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A coverage-dependent, mean-field microkinetic model of catalytic NO oxidation, NO+0.5 O2⇌NO2, at a Pt(111) surface has been developed, based on large supercell density functional theory (DFT) calculations. DFT is used to determine the overall energetics and activation energies of candidate reaction steps as a function of surface coverage. Surface coverage is found to have a significant but non-uniform effect on the energetics, pathways, and activation energies of reaction steps involving formation or cleavage of ONO and OO bonds, and inclusion of this coverage dependence is essential for obtaining a qualitatively correct representation of the catalysis. Correlations are used to express all reaction parameters in terms of a single coverage variable θ and steady-state solutions to the resultant mean-field models are obtained in the method of DeDonder relations. At conditions representative of NO oxidation catalysis, the surface coverage is predicted to be 0.25≤θ<0.4 ML and to be controlled by equilibrium between gas-phase NO and NO2 and chemisorbed O. O2 dissociative adsorption (O2(g)→2O*) is rate limiting in the model. The DFT-based mean-field model captures many features of the experimentally observed catalysis, and its short-comings point the way toward more robust models of coverage-dependent kinetics.
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Combustion of fuel in excess air can be up to 30 % more efficient than stoichiometric or rich combustion, but the NOx by-products of combustion are difficult to decompose or reduce to N2 in the highly oxidizing “lean” exhaust. Current NOx removal strategies, including the lean NOx trap1, 2 and selective catalytic reduction,3, 4 depend in part on catalytic oxidation of NO (the primary NOx product) to NO2 [Equation (1)]:
Supported Pt is the most common NO oxidation catalyst,5 but high Pt loadings are required to achieve satisfactory conversions and to resist oxidative deactivation.6–10 Understanding the function of NO oxidation catalysts would be invaluable in designing more robust and less costly catalytic materials.
NO oxidation [Equation (1)] is only modestly exothermic [ΔHo(298 K)=−57 kJ mol−1] and is thus equilibrium limited under conditions of practical interest.8 The rate per Pt site, or turnover frequency (TOF), increases with increasing Pt particle size,6, 11, 12 suggesting the low-index metal surfaces to be the most catalytically active. In fact, a single crystal Pt(111) surface has the highest reported NO oxidation TOF.13 Over supported particles, the reaction is approximately first order in both reactants and negative first order in NO2, so that the reaction is product-inhibited.6, 12, 14, 15 The activation energy is reported to be 80 kJ mol−1.6, 13, 14 To account for these observations, Mulla et al. proposed a kinetic model in which O2 adsorption [Equation (2)] is rate-limiting, and dissociation [Equation (3)] and ONO bond formation [Equation (4)] are both in quasi-equilibrium:
NO oxidation activity is associated with high surface coverage of oxygen on Pt.6–10, 14, 16, 17 The central importance of surface coverage is evident from simple energy arguments. A useful catalyst must bind oxygen strongly enough to promote O2 dissociation [Equations (2) and (3)] but weakly enough to allow ONO bond formation [Equation (4)]. The PtO bond energy must thus be greater than half of the OO bond energy (2.6 eV) but less than the ONO bond energy (3.2 eV). These constraints are not met by a Pt surface at low coverage.18, 19 At higher coverages, adsorbate–adsorbate interactions decrease surface-O binding,20−–22 and within some coverage window both O2 dissociation and NO2 formation can become energetically favorable. Previous estimates put this coverage in the range of 0.25–0.5 monolayers (ML) of chemisorbed O (1 ML=1 O per Pt).19, 18
Although a number of DFT-based NO oxidation studies have been reported,6, 8–10, 12–14, 17, 18, 23–26 none has explicitly considered the effect of O surface coverage on the reaction mechanism. Herein, taking advantage of large-supercell DFT simulations, we find that adsorption energies, sites, and even reaction pathways change in significant and distinct ways with surface coverage. We incorporate these DFT results into a coverage- dependent microkinetic model and solve over a range of conditions similar to those probed experimentally.13 The model shows steady-state coverages similar to those inferred from energy arguments, shows O2 dissociative adsorption to be rate limiting, and finds ONO bond formation to be in quasi-equilibrium. This last reaction is found to control O coverage and to introduce the inverse NO2 order observed experimentally. The results show that by modeling reaction steps at appropriate surface coverages, it is possible to account for many of the features observed over Pt(111) and supported Pt catalysts.
