Reliable electrochemical phase diagrams of magnetic transition metals and related compounds from high-throughput ab initio calculations
Reliable electrochemical phase diagrams of magnetic transition metals and related compounds from high- throughput ab initio calculations
James M. Rondinelli
ARTICLE Magnetic transition metals (mTM = Cr, Mn, Fe, Co, and Ni) and their complex compounds (oxides, hydroxides, and oxyhydroxides) are highly important material platforms for diverse technologies, where electrochemical phase diagrams with respect to electrode potential and solution pH can be used to effectively understand their corrosion and oxidation behaviors in relevant aqueous environments. Many previous decades-old mTM-Pourbaix diagrams are inconsistent with various direct electrochemical observations, because experimental complexities associated with extracting reliable free energies of formation (?fG) lead to inaccuracies in the data used for modeling. Here, we develop a high-throughput simulation approach based on density-functional theory (DFT), which quickly screens structures and compounds using efficient DFT methods and calculates accurate ?fG values, using high-level exchange-correlation functions to obtain ab initio Pourbaix diagrams in comprehensive and close agreement with various important electrochemical, geological, and biomagnetic observations reported over the last few decades. We also analyze the microscopic mechanisms governing the chemical trends among the ?fG values and Pourbaix diagrams to further understand the electrochemical behaviors of mTM-based materials. Last, we provide probability profiles at variable electrode potential and solution pH to show quantitatively the likely coexistence of multiple-phase areas and diffuse phase boundaries.
Magnetic transition metals (mTM = Cr, Mn, Fe, Co, and Ni) are
among the most important elements for human civilization.
Numerous mTM alloys and compounds have been applied broadly
and frequently throughout history. Various conventional
mTMbased structural alloys (e.g., Fe and Ni alloys) are widely used in
many low- and high-temperature fields, including civilian tools,
construction frameworks, biocompatible alloys,1?3 gas turbines,4?6
and nuclear-power equipment,7?9 owing to their excellent
mechanical properties, environmental benignity, and oxidation
and corrosion resistances. Superior mechanical properties are also
present in mTM-based high-entropy alloys,10?12 and their
corrosion resistance is under intensive investigation due to its
importance.13,14 A variety of mTM metals, oxides, and (oxy)
hydroxides are superior materials for photonic and
electrochemical catalyses (e.g., water splitting and pollutant decomposition).15?18
In addition, electrode materials based on mTM (hydr)oxides are
utilized in electrochemical capacitors19?21 and rechargable
lithium/sodium-ion batteries.22?24 Last, mTM oxides find
promising application in nonvolatile resistive random access
Electrochemical stability is among the most critical factors
determining the applications of materials in biological, marine,
and civilian fields. The stabilities of mTM metals and their
compounds against continuous corrosion and oxidation are one
of the prerequisites for systems-level integration of new materials
in aqueous and humid environments. On the other hand, the
synthesis and optimization of many functional materials (e.g.,
metal?organic frameworks,27 TiO2,28,29 and TM (hydr)oxides18) in
aqueous solutions require accurate knowledge about the
electrochemical behaviors of related compounds. The electrochemical
stabilities of materials can be effectively understood and predicted
from a Pourbaix diagram30?thermodynamic phase maps
indicating the equilibrium phases of a material system spanning a space
defined by electrode potential and solution pH.
Simulating a Pourbaix diagram requires the free energies of
formation (?fG) of all the involved species (the metal, its
compounds, and the associated aqueous ions). Although the
experimental ?fG (or chemical potentials) for the most common
aqueous ions (e.g., X2+ and X3+, X = Cr, Mn, Fe, Co, and Ni) usually
have a relatively small uncertainty (~0.02 eV/ion), those for many
solid compounds may have large uncertainties (e.g., ? ~ 1.6 eV per
formula unit (f.u.) for Co3O4) or are not available (e.g., ?fG of Cr3O4
is undetermined). Supplementary Information (part D) contains
our collected ?fG data of mTM compounds and their aqueous ions
from various databases. In addition, there are many unavoidable
technical or physical limitations imposed by experiment, for
example, fierce combustion, defect contamination,
uncontrollable/unmeasurable hydration, and solution filtering, which can
result in large uncertainties (inaccuracies) in the extracted
experimental ?fG values.31 Indeed, the Ti and Ni Pourbaix
diagrams simulated using experimental ?fG values are found to
be inconsistent with various direct electrochemical
1Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA and 2Key Laboratory of Marine Materials and Related Technologies,
Zhejiang Key Laboratory of Marine Materials and Protective Technologies, Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo
Correspondence: James M. Rondinelli ()
phenomena,31,32 which have been ascribed to the inaccuracies in
the free energies of formation. Furthermore, some of the
important compounds presenting potential and pH-dependent
stabilities may exhibit structures that are poorly or difficult to
characterize (e.g., NiOOH31,33) or remain to be determined by
experiment (e.g., Cr3O4, Cr(OH)2, FeO2, Co2O3, Co(OH)2, and Ni3O434).
Alternatively, state-of-the-art density-functional theory (DFT) can
be used as a reliable approach to assess the most stable structures
and determine accurate thermodynamic energies. Such structures
and thermodynamic energies can then be utilized to obtain
reliable Pourbaix diagrams that completely account for various
Here, we develop a high-throughput ab initio approach to
rapidly assess many possible structures and magnetic states of
mTM compounds, using efficient DFT methods, and then
accurately calculate ?fG data for the most stable phases, using
expensive high-level DFT methods to produce reliable
mTM?Pourbaix diagrams. We compare our DFT Pourbaix diagrams
with directly observed electrochemical, geological, and
biomagnetic phenomena on mTM-based metals and compounds to
comprehensively demonstrate the high accuracy of our fully ab
initio scheme. In addition, we reveal many chemical trends
appearing across both the ?fG data and Pourbaix diagrams.
Finally, we compute condition-dependent probability profiles for
the mTM metals, compounds, and aqueous ions to further
establish structure?electrochemical property relationships.
The electronic formation energy (?f?e) of a mTM compound
(XnOmHl) is calculated as
?f ?e ? ?e?XnOmHl?
where ?e is the total electronic energy obtained from DFT, and the
elemental X (=mTM), O2 molecule, and H2 molecule are the
reference species. At finite temperatures, the total free energy
(Gtot) of a solid or molecular gas is expressed as
Gtot?T ? ? ?e ? GT ?T ?;
where GT is the temperature-dependent component that consists
of contributions from atomic vibrations (including zero-point
energy), electronic excitation (exclusively in metals), and
molecular rotation and translation in the diatomic O2 and H2 gases.
Here, we use the standard state O2 and H2 gases at 298.15 K and
1.0 bar as the reference species, and their GT values are obtained
by summing their zero-point vibrational energies (calculated
herein from DFT) and free-energy drops from 0 to 298.15 K
(measured in experiment35). The GT values of the mTM
compounds are derived from the phonon spectra calculated
The standard free energy of formation (?fG) of a mTM
compound is calculated as
?f G?T ? ? ?f ?e ? ?GT ?T ?;
where the thermal correction ?GT is obtained in a similar manner
as given by Eq. (
), with the ?e values therein replaced by the
corresponding GT values here. The standard chemical potential of
a solid (?s) equals its ?fG value, i.e., ?s = ?fG.
For aqueous ions, the concentration-dependent chemical
potential of an aqueous ion I (?I) is calculated from its standard
chemical potential (?0, at 298.15 K, 1.0 M, and 1.0 bar, pH = 0)
?I ? ?0 ? RT ln??I ?;
where, R is the gas constant (8.314 J?mol?1?K?1), ?0 is obtained
from the experimental databases, and [I] is the aqueous-ion
activity that is approximated to be its concentration. In aqueous
environments, the relative electrochemical stabilities between
different species (e.g., metals, oxides, hydroxides, oxyhydroxides,
and aqueous ions) are described by their chemical potentials of
reaction (??), which are calculated using the reaction paths that
connect all of the considered species (see Supplementary
Information, part E).
Our high-throughput DFT approach is based on the following
considerations: first, efficient DFT methods can be used to quickly
determine the most stable structures among all the possible
structures. Second, precise but expensive DFT methods utilizing
higher-level density functionals can be used to calculate accurate
?fG values of these stable structures. Finally, the obtained ?fG
values can be used to construct Pourbaix diagrams. This
simulation scheme is implemented as a high-throughput ab initio
workflow, Fig. 1, consisting of five major parts described next.
Details of the DFT methods utilized in this work are described in
the Methods section, and include the use of the LDA, GGA (PBE
and PBEsol), metaGGA (RTPSS and MS2), and hybrid (HSE06)
The first major part is ?Fast DFT Structural Screening?, which
consists of five steps (boxes) described below: the boxes labeled
?Structure database? and ?Collect possible structures in Cr, Mn, Fe,
Co, Ni compounds? in Fig. 1 show that the many possible structures
for the mTM oxides, hydroxides, and oxyhydroxides are obtained
from the Inorganic Crystal Structure Database34 as well as the
literature.33 There are 26 different compound structures collected
(Supplementary Table S1), and the cations in these structures are
successively substituted by the 5 mTM elements to sample the
structure-composition space, resulting in 130 (=5 ? 26) structures
As indicated by the ?Set various initial magnetic states for each
structure? box, at least three initial magnetic configurations (e.g.,
nonmagnetic, ferromagnetic, and antiferromagnetic) are
considered to determine the ground-state magnetic structure for each
phase. In some compounds (e.g., X3O4 and XOOH), the finally
calculated magnetic configurations may be ferrimagnetic or
another complex order, for which we still simply use FM or AFM
to conveniently indicate the magnetic coupling character
between the neighboring cations therein. In some complex
structures (e.g., defective X3O4 and layered X2O3), there may be
at least two or three inequivalent AFM configurations requiring
sampling. Thus, there are ?4 magnetic configurations for each
structure on average, resulting in a total of ~520
structuremagnetism configurations to consider.
