Achiral and chiral ligands synergistically harness chiral self-assembly of inorganics

Abstract

Chiral structures and functions are essential natural components in biominerals and biological crystals. Chiral molecules direct inorganics through chiral growth of facets or screw dislocation of crystal clusters. As chirality promoters, they initiate an asymmetric hierarchical self-assembly in a quasi-thermodynamic steady state. However, achieving chiral assembly requires a delicate balance between intricate interactions. This complexity causes the roles of achiral-chiral and inorganic components in crystallization to remain ambiguous. Here, we elucidate a definitive mechanism using an achiral-chiral ligand strategy to assemble inorganics into hierarchical, self-organized superstructures. Achiral ligands cluster inorganic building blocks, while chiral ligands impart chiral rotation. Achiral and chiral ligands can flexibly modulate the chirality of superstructures by fully using their competition in coordination chemistry. This dual-ligand strategy offers a versatile framework for engineering chiroptical nanomaterials tailored to optical devices and metamaterials with optical activities across a broad wavelength range, with applications in imaging, detection, catalysis, and sensing.

Ligand chemistry tactfully steers chirality in inorganic self-assembly.

INTRODUCTION

Chirality is the property of a molecule or object that cannot superimpose on its mirror images through rotation or translation. While chirality is prominent in organic chemistry and biology (1–3), synthesizing and assembling chiral inorganics have attracted extensive interest in engineering materials (4–6). Typically, practical applications of chiral molecules are limited by their weak optical activity and narrow circular dichroism (CD) absorption bands (7, 8). However, inorganic crystals can achieve more flexible chirality through micro-/nanoscale modification or hierarchical self-assembly controlled by chiral additives (9–14). The chiroptical activity of these additives [circularly polarized light (7, 15, 16), chiral templates (17–20), and enantiomers (21, 22)] can be transmitted, amplified, and fixed within the inorganic superstructures. Specifically, chiral molecule–guided self-assembly has shown great promise for controlling achiral inorganics (23–25). Chiral inorganic superstructures, featuring enhanced and multiband CD absorption, demonstrate emerging applications in chiral resolution (26, 27), chiral catalysis (28–30), bioapplications (31–33), and chiral optics (34–36).

Activating ordered chiral stacking in inorganics is challenging (37, 38), as the self-assembly process requires a delicate balance between multiple interacting forces (39–43). This process involves molecular chirality transfer and superstructure self-organization (44, 45), functions that are interconnected yet often exhibit contradictory behaviors. Superstructural self-organization is driven by molecular coordination chemistry (3, 46–48). Chirality transfer is contingent on molecular stereochemistry (47, 49), constrained by molecular coordination. This complex interplay endows achiral molecules with the ability to regulate the chirality of inorganics through coordination competition (50). However, the practical implementation of this theoretical potential is hindered by the complexity of the required experimental procedures and the necessity for the precise selection of chiral molecules (45, 51, 52). These factors substantially restrict the ability to control and predict the outcomes of chirality induction, making it difficult to harness this process reliably. Consequently, despite these theoretical possibilities, the collective influence of achiral and chiral molecules on the chiral self-organization of inorganics during chemical crystallization remains unclear.

Here, we propose a dual-ligand approach for the chiral self-assembly of inorganic compounds. This strategy uses achiral carboxylic acids to drive the assembly process and chiral amino acids to act as chirality carriers, specifically targeting the synthesis of inorganic Cu(OH)₂. We used in situ CD spectroscopy and electron microscopy to monitor the evolution of chirality and morphology. The interactions and coordination chemistry of the dual ligands on the Cu(OH)₂ facets are analyzed using a combination of density functional theory (DFT) simulations and Fourier transform infrared (FTIR) spectroscopy. Molecular dynamics simulations and coarse-grained modeling were used to investigate the hierarchical assembly and chiral mechanism. These experiments and simulations provide insight into how molecular interactions and arrangements contribute to the chiral properties of the assembled structures.

RESULTS AND DISCUSSION

Chiral self-assembly driven by dual ligands

Cu-based complexes were constructed using achiral and chiral dual ligands under highly alkaline conditions (pH 12.2). We demonstrated controlled self-assembly by tuning the types and combinations of ligands. We first showed distinct Cu(OH)₂ superstructures trimmed without ligands, with citric acid, with chiral valine ligands, and with achiral-chiral dual ligands (Fig. 1A and fig. S1). The nanobelts were obtained by mixing copper and hydroxide ions (Fig. 1A and fig. S1A). Citrate bundles the nanosheets to create conical citrate nanobundles (Fig. 1A and fig. S1B). Conversely, chiral valine yielded only short l-/d-valine nanosheets (Fig. 1A and fig. S1C). However, the combination of achiral citric acid and chiral valine resulted in the formation of chiral citrate-l-/d-valine nanobundles (termed l-/d-nanobundles; Fig. 1A and fig. S1D). These results indicate that achiral and chiral ligands play distinct but cooperative roles in directing the self-assembly of Cu(OH)₂ building blocks (Fig. 1B and fig. S2). We adjusted the pH value from 12.2 to 12.8, resulting in the formation of chiral nanobundles (fig. S3). A pH value of 12.2 was used in this work.

Fig. 1. Chiral Cu(OH)₂ nanobundle.

(A) TEM images of Cu(OH)₂ assemblies and TEM reconstruction images of nanobelt, nanobundle, nanosheet, and l-nanobundle. (B) Chiral mechanism diagram for nanobundles using achiral-chiral dual ligands. (C) SEM images of as-synthesized chiral nanobundle and their (D) CD spectra and ultraviolet-visible (UV-vis) absorption spectra. (E) SEM images of calcined chiral nanobundle and their (F) CD spectra and UV-vis absorption spectra. L₁ (L₂), S₁ (S₂), and S_1,cal (S_2,cal) indicate the peaks from ligand-induced chirality and inorganic structural chirality. a.u., arbitrary units.

To investigate the role of valine enantiomers in inducing assembly chirality, we first analyzed the structures of the valine-based assemblies. The l-/d-valine nanosheets and l-/d-nanobundles exhibited identical characteristic reflections of orthorhombic Cu(OH)₂ (Joint Committee on Powder Diffraction Standards cards, no. 13-0420), as confirmed by powder x-ray diffraction spectra (fig. S4). This suggests that the incorporation of valine enantiomers did not alter the basic crystal structure of Cu(OH)₂. l- and d-valine nanosheets exhibited sheet-like structures with noticeable folding and shading (fig. S5, A and B). The nanosheets were 65 nm long, 9 nm wide, and 2 nm thick (fig. S5, C and D). In addition, the nanosheets exhibited ultraviolet-visible (UV-vis) bands below 400 nm and around 680 nm (fig. S5E). These valine nanosheets show no discernible CD signal, presumably because of their disordered arrangement. When citric acid is introduced into the reaction solution containing valine nanosheets, the nanosheets failed to assemble into nanobundles (fig. S6), suggesting that the nanosheets were not the fundamental building blocks for nanobundle assemblies.

