Thermodynamics of Lithium in TiO2 (B) from First Principles

Li insertion into spinel titanates occurs through a first-order phase ..... of Energy, Office of Science, Office of Basic Energy Sciences under Award ...
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Thermodynamics of Lithium in TiO2(B) from First Principles Andrew S. Dalton, Anna A. Belak, and Anton Van der Ven* Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States S Supporting Information *

ABSTRACT: We use first-principles density functional theory (DFT) calculations combined with statistical mechanical techniques based on the cluster expansion method and Monte Carlo simulations to predict the lithium site occupancies, voltage curves, and phase diagram for TiO2(B), a candidate anode material for lithium ion batteries. We find that Li intercalation is thermodynamically favorable up to a Li/Ti ratio of 1.25, higher than the theoretical maximum usually assumed for TiO2. The calculated phase diagram at 300 K contains three first-order phase transformations corresponding to major changes in the favored intercalation sites at increasing Li concentrations. Calculations based on DFT predict the stability of a new Li site at high Li concentrations in TiO2(B) and the occurrence of a dramatic site-inversion as Li is added to the host. KEYWORDS: lithium, intercalation, titanium dioxide, TiO2(B), cluster expansion, Monte Carlo

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INTRODUCTION The polymorphs of titanium dioxide are attractive candidates for anode intercalation compounds in lithium ion batteries. These materials have a theoretical capacity of one lithium ion per TiO2 formula unit (335 mAh/g), and their high voltage relative to metallic Li reduces the risk of electroplating, thereby making them safer materials than graphite.1 At least eight polymorphs of TiO2 are known in total,1 with rutile, anatase, and brookite occurring naturally. All TiO2 polymorphs consist of TiO6 octahedra joined at corners or edges, with open spaces that can accommodate Li ions. The Li sites typically have LiO6 (octahedral) or LiO4 (tetrahedral) coordination,2 but sometimes square pyramidal or square planar coordination as in TiO2(B).2,3 Spinel LiTi2O4, and especially its Li-excess variant,4,5 exhibit remarkable rate and cycling capabilities. Li insertion into spinel titanates occurs through a first-order phase transformation with negligible changes in lattice parameters, which, along with the high Li mobility in the spinel crystal structure,4 is likely responsible for their favorable electrochemical properties. The success of spinel titanates as a potential anode material has spurred interest in the many polymorphs of TiO2, which have a higher theoretical capacity than spinel Li1+xTi2O4 and its Liexcess variant. To date, though, the various polymorphs of TiO2 have proven more difficult to cycle electrochemically and require nanoscaling6,7 and doping8 to achieve acceptable charge/discharge rates and capacities. The TiO2(B) (bronze) polymorph9,10 has received attention recently because of its low density compared to other polymorphs,1 its high reversible capacity demonstrated in experiments,11−14 and its ability to be fabricated into nanowires11,13 and nanotubes.12,15 Calculations of Li site energies and diffusion barriers have been reported by Arrouvel et al.,3 and ordered phases obtained by neutron diffraction have been reported by Armstrong et al.16 © 2012 American Chemical Society

In this paper, we predict the thermodynamic properties of the insertion of Li into bulk TiO2(B) from first principles. We calculate the open circuit voltage, the temperature−composition phase diagram, and the occupancy of different Li sites as a function of Li concentration. Our approach combines firstprinciples total energy calculations with a cluster expansion for the configurational energy associated with Li-vacancy disorder and Monte Carlo simulations. This approach has proven successful in predicting the thermodynamic properties of a variety of Li intercalation compounds such as LiCoO2,17−20 LiFePO4,21 Lix(Ni0.5Mn0.5)O2,22 LixTiS2,23 and LiTi2O4.4,5 The approach enables a systematic determination of Li site preference and Li-vacancy ordering as a function of temperature and Li concentration. We find that Li insertion into bulk TiO2(B) leads to a series of intermediate ordered phases as well as first-order phase changes corresponding to large changes in the preferred sites occupied by Li. In addition to Li sites identified previously,3,16 we identify a new site that is energetically favorable at high Li concentration. We also predict that intercalation is thermodynamically possible up to a Li/Ti ratio of 1.25.

