Systematic Mapping of Binary Nanocrystal Superlattices: The Role of

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Systematic Mapping of Binary Nanocrystal Superlattices: The Role of Topology in Phase Selection Igor Coropceanu, Michael A. Boles, and Dmitri V. Talapin J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.8b12539 • Publication Date (Web): 14 Mar 2019 Downloaded from http://pubs.acs.org on March 14, 2019

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Journal of the American Chemical Society

Systematic Mapping of Binary Nanocrystal Superlattices: The Role of Topology in Phase Selection

Igor Coropceanu†#, Michael A. Boles†#, Dmitri V. Talapin†§* † Department of Chemistry and James Franck Institute, University of Chicago, Chicago, Illinois 60637, United States § Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439, United States

# I.C. and M.A.B. contributed equally to this work. * E-mail: [email protected]

Abstract The self-assembly of two sizes of spherical nanocrystals has revealed a surprisingly diverse library of structures. To date at least fifteen distinct binary nanocrystal superlattice (BNSL) structures have been identified. The stability of these binary phases cannot be fully explained using the traditional conceptual framework treating the assembly process as entropy-driven crystallization of rigid spherical particles. Such deviation from hard sphere behavior may be explained by the soft and deformable layer of ligands that envelops the nanocrystals, which contributes significantly to the overall size and shape of assembling particles. In this work we describe a set of experiments designed to elucidate the role of the ligand corona in shaping the thermodynamics and kinetics of BNSL assembly. Using hydrocarbon-capped Au and PbS nanocrystals as a model binary system, we systematically tuned the core radius (R) and ligand chain length (L) of particles and subsequently assembled them into binary superlattices. The resulting database of binary structures enabled a detailed analysis of the role of effective nanocrystal size ratio, as well as softness expressed as L/R, in directing the assembly of binary structures. This catalog of superlattices

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allowed us to not only study the frequency of different phases but to also systematically measure the geometric parameters of the BNSLs. This analysis allowed us to evaluate new theoretical models treating the co-crystallization of deformable spheres and to formulate new hypotheses about the factors affecting the nucleation and growth of the binary superlattices. Among other insights, our results suggest that the relative abundance of the binary phases observed may be explained not only by considerations of thermodynamic stability, but also by a postulated preordering of the binary fluid into local structures with icosahedral or polytetrahedral symmetry prior to nucleation.

Introduction Self-assembly of colloids into ordered arrangements, or superlattices, was first explored using micron-sized spherical beads comprised of a silica core and a thin organic stabilizer shell.1 The confinement of such particles in a small volume of solvent can result in an ordered arrangement that maximizes the free volume available to each particle. This process, called entropy-driven crystallization, favors the densest particle arrangements. In the case of single component systems of monodisperse spherical micrometer beads, this crystallization results in face-centered cubic (fcc) and hexagonally close-packed (hcp) superlattices, the densest sphere packings (both with density φ ≈ 0.74). Similarly, in a binary system consisting of micron-sized spherical beads of two different sizes, cocrystallization can produce AB, AB2, or AB13-type binary arrangements if sphere radius ratio (γ=RB/RA) allows for structures with a density approaching or exceeding the FCC limit.2 Such experiments lay the foundation for understanding colloidal (co)crystallization as a process which produces densest packings. In contrast to micrometer sized spheres, the co-assembly of two different sizes of spherical nanocrystals (NCs) capped with hydrocarbon surface ligands routinely results in a much richer set of binary phases (Fig. 1).3,4 The space-filling analysis which explains the binary assembly of micron-sized colloidal beads thus fails to capture important aspects of analogous experiments carried out at the nanoscale. This becomes clear when the sphere packing density for binary nanocrystal superlattices (BNSLs) is estimated within the same hard sphere (HS) formalism. In this scheme each NC is treated as an effective sphere where the inorganic radius (R) and the length


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of the hydrocarbon ligands (L) are combined into an effective hard sphere with radius (r) given by the optimal packing model (OPM) as shown in Fig. 1b.5 A key assumption of this model is that when two particles come into contact the ligand shell of each particle will become space filling only along the axis of contact as shown in Fig. 1b, top. This model allows for a simple description of a soft NC with an inorganic radius R and a ligand length L to be represented as a hard sphere with an effective radius r given by the following formula: 𝑟 = 𝑅(1 + 3𝜀𝐿/𝑅)1/3

(1)

