Progress in Accurate Chemical Kinetic Modeling, Simulations, and

Perspective Cite This: ACS Catal. 2019, 9, 6624−6647

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Progress in Accurate Chemical Kinetic Modeling, Simulations, and Parameter Estimation for Heterogeneous Catalysis Sebastian Matera,*,† William F. Schneider,*,‡ Andreas Heyden,*,§ and Aditya Savara*,∥ †

Fachbereich Mathematik and Informatik, Freie Universität, 14195 Berlin, Germany Department of Chemical and Biomolecular Engineering, University of Notre Dame, Notre Dame, Indiana 46556, United States § Department of Chemical Engineering, University of South Carolina, Columbia, South Carolina 29208, United States ∥ Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, United States Downloaded via BUFFALO STATE on July 18, 2019 at 14:06:27 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

‡

S Supporting Information *

ABSTRACT: Chemical kinetic modeling in heterogeneous catalysis is advancing in its ability to provide qualitatively or even quantitatively accurate prediction of real-world behavior because of new advances in the physical and chemical representations of catalytic systems, estimation of relevant kinetic parameters, and capabilities in kinetic modeling. This Perspective describes current trends and future areas of advancement in chemical kinetic modeling, simulation, and parameter estimation: ranging from elementary step calculations to multiscale modeling to the role of advanced statistical methods for incorporating uncertainties in predictions. Multiple new or growing methodologies are covered, examples are provided, and forward-looking topics for advancement are noted. KEYWORDS: kinetics, modeling, density functional theory, Bayesian statistics, kinetic Monte Carlo, multiscale modeling, cluster expansions, uncertainty

I. INTRODUCTION Chemical kinetic models relate the physical and chemical state of a system (concentrations, pressure, temperature, electric potential, etc.) to the rates of consumption/formation of species per unit time (such as molecules produced per unit time). Kinetic models are central to our understanding of catalysis and form the basis for insight-driven discovery of new catalytic materials.1,2 Chemical kinetic models can generally be categorized into: “phenomenological” models, in which the goal is to capture the essential features of observed rates without necessarily direct connections to the chemical details of a system; and “elementary step” models (including “microkinetic” models), in which the goal is to be physically correct with regard to the underlying elementary chemical steps. Elementary-step-based kinetic modeling is increasing in use, and recent advances have substantially expanded the capabilities and predictive accuracy of such models.2−8 Stateof-the-art models are increasingly able to account for the structure and composition of individual catalytic sites, as defined by the locally bonded molecules, as well as the influence of the chemical environment on the energetics at the catalytic site. By incorporating these complexities, simulations are increasingly able to produce results with qualitative or even semiquantitative agreement with experimental observations of absolute kinetics (rates, conversions, selectivities, etc.) and © XXXX American Chemical Society

capture how these vary with reaction conditions (temperatures, concentrations) and across different catalytic materials. We recognize that “sufficient accuracy/agreement” is a matter of context: for example, it could be correctly identifying the most important pathways in a complex reaction network, simulating the correct ranking order for the relative rates across a series of catalytic materials, or predicting absolute rates (turnover frequency) within 10% of observed values. Given that the ultimate goal in kinetic modeling is accuracy relative to realworld values or trends, elementary-step kinetic modeling plays a central role because of its ability to correctly describe chemistry and to achieve kinetic accuracy. Elementary-step-based kinetic models are usually developed either bottom-up or top-down. A bottom-up model starts from molecular-level information to obtain kinetic parameters that are then fed into simulations/calculations, and the predictions can then be compared to experimental observations. A topdown model starts with the experimental observations and then tries to back out kinetic parameters and molecular-level information via simulations/calculations and fitting or parameter estimation. In both approaches, it is recognized Received: March 25, 2019 Revised: June 3, 2019 Published: June 13, 2019 6624

DOI: 10.1021/acscatal.9b01234 ACS Catal. 2019, 9, 6624−6647

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Figure 1. Illustration of the connections between open heterogeneous catalytic reaction systems and the closed representations available from firstprinciples calculations. Connections between the two are made through statistical mechanical computations of free energies and elementary step modeling-based simulations of reaction rates.

role. Accurate models generally emerge from an interplay of bottom-up and top-down approaches,2,27 as illustrated in Figure 2. Knowledge about catalytic systems derives from an

that catalytic reactions occur on specific active sites, that there may be more than one type of active site present on a catalyst, and the goal is to figure out the correct molecular-level description of the system. Model inadequacies emerge from two types of errors: (a) deficiencies in the model itselfsuch as lack of inclusion of a reaction or active site; and (b) deficiencies in the physics/computationssuch as error in the estimated kinetic parameters for an elementary step. Today’s bottom-up approach starts with postulating/ enumerating a list of plausible elementary steps and activesite structures, after which the associated rate parameters (typically reaction energies, activation energies, and prefactors) are computed or estimated and employed in kinetic models (generally consisting of rate equations and mass balance equations). The kinetic models formulated are then evaluated to predict behavior.2 Today, density functional theory (DFT) is widely used to compute reaction pathways and rate parameters in the context of transition-state theory.9 However, most first-principles models probe energy landscapes in finitesized and closed systems, while catalytic systems are open, dynamic, and variable in time and space (Figure 1). Thus, achieving accuracy requires bridging the temperature, pressure, and complexity gaps between the first-principles world and catalytic reality, to create simulations that can match actual catalyst usage (“operando”10−15) conditions. Increases in computational power and tools have greatly advanced bottom-up capabilities, but accurate predictions remain challenging. A key issue is to quantify uncertainties in firstprinciples calculations and identify which parameters the predictions most sensitively depend on. Accordingly, substantial research in the past decade has gone into quantifying uncertainties and minimizing prediction errors.16−26 Today, the lines between a bottom up and top-down approach are blurred, with combined approaches playing a

