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Ab Initio Adsorption Thermodynamics of H2S and H2 on Ni(111): The Importance of Thermal Corrections and Multiple Reaction Equilibria Dayadeep S. Monder* and Kunal Karan Queen’s-RMC Fuel Cell Research Centre, Kingston, ON, Canada K7L 5L9, and Department of Chemical Engineering, Queen’s UniVersity, Kingston, ON, Canada K7L 3N6 ReceiVed: August 13, 2010; ReVised Manuscript ReceiVed: October 19, 2010
The presence of trace amounts of H2S in H2-rich fuel poisons Ni-based solid oxide fuel cell anodes, adversely affecting the electrochemical performance. This study uses density functional theory (DFT) to describe the competitive adsorption thermodynamics of H2S and H2 on Ni(111). Unlike previous DFT-based studies on the H2S-H2-Ni system, a vibrational analysis of the adsorbates is performed to calculate the thermal corrections to the enthalpy and entropy of the surface species. Parallel adsorption reactions of H2 on Ni explicitly accounting for coverage effects of the S and H adatoms on the Ni(111) surface are included in the analysis. The resulting equilibrium equations for the multiple adsorption/desorption reactions are then solved to calculate the S and H coverage over a wide range of T, PH2S, and PH2.This study illustrates the errors introduced in the predicted S coverage if H2 adsorption in parallel with H2S adsorption is neglected or if the thermal corrections to the enthalpy and entropy of reaction are not handled properly. H2 + 2 / a 2H/
Introduction The interaction of hydrogen sulfide with a nickel surface is of significant importance in industrial catalysis1 and in solid oxide fuel cell (SOFC) anode electrocatalysis.2 H2S at ppm levels poisons the Ni surface due to sulfur adsorption. At higher concentrations, bulk nickel sulfides are formed.3 Relevant work in the literature includes experimental studies of sulfur chemisorption on Ni surfaces exposed to H2-H2S,1,4,5 density functional theory (DFT) based theoretical studies of H2S-Ni interactions,6-8 and experimental observations of increased polarization losses in Ni-based SOFC anodes upon introduction of H2S in H2 -rich fuel.9-12 Currently available modelssboth empirical and theoreticals make various approximations while describing sulfur poisoning of Ni exposed to H2 - H2S mixtures. Proposed sulfur adsorption isotherms4,5 assume that reaction 1 adequately describes H2-H2S-Ni interactions such that the S coverage at any temperature is a function of only the ratio PH2S/PH2
H2S + / a H2 + S/
(1)
Hansen13 adopts the istotherm presented by Alstrup et al.5 to correlate the loss in SOFC performance upon introduction of H2S in the H2 feed. The DFT-based model of Wang and Liu6 also considers only reaction 1 in constructing a Ni-S adsorption phase diagram. Thus, these studies neglect the parallel and competitive adsorption of H2 (reaction 2). Further, the activities, and the thermal corrections for the adsorbed species are not considered. Entropy of all surface species is ignored in ref 6, while thermal corrections are made only for the gas phase species. The DFT-based model by Galea et al.7 includes H2-Ni interactions but does not calculate thermal corrections to reaction enthalpy and assumes that the translational and rotational entropy of the gas species is lost on adsorption * To whom correspondence should be addressed. E-mail: mishamonder@ gmail.com (D.S.M.);
[email protected] (K.K.).
(2)
In this work, we report an ab initio thermodynamic model to describe the competitive adsorption of H2S and H2 on the Ni(111) surface. We show that the aforementioned approximations, i.e., neglecting H2 adsorption, partial thermal corrections for surface species properties, and a complete neglect of the surface species activities, lead to substantial errors in the predicted surface coverage of sulfur. DFT calculations are used to compute the electronic energies and vibrational frequencies of the considered species. These are then used to obtain coverage-dependent reaction energetics that include the appropriate thermal corrections for the reaction enthalpies and entropies. The calculated Gibbs free energies for the reactions considered are used to build the equilibrium equations, which are solved simultaneously to obtain the surface coverage of each adsorbed species for specified operating conditions. This work illustrates how the assumptions used in the current models in the literature can lead to substantial errors in the calculated reaction free energies and, consequently, in the predicted adsorbed species surface coverages. Phatak et al.14 use a similar approach to calculate the surface coverages of the adsorbed species resulting from the adsorption of H2O and H2 on the (111) surfaces of 5 metals. However, they do not perform vibrational analyses for their systems and estimate the entropy of surface species by subtracting translational entropy from the corresponding gas species entropy. It is important to emphasize that actual Ni surfaces in industrial catalysts and SOFC anodes expose many different low and high index crystal faces. We focus on only the Ni(111) surface in this work since our objective is to better understand the underlying mechanism of sulfur poisoning at the most fundamental levels. Also, as bulk nickel sulfides are formed at PH2S/PH2 values that are much higher than those found in SOFC anodes, we do not consider bulk nickel sulfide formation in this work.
