Ligand-induced incompatible curvatures control ultrathin nanoplatelet polymorphism and chirality

Significance

Applying forces at the surface of thin objects enables the emergence of various chiral three-dimensional shapes such as helices, twists, and rolls. At the nanoscale, ultrathin nanoplatelets (NPL) are extended sheet-like crystalline particles coated with a layer of organic ligands that display such polymorphism, but the physical reasons behind such shape-selectivity are poorly understood. We show that the ligands impose surface stresses that differ between the top and bottom surfaces, eventually forcing the NPL to its deformed shape. We further identify a key parameter, the spontaneous curvature, that encapsulates all the molecular-level details of the ligand-crystal interaction and determines the shape of ligand-coated NPLs based on a small number of global physical quantities such as size, thickness, and lattice orientation.

Abstract

The ability of thin materials to shape-shift is a common occurrence that leads to dynamic pattern formation and function in natural and man-made structures. However, harnessing this concept to rationally design inorganic structures at the nanoscale has remained far from reach due to a lack of fundamental understanding of the essential physical components. Here, we show that the interaction between organic ligands and the nanocrystal surface is responsible for the full range of chiral shapes seen in colloidal nanoplatelets. The adsorption of ligands results in incompatible curvatures on the top and bottom surfaces of the NPL, causing them to deform into helicoïds, helical ribbons, or tubes depending on the lateral dimensions and crystallographic orientation of the NPL. We demonstrate that nanoplatelets belong to the broad class of geometrically frustrated assemblies and exhibit one of their hallmark features: a transition between helicoïds and helical ribbons at a critical width. The effective curvature $\overline{\kappa }$ is the single aggregate parameter that encodes the details of the ligand/surface interaction, determining the nanoplatelets’ geometry for a given width and crystallographic orientation. The conceptual framework described here will aid the rational design of dynamic, chiral nanostructures with high fundamental and practical relevance.

Helical structures are fascinating chiral motifs that are found at different length scales in Nature. Ranging from the molecular scale of DNA and proteins to large-scale structures like tornadoes or galaxies, helices can be stabilized by a broad range of driving forces. Beyond their aesthetic appeal, they can impart useful properties, which has inspired the design of novel chiral materials with specific mechanical or optoelectronic characteristics. At the nanometer scale, helical shapes are especially coveted (1) as their chirality can endow them with desirable properties including chirality-induced spin selectivity, circularly polarized luminescence or circular dichroïsm. In this realm, nanoplatelets (NPL), a class of ultrathin 2D nanoparticles coated with a monolayer of surfactants (2), are particularly relevant because they display both outstanding optical properties and can be deformed into helical shapes and assemblies. Aside from flat and tubular shapes, two kinds of helices have been observed in NPL: purely twisted helicoids with nonzero Gaussian curvature and a straight centerline and helical ribbons which have a cylindrical curvature and a helical centerline (3). The underlying physical mechanism which gives rise to the emergence of this polymorphism is still unknown. It is clear from previous work that ligands play a significant role in controlling NPL shape, since ligand exchange can induce large shape variations in CdSe and CdTe NPL (4–7). However, the interplay between the NPL width, thickness, pitch, radius of curvature, crystallographic structure/orientation, and the molecular interactions at and near the ligand/crystal interface needs further rationalization for predictive design to be possible. It is not yet clear which of these parameters are most relevant for shape selection and hence which knob one has to turn to achieve the desired structure. Moreover, the relevant parameters which trigger the transitions between these different shapes have not been identified, which calls for a unifying conceptual framework that can rationalize the complete design space.

We provide here such a framework that comprehensively explains the polymorphism of NPL and the emergence of chirality in these systems. The coexistence of tubes, helicoïds and helical ribbons in NPLs leads us to hypothesize that they belong to geometrically frustrated systems in which the bending and stretching energies compete to dictate the shape of thin ribbons along with geometrical constraints. Diverse systems at different length scales, such as surfactants or chiral seed pods obey such a formalism (8–11), with the common physical ingredients being a spontaneous curvature of microscopic origin and a slender geometry. Using extensive molecular dynamics simulations, along with theory and experiments, we show that the spontaneous curvature comes from interaction between the ligands and the NPL crystal surface, which induces directionally anisotropic stresses on the top and bottom surfaces and, hence, preferred curvature along different crystallographic axes. Chirality results from the angle between the edge of the NPL and the directions of preferred curvature, while the NPL’s specific shape is dictated by the scaling of the bending and stretching energies with respect to its width and thickness.

