First-Principles Analysis of Phase Stability in LayeredâLayered

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First-Principles Analysis of Phase Stability in Layered−Layered Composite Cathodes for Lithium-Ion Batteries Hakim Iddir and Roy Benedek* Argonne National Laboratory, Argonne, Illinois 60439, United States ABSTRACT: The atomic order in layered−layered composites with composition xLi2MnO3·(1 − x)LiCoO2 is investigated with first-principles calculations at the GGA+U level. This material, and others in its class, are often regarded as solid solutions; however, only a minute solubility of Li2MnO3 in a LiCoO2 host is predicted. Calculations of Co vacancy formation and migration energies in LiCoO2 are presented to elucidate the rate of vacancymediated ordering in the transition-metal-layer. The neutral Co vacancy formation energy is predicted to be in a range centered slightly above 1 eV but varies widely with the oxygen chemical potential. The calculated migration energy for the vacancy with charge q = −3e is approximately 1 eV. These values are small enough to be consistent with rapid ordering in the transition metal layer at typical synthesis temperatures and, therefore, separated Li2MnO3 and LiCoO2 phases. The relatively small (of the order of a few nm) Li2MnO3 domain sizes observed with TEM in some xLi2MnO3·(1 − x)LiMO2 composites may result from other factors, such as coherency strain, which perhaps block further domain coarsening in these materials.

I. INTRODUCTION Higher capacity and energy-density electrodes are prerequisite to extending the range of lithium-ion battery powered PHEVs and EVs.1 The introduction of layered−layered composites2 was a significant advance, essentially doubling the capacity available in the earlier generation of lithium-ion batteries based on LiCoO2, and has spurred a huge research effort on these materials. A key feature of the layered−layered materials, nominally of composition xLi2MnO3·(1 − x)LiMO2, is the structural compatibility of the monoclinic (Li2MnO3) and rhombohedral (LiMO2, where M = Mn, Ni, Co, etc.) crystal structures, which both exhibit (strained) cubic-close-packed oxygen sublattices with comparable lattice constants. This compatibility enables synthesis of the composite material with coherent interfaces or perhaps solid-solution-like ordering. Although extensive atomistic modeling of the individual phases has been performed3 (for example, see refs 4−6 for the monoclinic phase and refs 7−10 for the rhombohedral phase), few simulations of the lithium-rich composites11 have been attempted. We present here first principles modeling of layered−layered composites to elucidate their atomic ordering and phase equilibrium. The calculations are performed for the model system xLi2MnO3·(1 − x)LiCoO2, which has been extensively investigated experimentally, particularly for x = 0.5.12 Although the phase diagram of xLi2MnO3·(1 − x)LiMO2 composites has not previously been simulated, a wide miscibility gap might be anticipated, in view of the different crystal symmetries of the end members (x = 0, x = 1) at all temperatures (rhombohedral vs monoclinic). On the other hand, some measurements have been interpreted as showing solid-solution behavior.13−19 Cluster expansion methods20 provide a rigorous (but computationally demanding) approach to the modeling of © 2014 American Chemical Society

pseudobinary and ternary alloys that are amenable to a latticegas treatment and have been applied to analyze ordering in some layered lithium transition metal oxides.21 The clusterexpansion approach would be applicable in principle to xLi2MnO3·(1 − x)LiCoO2; however, we employ a more approximate treatment that we believe captures the essential features of the material in a simpler way, particularly the existence of a wide miscibility gap.

