J. Phys. Chem. B 2006, 110, 21833-21839
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Calculated Phase Diagrams for the Electrochemical Oxidation and Reduction of Water over Pt(111) Jan Rossmeisl* and Jens K. Nørskov Center for Atomic-scale Materials Physics, NanoDTU, Department of Physics, Technical UniVersity of Denmark, DK-2800 Lyngby, Denmark
Christopher D. Taylor,† Michael J. Janik,† and Matthew Neurock†,‡ Department of Chemical Engineering and Department of Chemistry, UniVersity of Virginia, CharlottesVille, Virginia 22904 ReceiVed: May 24, 2006; In Final Form: August 25, 2006
Ab initio density functional theory is used to calculate the electrochemical phase diagram for the oxidation and reduction of water over the Pt(111) surface. Three different schemes proposed in the literature are used to calculate the potential-dependent free energy of hydrogen, water, hydroxyl, and oxygen species adsorbed to the surface. Despite the different foundations for the models and their different complexity, they can be directly related to one another through a systematic Taylor series expansion of the Nernst equation. The simplest model, which includes the potential only as a shift in the chemical potential of the electrons, accounts very well for the thermochemical features determining the phase-diagram.
Introduction The atomic and molecular scale processes that occur at the interface between a metal surface and an aqueous solution ultimately dictate the overall electrochemical and electrocatalytic behavior.1,2 The electrochemical potential that forms is the result of ion gradients that are established across the solution phase, doping of the solid phase, or through potentiostatic control. Despite the long and well-established history of electrochemistry at metal surfaces, a clear theoretical description of atomic and electronic scale processes that occur on the electrode surface has yet to be established. The use of molecular dynamics simulations and Monte Carlo simulations with various levels of description of the electrode surface and the electrolyte species has helped to establish the basic concepts surrounding the Guoy-Chapman theory of ion distributions, emerging theories regarding water layering, and the structure and behavior of water close to the electrode.3-5 However, phenomena such as chemical bonding of species to the electrode (chemisorption), ion adsorption with partial discharge, and chemical reactivity require a more complete analysis of the changes of the electronic structure at the electrode surface as well as within the solution phase. The study of condensed matter interfaces, such as the electrodeelectrolyte interface, has been hindered thus far due to its complexity and the lack of clear conventions to describe electrochemical conditions. Recent work, however, indicates that such conventions are emerging.1,2,6-8 It has, for example, been shown that many of the conventions from heterogeneous catalysis and gas-phase reactions can be applied to electrochemistry.2 This requires a reference to connect the gas-phase surface calculations to those under electrochemical control. There are various possible choices, however, as to how one decides upon a good reference state. For example, in the cluster * Address correspondence to this author. † Department of Chemical Engineering, University of Virginia. ‡ Department of Chemistry, University of Virginia.
calculations performed by Anderson et al. the reference state was determined by estimating ∆G for a product and reactant state differing by n electrons (i.e., for a one electron transfer, n ) 1, this corresponds to the ionization potential for the cluster).7 The equilibrium potential was then calculated via
U/V NHE ) ∆G/nF - 4.6 V
(1)
where the term 4.6 V is introduced to account for the work function of the normal hydrogen electrode.9 In the proton/ electron-transfer reactions studied in ref 2, the product and reactant states differ by n(H+ + e), which allows a natural relationship to the normal hydrogen electrode, defined by the equilibrium
H+ + e- T 1/2H2
(2)
Thus the energy of n(H+ + e) can be directly replaced with the energy of H2. The reaction energies at subsequent potentials are then directly determined by raising or lowering the reaction energy in equilibrium with H2 according to the change in potential relative to the standard hydrogen electrode:
∆G ) ∆G(0) + nF(U NHE)
(3)
We will refer to this approach as Model 1. Model 2 is derived from Model 1, and differs only by the inclusion of an electric field that is used to simulate the change in surface interactions with the potential, U, for a slab that is initially charge neutral. Finally, a quite different approach has been adopted by considering the absolute potential of the interface under consideration as the work function of a periodic slab representation of the interface.1,6,10 The work function is therefore related to the normal hydrogen electrode, similar to eq 1, via the following equation
U/V NHE ) Φ/e - 4.6 V
10.1021/jp0631735 CCC: $33.50 © 2006 American Chemical Society Published on Web 10/07/2006
(4)
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Figure 1. Side and top views of the five calculated configurations. From the left: the water layer with the hydrogen pointing up, the water layer with hydrogen pointing down, chemisorbed atomic hydrogen at the surface with a water layer above, the half dissociated water layer where every second water molecule is substituted with HO*, and chemisorbed atomic oxygen at the surface with a water layer above. The coverage of H*, OH*, and O* is 1/3 of a monolayer. There are four water molecules per unit cell.
