Towards First Principles-Based Prediction of Highly Accurate

Article pubs.acs.org/JPCC

Towards First Principles-Based Prediction of Highly Accurate Electrochemical Pourbaix Diagrams Zhenhua Zeng,† Maria K. Y. Chan,‡ Zhi-Jian Zhao,† Joseph Kubal,† Dingxin Fan,† and Jeffrey Greeley*,† †

School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907, United States Center for Nanoscale Materials, Argonne National Laboratory, Lemont, Illinois 60439, United States

‡

S Supporting Information *

ABSTRACT: Electrochemical potential/pH (Pourbaix) diagrams underpin many aqueous electrochemical processes and are central to the identification of stable phases of metals for processes ranging from electrocatalysis to corrosion. Even though standard DFT calculations are potentially powerful tools for the prediction of such diagrams, inherent errors in the description of transition metal (hydroxy)oxides, together with neglect of van der Waals interactions, have limited the reliability of such predictions for even the simplest pure metal bulk compounds, and corresponding predictions for more complex alloy or surface structures are even more challenging. In the present work, through synergistic use of a Hubbard U correction, a state-of-the-art dispersion correction, and a water-based bulk reference state for the calculations, these errors are systematically corrected. The approach describes the weak binding that occurs between hydroxyl-containing functional groups in certain compounds in Pourbaix diagrams, corrects for self-interaction errors in transition metal compounds, and reduces residual errors on oxygen atoms by preserving a consistent oxidation state between the reference state, water, and the relevant bulk phases. The strong performance is illustrated on a series of bulk transition metal (Mn, Fe, Co, and Ni) hydroxides, oxyhydroxides, binary, and ternary oxides, where the corresponding thermodynamics of redox and (de)hydration are described with standard errors of 0.04 eV per reaction formula unit. The approach further preserves accurate descriptions of the overall thermodynamics of electrochemically relevant bulk reactions, such as water formation, which is an essential condition for facilitating accurate analysis of reaction energies for electrochemical processes on surfaces. The overall generality and transferability of the scheme suggests that it may find useful application in the construction of a broad array of electrochemical phase diagrams, including both bulk Pourbaix diagrams and surface phase diagrams of interest for corrosion and electrocatalysis.

1. INTRODUCTION Transition metal oxides, hydroxides, and oxyhydroxides oxyhydroxides (together termed “(hydroxy)oxides”) are active materials for a variety of electrochemical processes, including fuel cells, electrolyzers, photoelectrochemical cells, and batteries.1−8 Electrochemical potential/pH (Pourbaix) diagrams, in turn, provide a comprehensive thermodynamic description of the phase stability of these materials in aqueous electrochemical environments and, as such, are critical tools for understanding the durability or degradation of these systems as a function of electrode potential and pH.4,9−13 Although high quality data to construct Pourbaix diagrams are often available for pure metal bulk materials, such information is generally lacking for more complex alloys, nanostructures, and surfaces/ interfaces. Theoretical analyses based on density functional theory (DFT) calculations, however, have the tantalizing potential to yield predictions of these properties, provided © XXXX American Chemical Society

that calculations of formation energies with sufficient accuracy can be efficiently carried out. Calculation of reaction and formation energies in even simple 3d transition metal oxides is a challenge with standard DFT-GGA calculations, as inaccuracies are introduced both by self-interaction errors on d-orbitals of transition metals with different oxidation states and by the incorrect description of the gas phase O2 reference state,4,11,12,14−23 resulting in incomplete error cancellation when computing the differences between the compounds’ total energies and the total energies of their elemental constituents.19 To address the above problems, a variety of methodologies have been proposed.11,14,16−19,24−26 Some of these methods use Received: April 1, 2015 Revised: July 5, 2015

A

DOI: 10.1021/acs.jpcc.5b03169 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

Table 1. Compounds Included in the U Fitting and the Transition Metal Reference State Correction,a the Crystal Structures and the Magnetic Structures Used in the DFT Calculations, Resulting Absolute Magnetic Moments (MOM) on each Distinct Transition Metal Site at the Optimal U Value, Experimental and Predicted Standard Enthalpies of Formation, the Corresponding Deviation, and the Root Mean Squared Deviation (rmsd) crystal structure FeO Fe(OH)2 Fe3O4 α-FeOOH α-Fe2O3 CoO β-Co(OH)2 Co3O4 β-CoOOH LiCoO2 NiO β-Ni(OH)2 LiNiO2 BaNiO3 MnO Mn(OH)2 Mn3O4 α-Mn2O3 LiMn2O4 rmsd

