Size-dependent diffusion of supported metal nanoclusters: mean-field-type treatments and beyond for faceted clusters.

Nanostructured systems are intrinsically metastable and subject to coarsening. For supported 3D metal nanoclusters (NCs), coarsening can involve NC diffusion across the support and subsequent coalescence (as an alternative to Ostwald ripening). When used as catalysts, this leads to deactivation. The dependence of diffusivity, DN, on NC size, N (in atoms), controls coarsening kinetics. Traditional mean-field (MF) theory for DNversus N assumes that NC diffusion is mediated by independent random hopping of surface adatoms with low coordination, and predicts that DN ∼ hN-4/3neq. Here, h = ν exp[-Ed/(kBT)] denotes the hop rate, and neq = exp[-Eform/(kBT)] the density of those adatoms. The adatom formation energy, Eform, approaches a finite large-N limit, as does the effective barrier, Eeff = Ed + Eform, for NC diffusion. This MF theory is critically assessed for a realistic stochastic atomistic model for diffusion of faceted fcc metal NCs with a {100} facet epitaxially attached to a (100) support surface. First, the MF formulation is refined to account for distinct densities and hop rates for surface adatoms on different facets and along the base contact line, and to incorporate the exact values of Eform and neqversus N for our model. MF theory then captures the occurrence of local minima in DNversus N at closed-shell sizes, as shown by KMC simulation. However, the MF treatment also displays fundamental shortcomings due to the feature that diffusion of faceted NCs is actually dominated by a cooperative multi-step process involving disassembling and reforming of outer layers on side facets. This mechanism leads to an Eeff which is well above MF values, and which increases with N, features captured by a beyond-MF treatment.

Nucleation-mediated reshaping of facetted metallic nanocrystals: Breakdown of the classical free energy picture.

Shape stability is key to avoiding degradation of performance for metallic nanocrystals synthesized with facetted non-equilibrium shapes to optimize properties for catalysis, plasmonics, and so on. Reshaping of facetted nanocrystals is controlled by the surface diffusion-mediated nucleation and growth of new outer layers of atoms. Kinetic Monte Carlo (KMC) simulation of a realistic stochastic atomistic-level model is applied to precisely track the reshaping of Pd octahedra and nanocubes. Unexpectedly, separate constrained equilibrium Monte Carlo analysis of the free energy profile during reshaping reveals a fundamental failure of the classical nucleation theory (CNT) prediction for the reshaping barrier and rate. Why? Nucleation barriers can be relatively low for these processes, so the system is not locally equilibrated before crossing the barrier, as assumed in CNT. This claim is supported by an analysis of a first-passage problem for reshaping within a master equation framework for the model that reasonably captures the behavior in KMC simulations.

Shape Stability of Truncated Octahedral fcc Metal Nanocrystals.

Metallic nanocrystals (NCs) can be synthesized with tailored nonequilibrium shapes to enhance desired properties, e.g., octahedral fcc metal NCs optimize catalytic activity associated with {111} facets. However, maintenance of optimized properties requires stability against thermal reshaping. Thus, we analyze the reshaping of truncated fcc metal octahedra mediated by surface diffusion using a stochastic atomistic-level model with energetic input parameters for Pd. The model describes NC thermodynamics by an effective nearest-neighbor interaction and includes a realistic treatment of diffusive hopping for undercoordinated surface atoms. Kinetic Monte Carlo simulation reveals that the effective barrier, Eeff, for the initial stage of reshaping is strongly tied to the degree of truncation of the vertices in the synthesized initial octahedral shapes. This feature is elucidated via exact analytic determination of the energy variation along the optimal reshaping pathway at low-temperature (T), which involves transfer of atoms from truncated {100} vertex facets to form new layers on {111} side facets. Deviations from predictions of the low-T analysis due to entropic effects are more prominent for higher T and larger NC sizes.

Encapsulation of metal nanoparticles at the surface of a prototypical layered material.