II. SOLUBILITY AND COARSENING A. Solubility of Li2MnO3 in LiCoO2. We assume initially that the solubility of Li2MnO3 in LiCoO2 is small (dilute limit) and then show that this assumption leads to consistent results. With first-principles methods, we calculate the formation energies of (Li3[LiMn2]O6)n complexes (solutes), which may be regarded as point defects embedded in the LiCoO2 matrix. Such complexes are the smallest stoichiometric units of Li2MnO3 that, in the limit of large n, would assume the structure of the bulk monoclinic phase. Since the chemistry of the Li layers and O layers is identical in Li2MnO3 and LiCoO2, we focus on the substitution of [LiMn2]n complexes for Co3n clusters in a Co layer of LiCoO2. Examples of such complexes are illustrated in Figure 1 for n = 1 to 6. We employ a cell with two oxygen layers, a lithium layer, and a mixed Li−Mn−Co layer (Figure 1) in each periodic unit of our computational cell. Such a minimally thick periodic unit cell slab is a reasonable representation of the material, provided (a) interslab interactions between adjacent Li−Mn−Co layers are weak Received: October 3, 2013 Revised: March 4, 2014 Published: March 4, 2014 2407

dx.doi.org/10.1021/cm403256a | Chem. Mater. 2014, 26, 2407−2413

Chemistry of Materials

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indices28 of 222. Vibrational contributions29 to the free energy were not included. Our objective in the following is to demonstrate that the iono-covalent interactions in the composite materials result in formation energies well above thermal energies, which makes decomposition into separate phases thermodynamically favorable. We express the formation energy of an [LiMn2]n complex embedded in LiCoO2 as Ef (n) = Esup(n) − [3nEfu(Li4/3Mn2/3O2 ) + (48 − 3n) Efu(LiCoO2 )]

(1)

where Esup(n) is the total energy of the LiCoO2 supercell with a substituted [LiMn2]n complex (Figure 1), Efu(LiCoO2) is the energy per formula unit of LiCoO2, and Efu(Li4/3Mn2/3O2) the energy per Li4/3Mn2/3O2 unit of Li2MnO3. The quantity in brackets in eq 1 represents the reference energy that corresponds to separate Li2MnO3 and LiCoO2 phases. We note that eq 1 is appropriate for the dilute limit, that is, n ≪ 16. For pure LiCoO2, Ef(0) = 0. In the results presented below, the reference energy Efu(Li4/3Mn2/3O2) is taken to be Esup(n = 16) /(48); only small differences are found if Efu(Li4/3Mn2/3O2) is calculated based on the monoclinic C2/m structure (rather than R3̅m) of Li2MnO3. For each cluster size n, the supercell energy Esup(n) is obtained with lattice constants a and c relaxed to minimize the energy (see the caption to Figure 1). The optimized lattice constants show an approximately linear dependence on n (or x) for n = 0 to 6 (Figure 1), with d(ln a)/dx ≈ 0.02, and d(ln c)/dx ≈ 0.015. That these results lie in a physically reasonable range is indicated by comparison with a Vegard’s law analysis of experimental lattice constants for the end-members LiCoO212,30,31 and Li2MnO3,12,18,32 which yields d(ln a(expt))/dx ≈ 0.012, and d(ln c(expt))/dx ≈ 0.016. Calculated formation energies Ef(n) are plotted in Figure 2. The energy of the smallest complex (n = 1) is minimized for a

Figure 1. Compact [LiMn2]n clusters embedded in a Co layer. Large circles: Li. Small circles: Mn. Diamonds: Co. The clusters represent hypothetical fragments of Li2MnO3 dissolved in LiCoO2. Cell parameters optimized for each value of n. For n = 0: a = 2.833 Å, c = 14.176 Å. For n = 1: a = 2.836 Å, a = 14.183 Å. For n = 2: a = 2.839 Å, c = 14.190 Å. For n = 3: a = 2.845 Å, c = 14.209 Å. For n = 4: a = 2.847 Å, c = 14.232 Å. For n = 5: a = 2.85 Å, c = 14.231 Å; n = 6: a = 2.855 Å, c = 14.254 Å.