In the equation above 𝜀 is a scaling factor that accounts for non-ideality of the grafting density as described in SI Section 3, with the classical OPM result corresponding to the case 𝜀 = 1. Numerical estimates for the grafting densities for the particles used in our study are summarized in table S1. This result was first derived by considering the smallest radius (r) to which a particle can be confined such that the volume of a cone bounded by the footprint of the ligand (Ao) on the NC’s surface and ending at r becomes equal to the initial volume of the free ligand treated as a cylinder with a cross-sectional area Ao and a relaxed length L. The same result can also be derived by considering the increase in the radius of an inorganic particle by adding a homogenous layer that accommodates the total volume of the ligands (see Section 3 of the SI for both derivations). A convenient aspect of this result is that the minimum radius of approach for any particle can be calculated independently such that the minimum interparticle spacing between two particles (A, B) is immediately given by the simple additive relationship6: 𝑑𝐴𝐵 = 𝑟𝐴 + 𝑟𝐵

(2)

This treatment then makes it possible to calculate the theoretical packing fraction for a binary phase based on the effective size ratio between the two particles (γeff ≈ rB / rA), where A and B indexes are used for larger and smaller NCs, respectively, so that 0 ≤ γeff ≤ 1. Among the set of binary phases observed by our group and others over the last decade, the majority are predicted to fill space less efficiently than phase-separated fcc/hcp arrangements (the dotted trace). This puzzle has recently been addressed by both theoretical and experimental investigations. For example, a similarly rich set of binary phases is predicted to develop from ensembles of spherical particles with electrostatic surface charges7,8 or non-additive repulsive potentials.9 A systematic transmission electron microscopy (TEM) image analysis has indicated that the NC ligand shell can


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undergo significant deformation,10 leading to BNSLs with the lattice parameters that drastically vary from the bounds set by a simple hard sphere model. Such distortions of the soft capping layer allows NCs to pack as non-spherical units, which can invalidate the assumptions on which the space-filling analysis shown in Fig. 1c is based.

Figure 1. (a) TEM micrographs of the observed BNSL phases for the PbS/Au system discussed in this work (all scale bars are 20nm) and a representative FFT of a DDQC structure; (b) Definition of the key geometric parameters of the hard sphere model: R defines the inorganic radius of the A/B particle, L defines the length of the ligands on each particle and rA/B denotes the effective hard sphere radius of each particle; (c) Predicted density by sphere packing analysis of observed BNSL phases; the dashed line marks the fcc limit; points on the chart denote the BNSLs from section 2 of this work.

Moreover, while some of the binary NC phases, such as various Frank-Kasper phases, have little precedent in the hard sphere literature, they have been observed in soft matter systems ranging from liquid crystals11 to block copolymers12,13. Indeed, we can envision that NCs may occupy a space between the extremes of hard spheres and soft matter, allowing us to explore the transition between these two regimes. The challenge is then to develop a conceptual framework which describes the packing properties of such deformable objects. Along these lines, a model was recently proposed which treats ligand corona deformation as a set of topological defects (e.g.,


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vortices or disclinations) constrained by the geometry of the grafting surface.6,14 Combined experimental and theoretical efforts will be required to mutually advance the understanding of nanoscale colloidal crystallization beyond simple sphere packing arguments and offer a promising route to gaining control over the self-assembly process. Consistent observation of BNSL structures both within and well beyond the set of efficient sphere packings motivated us to explore the role of the hydrocarbon capping layer in driving twocomponent NC self-assembly. Using Au and PbS NCs as a model system, we systematically mapped out the phases formed by co-crystallization of these particles with various core diameters and ligand chain lengths. Our results indicate that in addition to effective NC radius ratio (γeff ≈ rAu / rPbS), the softness of each NC (defined by the ratio of the ligand length to the inorganic radius:  = L/R) also plays an important role in the phase behavior of two-component NC mixtures. These two parameters have a crucial impact both on the kinetics of the crystallization process as well as in the selection of the binary phases that are observed. This paper is structured in three parts. In the first section, we study the effect of the ligand length on the dynamics of the assembly process to identify the general conditions that promote the formation of well-ordered binary phases. In the second part we use the insight obtained from this initial survey to design a set of 70 assembly experiments with different combinations of inorganic core diameter and ligand lengths. We then extract the geometric parameters of the resulting binary structures and compare these to the predictions made by two theoretical models. The resulting analysis shows that while a simple hard sphere description is sufficient to explain the structure of certain phases, in other cases the packing fraction is found to be much higher than the model predicts. We find that we can account for this discrepancy by going beyond the hard sphere picture and instead using a model that explicitly describes the modes through which the ligand corona can deform to produce denser structures. Finally, in the last section we analyze the relative abundance of the different binary phases observed. Here we propose the hypothesis that certain phases tend to be more common because they have a lower nucleation barrier, which may in turn be related to the structural similarity between the final crystalline phase and the intermediate icosahedral/polytetrahedral local order of the fluid prior to nucleation.