Encapsulation of metal nanoparticles just below the surface of a prototypical layered material, graphite, is a recently discovered phenomenon. These encapsulation architectures have potential for tuning the properties of two-dimensional or layered materials, and additional applications might exploit the properties of the encapsulated metal nanoclusters themselves. The encapsulation process produces novel surface nanostructures and can be achieved for a variety of metals. Given that these studies of near-surface intercalation are in their infancy, these systems provide a rich area for future studies. This Review presents the current progress on the encapsulation, including experimental strategies and characterization, as well as theoretical understanding which leads to the development of predictive capability. The Review closes with future opportunities where further understanding of the encapsulation is desired to exploit its applications.

Equilibrium shapes of facetted 3D metal nanoclusters intercalated near the surface of layered materials.

Experimental studies indicate that 3D crystalline metal nanoclusters (NCs) intercalated under the surface of graphite have flat-topped equilibrated shapes. We characterize the shapes of these facetted NCs sandwiched between a blanketing graphene layer and the underlying graphite substrate. Specifically, we focus on the cases of fcc Cu and hcp Fe NCs. The analysis involves numerical minimization of the system energy for a specified NC volume and NC height, the latter corresponding to the separation between parallel top and bottom facets. Our numerical analysis quantifies how the distance of the side facet planes from center of the nanocluster varies linearly with a natural characteristic linear dimension of the nanocluster. Calculated shapes of fcc Cu and hcp Fe NCs are consistent with the hexagonal footprints observed in scanning tunneling microscopy studies.

Reshaping of Truncated Pd Nanocubes: Energetic and Kinetic Analysis Integrating Transmission Electron Microscopy with Atomistic-Level and Coarse-Grained Modeling.

Stability against reshaping of metallic fcc nanocrystals synthesized with tailored far-from-equilibrium shapes is key to maintaining optimal properties for applications such as catalysis. Yet Arrhenius analysis of experimental reshaping kinetics, and appropriate theory and simulation, is lacking. Thus, we use TEM to monitor the reshaping of Pd nanocubes of ∼25 nm side length between 410 °C (over ∼4.5 h) and 440 °C (over ∼0.25 h), extracting a high effective energy barrier of Eeff ≈ 4.6 eV. We also provide an analytic determination of the energy variation along the optimal pathway for reshaping that involves transfer of atoms across the nanocube surface from edges or corners to form new layers on side {100} facets. The effective barrier from this analysis is shown to increase strongly with the degree of truncation of edges and corners in the synthesized nanocube. Theory matches experiment for the appropriate degree of truncation. In addition, we perform simulations of a stochastic atomistic-level model incorporating a realistic description of diffusive hopping for undercoordinated surface atoms, thereby providing a visualization of the initial reshaping process.

Generalized hydrodynamic analysis of transport through a finite open nanopore for two-component single-file systems.

Single-file diffusion (SFD) in finite open nanopores is characterized by nonzero spatially varying tracer diffusion coefficients within a generalized hydrodynamic description. This contrasts with infinite SFD systems where tracer diffusivity vanishes. In standard tracer counterpermeation (TCP) analysis, two reservoirs, each containing a different species, are connected to opposite ends of a finite pore. We implement an extended TCP analysis to allow the two reservoirs to contain slightly different mixtures of the two species. Then, determination of diffusion fluxes through the pore allows extraction of diffusion coefficients for near-constant partial concentrations of the two species. This analysis is applied for a lattice-gas model describing two-component SFD through a finite linear pore represented by a one-dimensional array of cells. Two types of particles, A and B, can hop only to adjacent empty cells with generally different rates, h_{A} and h_{B}. Particles are noninteracting other than exclusion of multiple cell occupancy. Results reveal generalized hydrodynamic tracer diffusion coefficients which adopt small values inversely proportional to pore length in the pore center, but which are strongly enhanced near pore openings.

Complex oscillatory decrease with size in diffusivity of {100}-epitaxially supported 3D fcc metal nanoclusters.