Figure 2. Process flow between theory and experiments is connected by prior knowledge and simulations. In practice, these “gears” drive each other in a feedback loop, with different gears doing the driving at different times.

interplay of theory (electronic structure theory, kinetic theory, etc.), prior chemical knowledge, experimental observations, and modeling/simulation. Modeling and simulation have an integral role in this cycle: our confidence in our understanding of a system derives from our ability to describe measured results and to predict not-yet measured results. To fulfill their role in turning these gears, simulations and models must 6625


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ACS Catalysis balance physical and chemical reliability with practical accessibility, so that microscopic details must sometimes be sacrificed in the interest of utility. This Perspective highlights recent advances and forwardlooking needs in chemical kinetic modeling and simulation. This Perspective is primarily geared toward cases for which the elementary steps are well described by transition-state theory, though the methods discussed here can be applied to other systems as well. We note that these advances have become possible from coupling advances in computing and methodology. Specifically, advances in individual processor speeds and parallel processing, which have been invested in by institutions and governments to create high-performance computing infrastructure, have made it feasible to implement and apply more and more sophisticated electronic structure and kinetic models, as well as numerical methods to quantify and propagate uncertainties. A significant fraction of new discoveries is thus being made by innovative applications of previously existing theories combined with advanced computing power. In the sections below, we do not focus on this new infrastructure that is enabling scientific advances but on the science and methods that have now become accessible. In particular, we first focus on advances in multiscale modeling: starting from the active site and its environment and working outward. This then motivates the need for more accurate molecular-level models, at which point we discuss advances in free energy calculations of active sites and their environments. Given that the uncertainties remaining are still nontrivial even with the best computational methods, we then discuss Bayesian parameter estimation, Bayesian model selection, sensitivity analysis, and a future outlook. We believe these methodologies will each play important roles in the future of kinetic modeling and understanding. The field is acronym rich, and for the convenience of the reader we provide a list of acronyms referred to in the text in Table 1.

Table 1. List of Acronyms acronym 1p-KMC BEP relations BMS BPE BPNN CC CE CFD CPE DFT DFTB, PM7, xTB EB-MSI FF FT GGA HO HPC HPD HT hTST iBMS KMC MAP MCMC MD MF MF-MKM ODE OOBE PES QMC SA sBMS SDEs TPD ΨQC

II. MULTISCALE MODELING Macroscopic behavior originates from the molecular scale: thus, accurate calculations of macroscopic observations require coupling the knowledge of the molecular scale to that of the macroscopic scale.28 Often, the different scales are computed separately and coupled either strongly or weakly. The accuracy of the individual scales and propagations of uncertainty are thus an integral consideration in a multiscale approach.29−31 In modern modeling and simulation, microkinetic modeling based on elementary step kinetics based on atomic-scale energetics is often used to connect the scales of atomic motion and bond making/breaking with the macroscopic reactor scales of conversion, selectivity, heat, and so on.32 In a first-principles multiscale modeling strategy, the model starts at a quantum-chemical model to describe bond making and bond breaking, and the multiscale model then involves simulating multiple “layers” of scales to reach the macroscale (Figure 3). The electronic Schrödinger equation has a large number of degrees of freedom for polyatomic systemswhich is typical in heterogeneous catalysis because of the number of atoms required to describe an active site and its surroundings with chemical accuracy. Direct numerical evaluation using wave function quantum chemical calculations (ΨQC), for example, coupled cluster (CC) or quantum Monte Carlo (QMC), is therefore impractical. Density functional theory (DFT) thus serves as the current workhorse for calculating the

definition first-principles KMC Bell−Evans−Polanyi relations also known as Brønsted−Evans−Polanyi relations Bayesian model selection Bayesian parameter estimation back-propagation neural networks coupled cluster cluster expansion computational fluid dynamics conventional parameter estimation density functional theory tight-binding approaches error-based modified Shepard interpolation force fields free translator generalized gradient approximation harmonic oscillator high-performance computing highest probability density hindered translator harmonic transition-state theory iterative Bayesian model selection kinetic Monte Carlo maximum a posteriori Markov chain Monte Carlo molecular dynamics mean-field mean field microkinetic modeling ordinary differential equations binding energy for two oxygen atoms relative to gasphase O2 potential energy surface quantum Monte Carlo sensitivity analysis simple Bayesian model selection systems of differential equations temperature-programmed desorption wave function quantum chemical calculations