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Results and Discussion 1. Tailoring ligand length to enable nanocrystal assembly under quasi-equilibrium conditions To begin the investigation of the role of capping ligands in NC self-assembly, we prepared several sizes of Au and PbS NCs and exchanged the native ligands for those with a desired hydrocarbon chain length. For example, oleylamine-capped 5.1-nm Au NCs were stirred in hexane solution of excess n-alkanethiol (~1:1 mass ratio of ligand to dry NCs), washed twice with ethanol, and redispersed in octane. Using this ligand exchange approach, we first obtained Au NCs with hexanethiol (C6), nonanethiol (C9), dodecanethiol (C12), and pentadecanethiol (C15) capping ligands, spanning softness ratio 0.33 < < 0.89 (Fig. S1). Similarly, oleic acid (C18) capped 7.5nm PbS NCs were stirred in hexane solution of nonanoic acid (C9), myristic acid (C14), or erucic acid (C22), washed twice with ethanol, and redispersed in octane, providing a set of four PbS NCs with softness ratio spanning 0.32 <  < 0.68. We found that, in contrast to binary mixtures of colloidal beads (e.g., silica or polystyrene) whose phase behavior can be rationalized exclusively on the basis of a single parameter, namely the size ratio (γeff), the softness () of surfactant-capped NCs also plays an important role in binary NC self-assembly. In particular, for each Au/PbS pair assembled at constant size ratio, as the length of capping ligands was varied, striking differences in both microscopic and macroscopic structure of binary films was observed. For example, evaporating octane solution containing 7.5-nm PbSC9 and 5.1-nm Au-C6 (γeff = 0.68, PbS = 0.32 and Au = 0.33) NCs at 50°C over tilted TEM grid produced separate phases of single-component superlattices (SCSLs), with thin films of closepacked PbS NCs covering the carbon support and hexagonal/triangular platelets of Au NCs distributed across the surface (Fig. 2a). On the other hand, qualitatively different demixing was observed for the case of 7.5-nm PbS-C14 and 5.1-nm Au-C9 (γeff = 0.68, PbS = 0.48 and Au = 0.47). In this case shown in Fig. 2b, irregularly-shaped and highly intermixed domains of each component resembled the characteristic


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pattern produced by spinodal decomposition of molecular mixtures.15 Further increasing the softness of both building blocks at the same γeff ≈ 0.7 using 7.5-nm PbS-C18 and 5.1-nm Au-C12 (γeff = 0.69, PbS = 0.55 and Au = 0.61) produced MgZn2 and CaCu5-type BNSLs (Fig. 2c, as a side note, these two phases frequently appeared to coexist, as shown in Fig. S10). Finally, somewhat surprisingly, assembly of the softest pair at γeff ≈ 0.7, 7.5-nm PbS-C22 and 5.1-nm AuC15 (γeff = 0.70, PbS = 0.67 and Au = 0.75) produced mostly disordered binary films with small presence of ordered Au-C15 NCs (Fig. 2d). We extended this exploration to other pairs of Au and PbS NCs: 4.3-nm Au and 6.3-nm PbS (γeff ≈ 0.7, Fig. S6), 4.3-nm Au and 7.5-nm PbS (γeff ≈ 0.6, Fig. S7), and 3.1-nm Au and 7.5-nm PbS (γeff ≈ 0.5, Fig. S8), finding that intermediate-length ligands, regardless of size ratio, generally produced BNSLs (which for this first set of experiments included MgZn2, CaCu5, NaZn13, and A6B19 in one case). We have summarized the qualitative results of the 16 assembly experiments described above as a function of the average ligand length (expressed in terms of the number of carbon atoms) in Fig 3d. Once again, the key conclusion is that intermediate length ligands were key to enabling the formation of binary superstructures.


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Figure 2: Capping ligands determine macroscopic morphology and binary phase behavior of 7.5nm PbS and 5.1-nm Au nanocrystals at γeff ~ 0.7. (a) TEM overview (left) and zoom of typical local structure (right) of phase-separated PbS-C9 and Au-C6 NCs evaporated from octane solution over carbon support. Inset, Au-C6 dendritic SLs. (b) Overview (left) and zoom (right) of phaseseparated PbS-C14 and Au-C9 NCs. Inset, demixed SL domains resemble those produced via spinodal decomposition. (c) Overview (left) and zoom (right) of PbS-C18 and Au-C12 NCs cocrystallized into MgZn2 and CaCu5-type BNSLs. (d) Overview (left) and zoom (right) of PbSC22 and Au-C15 NCs which produce disordered binary films with fcc Au SLs present as minority component.