10.1021/jp1076774 2010 American Chemical Society Published on Web 12/06/2010
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Computational Methods Electronic Structure and Vibrational Mode Calculations. The electronic structure calculations are performed using GPAW,15 a real-space grid-based DFT code that uses the projector-augmented wave (PAW) method.16 The Ni (111) surface is modeled using a 2 × 2 Ni cell, periodic in the plane of the surface and three layers deep. (Choosing a unit cell for periodic DFT calculations involves a trade-off between computational expense and the accurate estimation of low coverage reaction energetics. While the results will change for a larger unit cell, our main arguments and the discussion around the thermal corrections to the energetics are not expected to change.) Exchange-correlation effects are treated using GGA and the RPBE functional.17 The gas-phase molecule calculations are done using spin-polarized calculations while spin-paired calculations are used for the surface species. (Although Ni is ferromagnetic, the Curie temperature for Ni is 354 °C, well below the temperature range we are interested in: 400-900 °C. Thus, spin-paired calculations may be more appropriate for simulating the reaction conditions for high temperature catalysis on Ni. In our calculations ∆Eads was 0.1-0.3 eV lower, adsorption was less favorable, for spin-polarized calculations.) The top layer of Ni atoms is allowed to relax, while the bottom two are fixed at the bulk Ni crystal lattice spacing. The calculated lattice constant (3.52 Å) is within 0.01 Å of the experimental value. Vacuum layers 6 Å thick are used above and below the slabs along with a 6 × 6 × 1 Monkhorst-Pack k point mesh. A grid point density of 0.2 Å (equivalent to a plane-wave energy cutoff of 470 eV) is used, and the convergence criterion for each selfconsistent field calculation is 10-6 eV/atom. While GPAW also allows localized basis set based calculations using linear combinations of atomic orbitals (LCAO) calculations,18 in this work we used the more accurate grid-based finite-difference calculations and did not use LCAO. The criterion for structure optimization is that the interatomic forces for the relaxed atoms are below 0.01 eV/Å. The zero point and thermal energy as well as entropy for the surface species are calculated from a vibrational analysis for the adsorbed atoms. Vibrational modes and frequencies for the adsorbate atoms of all surface species are calculated from a finite difference approximation of the Hessian matrix for the adsorbates using the method presented by Frederiksen.19 The energy minimization and frequency analysis is done using ASE,20 a Python based front end for several electronic structure and molecular modeling programs. The Thermodynamic Model. We consider the interactions between H2S, H2, and the Ni(111) surface using seven reactions (eqs 3a-3g). Each surface species (see Figure 1) is fully dissociated and is a combination of S and H adatoms on the 2 × 2 Ni unit cell. The surface cell is labeled C, and the number and identity of the adatoms for each species appear in the subscript. The configuration (binding sites and internuclear distances) of the adatoms for each species is the configuration with the lowest electronic energy. For example, species C2H has two H adatoms, one on a 3-fold hollow (face-centered cubic, fcc) site and the other on a 3-fold filled (hexagonal closed packed, hcp) site. This configuration is the lowest electronic energy (most stable) configuration with two H adatoms on a 2 × 2 Ni (111) surface cell. The surface is assumed to consist of a large number (∼Avogadro constant) of the above surface cells in equilibrium with the specified gas-phase conditions.
