Spontaneous Patterning of Binary Ligand Mixtures on CdSe Nanocrystals: From Random to Janus Packing

Binary compositions of surface ligands are known to improve the colloidal stability and fluorescence quantum yield of nanocrystals (NCs), due to ligand–ligand interactions and surface organization. Herein, we follow the thermodynamics of a ligand exchange reaction of CdSe NCs with alkylthiol mixtures. The effects of ligand polarity and length difference on ligand packing were investigated using isothermal titration calorimetry (ITC). The thermodynamic signature of the formation of mixed ligand shells was observed. Correlating the experimental results with thermodynamic mixing models has allowed us to calculate the interchain interactions and to infer the final ligand shell configuration. Our findings demonstrate that, in contrast to macroscopic surfaces, the small dimensions of the NCs and the subsequent increased interfacial region between dissimilar ligands allow the formation of a myriad of clustering patterns, controlled by the interligand interactions. This work provides a fundamental understanding of the parameters determining the ligand shell structure and should help guide smart surface design toward NC-based applications.


n-Alkylthiols reduction
To avoid inaccuracies in the ligands concentration, derived by S-S chain coupling bonds, the purchased ligands were reduced prior to use, as was discussed in detailed elsewhere. 1 .
Briefly, two equivalents of NaBH4 powder were added to a solution of alkylthiol in ethanol and TDW (1:4). After 12 hours of stirring at room temperature, the solution was extracted with chloroform (3x75 ml portions). Following that, the unified organic phase was dried over MgSO4 , filtered and then evaporated under vacuum in order to separate between the chloroform and the reduced alkylthiol. The reduced ligands were kept under an inert atmosphere with no exposure to UV light for future use. The yield of the reduction procedure is 70%, to give a final product with no more than 5% of disulfide.

Surface sites calculation
As described in our previous studies, 1,2 the number of Cd surface sites (Nsurface) was calculated based on a simple spherical model for the NCs with a lattice parameter of a=6.050Å (zinc blende). We assumed a uniform zinc blende CdSe layer on the surface, hence the number of Cd surface sites is where Ntotal is the total number of Cd atoms in the NC and Ninternal-sphere is the number of core Cd. Ntotal was calculated considering the volume of a spherical NC with a radius RNC, the density of CdSe (ρCdSe) and its molar mass (MwcdSe): Ninternal-sphere was calculated in a similar way to Ntotal excluding the outer layer of the surface Cd: For the investigated d=3.0 nm NC, 127 Cd surface sites are expected.
The results were compared with a pyramidal model for zinc blend CdSe NCs with four exposed (111) facets. The height of the pyramid, h, was taken as the calculated diameter of the NC, hence the edge length, c, is: The Cd atoms are spaced on the edge according to the nearest-neighbor distance, d, of the unit cell: Hence, the length of an edge, c, containing N atoms is: Using eq. (S4) in eq. (S6), we can determine the number of atoms on the edge: Since the zinc blend NCs are actually a truncated pyramid, the atoms of the outer edges were removed, and the new faces lost one atoms per line, per side. Therefore, the number of Cd surface atoms on a single face, Nface, with a base containing (N-2) atoms is calculated by: By using eq. (S7) in eq. (S8) and multiplying it by 4 (for the four faces), we find the total number of Cd surface atoms in all four faces as a function of the pyramid height: For the investigated d=3.0nm NC, 130 Cd surface sites are expected.
In addition, an atomistic model was also considered to verify the suggested models for surface sites. A semi-spherical NC was simulated from the bulk zinc blend CdSe crystal structure by removing atoms located beyond a distance that is greater than the desired radius ( Figure S2). The remaining atoms resulted in a non-stoichiometric ratio between Cd and Se atoms. The atoms in the outer layer were considered as surface sites. To account for the range of error in the size estimation, results for NC diameters of 2.8, 3.0 and 3.2 nm are presented in

