Kinetic Monte Carlo Algorithm Research Articles

A statistical approach to develop a detailed soot growth model using PAH characteristics

A detailed PAH growth model is developed, which is solved using a kinetic Monte Carlo algorithm. The model describes the structure and growth of planar PAH molecules, and is referred to as the kinetic Monte Carlo–aromatic site (KMC-ARS) model. A detailed PAH growth mechanism based on reactions at radical sites available in the literature, and additional reactions obtained from quantum chemistry calculations are used to model the PAH growth processes. New rates for the reactions involved in the cyclodehydrogenation process for the formation of 6-member rings on PAHs are calculated in this work based on density functional theory simulations. The KMC-ARS model is validated by comparing experimentally observed ensembles on PAHs with the computed ensembles for a C 2H 2 and a C 6H 6 flame at different heights above the burner. The motivation for this model is the development of a detailed soot particle population balance model which describes the evolution of an ensemble of soot particles based on their PAH structure. However, at present incorporating such a detailed model into a population balance is computationally unfeasible. Therefore, a simpler model referred to as the site-counting model has been developed, which replaces the structural information of the PAH molecules by their functional groups augmented with statistical closure expressions. This closure is obtained from the KMC-ARS model, which is used to develop correlations and statistics in different flame environments which describe such PAH structural information. These correlations and statistics are implemented in the site-counting model, and results from the site-counting model and the KMC-ARS model are in good agreement. Additionally the effect of steric hindrance in large PAH structures is investigated and correlations for sites unavailable for reaction are presented.

How Focal Adhesion Size Depends on Integrin Affinity

Understanding how the thermodynamics and kinetics of integrin receptor binding and clustering impact the formation of focal adhesions is important for understanding the mechanisms cells use to sense and respond to physical cues in their environment. Cells on chemically well-defined surfaces were observed to have distributions of focal adhesions shifted toward smaller sizes when presented with higher affinity ligands (Kato, M.; Mrksich, M. Biochemistry 2004, 43, 2699). In this paper, we account for this trend with a simple model in which integrins are treated as particles on a lattice, and their stochastic dynamics are simulated with a kinetic Monte Carlo algorithm. How the trend depends on force-coupled growth, membrane fluctuations, and heterogeneity of receptor-ligand interactions is analyzed. Predictions are made for substrates in which the ligands presented can vary in either space or time, so that the model can be validated experimentally.

Path Sampling Calculation of Methane Diffusivity in Natural Gas Hydrates from a Water-Vacancy Assisted Mechanism

Increased interest in natural gas hydrate formation and decomposition, coupled with experimental difficulties in diffusion measurements, makes estimating transport properties in hydrates an important technological challenge. This research uses an equilibrium path sampling method for free energy calculations [Radhakrishnan, R.; Schlick, T. J. Chem. Phys. 2004, 121, 2436] with reactive flux and kinetic Monte Carlo simulations to estimate the methane diffusivity within a structure I gas hydrate crystal. The calculations support a water-vacancy assisted diffusion mechanism where methane hops from an occupied "donor" cage to an adjacent "acceptor" cage. For pathways between cages that are separated by five-membered water rings, the free energy landscape has a high barrier with a shallow well at the top. For pathways between cages that are separated by six-membered water rings, the free energy calculations show a lower barrier with no stable intermediate. Reactive flux simulations confirm that many reactive trajectories become trapped in the shallow intermediate at the top of the barrier leading to a small transmission coefficient for these paths. Stable intermediate configurations are identified as doubly occupied off-pathway cages and methane occupying the position of a water vacancy. Rate constants are computed and used to simulate self-diffusion with a kinetic Monte Carlo algorithm. Self-diffusion rates were much slower than the Einstein estimate because of lattice connectivity and methane's preference for large cages over small cages. Specifically, the fastest pathways for methane hopping are arranged in parallel (nonintersecting) channels, so methane must hop via a slow pathway to escape the channel. From a computational perspective, this paper demonstrates that equilibrium path sampling can compute free energies for a broader class of coordinates than umbrella sampling with molecular dynamics. From a technological perspective, this paper provides one estimate for an important transport property that has been difficult to measure. In a hydrate I crystal at 250 K with nearly all cages occupied by methane, we estimate D approximately 7 x 10(-15) X m(2)/s where X is the fraction of unoccupied cages.