Coverage-Dependent Surface Reactions
To explore sensitivity to surface coverage, we compared adsorption energies and reactions of relevant NO oxidation intermediates including O, NO, NO2, and O2 at several O-precovered Pt(111) surfaces. We chose as coverage models six previously identified energetically stable chemisorbed O arrangements with coverages between 0 and 0.5 ML19 (Figure 1). O atoms strongly prefer to bind to Pt(111) in threefold, FCC sites up to 0.5 ML, and they prefer maximally separated configurations due to strong repulsive forces between adsorbed O.18, 19, 27, 28 Above 0.5 ML, the surface becomes unstable to reconstruction and partial oxidation.19, 29, 30 As shown below, the models predict steady-state coverages under reaction conditions at or below 0.4 ML, below that at which these reconstructions become significant.
We systematically searched adsorption configurations over each coverage model to identify preferred adsorption sites and corresponding adsorption energies. The most stable adsorbed NO, NO2, and O2 configurations at three different O coverages are shown in Figure 2. Corresponding adsorbate binding energies (negative energies correspond to exothermic adsorption) are plotted in Figure 3.
NO binds most strongly of all of the adsorbates considered. Both linear binding in a threefold FCC site and bent binding in a site atop a Pt atom are energetically accessible on an O-covered Pt surface, and the ability to shift between these two binding modes is important to maintaining the NO binding energy. The FCC site is preferred over the atop site up to 0.25 ML O. At higher coverage, where close contacts between NO and surface O become unavoidable, NO migrates from a linear FCC to a bent atop conformation to minimize sharing of bonds with surface Pt atoms. This preference is retained up to the highest coverages considered.
Even more so than NO, NO2 can adopt its adsorption configuration to optimize surface binding. At low coverage, NO2 preferentially binds along one NO bond, bridging two Pt, in a μ-N,O-nitrito configuration.18, 27 At 0.25 ML O, bridge bonding becomes disfavored due to multiple NO2O* adsorbate interactions that are relieved by relaxation into a one-fold, atop, and upright nitro configuration (Figure 2). NO2 maintains this geometry up to 0.5 ML O, at which point, like NO, its bonding is further weakened by sharing of one Pt with surface O. NO2 binding is weaker than NO at all coverages (Figure 3 a), and as shown below, under NO oxidation conditions, NO2 coverage is expected to be negligibly small.
O2 binding is the weakest of all the adsorbates of interest here (Figure 3 a). O2 adsorbs on clean Pt(111) in nearly isoenergetic top-bridge-top (TBT) and top-FCC-bridge (TFB) conformations,21, 31, 32 and these two remain nearly indistinguishable to higher coverage. As coverage increases, sharing of a surface Pt with preadsorbed O decreases binding by 0.36 and 0.30 eV for TBT and TFB, sites, respectively, accounting for the majority of the adsorption energy decrease. The OO separation decreases from 1.37 Å on the clean surface to 1.30 Å on 0.25 ML O, and the Pt–O2 distances similarly increase from 2.0 to 2.3 Å. Bader33 and density of states (DOS) analyses show that charge donation from Pt d to O2 π* states decreases with increasing O coverage. All of these results are consistent with weak O2 adsorption, becoming endothermic and thus metastable to O2 desorption at the highest O coverage considered.
Despite these variations in adsorption configuration with coverage, optimal adsorption energies track one another (Figure 3 a). NOx and O2 binding energies are plotted against atomic O binding energies in Figure 3 b. Nearly linear correlations are evident, and least-squares analysis yields R2 values of 0.92 and greater. The correlation for NO, the most strongly bound surface intermediate, is:((5))
where R2=0.99. The slopes for the various adsorbates differ significantly and for NO and NO2 are less than one, reflecting the ability of these adsorbates to adopt conformations that reduce unfavorable interactions with adsorbed O. The O2 slope, in contrast, is approximately one: OO and O2O interactions are nearly the same at the Pt surface.
Coverage-dependent activation energies
The GGA potential energy surface (PES) for the reaction of chemisorbed O and NO on a Pt(111) slab [Equation (4)] has been reported by several groups.18, 24, 25, 34–36 Figure 4 shows the results obtained here at several surface coverages. At low coverage, NO and O, bound initially in 1NN (1NN=first nearest neighbor) FCC, sites come together along a path to form bridge bound NO2.18, 35, 36 At the transition state, the NO and O fragments bind in bridge and atop sites, respectively, straddling an HCP site (Figure 4). The low-coverage GGA activation energy is 1.5 eV.36 Furthermore, the surface reaction is approximately 0.6 eV endothermic, even when referenced to 1NN reactants: PtO binding is stronger than ONO at this coverage.18, 19, 27, 36 The reaction energy and barrier increase by 0.2 eV when referenced to well-separated NO* and O*.