The next step in the screening scheme uses the efficient PBEsol
functional first to fully optimize the lattice constants and atomic
positions of all 520 structure-magnetism configurations, which are
followed by further optimizations at the MS2 level (?Calculate
PBEsol & MS2 energies,? Fig. 1). Thus, there are about 1,040 DFT
structural optimizations in the structural screening step. At a
specific chemical composition, we find that both PBEsol and MS2
give the same relative stabilities among different polymorphs
(Supplementary Table S1).
The final step in the screening procedure is to ?Select the most
stable structures? with the lowest electronic energy for each
composition. These phases are then used as inputs for additional
more advanced DFT calculations to obtain higher-accuracy
electronic energies and vibrational free energies. We are
conservative in our down-selection process, and also include
one to four metastable configurations for the next step, which
further justifies that our assignment of the relative stabilities
among polymorphs from PBEsol and MS2 is the same as those
from the higher-level hybrid functional HSE06.
Calculate ?f e
e Generate a discrete phase
od space spanned by V and pH
ic Calculate ? (V, pH) within
re the specified phase space
lN Generate Pourbaix diagram
Furthermore, we note that for transition-metal systems with
localized 3d orbitals, the electronic exchange potentials in
conventional DFT methods (e.g., LDA and GGA) may need
improvement due to the delocalization error therein.33,36,37 An
efficient alternative to the computationally expensive hybrid
functional with exact electronic exchange (e.g., HSE06 used here)
is the so-called DFT plus Hubbard U (DFT + U) method with a
static mean-field on-site Hartree?Fock approximation,36,37 which
has an efficiency equivalent to that of a GGA functional and is
quite useful for large-scale computation of lattice and
thermodynamic energies of complex materials. The DFT + U method,
however, usually requires the experimental compound free
energies of formation, ?fG values, as the energetic references to
fit both the tunable parameter U and the ad hoc energetic
correction to the O2 molecule,38?41 where the fitted U for a
transition-metal cation may also depend on its coordination
number and anion type.41
In simulating electrochemical phase diagrams, the DFT + U
method additionally requires ad hoc energetic corrections to the
important aqueous ions to reproduce the dissolution energies of
compounds, as derived from experimental characterizations.42?44
Therefore, when using the DFT + U method, it will involve
complicated ad hoc numerical processing, and the quality of the
simulated electrochemical phase diagrams will also rely on the
accuracy and availability of the experimental energies. For these
reasons, we do not consider the DFT + U method here, which was
frequently used in earlier simulations of Pourbaix diagrams and
are only as accurate as the experimental energies they
reproduce.42?44 This aspect makes assessment of calculated Pourbaix
diagrams to direct measurements important as we show here.
The second major part ?Accurate DFT Free Energies? consists of
three boxes as described here. We use the most stable structures
to recalculate the electronic formation energies ?f?e using HSE06
(?Calculate ?f?e using HSE06?). Phonon spectra for these phases are
computed from DFT, using the efficient PBEsol functional
(?Calculate ?Gvib using PBEsol?), which is sufficiently accurate to
obtain reliable vibrational free energies (Gvib) and vibrational
formation free energies (?Gvib) of solids.32,45 In addition, other DFT
methods (i.e., LDA, PBE, and RTPSS) are also used to calculate the
functional-dependent ?f?e values (Supplementary Tables S2?S6).
Accurate ?fG values at standard conditions (298.15 K and 1.0 bar)
are derived using the HSE06 ?f?e and PBEsol ?Gvib data (from Eq.
)). The obtained ?fG values are then assigned as the standard
chemical potentials (?s) of the mTM compounds (?Calculate ?s =
?fG?). The aforementioned DFT ?fG data appear in Supplementary
Tables S7 and S8. The experimental values are given in
Supplementary Tables S9?S14.
The third major part ?Experimental Data of Aqueous Ions?
consists of three steps. Rather than computing the chemical
potentials of the aqueous ions involved in the electrochemical
phase diagrams, we first ?List relevant aqueous ions? and then
tabulate their standard chemical potentials (?0) by collecting them
from various experimental databases (?Collect experimental
standard chemical potentials (?0)??, Fig. 1). Based on the collected ?0
values (Supplementary Tables S9?S14), the ?I values at any
specified [I] are calculated using Eq. (
) (?Calculate ?I at specified
The fourth major part ?Reaction Thermochemical Formula?
consists of two boxes. Apart from the calculated and collected
chemical potentials, the relevant formula and numerical solvers
are required to model the electrochemical thermodynamics.
Regarding the former aspect, we list the reaction paths that
connect all of the solids and aqueous species (?List relevant
reaction paths?), based on which the dependencies of the
chemical potentials of reaction (??) on electrode potential V
(with respect to the standard hydrogen electrode, i.e., VSHE) and
solution pH are formulated (?Formulate ??(V, pH) using Nernst
equation?). All ??(V, pH) formula for the considered reaction paths
are available in Supplementary Information (part E).
The fifth major part is the ?Final Numerical Modelling? consisting
of three steps. As indicated by the box of ?Generate a discrete
phase space spanned by V and pH?, prior to the diagram modelings,
a dense descrete numerical grid (400 ? 400) is used to precisely
describe the complete range of phase space of interest. Here, we
focus on VSHE ? [?2, 3] V and pH ? [?2, 16]. Next, the calculated ?s
and ?I values for all of the mTM species (i.e., metals, oxides,
hydroxides, oxyhydroxides, and aqueous ions) are used as inputs
into the ??(V, pH) formula, and their relative ?? values at each
numerical grid point are calculated (?Calculate ??(V, pH) within the
specified phase space?). From this assessment at each grid point,
the most stable chemical form is identified. Last, a Pourbaix
diagram is generated after scanning the complete numerical grid.
Although only mTM-based materials are studied here, this
highthroughput ab initio approach (as depicted in Fig. 1) can be
directly applied to simulate the ab initio Pourbaix diagrams of
many materials other than the mTM-based compounds. In
addition, for many nonmagnetic materials, (e.g., Ti metal and
oxides32), it is unnecessary to perform the complex magnetic-state
screening step required in this work.
The calculated electronic formation energies (?f?e) and free
energies of formation at the standard condition (?fG at 298.15 K
and 1.0 bar) for the mTM oxides, hydroxides, and oxyhydroxides
are shown in Fig. 2. The ?fG values per atom (Fig. 2b) indicate the
relative stabilities among the component compounds in a sample
with a fixed global composition. The ?fG values per cation (Fig. 2c)
can be used to derive the relative stabilities among the
compounds when in contact with a reactive environment, e.g.,
an O2 (and/or H2) atmosphere or an aqueous solution. Indeed, the
DFT-calculated ?fG values per cation are used in the simulation of
the mTM?Pourbaix diagrams (Fig. 3), which are also compared
with the diagrams simulated using the experimental (Expt) ?fG
In addition, we will use the mTM compound and aqueous-ion
?fG chemical trends (Figs 2c, 4) to explain the chemical trends in
the Pourbaix diagrams (Fig. 3). Detailed DFT and Expt ?fG data can
be found in Supplementary Information (part A?D). We also
consider any possible precipitation of metastable phases at
various electrochemical conditions by analyzing probability profiles
for all of the mTM species at variable pH and VSHE (Fig. 5). These
data are also useful for understanding the synthesis,
characterization, and application of the related materials in aqueous
The ?f?e and ?fG per atom data can indicate the intrinsic stabilities
of materials, and be readily explained using microscopic electronic
structure-based models. They are also widely used to describe the
thermodynamic stabilities of numerous alloys.46,47 The observable
difference of 0.1 ?0.3 eV per atom between any ?f?e (Fig. 2a) and
its counterpart ?fG (Fig. 2b) indicates the considerable
destabilizing thermal effects. Such thermal effects, however, do not alter
any chemical trends, i.e., the relative stabilities among different
compositions of a mTM or between different mTM compounds at
the same O (and H) compositions. Nonetheless, inclusion of the
thermal effects is required for the precise simulation of
thermodynamic and electrochemical phase diagrams.
Figure 2a, b shows that ?f?e (and ?fG) generally increase with
increasing the number of 3d electrons (n3d: Cr < Mn < Fe < Co <
Ni). MnO and Mn(OH)2, however, are an exception to this trend;
these two compounds are lower in energy than CrO and Cr(OH)2,
respectively. We ascribe the general decrease in compound
stability to the variation of the 3d-orbital physics47?49 as follows:
first, the 3d orbitals become more localized and lower in energy
with increasing nuclear charge; therefore, both the intra-atomic
orbital hybridization and interatomic electron transfer become
less energetically favorable, resulting in the decreased strength of
the covalent?ionic mTM?O bonds,33 i.e., decreased stability.
Second, the 3d orbitals are close to half filling for Cr and Mn.
The further addition of electrons will lead to the pairing of spins to
form spin singlets, which also makes the mTM atoms less sensitive
to changes in bonding.
Interestingly, MnO and Mn(OH)2 (Fig. 2b), as well as the aqueous
Mn2+ ion (Fig. 4d) have unexpectedly low ?fG values, which we
again ascribe to the orbital character of the Mn(II) cation. For an
atom with half-filled d orbitals (i.e., with the largest number of
unpaired 3d electrons), the electronic structure is especially
sensitive to the coordination environment, leading to
coordination flexibility and low stability. This is further supported by
another abnormal behavior of elemental Mn: it exhibits a distorted
BCC* structure (? phase34), which is not anticipated from the
structural trend (HCP?BCC?HCP?FCC) for the TM elements in the
same 3d row.47,49
Figure 2a, b also shows that upon going from Cr to Ni, the
nominal cation valence of the most stable oxide decreases from
+3 (Cr2O3), to +2.67 (Fe3O4 and Mn3O4) and +2.0 (CoO and NiO).
We attribute this behavior to the increased mTM electronegativity,
i.e., the less-favored interatomic electron transfer mentioned
above. The relative stability of the mTM oxyhydroxides (XOOH)
with respect to hydroxides (X(OH)2) also decreases from Cr to Ni,
due to the same increased mTM electronegativity.