Kinetic activation-relaxation technique: An off-lattice self-learning kinetic Monte Carlo algorithm

Many materials science phenomena are dominated by activated diffusion processes and occur on time scales that are well beyond the reach of standard molecular-dynamics simulations. Kinetic Monte Carlo (KMC) schemes make it possible to overcome this limitation and achieve experimental time scales. However, most KMC approaches proceed by discretizing the problem in space in order to identify, from the outset, a fixed set of barriers that are used throughout the simulations, limiting the range of problems that can be addressed. Here, we propose a flexible approach---the kinetic activation-relaxation technique ($k$-ART)---which lifts these constraints. Our method is based on an off-lattice, self-learning, on-the-fly identification and evaluation of activation barriers using ART and a topological description of events. Using this method, we demonstrate that elastic deformations are determinant to the diffusion kinetics of vacancies in Si and are responsible for their trapping.

Coupling kinetic dislocation model and Monte Carlo algorithm for recrystallized microstructure modeling of severely deformed copper

By coupling a kinetic dislocation model and Monte Carlo algorithm, the recrystallized microstructure of severely deformed Oxygen Free High Conductivity Copper (OFHC) is predicted at different strains imposed by Equal-Channel-Angular-Pressing (ECAP) and annealing temperatures. From a flow field model, the strain rate distribution during the ECAP of the material in a curved die is calculated. Then using the kinetic dislocation model, the total dislocation density and correspondingly the stored energy after each ECAP pass is estimated. Utilizing the Monte Carlo algorithm and the stored energy, the recrystallized microstructure is predicted. The results show that the recrystallized grain size is decreased rapidly from the strain of first to fourth pass and then it is decreased slowly. Also, it is achieved that with increasing the annealing temperature, the grain size is increased. Moreover, a good agreement is observed between the predicted results and experimental data.

Dislocation dynamics simulations of plasticity in Fe laths at low temperature

Plastic deformation in 16MND5 steel made of Fe laths is investigated using three-dimensional dislocation dynamics (DD) simulations, adapted to treat the body-centred cubic crystalline structure, strained in the ductile to brittle transition temperature range. In that regime, the edge segment velocity is proportional to the local effective resolved shear stress, whereas the screw segments follow a thermal activation scheme. The adopted cross-slip rules are derived from atomistic simulations, implemented in the DD code using a kinetic Monte Carlo algorithm. Specific loading and boundary conditions are worked out, with a view to accounting for the bainitic microstructure of the steel and its specific deformation mode. In these conditions, the implemented cross-slip behaviour is shown to play an essential role in the development of specific dislocation arrangements forming at different temperatures, also observed in 16MND5 steel. The presented results also provide insights on dislocation-based deformation mechanisms possibly involved in damage initiation.

A kinetic Monte Carlo study on the nucleation mechanisms of oil‐soluble initiators in the miniemulsion polymerization of styrene

AbstractThe use of oil‐soluble initiators in free‐radical miniemulsion polymerization has increased due to their ability to generate radicals primarily within monomer droplets. Existing theories concerning the nucleation mechanism for oil‐soluble initiators suggest that a single radical must be formed within a particle for propagation to occur, despite the fact that an oil‐soluble initiator molecule decomposes to form two radicals. According to existing theories, the primary source of nucleating radicals may be either within the particle or from the small amount of initiator in the aqueous phase. At the nanometer size of miniemulsion particles, concentration variables do not adequately describe the reactions in a particle, and use of such variables may lead to the inaccurate description of mechanisms within a particle. Application of the kinetic Monte Carlo algorithm allows for the simulation of individual reactive events, such as propagation or termination, while tracking macroscopic observables, such as conversion and molecular weight. Comparison of the simulation results to experimental data for styrene miniemulsion polymerization indicates that desorption of radicals may be more significant at smaller particle sizes and lower reaction temperatures. Oil‐soluble initiator radicals generated in the aqueous phase are found to be insignificant under typical conditions in the miniemulsion polymerization of styrene. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 6114–6128, 2008