Figure 3. Multiscale modeling involves modeling multiple scales with different methodology and using the inputs from one scale to another using either strong or weak coupling. Acronyms are defined in Table 1.

electronic structure (energetics of adsorption, activation energies, etc.).8,9,24,33 Because of methodological advances and more widely accessible high-performance computing (HPC) resources, such calculations are nowadays widespread 6626


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allow bonds to be made or broken during the trajectory and are less computationally demanding than semiempirical quantum chemical approaches by a factor of 10 to 100. Finally, potentials represented by back-propagation neural networks (BPNNs) can also be reactive, depending on the training data,95 and are potentially the most computationally efficient approach. Reactive force fields often have problems in properly describing chemical reactions, especially when it comes to problems that involve metals, although BPNN force fields after training do show promise.95 Tight-binding approaches require system-specific parameters to be developed but show promise for becoming widespread and chemically accurate for a greater variety of chemical systems in the coming years because of their low computational cost and potential for high accuracy.97−99 Regardless of these advances, MD will always have challenges when it comes to describing chemical reactions of systems of molecules, because reactions are inherently rare events on the time scales of MD. As a result, a simulation might spend millions (or more) MD steps in one metastable state before a chemical reaction appears that would enable the system to transition to another metastable state. Accelerated dynamics methods, such as forward-flux sampling, ParSplice,100 or varieties of metadynamics101 provide some means of extending simulation time scales or otherwise accessing rare event states. In many cases, the computational cost can be reduced through coarse-graining: the temporal sequences of metastable states can be approximated as Markov jump processes if there are no significant molecular-scale kinetic memory102,103 effects. In heterogeneous catalysis on surfaces, this coarse-graining is typically accomplished by mapping the metastable states onto a lattice of Langmuir-type104,105 adsorption sites and the use of harmonic transition-state theory (hTST) for determining transition rates.5,7 The resulting configuration space is typically high-dimensional (the simplest CO oxidation problem on a 20 × 20 lattice has 10190 metastable states), and therefore, statistical averages are practically obtained by simulating sample configuration trajectories using spatial kinetic Monte Carlo (KMC) methods.3,8,106 The main strength of KMC is that it is a purely numerical method;106 that is, once the mechanism and the rate constants are determined, KMC does not by itself introduce further approximationsunlike MF-MKM.35,36,50,107,108 Spatial KMC thus keeps the configurational information about the system and allows for incorporating adsorbate−adsorbate interactions, which can be important to include for accurate kinetic modeling on surfaces (see next section). As a result of these properties and the possibility to obtain the inputs practically from quantum chemical simulations, such KMC approaches are increasing in popularity in heterogeneous catalysis research.5,7,32,109−112 While KMC simulations were historically performed using code developed specifically for that application, the gains in KMC usage are accompanied by generalized and sophisticated software packages109,113−118 which are at the cusp of becoming widespread. The KMC methodology has been applied to gain insight into a number of catalytic systems,3,7,8 often in the combination with DFT-based calculation of mechanistic steps and Arrhenius hTST rate parameters, which is then termed first-principles kinetic Monte Carlo (1p-KMC). A particularly well-studied system is that of catalytic CO oxidation over RuO2(110), which involves adsorbed CO

for the elucidation of the energetics and geometries of intermediates on surfaces, as well as for the transition states, and for the simulation of chemical specific spectra (e.g., infrared, X-ray photon spectroscopy, etc.). Periodic boundary conditions implemented within plane wave, supercell DFT codes provide the most efficient and generally applicable framework for describing extended surfaces and are commonly employed in heterogeneous catalysis studies. A common and mature practice is the calculation of activation energies with DFT using harmonic transition-state theory (hTST) along with mean field microkinetic modeling (MF-MKM).34 That is, that the spatial configurations of adsorbates is neglected, leading to concentration-based rate equations.35−38 Ordinary differential equations (ODEs) are used in the MF-MKM approach, and ODE solvers capable of handling any numerical stiffness that arises are widespread, such as the case of Sundials used by the MKMCXX software.39 Methods for solving the ODEs with even lower computational costs have also been published.38,40−42 Well-mixed (mean field) problems can also be investigated using stochastic methods, but such methods are less widely used.43−48 Because of the low computational cost, large numbers of conditions can be explored with MF-MKM to obtain trends.39,49 The downside of MF-MKM is that perfect mixing of the adsorbates is assumed, which might introduce sizable and qualitative errorseven for seemingly simple problems.50−54 In addition, describing interactions between adsorbates is challenging, and attempts to include them in MF-MKM are generally performed in an ad hoc fashion.55−58 When steady-state solutions are of interest, the time-scales of ODEs can be further coarse-grained using energetic span models.59,60 The computational cost of evaluating ODEs is usually low enough that such coarse graining is not necessary: the real advantage of the energeticspan model is that fewer elementary steps need to be computed using DFTonly the most rate controlling59−61 steps. Large numbers of materials can now be screened with approximate rates calculated for individual elementary steps using DFT combined with Bell−Evans−Polanyi relations (BEP relations, also known as Brønsted−Evans−Polanyi relations),34,62−66 in conjunction with linear scaling relations25,33,34,64,66−71 or more advanced26,72−76 methods. To account for the shortcomings of MF-MKM in describing spatially correlated dynamics at the supramolecular mesoscale, a straightforward option would be to evaluate the problem at the atomic scale and to get time-resolved information via molecular dynamics (MD) simulations. MD computes the trajectories of atoms/molecules on a potential energy surface (PES) using Newtonian mechanics, frequently coupled with baths to impose constraints on temperature, pressure, or even chemical potential. The PES may be based on classical force fields (FF), for instance, to describe the dynamics of molecules in pores.77−86 However, dynamic simulations of chemical reactions with bond breaking and bond making require PES that are either explicitly or implicitly based on electronic structure models. So-called ab initio MD,87−89 in which the PES is computed on-the-fly explicitly with DFT, is computationally demanding and limited to modest system sizes and rather short time scales of typically less than a few hundred atoms and 102 and to even change the qualitative behavior of the reaction rate, including which species are dominant on the surface.21,112,183,209 Even second and third neighbor effects are non-negligible.6,206,208−211 Thus, it is crucial to include adsorbate interactions, and methods for doing so are still an active area of research.53,55,114,210,212−216 In the earliest lattice-based microkinetic models, these interactions between adsorbates were incorporated empirically through some combination of through-space and throughsurface terms.217−219 More recently cluster expansions (CEs) parametrized against DFT calculations have gained popularity.6,53,144,183,186,208,210,211,215,220,221 In these models, the energy of a configuration of adsorbates is written as an expansion in one-body, two-body, and higher-order interaction terms. Fitting CE models involves both selection of an optimal set of interaction terms (“clusters”) and determining the weighting factors on each term.211 Fitting is commonly done against DFT calculations through an iterative selection of training configurations. Figure 6 compares the ability of trained CEs to represent the formation energy of various configurations and coverages of O on a Pt(111) surface.222 The

