Density Functional Investigation of the Thermodynamic Stability of

ARTICLE pubs.acs.org/JPCC

Density Functional Investigation of the Thermodynamic Stability of Lithium Oxide Bulk Crystalline Structures as a Function of Oxygen Pressure Kah Chun Lau* and Larry A. Curtiss* Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, United States

Jeffrey Greeley Center for Nanoscale Materials Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, United States ABSTRACT: Density functional theory is used together with classical statistical mechanical analyses to investigate the thermodynamic stability of bulk crystalline LiO2, Li2O, and Li2O2 as a function of the oxygen environment. The results indicate that lithium peroxide (Li2O2(s)) and superoxide (LiO2(s)) are likely to be stable only under O2-rich conditions with high oxygen partial pressures (PΟ2), whereas Li2O is the most stable at ambient conditions. Additionally, the trends in the density functional calculated equilibrium potential for an ideal reversible Li
O2 couple can be described by an analytical equation as a function of pressure and temperature. As part of this work, we have also calculated the structure and thermodynamics for lithium superoxide. It is found to be stable with respect to lattice vibrations, with an O
O stretching vibration mode very similar to that of the isolated LiO2 molecule and to the O2
ion radical.

1. INTRODUCTION Unlocking the true energy capabilities of lithium-ion batteries has been mainly limited by the low capacity associated with intercalation and conversion reactions at the positive electrodes.1,2 However, these problems can potentially be overcome by allowing lithium to react directly with oxygen from the atmosphere in a lithium
air battery (Li
O2 battery), as proposed by Abraham et al.3 Theoretically, it is perhaps the highest energy density electrochemical power source that can be configured.3 To date, Li
O2 cells have exhibited good capacity utilization versus discharge rates, but their rechargeability has been rather limited, due in part to the limited cycle life of the cathode.4
11 During the electrochemical reduction of oxygen, the small yet highly reactive Li cations form oxides that may precipitate on the electrode surfaces. These surface species produced by the O2 reduction tend to passivate the electrode, shut down the reduction, and render the reaction irreversible. The products of the cathode reactions involve insoluble oxides of lithium that determine the kinetics, thermodynamics, and electrochemistry at the cathodes. The overall reactions involve the formation and decomposition of lithium oxide and peroxide as follows: (Li2O2)solid + 2Li+ + 2e
T 2(Li2O)solid with Erev = 2.87 V/Li, 2Li+ + 2e
+ O2 T (Li2O2)solid with Erev = 2.96 V/Li, and 4Li+ + 4e
+ O2 T 2(Li2O)solid with Erev = 2.91 V/Li, where Erev is the reversible cell voltages referenced versus Li/Li+.12 Besides the two-electron transfer process that involves the formation and decomposition of Li2O and Li2O2, Hummelshøj et al.13 have recently suggested, on the basis of first principles r 2011 American Chemical Society

calculations, that oxygen can be reduced by lithium metal via a one-electron transfer process forming an adsorbed LiO2 species on the surface. In addition, the formation and the presence of superoxide (O2
) with a finite lifetime observed in Li
O2 batteries14,15 has suggested that this metastable superoxide of lithium (LiO2) could be formed as a preferred pathway for oxygen reduction in a one-electron process, during charge/discharge cycles of Li
O2 cells, before chemically decomposing to Li2O2 and O2 through a disproportionation reaction. In general, we note that the electrochemical response of the O2/O2
redox couple on a particular electrode is dependent on the solvent and nature of the counterion,14,15 together with the nature of the electrode catalysts;5
9 which affect the formation and decomposition of both Li2O and Li2O2 in a nontrivial and poorly understand manner, thus hindering the progress of this new technology. The equilibrium potentials for the formation of both Li2O and Li2O2 are very similar and are both thermodynamically possible within the typical potential range of Li
O2 batteries. Thus, it is of fundamental importance to develop a correct qualitative understanding of the electrochemical reactions, chemical reversibility and thermodynamic properties for lithium metal
oxygen (Li
O2(g)) electrochemical couples to develop the Li
O2 cells. Most experiments to date on the most common rechargeable aprotic Li
O2 batteries have employed O2 rather than air to Received: July 16, 2011 Revised: September 27, 2011 Published: October 11, 2011 23625

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avoid unwanted parasitic reactions.10 In these systems, the discharge products may depend substantially on the oxygen pressure and are not well understood.11 Indeed, these changes may even lead to the formation of different bulk stoichiometries, such as lithium superoxide (LiO2), which have received little study, and remain largely unexplored by both theorists and experimentalists.16 The determination of the Gibbs free energy of formation for the bulk oxides of lithium (LixOy) as a function of oxygen pressure is a useful baseline study for developing understanding the products of Li
O2 electrochemical reactions that is needed for the development of effective catalysts and electrolytes for rechargeable Li
O2 cells. In this paper, we present a density functional study of the thermodynamics of bulk crystalline LiO2, Li2O, and Li2O2 at different oxygen chemical potentials. As part of this study, we have calculated the structure and thermodynamics for lithium superoxide, for which there is little experimental or theoretical information in the literature. In the next section, the methods employed are briefly introduced. The results are presented and discussed in section 3. Conclusions are presented in section 4.

