Pourbaix Diagrams for H2O Oxidation to Adsorbed ... - ACS Publications

This finding is based on calculations for the onset potentials over the pH range 0–14 for 15 water oxidation reactions forming OH(ads) + H+(aq) + eâ...
0 downloads 0 Views 3MB Size
Download PDF
Article Cite This: J. Phys. Chem. C 2018, 122, 9958−9964

pubs.acs.org/JPCC

Pourbaix Diagrams for H2O Oxidation to Adsorbed OH on Pt(111) and Why They Differ from Those for Bulk Solids Alfred B. Anderson*

J. Phys. Chem. C 2018.122:9958-9964. Downloaded from pubs.acs.org by NEWCASTLE UNIV on 08/06/18. For personal use only.

Department of Chemistry, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106, United States ABSTRACT: Density functional calculations show that Pourbaix diagrams for reactions involving surface species are not the same as for bulk solids. This finding is based on calculations for the onset potentials over the pH range 0−14 for 15 water oxidation reactions forming OH(ads) + H+(aq) + e− on Pt(111). The deviations are results of the dependencies of Gibbs energies of adsorbed reactants on the electrode potential, causing their activities to change with potential instead of being fixed at unity, which is the case for Pourbaix diagrams of solids. In a sense, the redox couple changes as the electrode potential changes because at each potential the adsorbed reactants and products have new structures.

■

INTRODUCTION Reversible potentials follow the Nernst equation for electrochemical reactions in aqueous electrolytes when protons are generated by oxidation of A−H bonds or are consumed by reduction of H+(aq). The electrode’s Fermi level determines the potential and vice versa.1 According to the Nernst equation for equilibrium conditions, the potential of the standard hydrogen electrode decreases at the rate of 0.0592 pH·V from 0.0 V at pH = 0 to −0.828 V at pH = 14 on the standard hydrogen electrode (SHE) scale. In concert, the Fermi level rises from −φSHE, where φSHE is the thermodynamic work function of the standard hydrogen electrode at pH = 0, to φSHE + 0.828 pH·eV at pH = 14. In the case of other reactions involving transfer of n electrons and n protons, the reaction’s standard reversible potential decreases from U0 at pH = 0 to U0 − 0.828 V at pH = 14, and the Fermi level rises in concert from its value at U0 by 0.828 eV. Figure 1 shows these shifts for the standard hydrogen electrode, both in standard hydrogen electrode (SHE) potentials and in reversible hydrogen electrode (RHE) potentials for measurements where the counter electrode electrolyte pH is the same for the hydrogen electrode. Included in the lower half of Figure 1 is a second reaction, OH(aq) + H+(aq) + e− ⇌ H 2O(l)

Figure 1. Relationships between electrode potential and Fermi energy for the standard hydrogen electrode and for OH(aq) reduction to H2O(l) in electrolytes of pH 0 and 14.

The activity of H2O(l), a pure substance, is equal to 1. Further details are available in standard physical chemistry texts.3 For decreasing concentrations, molalities and molarities become identical, and for generating Pourbaix diagrams, the concentration of species A is simply expressed [A], interpreted as molar concentration of A with assumed activity coefficients of 1 for concentrations up to 1 M H+(aq). This is done even though γ± begins to decrease from 1 as concentrations increase above about 10−3 M.4 Pourbaix diagrams are graphs of open-circuit electrode potentials as functions of pH when bulk- and solution-phase species are in equilibrium. They are often used to predict corrosion of solids.5,6

(1)

This is the reaction of interest in this article but with the OH and H2O adsorbed instead of being in the bulk solution. The Nernst equation for eq 1 at 298.15 K is U = 2.72V − 0.0592 log[a(H 2O)]/[(a(OH)a(H+)]V

(2)

0

The value 2.72 V is U for the reaction as determined from experimental measurements.2 The a is activities for H2O(l), OH(aq), and H+(aq). Activity is given as a product of mean activity coefficient for the ion and its counterion, γ±, and the molal concentration, m: a = γ±m (3) © 2018 American Chemical Society

