# 快速时时彩: Numerical simulations of electrocatalytic processes.pdf

Numerical simulations of electrocatalytic processes M. T. M. Koper Eindhoven University of Technology, Eindhoven, The Netherlands 1 INTRODUCTION Computer simulations are playing an increasingly impor- tant role in understanding the molecular basis of catalytic and electrocatalytic processes. The aim of this chapter is to present a few applications of the various computer simulation methods available to model electrocatalytic reac- tions. Our attention is restricted to simulation methods that include some sort of molecular detail. Computer simula- tion methods to treat phenomena such as reaction flow, mass transfer, and heat transfer normally do not include a molecular picture of the process taking place. They are discussed in Volume 1. It is important to realize what level of theory is involved when one is carrying out molecular computer simulations. Theoreticians or computational chemists usually resort to computer simulation when a problem that has been formu- lated in quantum mechanics or statistical mechanics cannot be solved analytically, or is too difficult to solve by existing mathematical techniques. The type of computer simulation chosen for solving a particular problem depends on the time-scale of the phenomenon to be modeled and the level of molecular detail one wishes to include. Given that the power of present day computers allows modeling of larger and more complicated systems, molecular computer simula- tion has become a field of physical sciences that can hardly be considered “theory” any longer. The complexity of the problem may render a computer simulation similar to an experiment. However, the control that the computer experi- menter has over his or her system is much greater than that in a real experiment, and it is precisely this feature that makes computer simulation such a powerful tool in gaining molecular-level insight. Application of modern molecular computer simulation techniques to electrochemical and electrocatalytic systems is of relatively recent vintage. Undoubtedly this is related to the complexity of modeling metal/liquid interfaces, let alone the modeling of reactions taking place at such inter- faces. In this chapter, the main principles of two important computer simulation techniques are discussed and illus- trated by some specific applications to electrocatalytic pro- cesses. These applications provide a unique molecular-level insight which, when used in conjunction with experimental investigations of well-defined systems, leads to an under- standing that transcends that obtained by experiment or theory alone. The emphasis in this chapter is on quantum chemical calculations of carbon monoxide and hydroxyl with metal and alloy surfaces, and Monte Carlo simula- tions of carbon monoxide oxidation on electrode surfaces, as these processes are of obvious interest for fuel cell catalysis. 2 QUANTUM-CHEMICAL CALCULATIONS 2.1 Density functional theory One very important development in “practical” ab initio quantum chemistry in the last two decades has been the application of so-called density-functional theory (DFT) methods to the calculation of properties of large ensem- bles of atoms. [1] The classical Hartree–Fock methods are Edited by Wolf Vielstich, Hubert A. Gasteiger, Arnold Lamm and Harumi Yokokawa. ? 2010 John Wiley open circles refer to binding to a Pt site. (Adapted with permission from Shubina and Koper (2002) [19] ? American Chemical Society.) 4 Methods in electrocatalysis Pt Sn CO: ?1.40 OH: ?2.32 CO: 0 OH: ?2.66 CO: ?0.08 OH: ?2.50 CO: ?1.82 OH: ?2.24 CO: ?1.14 OH: ?1.97 CO: ?0.49 OH: ?2.24 CO: ?1.35 OH: ?2.36 CO: ?1.64 OH: ?2.53 Figure 5. DFT/GGA chemisorption energies of CO and OH on the various adsorption sites of the Pt 3 Sn(111) surface. 2.3 Field-dependent binding of CO to metals A feature of particular interest in interfacial electrochem- istry is the potential dependence of both the binding energy and the vibrational properties of chemisorbed CO. This has been a subject of intense experimental and theoret- ical study ever since the first observations with infrared reflection spectroscopy of the variations in C–O stretching frequency with electrode potential (the so-called “interfa- cial Stark tuning effect”). In collaboration with the Weaver group at Purdue, we have recently undertaken detailed DFT calculations of the potential-dependent chemisorption of CO on platinum-group (111) surfaces, [9, 21, 22] modeled as clusters, for comparison with the extensive vibrational char- acterization of these systems as carried out by the Purdue group. [23, 24] The electrode potential in these studies was modeled as a variable external electric field applied across a metal cluster, an approach already taken by many others in the past. Twoissuesareofinterestinrelationtothepotential- dependent bonding of CO to transition metal surfaces: the potential-dependent binding energy and the result- ing potential-dependent site