Strong effect of the nonpolar solvent molecular structure on CdSe nanoplatelet stacking

Abstract

We report a drastic difference in stacking behavior of oleic acid-stabilized 4-monolayer (4 ML) CdSe nanoplatelets (NPLs) in toluene and methylcyclohexane (MCH), two nonpolar solvents that differ in the conformational flexibility of their molecules. Using liquid cell transmission electron microscopy (TEM) and small angle scattering (SAXS) techniques, we show that NPLs form microns-long ribbons consisting of 4 ML CdSe NPLs in toluene, the solvent widely used to form stable colloidal solutions of a broad range of quasi-spherical nanoparticles. In contrast, 4 ML CdSe NPLs are well dispersed in MCH. The difference in stacking behavior of NPLs in toluene and MCH suggests that the conformational flexibility of the solvent molecules, such as the ability to adopt multiple chair conformations, modulates nanoplatelet interactions. Molecular dynamics (MD) simulations reveal that solvent molecules subtly alter the structure of the organic ligand shell. These solvent-dependent changes propagate to the inorganic core, modulating the degree of CdSe nanoplatelet (NPL) twisting and, consequently, the properties of the nanoparticles. We show that toluene better solvates oleate ligands while MCH induces a bimodal oleate span distribution, which can lead to increased solubility of CdSe NPLs. In addition, the solvent can also influence the inorganic core, which, in turn, can modify the nanoparticle properties. We demonstrate that destabilization of toluene solution containing ribbons of 4 ML CdSe NPLs without CdS shells results in the formation of NPL assemblies with amplified spontaneous emission (ASE) with a low threshold of 14 µJ cm⁻² that is comparable with that of CdSe/CdS core/shell NPLs. Our results emphasize that the solvent plays a major role in mediating interactions between NPLs and hence their processability for fabrication of functional structures.

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Introduction

CdSe nanoplatelets (NPLs) reveal well-defined, thickness-tunable opto-electronic absorption features with ultranarrow emission bandwidth, high absorption cross-sections and high oscillator strengths.^1–4 They can therefore be used to design light-emitting diodes (LEDs), compact lasers, and other opto-electronic devices.^1–4 NPLs offer the advantage of solution processability,^5–7 enabling the integration of individual components into functional structures and devices.⁸ The particle–particle interactions dictate the distance and interactions^9–11 between neighboring NPLs and hence affect the optical properties. As a result, the behavior of NPLs in solution and its correlation with the structures formed via drying or by solution destabilization are important for designing functional structures.^3,12–15

Overall, significant progress has been made in understanding nanoparticle (NP) interactions by focusing on their inorganic cores, both in solution and within assembled structures.^16,17 However, the nanoparticle surface remains difficult to probe at the molecular level, limiting quantitative insight into monolayer structure.¹⁸ Nevertheless, a growing body of experimental evidence demonstrates that the nature of the ligands, their distribution, and their solvation^5,19–26 play a critical role, and that interparticle forces arising from ligand interactions can even exceed van der Waals interactions between the inorganic cores.^27,28

Solvent effects on NP interactions have been studied theoretically^{23,25,29–33} and experimentally,^{23,25,31,32,34–39} mainly for three-dimensional (3D) faceted and quasi-spherical NPs. It was shown that the colloidal stability of 3.4 nm Au NPs decreased in the order cyclohexane > heptane > nonane > decane > toluene and suggested that details of the molecular interactions between the solvent and ligand shell explain the difference.³⁹ It was shown that simple cubic and body-centered-tetragonal lattices were formed by cubic Pt NPs on evaporation of their toluene and hexane solutions, respectively.³⁴ Also, different crystalline structures of PbS NP superlattices were fabricated from toluene and hexane solutions, respectively.³⁷ The difference in the formed structures was attributed to the different solubilities of PbS NPs in toluene and hexane³⁷ that can be connected to the fact that the solvent itself affects the thickness of the ligand shell, as was recently demonstrated in detailed small angle X-ray and neutron scattering studies on PbS NPs.²⁸ It was also reported that ligand shell solvation changes in the presence of a non-solvent,^5,29 which, in turn, affects the self-assembly of NPs.⁴⁰ Additionally, the preferential solvation of the ligand shell by one of the components in the solvent mixtures was demonstrated.³¹ However, generally in the case of faceted or quasi-spherical NPs, it is assumed that NPs stabilized with organic molecules have good solubility in nonpolar solvents unless some manipulations with colloidal solutions (typically via addition of a different surface ligand or a non-solvent) are performed; and solvents such as, for example, toluene and hexane are used regularly and considered to be replaceable with each other. The same is expected for NPs with different morphology, such as NPLs.

To date, NPL superstructures have been reported to be formed by either destabilizing their colloidal solutions using polar solvents or solvent evaporation.^15,41–43 For example, multi-strand bundles of CdSe NPLs were observed upon addition of ethanol to CdSe NPLs in hexane.¹² These micrometer long, highly aligned needle-like superstructures exhibit polarized emission. Also, addition of ethanol to hexane solutions of CdSe NPLs resulted in the formation of stacks of NPLs with strong and long-range ultra efficient Förster resonance energy transfer capability.⁴⁴ Single-strand ribbons of stacked CdSe NPLs were formed by solvent evaporation of their colloidal solutions upon addition of oleic acid.^45,46 It was also shown that the addition of alkyl phosphonic acids¹⁵ or fatty acids⁴⁷ induced the assembly of CdSe NPLs into stacks in chloroform and hexane, respectively. In the case of NPLs, it is reasonable to expect a stronger effect of the ligands on the NPLs since the ratio of ligand shell/inorganic core is substantially larger as compared to 3D NPs⁴⁸ and hence the role of the ligand shell is more dramatic.⁴⁹ It is widely acknowledged that some solvents (e.g. hexane and MCH)^7,47,50–52 favor the “flat” orientation of CdSe NPLs upon fast solvent evaporation while pronounced stacking of CdSe NPLs can be noticed in samples deposited from chloroform.¹⁵ Moreover, the ligands were shown to induce surface strain, resulting in ligand-induced mechanical stress and distortion of thin NPLs, leading to the formation of chiral ribbons upon solvent evaporation⁴⁵ or the curling of NPLs into helices.^53,54 Nevertheless, regardless of whether the studies on CdSe NPLs focused on their assembly via ligand manipulation,^15,45–47 solvent composition manipulations,^12,44 assembly kinetic effects⁷ or cation exchange in CdSe NPLs,^51,55,56 the role of the nonpolar solvent was not considered.

