Molecular methods Kinetic Monte Carlo

Master equation methods are not tire only option for calculating tire kinetics of energy transfer and analytic approaches in general have certain drawbacks in not reflecting, for example, certain statistical aspects of coupled systems. Alternative approaches to tire calculation of energy migration dynamics in molecular ensembles are Monte Carlo calculations [18,19 and 20] and probability matrix iteration [21, 22], amongst otliers. 

Ab initio methods allow the nature of active sites to be elucidated and the influence of supports or solvents on the catalytic kinetics to be predicted. Neurock and coworkers have successfully coupled theory with atomic-scale simulations and have tracked the molecular transformations that occur over different surfaces to assess their catalytic activity and selectivity [95-98]. Relevant examples are the Pt-catalyzed NO decomposition and methanol oxidation. In case of NO decomposition, density functional theory calculations and kinetic Monte Carlo simulations substantially helped to optimize the composition of the nanocatalyst by alloying Pt with Au and creating a specific structure of the PtgAu7 particles. In catalytic methanol decomposition the elementary pathways were identified... 

An overreaching theme of the present chapter, besides broken ergodicity, has to do with the fact that most of the enhanced sampling methods that we shall discuss address situations in which one cannot clearly identify a reaction coordinate that can be conveniently used to describe the kinetic evolution of the system of interest. While methods for enhanced sampling are designed to yield accurate results faster than regular molecular dynamics or Monte Carlo (MC) methods, it is our belief that there is no perfect method, but that, rather, there are methods that perform better for particular applications. Moreover, it should be noted that, while in instances when a proper reaction coordinate can be identified methods described in other chapters are probably more efficient, they could still benefit by sampling in conformational directions perpendicular to the reaction coordinate. 

Figure 2 illustrates major modeling methods, i.e., ab initio molecular dynamic (AIMD), molecular dynamic (MD), kinetic Monte Carlo (KMC), and continuum methods in terms of their spatial and temporal scales. Models for microscopic and macroscopic components of a PEFC are placed in the figure in terms of their characteristic dimensions for comparison. While continuum models are successful in rationalizing the macroscopic behaviors based on a few assumptions and the average properties of the materials, molecular or atomistic modeling can evaluate the nanostructures or molecular structures and microscopic properties. In computational... 

The next section gives a brief overview of the main computational techniques currently applied to catalytic problems. These techniques include ab initio electronic structure calculations, (ab initio) molecular dynamics, and Monte Carlo methods. The next three sections are devoted to particular applications of these techniques to catalytic and electrocatalytic issues. We focus on the interaction of CO and hydrogen with metal and alloy surfaces, both from quantum-chemical and statistical-mechanical points of view, as these processes play an important role in fuel-cell catalysis. We also demonstrate the role of the solvent in electrocatalytic bondbreaking reactions, using molecular dynamics simulations as well as extensive electronic structure and ab initio molecular dynamics calculations. Monte Carlo simulations illustrate the importance of lateral interactions, mixing, and surface diffusion in obtaining a correct kinetic description of catalytic processes. Finally, we summarize the main conclusions and give an outlook of the role of computational chemistry in catalysis and electrocatalysis. 

The kinetic Monte Carlo (KMC) simulation method focuses on the state-to-state dynamic transitions and neglects the short-time system fluctuations. This approximation allows much longer timescales to be reached, without chemically relevant compromise in the resolution of the simulation, especially for solid-state systems. This is particularly important, since the diffusion of an oxygen ion on the surface of a YSZ electrolyte (among defect sites) requires approximately 1 ps, and the adsorption of one molecular oxygen onto the YSZ at 0.01 atm pressure requires approximately 0.5 ps [32]. Thus, deterministic simulation methods, like MD, are not easily able to capture this behavior, so other methods must be employed. 

Most molecular simulation techniques can be categorized as being among three main types (1) quantum mechanics, (2) molecular dynamics (MD) and (3) kinetic Monte Carlo (KMC) simulation. Quantum mechanics methods, which include ah initio, semi-empirical and density functional techniques, are useful for understanding chemical mechanisms and estimating chemical kinetic parameters for gas-phase... 