preference, and the potential- dependent vibrational properties, in particular the internal C–O stretching frequency. Figure 6 shows the effect of the applied field on the binding energy of CO in a onefold (atop) and multifold (hollow) coordination on a 13-atom Pt cluster. What is particularly significant in this figure is the change in site preference predicted by these calcula- tions, from atop coordination at positive fields to three- fold coordination at negative fields. Qualitatively, such a potential-dependent site switch has indeed been observed experimentally. [23, 24] By decomposing the binding energy into steric and orbital contributions, it can be appreci- ated that the main factor that drives this preference for multifold coordination towards negative field is the back donation contribution. This increasing importance of back donation with more negative field can be understood on the basis of the classical Blyholder picture. A more negative field implies an upward shift of the metal electronic lev- els with respect to the chemisorbate 2π ? orbital, leading to an enhanced back donation of metal electrons. Since the interaction of the metal with the 2π ? orbital is a bonding interaction, and bonding interactions tend to favor multi- fold coordination, CO will favor multifold adsorption sites at negative fields. The change in the intramolecular C–O stretching mode with electrode potential has also been a subject of long- standing theoretical interest, and has become known as the interfacial Stark effect. Important theoretical contributions in this field have been made by Lambert, Bagus, Illas, Head- Gordon, and Tully. [25–29] We have re-examined this issue in the context of a detailed comparison of DFT calculations with the spectroscopic data of CO and NO adsorbed on a variety of single-crystalline transition metal electrodes. [9] These calculations confirm that NO generally possesses a higher Stark tuning slope (i.e., change in intramolecular stretching mode with potential) than CO, and that multifold CO exhibits a higher Stark tuning slope than atop CO. In agreement with earlier calculations, a decomposition analysis shows that the lowering (blue shift) of the internal C–O stretching mode upon adsorption is mainly due to the back donation contribution, as metal electrons occupy the 2π ? antibonding orbital of CO. [30, 31] This back donation contribution offsets the steric contribution, also known as the “wall effect”, which by itself leads to a red shift upon chemisorption. Although the positive Stark tuning slope is mainly the result of the back donation contribution, i.e., the negative field leading to enhanced back donation and hence a lowering of the C–O stretching mode, the different slopes found for the different metals and coordination geometries are generally found to be the result of a subtle interplay ?1.2 ?1.1 ?1.0 Energies (eV) Total binding energy ?0.01 0.00 0.01 ?2 ?1 0 1 2 Steric Field (a.u.) ?0.01 0.00 0.01 ?2 ?1 0 1 Backdonation ?22 ?1 0 1 2 Donation atop hollow Figure 6. Field-dependent binding energy plots, and constituent steric, donation and back donation components, for CO adsorbed atop and hollow on a 13-atom Pt(111)-type cluster. The orbital components are plotted with respect to their zero-field values; 0.01 field a.u. corresponds to 0.514 V ? A ?1 . (Adapted from Koper and van Santen (1999) [20] with permission from Elsevier Science.) 5 Numerical simulations of electrocatalytic processes between back donation, donation, and steric effects. [9] Hence one should in general be careful with assigning a higher Stark tuning slope to an enhanced back donation. 3 DYNAMIC MONTE CARLO SIMULATIONS 3.1 Principles The methods described in the previous section allow one to obtain information on the energetics of single atoms or molecules, or a small ensemble of molecules, interacting with a surface. In principle, these methods can be combined with molecular dynamics to obtain dynamic information on a nanosecond time scale. However, a simulation of the overall dynamic behavior of an extended catalyst with many adsorbates interacting and reacting with each other is currently beyond the scope of any quantum chemical calculation. Still, such behavior is very relevant for understanding the macroscopic properties of (model) catalysts. One way to bridge this gap is to use a “coarse-graining” approach in which the surface reaction is modeled in terms of elementary steps on a lattice-like surface. The lattice points correspond to the adsorption sites on the catalyst surface. Adsorption energies and interaction energies can be in principle calculated from a DFT/GGA calculation, and rate constants can be estimated by using transition state theory. Alternatively, rate constants and energetics can be estimated by comparison with experiment or used as adjustable parameters to study the influence of their variation on the overall behavior. The overall dynamic behavior is computed from a dynamic Monte Carlo (DMC) simulation. The