Diffusion and coalescence of supported 3D metal nanoclusters (NCs) leads to Smoluchowski Ripening (SR), a key pathway for catalyst degradation. Variation of the NC diffusion coefficient, DN, with size N (in atoms) controls SR kinetics. Traditionally, a form DN∼N-ß was assumed consistent with mean-field analysis. However, KMC simulation of a stochastic model for diffusion of {100}-epitaxially supported fcc NCs mediated by surface diffusion reveals instead a complex oscillatory decrease of DN with N. Barriers for surface diffusion of metal atoms across and between facets, along step edges, etc., in this model are selected to accurately capture behavior for fcc metals. (This contrasts standard bond-breaking prescriptions which fail dramatically.) For strong adhesion, equilibrated NCs are truncated pyramids (TP). Local minima of DN sometimes but not always correspond to sizes, NTP, where these have a closed-shell structure. Local maxima generally correspond to N≈NTP + 3 for N = O(102). For weak adhesion, equilibrated NCs are truncated octahedra (TO), and local minima of DN occur for sizes close or equal to those of just a subset of closed-shell structures. Analytic characterization of energetics along the NC diffusion pathway (which involves dissolving and reforming outer layers of facets) provides fundamental insight into the behavior of DN, including the strong variation with N of the effective NC diffusion barrier.

Reshaping, Intermixing, and Coarsening for Metallic Nanocrystals: Nonequilibrium Statistical Mechanical and Coarse-Grained Modeling.

Self-assembly of supported 2D or 3D nanocrystals (NCs) by vacuum deposition and of 3D NCs by solution-phase synthesis (with possible subsequent transfer to a support) produces intrinsically nonequilibrium systems. Individual NCs can have far-from-equilibrium shapes and composition profiles. The free energy of NC ensembles is lowered by coarsening which can involve Ostwald ripening or Smoluchowski ripening (NC diffusion and coalescence). Preservation of individual NC structure and inhibition of coarsening are key, e.g., for avoiding catalyst degradation. This review focuses on postsynthesis evolution of metallic NCs. Atomistic-level modeling typically utilizes stochastic lattice-gas models to access appropriate time and length scales. However, predictive modeling requires incorporation of realistic rates for relaxation mechanisms, e.g., periphery diffusion and intermixing, in numerous local environments (rather than the use of generic prescriptions). Alternative coarse-grained modeling must also incorporate appropriate mechanisms and kinetics. At the level of individual NCs, we present analyses of reshaping, including sintering and pinch-off, and of compositional evolution in a vacuum environment. We also discuss modeling of coarsening including diffusion and decay of individual NCs and unconventional coarsening processes. We describe high-level modeling integrated with scanning tunneling microscopy (STM) studies for supported 2D epitaxial nanoclusters and developments in modeling for 3D NCs motivated by in situ transmission electron microscopy (TEM) studies.

Communication: Diverse nanoscale cluster dynamics: Diffusion of 2D epitaxial clusters.

The dynamics of nanoscale clusters can be distinct from macroscale behavior described by continuum formalisms. For diffusion of 2D clusters of N atoms in homoepitaxial systems mediated by edge atom hopping, macroscale theory predicts simple monotonic size scaling of the diffusion coefficient, DN ∼ N-ß, with ß = 3/2. However, modeling for nanoclusters on metal(100) surfaces reveals that slow nucleation-mediated diffusion displaying weak size scaling ß < 1 occurs for "perfect" sizes Np = L2 and L(L+1) for integer L = 3,4,… (with unique square or near-square ground state shapes), and also for Np+3, Np+4,…. In contrast, fast facile nucleation-free diffusion displaying strong size scaling ß ≈ 2.5 occurs for sizes Np+1 and Np+2. DN versus N oscillates strongly between the slowest branch (for Np+3) and the fastest branch (for Np+1). All branches merge for N = O(102), but macroscale behavior is only achieved for much larger N = O(103). This analysis reveals the unprecedented diversity of behavior on the nanoscale.