Electrochemical Barriers Made Simple - The Journal of Physical

Letter pubs.acs.org/JPCL

Electrochemical Barriers Made Simple Karen Chan† and Jens K. Nørskov*,†,‡ †

Department of Chemical Engineering, Stanford University, Stanford, California 94305, United States SLAC National Accelerator Laboratory, Menlo Park, California 94025, United States

‡

ABSTRACT: A major challenge in the theoretical treatment of electrochemical charge transfer barriers is that simulations are performed at constant charge, which leads to dramatic potential shifts along the reaction path. Real electrochemical systems, however, operate at constant potential, which corresponds to a hypothetical model system of infinite size. Previous studies of hydrogen evolution have relied on a computationally costly scheme that extrapolates the barriers calculated on increasingly larger cells, and extension of this scheme to more complex reactions would be prohibitively costly. We present a new method to determine constant potential reaction energetics for simple charge transfer reactions that requires only (1) a single barrier calculation in an electrochemical environment and (2) the corresponding surface charge at the initial, transition, and final states. This method allows for a tremendous reduction in the computational resources required to determine electrochemical barriers and paves the way for a rigorous DFT-based kinetic analysis of electrochemical reactions beyond hydrogen evolution.

E

corresponding interfacial charge at the initial, transition, and final states. Figure 1 illustrates the issue with a constant charge barrier calculation for a Heyrovsky (H* + H+ + e− → H2 + *) reaction barrier on Pt(111) in a 3 × 4 unit cell with a full monolayer of hydrogen in the fcc sites. In the bottom figure, the electronic energy barrier is shown in red, whereas the corresponding change in work function Φ is shown in blue. Φ is related to the absolute potential vs the standard hydrogen electrode USHE via

lectrochemical interfaces are considerably more complex than solid|gas interfaces. Solvent layers, ions, and the effect of potential are all very challenging to treat theoretically at the molecular level. In recent years, the computational hydrogen electrode model1 has provided an elegant way to explore reaction energetics without explicit treatment of the electrons and ions in solution.2−4 However, any consideration of kinetics and charge transfer barriers necessitates the inclusion of the solvent and charge in the model system. In recent years, there have been a variety of schemes proposed to model the charged electrochemical interface: where the electrode is immersed in a homogeneous background charge5 or an implicit solvent,6 where countercharge resides in a conductor7,8 or a localized planar region9 across from the electrode, or an approach that hypothesizes proton transfer barriers to be approximated by surface hydrogenation ones,10 all with their own assumptions and shortcomings.11,12 A full ab initio treatment of electrochemical interfaces is appealing in that it does not rely on empirical assumptions about the solvent or countercharge.11,13 The major challenge here is that simulations are done at constant charge, which means that the interfacial charge density and corresponding potential change along the reaction path.13,14 Real electrochemical systems, on the other hand, operate at constant potential, which corresponds to a hypothetical model system of infinite size. In previous studies of hydrogen evolution and oxidation, this issue has been circumvented by extrapolating the barriers calculated in model systems of increasing size.11,15 However, extension of this scheme to more complex reactions and systems would be extremely costly computationally. In this Letter, we demonstrate a simple method to determine constant potential reaction energetics for simple charge transfer reactions, requiring only a single barrier calculation and the © XXXX American Chemical Society

USHE =

Φ − ΦSHE e

(1)

and ΦSHE has been determined experimentally to be ∼4.4 eV.16 The potential in this model system therefore changes by over 2 V between initial and final states. The top figures show positive and negative charge density difference isosurfaces at the initial (IS), transition (TS), and final states (FS). The isosurfaces were calculated via Δρ = ρall − ρslab + water + ad − ρH or H − H

(2)

where H refers to the extra hydrogen in the water layer in the initial state (IS) and H−H to the two hydrogens forming H2 gas in the transition (TS) and final states (FS). They illustrate qualitatively the trend of more negative surface charge density (−qIS < −qTS < −qFS) corresponding to lower Φ (ΦIS < ΦTS < ΦFS). The principle of extrapolation using a series of increasingly larger cells11,15 is illustrated schematically in Figure 2. Figure 2a shows that as the unit cells increase in size, the change in interface charge density during a charge transfer reaction from Received: May 19, 2015 Accepted: June 22, 2015