The barrier to ONO bond formation is calculated to substantially decrease with increasing coverage (Figure 4). On P(2×2)O, we identify a path in which atop-bound NO combines with a 1NN O to form O-down atop-bound NO2.36 The reaction passes through a transition state in which the O atom has migrated to a bridge site, the PtNO distance increases to 3.0 Å, so that NO is no longer directly bound to the surface, and the ONO distance decreases from 2.79 to 1.69 Å. Frequency and NEB (NEB=nudged elastic band) analysis confirm that this transition state connects the reactant and product states. Despite the substantial loss of surface bonding, the reaction barrier is significantly (0.8 eV) less than on the clean surface; unfavorable adsorbate–adsorbate interactions are decreased along the reaction pathway, in part because of the adaptability of the reactants, products, and transition state to the surface coverage. This transition state has three fewer unfavorable 1NN interactions than would be found if the reaction proceeded along the same path as found at low coverage. Because the product configuration along the MEP is weakly bound, we expect facile NO2 desorption and for the reverse reaction to happen directly from the gas phase.
On the P(2×1)O model, the entire potential energy surface is pushed up in energy, so that the reaction barrier relative to NO and O is essentially unchanged. Again atop-bound NO combines with a 1NN O to form O-down, atop-bound NO2. However, to minimize adsorbate interactions, at the transition state surface binding is through a single, atop O (Figure 4). The TS geometry is nearly identical to that at 0.25 ML, except the ONO bond is 0.2 Å longer. The transition state experiences four fewer 1NN O interactions than the NO*+O* reactant state, and seven fewer than would be predicted by extrapolation from the low coverage pathway.
In summary, the NO*+O* reaction path itself changes with coverage: FCC-bound NO at low coverage reacts with a larger barrier than atop-bound NO, which is present at higher O coverage. In the higher coverage regime, the reactant, transition state, and product are all shifted up equally with coverage, and the barrier in this regime is 0.7 eV, independent of surface coverage. In the kinetic model we increase this reaction barrier by 0.4 eV to account for the energy cost of bringing FCC O and atop NO into 1NN proximity.
O2 dissociation on clean Pt(111) [Equation (3)] has also been studied extensively within DFT supercell models.35–38 The results shown in Figure 5 agree well with earlier reports. At low coverage, NEB calculations identify a transition state that connects bridge-bound O2 to two FCC-bound O atoms in 1NN sites. The barrier is −0.24 eV with respect to O2(g).36 Consistent with expectation, O2 dissociation is facile at low coverage.
Unlike the NOx adsorbates, the preferred chemisorbed O and O2 binding sites do not change with increasing coverage. We constructed NEB pathways from bridge-bound O2 to two 1NN FCC O atoms on the higher coverage model surfaces (Figure 5.) The TSs have O in atop and bridge sites and straddling HCP sites, similar to the low coverage ONO bond-formation TS, but at 0.25 and 0.5 ML coverage the TS geometries are slightly distorted because of repulsion with surface O.36 For example, at 0.25 ML coverage, the atop O maximizes its distance from surface O and thus minimizes its distance from bridging O, resulting in a smaller OO bond length than at 0 or 0.5 ML. Such binding is not possible on the crowded 0.5 ML surface. Because O2 dissociation increases the number of surface species, dissociation becomes progressively less exothermic with increasing coverage. Furthermore, the activation energy for O2 dissociation increases markedly, to 0.95 and 2.28 eV relative to O2(g) at 0.25 and 0.5 ML, respectively (Figure 5).36 The transition state has two and six 1NN O at these coverages, twice the number of O–O2 interactions in the reactant states and two fewer than the number of O–O interactions in the product states. The O2 dissociation rate is thus expected to be highly sensitive to surface coverage.
