A consistent hierarchy of generalized kinetic equation approximations to the master equation applied to surface catalysis

Abstract

We develop a hierarchy of approximations to the master equation for systems that exhibit translational invariance and finite-range spatial correlation. Each approximation within the hierarchy is a set of ordinary differential equations that considers spatial correlations of varying lattice distance; the assumption is that the full system will have finite spatial correlations and thus the behavior of the models within the hierarchy will approach that of the full system. We provide evidence of this convergence in the context of one-and twodimensional numerical examples. Lower levels within the hierarchy that consider shorter spatial correlations, are shown to be up to three orders of magnitude faster than traditional kinetic Monte Carlo methods (KMC) for one-dimensional systems, while predicting similar system dynamics and steady states as KMC methods. We then test the hierarchy on a two-dimensional model for the oxidation of CO on RuO2 (110), showing that low-order truncations of the hierarchy efficiently capture the essential system dynamics. By considering sequences of models in the hierarchy that account for longer spatial correlations, successive model predictions may be used to establish empirical approximation of error estimates. The hierarchy may be thought of as a class of generalized phenomenological kinetic models since each element of the hierarchy approximates the master equation and the lowest level in the hierarchy is identical to a simple existing phenomenological kinetic models.

A practical approach to the sensitivity analysis for kinetic Monte Carlo simulation of heterogeneous catalysis

“…2 corroborate this interpretation, but also the results on spatial correlation from refs. 19,20 point into this direction. Also, we can work with much smaller truncation limits M for small p CO than for large, although, in both cases, we are not close to the reaction conditions for which we expect a second order phase transition.…”

“…the refs. 3,19,27 ) for which we have already performed a sensitivity analysis 16 . This makes it well-suited for testing our scheme and we can concentrate on discussing the peculiarities of the employed sampling approaches.…”

“…In this study, we device a three-stage strategy for the estimation of stationary sensitivities, which is suitable for these problems. We illustrate this strategy on the model for the CO oxidation on RuO 2 (110) by Reuter 4 , which is a popular fruit-fly test problem 19,20 . We revisit the reaction conditions studied in ref.…”

Machine-Learned Corrections to Mean-Field Microkinetic Models at the Fast Diffusion Limit

“…Microkinetic models are commonly used in heterogeneous catalysis to relate atomic-scale information of the reaction system, i.e., thermochemistry of the intermediates and kinetics of the elementary steps, with measurable properties such as reaction rate, yield, selectivity, and conversion. − Such models are either (1) “mean-field” ordinary differential equations (ODE), which assume that the spatial distribution of surface intermediates is uncorrelated and can be characterized by an average quantity such as the coverage, , (2) spatial kinetic Monte Carlo (kMC) simulations that rigorously take into account the spatial distribution of intermediates on the surface and the general stochasticity of the reaction system, , (3) methods that explicitly capture the higher-order correlation within the adsorbates, such as quasi-chemical approximation , and the site ensemble method, or (4) methods that approximately solve the full master equation. , Kinetic Monte Carlo methods are, as expected, more accurate in capturing the underlying physics of the system; mean-field models, on the other hand, have a more convenient mathematical form, which is often the reason they are a common choice in catalysis modeling. , In particular, mean-field models are faster to solve because they deal with spatial averages (surface coverage), resulting in an ODE form of the microkinetic model, which are deterministic and easy to integrate. On the other hand, spatial kMC simulations deal with detailed adsorbate configuration; however, they can suffer from multiplicity (disparity) in the timescales of reaction and diffusion events and need additional sophistication to handle such issues. − While ODE-based models can also suffer from timescale multiplicities , because of a combination of fast/slow reactions (e.g., adsorption steps are fast, while bond activation steps are substantially slower), advanced techniques such as backward-difference methods, polynomial-based collocation, , and neural network (NN) approximators can potentially overcome these issues.…”

Local-metrics error-based Shepard interpolation as surrogate for highly non-linear material models in high dimensions

“…III B 2, we needed 10 000 results for reference, which with more complex multi-species models could cost vast amounts of computational time. On the other hand, the RuO 2 CO oxidation model has been characterized in detail both in (p CO , p O 2 , T )-space 40,[46][47][48][49] and in rate constant space [42][43][44] and its behavior is well understood. This makes it ideal for testing new theoretical developments such as the one presented here.…”