To validate the achiral-chiral dual-ligand strategy, we replaced chiral valine with racemic valine during the synthesis. The resulting conical citrate-racemic-valine nanobundles (denoted as racemic nanobundles) were achiral, exhibiting the same morphology and a lack of CD signal as the citrate nanobundles (fig. S7). In contrast, the transmission electron microscope (TEM) reconstruction images of l-nanobundle demonstrated chiral twisting (Fig. 1A and fig. S8), with a video illustrating the angle-dependent rotation (movie S1). The scanning electron microscope (SEM) images of l- and d-nanobundles displayed opposite rotations (Fig. 1C and fig. S9A) and antipodal CD signals (Fig. 1D). These phenomena are determined by the chirality of valine, confirming that the chirality of nanobundles originates from valine enantiomers (fig. S10). The nanosheets within chiral nanobundles had a length of 280 nm and a width of 10 nm (fig. S11). These dimensions were longer than the valine nanosheets with the same width. This phenomenon indicates that the ligand does not disrupt the crystallinity of Cu(OH)₂. The overall length and central width of chiral nanobundles were 325 and 70 nm, respectively (fig. S11).

The double CD bands of the chiral nanobundles at 295 nm (L₁) and 327 nm (L₂) were attributed to the ligand-to-metal charge transfer from the carboxylate group of valine to Cu²⁺ (Fig. 1D) (53). In addition, the shoulder peak at 575 nm (S₁) and the broad peak at 910 nm (S₂) can be identified in the integrated g factor curve and CD spectra from 200 to 1600 nm (fig. S12). These peaks originate from the chiral arrangement of the nanosheets within the nanobundles, which results in scattering-based optical activity at the chiral interface. When the chiral nanobundles were submerged in water, the scattering-based S₁ and S₂ signals exhibited varying levels of attenuation (figs. S13 and S14). Because the optical activity induced by the chiral shape was suppressed because of the reduced refractive index contrast between Cu(OH)₂ and water compared with Cu(OH)₂ and air (54).

After removing the surface valine via calcination, l-/d-valine nanosheets were transformed into cross-linked particles (fig. S15A). The large surface area of the nanosheets allowed the building nanoparticles within different nanosheets to merge into larger particles. Similarly, the nanosheets within the chiral nanobundles were transformed into interconnected particles (fig. S15, B and C). The calcined l- and d-typed nanobundles maintained opposite rotations (Fig. 1E and fig. S9B), whereas their crystal structure changed from orthorhombic Cu(OH)₂ to monoclinic CuO (figs. S4 and S15D). The CD bands below 450 nm, which originate from ligand-to-metal charge transfer, nearly vanished because of the decomposition of valine. In contrast, the scattering-based CD signal, S_1, and S₂ underwent redshifts of approximately 170 and 350 nm, respectively (Fig. 1F and fig. S12). This redshift is attributed to the increased particle size within the nanobundles and the change in the chiral interface from Cu(OH)₂/air to CuO/ air.

Thermogravimetric analysis revealed that chiral nanobundles and valine nanosheets contained estimated mass fraction of 17 and 19% of the surface organic ligands, respectively (fig. S16A). The absence of vibrations from the hydroxyl, amino, and carboxyl groups above 1300 cm⁻¹ in the FTIR spectra confirmed the complete decomposition of the organic ligands after calcination (fig. S16B). X-ray photoelectron spectroscopies revealed that the characteristic Cu 2P_3/2 absorption peak shifted from 934.8 eV for Cu(OH)₂ to 933.6 eV for CuO (fig. S17) (55). In addition, the O 1s peak shifted from 531.2 to 529.4 eV (fig. S18). The shifts in the absorption peaks of Cu 2P_3/2 and O 1s indicate the conversion of Cu(OH)₂ to CuO after calcination.

In situ chirality evolution

The in situ CD spectra of the chiral nanobundles enabled the visualization of chirality evolution with symmetric and continuous redshifts over time (Fig. 2, A and B). After 15 min, three emerging ligand-related CD signals appeared at 254 (L₀), 292 (L₁), and 330 nm (L₂). They eventually intensified and redshifted by 0, 2, and 6 nm, respectively, after 120 min. Correspondingly, the intensity of the in situ UV-vis absorption spectra gradually decreased between 200 and 270 nm (L₀) but increased between 270 and 500 nm (L₁ and L₂) (Fig. 2C).

Fig. 2. In situ CD test for nanosheet and chiral nanobundle.

CD spectra of l-nanobundles (A) and d-nanobundles (B). (C) In situ UV-vis absorption spectra. (D) TEM images of d-nanobundles captured at 120 min. (E) CD spectra of l- and d-valine nanosheets. (F) In situ UV-vis absorption spectra of d-valine nanosheets. L₀, L₁, and L₂ indicate the peaks of ligand-induced chirality.

The TEM images showed a large number of nanoparticles surrounding the nanobundles, indicating that the nanoparticles are assembly units to form chiral nanobundles (Fig. 2D). In addition, detailed TEM observations revealed that the hierarchical structure of the nanobundles included bundle structures, sheet structures, and building nanoparticles (figs. S19 and S20). As the reaction progressed, the size of the nanobundles increased because of the extension of nanosheets by nanoparticle-driven assembly (fig. S21). Therefore, in dual-ligand systems, achiral and chiral ligands jointly control the formation of nanoparticles. Chiral ligands induce misalignments in the arrangement of building nanoparticles, resulting in their extensive accumulation on the nanosheets (fig. S22). Deviations in the arrangement of building nanoparticles can be amplified through the attractive forces provided by achiral ligands, causing the nanosheets to have a multilayer assembly. The process ultimately results in a chiral rotation in the nanobundles. This mechanism illustrates how dual ligands cooperatively transfer, transmit, and amplify chirality in a hierarchical nanobundle assembly.