Here we studied solutions of CdSe NPLs in toluene and methylcyclohexane (MCH). These two nonpolar solvents have similar molecular weights but strongly differ in their molecular structures and hence conformational flexibilities. Toluene has a rigid, planar benzene ring with a methyl substituent, while MCH is a saturated flexible cycloalkane. Toluene is one of the most commonly used solvents to disperse different types of NPs, including NPLs.^55,57,58 In turn, MCH is often the empirically chosen solvent to fabricate optical devices.⁵⁰ Using liquid cell TEM and SAXS, we demonstrate the dependence of 4 ML CdSe NPLs on the solvent to form stacks or be fully dispersed, and we propose that this behavior depends on the conformational flexibility of the solvent molecules. Molecular dynamics (MD) simulations revealed important differences in the surface oleate shell in toluene and MCH indicating that the structure of the ligand shell can vary with the solvent even though they are both considered “good” solvents for NPs stabilized with oleic acid. We discuss several effects such as solvent structuring or NPL twisting which can relate the differences in stacking to the structure of the monolayer. Finally, we exploit the tendency of 4 ML CdSe NPLs to form stacks to fabricate self-assembled structures with a low lasing threshold of 14 µJ cm⁻², which is comparable with the values previously reported for structures assembled from more sophisticated CdSe/CdS core/shell NPLs. We also demonstrate that 3 ML and 5 ML CdSe NPLs reveal trends in toluene and MCH similar to those of 4 ML CdSe NPLs. Understanding the stacking behavior of NPLs in solvents is important since it affects the chemical reactivity of NPLs, self-assembly of NPLs into hierarchical structures and therefore the performance of NPL-based functional structures fabricated via solution processing.

Results and discussion

The 3, 4 and 5 ML (ML is defined with respect to the total Se layers^2,4) thin CdSe NPLs were synthesized according to the previously reported method.⁵⁹ The experimental details are presented in the SI (Fig. S1). We primarily used 4 ML CdSe NPLs, the most widely studied system, to examine solvent effects on NPL interactions and to connect our results with previously reported data. Additionally, we conducted experiments using 3 ML and 5 ML CdS NPLs in toluene and MCH. We dispersed equivalent amounts of the purified 4 ML CdSe NPLs with lateral dimensions of ∼24 nm × 8 nm (Fig. 1) in MCH and toluene. CdSe NPLs underwent exactly the same purification steps prior to their dissolution in MCH and toluene to have the same concentration of oleate ligands in both solvents in order to minimize the effects associated with the different ligand coverage.⁶⁰ Toluene and MCH were chosen as model systems of solvent molecules with different molecular geometries for two reasons: (i) toluene is one of the most common solvents for all NPs synthesized in nonpolar solvents and (ii) MCH and toluene have higher boiling temperatures (∼101 °C and 111 °C, respectively) than cyclohexane (∼81.74 °C) and hexane (∼69 °C), two other common solvents for colloidal NPs. The higher boiling points of MCH and toluene enable reproducible liquid cell TEM studies.

Fig. 1 Dark-field STEM images of 4 ML (monolayer) CdSe NPLs deposited from MCH (a) and toluene (b) solutions. Optical absorption (c) and PL emission (d) spectra for 4 ML CdSe NPLs dispersed in toluene and MCH. The SAXS spectra (e) measured for 4 ML CdSe NPLs dispersed in MCH, in toluene and a toluene mixture with 25% methanol (purple, pink and red spectra, respectively).

We first analyzed the 4 ML CdSe samples deposited via the fast evaporation of their toluene and MCH solutions using conventional TEM. Fig. 1a shows that in the sample obtained by drying the MCH solution, NPLs are randomly oriented but aligned parallel to the substrate. In contrast, periodic stacks of NPLs oriented perpendicular to the substrate are observed in the case of samples deposited from toluene (Fig. 1b). Previous studies suggest that in the case of NPs and NPLs with different morphologies, the assembly of both the NPs and NPLs is induced by solvent evaporation of the colloidal solutions of NPs randomly distributed in the solvent.^{12,37,45,47,61–64} The different drying behaviors of 4 ML CdSe NPLs from toluene vs. MCH as visualized with TEM point to solvent-dependent interparticle forces in solution. The optical absorbance data show no differences in the position of the heavy-hole (hh) excitons in toluene and MCH (Fig. 1c); however, the UV/vis spectrum in toluene has a higher background than that in MCH, which can be indicative of light scattering (Fig. 1c). Interestingly, the maximum of the PL spectrum of CdSe NPLs in toluene is red shifted by approximately 3 nm, which can indicate energy transfer or re-absorption effects from path length elongation from scattering (Fig. 1d). Moreover, small-angle X-ray scattering (SAXS) data show a substantial degree of ordering of CdSe NPLs in toluene dispersions, evidenced by the presence of two sharp peaks at the q of 0.117 Å⁻¹ and 0.233 Å⁻¹ in the corresponding SAXS diagrams (Fig. 1e). In contrast, the SAXS pattern in MCH shows only a hump around 0.115 Å⁻¹ indicating only traces of ordering (Fig. 1e). The same features are visible in the SAXS patterns of 5 ML CdSe NPLs in these two solvents. Stacking peaks at 0.11 and 0.22 Å⁻¹ in toluene indicate a lamellar ordering,⁴⁷ while the featureless SAXS pattern of 5 ML CdSe dissolved in MCH proves that the NPLs are randomly dispersed in this solvent (Fig. S2a). While the stacks observed in TEM for toluene in Fig. 1b could result from evaporation induced self-assembly on the TEM grid, optical and SAXS data (Fig. 1c–e) point towards stacking in solution prior to evaporation.

Motivated by the observation of the presence of long-range ordering in toluene solutions of CdSe NPLs and different packing of NPLs deposited from MCH and toluene (Fig. 1), we decided to conduct an in situ TEM study in liquid cells. In these experiments, a droplet of NPL solution was sandwiched between two amorphous carbon sheets. The sealing of the liquid cells was adjusted to enable evaporation in areas close to the seal. The in situ TEM experiments revealed well dispersed CdSe NPLs in MCH and the presence of tens of microns long ribbons consisting of 4 ML CdSe NPLs in toluene (Fig. 2). Interestingly, at the edge of the droplet (Fig. 2e and f), evaporation of the solvent was associated with the ribbons breaking into smaller fragments, as evidenced by the TEM images of the dried area of the liquid cell. Fig. S3 and S4 show similar trends in 4 ML CdSe NPLs from different batches. Other studies also reported similar images of the stacks of 4 ML CdSe NPLs deposited from toluene solutions.^5,34 Liquid cell TEM data are in agreement with the SAXS, UV/vis and TEM data obtained for dried samples: toluene induces strong stacking of the NPLs in solution while they are well dispersed in MCH.