Polymorphism of CdSe Nanoplatelets

We focus on zinc blende CdSe NPL (12, 13) coated, in their native state by carboxylate groups. NPL can display different geometries depending on their lateral dimensions, thickness, and surface functionality, as illustrated in Fig. 1. NPL with lateral dimensions from 5 to 15 nm can either be flat or display a twisted helicoidal shape with a nonzero Gaussian curvature in which the centerline corresponds to the [100] crystallographic axis. Tube-like geometries with a mean radius of curvature from 10 to 20 nm are observed when the NPLs have a larger lateral dimension (hundreds of nanometers). Several reports show tubular NPL with principal curvatures along either the [110] or [1$\overline{1}$0] axes (14–16). Finally, helical ribbons have been reported (17, 18). In this geometry, the Gaussian curvature is also zero, but the centerline is a helix that rotates along the [110] axis with a defined pitch, e.g., $p=25\mathrm{nm}$ (17) and $p=122\mathrm{nm}$ (18). This geometry is observed for intermediate widths $w=27\mathrm{nm}$ (18) when one of the lateral dimensions is much longer than the other. We thus observe a variety of shapes (helicoid, helical ribbons, and tubes) which depend on the NPL’s width and the crystal lattice’s orientation with respect to the edges. Furthermore, several reports provide evidence of shape changes when the surface chemistry of the NPL is modified via ligand exchange (5–7, 14, 15). This leads us to hypothesize that the interaction between the surface of the NPL and the organic ligands is critical in determining the shape of the NPL.

Fig. 1.

CdSe colloidal nanoplatelet polymorphism. Three different shapes have been observed for CdSe NPL: (A) helicoids with a nonzero Gaussian curvature and a straight centerline, (B) helical ribbons with a helical centerline and a zero Gaussian curvature, and (C) tubes with a circular centerline and a zero Gaussian curvature. For each case, an idealized shape (Top), a snapshot from an MD simulation (Center) and an electron microscopy image (Bottom) are shown. The MD simulation snapshots depict 3ML NPLs with a thickness of 1.1 nm and lateral dimensions of 150 × 10 nm. The TEM images are adapted from refs. 4, 18, and 14.

Ligand Induced Incompatible Curvatures

To better understand the key ingredients of the ligand/NPL system, we performed an extensive set of molecular dynamics (MD) simulations. We considered zinc-blende CdSe NPL whose top and bottom facets are cadmium-rich (001) crystallographic planes coated with acetate and/or oleate ligands (see Materials and Methods and SI Appendix for details). Along its thickness, an N monolayer (ML) NPL presents N+1 cadmium planes and N selenium planes, corresponding to a thickness t between 0.6nm (for 2 ML) and 3.5nm (for 11 ML). We first simulated square 3 ML CdSe NPL ($t=1.1\mathrm{nm}$, $L=10\mathrm{nm}$) with sufficient acetate ligands to provide charge neutrality ($3.7\text{ligands}/{\mathrm{nm}}^2$, close to the experimentally measured surface density of $4.8\text{ligands}/{\mathrm{nm}}^2$) (19). Ligands are first adsorbed on the flat NPL surface while the crystalline configuration of the NPL remains fixed. Fig. 2C shows a resulting configuration of ligands on the Top and Bottom surfaces of the NPL before (Left) and immediately (1ps) after (Right) relaxation of the crystal. For all of the monolayer compositions studied (acetate, oleate, and a 1:1 mixture of the two), the azimuthal distribution of the ligands is strongly peaked at 45^° and −45^° (Fig. 2E), meaning that most of the carboxylate moieties are oriented with their O-O axis perpendicular or parallel to the [110] axis of the crystal (Fig. 2D). The ligand tail length influences their polar orientation (Fig. 2F), with acetate ligands having a peaked $p(\theta )$ distribution. At the same time, monolayers that include oleic acid chains display a long tail distribution with a significant probability for their tail to be perpendicular to the surface. We found that acetate ligands bind to the Cd-rich (001) surfaces in two ways, shown in Fig. 2C. In the bridging mode i), the acetate molecule stands symmetrically on top of two Cd atoms. In contrast, in the tilting mode ii), the acetate straddles a Cd atom asymmetrically, with one of the oxygens on top of the Cd atom and the other in the space between two Cd atoms. These two positions have preferred orientations: The first is observed along the dense Cd-Se-Cd rows (which are perpendicular to each other on the Top and Bottom facets, see Fig. 2B). In contrast, the second position generally happens perpendicular to the dense rows. The dominant binding mode in our simulations is the tilting mode (ii), oriented perpendicular to the dense rows. This is also the preferred tilting mode predicted by DFT (20, 21), though ref. 21 argues that a chelating mode is globally preferred. As discussed below, the direction in which the NPL bends appears to be inherent to the crystal structure, with the overall shape determined by the crystal orientation and dimensions and the ligand type.