Figure 3. Capping ligand control over structure of binary nanocrystal films. (a) Illustration of phase separation of binary NC mixture. Equilibrium demixing may be observed if phase separation is favored in colloidal crystalline state (left), while kinetic demixing occurs if NC components assemble at different points in time during the assembly process (right). (b) Illustration of cocrystallization or vitrification of binary NC mixture. BNSLs form when such structures represent optimal packing in the colloidal crystal (left). Disordered binary films are


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produced when time required for colloidal crystallization exceeds time the system spends in dense, solvated state (right). (c) Schematic of entropic versus energetic pathways to NC self-assembly. If NCs remain dispersed in solution throughout solvent evaporation, entropic crystallization may occur in the dense suspension (blue trace). On the other hand, if interparticle attractions are significant, NCs may aggregate in a disordered fashion before reaching the entropic phase transition (red trace). (d) A summary of the survey experiments studying the role of softness described in section I showing the results of 16 assembly experiments as a function of the average ligand length (in terms of the number of carbon atoms on the A and B particles) on each NC and the γeff value of each pair. The results show that for short ligands the result was either macroscopic phase separation (MPS) or spinodal decomposition (SD) shown as black triangles, while for very long ligands amorphous jamming (AJ) was observed as denoted by black squares. Only for intermediate length ligands was successful cocrystalization (CC) observed as shown by red circles. The area shaded in red represents the region where we can expect successful co-crystallization of binary structures, the gradient denoting uncertainty in the exact position of the boundary of this region.

The TEM overview images presented in Fig. 2a,c reveal binary NC film morphology over one 400-mesh grid square, or approximately 1600 μm2 area (see Fig. S2-S5 for larger version of each panel). Such images provide complementary information to high-magnification images revealing superlattice structure, offering insights into the details of drying-mediated NC assembly (e.g., interparticle interactions, kinetic versus thermodynamic control, role of solvent) as discussed below. The intermixed and irregularly-shaped domains of phase-separated packings of Au-C9 and PbS-C14 NCs (Fig. 2b) resemble those produced by spinodal decomposition of a two-component mixture.15,16 In this process, quenching a binary liquid from a homogenous to a heterogeneous state produces a network of separated, equilibrium-phase domains, with typical domain size increasing with time via coarsening. It should be emphasized that spinodal decomposition cannot lead directly to spatially separated crystalline domains.17 Instead, the isotropic fluid first separates into disordered domains, followed by crystallization within each domain. The limited extent of domain coalescence at the point of NC immobilization on carbon support indicates that decomposition of the mixture occurs shortly before complete removal of solvent. This observation suggests that Au-C9 and PbS-C14 remain intermixed until the final stages of solvent evaporation, leading us to conclude that separately close-packed phases of pure A- and B-components in fact represent the thermodynamic ground state of this system (Fig. 3a, left). Kinetic phase separation, on the other hand, can take place if the ordering transition for each component occurs at different times during solvent evaporation (Fig. 3a, right). Evidence for such


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a process was found in several binary systems containing Au-C6 NCs. In this case, we found faceted or dendrite-like superlattices of Au NCs upon solvent evaporation (Fig. 2a), suggesting nucleation of Au NCs in solution before particles are condensed into sufficiently small solvent volume to initiate entropic colloidal crystallization (Fig. 3c). Flat, dendritic Au NC SLs likely represent the intermediate case in which a fraction of the Au NCs nucleate in the solution bulk while the remaining particles precipitate onto pre-formed single component superlattices of the first-to-crystallize phase when confined in the dense colloidal layer. Such nucleation in the solution bulk should deplete the binary system of the strongly-interacting Au component and can result in demixed state as a kinetic product. Similar analysis of large-area film morphology of the Au-C12 and PbS-C18 system (Fig. 2c) indicates that, like the C9/C14 case, assembly occurs under thermodynamic control, with both components intermixed at the late stages of solvent evaporation. However, instead of late-stage demixing as we found in the C9/C14 system, the C12/C18 combination allows for cocrystallization into BNSLs (Fig. 3b, left). The rounded, irregularly-shaped BNSL domains appear to tile the substrate surface, suggesting nucleation and growth of BNSLs from the disordered binary mixture until two growing domains meet at a grain boundary, seen as the ~0.5-μm gap between dry BNSLs (Fig. 2c). In addition, cracks in the center of the BNSLs indicate that domains are pinned to the carbon support while they still contain a significant volume of octane solvent. The wet BNSL dries by evaporation of solvent from the domain edge towards the interior, pinning the BNSL periphery to the carbon support and generating tensile strain ultimately relieved by tearing in the center of the domain. Such pinning and tearing of superlattice domains during late stages of solvent evaporation may be avoided by using a support that prevents superlattice adhesion (e.g., a liquid subphase such as diethylene glycol18), allowing for assembly of BNSLs with domain size approaching square centimeters. Somewhat surprisingly, our attempts to assemble Au-C15 + PbS-C22, the softest combination at γeff ≈ 0.7, typically resulted in disordered binary films (Fig. 2d) despite excellent colloidal stability of both these NCs in octane solvent. We observed significant improvement in ordering of PbS-C22 single-component films when the NC solution was evaporated over a water droplet instead of the tilted carbon support used in the other experiments presented in this work (Fig. S9). From these observations we conclude that relaxation of condensed, disordered NC solution to colloidal