and (b) all Li−Mn−Co layers have essentially the same composition (absence of staging). To investigate (a), calculations were performed of the cluster formation energy as a function of the relative translation of [LiMn2]n clusters in adjacent layers parallel to the layers; thus, only interlayer interactions are affected by the translation. This translation is achieved by addition or subtraction of hexagonal-layer lattice vectors to the cell vector A3, specified below. Only relatively small differences (hundredths of an electron volt, in the formation energies Ef(n) described below) are found as a function of the translation vector, and the neglect of interlayer interactions therefore appears reasonable. In general, slightly lower energies are found for [LiMn2]n complexes stacked essentially vertically, rather than staggered. Our first-principles calculations were based on the PAW representation22,23 at the GGA+U level of density functional theory,24 with PBE exchange-correlation potentials,25 as implemented in the VASP code.26 A periodic supercell with 6 × 8 sites per layer was built, with 48 formula units = 16 Co3 units in the case of n = 0, 4 × 48 = 192 ions in the supercell. The supercell vectors [A1= (6a, 0, 0), A2 = (0, 4 √3a, 0), A3 = (a/12, (√3a)/6, c/3), where a and c are the lattice parameters of the standard hexagonal cell] are consistent with the R3m ̅ symmetry of the LiCoO2 prototype. Effective Coulomb parameters Ueff(Co) = 5 eV and Ueff(Mn) = 4.84 eV were employed.27 In the absence of (Li or Co) vacancies, Co ions adopt a zero spin nonmagnetic state. Unless otherwise specified, the presented results correspond to single k-point (the Γ point) sampling; tests with larger k-point sets yielded similar results. Calculated migration energies, for example, changed by only hundredths of an electron volt for M−P

Figure 2. Formation energy Ef as a function of cluster size n.

linear Mn−Li−Mn configuration with the Li ion at the center (Figure 1). The lowest energy cluster of a given size n is compact, owing to the preference of Li ions for Mn nearest neighbors. The Li ions on the periphery of the clusters have fewer than jneigh = 6 Mn neighbors, and therefore, Δjneigh(n)= 6n − jneigh(n) > 0, where jneigh(n) is the number of Li−Mn neighbor pairs for the domain of size n. Energetically favorable cluster shapes minimize Δjneigh(n). 2408



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where norient (which equals 3 for the linear trimer) is the number of independent orientations of a given trimer centered on a particular site. For simplicity, we neglect higher order (n > 1) clusters, or nonlinear LiMn2 trimers. The boundary xsolvus between a singlephase solid solution and a two-phase region is then given by eq 5, which corresponds to a nominal composition xsolvusLi2MnO3· (1 − xsolvus)LiCoO2

The agglomeration of two separate clusters to form a single larger cluster is expected to lower the total energy because it results in a smaller deficiency Δjneigh(n). This behavior is illustrated in Figure 3, which shows cluster binding energies and

xsolvus = 2xs = 2norient exp(−Ef (1)/(kBT ))

in the dilute limit. (For a composition 2xsLi2MnO3·(1 − 2xs)LiMO2, there are xs Li ions in the transition metal layer, which accounts for the factor of 2 in eq 6). Layered−layered composite specimens are typically prepared at about 800 °C, or about 1100 K. For our calculated formation energy Ef(1) = 0.65 eV (Figure 2), we find xsolvus(1100 K) = 6 × 10−3. Corrections to this analysis, for example, by including higher order clusters, and other (nonlinear) trimer configurations for n = 1, would not appreciably change xsolvus. (Calculations with larger k-point sets also do not qualitatively change the results).

Figure 3. Binding energies and coarsening energies based on the formation energies plotted in Figure 2.