Fig. 2.

Ligand interaction with the NPL surface. (A) Crystallographic structure of CdSe zinc-blende NPL. The Top and Bottom facets correspond to the (001) planes. (B) View along the [001] axis showing how Cd and Se atoms in adjacent layers are bonded. (C) Simulation snapshots of acetate ligands binding onto the Top and Bottom surfaces of a CdSe NPL (1.1 nm thick) along the [001] direction. (D) Schematic of the COO⁻ binding moiety of acetate and oleate ligands on the surface of a CdSe NPL. The vector $\widehat{\mathbf{n}}$ is the unit normal to the plane formed by the carbon and two oxygen atoms, which makes an angle $\theta$ with the $\mathbf{z}$-axis and whose projection on the $\mathit{xy}$-plane makes an angle $\varphi$ with the $\mathbf{x}$-axis. The axes $\mathbf{x}$, $\mathbf{y}$, and $\mathbf{z}$ represent the [100], [010], and [001] crystallographic directions. z is the perpendicular distance of an atom of the binding moiety from the $\mathit{xy}$-plane. Probability density functions of the angles $\varphi$ (E) and $\theta$ (F) associated with the normal to the plane formed by the atoms of COO⁻ for different ligands coating a CdSe NPL. The labels denote the following: Ac—acetate coated NPL; Ol—oleate coated NPL; AcOl—1:1 mixture of acetate and oleate.

Elastic Theory of Thin Sheets

When the ligands reach their equilibrium configuration, the NPL crystal is relaxed yielding insight into how it deforms under the effect of surface stresses induced by the ligands. We first consider square NPL with a 10 nm width and edges parallel to the [110] and [$\overline{1}$10] directions (Fig. 3). The relaxed NPL displays a saddle shape, as can be seen in the MD snapshot (Fig. 3A) and in the displacement field in Fig. 3C. This characteristic shape has nonzero Gaussian curvature with a positive principal curvature direction along [110] and a negative principal curvature along [$\overline{1}$10]. Our simulations are thus consistent with the experimental observation of preferred bending along the [110] directions. To understand the origin of this deformation, we mapped the in-plane forces on the crystal atoms averaged over 100 to 300fs after the beginning of the crystal deformation (Fig. 3B). We track down the origin of the deformation to ligand-induced stresses on the top and bottom surfaces, which are directionally anisotropic (SI Appendix, Fig. S1). The anisotropy in stress can also be seen in SI Appendix, Fig. S3, particularly in the case of the midplane. On the top surface, the anisotropic stress tends to bend the crystal in the [110] direction hence imposing a preferred positive curvature, while on the bottom surface, the stress direction is orthogonal and thus yields a preferred curvature along the [$\overline{1}$10] axis. Due to surface stress, a strain gradient appears along the [001] axis of the NPL, generating a spontaneous curvature.

Fig. 3.

(A) Simulation snapshots of a square 3 ML CdSe NPL with 10nm edge length and thickness 1.1 nm. The two viewing directions illustrate the nonzero Gaussian curvature. (B) Spatial variation of in-plane forces (in pN) on the atoms of the midplane calculated by averaging over 100 to 300fs after the commencement of deformation. The colors of the arrows indicate the magnitude of the forces. (C) Equilibrium displacement field of the NPL midplane (in nm) along the [001] direction. (D) Simulation snapshots for square NPL of increasing thickness (3 to 11 ML, i.e., 1.1 to 3.5 nm in thickness) from Left to Right. (E) The radius of curvature (nm) as a function of the number of monolayers in experiments and MD simulations. Experimental points come from Po et al. (5). The dashed lines represent fits to Eq. S9 of the SI Appendix with fitted parameters $\mathrm{\Psi }{\kappa }_0=0.1328$ for experiments and 0.0712 for simulations and $e=2.752\mathrm{nm}$ for experiments and 2.456nm for simulations.