The molecular dynamics and Monte Carlo simulation methods differ in a variety of ways. The most obvious difference is that molecular dynamics provides information about the time dependence of the properties of the system whereas there is no temporal relationship between successive Monte Carlo configurations. In a Monte Carlo simulation the outcome of each trial move depends only upon its immediate predecessor, whereas in molecular dynamics it is possible to predict the configuration of the system at any time in the future - or indeed at any time in the past. Molecular dynamics has a kinetic energy contribution to the total energy whereas in a Monte Carlo simulation the total energy is determined directly from the potential energy function. The two simulation methods also sample from different ensembles. Molecular dynamics is traditionally performed under conditions of constant number of particles (N), volume (V) and energy (E) (the microcanonical or constant NVE ensemble) whereas a traditional Monte Carlo simulation samples from the canonical ensemble (constant N, V and temperature, T). Both the molecular dynamics and Monte Carlo techniques can be modified to sample from other ensembles for example, molecular dynamics can be adapted to simulate from the canonical ensemble. Two other ensembles are common ... 

Once a stochastic molecular dynamics-based framework is developed it is possible to modify it or use it as a building block in order to enhance sampling efficiency. One can for example incorporate these methods into kinetic Monte-Carlo, other metropolized schemes, or various advanced path sampling techniques. Thus the design of good molecular SDE methods is the cornerstone of much of modern molecular modelling. 

The theoretical methods to investigate the evolution kinetics of ordered microdomain structures are those in the atomic-scale including molecular dynamics simulations, Monte Carlo simulations, dynamic SCFT, dynamic density functional theory (DDFT), and those in the meso-scale including dissipative particle dynamics (DPD) simulations, etc. More details of these approaches can be found in the literatures. 

P. Kratzer, Monte Carlo and kinetic Monte Carlo methods—a tutorial, in Multiscale Simulation Methods in Molecular Sciences—Lecture Notes, NIC Series, ed. by J. Grotendorst, N. Attig, S. Bliigel and D. Marx, Forschungszentrum Jiilich, Jiilich, 2009. 

Chapters 18-21 discuss core-shell and advanced Pt alloy catalysts (which also can be considered to have a core-shell structure). Chapter 18 studies the fundamentals of Pt core-shell catalysts synthesized by selective removal of transition metals from transition metal-rich Pt alloys. Chapter 19 outlines the advances of core-shell catalysts synthesized by both electrochemical and chemical methods. The performance, durability, and challenges of core-shell catalyst in fuel cell applications are also discussed. Chapter 20 reviews the recent analyses of the various aspects intrinsic to the core-shell structure including surface segregation, metal dissolution, and catalytic activity, using DFT, molecular dynamics, and kinetic Monte Carlo. Chapter 21 presents the recent understanding of activity dependences on specific sites and local strains in the surface of bulk and core-shell nanoparticle based on DFT calculation results. 

Rate constants can be estimated by means of transition-state theory. In principle all thermodynamic data can be deduced from the partion function. The molecular data necessary for the calculation of the partion function can be either obtained from quantum mechanical calculations or spectroscopic data. Many of those data can be found in tables (e.g. JANAF). A very powerful tool to study the kinetics of reactions in heterogeneous catalysis is the dynamic Monte-Carlo approach (DMC), sometimes called kinetic Monte-Carlo (KMC). Starting from a paper by Ziff et al. [16], several investigations were executed by this method. Lombardo and Bell [17] review many of these simulations. The solution of the problem of the relation between a Monte-Carlo step and real time has been advanced considerably by Jansen [18,19] and Lukkien et al. [20] (see also Jansen and Lukkien [21]). First principle quantum chemical methods have advanced to the stage where they can now offer quantitative predictions of structure and energetics for adsorbates on surfaces. Cluster and periodic density functional quantum chemical methods are used to analyze chemisorption and catalytic surface reactivity [see e.g. 24,25]. 

Statistical mechanical Monte Carlo as well as classical molecular dynamic methods can be used to simulate structure, sorption, and, in some cases, even diffusion in heterogeneous systems. Kinetic Monte Carlo simulation is characteristically different in that the simulations follow elementary kinetic surface processes which include adsorption, desorption, surface diffusion, and reactivity . The elementary rate constants for each of the elementary steps can be calculated from ab initio methods. Simulations then proceed event by event. The surface structure as well as the time are updated after each event. As such, the simulations map out the temporal changes in the atomic structure that occur over time or with respect to processing conditions. 

A much more in-depth description of the full range of different ab initio quantum mechanical methods, free energy and kinetic Monte Carlo methods, molecular dynamics, ab initio molecular dynamics and linear scaling methods is given in the Appendix. 

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