The aforementioned chemical trends for mTM compounds can
help to predict and understand various thermodynamic and
electrochemical phenomena, and provide much insights for the
design, synthesis, and application of related materials. To fully
quantitatively understand the electronic mechanisms underlying
these chemical trends, an in-depth and comprehensive
investigation into the orbital properties (e.g., energy level, occupation,
hybridization, and bonding) of the TM atoms in the various
coordination environments is required. We do not address these
issues further, but rather focus on compound stability in aqueous
environments in the form of electrochemical phase diagrams.
In realistic situations, the compounds are always in contact with
a reactive environment, e.g., in an O2 (plus H2) atmosphere. Under
such conditions, the oxidation of a metal (X) is described by the
In this later scenario, the ?fG values are specified per cation, Fig. 2c,
rather than per atom to assess the relative stabilities among all of
the involved mTM species (metal, compounds, and aqueous ions).
Then the ?fG values per cation, electrode potential, and solution
pH should be simultaneously considered to clearly understand the
Pourbaix diagrams presented in the following section.
Pourbaix diagrams: general trends
The mTM?Pourbaix diagrams at a moderate [I] of 10?6 mol/L
(10?6 M) simulated using Expt and DFT ?fG values are compared
in Fig. 3 (left and center columns). In electrochemical experiments,
[I] is usually not controlled and characterized; thus, the DFT
Pourbaix diagrams at [I] = 10?2 M are also provided in Fig. 3 (right
column). The Pourbaix diagrams at [I] = 10?2 M are also useful for
the experimental synthesis of related compounds through
solution precipitation, where a relatively high [I] should be
required (DFT Pourbaix diagrams constructed for a broader range
of [I], from 10?8 to 10?2 M, are also provided in Supplementary
Fig. S2). In addition, some compounds such as Cr2O3, Mn2O3,
Fe2O3, FeO, Cr(OH)2, and Ni(OH)2 with secondary electrochemical
stabilities find importance in numerous realistic applications; for
that reason, we also calculate their phase domains by excluding
the more stable phases in the DFT Pourbaix diagrams found in the
center and right panels of Fig. 3.
In a Pourbaix diagram, the domains consisting of the metal, its
compounds, and aqueous ions are called immunity, passivation,
and corrosion domains,30 respectively. We find in the DFT Pourbaix
diagrams that the relative stabilities (i.e., phase domains) of the
passivating compounds increase with increasing [I] in Fig. 3a?e
(center and right columns), due to the decreased stabilities of the
aqueous ions (by Eq. (
)). The critical pH value at the left (right)
passivation-domain boundary generally decreases by ~2
(increases by ~4) with increasing [I] by 104 times. The phase
areas at low electrode potentials (VSHE) are always occupied by
mTM metals, and the mTM compounds and aqueous ions with
more oxidized mTM are stabilized with increasing VSHE. For
example, there is a phase transition of Co ? CoO/Co(OH)2 ?
Co3O4 ? CoOOH ? CoO2 (Fig. 3d). This electrochemical trend is
ascribed to the behavior of the positively (negatively) charged
electrode, which extracts electrons from (introduces electrons
into) the materials, making them more oxidized (reduced).
In the Pourbaix diagrams of Cr, Mn, and Fe (Fig. 3a?c), the
highVSHE areas are occupied by complex aqueous ions with highly
oxidized mTM (e.g., [XO4]2?). However, these aqueous ions are
absent in the Pourbaix diagrams of Co and Ni (Fig. 3d, e), because
the data for [CoO4]2? and [NiO4]2? ions are unavailable and thus
are not considered in our simulations. This lack of data could be
related to the decreased aqueous-ion stabilities upon going from
Cr to Ni (Supplementary Table S14), making it challenging or even
impossible to detect the [CoO4]2? and [NiO4]2? ions at high
voltages in electrochemical experiments. Nonetheless, the most
common aqueous ions (i.e., X2+ and X3+) can be used to
understand the energetic trend for aqueous ions from Cr to Ni.
Figure 4d depicts their ?fG values, revealing an obvious general
increasing trend, except for the abnormal dip at Mn2+. This
energetic trend for aqueous ions and the special behavior of Mn2+
are similar to those described above for the solid-state
compounds and thus are likely governed by the same electronic
Upon moving from Fe to Co, and Ni, the electrochemical phase
domains of XO and X(OH)2 expand toward higher VSHE (Fig. 3c?e).
This trend occurs because the stabilities of other compounds (e.g.,
X2O3 and XOOH) decrease much faster than those of XO and X
(OH)2 (Fig. 4a?c). The appearance of MnO and Mn(OH)2 in the Mn
Pourbaix diagrams (Fig. 3b) arises from their unexpected low ?fG
values (see Figs 2a, 4a). In the DFT mTM?Pourbaix diagrams, the
nominal cation charge of the most favored mTM compound at
VSHE ~ 0 generally increases, e.g., from CrOOH to Mn3O4, Fe3O4,
CoO, and NiO, due to the different destabilizing rates for the
compounds with different cation charges (Fig. 4a?c).
In the following sections, we demonstrate the advancement of
our high-throughput ab initio method in simulating accurate
mTM?Pourbaix diagrams ranging from Cr to Ni, which in some
cases have been lacking for over 50 years. Assessments are made
in detail with various electrochemical phenomena directly
observed in recent decades. An explicit comparison of the
improvement enabled by our approach is provided for the Ni
Pourbaix diagram, which is shown together with many
electrochemical observation results in Supplementary Fig. S3. As
described earlier, the mTM-based materials have been widely
used in numerous fields (e.g., structural materials, catalysts,
electrode materials, and electronic devices), where the materials
always closely contact with different aqueous environments
during their synthesis and exploitation. Thus, precisely knowing
their electrochemical phase stabilities can be highly helpful for
designing materials, optimizing the synthesis and application
conditions, and controlling material phases and properties.
Cr Pourbaix diagrams
The experimental (Expt) and our DFT Cr Pourbaix diagrams exhibit
quite similar phase domains for Cr2O3 (Fig. 3a, left and center
panels), owing to the closeness in the free energies of formation
(Fig. 4c). This good theory?experiment agreement is ascribed to
the high thermodynamic stability (large ?fG) of Cr2O3, which
serves to suppress defect generation during the combustion
process in thermodynamic experiments used for estimating ?fG
(as discussed in the Introduction). Thus, the contaminating effect
of defects is largely minimized in the experimental ?fG of Cr2O3.
Cr2O3 is a ubiquitous oxide readily formed on various alloys, e.g.,
steels,50 under atmospheric conditions. However, it is well known
that Cr2O3 never forms on Cr in aqueous solutions, but its hydrous
counterparts (CrOOH and Cr(OH)3) appear as the passivating
compounds, as detected in several electrochemical
experiments.51?53 Cr(OH)3 is a highly hydrated material consisting of
molecular Cr(OH)3 units, and its structure still has not been well
characterized. For these reasons, we do not considered it explicitly
in Fig. 3a. The reason Cr(OH)3 will form initially on Cr metal in
solutions is probably due to its higher kinetic activity, whereas
CrOOH will gradually grow beneath the outer Cr(OH)3 layer,53
indicating the higher electrochemical stability of CrOOH.
Therefore, CrOOH should be the stable phase in aqueous solutions, and
both Cr2O3 and Cr(OH)3 should appear as metastable phases. This
assessment is consistent with our DFT Cr Pourbaix diagrams (Fig.
3a, center and right panels), whereas the experimental Cr Pourbaix
diagram reveals a much smaller phase domain over which Cr2O3 is
stable (Fig. 3a, left panel).
CrOOH precipitates are widely observed in solutions at pH ?
3,53?57 which is consistent with the boundary between CrOOH and
Cr3+ at pH 2 ~ 3.5 and VSHE ~ 0 V in the DFT diagrams (Fig. 3a,
center and right panels). The hydrous Cr2O3 (e.g., CrOOH) formed
in aqueous solutions transforms into the anhydrous Cr2O3 only
upon heating at temperatures ?700 K.55,57 In addition, the
formation of anhydrous Cr2O3 (not CrOOH) underneath an outer
Cr(OH)3 layer has been observed on 254 MO stainless steels
(Fe?20%, Cr?18%, and Ni?6% Mo, in wt.%).58 These corrosion
processes likely originate from the kinetic and/or thermodynamic
effects of the other alloying elements on the electrochemical
stabilities of Cr2O3 and CrOOH. Such interactions still require
further detailed experimental and theoretical investigations to
understand the microscopic mechanisms governing the
appearance of these phases.