The directional stress anisotropy between the top and bottom layers is due to the interplay between the crystalline structure of the NPL and the cutting planes at their top and bottom. The NPL synthesis involves carboxylates, X-type ligands that bind preferably to cations (22). Hence, the top and bottom surfaces are cadmium-rich [001] surfaces. The improper rotation symmetry of the $\overline{4}$ axis in the zinc blende ($\overline{4}3m$) structure means that as we go from the top surface to the bottom, the Cd-Se bonds rotate by 90^° for each monolayer added (Fig. 2B). These bonds thus adopt a helical configuration whose axis is oriented along the [001] direction. In the end, the Cd-Se bonds at the top and bottom of the NPL are always oriented orthogonal to each other. Chirality arises when the edges of the NPL are misaligned with the directions of preferred curvature, inducing twisting, and one edge is longer than the other, such that one handedness of twisting has lower energy than the other. This is further discussed in the next section. Note that the mid-plane of the NPL is not a mirror symmetry plane since the reflection of a Se atom in the bottom part corresponds to a vacant tetrahedral site in the top part. There are also no odd–even effects in the number of monolayers, with the preferred curvatures always orthogonal, whatever the number of monolayers, as shown in detail in SI Appendix, Fig. S4. This is consistent with twisted geometries observed for 3, 4, and 5 ML CdSe NPLs (3, 23).

Equilibrium configurations of NPL thus result from the minimization of elastic energy, with curvatures imposed at the top and bottom layers by the surfactant monolayer. This elasto-geometrical problem can be solved using the incompatible elasticity theory of thin sheets, in which the total elastic energy is described as the sum of bending and stretching energies. The two terms can be expressed as a function of metric and curvature tensors and the material’s mechanical properties. This framework has proven accurate in describing a wide variety of microscopic and macroscopic systems ranging from surfactant assemblies (24) to chiral seed pods (25). In our case, the ligands dictate a curvature only at the top and bottom surfaces (SI Appendix, Fig. S1), while strains beyond the first atomic layers result from strain propagation along the NPL thickness toward the mid-plane. We developed a multi-layer model in the incompatible elasticity framework, which features these boundary conditions (Theory in SI Appendix). The derived energy functional shows that NPL can be described as ribbons with a pure spontaneous twist, where the effective curvature is given by

$$\begin{array}{c}\hfill \overline{\kappa }={\kappa }_0\mathrm{\Psi }\frac{{\mathrm{\Phi }}^3}{1+{\mathrm{\Phi }}^3},\end{array}$$ [1]

where ${\kappa }_0$ is the curvature at the NPL/ligand interface, $\mathrm{\Psi }=Y_{\mathrm{lig}}/Y_{\mathrm{bulk}}$ is the ratio between the interfacial and bulk Young’s moduli, and $\mathrm{\Phi }=e/t$ is the ratio between the effective thickness of the surfactant layer and the NPL thickness.

Comparison with Simulations and Experiments

To test the scaling between the effective curvature and the thickness, we simulated 10nm by 10nm square NPL and calculated their radii of curvature along the [110] and [$\overline{1}$10] directions as a function of NPL thickness (for acetate ligands). Results from these simulations, as well as similar experiments (5), are shown in Fig. 3E. While there are numerical differences between the two, this is not surprising given the experiments were done on much larger NPL that lie in the stretching-dominated regime and form tubes with cylindrical curvature rather than saddle shapes. Fitting the variation of curvature with thickness to Eq. 1 with two unknown parameters yields an excellent fit with very similar values for the effective thickness of the surfactant layer e, which is, as expected, of the same order of magnitude as the thickness of the NPL. The larger deviation observed for the ${\kappa }_0\mathrm{\Psi }$ prefactor between experiments and simulations can be explained by the difference in experimental and simulation size regimes, as noted above. Other possible contributions could be the accuracy of the force field used to describe the ligand-surface interactions or uneven ligand coverage between the NPL surfaces in experiments (26). Still, the overall agreement between theory, experiment, and MD simulations proves the relevance of our model in grasping the key physical ingredients at play.