Analysis of the self-assembly of the chiral nanobundles showed that valine lost its chirality. The chirality loss is due to changes in the asymmetric charge distribution resulting from complex formation with the Cu ions (fig. S10). However, the introduction of citric acid did not restore the chirality of valine. Consequently, no CD signals were observed during the early stages of nanobundle formation. In contrast, the valine nanosheets showed a weak L₀ CD signal with a diminished UV-vis signal within 200 and 270 nm (Fig. 2, E and F), which arose from the valine on the nanosheets. For the chiral nanobundles, the L₁ and L₂ signals were attributed to valine in different chemical environments, which resulted from the presence of the neighboring citric acid on their building nanosheets. We introduced isovaleric acid (without an NH₂), N-acetyl-d-valine (with NH₂ blocked by an acetyl group), and Boc-Val-OH (with NH₂ blocked by a tert-butoxycarbonyl group) into a dual-ligand system. These molecules, when mixed with achiral citrate, formed nanobundles (fig. S23). However, the nanobundles prepared by N-acetyl-d-valine and Boc-Val-OH were weaker than those prepared by valine due to the NH₂ blocking (Fig. 1D and fig. S23). The NH₂ blocking limits NH₂ availability, increases steric hindrance, and reduces hydrogen bonding potential, which reduces the interaction network with citrate or the Cu(OH)₂ surface. Therefore, while the NH₂ blocking does not affect the bundle structure, it will decrease the transmission of chirality.

Coordination of dual ligands and their competition

To gain insight into the coordination behavior of valine and citric acid on the Cu(OH)₂ surfaces, we performed DFT simulations. We considered two valine configurations, valine(I) and valine(II), which differed in the coordinates of the carboxylate group (fig. S24, A and B). We examined the interactions of both valine configurations with the {001}, {011}, and {010} facets of Cu(OH)₂. The results showed that valine(II) achieved stable coordination with the {010} and {001} facets, forming {010}-valine(II) and {001}-valine(II), respectively (Fig. 3A and fig. S24, C to F). In contrast, valine(I) did not form stable coordination structures on any of the investigated facets. On the basis of these findings, we chose the {010} facet to study citric acid coordination and molecular dynamics simulations.

Fig. 3. Coordination modes and chiral mechanism of achiral-chiral dual ligands for chiral nanobundles.

(A) Optimized coordination of valine on Cu(OH)₂ {010} facets for {010}-valine(II). Optimized coordination of citric acid on Cu(OH)₂ {010} facets for {010}-citrate-α (B) and {010}-citrate-β (C). (D) FTIR spectra of valine nanosheet. (E) FTIR spectra of chiral nanobundles. (F) Molecular dynamics simulation driven by citric acid. (G) Coarse-grain diagram for citric acid and valine coordinating with the surface of Cu(OH)₂. (H) Molecular mechanism of the structural chirality between nanosheets. Relationship between structural g value and quantity of citric acid with two chiral valine (I) and chiral valine with two citric acids (J).

Citric acid contains three carboxyl groups, including two α- and one β-carboxyl groups. We investigated the binding of citric acid to the Cu(OH)₂ {010} facet and identified five distinct coordination modes: {010}-citrate-α, {010}-citrate-αα, {010}-citrate-αβ, {010}-citrate-αβα, and {010}-citrate-β (Fig. 3, B and C, and fig. S25). These coordination modes differ in the number and type of carboxyl groups involved in binding, as well as in the coordination geometry (unidentate, bidentate, or tridentate). The DFT calculations revealed that the unidentate {010}-citrate-α and bidentate {010}-citrate-αβ have more stable binding energy, with values of −17.4 and −14.2 kcal/mol, respectively (fig. S25). The higher stability of these configurations can be attributed to the optimal balance between the number of Cu─O bonds formed and the preservation of intramolecular hydrogen bonds. Notably, the {010}-citrate-α and {010}-citrate-αβ configurations feature uncoordinated groups that can potentially bind to other Cu(OH)₂ building nanosheets within the nanobundle. This intersheet binding enhances the overall stability of the nanobundles and contributes to their unique hierarchical structures.

The DFT simulations provided valuable insights into the vibrational signatures of valine and citric acid when coordinated to the Cu(OH)₂ {010} facet. In the case of {010}-valine(II), the simulated vibrations of the carboxylate (ν_COO) were predicted at 1651 and 1225 cm⁻¹, respectively (fig. S26A). The low wave number of the ν_COO value indicates the carbonyl stretch where oxygen is covalently bonded to the Cu, namely, the C═O─Cu group. This coordination supports the strong influence of Cu on the stability of the {010}-valine(II) configuration.

Our simulations also identified three specific types of hydrogen bonds: N─H···O═C, N···H─O─Cu, and C═O···H─O─Cu. These hydrogen bonds collectively contribute to a complex vibrational band at 3319 cm⁻¹. These interactions highlight the role of hydrogen bonding in anchoring valine to the Cu(OH)₂ surface, thereby facilitating drive assembly units. For the {010}-citrate-α configuration, the carboxylate vibration at 1646 cm⁻¹ indicates effective carboxylate binding, with hydrogen bonds involving O─H···O═C and C═O···H─O─Cu further stabilizing the interaction (fig. S26B). These findings are consistent with the bands observed in {010}-citrate-β (fig. S26B) but with lower intensities and slightly different peak positions. These differences suggest that the hydrogen bonding network in {010}-citrate-β is weaker than that in {010}-citrate-α, which may explain the lower stability of the former configuration. Therefore, the DFT simulations provide a detailed picture of coordination environment of valine and citrate and their vibrational signatures of the Cu(OH)₂ {010} facet.

In the FTIR spectroscopy analysis, the sharp and broad bands at 3570 and 3299 cm⁻¹ in valine nanosheets were attributed to free NH and hydrogen-bonded NH stretching vibrations, respectively (Fig. 3D) (56). Similarly, in the citrate nanobundles, corresponding dual bands at 3570 and 3299 cm⁻¹ were attributed to the free OH and hydrogen-bonded OH (fig. S27). In the case of chiral l-/d-nanobundles, similar dual bands appeared at 3570 and 3299 cm⁻¹, suggesting the overlapping stretching vibrations of NH and OH groups (Fig. 3E). The broadband at 3299 cm⁻¹ reflects a complex network of hydrogen bonds, including N···H─O─Cu, N─H···O═C, O─H···O═C, and C═O···H─O─Cu. These interactions enabled citrate to act as a bridge between the building blocks, enhancing the structural stability of the nanobundles. Valine coordination plays a critical role in transferring chirality within this framework, as evidenced by chiroptical signals L₁ and L₂ (Fig. 2, A and B).