Mn Pourbaix diagrams
Mn oxides (e.g., Mn3O4, Mn2O3, and MnO2) have promising
application in water electrolysis, due to their favorable catalytic
reactivity under electrochemical conditions, and these oxides
always coexist in experimentally synthesized samples.59?63 This is
well explained by the calculated ?fG values per cation (Fig. 2c),
and their small differences indicate the similar thermochemical
stabilities of these oxides in contact with a dry/aqueous
environment. However, Mn (hydr)oxides present serious adverse
dissolution problems in aqueous environments,20 which should be
due to their low electrochemical stabilities, as indicated by their
relatively small or absent phase domains in both the experimental
and DFT Mn Pourbaix diagrams (Fig. 3b). In contrast, the layered
MnO2 compound is readily stabilized by intercalation with the
alkaline cations (e.g., Li, K, and Ca) from aqueous solutions.59,60,64
An early experiment using X-ray diffraction65 observed in
solutions with [I] ? 10?2 M that (
) at T ~ 298 K, only Mn3O4 is
present at pH 8.5?9; (
) at T ? 310 K, Mn3O4 appears at pH down to
) at T ? (283, 290) K, Mn3O4 and MnOOH coexist at pH 8.0?8.5;
) at T ~ 273.6 K, only MnOOH occurs at pH 8.5~9.0. These
observations were later confirmed using multiple probes,
including high-resolution X-ray diffraction, Raman spectroscopy, and
Xray photoelectron spectroscopy.66 In those experiments, MnOOH
or/and Mn(OH)2 was found to initially precipitate and then
transform into Mn3O4 in solutions with [I] ? 10?4 M and pH 10. It
should be noted that precisely distinguishing MnOOH from Mn
(OH)2 in experiment is nontrivial, due to their similar layered
structures and the uncontrollable degree of hydrogenation.33 The
initial formation of MnOOH/Mn(OH)2 is ascribed to the high kinetic
activity that may be due their layered structures?the oxidation of
metals in solutions always initiates with the adsorption of OH
We can conclude from these electrochemical observations that
MnOOH/Mn(OH)2 is an intermediate precipitate, and it will
spontaneously convert into more stable Mn3O4 at temperatures
?283 K, while this kinetic transition will be deactivated at lower
temperatures (<283 K).65 Indeed, the Mn3O4 domain appears as a
major field in the DFT Mn Pourbaix diagrams (Fig. 3b, center and
right panels). Furthermore, Mn3O4 is observed to be stable in
solutions with pH ? 8.5 and [I] 10?2?10?4 M,65,66 which also
strongly supports the DFT-determined phase boundary at pH
8.5~9.5 (VSHE ~ 0 V). In addition, an operando X-ray absorption
near-edge structure (XANES) spectroscopy63 has recently been
performed and observed the oxidation of Mn3O4 at 0.2?0.4 V in a
solution with pH = 14. This result is also consistent with the upper
phase boundary of Mn3O4 at 0.3?0.4 V in the DFT-simulated Mn
In the DFT Mn Pourbaix diagrams, Mn(OH)2 only has a negligibly
small phase domain at pH ~ 11.5 when [I] is as high as 10?2 M, and
MnOOH is totally absent. These aspects are consistent with the
experimentally observed intermediate roles played by Mn(OH)2
and MnOOH. According to Pourbaix diagrams obtained using
experimental free energies of formation, however, stable Mn(OH)2
is expected to persist at pH values >8.7 in a solution with [I] ?
10?2 M. In addition, MnOOH also presents an observable region of
phase stability at low anodic potentials ([I] = 10?6 M, Fig. 3b, left
panel). At a phase boundary, there is an inevitable thermodynamic
blurring (?pH ~ 1 at room temperature, discussed later in the
section Probability analysis) due to the nonzero probability for a
metastable phase at a finite temperature. Thus, Mn(OH)2
precipitates are expected to appear at a pH lower than the phase
boundary in the experimental Pourbaix diagram (at pH < 8.7).
These derived phenomena from the experimental Pourbaix
diagrams are clearly inconsistent with the above experimental
Fe Pourbaix diagrams
The most frequently observed Fe oxides are Fe3O4 and Fe2O3,
owing to their close thermodynamic stabilities (Fig. 2c) and the
fact that the less stable FeO phase is only possible under highly
reducing conditions, e.g., high H2 concentration.50,71 The major
discrepancy between the experimental and DFT Fe Pourbaix
diagrams appears as a difference in the relative stabilities between
Fe3O4 (magnetite) and Fe2O3 (hematite) as shown in Fig. 3c (left
and center panels). In addition, FeO is only observable in the DFT
diagrams (Fig. 3c, center and right panels).
Fe3O4 forms rapidly in various electrochemical experiments;
however, any direct formation of Fe2O3 has not been observed in
the Fe3O4 products.72?74 Fe2O3 is only obtained by further
oxidizing Fe3O4 using additional aeration plus heating.72,74 In
aqueous solutions, however, the formation of Fe2O3 may be
facilitated by other alloying elements or related oxides (e.g.,
stainless steel with Cr, Ni, and Mo58). It is the small difference in
the ?fG values of these phases (Fig. 2c) that results in their readily
variable relative stabilities in different environments. These
electrochemical observations are more consistent with the DFT
Fe Pourbaix diagrams (Fig. 3c, center and right panels), where
Fe3O4 is more stable than Fe2O3 at VSHE ~ 0 V. Furthermore, solid
Fe3O4 is reported to be stable at pH ? 3.5,72,73,75,76 which is also in
good agreement with its dissolution boundary at pH 4.5?3.0 in the
DFT diagrams. In contrast, the experimental Pourbaix diagrams
show that the dissolution boundary for the stable oxide (Fe2O3)
resides at pH 6.5?5.0, which is inconsistent with the observed
We also note that although Fe2O3 is more stable in the
experimental diagram (Fig. 3c, left panel), it has a phase domain
that is quite close to that of the metastable Fe2O3 in the DFT
diagram (Fig. 3c, center panel, dotted lines). This similarity
indicates that the experimental ?fG of Fe2O3 is likely quite
accurate, while that of Fe3O4 is somewhat underestimated. The
inaccurate experimental ?fG of Fe3O4 may be ascribed to the
uncontrollable defect concentration in defective Fe3O4 spinel
samples during combustion heat measurements.
Furthermore, regarding the relative electrochemical stabilities
between Fe3O4 and Fe2O3, available geological and biomagnetic
evidences also strongly support our DFT results. During the
longterm geological evolution of ultramafic rocks driven by aqueous
fluid, magnetite (Fe3O4) rather than hematite (Fe2O3) minerals
formed from primary olivine minerals.77 It should be the
electrochemically stable Fe3O4 in iron ore (i.e., lodestone, a
magnetite mineral) that is pervasive. Indeed, magnetite
nanocrystals have also been widely found in magnetotactic bacteria78,79
and under the upper-beak skin of homing pigeons.80 These
evidences clearly indicate the higher electrochemical stability of
Fe3O4 than that of Fe2O3, as revealed by our DFT Fe Pourbaix
diagrams. Our assessment here motivates additional
measurements to quantify the experimental accuracy of ?fG(Fe3O4).
During the electrochemical oxidation of pure Fe with increasing
electrode potential (scan rate 0.04 V/s) in solutions with pH =
14,69,70,75 the reaction starts with the adsorption of OH on Fe
surface at VSHE ~ ?0.9 V. Next, the formation of FeO (and/or Fe
(OH)2) occurs at ??0.7 V and the further oxidation of the outer
(inner) FeO layer into Fe3O4 and/or Fe2O3 occurs at ??0.5 V
(??0.2 V). From our DFT Pourbaix diagrams for Fe (Fig. 3c, center
and right panels), we find that the Fe?FeO boundary at pH 14
resides at VSHE ~ ?1.04 V, which is a little lower than the observed
initial-oxidation potential of ~?0.7 V. This small difference (by
?0.3 V) in oxidation potential is reasonable, because any effective
kinetic factor will tend to slow the oxidation process in
experiment. The different oxidation potentials for the inner and
outer FeO layers themselves are indicative of a
nonthermodynamic factor at play. In addition, the DFT FeO?Fe3O4
and FeO?Fe2O3 boundaries (at pH 14) reside at ?0.99 and ?0.5 V,
respectively, which are reasonably lower than the observed
oxidation potentials of FeO at ?0.2 ~ ?0.5 V. In these early
voltammetric experiments, it was either not possible or
exceedingly difficult to distinguish between FeO and Fe(OH)2 (Fe3O4 and
Fe2O3). For this reason, we suggest that additional in situ
characterization of the samples be performed, e.g., using X-ray
diffraction (XRD), X-ray photoelectron spectroscopy (XPS), or
Raman spectroscopy, to better differentiate the phases present.
In addition to the consistency between our DFT Fe Pourbaix
diagrams and various electrochemical/geological observations,
the knowledge gained from our DFT diagrams can also help
correctly understand the oxide formation on Fe samples with
corrosion-resistant polymer coatings.81,82 The Bragg peaks
observed in the XRD patterns on such corroded samples were
ascribed to the Fe substrate, Fe2O3, and FeOOH;82 however, our
analysis indicates that Fe3O4 should form as the stable oxide
under electrochemical conditions. To assess the interpretation in
refs. 81,82, we examined the XRD patterns for standard Fe, Fe2O3,
Fe3O4, and FeOOH samples83?85 in detail and found that for the
corroded Fe samples with polymer coatings,82 the XRD peaks, can
be ascribed solely to Fe and Fe3O4. When the polymer
composition changes,81 however, XRD peaks consistent with the
formation of FeOOH appear. The FeOOH phase that appears may
be retained after the adsorption of OH (described above). It
remains unknown whether there exist any additional chemical or
voltaic-cell effects at the Fe/Fe3O4?polymer interface that alter the
stability of Fe compounds (e.g., FeOOH). Therefore, our DFT Fe
Pourbaix diagrams are useful for reinterpreting experimental
results and should motivate further experimental and theoretical
studies on the corrosion mechanisms for Fe-based materials.
Co Pourbaix diagrams Based on various measurements using cyclic voltammetry, ellipsometry, X-ray photoelectron spectroscopy, and M?ssbauer spectroscopy on the oxidation of Co metal in solutions with pH
10 ?14.4,68,86?89 it is known that (
) Co(OH)2 initially forms on a Co
) CoO may grow underneath Co(OH)2, resulting in a
sandwich heterostructure of Co/CoO/Co(OH)2; and (
) CoO and Co
(OH)2 will transform into Co3O4 and/or CoOOH at VSHE at ?0.3 V.
Given these experimental observations, we now assess the
experimental and DFT-simulated Co Pourbaix diagrams.
In the DFT Co Pourbaix diagrams (Fig. 3d, center and right
panels), CoO and Co(OH)2 are stable against dissolution into Co2+
in alkaline solutions, which is consistent with their formations on
Co surface at pH ? 10 in experiment. In the experimental Co
Pourbaix diagram, however, the phase domain of CoO (and Co
(OH)2) is significantly undestimated to be within pH ? (10.3, 11.0),
which is inconsistent with the observed passivation domain at pH
values >~10. In the observed Co/CoO/Co(OH)2 sandwich structure,
the initial formation of Co(OH)2 should be ascribed to the higher
kinetic activity for its formation, and the later growth of CoO
underneath Co(OH)2 indicates that CoO is thermodynamically
more stable than Co(OH)2 in solution. These conclusions strongly
support our DFT electrochemical results, where Co(OH)2 is a
metastable phase relative to CoO. According to the DFT Co
Pourbaix diagrams, a VSHE ? ?0.1 V is required to activate the
CoO?Co3O4/CoOOH transition at pH ? 14, which is close to the
experimental value (?0.3 V). The deviation of ~0.4 V is reasonable
when considering the possible influence of kinetic effects.