An important parameter for shape selection in incompatible elastic ribbons is the angle between the principal direction of curvature and the ribbon’s longitudinal axis. Since we previously showed that ligands induce principal curvatures in the [110] and [$\overline{1}$10] directions, we define $\alpha$ as the angle between the long edge of the NPL and the [110] axis. Fig. 4A shows simulation snapshots for $\alpha$ varying from 0 to $\pi /2$ for NPL with a ribbon geometry ($L_x=10$ nm and $L_y=100$ nm) covered in a 1:1 mixture of oleate and acetate ligands. The helicoid is the favored geometry for $\alpha =\pi /4$. As the angle increases toward $\pi /2$ (or decreases toward 0), the shape unwinds to finally reach a circular tube-like shape for $\alpha =\pi /2$ (or 0). For chiral shapes, corresponding to $\alpha \ne \pi /2$ or 0, the two enantiomers are obtained for $\pi /2-\alpha$ and $\pi /2+\alpha$. We stress that shapes corresponding to $\pi /4-\alpha$ and $\pi /4+\alpha$ are not enantiomorphs since one can be obtained from the other by a simple rotation. From the equilibrated shapes, we extract the pitch p and radius r (Materials and Methods) and report them as a function of $\alpha$ in Fig. 4C. For narrow strips, i.e., in the limit where the width $w\to 0$, Armon et al. (27) derived the relations $\tilde{r}=cos2\alpha$ and $\tilde{p}=2\pi \sin 2\alpha$, where $\tilde{r}=\overline{\kappa }r$ and $\tilde{p}=\overline{\kappa }p$ are the reduced radius and pitch, respectively. Fitting the extracted values from the simulation to these relations yields an excellent fit with the only free parameter $\overline{\kappa }=3.35\times {10}^{-2}$ nm⁻¹.

Fig. 4.

(A) Snapshots of 3ML CdSe NPL conformations from MD simulations as a function of the angle $\alpha$ between the [110] crystallographic direction and the long edge of the NPL. The NPL are 200 nm long (except for $\alpha =\pi /4$, which is 150 nm) and 10 nm wide with a thickness of 1.1 nm, coated with a 1:1 mixture of acetate and oleate ligands (not shown for clarity). (B) Radius and pitch of the NPLs in (A) as a function of the angle $\alpha$, along with theoretical predictions (dashed lines) from Armon et al. (27) (C) Snapshots of CdSe NPL conformations from MD simulations for different widths at a fixed angle $\alpha ={45}^{°}$ between the [110] crystallographic direction and the long edge of the NPL. The NPLs are 150 nm long with a thickness of 1.1 nm and coated with acetate ligands (not shown for clarity). (D) Radius and pitch for NPL similar to those in (C) as a function of width at $\alpha =\pi /4$, along with theoretical predictions (dashed lines) from Grossman et al. (28) (see SI Appendix, section 1.4 for details).

The hallmark feature of ribbons bearing incompatible curvatures is a second-order phase transition between a helicoidal shape for small widths into a helical shape as the width increases. The stretching energy scales as $w^5$ whereas the bending energy scales like w. Hence, when the width reaches a critical value, it becomes energetically favorable to unwind the helicoid and transition to a helical shape (or spiral) where the Gaussian curvature is expelled. In this geometry, the stretching energy term dominates and NPL behave as ribbons with a single intrinsic curvature that would be obtained by stretching only one side (top or bottom). Fig. 4B shows the shape evolution with increasing width w at $\alpha =\pi /4$ for NPL with a ribbon geometry ($L_y=150$ nm) covered with acetate ligands. We observe a clear transition between helicoïdal ribbons at low w and helical ribbons as the width increases. From the simulated shapes, we extract the pitch p and radius r (Materials and Methods) and report them as a function of the reduced width $\tilde{w}=w/w_c$ in Fig. 4D. The unwinding transition occurs for a critical width $w_c\approx 23$ nm for $\nu =0.35$, $t=1.1$ nm with $\overline{\kappa }=4.65\times {10}^{-2}$ nm⁻¹ as the only fitting parameter for r and p as a function of w to equations in ref. 28. The small difference in $\overline{\kappa }$ between the two figures points toward the role of the ligand tails on the reference curvature with pure acetate ligands allowing a larger $\overline{\kappa }$ than mixed oleate/acetate. Furthermore, we retrieve in simulations the asymptotic behaviors predicted by theory: in the narrow limit the pitch ${\tilde{p}}^{†}=2\pi$, and in the wide limit the pitch ${\tilde{p}}^{‡}=2\pi /\left(1-\nu \right)$ and the radius ${\tilde{r}}^{‡}=1/\left(1-\nu \right)$.