Received: February 6, 2018 Revised: April 5, 2018 Published: April 19, 2018 9958

DOI: 10.1021/acs.jpcc.8b01315 J. Phys. Chem. C 2018, 122, 9958−9964

Article

The Journal of Physical Chemistry C

For each, the reversible potential was calculated for an oxidation reaction wherein one adsorbed water molecule was oxidized, yielding OH(ads) + H+(aq) + e−. Four of the Pourbaix diagrams shown here are from graphs used to predict the pH dependence of the double-layer width on Pt(111) electrodes in a recent article.9 During that study, data were calculated that are used in this article to make Pourbaix diagrams for 11 additional surface reactions. Predictions were made using the Interface code.9,10 With this code, the structures and stabilities of the model interfaces respond to shifts in the Fermi energy, Ef(q), caused by added surface charge, q. For the electrolyte, the code uses 1 M C+ cations and 1 M A− anions, both with radii 3.0 Å in a dielectric continuum. The electrolyte ionic strength is 1. A modified Poisson−Boltzmann theory determines changes in electrolyte polarization energies that occur in response to added surface charges. Using twodimensional density functional theory, the potential, U, of a single slab representing an electrode is then a function of charge added to the translational cell:

Such diagrams can include (1) vertical lines for equilibria involving no electron transfer at a given pH, (2) horizontal lines for equilibria involving electron transfer but no proton transfer, and (3) slanted lines for equilibria involving electron and proton transfer, the slope being given by the Nernst equation. This article examines the third case, but instead of bulk/solution equilibria, it addresses surface/solution equilibria. At issue is the question of whether Pourbaix diagrams for equilibria between solids and bulk solution species, with the factor 0.0592 in eq 2, apply to charge-transfer events with reactants or products adsorbed on electrode surfaces. This factor has been used in the Nernst equation for qualitative discussions of OnHm redox chemistry on metal surfaces.7,8 In that work, the adsorption energies were noted to depend on the interfacial electric field, itself dependent on the potential applied to the electrode. However, for simplicity, the authors neglected the field effects. The electric field is a property derived from the variationally optimized electron wave functions and the resulting self-consistent charge density functions obtained from them. However, knowledge of the electric field is not required for calculating the effects of changing electrode potential in quantum mechanical calculations. When a self-consistent theory including surface charging and double-layer polarization is used, calculated potentials, which are functions of the Fermi energy, are accurate within the accuracy of the computational model. Adsorbate properties calculated with such a theory are accurate. As the surface is charged, changing the potential and shifting the Fermi level, the structures and stabilities of solvated adsorbates change, and this affects the theoretically calculated Pourbaix plots. This article establishes the consequences to Pourbaix diagrams of imposing the Fermi level shifts that accompany pH changes for redox reactions with protons. Included are surfaces both saturated and unsaturated by mixtures of OH(ads) and H2O(ads).

U (q) = [−φSHE − Ef (q)]/F

(4)

To determine φSHE, the Gibbs energies of the left and right sides are set equal in the equilibrium reaction H+(aq) + e−( −φSHE) ⇌

1 H2 2

(5) +

Using energies calculated with this code for H (aq) and 1/2 H2, the electron energy at equilibrium is −4.454 eV, so that φSHE = 4.454 eV.

■

COMPUTATIONAL DETAILS Full details of the Interface code are in ref 10. In brief, the code includes the following. For core and valence electrons, the wave

Figure 2. Calculated Pourbaix diagrams for oxidizing saturated water structures on Pt(111) to form OH(ads) + H+ + e−. Initial water structures are (a) 2/3 ML H2O H-down (1); (b) 2/3 ML H2O H-down (2); (c) 2/3 ML H2O MD; and (d) 8/12 ML H2O in the H-down (1) structure. These structures are in Figure 3. 9959

DOI: 10.1021/acs.jpcc.8b01315 J. Phys. Chem. C 2018, 122, 9958−9964

Article

The Journal of Physical Chemistry C functions are taken as linear combinations of pseudoatomic orbitals (LCPAO).11 Norm-conserving pseudopotentials (NCPP) are used for effective core potentials.8,12 A generalized gradient approximation with revised Perdew−Burke−Ernzerhof (GGA-RPBE) functional13 and a basis set of double-ζ polarization (DZP)10