2. METHODOLOGY We employed the plane wave basis projector augmented wave (PAW) method17,18 in the framework of density functional theory (DFT). For the exchange-correlation functionals, the generalized gradient approximation (GGA), expressed in the Perdew
Burke
Ernzerhof (PBE)20 formulation as implemented in the Vienna Ab-initio Simulation Package (VASP)19 was used. The chosen plane wave kinetic energy cutoff was 430 eV, and a mesh of 9  9  9 was used for the integration of the first Brillouin zone (BZ) for all Li, Li2O, Li2O2, and LiO2 crystalline bulk calculations. The radial cutoffs of the PAW potentials of Li and O were 2.05 and 1.52 Å, respectively. In all cases, the 2s electrons for Li and the 2s and 2p electrons for O were treated as valence electrons, and the remaining electrons were kept frozen. The total energy convergence was set to 1  10
6 eV. To obtain the equilibrium structures, both cell parameters and internal atomic positions were optimized until the residual forces became less than 1  10
3 eV/Å. For the phonon calculations, the results were obtained based on the direct method21 as implemented in the FROPHO code22 within the harmonic approximation. To obtain the force constants for the phonon calculations, atomic displacements of 0.015 Å were employed, with all atomic displacements considered in all three Cartesian directions, instead of only considering symmetry inequivalent atomic displacements. To investigate the relative stability of different bulk lithium oxide crystals under different thermodynamic conditions at finite temperatures, we have employed an approximate scheme to calculate the Gibbs free energy within a thermodynamic model, as suggested by Reuter et al.23 and Bollinger et al.24 The thermodynamic potential to represent the system is the Gibbs free energy, G(T,P,NLi,NO), which depends on the number of Li (NLi), and O (NO) atoms in the sample. The thermodynamically most favorable system is then the one that minimizes the Gibbs free energy, γ(T,P) (or ΔG) of the bulk crystalline oxide, which is defined as23,24 γðT, PÞ ¼ GðT, P, NLi , NO Þ
NLi μLi ðT, PÞ
NO μO ðT, PÞ ð1Þ

Figure 1. Lattice structures of the oxides studied in this work: (a) lithia, Li2O (b) lithium peroxide, Li2O2, and (c) lithium superoxide, LiO2.

where μLi and μO are the chemical potentials of a Li atom and an O atom, respectively, in the system. For a Li
O2 battery that typically operates at room temperature and in an ∼1.0
10 atm O2-enriched gas environment,4,10,11 the μLi can be approximated as the Gibbs free energy of bulk lithium, whereas the μO can be approximated as μO(T,P) = (1/2)μO2 where thermodynamic and chemical equilibrium is assumed,23
25 and in which the μO2(T,P) can be written as μO2 ðT, PÞ ¼ ΔHO2 ðT, P0 Þ þ FOvib2 ðT ¼ 0 KÞ þ EO2

  P
TSO2 ðT, P Þ þ kB T ln 0 P 0

ð2Þ

where P0 = 1.0 atm; ΔHO2(T,P0) = HO2(T,P0)
HO2(T = 0 K, P0); Fvib O2 (T = 0 K) represents the sum of the zero-point vibration energies for the oxygen molecule; and EO2 is simply the total energy of an isolated O2 molecule from the DFT calculation. The values of the Gibbs free energies of the solid states in the system are approximated by their corresponding DFT energies. This approximation is justified because, for solids at low pressure and far below their melting points, the free energy correction of the volume variation due to pressure and temperature changes in the simulations can be ignored (i.e., G(T,P,NLi,NO) = F(T,P,NLi,NO) + PV(T,P,NLi,NO) ∼ F(T,V,NLi,NO) where the volume, V is a function of pressure, P). Thus, this leaves us with only the contributions of the lattice vibrations in the Gibbs free energy, similar to the Helmholtz free energy F(T,V,NLi,NO), as23
25 FðT, V , NLi , NO Þ ¼ Etotal ðV , NLi , NO Þ þ Evib ðT, V , NLi , NO Þ
TSvib ðT, V , NLi , NO Þ

ð3Þ

that depend on all the vibrational modes in the system.

