Potential of mean force for two nanocrystals: Core geometry and size, hydrocarbon unsaturation, and universality with respect to the force field

Theory and simulation of ligand functionalized nanoparticles - a pedagogical overview

Synthesizing reconfigurable nanoscale synthons with predictive control over shape, size, and interparticle interactions is a holy grail of bottom-up self-assembly. Grand challenges in their rational design, however, lie in both the large space of experimental synthetic parameters and proper understanding of the molecular mechanisms governing their formation. As such, computational and theoretical tools for predicting and modeling building block interactions have grown to become integral in modern day self-assembly research. In this review, we provide an in-depth discussion of the current state-of-the-art strategies available for modeling ligand functionalized nanoparticles. We focus on the critical role of how ligand interactions and surface distributions impact the emergent, pre-programmed behaviors between neighboring particles. To help build insights into the underlying physics, we first define an "ideal" limit - the short ligand, "hard" sphere approximation - and discuss all experimental handles through the lens of perturbations about this reference point. Finally, we identify theories that are capable of bridging interparticle interactions to nanoscale self-assembly and conclude by discussing exciting new directions for this field.

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.

Kinetic Growth of Multicomponent Microcompartment Shells

An important goal of systems and synthetic biology is to produce high value chemical species in large quantities. Microcompartments, which are protein nanoshells encapsulating catalytic enzyme cargo, could potentially function as tunable nanobioreactors inside and outside cells to generate these high value species. Modifying the morphology of microcompartments through genetic engineering of shell proteins is one viable strategy to tune cofactor and metabolite access to encapsulated enzymes. However, this is a difficult task without understanding how changing interactions between the many different types of shell proteins and enzymes affect microcompartment assembly and shape. Here, we use multiscale molecular dynamics and experimental data to describe assembly pathways available to microcompartments composed of multiple types of shell proteins with varied interactions. As the average interaction between the enzyme cargo and the multiple types of shell proteins is weakened, the shell assembly pathway transitions from (i) nucleating on the enzyme cargo to (ii) nucleating in the bulk and then binding the cargo as it grows to (iii) an empty shell. Atomistic simulations and experiments using the 1,2-propanediol utilization microcompartment system demonstrate that shell protein interactions are highly varied and consistent with our multicomponent, coarse-grained model. Furthermore, our results suggest that intrinsic bending angles control the size of these microcompartments. Overall, our simulations and experiments provide guidance to control microcomparmtent size and assembly by modulating the interactions between shell proteins.

Ligand Effects in Assembly of Cubic and Spherical Nanocrystals: Applications to Packing of Perovskite Nanocubes

We establish the formula representing cubic nanocrystals (NCs) as hard cubes taking into account the role of the ligands and describe how these results generalize to any other NC shapes. We derive the conditions under which the hard cube representation breaks down and provide explicit expressions for the effective size. We verify the results from the detailed potential of mean force calculations for two nanocubes in different orientations as well as with spherical nanocrystals. Our results explicitly demonstrate the relevance of certain ligand conformations, i.e., "vortices", and show that edges and corners provide natural sites for their emergence. We also provide both simulations and experimental results with single component cubic perovskite nanocrystals assembled into simple cubic superlattices, which further corroborate theoretical predictions. In this way, we extend the Orbifold Topological Model (OTM) accounting for the role of ligands beyond spherical nanocrystals and discuss its extension to arbitrary nanocrystal shapes. Our results provide detailed predictions for recent superlattices of perovskite nanocubes and spherical nanocrystals. Problems with existing united atom force fields are discussed.

The Crucial Role of Solvation Forces in the Steric Stabilization of Nanoplatelets

The steric stability of inorganic colloidal particles in an apolar solvent is usually described in terms of the balance between three contributions: the van der Waals attraction, the free energy of mixing, and the ligand compression. However, in the case of nanoparticles, the discrete nature of the ligand shell and the solvent has to be taken into account. Cadmium selenide nanoplatelets are a special case. They combine a weak van der Waals attraction and a large facet to particle size ratio. We use coarse grained molecular dynamics simulations of nanoplatelets in octane to demonstrate that solvation forces are strong enough to induce the formation of nanoplatelet stacks and by that have a crucial impact on the steric stability. In particular, we demonstrate that for sufficiently large nanoplatelets, solvation forces are proportional to the interacting facet area, and their strength is intrinsically tied to the softness of the ligand shell.

