Pair and many-body interactions between ligated Au nanoparticles

Three-body interaction of gold nanoparticles: the role of solvent density and ligand shell orientation

Molecular dynamics simulations are used to study the effective interactions of alkanethiol passivated gold nanoparticles in supercritical ethane at two- and three-particle levels with different solvent densities. Effective interaction is calculated as the potential of mean force (PMF) between two nanoparticles, and the three-body effect is estimated as the difference in PMFs calculated at the two- and three-particle levels. The variation in the three-body effect is examined as a function of solvent density. It is found that effective interaction, which is completely repulsive at very high solvent concentrations, progressively turns attractive as solvent density declines. On the other hand, the three-body effect turns out to be repulsive and increases exponentially with decreasing solvent density. Further, the structure of the ligand shell is analyzed as a function of nanoparticle separation, and its relationship with the three-body effect is investigated. It is observed that the three-body effect arises when the ligand shell begins to deform due to van der Waals repulsion between ligand shells. The study provides a deep insight into good understanding of the solvent evaporation-assisted nanoparticle self-assembly and can aid in experiments.

Coarse-Grained Many-Body Potentials of Ligand-Stabilized Nanoparticles from Machine-Learned Mean Forces

Colloidal nanoparticles self-assemble into a variety of superstructures with distinctive optical, structural, and electronic properties. These nanoparticles are usually stabilized by a capping layer of organic ligands to prevent aggregation in the solvent. When the ligands are sufficiently long compared to the dimensions of the nanocrystal cores, the effective coarse-grained forces between pairs of nanoparticles are largely affected by the presence of neighboring particles. In order to efficiently investigate the self-assembly behavior of these complex colloidal systems, we propose a machine-learning approach to construct effective coarse-grained many-body interaction potentials. The multiscale methodology presented in this work constitutes a general bottom-up coarse-graining strategy where the coarse-grained forces acting on coarse-grained sites are extracted from measuring the vectorial mean forces on these sites in reference fine-grained simulations. These effective coarse-grained forces, i.e., gradients of the potential of mean force or of the free-energy surface, are represented by a simple linear model in terms of gradients of structural descriptors, which are scalar functions that are rotationally invariant. In this way, we also directly obtain the free-energy surface of the coarse-grained model as a function of all coarse-grained coordinates. We expect that this simple yet accurate coarse-graining framework for the many-body potential of mean force will enable the characterization, understanding, and prediction of the structure and phase behavior of relevant soft-matter systems by direct simulations. The key advantage of this method is its generality, which allows it to be applicable to a broad range of systems. To demonstrate the generality of our method, we also apply it to a colloid-polymer model system, where coarse-grained many-body interactions are pronounced.

Molecular Theory: A Tool for Predicting the Outcome of Self-Assembly of Polymers, Nanoparticles, Amphiphiles, and Other Soft Materials

The supramolecular organization of soft materials, such as colloids, polymers, and amphiphiles, results from a subtle balance of weak intermolecular interactions and entropic forces. This competition can drive the self-organization of soft materials at the nano-/mesoscale. Modeling soft-matter self-assembly requires, therefore, considering a complex interplay of forces at the relevant length scales without sacrificing the molecular details that define the chemical identity of the system. This mini-review focuses on the application of a tool known as molecular theory to study self-assembly in different types of soft materials. This tool is based on extremizing an approximate free energy functional of the system, and, therefore, it provides a direct, computationally affordable estimation of the stability of different self-assembled morphologies. Moreover, the molecular theory explicitly incorporates structural details of the chemical species in the system, accounts for their conformational degrees of freedom, and explicitly includes their chemical equilibria. This mini-review introduces the general ideas behind the theoretical formalism and discusses its advantages and limitations compared with other theoretical tools commonly used to study self-assembled soft materials. Recent application examples are discussed: the self-patterning of polyelectrolyte brushes on planar and curved surfaces, the formation of nanoparticle (NP) superlattices, and the self-organization of amphiphiles into micelles of different shapes. Finally, prospective methodological improvements and extensions (also relevant for related theoretical tools) are analyzed.

Modeling of effective interactions between ligand coated nanoparticles through symmetry functions

Ligand coated nanoparticles are complex objects consisting of a metallic or semiconductor core with organic ligands grafted on their surface. These organic ligands provide stability to a nanoparticle suspension. In solutions, the effective interactions between such nanoparticles are mediated through a complex interplay of interactions between the nanoparticle cores, the surrounding ligands, and the solvent molecules. While it is possible to compute these interactions using fully atomistic molecular simulations, such computations are too expensive for studying self-assembly of a large number of nanoparticles. The problem can be made tractable by removing the degrees of freedom associated with the ligand chains and solvent molecules and using the potentials of mean force (PMF) between nanoparticles. In general, the functional dependence of the PMF on the inter-particle distance is unknown and can be quite complex. In this article, we present a method to model the two-body and three-body PMF between ligand coated nanoparticles through a linear combination of symmetry functions. The method is quite general and can be extended to model interactions between different types of macromolecules.