In addition, citric acid and valine manipulate the assembly of the building blocks through carboxylate (COO) chemical coordination. In situ and ex situ FTIR tests have shown that carboxylate bands remained at consistent peak positions at 1386 and 1551 cm⁻¹ (fig. S28), indicating that COO─Cu maintained the same coordination in the hierarchical building blocks. For the final nanobundles, the ν_as (asymmetric vibrations) and ν_s (symmetric vibrations) of carboxylate in valine nanosheets or chiral nanobundles were identified at 1588 and 1347 cm⁻¹, respectively, indicating unidentate coordination (fig. S27). The ν_as and ν_s of carboxylate observed at 1621 and 1324 cm⁻¹, respectively, were attributed to “pseudo-unidentate” coordination, where the carboxylate connects to a hydrogen acting as a pseudo-metal ion (57).

Chirality mechanism of chiral superstructure

To elucidate the chiral assembly mechanism, molecular dynamics simulations provided insights into the dynamic self-assembly of building blocks, which were guided by surface ligands, including valine, citrate, and a combination of both (Fig. 3F and fig. S29, A to C). The stacking angle θ of building blocks is described using the formula: θ = sign·arccos(μ₁ × μ₂/|μ₁||μ₂|), where μ₁ and μ₂ are vectors representing the long sides of two adjacent building blocks (fig. S29D). The configurations of building blocks coated with l-valine exhibited notable stacking angles, ranging between 60° and 90° (fig. S30A). Building blocks coated with citric acid and l-valine exhibited a broad range of stacking angles from 0° to 90° (fig. S30B). In contrast, the building blocks covered solely with citric acid displayed a parallel stacking pattern. These findings highlight the role of chiral molecules in influencing the local chiral orientation of assemblies by rotating the building block. A Monte Carlo simulation was used to depict coarse-grained specific interactions of the building blocks (figs. S31 and S32). The results revealed that citric acid linkers enhanced the aggregative interactions between the building blocks by two orders of magnitude, whereas chiral molecules did not. Achiral citric acid is essential for chiral valine to effectively impart chiral forces and activate the entire bottom-up pathway of chiral self-organization in inorganic compounds.

Coarse-grained modeling was used to further elucidate the chiral stacking relationship among the building nanosheets (Fig. 3G). A molecular model was used to visualize the interactions between the chiral molecule (valine, simplified to a central and four distinct end points) and an achiral linker (citric acid, a simplified two-point model with two symmetrical interaction sites) with the nanosheets. Consistent with the above findings, chiral molecules failed to bridge the nanosheets into a stable structure because of their insufficient aggregative interaction capabilities. Conversely, the linkers effectively connected different nanosheets, forming a stable assembly (fig. S33). Within achiral-chiral dual-ligand system, the achiral linker successfully bridged two nanosheets, one above the other. Chiral molecules were fixed on either the lower or upper nanosheets through carboxylate, thereby rotating the other nanosheets to initiate structural chirality (fig. S34). The position of the R group (point A in Fig. 3H) of the chiral molecule was determined by its spatial structure, which was influenced by the chiral carbon (P), carboxylate coordination (B), and flexible amino hydrogen bonds (C).

Considering the coordination competition between achiral linkers and chiral molecules, their contributions was further evaluated through coarse-grained modeling. An anisotropic factor for polarization rotation, defined as $g=-\overrightarrow{r}·(\overrightarrow{{\mathrm{\mu }}_1}\times \overrightarrow{{\mathrm{\mu }}_2})$, was used to quantify the chirality of nanosheets within nanobundles. On the basis of the refined model, adjusting the number of citric acid molecules from one to four, while keeping the quantity of valine constant, resulted in an initial increase in the g value (0.029), followed by a subsequent decrease (Fig. 3I). Similarly, altering the number of chiral valines from one to four led to an increase in the g value, which was correlated with an increased valine number (Fig. 3J).

Chirality modulation mediated by dual-ligand strategy

On the basis of our simulations, experimental investigations were conducted to elucidate the chirality changes in nanobundles by independently regulating the achiral-chiral dual ligands. When the citric acid dosage was increased from 6 to 14 mM, both chiral valine and achiral citrate led to a transition from nanosheets to nanobundles (fig. S35). Modulation by citric acid also caused the assembly to change from nanosheets to nanobundles in the absence of chiral valine (fig. S36), further demonstrating that the assembly structure is controlled by citrate. The nanosheets obtained using achiral-chiral ligands showed no CD signal, whereas the nanobundles displayed chiral signals (fig. S35A). This observation indicates that the chirality of inorganic compounds depends on the rotation of nanosheets within the nanobundles. The nanobundles showed an initial increase, followed by a decrease in CD signal (Fig. 4A). For the assembly controlled by achiral-chiral ligands, a chiral valine dosage from 0.5 to 10 mM led to the formation of nanobundles (fig. S37). A dose of 10 mM resulted in larger nanobundles, likely because sufficient valine promoted the nanosheet aggregation. In the absence of citric acid, 20 mM valine produced smaller nanobundles with nonantipodal CD signals (fig. S38). This demonstrates that valine can form chiral networks through chemical bonds and hydrogen bonds. However, in this case, valine enantiomers could not accurately balance the linking and rotating functions of the assembly units. The g values of the chiral nanobundles produced using achiral-chiral ligands, increased from 0.005 to 0.02, indicating a notable enhancement in chirality (Fig. 4B). Chiral rotation became increasingly noticeable with increasing valine concentration, particularly after the surface ligands were removed (fig. S39).

Fig. 4. Chirality control of nanobundles using achiral-chiral dual ligands.

(A) SEM images of l-nanobundle using different concentrations of citrate. Inset is TEM image. g factor of the assembly varies with the concentration of citrate under the use of 5 mM l-/d-valine. (B) SEM images of l-nanobundle using different concentrations of l-valine. g factor of the assembly varies with the concentration of l-/d-valine under the use of 10 mM citrate. (C) Contours of g factors between 400 and 600 nm for chiral nanobundle modulated by the dosage of citrate and chiral valine. (D) Chiral nanobundle assembly controlled by achiral-chiral ligands. (E) Solution NMR of the supernatant from l-nanobundle (green) and d-nanobundle (blue). ppm, parts per million. (F) Contours of the surface-citrate ratio of assemblies from modulating the concentration of citric acid and chiral valine.

The contours of the g factors provided a clear visualization of the chirality trend (Fig. 4C and D). Initially, the increase in citric acid dosage facilitated the formation of nanobundles and enhanced the chiroptical chirality owing to enhanced interactions between the building units. Subsequently, as the citric acid concentration continued to increase, the chiral valine began to be displaced, which resulted in a decrease in chirality. Conversely, increasing the chiral valine dosage promoted chirality transfer, which was supported by the structural integrity of the nanobundles maintained by citric acid. Therefore, the independent modulation of achiral and chiral ligands in the experimental and simulation settings demonstrated consistent trends (Fig. 3, I and J).