III. VACANCY KINETICS OF TRANSITION-METAL-LAYER ORDERING The preceding solubility analysis suggests that, at the synthesis temperature, thermodynamics favors phase separation of composites xLi2MnO3·(1 − x)LiCoO2, or at least the formation of large (LiMn2)n aggregates, except at very small x. The equilibration process, however, is governed by kinetics, and the adequacy of kinetic rates to achieve equilibration on laboratory time scales, therefore, comes into question. Point defect properties play an important role in equilibration and will be addressed in this section. It is assumed in the following analysis that only thermally activated, and not structural, vacancies occur for xLi2MnO3·(1 − x)LiCoO2 (recent work showed that TM-layer vacancies are present in some regions of the Li−Mn−Ni pseudoternary phase diagram,36 which would tend to accelerate ordering kinetics). M-vacancy diffusion limited kinetics, for thermally activated vacancies, is determined largely by the self-diffusion activation energy Q = Ef(VM) + Emig(VM) (vacancy formation and migration energies).37 For Q less than about 2 eV, kinetics should be reasonably fast at temperatures above about 1000 K.38 The properties of transition-metal vacancies in layered LiMO2, or in xLi2MnO3·(1 − x)LiMO2 composites, are generally poorly known, and no direct experimental measurements of M-vacancy properties have been reported. Effective reaction activation energies have been derived from analyses of synthesis and growth kinetics.39,40 For solid-state reaction synthesized LiCoO2, activation energies were found in the range 2−3 eV, varying as a function of conversion fraction.39 Lower activation energies were obtained from an analysis of growth kinetics of Li(Mn1/3Co1/3Ni1/3)O2.40 It is unclear, however, whether these effective activation energies are related to intrinsic point defect properties. Point defect simulations exist for Frenkel, Schottky, and other complex defects in layered LiMO2,10,8 as well as oxygen vacancies,41 but little attention has been given to M-vacancies. The point defect species that most readily assists atomic ordering of the transition metal layers of such materials has not yet been identified; however, because monovacancies dominate the ordering of many crystalline materials, we performed

coarsening energies calculated with the formation energies in Figure 2. The binding energy of a cluster of size n, relative to forming n monomers, is E B(n) = nEf (1) − Ef (n)

(2)

which is positive at all values of n considered. We define the coarsening energy as the energy gained in merging two smaller clusters of size n′ and n − n′ into a larger one of size n EC(n; n′, n − n′) = Ef (n′) + Ef (n − n′) − Ef (n)

(3)

Results for EC for n = 6 (Figure 3) indicate that coarsening lowers the energy in the size range considered. These analyses of EB(n) and EC relate to clustering (domain growth) only in a single layer. In general, domains in a given layer tend to align with those in adjacent layers; however, interlayer interactions are considerably weaker than intralayer interactions, as mentioned above, and detailed results on interlayer interactions are not presented here. Properties such as EB(n) and EC are expected to influence the kinetics of coarsening. The coarsening of two-phase alloys is an active and still developing field, particularly in the area of structural materials.33 An atomistic treatment of the coarsening of a model two-dimensional system with classical interatomic potentials34 illustrates the complexities. The issue of coarsening is discussed further below. B. Solvus Line. To estimate the position of the solvus line, the solubility limit, we consider a dilute (nonoverlapping) distribution of LiMn2 (n = 1) complexes embedded in the Co layers of LiCoO2. For a concentration xs (fraction of Co-layer sites occupied by the Li ion at the center of an LiMn2 trimer), the total energy per Co layer site is E(xs) = Efu(LiCoO2 ) + xsEf (1)

(4)

Minimization of the free energy, including the configurational (but neglecting vibrational) entropy, leads to an equilibrium concentration35 xs = norient exp(−Ef (1)/(kBT ))

(6)