Interaction between dilute water vapor and dodecane thiol ligated Au nanoparticles: Hydrated structure and pair potential of mean force

The interaction between two ligated nanoparticles depends on whether they are isolated or immersed in a liquid solvent. However, very little is known about the influence of solvent vapor on the interaction between two ligated nanoparticles. Recent experiments yield the surprising result that the cyclic exposure of solvent free suspended monolayers of dodecane thiol ligated gold nanoparticles (AuNPs) to water vapor and dry nitrogen generates reversible cyclic decreases and increases in Young's modulus of the monolayer, implying corresponding cyclic changes in the AuNP-AuNP interaction. We examine how water vapor interacts with an isolated dodecane thiol dressed AuNP and how water vapor affects the interaction between a pair of nanoparticles, using all-atom molecular-dynamics simulations. We find that there is condensation of water molecules onto the ligand shell of an AuNP in the form of clusters of 100-2000 molecules that partially cover the shell, with most of the water in a few large clusters. A water cluster bridges the AuNPs, with a sensibly constant number of water molecules for AuNP-AuNP separations from the edge-to-edge contact up to center-to-center separations of 100 Å. The wet AuNP-AuNP interaction has a slightly deeper and wider asymmetric well than does the dry interaction, a change that is qualitatively consistent with that implied by the observed water vapor induced change in Young's modulus of a monolayer of these AuNPs. We find that macroscopic analyses of water drop-deformable surface interactions and dynamics provide both guidance to understanding and qualitatively correct predictions of the phenomena observed in our simulations.

Body centered tetragonal nanoparticle superlattices: why and when they form?

Body centered tetragonal (BCT) phases are structural intermediates between body centered cubic (BCC) and face centered cubic (FCC) structures. However, BCC ↔ FCC transitions may or may not involve a stable BCT intermediate. Interestingly, nanoparticle superlattices usually crystallize in BCT structures, but this phase is much less frequent for colloidal crystals of micrometer-sized particles. Two origins have been proposed for the formation of BCT NPSLs: (i) the influence of the substrate on which the nanoparticle superlattice is deposited, and (ii) non-spherical nanoparticle shapes, combined with the fact that different crystal facets have different ligand organizations. Notably, none of these two mechanisms alone is able to explain the set of available experimental observations. In this work, these two hypotheses were independently tested using a recently developed molecular theory for nanoparticle superlattices that explicitly captures the degrees of freedom associated with the ligands on the nanoparticle surface and the crystallization solvent. We show that the presence of a substrate can stabilize the BCT structure for spherical nanoparticles, but only for very specific combinations of parameters. On the other hand, a truncated-octahedron nanoparticle shape strongly stabilizes BCT structures in a wide region of the phase diagram. In the latter case, we show that the stabilization of BCT results from the geometry of the system and it does not require different crystal facets to have different ligand properties, as previously proposed. These results shed light on the mechanisms of BCT stabilization in nanoparticle superlattices and provide guidelines to control its formation.

Understanding the binary interactions of noble metal and semiconductor nanoparticles

Molecular dynamics simulations are used to study the solvation and effective pair interactions of Au (1.2 nm) and CdSe (2.2 nm) nanoparticles passivated with alkanethiol and alkylamine ligands, respectively, for two different chain lengths in vacuum and n-hexane at 300 K. The solvation studies focus on quantifying the ligand and solvent shell structures, which are used to rationalize the interactions of nanoparticles in solution. To investigate the effective pair interactions, we compute the isotropic potential of mean forces (PMFs) between two nanoparticles and also analyze the anisotropy in the interactions that arises as a result of ligand shell fluctuations. Both isotropic and anisotropic contributions to the effective pair interactions between the two classes of nanoparticles are compared as a function of the ligand chain length and the solvent quality. It is demonstrated that the inclusion of the anisotropic aspect in the interparticle interactions is essential to properly describe the self-assembly thermodynamics of passivated nanoparticles. The implications of the coarse-grained modeling of the formation of binary nanocrystal superlattices (BNSLs) are considered.

The Phase Behavior of Nanoparticle Superlattices in the Presence of a Solvent

Superlattices of nanoparticles coated by alkyl-chain ligands are usually prepared from a stable solution by evaporation, therefore the pathway of superlattice self-assembly critically depends on the amount of solvent present within it. This work addresses the role of the solvent on the structure and the relative stability of the different supercrystalline phases of single-component superlattices (simple cubic, body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal close-packed). The study is performed with a molecular theory for nanoparticle superlattices introduced in this work, which predicts the structure and thermodynamics of the supercrystals explicitly treating the presence and molecular details of the solvent and the ligands. The theory predicts a FCC-BCC transition with decreasing solvent content due to the competition between the translational entropy of the solvent and the entropy and internal energy of the ligands. This result provides an explanation for recent experimental observations by in situ X-ray scattering, which reported a FCC-BCC transition during solvent evaporation. The theory also predicts the effects of the length and surface coverage of the ligands and the radius of the core on the phase behavior in agreement with experimental evidence and previous molecular dynamics simulations. These results validate the use of the dimensionless softness parameter λ (ratio of ligand length to core radius) to predict the phase behavior of wet superlattices. Our results stress the importance of explicitly considering the presence of the solvent in order to reach a complete picture of the mechanisms that mediate the self-assembly of nanoparticle superlattices.