In solution, nuclear magnetic resonance (NMR) spectroscopy, citric acid and valine exhibited distinct signals (fig. S40). The concentrations of the achiral-chiral dual ligands were quantified using dimethyl sulfoxide (Fig. 4E and fig. S41). Within the system of dual ligands, a citric acid ratio exceeding 60% ensured the formation of nanobundles, whereas a ratio below 55% predominantly resulted in the formation of nanosheets (Fig. 4F). Increasing the chiral valine ratio from 0 to 34% and reducing the citric acid ratio from 100 to 66% notably enhanced the chirality of the nanobundles.

To assess the general applicability of the dual-ligand strategy, we further investigated the substitutability of both chiral and achiral ligands, demonstrating their flexibility. Chiral ligands, such as valine, phenylalanine, penicillamine, and cysteine, along with achiral citric acid, were used to produce chiral nanobundles, resulting in decreased g values (Fig. 5A and fig. S42). From a coordination chemistry perspective, the chiroptical activity of the nanobundles using thiol-free valine and phenylalanine was notably stronger than that of the nanobundles using thiol-containing penicillamine and cysteine. This difference can be attributed to thiol coordination, which weakens the charge distribution responsible for the chiral properties of chiral ligands (fig. S43). Achiral ligands, such as citric, racemic tartaric, aspartic, and glutamic acids, along with chiral valine, were examined for their impact on nanobundle formation (Fig. 5B and fig. S44), with these ligands leading to a sequential decrease in g values. Nanobundle formation was enhanced by achiral ligands with multiple functional groups capable of forming network of chemical and hydrogen bonds. In addition, the combination of racemic tartaric acid and chiral phenylalanine successfully produced chiral nanobundles (fig. S45).

Fig. 5. Expansion of the achiral-chiral dual-ligand strategy.

Diagram, g factor, and SEM images of chiral nanobundles synthesized by both different chiral molecules and achiral citrate (A), chiral valine and different achiral linkers (B), and different chiral linkers (C).

Chiral multifunctional ligands, such as chiral tartaric, chiral aspartic, and chiral glutamic acids, can replace both chiral and achiral ligands. These compounds can sacrifice two or more functional groups to connect the building nanosheets into nanobundles while simultaneously transferring their chirality to the nanosheet arrangement (Fig. 5C and fig. S46). However, this process may weaken the transmission of chirality owing to the disrupted asymmetric charge distribution. Thus, the selection criteria for achiral ligands involve having at least two carboxyl groups capable of bridging the nanosheets for strong coordination. In contrast, the selection criteria for chiral ligands should ensure that one strong coordinating group is present, whereas other groups should maintain an asymmetric charge distribution around the chiral center.

We explored the feasibility of using different metal ions within the dual-ligand system. l- and d-CdS were nanobundles successfully synthesized using a combination of citric acid and valine (fig. S47, A and B). Previous studies reported the formation of chiral CuS and CdTe nanobundles (58, 59). Furthermore, we investigated the incorporation of various metal ions into the nanobundles. Nanobundles doped with Zn ions at concentrations of 3, 10, and 20% consistently exhibited Zn atomic content (fig. S47, C to E). Similarly, those doped with Fe ions at concentrations of 0.1, 1, and 3% displayed Fe atomic contents of 4, 25, and 70%, respectively (fig. S47, F to H), indicating an enhanced coordination capability between Fe ions and the dual ligands. This observation highlights the effectiveness of the dual-ligand system for doped cations.

We used achiral-chiral dual ligands to initiate the asymmetric hierarchical self-assembly of inorganic nanobundle superstructures via coordination chemistry. The relative proportions of achiral citric acid and chiral valine synergistically modulate the hierarchical assembly patterns of chiral nanobundles. Although these phenomena are governed by a delicate balance of intricate interactions, the competitive effects of chiral and achiral ligands in coordination chemistry can be adjusted, allowing their chiroptical activities to be synergistically modulated. We used DFT, molecular dynamics, and coarse-grained simulations to elucidate the bottom-up crystallization control of inorganic components using achiral-chiral ligands. The achiral-chiral dual ligands not only effectively address the challenges of chirality transfer, propagation, and amplification in the assembly of inorganic superstructures but also substantially enrich the selectivity for chiral and achiral molecules, expanding the array of adjustable tools for constructing chiral inorganic superstructures.

MATERIALS AND METHODS

Chemicals

Sodium hydroxide (NaOH; >98%), copper(II) chloride dihydrate (CuCl₂·2H₂O; 99%), and lead(II) perchlorate trihydrate (97%) were purchased from Macklin Reagent. Citric acid monohydrate (99%), sodium citrate (99%), l-valine (99%), d-valine (98%), l-penicillamine (>98%), d-penicillamine (>98%), l-cysteine hydrochloride monohydrate (99%), d-cysteine hydrochloride monohydrate (98%), l-phenylalanine (99%), d-phenylalanine (98%), l-glutamic acid (>98.5%), and d-glutamic acid (98%) were obtained from Aladdin Reagents. d-tartaric acid (>99%) and l-tartaric acid (>99%) were acquired from the Shanghai Yuanye Bio-Technology Co. d-aspartic acid (99%) and l-aspartic acid (99%) were sourced from Coolaber Reagents. All chemicals were used without further purification. Water was purified with a Millipore system.

Synthesis

Cu(OH)₂ nanobelt, l-valine nanosheet, d-valine nanosheet, citrate nanobundle, l-nanobundle, and d-nanobundle were synthesized by adding CuCl₂·2H₂O to water (10 mM) with vigorous stirring. The solutions were pre-cooled at 4°C for 1 hour before 1 M NaOH was added to adjust pH to 12.2. After stirring for 3 min, they were left to stand for 1 day at 4°C. Subsequently, the respective Cu(OH)₂ assemblies were collected through gentle centrifugation (4000 rpm for 3 min) and washed twice with deionized water. The synthesis of l-valine (d-valine) nanosheet involved adding l-valine (d-valine) (5 mM) before adding NaOH. Similarly, citric acid (10 mM) was added before the addition of NaOH for citrate nanobundle synthesis. For l-nanobundle (d-nanobundle) synthesis, l-valine (d-valine) (5 mM) and citric acid (10 mM) were added before adding NaOH.