1 H + C a CH 2 2
(3a)
H2 + C a C2H
(3b)
2H2 + C a C4H
(3c)
H2S + C a C2H,S
(3d)
C2H,S a H2 + CS
(3e)
H2S + CS a C2H,2S
(3f)
C2H,2S a H2 + C2S
(3g)
At constant T and P, the condition for chemical equilibrium21 dictates that the Gibbs free energy of reaction for each of the seven reactions is zero: eq 4. ∆Gj is computed using the molar free energies Gi, eq 5, of the species involved in reaction j. R is the ideal gas constant, T is the temperature and ai is the activity of species i.
∆Gj ) 0
j ∈ (1, ..., 7)
Gi ) Gio + RT ln ai
(4) (5)
Combining equations eqs 4 and 5 and rearranging the resulting equation leads to the definition for the thermodynamic equilibrium constant Kj as given by eq 6
∏ i
( )
aiνi,j ) exp
-∆Gjo ≡ Kj RT
(6)
νi,j is the stoichiometric coefficient of i in reaction j. The activity of gas species is given by the ratio of species partial pressure to standard pressure (1 bar), whereas that of the surface species is defined by eq 7. Ci is the molar surface concentration, and θi is the surface coverage of cell type i (Figure 1 lists all the surface species considered in this work). Although surface species activity has generally been ignored
Figure 1. Surface species considered in the thermodynamic analysis. The small white spheres are H atoms, the large yellow spheres S, and the green ones Ni. All adatoms on CH, C4H, C2H,S, CS, and C2H,2S occupy 3-fold hollow or fcc sites. On C2H and C2S one adatom occupies a 3-fold hollow while the second is in a 3-fold filled or hcp site.
Adsorption Thermodynamics of H2S and H2 on Ni(111)
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in ab initio adsorption thermodynamics,22,23 the definition in eq 7 has been employed extensively in microkinetic models24-26
ai )
Ci ) θi CT
(7)
The equilibrium condition is imposed by using eq 6 for each of the seven reactions 3a-3g. For example, the equilibrium equation for reaction 3d is given by eq 9. Note that this equation, as well as the equations for the other six reactions, is linear in the unknown θi
TABLE 1: Calculated Vibrational Frequencies for H2, H2S, and Surface Speciesa species, i
vibrational frequencies, ωi,k cm-1
H2 H2 S CH C2H C4H
4362 (4401)30 1170, 2625, 2641 (1183, 2615, 2626)31 829, 838, 1081 848, 849, 861, 866, 1097, 1115 951, 951, 954, 977, 983, 1047, 1050, 1050, 1128, 1130, 1133, 1159 209, 243, 328, 775, 859, 1111, 1115, 1182, 1214 209, 211, 335 141, 163, 171, 219, 221, 296, 967, 1062, 1098, 1137, 1239, 1398 219, 220, 236, 237, 295, 327
C2H,S CS C2S,2H C2S
θ2H,S ) K3d θVPH2S θ2H,S - K3dPH2SθV ) 0
(8)
(9)
An eighth equation is given by the summation constraint on the fractional surface coverages ∑iθi ) 1. These eight linear equations are solved simultaneously to obtain the eight θi for specified PH2, PH2S, and T using standard linear algebra in MATLAB.27 The S and H coverage per surface Ni atom (θ′H and θ′S) are then computed using eqs 10a-10c
1 1 θ′H ) θH + (θ2H + θ2H,S + θ2H,2S) + θ4H 4 2
a Experimentally observed frequencies for the gas species are given in parentheses along with the original references.
