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.

Recently, Gillespie introduced the tau-leap approximate, accelerated stochastic Monte Carlo method for well-mixed reacting systems [J. Chem. Phys. 115, 1716 (2001)]. In each time increment of that method, one executes a number of reaction events, selected randomly from a Poisson distribution, to enable simulation of long times. Here we introduce a binomial distribution tau-leap algorithm (abbreviated as BD-tau method). This method combines the bounded nature of the binomial distribution variable with the limiting reactant and constrained firing concepts to avoid negative populations encountered in the original tau-leap method of Gillespie for large time increments, and thus conserve mass. Simulations using prototype reaction networks show that the BD-tau method is more accurate than the original method for comparable coarse-graining in time.

The hydrogenation of phenol has been carried out in supercritical carbon dioxide (scCO 2 ) under very mild reaction conditions at the temperature of 50 8C over palladium supported Al-MCM-41 (metal loading~1%). This palladium catalyst is shown to be highly active and promotes the selective formation of cyclohexanone (~98%), an industrially important compound, in a "one-pot" way. The effects of different variables like carbon dioxide and hydrogen pressure, reaction time and also silica/alumina ratio of the MCM-41 support along with palladium dispersion are presented and discussed. The pressure effect of carbon dioxide is significantly prominent in terms of conversion and cyclohexanone selectivity. Moreover, the silica/alumina ratio was also found to be an important parameter to enhance the effectiveness of the catalyst as it exhibits a remarkable increase in phenol conversion from 20.6% to 98.4% as the support changes from only silica MCM-41 to Al-MCM-41. A plausible mechanism for the hydrogenation of phenol to cyclohexanone over the palladium catalyst has been proposed. The proposition is validated by transition state calculations using density functional theory (DFT), which reveal that cyclohexanone is a favorable product and stabilized by < 19 kcal mol À1 over cyclohexanol in scCO 2 medium. Under similar reaction conditions, phenol hydrogenation was also carried out with rhodium, supported on Al-MCM-41. In contrast to the palladium catalyst, a mixture of cyclohexanone (57.8%) and cyclohexanol (42.2%) was formed. Detailed characterization by X-ray diffraction and transmission electron microscopy confirmed the presence of metal nanoparticles (palladium and rhodium) between 10-20 nm. Both the catalysts exhibit strikingly different product distributions in solventless conditions compared to scCO 2 . This method can also be successfully applied to the other hydroxylated aromatic compounds.

We present a novel computational algorithm called the accelerated superbasin kinetic Monte Carlo (AS-KMC) method that enables a more efficient study of rare-event dynamics than the standard KMC method while maintaining control over the error. In AS-KMC, the rate constants for processes that are observed many times are lowered during the course of a simulation. As a result, rare processes are observed more frequently than in KMC and the time progresses faster. We first derive error estimates for AS-KMC when the rate constants are modified. These error estimates are next employed to develop a procedure for lowering process rates with control over the maximum error. Finally, numerical calculations are performed to demonstrate that the AS-KMC method captures the correct dynamics, while providing significant CPU savings over KMC in most cases. We show that the AS-KMC method can be employed with any KMC model, even when no time scale separation is present (although in such cases no computational speed-up is observed), without requiring the knowledge of various time scales present in the system.

While lattice kinetic Monte Carlo (KMC) methods provide insight into numerous complex physical systems governed by interatomic interactions, they are limited to relatively short length and time scales. Recently introduced coarse-grained Monte Carlo (CGMC) simulations can reach much larger length and time scales at considerably lower computational cost. In this paper we extend the CGMC methods to spatially adaptive meshes for the case of surface diffusion (canonical ensemble). We introduce a systematic methodology to derive the transition probabilities for the coarse-grained diffusion process that ensure the correct dynamics and noise, give the correct continuum mesoscopic equations, and satisfy detailed balance. Substantial savings in CPU time are demonstrated compared to microscopic KMC while retaining high accuracy.