For the synthesis of chiral nanobundles with adjusted dosages of citric acid or chiral valine and other chiral molecules, other linkers, and chiral linkers, the same synthesis procedure was used with specific variations in the components and quantities used. When synthesizing at variable pH levels, NaOH was gradually added to reach pH values of 12.2, 12.5, and 12.8, following the same fundamental steps.

Calcination of assemblies

After 12 hours of freeze-drying, the nanobundles and nanosheets underwent calcination through a gradient heating process under argon gas protection. We held the temperature at 80°C for 30 min to remove residual water, then raised the temperature from 80° to 350°C at a rate of 5°C/min, and, last, maintained the temperature at 350°C for 60 min.

Characterizations of assemblies

TEM images were captured using an FEI Talos F200S apparatus operating at 200 kV. We have used TEM tomography as an angle-dependent imaging technique to confirm the chiral features of nanobundles. Electron tomography tilt series were acquired on a FEI Talos F200s TEM operated at 200 kV and equipped with a complementary metal-oxide semiconductor, using the Serial-EM acquisition software. Tilt series were acquired by tilting the specimen from −68° to +62° in increments of 1°, and they were aligned using patch tracking and then reconstructed using weighted back projection with Inspect 3D and Avizo software. SEM images were obtained using the SU8010 under 5 kV under voltage and an electric current of 10 mA. CD spectra were recorded using a Chirascan Plus spectrophotometer with an integrating sphere. Powder samples were analyzed by transmitting light through a quartz plate, using a xenon lamp for wavelengths up to 1000 nm and a tungsten lamp for wavelengths between 1000 and 1600 nm. FTIR spectra were acquired using a Tensor II spectrometer equipped with an attenuated total reflection accessory. X-ray diffraction spectra for powder samples were measured using a Bruker D8 Advance spectrometer using Cu Kα radiation (λ = 0.15418 nm). NMR experiments were performed on a QUANTUM-I-400 MHz from Qone Instruments, with the sample dissolved in deuteroxide.

X-ray photoelectron spectroscopy analysis was performed using an ESCALab 250Xi electron spectrometer from Thermo Fisher Scientific. Atomic force microscope measurements were carried out using a Bruker Dimension ICON instrument. Thermogravimetric analysis was measured on a PerkinElmer TGA8000. In situ CD tests were kept in a water bath at 4°C. An electronic temperature control system was used to regulate the temperature of the sample holder more precisely and stably. For chiral nanobundles, in situ CD tests were performed at the following time intervals: 3, 6, 10, 15, 20, 30, 40, 50, 60, 80, 100, and 120 min. Valine nanosheets underwent in situ CD tests at the following time intervals: 2, 4, 6, 8, 10, 12, 15, 20, 25, 30, 35, 40, 52, and 65 min.

Calculation of the g factor

The g factor (g value), as an anisotropic parameter for polarization rotation, is an independent concentration parameter. It is determined using the following formula, which relies on the differential absorbance (ΔA) of left-handed and right-handed circularly polarized light (A_LCP and A_RCP) compared to the total absorbance (A_total) (60, 61)

$$g_{\text{factor}}=\frac{A_{\text{LCP}}-A_{\text{RCP}}}{A_{\text{LCP}}+A_{\text{RCP}}}=\frac{\mathrm{\Delta }A}{A_{\text{total}}}\approx \frac{\text{CD} \left(\text{mdeg}\right)}{32,980\times \text{Abs}}$$ (1)

where Abs represents the absorbance.

DFT simulations

DFT simulations were performed using CP2K (version 2023.1) (62) based on the mixed Gaussian and plane-wave scheme (63) and the Quickstep module (64). The geometry optimization and phonon calculation used Perdew-Burke-Ernzerhof exchange correlation functional (65) and molecularly optimized short-range Double-Zeta-Valence plus Polarization basis set (66) with Goedecker-Teter-Hutter pseudo-potentials (67). The plane-wave energy cutoff was set to 700 rydbergs (Ry). The calculation was performed at the gamma point only without symmetry constraint. Structural optimization was conducted using the Broyden-Fletcher-Goldfarb-Shannon optimizer until the maximum force fell below 0.00045 Ry/bohr (0.011 eV/Å). The finite displacement method was used for the phonon calculation, with an incremental displacement of 0.01 bohr (0.0053 Å). The phonon spectra were analyzed using the Multiwfn (version 3.8 dev) package (68). Details for the coordination simulation of valine or citric acid and Cu(OH)₂ facets can be found in the Supplementary Materials.

Molecular dynamics simulation

The molecular dynamics simulation was performed using the GROMACS package (version 2020.6) (69). The Cu(OH)₂ nanosheet with surface-adsorbed valine was obtained by cleaving by {010} facet of the original crystal, resulting in a thickness of 0.5 unit cell. It was then expanded to form a 3 × 2 supercell. Similarly, the l-nanobundle was prepared with a molar ratio of citrate to l-valine being 3:1. It involved cleaving the {010} facet of the original crystal to a thickness of 0.5 unit cell and expanding it to an 8 × 1 supercell. In addition, citrate nanobundle adsorbed by citrate for simulation was cleaved by the {010} facet of the original crystal, with a thickness of 0.5 unit cell, and expanded to an 8 × 1 supercell.

The force field of all these nanosystems was generated by the Sobtop 1.0 (dev2) package (70). Cu was assigned universal force field atom types, and the other elements were assigned general amber force field (GAFF) atom types. The bonded parameters were initially built using the GAFF force field. The remaining parameters were determined through established methods. The atomic charges of the Cu(OH)₂ crystal motif were manually modified to 0.40 (Cu), −0.50 (O), and 0.30 (H). The atomic charges of the ligands (citrate and valine) were calculated using DFT calculations under the restrained electrostatic potential formalism. The valine nanosheet systems for molecular dynamics simulation consisted of eight randomly distributed building blocks in a water box (11 nm by 11 nm by 11 nm). The citrate nanobundle and l-nanobundle systems followed a similar molecular dynamics simulation. All three systems were first minimized using the conjugate-gradient algorithm and then equilibrated through running for 100-ps NVT simulations. The production runs in the NVT ensemble at 298 K were performed for 30 ns using the leapfrog algorithm with a time step of 2 fs to integrate the equations of motion. Electrostatic forces were treated with the particle-mesh Ewald approach. The cutoff values of van der Waals and electrostatic forces were set to 1.2 nm. The LINCS algorithm was used to preserve bonds.