Figure 6. Cluster expansion fitted and DFT formation energies of four CEs of increasing size. Reprinted with permission from ref 222. Copyright 2018 American Chemical Society.

number of “Figures” listed in each panel reflects the number of terms included in the expansion. The convex formation energy, shown as a solid or dashed line for each series in Figure 6, reflects repulsive interactions between adsorbates that can be captured qualitatively in a CE that includes only nearestneighbor interactions (panel a, 3 Figures) but requires a larger number of two-body and even 3-body terms (panel c, 8 Figures) to approach quantitative reliability (precision on the order of 0.1 eV). If the activation energy of a particular reaction step can be correlated with a reaction energy, through a Brønsted−Evans− Polanyi type relationship, an adsorbate cluster expansion can be used directly to predict site-specific rates. This strategy was used in ref 210, and Figure 7 illustrates the catalytic consequences of adsorbate−adsorbate interactions in the context of a simple kinetic model for catalytic oxidation of NO to NO2 over Pt(111). In this example, O is assumed to be the exclusive surface species, and O2 dissociation is taken to be rate-limiting.114,210,220 Oxygen configurations relevant to a given reaction condition are generated by Monte Carlo using a CE to represent the configuration-dependent energy of O adsorbates on Pt(111). The bottom of the figure illustrates a selection of neighbor vacancy pairs observed during the MC simulation and the corresponding energy to fill that vacancy by dissociating O2 at that site. The more than 1 eV difference in dissociation energies (∼100 kJ mol−1) reflects the strong and nonuniform influence of adsorbate interactions at different sites. The top left of the Figure shows a snapshot extracted from the MC simulations in which binding site pairs are colorcoded by the net binding energy of two O atoms referenced to gas-phase O2 (abbreviated as OOBE). A color-coded histogram collected from many such snapshots is shown in the top right. The consequence of adsorbate interactions at finite temperatures and coverages is to create a distribution of sites of differing potential reactivity, and observed reaction rates become an ensemble average over this distribution. If, as in the case of O2 dissociation, activation energies are lowest and rates greatest at the lowest-energy sites, then accurate description of the minority of sites at the low-energy tail of the histograms becomes key to reliable predictions of ensemble-averaged rates. As noted, for catalytic systems, these effects affect the rates of reactions by multiple orders of magnitude, and change the 6631


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Figure 7. (a) A representative snapshot of GCMC-generated O configurations at 600 K with μO = −0.17 eV and a coverage of 0.36 monolayers of oxygen. Oxygen atoms are shown as black, and vacant FCC site pairs are rainbow color-coded by OOBE. The red side of the spectrum indicates more exothermic adsorption and the blue side of the spectrum indicates less exothermic adsorption. (b) A rainbow color-coded OOBE histogram from many snapshots under these conditions, grouped in 10 meV bins and normalized to the number of basis sites at zero coverage (N ∼ 0). (c) Local configurations extracted from snapshots for various strengths of divacancy sites on the same color scale. Adapted with permission from ref 210. Copyright 2012 Elsevier.