Effective Curvature

The fact that all the simulated data can be fitted with only one free parameter $\overline{\kappa }$, both for variations in $\alpha$ and w, is remarkable. This unique parameter encodes the interaction between the ligands and the crystal surface. Different ligand head groups and binding motifs affect the imposed curvature at the interface as shown experimentally for CdSe (6) and CdTe (7). This effect is also observed in our MD simulations, where 5ML NPL coated with butanethiol (SI Appendix, Fig. S2) displays a much smaller radius of curvature (28 nm) than the same NPL coated with acetate (77 nm, Fig. 3E). The greater curvature induced by the thiol ligands in our simulations is consistent with the experimental observation that flat 5 to 6 ML CdTe NPL roll up when their oleic acid ligands are replaced with hexadecanethiol (7).

More surprisingly, $\overline{\kappa }$ also depends on the interactions between the ligand tails, which includes enthalpic contributions due to van der Waals interactions and entropic effects related to the free volume available for the ligand chains in the monolayer to undergo conformational changes. In simulations, the critical role of repulsive entropic interactions between ligands is evidenced by curvature variations observed when the $ϵ$ parameter (which tunes the strength of vdW attraction between the tails) is changed. For example, for 3 ML 20 × 10nm CdSe NPLs at 100K, on reducing $ϵ$ from 100 to 50%, the radius of curvature of the NPL midline changes from 43.5 to 36.9nm for octanoate ligands. In addition, when the chain length of linear thiolate ligands increases from 4 to 18 carbons, we observe a significant decrease in curvature in MD simulations, in semi-quantitative agreement with experiments (SI Appendix, Fig. S2). Shapeshifting is even more dramatic when a branched ligand is used. Helical NPL initially coated with carboxylates undergo complete unfolding when functionalized with 2-ethyl-1-hexane thiol (SI Appendix, Fig. S5), showing that entropic interactions between ligand tails play a key role in determining NPL shape.

Conclusion and Perspectives

In this study, we demonstrated that the natural curvature of CdSe NPL is due to interaction between ligands and the NPL crystal. This interaction induces directionally anisotropic surface stresses, leading to preferred curvature along different crystallographic axes on the top and bottom surfaces. The angle between the NPL edge and these axes defines its chirality, while the balance of bending and stretching energies relative to the NPL’s width and thickness determines its shape.

We traced back this effect’s origin to the NPL’s crystallographic structure. Both top and bottom surfaces of the NPL are cadmium-rich (001) planes and the symmetry elements of the zinc-blende structure lead to orthogonally oriented Cd-Se bonds on these surfaces. The shape of NPLs composed of other materials could be explained by this theoretical framework if they exhibit similar structural features. For example, other zinc-blende NPLs may exhibit frustrated curvature effects (7). More generally, we predict that NPLs that have a crystal structure with a $\overline{6}$, $\overline{4}$ or $\overline{3}$ symmetry axis suitable oriented to their cutting planes have the potential to exhibit such behavior. This prediction is consistent with a recent report of chiral Gd₂O₃ NPLs (29). Cubic Gd₂O₃ belongs to the Ia3 space-group which has a $\overline{3}$ symmetry axis and, interestingly, these NPL exhibit helical geometry in some instances. Chirality can also be achieved for more symmetric crystal structures via the use of chiral ligands. For example, gold has been shown to display twisted ribbon geometries when functionalized with chiral thiols (1).

Another intriguing possibility is to use this insight to design NPL that can undergo large shape changes in solution with marginal changes in the structure of the NPL/monolayer interface. Close to the unwinding transition ($w\simeq w_c$), dramatic shape changes are expected to occur upon small $\overline{\kappa }$ changes. Since this parameter depends on the molecular details of the self-assembled monolayer, we expect that changing the chemical identity of the surfactants, e.g., via ligand exchange, could drive such a shape change. Another feature of this class of systems is bistability (30), meaning that two geometrically different conformations can have the same energy. It should thus be possible to go from one configuration to the other with minimal energy input to the system. This feature could be exploited in stimuli-responsive nanomaterials by applying the theory and physical insights described in this paper, linking the molecular structure of ligands with the morphology of inorganic NPL.