2663

DOI: 10.1021/acs.jpclett.5b01043 J. Phys. Chem. Lett. 2015, 6, 2663−2668

Letter

The Journal of Physical Chemistry Letters

a simple capacitor. The energy changes at increasing cell sizes can thus be extrapolated to that at infinite cell size a∞, the limit where charge density and potential from IS to TS to FS remains constant. The major issue with this scheme is its computational cost, given that density functional theory calculations scale with O(n3e ), where ne is the number of electrons. Assuming that at minimum three to four cell sizes are required, that is, unit cells of size ao, 2ao, 3ao, or 4ao, this leads to a total computational time ∼2 orders of magnitude more costly than that of a single barrier calculation in the smallest cell of size ao. In systems containing large adsorbates or heavily solvated ions that require more than one water bilayer, such an extrapolation scheme would be prohibitively costly. Here, using simple electrostatic arguments, we show that we can obtain constant potential reaction energetics using a single barrier calculation and the interfacial charge at the IS, TS, and FS. This method is premised upon the idea that, for a given charge transfer across the electrochemical interface, the “chemical” and electrostatic contributions to the change in energy are separable

E = Echem + Eel

Figure 1. Heyrovsky reaction barrier on Pt in a (3 × 4) unit cell with a full monolayer of *H. Top: charge density difference isosurfaces for the initial, transition, and final states; magenta corresponds to an isosurface of ρ = 0.001 e Bohr−3 and blue to −0.001 e Bohr−3. H atoms corresponding to the final H2 are highlighted in yellow. Bottom: the electronic energy E and corresponding work function Φ along the reaction path (dotted lines connecting work function data are only there to guide the eye).

(3)

For simple proton transfers, without adsorbates with strong dipoles that reorient dramatically or heavy solvent reorganization, the electrostatic component is purely capacitive, Eel = Ecap. With surface charge density θ = q/N, where q is the number of charged ions and N the number of surface atoms in the unit cell, the capacitance and capacitor energy per surface atom are, respectively

IS to TS to FS becomes smaller. Figure 2b illustrates a series of hypothetical reaction and activation energies, calculated at different cell sizes ai, as a function of the corresponding potential change. As detailed below, they should depend linearly on potential change, assuming the interface functions as

C=−

eθ U − Upzc

Ecap =

e 2θ 2 2C

(4)

where Upzc is the potential of zero charge. For a given proton transfer from state 1 to 2, where states 1 and 2 correspond to,

Figure 2. (a) Schematic showing proton transfer to a surface calculated in hypothetical unit cells of different size, ao and a1. As the unit cell increases in size, the changes in interface charge density and the corresponding potential becomes smaller. (b) Extrapolation of reaction and activation energies calculated at different cell sizes ai and extrapolated to that at infinite cell size a∞, the constant potential limit.11,15 2664


Letter


Table 1. Model Systems Considered: Representative Images, Cell-Extrapolation Details, and Interface Structural Details

for example, IS, TS, or FS, the capacitive contribution to the energy change in a cell with N surface atoms is therefore 2 ⎡ e 2 ⎛ q ⎞2 e 2 ⎜⎛ q1 ⎟⎞ ⎤ ⎥ Ecap,2 − Ecap,1 = N ⎢ ⎜ 2 ⎟ − 2C ⎝ N ⎠ ⎦ ⎣ 2C ⎝ N ⎠

e2 = [2(q2 − q1)q1 + (q2 − q1)2 ] 2CN

or, in terms of work functions (directly output from a DFT calculation without the need for an experimental ΦSHE reference), and rearranging, E2(Φ1) − E1(Φ1) = E2(Φ2) − E1(Φ1)

(5)

+ (6)

(7)

(8)

where q2 − q1 is the change in charge from state 1 to 2. In eq 8, the second term in the square brackets, ((U2 − U1)/2), gives the finite cell size contribution; as the cell size approaches infinity, both U2 − U1 and q2 − q1 approach zero. The first term therefore refers to the electrostatic contribution to the energy change between state 1 and 2 at the constant potential U1. The total energy change from state 1 to 2 is therefore

−

(q2 − q1)(Φ2 − Φ1) 2

(11)