Brønsted-Evans-Polanyi (BEP) relationships are a convenient means of correlating the activation energies of structurally similar transition states.39, 40 Figure 6 plots the computed O2 dissociation barrier against the computed dissociative adsorption energy to 1NN products, both energies taken with respect to O2 gas. Despite the differences in OO bond lengths in the TS geometries (Figure 5), the two energies are linearly correlated with unit slope, because the relationships between the number of O interactions in the transition and product states at these values of θ are identical.((6))
In contrast to the ONO bond formation reaction, a lateral interaction model thus appropriately describes the coverage dependence of O2 dissociation. The binding energy of an OO nearest neighbor pair is to a good approximation related to that of an individual O adsorbate by Δ(θ)=2Δ(θ)+1NN interaction energy, where the O–O 1NN interaction energy is 0.2 eV. By using this relationship, the O2 dissociation activation energy can be written as:((7))
One way to circumvent the increasingly unfavorable energetics of O2 dissociation with increasing surface O coverage is through an “NO-assisted” dissociation pathway:13, 19((8)), ((9))
Such a pathway is relevant if the energetics of one O reductant and one OM bond are energetically favorable over the formation of two OM bonds. Stable OONO adsorbates have been calculated to be more stable than NO*+2O* for θ>0.5 ML.13 On the low coverage surface, we construct a pathway starting from FCC bound NO and a 2NN TBT O2. The products are an O-down atop NO2 adsorbate, identical to the one used to model O-NO bond formation, and FCC O. The TS is a trans-OONO structure bound approximately in a bridge site (Figure 7). The TS geometry is independent of θ and similar to those of stable cis- and trans-OONO species,13 but its OONO bond is longer (1.7 vs 1.3 Å) and its OONO bond is shorter (1.4 vs 1.8 Å), consistent with simultaneous OO dissociation and ONO bond formation. It has zero, zero, and one 1NN O at 0, 0.25, and 0.5 ML O, respectively, identical to NO, and dissociation barriers relative to NO*+O2 are independent of θ (Figure 7):((10))
indicating the O–atop NO and O–OONO 1NN interaction are similar. Like NO and NO2, the OONO transition state has a one-fold atop binding geometry that minimizes interactions with FCC O. O2 dissociation through this pathway could avoid the high barrier associated with independent O2 dissociation at high θ.
Coverage-Dependent Microkinetic NO Oxidation Model
These DFT results show that binding energies, reaction sites and even reaction pathways are sensitive to surface coverage, but vary in distinct ways with it. In the low coverage model, ONO bond formation [Equation (4)] is both endothermic and highly activated, whereas O2 dissociation [Equation (3)] is facile. In contrast, the relative energetics and barriers of both reactions switch with increasing surface coverage, and only at these higher coverages are the relative order of predicted rates consistent with observations. These features are consistent with rate-limiting O2 dissociation and relatively rapid NO2 formation and dissociation.
We used these results to parameterize a mean field kinetic model with coverage-dependent reaction parameters. The key model parameter is the coverage-dependent O binding energy, Δ(θ), or the energy to add an oxygen atom to a Pt surface at total O coverage θ. Previous equilibrium19 and preliminary kinetic models showed that the 0.25–0.5 ML coverage range is the most relevant to catalytic NO oxidation on Pt(111). We use a previously reported DFT database of coverage- dependent O adsorption energies19 and fit as a function of θ. Adsorption is found to weaken nearly linearly with θ in the range of interest, consistent with adsorbate-adsorbate interactions dominated by pairwise repulsions. We model the O binding energy as:((11))
This coverage dependence is similar to a recent report of GGA-calculated binding energies on Pt(111).41
By taking advantage of the correlations noted above between the adsorption energies of O and other adsorbates and transition states, Equation (11) can be used to write all relevant reaction parameters in terms of θO within the coverage range 0.25≤θO<0.5 ML (Table 1). The ONO bond formation and NO-assisted O2 dissociation barriers are taken to be independent of coverage between 0.25≤θ<0.5 ML, consistent with the computed reaction profiles.
Table 1. DFT-parameterized coverage-dependent reaction (ΔE) and activation (ΔEact) energies for 0.25≤θ<0.5. Energies are parameterized against total coverage, θ=θO+ θNO.
[a] PW91-GGA energies; [b] corrected for overall reaction energy error.
These correlations are based on the calculated interactions between an adsorbate or transition state and adsorbed O. As shown below, at large PNO/P ratios, NO coverage can become appreciable. We calculate a two-body O–FCC NO interaction of 0.23 eV per pair, nearly identical to an O–O interaction.18, 19, 28, 41 To within a small error, the θO-dependent energies are a reasonable representation of binding and activation energies at θ= θNO+θO.