Fig. 2 Dark-field STEM images of 4 ML CdSe NPLs in MCH (a–c) and toluene (d–f) obtained in a liquid cell upon drying of the corresponding solutions.

Toluene and MCH have similar Hansen solubility parameters (HSPs): 18.165 MPa^1/2 and 16 MPa^1/2, respectively.⁶⁵ Recent studies report that the HSP of oleic acid is between 17.21 MPa^1/2 and 18.74 MPa^1/2,^66,67 higher values than the earlier reported 15.6 MPa^1/2.⁶⁸ The smaller the difference between the Hansen solubility parameters of two substances, the better their solubility.⁵ From this perspective, the oleic acid shell should be slightly more solvated in toluene even though the difference is not that pronounced. However, the HSP considers solubility in bulk, which includes the interactions between the headgroup and the solvent. In our case, the carboxylate group is bound to the NPL surface, and the observed difference likely comes from the interplay between the carbon ligand backbone of the ligand chain and the solvents.

Previously, it was computationally shown that hexane and toluene are both “good” solvents of 4 nm cuboctahedra PbSe NPs stabilized with oleic acid and penetrate the ligand shell all the way to the NP surface with ∼3 ligands per nm² grafting densitity.²⁵ The study revealed slightly more toluene molecules than hexane for a given radial distance from the center in the oleate shell.²⁵ To gain more insight into the effect of the solvent on the oleate ligand shell of CdSe NPLs, we performed MD simulations on 4 ML CdSe NPs capped with oleates with a grafting density of ∼3.27 ligands per nm² in MCH and toluene (see the Methods section for details). We described the solvent explicitly to account for the molecular details of the interaction between the ligand brush and the solvent even though explicitly modeling the solvent is computationally demanding because it requires many additional atoms to be accounted for; however, as shown below, it yields valuable insights.

Fig. 3a depicts the relevant parameters that capture the conformation of the oleate ligand at the surface of the NPLs. The span is defined as the vertical distance between the atoms in the molecule that are farthest apart and R_ee is the end-to-end distance, i.e., the distance between the atom attached to the surface and the last atom of the carbon chain C₁₈. These parameters provide a quantitative description of the ligand molecule's conformation at the surface. Fig. 3b shows the density profiles of the ligand layer in the absence of a solvent and in toluene and MCH. The density profiles for toluene and MCH are similar, and very close to that in the absence of a solvent. This is expected, as the ligands are typically tightly bound to the surface Cd atoms. There are several sharp peaks close to the NPL surface, which, based on their locations, likely originate from the atoms of the binding group. However, our calculations revealed more extended ligand shells in toluene than in MCH with the maximum of the distribution of the span and R_ee at larger distances in toluene: the R_ee parameter of oleates at the CdSe NPLs’ surface is longer by ∼0.53 Å in toluene than in MCH. The difference in the maximum span parameters of the oleate ligand in toluene and MCH is ∼0.43 Å. Therefore, the span and R_ee parameters are indicative of a slightly more extended ligand shell at the surface of the CdSe NPLs in toluene.

Fig. 3 (a) Depiction of the ligand at the surface of NPLs demonstrating the characteristic parameters of the ligand such as the end-to-end distance parameter (R_ee) and span; (b) density profiles of oleate ligands normal to the 4 ML thick CdSe NPL surface in the absence of any solvent and in the presence of toluene and MCH. (c) Span of oleate ligands as a function of distance from the 4 ML thick CdSe NPL's surface. (d) Dependence of the parameter characterizing the distance between the end group of the ligand and the group attached to the NPL's surface (R_ee). (e) Density profiles of solvents in oleate ligands. (f) MD simulation snapshot of a 4 ML 24 nm × 8 nm rectangular NPL in toluene with stripped ligands demonstrating the arrangement of toluene molecules with respect to the inorganic 4 ML CdSe NPLs (viewed along the short in-plane dimension); MD simulation snapshot of a 4 ML 24 nm × 8 nm rectangular NPL in toluene (left) and MCH (right) coated with oleate ligands and with only the CdSe core shown (g and h, respectively, viewed along the long-in-plane dimension). (i) HRTEM image of 4 ML CdSe NPLs deposited from toluene.

It is worth noting that a larger ligand shell thickness in toluene than in cyclohexane (15.6 Å and 14.8 Å, respectively) and larger volume fractions of solvent within the monolayer for toluene than for cyclohexane were recently reported for oleyl-capped PbS spherical NPs.²⁸ These results agree with our atomic simulations. Indeed, the density profile of the solvent inside the monolayer displays two peaks in the case of toluene: one sharp peak with a maximum very close to the NPL surface at 2 Å and a broader shoulder with a maximum at 7 Å. In contrast, the solvent density increases monotonically for MCH. We believe this is due to the different steric profiles of the two molecules: toluene can easily slip between the hydrocarbon chains of the monolayer, while the chair conformation of MCH is bulkier. It is also possible that the moderate polarity of toluene makes it more compatible with the interfacial region which bares carboxylate groups. A surprising feature of our simulations is the bimodal character of the span and R_ee probability distribution. For MCH, the span distribution displays two maxima with the same height at 14.6 and 15.2 Å (Fig. 3c). The R_ee distribution shows a dominant primary mode at 17.1 Å and a secondary mode of smaller magnitude at 16.8 Å (Fig. 3d). The same features are observed with toluene except that the two peaks in the span distribution are closer and appear like a broader monomodal distribution with a plateau between 15.6 and 16 Å.