Tunable interaction potentials and morphology of polymer-nanoparticle blends

We present the results of molecular dynamics simulations of a family of polymer nanocomposite systems. The polymer is described by a generic bead-and-spring model, while the polymer chains and the nanoparticles (NPs) interact by Hamaker-style potentials. The potential describing NP-NP interactions is modified by a tuning parameter f, which can be changed continuously between f = 0 (for fully developed van der Waals attractions between the NPs) and f = 1 (for completely repulsive interparticle interactions). We explore systematically the effect of the f parameter on the blend morphologies, for two representative NP sizes. When the polymer-NP attractions are decreased, the systems undergo a transition from dispersed to aggregated morphologies. The sharpness of the transition gradually increases with the interparticle attractions (i.e., decreasing f).

The influence of fractional surface coverage on the core-core separation in ordered monolayers of thiol-ligated Au nanoparticles

We report the results of grazing incidence X-ray diffraction (GIXD) measurements from water supported Langmuir monolayers of gold nanoparticles ligated with dodecanethiol (12 carbons), tetradecanethiol (14 carbons), hexadecanethiol (16 carbons), and octadecanethiol (18 carbons). These monolayers are formed from solutions with varying concentrations of the respective thiols. We show that equilibrium between adsorbed thiol molecules and the thiols in the bulk solution implies fractional coverage of the Au nanoparticle core. We also show that the nanoparticle-nanoparticle separation and the correlation length of particles in these ordered films increases with thiol concentration in the parent solution, and that excess thiol can be found in the space between particles as well as in islands away from the particles. Using the equilibrium constant relating ligand solution concentration and nanoparticle surface coverage of the gold core by the ligand molecules, we interpret the way in which varying thiol concentration affects the nanoparticle-nanoparticle separation as a function of surface coverage of the gold core by the ligand molecules.

Interaction between capped tetrahedral gold nanocrystals: dependence on effective softness

Atomistic molecular dynamics simulations are performed to explore the interaction between two alkylthiol-capped tetrahedral gold nanocrystals (NCs) in a vacuum. The results highlight the influential role of the effective softness of the ligated NCs, i.e. the ratio of the ligand length to the core size. For sufficiently large softness, the relatively long ligand molecules round the shape of the NCs, causing their interaction to be nearly isotropic. For small effective softness, the relative shortness of the ligand molecules leads to a geometrically asymmetric morphology of the NCs, so that the interaction is orientation-dependent and is the strongest when the two NCs face each other with (111) facets. These findings are helpful for the understanding of interaction and structure formation in superlattices self-assembled from non-spherical ligand-capped NCs.

Coexistence of hcp and bct Phases during In Situ Superlattice Assembly from Faceted Colloidal Nanocrystals

We study the in situ self-assembly of faceted PbS nanocrystals from colloidal suspensions upon controlled solvent evaporation using time-resolved small-angle X-ray scattering and X-ray cross-correlation analysis. In our bulk-sensitive experiment in transmission geometry, the superlattice crystallization is observed in real time, revealing a hexagonal closed-packed (hcp) structure followed by formation of a body-centered cubic (bcc) superlattice. The bcc superlattice undergoes continuous tetragonal distortion in the solvated state shortly after its formation, resulting in the body-centered tetragonal (bct) structure. Upon solvent evaporation, the bct superstructure becomes more pronounced with the still coexisting hcp phase. These findings corroborate the existing simulations of assembling cuboctahedral-shaped particles and illustrate that we observed the predicted equilibrium states. This work is essential for a deeper understanding of the fundamental forces that direct nanocrystal assembly including nanocrystal shape and ligand coverage.

Dictating Nanoparticle Assembly via Systems-Level Control of Molecular Multivalency

Nanoparticle assembly can be controlled by multivalent binding interactions between surface ligands, indicating that more precise control over these interactions is important to design complex nanoscale architectures. It has been well-established in natural materials that the arrangement of different molecular species in three dimensions can affect the ability of individual supramolecular units to coordinate their binding, thereby regulating the strength and specificity of their collective molecular interactions. However, in artificial systems, limited examples exist that quantitatively demonstrate how changes in nanoscale geometry can be used to rationally modulate the thermodynamics of individual molecular binding interactions. As a result, the use of nanoscale design features to regulate molecular bonding remains an underutilized design handle to control nanomaterials synthesis. Here we demonstrate a polymer-coated nanoparticle material where supramolecular bonding and nanoscale structure are used in conjunction to dictate the thermodynamics of their multivalent interactions, resulting in emergent bundling of supramolecular binding groups that would not be expected on the basis of the molecular structures alone. Additionally, we show that these emergent phenomena can controllably alter the superlattice symmetry by using the mesoscale particle arrangement to alter the thermodynamics of the supramolecular bonding behavior. The ability to rationally program molecular multivalency via a systems-level approach therefore provides a major step forward in the assembly of complex artificial structures, with implications for future designs of both nanoparticle- and supramolecular-based materials.