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We describe experimental observations and theoretical analysis of the coarsening of distributions of twodimensional nanoclusters, either adatom islands or vacancy pits, on metal surfaces. A detailed analyses is provided for Ag(111) and Ag(100) surfaces, although we also discuss corresponding behavior for Cu(111) and Cu(100) surfaces. The dominant kinetic pathway for coarsening can be either Ostwald ripening (OR), i.e., growth of larger clusters at the expense of smaller ones, or Smoluchowski ripening (SR), i.e., diffusion and coalescence of clusters. First, for pristine additive-free surfaces, we elucidate the factors which control the dominant pathway. OR kinetics generally follows the predictions of mesoscale continuum theories. SR kinetics is controlled by the size-dependence of cluster diffusion. However, this size-dependence, together with that of nanostructure shape relaxation upon coalescence, often deviates from mesoscale predictions as a direct consequence of the nanoscale dimension of the clusters. Second, we describe examples for the above systems where trace amounts of a chemical additive lead to dramatic enhancement of coarsening. We focus on the scenario where "facile reaction" of metal and additive atoms leads to the formation of mobile additivemetal complexes which can efficiently transport metal across the surface, i.e., additive-enhanced OR. A suitable reaction-diffusion equation formulation is developed to describe this behavior. ReceiVed: July 19, 2008; ReVised Manuscript ReceiVed: December 25, 2008 We describe experimental observations and theoretical analysis of the coarsening of distributions of twodimensional nanoclusters, either adatom islands or vacancy pits, on metal surfaces. A detailed analyses is provided for Ag(111) and Ag(100) surfaces, although we also discuss corresponding behavior for Cu (111) and Cu(100) surfaces. The dominant kinetic pathway for coarsening can be either Ostwald ripening (OR), i.e., growth of larger clusters at the expense of smaller ones, or Smoluchowski ripening (SR), i.e., diffusion and coalescence of clusters. First, for pristine additive-free surfaces, we elucidate the factors which control the dominant pathway. OR kinetics generally follows the predictions of mesoscale continuum theories. SR kinetics is controlled by the size-dependence of cluster diffusion. However, this size-dependence, together with that of nanostructure shape relaxation upon coalescence, often deviates from mesoscale predictions as a direct consequence of the nanoscale dimension of the clusters. Second, we describe examples for the above systems where trace amounts of a chemical additive lead to dramatic enhancement of coarsening. We focus on the scenario where "facile reaction" of metal and additive atoms leads to the formation of mobile additivemetal complexes which can efficiently transport metal across the surface, i.e., additive-enhanced OR. A suitable reaction-diffusion equation formulation is developed to describe this behavior.

Traditional mean-field rate equations of chemical kinetics for spatially uniform systems1−3 and the corresponding reaction−diffusion equations describing spatial heterogeneity4−6 have proved immensely useful in elucidating catalytic processes. However, it is well-recognized that standard mean-field rate expressions neglect spatial correlations in the reactant and/or product distribution. It is less well appreciated that the standard treatment of diffusion is generally applicable only at low concentrations and in unrestricted environments.

Self-assembly of ensembles of supported 2D or 3D nanoclusters (NCs) by surface deposition, and of unsupported 3D NCs by solution-phase synthesis, produces intrinsically nonequilibrium systems. Individual nanoclusters can have far-from-equilibrium shapes and composition profiles. The free energy of the ensemble can be lowered by coarsening which can involve Ostwald ripening or Smoluchowski ripening (NC diffusion and coalescence). Preservation of individual NC structure and inhibition of coarsening is key for, e.g., avoiding catalyst degradation. In this review, we focus on crystalline metallic NCs. Atomistic-level modeling of equilibration processes 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. We also discuss modeling of coarsening including diffusion and decay of individual NCs, and unconventional coarsening processes. We describe high-level modeling integrated with STM studies for 2D epitaxial NCs, and developments in modeling for supported and unsupported 3D NCs motivated by in situ TEM studies.