Filhol and Neurock also introduced a scheme to tune the electrode potential by modifying the surface charge density.1 A second reference is then used to monitor the change in this potential relative to the potential at some distance from the electrode in the solution phase. This approach is referred to herein as Model 3. In this paper we calculate the phase diagram of Pt(111) in water as a function of the applied potential for all three different models to understand the similarities as well as the differences between them. Only the thermochemistry is considered in this work, meaning that we do not investigate the mechanisms or the activation barriers for the phase transitions. In the first scheme, Model 1, the adsorption energies are treated independently of the electrochemical potential, as in the earlier papers2,11 and similar to the scheme developed by Anderson et al.7 We then compare the reaction energies reported using this model to simulations in which the adsorption energy is allowed to vary with the electrochemical potential. In this second scheme (Model 2) the potential dependence is estimated by monitoring the change in adsorption energy of water and its dissociation products as a function of the electric field. The potential is then subsequently determined by estimating the double-layer width and multiplying accordingly. Finally, we compare with the results obtained using the approach from refs 1, 6, and 10 for including the electrode potential and a more extensive model of the electrolyte region, Model 3. These models are described in more detail in subsequent sections. Computational Details For Models 1 and 2 density functional (DFT) calculations using a plane wave implementation12 on a 3-layer 3 × x3 Pt(111) slab were performed on the level of the generalized gradient approximation Revised-Perdew-Burke-Ernzerhof (RPBE)13 for the exchange-correlation term. A 3 × 4 × 1 Monkhorst-Pack mesh was used to sample the first Brillouin zone. Ultrasoft pseudopotentials were used to model the ion cores.14 The electronic wave functions can therefore be represented on plane wave basis set with a cutoff energy of 340 eV. The electron density is treated on a grid corresponding to a plane wave cutoff at 500 eV. A Fermi smearing of 0.1 eV and Pulay mixing is used to ensure a fast convergence of the self-consistent electron density. The atomic positions were relaxed until the sum of the absolute forces was less than 0.05 eV/Å. All calculations were performed with the ASE simulation package.15 Model 3 was examined with the Vienna Ab initio Software Package,16-19 with a 3 × 3 × 1 k-point Monkhorst-Pack20 mesh
of the first Brillouin zone and a plane-wave basis with a cutoff energy of 396.0 eV. Methfessel-Paxton smearing21 of order 2 with a value of σ of 0.2 eV was applied to aid convergence. The model utilized a 3 × 3 periodic simulation cell to capture the surface periodicity and periodicity of the ice-like water overlayer at the interface. An interslab spacing of 14.64 Å was used. Twenty-four water molecules were introduced in this interlayer volume to give a density of 1 g/cm3. Molecular dynamics simulations as described in the paper by Filhol and Neurock1 were used to anneal an appropriate water structure for simulation of the aqueous region. Results and Discussion Model 1. The cathodic and anodic activation of water can lead to various surface intermediates, which we may describe as different surface phases. Four different states of the surface are considered herein: the water covered surface (that is, the clean surface with a water overlayer), the hydrogen covered surface, the hydroxyl covered surface, and the oxygen covered surface. A water layer is introduced as described in refs 3 and 22 in order to model the aqueous/metal interphase. The water layer can point the hydrogen atoms either away from the surface or toward the surface; these two configurations are very close in energy. The surface structures are shown in Figure 1. Surface phase changes may occur as a consequence of proton transfer between water and the adsorbents and electron transfer between the absorbents and the electrode. The reactions connecting the different states of the surface can be written such as
H2O + * T OH* + e- + H+
(5)
OH* + * T O* + e- + H+
(6)
* + e- + H+ T H*
(7)