Fm3m ̅ (refs 14, 47) P3̅m1 (ref 51) Fm3̅m (ref 14) Pnma (refs 54, 55) R3̅c (ref 14) Fm3̅m (refs 14, 47) P3̅m1 (ref 51) Fm3m ̅ (ref 14) R3m (ref 59) R3m (ref 60.) Fm3̅m (refs 14, 47) P3̅m1 (ref 51) R3m (ref 65) P63/mmc (ref 66) Fm3̅m (refs 14, 47) P3̅m1 (ref 51) I41/amd (refs 14, 16) Pbca (refs 14, 16) Fd3m (ref 72)

magnetic structure AFM (refs 14, 47) AFM (ref 52) FiM (ref 53) AFM (refs 54, 56) AFM (ref 14) AFM (refs 14, 47) AFM (ref 58) AFM (ref 14) DM (ref 60) DM (ref 60) AFM (refs 14, 47) AFM (ref 63) AFM (ref 65) DM (refs 66, 67) AFM (refs 14, 47) AFM (ref 69) FiM (ref 71) FM (refs 14, 16) AFM (ref 73)

MOM (μB) 3.58 3.71 3.90,b 3.80,c 3.91c 4.06 3.98 2.66 2.75 2.65,b 0.0c 0.0 0.0 1.68 1.77 1.29 0.0 4.57 4.65 4.54,b 3.75,c 3.91c 3.86 3.87,c 3.15c

∇fH0298 (eV, exp) −2.78 (refs 41−43) −5.95 (refs 41, 42) −11.57 (refs 41−43) −5.82 (refs 42, 43, 57) −8.54 (refs 41−43) −2.78 (refs 41−43) −5.69 (refs 42, 43) −9.43 (refs 41−43) −4.73 (ref 61) −7.03 (refs 43, 62) −2.78 (refs 42, 43) −5.74 (ref 64) −6.15 (refs 43, 62) −9.52 (ref 68) −2.78 (refs 41−43) −7.24 (ref 70) −14.38 (refs 41−43) −9.94 (refs 41−43) −14.31 (ref 72)

∇fH0298 (eV, calc) a −5.99 −11.61 −5.84 −8.52 a −5.67 −9.48 −4.83 −7.03 a −5.69 −6.15 −9.50 a −7.27 −14.34 −10.01 −14.34

deviation (eV) −0.04 −0.05 −0.02 0.02 −0.06 −0.05 −0.10 −0.01 0.04 0.00 0.02 −0.03 0.03 −0.07 −0.03 0.04

a

The experimental formation enthalpies of the monoxides for each metal were used to determine the corresponding metallic reference energy; therefore, no calculated values of the formation enthalpies are reported for these monoxides. See text for additional details. bA sites in spinel structure AB2O4. cB sites in spinel structure AB2O4.

hybrid functionals,16,17 while many others employ the wellknown Hubbard U correction,27 which seeks to correct for incomplete cancellations in self-interaction errors that arise from changes in oxidation states and often implies some amount of fitting to experimental data.11,14,17−19,23,24 As an example in the latter category, a strategy for prediction of formation energies of a variety of bulk oxides involves adjusting the total energy of the gas phase O2 molecule through a fit to non-transition metal oxides, fitting Hubbard U values based on the experimental oxidation energies of transition metal oxides, and correcting the elemental transition metal energies by fitting to binary oxide formation enthalpies.14,18 The shifted O2, fitted U, and corrected elemental metal energies are then used to calculate formation enthalpies of other related compounds that were not included in the original data set. In attempts to extend this strategy to aqueous electrochemical systems involving reactivity of hydrides, hydroxides, and/or oxyhydroxides, an extra shift of the H2 total energy has also been considered, by fitting either to the experimental H2O formation energy or to non-transition metal hydride formation energies.4,11,24 These approaches have achieved success in predicting the formation enthalpies of subsets of hydrides, oxides, hydroxides, and oxyhydroxides,4,11,24 but the techniques do not generally describe all of these classes of materials simultaneously, and they may not provide consistent descriptions of either the bulk thermodynamics of liquid or gas reactions or of the gas phase atomization energies of O2 and H2. An alternative strategy, involving use of hybrid functionals, yields accurate predictions of gas phase thermodynamics and also avoids the U fitting process,28 but errors in the formation energy of solid compounds are still relatively large in comparison to errors from many U schemes (see Supporting Information).16,17 Given these considerations, to provide accurate descriptions of