The MD calculations indicate slightly more extended ligand conformations for oleate molecules on 4 ML CdSe NPLs in toluene, as reflected by larger spans and end-to-end distances (R_ee) (Fig. 3c and d). Previous studies have reported a correlation between ligand chain length and NP solubility in nonpolar solvents. For example, Au NPs stabilized with longer-chain ligands (dioctadecylamine) exhibit higher dispersion stability than those capped with shorter ligands such as didodecylamine and dioctylamine.⁶⁹ Similarly, CdSe NPs capped with undec-10-enoic acid show lower solubility in toluene compared to those stabilized with the longer oleic acid.⁷⁰

However, this trend is not universal. MD simulations have shown that 5 nm Au NPs stabilized with C₁₈ alkanethiol can exhibit lower stability against aggregation at 300 K compared to those capped with the shorter C₁₂ alkanethiol.³² Notably, in that study the ligand grafting density (∼5.5 ligands per nm²) was sufficiently high to limit solvent penetration into the ligand shell.^25,32 It has previously been demonstrated that at such high grafting densities (>4.5 ligands per nm²), solvent accessibility is significantly reduced, which alters interparticle interactions.²⁵ Therefore, discrepancies in reported trends can be attributed to variations in grafting density, nanoparticle size, ligand chemistry, and solvent accessibility.

Consequently, the trends in ligand extension observed in Fig. 3c and d cannot be directly mapped onto prior studies, and the experimentally observed stacking of 4 ML CdSe NPLs in toluene cannot be explained solely by ligand length. Instead, we observe a pronounced difference in the distribution of oleate spans between methylcyclohexane (MCH) and toluene (Fig. 3d). Importantly, previous studies have shown that areal partitioning of ligands with different chain lengths (e.g., myristate (C₁₄) and hexanoate (C₆)) on nanoparticle surfaces, associated with increased rotational freedom, can enhance nanoparticle solubility by up to six orders of magnitude.⁷⁰

Although solvent-induced fluctuations in ligand spans are unlikely to produce a repulsive barrier comparable to that arising from chemically defined heterogeneity in mixed-ligand systems,⁷⁰ the dramatic solubility enhancement observed in such systems suggests that even subtle variations in ligand conformations may play a role. In this context, a modest difference in ligand spans (∼0.7 Å) may still influence the solubility of CdSe NPLs. We, therefore, propose that solvent-induced fluctuations leading to a bimodal distribution of ligand spans, even with a small amplitude (∼0.7 Å) in span observed in MCH, may contribute to the experimentally observed higher solubility of 4 ML CdSe NPLs in MCH compared to toluene.

While agglomeration of larger NPs is primarily driven by van der Waals attractions between inorganic cores, ligand–ligand interactions are recognized as the dominant attractive forces at smaller sizes.^71–74 Studies on Au NPs showed a crossover near ∼7.4 nm, below which interparticle interactions are ligand-shell dominated. For few-monolayer-thick CdSe NPLs, the effect is even more pronounced: the Hamaker constant of CdSe is approximately one-third that of Au,⁷⁵ making the ligand contribution especially significant.⁷⁶ Consequently, solvent-induced changes in ligand solvation can play a decisive role in governing interactions and assembly. Previously, MD simulations on cube-octahedral-shaped NPs with oleic acid shells also revealed more toluene molecules than hexane in the oleic acid corona.²⁵ It was proposed that the π–π interactions between the toluene rings are stronger than the ligand toluene interaction, resulting in stacking of the toluene molecules in order to maximize their interactions.²⁵ More rigid toluene molecules, that have only limited flexibility due to their methyl groups, “lock” the surface ligands, decreasing their conformational entropy. This explanation aligns with previously reported demonstration of the correlation between the rotational freedom of the surface ligands and the solubility of NPs.⁷⁷ Our results agree with the hypothesis that more toluene molecules can be accommodated within the oleate ligand corona²⁵ as compared to solvent molecules with more flexible structures (e.g. MCH or hexane). It is worth noting that the TEM results reported for purified NPLs deposited from hexane,⁴⁵ whose molecules also exhibit high conformational flexibility, are similar to those we obtained from MCH (Fig. 1a): in both cases, the NPLs are randomly distributed and oriented parallel to the substrate.⁷⁸

MD simulations revealed another interesting feature that can be relevant to stacking NPLs: they are not flat but twisted. Significant deformations are observed for 4 ML CdSe (Fig. 3g, h and Fig. S5–S8) in MD simulations for both toluene and MCH but with a larger magnitude for toluene than for MCH as quantified by the average pitch of the helix, which is smaller for toluene (259 nm) than for MCH (452 nm). A MD simulation snapshot visualizing how toluene molecules arrange themselves relative to the surface of 4 ML CdSe NPLs is shown in Fig. 3f. The structure of 4 ML CdSe NPLs has been extensively studied and is generally assigned to the zinc blende phase.^79,80 Twisting of zinc blende 4 ML CdSe NPLs with a pitch of approximately 400 nm has previously been observed in “dry” samples prepared in the presence of oleic acid ligands.⁴⁵ Fig. 3i and Fig. S9 present HRTEM images of 4 ML CdSe NPLs deposited from toluene, which preferentially orient on their edges, revealing pronounced twisting and associated lattice distortions. These observations are in good agreement with the trends observed in MD simulations.

Such a large effect of the solvent on the shape of the crystal is surprising at first glance but can be rationalized within the framework of incompatible curvatures developed by Monego et al.⁵⁴ It was shown that NPLs could be described as elastic ribbons whose effective curvature is proportional to the curvature imposed at the ligand–NPL interface.⁵⁴ In other words, the ligand monolayer imposes a curvature at the top and bottom surfaces of the NPL that depends both on the NPL/ligand interactions and on the lateral interaction between ligand tails.⁵⁴ Solvent penetration can significantly modify ligand–ligand interactions, thereby affecting the NPL curvature. A solvent that penetrates more deeply into the ligand layer, for instance, pushes the ligand tails further apart, increasing the lateral strain at the interface and ultimately driving a larger curvature of the NPL. This trend is confirmed experimentally by measuring the curvature of thinner NPLs in solution for MCH and toluene.

The small lateral extension of the 4 ML NPLs precludes the measurement of curvature using SAXS.^81,82 However, we measured in both solvents the SAXS pattern of 3 ML NPLs with larger lateral dimensions (Fig. S2b). The oscillations in the SAXS intensity are shifted towards lower wave vectors in MCH than in toluene. As established in an earlier study,⁸¹ a shift toward lower wavevectors in the SAXS oscillations corresponds to an increase in the radius of curvature, i.e. flatter NPLs. The 3 ML NPLs are, therefore, flatter in MCH than in toluene, which is qualitatively consistent with our MD simulation results. The solvent effect on curvature is likely more pronounced for 3 ML NPLs than for 4 ML NPLs owing to their smaller thickness.