Effective Temperature in Relaxation of Coulomb Glasses

We study relaxation in two-dimensional Coulomb glasses up to macroscopic times. We use a kinetic Monte Carlo algorithm especially designed to escape efficiently from deep valleys around metastable states. We find that, during the relaxation process, the site occupancy follows a Fermi-Dirac distribution with an effective temperature much higher than the real temperature T. Long electron-hole excitations are characterized by T(eff), while short ones are thermalized at T. We argue that the density of states at the Fermi level is proportional to T(eff) and is a good thermometer to measure it. T(eff) decreases extremely slowly, roughly as the inverse of the logarithm of time, and it should affect hopping conductance in many experimental circumstances.

Accelerated kinetic Monte Carlo algorithm for diffusion-limited kinetics

If a stochastic system during some periods of its evolution can be divided into noninteracting parts, the kinetics of each part can be simulated independently. We show that this can be used in the development of efficient Monte Carlo algorithms. As an illustrative example, the simulation of irreversible growth of extended one-dimensional islands is considered. The approach allowed us to simulate the systems characterized by parameters superior to those used in previous simulations.

Aromatic site description of soot particles

A new, advanced soot particle model is developed that describes soot particles by their aromatic structure, including functional site descriptions and a detailed surface chemistry mechanism. A methodology is presented for the description of polyaromatic hydrocarbon (PAH) structures by their functional sites. The model is based on statistics that describe aromatic structural information in the form of easily computed correlations, which were generated using a kinetic Monte Carlo algorithm to study the growth of single PAH molecules. A comprehensive surface reaction mechanism is presented to describe the growth and desorption of aromatic rings on PAHs. The model is capable of simulating whole particle ensembles which allows bulk properties such as soot volume fraction and number density to be found, as well as joint particle size and surface area distributions. The model is compared to the literature-standard soot model [J. Appel, H. Bockhorn, M. Frenklach, Combust. Flame 121 (2000) 122–136] in a plug-flow reactor and is shown to predict well the experimental results of soot mass, average particle size, and particle size distributions at different flow times. Finally, the carbon/hydrogen ratio and the distribution of average PAH sizes in the ensemble, as predicted by the model, are discussed.

A constant-time kinetic Monte Carlo algorithm for simulation of large biochemical reaction networks

The time evolution of species concentrations in biochemical reaction networks is often modeled using the stochastic simulation algorithm (SSA) [Gillespie, J. Phys. Chem. 81, 2340 (1977)]. The computational cost of the original SSA scaled linearly with the number of reactions in the network. Gibson and Bruck developed a logarithmic scaling version of the SSA which uses a priority queue or binary tree for more efficient reaction selection [Gibson and Bruck, J. Phys. Chem. A 104, 1876 (2000)]. More generally, this problem is one of dynamic discrete random variate generation which finds many uses in kinetic Monte Carlo and discrete event simulation. We present here a constant-time algorithm, whose cost is independent of the number of reactions, enabled by a slightly more complex underlying data structure. While applicable to kinetic Monte Carlo simulations in general, we describe the algorithm in the context of biochemical simulations and demonstrate its competitive performance on small- and medium-size networks, as well as its superior constant-time performance on very large networks, which are becoming necessary to represent the increasing complexity of biochemical data for pathways that mediate cell function.