In a recent experiment,90 in situ Raman spectroscopy was used
to characterize a Co electrode with deposited Co3O4, which was
then immersed into a solution at pH ~ 13. Here, it was found that
Co3O4 coexists with CoOOH at an initial 0.16 V, and a
Co3O4?CoOOH transition occurred upon increasing the anodic
potential. This experimental observation is also consistent with the
boundary at VSHE ~ 0 V in our DFT Co Pourbaix diagrams. In
another experimental measurement using extended X-ray
absorption fine-structure (EXAFS) spectroscopy,91 CoOOH was observed
to be stable at VSHE ? 0.75 V in a solution at pH 7, which is
consistent with the stability of CoOOH at >0.3 V in the DFT
In Yeo?s measurement,90 there is no evidence of Co(IV) (e.g.,
CoO2) up to 0.86 V. In an earlier measurement, however, using
M?ssbauer spectroscopy,92 stable CoO2 (with intercalated Fe) was
observed at VSHE ? 1.1 V and pH 8.5, which was further confirmed
by Kanan?s EXAFS measurement,91 i.e., a stable Co(IV) state exists
at VSHE ? 1.25 V and pH 7. These measured VSHE values for stable
CoO2 are obviously lower than the DFT ones by about 1.0 V, which
may be due to the stabilizing effects of some aqueous ions (e.g.,
Fe and Na) that can readily intercalate into the layered CoO2
structure. To that end, the effects of electrolyte composition and
possible intercalation for CoO2 are interesting topics for future
experimental and theoretical studies.
Ni Pourbaix diagrams
NiO and Ni(OH)2 are especially important for Ni-based
corrosionresistant alloys, electrodes, and catalysts, and there are numerous
experimental observations reported that can be used to assess the
DFT Ni Pourbaix diagrams presented here. In the DFT Ni Pourbaix
diagrams (Fig. 3e, center and right panels), NiO exhibits a slightly
larger phase domain than the metastable Ni(OH)2, which explains
their ubiquitous experimentally observed coexistence in
solutions.31,67,93?96 Similar to the situation described above for Co,
initially, Ni surfaces will be passivated by Ni(OH)2, likely due to the
higher kinetic activity for its formation, which is followed by the
growth of NiO underneath,94?96 indicating the higher
thermodynamic stability of NiO as our DFT results show. In addition, NiO
and/or Ni(OH)2 are stable experimentally against dissolution at pH
? 4,31,67,93?106 which is consistent with their dissolution
boundaries in the DFT Ni Pourbaix diagrams at pH 3?5 and 4?6,
respectively. However, in the experimental Ni Pourbaix diagram
(Fig. 3e, left panel), their phase stabilities are highly
underestimated, with Ni(OH)2 and NiO only stable over the pH
range from 9 to 12.
In various alkaline solutions at pH 13?15, the Ni(OH)2?NiOOH
transition is observed at VSHE 0.5~1.0 V, which is highly consistent
with their phase boundary at ~0.75 V in the DFT Ni Pourbaix
diagrams. In contrast, the experimental Ni Pourbaix diagram (Fig. 3e,
left panel), shows that both Ni(OH)2 and NiOOH are unstable at pH
of ~14, and the upper boundary of Ni(OH)2 resides only at ?0.3 V.
To better reveal more of the electrochemical subtleties of the
mTMs, their compounds, and aqueous ions, we calculate
probabilities with respect to VSHE and pH as31
Pj exp? kBT?
where kB is Boltzmann constant, i and j index the species, and ??
depends on VSHE, pH, and [I] (fixed at 10?2 M here). Two types of
electrochemical conditions are considered here: first variable pH at
fixed VSHE (=0 V) and second, variable VSHE at fixed pH (=7) as
shown in Fig. 5. The calculated probabilities for the Cr, Mn, Fe, Co,
and Ni species within these two conditions at values ?10?6 would
indicate possible precipitation of the metastable species. It should
be noted that a decrease in probability by one order of magnitude
corresponds to an increase in ?? by about 0.06 eV. Two
observations from these probability profiles can be discerned
that may be important for the mTM compounds under various
electrochemical conditions: phase-boundary blurring and
coexistence of multiple (stable and metastable) phases.
First, at finite temperatures (e.g., 298.15 K), the probability P of a
stable phase exponentially decreases from 1 down to 0 upon
traversing a phase boundary from one stable domain to another.
Rather than an abrupt transition, finite temperature effects result
in a diffuse crossover or ?thermodynamic blurring? of the phase
boundary (Fig. 5). This indicates that a detectable precipitation of
a metal/compound may occur at an electrochemical condition
beyond its domain of stability. If P = 1% is used as an approximate
cutoff criterion, then the mTM species generally exhibit the
aforementioned phase-boundary blurring effects (?) of the order
?pH ? 1 and ?VSHE ? 0.1 V. The thermodynamic blurring in VSHE is
much smaller, because the electrode potential more significantly
affects the reaction thermodynamics, especially when species with
different cation-charge states are involved. CoO, Co3O4, and
CoOOH have exceptionally large ?pH values (?2), because they
are quite close in electrochemical stability (Fig. 4a?c), resulting in
their comparable probabilities within a relatively large pH range
Second, within a phase domain or at a phase boundary, many
secondary metastable phases with observable probabilities can be
found (Fig. 5). Here, we list the domains with significant phase
) Cr2O3 in the CrOOH domain; (
) MnO, Mn2O3, and
MnO2 in the domain (and at the domain boundaries) of Mn3O4; (
FeO, Fe(OH)2, and Fe2O3 in the domain (and at the domain
boundaries) of Fe3O4; (
) Co(OH)2 and Co3O4 in the domains (and
at the domain boundaries) of CoO and CoOOH; (
) Ni(OH)2 in the
NiO domain; (
) Ni3O4 and Ni2O3 at the NiO?NiOOH boundary.
In summary, we proposed a first-principles high-throughput
method to calculate reliable Pourbaix diagrams, which have
largely motivated us to understand various electrochemical
behaviors of mTMs (Cr, Mn, Fe, Co, and Ni) and their complex
compounds (e.g., oxides, hydroxides, and oxyhydroxides). In our
approach, a large number of structure-magnetism configurations
are screened using efficient DFT methods, and precise ?fG values
are calculated using high-level DFT methods for down-selected
configurations. Many chemical trends in the calculated ?fG values
and Pourbaix diagrams were also uncovered. Last, we calculated
the probability profiles for mTM metals, their compounds, and
aqueous ions at different conditions, which has advanced our
understanding of electrochemical transformations.
The high accuracy of our DFT Pourbaix diagrams was
comprehensively supported by various electrochemical
phenomena observed over the past several decades, which justifies the
scheme implemented and motivates a careful reassessment of
some experimental energies used in the construction of
electrochemical phase diagrams. We anticipate the wide applications of
these DFT Pourbaix diagrams in the design, characterization,
synthesis, and application of various mTM-based materials in
many future technological fields. Furthermore, there may be many
factors other than the bulk thermodynamics determining the
formations of mTM compounds in realistic (complex) situations,
e.g., kinetic activity, hydration, size effects, and substrate-induced
phenomena. We anticipate that the information obtained from
our ab initio Pourbaix diagrams and these thermodynamic
probability profiles will be especially helpful to qualitatively and
quantitatively determine the influence of such additional factors in
There exist various density functionals to approximate the exact electronic
exchange-correlation interaction, and according to Jacob?s ladder for
DFT,107 they can be categorized into four tiers: (
approximations (LDA), (
) generalized-gradient approximations (GGA), (
metaGGA functionals, which additionally include electronic kinetic energy
to improve the description of nonlocal electronic exchange, and (
functionals, which partially incorporate the exact nonlocal electronic
exchange. The rank-order increasing accuracy of these DFT methods is
LDA, GGA, and metaGGA, followed by hybrids at the expense of
computational efficiency in a decreasing rank-order LDA, GGA, metaGGA,
The increased accuracy from the semilocal LDA and GGA functionals to
the nonlocal metaGGA and hybrid functionals is mainly ascribed to the
better approximated exchange potentials, e.g., the enhanced nonlocal
nature and remedied self-interaction error.33,108?114 To simultaneously
exploit both the accuracy and efficiency of DFT, different approximations
for the electronic exchange and correlation to DFT can be used to solve
different aspects of these complex problems. In addition, by comparing
the results from different DFT methods, it is possible to build insight into
both the performance of these methods and the underlying electronic
interactions in a material.
The DFT calculations for the structures and electronic energies of
magnetic transition metals and their compounds are performed using the
VASP code,115 where projector-augmented wave (PAW)
pseudopotentials116 are used to describe the electronic wavefunctions and potentials. In
the PAW pseudopotentials for the mTM-based materials with localized 3d
orbitals, we include the aspherical (ASPH) gradient corrections to the PAW
spheres to reduce the large energetic errors that may occur (e.g., ? ?
1.0 eV/f.u. in NiO). The cutoff energy for the plane-wave expansion is
600 eV. The reciprocal k grid is \2a00 ? 2b00 ? 2c00 for the mTM compounds (a0, b0,
and c0 are the lattice constants scaled by unit of angstrom) and 12 ? 12 ?
12 for the mTMs. Phonon spectra are calculated using the PHONOPY
code,117 where the small-displacement method118 is implemented. The
energy and atomic-force convergence thresholds for the self-consistent
DFT calculations are 10?7 eV and 10?3 eV/?, respectively.
In this work, the LDA functional,119,120 GGA functionals (PBE121 and
PBEsol122), metaGGA functionals (RTPSS123 and MS2109), and hybrid
functional (HSE06113) are considered. In HSE06, 25% PBE electronic
exchange is replaced by a screened nonlocal Fock exchange (screening
length ~10 ?). Due to the two-body electronic exchange potential in
HSE06, two sets of reciprocal grids (e.g., k and k* grids) are required in the
calculations, and the second k* grid is set to be approximately 1a00 ? 1b00 ? 1c00.