The curvature of NPLs can affect their self-assembly by inducing large permanent dipole moments. Although the zinc blende structure of CdSe is nonpolar in the bulk, atomic displacements away from equilibrium lattice positions break the local symmetry and create a charge imbalance, endowing curved NPLs with a permanent ground-state dipole.⁸³ This dipole moment grows with curvature, as more pronounced bending stretches the crystal lattice further from its equilibrium configuration. Stronger dipole moments, in turn, enhance inter-NPL attraction through Keesom dipole–dipole interactions, which have been shown to be of comparable magnitude to other interactions in NP systems.⁸⁴ We estimated dipole moments from the atomic positions and partial charges extracted from the MD simulations. As expected, flat NPLs prior to crystal relaxation yield a dipole of exactly zero, since all partial charges compensate perfectly. After relaxation, the calculated dipole moments are 1063 ± 230 D in toluene and 678 ± 136 D in MCH, reflecting the larger curvature adopted by NPLs in toluene. These values correspond to dipole densities of 2–3 D nm⁻³, which are in good agreement with the order of magnitude measured experimentally via transient birefringence.⁸³

Therefore, both entropic effects induced by solvents in ligand shells and bending of CdSe NPLs can contribute to the difference in stacking behavior observed in MCH and toluene. While additional studies are required to quantify these contributions and their coupling, both drive 4 ML CdSe NPLs toward stacking, with toluene preferentially promoting stack formation.

The propensity of CdSe NPLs to form micrometer-long stacks in toluene motivated us to explore how this behavior can be harnessed for device fabrication. Previous work has pursued solution-processed strategies to build functional architectures, particularly low-threshold lasing structures, including tightly packed CdSe-based core–shell NPLs that reduce optical scattering and raise the effective refractive index.^85–90 Because 4 ML CdSe NPLs readily stack in toluene yet remain colloidally stable for weeks without precipitation, we investigated their use in forming three-dimensional optically active structures.

We applied a controlled oversaturation approach that was previously utilized for quasi-spherical and rod-like NPs,^5,41–43 by destabilizing toluene dispersions of 4 ML CdSe NPLs with methanol. It is worth mentioning that in the control experiment introducing 25 vol% methanol yielded sharp lamellar reflections in the SAXS pattern, confirming NPL stacking (Fig. 1e). We also observed a shift of the primary peak from q ≈ 0.117 Å⁻¹ to 0.119 Å⁻¹ indicating a decrease in inter-NPL spacing in the presence of methanol (Fig. 1e)

To grow superlattices, a 2 mL solution of CdSe NPLs in toluene (∼7 mg mL⁻¹) was placed in a crystallization tube containing a glass strip. A 2 mL layer of methanol was carefully added to avoid intermixing, and the tube was sealed. The density of toluene (0.867 g cm⁻³) is higher than the density of methanol (0.792 g cm⁻³) and therefore the formation of the aggregates took place mainly below the toluene/methanol interface. After two weeks, a uniform solid film of CdSe NPLs formed along the tube walls and the glass strip below the toluene/methanol interface. Large film fragments of micron size were easily delaminated using a metal needle (Fig. 4a and b). Note, that the destabilization of the MCH solution of 4 ML CdSe NPLs by either methanol or isopropanol also results in the formation of stacks (Fig. S10). This observation agrees with the previously reported assembly of CdSe NPLs in hexane induced by methanol or ethanol.^12,44 Note that the MCH solution of 4 ML CdSe NPs was placed above the non-solvents since the density of MCH (0.770 g cm⁻³) is lower than the density of both methanol and isopropanol (0.786 g cm⁻³) (Fig. S10). It is also worth noting that these stacks form a gel-like structure that makes them less suitable for optical studies as compared to solid assemblies prepared by destabilization of the toluene solution.

Fig. 4 Optical (a) and photoluminescence (b) images of the film formed via destabilization of the toluene solution of 4 ML CdSe NPLs by methanol; (c) two-dimensional SAXS pattern measured on self-assembled 4 ML CdSe NPLs shown in (a); (d) depiction of possible arrangement of 4 ML CdSe NPLs in the film; (e) emission spectra of 4 ML CdSe NPL film at different pump fluences at room temperature; (f) normalized integrated emission intensity from the 4 ML CdSe NPL film as a function of pump fluence at room temperature.

SAXS data from the films on glass strips deposited by stabilization of the toluene solution of 4 ML CdSe NPLs with methanol were collected under reflection geometry, known as grazing incidence SAXS (GISAXS). The incident angles used in this work were 0.2–0.5 degrees, which are higher than the critical angle of the substrate, to avoid X-ray refraction while taking advantage of the long footprint. For simplicity, we will refer GISAXS to SAXS since our data do not present any grazing incident effect.

The SAXS data shown in Fig. 4c demonstrate well-resolved diffraction pattern characteristics of structures with the preferred orientation of the CdSe NPLs; however, the overall crystalline character is more random as compared to superlattices formed by quasi-spherical NPs⁹¹ as is evident from the presence of rather broad arcs in the SAXS pattern. The center of the broad azimuthal arc is not in the lateral direction, or on the q_‖ axis, but it is tilted more than 10 degrees. In addition to those peaks due to the NPL stacking, there is another arc centered at 90 degrees in the smaller q region at a q of ∼0.0583 Å⁻¹, indicating the layering of NPL stacks along a direction normal to the film. No lateral peaks are found, indicating that the ribbons are not oriented parallel to each other so well in each layer. Fig. 4d depicts the possible arrangements of 4 ML CdSe NPLs within the film that can be reconstructed based on the SAXS data. The CdSe NPLs are arranged on the edge tilted by ∼10.5 degrees with respect to the substrate. The center-to-center distance is ∼45.9 Å and therefore the interparticle spacing is ∼32.9 Å given the thickness of 4 ML CdSe NPLs is ∼13 Å. From the peak width along the q direction, the coherent length in the self-assembled sample is estimated to be ∼6 layers. The center-to-center distance between neighboring NPLs in films is smaller than that in free-standing ribbons in toluene in the presence of methanol (∼52.7 Å).