The data in Table 1 provide all the information necessary to parameterize a mean-field microkinetic model for Pt(111)-catalyzed NO oxidation between 0.25 and 0.5 ML total coverage. The mean-field model assumes the surface O and free site configurations are completely uniform at all coverages, mirroring those in Figure 1, with intermediate coverages represented by linear interpolations. It thus neglects disorder in the O and free site configurations and thus regions where O and free sites may cluster. Standard-state reaction free energies, Δ(θ,T), are obtained by including reactant and product vibrational zero-point energies (EZP), the finite temperature vibrational free energy of the adsorbates (ΔFvib), and the finite temperature total free energy of any gas-phase species (ΔGo(T)).19 For NO adsorption (Table 1, reaction B), for instance, we have((12))
This formulation neglects the small contribution of configurational entropy. Adsorbate harmonic vibrational spectra are calculated at low coverage.13, 27 The change in free energy of gas-phase species, which we take from experimental tabulations,42 is by far the largest contribution to the temperature dependence.
In a first set of numerical calculations we estimated the equilibrium coverages of the various surface intermediates at conditions representative of NO oxidation, including P=0.1 MPa, y=10 % (y=mole fraction), y=100 ppm, and T=573 K. Total coverage θ>0.25 ML and is dominated by adsorbed O and NO. In this coverage range the mean-field-predicted rate of O2 dissociation is controlled by reaction A (Table 1). Calculated θ and θ assuming immobile adsorbates are less than 10−7 ML at all considered conditions and thus their effects on the site balance are neglected. O2 associative adsorption to a mobile precursor has been suggested to be rate limiting in Pt-catalyzed NO oxidation,12, 14 based primarily on the observed linear rate dependence on free site density. Within the DFT-derived parameters at higher coverage, kinetic results including a mobile O2 precursor [Equations (2) and (3)] are identical to a model in which O2 dissociates directly from the gas phase through a mobile precursor43 (Table 1, reaction A), because the O2 dissociation transition state is the highest point on the computed free energy surface. We thus model O2 dissociation using reaction A. As we show below, this approach gives results in good agreement with observed reaction rates.
To capture equilibrium bounds on individual steps and overall NO oxidation, net reaction rates are written as a function of reversibility βj:44, 45((13))
The reversibilities are equal to the reaction quotient divided by the equilibrium constant and are related to the reversibility of the overall gas reaction, β:((14))
where γj is the stoichiometric reaction coefficient. By using these relations and site conservation, the steady-state NO, O, and free site coverages can be calculated:((15)), ((16)), ((17))
NO adsorption is taken to be barrierless and thus equilibrated; the O and free site coverages depend on the reversibilities of the other reactions. Furthermore, overall reaction rates depend only on the rates of O2 dissociation and ONO bond formation.
The coverage-dependent mean-field rates of the four surface reactions are written using the calculated kinetic parameters:((18)), ((19)), ((20)), ((21))
For a given set of reaction conditions and β, the steady-state solutions of these equations are the βA and θ that make 2rA= and rD=. Here “indep” and “assist” indicate the two different mechanisms explored in this work, based on independent and assisted O2 dissociation, respectively (the mechanisms are explained in Section 3.2). The value of rC for these two mechanisms differs only in how βA is related to βC [Equation (14)]. The solutions are obtained iteratively by simultaneously solving the steady-state equations and the site balance, θ*+θO+θNO=1.
The free site dependence in Equation (18) is determined by the statistical availability of O2 dissociation sites at a particular coverage. The arbitrarily assumed dependence is appropriate for diatomic dissociation in a random, uncorrelated distribution of adsorbates, as is typically assumed in a mean-field model. As shown above, adsorbates interact and thus their surface distribution is not random. The inability to predict this dependence is a weakness of the mean field approach.
The DFT-GGA model overestimates the gas-phase NO oxidation energy by 50 % [Equation (1)]; the −0.6 eV experimental 0 K energy can be compared with the PW91-GGA calculated energy of −1.18 eV. Equivalently, the ONO bond energy is exaggerated 0.6 eV with respect to the OO by the GGA. The unmodified GGA parameters thus predict the NO oxidation equilibrium to favor products to artificially high temperatures, and this error is propagated in predicted coverages, rates, and conversions. To correct for this GGA error, we adjust the overall reaction energy to be thermodynamically consistent with the experimental results. The results below are for a model in which the NO2 adsorption energy is increased by 0.6 eV (as denoted in Figures 4 and 7 as “NO2 expt′l”). This adjustment propagates into reactions C and D (Table 1):((22)), ((23))
Adjusted reaction energies are given in Table 1. Because adsorbed NO2 does not play a role in the overall kinetics, this correction ultimately sets the gas-phase oxygen potential to the correct experimental value but otherwise has a negligible effect on the model results.