3. RESULTS AND DISCUSSION a. Structural Properties of Bulk Crystalline Phases of Li2O, Li2O2, and LiO2. In order to have a uniform basis for the

thermodynamic stability studies in section 3.c, we have calculated the structures of bulk LiO2, Li2O, and Li2O2 at a uniform level of theory. The calculated structures are confirmed by comparing with experiment, including the measured lattice constant of Li2O and the measured X-ray diffraction (XRD) patterns for Li2O2 and Li2O. There are no experimental or theoretical results for the structure of LiO2. Li2O. Among the bulk crystalline oxides of lithium found in the literature, one of the most commonly known crystalline compounds in this binary system is lithia (i.e., lithium oxide, Li2O26
30). At ambient pressure, it exists in the antifluorite structure α-Li2O Fm3m space group, characterized by oxygen (O2
) ions arranged 23626

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in a face-centered cubic (fcc) sublattice with lithium (Li+) ions in tetrahedral interstitial sites (Figure 1). As confirmed by experiments and theory,26,27 this antifluorite phase is found to be stable up to ∼45 GPa before being transformed into an orthorhombic anticotunnite structure (Pnma). For this antifluorite structure, Table 1. The Basic Lattice Properties of Bulk Lithium and Its Oxides (i.e., Li2O, Li2O2, and LiO2), Including Lattice Constants and Their Gibbs Formation Energies Per Li Atom, with Respect to Lithium and Molecular Oxygen (with (ΔG0 in eV/ Li) and without (ΔG in eV/Li) the Phonon Contributions from the Oxides) at Zero Temperature

GGA-PBE

a (Å)

(this work)

b/a

Li

Li2O

Li2O2

3.45

4.62

3.14(3.13)a

ΔG ΔG0 a (Å)

3.95 1.25

2.43 (2.43)a

0.75


3.16


3.26


3.28


3.04


3.13


3.10

c/a

GGA-PBE

LiO2d

3.44(3.49)b

4.66(4.62)c 3.19 (F€oppl)

(Y. Xu et.al.)36 b/a c/a a

2.41 (F€oppl)

Initial configuration in F€oppl structure (in P6 representation). b Experimental values from ref 37. c Experimental values from ref 38. d The LiO2 structure is in the orthorhombic phase (in Pnnm representation). The LiO2 in pyrite phase (i.e., in Pa3 symmetry) is metastable, due to several imaginary phonon frequencies found in the phonon spectra.

the DFT-optimized lattice constant a is given in Table 1 and is comparable to reported28,29 values of ∼4.53
4.60 Å in Table 1. In this lattice structure, the oxygen atoms are well separated. The O
O minimum distance is typically ∼3.25 Å, whereas the Li
Li minimum distance is typically ∼2.30 Å. For the strong Li
O cation
anion bonds, however, the anion coordination number is 8, with distances of ∼1.99 Å, comparable to previous theoretical findings.28 Li2O2. Compared to lithia (Li2O), which is the only compound in the lithium
oxygen binary system that melts at atmospheric pressure without decomposition,16 the bulk crystalline phases of lithium peroxide (Li2O2) and superoxide (LiO2) have seemingly not been sufficiently studied, to the point that even their crystalline structures have not been unambiguously determined,16 especially for the LiO2 crystalline phase. Instead of preferring a cubic lattice, as is the case with Li2O, the crystal phase of Li2O2 has been proposed to be a hexagonal crystal lattice31,32 with a space group of P6.32 However, it has been recently reinvestigated by Cota et al.33 and reported to be in space group P63/mmc with higher symmetry, based on first-principles calculations. In this work, we have investigated both geometries, and after full geometry relaxation, we found that the two structures are almost degenerate within the accuracy of our calculation, with the structure in P63/mmc being slightly more stable than P6, with ΔEtotal ∼ 0.002 eV. To compare our calculated Li2O2 crystal lattice with Li2O2 compounds that are suggested to be the dominant discharge product in a Li
O2 battery,3,4 the XRD powder patterns of the

Figure 2. The simulated XRD of a perfect Li2O (top left) and Li2O2 (top right) crystal, and the experimental powder XRD for α-Li2O at 34.3 GPa26 (bottom left) and for Li2O2 from a Li
O2 battery electrode before charging in experiment4 (bottom right). The simulated XRD of Li2O is calculated using an X-ray wavelength λ = 0.41285 Å) as suggested in experiment.26 The inset (top left) shows the dominant XRD peaks found in the simulated XRD at 2θ < 24°, to be compared with the experimental XRD of α-Li2O (bottom left) found at the regime of 2θ < 22°. (Adapted figure reproduced fom ref 26, Figure 2. Copyright 2006 American Physical Society.) 23627