Inspired by previous studies reporting lasing on CdSe NPLs assemblies,^1–4,89 we investigated if we could observe the amplified spontaneous emission (ASE) phenomenon in our structures. The films of 4 ML CdSe NPLs were excited with frequency-doubled pump pulses from an amplified Ti:sapphire laser system, focused on a stripe along the sample. When the pump fluence exceeded 14 µJ cm⁻², a narrow peak appeared due to ASE (Fig. 4e). Lower ASE threshold values were previously reported only for more sophisticated NPLs such as CdSe/CdS core/shell NPLs.^{2,85,89,92,93} The intensity of the ASE peak increases rapidly with pump fluence (Fig. 4f). The CdSe NPLs have well-studied polarized emission related to their transient dipoles.^6,83,87,94 We believe that destabilization of the toluene solution containing stacks of 4 ML CdSe NPLs by methanol can generate structures with lower inhomogeneity of the amplified wave. Since the stacks of NPLs are a structure with aligned dipoles, it is reasonable to suggest that dipole alignment can be responsible for the lasing in the self-assembled structures. This observation agrees with recently published data on the role of dipole alignment of NPs in the generation of polarized emission.⁹⁵ The alignment of the dipole moments of CsPbBr₃ NPs in the superlattice structure was reported to result in a macroscopic polarization effect producing a net polarization in the direction of the alignment, resulting in spontaneous linear polarized emission.⁹⁵ After ∼6 months of storage at room temperature under nitrogen, the self-assembled films stopped emitting ASE, and the SAXS data revealed a significant change in the ordering of NPLs (Fig. S6). Instead of one peak at a q of ∼0.137 Å⁻¹, the samples started to reveal two broad split peaks at ∼0.131 Å⁻¹ and ∼0.155 Å⁻¹, respectively without any preferred orientation, and the peak from the packing of ribbons was no longer observed (Fig. S11). These changes are indicative of the rearrangement of the CdSe NPLs in the self-assembled structures. Because the GISAXS measurements were performed on oriented films, we cannot exclude a geometric (projection) effect. For example, when a stack of d-spacing 48 Å, corresponding to a q of ∼0.131 Å⁻¹, is tilted by 32 degrees, its d-spacing will decrease to 40.5 Å, corresponding to 0.155 Å⁻¹. The SAXS pattern of the stored samples demonstrates stacks of smaller numbers of NPLs, randomly oriented without any stack-to-stack correlation (Fig. S11). The peak broadening suggests a distribution of spacings and/or orientations, which may arise from bending, fragmentation, or slipping between neighboring CdSe NPLs. An increase in the interparticle spacing may originate from the solvent loss from the ligand shell. Interestingly, as indicated by the MD simulations, in toluene the maximum end-to-end distance (R_ee ∼ 17.4 Å) of the oleate molecules exceeds the corresponding span (∼15.7 Å) (Fig. 3c and d). This suggests that solvent evaporation may be accompanied by relaxation of the ligand chains into more extended conformations, leading to an expansion of the effective ligand shell. Consistent with this interpretation, we have previously demonstrated for PbS NPs capped with oleic acid that expulsion of toluene results in a more extended ligand shell.⁵ In line with this structural perturbation, no ASE was observed following the transformation that directly points to the correlation between the structure of the assembled CdSe NPLs and ASE.

It is worth noting that we also did not observe ASE in samples prepared by fast drying of the toluene solutions. Such samples are characterized by larger interparticle spacing, as indicated by a signal with a q of ∼0.129 Å⁻¹ and a smaller coherent length (∼4 layers). Also, the stack layering peak was not observed. The representative SAXS pattern of the sample prepared via fast drying of the toluene solution is shown in Fig. S12. Even though the SAXS pattern shows a very sharp diffraction peak characteristic of NPL stacking (Fig. S12), it does not show ribbon packing or the layering peak that was present in the SAXS spectrum of the self-assembled structures, showing lasing.

Conclusions

Our results indicate that the solvation of the ligand shell itself can play a significant role, and this effect can go beyond just the effect of increased or decreased concentrations of the solvent molecules in the ligand shell. We attribute this difference to the higher flexibility of MCH molecules as compared to toluene molecules. The cyclohexane ring in MCH introduces conformational flexibility allowing different chair conformations to be adopted and increasing the number of possible molecular arrangements. In contrast, planar toluene molecules with rigid aromatic structures do not exhibit the same level of conformational freedom as MCH molecules. We demonstrated that in toluene, the 4 ML CdSe NPLs form long stacks that tend to collapse into smaller fragments upon fast evaporation of the solvent, while in MCH, they remain well-dispersed. This behavior can substantially influence the solution processability of the individual components and affect the properties of the functional structures and devices. It can affect the structure and its homogeneity of the deposited NPL assemblies and roughness, which significantly impacts, for example, the polarization of the ASE.⁹⁶ It can also impact the efficiency of the post-preparative chemical processes involved in chemical manipulations, for example, surface modification or synthesis via cation exchange reactions.⁵⁵

While oleic acid stabilized quasi-spherical NPs form stable colloidal solutions in toluene,⁵ which is one of the most common solvents for a broad range of colloidal NPs synthesized in nonpolar solvents, CdSe NPLs are found to form micron long ribbons in toluene. Through MD simulations, we identified that toluene results in more extended and more solvated ligand shells and a bimodal distribution of spans of the surface oleate in MCH. The drastic difference in the stacking behavior of CdSe NPLs in toluene and MCH opens the possibility of controlling the stacking by adjusting the composition and the solvent ratio in solvent mixtures. We also demonstrate that solvation of the ligand shell can influence the deformation of CdSe NPLs, which, in turn, can induce strain. In the case of NPLs, this strain can contribute to the formation of a dipole moment, ultimately affecting interparticle interactions. The strains can also affect the chemical activity of the NPLs.^97,98 Furthermore, we speculate that self-assembled configurations may enhance the material's optical performance, particularly in ASE. Notably, CdSe NPLs assembled from toluene exhibit a low ASE threshold of 14 µJ cm⁻² that is comparable with the value obtained on assemblies formed from more sophisticated CdSe/CdS core/shell NPLs. The samples in which the NPL arrangement was altered during storage do not exhibit ASE, thereby directly highlighting the importance of structural order for ASE values. This study further emphasizes the need to view the ligand shell as a composite ligand–solvent corona, not a dry monolayer of ligands, and offers valuable insights into the solvent-dependent behavior of CdSe NPLs, which can inform the design of functional optical structures for applications such as lasing and optoelectronic devices as well as guide the solvent choice for cation exchange reactions in NPLs.

Methods

Chemical reagents

Sodium myristate (≥99%), cadmium(II) acetate (Cd(OAc)₂; 99.995%), cadmium acetate dihydrate (Cd(OAc)₂·2H₂O), selenium powder (Se, ≥99.5%), octadecene (ODE; 90%), oleic acid (90%), toluene (98%), anhydrous isopropanol (99.8%), and anhydrous methanol were purchased from Sigma Aldrich. All chemicals were used without further purification.