Materials and Methods

Synthesis and Functionalization of CdSe Nanoplatelets.

Synthesis of 3ML CdSe NPLs.

In a typical synthesis, 140 mL ODE, 2.22 g Cd(Acetate)2·2H₂O, and 2.23 mL oleic acid are introduced in a 250-mL three-neck flask. The temperature of the solution is raised to 200 ^°C under Argon flux. After injection of 3.6 mL of TOPSe 1M, the reaction mixture is annealed at the same temperature for 10 min, forming 3 MLs CdSe NPLs. At the end of the reaction, 10 mL of oleic acid is added before the solution cools down to room temperature. NPLs are isolated from the reaction mixture by centrifugation; they are then dispersed in hexane and precipitated one more time by adding ethanol. The final product is dispersed in hexane.

Functionalization by thiols.

200 $\mathrm{\mu }$L of the previous 3ML CdSe NPL dispersion are added to 1 mL of toluene. 1 mmol (large excess amount) of the appropriate thiol is added, and the solution is heated for 3 d at 65 ^°C. The NPLs are precipitated by adding ethanol, the dispersion is centrifuged at 9,000 rpm for 3 min and the precipitate is redispersed in toluene. The purification step is repeated once.

Molecular Dynamics Simulations.

Molecular dynamics (MD) simulations of isolated ligand-coated CdSe NPL were performed at constant volume and temperature (100K for thiol-coated NPL and 300K for the remaining systems), maintained via a Nosé-Hoover thermostat, using the LAMMPS software package (31). The NPL and ligands were modeled with explicit atoms, while the solvent was treated implicitly, in order to limit computational costs, by reducing the interaction strength between ligand chains to 85% (3ML oleate-coated NPL) and 50% (5 ML thiol-coated NPL) of their original value. These values were chosen based on results from previous work with alkanethiols coating spherical nanoparticles (32) and on the fact that ligand chains remained disordered and mobile at the chosen temperature. Following Rabani (33), the interaction potential for CdSe consists of a short-ranged 12 to 6 Lennard-Jones potential combined with a long-range Coulombic contribution, with parameters given in ref. 33. Ligand interactions were modeled using the OPLS-AA force field (34). The set of parameters for each type of ligand were obtained from the LigParGen web server (35–37). The MD equations of motion were integrated using the velocity-Verlet algorithm with a time step of 1fs.

CdSe NPL with Cd-terminated [001] polar surfaces (Fig. 2A) were constructed by first placing the Cd and Se atoms on a zinc blende lattice (see SI Appendix for calculation of the lattice parameter). The structures thus obtained were cut by a set of parallel planes along the crystallographic axes to obtain the desired sizes and the number of monolayers. For NPL with different orientations of the [110] axis, the lattice was rotated by appropriate angles prior to cutting. Next, the crystalline core was coated with either acetate, oleate, or thiolate ligands at a density of 3.27 ligands nm², which ensured charge neutrality of the NPL. The coating was performed by randomly placing the ligands within a thin shell encompassing the core and then optimizing their positions using the software package PACKMOL (38).

The resulting ligand-coated core was placed at the center of a simulation box, with at least 50Å of vacuum added on all sides. In almost all cases, the simulation box was a rectangular parallelepiped, where the longest and shortest dimensions were of the order of the associated NPL dimensions in their undeformed state. This ensured that in practice, after equilibration, the box dimensions remained significantly larger than the NPL size. Periodic boundary conditions were applied along all three dimensions.

Equilibration was performed via a sequence of stages. First, an energy minimization run was carried out to remove any significant overlap of the atoms. Second, the ligands were relaxed at constant energy with a maximum atomic displacement of 0.1Å per time step for a duration of 50ps. Third, the ligands were relaxed at the desired temperature for another 100ps. Note that during the above three stages, only the atoms belonging to the ligands were displaced and the CdSe core was held static. Finally, both the ligands and the crystalline core were allowed to equilibrate together at the chosen temperature for 6ns. Equilibration was followed by a production run of 3 to 25 ns, depending on the NPL size and whether significant fluctuations persisted in the resulting configurations. The configurations were sampled every 10ps. All results presented here have been averaged over all the sampled configurations.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Molecular Dynamics lammps code, relaxed structures data have been deposited in Zenodo (https://doi.org/10.5281/zenodo.10458753) (39).