Pourbaix diagrams and related electrochemical stability phenomena, it is important to seek methods that overcome some of these challenges. In the present study, we introduce a synergistic approach that provides accurate predictions of Pourbaix diagrams for bulk transition metal oxides, hydroxides, and oxyhydroxides while simultaneously preserving good descriptions of related solid, liquid, and gas phase thermodynamics. The strategy combines Hubbard U corrections,4,12,14,27 use of van der Waals corrected density functionals, 2 9 and a water-based reference state.10,20,30−32 In addition to the crucial role of Hubbard U corrections in reducing self-interaction errors, the use of van der Waals functional provides a good description of weak intermolecular forces, such as intermolecular OH bonds, that are present in some transition metal hydroxide and oxyhydroxides. Further, the use of the water reference state allows overall liquid phase reaction thermodynamics to be accurately described, in turn facilitating subsequent electrocatalytic calculations that might be performed using the method, and also preserves a consistent oxidation state for O atoms between the reference state and most bulk phases. Indeed, for any species in the Pourbaix diagram that has the same oxidation state for oxygen as in water, the approach predicts a series of reaction enthalpies with root-mean-square deviations (rmsd) of 0.04 eV per formula unit compared to high quality experimental data. We demonstrate that the scheme is general and transferable and could therefore be used for the construction of a broad range of electrochemical Pourbaix diagrams, with potential applications to improved predictions of corrosion and electrocatalysis. B


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we refer all energetics to gas-phase H2O and H2,10,20,30 and we note, in passing, that the energetics of bulk liquid water, which are needed for generation of Pourbaix diagrams, can be derived from an entropic correction to this gas phase reference, as discussed below and in the Supporting Information. To further provide a reasonable description of van der Waals interactions in solids, which are expected to be particularly significant for layered Fe(OH)2 and FeOOH compounds, we use the selfconsistent van der Waals density functional (vdW-DF) optPBE, since this modified functional has shown good performance in solids.29 To determine the value of the Hubbard U parameter, which is applied to the d-orbitals to provide a better description of on-site Coulomb interactions,4,12,14,27 we generate four linearly independent reactions involving the five bulk compounds described above, together with gaseous H2 and H2O (see Figure 1a):

2. COMPUTATIONAL DETAILS Spin-polarized density functional theory (DFT), DFT+U, and hybrid HSE0633 calculations are performed based on the projected augmented wave (PAW) method34−36 as implemented in the Vienna Ab-initio Simulation Package (VASP).37,38 For DFT+U calculations, U−J values ranging from 0 to 7 eV, with an interval of 0.5 eV, are applied to dorbitals of transition metal elements,27 and the total energies are evaluated both with the generalized gradient approximation (GGA-PBE39) and with the van der Waals density functional (vdW-DF) optPBE.29 The unit cell shape, volume, and internal structural parameters are self-consistently optimized by minimizing the stress tensor and atomic forces, and all results are reported for geometries that have been self-consistently optimized with the corresponding functional and U values. A cutoff energy of 500 eV is used for the planewave expansion in all calculations. Monkhorst−Pack k-point grids are used for Brillouin zone integration. For antiferromagnetic (AFM) monoxides with two formula fcc unit cells, we use a (6 × 6 × 6) grid. For the other (hydroxy)oxides, we use equivalent or denser k-point grids. An orthorhombic box (14 × 15 × 16) Å3 and a single k-point (0.25, 0.25, 0.25) for the Brillouin zone sampling are employed for gas phase species. The equilibrium geometries are obtained when the maximum atomic forces are smaller than 0.01 eV/Å and by employing a total energy convergence of 10−5 eV for the electronic self-consistent field loop. For transitional metal (hydroxy)oxides, the aspherical contributions (LASPH) from the gradient corrections inside the PAW spheres, which we find to have some influence on the determination of total energies in the present study, particularly on Ni and Co (hydroxy)oxides, have been self-consistently included in the calculations (examples of the impact of LASPH on total energies and U values are provided in the Supporting Information, including Figures S1−S4 and Table S8). For those compounds whose crystal structure and/or magnetic structure has not been fully elucidated, we have performed an extensive analysis starting both from corresponding low energy structures in the Materials Project database40 and from well-known low energy structures of the other compounds with the same stoichiometry in the present study, together with magnetic structures with all possible linear combinations. The free energies of Fe-containing compounds and gas phase species are evaluated by considering contributions from zero point energy, heat capacity, and entropy at finite temperature. By further decomposing these thermodynamic data of Fe compounds into the contribution of each constituent elemental species, we identify more efficient, simplified strategies that are in turn used to evaluate the free energies of Mn, Co, and Ni-containing compounds. Details of this approach are provided in the text and in the Supporting Information.