Effects of wetting and misfit strain on the pattern formation of heteroepitaxially grown thin films

We investigate surface patterns of heteroepitaxially grown thin films using a three-dimensional kinetic Monte Carlo algorithm. Both the effects of elastic strain and wetting energy are taken into account. For the calculation of elastic strain energy induced by the lattice mismatch between the film and the substrate, a lattice discrete Green’s function method is used and numerically evaluated. For the wetting effect, different function forms of wetting energy have been proposed to model the wetting intensity between thin film and the substrate materials. The effects of wetting and misfit strain together with other growth conditions such as deposition rate and temperature of the substrate are thus investigated. Three growth modes, i.e., Layer-By-Layer, Stranski–Krastanow (S–K), and Volmer–Weber (V–W) of thin films are demonstrated clearly in our investigations. The occurrence of S–K growth mode is affected by the competition between the wetting effect and the strain effect.

Numerical study of relaxation in Coulomb glasses

AbstractWe present a kinetic Monte Carlo algorithm to study relaxation of Coulomb glasses at low temperatures. The algorithm can integrate out exactly fast transitions avoiding repetitive hops characteristic of metastable states, so that it does not get stuck in most of them. The present implementation of the method can follow the dynamics of systems with up to 1000 sites to macroscopic times. This is a general procedure that can be extended to most slow relaxation systems. We present results for energy relaxation. We also discuss a temperature definition during the non equilibrium relaxation process and show preliminary results on aging. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Finite State Projection Approach for the Analysis of Stochastic Noise in Gene Networks

In order to capture important subcellular dynamics, researchers in computational biology have begun to turn to mesoscopic models in which molecular interactions at the gene level behave as discrete stochastic events. While the trajectories of such models cannot be described with deterministic expressions, the probability distributions of these trajectories can be described by the set of linear ordinary differential equations known as the chemical master equation (CME). Until recently, it has been believed that the CME could only be solved analytically in the most trivial of problems, and the CME has been analyzed almost exclusively with kinetic Monte Carlo (KMC) algorithms. However, concepts from linear systems theory have enabled the finite state projection (FSP) approach and have significantly enhanced our ability to solve the CME without resorting to KMC simulations. In this paper, we review the FSP approach and introduce a variety of systems-theory-based modifications and enhancements to the FSP algorithm. Notions such as observability, controllability, and minimal realizations enable large reductions and increase efficiency with little to no loss in accuracy. Model reduction techniques based upon linear perturbation theory allow for the systematic projection of multiple time-scale dynamics onto a slowly varying manifold of much smaller dimension. We also present a powerful new reduction approach, in which we perform computations on a small subset of configuration grid points and then interpolate to find the distribution on the full set. The power of the FSP and its various reduction approaches is illustrated on few important models of genetic regulatory networks.

Synchronous parallel kinetic Monte Carlo for continuum diffusion-reaction systems

A novel parallel kinetic Monte Carlo (kMC) algorithm formulated on the basis of perfect time synchronicity is presented. The algorithm is intended as a generalization of the standard n-fold kMC method, and is trivially implemented in parallel architectures. In its present form, the algorithm is not rigorous in the sense that boundary conflicts are ignored. We demonstrate, however, that, in their absence, or if they were correctly accounted for, our algorithm solves the same master equation as the serial method. We test the validity and parallel performance of the method by solving several pure diffusion problems ( i.e. with no particle interactions) with known analytical solution. We also study diffusion-reaction systems with known asymptotic behavior and find that, for large systems with interaction radii smaller than the typical diffusion length, boundary conflicts are negligible and do not affect the global kinetic evolution, which is seen to agree with the expected analytical behavior. Our method is a controlled approximation in the sense that the error incurred by ignoring boundary conflicts can be quantified intrinsically, during the course of a simulation, and decreased arbitrarily (controlled) by modifying a few problem-dependent simulation parameters.