involving more than about two surface species.215 Given their expense, CE-based models are unlikely to obviate the need for mean-field models that implicitly incorporate adsorbate interactions in coverage-dependent (activation) energies.53 There are opportunities to leverage the explicit, CE-based kinetic model results to guide the improvement of and to characterize the reliability of coarse-graining the environment around the binding site214,215 or to guide machine learning216 to bridge the computational accessibility gap between explicit inclusion of local adsorbate configurations and mean field microkinetic models. A number of studies have been published that incorporate molecular-scale adsorbate interactions in reaction network models,55−57 including multiscale modeling21,112 examples. Developing computationally accessible approximations that account for the free energy consequences of interadsorbate interactions with accuracy commensurate to other model assumptions is an important present challenge in the field.

concentration distribution of intermediates on the surface not only quantitatively but also qualitatively. The “silver lining” in these observations is that improvements in the predictive reliability of the model can in principle be obtained by improving predictions of rates only at the kinetically important sites and configurations, which may be a minority. At present, sensitivity analyses (Section VI) can identify the subset of sites that dominate the catalytic rates in a given simulation, but efficient approaches to predict that subset a priori are not available. Methods to predict relevant sites and configurations could both accelerate KMC-based models and improve the accuracy of MF-MKM models. The simple example of oxygen adsorption highlights the role of adsorbate interactions in influencing the state of a reacting surface. Explicit CE representations can be extended to include more than one adsorbate, with an increase in complexity in construction and parametrization.215 Advances are needed in CE construction that balance accuracy with parametrization and application cost. One promising strategy is to relate interaction parameters with simpler physical quantities, to develop less expensive correlations.207,213 Approaches that augment the local interactions captured in a traditional CE with contributions to interactions that are long-ranged, such as dipolar interactions, may be necessary for some systems. There has been progress as well in extending the CE interaction concepts to lower symmetry surfaces and to multifaceted particle morphologies.223 While most effort has focused on the effect of adsorbate interactions on adsorbate energies, the CE concept has been extended to transition states,224,225 and such approaches may be necessary for high-accuracy kinetic models. Cluster expansions are presently the only computationally accessible methodology that can achieve better than 0.1 eV precision for predicting the effects of adsorbate−adsorbate interactions of nonexplicitly calculated configurations. However, cluster expansions are presently impractical for cases

IV. BAYESIAN PARAMETER ESTIMATION While kinetic modeling can be performed completely from first-principles, most bottom-up models are not quantitatively predictive, mainly because: (I) the active-site models used are, by design, characteristic models that do not contain the full complexity of an experimental catalyst sample; and (II) the theoretical models used for estimating elementary rate constants contain significant uncertainties related to an underlying PES typically determined by DFT and a reaction rate theory based on hTST. The most quantitatively predictive models are generally those that are based on fitting experimental data for parameter estimation (top-down modeling), which is essentially a form of interpolating between the experimental data. If the desire is to use models to extrapolate outside the domain of experimental observation, the goal of parameter estimation in chemical kinetics is 6632


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day DFT-computed adsorption energy would be incorrect by 200 kJ/mol: such a large discrepancy would suggest some other source of error (such as performing a calculation with the wrong adsorption site). Rather than discarding existing knowledge of uncertainties, we advocate including such information during kinetic parameter estimation during combined top-down and bottom-up modeling. Ideally, the correlation structure of parameter uncertainties should also be considered whenever quantifiable,19 because this correlation structure significantly reduces the flexibility of microkinetic models. While it is in principle possible to include these uncertainties in CPE by biasing the objective function, there is no way to do so that is accepted as correct for all problems, and there can be no universally correct method for doing so because the gradients and shapes of parameter-response surfaces will be different for different problems.244,245 A more general and established method for including these uncertainties during parameter estimation is through BPE. BPE changes the question slightly, to a probabilistic question: “What is the probability of certain parameter values being true given the evidence observed and our prior beliefs of what values the parameters are likely to have?” The highest probability is then the maximum of the a posteriori distribution (MAP), which corresponds to the maximum value from Bayes Theorem (eq 1). Bayes Theorem is provided here with the variable symbols chosen in accordance with the conventions of Bayesian statistics variable naming for the present context.

