Voltage

The energy density and specific energy of a battery are the products of its voltage and its volumetric or gravimetric charge capacity (i.e., the amount of lithium ions that can be reversibly removed and replaced in the materials during charge and discharge). However, one cannot simply increase the voltage without considering the limitations of other battery components. Electrolytes currently used in commercial cells limit the usable voltage to less than about 4.5 V versus Li/Li⁺, though the accessible stability up to 5 V is good enough to test new materials in laboratories.

The average equilibrium voltage of a cathode can be related to the difference in the Gibbs free energy between its charged state (delithiated phase) and discharged state (lithiated phase), taking into account the reference state of lithium on the anode.Reference Aydinol, Kohan, Ceder, Cho and Joannopoulos¹ Because the contributions of entropy and volume effects to the Gibbs free energy difference are small, one may approximate the voltage using energy differences computed at zero temperature and pressure.

Standard approximations to density functional theory (DFT), such as the local density approximation (LDA) and generalized gradient approximation (GGA), generally lead to large errors in predicted voltages.Reference Zhou, Cococcioni, Marianetti, Morgan and Ceder² This failure can be attributed to the incomplete cancellation of the non-physical electron self-interaction when an electron is transferred between significantly different environments. This prediction error is particularly significant in redox processes where electron transfer occurs between distinct orbitals. For example, in a typical Li-ion cell, electrons are transferred from the anode material (either Li metal or Li in carbon) to the localized d orbitals of an intercalation oxide. The self-interaction error in voltage predictions can be overcome with methods such as DFT+U (Reference 3) or hybrid functionals.Reference Heyd, Scuseria and Ernzerhof^4–Reference Chevrier, Ong, Armiento, Chan and Ceder⁶ Although the U parameter in DFT+U can be determined by self-consistent linear response theory (i.e., by calculating how the occupation of the transition metal orbitals change with the local potential),Reference Zhou, Cococcioni, Marianetti, Morgan and Ceder^2, Reference Kulik, Cococcioni, Scherlis and Marzari⁷ it is often fitted for each transition metal element to experimental oxidation energies.Reference Wang, Maxisch and Ceder⁹ In hybrid functionals, some self-interaction is removed without relying on an adjustable parameter by incorporating a fraction of the exact Hartree-Fock exchange (the non-local effect of the Pauli exclusion principle) though at considerably greater computational expense. Figure 2 demonstrates the good agreement of voltages computed using GGA+U and the Heyd-Scuseria-Ernzerhof (HSE) functional, with experimental voltages for a variety of cathode materials, with differences on the order of only a few hundred mV.

Figure 2.

Comparison between experimental and computed (using generalized gradient approximation [GGA]+U or Heyd-Scuseria-Ernzerhof [HSE]) voltages. As titanium does not require a U value to reproduce accurate energies, no U value was used for Li₂Ti₂O₄ and LiTiS₂. Data from Chevrier et al.Reference Chevrier, Ong, Armiento, Chan and Ceder⁶

Screening materials through voltage prediction before any experimental work is conducted is now quite common. For instance, Zhou et al. predicted LiNiPO₄ to have a voltage of 5.1 V,Reference Zhou, Cococcioni, Kang and Ceder¹⁰ which was subsequently measured experimentally to lie between 5.1 V and 5.3 V.Reference Wolfenstine and Allen¹¹ Likewise, the Co^2+/3+ redox couple in Li₂CoSiO₄ was predicted to occur at a voltage of 4.4 VReference Arroyo-de Dompablo, Armand, Tarascon and Amador¹² and confirmed at 4.3 V experimentally.Reference Lyness, Delobel, Armstrong and Bruce¹³

While the previously described approach gives the average voltage as Li is exchanged, for most materials, the voltage changes as Li is inserted into a crystal structure. The actual voltage profile of a material can be obtained in great detail by predicting the Li configurations in the structure and their associated energy as a function of Li content by subdividing the lithiation interval and predicting the local average voltage within each interval,Reference Bréger, Meng, Hinuma, Kumar, Kang, Shao-Horn, Ceder and Grey^14–Reference Hwang, Tsai, Carlier and Ceder¹⁷ or through more advanced methods such as the cluster expansion.Reference Van Der Ven, Aydinol and Ceder^18, Reference Bhattacharya and Van Der Ven¹⁹ The cluster expansion method consists of fitting a model Hamiltonian to a large number of ab initio computations. This model Hamiltonian can subsequently be used to evaluate the energy of any lithium-vacancy arrangement at a much lower computational cost. By combining a cluster expansion with Monte Carlo simulation, configurational entropy can be accounted for to generate an accurate voltage profile.

