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Adsorbate Entropies with Complete Potential Energy Sampling in Microkinetic Modeling Mikkel Jørgensen* and Henrik Grönbeck* Department of Physics and Competence Centre for Catalysis, Chalmers University of Technology, 412 96 Göteborg, Sweden S Supporting Information *

ABSTRACT: The influence of different approximations on adsorbate entropies is investigated for density functional theory based mean-field kinetic modeling. Using CO oxidation over Pt(111) as a prototypical reaction, we compare four approximations: the harmonic approximation, the hindered translator, the free translator, and complete potential energy sampling (CPES). The CPES method results in particularly good agreement with previously measured experimental data. Given its general applicability and moderate computational cost, the CPES method stands out as a preferable option to describe adsorbate entropies.

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INTRODUCTION Heterogeneous catalysis is an enabling technology, which is crucial for modern society. About 90% of the chemical industry is based on catalytic processes,1 and catalysis is key in emission control and sustainable energy systems. This has motivated the large efforts to enhance catalytic performance, as slight improvements can have major impacts on a global scale. One route for catalyst development is using the fundamental understanding of elementary processes on the atomic scale. During the past decades, kinetic modeling with parameters from density functional theory (DFT)2,3 has emerged as a viable tool to obtain a detailed understanding of governing reaction mechanisms and kinetic behavior.4−6 The success of DFT-based kinetic models depends on an accurate potential energy landscape, which is currently limited by the accuracy of the exchange-correlation functionals. However, the development of semilocal functionals for reactions on metals7,8 and schemes to reduce the selfinteraction error for oxides9−11 has enabled calculations of adsorption and reaction energies with reasonable accuracy. Given the energy landscape, the kinetics are often crucially dependent on sticking coefficients, adsorbate−adsorbate interactions, and entropy changes along the reaction path. When considering the entropy of the adsorbates, the extreme limits are to model the adsorbates either as immobile or as a two-dimensional gas. The entropies of immobile species are © XXXX American Chemical Society

generally described within the harmonic approximation (HA), where adsorbates are subject to a parabolic potential and translations are modeled as frustrated vibrations. In the limit of a two-dimensional gas, the entropy is instead modeled as a free translator (FT). In general, a proper description of the adsorbate entropies could require different models depending on the diffusion barriers. This was recently emphasized in ref 12 where entropies were estimated with the so-called hindered translator (HT) and hindered rotor models.13 The hindered translator description was found to improve the entropy estimate considerably for closed-shell molecules, such as methane and propane.12 Although the Hindered translator models were shown to match experimental adsorbate entropies better than the harmonic approximation,12 the implications of going beyond the harmonic approximation in microkinetic modeling has yet to be investigated. The question is to what extent alternative methods to describe adsorbate entropies affect reaction kinetics. In this paper, we explore the influence of adsorbate entropies on mean-field microkinetic models that are derived from DFT calculations. Four approximations are compared: the harmonic approximation, the hindered translator, the free translator, and Received: November 15, 2016 Revised: February 24, 2017 Published: March 13, 2017 A

DOI: 10.1021/acs.jpcc.6b11487 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

as for the geometry optimizations. Initial interpolations between the NEB images are done with the image dependent pair potential method.35 Vibrational analysis of the transition state for CO oxidation is used to verify that the structure represents a genuine saddle point on the potential energy surface. The activation energy is evaluated as the energy difference between the TS and the adsorbates in separate cells. Mircokinetic Model. The model is based on the mean-field assumption, which is valid for a homogeneous surface with complete mixing of the adsorbates. This is a suitable approximation for CO oxidation over Pt(111) as the adsorbate−adsorbate interactions are repulsive. The microkinetic model is constructed from the simple scheme in eq 2, which has been used extensively in the past.15,36,37

complete potential energy sampling (CPES). As a model reaction, we use CO oxidation over Pt(111), which has been extensively investigated, experimentally14−21 and theoretically.15,21−26 The CO oxidation reaction over Pt(111) involves one reactant that is mobile (CO) and another reactant that is immobile (O). Thus, this reaction provides a way to assess the effect of the entropy approximations for different types of adsorbates. Furthermore, the CO oxidation reaction over platinum is technologically important and exhibits complex kinetic behavior such as bistability.15,27 We find particularly good agreement with experiments for the light-off temperature and the kinetic bistability region when the CPES method is applied. Given the general applicability and moderate computational cost, the CPES formalism stands out as a preferable, accurate, and unbiased option for describing adsorbate entropies.