generally to obtain accurate elementary kinetic constants/ parameters and functional forms that reflect the underlying elementary processes. There can be thousands of reactions in complex reaction networks,226−228 and generating the reaction networks24,25,91,228−233 is a field of work in its own right. For the purposes of this section, we assume that the reaction network has already been developed, including the elementary steps of which adsorbate intermediates are possible. For heterogeneous catalysis, a complete description of the elementary kinetics involves parameters not only for the activation energies and pre-exponentials in the absence of coadsorbates but also for the terms associated with nonidealities at finite coverage, such as those due to adsorbate interactions, thus leading to a large space for parameter estimation.55,197,234−238 When chemical knowledge is not available, the best parameter estimates are based on fits using objective functions (“metrics”) such as weighted sum of square residuals, and these residual-based methods are forms of conventional parameter estimation (CPE). However, when chemico-physical knowledge is available, such as realistic expected ranges, then such prior knowledge can be included by using Bayesian parameter estimation (BPE) to find credible values.22,234,239 We will point out differences between CPE and BPE and provide an example for demonstration. In this section, we intentionally distinguish estimation from optimization. This wording distinction is to emphasize that BPE facilitates finding the most credible/physically realistic description rather than finding the “optimal” fit.240 We believe there is room for increased usage of BPE, going forward. CPE occurs by optimization of a fit based on an objective function, through one or more of a variety of methods, such as gradient methods, evolutionary/genetic algorithms, and brute force search. These CPE methods seek to find an optimum of the objective function (typically the minimum of the sum of squared residuals), which requires an appropriate choice of the objective function.2,236,241,242 It is common to include weighting by the errors of the data during fitting but not to incorporate prior knowledge about physically realistic ranges for the parameters being estimated. Many studies using CPE directly fit the data. In heterogeneous catalysis, the microkinetic models can be very flexible, and the number of parameters can be as much as an order of magnitude larger than the number of uncorrelated high-quality experimental data points. As a result, there is often not an unique optimal solution. Furthermore, the physical insight of the microkinetic model and active-site model is lost during the top-down fitting process. Parameter estimation based solely on observed data does not incorporate the external information available about the energetics of relevant states external methods (such as from density functional theory, bond-order conservation, microcalorimetry, linear energy relations, etc.). One strategy is to use theoretical knowledge, such as DFT calculations, to provide initial guesses for CPE, which effectively combines top-down and bottom-up modeling: the experimental data is macroscopic while the theory estimates are from the atomic information.2 However, it is less appreciated that many external sources of initial guesseswhether empirical correlations or theoretical estimatesoften have associated estimable distributions of uncertainty that can be used to improve the final estimates. For example, current DFT calculations of adsorption energies and activation energies typically have on the order of 20 kJ/ mol uncertainties.19,168,212,243 It is thus unlikely that a present

P(θ|D) =

P(D|θ )*P(θ ) P(D|θ )*P(θ ) = P(D) constant

(1)

D represents the observed data (typically a vector of dimensionality ≥1), and θ represents a vector of all the parameters that would be required to simulate the data (e.g., activation energies, pre-exponential factors, adsorbate interaction terms, etc.). BPE can be used to obtain point estimates (such as discrete kinetic parameter values) via the maximum of the a posteriori distribution (MAP),246,247 or to obtain the region of parameter space with the highest probability density (HPD).248,249 When the HPD has been sufficiently sampled, the expected value (mean) can be used as a point estimate,250 rather than the MAP. BPE also provides a distribution of the estimated parameter values that correctly propagates the initial input uncertainties, and these distributions can then be used in further uncertainty quantification and sensitivity analysis. To perform BPE, eq 1 requires prior distributions: here, this means an initial guess for each parameter (often from theory) along with uncertainties that define probabilities surrounding that initial guess. Inclusion of priors enables a connection between the experimental errors and the theory uncertainty. This connection is in contrast to what is typically possible during CPE (which is one directional, either simulation or fitting, as shown in Figure 8) versus what is possible with BPE (which accounts for sources of error/uncertainty in both directions, by propagating the uncertainties as shown in Figure 9 in explained below). The steps involved in BPE are as denoted by gray circles in Figure 9: 1. Propose a parametric model that is hypothesized to be able to produce the real behavior if given the physically correct parameter values. 2. Define the prior distribution, P(θ), that encodes initial beliefs about each parameter’s expected value and 6633


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such as to perform model selection as described in Section V. For both steps 1 and 2, underestimates of uncertainty can easily lead to false or misleading predictions. To show the utility of BPE, we compare the simulated outputs using parameter value point estimates from CPE versus BPE for a simple example using MF-MKM and experimental data. The data are for ethanal temperatureprogrammed desorption (TPD) from CeO2(111).251−253 Most typical values for Arrhenius parameters of molecular desorption have activation energies, Ea, of 70 000 to 150 000 J mol−1 and pre-exponentials of 109 to 1016 s−1.37,105,254,255 However, more knowledge is available for this system: DFT calculations and theory provide estimates for the activation energy and pre-exponential of ethanal desorption of 41 000 J mol−1 and 1013 s−1.252,256 These estimates provide values for the priors in eq 1: an activation energy of 41 500 ± 20 000 J mol−1 and a pre-exponential factor of 1013±2 s−1, where the uncertainties represent standard deviations in a normal distribution. The BPE point estimates below are Bayesian MAP estimates,246,247 which take into account the information of the prior knowledge as well as the evidence (the observed data). As mentioned earlier, the MAP is the point in parameter space with the highest probability according to eq 1. Figure 10a shows the experimentally monitored252 rate of ethanal desorption from CeO2(111) as a function of temperature, as measured during temperature-programmed desorption, following adsorption of ∼1 ML of ethanal at 100. 4. Next, a third separate experimental data set is used to perform a predictive validation and assess consistency between the model predictions and experimental data. This step is required since Bayesian model calibration does not guarantee consistency between model prediction and experimental data. This step permits a further reduction in the number of active-site model structures that are able to describe the experimentally observed complexity, even when the evidence for or against a specific active-site model structure is weak or nonexistent, based on the Bayes factor. Walker et al.234 used the Mahalanobis distance as a consistency check to test whether the experimental data set is a possible outcome of the active-site model considering all uncertainties. 5. (Considering all permutations): Knowing that Bayesian model selection is highly sensitive to the prior distribution, the potential for bias can be mitigated when using the approach described above by using all possible permutations of the different data sets to inform, rank, and validate. That is, in this example, steps 2, 3, and 4 were performed on three separate models. To avoid bias, Steps 2, 3, and 4 were performed with models A, B, and C in each permutation possible: A, B, C; A, C, B; B, A, C; B, C, A; C, A, B; C, B, A.