Here * denotes a free site on the surface, e- is an electron in the electrode, and H+ denotes a proton solvated in the electrolyte. The chemical potential of the protons in the aqueous phase and the electrochemical potential of the electrons are variable parameters that can be used to determine the phase of the system. The chemical potential of the protons is given by the pH of the solution. The potential of the electrons is determined by the bias between the electrode under study and a reference electrode. The standard hydrogen electrode is used here as the reference; as such the potential is zero at pH 0, when
Electrochemical Oxidation and Reduction of Water over Pt(111)
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TABLE 1: The Binding Enthalpies at U ) 0 V for Absorbents without Water from References 2 and 23 and Enthalpies and Free Energy Changes for Pt with Water As Shown in Figure 1a OH* + e - + H+
O* + 2(e- + H+)
H* (e- + H+)
1.05 eV 0.46 eV 0.81 eV
1.57 eV 1.54 eV 1.59 eV
-0.33 eV -0.33 eV -0.09 eV
0.13 eV -0.11 eV
0.34 eV 0.36 eV
-0.51 eV -0.27 eV
1.49 eV 1.25 eV
2.75 eV 2.77 eV
0.21 eV 0.45 eV
Pt ∆E ∆Ew.water ∆Gw.water(Figure 1) Ni ∆E ∆Gw.water Au ∆E ∆Gw.water a
The free energies for the adsorbents on Ni (111) and Au(111) are obtained by assuming that the corrections due to zero point energy, entropy, and the interaction with water are the same as those for Pt.
protons and electrons are in equilibrium with hydrogen gas at standard conditions. The chemical potential for electrons at the electrode is then eU, where U is the bias between the electrodes as discussed above. Model 1 includes the effect of solvation and bias according to the following prescription: 1. By setting the reference potential to that of the standard hydrogen electrode, we can relate the chemical potential (the free energy per H) for H+ + e- in solution to that of 1/2H2 in the gas phase. At pH 0 in the electrolyte and 1 bar of H2 in the gas phase at 298 K the reaction free energy of 1/2H2 f H+ + e- is zero at an electrode potential of U ) 0. At standard conditions, the free energy ∆G0 ) ∆G(U ) 0, pH 0, p ) 1 bar, T ) 298 K) for the reaction of *AH f *A + H+ + e- can therefore be calculated as the free energy of the reaction *AH f *A + 1/2H2. 2. To model the water environment of the electrochemical cell we calculate the binding enthalpies, ∆Ew.water, in the presence of a layer of water, see the discussion above. (If the water layer is not included we denote the binding enthalpy ∆E.) 3. ∆Gw.water ) ∆Ew.water + ∆ZPE - T∆S is calculated as follows: The reaction energy, which refers here to the change in the electronic energy between the products and the reactants (∆E for Ni and Au, or ∆Ew.water for Pt), is calculated directly from the DFT results. The effect of water is subsequently taken into account for the Ni, Au systems by including the interaction term derived from the Pt calculations. The difference in zero point energies due to the reaction, ∆ZPE, and the change in entropy ∆S are determined by using DFT-calculated vibrational frequencies23 and standard tables for the gas-phase molecule.24 4. We include the effect of a bias on all states involving an electron in the electrode, by shifting the energy of this state by ∆GU ) -eU, where U is the electrode potential. 5. At a pH that is different from 0, we can correct for the free energy of H+ ions by the concentration dependence of the entropy or free energy: ∆GpH(pH) ) -kT ln[H+] ) kT(ln 10)pH. In this study we use a pH value of 0. The reaction free energy is then calculated as
∆G(U, pH, pH2 )1 bar, T ) 298 K) ) ∆Gw.water + ∆GU + ∆GpH (8) This method is also described in previous publications.2,11,23 In Table 1 the binding enthalpies relative to gas-phase water and hydrogen are listed, together with the calculated free energies. Ni(111) and Au(111) are included for comparison. The
Figure 2. The phase diagram showing the free energy for different surface structures for water at pH 0 in contact with Au(111), Pt(111), and Ni(111). The figure is based on the free energy values in Table 1 and represents the results of Model 1. The lowest line represents the thermochemically most stable phase. The crossing of the two bottom lines indicates a phase change. The free energy for liquid water and hydrogen gas at standard conditions is defined as ∆G ) 0. The lines with a slope of 1 eV/V are related to H*: ∆GH*w.water(U) ) ∆GH*w.water(0) + eU. The lines with a slope of -1 eV/V are related to OH*: ∆GOH*w.water(U) ) ∆GOH*w.water(0) - eU. The lines with a slope of -2 eV/V are related to O*: ∆GO*w.water(U) ) ∆GO*w.water(0) - 2eU.