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COMPUTATIONAL METHOD Density Functional Theory Calculations. DFT is used to calculate the energy landscape using the plane-wave Vienna abinito simulation package (VASP).28−30 The valence−core electronic interactions are described by the projectoraugmented wave (PAW) scheme.31 The numbers of electrons treated explicitly are Pt (10), O (6), and C (4). The calculations are performed within the generalized gradient approximation (GGA) using the RPBE7 exchange-correlation functional. Oxygen chemisorption in a (2 × 2) supercell is used to test convergence with respect to plane-wave cutoff, k-point density, and vacuum layer width. The oxygen adsorption energy is converged within 0.05 eV with a plane-wave cutoff of 450 eV, a (6 × 6 × 1) k-point grid, and a 12 Å vacuum layer separating periodic images of the slab. The surface slabs are represented by four atomic layers, which is sufficient for converged surface energies. The surfaces are constructed from the theoretical lattice constant, which is found to be 4.00 Å using a (12 × 12 × 12) k-point grid for the primitive unit cell. The construction of surface models, structural optimizations, and vibrational calculations are performed using the atomistic simulation environment (ASE).32 Structures are considered to be optimized when all forces are smaller than 0.05 eV/Å. In the calculations, the two bottom layers are kept fixed to emulate a bulk surface. The energies of gas-phase molecules are calculated in a cubic (30 Å × 30 Å × 30 Å) cell in appropriate spin states (triplet for O2 and singlet for CO and CO2). Adsorbed species are treated as closed shell systems. The adsorption energy (EX) of an adsorbate X is calculated by E X = Eslab + E X (g ) − E X/slab

CO(g) + ∗ ↔

CO*

O2 (g) + 2∗ ↔

2O*

CO* + O* → CO2 (g) + 2∗

(2)

The catalytic activity in the mean-field picture is obtained by solving a set of coupled differential equations for the surface coverages {θi}. dθCO = k1f pCO θ − k1bθCO − k 3f θCOθO * dt dθO = 2k 2f pO θ 2 − 2k 2bθO2 − k 3f θCOθO 2 * dt θ 1 − θO − θCO = *

(3)

Here kf and kb are the rate constants for the forward and backward rate of reaction, respectively, pCO and pO2 are the pressures, and θ∗ denotes the fraction of empty sites on the surface. In this model, a site is connected to one Pt atom and describes different types of binding geometries, i.e., on-top, fcc, hcp, and bridge. The differential equations are integrated numerically until steady state is reached using SciPy38 with the VODE solver for stiff problems. The rate constant k for the CO + O reaction is calculated by use of harmonic transition state theory39 (TST): k=

⎛ E ⎞ Zts kBT exp⎜ − a ⎟ Z ini h ⎝ kBT ⎠

(4)

where Ea is the activation energy, Zts is the partition function of the transition state, and Zini is the partition function of the initial state. The ratio between the partition functions is dependent on the approximation to the adsorbate entropy. The rate constants of the adsorption reactions are calculated according to ref 1(page 120):