(5) 2409



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migration path. DFT calculations confirm that LiCoO2 with a single Co vacancy, charge state q = −3e, and ions close to their ideal lattice sites, does indeed have a stable solution with trivalent and nonmagnetic Co ions, as in the perfect crystal. As discussed below, however, a Co ion migrating from its lattice site toward the vacancy lowers its energy by disproportionation to 4+. The tetravalent state is maintained over most of the migration path. In a migration step, a Co ion traverses a triangular face of the octahedron in which it initially resides and then enters the octahedron of a vacant neighboring site by squeezing through a similar triangular face (which shares an edge with the octahedron originally occupied by the Co ion). Simulations indicate that Co−O bond lengths are compressed by up to about 5% as the migrating ion moves through the triangular oxygen-ion bottlenecks. We simulated the migration of a Co ion in LiCo1−yO2 with a single Co vacancy (y = 1/48). As mentioned above, we focus primarily on the charge state q = −3e. Two different simulation procedures were employed: a constrained relaxation method and the nudged elastic band (NEB) method, using the climbing image method,44 with all atoms in the cell free to relax. As shown below, both methods yield consistent results. The constrained relaxation was performed for a series of positions of the migrating Co ion along a path between two lattice sites, one of which is initially vacant. At each position along the path, a constrained energy minimization is performed. Only one of the migrating-Co coordinates is constrained so that the path is not forced to lie along a straight line between the migrating ion’s initial and final sites. Some oxygen ion coordinates far from the migrating ion are fixed to avoid a rigid translation of the entire cell to compensate for the displacement of the migrating Co ion. The supercell basis vectors were held constant during the vacancy migration calculations. Since the transition-metal ordering processes for layered− layered composites take place primarily at the synthesis temperatures of about 1100 K (≈ 800 °C), simulations of diffusion-related properties should take into account thermal expansion of the lattice. Migration energies of Li ions, for example, are sensitive to the O−Li−O slab thickness in LiMO2,45 which is proportional to c. Similarly, the migration energy of Co ions in LiCoO2 is expected to be sensitive to the O−Co−O slab thickness and, thus, to c. Initially, we considered lattice constant expansions of LiCoO2 of 0.5% for a and 3.5% for c, which would be appropriate for a graphite-like material at elevated temperatures.46 The results of this analysis are shown in Figure 4, discussed in the following paragraph. The interlayer bonding in LiCoO2, however, is stronger than that in graphite, and its thermal expansion is therefore expected to be less anisotropic. Recently, XRD measurements have been made on LiCoO2 at elevated temperatures.47 The measured values at 800 °C are a = 2.907 Å and c = 14.58 Å. Relative to room temperature, a is expanded by about 3.5% and c by 4%. We return to the effect of thermal expansion on the Co ion migration energy in section C below. Prior to the migration energy simulations (for a = 1.005a0 and c = 1.035c0, where a0 and c0 are the simulated low temperature values), atomic positions in the 191-atom (192 sites, 1 Co vacancy) cell were relaxed to equilibrate in the presence of the vacancy. All Co ions assume zero magnetic moment, with t2g = (3↑, 3↓), in the equilibrium state. If the migrating Co ion maintained a nonmagnetic state along its

simulations to establish the order of magnitude of monovacancy properties in LiCoO2. A. Vacancy Formation Energy. The formulation of point defect formation energies in terms of parameters accessible from first-principles numerical calculations for semiconductors and insulators is well established.35,42,43 For an M vacancy with charge q Ef [VM (q)] = Etot[VM (q)] − Etot[bulk] + μM + qA

(7)

where Etot[VM(q)] is the energy of a supercell that contains the vacancy, Etot[bulk] is the energy of the defect-free supercell, μM is the M-atom chemical potential, and A contains the electron chemical potential and a band-alignment correction. The formation energy calculations presented here are based on low-temperature lattice constants. Possible charge states for a covacancy include q = 0, −1, −2, and −3 in units of the electron charge e. The first of these (q = 0) is the neutral vacancy, and q = −3 corresponds to the removal of a Co3+ ion, without removing the three donor electrons for Co ions in stoichiometric LiCoO2. The neutral (q = 0) Co vacancy in LiCoO2 is considered as an illustration. Calculation of Ef [VCo(q)] from eq 7 requires the chemical potential of Co, μCo, which is related to that of O by μ LiCoO2 = μLi + μCo + 2μO

(8)