ITC measurements and analysis
4.1. Derivation of a single-site ligand exchange model As described in our previous studies, 1,2 the ligand exchange model is based on the wellknown "single set of independent binding sites" model. 4 We modified the known "binding" model, which considers only the attachment of the new ligand, in order to take into account also the detachment of the native ligand. This "exchange" model is necessary for ligand exchange reactions since both processes, detachment and attachment of the ligands, release heat which is measured by the ITC instrument.
The ligand exchange reaction between the native ligand ′ and the exchanged ligand for a single surface site is The equilibrium constant is defined as Assuming that each native ligand L' is exchanged with a single new ligand L and no free L' is present initially 5 (supported by TGA data, see in the next section), we get: While for the exchanged ligands: And for all surface sites: In the previous equations, [ ] 0 is the total number of surface sites on the NC (based on a spherical model, as explained before), n is the ratio between the actually exchanged ligands and the available surface sites (i.e., the reaction stoichiometry coefficient), and [ ] 0 is the total added exchanged ligand.
Given the expressions above, the equilibrium constant can be written as: We define as the NC surface coverage, and hence, Given eq. (S12), eq. (S15) can be rewritten as During an ITC experiment, we measure the total amount of heat released per injection of ligand, which is correlated with the enthalpy change of the reaction By using the solution for the quadratic equation(S17), eq. (S18) can be written as and the heat released per injection of ligand is By implanting eq. (S21) into eq. (S20), we get the final equation for fitting The other thermodynamics parameters Δ and Δ are calculated by using the known thermodynamics relations

Experimental data and fitting
All fittings were done in NanoAnalyze Software v 3.10.0 (TA instrument).   Figure S3. Errors were calculated based on the quality of the fitting.

Error analysis:
Errors of the extracted thermodynamics parameters were determined by the quality of the fitting. In addition, we considered the reproducibility of the measurement by performing the same experiments three times and calculating the standard deviation of each parameter.
The error in the surface site's concentration was calculated by a triple measurement of the absorption. Figure S4. Real Figure S7.   Figure S8.   Figure S9.  Table   S4 and

Additional surface characterization
Thermogravimetric analysis (TGA) was used to quantify the changes in the organic coverage of the NCs upon ligand exchange, where the differences in the overall mass loss and in the shape of the TGA thermograms indicate the changes in the surface ligand layer composition. Representative TGA results for the NCs with their native oleate ligands and upon ligand exchange with pure C6SH, pure C10SH as well as with C6SH0.5:C10SH0.5 and C6SH0.66:C10SH0.34 binary compositions are presented in Figure S11. The initial oleate coverage of the NCs was determined via the mass loss up to 500 o C, which is attributed to any organic species present in the sample. According to the thermogram, 44% of the total mass was organic, consistent with a full surface coverage and a 1:1 binding ratio (all Cd surface sites are bound to a single oleate ligand).
The post-ITC samples were analyzed similar to our previous report. 2 Prior to the analysis, the post-ITC samples were purified from excess free ligands by multiple cycles of precipitation and re-dispersion process using toluene (solvent) and ethanol (anti-solvent). As mentioned elsewhere, 2 the ITC conditions allow only partial ligand exchange, due to the small amounts of alkylthiol ligands added to the NCs in each titration point, resulting in an overall excess of alkylthiols to oleate of 1.5-2, which is insufficient to induce complete ligand exchange. This stands in contrast to the conditions for full ligand exchange which require an immediate disturbance to the system by the quick addition of a large excess of the exchanging ligand.
Hence, the final surface coverage is slightly more complex, as more than one ligand type is involved. As can be observed from Figure S11, for the oleate coated NCs (red), the main mass loss is above 310 o C, while the mass loss of the alkylthiolate in the post-ITC samples is mostly lower than that. Thus, similar to our previous report, the thermograms of the post-ITC samples were divided into 2 regions: 110-315 o C, which is attributed to bound alkylthiolate ( Figure S11, blue region), and 315-500 o C, which is attributed to bound oleate ( Figure S11, red region). The   Figure   S11.
As described in the main text, the ITC data analysis in this work assumes that the ligand ratio on the NCs was as in the titrant added to the solution. To estimate the ligand ratio on the NC surface, we used the Langmuir model for competitive adsorption. 6 According to this model, the ratio of the surface absorbed molecules can be derived using G values extracted from the experimental ITC curves. Specifically, the surface coverage of each component is given by: . According to this model, we can estimate the ratio of the surface absorbed molecules,

Ideal mixture model
We define an ideal mixture as one where there is no excess enthalpy upon mixing (ΔHmix=0), and the mixing process is spontaneous due only to the increase in the configurational entropy.
The configurational entropy is based on the enumeration of all the available states for spatially organizing the mixture. For an ideal binary mixture on a lattice with total LA+LB sites (LA and LB are the numbers of each ligand), the ideal configurational entropy of mixing is: where XLA and XLB are the molar fractions of the two components.