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METHOD The free energy of an intercalation compound determines many of its electrochemical and thermodynamic properties. The voltage, for example, is linearly related to the Li chemical potential according to the Nernst equation V (x) = −[μLi (x) − μLireference ]/e

where μLi(x), the Li chemical potential of the intercalation compound, is the derivative of the free energy with respect to Received: November 2, 2011 Revised: April 10, 2012 Published: April 11, 2012 1568

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Li concentration x, μreference is the chemical potential of the Li reference anode (a constant for metallic Li), and e is the charge of an electron. Phase stability of the intercalation compound as a function of Li concentration is determined by a minimum of the free energy. Well-established statistical mechanical techniques have been developed to predict the free energies of intercalation compounds from first principles.24−27 Although electronic and vibrational excitations play a role in determining thermodynamic properties, the most important contribution to the free energy of an intercalation compound arises from configurational degrees of freedom associated with all the possible ways of distributing Li ions and vacancies over the interstitial sites of the host. These configurational excitations can be sampled with Monte Carlo simulations applied to a lattice model Hamiltonian based on a cluster expansion. A cluster expansion24 in this context describes the dependence of the total energy of the intercalation compound on the configuration of Li and vacancies represented as σ⃗, which denotes the collection of occupation variables σi assigned to each interstitial site i, with the value +1 if occupied and −1 if vacant. In terms of these occupation variables, the total energy of the crystal can be expressed as an expansion of cluster functions ϕα(σ⃗) according to24 E(σ ⃗) = V0 +

free energy becomes equal to the energy of the intercalation compound. The grand canonical free energy, defined as Φ = G − μLix can also be calculated through free energy integration according to β Φ(μLi , T ) = βoΦ(μLi , To) +

where the discrete polynomials

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∏ σi i∈α

RESULTS A. Structure. Figure 1a illustrates the TiO2(B) crystal structure. It has monoclinic symmetry and belongs to the C2/m space group9,10 with cell parameters a = 12.1787 Å, b = 3.7412 Å, c = 6.5249 Å, and β = 107.054°.10 The structure contains an array of open tunnels along the b-direction.3 The unit cell of TiO2(B) contains eight TiO2 formula units with two crystallographically distinct Ti sites and four distinct

each correspond to the product of occupation variables belonging to a cluster of Li sites (point cluster, pair cluster, triplet cluster, etc.) designated by α. The coefficients Vα, referred to as effective cluster interactions (ECI), are expansion coefficients and can be determined by fitting to the firstprinciples energies of a subset of Li-vacancy configurations within the host. The energies used to fit the expansion coefficients are typically calculated with approximations to density functional theory (DFT). Once a cluster expansion has been constructed for a particular intercalation compound, it is possible to calculate finite temperature thermodynamic properties with Monte Carlo simulations, which explicitly sample Li-vacancy configurations with the probability distribution of statistical mechanics. Monte Carlo simulations within the grand canonical ensemble enable the calculation of the Li concentration x and the average grand canonical energy Ω = E − μLix as a function of Li chemical potential and temperature (both the average grand canonical energy Ω and the average energy E are per LixTiO2 formula unit). Inserting the calculated relationship between μLi and x into the Nernst equation then yields the voltage profile (we set μreference equal to the energy of bcc Li). Free energies can be Li obtained by integrating grand canonical Monte Carlo data from convenient reference states at which the free energy is known. For example, the Gibbs free energy, G, as a function of Li concentration x at constant temperature T can be written as

∫x

Ωdβ

where β = 1/(kBT). Convenient reference states for these integrals are well ordered phases at low temperature where Φ(μ Li ,T o ) can be calculated with a low temperature expansion.28,29 In this work, first-principles energies were calculated with DFT as implemented in the VASP program,30−33 using projector augmented wave (PAW) pseudopotentials34,35 and the generalized gradient approximation (GGA) to DFT as parametrized by Perdew, Burke, and Ernzerhof.36 The atomic positions of each configuration were relaxed to minimize the total energy using the conjugate gradient method, allowing for changes in cell shape and volume. The configurations were then rerelaxed to account for changes in the plane wave basis set due to changes in volume during the first relaxation. We used an energy cutoff of 400 eV for our plane wave basis set. For the unit cell, a k-point grid of 4 × 13 × 8 was sufficient to converge the calculated energies to within 1 meV per formula unit. We performed our DFT calculations without spin polarization, because the energies as calculated with and without spin polarization for several low-energy structures (corresponding to ground states or near ground states) were, within the numerical accuracy of the VASP calculations, identical.