For extrapolation of activation energies, an accurate determination of the TS and corresponding qTS is required; to do so, we have used well-converged climbing-image nudged elastic band calculations.17 The dependence of the reaction energetics with potential can be determined by examining cells with different water dipoles or ion concentrations. We note that the enthalpic contributions to solvent reorganization upon proton

e(q2 − q1)(U2 − U1) 2

(10)

E2(Φ2) − E1(Φ2) = E2(Φ2) − E1(Φ1)

E2(U2) − E1(U1) = [E2(U1) − E1(U1)] −

2

We can estimate q2 − q1 from our DFT results using a Bader analysis; therefore, the total energy change at constant Φ1, E2(Φ1) − E1(Φ1), can be obtained from a single barrier calculation. This simple analysis is applicable to both activation energies Ea and reaction energies ΔE. For any energy change, the extrapolation can be done to the work function of either state 1 or 2; interchanging state 1 and 2 in eq 10 and rearranging

⎡ 2CN (U1 − Upzc) CN (U2 − U1) ⎤ e ⎥ (q2 − q1)⎢ − = − ⎥⎦ ⎢⎣ 2CN e e 2

⎡ U − U1 ⎤ =e(q2 − q1)⎢ −(U1 − Upzc) − 2 ⎥⎦ ⎣ 2

(q2 − q1)(Φ2 − Φ1)

(9) 2665


Letter


case. Extrapolation of Ea was not done for systems where the reaction is barrierless or where Ea simply equals ΔE. Where extrapolation is done toward the FS (see Table 1), Ea refers to the barrier for the reverse reaction, ETS − EFS; for reaction energies, ΔE always refers to EFS − EIS. In the extrapolation of the Volmer barrier, only two points are shown, because the reaction was barrierless in larger cells. In general, both and ΔE and Ea show the linear behavior expected from eq 10. Scatter in the data for the Volmer reaction at 1 ML H* can be attributed to slight differences in the hydrogen bonding environment amongst different cell sizes, which results in slight variations in Echem in eq 3. Table 2 shows the changes in q, E, and Φ used to obtain the charge-extrapolated values. We note that for all proton transfer reactions, the charge transferred from IS to FS lies around 0.7; this may result from a DFT electron delocalization error or a physical, partial transfer of electronic charge from the metal toward the hydronium ions in the Helmholtz plane. A comparison of charge- and cell-extrapolated ΔE and Ea at constant potential is shown in the last two columns for each case in Table 2, as well as in the parity plot in Figure 4. Regardless of the unit cell size used, the simple chargeextrapolated scheme gave, within usual GGA-DFT error,21 nearly identical constant potential reaction energetics to that from the previous, dramatically costlier cell-extrapolation scheme. In summary, we have presented a simple method to determine electrochemical reaction energetics at constant potential that require only (1) a single barrier calculation in an electrochemical environment and (2) the corresponding surface charge at the initial, transition, and final states. This method allows for a substantial reduction in computational effort and paves the way for a rigorous DFT-based kinetic analysis of reactions beyond hydrogen evolution and oxidation. We note that although this analysis has been applied to simple model systems with a single water layer, the resulting reaction energetics can be applied in continuum interfacial models that account for the impact of diffuse charge and mass transport, phenomena at length scales too large to treat from an ab initio perspective. Future work will extend this to more complex electrochemical reactions involving adsorbate dipole and solvent reorientation, which will provide an additional contribution to the electrostatic component in reaction energies.

transfer have been included via the water layer. Entropic contributions on the other hand require large scale, dynamics simulations,18 which as a first approximation can be considered to add an additional 0.2−0.3 eV to the overall calculated barrier.19 In what follows, we show the comparison between the values obtained using the previous cell-extrapolated and the present charge-extrapolated schemes for three simple elementary proton transfer reactions, Volmer, Heyrovsky, and OH reduction to H2O * + H+ + e− → H*

(12)

H* + H+ + e− → H 2 + *

(13)

OH* + H+ + e− → H 2O + *

(14)

Table 1 lists the details of the systems considered: • Representative images of the IS, FS, andwhere barriers were calculatedTS. • The state and Φ to which extrapolation is performed (state 1 with Φ1 in eq 10 or state 2 with Φ2 in eq 11; in the hypothetical example of Figure 2, this would refer to the IS).20 • The adsorbate coverage and orientation of the water layer. Figure 3 shows the ΔE and Ea vs ΔΦ at various cell sizes, and the extrapolation of ΔE and Ea to the infinite cell size, ΔΦ = 0