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The Journal of Physical Chemistry C optimized Li2O2 crystal in P63/mmc representation are computed (Figure 2). With peak broadening by Gaussian functions37 using the same X-ray wavelength (λ = 1.936 Å) as is employed in Li
O2 battery experiments (i.e., Fe Kα1 radiation),4 the simulated XRD patterns show good agreement with experiment. Compared to the XRD of a perfect Li2O crystal (Figure 2), an obvious difference between the XRD of Li2O and Li2O2 can be found. The most intense XRD peaks in the simulated Li2O2 spectrum are similar to the experimental peaks for Li2O2 compounds, i.e., ball-milled Li2O2 with an average particle size of ∼100 nm in the Li
O2 battery electrode,4 and they are found within the range of diffraction angles, i.e., 30.0° e 2θ e 80.0°, as shown (Figure 2). For the Li2O crystal, the dominant peaks are from (111) and (220) planes at 2θ e 9.0° and 14.0°, consistent with the experimental observation from the literature26 (Figure 2). For a perfect crystal of Li2O2 from simulation (Figure 2, top right), the most intense peak is from the crystal plane (111) at 2θ ∼ 44.5°, as shown in the simulated XRD. Interestingly, the most intense powder XRD peak can also be found at 2θ ∼ 45° in the electrode before charging the Li
O2 battery.4 In both simulated and experimentally measured XRD, the next intense peaks are located in the vicinity of 2θ ∼ 40.0° and 80.0°, where the signatures can be attributed to crystallographic planes present in (111) and (210). For other less intense peaks found in the experimentally measured XRD at 2θ ∼ 52.0° and 62.0°, correspondence with the crystal planes (102) and (103) is found. Here, the only difference between the simulated XRD from a perfect Li2O2 crystal and the experimental XRD from Li2O2 nanoparticles is the signature of the (00l) plane, which only appears in a perfect crystal. In contrast to Li2O2 nanoparticles residing on the electrode, the signature of the (002) and (004) planes is at 2θ ∼ 29° and 61°, indicating that the (00l) plane growth along the c-axis within the hexagonal facets from the symmetry of a crystal might be missing for these Li2O2 nanoparticles after ball-milling. However, in general, the good agreement among the simulated XRD patterns (for a perfect crystal) and experimental observations (for nanoparticles) confirms that the Li2O2 crystalline phase should be in its distinct hexagonal morphology. From the relaxed geometry of Li2O2, our predicted cell parameters are comfortably within those values reported in the literature,33
36 i.e., a ∼ 3.14 Å and c/a ∼ 2.43. In the crystal, while the oxygen atoms are most likely in the form of O2 ions with an O
O distance of ∼1.55 Å, close to a previous theoretical (i.e., O
O bond ∼1.57 Å)35 and experimental finding (i.e., O
O bond ∼1.50 Å).32 For the Li
O cation
anion distances, two types of Li cations can be found in this system, with reported values35 of 1.93 and 2.06 Å. In both cases, the cations are surrounded by 6 O anions. Half of the cations, which are at the center of a triangular prism in the lattices, have a Li
O distance of ∼1.96 Å. The other half of the cations lie in an octahedral site, and their Li
O distance is longer, i.e., 2.13 Å. LiO2. Relative to lithium oxide (Li2O) and peroxide (Li2O2), the existence of a bulk crystalline phase of lithium superoxide, LiO2, is even less well studied and requires further experimental and theoretical confirmation.16 The first evidence of the possibility of the existence of a lithium superoxide (LiO2) compound was obtained by the reaction of lithium peroxide (Li2O2) with ozone.39 The LiO2 compound (i.e., ∼ 47% of LiO2) apparently is unstable at room temperature, and its powder XRD data have only been characterized at liquid nitrogen temperatures.16,40 In addition to the XRD results,40 its existence has been indirectly proven by a recent photoemission study on a compound with