functions are used. A modified Poisson−Boltzmann theory (MPB)14,15 is incorporated in the code to accommodate potentialdependent changes in electrolyte polarization and ionic distribution caused by surface charging. A dielectric continuum14 is used to represent the interaction energy of the interface components with bulk water. In this study, three-layer Pt(111) slabs were employed and two-dimensional Interface band calculations were performed. Adsorption was studied on the top layer of atoms, and the top two layers of the slab were relaxed while the bottom layer faced vacuum and was fixed at the calculated lattice constant of 4.03 Å. A 3 × √3 translational cell containing six Pt atoms per layer was used, and the 3 × 6 × 1 k-point mesh for Monkhorst−Pack sampling16 was applied. Reference 10 shows that errors in PtO bond strengths calculated with the Interface code are around 0.1 eV for the PtO molecule and for O adsorption at the potential of zero charge. Theoretical Determination of Reversible Potentials at Different pH Values. For solid−solution equilibria, the trend of Pourbaix plots given by the Nernst equation is a decrease in equilibrium potential with increasing pH at the rate of −0.0592 V per unit change in pH. With the Interface code, a first-principles Nernst equation with a slightly different rate of −0.0604 V per unit increase in pH was calculated.17 The −0.0604 factor resulted from the theoretically calculated −log of the water constant being 0.283 larger than the experimental value 14.00. The overestimate is caused by the calculated value of ΔwG0 being 0.017 eV more than the experimental value. Using the definition pH = −log[H+(aq)] and the −0.604 factor, over the 0−14 pH range, the change in the reduction equilibrium potential is 0.845 V, compared to the experimental value 0.828 V. This approximation affects the Pourbaix-like diagrams in this study only slightly and therefore does not affect the conclusions. The equilibrium reactions considered in this article all have the form Ox(U ) + H+(aq) + e−(U ) ⇌ Red(U )

When this reaction is at equilibrium, the following holds

(6) 17

G{Red(U )} = G{Ox(U )} + G{H+(aq)} − 4.454eV − FU + (0.0604V)pH

Figure 3. From top to bottom, 2/3 ML H2O in H-down (1), in H-down (2), and in MD structures, and 1/6 ML OH + 1/2 ML H2O. Charges added to the translation cell are in e units, and potentials are in V. Adapted with permission from Zhao, M.; Anderson, A. B. Predicting the Double Layer Width on Pt(111) in Acid and Base with Theory and Extracting It from Experimental Voltammograms. J. Phys. Chem. C 2017, 121, 28051−28064. Copyright 2017 American Chemical Society.

(7)

Equation 4 provides the equilibrium potentials for the water oxidation reactions as functions of pH, used to generate the Pourbaix diagrams for this article. In it, G{Red(U)} and G{Ox(U)} are the calculated Gibbs energies of the reduced and oxidized (minus the H+(aq) energy) systems, and G{H+(aq)} is the calculated Gibbs

Figure 4. Calculated Pourbaix diagrams for oxidizing water at 1/2 ML coverage at two structures on Pt(111) to form OH(ads) + H+ + e− with 1/4 ML H2O(aq) water added. In panel a, water molecules bonded by lone-pair donation to the surface become, in panel b, hydrogen bonded to the surface.9 9960

DOI: 10.1021/acs.jpcc.8b01315 J. Phys. Chem. C 2018, 122, 9958−9964

Article

The Journal of Physical Chemistry C

from 1/3 ML OH(ads), and (3) 2/3 ML OH(ads) coverage from 1/2 ML OH(ads). Graphs of onset potentials for four cases with initial saturation coverage by water versus pH are in panels a−d, Figure 2, with polynomial functions fitted to the calculated data points. The onset potentials are equilibrium potentials above which oxidized species dominate and below which reduced species dominate. For their comparison with experiment, see ref 9. Average slopes, calculated using the difference between the highest and the lowest potentials divided by 14, are scattered from 0.043 V/pH to 0.089 V/pH. For brevity, in the following, slopes are given with units omitted. For the two “H-down” water structures, the average slopes are −0.051 and −0.041, and both are less than the −0.060 slope from the first-principles Nernst equation for solid− solution equilibria. The third structure, called MD, has no OH bonds directed toward the surface, and its slope is −0.081. These three 2/3 ML water structures and their potential dependencies

energy of the hydrated hydronium ion at 1 M concentration. In this work, the oxidized species is OH(ads) and the reduced species is H2O(ads), and it is shown that their activities vary from 1, which is the value for standard solutions and pure solids used in previous Pourbaix diagrams for surfaces in refs 7 and 8.