We use a one-dimensional step model to study quantitatively the growth of step bunches on Si(111) surfaces induced by a direct heating current. Parameters in the model are fixed from experimental measurements near 900 • C under the assumption that there is local mass transport through surface diffusion and that step motion is limited by the attachment rate of adatoms to step edges. The direct heating current is treated as an external driving force acting on each adatom. Numerical calculations show both qualitative and quantitative agreement with experiment. A force in the step down direction will destabilize the uniform step train towards step bunching. The average size of the step bunches grows with electromigration time t as t β , with β ≈ 0.5, in agreement with experiment and with an analytical treatment of the steady states. The model is extended to include the effect of direct hopping of adatoms between different terraces. Monte-Carlo simulations of a solid-on-solid model, using physically motivated assumptions about the dynamics of surface diffusion and attachment at step edges, are carried out to study two dimensional features that are left out of the present step model and to test its validity. These simulations give much better agreement with experiment than previous work. We find a new step bending instability when the driving force is along the step edge direction. This instability causes the formation of step bunches and antisteps that is similar to that observed in experiment. 68.35.Ja,68.10.Jy,68.55.Jk,05.70.Ln

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Using scanning tunneling microscopy, we observe an adlayer structure that is dominated by short rows of S atoms, on unreconstructed regions of a Au(111) surface. This structure forms upon adsorption of low S coverage (less than 0.1 monolayer) on a fully reconstructed clean surface at 300 K, then cooling to 5 K for observation. The rows adopt one of three orientations that are rotated by 30 • from the close-packed directions of the Au(111) substrate, and adjacent S atoms in the rows are separated by √ 3 times the surface lattice constant, a. Monte Carlo simulations are performed on lattice-gas models, derived using a limited cluster expansion based on density functional theory energetics. Models which include long-range pairwise interactions (extending to 5a), plus selected trio interactions, successfully reproduce the linear rows of S atoms at reasonable temperatures.

We analyze the spatiotemporal behavior of species concentrations in a diffusion-mediated conversion reaction which occurs at catalytic sites within linear pores of nanometer diameter. Diffusion within the pores is subject to a strict single-file (no passing) constraint. Both transient and steady-state behavior is precisely characterized by kinetic Monte Carlo simulations of a spatially discrete lattice-gas model for this reaction-diffusion process considering various distributions of catalytic sites. Exact hierarchical master equations can also be developed for this model. Their analysis, after application of mean-field type truncation approximations, produces discrete reaction-diffusion type equations (mf-RDE). For slowly varying concentrations, we further develop coarse-grained continuum hydrodynamic reaction-diffusion equations (h-RDE) incorporating a precise treatment of single-file diffusion in this multispecies system. The h-RDE successfully describe nontrivial aspects of transient behavior, in contrast to the mf-RDE, and also correctly capture unreactive steady-state behavior in the pore interior. However, steady-state reactivity, which is localized near the pore ends when those regions are catalytic, is controlled by fluctuations not incorporated into the hydrodynamic treatment. The mf-RDE partly capture these fluctuation effects, but cannot describe scaling behavior of the reactivity. Keywordscatalytic conversion, catalytic sites, continuum hydrodynamics, conversion reactions, discrete lattices, hydrodynamic treatment, kinetic Monte Carlo simulation, master equations, reaction diffusion, scaling behavior, single file diffusion, spatiotemporal behaviors, fluid dynamics Disciplines Mathematics | Physical Chemistry | Physics CommentsThe following article appeared in Journal of Chemical Physics 134,11 (2011) We analyze the spatiotemporal behavior of species concentrations in a diffusion-mediated conversion reaction which occurs at catalytic sites within linear pores of nanometer diameter. Diffusion within the pores is subject to a strict single-file (no passing) constraint. Both transient and steady-state behavior is precisely characterized by kinetic Monte Carlo simulations of a spatially discrete lattice-gas model for this reaction-diffusion process considering various distributions of catalytic sites. Exact hierarchical master equations can also be developed for this model. Their analysis, after application of mean-field type truncation approximations, produces discrete reaction-diffusion type equations (mf-RDE). For slowly varying concentrations, we further develop coarse-grained continuum hydrodynamic reaction-diffusion equations (h-RDE) incorporating a precise treatment of single-file diffusion in this multispecies system. The h-RDE successfully describe nontrivial aspects of transient behavior, in contrast to the mf-RDE, and also correctly capture unreactive steady-state behavior in the pore interior. However, steady-state reactivity, which is localized near the pore ends when those regions a...