The chemical potential of Li in eq 8 can be obtained from the initial cell voltage during charge, V0 = 3.9 eV:12 μLi(LiCoO2) = −V0 + μLi(metal) = −5.8 eV. Since the oxygen chemical potential, μO, is not fixed and varies with the synthesis conditions, however, the Co chemical potential, μCo, and therefore Ef [VCo(q)], are not unique. To estimate a plausible range within which Ef [VCo(q)] is likely to lie, we hypothesize that the relevant conditions for LiCoO2 synthesis are less oxidizing than those at which Co3O4 would be oxidized to form LiCoO2 1 1 Co3O4 + Li + O2 ↔ LiCoO2 3 3

(9)

and less reducing than those at which CoO2 would be reduced to LiCoO2 by the reaction 1 1 Li 2O + CoO2 ↔ O2 + LiCoO2 2 4

(10)

Substituting DFT calculated values for properties that enter eqs 7 and 8, we find Ef [VCo(q = 0)] = −8.99eV − 2μO

(11)

Using DFT formation energies calculated for LiCoO2, Co3O4, Li2O, and CoO2, we find μO = −4.7 and −5.55 from eq 9 and eq 10, which correspond to Ef [VCo(q = 0)] = 0.4 and 2.1 eV, respectively. If we assume that these two values represent lower and upper bounds, then the operative range of the vacancy formation energy may be approximately Ef [VCo(q = 0)] = 1.25 ± 0.85eV

(12)

B. Vacancy Migration Energies. To illustrate vacancy migration in LiCoO2, we present detailed results for the charge state q = −3e (Emig(VCo) for q = 0 is found to be of similar magnitude), where we add back the three donor electrons for a Co3+ ion, to enable oxygen ions to retain their ideal oxidation state 2− in the presence of the vacancy. This choice is expected to give the least complicated electronic structure along the 2410



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= 0.95 eV. This value is slightly smaller than the migration energy obtained from Figure 4, for which a graphite-like expansion was assumed. We note that dynamical effects, beyond transition state theory,48 may decrease the effective migration energy relative to the static energy barrier calculated in this work. D. Self-Diffusion Activation Energy. In the simplest model of atomic mixing, the atomic exchange rate (exchanges per atom per second) is controlled primarily by the selfdiffusion activation energy Q = Ef(VCo) + Emig(VCo). The present results suggest an activation energy in the vicinity of 2 eV. A value of Q in this range is low enough that extensive ordering would occur at 800 °C or above38 within laboratory time scales.

IV. SELECTIVE REVIEW OF EXPERIMENT Several experimental probes are sensitive, in principle, to domain size and atomic order in xLi2MnO3·(1 − x)LiMO2 composite materials, including TEM,49−51 X-ray52 and neutron diffraction,53 Li NMR magic-angle spinning spectroscopy,54 and others. Only a selective discussion of the enormous literature on this subject is possible here. In general, the domain size distribution is not well established for any of the composites. XRD measurements by Dahn’s group13−17 as a function of x for xLi2MnO3·(1 − x)LiMO2, with M = Ni0.5Mn0.5 and M = Cr0.5Mn0.5, followed Vegard’s law behavior as would be expected for solid solutions. These XRD results, therefore, show no clear evidence of domains of a second phase. A distribution of small isolated domains, on the scale of a few nanometers, and perhaps other domain topologies as well, however, may also exhibit Vegard’s law behavior and, therefore, be indistinguishable from solid solutions by XRD. Besides the experimental probes mentioned above, initial first-charge cell voltages also conveys indirect information about domain size distribution. Removal of a Li ion from the Li layer of xLi2MnO3·(1 − x)LiCoO2 creates a (Li-vacancy)-(Copolaron) complex,55 that is, VLi-Co4+. The corresponding (initial first charge) cell voltage is

Figure 4. Energy along path of Co ion migration from a site near x/a = 0 toward a Co vacancy at x/a = 1.0. Insert shows the octahedral coordination of two nearest neighbor Co.

