Many-Body Effects in Nanocrystal Superlattices: Departure from Sphere Packing Explains Stability of Binary Phases

Abstract

This work analyzes the role of hydrocarbon ligands in the self-assembly of nanocrystal (NC) superlattices. Typical NCs, composed of an inorganic core of radius R and a layer of capping ligands with length L, can be described as soft spheres with softness parameter L/R. Using particle tracking measurements of transmission electron microscopy images, we find that close-packed NCs, like their hard-sphere counterparts, fill space at approximately 74% density independent of softness. We uncover deformability of the ligand capping layer that leads to variable effective NC size in response to the coordination environment. This effect plays an important role in the packing of particles in binary nanocrystal superlattices (BNSLs). Measurements on BNSLs composed of NCs of varying softness in several coordination geometries indicate that NCs deform to produce dense BNSLs that would otherwise be low-density arrangements if the particles remained spherical. Consequently, rationalizing the mixing of two NC species during BNSL self-assembly need not employ complex energetic interactions. We summarize our analysis in a set of packing rules. These findings contribute to a general understanding of entropic effects during crystallization of deformable objects (e.g., nanoparticles, micelles, globular proteins) that can adapt their shape to the local coordination environment.

Dendron-Mediated Engineering of Interparticle Separation and Self-Assembly in Dendronized Gold Nanoparticles Superlattices

“…The failure of simple hard sphere models may be explained by the fact that ligand-coated NPs are, on close inspection, neither hard nor spherical. Recent work by Boles and Talapin 64 suggests that "softness", or the relative volume of the organic shell of hydrophobic NPs, alters and enhances co-assembly compared to hard systems by increasing the total filling fraction beyond that which is expected from a hard-sphere model. This has been used to explain, for example, the observation of both fcc and bcc single-component structures, although we did not observe bcc structures in any of the Au@L hybrids.…”

Binary nanoparticle superlattices of soft-particle systems

“…The NP radius (defined from the 2D hexagonal separation) was taken from ref. 2, table 1 or by the Optimal Packing Model formula (19,31), depending on whether the hydrocarbon capping ligands are unsaturated or saturated. The plots in Fig.…”

“…19), to consider a soft potential, we examine particles of different diameters interacting with an inverse power law (IPL):…”

“…However, HS models do not provide a satisfactory description of experiments, as clear from the fact that crystals isomorphic to MgZn 2 , CaCu 5 (1), body centered cube AB6 (bcc-AB 6 ) (4) [also known as Cs 6 C 60 (7)], or quasi-crystals (16), just to name a few, have not been reported as equilibrium phases for HS (17,18). It has also been observed that different binary systems with the same NP hydrodynamic radius (with different hydrocarbon chain length, for example) do not exhibit the same equilibrium phase (19), clearly pointing to a phase diagram that depends on other parameters besides the ratio of the two NPs diameters, which completely determines the phase diagram of the HS system (15).…”

The nature and implications of uniformity in the hierarchical organization of nanomaterials

“…The forces involved in these interactions can occur at the whole-nanoparticle level (e.g., entropic forces that drive the packing of hard spheres into a dense lattice) (8,62) or in localized regions (e.g., the hybridization of complementary DNA molecules attached to particles) (52), and often these levels function cooperatively (63,64). Whereas most interactions are relatively short ranged at the nanoscale and rarely influence next-nearest neighbor interactions, many-body effects and population-level interactions become important in crystalline assemblies, given the large number of particles, and thus we focus on these effects here.…”