■

RESULTS AND DISCUSSION Pourbaix Diagrams for Forming OH(ads) on Pt(111). Calculated values of G{Red(U)} and G{Ox(U)} for 15 different initial adsorbate structures forming OH(ads) upon oxidation, when put into eq 4, yielded values of reversible potentials Urev at pH = 1, 2, 3, ..., 14. They appear in graphs presented in the figures. Twelve of the reactions form OH(ads) from H2O(ads) with different surface H2O amounts and structures. Eleven of these form 1/6 monolayer (ML) OH(ads), and one forms 1/12 ML OH(ads). Three additional reactions form (1) 1/3 ML OH(ads) following the 1/6 ML coverage, (2) 1/2 ML OH(ads) coverage

Figure 5. Calculated Pourbaix diagrams for oxidizing unsaturated water structures on Pt(111) to form OH(ads) + H+ + e− at coverages shown. In each case, 1/6 ML OH(ads) forms upon oxidation. Panel a: 1/6 ML H2O(ads); panel b: 1/3 ML H2O. Panels c and d: water molecules bonded by lone-pair donation to the surface in panel c detach from the surface in panel d. Panels e and f show, respectively, what happens when 1/4 ML H2O(aq) is adsorbed with the OH(ads) product formed from 1/6 ML H2O(ads) and 1/3 ML H2O(ads). Reference9 has additional structure information. 9961

DOI: 10.1021/acs.jpcc.8b01315 J. Phys. Chem. C 2018, 122, 9958−9964

Article

The Journal of Physical Chemistry C

Figure 6. Calculated Pourbaix diagrams for oxidizing various water and OH(ads)structures on Pt(111) at saturation coverages to form saturated surfaces with increasing amounts of OH(ads). Panel a shows 1/3 ML OH(ads) in equilibrium with 1/6 ML OH(ads), panel b shows equilibria for 1/2 and1/3 ML coverages, and panel c shows equilibria for 2/3 and 1/2 ML coverages.

energy scale of the Red(U) curve depends on pH in accordance with eq 4. The positions of the curves and their curvatures are different for each reaction, which is why these surface reactions do not obey the Nernst equation for solid−solution equilibria. The presence of the different slopes and curvatures is perhaps not surprising because bonding interactions with the surface include lone-pair donation between oxygen atoms in OH and H2O and the surface and hydrogen bonding between H2O with the surface, as well as hydrogen-bonding interactions between adsorbates. As may be seen in Figure 3, taken from ref 9, the adsorbate structures change with potential, in accordance with expectations for donor/acceptor interactions with the Pt surface as the Fermi level moves up or down in energy. Particularly visible is how for structure 1 the central water molecule settles on the surface as the electrode potential increases, a result of increasing lone-pair donation bonding to the surface as the Fermi energy drops. Structure 2 and MD change less notably with increasing potential, but two of the three water molecules in the 1/2 ML H2O(ads) + OH(ads) oxidation product settle down onto the surface as the potential increases in Figure 3. The different slopes and curvatures, determined by minimizing the system energies, are associated with the varying numbers and structures of lonepair donation and hydrogen-bonding interactions and the ways in which they depend on the Fermi energy or, equivalently, the electrode potential change.

are shown in Figure 3 along with the oxidation product, 1/2 ML H2O + OH(ads). When the H-down (1) structure is used with a larger translational cell to form 1/12 ML OH(ads), the slope is −0.046, not far from the −0.051 slope for forming 1/6 ML OH(ads). There might be water vacancies on the Pt surface at ambient temperatures. Vacancies created by placing an adsorbed water molecule in the H-down (1) structure and in the H-down (2) structure in solution give corresponding slopes shown in Figure 4. As seen in panels a and b, the slopes are −0.058 and −0.043. They are close to the slopes for these structures given earlier for the saturated surface, which are 0.051 and 0.041, respectively. Other H2O vacancy structures were 1/6, 1/3, and 1/2 ML H2O structures (1) and (2), in Figure 5, panels a−d. They have corresponding slopes −0.088, −0.081, −0.065, and −0.048, all different from 0.060. When 1/6 ML of aqueous H2O is added, the first two slopes become 0.074 and 0.070, as shown in panels e and f. Calculations to generate higher OH(ads) coverages for saturated surfaces shown in panels a, b, and c in Figure 6 also yielded scattered values for their slopes. For forming 1/3, 1/2, and 2/3 ML OH(ads) the slopes are −0.058, −0.089, and −0.087, respectively. The slopes and curvatures in Figures 2 and 4−6 are dependent on the potential dependencies of the Gibbs energies of Ox(U) + H+(aq) + e−(U) and of Red(U). Second-order polynomials of the form aU2 + bU + U0 fit all of the calculated Gibbs energy points in the potential range of interest. Parameters obtained in the fitting are in Table 1, where it is seen that coefficient a is less than 0 so that all curvatures are negative. The parameter U0 is the curve maximum and is close to the potential of zero charge for reduced species.18 The equilibrium potential is the potential where these curves intersect. Figure 7 schematically illustrates the intersections for a reaction pH = 0 and pH = 14. The position on the