entire migration path between its initial site and the vacant site, then the migration energy would be greater than 2 eV. The nonmagnetic state of the migrating Co ion, however, is stable only close (within about 0.08a) to the initial site of the migrating ion. As mentioned above, the Co ion becomes tetravalent for most of the migration path. Results for the energy along the migration path are plotted in Figure 4. The states close to x/a = 0 and 1.0 are trivalent and nonmagnetic, whereas the interior points (x/a = 0.08−0.92) correspond to the smaller Co4+ ion, which lowers the short-range Co−O repulsion. (Charge compensation for the oxidation of Co to its tetravalent state occurs on neighboring oxygen ions.) The two computational procedures (constrained relaxation and NEB) are seen to yield similar results. The barrier between the local energy minima at x = 0 and x/ a = 0.08 is only about 0.1 eV, of the order of kT at 800 °C, so the exchange between these two states would be relatively rapid. Migration of the Co ion from this minimum-energy site across the energy maximum (corresponding to the transition state at x/a = 0.5) to a similar site near the vacancy (at x/a = 0.0) requires surmounting an activation barrier slightly larger than 1 eV. The E(x/a) curve shows points of inflection on either side of the maximum. These points of inflection correspond to the migrating ion crossing the face of the oxygen octahedron of the original site and that of the site to which the Co ion migrates (the octahedra, projected onto the x−c plane, are depicted in the inset to Figure 4). C. Effect of Thermal Expansion on Migration Energy. The measured lattice constants of LiCoO2 at 800 °C show an approximately isotropic expansion of 4% relative to low temperature values.47 To estimate the effect of thermal expansion on static energy barriers for vacancy migration, we performed simulations for expanded lattice constants a/a0 = c/ c0 = 1 + ε as a function of ε. The energy difference between the minimum energy configuration (at x/a = 0.08 in Figure 4) and the transition state configuration (at x/a = 0.5) is found to be approximately Emig (ε) = 1.25 − 8εeV

Ecell(x Li 2MnO3·(1 − x)LiCoO2 ) = Ecell(LiCoO2 ) + ΔEf (VLi‐Co4 +)

(14)

4+

where Ef(VLi-Co ) is the formation energy of the VLi-Co4+ complex upon removal of a Li atom. Simulations indicate that the formation energy Ef(VLi-Co4+) is minimized when the polaron site is at the periphery of an [(LiMn2)n] domain (such as those illustrated in Figure 1) at a nearest-neighbor site to a domain-boundary Li. It then effectively decreases Δjneigh(n) [cf. discussion in section IIA]. For the smallest domains (n = 1), the binding energy of the complex, that is, the difference ΔEf(VLi-Co4+) between the formation energy at a low-energy site in xLi2MnO3·(1 − x)LiCoO2 and that in pure LiCoO2 is of order 200 meV. Thus, in a specimen with a high concentration of n = 1 (Figure 1a) domains, one would expect an initial (charge) voltage measurably lower than that in LiCoO2 by the binding energy ΔEf(VLi-Co4+). That no appreciable voltage diminution has actually been observed12 is an indication that the concentration of domains of minimal size is small for annealed specimens. Also sensitive to domain sizes on the subnanometer scale is Li NMR spectroscopy of the composite materials, which is dominated by the highly local Fermi-contact interaction.56 NMR measurements, in conjunction with bond-pathway

(13)

On the basis of eq 13, the migration energy that corresponds to the observed thermal expansion of about 4% (ε = 0.04) is Emig 2411