(1)

where Eslab, EX(g), and EX/slab are the electronic energies for the bare slab, the gas phase adsorbate, and the slab with the adsorbate. A positive energy indicates exoergic adsorption. The vibrational modes are calculated using ASE at the optimized geometries using the harmonic approximation and finite differences (two point difference with steps of 0. 01 Å). For adsorbates, all surface atoms are fixed during the vibrational analysis. The vibrational energies are used to correct adsorbate and transition state (TS) energies for zero-point motion. The activation energy for CO2 formation from adsorbed CO and O is obtained with the climbing image nudged elastic band (NEB)33 method using the VTST tools34 with seven intermediate images and the same force convergence criterion

kads =

ps0A site 2πmkBT

(5)

where p is the pressure, s0 is the sticking coefficient, and Asite is the area of one site. The rate constants for adsorption are calculated using experimental sticking coefficients, i.e., 0.9 for CO and 0.1 for O 2.15,40,41 The experimental sticking coefficients account for the fact that CO and O2 adsorb via molecular precursors.42 The rate constants for desorption are calculated from the adsorption rate constants and the equilibrium constant K: B


Article

The Journal of Physical Chemistry C kdes =

kads , K

are determined by visual inspection. The hindered translator method is derived considering an adsorbate in a sinusoidal potential with a period b reflecting the distance between sites on the surface, which is assumed to have a square symmetry.12,13,44 The translational entropy in the two equivalent directions is given by12

K = exp( −ΔH /kBT ) exp[(Ssurf − Sgas)/kB] (6)

where ΔH is the zero-point corrected enthalpy of adsorption, Ssurf is the entropy of the adsorbate, and Sgas is the entropy of the molecule in the gas phase. The adsorption energies in the microkinetic model are given repulsive coverage dependencies. The CO−CO and O−O interactions are obtained by fitting the differential adsorption energies to E0 − a exp(bθi). The parameters for CO are E0 = 1.438 eV, a = 0.048 eV, and b = 3.725. The corresponding values for O are E0 = 1.197 eV, a = 0.181 eV, and b = 2.989. The CO−O interactions are considered to have a linear dependence where the CO adsorption energy is reduced as a function of the oxygen coverage with a slope of βCO−O = −1.15 eV, and the O adsorption energy is reduced as a function of CO coverage with a slope of βO−CO = −2.29 eV. The coverage dependences are described further in the Supporting Information. Adsorbate Entropy Calculations. Calculation of the reaction rates from first-principles requires estimates of changes in entropies along the reaction path. Here we compare four different schemes for calculating the entropy: the harmonic approximation (HA), the hindered translator (HT), the free translator (FT), and complete potential energy sampling (CPES). The models assume a canonical (T,V,N) ensemble where the volume is obtained from a constant pressure via the ideal-gas equation of state. The entropy is in the canonical ensemble given by the partition function (Z) according to43 S=−

)I ( ) exp⎛⎜ 2 ⎞⎟ ⎡ ⎤ ⎝ (2 + 16r )T ⎠ ⎢⎣1 − exp(− )⎥⎦

πr

Z THT =

x

Z TFT = A

i=1

e−ℏωi /2kBT 1 − e−ℏωi / kBT

2

x

x

2πmkBT h2

(11)

In this approximation, the remaining degrees of freedom are treated as vibrations. In the complete potential energy sampling (CPES) approach, the translational entropy is estimated by explicit calculations of the potential energy experienced by each adsorbate. The semiclassical canonical partition function45 is given by

(8)

Z TCPES =

The four investigated methods are different only in the way the translational entropy is treated. In the harmonic approximation the adsorbate is treated as a quantum harmonic oscillator that is confined in a harmonic potential having vibrational modes with energies ℏωi. In this approximation, rotations and translations are treated as frustrated vibrations. The partition function for an adsorbate with N atoms is given by43 3N

rx 2Tx

Tx = kBT/hνx is the ratio between the thermal energy and the vibrational energy and rx = Wx/hνx is the ratio between the diffusion barrier and the vibrational energy, where νx = (Wx/ 2mb2)1/2 is the vibrational frequency of the adsorbate. Nsites is the total number of surface sites, and I0 is the zero-order modified Bessel function of the first kind, arising from the periodicity of the potential. The factor Nsites grows with the surface area, as one expects from a two-dimensional gas. The last exponential factor in eq 10 was introduced in ref 12 to account for zero-point energy corrections, which may be important at low temperatures. The hindered translator method is designed to coincide with the harmonic approximation result when kBT is lower than the diffusion barrier. At conditions where kBT is well above the diffusion barrier, the entropy approaches the result of a 2-dimensional gas. With the free translator partition function, the translational entropy is calculated by treating the adsorbates as a twodimensional gas. The translational partition function for a gas of one molecule that is confined to an area A is given by43