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LiO2 stoichiometry at ∼25 K; in spite of these results, however, the actual bulk crystalline structure of LiO2 has not yet been fully resolved.41 Several different possible configurations of LiO2 have been considered in our calculations. Assuming that the superoxide ion O2
is located at the anion position, which is probably dynamic in nature and analogous to NaO2 with various polymorphisms,42,44 a simple crystal structure optimization, together with phonon vibration frequency calculations on these structures, were performed to assess their relative stability. In the case of lithium, the existence of a superoxide ion O2
in the crystal lattice is energetically preferred to a nonzero spin-polarized (S = 1) solution, in contrast to the calculated results for oxides (S = 0) and peroxides (S = 0) of lithium. Among the configurations we studied, the most stable configuration with no imaginary phonon frequency is found to be the orthorhombic structure (i.e., in Pnnm symmetry) (Table 1). We note that the LiO2 in pyrite structure (i.e., in Pa3 symmetry) suggested by Seriani et.al.,35 is found to be ∼0.14 eV/Li less stable than the orthorhombic phase and is consistent with their findings.35 Energetically, the LiO2 in orthorhombic phase is slightly less favorable compared with Li2O2 at zero temperature. The O
O distance, 1.34 Å, is analogous to O2
ions in NaO2 (i.e., ∼ 1.31 Å)43 and is shorter than the O
O distance (∼ 1.55 Å) of O22
-like ions found in peroxide. The Li
O distance is ∼2.10 Å, slightly larger than in both oxide and peroxide. b. Phonon Spectra for Li2O, Li2O2, and LiO2. The phonon spectra of Li2O, Li2O2, and LiO2 were calculated for use in the thermodynamic stability studies in section 3.b using the structures obtained from section 3.a. These calculations also provide information on the lattice structure of these bulk crystalline materials, especially for LiO2, where there is no previous experimental data. The calculated phonon dispersion curves ω(q) in q-space of these compounds were calculated and are shown in Figure 3. For local density approximation (LDA) and GGA phonon calculations, we found that they agree very well qualitatively. Thus, only the results of GGA/PBE calculation are given. Li2O. For Li2O in the Fm3m space group, it has been known that this compound is thermodynamically stable at ambient conditions; therefore it is expected to be dynamically stable at zero temperature throughout the BZ. As shown in Figure 3, all the phonons were found to be stable across the BZ at Γ-, X-, Wand L-points, as defined in ref 45. For this low pressure α-phase, only one Raman active optical phonon mode, T2g, which describes the motion of the Li sublattice, has recently been experimentally verified.26 The signature Raman mode is found to be ∼575 cm
1 (∼ 17.24 THz) and is consistent with the partial phonon density of states of Li as shown in Figure 3. Li2O2. For the peroxide Li2O2, the shorter O
O distance compared to Li2O in the crystal lattice yields a rather different phonon character. Throughout the phonon range of ∼4
24 THz (∼131
800 cm
1), the flat phonon dispersion along the wave vectors q in BZ is well represented by their comparatively localized phonon density of states. The phonon branches examined in our study (Figure 3) include Σ [0ζ0], U [0(1/ 2)ζ], R [0ζ(1/2)] and Δ [00ζ] across Γ-, M-, L- and A-points. Since no imaginary vibrational modes can be found at the zone center and at the other direction in the BZ, this peroxide crystalline phase is supposed to be dynamically stable at the ground state. In contrast to Li2O, the motions of oxygen ions dominate the vibrational states in the low and high frequency regimes, instead of the Li cations. For the highest vibration phonon modes (i.e., ωmax ∼ 799.78 cm
1) (∼ 23.98 THz), the sharp and strong peak of the oxygen partial phonon density of 23628

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Figure 3. Phonon dispersion, total phonon density of states (orange), and partial phonon density of states (in blue and green color) of (top) lithia, Li2O (center) lithium peroxide, Li2O2, and (bottom) lithium superoxide, LiO2. The lithium and oxygen partial phonon densities of states are shown in blue and green.

states appeared to be due to the O
O stretching mode, a signature of the existence of a peroxy O
O bond, which is stoichiometrically analogous to O22
-like ions in the crystal lattice. This value is close to the reported Raman spectrum of lithium peroxide, with a strong broad band at ∼790 cm
1 from the O
O stretch vibration, yet still slightly lower than the O
O

stretch of ∼843
873 cm
1 from the Raman spectra of the barium and strontium oxide peroxide series.46 LiO2. With the oxygen-rich stoichiometry of LiO2, the contribution of oxygen in the lattice dynamics is expected to be significant. As shown in Figure 3, no imaginary vibrational modes appear in the calculated phonon dispersion curves of LiO2 in BZ 23629

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Figure 4. The calculated Gibbs free energy of formation (per unit formula) ΔG (eV/Li) of Li2O (red), Li2O2 (orange), and LiO2 (green) with (filled circles) and without (hollow circles) oxide compound phonon contributions at PO2 = 1.0 atm as a function of temperature, T. (Right) The calculated phonon entropy S (per formula unit in J/K mol) of Li2O (red), Li2O2 (orange), and LiO2 (green), together with the reported Li2O entropy contribution from experiment (blue lines).12

that we explored. Therefore, in terms of dynamic stability, the existence of this low temperature phase might be equally probable as compared to the other oxides of lithium. Along the Δ[̅ξ00], C[(1/2)ζ0], D[ξ(1/2)0], and Σ[0ζ0] branches, all the phonon dispersion is flat with a large frequency gap between ∼11 and 31 THz. Compared to oxide and peroxide, the overlap contribution of phonons among the cations and anions appeared to be minimal. In the low frequency regime, the vibrations are mostly attributed to the anions, while in the intermediate range it is dominated by the coupling of Li
O pairs in the lattices. For the high frequency phonon modes, the motion is determined by O2 anions with an O
O distance of 1.34 Å as represented by the sharp peak in the oxygen partial phonon density of states (Figure 3). As a sign of an ionic molecular solid, the ωmax of this compound is found to be ∼36 THz (or 1196 cm
1), smaller than a typical O2 diatomic stretching (∼1580 cm
1),12 close to the O2
superoxide ion radical stretching mode (∼1089 cm
1),12 and comparable to its basic stoichiometric unit, i.e., the isolated LiO2 molecule with a O
O stretching frequency at ∼1094 cm
1 and with a bond distance of 1.33 Å in the gas phase.47 c. Thermodynamic Properties of Li2O, Li2O2, and LiO2. Gibbs Free Energies of Formation at Different Temperatures. We have calculated the equilibrium Gibbs free energy, G as defined in eq 1, over a temperature range of 0
300 K for the three oxides, neglecting the energy correction due to applied pressure. Specifically, to account for the possible entropy changes via lattice dynamics due to the temperature, the phonon contribution of entropy in the free energy, G, of the solids, defined by    pωðq, sÞ ln 1
exp
S ¼
kB kB T q, s