∑ Vαφα(σ ⃗)

G(x , T ) = G(xo , T ) +

β

o

α

ϕα(σ ⃗) =

∫β

x

μLi dx

o

A convenient reference concentration is, for example, xo = 0, where because of the absence of configurational entropy, the

Figure 1. Relaxed configurations for TiO2(B) (a) without Li and (b) fully lithiated, showing the positions of the A1, A2, C, and C′ Li sites. 1569

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O sites10 that vary in their coordination to Ti atoms.3 The oxygen sites, in the order given by Feist and Davies,10 are the following: (1) atoms with nearly linear 2-fold coordination that lie in the (0 0 1) plane and join TiO6 octahedra at their corners, (2) atoms with tetrahedral 4-fold coordination, (3) atoms with planar 3-fold coordination parallel to the ac-plane, and (4) atoms with planar 3-fold coordination parallel to the ab-plane. These oxygen atoms are labeled by Arrouvel et al.,3 respectively, as Obr (bridging), O4f, and O3f (which includes both types of 3fold coordinated atoms), as shown in Figure 1a. Three distinct sites for Li intercalation in TiO2(B) have been identified previously3,37 and have been labeled A1, A2, and C. The unit cell contains four A1 sites, which are 5-fold coordinated to oxygen atoms and sit in the two (0 0 3) planes not coinciding with (0 0 1); four A2 sites, which are also 5-fold coordinated to oxygen and sit in the (0 0 1) planes; and two C sites, which have planar 4-fold coordination to four Obr atoms and lie in the (0 0 1) planes at the center position between opposite pairs of A1 and A2 sites. The total of ten Li sites means that the TiO2(B) crystal can potentially hold 1.25 Li atoms for each TiO2 formula unit. When Li is relatively dilute (one Li per unit cell), the fractional position of the C site is displaced by approximately 0.1 in the b-direction3 from the ideal coordinates of (0.0, 0.5, 0.0) and Wyckoff position 2b, to coordinates of (0.0, 0.6, 0.0). At high Li concentrations, when the neighboring A1 and A2 sites are (partially) occupied by Li, we find that the stable position of the C site is displaced in the b-direction, having coordinates of (0.0, 0.0, 0.0) and Wyckoff position 2a. In this position, a Li atom has planar 4-fold coordination between two Obr atoms and a pair of O3f atoms above and below the (0 0 1) plane. This displacement of the C site with increasing Li concentration was also observed in calculations by Koudriachova.38 To avoid confusion with the C-site close to the 2b Wyckoff position, we label this new displaced site in the 2a position as C′. The fully lithiated TiO2(B) structure after relaxation is shown in Figure 1b. The intercalation of Li causes the originally distorted octahedra of O surrounding Ti to straighten, and all of the O−Ti−O and Ti−O−Ti bond angles become close to 90° or 180°. Also shown in Figure 1b are the C sites and their relation to the C′ sites. B. Energy, Li-Site Preferences, and Li-Vacancy Ordering. The construction of a cluster expansion for the configurational energy requires the identification of a network of intercalation sites for Li occupancy within the TiO2(B) host that are likely to be occupied as the Li composition and temperature is varied. We explored the Li site preference in the dilute limit by calculating the energy of a single Li ion in the A1, A2, and C sites, respectively, of a unit cell of TiO2(B) (Li concentration corresponding to x = 0.125). Our calculations indicate that in the dilute limit, the A1 site has the lowest energy, followed by the A2 (32 meV higher) and the C (113 meV higher) sites (note that Li in the C-site is slightly shifted as described above and found previously).3 Our calculations, therefore, predict the C site to be the least stable of the three in the dilute limit. This sequence of Li site preference contradicts previous reports based on first-principles studies using similar methods.3,16 The same trend is predicted when two Li ions are present per unit cell (x = 0.25), the maximum number of Li ions per unit cell that can be accommodated by the C-sites exclusively. Above x = 0.25, Li must be accommodated, at least partially, by either the A1 or A2 sites, and we found that at