conceptual basis of the DMC method is the Master Equation, which describes the time evolution of the system. Specifically, the Master Equation is a differential equation for the probabilities P α of finding the system in a certain configuration α: dP α dt = summationdisplay β W βα P β ? W αβ P α (2) where W βα are the transition probabilities per unit time for the system changing from configuration β to α.The probabilities and transition probabilities change over time due to the changes taking place on the surface. A numerical simulation method that keeps track of all changes and possible changes taking place on the surface as defined by the reactions that can take place on the surface was formulated by Gillespie [32] and implemented by many others. [33–35] It simulates solutions of the Master Equation by a method using random number generation, in a spirit very similar to the better known Metropolis Monte Carlo method. Since Gillespie’s method involves an exact introduction of time into the Monte Carlo method, these simulations are often referred to as Dynamic Monte Carlo or Kinetic Monte Carlo. Although there are various ways to implement the method, the most popular one is the First Reaction Method. Briefly, the First Reaction Method involves making a list of all possible changes that can take place on the surface at a certain moment. Then, a time from the probability distribution for each possible change as defined by the transition probabilities (i.e., rate constants) is randomly picked and the system is updated by the configuration change with the minimum time. Another reason for doing Dynamic Monte Carlo simula- tions is a more conceptual one. The standard approach to kinetic modeling of surface-catalyzed reactions is to express all reaction rates in terms of average coverages θ. [35] Such an approach is always an approximation as it ignores any existing local correlations that may exist on the surface. Basically, it amounts to assuming a perfect mixing of all reaction partners and hence neglects effects of ordering, island formation, or inhomogeneous surface properties. In statistical mechanics literature, this approximation is known as the mean-field approximation. In general, little is known about the accuracy of the mean-field approximation and presumably the best way to test its validity is by compari- son with Monte Carlo simulations, which always give the exact outcome provided a good statistical sampling of all the possible configurations is carried out. 3.2 Application to CO oxidation One issue that is particularly well suited to a DMC simulation study is the influence of CO surface diffusion on the overall CO oxidation kinetics. This has been a subject of considerable interest in many recent experimental studies of CO oxidation, and is of obvious importance for understanding CO oxidation on bifunctional bimetallic catalysts (as illustrated below). CO oxidation on a Pt electrode is generally assumed to take place by a Langmuir–Hinshelwood mechanism between adsorbed CO and an oxygen-containing species that is formed from water, usually assumed to be adsorbed OH. The OH is formed on a free site “?” by the dissociative adsorption of water, and then reacts with a CO adsorbed on a neighboring site leaving behind two free sites: H 2 O + ? ←?→ OH ads + H + + e ? (3) CO ads + OH ads ???→ CO 2 + H + + e ? + 2 ? (4) 6 Methods in electrocatalysis This mechanism, when considered in the mean-field approximation, can explain quite a few experimental observations. It reproduces, for instance, the shape of the chronoamperometric transient in a stripping experiment, [36, 37] the negative reaction order in the CO bulk concentration in a continuous oxidation experiment, [38] and the hysteresis and S-shaped form of the polarization curve of continuous CO oxidation experiments. [39] Dynamic Monte Carlo simulations have been carried out to assess the influence of a finite CO surface mobility on the stripping voltammetry and the chronoamperometric oxidation transient. [40] The surface was modeled as a square lattice of surface sites. The site is either empty or occupied by OH or CO. The surface hydroxyl is formed by reaction (3). As we are considering a CO stripping experiment, CO is pre-adsorbed up to a certain maximum coverage. Typically 99% of the surface is initially covered with CO to model a saturated CO adlayer. Reaction (4) can take place with a certain potential-dependent reaction probability when CO and OH are adsorbed on neighboring sites. The overall reaction rate, and hence electric current, will depend on the degree of mixing, which is primarily influenced by the CO diffusion rate as the OH formation can take place reversibly on any part of the surface. The CO diffusion is modeled as the CO exchanging places with a neighboring empty site: CO + ? ???→ ? + CO (5) The rate of this process is D s ?1 , which corresponds to a surface diffusion coefficient of ca. D × 10 ?15 cm 2 s ?1 . Figure 7 compares the stripping voltammetry