∑




1 T

∑ q, s

pωðq, sÞ  pωðq, sÞ exp
1 kB T 

ð4Þ

has been used.22 As the temperature is far from the bulk melting temperature, we expect the system entropy will be mostly due to phonons, and this assumption works particularly well for the lithia (Li2O) crystal (Figure 4), which has a high melting point (i.e., Tm ∼ 1700 K).12 For the bulk crystalline phase of lithium metal, only the body-centered cubic (bcc) structure has been considered since it is the most stable phase at room temperature, compared to the fcc, hexagonal close-packed (hcp), and 9R phases that are only energetically stable at low temperatures.48 For bcc lithium, the calculated lattice constant is within ∼1.1% of the experimental value of a ∼ 3.49 Å, and the calculated cohesive energy of bcc lithium is in agreement with the experimental cohesive energy (∼1.63 eV/Li) of Li bulk solid to within ∼1.7% at zero temperature.49 To estimate the free energy of gas phase O2, the approximation defined23,24 in eq 2 has been made. The results for the free energies are given in Table 1 and shown in Figure 4. All three oxides of lithium are thermodynamically stable (ΔG < 0) with respect to bulk lithium at zero temperature. Thus, the formation of these oxides will occur spontaneously at sufficiently low temperatures under appropriate thermodynamic conditions. Compared to the peroxide (Li2O2) and superoxide (LiO2), the antifluorite lithia (Li2O) is energetically slightly less preferable at low temperatures. However, if the phonon contribution is included, a transition in thermodynamic stability will take place as temperature is increased. Compared to Li2O, the Li2O2 has a relatively high zero-point energy and phonon contribution to the free energy, and this leads to stabilization of Li2O. As shown in Figure 4, the superoxide is generally the least stable. In fact, it is only the most stable phase at very low temperatures (i.e., below 50 K) if the phonon contributions are ignored, while it will never be the most stable phase if phonon contributions are included. When the phonon contributions are included, together with the value of EO2 =
9.85 eV50 as defined in eq 2 (at T = 298 K and PO2 = 1.0 atm) at standard conditions, the calculated Gibbs free energy of formation of Li2O2 is
2.86 eV/Li atom, 23630

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Figure 5. (a) The calculated Gibbs free energy of formation (per unit formula) ΔG (eV/Li) of Li2O (red), Li2O2 (orange), and LiO2 (green) with (filled circles) and without (hollow circles) oxide compound phonon contributions at T = 298 K as a function of oxygen partial pressure (in atm), within the range of PO2 = 1.0  10
15 to 10 atm, where the approximations of eqs 1
3 hold. (b) The predicted electromotive force, e.m.f. (E in V), as a function of oxygen partial pressure, PO2 (in atm) within a range of PO2 = 1.0  10
15 to 10 atm, at room temperature (T = 298 K), derived from ΔG with (filled circles) and without (hollow circles) oxide compound phonon contributions, relative to the values that obtained from the analytical equation (diamonds), i.e., E = E0
(RT/nF) ln(Qa), where E0 is the DFT values with phonon contributions. The range covered by the dotted blue lines corresponds to the reported open-circuit voltages, V0 of 2.9
3.0 V measured experimentally in Li
O2 cells.4,10 The E0 (in V) is the computed equilibrium potential for Li2O (E0 = 2.88 V), Li2O2 (E0 = 2.86 V), and Li2O2 (E0 = 2.61 V) at T = 298 K and PO2 = 1.0 atm based on DFT values with phonon contributions that follow eqs 1
3.