■

CONCLUSIONS This article shows, using density functional theory, that when reactants or products adsorbed on an electrode in aqueous electrolyte consume or discharge protons, the reversible potential does not depend on pH in the same way as it does for solid− solution interfaces. It is because the Gibbs energies of adsorbed reactants depend on the electrode potential, making their activities 9962

DOI: 10.1021/acs.jpcc.8b01315 J. Phys. Chem. C 2018, 122, 9958−9964

Article

The Journal of Physical Chemistry C Table 1. Fifteen Oxidation Reactions of H2O(ads) to OH(ads) + H+(aq) + e−a systems

adsorbates

parameter before x2 in second-order fit (G vs U)

a

8/12 ML H2O(ads) 1/12 ML OH(ads) + 7/12 ML H2O(ads) + H+ + e− 1/6 ML H2O(ads) 1/6 ML OH(ads) + H+ + e− 1/6 ML H2O(ads) + 1/4(H2O)4 1/6 ML OH(ads) + 1/6 ML H2O(ads) + H+ + e− 1/3 ML H2O(ads) 1/6 ML OH(ads) + 1/6 ML H2O(ads) + H+ + e− 1/3 ML H2O(ads) + 1/4(H2O)4(aq) 1/6 ML OH(ads) + 1/3 ML H2O(ads) + H+ + e− 1/2 ML H2O(ads) structure 1 1/6 ML OH(ads) + 1/3 ML H2O(ads) + H+ + e− 1/2 ML H2O(ads) structure 2 1/6 ML OH(ads) + 1/3 ML H2O(ads) + H+ + e− 1/2 ML H2O(ads) structure 1 + 1/4(H2O)4(aq) 1/6 ML OH(ads) + 1/2 ML H2O(ads) + H+ + e− 1/2 ML H2O(ads) structure 2 + 1/4(H2O)4(aq) 1/6 ML OH(ads) + 1/2 ML H2O(ads) + H+ + e− 2/3 ML H2O(ads) H-down (1) 1/6 ML OH(ads) + 1/2 ML H2O(ads) + H+ + e− 2/3 ML H2O(ads) H-down (2) 1/6 ML OH(ads) + 1/2 ML H2O(ads) + H+ + e− 2/3 ML H2O(ads) MD 1/6 ML OH(ads) + 1/2 ML H2O(ads) + H+ + e− 1/6 ML OH(ads) + 1/2 ML H2O(ads) 1/3 ML OH(ads) + 1/3 ML H2O(ads) + H+ + e− 1/3 ML OH(ads) + 1/3 ML H2O(ads) 1/2 ML OH(ads) + 1/6 ML H2O(ads) + H+ + e− 1/2 ML OH(ads) + 1/6 ML H2O(ads) 2/3 ML OH(ads) + H+ + e−

−0.49897 −0.35959 −0.26119 −0.2186 −0.26119 −0.21054 −0.29855 −0.21054 −0.29855 −0.21785 −0.2958 −0.21785 −0.22223 −0.21785 −0.2958 −0.33891 −0.22223 −0.33436 −0.2333 −0.33436 −0.11201 −0.33436 −0.19576 −0.33436 −0.43249 −0.21144 −0.21053 −0.19259 −0.1926 −0.20496

b c d e f g h i j k l m n o a

parameter before x in second-order fit (G vs U)

potential at the maximum point/V (SHE)

0.8531 −0.5021 0.29881 −0.4007 0.29881 −0.5786 0.24654 −0.5786 0.24654 −0.64687 0.32449 −0.64687 0.62676 −0.64687 0.32449 −0.72743 0.62676 −0.71565 0.42025 −0.71565 0.50015 −0.71565 0.04362 −0.71565 0.33052 −0.78775 0.20598 −0.50636 0.49367 −0.18111

0.855 −0.698 0.572 −0.917 0.572 −1.374 0.413 −1.374 0.413 −1.485 0.548 −1.485 1.410 −1.485 0.548 −1.073 1.410 −1.070 0.901 −1.070 2.233 −1.070 0.111 −1.070 0.382 −1.863 0.489 −1.315 1.282 −0.442

Parameters are for second-order fits to the calculated potential-dependent reantant and product Gibbs energies.

Notes

The author declares no competing financial interest.

■

ACKNOWLEDGMENTS The author thanks Meng Zhao for providing the table and figures used in this article.