Ionic and electronic conductivities

Charge and discharge of a lithium-ion battery consist of a series of events: during discharge, Li⁺ ions diffuse out of the anode material, solvate in the electrolyte, transfer through the electrolyte in the porous separator, and the process reverses at the cathode where Li⁺ desolvates and diffuses into the cathode. Electrons follow a different path, migrating from the anode material to the current collector, through the external circuit where they perform work, and through the carbon black in the cathode until they reach the intercalation cathode material, where they must reside on a transition metal ion close to the Li⁺ ion that was inserted from the electrolyte. Although all of these events can, in principle, be rate limiting, researchers have focused mostly on understanding and improving the electron and ion transport limitations in the cathode active materials.

Ionic conductivity

Li⁺ diffusion in electrode materials has been described in great detail by Van der Ven.Reference Van Der Ven, Aydinol and Ceder^18, Reference Morgan, Van Der Ven and Ceder^20, Reference Van Der Ven and Ceder²¹ Diffusion requires a low enough migration barrier for the ion as well as the presence of vacancies. The latter is usually not a problem, as the process of delithiation (battery charging) creates the necessary vacancies for Li motion. While dilute diffusion theory can be a reasonable guide in estimating the rate capability of a material, it needs to be realized that upon delithiation of the cathode, the concentration of vacancies and lithium are similar, and complex ordering phenomena or other phase transformations can occur.Reference Van Der Ven, Aydinol and Ceder¹⁸ Diffusion in such non-dilute limits has been studied with first-principles methods by combining kinetic Monte Carlo with cluster expansions and activation energies obtained from DFT.Reference Van Der Ven and Ceder²¹ Results on LiCoO₂ showed that Li diffusion for layered materials actually occurs through a di-vacancy mechanism, underscoring the danger of relying too much on simplified diffusion models.

To obtain reasonable rate performance at room temperature, the barrier for Li migration in a crystal has to be below 500–600 meV. The Li migration barrier can be computed using the nudged elastic band method.Reference Jonsson, Mills, Jacobsen, Berne, Ciccotti and Coker²² This method approximates the transition state by relaxing a set of initial guesses; the relaxations are constrained by a spring force to prevent transition states from settling into equilibrium and to keep them well-spaced from one another. Systematic studies of the Li activation barrier through computations have given considerable insight into mechanisms to improve Li mobility.

Kang et al. found that Li mobility in layered cathodes such as LiCoO₂ and Li(Ni,Mn,Co)O₂ is controlled by layer spacing, thereby explaining the very large variation of diffusion constant with Li concentration in LiCoO₂.Reference Levi, Salitra, Markovsky, Teller, Aurbach, Heider and Heider^25–Reference McGraw, Bahn, Parilla, Perkins, Readey and Ginley²⁷ This insight led to ways of improving the rate capability of layered Li(Ni_0.5Mn_0.5)O₂ through crystal structure engineering.Reference Kang, Meng, Bréger, Grey and Ceder²⁴ The diffusion mechanisms and defect energies in LiCoO₂ were also comprehensively probed using atomistic methods by Fisher et al.Reference Fisher, Islam and Moriwake²⁸ For graphite (the dominant commercial anode material), Toyoura et al. were able to calculate the diffusive prefactor,Reference Toyoura, Koyama, Kuwabara, Oba and Tanaka²⁹ and Persson et al. subsequently showed that diffusion within the two-dimensional graphite layers was intrinsically high-rateReference Persson, Sethuraman and Hardwick³⁰ while significantly more sluggish in other directions. This finding elucidated the wide range of experimental diffusion coefficients reported in the literature.