these higher Li concentrations, Li occupancy of the C site is not stable because Li relaxes to the neighboring C′ sites described above. Having established that the C-site is the least stable in the dilute limit and not even a local minimum at higher Li concentrations, we excluded it from our cluster expansion description. Instead, we use a network of Li-sites consisting of the A1, A2, and C′ sites to sample configurational degrees of freedom within the TiO2(B) host. First-principles energies (DFT-GGA) and relaxed atomic positions were calculated for 83 symmetrically distinct configurations of Li ions in TiO2(B). These included 52 configurations with the periodicity of the unit cell and 31 configurations within various supercells containing two unit cells. All of the configurations were stable; none of the atoms relaxed to positions equivalent to a different configuration. The formation energies (relative to the unit cell configurations Ti8O16 and Li10Ti8O16) of the 83 configurations considered in this work, as well as two additional configurations with Li occupying C sites, are shown in Figure 2. The ground

Figure 2. Formation energies per unit cell (Li8xTi8O16) calculated by DFT for 85 Li-vacancy configurations in TiO2(B). Open circles indicate structures containing C sites (4g), filled diamonds indicate structures containing C′ sites (2a), and open diamonds indicate structures containing only A1 and A2 sites.

states, corresponding to the configurations with formation energies on the convex hull, include four structures of intermediate Li concentration at x = 0.25, x = 0.5, x = 0.75, and x = 1. The relaxed atomic positions of these ground states are shown in Figure 3; coordinates (in cif format) can be found in Supporting Information. The structure at x = 0.25 consists of two TiO2(B) unit cells layered in the c-direction, each having a different A1 plane filled. At x = 0.5, all of the A1 sites are filled. At x = 0.75, there is an inversion of the Li-site occupancy, and the filled sites are A2 and C′, all lying in the (0 0 1) plane. The structure at x = 1 is also layered in the c-direction, similar to the cell for x = 0.25 but with all A2 and C′ sites filled. Intercalation of Li causes changes in the lattice parameters and a general increase in volume, as shown in Figure 4. The increase in the b-axis is mainly responsible for the increased volume, with some contribution from the a-axis at intermediate concentrations. C. Finite Temperature Thermodynamics. The formation energies in Figure 2 were used to determine the coefficients of a truncated cluster expansion using a least-squares fit. The clusters of the truncated expansion were chosen with the help of a genetic algorithm39 by minimizing a cross-validation score.40 Several clusters were also selected manually to ensure that the DFT ground states are reproduced among the 83 1570

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Figure 3. Ground-state structures for Li concentrations of (a) 0.25, (b) 0.5, (c) 0.75, and (d) 1.0.

Figure 4. Changes in volume and lattice vectors as a function of Li concentration, relative to their values at x = 0.

Figure 5. Calculated temperature−composition phase diagram for Li in TiO2(B).

configurations used in the fit. Our cluster expansion for TiO2(B) contains 26 ECI that correspond to the empty cluster, three point clusters, 16 pair clusters, and 6 triplet clusters. The root-mean-squared error of the fit is 4 meV per TiO2 formula unit, while the cross-validation score is 7 meV per formula unit. Our final cluster expansion does not predict any new ground states. Grand canonical Monte Carlo simulations were applied to the cluster expansion to calculate thermodynamic properties at nonzero temperatures. The Monte Carlo runs used a 12 × 12 × 12 supercell with 1000 passes for equilibration, followed by 3000 passes for data collection. Free energy integration, combined with the low-temperature expansion27,28,41 below 100 K, was used for each of the ordered phases for use in determining first-order phase boundaries within the phase diagram. Figure 5 shows the calculated temperature−composition phase diagram of LixTiO2(B). It contains a variety of intermediate ordered phases at x = 0.25, 0.5, 0.75, and 1.0. At x = 0.25 (the α-phase), the Li ions order over half of the A1 sites, as illustrated in Figure 3a. This phase disorders through a