where kB is Boltzmann’s constant. The total partition function of an adsorbate consists of the partition function for translations (ZT), rotations (ZR), and vibrations (Zvib). Assuming that the degrees of freedom are independent, the total partition function is

∏

2

0

(10)

(7)

Z=

rx + 1 Tx

1 Tx

1 ⎛ ∂Z ⎞ ∂ ( −kBT ln Z)V , N = kB ln Z + kBT ⎜ ⎟ Z ⎝ ∂T ⎠V , N ∂T

Z = Z TZ R Zvib

(

Nsites T x exp −

2πmkBT 2

h

⎛ − V (x , y ) ⎞ ⎟ dx dy kBT ⎠

∬ exp⎜⎝

(12)

where the integration is performed numerically over a surface cell, and V(x, y) is the potential energy as a function of position (x, y) as obtained by first-principles calculations. The potential energy V(x, y) is obtained in practice by interpolation between a set of calculated points in the surface cell. As for the hindered translator and free translator methods, the remaining degrees of freedom are treated as vibrations. The partition function of a gas-phase molecule is calculated as a three-dimensional free translator confined to a volume given by the pressure through the ideal gas equation of state pV = kBT. The rotational partition function of a linear molecule is given by43

(9)

The partition function of the transition state in the present model is treated within the harmonic approximation irrespective of how the partition functions of the adsorbate are modeled. In the hindered translator approximation,12 vibrational entropy is calculated as in the harmonic approximation, whereas translational entropy is calculated separately using the diffusion barriers (Wx) of the adsorbates. The two modes in the vibrational analysis that should be assigned to translations

Zrot =

8π 2IkBT σh2

(13)

where I is the moment of inertia of the molecule and σ is the symmetry number. The remaining degrees of freedom are vibrational and are treated within the harmonic approximation. C


Article

The Journal of Physical Chemistry C The four methods to treat the translational entropy are related to different approximations to the potential energy surface of the adsorbates. In the harmonic approximation, the potential is assumed to be parabolic, in the hindered translator model it is sinusoidal, in the free translator approximation it is constant, and in CPES it is the potential obtained by firstprinciples calculations. The hindered translator improves on the harmonic approximation in the case of very flat energy landscapes. It should be stressed that hindered translator assumes two independent directions of translation, which is equivalent to assume that ZT(x,y) = ZT(x)ZT(y). This approximation is not valid unless the energy landscape is very flat. For a surface with a fcc (111) symmetry, the diffusion directions are not independent, and the approximation should merely be viewed as a way to improve on the harmonic approximation, which assumes that adsorbates are pinned to one site. In the CPES method, the detailed energy landscape is taken into account, and no assumptions are made on preferred directions of diffusion. We note that the CPES expression for the entropy will approach that of a free translator for V(x, y)/ kBT → 0. Thus, the free translator provides an upper bound to the translational part of the entropy. The computational cost of CPES is similar to the hindered translator as it does not involve calculations of diffusion barriers but instead requires a set of local relaxations.