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crystalline state can be impeded by strong van der Waals interaction between long capping ligands and amorphous carbon TEM support as well as kinetic entanglement between C22 ligands on neighboring particles in dense NC solution (Fig. 3b, right). It is worth noting that other computational and experimental studies have also highlighted the important role played by such short range interactions mediated by surface ligands in directing the assembly process of colloidal nanocrystals as the solution becomes concentrated upon solvent evaporation.19,20 Several attempts to anneal such films in solvent vapor, however, did not significantly improve ordering in our case. To recapitulate the main points of this section, our goal was to establish general criteria for the selection of particle sizes and ligands in the Au/PbS system that would promote the formation of binary superlattices. We determined that a critical parameter affecting the general outcome of the assembly experiments was the length of the ligands for each combination of particles. On one extreme, very short ligands tended to promote rapid crystallization of a single component phase, generally of the smaller B particles (in this case Au). Since this process depleted the solution of one population of nanocrystals, the net outcome was usually phase separation. At the other extreme, very long ligands tended to jam particles together during the assembly process, and thus prevented them from exploring the favored local configurations that would give rise to orderly crystals. However, under the conditions studied, there was a favored position in the middle where intermediary ligand lengths modulated the timescales associate with each of these processes in such a way that co-crystallization became possible. Having established that both phase separation and BNSL assembly can occur under thermodynamic control for the intermediate-length (e.g., Au-C9 + PbS-C14 and Au-C12 + PbS-C18) capping ligands, we seek to understand why some combinations promote cocrystallization and others lead to equilibrium demixing and in the case of the former why we observe certain phase especially frequently.

2. Understanding the geometrical parameters and packing densities of the observed BNSL phases Based on the insight from the experiments described in the previous section, we next sought to more systematically elucidate the formation of various BNSL phases by compiling a large


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experimental database of BNSLs assembled from building blocks with a range of sizes of the inorganic cores and the softness of each nanocrystal. First, we prepared monodisperse sets of Au and PbS NCs with a range of core sizes. We subsequently exchanged native surface ligands for alkanethiols (Au) or carboxylic acids (PbS) with ligands varying from 6 to 18 carbons in length in order to fabricate a series of Au NCs with a softness range 0.47 <  < 1.47 and PbS NCs in the range 0.31 <  < 0.66. Using the resulting NC library, we assembled a new set of 70 binary pairs, more than half of which yielded BNSL thin films as determined by TEM. To minimize uncontrolled run-to-run variations, all assembly experiments have been carried out over a short period of time using same setup, temperature and pressure. The full results are provided in Table S2 and figures S16-S26 and a more compact form of the table is shown in Fig. 4a, where red marks the experiments for which large BNSL domains were observed while black marks those experiments where the result was either disorder or phase separation with at most small (0) texture on either or both the A and B particles is compatible with the geometry of the crystal lattice for a given range of γeff, OTM branches can arise with a higher density than the trivial HS branch as shown in Fig. 6c in the case of Li3Bi. As a final note, the introduction of additional vortices has an associated entropic cost by reducing degrees of freedom available to the ligands. For this reason, the most common binary phases observed can be described in terms of 3 or 4 vortices, which give rise to the textures shown in Fig. 6b. For example, in the case of Li3Bi, the OTM branch corresponds to a configuration of 4 vortices on the B particles located on Wyckoff position 8c, giving rise to a tetragonal texture. As Figure 6c shows, the OTM description proved very successful in reconciling the experimental packing fraction measured for Li3Bi. To use the same example as used above, the OTM model predicts a lattice constant of 18.2 nm and a packing fraction of 0.84, in close agreement to the experimental data. In order to confirm that the apparent lattice contraction was not due to additional deformations along other crystallographic axes, we took TEM images at different tilting angles to measure the unit cell along different stereographic projections as summarized in Fig. S11. These results confirmed that the lattice constant was identical along all projections. As a result, the higher packing fraction observed is in fact due to a reduced lattice constant that simply cannot be accommodated by the HS model, but which follows directly from the prediction of the OTM model. As shown in Fig. 7b and as summarized in the SI, we found similar results for other phases as well, most notable NaZn13 and MgZn2. In general, when the OTM prediction of the packing fraction differed significantly from the hard sphere model, we found the experimental value to also be higher than the HS prediction.