The dimensionality of diffusion is important for the macroscopic transport behavior. Some of the highest rate cathode materials have two- and three-dimensional diffusion. In layered materials, such as LiCoO₂, diffusion is two-dimensional, and Li⁺ moves rapidly within a single layer. Removal of too much lithium from the layer can lead to a contraction of the layer spacing, thereby increasing the activation barrier for motion. In three-dimensional diffusers such as spinel oxides, the Li diffusion constant varies less with Li content, as the framework of the structure remains intact in all dimensions as Li is removed. Spinels are therefore some of the highest rate cathode materials available.Reference Ma, Kang and Ceder^31, Reference Nakahara, Nakajima, Matsushima and Majima³²

One-dimensional diffusion is somewhat pathological; even when high lithium mobility can be achieved in a channel, blocking of channels by defects limits the distance over which such high transport rate can be achieved, and may explain why a material such as LiFePO₄ (which has one-dimensional diffusion channels) only functions well as a nanomaterial.Reference Malik, Burch, Bazant and Ceder³³ Indeed, while first-principles computationsReference Morgan, Van Der Ven and Ceder²⁰ predict that the migration barrier for Li in the channels of LiFePO₄ is only about 200 meV, indicating ultrafast one-dimensional diffusion, measurements on large single crystalsReference Amin, Maier, Balaya, Chen and Lin^34, Reference Amin, Balaya and Maier³⁵ indicate somewhat lower three-dimensional diffusion. The paradox of this size-dependent diffusion was recently resolved by incorporating channel blocking defects into the diffusion modelReference Malik, Burch, Bazant and Ceder³³ and confirmed the prediction that nanoscale LiFePO₄ would have extremely high charge and discharge capability.

Based on the first-principles predictions, a high-rate LiFePO₄ has been designed with the help of first-principles phase diagrams.Reference Ong, Wang, Kang and Ceder³⁶ Using off-stoichiometry that creates Li-conducting surface phases,Reference Kang and Ceder^37, Reference Kayyar, Qian and Luo³⁸ materials could be made that, when embedded in optimized electrode configurations, achieved charge and discharge times approaching only a few minutes (Figure 3) or even seconds.Reference Kang and Ceder³⁷

Figure 3.

(a) Calculated phase diagram and (b) experimental electrochemical performance of an off-stoichiometric LiFePO₄. Off-stoichiometric experiments are typically performed with Li-excess or Li-deficiencies (directions B and C), but these directions do not improve electrochemical performance in LiFePO₄. The calculated phase diagram shows that off-stoichiometry in an unconventional direction (mixed Li/P excess, direction A) leads to equilibrium with the highly conductive glass Li₄P₂O₇. Li₄P₂O₇ forms a surface coating that facilitates Li⁺ transfer into the cathode, thus allowing for very high rates (up to 44 C, or full discharge in less than 1.5 minutes) to be achieved in laboratory experiments (b). Adapted from Reference 37.

Phase stability

Several important properties of electrode materials are highly dependent on structure: diffusion, voltage, and stability can vary significantly from one crystal structure to another. DFT has been used early on with a high level of success to predict structures and to provide insights into the structural stability of cathode materials: for instance, LiMnO₂ forms an orthorhombic structure instead of the more common layered α-NaFeO₂ structure because of magnetic effects.Reference Mishra and Ceder⁴⁴ More recently, DFT has been used to predict stable new electrode materials. Arroyo et al. predicted a new high-pressure polymorph for Li₂MnSiO₄, confirmed subsequently by experiments.Reference Arroyo-de Dompablo, Dominko, Gallardo-Amores, Dupont, Mali, Ehrenberg, Jamnik and Morán⁴⁵ Recent developments in crystal structure predictionReference Woodley and Catlow^46–Reference Oganov and Glass⁴⁹ are likely to lead to more new cathode materials guided by computations.