FeO + H 2O → FeOOH + 0.5H 2

(1a)

2FeO + H 2O → Fe2O3 + H 2

(2a)

3FeO + H 2O → Fe3O4 + H 2

(3a)

0.5Fe3O4 + H 2O + 0.5H 2 → 1.5Fe(OH)2

(4a)

To provide maximum sensitivity to changes in U, we require that these reactions involve a change in iron oxidation state. Reactions that involve such a change will have incomplete cancellation of self-interaction errors on the iron orbitals, and the reaction energies will thus be more likely to vary substantially as a function of U. This variation, in turn, facilitates the identification of the U value that provides the best agreement with the experimental data for all considered reactions. To compare the experimental reaction enthalpies at room temperature, we have taken into account the zero point energy (ZPE) and the integrated heat capacity up to 298 K (δH 0→298K ); these quantities are taken from standard thermodynamic tables for the gas phase41 and are calculated using Γ-point phonons for the solids (see Supporting Information). As the experimental enthalpies from different sources differ by up to 9 kJ/mol for some compounds in the present study,41−43 we have averaged the reported experimental values. Additionally, the reaction enthalpy is calculated on a per H2O basis (and per 0.5O2 basis for the O2 shift methods, as discussed below) in order to make it commensurate for different reaction stoichiometries. Finally, using this procedure, we evaluate the root mean square deviation (rmsd) of the calculated reaction enthalpies with respect to the experimental values. As seen in Figure 1, the minimum in rmsd, 0.03 eV, occurs for a value of U = 2.56 eV. At this value of U, the ordering of the calculated reaction enthalpies is identical to that of the experimental results. The excellent agreement with experimentally measured reaction enthalpies in the present approach compares favorably with alternative strategies for reaction enthalpy prediction.14,18,19,24 Although these alternative techniques have been highly successful in their chosen materials areas, they have generally been developed for subsets of the total materials space that must be examined to describe complete Pourbaix diagrams, and they may, therefore, not describe all regions of this space with comparable accuracy. For example, as mentioned above, a typical strategy involves combining DFT-PBE+U calculations with shifts of the gas phase O2 energy determined by comparison to experimental formation enthalpies in non-

3. RESULTS AND DISCUSSION Iron oxides (FeO, Fe2O3, Fe3O4), hydroxides (Fe(OH)2), and oxyhydroxides (FeOOH) have well-defined crystal structures, known magnetic structures, and high quality experimental thermodynamic data (Table 1), and as such, they constitute a convenient system for the description of our synergistic approach: the combination of a Hubbard U correction, a selfconsistent dispersion correction, and a proper bulk reference state in the prediction of electrochemical Pourbaix diagrams of transitional metal (hydroxy)oxides. To avoid well-known problems with DFT in describing the gas phase O2 energy, C


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(2b)

3FeO + 0.5O2 → Fe3O4

(3b)

0.5Fe3O4 + 0.5O2 + 1.5H 2 → 1.5Fe(OH)2

(4b)