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Figure 7: Packing fraction analysis of different phases. (a) Phases that can generally be described within the HS model. (b) Phases where HS models proved insufficient to describe all data points: circles correspond to experimental data points, solid lines correspond to the theoretical hard sphere packing density, while dashed lines correspond to OTM branches.

There was one phase that proved not to be well described by any of the models, namely CuAu. For this phase the packing fractions proved unphysically high, with the estimates based on the original grafting density of particles sometimes giving packing fractions greater than 1. This anomalous result can be rationalized by the previously observed behavior that formation of the CuAu phase tends to be accompanied with depletion of the ligands when lead chalcogenides are used as one of the building blocks. Indeed, the small lattice constants observed simply cannot be accommodated if the original ligand density in the particles were maintained. The particular details associated with the formation of the CuAu phase will be discussed in a separate publication. The last point brings an important caveat to the analysis shown in this work. The discussion of the packing density presented in the section above is based on the premise that the size and grafting density of each population of particles is the same in the original liquid solution and in the final BNSL structures. However, as the case of CuAu indicates, we cannot exclude the possibility of ligand loss or rearrangement during the assembly process. While it is difficult to make any quantitative predictions about the extent of such processes, we can outline some general considerations of the problem. In terms of ligand loss, we can expect such an effect to more severe for the rather labile carboxylate ligands bound to PbS. By contrast, in the case of the Au nanoparticles the large strength of the Au-S bond can be expected to minimize ligand detachment.


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Moreover, Au has a much higher affinity for thiolate groups compared to carboxylate. These two factors taken together suggest that cross-exchange of ligands between the two populations can be expected to be limited in extent. By contrast, ligand rearrangement is more likely to occur within each population of nanocrystals, especially in the case of PbS. Moreover, to the extent that free ligands are present in solution, they may also become weakly associated with nanocrystals and incorporated into the final BNSL structure. While it is difficult to gauge the importance of these secondary effects, they are certainly worth exploring in more detailed follow-up studies. In conclusion to this section, we created a systematic dataset in order to quantitively study how the abundance and geometric properties of BNSL structures changes as a function of both the inorganic radius and the ligand length of the constituent A/B particles. We then used this dataset in order to quantitively verify predictions made by different theoretical packing models. We were able to confirm that for a wide range of structures a simple hard sphere packing model was largely sufficient to explain all of the experimental results. However, for certain structures, especially those belonging to the Li3Bi phase, the hard sphere packing model proved insufficient. In these cases, a more general model that explicitly accounted for collective deformation of the ligand corona (OTM) was in many cases able to give much better quantitative agreement with experimental observables. We expect that the quantitative dataset presented in this work may aid in the further refinement of these theoretical models. 3. Rationalizing the relative abundance of the BNSL phases with respect to considerations of thermodynamic stability and probability of nucleation. While more sophisticated models of the ligand corona can be very successful in explaining the geometric properties of the BNSL structures observed, they are still not sufficient to explain the relative frequency of individual phases (see Fig. 8a and Table III in the SI). A natural starting point to rationalize the presence of specific phases would be to use the density as a proxy for the thermodynamic stability of the structure in question. As already discussed in the introduction, from an entropic perspective, a higher density is favorable as such an arrangement maximizes the average free volume available to each particle. Moreover, because nanocrystals also experience a net attraction due to van der Waals forces, a higher density also leads to an additional enthalpic stabilization via a higher cohesive energy. Nevertheless, as can be seen in Fig. S15, for a given value of γeff, it is far from clear that the phases that formed had the densest possible structure, a


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fact seen even more clearly in Fig. 8b. As the discussion above strongly suggests, we need to go beyond considerations of the density in order to try to understand why certain phases tend to be more favored than others. One hypothesis is that certain phases form more easily than others, e.g. by having a lower barrier for nucleation. This hypothesis is strengthened when analyzing not just samples where large binary phases were observed, but also those experiments that yielded fragments of binary phases. An interesting observation is that in the latter case one often encounters not just a few isolated fragments of different phases, but often a large number of separate fragments of the same phase. This situation was especially commonly observed for CaCu5, NaZn13, and AlB2 phases as shown in Figs. 9 and S12. Indeed, even in experiments where large BNSL domains were observed, one could still often detect numerous fragments of the phases listed above. The large number of such disconnected fragments strongly suggests that in these experiments the assembly proceeded through multiple independent nucleation events, which again appears to imply that such BNSL phases have an easier nucleation pathway.