Mechanisms and rate-determining steps
We separately determine steady-state solutions to an independent O2 dissociation model, including reactions A–C (Table 1), and an assisted O2 dissociation model in which reaction A is replaced by reaction D. Steady-state rates and coverage are a function of temperature and the partial pressures of reactants and products. For analysis, it is useful to set the total NOx concentration and display results as a function of conversion XNO with respect to 100 % NO. Rates and coverages at P=0.1 MPa, y=10 %, y=100 ppm, and T=573 K are shown in Figure 8, plotted over the range XNO≈0–0.93.
The independent O2 dissociation TOF is greater than the NO-assisted O2 dissociation TOF at all conversions (Figure 8 a). The difference is maximized near XNO=0, where the independent route is more than 100 times faster than the assisted one, but the difference decreases with increasing conversion. The computed TOF for the independent mechanism is 0.9 site−1 s−1 near XNO=0, and decreases with XNO to 0.004 site−1 s−1 at XNO=0.93. This initial TOF agrees remarkably well with the experimental value of 0.4 site−1 s−1 at the same conditions on Pt(111).13
NO and O coverages are the same for both models (Figure 8 b). Under these reaction conditions, total surface coverage increases from 0.25 ML at low conversion up to 0.4 ML near XNO=1. NO is the most abundant surface species at low values of XNO, and p/pNO is small. θNO<0.25 ML for all values of XNO. Oxygen becomes the most abundant surface species at XNO>0.25, and θNO is less than 0.1 ML for XNO>0.33. Other ways of distributing the gas-phase reaction energy correction yield different relative but similar total coverages of NO and O, and thus nearly identical solutions.
At low XNO, where θ is at its minimum, the NO-assisted barrier is up to 0.15 eV greater than the independent O2. At higher XNO, the independent O2 dissociation barrier becomes at least as large as the NO-assisted one, but the NO coverage and thus the NO-assisted O2 dissociation rate become small. NO oxidation thus proceeds via independent O2 dissociation at all values of XNO. The competition between independent and assisted dissociation pathways is sensitive to the relative pressures of NO and O2. Higher NOx partial pressures produce higher NO coverages and, as the assisted step is modeled as a reaction between adsorbed NO and gas-phase O2, the assisted rate increases. Furthermore, these higher coverages inhibit direct O2 dissociation. Microkinetic modeling results at P=100 kPa, y=10 %, y=1 %, and T=573 K are shown in Figure 9. NO-assisted O2 dissociation is found to dominate the overall reaction at these very high y.
In the DeDonder formalism, reaction steps with βj near unity are in quasi-equilibrium, and steps with βj≈β are rate limiting.44 At y=100 ppm, we find βA=β and βC=1 at all values of XNO, so that the NO⇌NO2 reaction C (Table 1) is quasi-equilibrated and O2 dissociation is rate determining. The results are identical for the NO-assisted mechanism, since βD=βAβBβC. The “degree of rate control,”46–48 defined as the normalized partial derivative of the overall rate with respect to a rate constant, holding constant the reaction step free energy, is an alternative measure of kinetic control:((24))
We calculate these derivatives by finite difference. χrc,A=1 at all values of XNO, and χrc,C=0, indicating that O2 dissociation has a large positive control on overall rate.
Both the methods of DeDonder relations and the degree of rate control indicate O2 dissociation is rate limiting in this model. These results are a direct consequence of the fact that the rate of O2 dissociation is much more coverage sensitive than the NO reaction to NO2. Because O2 dissociation is rate limiting, the O2 reaction order is positive and, from the form of Equation (18), expected to be near unity.
O coverage is governed by the NO2/NO equilibrium [Equation (16)]6, 12–14, 36 and increases with XNO. High NO2/NO ratios thus suppress net oxidation by driving up O* coverage, leading to a negative rate-order dependence in NO2 and, under conditions in which θNO is small, a positive rate-order dependence in NO. The predicted rate orders in this mean field model will depend on reaction conditions, reaction parameters, and on the statistical availability of dissociation sites as a function of coverage. This site availability cannot be predicted from a mean-field model but must be determined from the distribution of adsorbates and dissociation sites as a function of coverage, which is sensitive to the exact surface being modeled.