Monovalent ion-mediated charge-charge interactions drive aggregation of surface-functionalized gold nanoparticles

Monolayer-protected metal nanoparticles (NPs) are not only promising materials with a wide range of potential industrial and biological applications, but they are also a powerful tool to investigate the behaviour of matter at nanoscopic scales, including the stability of dispersions and colloidal systems. This stability is dependent on a delicate balance between attractive and repulsive interactions that occur in the solution, and it is described in quantitative terms by the classic Derjaguin-Landau-Vewey-Overbeek (DLVO) theory, that posits that aggregation between NPs is driven by van der Waals interactions and opposed by electrostatic interactions. To investigate the limits of this theory at the nanoscale, where the continuum assumptions required by the DLVO theory break down, here we investigate NP dimerization by computing the Potential of Mean Force (PMF) of this process using fully atomistic MD simulations. Serendipitously, we find that electrostatic interactions can lead to the formation of metastable NP dimers at physiological ion concentrations. These dimers are stabilized by complexes formed by negatively charged ligands belonging to distinct NPs that are bridged by positively charged monovalent ions present in solution. We validate our findings by collecting tomographic EM images of NPs in solution and by quantifying their radial distribution function, that shows a marked peak at interparticle distance comparable with that of MD simulations. Taken together, our results suggest that not only van der Waals interactions, but also electrostatic interactions mediated by monovalent ions at physiological concentrations, contribute to attraction between nano-sized charged objects at very short length scales.

Reversible assembly of nanoparticles: theory, strategies and computational simulations

The significant advances in synthesis and functionalization have enabled the preparation of high-quality nanoparticles that have found a plethora of successful applications. The unique physicochemical properties of nanoparticles can be manipulated through the control of size, shape, composition, and surface chemistry, but their technological application possibilities can be further expanded by exploiting the properties that emerge from their assembly. The ability to control the assembly of nanoparticles not only is required for many real technological applications, but allows the combination of the intrinsic properties of nanoparticles and opens the way to the exploitation of their complex interplay, giving access to collective properties. Significant advances and knowledge gained over the past few decades on nanoparticle assembly have made it possible to implement a growing number of strategies for reversible assembly of nanoparticles. In addition to being of interest for basic studies, such advances further broaden the range of applications and the possibility of developing innovative devices using nanoparticles. This review focuses on the reversible assembly of nanoparticles and includes the theoretical aspects related to the concept of reversibility, an up-to-date assessment of the experimental approaches applied to this field and the advanced computational schemes that offer key insights into the assembly mechanisms. We aim to provide readers with a comprehensive guide to address the challenges in assembling reversible nanoparticles and promote their applications.

Ligand structure and adsorption free energy of nanocrystals on solid substrates

We present an investigation on the absorption of alkylthiolated nanocrystals on a solid substrate. We calculate adsorption free energies and report a number of effects induced by the substrate. Nearest neighbor distances and bonding free energies are significantly different than for a free floating case, there is a weakening of bonding free energies among nanocrystals, and the adsorption is manifestly anisotropic, i.e., stronger along certain directions of the nanocrystal core. We contend that this last result accounts for the Bain transition (fcc → bcc) observed in experimental results. We report the presence of vortices induced by the substrate, which explain the increased nearest neighbor distance among nanocrystals, which is in excellent quantitative agreement with experimental results and with the predictions of the Orbifold Topological Model. Implications for the assembly of nanostructures and future experiments are also discussed.

Assembly of nanocrystal clusters by solvent evaporation: icosahedral order and the breakdown of the Maxwell regime

We carry out molecular dynamics simulations of N gold alkylthiolated nanocrystals (0 ≤ N ≤ 29) contained in liquid droplets of octane, nonane and decane coexisting with its vapor. The equilibrium structures that result when all the solvent dries up consist of highly symmetric nanocrystal clusters with different degrees of icosahedral order that are thoroughly characterized. We show that the relaxation times follow two regimes, a first for small nanocrystal packing fraction, dominated by the diffusion of vapor molecules (Maxwell regime, relaxation times independent of N) and another, for larger packing fractions, where the solvent diffuses through the cluster (with relaxation times growing like N2/3). We discuss the connection to the assembly of superlattices, prediction of lattice constants and evaporation models.