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RESULTS AND DISCUSSION Potential Energy Surface and Adsorbate Entropies. The potential energy surfaces for CO and O on Pt(111) are reported in Figure 1 together with a surface model. Adsorbate positions not representing local minima are sampled by structural relaxations only in the direction perpendicular to the surface. At 0.25 coverage, CO is preferably bonded in a bridge position and has an adsorption energy of 1.42 eV. However, the potential energy surface is flat, and the alternative adsorption sites (hcp, fcc, and on-top) are within 0.06 eV. By adding the zero-point energy corrections, the preferred site changes to the fcc position and the adsorption energy decreases to 1.36 eV. The vibrational energies for CO in the fcc position are [17.8, 18.3, 36.3, 36.3, 39.7, 216.8] meV. The modes were characterized by visual inspection.46 The fcc, hcp, and bridge sites are virtually isoenergetic while the on-top site is separated from the other sites by only 0.07 eV. We find a barrier of 0.08 eV for CO diffusion from the fcc site to the bridge site, evaluated in a (4 × 4) cell. It is well-known47,48 that gradient corrected functionals, such as RPBE, do not reproduce the experimentally observed adsorption site, which is the on-top site49 at low coverage. However, the site preference is sensitive to the coverage, and a mixture of sites are occupied at higher coverages.48 The lateral interactions used in the microkinetic model take this change in site preference into account (see Supporting Information). The potential energy surface of O is markedly different from that of CO. O prefers binding to the fcc site with an adsorption energy of 0.97 eV with respect to molecular oxygen in the gas phase. The fcc site is preferred over the hcp and bridge site by 0.40 and 0.65 eV, respectively. The on-top site is clearly separated from hollow and bridge sites with an endothermic adsorption energy with respect to molecular oxygen. Inclusion of zero-point corrections does not change the energy preference, and the adsorption energy at the fcc site is 0.95 eV after corrections. The vibrational energies for O in the fcc

Figure 1. Potential energy landscape for CO (top) and O (middle) adsorbed on Pt(111). The energies are given with respect to the energy minima for CO and O, respectively. The small black dots are points in between sites for which the local relaxations were performed. The Pt(111) surface cell is shown at the bottom. The symbols indicate surface sites according to ○ = on-top, △ = bridge, ⬡ = fcc, and ★ = hcp.

position are [47.3, 47.6, 55.1] meV.50 The barrier for diffusion between fcc and hcp sites is 0.58 eV including zero point corrections, which is in agreement with ref 51. The entropy of adsorbed CO and O in the four approximations is shown as a function of temperature in Figure 2. The entropy is consistently lowest for the harmonic approximation, whereas the free translator yields the highest entropy. The high entropy predicted by the free translator description is expected as it assumes a completely flat energy landscape and serves as an upper bound of adsorbate entropies. Given the potential energy surfaces in Figure 1, it can be assumed that CO is fairly mobile, whereas O is localized on the hollow adsorption sites. Therefore, the free translator description could be appropriate for CO but not for O. The potential energy surfaces also indicate that the harmonic approximation could be reasonable for O, whereas the entropy for CO is underestimated in this approximation for temperatures at reaction conditions. The hindered translator yields entropies between the harmonic approximation and the free translator. However, for both adsorbates it is well above the D


Article


Figure 3. Reaction free energy landscape for CO oxidation on Pt(111). Zero energy corresponds to CO(g) + O2(g). The landscape is reported without zero point corrections.