The l-valine–coated Cu(OH)₂ crystal with dimensions of 10 by 0.5 by 6 cells (3.0 nm by 1.4 nm by 3.0 nm) was used as a building block coated by 60 valine through {010}-valine(II). After 30-ns kinetic simulations, the building blocks formed a one-dimensional assembly and showed chiral stacking due to the hydrophobic interaction of valine. The Cu(OH)₂ crystal with dimensions of 15 by 0.5 by 2 cells (4.4 nm by 1.8 nm by 1.0 nm) is coated by 8 l-valine and 24 citric acid using {010}-citrate-α and {010}-valine(II) as a building block. After 25 ns, the building blocks were predominantly arranged in a right-handed spiral pattern. The Cu(OH)₂ crystal with dimensions of 15 by 0.5 by 2 cells (4.4 nm by 1.8 nm by 1.0 nm) was coated by 32 citric acid as a building block. After 30 ns, the building blocks were predominantly arranged in a parallel manner, giving rise to a compact bundled nanostructure.

The coarse-grained model

Model 1

The Cu(OH)₂ nanoparticles were simplified as a cube with a side length of 3 nm, exhibiting nonspecific interaction. The system contains 100 nanoparticles, with the concentration of Cu(OH)₂ matching that of the experimental setting. The nonspecific pairwise interaction energy between two nanoparticles at a center-to-center distance of r is given by

$$F_{\text{ns}}=\left\{\begin{array}{ll}{\mathrm{\gamma }}_{\text{ns}}{\left(\frac{d}{r}\right)}^6,& r \le 1.3d\\ 0,& r>1.3d\end{array}\right.$$ (2)

where d is the side length of the nanoparticle. γ_ns is the interaction parameter between two nanoparticles, which is derived from nonspecific attractions and is relatively weak. We set its value to 8 k_BT, where k_B denotes Boltzmann’s constant and T is the temperature. The design of this interaction potential is inspired by the previous work (71).

We coarse-grained the linker and chiral molecules (such as citric acid and chiral valine) into the specific interaction between Cu(OH)₂ nanoparticles. The linker is simplified to the spring potential related only to the center-to-center distance of interacting faces on two particles. When the two particles with linkers face each other, the interaction is implemented as

$$F_{\text{lk}}=\left\{\begin{array}{cc}{\mathrm{\Gamma }}_{\text{lk}}+\frac{1}{2}{k}_{\text{lk}}{(r_{\mathrm{f}}-r_0)}^2,& \mid r_{\mathrm{f}}-r_0\mid \le 0.1 \text{nm}\\ 0,& \mid r_{\mathrm{f}}-r_0\mid >0.1 \text{nm}\end{array}\right.$$ (3)

where Γ_lk = 200 k_BT is the bond energy to simulate the chemical bond induced by the linker. k_lk, r_f, and r₀ are the force constants, the center-to-center distance of two interaction faces, and the reference distance, respectively. Because nanoparticles are linked by the linker rather than directly by a chemical bond, we choose a relatively large stiffness coefficient (5 × 10² kJ/mol per square nanometer) without being required to approach the chemical bond strength (10⁵ kJ/mol per square nanometer). A reference distance r_f = 0.5 nm is equivalent to the size of the linker.

The chiral molecule is simplified to the chiral potential related to the centroid and crystal plane orientation between particles. When the two particles with chiral molecules face each other, the interaction is implemented as follows

$$F_{\text{cr}}=\left\{\begin{array}{cc}{\mathrm{\eta }}_{\mathit{ij}}{\mathrm{\Gamma }}_{\text{cr}}{\text{cos}}^2({\mathrm{\theta }}_{\mathrm{f}}-{\mathrm{\theta }}_0),& \mid r_{\mathrm{f}}-r_0\mid \le 0.1 \text{nm}\\ 0,& \mid r_{\mathrm{f}}-r_0\mid >0.1 \text{nm}\end{array}\right.$$ (4)

where Γ_cr, θ_f, and θ₀ are the interaction parameters, the included angle of the in-plane direction vector of two interaction facets, and the reference angle, respectively. Γ_cr is 8 k_BT, and θ₀ is $\frac{\mathrm{\pi }}{6}$. η_ij is the chirality parameter, defined as

$${\mathrm{\eta }}_{\mathit{ij}}=\left\{\begin{array}{c}1,(\overrightarrow{{\mathrm{\nu }}_i}\times \overrightarrow{{\mathrm{\nu }}_j})\cdot (\overrightarrow{n_i}-\overrightarrow{n_j})\le 0\\ -1,(\overrightarrow{{\mathrm{\nu }}_i}\times \overrightarrow{{\mathrm{\nu }}_j})\cdot (\overrightarrow{n_i}-\overrightarrow{n_j})>0\end{array}\right.$$ (5)

Six normal vectors on the six faces of the cubic nanoparticles are defined as

$$\overrightarrow{n_1}=(1,0,0),\overrightarrow{n_2}=(0,1,0),\overrightarrow{n_3}=(0,0,1)$$

$$\overrightarrow{n_4}=(-1,0,0),\overrightarrow{n_5}=(0,-1,0),\overrightarrow{n_6}=(0,0,-1)$$ (6)

Six in-plane direction vectors on the six faces of the cubic nanoparticles are defined as

$$\begin{array}{c}\overrightarrow{v_1}=(0,0,-1),\overrightarrow{v_2}=(1,0,0),\overrightarrow{v_3}=(1,0,0),\\ \overrightarrow{v_4}=(0,0,-1),\overrightarrow{v_5}=(1,0,0),\overrightarrow{v_6}=(1,0,0)\end{array}$$ (7)

We used a Monte Carlo simulation to coarse-grain the linker and chiral molecule, representing their specific interactions among primary particles during self-assembly. The linker was modeled as a spring potential, dependent on the distance between interacting facets of two particles. The chiral molecule was simplified into a chiral potential considering the centroid and crystal plane orientation between particles. In addition, there is a nonspecific attraction between the particles.

Model 2

Self-assembly of chiral Cu(OH)₂ nanobundle is efficiently carried out on the basis of the dual ligand of citric acid (linker) and chiral valine (chiral molecules). All parameters are set concerning previous work considering the specific interaction type (72). The volume of the building nanosheet in the nanobundle is 270 nm by 10 nm by 1.3 nm. This model simplified the crystal plane as 0.5 nm by 1.5 nm, where several random sites can interact with both citric acid and chiral valine. Coarse-grained modeling described the interactions between the linker and building nanosheets. The most stable configuration is α-carboxyl groups of citric acid coordinating with the Cu(OH)₂ facet. It allows other carboxyl groups to interact with adjacent building nanosheets. The linker is simplified as a spherocylinder with dimensions of 0.7 nm in length (L_lm) and 0.2 nm in width (D_lm), having one binding site on each end. Both binding sites have the potential to coordinate with the building nanosheets.