Hybrid method for modeling epitaxial growth: Kinetic Monte Carlo plus molecular dynamics

We propose a concurrently coupled hybrid molecular dynamics (MD) and kinetic Monte Carlo (KMC) algorithm to simulate the motion of grain boundaries between fcc and hcp islands during epitaxial growth on a fcc (111) surface. The method combines MD and KMC in an adaptive spatial domain decomposition, so that near the grain boundary, atoms are treated using MD but away from the boundary atoms are simulated by KMC. The method allows the grain boundary to interact with structures that form on spatial scales significantly larger than that of the MD domain but with a negligible increase in computational cost.

Efficient kinetic Monte Carlo simulation

This paper concerns kinetic Monte Carlo (KMC) algorithms that have a single-event execution time independent of the system size. Two methods are presented—one that combines the use of inverted-list data structures with rejection Monte Carlo and a second that combines inverted lists with the Marsaglia–Norman–Cannon algorithm. The resulting algorithms apply to models with rates that are determined by the local environment but are otherwise arbitrary, time-dependent and spatially heterogeneous. While especially useful for crystal growth simulation, the algorithms are presented from the point of view that KMC is the numerical task of simulating a single realization of a Markov process, allowing application to a broad range of areas where heterogeneous random walks are the dominate simulation cost.

Kinetics of the Migration and Clustering of Extrinsic Gas in bcc Metals

Abstract We study the mechanisms by which gas atoms such as helium and hydrogen diffuse and interact with other defects in bcc metals and investigate the effect of these mechanisms on the nucleation of embryonic gas bubbles. Large quantities of helium and hydrogen are produced due to spallation and transmutation in structural materials in fusion and accelerator-driven reactors. The long time evolution of the extrinsic gas atoms and their accumulation at vacancies is studied using a kinetic Monte Carlo algorithm that is parameterized by the migration energies of the point defect entities. First-order reaction kinetics are observed when gas clusters with vacancies. If gas-gas clustering is allowed, mixed-order diffusion limited kinetics are observed. When dissociation of gas from clusters is allowed, gas-vacancy clusters survive to steady state while gas-gas clusters dissolve. We obtain cluster size distributions and reaction rate constants that can be used to quantify microstructural evolution of the irradiated metal.

Path ensembles and path sampling in nonequilibrium stochastic systems

Markovian models based on the stochastic master equation are often encountered in single molecule dynamics, reaction networks, and nonequilibrium problems in chemistry, physics, and biology. An efficient and convenient method to simulate these systems is the kinetic Monte Carlo algorithm which generates continuous-time stochastic trajectories. We discuss an alternative simulation method based on sampling of stochastic paths. Utilizing known probabilities of stochastic paths, it is possible to apply Metropolis Monte Carlo in path space to generate a desired ensemble of stochastic paths. The method is a generalization of the path sampling idea to stochastic dynamics, and is especially suited for the analysis of rare paths which are not often produced in the standard kinetic Monte Carlo procedure. Two generic examples are presented to illustrate the methodology.

Multiscale model of the manipulation of single atoms on insulating surfaces using an atomic force microscope tip

We present the results of the multiscale modeling of the process of lateral manipulation of a Pd adatom adsorbed on the MgO (001) surface using a noncontact atomic force microscope (AFM) at finite temperature and in real time as a tip moves above the surface. We show that the stochastic motion of Pd adatoms can be controlled by localized forces from an oscillating tip and demonstrate how this can be achieved in practice. The energy barriers for manipulation as a function of tip position in three dimensions above the surface are determined from atomistic calculations and then used in a kinetic Monte Carlo algorithm to determine the evolution of the system at a finite temperature and in real time for a realistic trajectory of the tip, which is in turn governed by a complete numerical simulation of the instrument including the response of the feedback loops. We can then predict the probability of a successful manipulation event for a given procedure. The multiscale modeling technique developed in this work can be used to determine optimum experimental protocols for controlled single-atom manipulation using noncontact AFM. © 2007 The American Physical Society.