■

COMPUTATIONAL METHODS Density functional theory calculations were carried out with the GPAW code and integrated with the atomic simulation environment,22−24 and the RPBE functional was used for exchange and correlation. The grid spacing for GPAW realspace calculations was 0.2 Å and the Fermi smearing 0.1 eV; energies were extrapolated to an electronic temperature of 0 K. As in previous work,11,13,15 all systems contained a periodic three-layer Pt(111) slab, one ice-like water bilayer,25 various adsorbates and charged H3O+, and at least 12 Å vacuum in the direction perpendicular to the surface. Previous work found little difference in the interfacial field between systems with one, two, or three water layers.11 Unit cells of sizes (3 × 2), (3 × 3), (3 × 4), (3 × 6), and (6 × 4) were sampled with Monkhorst−Pack k-point grids (4 × 6), (4 × 4), (4 × 3), (4 × 2), and (2 × 3). All systems were electroneutral with no homogeneous background charge added; excess H in the water layer spontaneously transferred electrons to the slab upon

Figure 3. Cell extrapolation scheme of refs 11 and 15 as applied to various simple proton transfer reactions; infinite sized cell/constant potential ΔE and Ea are marked with an “×” and correspond to extrapolated values at ΔΦ = 0. 2666


Letter

The Journal of Physical Chemistry Letters Table 2. Tabulated Changes in q, E, and Φ and Resultant Extrapolated ΔE and Ea at Constant Potentiala Volmer, to IS EFS − EIS

cell × × × ×

3 3 3 6

−0.03 −0.52 −0.52 −0.63

2 4 6 4

ΦFS − Φ

IS

2.37 0.91 0.62 0.37

ETS − EIS

ΦTS − Φ

0.14 0.01

IS

qFS − qIS

qTS − qIS

−0.57 −0.01 −0.49 0.02 −0.46 −0.44 cell-extrapolated:

0.19 −0.12

ΔE

Ea

−0.70 −0.75 −0.67 −0.71 −0.83

0.14 0.01

0.06

Volmer, 1 ML *H EFS − EIS

cell 3 3 3 6

cell 3 3 3 6

× × × ×

3 4 6 4

× × × ×

3 4 6 4

ΦFS − ΦIS

0.06 0.08 0.26 0.46

cell

EFS − EIS

3×3 3×4 3×6

0.34 0.54 0.71

1.86 1.45 1.02 0.92 cell-extrapolated: Volmer, to FS ΦFS − ΦIS 2.01 1.54 0.98 cell-extrapolated: Heyrovsky

EFS − EIS

ΦFS − ΦIS

ETS − EFS

−0.08 0.20 0.43 0.67

2.92 2.07 1.28 1.00

0.71 0.62 0.56

ΦTS − Φ

FS

−1.55 −0.76 −0.25

qFS − qIS

ΔE

−0.68 −0.67 −0.81 −0.81

0.69 0.57 0.68 0.83 0.72

qFS − qIS

ΔE

−0.62 −0.68 −0.69

0.96 1.06 1.05 1.08

qFS − qIS −0.75 −0.78 −0.81 −0.84 cell-extrapolated:

qTS − qFS

ΔE

Ea

0.31 0.28 0.26

1.01 1.00 0.94 1.09 0.96

0.48 0.52 0.53 0.54

OH to H2O

a

cell

EFS − EIS

3×3 3×4 3×6

−0.37 −0.23 −0.13

ΦFS − Φ

IS

1.64 1.22 0.80

ETS − EFS

ΦTS − ΦFS

0.46 0.31 0.18

−1.50 −1.02 −0.67

qFS − qIS −0.74 −0.75 −0.75 cell-extrapolated:

qTS − qFS

ΔE

Ea

0.74 0.76 0.76

0.24 0.23 0.17 0.10

−0.10 −0.08 −0.07 −0.04

All E and Φ are listed in eV, q in atomic units e.

activation barriers were calculated using the climbing image, nudged elastic band method with the forces on the climbing image converged to less than 0.02 eV/Å.17 For all systems, we have calculated the charge q using a Bader analysis27,28 of the charge of the Pt slab plus all adsorbates. For instance, in the case of the Heyrovsky reaction, qIS (see image in Table 1) includes the Pt slab, the full monolayer of H* in the fcc sites, as well the H* adsorbed on the top site, which becomes a part of the H2 molecule in the FS. This H is not included in the TS or FS, where it has desorbed.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

Figure 4. Parity plot of reaction energies and barriers obtained using the cell-extrapolation scheme from refs 11 and 15 shown in Figure 3 and the charge-extrapolated values.