VI. SENSITIVITY ANALYSIS AND UNCERTAINTY QUANTIFICATION The sizes of uncertainties of parameters obtained at the individual scales (e.g., electronic structure, supramolecular, etc.) are important, but for many systems, it is even more important to identify which of the uncertain parameters most control the output error. This sensitivity information can be obtained through sensitivity analysis (SA) and then serve to direct computational or experimental resources for the improvement of these parameters. This might also be employed for hierarchical model construction, where one starts with very approximate estimates (such as correlations) and performs increasingly expensive calculations (such as DFT) to reduce the errors in the parts of the model which have the most impact.25 SA allows one to draw qualitative conclusions, even in the presence of large uncertainties. The simplest SA approaches are local, which probe the partial derivative of the response(s) with respect to parameters. The most common local SA in heterogeneous catalysis is the degree of rate control.61 While other methods of local SA exist, recent years have seen rising awareness of the degree of rate control concept because of its practical usefulness and relatively intuitive appeal. Local SA is rather straightforward for MF-MKM models,34,61,284,285 but for KMC, it can require a

A key aspect is estimating the correlation structure in the DFT energies, for which there is more than one plausible method. For example, the correlation structure in DFT energies can be estimated by the BEEF and mBEEF functionals. A challenge with these functionals is that they only provide a correlation structure of energies within a specific class of functionals, which might significantly underestimate the uncertainty if the electronic structure of the active-site model and states in the reaction mechanism are not welldescribed by that specific class of functionals. Also, BEEF and mBEEF presently provide the same weight to all functionals, and it is not clear that this is the most accurate choice 6638


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ACS Catalysis major effort108,197,286 (more recent KMC local SA studies are cited below). Developing corresponding methodologies for KMC is an active area of research. However, most studies for local SA for KMC focus on well-mixed problems (which can be described without accounting for spatial heterogeneity).287,288 The strategies for a local SA for KMC, which are generally applicable to or have been intentionally designed for spatial problems, include coupled finite differences,167,287,289,290 loglikelihood estimators,129,166,290−292 relative entropy,167,288 and linear response estimators.167 Even with these advanced techniques, computational costs of local SA can remain high compared with the KMC estimation of turnovers or coverages, and the SA can be further complicated for systems displaying numerical stiffness. However, considering the sizable DFT uncertainties, the assumption of linearity behind local SA is likely to fail. In recent years, global approaches have therefore attracted attention and have been applied in first-principles-based modeling of heterogeneous catalysis mostly for MF-MKM models19,22,164,170,234 but also for KMC.16 While these studies found possible huge uncertainties and qualitative failures of local SA, the investigated problems showed significant impact on the results by only a few parameters (i.e., only a few atomic scale aspects were actually rate controlling). Parametric global SA approaches address the impact of finite errors by averaging over the space of the kinetic parameters for a fixed set of conditions.234 Global SA can thus require a very large number of simulations, each run at a different set of input parameters, and these costs can be substantially larger than those of local SA. On the other hand, they are based on integral formulations, and therefore, they are less affected by noise in KMC simulations relative to local SA analyses. In fact, the global SA in ref 16 came at a comparable cost and with comparable accuracy relative to the advanced local SA in ref 167 for the same KMC model. Another advantage of global approaches is that they are trivially scalable on HPC systems because every model evaluation is independent. Because of these properties and the foreseeable continued high computational costs of accurate electronic structure methods, we will likely see a rise in popularity of global uncertainty quantification and SA methods in the nearer future. These methods not only enable model improvement but also aid in extracting chemically meaningful information from highdimensional chemical kinetics problems, such as which reactions are most rate-determining. Uncertainty quantification and SA require three critical ingredients. The first is a physically sufficient and accurate base model, that is, the model should be able to capture the kinetic behavior and deviations from reality should be only due to parametric dependence: in our case, due to the uncertainty of the DFT energetics or other sources of kinetic parameter predictions. Qualitative sources of errors in the physical model can generally not be addressed by these methods, such as errors due to unknowingly failing to include important adsorbate interactions. The second is a probability distribution for the errors, encoding our gained knowledge on the possible parameter errors. The probability distribution might be estimated based on experience; for example, the typical value of 0.2 eV uncertainty in DFT calculations of energies might be interpreted as bounds leading to a multivariate uniform distribution of the respective energies on the corresponding hypercubic parameter space. If we know more about the correlation of the errors or have corresponding experimental

data, this can be included using Bayesian modeling (as discussed in Section IV). For example, the approaches mentioned in Section IV for obtaining the correlation of structure of errors, that is, using the correlation from the (m)BEEF functional or an ensemble of functionals together with a probabilistic latent variable model, could be used. Finally, we need a methodology to efficiently explore the usually high-dimensional parameter space. When the error distribution can be mapped on a hypercube, methods like Quasi Monte Carlo16,293 or (multilevel) adaptive Sparse Grids163,164,294 offer a good alternative to standard Monte Carlo because of their higher asymptotic convergence rates. Overall, we expect significant process in the near future on performing sensitivity analysis under uncertainty.