As shown in Figure 1b, plotting the resulting reaction enthalpies as a function of U indicates that a U value of approximately 3.0 eV provides an excellent description of the energetics of reactions involving only oxides, including 2FeO + 0.5O2 → Fe2O3 and 3FeO + 0.5O2 → Fe3O4. However, since the technique does not explicitly incorporate vdW effects or reference state corrections for hydrogen-containing moieties, the description is less accurate for reactions containing hydroxides or oxyhydroxides. Indeed, the ranking of calculated reaction energies is qualitatively different from the experimental results, and no single U value can accurately describe all of these reaction energies. To address the challenge related to the reference states of the hydrogen-containing species, two H2 correction strategies have been proposed, involving either preserving the experimental H2O formation enthalpy or correcting the H2 energy by fitting to experimental formation enthalpies in non-transition metal hydrides.4,11,24 The former strategy preserves the bulk water formation energy and is effective in generating Pourbaix diagrams of intermediate levels of quantitative accuracy (Figure 1 and Supporting Information). The approach does, however, overcorrect the H2 gas phase bond energy, and consequently hydride formation enthalpies (0.80 eV/H2 on average; see Supporting Information), which are in fact reasonably well described by standard DFT calculations.15,24,42,44 As a result, use of the approach to describe surfaces, where the hydrogen coverage may vary substantially as a function of pH and potential, may involve non-trivial errors. The latter strategy, in turn, works very well for hydride-based materials, but it has the disadvantage of not guaranteeing a reasonable formation energy of water from H2 and O2 (−0.77 eV error per H2O), leading to difficulties for both hydroxide and oxyhydroxide bulk systems (see Supporting Information) and for surface calculations where the coverage of O or OH varies substantially as a function of conditions. To generate full electrochemical Pourbaix diagrams from our results, the data at optimized U are used to calculate standard enthalpies of formation (see Table 1 and Table 2) with respect to a metallic Fe reference state whose enthalpy (including ZPE and δH terms) is, in turn, fit to the experimental enthalpy of the Fe + H2O → FeO + H2 reaction; this procedure is effectively identical to determining a separate value of U for metallic Fe (see Table 3 and Supporting Information), as described in the literature.45−47 To construct the Pourbaix diagrams, the free energies of formation are calculated by considering the entropies of solid and gas phase species (see Supporting Information), and the chemical potential of liquid water is determined from gas phase H2O at a partial pressure of 0.035 bar, the vapor pressure of water at 298 K. The Pourbaix diagrams at an alkaline pH of 13 are shown in Figure 2 on a reversible hydrogen electrode (RHE) scale, where HFeO2−, the likely alkaline counterion, is set to a typical concentration 1 × 10−6 M.9 Both the calculated diagram, using the water reference state with vdW correction, and the corresponding experimental results show a metallic region at low voltages, followed by a small region of stability for Fe(OH)2. As the potential continues to increase, Fe(OH)2 transforms to Fe3O4 at around −0.1 V and finally to Fe2O3 or FeOOH at around 0.15 V. For comparison, we show the full Pourbaix diagrams (potential−pH

Figure 1. Reaction enthalpies as a function of Hubbard U: (a) a water reference state with vdW corrections (H2O reference + vdW) and (b) a standard O2 reference state shifted based on the formation energy of non-transition metal oxides without vdW corrections (O2 shift). The experimental reaction enthalpies are indicated by horizontal dashed lines with the same color scheme. (c) The rmsd profile as a function of U with different reference state schemes. In addition to the O2 shift and H2O reference + vdW approaches, comparisons are made to a strategy using the O2 shift plus a H2 shift based on the formation energy of non-transition metal hydrides (O2 shift + H2 shift (MHx base)), an H2 shift determined by preserving the H2O formation energy (O2 shift + H2 shift (H2O base)), and an H2O reference scheme without vdW corrections (H2O reference); see text and Supporting Information for details. In addition to the reactions involving a change in iron oxidation state, which are used in the U fitting process, two reactions that do not involve a change in iron oxidation state (FeO → Fe(OH)2 and Fe2O3 → FeOOH) are also plotted to show the performance of the water reference scheme with vdW corrections.

transition metal oxides,14 where self-interaction errors are not significant. We have applied this so-called O2 shift strategy to the following reactions of Fe-containing compounds, which are modified from reactions 1a−4a to account for the change in reference state, using the PBE functional. FeO + 0.5O2 + 0.5H 2 → FeOOH

2FeO + 0.5O2 → Fe2O3

(1b) D


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Table 2. Compounds Not Included in the U Fitting, Crystal Structures and Magnetic Structures Used in the DFT Calculations, Resulting Absolute Magnetic Moments (MOM) on Each Distinct Transition Metal Site at the Optimal U Value, Experimental and Predicted Standard Enthalpies of Formation, and the Corresponding Deviation

Co2O3 CoO2 Ni3O4 Ni2O3 β-NiOOH NiO2 γ-MnOOH β-MnO2 CoFe2O4 NiFe2O4 MnFe2O4 a

crystal structure

magnetic structure

MOM (μB)

R3̅c P3̅m1 (ref 74) Fm3̅m R3̅c P1 C2m/R3m (refs 74, 75) P21/c (ref 76) P42/mnm (refs 14, 16) Imma (ref 78) Fd3m (ref 79) Fd3m (ref 80)

DM (ref 60) FM FiM AFM FM DM (ref 67) AFM (ref 77) AFM (refs 14, 16) FiM (ref 78) FiM (ref 79) FiM (ref 80)

0 1.20 1.89,a 1.42b 1.65,c 0.17d 1.07 0.0 3.83 1.94 2.58b (Co), 3.99a (Fe), 4.11b (Fe) 1.67a (Ni), 4.14b (Fe) 4.40a (Mn), 4.11b (Fe)

∇fH0298 (eV, calc)

deviation (eV)

−6.56 (ref 48), −6.36 (ref 49) −5.40 (ref 6.) −11.28 (ref 42) −11.24 (ref 42) −11.74 (ref 42)

−6.21 −2.89 −7.28 −4.90 −4.11 −2.25 −6.62 −5.30 −11.28 −11.23 −11.74

0.10 0.00 0.01 −0.01

A sites in spinel structure AB2O4. bB sites in spinel structure AB2O4. cWith short Ni−O bond length. dWith longer Ni−O bond length.