The authors declare that the data supporting the findings of this study are available
within the paper and its Supplementary Information files. The simulation output files
are available upon reasonable request. They are not publicly available due to the very
large file sizes. Parameters of the input files are described in the computational
methods. Atomic structures for the simulation cells are available upon request.
The Vienna Ab Initio Simulation Package (VASP) is a proprietary software available for
purchase at https://www.vasp.at/. Data processing scripts written to process output
files and create figures are available upon request.
L.-F.H. and J.M.R. were supported by the Office of Naval Research under Award no.
N00014-16-1-2280. Calculations were performed using the QUEST HPC Facility at
Northwestern University, the HPCMP facilities at the Navy DSRC, the Extreme Science
and Engineering Discovery Environment (XSEDE) supported by the National Science
Foundation (NSF) under award number ACI-1548562, and the Center for Nanoscale
Materials (Carbon Cluster). Use of the Center for Nanoscale Materials, an Office of
Science user facility, was supported by the U.S. Department of Energy, Office of
Science, and Office of Basic Energy Sciences, under Contract No.
L.-F.H. and J.M.R. designed the theoretical method, performed ab initio calculations,
analyzed the results, and wrote the paper.
Supplementary Information accompanies the paper on the npj Materials
Degradation website (https://doi.org/10.1038/s41529-019-0088-z).
Competing interests: The authors declare no competing interests.
Publisher?s note: Springer Nature remains neutral with regard to jurisdictional claims
in published maps and institutional affiliations.
Open Access This article is licensed under a Creative Commons
Attribution 4.0 International License, which permits use, sharing,
adaptation, distribution and reproduction in any medium or format, as long as you give
appropriate credit to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made. The images or other third party
material in this article are included in the article?s Creative Commons license, unless
indicated otherwise in a credit line to the material. If material is not included in the
article?s Creative Commons license and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly
from the copyright holder. To view a copy of this license, visit http://creativecommons.
1. Hoar , T. P. , Mears , D. C. & Evans , U. R. Corrosion-resistant alloys in chloride solutions: materials for surgical implants . Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci . 294 , 486 - 510 ( 1966 ).
2. Pourbaix , M. Electrochemical corrosion of metallic biomaterials . Biomaterials 5 , 122 - 134 ( 1984 ).
3. Jacobs , J. J. , Gilbert , J. L. & Urban , R. M. Current concepts review-corrosion of metal orthopaedic implants . J. Bone Jt. Surg . 80 , 268 - 282 ( 1998 ).
4. Eliaz , N. , Shemesh , G. & Latanision , R. M. Hot corrosion in gas turbine components . Eng. Fail. Anal. 9 , 31 - 43 ( 2002 ).
5. Padture , N. P. , Gell , M. & Jordan , E. H. Thermal barrier coatings for gas-turbine engine applications . Science 296 , 280 - 284 ( 2002 ).
6. Clarke , D. R. & Levi , C. G. Materials design for the next generation thermal barrier coatings . Ann. Rev. Mater. Res . 33 , 383 - 417 ( 2003 ).
7. Zinkle , S. J. & Snead , L. L. Design ingradiation resistance in materials for fusion energy . Annu. Rev. Mater. Res . 44 , 241 - 267 ( 2014 ).
8. Rodriguez , D. & Chidambaram , D. Accelerated estimation of corrosion rate in supercritical and ultra-supercritical water . npj Mater. Degrad . 1 , 14 ( 2017 ).
9. Carranza , R. M. & Rodr?guez , M. A. Crevice corrosion of nickel-based alloys considered as engineering barriers of geological repositories . npj Mater. Degrad. 1 , 9 ( 2017 ).
10. Yeh , J.-W. et al. Nanostructured high-entropy alloys with multiple principal elements: novel alloy design concepts and outcomes . Adv. Eng. Mater . 6 , 299 - 303 ( 2004 ).
11. Zhang , Y. et al. Microstructures and properties of high-entropy alloys . Prog. Mater. Sci. 61 , 1 - 93 ( 2014a ).
12. Li , Z. & Raabe , D. Strong and ductile non-equiatomic high-entropy alloys: design, processing, microstructure, and mechanical properties . JOM 69 , 2099 - 2106 ( 2017 ).
13. Qiu , Y. , Thomas , S. , Gibson , M. A. , Fraser , H. L. & Birbilis , N. Corrosion of high entropy alloys . npj Mater. Degrad . 1 , 15 ( 2017 ).
14. Shi , Y. , Yang , B. & Liaw , P. K. Corrosion-resistant high-entropy alloys: a review . Metals 7 , 43 ( 2017 ).
15. Sivula , K. , Le Formal , F. & Gr?tzel , M. Solar water splitting: progress using hematite (?-Fe2O3) photoelectrodes . ChemSusChem. 4 , ( 432 - 449 ( 2011 ).
16. Lin , H. , Zhang, Y. , Wang , G. & Li , J .-B. Cobalt-based layered double hydroxides as oxygen evolving electrocatalysts in neutral electrolyte . Front. Mater. Sci. 6 , 142 - 148 ( 2012 ).
17. Subbaraman , R. et al. Trends in activity for the water electrolyser reactions on 3d M (Ni , Co,Fe, Mn) hydr (oxy) oxide catalysts . Nat. Mater . 11 , 550 - 557 ( 2012 ).
18. Burke , M. S. , Enman , L. J. , Batchellor , A. S. , Zou , S. & Boettcher , S. W. Oxygen evolution reaction electrocatalysis on transition metal oxides and (oxy)hydroxides: activity trends and design principles . Chem. Mater . 27 , 7549 - 7558 ( 2015 ).
19. Deng , W. , Ji , X. , Chen , Q. & Banks , C. E. Electrochemical capacitors utilising transition metaloxides: an update of recent developments . RSC Adv . 1 , 1171 - 1178 ( 2011 ).
20. Wang , G. , Zhang, L. & Zhang , J. A review of electrode materials for electrochemical supercapacitors . Chem. Soc. Rev . 41 , 797 - 828 ( 2012 ).
21. Cheng, J. P. , Zhang, J. & Liu , F. Recent development of metal hydroxidesas electrode material of electrochemical capacitors . RSC Adv . 4 , 38893 - 38917 ( 2014 ).
22. Fergus , J. W. Recent developments in cathode materials for lithium ion batteries . J. Power Sources 195 , 939 - 954 ( 2010 ).
23. Kim , S.-W., Seo , D.-H., Ma , X. , Ceder , G. & Kang , K. Electrode materials for rechargeable sodium-ion batteries: potential alternatives to current lithium-ion batteries . Adv. Energy Mater . 2 , 710 - 721 ( 2012 ).
24. Yuan , C. , Wu , H. B. , Xie , Y. & Lou , X. W. Mixed transition-metal oxides: design, synthesis, and energy-related applications . Angew. Chem . Int. Ed. 53 , 1488 - 1504 ( 2014 ).
25. Waser , R. & Aono , M. Nanoionics-based resistive switching memories . Nat. Mater . 6 , 833 - 840 ( 2007 ).
26. Sawa , A. Resistive switching in transition metal oxides . Mater. Today 11 , 28 - 36 ( 2008 ).
27. Devic , T. & Serre , C. High valence 3p and transition metal based MOFs . Chem. Soc. Rev . 43 , 6097 - 6115 ( 2014 ).
28. Kumar , S. G. & Devi , L. G. Review on modified TiO2 photocatalysis under UV/ Visible light: selected results and related mechanisms on interfacial charge carrier transfer dynamics . J. Phys. Chem. A 115 , 13211 - 13241 ( 2011 ).
29. Macwan , D. P. , Dave , P. N. & Chaturvedi , S. A review on nano-TiO2 sol-gel type syntheses and its applications . J. Mater. Sci . 46 , 3669 - 3686 ( 2011 ).
30. Pourbaix , M. ATLAS of Electrochemical Equilibria in Aqueous Solutions . (Pergamon Press, Oxford, 1966 ).
31. Huang , L.-F. , Hutchison , M. J. , Santucci , R. J. Jr. , Scully , J. R. & Rondinelli , J. M. Improved electrochemical phase diagrams from theory and experiment: the Niwater system and its complex compounds . J. Phys. Chem . C. 121 , 9782 - 9789 ( 2017 ).
32. Huang , L.-F. & Rondinelli , J. M. Electrochemical phase diagrams for Ti oxides from density functional calculations . Phys. Rev. B 92 , 245126 ( 2015 ).
33. Huang , L.-F. & Rondinelli , J. M. Electrochemical phase diagrams of Ni from ab initio simulations: role of exchange interactions on accuracy . J. Phys.: Condens. Matter 29 , 475501 ( 2017 ).
34. Belsky , A. , Hellenbrandt , M. , Karen , V. L. & Luksch , P. New developments in the inorganic crystal structure database (ICSD): accessibility in support of materials research and design . Acta Crystallogr. Sect. B 58 , 364 - 369 ( 2002 ).
35. Chase , M. W. NIST-JANAF Thermochemical Tables . 4th edn (American Institute of Physics, New York, 1998 ).
36. Anisimov , V. I. , Aryasetiawan , F. & Lichtenstein , A. I. First-principles calculations of the electronic structure and spectra of strongly correlated systems: the LDA + U method . J. Phys.: Condens. Matter 9 , 767 ( 1997 ).
37. Campo , V. L. Jr . & Cococcioni , M. Extended DFT + U + V method with on-site and inter-site electronic interactions . J. Phys.: Condens. Matter 22 , 055602 ( 2010 ).
38. Wang , L. , Maxisch , T. & Ceder , G. Oxidation energies of transition metaloxides within the GGA + U framework . Phys. Rev. B 73 , 195107 ( 2006 ).
39. Jain , A. et al. Formation enthalpies by mixing GGA and GGA + U calculations . Phys. Rev. B 84 , 045115 ( 2011 ).
40. Lutfalla , S. , Shapovalov , V. & Bell , A. T. Calibration of the DFT/GGA+U method for determination of reduction energies for transition and rare earth metal oxides of Ti, V, Mo, and Ce. J. Chem . Theory Comput . 7 , 2218 - 2223 ( 2011 ).