respectively, where the adsorption energy for O is calculated with respect to O2(g). The reaction is exothermic with an activation energy of 1.06 eV. Adding the entropy contributions at 450 K, it is clear that the different approximations result in a sizable spread of the free energy. The configuration with coadsorbed CO and two O has a spread of 0.51 eV. The effect of the entropy approximation on the preexponential factors is shown in Table 1. The pre-exponential factor for adsorption of CO is one order of magnitude higher than for O. This is mainly due to the difference in sticking coefficients of CO and O 2 adsorption. The different approximations to the entropy affect the pre-exponential factors for desorption and oxidation. The effect is substantial with, for example, a difference in four orders of magnitude between the free translator and the harmonic approximation for the oxidation step. It is interesting to compare our calculated kinetic parameters to experimentally measured values, which are determined using temperature-programmed desorption (TPD).40,52 The heat of adsorption was measued to be 1.44 eV for CO40 and 2.21 eV for oxygen52 at low coverages. Our calculated adsorption energy of CO is in very good agreement with the experiments, but the value for adsorption of oxygen is slightly lower. Our calculated pre-exponential factors for CO desorption (A1b in Table 1) are in good agreement with the experimental value40 of 4 × 1015 s−1. The pre-exponential factor for oxygen desorption was measured experimentally to be 2.4 × 10−2 s−1 cm−2.40 The unit of this pre-exponential factor stems from the second-order nature of oxygen desorption. If we assume a site density of 2.9 × 1015 sites cm−2, the experimental preexponential factor is equal to 6.9 × 1013 s−1, which is close to the calculated values. It should be noted that uncertainties exist when comparing to experiments where for example inclusion of adsorbate−adsorbate interactions and precursor states in the analysis would increase the pre-exponential factors for desorption. The pre-exponential factors derived in refs 40 and 52 neglect coverage dependencies, which is a limitation as the pre-exponential factor for CO desorption has been measured to vary between 1014.3 and 107.5 s−1 for low and high coverage, respectively.53 Turnover Frequencies and Surface Coverages. We calculate the turnover frequencies (TOFs) in order to assess the influence of the four approximations on the adsorbate entropy (see Figure 4). To make the differences between the methods clear, the results are shown on a logarithmic scale. For all temperatures, the calculations are initiated with a CO

Figure 2. Entropies of CO (top) and O (bottom) on Pt(111) in the four approximations: harmonic approximation (HA), hindered translator (HT), free translator (FT), and complete potential energy sampling (CPES).

harmonic approximation at all relevant temperatures. The adsorbate entropy approaches the free translator at a lower temperature for CO than for O, which is an effect of the low CO diffusion barrier. The CPES method predicts entropies between the harmonic approximation and the hindered translator which is a consequence of taking the full potential energy landscape into account. For CO, the CPES approaches the free translator slower than the hindered translator. The faster approach for the hindered translator is a consequence of assuming only the lowest energy pathway for diffusion within this method. For O, the CPES method predicts an entropy closer to the harmonic approximation than the hindered translator. This is reasonable given the corrugated potential energy surface for oxygen. The comparison between the four methods shows that the choice of approximation is important. Among different schemes, the CPES method is the one that captures the potential energy surface in the most accurate way and presumably also gives the best description of the entropy. Comparing the CO entropies, the harmonic approximation underestimates the entropy throughout the temperature range by about 14%. The overestimation of the free translator is highest at low temperatures and only 4% at 1000 K. The trend is reversed for oxygen where the harmonic approximation is underestimating the entropy by 11% at 1000 K, whereas the free translator overestimates the entropy by 59%. The entropies evaluated with the CPES and the hindered translator approach the free translator values in the limit of high temperatures. For the CPES, the entropies are within 0.05 meV/K at temperatures above 970 and 2380 K for CO and O, respectively. The corresponding temperatures are 650 and 2760 K for the hindered translator. The Reaction Energy Landscape. The energy landscape of the reaction is shown in Figure 3. CO and O are strongly bound to the surface with adsorption energies 1.42 and 0.97 eV, E


Article

The Journal of Physical Chemistry C Table 1. Calculated Kinetic Parametersa reaction

CO adsorption A1f [Pa−1 s−1]

CO desorption A1b [s−1]

O2 adsorption A2f [Pa−1 s−1]

O2 desorption A2b [s−1]

HA HT FT CPES

× × × ×

× × × ×

× × × ×

× × × ×

1.6 1.6 1.6 1.6

3

10 103 103 103

3.4 2.3 1.4 4.2

15

10 1014 1014 1014

1.6 1.6 1.6 1.6

2

10 102 102 102

8.5 3.5 1.1 1.3

15

10 1014 1012 1015

CO oxidation A3f [s−1] 5.1 6.3 2.2 3.3

× × × ×

1012 1010 108 1010

a

Pre-exponential factors (A) at 400 K, evaluated at zero coverage. Harmonic approximation (HA), hindered translator (HT), free translator (FT), and complete potential energy sampling (CPES).

