unit cells derived from the SAXS analysis.\n• In Figure 3d, the SAXS profile is shown. The grey lines and black curve appear to\ndisplay some features in between the (100) and (110) peaks. It would be good if these\nfeatures can be commented on as they diverge from the modeled intensity profile.\nResponse: We thank the reviewer for this observation. In the main text, we in fact\ncomment on these weak peaks: “Small satellite peaks near the prominent (110)\nreflection indicate systematic defects, as discussed in detail in the SI”. The additional\nfeatures near the (110) peak correspond to satellite peaks that arise from lattice\nelongation or substitution effects, as discussed in detail in the Supporting Information\n(Figure S23, Lattice Distortion section). We have also included simulations of lattice\ndistortion in the SI.\nAction taken: A clarifying sentence has now been added to the revised manuscript to address\nthis point explicitly. “Small satellite peaks near the prominent (110) reflection indicate systematic\ndefects or lattice elongation as discussed in detail in the SI.”\n• While I believe this work is exciting, it is spiritually similar to the paper\nhttps://doi.org/10c.1073/pnas.2302037120. This paper should be cited and discussed.\nAdditionally, the first article cited 10.1126/science.121049 also reports an “open” crystal\ntype of an NaCl crystal. Thus, the novelty claim that this work “introduces a new\nparadigm” should be toned down.\nResponse: We thank the Reviewer for bringing this reference to our attention. We agree\nthat there are similarities between our approach and the cited work. Particularly relevant\nis the modulation of repulsion, which stabilizes open lattices, as observed in our\nexperiments as well. There are, however, important distinctions: in our system, a\ndiamond structure is obtained even in the absence of repulsion, whereas in the cited\nwork, such a phase is only predicted at a very narrow range of repulsion parameters,\nmaking it challenging to realize experimentally, and indeed resulting in a disordered\ndiamond lattice. In our case, repulsion broadens the stability window of an already stable\ndiamond structure. We are now mentioning the cited work in our introduction.\nWe also note that NaCl is not strictly an “open” lattice, whereas the simple cubic and\ndiamond are. In the revised manuscript, we have added a discussion that provides a\nproper context for the novelty of our results relative to this work. Importantly, we are not\nclaiming a paradigm shift in the assembly of NP superlattices; rather, we present a\nrobust and straightforward design strategy, accompanied by clear guidelines, that we\n13\nbelieve will enable the broader community to reproducibly explore and extend these\nsuperstructures.\nAction taken: The Following statement has been added to the Introduction with citing the\nReviewer’s mentioned work: “Low-coordination cubic lattices have also been realized through\nDNA-programmed NPs.” Additionally, we have toned down the statement “introduces a new\nparadigm” to “a broadly applicable framework.”\nMinor Comments:\n• The figures should clearly delineate between the experimental curves (in gray, I\nassume) and the experimental fit (the black curves). Right now, the figures only point\nout the blue curve as the model curve.\nResponse: Thank you for bringing this to our attention. In the revised manuscript figure\ncaptions, we have clarified that the solid black lines passing through the experimental\ndata points represent structure-factor-Lorentzian fits to the experimental SAXS profiles.\nThe experimental data are shown as markers, while the fitted curves are overlaid for\ndirect comparison.\nAction taken: Figure captions of Fig. 2 and 3 revised to include following clarification “The solid\nline overlaid on the experimental data points (open circles) represents the fitted structure factor\ncorresponding to each lattice.” and “Experimental data are shown as open circles; structure-\nfactor-Lorentzian fits are drawn as solid black lines through the data; and modeled intensity\nprofiles from structural models are plotted below as blue solid lines with individual peak\ncontributions highlighted in color.”\n• Figure 3c. I am unsure as to why the lattice spacing selected includes more than is\nshown for the simple cubic lattice. Additionally, since the lattice spacing is much higher,\nis it possible that the SAXS pattern is that of the simple cubic, just shifted to the left?\nResponse: We thank the Reviewer for raising this point. The lattice shown in Figure 3c\ncorresponds to a KCl-type structure. The characteristic (111) pre-peak in KCl arises\nfrom the slight electron density contrast between K+ and Cl−, which are otherwise\nelectronically similar. In our system, both components are AuNPs of nominally the same\nsize (~10 nm), and we rationalize the presence of this feature as a consequence of\nfractionalization, as discussed in detail in the main manuscript and SI.\nWhen the electron densities become effectively identical, the (111) pre-peak is\nsuppressed, and the (200) reflection of the KCl lattice is relabeled as the (100) reflection\nof a simple cubic lattice. Accordingly, the simple cubic phases reported here can be\nconsidered limiting cases of NaCl/KCl-type lattices with negligible electron density\ncontrast. In this situation, the lattice constant of the simple cubic structure is\n14\napproximately half that of the corresponding NaCl/KCl lattice. This clarification has been\nadded to the Supporting Information section “Derived simple cubic structure.”\nAction Taken: Clarifications have been added to the Supporting Information section “Derived\nsimple cubic structure.”\n• While it’s possible for the reader to calculate the nearest neighbor distances, it would\nbe nice to have them calculated by the authors and displayed/compared against\nexpected values, given the length of the PEG and core AuNP size.\nResponse: The nearest-neighbor (NN) distances for both A–A and A–B particle pairs\nare summarized in the Supporting Information (Figures S7–S20) in each figure. In\ncomparison with the expected values, the extracted NN distances are consistent with\nthe average hydrodynamic sizes of the respective nanoparticles. The observed\nvariations (±5 nm) likely arise from differences in polymer conformation, lattice packing,\nthe presence of trapped counterions, and/or excess water within the assemblies.\nAction Taken: We have added the following statement to the manuscript to guide readers: “For\ncompleteness, the Supporting Information provides the detailed lattice parameters, nearest-\nneighbor distances in Figs. S7–S20, and peak line widths (FWHM) in Table S8. The FWHM values\nwere used to estimate the sizes of the crystalline domains, which are on the order of 1 μm or\nlarger for the highest-quality crystals, while lower-quality domains are typically in the\nsubmicrometer range.”\n15"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have made proper modifications, and I believe that the manuscript can be accepted for publication in Nature Communication in the current form.", "reviewer_2": "The authors have addressed all my comments and I now recommend its publication on Nature Communications.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "My comments were addressed and I recommend the publication."}, "author_response": "Reply: We thank the reviewer for the positive feedback and for acknowledging the quality of our revision. We appreciate your recommendation for publication.\nReply: We are grateful to the reviewer for the encouraging assessment and for recommending our work for publication.\nReply: We thank the reviewer for the supportive evaluation and for confirming that no further revisions are required. We greatly appreciate your recommendation for acceptance.\nReply: We thank the reviewer for recommending our work for publication."}]

Decision letter