second-order transition upon heating. The second-order transitions were determined by locating the divergence of the heat capacity in Monte Carlo runs with decreasing temperature at constant chemical potential. In the stoichiometric phase at x = 0.5, Li ions fill all available A1 sites and there is no thermodynamic transition separating it from the high temperature solid solution at low Li concentrations (x < 0.5). This is thermodynamically possible because the symmetry of the low temperature ordering at x = 0.5 (all A1 sites filled) is identical to that of the disordered solid solution (Li distributed over A1, A2 and C′ sites). The β-phase at x = 0.75 has Li occupying the A2 and C′ sites, while the γ-phase at x = 1.0 has full occupancy of A2 and C′ sites but partial occupancy of the A1 sites. The vacancy ordering over the A1 sites at x = 1.0 is identical to the Li ordering of the A1 sites at x = 0.25. Large two-phase regions separate the various ordered phases, which correspond to first-order phase transformations when varying the Li concentration, holding the temperature constant. The first-order transitions were determined by the intersection of the grand canonical free energies integrated from the separate ground states. The large two-phase region between x = 0.5 and 0.75 arises from a complete site inversion. The site 1571

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DISCUSSION Experimental characterization of the true thermodynamic voltage curve and the Li-site occupancy of lithiated TiO2(B) has proven elusive. Often samples of TiO2(B) contain substantial impurity phases, most notably anatase TiO2, making it difficult to isolate features in the voltage curve arising from Li insertion into TiO2(B).14 Furthermore, XRD diffraction peaks of TiO2(B) and anatase are difficult to disentangle, especially when the crystallites have nanoscale dimensions that result in peak broadening.14 Cyclic voltammograms of TiO2(B)-Li half cells typically exhibit three peaks at approximately 1.7 V, 1.55 V, and 1.5 V during discharge and 1.95 V, 1.6, and 1.55 V on charge.14,16 The highest voltage peak has been attributed to Li intercalation of the anatase impurity phase, while the two other peaks at lower voltage are present even in TiO2(B) phase pure samples.14 Measured voltage profiles exhibit structure indicative of phase transformations upon Li insertion/removal from TiO2(B), though unambiguous plateaus and steps are difficult to discern with additional uncertainties as to the actual Li concentration as a function of the state of charge/discharge.16 Furthermore, most studied samples of TiO2(B) consist of nanoscale crystallites, typically nanorods ranging between 100 to 30 nm in diameter.16 Nanoscaling of electrode particles is likely to alter the thermodynamics of the bulk.42,43 The broken symmetry at the surfaces along with space charge effects44,45 that emerge at electrode−electrolyte interfaces can potentially affect the stability of Li ordering within the interior of very small electrode particles, possibly promoting disorder by decreasing order−disorder temperatures. This will have the effect of smoothing out the voltage profile, not unlike what occurs with increased thermal excitation when the temperature is raised. It is with all of these uncertainties in mind that we compare the calculated voltage curve and Li-site occupancies with measured ones. The calculated phase diagram should only be considered as a qualitative prediction of phase stability at finite temperature in LixTiO2(B). Because of errors intrinsic to the generalized gradient approximation to DFT as well as numerical errors due to truncation of the cluster expansion and the neglect of vibrational excitations, predicted transition temperatures can be under or overestimated by 100 K or more. Furthermore, it is well-known that DFT systematically under-predicts voltages for Li insertion into transition metal oxides and that only qualitative comparisons with experimental voltages should be made.46 Two distinct plateaus due to first-order phase transitions are present in the calculated voltage profile between x = 0 and x = 1 at 300 K (Figure 7). The first plateau is very pronounced, even at elevated temperature, and occurs between x = 0.5 and 0.75 due to the site inversion from A1 occupancy at x = 0.5 to A2 and C′ occupancy at x = 0.75. This plateau should correspond to one of the peaks measured in cyclic voltammogram experiments14 if the site inversion from A1 to A2 and C′ is an accurate prediction of GGA-DFT. The second plateau occurs between x = 0.75 and 1.0. The calculated voltage profiles also indicate that Li can be accommodated in LixTiO2 beyond x = 1. A plateau is predicted between x = 1.0 and 1.25, but occurs at a significantly lower voltage. Nevertheless, the voltage of this plateau remains positive such that Li intercalation up to a stoichiometry of Li1.25TiO2 is stable relative to metallic Li. Experiments with single crystal TiO2(B) nanowires by Liu et al.13 show a Li/Ti ratio of 1.17 for the initial discharge. The authors attribute this