Cu(OH)₂ building nanosheet (the gray-green plane in Fig. 3G), citric acid (spherocylinder-shaped rod), and chiral valine (tetrad model) are the three components in the chiral system. To account for the excluded volume effects, a hard spherical potential was applied. The long and short axis of the Cu(OH)₂ plane are denoted by $\overrightarrow{\mathrm{\mu }}$ and $\overrightarrow{\mathrm{\tau }}$, respectively. Two end sites of the citric acid denote the carboxyl groups that bind to the surface of the building nanosheets, and their linkage is marked as $\overrightarrow{r}$. In addition, the chiral valine with four groups has different interactions with nanosheet surfaces, such as a chemical bond, hydrogen bond, Van der Waals force, and steric hindrance effects. The chiral carbon is denoted as reference point P. Vectors ($\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$) point from P to the mass center of different valine groups (Fig. 3G). The end points of these vectors are labeled as A, B, and C.

The nanobundle system with just two building nanosheets and one linker was built. Considering symmetry, the energetically favorable structure involves the linker molecule vertically connecting the two building nanosheets. To establish a reference coordinate system, taking one site on Cu(OH)₂ surfaces that bond with a linker as the origin, the short axis ($\overrightarrow{{\mathrm{\tau }}_1}$) of the crystal plane as the x axis, and the long axis ($\overrightarrow{{\mathrm{\mu }}_1}$) as the y axis. The z axis is defined as the normal vector direction ($\overrightarrow{{\mathrm{\tau }}_1}\times \overrightarrow{{\mathrm{\mu }}_1}$). The long-axis direction $\overrightarrow{r}$ of linker points from the origin to the binding site on the other building nanosheet.

The system energy induced by the chemical bond is described as follows, regardless of the translation and rotation of the mass center in the system.

$$H_1=E_{\text{bond}1}+E_{\text{bond}2}=\frac{1}{2}k{d}_1^2+\frac{1}{2}\mathrm{\epsilon }{{\mathrm{\theta }}_1}^2+\frac{1}{2}k{d}_2^2+\frac{1}{2}\mathrm{\epsilon }{{\mathrm{\theta }}_2}^2$$ (8)

Here, $\frac{1}{2}k{d}^2$ and $\frac{1}{2}\mathrm{\epsilon }{\mathrm{\theta }}^2$ are interactions between the linker molecule and Cu(OH)₂ crystal surfaces at the origin. It includes constraints on bond length and bond angle. Subscripts 1 and 2 distinguish the upper and lower nanosheets. θ is the angle between the normal vector of Cu(OH)₂ crystal surfaces ($\overrightarrow{{\mathrm{\tau }}_1}\times \overrightarrow{{\mathrm{\mu }}_1}$) and orientation vectors ($\overrightarrow{r}$). The distance between two sites on the linker molecule and the corresponding site on the Cu(OH)₂ crystal surfaces is denoted by d₁ and d₂. The force constants k₁ and k₂ are related to bond length and have a value of 5 × 10⁵ kJ/mol per square nanometer. The two force constants are associated with the same bond angle of 500 kJ/mol per square radian.

The dual-ligand system was further investigated. The chiral carbon of amino acids was chosen as the reference point P. Four branches were used to represent the different groups of chiral carbon. Each branch was described as a spherocylinder-shaped structure with a length of 0.5 nm (L_cm) and a diameter of 0.1 nm (D_cm). Three branches on a chiral carbon are sufficient to describe the molecule’s chirality. The structure of chiral valine used was simplified as above. The carboxylate of chiral valine formed a chemical bond with Cu²⁺ of Cu(OH)₂ crystal surfaces. It is simplified as the chemical bond between B point and the Cu(OH)₂ surface site. The chiral molecule could also form weak bonds with the crystal plane by NH₂ or OH. They are simplified as the weak bond between C and the Cu(OH)₂ surface site. R groups of chiral amino acids, such as methyl and the benzene ring, may interact with Cu(OH)₂ crystal surfaces. It represents nonspecific interactions between A and Cu(OH)₂ crystal surfaces.

The energy of the interaction between the chiral molecule and the crystal plane is as follows

$$\begin{array}{c}{H}_2=E_{\text{chemica}{\mathrm{l}}_{\text{bond}}}+E_{\text{wea}{\mathrm{k}}_{\text{bond}}}+E_{\text{nonspecific}}\\ =\frac{1}{2}{k}_{\mathrm{c}}{d}_{\mathrm{c}}^2+\frac{1}{2}{\mathrm{\epsilon }}_{\mathrm{c}}{({\mathrm{\theta }}_{\mathrm{c}}-{\mathrm{\theta }}_{\mathrm{c}0})}^2+\frac{1}{2}{k}_{\mathrm{b}}{d}_{\mathrm{b}}^2+\frac{1}{2}{\mathrm{\epsilon }}_{\mathrm{b}}{({\mathrm{\theta }}_{\mathrm{b}}-{\mathrm{\theta }}_{\mathrm{b}0})}^2-{\mathrm{\omega }}_{\mathrm{a}}{\left(\frac{D_{\text{cm}}}{d_{\mathrm{a}}}\right)}^6\end{array}$$ (9)

Here, the distances between B or C and their corresponding locations on Cu(OH)₂ crystal surfaces are d_b and d_c, respectively. The minimal distance between A and the crystal plane is d_a. k_b (2 × 10⁵ kJ/mol per square nanometer) and k_c (2 × 10³ kJ/mol per square nanometer) are force constants corresponding to point B (chemical bond) and C (weak bond), respectively. Angular displacements of $\overrightarrow{b}$ and $\overrightarrow{c}$ are (θ_b − θ_b0) and (θ_c − θ_c0), respectively. The force constant associated with the bond angle are k_b = k_c = 200 kJ/mol per square radian. k_a is the 5 kJ/mol interaction constant corresponding to point A.

A Monte Carlo scheme is used to simulate the mechanism of chiral nanobundle induced by the dual-ligand strategy. It includes translational and rotational movements of individual particles. The acceptance probability of that movement is p = min {1, e^−Δμ/k_BT}, where Δμ denotes the energy difference between the two states before and after the move. The simulations were run in a cubic simulation box with periodic x and y bounds. The box’s height (L) is kept constant at 50 nm.

Supplementary Materials

The PDF file includes:

Figs. S1 to S47

Legend for movie S1

Other Supplementary Material for this manuscript includes the following:

Movie S1