Figure 8: (a) Relative frequency of large BNSL domains identified vs. the effective size ratio (γeff): solid triangles mark cases where BNSL successfully formed while empty triangles show cases where only BNSL fragments of that particular phase were detected. (b) Plot of the observed binary phases as a function of the degree of icosahedral order (fico) and the average theoretical density of the structures belonging to each phase (calculated by taking the highest packing fraction predicted by the OPM model or OTM model); the size of each circle is scaled by the number of structures (including fragments) observed. Because of the anomalous behavior of the CuAu phase (related to ligand loss), we omitted this phase from the figure in panel b.


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The immediate question then becomes what structural features distinguishes those phases that appear to have an especially easy pathway for nucleation. A compelling clue at this point is that many of the phases observed are characterized by structural features that locally exhibit a generalized form of icosahedral symmetry. We can put this idea on a more general and quantitative footing by adopting the notion of the icosahedral order (fico) using a figure of merit previously introduced by Travesset et al.27 This quantity seeks to capture the degree to which the unit cell of the binary phase deviates from the ideal {3,3,5} polytope through a network of disclinations, which act as effective defects from icosahedral symmetry. As Fig. 8a shows, the majority of successful BNSL experiments observed in our dataset exhibit phases with a high degree of icosahedral order. Moreover, phases with the highest degree of icosahedral order such as NaZn13 and CaCu5 were also especially prolific in producing a large number of small isolated fragments as shown in Fig 9a and Fig 9b. This behavior stands in contrast to phases with a low degree of icosahedral order, such as Li3Bi, which were not observed to give rise to a large number of separate domains, but instead were often found in large single crystalline structures as shown in Fig. 9c. We can analyze this discrepancy more systematically by plotting the highest density of discrete BNSL domains that was observed for each phase versus fico as shown in Fig. 9d. While this treatment is not fully quantitative, it nevertheless captures an important qualitative trend, namely that the higher the value of fico was for a certain phase, the more likely it was to detect multiple discrete domains of that phase in close proximity to each other. If we take this density as a proxy for the frequency of nucleation events, then there appears to be a strong correlation between the degree of icosahedral order of a binary phase and its rate of nucleation. To summarize, our data suggests that phases with a higher degree of icosahedral order are both 1) especially likely to form for a wide range of conditions and 2) often produce many discrete domains. These two observations taken together motivated us to try to establish a mechanistic link between the degree of icosahedral order of a binary structure and the apparent ease of nucleating a crystal of that phase


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Figure 9: Analysis of BNSL fragments: Presence of multiple disconnected (a) NaZn13 and (b) CaCu5 fragments in close proximity (inset shows a magnified view of each phase) (c) the tendency of Li3Bi to form in large single crystalline domains, and (d) maximum density of discrete domains observed for each binary phase vs. the degree of icosahedral order of the phase.

The key to such a connection may lie in the structure of the fluid prior to nucleation. It has long been established both experimentally and theoretically that for many systems ranging from colloids to metallic glasses nucleation doesn’t directly proceed from a homogenous liquid. Rather, the liquid often tends to acquire a certain degree of short or medium range order before nucleation starts.28 Moreover, this local structure is often found to be dominated by an icosahedral or polytetrahedral order.29–33 Such a situation was first formulated as a hypothesis by Frank to explain supercooling in molten metals.34 Frank argued that the reason that such liquids did not immediately solidify when cooled below the melting temperature was because the liquid was locally ordered with an icosahedral symmetry. Such an icosahedral arrangement is energetically favorable locally


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because it allows a central particle to maximize its interaction with twelve neighboring particles. For example, in the case of a system of 13 atoms described by a Lennard-Jones potential the binding energy is 8.4% higher when the atoms are arranged as an icosahedron rather than in a cubic or hexagonal close packed configurations. An identical result holds for NCs as well.35 However, while an icosahedral arrangement is preferred locally, such a structure is incompatible with extended tiling in real space. This mismatch between locally and globally favored structures gives rise to a geometric frustration that tends to inhibit the nucleation of crystal phases.36–38 Since that seminal work the importance of icosahedral and polytetrahedral short range order has been highlighted both through simulations and experiments in systems ranging from liquid metals to polymers, and from mixtures described by soft Lennard-Jones potentials to hard sphere potentials in the presence of geometric confinement.29,39–42

… Figure 10: Evolution of a binary solution of NCs from an isotropic solution at t0 (left) to a denser solution containing units or fragments with icosahedral or polytetrahedral order at a later time ti, (middle) which can finally nucleate into a solid phase for which the unit cell resembles the structure of the structured liquid to varying degrees as shown for the examples of the coordination of the B particles in NaZn13 and CaCu5 (right) phases.