Figure 10 a shows the experimental gas-phase equilibrium conversion of NO to NO2 vs temperature, starting from 10 Pa NO and 10 kPa O2 in 100 kPa total pressure. To illustrate the predicted catalyst light-off behavior in a reactor, we calculate conversion using the independent O2 microkinetic model, assuming a continuous stirred tank reactor (CSTR) with a residence time τ=100 s, site density LV−1=1.5×1015 sites per unit reactor volume, and constant inlet NO concentration CNO,0=10 Pa/RT:((25))
where rmicro are model-calculated rates and NA is Avogadro’s number. Equation (25) must be solved iteratively, and the CSTR model is computationally convenient, but the qualitative results are insensitive to the form of the macroscopic model. The superficial features of the catalysis are well described, including the light-off near 300 °C and the turnover in conversion due to equilibrium. Above 500 °C, the calculated coverage drops below 0.25 ML, outside the parameterization range of the model. A model based on unmodified GGA energy parameters yields similar conversions at low temperature but fails to reproduce the equilibrium limitation.
Because the NO oxidation reaction rate depends on both reactant and product concentrations and is equilibrium limited in the temperature range of interest, there is some arbitrariness in how to calculate an apparent activation energy. In Figure 10 b–d, we show the calculated steady-state reaction rate, coverages, and βi values for the independent O2 dissociation mechanism as a function of temperature holding the overall β constant. The rate is seen to increase and coverage slowly decrease with increasing temperature; reaction C (Table 1) becomes less reversible and reaction A more so, as the coverage and NO2/NO ratios decrease. An Arrhenius plot of the data calculated in this manner (Figure 10 b) yields an apparent activation energy of 2 eV. Under the conditions in which O2 dissociation is rate limiting and βA→0, the apparent activation energy can be written from Equation (18) as((26))
The first term contributes negligibly to the activation energy and, at coverages on the order of 0.3 ML, the second term contributes approximately 0.7 eV. The last two terms depend on the temperature dependence of the adsorbate (or vacant site) coverage and thus are ultimately tied to the O and NO adsorption energies. This derivative is sensitive to the conditions at which it is taken. Between 250 and 360oC at p=10 kPa, pNO=2.5 Pa, and β=0.03, conditions at which θNO is small and thus where sensitivity to pNO and are minimized, the average value of ∂θ/∂(1/T) is 288 K. The coverage dependence of is 2.32 eV (Table 1), and at all temperatures of interest the second to last term is an order of magnitude greater than the last. Combining factors, the third term contributes approximately 1.2 eV to the total and ΔEapp=2 eV.
Modeling reaction rates with an explicitly coverage- dependent activation energy thus considerably overestimates the 0.8 eV NO oxidation apparent activation energy found on Pt(111).13 This failing can be traced to the mean-field assumption that all O2 dissociation sites are equivalent and grow monotonically in activation energy with coverage. These assumptions overlook the heterogeneity of the reacting surface at finite coverages and the possibility that the low energy dissociation sites dominant at 0.25 ML retain some finite surface concentration at higher coverages. In fact, in the limit where =0, ΔEapp is 0.8 eV, much closer to experimental observation.
The modest enthalpy of NO oxidation to NO2 has several consequences for both experimental investigation and theoretical modeling of the Pt-catalyzed reaction. The catalysis must be carried out at high O2 chemical potentials in order to promote NO2 formation. Furthermore, due to the relatively strong bonding of both NO and O to the Pt surface, catalytic activity is intimately tied to partial passivation of the metal surface at high adsorbate coverages,49 and activity is associated with only a narrow coverage window. Lastly, the reaction becomes equilibrium-limited at modest temperatures, so that both kinetics and thermodynamics control reactivity. Capturing the coupling between reaction environment, surface coverage, and mechanism is thus key to a successful description of the catalysis.
In this work we have used plane-wave DFT-GGA calculations to determine relevant adsorption and activation energies as an explicit function of coverage of chemisorbed oxygen, the most abundant surface species under most catalytic conditions. Both adsorption and activation energies for key intermediates, including O, O2, NO, and NO2, were found to exhibit strong coverage sensitivities, but NO and NO2 were more adaptable to interactions with nearby adsorbates and thus less coverage-sensitive than the oxygen species. The barrier to O2 dissociation increased rapidly with coverage. The barrier to an “assisted” O2 dissociation pathway, in which NO and O2 interact directly at the catalyst surface, was much less coverage sensitive but was limited by the weak binding and low coverage of molecular pairs. In contrast to O2 dissociation, the ONO bond-forming reaction [Equation (4)] could adapt to differing surface coverages and its barrier was predicted to have a modest coverage sensitivity.