TABLE 2: Reaction Energies and Temperature Corrections at 700°C for Eqs 3a-3ga -T∆Sjo reaction
∆Eel,j
gas ∆Hth,j
3a 3b 3c 3d 3e 3f 3g
-0.50 -1.06 -1.84 -1.91 0.15 0.39 -0.31
-0.28 -0.57 -1.13 -0.78 0.57 -0.78 0.57
surf ∆Hth,j
∆Hj0
0.29 0.58 1.20 0.86 -0.60 0.87 -0.62
-0.50 -1.04 -1.78 -1.84 0.12 0.49 -0.36
S
surf
)0
Ssurf(ν,T)
0.83 1.67 3.34 2.54 -1.67 2.54 -1.67
0.64 1.29 2.67 1.70 -1.34 1.62 -1.24
a
(10a) 1 1 θ′S ) (θS + θ2H,S) + (θ2S + θ2H,2S) 4 2
(10b)
θ′V ) 1 - θ′H - θ′S
(10c)
The free energy of species i, G0i is computed according to eq 11
Gi0 ) Hi0 - TSi0
(11)
Hi0 ) Eel,i + Ezpv,i + Eth,i + PVm,i
(12)
H0i is the molar enthalpy, Eel,i is the 0 K electronic energy, Ezpv,i is the zero point vibrational energy, Eth,i is the thermal (translational, rotational, vibrational, electronic) energy due to finite temperature, and S0i is the molar entropy of species i. Vm,i is the molar volume of i. In eq 12, PVm,i can be replaced by RT for gas species (assuming ideal gas). The PVm,i term for the surface species can be ignored as the molar volume of surface species is negligible. Eel,i is obtained directly from DFT, and we calculate Ezpv,i, Eth,i, S0i using standard relationships from statistical thermodynamics.28 These properties are computed by considering the translational, rotational, and vibrational modes for the gas species, and the vibrational modes/frequencies for the adatoms on the surface species. For the gas species, our calculated values for Ezpv,i + Eth,i and S0i agree within 0.002 eV (0.5%) and 0.05 J/mol/K (0.02%), respectively, at 700 °C compared to values given in standard thermodynamic tables.29 Ezpv,i + Eth,i and S0i for the surface species (see eqs 13 and 14) are calculated using the vibrational
All energies are in electronvolts. Column 1 gives reaction electronic energy at 0 K; column 2 gives the T correction to enthalpy of reaction for gas species only (includes ∆Ezpv,j); column 3 gives the T correction to reaction enthalpy for surface species only (includes ∆Ezpv,j); column 4 gives reaction enthalpy calculated by adding columns 1, 2, and 3 (include all zero point and thermal corrections); column 5 gives the entropy of reaction assuming surface species have no entropy; column 6 gives entropy of reaction with surface entropy calculated from adsorbate vibrational frequencies.
modes/frequencies for the adsorbed atoms (there are no rotational modes available to the fully dissociated adsorbates). Configurational entropy of the surface species is not considered in this work.
Ezpv,i + Eth,i ) R
∑ k
Si0 ) R
∑ k
(
(
Θk Θk + 2 exp(Θk/T) - 1
)
(13)
)
Θk/T - log[1 - exp(-Θk/T)] exp(Θk/T) - 1
(14)
Θk ) hνi,k/kB is the vibrational temperature for the kth vibrational frequency νi,k for species i, h the Plank constant, and kB the Boltzmann constant. Results and discussion Table 1 gives the calculated values for the fundamental vibrational frequencies for all the gas and surface species considered. The calculated values for the gas species are within 1% of the experimentally observed frequencies. Table 2 presents the energetics for reactions eqs 3a-3g. The ∆Eel,j values (0 K, without zero point corrections) and the ∆H0j values (all temperature corrections for both gas and surface
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Figure 2. Surface coverage of sulfur (a) and hydrogen (b) as functions of PH2S/PH2 and T. PH2 ) 1 - PH2S = 1 atm. The inset (c) shows how the hydrogen coverage is related to the sulfur coverage. The data points in (c) are obtained by plotting the normalized θ′H in (b) against θ′S in (a) for identical PH2S/PH2 and T. The H coverage θ′H is normalized by θH′0, the coverage for PH2S ) 0 at the T considered. The highlighted section in (c) corresponds to the conditions where Matsuzaki and Yasuda9 first observe an increase in polarization resistance (see text).