Notes

The authors declare no competing financial interest.

■

geometry optimization. Water layers and adsorbates were placed on only one side of the slab and a dipole correction was applied along the axis perpendicular to the slab.26 The two bottom Pt layers were constrained and all other atoms relaxed until the forces on them were less than 0.02 eV/Å. All

ACKNOWLEDGMENTS

This material is based on work supported by the Air Force Office of Scientific Research through the MURI program under AFOSR Award No. FA9550-10-1-0572. 2667


Letter


■

avoided by using only cells that are multiples of the smallest cell (e.g., 3 × 3, 3 × 6, 3 × 9, etc., see Figure 2 of main text), however, it would come with a further increase in computational cost. (21) Medford, A. J.; Wellendorff, J.; Vojvodic, A.; Studt, F.; AbildPedersen, F.; Jacobsen, K. W.; Bligaard, T.; Nørskov, J. K. Assessing the Reliability of Calculated Catalytic Ammonia Synthesis Rates. Science 2014, 345, 197−200. (22) Mortensen, J. J.; Hansen, L. B.; Jacobsen, K. W. Real-Space Grid Implementation of the Projector Augmented Wave Method. Phys. Rev. B 2005, 71, 035109. (23) Enkovaara, J.; Rostgaard, C.; Mortensen, J. J.; Chen, J.; Dułak, M.; Ferrighi, L.; Gavnholt, J.; Glinsvad, C.; Haikola, V.; Hansen, H. A.; et al. Electronic Structure Calculations with GPAW: a Real-Space Implementation of the Projector Augmented-Wave Method. J. Phys.: Condens. Matter 2010, 22, 253202. (24) Bahn, S.; Jacobsen, K. W. An Object-Oriented Scripting Interface to a Legacy Electronic Structure Code. Comput. Sci. Eng. 2002, 4, 56−66. (25) Ogasawara, H.; Brena, B.; Nordlund, D.; Nyberg, M.; Pelmenschikov, A.; Pettersson, L. G. M.; Nilsson, A. Structure and Bonding of Water on Pt(111). Phys. Rev. Lett. 2002, 89, 276102. (26) Bengtsson, L. Dipole Correction for Surface Supercell Calculations. Phys. Rev. B 1999, 59, 12301−12304. (27) Bader, R. F. W. A Quantum Theory of Molecular Structure and Its Applications. Chem. Rev. 1991, 91, 893−928. (28) Henkelman, G.; Arnaldsson, A.; Jónsson, H. A Fast and Robust Algorithm for Bader Decomposition of Charge Density. Comput. Mater. Sci. 2006, 36, 354−360.