Table 3. Summary of Determined U Values for Transition Metal (Hydroxy)oxides, and the Corresponding Values for Metallic Reference State Determined from the Reaction M + H2O → MO + H2 by Preserving the Experimental Reaction Enthalpiesa

plots, without free energy data) in later panels in Figure 2. In all cases, both the qualitative and quantitative agreement of our approach with the experimental data are excellent. For reference, we also plot corresponding Pourbaix diagrams that would be generated by the various alternative approaches described above. These plots show either increased quantitative errors compared to experiment or, in some cases, qualitative discrepancies (Figure 2a and Supporting Information). To test the generality of the present scheme, we have extended the analysis to other transition metal systems, including Ni, Co, and Mn oxides, hydroxides, and oxyhydroxides. As with Fe, we choose only compounds for which unambiguously accurate experimental data are available,

U (eV) M MOxHy a

∇fH0298 (eV, exp)

Mn

Fe

Co

Ni

1.89 3.70

1.56 2.56

1.94 3.50

2.89 5.20

See Supporting Information for more details.

Figure 2. Electrochemical phase diagrams at pH = 13 and HFeO2− concentration of 1 × 10−6 M generated with data from (a) O2 shift scheme (PBE, with no corrections for the hydrogen reference and no vdW corrections), (b) H2O reference + vdW correction scheme (present analysis), and (c) experiment (see also Supporting Information). The corresponding full potential−pH diagrams (d−f), which do not have absolute free energy information, are provided for comparison. The red dashed lines indicate the two-dimensional cuts displayed in (a−c). E


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The Journal of Physical Chemistry C and we restrict our analysis to linearly independent reactions that involve a change in transition metal oxidation state. We consider only compounds where the nominal oxidation state of oxygen is 2, and to increase the statistical space of our analysis, we include data from alkaline oxides of these metals. We note that, for simplification, this analysis includes only ZPE and δH corrections from H2, and only ZPE corrections from hydroxyl hydrogen atoms for various solid compounds; these specific corrections are shown, in the case of Fe, to be the only terms that contribute significantly to the reaction enthalpies (see Supporting Information). For Ni, we analyze the compounds NiO, Ni(OH)2, LiNiO2, and BaNiO3, which yields a U value of 5.2 eV with a rmsd of 0.02 eV (Figure 3). For Co, we analyze

Figure 4. Reaction enthalpies (a) and corresponding rmsd (b) profiles of Co (hydroxy)oxides as a function of U with H2O reference + vdW scheme. Note that O2 used in the reactions here has been corrected based on the H2O reference (see section S1.3 for more details).

Figure 3. Reaction enthalpies (a) and corresponding rmsd (b) profiles of Ni (hydroxy)oxides as a function of U with H2O reference + vdW scheme. Note that O2 used in the reactions here has been corrected based on the H2O reference (see section S1.3 for more details).

CoO, Co(OH)2, Co3O4, CoOOH, and LiCoO2, with a U value of 3.5 eV and a rmsd of 0.06 eV (Figure 4). Finally, for Mn, we consider MnO, Mn(OH)2, Mn3O4, Mn2O3, and LiMn2O4. We note that β-MnO2 has a highly nontrivial helimagnetic structure, which is beyond the scope of the present study to treat, and we therefore exclude it from consideration; similarly, experimental data for MnOOH are not unambiguous (∼0.2 eV uncertainty) and are therefore not considered.48,49 As with Ni, Co, and Fe, the results for Mn are highly encouraging, with an optimal U value of 3.7 eV and a standard error of 0.06 eV (see Figure 5). On the other hand, with the O2 shift strategy and the PBE functional, as described in Figure 1b, the rmsd values are 0.23, 0.25, and 0.31 eV for Ni, Co, and Mn, respectively, implying that the use of the vdW-corrected functional, together with use of the bulk water reference states, gives better performance for these classes of systems. We note, in passing, that while the proposed approach provides strong performance for groups of compounds involving mixed stoichiometries and

Figure 5. Reaction enthalpies (a) and corresponding rmsd (b) profiles of Mn (hydroxy)oxides as a function of U with H2O reference + vdW scheme. Note that O2 used in the reactions here has been corrected based on the H2O reference (see section S1.3 for details). F


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Figure 6. (a) Ni electrochemical phase diagrams at pH = 13 and a Ni(OH)3− concentration of 1 × 10−6 M with the calculated free energies of formation (Supporting Information, Table S12) and (b) the corresponding full potential−pH diagrams. The red dashed lines indicate the twodimensional cuts displayed in (a).