phase diagrams for the three metal surfaces at 298 K and pH 0 are shown in Figure 2. At very negative biases all the metals will adsorb hydrogen, and at high positive potentials only oxygen will be present on the surface. However, the potentials at which the phases change are different for the different metals. Gold is noble and therefore it needs quite extreme potentials to bind any of the adsorbates. Nickel on the other hand always has an adsorbate bound to the surface, whereas platinum is somewhere in between. Hydrogen starts to adsorb at platinum at a potential close to U ) 0.09 V, which is consistent with the fact that platinum is a good catalyst for hydrogen evolution under acidic conditions.25 According to the model, the formation of OH* is down hill in free energy for potentials above 0.79 V. This value is not trivially compared to voltammograms on Pt(111). The theoretical value denotes the potential at which reaction 5 is in equilibrium, this means that it should be compared with the reversible potential from the voltammogram. Underpotential deposited species seen in experiments are most likely due to adsorption on defects such as steps, and are therefore not relevant for comparison. The reversible potential for OH adsorption from experiment is ∼0.8 V,26 which is in excellent agreement with the value prediction with the model of 0.79 V. We therefore conclude that Model 1 is in excellent qualitative and quantitative agreement with experimental observations on these points. At the water decomposition potential surface hydroxyl and surface oxygen are very close in energy. Both phases, therefore, will probably coexist. Higher potentials favor the surface oxide. On the Au substrate, water appears to be the thermodynamically favored state over the entire range of potentials examined whereas on Ni(111) water is never the most stable structure. On Ni there is a phase transition from the hydride phase directly to the hydroxyl phase at 0.19 V. The hydroxyl phase, however, is only stable within a very narrow window (0.19-0.21 V) as it further oxidizes to form O*. Model 2. In Model 1 the only way in which the bias enters the calculation is through the chemical potential of the electrons (the -eU term in the free energy). This neglects the fact that the bias will also give rise to electrical fields at the watermetal interface and will change the magnitude and structure of
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Figure 3. The total energy change for the Pt(111) slab as a function of the external field, Eslab(). The clean Pt surface does not have a permanent dipole moment, so in this case only the polarization term has a contribution to the total energy. The calculated points are fit to a parabola represented with the solid line. This contribution to the total energy is subtracted in the binding energies.
the electronic charge on the surface. Because the liquid phase has ions and is electrically conducting, this change in charge on the electrode will be screened by a charge of the same magnitude but opposite in sign from ions attracted to the nearsurface region. The potential drop outside the electrode is therefore restricted to this double layer of opposite charges, and appreciable electrical field strengths can build up in this region. If the average width of the double layer is d, then the average change in the field is of the order δ ∼ δU/d. Typically d is believed to be of the order of a few angstroms, here it is assumed to be 3 Å. The simplest way of accounting for the local field effects induced by a change in bias is to approximate it by a constant field given by the bias as ) U/d. Such a constant field can be imposed in the DFT calculation and the electrostatic effects estimated in this way.27 Including this correction comprises Model 2. The static response of any system to an electric field can be expanded in the field strength. The energy change of a slab with adsorbates and a water layer due to an external field, , can be written
∆E() ) µ - 1/2R 2+ ...
(9)
where µ is the dipole moment of the slab in the direction of the field, and R denotes the static polarizability. We have calculated the energy change due to an external field on the clean slab, and the results are presented as a function of the applied field in Figure 3. Since the clean slab has inversion symmetry, there is no permanent dipole moment and only the polarizability contributes to the energy change. Next we consider the response of the metal slab with the water layer to variation of the surface field. Figure 4 depicts the field-induced change in the energy of a slab in contact with a water layer in which the H atoms point up or down (as in the structures shown in Figure 1). Here we subtract the response from the clean slab to obtain the response of the water overlayer alone. The two different water configurations have dipole moments in different directions, and therefore the µ term is different for the two configurations. It should be noted that the hydrogen-end down structure for adsorbed water is stable up until about +0.1 V. At higher potentials, water is oriented with its oxygen end toward the surface. This change in water structure as a function of potential is known as the water flip-flop
Figure 4. The adsorption enthalpy of the water layers (per four water molecules) as a function of the applied field, ∆Ewater layer(). Red is the water with hydrogen pointing up away from the metal surface and black is the water with hydrogen pointing toward the surface. The structures are allowed to relax, but only a minor relaxation is taking place. The potential of the electrode we estimate by assuming a 3.0 Å thick double layer, U is then 3 Å times the field. The estimated values of the potential are shown on the upper x-axis.