occupancy as a function of Li concentration is shown in Figure 6. Below x = 0.5, only A1 sites are occupied with the A2 and C′

Figure 6. Occupancies of the A1, A2, and C′ sites as a function of Li concentration at 300 K.

sites vacant, whereas above x = 0.75, the A2 and C′ sites are always occupied. Further Li insertion above x = 0.75 is accommodated by the A1 sites, with the ordered phase at x = 1.0 having all A2 and C′ sites filled but only half the A1 sites filled. Figure 7 shows the calculated equilibrium voltage curve at 300 K when LixTiO2(B) is used as a cathode relative to a

Figure 7. Calculated equilibrium voltage curve for TiO2(B) at 300 K.

metallic lithium reference anode. The predicted voltage curve has pronounced steps and plateaus as a result of the intermediate ordered phases and two-phase regions in the equilibrium phase diagram. The ordering over the A1 sites results in a step at x = 0.25 in the voltage curve. The portions of the voltage profile adjacent to the step, while not exactly plateaus, are relatively flat due to the occurrence of second order phase transitions. The complete filling of A1 sites at x = 0.5 results in a step in the voltage curve followed by a broad plateau corresponding to the two-phase region between x = 0.5 and 0.75 whereby Li in A1 sites and additional Li are displaced to A2 and C′ sites. Two additional plateaus appear separated by a step at x = 1.0 as Li progressively fills A1 sites. 1572

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implies that any under-prediction of the a lattice parameter with DFT-GGA could result in an incorrect prediction of site preference at low Li concentration in LixTiO2(B). The study by Armstrong et al.16 indicated Li occupancy of A1 sites at x = 0.5, consistent with our predictions reported here. At higher Li concentrations, Armstrong et al.16 concluded a combination of A1 and A2 occupancy. They did not, however, report that they considered the possibility of Li occupancy in the C′ site in their refinements nor in their first-principles calculations of total energies. Our DFT-GGA calculations indicate that configurations with C′ occupancy above x = 0.75 have lower energies than configurations with only A1 and A2 occupancy. We note that DFT within the generalized gradient approximation can fail to predict phase stability and Li-site preferences accurately. DFT within GGA, for example, fails to predict the miscibility gap in LixFePO4 between FePO4 and LiFePO4.48 Even hybrid DFT−Hartree-Fock approaches fail to predict a miscibility gap in the LixFePO4 system.49 It is only with the use of GGA+U, which in effect corrects for the selfinteraction present in approximations to DFT, that the correct phase stability as a function of Li concentration in LixFePO4 is predicted.48 Nevertheless, GGA+U is not suited for all transition metal oxides. To explore the sensitivity of our results to the particular firstprinciples approximation, we calculated the energies of a variety of key Li-vacancy configurations using GGA+U50 and a hybrid DFT−Hartree-Fock method.51−54 Formation energies are shown in the Supporting Information (Figures S1 and S2). Neither method predicts the C site to be the most stable at x = 0.125, although GGA+U favors the A2 site over A1 for all values of U tested. Both methods predict the A1/A2 full occupancy to be more stable at x = 1 than the DFT-GGA ground state configuration with A2, C′, and partial A1 occupancy. With the exception of GGA+U with the lowest value of U considered in this work, these methods all would predict voltage curves with very large plateaus, in qualitative disagreement with experiments.