species) are within 0.1 eV of each other because the T corrections for the gas and surface species enthalpy roughly cancel out. However, including the T corrections for the gas species enthalpy alone can lead to errors of ∼1 eV in ∆H0j and ∆G0j . Ignoring the vibrational entropy of the surface species can lead to differences of 0.2-0.9 eV in ∆G0j . The assumptions used by Wang and Liu,6 neglecting the entropy and the thermal corrections for enthalpy of the surface species, give ∆G0 ) -1.11 eV for reaction 1 compared to a T corrected value of -1.36 eV. In this case, the incomplete thermal corrections for reaction enthalpy partially cancel out the larger error in entropy of reaction. We obtain a value of ∆H0(300 K) ) -1.01 eV/mol for H2 adsorption which compares very well with the experimentally obtained value of -0.98 eV for Ni(111).32,33 Our value of ∆H0(750 °C) ) -1.72 eV for the overall S adsorption reaction, eq 1, lies just outside the range of -1.39 to -1.65 eV reported in the experimental literature for various supported and unsupported Ni surfaces.1 The effective adsorption energy on a polycrystalline surface examined in experiments is expected to be somewhat different from that on a single facet of a perfect crystal Ni, which could explain the difference above. Parts a and b of Figure 2 present the calculated sulfur and hydrogen surface coverages as a function of PH2S/PH2 and T. The S coverage increases with increasing PH2S and saturates at a maximum S/Ni ratio of 1/4 on the Ni(111) surface. Interestingly, the relative decrease in the coverage of H is directly proportional to the increase in S coverage as seen in Figure 2c. The H coverage θ′H does not go to zero at θ′S ) 1/4 but is greatly reduced, e.g., θ′H is reduced to 1% of the H2S free value when 10 ppm H2S is introduced in H2 at 700 °C. This strongly suggests that the poisoning of Ni by S adatoms is an electronic effect as opposed to a physical blocking of Ni atoms on the surface. A linear decrease in H2 adsorption with the increase in S on the surface has been experimentally observed1 for
Figure 3. Performance loss observed experimentally in Zha et al.11 vs predicted surface coverage of sulfur. The authors define performance loss as the change in current density at a constant cell voltage (0.7 V in the data plotted here) upon introduction of H2S into the H2 feed. The line is a linear fit to data with R2 ) 0.964.
polycrystalline Ni although the reported saturation coverage of S was higher. In experiments on sulfur poisoning of SOFC Ni-YSZ anodes, Matsuzaki and Yasuda9 observed that the minimum H2S concentration where the polarization resistance increased (onset of poisoning) was 2, 0.5, and 0.05 ppm at 1000, 900, and 750 °C, respectively. Our model predicts values of θ′S ∼ 4% and θ′H/θH′0 ∼ 85% at these three data points (see Figure 2c). From this plot, it would appear that sulfur poisoning is only observed when the coverage of hydrogen (the electrochemically active species) decreases below a threshold value. In Figures 3 and 4, we further examine the correlation of the S coverage predicted by our model to experimentally observed performance loss in an SOFC due to sulfur poisoning reported by Zha et al.11 and Cheng et al.12 The points in Figures 3 and
Adsorption Thermodynamics of H2S and H2 on Ni(111)
Figure 4. Increase in polarization resistance observed experimentally in Cheng et al.12 vs predicted surface coverage of sulfur. T ) 800 °C. G1 and G2 represent data collected at two different current densities, while P1 and P2 represent data collected at two different cell voltages. The R2 values for the linear fits to data - G1: 0.976, G2: 0.977, P1: 0.974, P2: 0.895.