Figure 7. (a) Co electrochemical phase diagrams at pH = 13 and a HCoO2− concentration of 1 × 10−6 M with the calculated free energies of formation (Supporting Information) and (b) the corresponding full potential−pH diagrams. The red dashed lines indicate the two-dimensional cuts displayed in (a).

Figure 8. (a) Mn electrochemical phase diagrams at pH = 13 and a HMnO2− concentration of 1 × 10−6 M with the calculated free energies of formation (Supporting Information) and (b) the corresponding full potential−pH diagrams. The red dashed lines indicate the two-dimensional cuts displayed in (a).

oxidation states, the relative energies for different phases of a material with a given stoichiometry do not appear to be particularly sensitive to the chosen electronic structure method. The full set of calculated and experimental formation enthalpies are summarized in Table 1 and Table 2. The fit U values are provided in Table 3. The encouraging results for the diverse systems described above strongly imply that the present method will produce theoretical electrochemical Pourbaix diagrams of high quality.

Further, the method can be used to predict energetics for oxide, hydroxide, or oxyhydroxide phases for which available experimental data are either controversial or nonexistent, resulting in more complete phase maps for these electrochemical systems. An example of the power of this approach is given by an analysis of MnOOH, whose absolute stability and relative stability with respect to Mn2O3 are still controversial. The standard enthalpy of formation in ref 48, −6.54 eV (−5.78 eV for the free energy of formation), would predict the G


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Table 4. Performance of H2O Reference Scheme on the Reaction Enthalpies (ΔH) between Non-Transition Metal Oxides and Hydroxides PBE Na2O + H2O → 2NaOH K2O + H2O → 2KOH Rb2O + H2O → 2RbOH Cs2O + H2O → 2CsOH MgO + H2O → Mg(OH)2 CaO + H2O → Ca(OH)2 SrO + H2O → Sr(OH)2 BaO + H2O → Ba(OH)2 Al2O3 + H2O → 2AlOOH rmsd (eV)

vdW-optPBE

exp, ΔH (eV)

ΔH (eV)

deviation (eV)

ΔH (eV)

deviation (eV)

−1.99 −2.53 −2.66 −2.55 −0.85 −1.13 −1.40 −1.62 −0.90

−1.77 −2.30 −2.43 −2.20 −0.72 −1.00 −1.21 −1.44 −0.82

0.22 0.23 0.23 0.35 0.12 0.13 0.19 0.18 0.07 0.21

−1.99 −2.55 −2.67 −2.54 −0.92 −1.17 −1.38 −1.64 −0.91

0.00 −0.01 0.00 0.01 −0.07 −0.03 0.02 −0.02 −0.01 0.03

= Ni, Co, Fe, and Mn. As seen in Figure 1, such reactions show very weak sensitivity to the value of U and hence provide a test case with which to evaluate the effects of the chosen gas phase reference states and vdW corrections in the absence of significant self-interaction errors. The rmsd of 0.04 eV for the reaction enthalpies at the optimized values of U for each compound confirms the advantages of this approach. The synergy of the combined vdW and water reference scheme is further confirmed by similar analyses on non-transition metal (hydroxy)oxides, where self-interaction errors are generally negligible and Hubbard U corrections are hence unnecessary; as is shown in Table 4, the rmsd for a series of nine reactions involving alkaline, alkaline earth, and Al compounds is approximately 0.03 eV. Finally, after accounting for the improved reference state and vdW interaction, Hubbard U corrections correctly describe self-interaction errors in transition metal compounds. Indeed, the fact that a single U describes well changes in metal oxidation states between two and four is evidence of the robustness of the method. The combined strategies described above, including use of water, instead of O2, as the reference state, use of van der Waals corrections, and employment of Hubbard U corrections, provide accurate descriptions of electrochemical thermodynamics and consequently yield theoretical Pourbaix diagrams of nearly quantitative accuracy. Although these methods are benchmarked against a series of bulk transition metal oxide, hydroxide, and oxyhydroxide compounds, they also yield excellent predictions for multimetallic compounds of varying oxidation states not included in the original training set (Table 2), showing the transferability and predictive power of this approach. Given these characteristics, it is likely that the method will also provide reliable predictions of energetics for surface and bulk vacancies, for hybrid multimetallic systems of variable oxidation states, for thin films and islands, and for electrocatalytic processes that involve changes in surface concentrations of hydrogen and oxygen. Such predictions, in turn, could provide novel information about mechanisms of surface corrosion and electrocatalysis that cannot be straightforwardly accessed using either standard DFT calculations or exclusively experimental strategies.