We can adapt the picture above to our specific case of the formation of binary nanoparticle phases during solvent evaporation. The steps along BNSL nucleation pathway can be summarized in a schematic way in Figure 10. At the beginning of the assembly (left panel) the NCs are well dispersed in the solution with only weak structural correlations. As the system is reduced in size (as the solvent evaporates), local order can start to arise as particles begin to form either transient


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or longer lived tetrahedral and icosahedral clusters or fragments of such clusters (middle panel). Finally, at an even later time binary phases start to nucleate in this structured liquid (right panel). At a purely qualitative level we can expect that binary phases for which the unit cells more strongly resemble the structure of the liquid can nucleate more easily. This model makes it easy to then rationalize why phases such as NaZn13, or CaCu5 appeared to nucleate so easily given the common structural motifs of their unit cell and the postulated local structure of the pre-nucleation liquid. Similar considerations would also then explain the formation of various quasicrystalline binary phases, which were observed both in our current library as well as in other binary NC systems.43,44 Conversely, the same hypothesis could explain the relative scarcity of phases like NaCl for which nucleation would require a large degree of rearrangement. Finally, it is worth emphasizing that a similar propensity for the formation of Frank-Kasper, quasiFrank Kasper, or quasicrystalline phases have been observed in hydrocarbon-capped NCs and spherical micelles comprised of organic dendrimers45,46, surfactants47, and block copolymers.12,48 This apparent universality strongly suggests a common mechanism related to the emergence of local icosahedral/polytetrahedral order prior to nucleation. Due to the ease of tuning their softness and interparticle potentials, NCs may offer an ideal test system to further study the key factors that influence the structure of the liquid during the assembly process and the related pathways that lead to the formation of different crystalline phases. In this section we sought to establish a general framework to explain why certain phases tended to be observed especially frequently, even for size ranges where structures of a different phase had a higher theoretical density. Our tentative hypothesis is that the rate of nucleation of a particular BNSL phase depends on the symmetry of the unit cell. In particular, structures with a high degree of icosahedral symmetry appear to nucleate more easily, suggesting that the liquid solution from which these structures emerge has a similar local ordering. This hypothesis may help explain why density alone seems to be insufficient to explain the relative abundance of the different phases observed. While thermodynamic considerations may favor the formation of the densest structures possible, the kinetics of the nucleation may instead favor the formation of structures with a high degree of icosahedral order, even if their density is lower. The distribution of the BNSL phases observed then reflects the tension of these two effects under the specific conditions under which the assembly is carried out.


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Conclusion In summary, we have used PbS and Au NCs as a model system to elucidate the factors affecting the self-assembly and phase selection of BNSLs. We observed that both the size of the inorganic cores as well as the length of the ligands played a critical role in controlling the kinetics of the assembly process and in determining which binary phases successfully formed. We found that the shortest and longest ligands kinetically hinder cocrystallization, while intermediate ligand lengths can give rise to BNSL equilibrium phases. The BNSL structures that were observed for this simple binary system encompassed a rich set of phases that cannot be predicted from simple hard-sphere considerations. This apparent discrepancy can in part be rationalized by explicitly considering the deformable nature of the ligand shell. This description of the particles as deformable spheres makes it possible to theoretically rationalize the much higher packing fractions observed experimentally, which cannot be reconciled within the hard sphere framework. Finally, by systematically analyzing the relative abundance of different phase and fragments, we established a tentative link between the icosahedral order of a phase and the ease of nucleation. This insight may highlight the importance of short-range ordering of the solution prior to nucleation. These results shed light on the role of soft capping ligand corona in enabling the rich phase diagram of BNSLs as compared with analogous colloidal beads and lay the groundwork for targeted self-assembly of desired NC structures. Finally, the generation of a systematic libraries also offers a useful opportunity for data mining to recognize and explore additional trends and patterns.

Associated Content: Supporting Information. Experimental methods, calculation of the ligand length and particle softness, derivation of the hard sphere radius, calculation of the experimentally measured packing fraction, supplementary figures.

Acknowledgements:


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We thank Alex Travesset for stimulating discussions and providing access to HOODLT package. I.C. was supported by MICCoM as part of the Computational Materials Sciences Program funded by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Sciences and Engineering Division (5J-30161-0010A). M. B. was supported by the University of Chicago Materials Research Science and Engineering Center, funded by NSF under award no. DMR-1420709. D.V.T. acknowledges support by NSF under award number DMR-1611371 and The David and Lucile Packard Foundation. Use of the Center for Nanoscale Materials, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Argonne National Laboratory, was supported by the U.S. DOE under Contract No. DE-AC02-06CH11357.


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Systematic Mapping of Binary Nanocrystal Superlattices: The Role of

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