mechanism (also found for water over Pd1 and with molecular dynamics simulations28 and cluster calculations29 and X-ray surface studies by Toney30). The adsorption energy of the water layer is calculated as follows: ∆Ewater layer() ) Etotal() - Eslab() - Eslab - 4Ewater in a vacuum, where Etotal() is the total energy of the water covered surface, Eslab() + Eslab denotes the total energy of the clean surface at a given field strength, and Ewater in a vacuum is the energy of a water molecule in a vacuum. Finally we have included the adsorbed species, and calculated the corresponding field-induced change in the adsorption energy. In these calculations we have subtracted the response of the water overlayer, since it is present in all the surface phases. The exception is adsorbed OH* where no polarization of the water is corrected for, since the water molecules’ dipoles are close to parallel with the surface. The binding enthalpy at a given field strength is calculated by using the equation
∆Ew.water() ) Etotal() - Etotal(0) + ∆Ew.water(0) ∆Ewater layer() + ∆Ewater layer(0) - Eslab() (10) where Etotal() is the total energy of the system, ∆Ewater layer() is the adsorption enthalpy of the water layer, shown in Figure 4, and Eslab() denotes the total energy of the clean platinum slab. The changes in Ew.water() are shown in Figure 5 for H*, OH*, and O*. The results in Figure 5 indicate that the effect of field strength on the adsorption of H, OH, and O is small. In Model 2 we include the effect of the electrical field assuming that the width of the double layer is d ) 3 Å. The corresponding phase diagram for Pt(111) in shown in Figure 6. The results presented in Figure 6 show that Model 2 closely replicates Model 1, except for the stabilization of adsorbed OH, so that there is now a window of potentials where OH* is the most stable surface species. Model 3. The third model presented in this paper is based upon the strategy developed for considering the activation of water on Pd(111).1 An electric field is induced at the interface by changing the number of electrons available to the metal of the electrochemical system. This effectively introduces a surface charge density of q/2A, where q is the charge applied and A is the area of the slab face. To maintain cell neutrality, and thereby meet the requirements for periodic calculations, a homogeneous
Electrochemical Oxidation and Reduction of Water over Pt(111)
Figure 5. The adsorption enthalpy of the intermediates as functions of the applied field, calculated in the presence of water, ∆Ew.water(). The potential of the electrode, U, was estimated by multiplying the thickness of the double layer (assumed here to be 3.0 Å) by the field that is reported here in terms of V/Å. The estimated values of the potential are shown on the upper x-axis.
Figure 6. The electrochemical phase diagram for the reduction and oxidation of water over Pt(111) at pH 0 as a function of bias. The red lines are the results from Model 2, while the black dotted lines represent the results from Model 1 also shown in Figure 2 (middle section). There is a very good agreement between the two model phase diagrams, which indicates that the adsorption enthalpies are almost conserved when the potential is changed. However, OH* is stabilized so that it is present on the surface around 0.8 V. The dashed lines that result from Model 1 are the following: GH(U) ) EH(0) + 0.24 eV + eU, GOH(U) ) EOH(0) - 0.24 eV - eU, and GO(U) ) EO(0) + 0.05 eV - 2eU. The red lines which refer to Model 2 ar ethe following: GH(U) ) EH(U) + 0.24 eV + eU, GOH(U) ) EOH(U) - 0.24 eV - eU, and GO(U) ) EO(U) + 0.05 eV - 2eU.