translator method is consistently overestimating the light-off temperature. The closest agreement between simulations and experiments is obtained with the hindered translator and CPES methods. The main reason for the low light-off temperature with the harmonic approximation is the underestimation of the CO entropy. Similarly, the high light-off temperature in the free translator model is caused by the overestimation of the CO entropy. In this case, the light-off temperature is overestimated by more than 100 K. There is a sudden drop in the TOF after the light-off, which is most pronounced in the free translator model. This drop is caused by oxygen blocking CO adsorption sites. The declining TOFs after the light-off signals that CO adsorption limits the rate. The temperature-dependent adsorbate coverages are shown in Figure 5. CO poisons the Figure 4. CO oxidation turnover frequencies on Pt(111) for the four approximations. The pressures are pCO = 6.6 ×10−6 mbar and pO2 = 1.3 ×10−5 mbar as in ref 15. Harmonic approximation (HA), hindered translator (HT), free translator (FT), and complete potential energy sampling (CPES).

covered surface. The TOF has a light-off when CO desorbs, giving free sites for oxygen to adsorb. Here the light-off temperature is defined as the temperature where the TOF reaches 50% of its maximum. At temperatures above the lightoff, the reaction proceeds in a regime where CO adsorption limits the rate.14−16,54 The different treatments of the entropy have clear effects on the TOF. The harmonic approximation gives the largest TOF at all temperatures, whereas the free translator predicts the lowest TOF. The light-off temperature is a characteristic measure of the CO oxidation reaction. The light-off temperatures are compared to experimental results on metallic Pt(111) in Table 2 for a set of different reaction conditions. In general, higher pressures result in higher light-off temperature. Describing the adsorbate entropies within the harmonic approximation results in a lower light-off temperature than the other approximations, and it is also underestimating the light-off temperature as compared to the experiments. The free Figure 5. Coverage of CO (top) and O (bottom) on Pt(111) as a function of temperature for the four approximations. The pressures are pCO = 6.6 ×10−6 mbar and pO2 = 1.3 ×10−5 mbar. Harmonic approximation (HA), hindered translator (HT), free translator (FT), and complete potential energy sampling (CPES).

Table 2. Experimental and Theoretical Light-Off Temperaturesa THA c

pressure

THT c

TFT c

TCPES c

[mbar]

pCO/pO2

[K]

[K]

[K]

[K]

10−7 10−6 10−2 10−1 100

0.22 0.5 1 2 2

417 449 570 634 708

454 487 630 707 794

543 599 754 808 895

449 481 626 703 789

TExp c [K] 480 485 675 700 765

(ref (ref (ref (ref (ref

16) 15) 17) 56) 56)

surface at low temperatures, whereas the coverage is low when the surface is active. An appreciable oxygen coverage is reached only after the light-off. The coverages for both CO and O are ordered according to the entropy. The lowest coverage is predicted for the harmonic approximation and the highest for the free translator method. The smallest differences are present at low temperatures as this is where the entropic part of the free energy is least significant. In the limit of high temperatures, the

a

Light-off temperatures for the different entropy approximations. Harmonic approximation (HA), hindered translator (HT), free translator (FT), and complete potential energy sampling (CPES). The simulations are compared to experimental reports (Exp). F