While the as-synthesized structures of electrode materials can often be accurately determined with diffraction techniques, it is important to understand the changes in structure that occur upon lithium removal. For instance, a new phase of Li_xCoO₂ present at high states of charge was detected through first principles computationsReference Van Der Ven, Aydinol and Ceder¹⁸ and later confirmed in experiments.Reference Ménétrier, Saadoune, Levasseur and Delmas⁴² In most cases, partially delithiated states present a unique electronic structure problem, as Li removal leads to mixed valence on the transition metal ions. A material such as Li_xFePO₄ should therefore be written as (Li_x,V_Li(1–x))(Fe³⁺_(1–x)Fe²⁺_x)PO₄, where V_Li represents a Li vacancy. The coupling between the Fe²⁺/Fe³⁺ charge ordering and the Li-vacancy arrangement has been found to be crucial in reproducing the Li_xFePO₄ phase diagram. Zhou et al. found the DFT+U scheme could localize electrons and holes on the different Fe sites and explain the unusual shape of the experimental phase diagram by modeling the electronic entropy of the system (see Figure 4).Reference Zhou, Maxisch and Ceder⁵⁰ Recently, the approach used in Zhou’s work has been extended to study the more complex Li(Mn,Fe)PO₄ system.Reference Malik, Zhou and Ceder⁵¹ The use of standard LDA or GGA creates a single Fe type of intermediate valence by delocalizing Fe²⁺ and Fe³⁺, leading to a qualitative failure in predicting the phase diagram of this material;Reference Zhou, Maxisch and Ceder⁵⁰ in LDA and GGA, the system is predicted to form a solid solution with intermediate Li-vacancy ordering rather than a two-phase miscibility gap. Given the fundamental difference in how such systems would lithiate and delithiate (two-phase reaction versus solid solution), a proper treatment of electron correlation is essential for making relevant predictions on battery materials. Only in Li_xCoO₂, which becomes metallic upon delithiation, is the delocalized description of GGA and LDA appropriate.

Figure 4.

Comparison between the Li_xFePO₄ experimental phase diagram from Delacourt et al.Reference Delacourt, Poizot, Tarascon and Masquelier⁵² and Dodd et al.Reference Dodd, Yazami and Fultz⁵³ and computed phase diagrams obtained with electronic entropic contribution (middle) and without (bottom).Reference Zhou, Maxisch and Ceder⁵⁰ The triphylite LiFePO₄ is referred to by T; the fully delithiated FePO₄ phase, heterosite, by H; and the solid solution region is indicated by SS. Only the introduction of the electronic entropy contribution reproduces the experimentally observed solid solution region. © 2006, American Physical Society.

While most of the work on lithium battery electrodes has concerned intercalation, electrodes operating without preserving a host framework are currently of interest due to their much larger theoretical capacities.Reference Cabana, Monconduit, Larcher and Palacín⁵⁵ For instance, FeF₃ has been demonstrated experimentally as a potential electrode following the reaction: FeF₃ + Li → 3LiF + Fe.Reference Badway, Pereira, Cosandey and Amatucci⁵⁶ In this so-called conversion mechanism, the charged and discharged products are not structurally related. First-principles computations also have been used to investigate conversion cathodes such as FeF₃ and BiF₃.Reference Doe, Persson, Hautier and Ceder^57, Reference Doe, Persson, Meng and Ceder⁵⁸

Studying conversion batteries by first-principles computations presents unique challenges. DFT errors in reaction energies are typically smaller when the reaction involves phases similar in chemical bonding. As conversion reactions may involve very different chemical phases (e.g., metals and fluorides), their associated computed voltage can show larger discrepancies with experiments than insertion voltages. In addition, the kinetics of ionic rearrangements are typically more difficult to model than diffusion in intercalation compounds, as reaction paths are largely unknown. The understanding and design of conversion electrodes will require significant advances in the currently available modeling techniques.