countercharge of uniform charge density -q/AL is distributed across the entire supercell, where L is the cell height. The interlayer space is then completely saturated with H2O at a density of 1 g/cm3. In this way the homogeneous countercharge is embedded within a dielectric environment, and therefore has a screened electrostatic interaction with the slab. Previous calculations showed that the field which is induced in this way closely replicates the double layer environment that results from a true polarization by actual ions in the outer Helmholtz plane, assuming that the unit cell that is used is of appropriate size.6 The supercell geometry that is shown in Figure 7 is used here to model the activation of water over Pt(111). Panels a-d in Figure 8 depict the optimized adsorption geometries for the surface hydrogen, surface hydroxyl, and surface oxygen phases. In the absence of an applied potential, water and hydroxyl adsorb at the atop sites, as has been shown previously31 for water and hydroxyl adsorption on Pt(111), whereas oxygen and hydrogen
J. Phys. Chem. B, Vol. 110, No. 43, 2006 21837
Figure 7. Supercell system used to describe the metal/solution interface in Model 3. Three supercells are shown.
adsorb at 3-fold hollow sites.32 As in Model 2, the hydrogen atom was placed at the Pt(111)/water interface where the water has its hydrogen atoms directed toward the metal surface. The energies of the systems were then calculated at potentials corresponding to surface charges of -15 through to +15 µC/ cm2. This is accomplished by either adding or subtracting between 0, 0.5, or 1 electrons to or from the number of electrons required for neutrality within the unit cell. As in Models 1 and 2, the difference of a proton and electron between the phases is remedied by the addition of H2 energies where appropriate, modified by the electrochemical potential of the phase (eqs 2 and 3). In this model (Model 3), the electrochemical potential is determined by the direct reference to the electrostatic potential drop observed across the interface, as obtained by Poisson’s equation applied to the self-consistent electron density of the system. In this way the work function is determined for the metal/electrolyte system, which can then be straightforwardly related to the electrochemical potential via
U0 ) ∆Φ|slab,vac - 4.6 V NHE
(11)
where ∆Φ|slab,vac refers to the work function of the metal in a vaccuum. The potentials calculated in this way are clearly dependent upon the exact configuration of the water molecules and adsorbate species within the unit cell. Variations in the configurations used indicate that the total error introduced in the potential is in fact considerable and of the order of 0.3 V. For charged phases, it is not possible to make such a direct work function measurement, due to the presence of electric field effects throughout the vacuum region within the periodic simulation. Equation 5 therefore only applies to the neutral slab case. The potential for the charged slabs is determined consequently by reference to the potential of the solution layer in the neutral case:
Uq ) ∆Φ|slab,solution - Φsoln,0 + U0
(12)
The energy of the slab is then corrected to remove the energy contribution from the background countercharge, and the excess electrons such that the energies of the variously charged slabs are directly comparable.
E ) Eslab -
∫0q 〈V(q)〉 dq + qU
(13)
More details on this model can be found in a separate publication.6
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Figure 8. Snapshots indicating the adsorption geometry of (a) water, (b) oxygen, (c) hydroxyl, and (d) hydrogen for the neutral slabs utilized in Model 3.
Figure 9. Potential-dependent energies for water and hydroxyl dissociation and hydrogen ion adsorption on Pt(111) as determined with Model 3. Phase energies are shown in panel a, and reaction energies, relative to the inert H2O phase, in panel b. Dashed lines in panel b indicate energies determined with Model 1, for the surface/solvent ensembles treated in Model 3.