■

(1) Reiss, H. The Fermi Level and Redox Potential. J. Phys. Chem. 1985, 89, 3783−3791. (2) Stanbury, D. M. Reduction Potentials Involving Inorganic Free Radicals in Aqueous Solution. Adv. Inorg. Chem. 1989, 33, 69−138. (3) Atkins, P.; de Paula, J. Physical Chemistry, 9th ed.; W. H. Freeman: New York, 2012. (4) Schneider, A. C.; Pasel, C.; Luckas, M.; Schmidt, K. G.; Herbell, J.D. Determination of Hydrous Single Ion Activity Coefficients in Aqueous HCl solutions at 25 °C. J. Solution Chem. 2004, 33, 257−273. (5) Gileadi, E. Electrode Kinetics for Chemists, Chemical Engineers, and Materials Scientists; Wiley-VCH, Inc: New York, 1993. (6) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry, 2nd ed; Kluwer Academic/Plenum: New York, 2000; Vol. 2b. (7) Hansen, H. A.; Rossmeisl, J.; Norskov, J. K. Surface Pourbaix Diagrams and Oxygen Reduction Activity of Pt, Ag and Ni(111) Surfaces Studied by DFT. Phys. Chem. Chem. Phys. 2008, 10, 3722− 3730. (8) Kondov, I.; Faubert, P.; Muller, C. Activity and Electrochemical Stability of a Chromium Modified Nickel Catalyst for Oxygen Reduction Reaction. Electrochim. Acta 2017, 236, 260−272.

Figure 7. This shows solving eq 4 in the text graphically for reversible potentials at pH = 0 and 14.

potential-dependent instead of being unity, as they are for solid− solution equilibria in Pourbaix diagrams. In a sense, the redox couple changes as the electrode potential changes because at each potential the adsorbed reactants and products have new structures.

■

REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone 216-368-5044. ORCID

Alfred B. Anderson: 0000-0003-3691-0439 9963

DOI: 10.1021/acs.jpcc.8b01315 J. Phys. Chem. C 2018, 122, 9958−9964

Article

The Journal of Physical Chemistry C (9) Zhao, M.; Anderson, A. B. Predicting the Double Layer Width on Pt(111) in Acid and Base with Theory and Extracting It from Experimental Voltammograms. J. Phys. Chem. C 2017, 121, 28051− 28064. (10) Jinnouchi, R.; Anderson, A. B. Electronic Structure Calculations of Liquid-Solid Interfaces: Combination of Density Functional Theory and Modified Poisson-Boltzmann Theory. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 77, 245417−1−18. (11) Soler, J. M.; Artacho, E.; Gale, J. D.; Garcia, A.; Junquera, J.; Ordejon, P.; Sanchez-Portal, D. The SIESTA Method for Ab Initio Order-N Materials Simulation. J. Phys.: Condens. Matter 2002, 14, 2745− 2779. (12) Troullier, N.; Martins, J. L. Efficient Pseudopotentials for Planewave Calculations. Phys. Rev. B: Condens. Matter Mater. Phys. 1991, 43, 1993−2006. (13) Hammer, B.; Hansen, L. B.; Norskov, J. K. Improved Adsorption Energetics within Density-functional Theory using Revised PerdewBurke-Ernzerhof Functionals. Phys. Rev. B: Condens. Matter Mater. Phys. 1999, 59, 7413−7421. (14) Borukhov, I.; Andelman, D.; Orland, H. Steric Effects in Electrolytes: A Modified Poisson-Boltzmann Equation. Phys. Rev. Lett. 1997, 79, 435−438. (15) Otani, M.; Sugino, O. First-principles Calculations of Charged Surfaces and Interfaces: A Plane-wave Nonrepeated Slab Approach. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 73, 115407−1−11. (16) Monkhorst, H. J.; Pack, J. D. Special points for Brillouin-zone integrations. Phys. Rev. B 1976, 13, 5188−5192. (17) Zhao, M.; Anderson, A. B. Predicting pH Dependencies of Electrode Surface Reactions in Electrocatalysis. Electrochem. Commun. 2016, 69, 64−67. (18) Tian, F.; Anderson, A. B. Effective Reversible Potential, Energy Loss, and Overpotential on Platinun Fuel Cell Cathodes. J. Phys. Chem. C 2011, 115, 4076−4088.

9964

DOI: 10.1021/acs.jpcc.8b01315 J. Phys. Chem. C 2018, 122, 9958−9964