additional capacity to defect sites, but our results here suggest that at least part of the additional Li may have been incorporated into normal lattice sites. A recent neutron diffraction study by Armstrong et al.16 investigated site occupancy as a function of Li concentration in TiO 2 (B). The neutron diffraction patterns at low Li concentration were interpreted as arising from a two-phase mixture consisting of TiO2(B) and Li0.25TiO2(B) (obtained after chemical lithiation).16 The refinements of the diffraction patterns led the authors to conclude C-site occupancy in Li0.25TiO2(B). For fixed lattice parameters, C-site versus A1-site occupancy can only be distinguished by differences in intensity between diffraction peaks and not by the existence or absence of particular peaks. Furthermore, it was not reported if the presence of anatase as a minority phase was explicitly considered in the refinements. Anatase TiO2 incorporates Li through a two-phase reaction to form Li0.5TiO2 with an orthorhombic symmetry.47 Anatase TiO2 and its orthorhombic lithiated variant, Li0.5TiO2, exhibit very similar diffraction patterns to TiO2(B), making it difficult to distinguish the two polymorphs in diffraction studies.13,15 The same study by Armstrong et al.16 also reported on firstprinciples DFT calculations that predicted C site occupancy to be energetically preferred at low Li concentration.3,16 Our present study, relying on DFT-GGA calculations, predict that the C site is energetically least favored when compared to A1 and A2 occupancy at or below x = 0.25. Our DFT-GGA calculations were performed allowing all atomic positions and cell parameters to relax fully. The DFT calculations by Arrouvel et al.3 reported on by Armstrong et al.,16 in contrast, were performed by fixing ratios among the lattice parameters to those observed experimentally. To be a true and unbiased prediction of DFT, however, full relaxations of all coordinate and cell parameters should be performed. We did find, however, that the relative site stability is sensitive to the unit cell dimensions. Figure 8, for example, shows the variation in

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CONCLUSION From first principles, we find that all ten of the Li intercalation sites in the unit cell of TiO2(B) can be filled, leading to a general stoichiometry of LixTiO2 where the Li concentration x ranges from 0 to 1.25. We identify an additional Li site, C′, that is predicted to be stable with DFT-GGA at high Li concentrations. The preferred sites occupied by Li are A1 for x ≤ 0.5, A2 and C′ for x = 0.75, and all sites (with increasing A1 occupation) at higher concentrations. The calculated phased diagram and voltage curve based on Monte Carlo simulations show three first-order phase transitions at 300 K, the most pronounced occurring between x = 0.5 to x = 0.75 because of a site inversion from A1 occupancy to C′ and A2 occupancy. DFT calculations show that for C′ occupancy to be energetically stable, the A2 sites should also be occupied, resulting in fully filled (0 0 1) sheets within TiO2(B).

Figure 8. Variation of the energy of Li0.25TiO2 (B) with A1 and C occupancy as the unit cell dimensions are varied from the fully relaxed DFT-GGA values to the experimental values.

energy between A1 site occupancy and C site occupancy at x = 0.25 as the unit cell dimensions are linearly varied between the DFT-GGA relaxed unit cell parameters and those reported as the experimental parameters.16 The difference in DFT and experimental volumes is primarily along the a lattice parameter, which is predicted to be smaller than the experimental value. As is clear in Figure 8, the relative stability between A1 and C is especially sensitive to variations in the a lattice parameter, with the A1 site preferred at the smaller DFT-GGA a lattice parameter and the C site preferred at larger values for a. This

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ASSOCIATED CONTENT

S Supporting Information *

Energies as calculated by GGA+U and hybrid DFT−HartreeFock for selected Li-vacancy configurations, crystallographic information files (CIF) of ground state configurations (PDF). This material is available free of charge via the Internet at http://pubs.acs.org. 1573

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon work supported as part of the Northeastern Center for Chemical Energy Storage, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award DESC0001294. Computational resources provided by TERAGRID DMR100093 are also gratefully acknowledged. Images of crystal structures were produced with the VESTA program.55

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dx.doi.org/10.1021/cm203283v | Chem. Mater. 2012, 24, 1568−1574