REFERENCES

(1) Nørskov, J. K.; Rossmeisl, J.; Logadottir, A.; Lindqvist, L.; Kitchin, J. R.; Bligaard, T.; Jónsson, H. Origin of the Overpotential for Oxygen Reduction at a Fuel-Cell Cathode. J. Phys. Chem. B 2004, 108, 17886−17892. (2) Nørskov, J. K.; Bligaard, T.; Rossmeisl, J.; Christensen, C. H. Towards the Computational Design of Solid Catalysts. Nat. Chem. 2009, 1, 37−46. (3) Peterson, A. A.; Abild-Pedersen, F.; Studt, F.; Rossmeisl, J.; Nørskov, J. K. How Copper Catalyzes the Electroreduction of Carbon Dioxide into Hydrocarbon Fuels. Energy Environ. Sci. 2010, 3, 1311− 1315. (4) Calle-Vallejo, F.; Koper, M. T. M. Theoretical Considerations on the Electroreduction of CO to C2 Species on Cu(100) Electrodes. Angew. Chem., Int. Ed. 2013, 52, 7282−7285. (5) Taylor, C. D.; Wasileski, S. A.; Filhol, J.-S.; Neurock, M. First Principles Reaction Modeling of the Electrochemical Interface: Consideration and Calculation of a Tunable Surface Potential from Atomic and Electronic Structure. Phys. Rev. B 2006, 73, 165402. (6) Dabo, I., Li, Y., Bonnet, N., Marzari, N. In Fuel Cell Science; Wieckowski, A., Nørskov, J. K., Eds.; John Wiley & Sons, Inc.: Hoboken, NJ, 2010; pp 415−431. (7) Otani, M.; Sugino, O. First-Principles Calculations of Charged Surfaces and Interfaces: A Plane-Wave Nonrepeated Slab Approach. Phys. Rev. B 2006, 73, 115407. (8) Sugino, O.; Hamada, I.; Otani, M.; Morikawa, Y.; Ikeshoji, T.; Okamoto, Y. First-Principles Molecular Dynamics Simulation of Biased Electrode/Solution Interface. Surf. Sci. 2007, 601, 5237−5240. (9) Schnur, S.; Groß, A. Challenges in the First-Principles Description of Reactions in Electrocatalysis. Catal. Today 2011, 165, 129−137. (10) Nie, X.; Esopi, M. R.; Janik, M. J.; Asthagiri, A. Selectivity of CO2 Reduction on Copper Electrodes: The Role of the Kinetics of Elementary Steps. Angew. Chem., Int. Ed. 2013, 52, 2459−2462. (11) Rossmeisl, J.; Skúlason, E.; Björketun, M. E.; Tripkovic, V.; Nørskov, J. K. Modeling the Electrified Solid-Liquid Interface. Chem. Phys. Lett. 2008, 466, 68−71. (12) Groß, Axel, Highlight #125: Theory of Solid/Electrolyte Interfaces. Psi-k Highlight. http://www.psi-k.org/newsletters/News_ 125/Highlight_125.pdf (accessed May 2015). (13) Skúlason, E.; Karlberg, G. S.; Rossmeisl, J.; Bligaard, T.; Greeley, J.; Jónsson, H.; Nørskov, J. K. Density Functional Theory Calculations for the Hydrogen Evolution Reaction in an Electrochemical Double layer on the Pt(111) Electrode. Phys. Chem. Chem. Phys. 2007, 9, 3241−3250. (14) Wasileski, S. A.; Janik, M. J. A First-Principles Study of Molecular Oxygen Dissociation at an Electrode Surface: a Comparison of Potential Variation and Coadsorption Effects. Phys. Chem. Chem. Phys. 2008, 10, 3613−3627. (15) Skúlason, E.; Tripkovic, V.; Böjrketun, M. E.; Gudmundsdóttir, S.; Karlberg, G.; Rossmeisl, J.; Bligaard, T.; Jønsson, H.; Nørskov, J. K. Modeling the Electrochemical Hydrogen Oxidation and Evolution Reactions on the Basis of Density Functional Theory Calculations. J. Phys. Chem. C 2010, 114, 18182−18197. (16) Trasatti, S. The Absolute Electrode Potential: an Explanatory Note (Recommendations 1986). Pure Appl. Chem. 1986, 58, 955−966. (17) Henkelman, G.; Uberuaga, B. P.; Jónsson, H. A Climbing Image Nudged Elastic Band Method for Finding Saddle Points and Minimum Energy Paths. J. Chem. Phys. 2000, 113, 9901−9904. (18) Limmer, D. T.; Willard, A. P.; Madden, P.; Chandler, D. Hydration of Metal Surfaces Can Be Dynamically Heterogeneous and Hydrophobic. Proc. Natl. Acad. Sci. U.S.A. 2013, 110, 4200−4205. (19) Hansen, H. A.; Viswanathan, V.; Nørskov, J. K. Unifying Kinetic and Thermodynamic Analysis of 2 e− and 4 e− Reduction of Oxygen on Metal Surfaces. J. Phys. Chem. C 2014, 118, 6706−6718. (20) The work function Φ1 (eq 10 of the main text) to which extrapolation is done is given as an average from all calculations, as small shifts in adsorbate coverage between unit cells led to a slight variation in the work function. In principle, this variation can be 2668


Electrochemical Barriers Made Simple - The Journal of Physical

Recommend Documents