Regular mixture model
The regular mixture model is the simplest mean field model that allows to describe non-ideal mixing between two or more components. The model assumes an excess enthalpy and entropy of mixing (ΔHmix and ΔSmix, respectively) that are determined by the averaged environment of the components in the mixture. The excess enthalpy is modeled using an interaction parameter where , the integration parameter, represents the path for changing the Hamiltonian from a reference system (=0) to the system of interest (=1), and is the system's free energy, calculated as the thermal average of ℋ. Note that here our reference point is chosen to be the ideally mixed state, for which the configurational mixing entropy is described by eq.(S26): where T is the experimental temperature (303K). Hence, the total ΔGmix corresponding to the experimentally measured value ΔGmix=0 is given by summing eqs. (S32) and (S33).
To simulate the NC surface, a square 11x11 grid (121 sites) with periodic boundary conditions was used to fit the total number of experimentally known surface sites (127, similar results were achieved for 12x12 grid). For a given number of LA and LB, the system was first initialized to a randomly mixed state (similar results were achieved when starting with a phase separated state). Then, the system was allowed to reach equilibrium by performing a large number of Monte Carlo steps. According to the Metropolis algorithm, 8 at each step, a trial swap between two ligands was suggested and the change in the system energy upon switching, U, was calculated whereby each ligand interacts with its 4 neighbors: similar ligand pairs interact with 0 energy (G,AA=G,AB=0), and non-similar ligands interact with G. Swapping was allowed if it resulted in a decrease in the free energy, weighted by the Boltzmann probability, (S34) ℎ ∝ exp(Δ / ) After multiple steps, equilibrium was achieved, and the total energy of the system was calculated to give () for a specific . The procedure was repeated multiple times to give an averaged (), which was later used for calculating ΔGmix as described above.
The simulation was performed for several ratios of LA and LB, to give ΔGmix over the full range of ligand molar fractions. As detailed in the main text, the interaction parameter G which correspond to ΔGmix=0 is 3.4RT, and this interaction parameter was used for simulating the ligand shell structure at equilibrium, using the Monte Carlo algorithm described here. The number of non-similar pairs ( Figure 2 in the main text) and the average cluster area shown in Figure S13 were extracted from the simulated grid at equilibrium. Results are presented for the NC sized grid (11x11), which exhibit a gradual change, and for the macroscopic system approaching the thermodynamic limit (111x111).Cluster area changes gradually for the NC, but increases abruptly around the reported phase transition point (~3.5RT) 9,10 in the large system.
Fits for the enthalpy and the entropy (for each binary ligand set) are derived from the resulting G. To resolve the entropic contribution to the free energy, we used the same thermodynamic integration methodology described above, but this time ΔGmix was calculated for a temperature range of 297K to 309K (around the experimental temperature of 303K). The temperature dependence of G was considered according to eq. (S31), allowing to determine H and S that best reproduce the experimental results. The enthalpy and the entropy for mixing were calculated from the temperature dependent simulation according to the van't Hoff relation: (S35) Δ = − ; Δ = Δ + Δ As described in the main text, ΔSmix is composed of configurational and non-configurational terms. ΔS is extracted by calculating the interactions (derived from S) between all LALB pairs at equilibrium, for all simulated XC6SH. Then, ΔS , is extracted by subtracting ΔS from the total ΔSmix. Since ΔS represents the entropy of the ligand organization, which is directly derived from G, systems with similar G exhibit similar ΔS mix conf .
For the C6SH:C6SH(F) mixture, a similar procedure was applied with the requirement of positive total ΔGmix so as to match the ITC results.
For the C14SH:C18SH mixture, the simulation included in addition a linear dependence of H and S on ligand's molar fractions, as detailed in the main text. For completeness, we applied similar simulations also for the other investigated binary mixtures of C6SH:C10H, C6SH:C14SH, and C6SH:C18SH ( Figure S14), which exhibit deviations from the fitting provided by constant H and S. However, we note that since the deviations from the original model were minor, the new model should be analyzed with caution so as to avoid overinterpretation. Comparing the fitting-extracted ω and η ( Figure S14g and S14h, respectively), it is noticeable that the long ligands at each binary system induce loss of interaction energy and gain in entropy upon mixing. The parameters of C14SH, which represent the long ligand in the C6SH:C14SH system and the short ligand in the C14:SH:C18SH system, switch sign between both system, as the C14SH gain (loss) interactions and loss (gain) entropy upon mixing with the longer C18SH (shorter C6SH). The parameters for the C10SH and C6SH ligands are low and may vary due to using multiply fitting parameters to a system that could already be reasonably fitted with a single one (constant ).