compared to
2.88 eV/Li for Li2O. For LiO2, the predicted ΔG at ambient conditions is found to be
2.61 eV/Li, and thus is thermodynamically much less preferable compared to Li2O and Li2O2 as the temperature is raised (Figure 4). The relative thermodynamic stability of the bulk oxides of lithium are found to be modestly dependent on the phonon contributions to the free energy, as shown in Figure 4. Compared to peroxide (Li2O2) and superoxide (LiO2), the highly stable Li2O phase in the antifluorite structure has fewer phonon contributions to its Gibbs free energy of formation, leading to its modest thermal stability over the other compounds. From Figure 4, the thermodynamic instability of the LiO2 is shown to be driven by the entropy contribution from its crystal lattice vibrations. A similar trend is also found in Li2O2 (Figure 4), and this common characteristic might be attributed to the higher average phonon frequencies coming from the O
O contribution to their phonon density of states (Figure 3). This suggests that the existence of O2 molecular features in Li2O2 and LiO2 will lead to thermodynamic instability in these ionic molecular solids at ambient pressures if the system temperature is increased. This might be the main reason why the existence of the oxygen-rich Li2O2 and LiO2 solid phases is not preferred at high temperature, which is consistent with experimental observation of the thermal decomposition of Li2O2 into Li2O at ∼570 K.52 In the absence of a catalyst, we suspect that the O2 molecular feature, which has substantially larger phonon energy (Figure 4) especially in LiO2 phase, will ultimately hinder the reaction rate for the natural formation of this crystalline phase at ambient conditions. Free Energies of Formation and Equilibrium Potentials of a Reversible Li
O2 Cell at Variable Oxygen Pressures. Overall, we

find that the trends in ΔG of these LixOy(s) solids due to the changes in PO2 (Figure 5a), together with system temperature, can be described qualitatively by Le Ch^atelier’s principle. Changes in the oxygen gas concentration or oxygen gas partial pressure of the reactants subsequently will alter the thermodynamic equilibrium of a system, as defined in eq 5. According to Le Ch^atelier’s principle, an increase in PO2 (i.e., an increase in applied pressure or an increase in oxygen concentration) will shift the equilibrium to the side that would reduce that change in the system, and is thus thermodynamically preferable toward the formation of these LixOy(s) solids with higher oxygen content, as defined in eq 5. Similarly, this also holds for the reverse process (i.e., a decrease in PO2). As shown in Figure 5a, the thermally less favorable oxygenrich compounds (Li2O2 and LiO2) are particularly more sensitive to the changes in oxygen chemical potentials (μO2) through the variation in O2 partial pressure at any given temperature. In an O-poor environment (i.e., low oxygen chemical potential), the Li2O(s) is the most preferable condensed solid phase at room temperature (Figure 5a). Assuming a negligible change in volume at a realistic range of oxygen pressures, all the oxides (Li2O(s), Li2O2(s), and LiO2(s)) are found to be thermodynamically more favorable to form at high oxygen chemical potential (i.e., O-rich environment at high PO2 with PO2 > 1.0 atm). Despite the induced instability of the O
O phonons (Figure 4), an excess of oxygen in the stoichiometry for Li2O2(s) and LiO2(s) is stabilized at high oxygen chemical potentials (O-rich environment). At T = 298 K, the cross over between Li2O(s) and Li2O2(s) is found to take place at PO2 ∼ 10 atm (Figure 5a). Consequently, the stability of Li2O2(s) would be enhanced over the Li2O(s) as oxygen pressure is shifted toward the O-rich region, while for LiO2(s), the cross over 23631

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The Journal of Physical Chemistry C

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between LiO2(s) and Li2O(s) is unlikely at PO2 < 10 atm. We thus predict that for lithium metal
oxygen (Li(s)
O2(g)) electrochemical couples, the dominant chemical reactions are limited to the formation of Li2O(s) and Li2O2(s) within a realistic T and PO2 range, but not the LiO2(s). To derive the equilibrium potentials of an ideal Li
O2 cell, one has to consider the overall reactions of lithium metal
oxygen (Li
O2(g)) electrochemical couples as follows: 1 2LiðsÞ þ O2ðgÞ T Li2 OðsÞ , 2