Preparation of cadmium myristate

Cadmium nitrate tetrahydrate (1.23 g) was dissolved in 40 mL of methanol and 3.13 g of sodium myristate was dissolved in 250 mL of methanol under strong stirring; these solutions were then combined and stirred for approximately one hour. The whitish product was centrifuged at 6000 rpm for 10 minutes, the supernatant was discarded, and the white precipitate part was dissolved in 20 mL of methanol. The resulting precipitate of cadmium myristate was filtered and washed several times with methanol for the removal of excess precursors and the finally obtained precipitate was dried for 24 h under vacuum.

Synthesis of 4 ML CdSe NPLs

The synthesis was performed according to the literature method.⁵ Cadmium myristate (340 mg), 24 mg of Se, and ODE (30 mL) were added to a three-neck 100 mL flask. The solution was degassed under vacuum at 95 °C for 1.5 h. Then, the temperature of the solution was set at 240 °C under argon flow with a 30 °C min⁻¹ heating ramp rate. When the temperature reached 190 °C, 120 mg of cadmium acetate dihydrate powder was swiftly added. After 10 min of growth at 240 °C, the solution was cooled to room temperature by removing the heating mantle and using a heat gun in ambient/cold mode to quickly bring down the temperature. When the temperature had dropped to 100 °C, 2 mL of oleic acid was added to the flask.

Purification of post-synthesis byproducts and size selection of NPLs

After the synthesis, the reaction solution contained small fractions of 3 ML and 5 ML CdSe NPLs and therefore, we applied additional purification steps to remove the by-products of 4 ML CdSe NPLs. For that, when the solution reached room temperature, 15 mL of hexane was added. The mixture was then centrifuged at 5000 rpm for 10 min. The precipitate containing the NPLs was resuspended in 40 mL of toluene and hexane in a 1 : 1 ratio and allowed to sit for 2 h. Most of the quantum dots (red, emitting at 550 nm) were removed through another centrifugation procedure at 5000 rpm for 10 minutes as a reddish supernatant while the 3 ML and 4 ML NPLs were collected as precipitates. The resultant dispersion was subjected to centrifugation twice at 5000 rpm for 10 minutes, to remove the reaction byproducts.

For size selection between the 4 ML and 3 ML CdSe NPLs, the precipitate from the earlier steps was redispersed in 15 mL of n-hexane with 2 mL of ethanol and centrifuged again at 3000 rpm for 20 minutes. The yellow supernatant, containing 4 ML CdSe NPLs, was collected and stored at room temperature until the remaining 3 ML NPLs precipitated from the n-hexane dispersion, within approximately 1–2 weeks. Then, the agglomerated 3 ML CdSe NPLs were removed by centrifugation at 5000 rpm for 5 minutes. 4 ML CdSe NPLs in the supernatant were precipitated to remove the remaining 1-octadecene by adding 5 mL of methyl acetate and storing the dispersion in a fridge at 5 °C for 4 hours. After another centrifugation step at 5000 rpm for 5 minutes with a 10% volume fraction of ethanol added to hexane, the 4 ML CdSe NPLs were collected as a precipitate, redispersed in 20 mL of toluene, and then stored at room temperature.

Synthesis of 3 ML CdSe NPLs

The synthesis was performed according to the literature method.⁹⁹ In a 100 mL three-neck flask, 14 mL of 1–octadecene (ODE), 185 mg of cadmium acetate dihydrate, and 190 µL of oleic acid were mixed and degassed for 30 min. The solution was heated at 220 °C under argon. At the same temperature, 300 µL of 1 M TOP–Se was injected. The reaction mixture was kept at 220 °C for 10 min, resulting in the formation of 3 ML CdSe NPLs. At the end of the growth, 2 mL of oleic acid was added, and then the mixture was cooled to room temperature. The NPLs were purified by centrifugation, redispersed in hexane, precipitated with ethanol, and finally dispersed in toluene or MCH.

Synthesis of 5 ML CdSe NPLs

The synthesis was performed according to the literature method.⁹⁹ 340 mg of cadmium myristate and 28 mL of ODE were loaded into a 50 mL three-neck flask. The solution was stirred and degassed at room temperature for 30 min and at 95 °C for an hour under vacuum, respectively. After this, the heater was set at 250 °C, the vacuum was broken at 100 °C and the flask was filled with argon gas. When the temperature of the solution reached 250 °C, a pre-prepared solution of 24 mg of Se dispersed in 2 mL of ODE was swiftly injected into the hot solution. When the color of the solution became orange, 240 mg of cadmium acetate dehydrate was rapidly introduced. Then, the solution was kept at 250 °C for 10 minutes and 1 mL of OA was injected before cooling down to room temperature using a water bath. After size-selective precipitation, the 5 ML NPLs were dissolved and stored in toluene or MCH.

Optical absorption and photoluminescence spectroscopy

Absorption spectra were recorded using an Agilent Technologies Cary 60 UV-vis spectrophotometer. Photoluminescence spectra were recorded using a Fluorolog iHR 320 Horiba Jobin Yvon spectrofluorometer equipped with a PMT detector. The excitation wavelength was set at 420 nm.

Self-assembly experiments

2 mL of CdSe NPLs in toluene solution was added to the crystallization tube. Then, 2 mL of methanol was carefully added on the top of the toluene solution, avoiding the mixing of their interface. In ∼2 weeks, the CdSe NPLs completely precipitated in the form of a continuous film on the glass surface of the crystallization tube and inserted glass strip.

Small-angle X-ray scattering

SAXS data were measured for the solution samples in a capillary of 1.5 mm diameter or for self-assembled structures adhered to glass cover slips at the 12-ID-B beamline at the Advanced Photon Source in the Argonne National Laboratory. For solution measurements, an X-ray beam with an energy of 14 keV, or wavelength λ = 0.8856 Å, and a size of 300 × 100 μm² was used, with each sample typically exposed to the beam for a few seconds. The scattering data were collected using a Pilatus2M detector located about 2 m away from the samples. Absolute intensity was calculated using water as a standard.