The phase diagram for water over Pt(111) derived with Model 3 is presented in terms of absolute (atomization) energies in Figure 9a, and in terms of binding energies in Figure 9b. The free energy corrections based on the zero point energies and entropic terms used in the previous two models have been applied.2 The equilibrium potential for the adsorption of hydrogen is +0.16 V and that for oxygen adsorption is +0.5 V NHE. As observed in Models 1 and 2, the regions for oxygen and hydroxyl adsorption are quite close, and water activation to form the hydroxyl overlayer is not exothermic until 0.63 V NHE. These observations are qualitatively similar to those provided by Model 2 [recall 0.08 V (H*), 0.74 V (OH*), and 0.82 V (O*)], but contain shifts which are likely due to the difference in exchange-correlation functional and in interactions between the environment water and the adsorbates between the two approaches. For example, as in Model 1 above, Model 3 predicts that O formation will thermodynamically precede OH
formation on Pt(111), although since the formation of O is expected to be a stepwise procedure, realistically the two species will be present on the surface at the same time, depending on the kinetics of OH dissociation over the Pt(111) surface. Furthermore, changes in the number of hydrogen bonds between Models 1, 2, and 3 can modify adsorption energies by as much as -0.2 eV per additional hydrogen bond. A close examination of the many configurational arrangements of water possible about the adsorbate species is required to discern these subtle effects. The best approach may therefore involve the coupling of one or more of the models in this paper with a molecular dynamics simulation to determine statistics-based estimates of the adsorption free energies. Molecular dynamics simulations will also improve the sampling of the zero charge potential used as the reference potential for each phase. It is also instructive to directly compare the results of Model 3 with those obtained by the computationally simpler approach of Model 1. Since this model utilizes a different exchangecorrelation functional and a different solvation environment for the adsorbed species than Models 1 and 2, we have reapplied the methodology of Model 1 to calculating a phase diagram based on the atomic configuration optimized under neutral slab conditions as displayed in Figure 8. These energies are shown as the dotted lines appearing in Figure 9b. The phase diagram of Model 1 closely resembles that of Model 3. However, a small linear deviation from the unity slope predicted by the Nernst law occurs due to the interaction between the dipole of adsorbents and the field, as also seen in Model 2. There is qualitative agreement between the slopes found in Model 2 and Model 3 except for hydroxyl on the surface. This disagreement is probably due to the small number of water molecules included in Models 1 and 2. Model 2 shows that the nonunity slope is a consequence of the interactions between the field and the dipole moment (see eq 9). This means that whenever a nonunity slope is observed it need not indicate a partial charge transfer, but rather a dipole moment. Gibb’s relation can be written as
F(n +∆µ/d) ) ∆G/∆U
(14)
as ) ∆U/d as given above (recall that d is the width of the double layer, is the applied electric field, and ∆µ is the change in dipole moment for the reaction). We note that the dipole moment is a physical observable that can be measured as a work-function change or calculated directly as described above. The surface energy profiles in Figure 9a are well-approximated by a quadratic expansion about the potential of zero charge, with the second-order coefficient given by the capacitance of the interface, C ) -∂2G/∂U2:
G ) G0 - C/2(U - U0)2
(15)
The reaction energy plots in Figure 9b are linear, however, which suggests that the second-order term, the capacitance, is
Electrochemical Oxidation and Reduction of Water over Pt(111) close to being constant between the phases (the inferred capacitances are all close to 17 µF/cm2). If this were not the case, a second-order expansion of the Nernst equation would be required:
∆G(U) ) ∆G(U0) + F(n + ∆µ/d)(U - U0) + 1
/2∆CF(U - U0)2 (16)
Such an expansion may be required when a reaction involves considerable change in the surface dipole, either by the rearrangement of ions at the interface, or by species containing zwitterionic forms that are altered upon adsorption (such as amino acids). It can also be seen that this expansion is the equivalent of eq 9, but expanded in terms of the electrochemical potential, rather than the electric field. Conclusion We have presented three different models for the description of the complex thermochemistry of water interacting with a Pt(111) surface as a function of electrical bias. The three models differ considerably in complexity and in the amount of computational resources necessary to model and it is therefore important to compare the results of the different models to obtain an idea of the strengths and weaknesses. It is clear that for the systems considered here, the three models give nearly the same results. Polarization of the interface by using either Model 2 or Model 3 leads to only small changes in the adsorption energy of OH, H, and O relative to H2O. Since Model 1 is much simpler to apply, this suggests that it is a good starting point for these kinds of investigations, in particular for screening and trend studies. We anticipate that Model 1 will fail for systems (adsorbates) with large dipole moments. Here Model 2 or 3 must be applied. Quantitative differences between the results of Model 2 and Model 3 presented here are likely due to the difference in net water structure at the interface used in these static calculation methods and different exchange-correlation functionals used. Therefore, further research is needed to assess the complex role of hydrogen bonding to stabilize surface intermediates and to improve the estimation of these effects by static methods. For this kind of investigation, and investigations of the kinetics of surface processes involving a charge transfer between the slab and the electrolyte, Models 1 and 2 will not be sufficient, since they can only describe the thermochemistry. Here Model 3 or some other model that properly treats the charge state of the surface must be used. Acknowledgment. This work was supported by the Director, Office of Science, Office of Basic Energy Sciences, Division
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