VII. OUTLOOK ON THE FUTURE OF KINETIC MODELING IN HETEROGENEOUS CATALYSIS Future growth is foreseeable in multiple directions in the next 5 to 10 years. We group these into several thematic areas that are interconnected with the sections of this Perspective: 1. Advances in Elementary Step Parameter Computations. This will be accomplished by several means: (a) more reliable, calibrated electronic structure models, through leveraging of DFT and/or wave function approaches with better defined reliabilities; (b) more robust free energy models, exploiting “smarter” sampling methods for local configurations; (c) identification of reliable fingerprints of surface species and transition states that will enable the application of machine learning algorithms for predicting free energies in different local environments and on different materials; and (d) more robust rate parameters, beyond the harmonic TST approximation, including the possibility of on-the-fly (or precomputed) first-principles trajectories in dynamics simulations. 2. Advances in Structural Models. Future work will continue to push the boundaries of achieving more realistic, chemically faithful representations of catalytic structures and/ or ensembles of active-site structures. Approaches that may contribute to this effort include (a) mapping of first-principles energies onto appropriate, potentially machine-learned force fields (that is, to decrease the cost of complex models by using surrogates); and (b) increased attention on the influence of reaction conditions, finite temperature dynamics, and explicit inclusion of solvents on reaction mechanismsor more generally, to better account for complex environments. 3. Increased Use of Microkinetic Models and Molecular-Scale Kinetics. Microkinetic modeling is a mature approach, which is seeing increased use. We envision it becoming a “standard” to include microkinetic models as part of all mechanistic calculations. Going forward, there will be increased emphasis on going beyond mean-field models. There will be increased incorporation of local configuration heterogeneitystructural, adsorbate-induced, etc.into models, including by use of lattice KMC and MD methods. These efforts will result in more accuracy when combined with increased usage of uncertainty quantification and sensitivity analysis approaches, such as the degree of rate control. More accurate assessments will be obtained through the use of uncertainty propagation and quantification, while Bayesian model calibration will enable tighter integration of models with experimental data. 4. Increased Accuracy in Coarse-Grained Models. There is presently an increased focus on connecting microkinetic models (both MF and KMC) to transport models and 6639


Perspective

ACS Catalysis classical reaction engineering. These capabilities will enable the connection of catalytic and noncatalytic gas-phase processes and more accurate calculations of mass-transfer in environments without fast mixing, such as porous materials. Also, these capabilities will enable a more faithful representation of the chemistry in processes that involve significant heat and momentum gradients relevant for strongly exothermic processes, for example. 5. Improved Connections to Experiment. At present, many kinetic modeling studies are limited to predicting trends. There will continue to be an increased ability/emphasis on experimentally verifiable predictions with quantitative or semiquantitative accuracy via models that better incorporate the complexity of relevant experimental conditions. In this context, the use of Bayesian model calibration and design of experiment based on calibrated models will likely continue to increase. In considering the outlook of the future, open source and easily operable software will need to be deployed for newly developed methods to become widespread. History has shown that if the knowledge is not easy to use, other practitioners will not use it. For example, Redhead’s desorption peak temperature method295,296 is the most widespread method for estimating activation energies during temperature-programmed desorption, despite more sophisticated analyses236,295,297−304 being published. Similarly, BEP correlations have become widely used, while the Blowers−Masel equation305 and the bond-order-conservation type approximations173−178 have not become widely used. Recent requirements by some funding agencies to make software developed using funding both open-source and openly available will spur some “community” advancement. However, in many cases, a dedicated effort may be required to create tools that are generalizable beyond a single problem and user-friendly for others to use. Toward this end, the programs that involve funding that is dedicated toward software tool development will also play a significant role in the pace of community advancement. We favor efforts by principal investigators and funding agencies to develop and make available the tools that will be needed to investigate systems with ever increasing complexity at ever increasing levels of accuracy.27 We have an optimistic view of future progress in the field and expect rapid progress to be made during the next decade.

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Andreas Heyden: 0000-0002-4939-7489 Aditya Savara: 0000-0002-1937-2571 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.S. thanks Sonya D. Sawtelle for aid in overcoming challenges in the programming of, implementation of, and plotting of the Bayesian parameter estimation example shown in Figures 10 and 11. A.S. thanks Stephan Irle for feedback on several sentences in section II. Work by A.S. was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division. A.H. acknowledges financial support from the National Science Foundation under Grant No. DMREF1534260. The work by S.M. was carried out in the framework of Matheon supported by the Einstein Foundation Berlin. W.F.S. was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award DE-FG02−06ER15839. This manuscript has been authored, in part, by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acscatal.9b01234. Model details and parameter estimates for temperatureprogrammed desorption of ethanal from CeO2(111), as well as additional conceptual explanation of Figure 11. (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: *E-mail: *E-mail: *E-mail:

REFERENCES

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ORCID

William F. Schneider: 0000-0003-0664-2138 6640


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