Transmission electron microscopy imaging

TEM images of the CdSe NPLs were recorded using a Talos microscope at an acceleration voltage of 200 kV. Samples for the ex situ studies were prepared by drop-casting NPLs in solution onto ultrathin carbon 300-mesh Au/400-mesh Ni grids from Ted Pella followed by drying under vacuum overnight. For liquid cell experiments, a drop of the solvent (toluene, methylcyclohexane or 1-butanol) containing the 4 ML NPLs was loaded into two graphene TEM grids and directly mounted onto the Talos double-tilt TEM holder. TEM imaging was carried out in the STEM mode, and videos were captured using the Velos software.

Molecular dynamics (MD) simulations

Molecular dynamics (MD) simulations of CdSe nanoplatelets (NPLs) coated with oleate ligands were performed using the LAMMPS software package. All ligand and solvent atoms were explicitly included in the molecular model.¹⁰⁰ The OPLS-AA force field,¹⁰¹ with parameters obtained from the LigParGen web server,^102–104 was used to model the interactions for both ligand and solvent atoms. For CdSe, the interaction potential consisted of a short-range 12–6 Lennard-Jones term combined with a long-range coulombic contribution, with parameters taken from Rabani's work.¹⁰⁵ The Lennard-Jones parameters between the CdSe atoms and the atoms from the ligands and the solvent were obtained using the geometric mixing rule. CdSe NPLs with Cd-terminated [001] polar surfaces were constructed by first placing the Cd and Se atoms on a zinc blende lattice and then cutting the structures with a set of parallel planes to obtain the desired number of monolayers (MLs) and lateral dimensions. In this work, we have considered 4 ML NPLs with lateral extents of 24 × 8 nm (rectangular), such that the (110) direction made an angle of 45° with the NPL edge. A sufficient number of oleate ligands were placed randomly within a thin shell surrounding the NPL to ensure charge-neutrality (∼3.27 ligands per nm²), and the resulting structure was placed at the center of a simulation box filled with solvent molecules. The arrangement of the ligand as well as the solvent molecules was carried out using the PACKMOL software package.¹⁰⁶ The simulation box size was chosen to be big enough such that all NPL edges were at least 7 nm away from the boundary. For example, for a 4 ML 24 × 8 nm NPL, the box size was 38 × 11 × 11 nm; with methyl cyclohexane as the solvent, that would bring the system size to ∼1.9 million atoms. Periodic boundary conditions were applied along all three dimensions. The initial configuration was relaxed via a series of steps. Holding the positions of the crystal atoms fixed, we performed: (i) a step-limited constant energy run for 40 ps followed by energy minimization to remove any significant overlap of the atoms; (ii) a temperature ramp-up to 298 K over 30 ps; (iii) further relaxation at 298 K for 30 ps; and (iv) relaxation at 298 K and 1 atm pressure for 30 ps. During these steps, the temperature was controlled using a Langevin thermostat and the pressure using a Berendsen barostat. Next, the entire system (including the crystal) was relaxed at 298 K and 1 atm using a Nosé–Hoover thermostat and barostat for 200 ps, and finally followed by a production run of at least 5 ns, depending on the NPL size and whether significant fluctuations persisted in the resulting configurations. The configurations were sampled every 10 ps. All results presented here have been averaged over all the sampled configurations. The equations of motion¹⁰⁷ were integrated using a Verlet-like scheme¹⁰⁸ with time steps of 1 fs during steps (i)–(iv) and 0.25 fs for the rest. Simulations performed to determine the ligand and solvent density profiles held the crystal atom positions fixed throughout. Some runs were also performed in the absence of any solvent for comparison purposes. For simulation runs where the crystal atom positions were allowed to evolve in time, the NPLs exhibited a helicoidal configuration after relaxation. The pitch corresponding to the helicoid twist was calculated based on the linear relation between the angular displacement of the midplane surface normal with the distance along the centerline. The method for reconstructing the midplane from the atom positions is discussed in ref. 54.

Author contributions

P. B., B. T. D., P. C. d. S. C., B. W. and R. V. performed the synthesis. P. B. conducted the UV-vis, PL, and dry state TEM characterization of samples. P. B. and E. V. S. performed data analyses and wrote the manuscript. S. D. performed MD simulations and data analysis, and B. A. supervised the computational part of the study and the 3 and 5 ML work. Y. L., J. G. W. and P. B. conducted the liquid cell and aberration corrected microscopy studies. C. E. R. and R. S. conducted lasing experiments and analyzed data. P. B., E. V. S., X. Z., R. V., B. A. and B. L. conducted SAXS experiments. B. L. and B. A. provided analysis of SAXS data. X. M. L., E. V. S. provided project guidance. P. B. and E. V. S. prepared the manuscript with the input of all co-authors. All authors have given approval to the final version of the manuscript.

Conflicts of interest

Authors have no competing interests to declare.

Data availability

The data supporting this article have been included as part of the supplementary information (SI). Supplementary information: size-selection protocol for 4 ML CdSe NPLs; the SAXS spectra measured for 5 ML and 3 ML CdSe NPLs dispersed in MCH and in toluene; dark-field STEM images of different 4 ML CdSe NPL samples in toluene loaded in a liquid cell; high resolution images of the MD snapshots for 4 ML CdSe NPLs with oleic acid ligands and with ligand stripped; HRTEM images of 4 ML CdSe NPLs deposited from toluene solutions; summary of the SAXS data on destabilization of 4 ML CdSe NPLs in MCH by isopropanol and methanol; two-dimensional SAXS pattern measured on self-assembled 4 ML CdSe NPLs after 6 months of storage in a glovebox; two-dimensional SAXS pattern measured on the film prepared by the evaporation of the toluene solution of 4 ML CdSe NPLs. See DOI: https://doi.org/10.1039/d6nr00826g.

Acknowledgements

Work performed at the Center for Nanoscale Materials, a U.S. Department of Energy Office of Science User Facility, was supported by the U.S. DOE, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. This article is part of a project that has received funding from the European Research Council under the European Union's Horizon 2020 research and innovation program (Grant agreement No. 865995). This work was granted access to the HPC resources of Institut du Développement et des Ressources en Informatique Scientifique (IDRIS) under the allocation 2024-AD010913529R2 made by Grand Équipment National de Calcul Intensif (GENCI) and was supported by the LABEX iMUST of the University of Lyon (ANR-10-LABX-0064), created within the program “Investissements d'Avenir” set up by the French government and managed by the French National Research Agency (ANR). We also gratefully acknowledge support from the PSMN (Pôle Scientifique de Modélisation Numérique) computing centre of the École Normale Supérieure de Lyon.

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