Phase Diagram and Structure Map of Binary Nanoparticle Superlattices from a Lennard-Jones Model

A first-principles prediction of the binary nanoparticle phase diagram assembled by solvent evaporation has eluded theoretical approaches. In this paper, we show that a binary system interacting through the Lennard-Jones (LJ) potential contains all experimental phases in which nanoparticles are effectively described as quasi hard spheres. We report a phase diagram consisting of 53 equilibrium phases, whose stability is quite insensitive to the microscopic details of the potentials, thus giving rise to some type of universality. Furthermore, we show that binary lattices may be understood as consisting of certain particle clusters, i.e., motifs, that provide a generalization of the four conventional Frank-Kasper polyhedral units. Our results show that metastable phases share the very same motifs as equilibrium phases. We discuss the connection with packing models, phase diagrams with repulsive potentials, and the prediction of likely experimental superlattices.

When Like Destabilizes Like: Inverted Solvent Effects in Apolar Nanoparticle Dispersions

We report on the colloidal stability of nanoparticles with alkanethiol shells in apolar solvents. Small-angle X-ray scattering and molecular dynamics simulations were used to characterize the interaction between nanoparticles in linear alkane solvents ranging from hexane to hexadecane, including 4 nm gold cores with hexadecanethiol shells and 6 nm cadmium selenide cores with octadecanethiol shells. We find that the agglomeration is enthalpically driven and that, contrary to what one would expect from classical colloid theory, the temperature at which the particles agglomerate increases with increasing solvent chain length. We demonstrate that the inverted trend correlates with the temperatures at which the ligands order in the different solvents and show that the inversion is due to a combination of enthalpic and entropic effects that enhance the stability of the ordered ligand state as the solvent length increases. We also explain why cyclohexane is a better solvent than hexadecane despite the two having very similar solvation parameters.

Assembly by solvent evaporation: equilibrium structures and relaxation times

We present a study describing the dynamics and equilibrium of the assembly of nanostructures by solvent evaporation. We first consider N nanocrystals stabilized by capping ligands in a spherical droplet of liquid solvent coexisting with its gas and show that, as the liquid solvent evaporates slowly, NCs crystallize into clusters of high symmetry based on tetrahedral and octahedral units: tetrahedron (N = 4), octahedron (N = 6), icosahedron (N = 13), Archimedean truncated tetrahedron (N = 16) and Z₂₀ (N = 21). We derive explicit formulas for the process and rigorously compute relaxation times, which drastically increase when the packing parameter reaches the hard-sphere liquid-solid transition η = 0.49. This result shows that contrary to what occurs in an evaporation of a single component system, the relaxation times are not determined by the diffusion constant of the vapor, but rather, are dominated by the residence time of solvent molecules trapped within the capping ligands. Our theory provides a number of predictions that enable the design of new structures while improving the control and quality of their assembly.

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.

Colloidal Stability of Apolar Nanoparticles: Role of Ligand Length

Inorganic nanoparticle cores are often coated with organic ligands to render them dispersible in apolar solvents. However, the effect of the ligand shell on the colloidal stability of the overall hybrid particle is not fully understood. In particular, it is not known how the length of an apolar alkyl ligand chain affects the stability of a nanoparticle dispersion against agglomeration. Here, small-angle X-ray scattering and molecular dynamics simulations have been used to study the interactions between gold nanoparticles and between cadmium selenide nanoparticles passivated by alkanethiol ligands with 12-18 carbons in the solvent decane. We find that increasing the ligand length increases colloidal stability in the core-dominated regime but decreases it in the ligand-dominated regime. This unexpected inversion is connected to the transition from ligand-dominated to core-dominated agglomeration when the core diameter increases at constant ligand length. Our results provide a microscopic picture of the forces that determine the colloidal stability of apolar nanoparticles and explain why classical colloid theory fails.