41. Aykol , M. & Wolverton , C. Local environment dependent GGA + U method for accurate thermochemistry of transition metal compounds . Phys. Rev. B 90 , 115105 ( 2014 ).
42. Persson , K. A. , Waldwick , B. , Lazic , P. & Ceder , G. Prediction of solid-aqueous equilibria: scheme to combine first-principles calculations of solids with experimental aqueous states . Phys. Rev. B 85 , 235438 ( 2012 ).
43. Zeng , Z. et al. Towards first principles-based prediction of highly accurate electrochemical pourbaix diagrams . J. Phys. Chem . C. 119 , 18177 - 18187 ( 2015 ).
44. Huang , L.-F. , Scully , J. R. & Rondinelli , J. M. Modeling corrosion with firstprinciples electrochemical phase diagrams . Annu. Rev. Mater. Res . 49 ( 2019 ). https://doi.org/10.1146/annurev-matsci- 070218 -010105.
45. Huang , L.-F. , Lu , X.-Z. , Tennessen , E. & Rondinelli , J. M. An efficient ab-initio quasiharmonic approach for the thermodynamics of solids . Comput. Mater. Sci . 120 , 84 - 93 ( 2016 ).
46. M?ller , S. Bulk and surface ordering phenomena in binary metal alloys . J. Phys.: Condens. Matter 15 , R1429 ( 2003 ).
47. Huang , L.-F. et al. From electronic structure to phase diagrams: a bottom-up approach to understand the stability of titanium-transition metal alloys . Acta Mater . 113 , 311 - 319 ( 2016 ).
48. Landrum , G. A. & Dronskowski , R. The orbital origins of magnetism: from atoms to molecules to ferromagnetic alloys . Angew. Chem . Int. Ed. 39 , 1560 - 1585 ( 2000 ).
49. Huang , L.-F. , Grabowski , B. , McEniry , E. , Trinkle , D. R. & Neugebauer , J. Importance of coordination number and bond length in titanium revealed by electronic structure investigations . Phys. Status Solidi B 252 , 1907 - 1924 ( 2015 ).
50. Young , D. J . High Temperature Oxidation and Corrosion of Metals , Vol. 1 ( Elsevier , Amsterdam, 2008 ).
51. Piazza , C. S. & Di Quarto , Fo Photocurrent spectroscopic investigations of passive films on chromium . J. Electrochem. Soc . 137 , 2411 - 2417 ( 1990 ).
52. Maurice , V. , Yang , W. P. & Marcus , P. XPS and STM investigation of the passive film formed on Cr(110) single?crystal surfaces . J. Electrochem. Soc . 141 , 3016 - 3027 ( 1994 ).
53. Kim , J. , Cho , E. & Kwon , H. Photo-electrochemical analysis of passive film formed on (cr in ph 8.5) buffer solution . Electrochim. Acta 47 , 415 - 421 ( 2001 ).
54. Olazabal , M. A. , Nikolaidis , N. P. , Suib , S. A. & Madariaga , J. M. Precipitation equilibria of the chromium(vi)/iron(iii) system and spectrospcopic characterization of the precipitates . Environ. Sci. Technol . 31 , 2898 - 2902 ( 1997 ).
55. Abd El-sadek, M. S. & Babu , S. M. A controlled approach for synthesizing CdTe@CrOOH (core-shell) composite nanoparticles . Curr. Appl. Phys . 11 , 926 - 932 ( 2011 ).
56. Collins, R. N. , Clark , M. W. & Payne , T. E. Solid phases responsible for MnII, CrIII , CoII, Ni, CuII and Zn immobilization by a modified bauxite refinery residue (red mud) at pH 7.5 . Chem . Eng. J. 236 , 419 - 429 ( 2014 ).
57. Pardo , P. , Calatayud , J. M. & Alarc?n , J. Chromium oxide nanoparticles with controlled size prepared from hydrothermal chromium oxyhydroxide precursors . Ceram. Int . 43 , 2756 - 2764 ( 2017 ).
58. Liu , C. T. & Wu , J. K. Influence of pH on the passivation behavior of 254 SMO stainless steel in 3.5 NaCl solution . Corros. Sci . 49 , 2198 - 2209 ( 2007 ).
59. Jiao , F. & Frei , H. Nanostructured cobalt and manganese oxide clusters as efficient water oxidation catalysts . Energy Environ. Sci. 3 , 1018 - 1027 ( 2010 ).
60. Najafpour , M. M. , Haghighi , B. , Sedigh , D. J. & Ghobadi , M. Z. Conversions of Mn oxides to nanolayered Mn oxide in electrochemical water oxidation at near neutral pH, all to a better catalyst: catalyst evolution . Dalton Trans . 42 , 16683 - 16686 ( 2013 ).
61. Ram?rez , A. et al. Evaluation of MnOx, Mn2O3, and Mn3O4 electrodeposited films for the oxygen evolution reaction of water . J. Phys. Chem . C. 118 , 14073 - 14081 ( 2014b ).
62. Gorlin , Y. et al. Understanding interactions between manganese oxide and gold that lead to enhanced activity for electrocatalytic water oxidation . J. Am. Chem. Soc . 136 , 4920 - 4926 ( 2014c ).
63. Frydendal , R. et al. Operando investigation of Au-MnOx thin films with improved activity for the oxygen evolution reaction . Electrochim. Acta 230 , 22 - 28 ( 2017a ).
64. Takashima , T. , Hashimoto , K. & Nakamura , R. Mechanisms of pH-dependent activity for water oxidation to molecular oxygen by MnO2 electrocatalysts . J. Am. Chem. Soc . 134 , 1519 - 1527 ( 2012 ).
65. Hem , J. D. Rates of manganese oxidation in aqueous systems . Geochim. Cosmochim. Acta 45 , 1369 - 1374 ( 1981 ).
66. Jung , H. & Jun , Y.-S. Ionic strength -controlled mn(hydr)oxide nanoparticle nucleation on quartz: effect of aqueous Mn(OH)2 . Environ . Sci. Technol . 50 , 105 - 113 ( 2016 ).
67. Lyons , M. E. G. & Brandon , M. P. The oxygen evolution reaction on passive oxide covered transition metal electrodes in aqueous alkaline solution. part I - nickel . Int. J. Electrochem. Sci. 3 , 1386 - 1424 ( 2008 ).
68. Lyons , M. E. G. & Brandon , M. P. The oxygen evolution reaction on passive oxide covered transition metal electrodes in alkaline solution part II - cobalt . Int. J. Electrochem. Sci. 3 , 1425 - 1462 ( 2008 ).
69. Lyons , M. E. G. & Brandon , M. P. The oxygen evolution reaction on passive oxide covered transition metal electrodes in alkaline solution. part III - iron . Int. J. Electrochem. Sci. 3 , 1463 - 1503 ( 2008 ).
70. Lyons , M. E. G. & Brandon , M. P. A comparative study of the oxygen evolution reaction on oxidised nickel, cobalt and iron electrodes in base . J. Electroanal. Chem . 641 , 119 - 130 ( 2010 ).
71. Cornell , R. M. & Schwertmann , U. The Iron Oxides: Structure, Properties , Reactions, Occurrences and Uses (John Wiley & Sons, Weinheim, 2003 ).
72. Kang , Y. S. , Risbud , S. , Rabolt , J. F. & Stroeve , P. Synthesis and characterization of nanometer-size Fe3O4 and ?-Fe2O3 particles . Chem. Mater . 8 , 2209 - 2211 ( 1996 ).
73. Jolivet , J.-P. , Chan?ac , C. & Tronc , E. Iron oxide chemistry . From molecular clusters to extended solid networks . Chem. Commun . 481 - 483 ( 2004 ).
74. Petcharoen , K. & Sirivat , A. Synthesis and characterization of magnetite nanoparticles via the chemical co-precipitation method . Mater. Sci. Eng. B 177 , 421 - 427 ( 2012 ).
75. Burke , L. D. & Lyons , M. E. G. The formation andstability of hydrous oxide films on iron under potential cycling conditions in aqueous solution at high pH . J. Electroanal. Chem . 198 , 347 - 368 ( 1986 ).
76. Lee , J. , Isobe , T. & Senna , M. Preparation of ultrafine Fe3O4 particles by precipitation in the presence of PVA at high pH . J. Colloid Interface Sci . 177 , 490 - 494 ( 1996 ).
77. Beinlich , A. et al. Peridotite weathering is the missing ingredient of Earth's continental crust composition . Nat. Commun . 9 , 634 ( 2018a ).
78. Frankel , R. B. , Papaefthymiou , G. C. , Blakemore , R. P. & O'Brien , W. Fe3O4 precipitation in magnetotactic bacteria . Biochim. et. Biophys. Acta 763 , 147 - 159 ( 1983 ).
79. Mandernack , K. W. , Bazylinski , D. A. , Shanks , W. C. & Bullen , T. D. Oxygen and iron isotope studies of magnetite produced by magnetotactic bacteria . Science 285 , 1892 - 1896 ( 1999 ).
80. Hanzlik , M. et al. Superparamagnetic magnetite in the upper beak tissue of homing pigeons . Biometals 13 , 325 - 331 ( 2000 ).
81. Chen , C. et al. Achieving high performance corrosion and wear resistant epoxy coatings via incorporation of noncovalent functionalized graphene . Carbon 114 , 356 - 366 ( 2017b ).
82. Cui , M. et al. Processable poly(2-butylaniline)/hexagonal boron nitride nanohybrids for synergetic anticorrosive reinforcement of epoxy coating . Corros. Sci . 131 , 187 - 198 ( 2018b ).
83. Han, R . et al. 1D magnetic materials of Fe3O4 and Fe with high performance of microwave absorption fabricated by electrospinning method . Sci. Rep . 4 , 7493 ( 2014d ).
84. Wei , Y. et al. Facile synthesisof self-assembled ultrathin ?-FeOOH nanorod/ graphene oxide composites for supercapacitors . J. Colloid Interface Sci . 504 , 593 - 602 ( 2017c ).
