Three-body effects in triplets of capped gold nanocrystals

High-Precision Calculation of Nanoparticle (Nanocrystal) Potentials of Mean Force and Internal Energies

Thermodynamic stability assessment of nanocrystal systems requires precise free energy calculations. This study highlights the importance of meticulous control over various factors, including the thermostat, time step, potential cutoff, initial configuration, sampling method, and overall simulation duration. Free energy computations in dry (solvent-free) systems are on the order of several hundred kBT but can be obtained with consistent accuracy. However, calculation of internal energies becomes challenging, as they are typically much larger in magnitude than free energies and exhibit significant noise and reduced reliability. To address this limitation, we propose a new internal energy estimate that drastically reduces the noise. We also present formulas that enable the optimization of the parameters of the harmonic bias potential for optimal convergence. Finally, we discuss the implications of these findings for the computation of free energies in nanocrystal clusters and superlattices.

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.

Deconstructing Electrostatics of Functionalized Metal Nanoparticles from Molecular Dynamics Simulations

Gold nanoparticles (NPs) with different surface functionalizations can selectively interact with specific proteins, allowing a wide range of possible applications in biotechnology and biomedicine. To prevent their tendency to aggregate and to modulate their interaction with charged biomolecules or substrates (e.g., for biosensing applications), they can be functionalized with charged groups, introducing a mutual interaction which can be modulated by changing the ionic strength of the solvent. In silico modeling of these systems is often addressed with low-resolution models, which must account for these effects in the, often implicit, solvent representation. Here, we present a systematic conformational dynamic characterization of ligand-coated gold nanoparticles with different sizes, charges, and functionalizations by means of atomistic molecular dynamics simulations. Based on these, we deconstruct their electrostatic properties and propose a general representation of their average-long-range interactions extendable to different sizes, charges, and ionic strengths. This study clarifies in detail the role of the different features of the NP (charge, size, structure) and of the ionic strength in determining the details of the interparticle interaction and represents the first step toward a general strategy for the parametrization of NP coarse-grained models able to account for varying ionic strengths.

Machine-learning effective many-body potentials for anisotropic particles using orientation-dependent symmetry functions

Spherically-symmetric atom-centered descriptors of atomic environments have been widely used for constructing potential or free energy surfaces of atomistic and colloidal systems and to characterize local structures using machine learning techniques. However, when particle shapes are non-spherical, as in the case of rods and ellipsoids, standard spherically-symmetric structure functions alone produce imprecise descriptions of local environments. In order to account for the effects of orientation, we introduce two- and three-body orientation-dependent particle-centered descriptors for systems composed of rod-like particles. To demonstrate the suitability of the proposed functions, we use an efficient feature selection scheme and simple linear regression to construct coarse-grained many-body interaction potentials for computationally-efficient simulations of model systems consisting of colloidal particles with anisotropic shape: mixtures of colloidal rods and nonadsorbing polymer, hard rods enclosed by an elastic microgel shell, and ligand-stabilized nanorods. We validate the machine-learning (ML) effective many-body potentials based on orientation-dependent symmetry functions by using them in direct coexistence simulations to map out the phase behavior of colloidal rods and non-adsorbing polymer. We find good agreement with results obtained from simulations of the true binary mixture, demonstrating that the effective interactions are well-described by the orientation-dependent ML potentials.

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.

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.

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.

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.

Molecular interaction between asymmetric ligand-capped gold nanocrystals

Atomistic molecular dynamics simulations are performed to study the potential of mean force (PMF) between two asymmetric gold nanocrystals (NCs) capped by alkylthiols in a vacuum. We systematically investigate the dependence of the PMF on the sizes and capping ligand lengths of two NCs. The results show that the potential well depth scales linearly with increasing total length of two capping ligands on asymmetric dimers, but it hardly depends on the NC size. The predicted equilibrium distance between two asymmetric NCs grows significantly and linearly with the total size of two NCs and exhibits only a slight increase with increasing total ligand length. These findings are explained in terms of the amount of ligand interdigitation between NC surfaces as well as its alterations caused by the change in ligand length and NC size. Furthermore, we introduce a simple formula to estimate the equilibrium distance of two asymmetric NCs. On the basis of the computed PMFs, we propose an empirical two-body potential between asymmetric capped gold NCs.

Many Body Effects and Icosahedral Order in Superlattice Self-Assembly

We elucidate how nanocrystals "bond" to form ordered structures. For that purpose we consider nanocrystal configurations consisting of regular polygons and polyhedra, which are the motifs that constitute single component and binary nanocrystal superlattices, and simulate them using united atom models. We compute the free energy and quantify many body effects, i.e., those that cannot be accounted for by pair potential (two-body) interactions, further showing that they arise from coalescing vortices of capping ligands. We find that such vortex textures exist for configurations with local coordination number ≤6. For higher coordination numbers, vortices are expelled and nanocrystals arrange in configurations with tetrahedral or icosahedral order. We provide explicit formulas for the optimal separations between nanocrystals, which correspond to the minima of the free energies. Our results quantitatively explain the structure of superlattice nanocrystals as reported in experiments and reveal how packing arguments, extended to include soft components, predict ordered nanocrystal aggregation.

Capping Ligand Vortices as "Atomic Orbitals" in Nanocrystal Self-Assembly

We present a detailed analysis of the interaction between two nanocrystals capped with ligands consisting of hydrocarbon chains by united atom molecular dynamics simulations. We show that the bonding of two nanocrystals is characterized by ligand textures in the form of vortices. These results are generalized to nanocrystals of different types (differing core and ligand sizes) where the structure of the vortices depends on the softness asymmetry. We provide rigorous calculations for the binding free energy, show that these energies are independent of the chemical composition of the cores, and derive analytical formulas for the equilibrium separation. We discuss the implications of our results for the self-assembly of single-component and binary nanoparticle superlattices. Overall, our results show that the structure of the ligands completely determines the bonding of nanocrystals, fully supporting the predictions of the recently proposed Orbifold topological model.