Thermal stability

Safety is a major concern of lithium batteries, and developing computational approaches toward predicting safety has been a major focus of several federally supported energy programs. At elevated temperatures, cathode materials may potentially release oxygen, which can combust the electrolyte and lead to runaway reactions and fire. While the discharged cathode is usually relatively stable with respect to oxygen release, cathodes based on layered LiCoO₂, LiNiO₂, and spinel LiMn₂O₄ decompose with oxygen evolutionReference Chen and Richardson^59, Reference Dahn, Fuller, Obrovac and Vonsacken⁶⁰ when heated in the highly oxidized charged state. Olivine LiFePO₄,Reference Padhi, Nanjundaswamya and Goodenough⁶¹ on the other hand, offers much better thermal stability, and it has been tacitly assumed that all materials based on the phosphate polyanion would share this advantage.Reference Yamada and Chung^62, Reference Martha, Markovsky, Grinblat, Gofer, Haik, Zinigrad, Aurbach, Drezen, Wang, Deghenghi and Exnar⁶³ With the increasing application of lithium rechargeable batteries to large scale applications such as hybrid electric vehicles and fully electric vehicles, the thermal stability of the battery system is becoming ever more important.

Recent computational thermodynamics approaches can approximate the intrinsic thermal stability of an electrode material. Wang et al.Reference Wang, Maxisch and Ceder⁶⁴ first studied the thermal stability of layered LiCoO₂ and LiNiO₂ and spinel LiMn₂O₄ by computing their first-principles phase diagrams and calculating the heats for structural transformation and decomposition reactions of the delithiated states. Subsequently, Ong et al. developed an oxygen grand potential phase diagram approach to determine the oxygen chemical potential at which an electrode decomposes to evolve oxygen.Reference Ong, Jain, Hautier, Kang and Ceder⁶⁵ Using this approach, it has been shown that delithiated MnPO₄ has significantly worse thermal stability than delithiated FePO₄ (see Figure 5), both in terms of the critical oxygen evolution temperature as well as the amount of oxygen evolved. The calculations agree with previous experimental resultsReference Chen and Richardson^59, Reference Kim, Kim, Gwon and Kang⁶⁶ and have cast doubt on the assumption that all phosphate polyanion compounds share the excellent thermal stability of LiFePO₄.

Figure 5.

Calculated thermal stabilities of delithiated MnPO₄ and FePO₄ (represented by MPO₄ on the y-axis). The figure shows the number of moles of O₂ evolved per mole of cathode material as a function of temperature. It can be seen that the onset of O₂ evolution is at a much higher temperature (~700°C) for FePO₄ compared to MnPO₄ (~370°C), and the initial amount of O₂ evolved is also much less. This difference in thermal behaviors can be explained by the fact that Fe³⁺ and Mn²⁺ both have the exchange-stabilized 3d ⁵ electron configuration, which suggest that the decomposition of FePO₄ (requiring the reduction of Fe³⁺ to Fe²⁺) would be relatively more difficult than the decomposition of MnPO₄ (requiring the reduction of Mn³⁺ to Mn²⁺). Adapted from Reference Reference Ong, Jain, Hautier, Kang and Ceder65.

Electrolytes

The electrolyte functions as an ionic conductor and comprises a solvent, typically a mixture of a cyclic carbonate, such as ethylene carbonate (EC), and a linear carbonate, such as dimethyl carbonate (DMC), and a lithium salt, typically LiPF₆. An ideal electrolyte should have a wide enough electrochemical window to accommodate the operating voltage of the battery, chemical stability against the electrodes and other components, a high ionic conductivity with extremely low electronic conductivity, low flammability, and low cost. Current EC/DMC + LiPF₆ electrolytes are an imperfect compromise of these objectives. The electrochemical windows of EC/DMC + LiPF₆ electrolytes of up to approximately 5 V versus Li/Li⁺Reference Xu⁶⁷ are sufficient for current electrode systems but would be severely tested by higher voltage cathodes (e.g., LiNiPO₄, which has a predicted voltage of 5.1 V).Reference Zhou, Cococcioni, Kang and Ceder¹⁰ Current electrolytes are also unstable against the graphitic anode and function by forming a passivating solid-electrolyte interphase whose composition, structure, and transport mechanisms are still not well understood.Reference Xu⁶⁷