Article

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CONCLUSIONS We have investigated the effect of different approximations to adsorbate entropy within density functional theory based microkinetic modeling using CO oxidation over Pt(111) as a model reaction. Four methods were studied: the harmonic approximation, the hindered translator, the free translator, and complete potential energy sampling (CPES). To our knowledge, this is the first time the CPES method has been used to estimate adsorbate entropies within DFT based modeling. The choice of entropy approximation was shown to significantly affect turnover frequencies, coverages, light-off temperatures, and bistability regions. It was found that CPES overall compares favorably with experiments. The commonly used harmonic approximation was found to predict rather low entropies for both CO and O, which results in a too low lightoff temperature. Treating the adsorbates as free translators yields too high entropies and a poor agreement with experiments. The underestimation of adsorbate entropy by the harmonic approximation was recently discussed in ref 12, and the hindered translator was suggested as a possible improvement. We find that the hindered translator performs well for the CO oxidation reaction. The performance of the CPES method is for the current reaction similar to the hindered translator. However, the CPES method is more general as it takes into account the detailed energy landscape. In addition, the computational cost for the two methods is similar. Energy barriers have to be evaluated for the hindered translator, whereas local structural relaxations need to be performed for the CPES method. To reduce the computational cost, it is possible to apply the CPES method only to abundant species and treat other adsorbates in the harmonic approximation. Given the development of exchange-correlation functionals and increasingly accurate potential energy surfaces, it has become important to improve the description of the entropy terms. The CPES method appears to be an attractive way to improve the accuracy of first-principles-based microkinetic models.

oxygen coverages are similar in the harmonic approximation and the CPES method, which is consistent with the temperature dependence of the entropy in Figure 2. The CO coverage can be compared to the low-energy electron diffraction experiments of ref 55. With CO pressures of about 10−10 mbar, the saturation coverage was reported to be 0.47−0.49 at 300 K. At these conditions, the coverage is calculated to be 0.33, 0.40, 0.41, and 0.43 in the harmonic approximation, CPES method, hindered translator, and free translator, respectively. Note that the obtained saturation coverages are sensitive to the temperature. Lowering the temperature by 10 K yields an increase in CO coverage of ca. 5%. Bistability Region. CO oxidation over Pt(111) has a bistability region where the reaction can proceed with two different rates depending on the history of the reaction. This gives rise to hysteresis in the TOF during temperature ramps. The bistability arises from CO poisoning at low temperatures. Here we investigate how the different descriptions of the adsorbate entropies affect this kinetic phenomenon. The bistability was studied by performing simulations at a constant oxygen pressure of 1.3 × 10−5 mbar and varying the CO pressure. These conditions were chosen to be identical to the experiments in ref 15. The results are presented in Figure 6

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ASSOCIATED CONTENT

S Supporting Information *

Figure 6. Kinetic phase diagram for the bistability region. The reaction is bistable in the area between the lines. The oxygen pressure is 1.3 × 10−5 mbar. The shaded region is the experimental observations of Vogel et al.15 Harmonic approximation (HA), hindered translator (HT), free translator (FT), and complete potential energy sampling (CPES).

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b11487. Details on coverage dependence of adsorbate energies (PDF) Adsorbate and transition state structures (.xyz files) (ZIP)

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together with the experimental data.15 The critical CO pressure where the bistability window closes at high temperatures is sensitive to the description of the adsorbate entropies. The highest pressure is calculated for the free translator model, which severely overestimates the pressure as compared to the experiments. However, a fair agreement is obtained using the other approximations. The hindered translator and CPES models are within 77 K from the experimental critical temperature, while the harmonic approximation underestimates this temperature by 118 K. From the comparison with the experiments, we conclude that the hindered translator and CPES models show good agreements with the experimental measurements in both pressure and temperature. The deviations that exist could be caused by not accounting for coverage and temperature dependence of the sticking coefficients.

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (M.J.). *E-mail: [email protected] (H.G.). ORCID

Mikkel Jørgensen: 0000-0002-5083-528X Henrik Grönbeck: 0000-0002-8709-2889 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Vladimir Zhdanov for valuable comments. Financial support is acknowledged from the Swedish Research Council and Chalmers Areas of Advance Nanoscience and NanoG


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technology and Transport. The calculations were performed at PDC (Stockholm) via a SNIC grant. The Competence Centre for Catalysis (KCK) is hosted by Chalmers University of Technology and is financially supported by the Swedish Energy Agency and the member companies AB Volvo, ECAPS AB, Haldor Topsøe A/S, Scania CV AB, Volvo Car Corporation AB, and Wärtsilä Finland Oy.

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