existence of a MnOOH phase which has identical stability to Mn2O3. This result is consistent with observation in aqueous solution under ambient conditions.48,50 On the other hand, some calorimetric experiments have reported a value of 0.18 eV higher.49 Though the difference is not quantitatively large, it leads to qualitative differences in the phase predictions and to the disappearance of MnOOH from the Pourbaix diagram (see Supporting Information). The value from the present method, −6.62 eV, suggests that the former value is more reliable. Additional examples for predicted energetics for Mn, Fe, Co, and Ni-based compounds are given in Table 2. The full set of calculated free energies of formation are given in the Supporting Information along with available experimental data, and the predicted Pourbaix diagrams at pH = 13, together with the full potential-pH diagrams, are given in Figures 6−8. The success of the present approach on transition metal oxides, hydroxides, and oxyhydroxides is the result of a synergistic interplay among the chosen reference state for bulk phases (water), the use of van der Waals corrections, and Hubbard U corrections for self-interaction errors. Use of water as a reference state is a natural choice for electrochemical reactions in aqueous environments, and this choice both provides a consistent oxidation state for elemental oxygen between the reference state and the solid state compounds of interest, providing significant cancelation of errors, and also ensures that no inconsistencies in the water formation energies are introduced into the analysis. The correct thermodynamics of water formation, together with the reasonable O2 and H2 gas phase bond energies that the method yields (Supporting Information), is particularly useful if additional calculations of electrochemical reaction energetics, involving water, hydrogen, or oxygen as either reactants or products, are combined with calculations of electrochemical phase stability. We note that choosing appropriate references to maintain consistent oxidation states between the reference and the compounds of interest would likely be important for the highly accurate description of other species, such as peroxides, carbides, nitrides, sulfides, and halides. Use of vdW corrections, in turn, provides a greatly improved description of weak interactions between species within compounds of interest; such corrections are especially significant, for example, for hydrogen bonds in the layered compounds considered in this study. Indeed, the interlayer interaction in CrI2-type M(OH)2 is increased by approximately 0.2 eV per formula unit, on average, by introducing vdW corrections with the optPBE functional. This leads to an overall improvement of 0.2 eV on the reaction enthalpies for the series MO + 2H2O → M(OH)2 + H2, with M

4. CONCLUSIONS We have described a strategy to determine the thermodynamics and associated Pourbaix diagrams of a series of strongly correlated transition metal oxides, hydroxides, and oxyhydroxides. The approach uses a H2O gas phase reference state and explicitly includes van der Waals effects and self-interaction H


Article


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corrections through a Hubbard U description. A series of reaction free energies involving these classes of materials are described, with an average error of 0.04 eV per reaction formula unit, leading to excellent prediction of bulk Pourbaix diagrams. These promising predictions, which we explicitly demonstrate for Ni, Co, Fe, and Mn-based compounds, result from the synergy of several factors. The use of explicit van der Waals corrections considerably improves the accuracy of descriptions of layered hydroxide compounds and bulk oxyhydroxides, while the use of a water reference state to effectively fit the O2 energy is useful because the oxidation state of oxygen atoms does not change between this reference and the solid-state compounds being modeled. Careful fitting of U values for each system significantly reduces errors from self-interaction effects, and attractively, bonding energies of gas phase H2 and O2 are well described with the approach. The latter benefit, combined with the good descriptions of gas phase water formation energies, will facilitate use of the approach for calculations involving surface electrocatalysis. This scheme can thus be used not only to accurately predict the electrochemical Pourbaix diagrams of simple bulk oxides, hydroxides, and oxyhydroxides, but also to analyze a variety of complex bulk compounds and electrochemical reactions.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b03169. Thermodynamics and chemical potential evaluation; electrochemical redox, corrosion, and Pourbaix diagrams (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported through a DOE Early Career Award of the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy. Use of the Center for Nanoscale Materials was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract DE-AC02-06CH11357. Use of computational resources through the National Energy Research Scientific Computing Center (NERSC) is gratefully acknowledged.

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