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analyses54,57 hold promise for the identification of atomic arrangements in the vicinity of the Li ions in xLi2MnO3·(1 − x)LiMO2 composites, particularly for M = Co, small x, and short annealing times. Such a program is in progress in our laboratory,58 and results will be reported elsewhere. Reports of TEM observations of domains (for composites close to the composition M = Ni0.5Mn0.5) with monoclinic structures50 appear to confirm the decomposition into two phases, with domains in the nanometer size range. Other TEM observations,49 however, have been interpreted as showing solid solution behavior. In general, ambiguities in the interpretation of TEM, associated with the column approximation, are difficult to avoid. Two-phase Rietveld refinement of XRD measurements52 on a layered−layered material xLi2MnO3·(1 − x)LiMO2 (with x = 0.5 and M = Mn0.42Ni0.42Co0.16) yielded lattice constant values of the putative phases that lie within the interval spanned by those of the end members (LiMO2 and Li2MnO3), as would be expected in a coherent composite with misfit strain. This result is at least consistent with a two-phase coherent composite model.

short annealing time (at temperatures in the range 800−1000 °C, say) with little additional coarsening occurring thereafter.

VI. CONCLUSIONS The literature refers to materials with composition xLi2MnO3· (1 − x)LiMO2 alternatively as either solid solutions or as twophase composites. If the distinction between solid solution and phase separation behavior is to be made based on experiment alone, ambiguity may be inevitable. The model presented in this article predicts a phase-separated thermodynamic equilibrium state. The observed microstructure of such a composite material, however, is determined by kinetic considerations, which may block Li2MnO3 domain growth beyond a certain size.

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AUTHOR INFORMATION

Corresponding Author

*R. Benedek. E-mail: [email protected]. Author Contributions

All authors contributed equally. Notes

The authors declare no competing financial interest.

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V. DISCUSSION We have performed in this work atomic scale simulations of the model system xLi2MnO3·(1 − x)LiCoO2, in its pristine state, before Li insertion-extraction cycling. It is expected that the qualitative features apply also to other choices of M, particularly MnNiCo alloys. A simplified model of the solubility shows a miscibility gap that extends over all temperatures and compositions of practical interest. We would not expect more elaborate treatments based, for example, on the cluster expansion method, to give substantially different results. TEM observations of layered−layered composites xLi2MnO3·(1 − x)LiMO2 show domains a few nanometers in size for the compositions and synthesis procedures normally employed. This behavior raises the question of why the decomposition apparently does not result in larger monoclinic domains. In general, the kinetics of decomposition of supersaturated solutions into separate phases may be influenced by diverse factors, and the evolution of supersaturated materials is still an active area of materials science. Our analysis of Co vacancy properties in section III suggests that the migration energy component of Q, Emig(VCo), for LiCoO2, is on the order of 1 eV. If the formation energy Ef(VM), which has a wider range of uncertainty, were of a similar magnitude to Emig(VM), so that Q = 2 eV, atomicexchange kinetics would be relatively fast at 1100 K, a typical synthesis temperature.38 The presence of Li in the transition metal layer, which is expected to have lower migration energies than Co or Mn, might accelerate ordering processes further. Atomic rearrangements, driven by capillarity, could then, in principle, coarsen domains and lead to decomposition, large scale or spinodal,59 within laboratory time scales. Other kinds of barriers, however, may impede the coarsening. Classical models of precipitation (e.g., LSW), which predict mean precipitate volumes that increase linearly with time, are typically based on the assumption of noninteracting precipitates.60 Interprecipitate interactions61 and misfit elastic strain effects,62 however, may slowor even arrestcoarsening. A possible scenario is that the coherency strain limits the ability of the domains to grow. In that picture, the evolution of the domain size distribution effectively ceases after only a relatively

ACKNOWLEDGMENTS We are grateful to Javier Bareño, Mali Balasubramanian, Jason Croy, Yang Ren, Michael Thackeray, and others in the Argonne Voltage Fade team for helpful discussions. The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (“Argonne”). Argonne, a U.S. Department of Energy Office of Science Laboratory, is operated under contract no. DE-AC0206CH11357. Computer time allocations at the Fusion Computer Facility, Argonne National Laboratory, and at EMSL Pacific Northwest National Laboratory are gratefully acknowledged. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under contract no. DE-AC02-05CH11231. The work was supported by the Applied Battery Research Program of the Office of Vehicle Technologies, U.S. Department of Energy.

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