4 were generated by plotting the reported performance loss11 or increase in polarization resistance12 against the sulfur coverages, which are computed by our model using the specified feed composition and cell temperature. These plots show that the loss in performance is clearly related to the predicted S coverage. Hansen13 uses the isotherm of Alstrup et al.5 for the same data sets and gets marginally better linear fits. However, as illustrated in Figure 4, a single linear correlation does not fit the data for different operating conditions. In general, the loss in performance is lower at higher current (lower cell voltage) for the same inlet gas composition. This overall effect can be attributed to the local electrochemical environment (gas-phase and surface compositions as well as local electrode potential) at the reaction sites being a function of the electrical current being drawn from the cell. To fully explain this effect, one would have to use a model that accurately simulates the highly coupled transport and reaction processes at the anode. Thus, although the predicted S coverage and the observed loss in performance due to poisoning both increase with increasing PH2S and decreasing T, linear correlations such as those given in Hansen13 and Figures 3 and 4 here do not give additional insights into the fundamental mechanisms of sulfur poisoning of SOFC anodes. As noted earlier, the Alstrup isotherm used by Hansen calculates the S coverage assuming reaction 1 completely describes the interaction of H2S and H2 with Ni thus ignoring the competitive adsorption of H2 on Ni. Next, we illustrate the impact of this assumption on the predicted S coverage. The importance of considering H2-Ni interactions while quantifying the sulfur poisoning of Ni exposed to a H2S-H2 mixture is depicted in Figure 5, which shows how θ′S, or the extent of poisoning, is a function of both PH2S and PH2 instead of only the ratio PH2S/PH2. θ′S is higher for lower PH2 at identical values of PH2S/PH2. As seen in Figure 5, this effect is stronger at lower temperatures. This effect, where PH2 influences the sulfur coverage directly instead of just through the ratio PH2S/ PH2, can only be captured by modeling the competitive adsorption of both H2 and H2S. As mentioned in the Introduction, this effect is usually overlooked in the experimental as well as modeling work on S posioning. Figure 6 illustrates the effects of not including the appropriate thermal corrections to the calculated thermodynamic properties
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Figure 5. Surface coverage of sulfur and the effect of neglecting H2-Ni interactions. PH2 ) 1 - PH2S atm (solid lines) vs PH2 ) 0.1 atm (dashed lines).
Figure 6. Surface coverage of sulfur and the effect of simplifying assumptions about the thermodynamic properties of the surface species. The two curves (T ) 800 °C) on the right-hand side are computed using the simplifying assumptions for the thermal corrections of the surface species free energies given in refs 6 and 7 (see text).
on the predicted sulfur coverage. The three cases considered are (i) all appropriate thermal corrections as discussed earlier, (ii) the assumptions in Galea et al.7 where ∆H0j ) ∆Eel,j and gas molecules lose translational and rotational entropy on adsorption, and (iii) the assumptions in Wang and Liu6 where ∆H0j includes the thermal corrections for the gas species only and gas molecules lose all entropy on adsorption. The sulfur coverage predicted for 5 ppm H2S in H2 at 800 °C for the three cases is 0.22, 0.01, and 0.05 respectively. Thus, the usual assumptions used in the literature for reaction enthalpies and entropies would predict lower θ′S or higher tolerance to H2S. The improvement in the estimate for the last set of assumptions is the consequence of error cancellation for the calculated ∆Go where the error due to incomplete thermal corrections for the reaction enthalpies partially cancels out the error due to the neglected surface species entropies. Conclusion This study reports an ab initio thermodynamics model for the competitive adsorption of H2 and H2S on Ni(111). Our model explicitly considers the interaction of both H2 and H2S with a Ni surface while including surface coverage effects. Unlike previous models, our model applies all appropriate thermal
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corrections to the thermodynamic properties of the surface species considered. We perform vibrational analyses for the adsorbed atoms of the surface species and use standard relationships from statistical thermodynamics to calculate these thermodynamic properties. We use the above model to calculate S and H coverages over a wide range of T, PH2, and PH2S. Our results clearly demonstrate that significant errors are introduced in the predicted S coverage if H2 adsorption in parallel with H2S adsorption, and the thermal corrections to the enthalpy and entropy of reaction are not accounted for properly. These effects must be included to build a better theoretical understanding of S poisoning. From a fuel cell perspective, this understanding is important as: (1) Ni remains the anode electrocatalyst of choice, and (2) the operating temperature of SOFCs is being driven down from 750-1000 °C to 550-700 °C where sulfur poisoning poses a bigger problem. Acknowledgment. This research was supported through funding to the NSERC Solid Oxide Fuel Cell Canada Strategic Research Network from the Natural Science and Engineering Research Council (NSERC). We thank Prof. Tom Ziegler and Dr. Maxim Shishkin at the University of Calgary and members of the GPAW development and user communities for insightful discussions that lead to improvements in this work. References and Notes (1) Bartholomew, C.; Agrawal, P.; Katzer, J. AdV. Catal. 1982, 31, 135–242. (2) Hansen, J. B.; Rostrup-Nielsen, J. Sulfur poisoning on Ni catalyst and anodes. In Handbook of Fuel Cells: Fundamentals, Technology, Applications; Vielstich, W. , Lamm, A. H. A. G., Eds.; Wiley: New York, 2009. (3) Rosenqvist, T. J. Iron Steel Institute 1954, 176, 37–57. (4) Rostrup-Nielsen, J. J. Catal. 1968, 11, 220. (5) Alstrup, I.; Rostrup-Nielsen, J.; Roen, S. Appl. Catal. 1981, 1, 303– 314. (6) Wang, J.-H.; Liu, M. Electrochem. Commun. 2007, 9, 2212–2217. (7) Galea, N. M.; Kadantsev, E. S.; Ziegler, T. J. Phys. Chem. C 2007, 111, 14457–14468.