We have used these results to parameterize a coverage- dependent mean-field NO oxidation model. Observed linear correlations between coverage, adsorption, and activation energies are used to relate all the key reaction parameters to a single coverage variable. To capture the equilibrium limitations on the reaction, reaction rates are written reversibly using DeDonder relations, and the overall reaction energy is shifted to match experiment. Steady-state solutions for the coverages, reversibilities, and overall rate are obtained as a function of reaction conditions. Solutions of the model under relevant conditions show:
• Activity is associated with adsorbate coverages between 0.25 and 0.4 ML.
• Dissociative O2 adsorption from the gas-phase is rate controlling, yields a positive O2 order, and controls the apparent activation energy.
• Surface oxygen coverage is controlled by the equilibrium between gas-phase NO2 and NO and surface O* and thus by the NO2/NO ratio. Net NO oxidation is negative order in NO2 and, when NO coverage is low, positive over in NO.
• Pt lights off for NO oxidation at the same temperatures as the equilibrium switches from products to reactants.
This mean-field model goes beyond previous work in several key respects, including the explicit treatment of the nonlinear effects of coverage on the relative surface reaction rates, an unbiased solution for rate-determining steps, and incorporation of a proper treatment of the gas phase oxygen chemical potential. The model approach has some limitations as well: it cannot be used to compute exact rate orders, it exaggerates the apparent activation energy, and it of course assumes that microscopic reaction parameters are simple monotonic functions of coverage. To go beyond the assumption of a homogeneous reacting surface requires kinetic models that explicitly incorporate surface configurational effects, as in lattice-based kinetic Monte Carlo. The results presented herein provide key information to parameterize such models.
DFT calculations were performed with the periodic supercell planewave basis approach implemented in the Vasp code.50 The interaction between frozen core and valence electrons is described using projector augmented waves (PAW),51, 52 and electron-electron exchange and correlation are treated using the PW91 implementation of the generalized gradient approximation (GGA).53, 54
The close-packed Pt(111) surface was modeled using four layer 4 Pt×4 Pt supercells separated from vertical images by 14 Å vacuum space. A 6×6×1 Monkhorst–Pack55 k-point mesh was used to sample the first Brillouin zone. Pt atoms in the bottommost layer were held fixed in the calculated bulk positions and the remaining atoms including any adsorbates were allowed to relax. Structures were relaxed until the forces on all atoms were less than 0.03 eV Å−1. Spin polarization is included as appropriate for the gas-phase molecules and is negligible for adsorbates. Other details about DFT and energy calculations are given in Refs. 13, 19, 27, 36.
Transition states (TS) were located with a combination of the Climbing Image Nudged Elastic Band (CI-NEB)56–58 and dimer59 methods. Images were relaxed until forces perpendicular to the reaction path fell below 0.05 eV Å−1. The maximum root mean squared separation between images was kept to less than 1 Å. The vibrational spectra of all TSs were computed to ensure the presence of only one imaginary mode in the correct degree of freedom.
Coverage-dependent surface reaction rate constants were described using transition state theory (TST)43, 60((27))
where E are modeled coverage-dependent energies and ≠ indicates the transition state complex. The change in vibrational entropy from reactant to transition state is small in the reactions studied here and thus the partition function ratio is always close to unity. Taking the ratio to be one has negligible effect on the model results. Rate constants for molecule/surface reactions were parameterized using collision theory43, 60, 61 and are written as the product of the molecular flux impinging on the surface and a sticking coefficient:((28))
The molecular flux is calculated from the translational partition function of the impinging gas,43, 60, 61((29))
where [L] is the concentration of surface sites, and F has units of s−1 site−1. For unactivated adsorption the sticking coefficient is taken to be unity and for activated adsorption is described using an Arrhenius model:((30))
The activation energy, ΔE≠, of adsorption is equal to the energy of the adsorption transition state relative to the energy of the gas phase.60
We thank Dr. Abhijit Phatak for his helpful assistance and Dr. Fabio Ribeiro, Dr. Nick Delgass, and Dr. Andy Smeltz for helpful conversations and sharing of experimental results. Portions of this work were performed using the Northwest Indiana Computational Grid at the University of Notre Dame, the MSCF in EMSL at Pacific Northwest National Lab, and the Center for Research Computing at the University of Notre Dame. This work was supported by the University of Notre Dame and the Department of Energy, Basic Energy Sciences, under the Grant DE-FG02-06ER15830-001.