ΔGLi2 O ðT, PO2 Þ

2LiðsÞ þ O2ðgÞ T Li2 O2ðsÞ ,

ΔGLi2 O2 ðT, PO2 Þ

LiðsÞ þ O2ðgÞ T LiO2ðsÞ ,

ΔGLiO2 ðT, PO2 Þ

ð5Þ

where the Gibbs free energy of formation, ΔG, of these oxides can be a function of temperature (T) and oxygen partial pressure (PO2) when interacting with two bulk reservoirs, Li-metal and an ideal gas-like O2 environment. In thermodynamic equilibrium, the electromotive force (e.m.f., E), or the open-circuit voltage of the cell reactions in Li
O2 batteries, can be correlated with the equilibrium cell potentials (E0) of Li(s)
O2(g) electrochemical couples at standard conditions according to the Nernst equation, E0 =
(ΔG0/ne), where n is the number of electrons (e) transferred in the process. At ambient conditions (i.e., T = 298 K, 1.0 atm), the overall formation of Li2O(s), Li2O2(s), and LiO2(s) are found to be strongly exothermic (i.e., ΔG < 0) (Figure 4) with corresponding E0 values of 2.88, 2.86, and 2.61 V, respectively. For E0 of Li2O2(s) (i.e., 2.86 V), our value is larger than the values reported by Hummerlshøj et al.13 and Xu et al.,36 i.e., 2.47 and 2.70 V, respectively. Our theoretical values are in reasonable agreement with known standard thermodynamic data,12 2.91 V (Li2O(s)) and 2.96 V (Li2O2(s)), and with the experimental opencircuit voltage (V0) of ∼2.9
3.0 V that is typically measured in a Li
O2 battery operated at 1.0 atm O2 pressure.4,10 With this agreement, it is fairly reasonable to assert that the equilibrium potentials (or the open-circuit potential) measured in the experiment4,6,10 are very likely due to the formation of Li2O(s) and Li2O2(s), rather than the formation of LiO2(s). To predict the oxygen pressure dependence of the e.m.f. or open-circuit voltage of a reversible Li
O2 cell at thermodynamic equilibrium, eqs 1
3, combined with the reaction stoichiometries given in eqs 5, are consistent with the following simple analytical relation:53 E = E0
(RT/nF) ln(Qa). Qa, which defines the reaction quotient in terms of activities from the above (Li2O) reactions (eqs 5), can be simplified to Qa ∼ 1/P(1/2) O2 and Qa ∼ 1/PO2 (Li2O2 and LiO2). Here, the Li is assumed to be present in the solid state, so the activities are unity, and the activities of the O2 gas can be represented by the partial pressure if the gas pressure is low in the ideal Li
O2 cell. Thus, by knowing the E0 of a system at a given temperature and pressure (see the DFT + phonon predictions above), the e.m.f. of a reversible Li
O2 cell accompanying the changes in oxygen pressure can be predicted. Overall, the e.m.f of a reversible Li
O2 cell is predicted to be increased (Figure 5b) as PO2 is increased, due to the enhanced stability of Li2O(s), Li2O2(s), and LiO2(s) in oxygen-rich environments. Relative to Li2O2(s) and LiO2(s), the e.m.f of Li2O(s) is less sensitive to the variation of oxygen pressure. For the DFT results, which include the oxide compounds’ phonon contributions, the e.m.f values are overall smaller than the values without phonon

contributions. In addition, as shown in Figure 5b, one can safely disregard LiO2 as the discharge product due to its substantially lower in e.m.f values. On the other hand, this suggests that both Li2O and Li2O2 are the most probable chemical reaction products in the Li
O2 cell at thermodynamic equilibrium, since both predicted e.m.f values are within the range of reported V0 at T = 298 K and PO2 = 1.0 atm. However, since this is only a simple thermodynamic model, the ultimate determination of the actual discharge products of operating Li
O2 cell remains a subject for further studies, as the influence of catalyst, electrolyte, and heterogeneous triple boundary interfaces of electrolyte, electrodes, and gas have yet to be addressed.

4. SUMMARY In conclusion, DFT is used together with classical statistical mechanical analyses to investigate the thermodynamic stability of bulk crystalline LiO2, Li2O, and Li2O2 as a function of the oxygen environment. This is relevant to the basic bulk lithium metal
oxygen gas electrochemical couples found in the lithium
air (Li
O2) battery. The following important conclusions can be drawn: 1. The results indicate that lithium peroxide (Li2O2(s)) and superoxide (LiO2(s)) are likely to be stable only under O2rich conditions with high oxygen partial pressures (PO2), whereas Li2O is the most stable at ambient conditions. The trends in the density functional calculated equilibrium potential for of an ideal reversible Li
O2 couple can be described by an analytical equation as a function of pressure and temperature. 2. As part of this work, we have calculated the structure and thermodynamics for lithium superoxide. It is found be stable with respect to lattice vibrations with an O
O stretching vibration mode very similar to that of the isolated LiO2 molecule and the O2
ion radical, suggesting that bulk LiO2 is an ionic molecular solid. 3. On the basis of comparison of our calculated open-circuit voltage with the experimental open-circuit voltages found in Li
air cells, we find the most probable chemical reaction products of an operating Li
O2 cell are Li2O and Li2O2, but not LiO2. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected] (K.C.L.); [email protected] (L.A.C.).

’ ACKNOWLEDGMENT This work was supported by the U.S. Department of Energy under Contract DEAC0206CH11357. This material is based upon work supported as part of the Tailored Interfaces for Energy Storage, an Energy Frontier Research Center, and an Early Career Award (J.G.), both funded by the U.S. Department of Energy, Office of Science, and Office of Basic Energy Sciences. We gratefully acknowledge grants of computer time from EMSL, a national scientific user facility located at Pacific Northwest National Laboratory, the ANL Laboratory Computing Resource Center (LCRC), and the ANL Center of Nanoscale Materials. ’ REFERENCES (1) Armand, M; Tarascon, J. M. Nature 2008, 451, 652. (2) Tarascon, J. M.; Armand, M. Nature 2001, 414, 359. 23632

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