A sizable proportion of the work on electrolytes has been focused on understanding the reductive and oxidative stabilities and decomposition mechanisms of EC, DMC, and other solvent additives, with and without coordination with Li⁺ or PF₆^–.Reference Xing, Li, Wang, Gu, Xu, Tan and Yi^68–Reference Tasaki⁷¹ In some cases, a dielectric continuum solvation model,Reference Tomasi, Mennucci and Cammi⁷² where the complex system of the salt and solvent is modeled as a “solute” embedded in a dielectric continuum, was used. Another body of work involves using molecular dynamics simulations with only a few repulsion-dispersion parameter types to obtain excellent descriptions of the heats of vaporizations, densities, self-diffusion coefficients, structural factors, and conformational dynamics of solvents.Reference Borodin and Smith^73, Reference Borodin and Smith⁷⁴ A few computational studies have also been devoted to the study of the lithium salt, especially with regard to the oxidative stability of the anions, ion conductivities, and dissociation energies.Reference Johansson^75, Reference Ue, Murakami and Nakamura⁷⁶

In recent years, room-temperature ionic liquids (ILs), which are comprised entirely of ions, have emerged as promising replacement electrolytes due to their wide electrochemical windows, low flammability, and low vapor pressure.Reference Galinski, Lewandowski and Stepniak^77, Reference Garcia, Lavallee, Perron, Michot and Armand⁷⁸ The properties of ILs can be tuned by the suitable modification of the functional groups attached to the cation and anion, and therefore, ILs present an extremely large chemical design space that is amenable to computational design approaches. Ong et al. recently showed that qualitative trends in cation electron affinities and anion ionization energies are in good agreement with experimental data and found that electron-donating groups stabilize cations (leading to better reductive stability), and electron-withdrawing groups stabilize the anions (leading to better oxidation stability).Reference Ong and Ceder⁷⁹ However, it should be pointed out that the complex nature of the interactions in ILs presents a serious challenge for standard computational methods. For instance, Izgorodina et al. showed that the errors in the ion binding energies in ILs were predicted by many standard DFT functionals to be fairly large (above 12 kJ mol^–1).Reference Izgorodina, Bernard and MacFarlane⁸⁰ Nonetheless, computational approaches have been applied successfully to the investigation of the dielectric constants,Reference Izgorodina, Forsythb and MacFarlane⁸¹ decomposition mechanisms,Reference Howlett, Izgorodina, Forsyth and Macfarlane⁸² and formation energies and stabilities of ILs.Reference Borodin^83, Reference Gutowski, Holbrey, Rogers and Dixon⁸⁴ The development of force fields designed especially for ILs also has resulted in burgeoning research in the transport properties of these complex systems.Reference Borodin and Smith^85–Reference Monteiro, Bazito, Siqueira, Ribeiro and Torresi⁸⁹

In addition to the electrolyte solvent and salt, computational studies have shown that the oxidation potentials of redox shuttle additives, which provide overcharge and overdischarge protection for lithium batteries, can be estimated to a high degree of accuracy using first-principles calculations coupled with a dielectric continuum model.Reference Wang and Dahn^90, Reference Wang, Buhrmester and Dahn⁹¹

Computational resources

First-principles methods are more accessible than ever due to more efficient codes and reductions in the cost of computing over the last two decades, and most individual computations presented in this article can be run with 8–32 computing cores. A typical DFT calculation, including structural relaxations on a cathode material, may take 100–200 CPU hours on such resources. A voltage prediction encompasses two such calculations, and the accurate evaluation of voltage as a function of lithium content requires about 50–200 DFT calculations (7500–30000 CPU hours) in order to evaluate many possible Li vacancy arrangements at multiple intermediate compositions. Such a study requires at least a small computing cluster. In terms of computing activation barriers for Li diffusion, a single nudged elastic band calculation gives the barrier for one hop at one lithiation level and typically consumes about 5000 CPU hours. Because diffusivity may vary with the amount of lithium in the electrode, or “state of charge,” a comprehensive study of diffusion in a material may calculate activation barriers for several paths at several lithiation levels, thus multiplying this number by about a factor of 10, or 50000 CPU hours. Evaluating phase stability and thermal stability is typically less computationally expensive; a quaternary phase diagram may contain about 50–100 phases, which each require a DFT computation, and such a phase diagram can often be recycled to evaluate other materials in that chemical space.