85. Song , H. , Xia , L. , Jia , X. & Yang , W. Polyhedral ?-Fe2O3 crystals at RGO nanocomposites: synthesis, characterization, and application in gassensing . J. Alloy. Compd . 732 , 191 - 200 ( 2018 ).
86. Meier , H. G. , Vilche , J. R. & Arv?a , A. J. The electrochemical behaviour of cobalt in alkaline solutions part II. the potentiodynamic response of Co(OH)2 electrodes . J. Electroanal. Chem . 138 , 367 - 379 ( 1982 ).
87. Ohtsuka , T. & Sato , N. Anodic oxide film on cobalt in weakly alkaline solution . J. Electroanal. Chem . 147 , 167 - 179 ( 1983 ).
88. Foelske , A. & Strehblow , H.-H. Structure and composition of electrochemically prepared oxide layers on Co in alkaline solutions studied by XPS . Surf. Interface Anal . 34 , 125 - 129 ( 2002 ).
89. Bewick , A. , Guti?rrez , C. & Larramona , G. An in-situ IR spectroscopic study of the anodic oxide film on cobalt in alkaline solutions . J. Electroanal. Chem . 333 , 165 - 175 ( 1992 ).
90. Yeo , B. S. & Bell , A. T. Enhanced activity of gold-supported cobalt oxide for the electrochemical evolution of oxygen . J. Am. Chem. Soc . 133 , 5587 - 5593 ( 2011 ).
91. Kanan , M. W. et al. Structure and valency of a cobalt?phosphate water oxidation catalyst determined by in situ x-ray spectroscopy . J. Am. Chem. Soc . 132 , 13692 - 13701 ( 2010 ).
92. Simmons , G. W. , Vertes , A. , Varsanyi , M. L. & Leidheiser , H. Emission M?ssbauer studies of anodically formed CoO2 . J. Electrochem. Soc . 126 , 187 - 189 ( 1979 ).
93. Lo , Y. L. & Hwang , B. J. In situ Raman studies on cathodically deposited nickel hydroxide films and electroless Ni-P electrodes in 1 M KOH solution . Langmuir 14 , 944 - 950 ( 1998 ).
94. Hoppe , H.-W. & Strehblow , H.-H. XPS and UPS examinations of passive layers on Ni and Fe53Ni alloys . Corros. Sci . 31 , 167 - 177 ( 1990 ).
95. Druska , P. , Strehblow , H. H. & Golledge , S. A surface analytical examination of passive layers on CuNi alloys: part I. alkaline solution . Corros. Sci . 38 , 835 - 851 ( 1996 ).
96. Druska , P. & Strehblow , H.-H. Surface analytical examination of passive layers on Cu-Ni alloys Part II. acidic solutions . Corros. Sci . 38 , 1369 - 1383 ( 1996 ).
97. Schrebler Guzm?n , R. S. , Vilche , J. R. & Arv?a , A. J. The kinetics and mechanism of the nickel electrode III. The potentiodynamic response of nickel electrodes in alkaline solutions in the potential region of Ni(OH)2 formation . Corros. Sci. 18 , 765 - 778 ( 1978 ).
98. Melendres , C. A. & Xu , S. In situ laser Raman spectroscopic study of anodic corrosion films on nickel and cobalt . J. Electrochem. Soc . 131 , 2239 - 2243 ( 1984 ).
99. Lillard , R. S. & Scully , J. R. Electrochemical passivation of ordered NiAl . J. Electrochem. Soc. 145 , 2024 - 2032 ( 1998 ).
100. Han , S. Y. et al. The growth mechanism of nickel oxide thin films by roomtemperature chemical bath deposition . J. Electrochem. Soc . 153 , C382 - C386 ( 2006 ).
101. Yeo , B. S. & Bell , A. T. In situ Raman study of nickel oxide and gold-supported nickel oxide catalysts for the electrochemical evolution of oxygen . J. Phys. Chem . C. 116 , 8394 - 8400 ( 2012 ).
102. Trotochaud , L. , Young , S. L. , Ranney , J. K. & Boettcher , S. W. Nickel-iron oxyhydroxide oxygen-evolution electrocatalysts: the role of intentional and incidental iron incorporation . J. Am. Chem. Soc . 136 , 6744 - 6753 ( 2014 ).
103. Klaus , S. , Cai , Y. , Louie , M. W. , Trotochaud , L. & Bell , A. T. Effects of Fe electrolyte impurities on Ni(OH)2/NiOOH structure and oxygen evolution activity . J. Phys. Chem . C. 119 , 7243 - 7254 ( 2015 ).
104. Chen , J. S. , Gui , Y. & Blackwood , D. J. A versatile ionic liquid-assisted approach to synthesize hierarchical structures of?-Ni(OH)2 nanosheets for high performance pseudocapacitor . Electrochim. Acta 188 , 863 - 870 ( 2016 ).
105. Abbas , S. A. & Jung , K. D. Preparation of mesoporous microspheres of nio with high surface area and analysis on their pseudocapacitive behavior . Electrochim. Acta 193 , 145 - 153 ( 2016 ).
106. Jung , S. C. et al. Synthesis of nanostructured ?-ni(oh)2 by electrochemical dissolution-precipitation and its application as a water oxidation catalyst . Nanotechnology 27 , 275401 ( 2016b ).
107. Perdew , J. P. & Schmidt , K. Jacob's ladder of density functional approximations for the exchange-correlation energy . AIP Conf. Proc. 577 , 1 - 20 ( 2001 ).
108. Sun , J. et al. Semilocal and hybrid meta-generalized gradient approximations based on the understanding of the kinetic-energy-density dependence . J. Chem. Phys . 138 , 044113 ( 2013a ).
109. Sun , J. et al. Density functionals that recognize covalent, metallic, and weak bonds . Phys. Rev. Lett . 111 , 106401 ( 2013b ).
110. Heyd , J. , Scuseria , G. E. & Ernzerhof , M. Hybrid functionals based on a screened coulomb potential . J. Chem. Phys . 118 , 8207 - 8215 ( 2003 ).
111. Heyd , J. & Scuseria , G. E. Assessment and validation of a screened coulomb hybrid density functional . J. Chem. Phys . 120 , 7274 - 7280 ( 2004a ).
112. Heyd , J. & Scuseria , G. E. Efficient hybrid density functional calculations in solids: assessment of theheyd-scuseria-ernzerhof screened coulomb hybrid functional . J. Chem. Phys . 121 , 1187 - 1192 ( 2004b ).
113. Heyd , J. , Scuseria , G. E. & Ernzerhof , M. Erratum: ?hybrid functionals based on a screened coulomb potential? [ J. Chem . Phys. 118 , 8207 ( 2003 ) ] . J. Chem. Phys . 124 , 219906 ( 2006 ).
114. Vydrov , O. A. , Heyd , J. , Krukau , A. V. & Scuseria , G. E. Importance of short-range versus long-range hartree-fock exchange for the performance of hybrid density functionals . J. Chem. Phys . 125 , 074106 ( 2006 ).
115. Hafner , J. Ab-initio simulations of materials using VASP: density-functional theory and beyond . J. Comput. Chem . 29 , 2044 - 2078 ( 2008 ).
116. Bl?chl , P. E. Projector augmented-wave method . Phys. Rev. B 50 , 17953 - 17979 ( 1994 ).
117. Togo , A. & Tanaka , I. First principles phonon calculations in materials science . Scr. Mater . 108 , 1 - 5 ( 2015 ).
118. Parlinski , K. , Li , Z. Q. & Kawazoe , Y. First-principles determination of the soft mode in cubic ZrO2 . Phys. Rev. Lett . 78 , 4063 - 4066 ( 1997 ).
119. Ceperley , D. M. & Alder , B. J. Ground state of the electron gas by a stochastic method . Phys. Rev. Lett . 45 , 566 - 569 ( 1980 ).
120. Perdew , J. P. & Zunger , A. Self-interaction correction to density-functional approximations for many-electron systems . Phys. Rev. B 23 , 5048 - 5079 ( 1981 ).
121. Perdew , J. P. , Burke , K. & Ernzerhof , M. Generalized gradient approximation made simple . Phys. Rev. Lett . 77 , 3865 - 3868 ( 1996 ).
122. Perdew , J. P. et al. Restoring the density-gradient expansion for exchange in solids and surfaces . Phys. Rev. Lett . 100 , 136406 ( 2008 ).
123. Perdew , J. P. , Ruzsinszky , A. , Csonka , G. I. , Constantin , L. A. & Sun , J. Workhorse semilocal density functional for condensed matter physics and quantum chemistry . Phys. Rev. Lett . 103 , 026403 ( 2009 ).
124. Samsonov , G. V. The Oxide Handbook . (IFI/Plenum, New York, 1973 ).
125. Burgess , J. Metal Ions in Solution. (Ellis Horwood Limited , Chichester, 1978 ).
126. Bard , A. J. , Parsons , R. & Jordan , J. Standard Potentials in Aqueous Solution. (Marcel Dekker , New York, 1985 ).
127. Kubaschewski , O. , Alcock , C. B. & Spencer , P. J. Materials Thermochemistry. (Pergamon Press, Oxford, 1993 ).
128. Beverskog , B. & Puigdomenech , I. Revised pourbaix diagrams for iron at 25-300 ?c . Corros. Sci . 38 , 2121 - 2135 ( 1996 ).
129. Beverskog , B. & Puigdomenech , I. Revised pourbaix diagrams for nickel at 25-300 ?C. Corros . Sci. 39 , 969 - 980 ( 1997 ).
130. Beverskog , B. & Puigdomenech , I. Revised pourbaix diagrams for chromium at 25-300 ?c . Corros. Sci . 39 , 43 - 57 ( 1997 ).
131. Chivot , J. , Mendoza , L. , Mansour , C. , Pauport? , T. & Cassir , M. New insight in the behavior of Co-H2O system at 25-150 ?C, based on revised pourbaix diagrams . Corros. Sci . 50 , 62 - 69 ( 2008 ).
? The Author(s) 2019