Monder and Karan (8) Alfonso, D. R. Surf. Sci. 2008, 602, 2758–2768. (9) Matsuzaki, Y.; Yasuda, I. Solid State Ionics 2000, 132, 261–269. (10) Rasmussen, J. F. B.; Hagen, A. J. Power Sources 2009, 191, 534– 541. (11) Zha, S.; Cheng, Z.; Liu, M. J. Electrochem. Soc. 2007, 154, B201– B206. (12) Cheng, Z.; Zha, S.; Liu, M. J. Power Sources 2007, 172, 688–693. (13) Hansen, J. B. Electrochem. Solid State Lett. 2008, 11, B178–B180. (14) Phatak, A. A.; Delgass, W. N.; Ribeiro, F. H.; Schneider, W. F. J. Phys. Chem. C 2009, 113, 7269–7276. (15) Enkovaara, J. J. Phys.: Condens. Matter 2010, 22, 253202. (16) Blochl, P. E. Phys. ReV. B 1994, 50, 17953–17979. (17) Hammer, B.; Hansen, L.; Norskov, J. Phys. ReV. B 1999, 59, 7413– 7421. (18) Larsen, A. H.; Vanin, M.; Mortensen, J. J.; Thygesen, K. S.; Jacobsen, K. W. Phys. ReV. B 2009, 80, 195112. (19) Frederiksen, T.; Paulsson, M.; Brandbyge, M.; Jauho, A.-P. Phys. ReV. B 2007, 75, 205413. (20) Bahn, S. R.; Jacobsen, K. W. Comput. Sci. Eng. 2002, 4, 56–66. (21) Smith, W. R.; Missen, R. W. Chemical Reaction Equilibrium Analysis: Theory and Algorithms; John Wiley & Sons: New York, 1982. (22) Reuter, K.; Scheffler, M. Phys. ReV. Lett. 2003, 90, 046103-1. (23) Loffreda, D. Surf. Sci. 2006, 600, 2103–2112. (24) Cortright, R.; Dumesic, J. Kinetics of heterogeneous catalytic reactions: Analysis of reaction schemes. In AdVances in Catalysis; Academic Press: San Diego, 2001; Vol. 46. (25) Kandoi, S.; Greeley, J.; Sanchez-Castillo, M.; Evans, S.; Gokhale, A.; Dumesic, J.; Mavrikakis, M. Top. Catal. 2006, 37, 17–28. (26) Blaylock, D. W.; Ogura, T.; Green, W. H.; Beran, G. J. O. J. Phys. Chem. C 2009, 113, 4898–4908. (27) The Mathworks, “Getting Started with MATLAB: Version 7”, The Mathworks Inc., Natick, MA, 2005. (28) McQuarrie, D. A.; Simon, J. D. Molecular Thermodynamics; University Science Books: 1999. (29) McBride, B. J.; Zehe, M. J.; Gordon, S. NASA Glenn Coefficients for Calculating Thermodynamic Properties of Individual Species (NASA report TP-2002-211556), 2002. (30) Herzberg, G.; Huber, K. P. Molecular Spectra and Molecular Structure IV. Constants of diatomic molecules; Van Nostrand Reinhold: New York, 1979. (31) Shimanouchi, T. Tables of Molecular Vibrational Frequencies Part 5. J. Phys. Chem. Ref. Data 1972, 1, 189–216. (32) Lapujoulade, J.; Neil, K. J. Chem. Phys. 1972, 57, 3535–3545. (33) Christmann, K.; Schober, O.; Ertl, G.; Neumann, M. J. Chem. Phys. 1974, 60, 4528–4540.
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