The stochastic simulation algorithm (SSA) is an essentially exact procedure for numerically simulating the time evolution of a well-stirred chemically reacting system. Despite recent major improvements in the efficiency of the SSA, its drawback remains the great amount of computer time that is often required to simulate a desired amount of system time. Presented here is the “τ-leap” method, an approximate procedure that in some circumstances can produce significant gains in simulation speed with acceptable losses in accuracy. Some primitive strategies for control parameter selection and error mitigation for the τ-leap method are described, and simulation results for two simple model systems are exhibited. With further refinement, the τ-leap method should provide a viable way of segueing from the exact SSA to the approximate chemical Langevin equation, and thence to the conventional deterministic reaction rate equation, as the system size becomes larger.

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Approximate accelerated stochastic simulation of chemically reacting systems

There are two fundamental ways to view coupled systems of chemical equations:  as continuous, represented by differential equations whose variables are concentrations, or as discrete, represented by stochastic processes whose variables are numbers of molecules. Although the former is by far more common, systems with very small numbers of molecules are important in some applications (e.g., in small biological cells or in surface processes). In both views, most complicated systems with multiple reaction channels and multiple chemical species cannot be solved analytically. There are exact numerical simulation methods to simulate trajectories of discrete, stochastic systems, (methods that are rigorously equivalent to the Master Equation approach) but these do not scale well to systems with many reaction pathways. This paper presents the Next Reaction Method, an exact algorithm to simulate coupled chemical reactions that is also efficient:  it (a) uses only a single random number per simulation event, and (b) ...

Efficient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels

The active sites over commercial copper/zinc oxide/aluminum oxide (Cu/ZnO/Al2O3) catalysts for carbon dioxide (CO2) hydrogenation to methanol, the Zn-Cu bimetallic sites or ZnO-Cu interfacial sites, have recently been the subject of intense debate. We report a direct comparison between the activity of ZnCu and ZnO/Cu model catalysts for methanol synthesis. By combining x-ray photoemission spectroscopy, density functional theory, and kinetic Monte Carlo simulations, we can identify and characterize the reactivity of each catalyst. Both experimental and theoretical results agree that ZnCu undergoes surface oxidation under the reaction conditions so that surface Zn transforms into ZnO and allows ZnCu to reach the activity of ZnO/Cu with the same Zn coverage. Our results highlight a synergy of Cu and ZnO at the interface that facilitates methanol synthesis via formate intermediates.

http://science.sciencemag.org/content/sci/355/6331/1296.full.pdf

Active sites for CO2 hydrogenation to methanol on Cu/ZnO catalysts

Rational optimization of catalytic performance has been one of the major challenges in catalysis. Here we report a bottom-up study on the ability of TiO2 and ZrO2 to optimize the CO2 conversion to methanol on Cu, using combined density functional theory (DFT) calculations, kinetic Monte Carlo (KMC) simulations, in situ diffuse reflectance infrared Fourier transform spectroscopy (DRIFTS) measurements, and steady-state flow reactor tests. The theoretical results from DFT and KMC agree with in situ DRIFTS measurements, showing that both TiO2 and ZrO2 help to promote methanol synthesis on Cu via carboxyl intermediates and the reverse water–gas-shift (RWGS) pathway; the formate intermediates, on the other hand, likely act as a spectator eventually. The origin of the superior promoting effect of ZrO2 is associated with the fine-tuning capability of reduced Zr3+ at the interface, being able to bind the key reaction intermediates, e.g. *CO2, *CO, *HCO, and *H2CO, moderately to facilitate methanol formation. This ...

/pdf/optimizing-binding-energies-of-key-intermediates-for-co2-4x8hdwqf94.pdf

Optimizing Binding Energies of Key Intermediates for CO2 Hydrogenation to Methanol over Oxide-Supported Copper.

The microscopic spatial kinetic Monte Carlo (KMC) method has been employed extensively in materials modeling. In this review paper, we focus on different traditional and multiscale KMC algorithms, challenges associated with their implementation, and methods developed to overcome these challenges. In the first part of the paper, we compare the implementation and computational cost of the null-event and rejection-free microscopic KMC algorithms. A firmer and more general foundation of the null-event KMC algorithm is presented. Statistical equivalence between the null-event and rejection-free KMC algorithms is also demonstrated. Implementation and efficiency of various search and update algorithms, which are at the heart of all spatial KMC simulations, are outlined and compared via numerical examples. In the second half of the paper, we review various spatial and temporal multiscale KMC methods, namely, the coarse-grained Monte Carlo (CGMC), the stochastic singular perturbation approximation, and the τ-leap methods, introduced recently to overcome the disparity of length and time scales and the one-at-a time execution of events. The concepts of the CGMC and the τ-leap methods, stochastic closures, multigrid methods, error associated with coarse-graining, a posteriori error estimates for generating spatially adaptive coarse-grained lattices, and computational speed-up upon coarse-graining are illustrated through simple examples from crystal growth, defect dynamics, adsorption–desorption, surface diffusion, and phase transitions.

/pdf/an-overview-of-spatial-microscopic-and-accelerated-kinetic-4sih54fo7m.pdf

An overview of spatial microscopic and accelerated kinetic Monte Carlo methods

Monte Carlo methods for the simulation of the dynamic behavior of surface reactions are developed, based on the chemical master equation. The methods are stated in a general framework which makes them applicable to a variety of models. Three methods are developed. A comparative analysis of the performance of the three methods, both theoretically and empirically, is included.

Efficient Monte Carlo methods for the simulation of catalytic surface reactions

Results of dynamic Monte Carlo simulations of a model for CO oxidation on a reconstructing Pt(100) surface are presented. A comparison is made between simulations that explicitly include surface diffusion of adsorbed CO and simulations without diffusion. Oscillatory behavior as well as spatio-temporal pattern formation are studied as a function of system size. In the absence of diffusion the amplitude of kinetic oscillations decreases with grid size and oscillations are not stable. Spatio-temporal patterns appear, as expected for an excitable medium. Such patterns become stabilized by structural substrate defects. The length scale of the patterns is in the order of 10–100 nm, the temporal period of the oscillations is around 200 seconds. Inclusion of diffusion stabilizes and synchronizes oscillations. Spatio-temporal features now appear with larger spatial dimensions.

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Monte Carlo simulations of a surface reaction model showing spatio-temporal pattern formations and oscillations

Dynamic Monte Carlo simulations of oscillatory heterogeneous catalytic reactions

/pdf/an-introduction-to-molecular-heterogeneous-catalysis-25a4m2kwxq.pdf

RJ Gelten

Papers

Efficient Monte Carlo methods for the simulation of catalytic surface reactions

Monte Carlo simulations of a surface reaction model showing spatio-temporal pattern formations and oscillations

Dynamic Monte Carlo simulations of oscillatory heterogeneous catalytic reactions

An introduction to molecular heterogeneous catalysis